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Toni Bakhtiar

H2 Control PerformanceLimitations for SIMOFeedback Control Systems

Ph.D. Thesis

September 2006

The University of Tokyo

H2 Control Performance Limitations forSIMO Feedback Control Systems

Toni Bakhtiar

A thesis submitted in partial fulfillment ofthe requirements for the degree of

Doctor of Philosophy

DEPARTMENT OF INFORMATION PHYSICS AND COMPUTINGGRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY

THE UNIVERSITY OF TOKYO

7-3-1 HONGO, BUNKYO-KU, TOKYO 113-8656JAPAN

September 2006

Committee Members:

Professor Shigeki Sagayama The University of TokyoAssociate Professor Koji Tsumura The University of TokyoAssistant Professor Yasuaki Oishi The University of TokyoAssistant Professor Nobutaka Ono The University of TokyoProfessor Shinji Hara The University of Tokyo

Copyright c© 2006 by Toni Bakhtiar, Department of Information Physics andComputing, Graduate School of Information Science and Technology, theUniversity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.

This thesis was typeset by the author using MiKTEX 2.3 and WinEdt 5.3. Thetypeface used for the main text is Palatino.

Dedicated to:Jamil Karim and Nasrukan,

from whom I owe mathematics.

Abstract

This thesis is devoted to a research area that studies the fundamental per-formance limitation and trade-off of feedback control, a subject intensivelydeveloped in the linear time-invariant feedback systems, beginning with theclassical work of Bode in the 1940s on logarithmic sensitivity integrals. Inmodern control design, the studies on performance limitation serve as anappendage tool since they help a control system designer specifies reason-able control objectives, and understands the intrinsic limits and the trade-offbetween conflicting design considerations.

In this thesis we quantify and characterize the fundamental performancelimitations arise in H2 optimal tracking and regulation control problemsof single-input multiple-output (SIMO) linear time-invariant (LTI) feedbackcontrol systems. In tracking problem, the control performance is measuredby the tracking error response, possibly under control input constraint, withrespect to a step reference input. While in regulation problem, the perfor-mance is measured by the energy of the measurement output simultaneouslywith that of the control input and sensitivity constraints, against an impul-sive disturbance input.

Our primary interest is not on how to find the optimal or robust con-troller. Rather, we are interesting in relating the optimal performance withsome simple characteristics of the plant to be controlled. In other words, weprovide the analytical closed-form expressions of the optimal performancein terms of dynamics and structure of the plant. The analytical expressions,however, constitute guidelines for designing an easily controllable plant inpractical situations, from which the control system designer may rely in de-termining the optimal design parameters and reasonable control strategies.

We mostly focus our attention on tracking and regulation problems ofdiscrete-time systems. Toward the existing results of continuous-time sys-tems, we make small corrections and perform a few extensions. We then re-formulate and resolve both problems in delta domain. An analysis on thecontinuity property shows that we can completely recover the continuous-time expressions from the delta domain expressions stand point as sampling

ii

time approaches zero. Frankly speaking, we provide comprehensive and uni-fied expressions on the characterization of the control performance limita-tions in the H2 tracking and regulation problems. Furthermore, our analyti-cal expressions show that the optimal tracking and regulation performancesare explicitly characterized by the plant’s non-minimum phase zeros and un-stable poles as well as the plant gain. We confirm the effectiveness of the de-rived expressions by several illustrative examples. We also show how to ap-ply the analytical expressions to practical applications including the controlof three-disk torsional system, the determination of the optimal parametersin inverted pendulum system, and the sensor selection in magnetic bearingsystem. In addition, by exploiting the delta domain expressions we derivethe analytical closed-form expressions of the optimal tracking and regula-tion performances for delay-time systems and by invoking our discrete-timeLTI results we provide the similar expressions to approximate the optimaltracking performance for sampled-data systems.

Acknowledgement

This thesis is the result of three years of work at Department of InformationPhysics and Computing, the University of Tokyo, whereby I have been as-sisted and supported by many people. It is now my pleasure to express ahuge debt of gratitude to all of them.

My first and most earnest acknowledgement must go to my honorific ad-visor Prof. Shinji Hara, for his constant guidance, patience, and encourage-ment throughout my graduate studies. He has shown me stimulating waysto approach a research problem and the need to be persistent to accomplishany goal and has taught me innumerable lessons and insights on the work-ing of academic research in general. In every sense, none of this work wouldhave been possible without him.

My thanks also go to the committee members, Prof. Shigeki Sagayama,Prof. Koji Tsumura, Prof. Yasuaki Oishi, and Prof. Nobutaka Ono for readingprevious drafts of this thesis and providing many valuable comments thatimproved the presentation and contents of this thesis. My special thanks dueto Prof. Koji Tsumura for his critical suggestion and insights since the earlystep of my work. Pointing-out the application of the Poisson-Jensen formulais one of his remarkable breakthrough.

This research has been fully supported and funded by Ministry of Edu-cation, Culture, Sports, Science and Technology of Japan. Without their gen-erous aid, it would not have been possible for me to pursue and to com-plete this Ph.D. work successfully. I am also indebted to my home institu-tion, Department of Mathematics, Bogor Agricultural University, Indonesia,for providing an ideal environment in which I felt free and could focus on myresearch study. In particular, I thank Dr. Siswadi, Dr. Amril Aman, Dr. D.S.Priyarsono, Dr. Budi Suharjo, and Dr. Berlian Setiawaty for their encouragingsupport.

My sincere gratitude goes to all of the present and old members of thelaboratory for their amity and cordiality. I am really fortunate and proud tobe a part of this prestigious laboratory. I am particularly grateful to Prof. Hi-denori Kimura for ’opening’ the first gate to enter the University of Tokyo,

iv

Dr. Hideaki Ishii, Dr. Tomohisa Hayakawa, and Dr. Masaaki Kanno for theirguidance and encouragement, Daijiro Koga and Masahiro Senba for their un-selfish help, Yohei Kuroiwa and Dr. Masato Ishikawa for their assistance inmy early period of living and studying in Tokyo. I am also thankful to the de-partment office staff for dealing with my administrative business during mystudies. Especially, I thank Chikako Sakurai, Yokosaka, and Keiko Nakagakifor their kind help.

I sincerely would like to thank Masasuke Kurokawa and Sachiko Onofrom the Kurokawa Kindergarten in Tokyo, for their continuous assistanceand patience. My gratitude goes also to my friends in Tuban and Tokyo, fortheir companionship and huge support.

I would like to convey my heartfelt thanks to my teachers in the past, inTuban and Bogor. I owe them my immense gratitude for their generosity andgraciousness. I will never find enough words to express my deepest gratitudeto my late father Kurdianto and my mother Mamlu’ah. Their tender love andaffection has always been the extraordinary force for gaining the blessingroad of my life. They deserve far more credit than I can ever give them. I alsoowe a huge debt of gratitude to my brothers Istiko, Budi, and Benny, and myparents in law.

My final and most heartfelt acknowledgement must go to my wonderfulson Nauval and my beloved wife Nuril. For their understanding, patience,and encouragement, they deserve my everlasting love. Thank for being ev-erything I am not.

Tokyo, TONI BAKHTIARSeptember 2006

Ucapan Terima Kasih

Saya merasa sangat beruntung mendapatkan kesempatan yang tidak se-mua orang pernah mengalaminya: menyaksikan indah dan luasnya ilmu.Semakin saya menikmati, semakin saya terpesona. Semakin saya belajar,semakin saya sadar betapa banyak yang saya tidak tahu. Tuhan, terimakasih telah membawaku ke sini. Terima kasih atas campurtangan-Mu. Te-rima kasih atas segala kemurahan-Mu.

Izinkanlah saya memulainya dari Tuban, kota yang telah merampas ba-nyak ruang kenangan. Terima kasih kepada (alm) Bapak dan Ibu yang telahmerawat dan membesarkan dengan penuh kasih sayang, yang selalu men-dorong agar terus bersekolah, dan yang tidak pernah lupa melantunkansegenap doa. Semoga Tuhan selalu menyayangi panjenengan berdua. Terimakasih kepada Mas Is, Mas Budi, dan Benny atas kebaikannya. Semoga kitasenantiasa hidup dalam kebersamaan dan kerukunan. Terima kasih kepadakeluarga di Kutorejo dan Sendangharjo atas kasih sayang dan doanya. Sayaselalu teringat kebaikan dan kehangatan (alm) De Cuk, (alm) De Yah, DeOes, dan Lek Fik. Saya juga sangat berterimakasih kepada Bapak dan IbuSiyono dan keluarga di Pontianak atas semua bantuannya.

Saya sangat berhutang budi kepada Bu Yati dan Bu Sri di TK IWKATuban; Bu Karning, Bu Win, Bu Yati, dan semua guru di SD Negeri Kutorejo3 Tuban; Pak Jamil, Pak Rochman, dan semua guru di SMP Negeri 3 Tuban;Pak Nasrukan, Bu Widi, dan semua guru di SMA Negeri 1 Tuban. Teris-timewa Bu Widi, terima kasih atas pemberian kacamatanya. Bukan hanyateduh wajah Ibu yang bisa saya lihat dengannya, tapi juga terang jalan pan-jang di depan saya.

Saya sangat berbahagia selama bersekolah di Tuban bertemu denganbanyak teman baik, yang akhirnya menjadi sahabat dan bahkan saudara.Syukron katsir kepada Mbah Amin, Bancoce, Dhakim, Domy, Godar, Kemul,Mobik, Sepeh, Tekuk, Wiji Klomoh, dan seluruh anggota Keluarga Kaspo.Juga pada Boman, Keru, dan Soni. Terima kasih atas persahabatan dan per-saudaraan yang tulus. Saya bahagia ada di dalam keluarga ini. Saya ingindapat sering pulang ke Tuban untuk kembali berkumpul, menyegarkan hati.

vi

Saya sangat berterima kasih kepada Dr. Siswadi atas bantuan dan bim-bingannya, Dr. Amril Aman, Dr. D.S. Priyarsono, Dr. Budi Suharjo, dan Dr.Berlian Setiawaty atas dorongan dan dukungannya, dan juga kepada semuaguru dan rekan di Departemen Matematika, Institut Pertanian Bogor, ataskebaikan dan doanya. Kepada Benny, Heri, Dhito, dan Getsa, terima kasihtelah merawat rumah dengan baik.

Akhirnya saya tiba di Tokyo, kota sangat besar yang membuat saya tam-pak semakin ndeso dan takut. Untunglah saya menjumpai banyak orangbaik di sini. Terima kasih kepada Anwar Sanusi Kariem, M. Farid Ma’ruf,Khariri Ma’mun, dan Padang Wicaksono atas nggedobos dan nggedabrus-nya. Terima kasih kepada teman-teman di PPI Universitas Tokyo atas ke-baikan, kehangatan, dan kebersamaannya. Ungkapan yang sama juga ter-tuju kepada Rukun Tetangga Higashi Ayase dan Kita Ayase. Indahnya sakuramungkin bisa bercerita lebih tentang kita.

Di kanan saya ada Nauval dan di kiri saya ada Nuril. Alangkah baha-gianya dikelilingi mereka berdua.

Tokyo, TONI BAKHTIARSeptember 2006

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Ucapan Terima Kasih . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fundamental Performance Limitations . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Bode Integral Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Contribution of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Plant Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Coprime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Inner-outer Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Delta Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Tracking Performance Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Tracking Performance Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Tracking Error Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.2 Tracking Error Problem under Control Input Penalty . . 213.1.3 Plant Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Continuous-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Two Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Tracking Error Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

viii Contents

3.2.3 Tracking Error Problem under Control Input Penalty . . 283.3 Discrete-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Two Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Tracking Error Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.3 Tracking Error Problem under Control Input Penalty . . 41

3.4 Delta Domain Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.1 Two Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.2 Tracking Error Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.3 Tracking Error Problem under Control Input Penalty . . 483.4.4 Continuity Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Delay-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.6 Sampled-data Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.1 Fast Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.6.2 Approximation of the Optimal Performance . . . . . . . . . . 57

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Regulation Performance Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1 Regulation Performance Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Integral Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1.2 Energy Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 664.1.3 Output Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Continuous-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2.1 Energy Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.2 Output Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Discrete-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.1 Energy Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.2 Output Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 774.3.3 Output Regulation Problem with Noise . . . . . . . . . . . . . . 79

4.4 Delta Domain Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4.1 Energy Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.2 Output Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.3 Continuity Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.5 Delay-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Three-disk Torsional System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1.2 Tracking Performance Limitation . . . . . . . . . . . . . . . . . . . . 975.1.3 Regulation Performance Limitation . . . . . . . . . . . . . . . . . . 97

5.2 Inverted Pendulum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.2 Tracking Error Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2.3 Tracking Error Problem under Control Penalty . . . . . . . . 102

5.3 Magnetic Bearing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Contents ix

5.3.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.2 Regulation Problem: Continuous-time Case . . . . . . . . . . 1065.3.3 Regulation Problem: Discrete-time Case . . . . . . . . . . . . . . 107

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A Some Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.1 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.3 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118A.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.5 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

1

Introduction

1.1 Background

In classical paradigm, the optimal control problem reflects its main concernon the controller design. A bunch of efficient algorithms and methods forfinding the optimal or robust controller which stabilizes the feedback inter-connection subject to some predetermined specifications is the definite resultof this approach. The best controller is usually characterized by a set of Ric-cati equations or linear matrix inequalities. In tracking and regulation con-trol problems of minimum phase and stable systems, it is well-known thatthe non-minimum phase zeros and unstable poles of the open-loop systemlimit the best achievable control performance. In other words, the existenceof such kind of zeros and poles affects the easiness of the plant to be con-trolled.

In recent control applications more challenging control problems havebeen emerged, since they have to be solved by taking into account somephysical constraints such as measurement accuracy, control input effort, sam-pling period, delay time, and channel capacity. Hence, it is very interestingto investigate the control performance accomplishment in feedback controlsystem under some physical constraints. The importance of this topic lies inits ability to provide answers to a fundamental question: Is any minimumphase and stable plant always easy to control under physical constraints inpractical applications?

Under this perspective, since last decade there is a growing research ac-tivity on solving the optimal control problem by directing the main objectiveon the plant design instead of the controller design. This new paradigm as-sumes that the best controller is always available for a given plant. Hence, amore desirable plant, i.e., an easily controllable plant that satisfies some spec-ifications such as having minimal tracking error, minimal energy regulation,its poles lie in a prescribed region, etc., are requested. Obviously, instead ofhaving Riccati equations or linear matrix inequalities, existence of the ana-lytical closed-form expressions of the pursuing performance in terms of the

2 1 Introduction

plant dynamics and structure are inevitable. The analytical closed-form ex-pression characterized by plant dynamics and structure, however, reveals thefundamental performance limitations inherently affect the respecting controlproblem and subsequently constitute a class of easily controllable plants.

In modern control design, the knowledge on performance limitation con-stitutes an extremity, since it enables the control system designer specify thereasonable control objectives and understand the performance limitationsinherently affect the design. The understanding of performance limitationsthus promises to be of intrinsic as well as practical value. In practical situa-tions, the characterization of the fundamental performance limitations aidsthe designer to achieve the control performance by determining the opti-mal design parameters, e.g., providing the optimal length of pendulum ininverted pendulum system, and in selecting the reasonable control strate-gies such as selecting the sensor in magnetic bearing system. Wider plantdesign changes can also be considered based on the characterization includ-ing changing the apparatus, relocating actuators and sensors, adding newequipment to dampen the disturbance, adding new actuators and sensors,and changing the control objectives.

1.2 Fundamental Performance Limitations

In practical control design, it is often important to know how easy the plantcan be controlled, what control structure should be used, and how might theprocess be changed to improve control [54]. These three questions are relatedto the inherent control characteristics of the process itself. These characteris-tics comprise some intrinsic limitations of the process to be controllable, i.e.,to achieve acceptable control performance in maintaining the outputs withinspecified bounds from their reference inputs, in spite of unknown variations,using available inputs and measurements. Therefore, the performance limi-tation has the same essential attributes as the controllability:

• It is independent of the controller.• It is a fundamental property of the plant (or process) alone.• It can only be affected by plant design changes.

Fundamental performance limitations are central for all design problemsbut, due to Bode, often neglected in control works. A classic example to thisfact is the design of the flight controller for the X-29 experimental aircraft [1].Much design effort was done with many methods and much cost to fulfillthe design criteria: The phase margin should be greater than 45 for all flightconditions. An argument based on fundamental laws shows that this speci-fication is unfeasible.

Control system design always involves performance considerations andphysical limitations, which subsequently emerges a trade-off between the

1.2 Fundamental Performance Limitations 3

Fig. 1.1. Feedback control systems.

high performance objective and and the need for meeting hard design con-straints. Thus, having insights into fundamental performance limitations areuseful for control system, since we can quantify how various constraints maylimit the level of achievable performance leading to some realistic design ob-jectives. The characterization of the best achievable performance in terms ofthe dynamics and structure of the system to be controlled reveals what is and,conversely, what is not achievable prior to applying any specific techniquefor solving the control problem. A qualitative method, i.e., simulation ap-proach, may not satisfactorily tackle this problem, since one can never knowif the result is a fundamental property of the plant, or if it depends on someset-points.

Nowadays, there are two main research directions in the study of fun-damental performance limitations. First direction lies in the extensions ofthe well-known Bode’s integral theorem [3] to assess design constraints andperformance limitations via logarithmic-type integrals. Second direction fo-cuses on the formulations of optimal control problems to quantify and char-acterize the fundamental performance limits. Roughly speaking, after 1980there have been steady stream results in studying the fundamental limita-tions in feedback control system. Latest results on this studies can be seen ina special issue of the IEEE Transactions on Automatic Control in August 2003and a book that presents a comprehensive account of the modern frequencydomain/input-output results on limits to performance [53].

1.2.1 Bode Integral Relation

Consider the feedback control configuration depicted by Fig. 1.1, where Pis the plant and K is the stabilizing controller. The signals r, d, u, y, ande := r − y represent the reference input, disturbance input, control input,system output, and error, respectively. The sensitivity function S and thecomplementary sensitivity function T are defined by

S(s) :=1

1 + P (s)K(s), T (s) :=

P (s)K(s)1 + P (s)K(s)

. (1.1)

It is well-known that the sensitivity function and the complementary sensi-tivity function tell much about feedback loop since they describe the effectsof process variations. It is advantageous to have a small value of S to obtainsmall control error for commands and disturbance and a small value of T to

4 1 Introduction

allow large process uncertainty. Their definitions say that S(s) + T (s) ≡ 1holds for any choice of P (s) and K(s), which means that S and T cannot bemade small simultaneously. The loop transfer function PK is typically largefor small values of s = jω and it goes to zero as ω goes to infinity. This meansthat S is typically small for small s = jω and close to 1 for large ω. The com-plementary sensitivity function is close to 1 for small s = jω and it goes to 0as ω goes to infinity. A basic problem is to investigate if S can be made smallover a large frequency range. There are unfortunately severe constraints onthe sensitivity function.

The pioneering work of Bode in 1945 [3, pp. 285], which is originallyrelated to feedback amplifier design, reveals that the sensitivity reductioncannot be accomplished once a time at all frequency range of the imaginaryaxis. In other words, sensitivity reduction smaller than one over a particularfrequency range will contribute sensitivity expansion greater than one oversome other frequency range. Technically speaking, the sensitivity function,S, must obey the following integral relation for a linear time-invariant stableopen-loop plant ∫ ∞

0

log |S(jω)| dω = 0. (1.2)

In 1963, Bode’s theorem has been applied to the feedback control problemfor the very first time by Horowitz [33]. The result of Bode has played afundamental role in feedback design and have received renewed interest inrecent years. The utility and importance of this result has motivated and ledto several extensions to Bode’s theorem. It is pointed-out by Kwakernaak andSivan [36, pp. 440–441] that if the open-loop system is asymptotically stable,then the integral could be zero, finite, and infinite, depending on the degreedifference between numerator and denominator of the open-loop transferfunction. Important result has been achieved by Freudenberg and Looze [22],where the Bode’s integral has been extended to the open-loop unstable case,i.e., (1.2) has been generalized to

∫ ∞

0

log |S(jω)| dω = π

np∑

k=1

pk, (1.3)

where pk (k = 1, 2, . . . , np) are the unstable poles of the open-loop plant.It is showed by (1.3) that the integral is proportional to the unstable open-loop plant. The discrete-time versions of the sensitivity integral were fur-ther derived by Maciecjowski [38], Middleton [40], Mohtadi [45], Sung andHara [55]. The generalized Bode’s theorems for multi-variable systems havebeen carried-out by Chen and Nett [16] and Chen [8]. Moreover, there aresome extended versions of Bode’s theorem for different class of systems, in-cluding delay-time systems [23,27], linear time-varying systems [34], nonlin-ear systems [65], and linear time-periodic systems [52].

The progress of the researches has also led to the development of Bodeand Poisson type integrals [7, 9, 10, 32], making available a series of integral

1.2 Fundamental Performance Limitations 5

constraints, in either equality or inequality, on the sensitivity and comple-mentary sensitivity functions applicable to multi-variable systems. Other rel-evant extensions have been pursued in [26] and [28], toward problems per-taining to sampled-data systems and filter design.

Overall, the sensitivity and complementary sensitivity integrals charac-terize how certain plant properties such as non-minimum phase zeros andunstable poles in the open-loop transfer function may impose constraintsupon feedback design. They also serve as a benchmark for evaluating thesystem’s performance prior to and after controller design.

1.2.2 Optimal Control Problem

The classical optimal control methods give the optimal performance withthe specified criteria and constraints if the control objectives are achievable.They do not tell what to do if the objectives cannot be accomplished andthey seldom give useful insight into the mechanism that cause the limita-tions. It is therefore in the recent years a new paradigm of control problembased on the plant design rather than controller design has received muchinterest. The primary objective of this control paradigm is to relate the opti-mal performance with the plant properties. This technique gives insight intothe factors that fundamentally limit the achievable performance of a controlsystem, since it provides the analytical closed-form expression of the optimalperformance in terms of the plant dynamics and structure.

The studies on fundamental performance limitations in optimal controlproblem under the new paradigm have been extensively carried-out sincethe beginning of this decade. Most of the studies are performed in H2 opti-mal control setting and by using the transfer function approach, and only afew numbers that executed in H∞ criterion or by adopting the state spaceapproach (e.g., see [43,64]). Two among the topic studies which have earnedmuch attentions until the current state are the H2 optimal tracking and reg-ulation performance problems.

H2 Tracking Performance Problem

The tracking performance is usually quantified as the integral-square (forcontinuous-time case) or summation-square (for discrete-time case) of theerror between the output signal of the plant and the reference input signal,where the reference input r(t) is the unit step function. The best trackingability is then measured by the minimum tracking performance achievableby all controllers that stabilize the plant:

J∗ = infK∈K

∫ ∞

0

|e(t)|2 dt, (1.4)

where K is the set of all stabilizing controllers. For many years, investigationon the tracking performance of the feedback system has become an impor-tant problem. In linear time-invariant (LTI) context, the first result on the

6 1 Introduction

tracking step reference input of SISO stable systems can be found in [46],which show that the non-minimum phase zeros of the plant completely char-acterize the optimal tracking performance. Some generalization then havebeen made to unstable cases applicable for SIMO system [12] and MIMOsystem [17], including one for discrete-time system [58]. The latter has alsoextended the problem by considering different type of reference input signalssuch as sinusoidal and ramp signals. All the results suggest that in multi-variable systems, the spatial properties such as the direction of the refer-ence input give additional limitations on the optimal performance. While,other results have regarded the preview control as the control strategy [18]and considered the sampled-data system as the platform analysis [15]. Thereare also a number of results which accommodate the robustness issue in thetracking problem [6, 30, 50].

As one of the performance objectives, tracking accuracy is often mutuallyconflicting with other performance objectives, e.g., minimizing the energy ofthe control input:

J∗ = infK∈K

∫ ∞

0

(|e(t)|2 + |u(t)|2) dt. (1.5)

It means that the tracking error problem has been reexamined under con-dition that only finite control input energy is available. Therefore, under-standing a complex trade-off among those becomes fundamental from thecontrol performance achievability point of view. The accomplishment in thisextension consist of the results for SIMO systems [31] and MIMO stable sys-tems [14].

Frankly speaking, the investigation of the H2 tracking performance lim-itation for LTI systems is almost complete except that we do not have anyresults for SIMO LTI discrete-time system, since the MIMO result is validonly for right-invertible plant, where the number of input is greater or equalto that of output. New directions related to the research in this area may takeform of extending the analysis beyond LTI system, i.e., we may consider non-linear or time-varying system, and more complex control design than unityfeedback or expand the known results by new problem and application area,or by developments in novel design techniques and methods.

H2 Regulation Performance Problem

In optimal regulation performance problem, the control objective is to mini-mize the energy of the control input, or to minimize the energy of the controlinput simultaneously with the energy of the system output against an im-pulse disturbance input d(t), i.e.,

E∗ = infK∈K

∫ ∞

0

|u(t)|2 dt, (1.6)

1.4 Contribution of the Thesis 7

orE∗ = inf

K∈K

∫ ∞

0

(|y(t)|2 + |u(t)|2) dt. (1.7)

We call the former the energy regulation problem and the latter the outputregulation problem. Results on H2 energy regulation problem can be foundin [31] which is conducted for unstable/non-minimum phase SISO/SIMOplants. Equivalent results in SISO systems but articulated in term of signal-to-noise ratio constrained channels are found in [4, 42]. Meanwhile, resultson H2 output regulation problem of minimum phase SISO/MIMO systemsare presented in [14]. Note that the linear time-varying feedback stabilizationhas been discussed in [42]. It is well-known that the regulation performanceis not only imposed by non-minimum phase zeros and unstable poles of theplant, but also by the plant gain.

Summarizing the existing results, the investigation for the minimumphase plant is almost complete, while the researches for unstable and non-minimum phase plants are not complete. Especially further investigationsare required for the LTI discrete-time system.

1.3 Objectives

The primary objective of this research is two-fold. Firstly, we quantify andcharacterize the fundamental performance limitations may arise in the H2

optimal tracking and regulation performance problems pertaining to SIMOLTI feedback control system. We consider SIMO system with the reasonsthat MIMO system is theoretically hard and SIMO system itself is practicallymeaningful, since it consists of systems with single actuator and more thanone sensors, which commonly appear in typical situation of control prob-lems. Theory of fundamental performance limitations for the important spe-cial case of a single-input two-output (SITO) system can be found in [25, 60].Secondly, we provide guidelines of plant design from the view point of con-trol as a set of easily controllable plants in practical applications. We accom-plish the research objectives by deriving the analytical closed-form expres-sions of the best achievable performance.

1.4 Contribution of the Thesis

In this thesis we examine two kinds of optimal control problem in SIMO LTIfeedback control system, namely H2 optimal tracking and regulation controlproblems. Instead of proposing a novel algorithm to obtain the optimal or ro-bust controller, we investigate the fundamental performance limitations mayarise in the control problems, where the limits are represented by analyticalclosed-form expressions in terms of the plant properties. Analytical expres-sions rather than numerical solutions are quite useful not only to understand

8 1 Introduction

feedback control systems but also to characterize a set of easily controllableplants in practical situations, where the formulae can provide guidelines forplant design from the view point of control.

The main contribution of this thesis dwells on the result of H2 optimalregulation performance problem for SIMO discrete-time systems. Overall,we supply a comprehensive and unified expressions by deriving the ana-lytical expressions of the optimal tracking and regulation performances forcontinuous-time, discrete-time, and delta domain cases. To deal with as gen-eral as possible situation, we consider a class of non-minimum phase andunstable systems.

(i) We provided the analytical closed-form expressions of the optimal per-formance in tracking step input signal, where the tracking ability is mea-sured by the error between the input reference signal and the measure-ment output signal, possibly under control input constraint.• Continuous-time Case: The existing results of continuous-time sys-

tem contain some mistakes. We correct them by explicitly accountingan additional effect caused by the plant’s unstable poles.

• Discrete-time Case: We derive the analytical closed-form expressionsof the optimal tracking performance for discrete-time system. The keyidea proposed in this part is the application of the bilinear transfor-mation to obtain two key lemmas from the continuous-time counter-parts.

• Delta Domain Case: We reformulate and resolve the tracking perfor-mance problem of discrete-time system in terms of the delta operator.The existence of the sampling time explicitly in the expression enablesus to unify the all results.

• Delay-time Case: We demonstrate that by exploiting the delta do-main expression we can re-derive the existing result on the trackingperformance of the delay-time systems.

• Sampled-data Case: We provide a tiny contribution on tracking SISOsampled-data system. We derive the analytical closed-form expres-sion of the optimal tracking performance by implementing the fastsampling technique.

(ii) We provide the analytical closed-form expressions of the optimal regu-lation performance against an impulsive disturbance input, where theregulation performance is measured by the energy of the measurementoutput, possibly under control input and sensitivity constraints.• Continuous-time Case: We complete the existing results in continuous-

time case by deriving the analytical closed-form expression for non-minimum phase system and extend the problem to one with sensitiv-ity penalty.

• Discrete-time Case: We derive the analytical closed-form expressionsof the optimal regulation performance for discrete-time case. How-

1.5 Chapter Organization 9

ever, this will be the first outcome on regulation performance limita-tions of the discrete-time system.

• Delta Domain Case: We reformulate and resolve the regulation per-formance problem of discrete-time system in terms of the delta op-erator. We recover the continuous-time expressions by exhibiting thecontinuity property.

• Delay-time Case: Another small contribution is that we provide theanalytical closed-form expression of the optimal energy regulationperformance for simple SIMO delay-time system, where the plantis non-minimum phase and has only single unstable pole with puretime delay in the input port.

Regarding the continuity property of the delta domain solution, we showfor both optimal control problems that the limiting zeros of the discretizedsystems will not contribute any effects on the optimal performance providedthe sampling time is sufficiently small.

1.5 Chapter Organization

The remainder of this report is organized as follows. In Chapter 2 we statesome preliminaries. Here we introduce the notation used throughout this re-port and describe the considered feedback control system configuration. De-scription about plant decompositions and the set of all stabilizing controllerscan be found in this chapter. We put also in this chapter, the explanationabout delta operator and delta transform. Chapter 3 deals with the trackingperformance problem. We first describe the problem formulation and thenprovide the analytical closed-form expressions of the optimal tracking per-formance for continuous-time, discrete-time, delta domain, delay-time, andsampled-data cases, respectively. Chapter 4 is devoted to the optimal regu-lation problems, where we provide the analytical closed form expressions ofthe optimal regulation performance for continuous-time, discrete-time, deltadomain, and delay-time cases, respectively. We consider some application is-sues in Chapter 5. Here, we apply our results into three physical systems:three-disk torsional system, inverted pendulum system, and magnetic bear-ing system. Finally, some concluding statements are drawn in Chapter 6.

2

Preliminaries

2.1 Notation

We give a brief description of the notation used throughout this report. Wedenote the real set by R and the complex set by C. For any c ∈ C, its complexconjugate is denoted by c. For any vector u we shall use uT, uH, and ‖u‖ asits transpose, conjugate transpose, and Euclidean norm, respectively. We callthe one-dimensional subspace spanned by u the direction of u. For any twovectors u, v ∈ Cn, the angle between their directions is defined as

∠(u, v) = arccos|uHv|‖u‖‖v‖ .

For any matrix A ∈ Cm×n, we denote its conjugate transpose by AH andits column space by R[A]. The cardinality of a set S is denoted by #S. Ins-domain analysis, i.e., continuous-time case, let the open left half plane bedenoted byC− := s ∈ C : Re s < 0, the open right half plane byC+ := s ∈C : Re s > 0, and the imaginary axis by C0. And for any matrix functionf ∈ Cm×n we define f∼(s) := fT(−s). For any signal x(t), t > 0, we defineits Laplace transform x(s) by

x(s) = Lx(t) :=∫ ∞

0

x(t)e−st dt.

While in z-domain analysis, i.e., discrete-time case, the unit circle is denotedby ∂D := z ∈ C : |z| = 1. We also define the following sets: D := z ∈ C :|z| < 1, Dc := z ∈ C : |z| ≥ 1, and Dc := z ∈ C : |z| > 1. Clearly,D and Dc respectively can be seen as the regions inside and outside unitcircle. Furthermore, we define f∼(z) := fT(z−1). For any sequence x(k), k =0, 1, . . ., we define its Z transform x(z) by

x(z) = Zx(k) :=∞∑

k=0

x(k)z−k.

12 2 Preliminaries

Moreover, for g(s) measurable in C0 let define the Hilbert space

L2(C0) :=

g : ‖g‖22 :=12π

∫ ∞

−∞‖g(jω)‖2dω < ∞

,

in which the inner product is defined as

〈g1, g2〉 :=12π

∫ ∞

−∞gH1 (jω)g2(jω)dω. (2.1)

Similarly, for f(z) measurable in ∂D let define

L2(∂D) :=

f : ‖f‖22 :=12π

∫ π

−π

‖f(ejθ)‖2dθ < ∞

,

a Hilbert space with an inner product

〈f1, f2〉 :=12π

∫ π

−π

fH1 (ejθ)f2(ejθ) dθ. (2.2)

Next, define for g1(s) analytic in C+ and g2(s) analytic in C−,

H2(C0) :=

g1 : ‖g1‖22 := supσ>0

12π

∫ ∞

−∞‖g1(σ + jω)‖2dω < ∞

,

H⊥2 (C0) :=

g2 : ‖g2‖22 := supσ<0

12π

∫ ∞

−∞‖g2(σ + jω)‖2dω < ∞

,

and for f1(z) analytic in Dc and f2(z) analytic in D,

H2(∂D) :=

f1 : ‖f1‖22 := supr>1

12π

∫ π

−π

‖f1(rejθ)‖2dθ < ∞

,

H⊥2 (∂D) :=

f2 : ‖f2‖22 := supr<1

12π

∫ π

−π

‖f2(rejθ)‖2dθ < ∞

.

Then it is well-known that H2 and H⊥2 are subspaces of L2 containing func-tions that are analytic in C+ or Dc and C− or D, respectively. Also, they forman orthogonal pair of L2, i.e., for any h1 ∈ H2 and h2 ∈ H⊥2 , 〈h1, h2〉 = 0.Finally we denote by RH∞ the class of all stable and proper rational transferfunction matrices, or in discrete-time sense, the class of all rational matrixfunctions which are bounded and analytic in Dc.

2.2 Feedback Control Systems

In this present work, we shall consider the generic feedback configurationof finite dimensional LTI systems depicted in Fig. 2.1, which represents the

2.3 Plant Factorization 13

Fig. 2.1. The feedback control configuration.

standard unity feedback and one parameter control scheme. In this setup, Pdenotes the SIMO LTI plant and K the stabilizing controller. The plant canbe written as

P =

P1

P2

...Pm

, (2.3)

where Pi (i = 1, . . . ,m) are scalar transfer functions. We assume that thesystem is initially at rest and the feedback system is stable. We shall denoteby P (s) and P (z), the transfer functions of the plant P in s-domain and z-domain, respectively. More generally, henceforth we shall use the same sym-bol to denote a system and its transfer function, and whenever convenient, toomit the dependence on the frequency variables s and z. The signals r ∈ Rm,d ∈ R, u ∈ R, y ∈ Rm, and n ∈ Rm are the reference input, the disturbance in-put, the control input, the measurement output, and the measurement noise,respectively. The signal e := r − y ∈ Rm is the error. Hereafter, it will be as-sumed that all vectors and matrices involved in the sequel have compatibledimensions, and for simplicity their dimensions will be omitted.

A complex number z is said to be a zero of P if Pi(z) = 0. In addition,if z lies either in C+ for s-domain or Dc for z-domain then z is said to be anon-minimum phase zero. P is said to be minimum phase if it has no non-minimum phase zero; otherwise, it is said to be non-minimum phase. Onthe other hand, a complex number p is said to be a pole of P if P (p) is un-bounded. A pole p is said to be unstable if it lies in C+ or Dc. P is said to bestable if it has no unstable pole; otherwise, unstable. For technical reasons, itis assumed that the plant does not have zeros and poles at the same location.

2.3 Plant Factorization

2.3.1 Coprime Factorization

Two transfer function matrices F, G ∈ RH∞ are said to be right-coprime ifthey have equal number of columns and there exist matrices X,Y ∈ RH∞such that (

X Y) (

FG

)= XF + Y G = I.

14 2 Preliminaries

Or, in other words, the matrix (F, G)T is left-invertible inRH∞. Analogously,two transfer function matrices F,G ∈ RH∞ are said to be left-coprime if theyhave equal number of rows and there exist matrices X, Y ∈ RH∞ such that

(F G

)(XY

)= FX + GY = I,

or equivalently, the matrix ( F, G ) is right-invertible in RH∞.A plant transfer function P ∈ RH∞ is said to have right-coprime factor-

ization if P = NM−1, where N and M are right-coprime in RH∞. Similarly,P ∈ RH∞ is said to have left-coprime factorization if P = M−1N , where N

and M are left-coprime in RH∞. Here, it is tacitly assumed that M and Mbe square and non-singular. Then, P ∈ RH∞ is said to have doubly-coprimefactorizations if

P = NM−1 = M−1N , (2.4)

where N, M, N, M ∈ RH∞, and there exist X, Y, X, Y ∈ RH∞ that satisfythe double Bezout identity

[X −Y

−N M

] [M YN X

]= I, (2.5)

or equivalently, [M YN X

] [X −Y

−N M

]= I. (2.6)

If the transfer function of P is determined by

P =[

A BC D

], (2.7)

then the transfer function of the eight matrices are determined by

N =[

A + BF BC + DF D

], M =

[A + BF B

F I

],

N =[

A + HC B + HDC D

], M =

[A + HC H

C I

],

X =[

A + BF −HC + DF I

], Y =

[A + BF −H

F 0

],

X =[

A + HC −(B + HD)F I

], Y =

[A + HC −H

F 0

],

where matrix F is chosen such that (A + BF ) is stable and matrix H is se-lected such that (A + HC) is stable.

Based on the coprime factorization approach, the set of all controllers Kwhich stabilizes P is characterized by Youla parameterization [21, 59]

2.3 Plant Factorization 15

K := K : K = (Y −MQ)(NQ−X)−1

= (QN − X)−1(Y −QM); Q ∈ RH∞. (2.8)

Note that if P is tall, i.e., P is an SIMO plant, then N and N are tall, M andX are scalar, M and X are square, Y and Y are fat. Obviously, K and Q arefat.

For our analysis, we denote

N =

N1

N2

...Nm

, (2.9)

where Ni (i = 1, . . . , m) are scalar transfer functions. Furthermore, we mayalso define that p is the pole of P if and only if there is a unitary vector w suchthat M(p)w = 0. The unitary vector w is called the pole direction associatedwith p.

2.3.2 Inner-outer Factorization

A transfer function N , not necessarily square, is called an inner if N is inRH∞ and N∼N = I for all s = jω or z = ejθ and is called co-inner if N ∈RH∞ and NN∼ = I . A transfer function M is called outer if M is in RH∞and has a right inverse which is analytic in C+ or Dc. For an arbitrary P ∈RH∞,

P = ΘiΘo, (2.10)

where Θi is inner and Θo is outer, is defined as an inner-outer factorization ofP . We call Θi the inner factor and Θo the outer factor. If the transfer functionof P is given by (2.7), then the transfer functions of the discrete-time innerand outer factors, respectively, are determined by [35]

Θi(z) =[

A−BF BD−1s

C −DF DD−1s

], (2.11)

Θo(z) =[

A BDsF Ds

], (2.12)

where Ds be an appropriate surjective matrix satisfying DTs Ds = DTD +

BTPB, and F = (R+ BTPB)−1(BTPA + ST), with P is the solution of thediscrete-time algebraic Riccati equation

P = ATPA− (ATPB + S)(R+ BTPB)−1(BTPA + ST) +Q,

and Q = CTC, R = DTD, S = CTD.

16 2 Preliminaries

2.4 Delta Transforms

A book that provides a comprehensive account on delta operator is [44]. Thedelta operator δ is define as the following forward difference

δ , q − 1T

,

where q is the forward shift operator commonly used in discrete-time caseand T > 0 is the sampling time. For any sequence x(k), k = 1, 2, . . ., deltaoperator gives

δx(k) =(q − 1)x(k)

T=

qx(k)− x(k)T

=x(k + 1)− x(k)

T. (2.13)

By taking the Z transform of above equation we obtain

δx(z) =z − 1

Tx(z). (2.14)

We may say that in z-plane the delta operator will translate a point z ∈ Cone unit to the left and then scale it by factor of 1

T . Later, the variable δ isused as the delta operator variable and is analogous to the Laplace variable sfor continuous-time systems and the Z transform variable z for discrete-timesystems. We then obtain the relationship between variable z and variable δas follows,

δ =z − 1

T, (2.15)

z = Tδ + 1. (2.16)

For any sequence x(k) we define its delta transform by

Dx(k) = xT (δ) := T

∞∑

k=0

x(k)(Tδ + 1)−k, (2.17)

or equivalently,xT (δ) = T x(z)|z=Tδ+1. (2.18)

For T > 0 we define the following sets: ∂DT = δ ∈ C : |Tδ + 1| = 1,DT := δ ∈ C : |Tδ + 1| < 1, Dc

T := δ ∈ C : |Tδ + 1| ≥ 1, and DcT :=

δ ∈ C : |Tδ + 1| > 1. It is obvious that ∂DT can be seen as a circle centeredat δ = (− 1

T , 0) with radius 1T . Respectively, DT and Dc

T can be interpreted asregions inside and outside the circle. See Fig. 2.2.

For h(δ) measurable in ∂DT a Hilbert space L2 is defined as

L2(∂DT ) :=

h : ‖h‖22 :=

12π

∫ π/T

−π/T

∥∥∥∥h

(ejωT − 1

T

)∥∥∥∥2

dω < ∞

,

2.4 Delta Transforms 17

Fig. 2.2. The delta domain.

equipped with inner product

〈h1, h2〉 :=12π

∫ π/T

−π/T

hH1

(ejωT − 1

T

)h2

(ejωT − 1

T

)dω. (2.19)

In a similar manner we may construct the orthogonal pairs H2(∂DT ) andH⊥2 (∂DT ). The well-known Parseval’s identity is also derived in this domain:‖xT (k)‖22 = ‖xT (δ)‖22, where xT (k) := x(kT ) and ‖xT (k)‖22 , T

∑∞k=0 |xT (k)|2.

Let F (z) be given and define GT (δ) := F (Tδ+1). Then by setting θ = ωT ,it is not difficult to verify that the following H2 norms relation holds:

‖GT (δ)‖22 =‖F (z)‖22

T. (2.20)

This norm property plays an important role in our subsequent derivation,particularly in the derivation of the optimal tracking and regulation perfor-mances in delta domain.

Finally, for any matrix function f ∈ Cm×n we define f∼(δ) := fT( −δTδ+1 ).

3

Tracking Performance Limitations

In this chapter we formulate and solve the optimal tracking performanceproblems for SIMO LTI continuous-time and discrete-time systems. We studyfirst the tracking error problem and then extend the results to the trackingerror problem under control input penalty, possibly for stable and unstablesystems. Our primary attention in this work is on discrete-time case sincethe continuous-time results are already available. Existing results on trackingperformance limitation of continuous-time systems are initially establishedin [12, 31]. We shall cite these results with corrections since in some partsthey contain misleading. To solve the tracking error problem for discrete-time system we mainly encouraged by the continuous-time results of [12].The approach used in [12] can be explained as follows. Based on Youla pa-rameterization of all stabilizing controllers and the inner-outer factorizationof the plant, an H2 optimal control problem is built. By invoking a lemma,the general formula for optimal tracking error in terms of inner-outer factorsis derived. And by applying another lemma, the analytical closed-form ex-pressions of the optimal tracking performance are then obtained. Since weintend to bring this approach into discrete-time setting, the existence of twokey lemmas in z-domain is inevitable.

Subsequently, we reformulate and resolve the tracking problem in thedelta domain, from which we then show the continuity property, i.e., weshow that we can completely recover the continuous-time result from thedelta domain result stand point by approaching the sampling time to zero.Additionally, we also benefit our results for SIMO discrete-time systems toderive the optimal tracking performance of sampled-data systems. Appli-cations of our results on the tracking problem of three-disk torsional sys-tem and inverted pendulum system can be found in Chapter 5. In the lattercase we show that how the analytical closed-form expression of the optimalperformance can be exploited in determining the optimal parameters of in-verted pendulum system. We start this chapter by describing the trackingproblem formulation and then we provide our results in continuous-timeand discrete-time, respectively.

20 3 Tracking Performance Limitations

Fig. 3.1. The tracking setup.

3.1 Tracking Performance Problem

In this section we provide the formulation of theH2 optimal tracking perfor-mance problems, which consist of the tracking error problem and that undercontrol input penalty. For both problems we consider the feedback controlconfiguration depicted by Fig. 3.1, where we assume that the system’s out-put y is resulted solely from the reference input r. Let the input and outputsensitivity functions be defined by

Si := (1 + KP )−1, (3.1)So := (I + PK)−1, (3.2)

respectively. We consider step functions as the reference input signal, definedby

r(t) =

ν, t ≥ 00, t < 0 , r(s) =

ν

s, (3.3)

r(k) =

ν, k ≥ 00, k < 0 , r(z) =

zν

z − 1. (3.4)

where ν = (ν1, ν2, . . . , νm)T is a constant unitary vector and specifies thedirection of the reference input.

3.1.1 Tracking Error Problem

For a given input signal r, we define the tracking error ability as

Jc :=∫ ∞

0

‖r(t)− y(t)‖2 dt =∫ ∞

0

‖e(t)‖2 dt, (3.5)

Jd :=∞∑

k=0

‖r(k)− y(k)‖2 =∞∑

k=0

‖e(k)‖2, (3.6)

for continuous-time and discrete-time cases, respectively. Since e = Sor,where So is the output sensitivity function defined in (3.2), it follows fromthe well-known Parseval identity that

Jc = ‖So(s)r(s)‖22 =12π

∫ ∞

−∞‖So(jω)r(jω)‖2 dω,

Jd = ‖So(z)r(z)‖22 =12π

∫ π

−π

‖So(ejθ)r(ejθ)‖2 dθ.

3.1 Tracking Performance Problem 21

We can see that the tracking abilities are measured by the H2 norms of thetracking error. The best achievable tracking performances by all stabilizingcontrollers in K are then determined by

J∗c = infK∈K

Jc,

J∗d = infK∈K

Jd.

Since So = (X −NQ)M , we can further deduce

J∗c = infQ∈RH∞

∥∥∥(X −NQ)Mν

s

∥∥∥2

2, (3.7)

J∗d = infQ∈RH∞

∥∥∥∥(X −NQ)Mν

z − 1

∥∥∥∥2

2

, (3.8)

where N and M are the coprime factors of P governed in (2.4) and X is thecorresponding Bezout matrix given in (2.5). Note that (3.8) holds by takinginto account that z is an inner function.

3.1.2 Tracking Error Problem under Control Input Penalty

We may extend the preceding problem to the H2 tracking error problem un-der control input penalty. This problem is more realistic than one withoutpenalty on the control input, since the controller could not produce an outputbeyond the capability of the actuator. In this case we consider the followingperformance indexes

Jc :=∫ ∞

0

(‖e(t)‖2 + |uw(t)|2)dt, (3.9)

Jd :=∞∑

k=0

(‖e(k)‖2 + |uw(k)|2) , (3.10)

for continuous-time and discrete-time cases, respectively. Here uw is theweighted control input, i.e.,

uw(t) = L−1Wu(s)u(s),uw(k) = Z−1Wu(z)u(z),

with stable and minimum phase weighting function Wu. Note that if Wu = 0the problem then reduces to a tracking error one, which has been discussedin previous subsection. It follows from the Parseval’s identity that

J = ‖e‖22 + ‖uw‖22where J stands either Jc for continuous-time case or Jd for discrete-time case.Further, we can write


J = ‖Sor‖22 + ‖WuKSor‖22.

Using coprime factorizations (2.4), double Bezout identity (2.5), and Youlaparameterization (2.8) yields

J =∥∥∥(X −NQ)M r

∥∥∥2

2+

∥∥∥Wu(Y −MQ)M r∥∥∥

2

2

for a free parameter Q ∈ RH∞. The optimal performance achievable by allstabilizing controllers then can be determined by

J∗ = infQ∈RH∞

∥∥∥∥[

WuYX

]−

[WuM

N

]Q

M r

∥∥∥∥2

2

, (3.11)

where J∗ stands either J∗c for continuous-time case or J∗d for discrete-timecase.

3.1.3 Plant Augmentation

To solve the tracking error problem under control penalty, we adopt the keyidea of augmented plant initially introduced in [31]. An augmented plant Pa

is defined as

Pa =[

Wu

P

], (3.12)

from which we then slightly modify the feedback control configuration de-picted by Fig. 3.1 to one given by Fig. 3.2. We then obtain the correspondingstep input signal ra with direction νa = (0, νT)T and the performance indexes

Jca :=∫ ∞

0

‖ea(t)‖2 dt, (3.13)

Jda :=∞∑

k=0

‖ea(k)‖2, (3.14)

where

ea :=[

0r

]−

[uw

y

].

One of the key points addressed by this strategy is that the performanceindexes do not explicitly include the control input penalty u. Furthermore,the corresponding right and left coprime factorizations of Pa are derived as

Pa = NaM−1a = M−1

a Na, (3.15)

where

Na =[

WuMN

], Ma = M, Ma =

[1 00 M

], Na =

[Wu

N

],

3.2 Continuous-time Case 23

Fig. 3.2. The tracking configuration with augmented plant.

and the corresponding double Bezout identity is written as[

Xa −Ya

−Na Ma

] [Ma Ya

Na Xa

]= I, (3.16)

where

Xa =[

1 WuY0 X

], Ya = (0, Y ), Xa = X, Ya = (0, Y ).

For a free parameter Qa = (Q1, Q2) ∈ RH∞, the optimal performance indexJ∗a , which is either J∗ca or J∗da, can be expressed as

J∗a = infQa∈RH∞

‖(Xa −NaQa)Mara‖22, (3.17)

and then

J∗a = infQ2∈RH∞

∥∥∥∥[

WuYX

]−

[WuM

N

]Q2

M r

∥∥∥∥2

2

. (3.18)

The expression of J∗a in (3.18) is exactly equivalent with that of J∗ in (3.11)for the original plant P . By taking into account that there is no penalty tothe control input to be imposed, we can immediately follow the result of thetracking error problem.

3.2 Continuous-time Case

Results on tracking performance limitations of continuous-time systems canbe found in [12, 14, 17, 31]. The results presented in this section mainly citedfrom those references. Especially for unstable case, we make essential correc-tion on the results of [12, 31].

3.2.1 Two Lemmas

We state two lemmas which play important roles in the derivation. For anyfunction g(s) analytic in C+ we define the following class of function

F :=

g : limR→∞

maxθ∈[−π/2,π/2]

|g(Rejθ)|R

= 0

. (3.19)


The above class consists of functions with restricted behavior at infinity. Bythis, we intend to deal with integration over a contour that becomes arbitrar-ily long [53]. Generally speaking, if g is analytic and bounded magnitude inC+, then g is of class F. The following two lemmas can be found in [12].

Lemma 3.1 Let g(s) ∈ F be an analytic function in C+. Denote that g(jω) =g1(ω) + jg2(ω). Suppose that g(s) is conjugate symmetric, i.e., g(s) = g(s). Then

g′(0) =1π

∫ ∞

−∞

g1(ω)− g1(0)ω2

dω. (3.20)

Proof. See [53, pp. 50]. ¥

Lemma 3.2 Let g(s) be a meromorphic function in C+ and has no zero or pole onjω-axis. Suppose that g(s) is conjugate symmetric and log g(s) ∈ F. Also, supposethat zi ∈ C+ (i = 1, . . . , nz) and pk ∈ C+ (k = 1, . . . , np) are, respectively, non-minimum phase zeros and unstable poles of g(s), all counting their multiplicities.Provided that g(0) 6= 0, then

1π

∫ ∞

−∞log

∣∣∣∣g(ω)g(0)

∣∣∣∣dω

ω2=

nz∑

i=1

2zi−

np∑

k=1

2pk

+g′(0)g(0)

. (3.21)

Proof. See [13]. ¥

Remark: A meromorphic function on an open subset of the complex planeis a function that is analytic in all except a set of isolated points, which arepoles for the function.


Now we provide the analytical closed-form expressions of the optimal track-ing error performance [12], i.e., we derive the minimal value of the trackingerror performance (3.5):

J∗c = infK∈K

∫ ∞

0

‖e(t)‖2 dt,

which is further given by (3.7). First we give the result for stable case and thenwe extend it to unstable case. For the finiteness of Jc we made the followingstandard assumption.

Assumption 3.1 N(0) 6= 0.

Assumption 3.2 For r(t) in (3.3), ν ∈ R[N(0)].


In order for Jc to be finite, it is obvious the output sensitivity function So(s)must have a zero at s = 0 with input zero direction ν, i.e. So(0)ν = 0. Condi-tion N(0) 6= 0 is then required to avoid any hidden pole-zero cancellation ats = 0 so that the open loop system has an integrator. Condition ν ∈ R[N(0)]requires that the input signal must enter from direction lying in the columnspace of N(0) and gives the condition of step reference signal r(t) that a non-right invertible plant P (s) may track.

We denote by pk ∈ C+ (k = 1, . . . , np) the unstable poles of P (s) and byzij ∈ C+ (i = 1, . . . , m, j = 1, . . . , ni) the non-minimum phase zeros of Pi(s),and define the following index sets:

Kz := i : Ni(0) 6= 0,Kp := k : M(pk)ν = 0,Kpi := k : Ni(pk) = 0 (i = 1, . . . ,m).

Note that Kp contains the index of unstable poles whose direction is coin-cident with that of step input signal r(t). While, since N = PM then Kpi

contains the index of unstable poles of P (s) but not those of Pi(s). The indexset Kpi will play a key role for correcting an error in existing result shownin [12, 31].

Stable Plants

Since P (s) is stable, then we may choose N = N = P , X = M = 1, X =M = I , and Y = Y = 0. We define the inner-outer factorization such that

P (s) = Θi(s)Θo(s). (3.22)

We then may write (3.7) as

J∗c = infQ∈RH∞

∥∥∥(I −ΘiΘoQ)ν

s

∥∥∥2

2. (3.23)

Theorem 3.1 (Stable Plant [12]) Suppose that the SIMO plant P (s) given in(2.3) is stable and Pi(s) has non-minimum phase zeros zij (i = 1, . . . , m, j =1, . . . , ni). Then, under Assumptions 3.1 and 3.2, the optimal tracking performanceis given by

J∗c =∑

i∈Kz

ν2i

ni∑

j=1

2Re zij

|zij |2 +1π

∑

i∈Kz

ν2i

∫ ∞

0

log[ |Pi(0)|2‖P (0)‖2

‖P (jω)‖2|Pi(jω)|2

]dω

ω2. (3.24)

Proof. See [12]. ¥


Unstable Plants

We now extend the problem to an unstable plant. We define the inner-outerfactorization such that

N(s) = Θi(s)Θo(s). (3.25)

Theorem 3.2 (Unstable Plant) Suppose that the SIMO plant P (s) given in (2.3)has unstable poles pk (k = 1, . . . , np) and Pi(s) has non-minimum phase zeroszij (i = 1, . . . , m, j = 1, . . . , ni). Then, under Assumptions 3.1 and 3.2, the optimaltracking performance is given by

J∗c = J∗cs + J∗cu, (3.26)

where

J∗cs = Jcs1 + Jcs2,

J∗cu = Jcu1 + Jcu2

with

Jcs1 :=∑

i∈Kz

ν2i

ni∑

j=1

2Re zij

|zij |2 ,

Jcs2 :=1π

∑

i∈Kz

ν2i

∫ ∞

0

log[ |Pi(0)|2‖P (0)‖2

‖P (jω)‖2|Pi(jω)|2

]dω

ω2,

Jcu1 :=∑

i∈Kz

ν2i

∑

k∈Kpi

2Re pk

|pk|2 ,

Jcu2 :=∑

k,`∈Kp

4Re pk Re p`

(pk + p`)pkp`σkσ`(1−Θ∼i (pk)Θi(0))(1−Θ∼i (p`)Θi(0)),

and

σk :=

1 ; #Kp = 1∏

`∈Kp,` 6=k

p` − pk

p` + pk; #Kp ≥ 2.

Proof. The proof completely follows that of [12, Theorem 3.3] except for fewpoints. Let Θi in (3.25) is represented by Θi(s) = [w1(s), . . . , wm(s)]T. Theessential correction should be made in the evaluation of the set of non-minimum phase zeros of wi (i = 1, . . . ,m). Since wi is the element of theinner factor Θi, we can readily see that the set of non-minimum phase zerosof wi(s) is same as that of Ni(s), which includes the unstable poles of P (s)but not those of Pi(s) as well as non-minimum phase zeros of Pi(s), i.e., it isimmediate from Lemma 3.2 that

w′i(0)wi(0)

= −ni∑

j=1

2Re zij

|zij |2 −∑

k∈Kpi

2Re pk

|pk|2 +1π

∫ ∞

0

log∣∣∣∣wi(jω)wi(0)

∣∣∣∣dω

ω2.


The last point was not properly recognized in the proof of [12, Theorem 3.3],and hence the term Jcu1 caused by unstable poles is missing in the theorem.Once the correction above is made, the final formula can be derived in thesame way as in [12]. ¥

Remark: We make a couple of remarks on Theorem 3.2.

• In [12, Theorem 3.3], the first three terms in Theorem 3.2 are denoted bya single quantity J∗s , which is the optimal tracking error correspondingto the stable part of the plant Ps. One may presume that the closed-formexpression for J∗s is given previously by [12, Theorem 3.2], which is de-voted to the stable case. However, the inner-outer factorization in stablecase is given by P = ΘiΘo and that in unstable case by Ps = ΘiΘo. Itindicates that the inner factors in the two cases produce different sets ofnon-minimum phase zeros. Theorem 3.2 provides more accurate expres-sions by counting separately the effects caused by non-minimum phasezeros of the plant and those by other factors, i.e., some unstable poles ofthe plant.

• The expression in the theorem is not complete for SIMO unstable plantsince it includes an inner factor Θi(s) in Jcu2. We can obtain the closed-form expression of Θi(s) only for the SISO case and some simple/specialcases. See Corollary 3.1, and for discrete-time case see Corollary 3.4.

• There exists a special case where we can see the term Jcu2 caused by un-stable poles is zero even if the plant is unstable. See Corollary 3.3 for thediscrete-time case.

• We can also show that Jcu = Jcu1 + Jcu2 is zero when the sets of all un-stable poles of Pi(s) (i = 1, . . . ,m) are completely the same, since we cansee Kpi is empty for all i. Moreover, Kp is also empty for such a case.The case often happens for practical applications where we have onlyone actuator but we may add one or more extra sensors. The extra sensorcan dramatically improve the tracking performance for unstable and non-minimum phase plants as seen in an example of an inverted pendulumin Section 5.2.

• As noted above, the setKp contains the indices of unstable poles in whichsatisfy M(pk)ν = 0. In some cases of SIMO system, it is not so easy toobtain M matrix such that we can obtain the pole direction associatedwith pk. In SISO case, without loss of generality, it can be appointed suchthat

M(s) = M(s) :=np∏

k=1

s− pk

s + pk.

It is obvious that M(pk) = 0. Also note that if P (s) is an SISO plant, thenJcs2 = 0 and Jcu1 = 0.

In general, it is difficult to find the analytical expression of Θi for SIMOplant. But, in SISO case which plant has non-minimum phase zeros zi (i =


1, . . . , nz), the inner factor Θi in (3.25), without loss of generality, can be fixedas

Θi(s) =nz∏

i=1

zi − s

zi + s,

from which we get Θi(0) = 1. Let define ϕ(s) := Θ∼i (s)Θi(0), i.e.,

ϕ(s) :=nz∏

i=1

zi + s

zi − s,

then we state the tracking performance limitations for scalar systems in thefollowing result.

Corollary 3.1 Let P (s) be an SISO plant which has non-minimum phase zeroszi (i = 1, . . . , nz) and unstable poles pk (k = 1, . . . , np). Then,

J∗c =nz∑

i=1

2Re zi

|zi|2 +np∑

k,`=1

4 Re pk Re p`(1− ϕ(pk))(1− ϕ(p`))(pk + p`)pkp`σkσ`

. (3.27)

Example 3.1 We pick a simple example to illustrate the preceding result. We con-sider a scalar system whose transfer function is given by

P (s) =s− z

s− p.

Firstly we fix p = −1, i.e., we consider a stable plant, and compute the optimalperformance for z from 1 to 2. Fig. 3.3 confirms that non-minimum phase zero closedto imaginary axis deteriorates the performance. Secondly, we fix z = 1 and vary pfrom 0.1 to 2. Fig. 3.4 shows that whenever p closes to 1 then the optimal performancebecomes worse since it happens almost unstable pole-zero cancelation.

For both cases, we compute the optimal performance in two manners: by usingMATLAB toolbox and by using analytical closed-form expression in Corollary 3.1.We see that these two computations match well. In general, the optimal performanceof a system with one unstable pole p and one non-minimum phase zero z is governedby

J∗c =2z

+8p

(z − p)2.


In this part we consider the tracking error problem under control penalty,which has been studied in [31]. We note that the result presented in [31] alsocontains a small mistake and we correct it here. We provide the minimalvalue of the performance index (3.9), i.e.,

J∗c = infK∈K

∫ ∞

0

(‖e(t)‖2 + |uw(t)|2) dt,


−1 −0.5 0 0.5 1 1.5 20

5

10

15

20

25

z

J* c

by Matlab Toolboxby analytical expression

Fig. 3.3. The tracking error performance for stable continuous-time system (Example3.1).

0 0.5 1 1.5 20

100

200

300

400

500

600

700

800

900

p

J* c


Fig. 3.4. The tracking error performance for unstable continuous-time system (Exam-ple 3.1).

which is also given by (3.11). By adopting the augmented plant strategy wemay easily solve the problem since there is no penalty to the control inputto be imposed. Hence, we can immediately invoke the result of the trackingerror problem, i.e., Theorem 3.2.

We make an additional assumption for system plant as follow.

Assumption 3.3 P (s) has a pole at s = 0.

It is obvious that in order to make the steady state zero, the open-loop trans-fer function P (s)K(s) must contain an integrator. Consequently, plant P (s)must have an integrator instead of compensator K(s) may have an integratorto maintain a finite control energy cost. Assumption 3.3 is then necessary.

Note that, by plant augmentation strategy, the optimal performance (3.11)can be further expressed as


J∗c = infQa∈RH∞

∥∥∥[I + Na(Ya −QaMa)]νa

s

∥∥∥2

2. (3.28)

Now we provide the analytical closed-form solution for unstable case. Wedefine an inner-outer factorization such that

Na(s) :=[

Wu(s)M(s)N(s)

]= Θi(s)Θo(s). (3.29)

Theorem 3.3 Suppose that the SIMO plant P (s) given in (2.3) has unstable polespk (k = 1, . . . , np) and Pi(s) has non-minimum phase zeros zij (i = 1, . . . ,m, j =1, . . . , ni). Then, under Assumptions 3.1–3.3, the optimal tracking performance un-der control input penalty is given by

J∗c = J∗cs + J∗cu, (3.30)

where

J∗cs = Jcs1 + Jcs2,

J∗cu = Jcu1 + Jcu2

with

Jcs1 :=∑

i∈Kz

ν2i

ni∑

j=1

2Re zij

|zij |2 ,

Jcs2 :=1π

∑

i∈Kz

ν2i

∫ ∞

0

log[ |Pi(0)|2‖P (0)‖2

‖P (jω)‖2 + |Wu(jω)|2|Pi(jω)|2

]dω

ω2,

Jcu1 :=∑

i∈Kz

ν2i

∑

k∈Kpi

2Re pk

|pk|2 ,

Jcu2 :=∑

k,`∈Kp

4Re pk Re p`

(pk + p`)pkp`σkσ`(1−Θ∼i (pk)Θi(0))(1−Θ∼i (p`)Θi(0)),

and

σk :=

1 ; #Kp = 1∏

`∈Kp,` 6=k

p` − pk

p` + pk; #Kp ≥ 2.

Proof. We correct a small mistake made in [31]. We apply Theorem 3.2 toPa(s) instead of P (s). Since the first element of νa is equal to zero, then thereexists no extra term in J∗c related to Wu(s). By Assumption 3.6 we concludethat |Pi(0)| and ‖P (0)‖ are infinite but |Wu(0)| is finite. Then the following

|Pi(0)|2‖Pa(0)‖2 =

|Pi(0)|2‖P (0)‖2 + |Wu(0)|2 =

|Pi(0)|2‖P (0)‖2

holds. Also note that


‖Pa(jω)‖2|Pi(jω)|2 =

‖P (jω)‖2 + |Wu(jω)|2|Pi(jω)|2 .

The proofs for Jcs1, Jcu1, and Jcu2 are similar as those of Theorem 3.2. ¥

Problem of minimizing the tracking error under control input penaltygenerally provides additional limits imposed by the weighting function Wu,which appears in the logarithmic term in Jcs2 and in the inner factor Θi inJcu2. In general, we do not know the closed-form expression for Θi eventhough the plant P is scalar. Note that if we set Wu(s) = 0, i.e., non-penaltycase, then we can completely recover Theorem 3.2.

The expression in Theorem 3.3 is complete for SIMO marginally stableplants in a sense that the best achievable tracking performance with con-trol input penalty is characterized by non-minimum phase zeros and gain ofthe plant without using any inner-outer factorization or solving any Riccatiequation. See Corollary 3.2 below for the SISO case.

For scalar system, Theorem 3.3 can be further simplified as shown in thefollowing corollary.

Corollary 3.2 Suppose that the SISO plant P (s) is marginally stable and has non-minimum phase zeros zi (i = 1, . . . , nz). Then, under Assumptions 3.1–3.3, theoptimal tracking performance under control penalty is given by

J∗c =nz∑

i=1

2Re zi

|zi|2 +1π

∫ ∞

0

log[1 +

|Wu(jω)|2|P (jω)|2

]dω

ω2. (3.31)

For some simple cases, we may still obtain the analytical closed-form ex-pression of J∗c even the plant P (s) is unstable as shown in the followingexample.

Example 3.2 We consider an SISO plant

P (s) =s− 1

s(s− p).

The plant has one non-minimum phase zero at z = 1 and possibly one unstable poleat z = p, provided p > 0. We calculate the tracking performance under control inputpenalty with Wu(s) = 1. In the inner-outer factorization (3.29), fortunately we canobtain the closed-form expression of Θi as follows

Θi(s) =1

s2 +√

3 + p2s + 1

[s(s− p)s− 1

].

Hence, the optimal tracking performance is then given by

J∗c = 2 +1π

∫ ∞

0

log[1 +

1|P (jω)|2

]dω

ω2+

2p

[1− p + 1

p2 − p√

3 + p2 + 1

]2

.


0 0.5 1 1.5 20

500

1000

1500

2000

2500

3000

3500

4000

p

J* c


Fig. 3.5. The tracking error performance with control input penalty of unstablecontinuous-time system (Example 3.2).

Fig. 3.5 plots the optimal performance versus the location of unstable pole p. It isconfirmed that whenever p approaches 1 then the performance blows up. It is alsointeresting to note that the optimal tracking error performance with control inputpenalty in general is much greater than that without penalty, see Fig. 3.4 for com-parison.

3.3 Discrete-time Case

In this section we provide the discrete-time solutions of the optimal track-ing performance for discrete-time systems. The derivations performed in thiscase are parallel with that of continuous-time case.

3.3.1 Two Lemmas

The following two lemmas which are counterparts with Lemmas 3.1 and3.2 play inevitable roles in the derivation of the optimal performance fordiscrete-time system. Recall the class of function F defined in (3.19).

Lemma 3.3 Let f(z) ∈ F be an analytic function in Dc. Denote that f(ejθ) =f1(θ) + jf2(θ). Suppose that f(z) is conjugate symmetric, i.e., f(z) = f(z). Then

f ′(1) =12π

∫ π

−π

f1(θ)− f1(0)1− cos θ

dθ. (3.32)

Proof. Consider Lemma 3.1. The key idea of this proof emerges from a factthat for every stable rational transfer function g(s), analytic in C+, then thefunction

3.3 Discrete-time Case 33

f(z) := g

(z − 1z + 1

)(3.33)

is analytic in Dc. In other words, we consider the well-known bilinear orTustin transformation

s :=z − 1z + 1

. (3.34)

On the boundary this transformation is

jω :=ejθ − 1ejθ + 1

,

from which we get the correspondence between f(ejθ) = f1(θ) + jf1(θ) andg(jω) = g1(ω) + jg2(ω). We then may express ω, ω2, and dω as functions of θ,respectively, as follow

ω =sin θ

1 + cos θ, ω2 =

1− cos θ

1 + cos θ, dω =

11 + cos θ

dθ.

From (3.33), we obtain g′(0) = 2f ′(1). The proof is then completed by substi-tuting all appropriate terms to (3.20). ¥Lemma 3.4 Let f(z) be a meromorphic function in Dc and has no zero or pole on∂D. Suppose that f(z) is conjugate symmetric and log f(z) ∈ F. Also, suppose thatηi ∈ Dc (i = 1, . . . , nη) and λk ∈ Dc (k = 1, . . . , nλ) are, respectively, the non-minimum phase zeros and unstable poles of f(z), all counting their multiplicities.Provided that f(1) 6= 0, then

12π

∫ π

−π

log∣∣∣∣f(ejθ)f(1)

∣∣∣∣dθ

1− cos θ=

nη∑

i=1

|ηi|2 − 1|ηi − 1|2 −

nλ∑

k=1

|λk|2 − 1|λk − 1|2 +

f ′(1)f(1)

. (3.35)

Proof. The proof of this lemma is similar to that of Lemma 3.3. ¥Remark: The bilinear transformation, or Tustin transformation, is provedvery useful in relating continuous-time variable s and discrete-time variablez, especially in producing the discrete-time counterpart of an continuous-time lemma. Here we present two other lemmas obtained by invoking bilin-ear transformation to their continuous-time counterparts: Lemmas 4.3 and4.4, respectively.

Lemma 3.5 Let f(z) ∈ F be an analytic function in Dc. Suppose that f(z) is con-jugate symmetric and denote f(ejθ) = f1(θ) + jf2(θ). Then

−2f ′(−1) =1π

∫ π

−π

f1(θ)− f1(π)1 + cos θ

dθ. (3.36)

Lemma 3.6 Consider a conjugate symmetric function f(z) ∈ F. Suppose that f(z)is analytic and has no zero in Dc except possibly at z = ∞ with multiplicity v.Provided that f(−1) 6= 0, then

−2f ′(−1)f(−1)

+ 2v =1π

∫ π

−π

log∣∣∣∣f(ejθ)f(−1)

∣∣∣∣dθ

1 + cos θ. (3.37)



The tracking error problem in discrete-time setting can be stated as minimiz-ing the performance index (3.6), i.e., we determine the analytical closed-formexpression of

J∗d = infK∈K

∞∑

k=0

‖e(k)‖2,

which can further be expressed as (3.8). For the finiteness of Jd, we make thefollowing standard assumptions.


Assumption 3.5 For r(k) in (3.4), ν ∈ R[N(1)].

We can see that these two assumptions are the counterparts of Assumptions3.1 and 3.2, respectively. Assumption 3.4 requires the open loop system hasan integrator and Assumption 3.5 gives a condition that a non-right invert-ible plant P (s) may track the step input signal r(k).

We denote by λk (k = 1, . . . , nλ) the unstable poles of P (z) and by ηij (i =1, . . . , m, j = 1, . . . , ni) the non-minimum phase zeros of Pi(z). To facilitateour analysis we define the following index sets:

Jz := i : Ni(1) 6= 0Jp := k : M(λk)ν = 0Jpi := k : Ni(λk) = 0 (i = 1, . . . , m).

Stable Plants

For a given stable plant P (z), let introduce the inner-outer factorization ofthe plant P (z) as follows,

P (z) = Θi(z)Θo(z). (3.38)

Then we may write (3.8) as


∥∥∥∥(I −ΘiΘoQ)ν

z − 1

∥∥∥∥2

2

. (3.39)

Theorem 3.4 (Stable Plant) Suppose that the SIMO plant P (z) given in (2.3) isstable and Pi(z) has non-minimum phase zeros ηij (i = 1, . . . , m, j = 1, . . . , ni).Then, under Assumptions 3.4 and 3.5, the optimal tracking performance is given by

J∗d =∑

i∈Jzν2

i

ni∑

j=1

|ηij |2 − 1|ηij − 1|2 +

12π

∑

i∈Jzν2

i

∫ π

0

log[ |Pi(1)|2‖P (1)‖2

‖P (ejθ)‖2|Pi(ejθ)|2

]dω

1− cos θ.

(3.40)


Proof. The proof of this theorem is almost parallel with that of Theorem 3.1,which is given in [12]. In (3.39), matrix Q is to be selected such that [I −Θi(1)Θo(1)Q(1)]ν = 0 in order for J∗d to be finite. Define

Ψ(z) :=[

Θ∼i (z)

I −Θi(z)Θ∼i (z)

]. (3.41)

We can easily verify that Ψ is a norm preserving function, i.e., Ψ∼(ejθ)Ψ(ejθ) =I . So that by pre-multiplying Ψ to (3.39) we have


∥∥∥∥Ψ(I −ΘiΘoQ)ν

z − 1

∥∥∥∥2

2

= infQ∈RH∞

∥∥∥∥(Θ∼

i −ΘoQ)νz − 1

∥∥∥∥2

2

+∥∥∥∥

(I −ΘiΘ∼i )ν

z − 1

∥∥∥∥2

2

= infQ∈RH∞

∥∥∥∥(Θ∼

i −Θ∼i (1)) + (Θ∼i (1)−ΘoQ)ν

z − 1

∥∥∥∥2

2

+∥∥∥∥

(I −ΘiΘ∼i )ν

z − 1

∥∥∥∥2

2

.

By noting that(Θ∼

i −Θ∼i (1))ν

z − 1∈ H⊥2 ,

we may select Q ∈ RH∞ such that Θ∼i (1)−Θo(1)Q(1) = 0, and therefore

(Θ∼i (1)−ΘoQ)νz − 1

∈ H2.

As a result


∥∥∥∥(Θ∼

i (1)−ΘoQ)νz − 1

∥∥∥∥2

2

+∥∥∥∥

(Θ∼i −Θ∼i (1))νz − 1

∥∥∥∥2

2

+∥∥∥∥

(I −ΘiΘ∼i )ν

z − 1

∥∥∥∥2

2

.

Because Θo is outer, we can always select a Q such that

infQ∈RH∞

∥∥∥∥(Θ∼i (1)−ΘoQ)ν

z − 1

∥∥∥∥2

2

= 0.

And then by using H2 norm definition (2.2) we obtain

J∗d =∥∥∥∥

(Θ∼i −Θ∼

i (1))νz − 1

∥∥∥∥2

2

+∥∥∥∥

(I −ΘiΘ∼i )ν

z − 1

∥∥∥∥2

2

= − 12π

∫ π

−π

ReνHΘi(ejθ)Θ∼i (1)ν

− 11− cos θ

dθ.

Let us define f(z) := νHΘi(z)Θ∼i (1)ν. Under Assumption 3.5, we obtain

f(1) = 1. Applying Lemma 3.3 then yields

J∗d = −f ′(1) = −νHΘ′i(1)Θ∼i (1)ν. (3.42)


Denote the inner factor Θi(z) as follows,

Θi(z) =(w1(z), w2(z), . . . , wm(z)

)T.

According to Assumption 3.5, we may select ν = Θi(1) without loss of gen-erality. Then (3.42) becomes

J∗d = −m∑

i=1

wi(1)w′i(1) = −∑

i∈Jzν2

i

w′i(1)wi(1)

.

Note that i ∈ Jz assures that wi(1) 6= 0. Since wi(z) is element of inner factorΘi(z), then it has the same set of non-minimum phase zeros as Pi(z). Hence,by invoking Lemma 3.3 we have

w′i(1)wi(1)

= −ni∑

j=1

|ηij |2 − 1|ηij − 1|2 +

12π

∫ π

−π

log∣∣∣∣wi(ejθ)wi(1)

∣∣∣∣dθ

1− cos θ. (3.43)

And by noting that |wi(ejθ)| = |Pi(ejθ)|/‖P (ejθ)‖ we obtain


∣∣∣∣ = −12

log[ |Pi(1)|2‖P (1)‖2


].

This completes the proof. ¥

Theorem 3.4 demonstrates that the tracking performance for stable plantwith respect to step input signal generally does not only depend on the plantnon-minimum phase zeros but also on the plant gain. The first term of (3.4)denotes the effects contributed by the non-minimum phase zeros and thesecond term, i.e., integral term, denotes those of the plant direction. Notethat the expression shares some similarities with that of continuous-time casein Theorem 3.1. Beyond already known in SISO/MIMO cases [58], the non-minimum phase zeros of SIMO plant may contribute their effects in an un-usual fact depending on the input direction νi. To explain this, let consider asimple stable single-input two-output (SITO) plant as follows,

P (z) =[

P1(z)P2(z)

]=

1z − 1

2

[z − η

z

], |η| > 1.

Assumption 3.5 requires that the step input direction ν must be in the columnspace of P (1), i.e.,

ν =1√

(1− η)2 + 1

[1− η

1

],

which is a unitary vector. Then the first term of (3.40), denote by J∗dη , can bewritten as

J∗dη =(1− η)2

(1− η)2 + 1η2 − 1

(η − 1)2=

η2 − 1(1− η)2 + 1

,


which led tolimη→1

J∗dη = 0.

This simple case thus reveals a fact different from that already known in SISOand MIMO systems, i.e, in SIMO system a non-minimum phase zero close tothe point z = 1 may not give an effect.

An interesting fact may also be drawn concerning with the zero at infin-ity. In discrete-time system, one zero of this type corresponds to one sampledelay, in which instantaneously gives one penalty, since

|ηij |2 − 1|ηij − 1|2 → 1,

as ηij →∞.The interpretation of integral term is rather similar to that of Theorem

3.1, in which it concerns with the effect caused by changes in plant direction.Specifically, due to the weighting factor 1 − cos θ in the denominator, it isreinforced that zeros close to z = 1 may have more negative effect. This extraintegral term, however, points to the main difference between MIMO andSIMO systems.

Theorem 3.4 can also be applied to recover the counterpart result forright-invertible MIMO systems provided in [58, Th. 3.1]. Suppose that theright invertible plant P (z) has non-minimum phase zeros ηi (i = 1, . . . , nη)and its inner factor may be formed as

Θi(z) =nη∏

i=1

Li(z),

whereLi(z) =

z − ηi

1− ηizuiu

Hi + UiU

Hi

with ui are unitary vectors iteratively computed from the zero input directionvector of P (z) one at a time, and Ui are matrices whose columns, togetherwith ui, form an orthonormal basis of the corresponding Euclidean space,i.e., uiu

Hi + UiU

Hi = I . We can easily verify that

Θi(1) = 1,

Θ′i(1) =nη∑

i=1

1− |ηi|2|ηi − 1|2 uiu

Hi

Hence, by invoking (3.42), we get

J∗d = −nη∑

i=1

1− |ηi|2|ηi − 1|2 νuiu

Hi νH =

nη∑

i=1

|ηi|2 − 1|ηi − 1|2 cos2 ∠(uH

i , ν).


Unstable Plants

For unstable plant P (z) whose coprime factorizations are given by (2.4), letperform an inner-outer factorization such that

N(z) = Θi(z)Θo(z). (3.44)

From Bezout identity (2.5) we obtain MX − NY = I and then XM = I +NM−1Y M . And from its equivalence (2.6) we have Y M −MY = 0 and thenM−1Y M = Y . Consequently, (3.8) becomes


∥∥∥∥∥[I + ΘiΘo(Y −QM)]ν

z − 1

∥∥∥∥∥

2

2

. (3.45)

We provide the optimal tracking performance for discrete-time unstableplants in the following theorem.

Theorem 3.5 Suppose that the SIMO plant P (z) given in (2.3) has unstable polesλk (k = 1, . . . , nλ) and Pi(z) has non-minimum phase ηij (i = 1, . . . , m, j =1, . . . , ni). Then, under Assumptions 3.4 and 3.5, the optimal tracking performanceis given by

J∗d = J∗ds + J∗du, (3.46)

where

J∗ds = Jds1 + Jds2,

J∗du = Jdu1 + Jdu2

with

Jds1 :=∑

i∈Jzν2

i

ni∑

j=1

|ηij |2 − 1|ηij − 1|2 ,

Jds2 :=12π

∑

i∈Jzν2

i

∫ π

0

log[ |Pi(1)|2‖P (1)‖2


]dθ

1− cos θ,

Jdu1 :=∑

i∈Jzν2

i

∑

k∈Jpi

|λk|2 − 1|λk − 1|2 ,

Jdu2 :=∑

k,`∈Jp

(|λk|2 − 1)(|λ`|2 − 1)(1−Θ∼i (λk)Θi(1))(1−Θ∼i (λ`)Θi(1))hkh`(λk − 1)(λ` − 1)(λkλ` − 1)

,

and

hk :=

1 ; #Jp = 1∏

`∈Jp,` 6=k

λk − λ`

1− λ`λk; #Jp ≥ 2.


Proof. See Appendix A.1. ¥

In this theorem we call Jdu1 and Jdu2 the deterioration contributed bythe plant unstable poles. We remark that Jdu1 = 0 if P (z) is either a scalarplant or a SIMO plant with all unstable poles of Pi(z) (i = 1, . . . , m) arecompletely the same, since we can see Jpi is empty for all i. Furthermore,condition M(λk)ν = 0 indicates that an unstable pole will not affect throughJdu2 unless its direction coincides with that of the input signal, i.e., R[N(1)].Particulary, λk (k /∈ Jp) does not mean that λk has no effect at all since there isstill a possibility that k ∈ Jpi, i.e., λk gives its contribution through Jdu1 withthe similar manner the non-minimum phase zeros do. This fact, however, hasimproved our understanding on the roles of unstable poles in performancelimitations. Beyond this, we can also make similar remarks as we we havebelow Theorem 3.2.

We now consider two specific cases for illustrating the implication of The-orem 3.5. The proofs of the following corollaries are straightforward, thusomitted.

Corollary 3.3 Suppose the plant P (z) satisfies P (1) = [P1(1), 0, . . . , 0]T, and letthe input signal r be given by (3.4) with ν = [1, 0, . . . , 0]T. Suppose that P1(z)is stable and has non-minimum phase zeros at η1j (j = 1, . . . , n1) and P (z) hasunstable poles λk (k = 1, . . . , nλ). Then

J∗d =n1∑

j=1

|η1j |2 − 1|η1j − 1|2 +

∑

k∈Jp1

|λk|2 − 1|λk − 1|2 +

12π

∫ π

0

log[‖P (ejθ)‖2|P1(ejθ)|2

]dθ

1− cos θ.

(3.47)

Remark: Note that in the above corollary we do not restrict Pi(z) (i ≥ 2) tobe stable. The unstable poles of P (z) contributed by Pi(z) (i ≥ 2), if any, willnot give effects through Jdu2, i.e., Jdu2 = 0, since the pole directions do notcoincide with that of the input signal, that is M(λk)ν 6= 0.

In SISO case which plant has non-minimum phase zeros ηi(i = 1, . . . , nη),the inner factor in (3.44), without loss of generality, can be fixed as

Θi(z) =nη∏

i=1

z − ηi

1− ηiz,

from which we get Θi(1) = 1. Let define φ(z) := Θ∼i (z)Θi(1), i.e.,

φ(z) :=nη∏

i=1

1− ηiz

z − ηi,

then we state the tracking performance limitations for scalar systems in thefollowing corollary.


−2 −1 0 1 2 33

4

5

6

7

8

9

10

η

J* d

for k = 1for k = 4

Fig. 3.6. The tracking error performance for SIMO discrete-time systems (Example3.3).

Corollary 3.4 Let P (z) be an SISO plant which has non-minimum phase zerosηi (i = 1, . . . , nη) and unstable poles λk (k = 1, . . . , nλ). Then,

J∗d =nη∑

i=1

|ηi|2 − 1|ηi − 1|2 +

nλ∑

k,`=1

(|λk|2 − 1)(|λ`|2 − 1)(1− φ(λk))(1− φ(λ`))hkh`(λk − 1)(λ` − 1)(λkλ` − 1)

. (3.48)

Example 3.3 Consider the single-input two-output plant P (z) given by

P (z) =[

P1(z)P2(z)

]=

[z−η

z(1−2z)

k z−1z(z−λ)

],

with η 6= 1, λ 6= 1, k ≥ 0. Note that P (1) = [η − 1, 0]T and P1(z) is stable,i.e., the plant is in the case of Corollary 3.3. P (z) has possibly one non-minimumphase zero at z = η and one unstable pole at z = λ with pole direction vectorw = [0, 1]T . We vary η from −2 to 3 and take ν = [1, 0]T . Then we calculate theoptimal tracking performance J∗d by using Corollary 3.3 for λ = 3. Note that Jdu2 isalways zero regardless the value of λ since the pole direction does not coincide withthat of input signal, i.e., w /∈ R[P (1)]. The calculations of J∗d are plotted in Fig. 3.6,which confirms that J∗d increases with k and J∗d is unbounded as η = 1.

Example 3.4 We consider an SISO system represented by

P (z) =z − η

z(z − λ).

Note that P has relative degree 1. First we fix λ = 12 , i.e., we consider a stable plant,

and compute the optimal performance for η from −2 to 2. Fig. 3.7 confirms thatin SISO case non-minimum phase zero closed to unit circle deteriorates the perfor-mance. Second we fix η = 2 and vary λ from 0 to 4. Fig. 3.8 shows that whenever λ


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

5

10

15

20

25

η

J* d


Fig. 3.7. The tracking error performance for stable discrete-time system (Example 3.4).

0 0.5 1 1.5 2 2.5 3 3.5 40

200

400

600

800

1000

1200

1400

λ

J* d


Fig. 3.8. The tracking error performance for unstable discrete-time system (Example3.4).

closes to 2 then the optimal performance blows up since it happens almost unstablepole-zero cancellation. For both cases, we compute the optimal performance in twomanners: by using MATLAB toolbox and by using analytical closed-form expres-sion in Corollary 3.4. We see that these two computations match well. In general,the optimal performance of a discrete-time system with one unstable pole λ and twonon-minimum phase zero η and infinity is governed by

J∗d = 1 +η + 1η − 1

+η2(λ2 − 1)(λ + 1)2

(λ− η)2.


Now we extend the problem by minimizing the tracking error simultane-ously with the energy of the control input. We consider the control objective


(3.10), where its optimal value is given by (3.11). We implement the idea ofplant augmentation which is quite helpful in solving the problem.

In addition to Assumptions 3.4 and 3.5, we impose the following assump-tion.

Assumption 3.6 P (z) has a pole at z = 1.

This assumption requires that the plant P (s) should have an integrator toassure the finiteness of the control energy cost. Note that the optimal perfor-mance (3.17) can be further expressed as

J∗d = infQa∈RH∞

∥∥∥∥[I + Na(Ya −QaMa)]νa

z − 1

∥∥∥∥2

2

, (3.49)

We state our result by introducing an inner-outer factorization such that

Na(z) :=[

Wu(z)M(z)N(z)

]= Θi(z)Θo(z). (3.50)

Theorem 3.6 Suppose that the SIMO plant P (z) given in (2.3) has unstable polesλk (k = 1, . . . , nλ) and Pi(z) has non-minimum phase zeros ηij (i = 1, . . . , m, j =1, . . . , ni). Then, under Assumptions 3.4–3.6, the optimal tracking performance un-der control input penalty is given by

J∗d = J∗ds + J∗du, (3.51)

where

J∗ds = Jds1 + Jds2,

J∗du = Jdu1 + Jdu2

with

Jds1 :=∑

i∈Jzν2

i

ni∑

j=1

|ηij |2 − 1|ηij − 1|2 ,

Jds2 :=12π

∑

i∈Jzν2

i

∫ π

0

log[ |Pi(1)|2‖P (1)‖2

‖P (ejθ)‖2 + |Wu(ejθ)|2|Pi(ejθ)|2

]dθ

1− cos θ,

Jdu1 :=∑

i∈Jzν2

i

∑

k∈Jpi

|λk|2 − 1|λk − 1|2 ,

Jdu2 :=∑

k,`∈Jp

(|λk|2 − 1)(|λ`|2 − 1)(1−Θ∼i (λk)Θi(1))(1−Θ∼i (λ`)Θi(1))hkh`(λk − 1)(λ` − 1)(λkλ` − 1)

,

and

hk :=

1 ; #Jp = 1∏

`∈Jp,` 6=k

λk − λ`

1− λ`λk; #Jp ≥ 2.

3.4 Delta Domain Case 43

Proof. We apply Theorem 3.5 to Pa(z) instead of P (z). Since the first elementof νa is equal to zero, then there exists no extra term in J∗d related to Wu(z). ByAssumption 3.6 we conclude that |Pi(1)| and ‖P (1)‖ are infinite but |Wu(1)|is finite. Then the following

|Pi(1)|2‖Pa(1)‖2 =

|Pi(1)|2‖P (1)‖2 + |Wu(1)|2 =

|Pi(1)|2‖P (1)‖2

holds. Also note that

‖Pa(ejθ)‖2|Pi(ejθ)|2 =

‖P (ejθ)‖2 + |Wu(ejθ)|2|Pi(ejθ)|2 .

The proofs for Jds1, Jdu1, and Jdu2 are similar as those of Theorem 3.5. ¥

Corollary 3.5 Suppose that the SISO plant P (z) is marginally stable and has non-minimum phase zeros ηi (i = 1, . . . , nη). Then, under Assumptions 3.4–3.6, theoptimal tracking performance under control input penalty is given by

J∗d =nη∑

i=1

|ηi|2 − 1|ηi − 1|2 +

12π

∫ π

0

log[1 +

|Wu(ejθ)|2|P (ejθ)|2

]dθ

1− cos θ. (3.52)

Corollary 3.5 reveals that whenever the SISO plant is marginally stable wehave to pay extra cost, which is represented by integral term, to compensatethe deterioration in performance caused by the control input constraint. Thisfact, however, confirms that taking the control input penalty into accountgenerally worsens the performance.

Example 3.5 In this example we consider the following SISO marginally stablediscrete-time plant

P (z) =z − η

(z − 1)(z2 + 12 )

.

This plant has non-minimum phase zero at z = η provided η ∈ Dc and relativedegree 2. Fig. 3.9 shows the optimal performance J∗d , which is calculated by Corollary3.5 and by MATLAB toolbox, with respect to the location of zero η. If η tends to 1then it happens an almost hidden pole-zero cancellation. Consequently, the optimaltracking performance grows up as warned by Assumption 3.6.

3.4 Delta Domain Case

Our motivation revisiting the tracking (and regulation) problems in δ-domainis that the relationship between continuous-time and discrete-time results isnot quite clear. For example, recall the analytical closed-form expression ofthe optimal tracking error for SISO stable continuous-time plant P (s) whichhas non-minimum phase zeros zi (i = 1, . . . , nz) in Corollary 3.1:


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

5

10

15

20

25

30

35

η

J* d


Fig. 3.9. The tracking error performance with control input penalty of a marginallystable discrete-time system (Example 3.5).

J∗c =nz∑

i=1

2Re zi

|zi|2 ,

and compare it with its discrete-time counterpart plant P (z) which has non-minimum phase zeros ηi (i = 1, . . . , nη) in Corollary 3.4:

J∗d =nη∑

i=1

|ηi|2 − 1|ηi − 1|2 .

It is not easy to understand how the non-minimum phase zeros of continuous-time and discrete-time plants give their contributions in completely differ-ent ways. We implement the so-called delta operator to investigate this phe-nomenon. In the last few years, it has been extensively demonstrated that thedelta operator is superior to the shift operator in unifying the continuous-time and discrete-time expressions. Recall Section 2.4 for short introductionon delta operator.

3.4.1 Two Lemmas

We begin this part by reformulating the two key lemmas, i.e., Lemmas 3.3and 3.4, in the delta domain. Recall the set F defined in (3.19).

Lemma 3.7 Let h ∈ F be an analytic function in DcT . Denote that h( ejωT−1

T ) =h1(ω) + jh2(ω). Suppose that h is conjugate symmetric, i.e. h(δ) = h(δ). Then

h′(0) =T 2

2π

∫ π/T

−π/T

h1(ω)− h1(0)1− cosωT

dω. (3.53)


Proof. Consider the relationship h(δ) = f(Tδ + 1), where f is defined inLemma 3.3. Since f is analytic in Dc, then h is analytic in Dc

T . We can easilyverify that h′(0) = Tf ′(1). ¥

Lemma 3.8 Let h be a meromorphic function in DcT and has no zero or pole on

∂DT . Suppose that h is conjugate symmetric and log h ∈ F. Also, suppose that ζi ∈Dc

T (i = 1, . . . , nζ), and ρk ∈ DcT (k = 1, . . . , nρ) are, respectively, non-minimum

phase zeros and unstable poles of h, all counting their multiplicities. Provided thath(0) 6= 0, then

T 2

2π

∫ π/T

−π/T

log

∣∣∣∣∣h( ejωT−1

T )h(0)

∣∣∣∣∣dω

1− cosωT=

nζ∑

i=1

(2Re ζi

|ζi|2 + T

)−

nρ∑

k=1

(2Re ρk

|ρk|2 + T

)+

h′(0)h(0)

. (3.54)

Proof. Use relationships ηi = Tζi + 1 and λk = Tρk + 1 from (2.16). ¥


In this part we reformulate and solve the tracking error problem in term ofdelta operator. We consider the following tracking measure

Jδ := T

∞∑

k=0

‖e(k)‖2, (3.55)

which can be further expressed as

Jδ = ‖So(δ)rT (δ)‖22.

As the reference input we consider the step function (3.4) whose delta trans-form is given by

rT (δ) =Tδ + 1

δν. (3.56)

We include the sampling time T in (3.55) under the following explanation.Consider the operation of a zero-order hold embedded in sampled-data sys-tem which maps a discrete sequence e(k) into continuous-time signal eh(t).The process can be written as

eh(t) , e(k), kT ≤ t < (k + 1)T,

from which we may define the following measure:

Jh :=∫ ∞

0

‖eh(t)‖2 dt.


Since in this case we do vary the sampling time T instead of fix it then thetracking measure of the corresponding continuous-time signal

Jc :=∫ ∞

0

‖e(t)‖2 dt

can be fully recovered by evaluating Jh as T → 0. We can easily verify that

Jh =∞∑

k=0

∫ (k+1)T

kT

‖eh(t)‖2 dt = T

∞∑

k=0

‖e(k)‖2 = TJd,

from which we deduce (3.55). For the finiteness of Jδ we impose the follow-ing standard assumptions on P (δ) and ν.


Assumption 3.8 For r(k) in (3.4), ν ∈ R[N(0)].

We denote by ρk (k = 1, . . . , nρ) the unstable poles of P (δ) and by ζij (i =1, . . . , m, j = 1, . . . , ni) the non-minimum phase zeros of Pi(δ) and introducethe following index sets:

Iz := i : Ni(0) 6= 0Ip := k : M(ρk)ν = 0Ipi := k : Ni(ρk) = 0 (i = 1, . . . , m).

We state our result for unstable plant. We define an inner-outer factorizationsuch that

N(δ) = Θi(δ)Θo(δ), (3.57)

where we denote Θ∼i (δ) := ΘT

i ( −δTδ+1 ). Note that the optimal tracking perfor-

mance is determined by

J∗δ = infQ∈RH∞

∥∥∥[I + ΘiΘo(Y −QM)]ν

δ

∥∥∥2

2. (3.58)

Theorem 3.7 Suppose that the SIMO plant P (δ) given in (2.3) has unstable polesρk (k = 1, . . . , nρ) and Pi(δ) has non-minimum phase ζij (i = 1, . . . , m, j =1, . . . , ni). Then, under Assumptions 3.7 and 3.8, the optimal tracking performanceis given by

J∗δ = J∗δs + J∗δu, (3.59)

where

J∗δs = Jδs1 + Jδs2,

J∗δu = Jδu1 + Jδu2

with


Jδs1 :=∑

i∈Izν2

i

ni∑

j=1

(2Re ζij

|ζij |2 + T

),

Jδs2 :=T 2

2π

∑

i∈Izν2

i

∫ π/T

0

log

[|Pi(0)|2‖P (0)‖2

‖P ( ejωT−1T )‖2

|Pi( ejωT−1T )|2

]dω

1− cos ωT,

Jδu1 :=∑

i∈Izν2

i

∑

k∈Ipi

(2Re ρk

|ρk|2 + T

),

Jδu2 :=∑

k,`∈Ip

(T |ρk|2 + 2 Re ρk)(T |ρ`|2 + 2Re ρ`)qkq`ρkρ`(T ρkρ` + ρk + ρ`)

×

(1−Θ∼i (ρk)Θi(0))(1−Θ∼

i (ρ`)Θi(0)),

and

qk :=

1 ; #Ip = 1∏

`∈Ip,` 6=k

ρ` − ρk

T ρ`ρk + ρ` + ρk; #Ip ≥ 2.

Proof. In the light of the proof of Theorem 3.5 we can immediately write J∗δ =J∗1 + J∗2 , where

J∗1 =∥∥∥(Θi −Θ∼i (1))

ν

δ

∥∥∥2

2+

∥∥∥(I −ΘiΘ∼i )

ν

δ

∥∥∥2

2,

J∗2 = infQ∈RH∞

∥∥∥[Θ∼i (1) + Θo(Y −QM)]ν

δ

∥∥∥2

2.

Directly calculating the H2 norm in δ-domain we obtain

J∗1 = −T 2

2π

∫ π/T

−π/T

Re

νHΘi( ejωT−1T )ΘH

i (0)ν− 1

1− cos ωTdω.

Application of Lemma 3.7 provides J∗1 = Jδs1 + Jδs2 + Jδu1. The closed-formexpression of J∗2 can be obtained by performing standard partial fraction ex-pansion. In this process we define

q(δ) :=∏

k∈Ip

(Tδ + 1)− λk

1− λk(Tδ + 1),

qρ(δ) :=(Tδ + 1)− λk

1− λk(Tδ + 1),

qk :=∏

`∈Ip,` 6=k

λk − λ`

1− λ`λk,

where λk = Tρk + 1. We then obtain(

1qρ− 1− λk

1− λk

)1δ

=T (|λk|2 − 1)

(1− λk)(Tδ + 1− λk).


And also by taking into account the norms relation (2.20) we get∥∥∥∥

1Tδ + 1− λk

∥∥∥∥2

2

=1T

∥∥∥∥1

z − λk

∥∥∥∥2

2

=1T

1|λk|2 − 1

.

This can be utilized to show J∗2 = Jδu2. ¥


We can easily extend the result to one with control input penalty, i.e., weconsider the following tracking measure:

Jδ := T

∞∑

k=0

(‖e(k)‖2 + |uw(k)|2) , (3.60)

where uw(k) = D−1Wu(δ)u(δ). The optimal performance is then providedby

J∗δ = infQa∈RH∞

∥∥∥[I + Na(Ya −QaMa)]νa

δ

∥∥∥2

2, (3.61)

In addition to Assumption 3.7 and 3.8, we make the following assumption.

Assumption 3.9 P (δ) has a pole at δ = 0.

We also define an inner-outer factorization such that

Na(δ) =[

Wu(δ)M(δ)N(δ)

]= Θi(δ)Θo(δ). (3.62)

Theorem 3.8 Suppose that the SIMO plant P (δ) given in (2.3) has unstable polesρk (k = 1, . . . , nρ) and Pi(δ) has non-minimum phase ζij (i = 1, . . . , m, j =1, . . . , ni). Then, under Assumptions 3.7–3.9, the optimal tracking performance un-der control penalty is given by

J∗δ = J∗δs + J∗δu, (3.63)

where

J∗δs = Jδs1 + Jδs2,

J∗δu = Jδu1 + Jδu2

with

Jδs1 :=∑

i∈Izν2

i

ni∑

j=1

(2Re ζij

|ζij |2 + T

),

Jδs2 :=T 2

2π

∑

i∈Izν2

i

∫ π/T

0

log[|Pi(0)|2‖P (0)‖2

‖P ( ejωT−1T )‖2+|Wu( ejωT−1

T )|2|Pi(

ejωT−1T )|2

]

1− cosωTdω,


Jδu1 :=∑

i∈Izν2

i

∑

k∈Ipi

(2Re ρk

|ρk|2 + T

),

Jδu2 :=∑

k,`∈Ip

(T |ρk|2 + 2Re ρk)(T |ρ`|2 + 2 Re ρ`)qkq`ρkρ`(T ρkρ` + ρk + ρ`)

×

(1−Θ∼i (ρk)Θi(0))(1−Θ∼i (ρ`)Θi(0)),

and

qk :=

1 ; #Ip = 1∏

`∈Ip,` 6=k

ρ` − ρk

T ρ`ρk + ρ` + ρk; #Ip ≥ 2.

Proof. The proof is similar to that of Theorem 3.6. ¥

3.4.4 Continuity Property

Plant Transformation

Consider a continuous-time plant Gc(s) with the following state space rep-resentation:

Gc(s) =[

Ac Bc

Cc Dc

]= Cc(sI −Ac)−1Bc + Dc.

Suppose that the sampling interval is T . By using the step invariance trans-form or the zero-order hold (ZOH), i.e., x(t) = x(kT ), kT ≤ t < (k + 1)T ,discretizing the plant Gc(s) gives a corresponding discrete-time plant

Gd(z) =[

Ad Bd

Cc Dc

]= Cc(zI −Ad)−1Bd + Dc,

where

Ad := eAcT , Bd :=∫ T

0

eAct dtBc.

Subsequently, by using the delta operator described in Section 2.4 we obtainthe corresponding delta domain plant GT (δ) as follows,

GT (δ) =[

AT BT

Cc Dc

]= Cc(δI −AT )−1BT + Dc,

where

AT :=eAcT − I

T, BT :=

1T

∫ T

0

eAct dtBc.

As T tends to zero, it can be readily seen that AT tends to Ac and BT tendsto Bc. Therefore, by taking into account the relation δ = (esT − 1)/T , we seethat PT (δ) tends to Pc(s).


Pole-Zero Transformation

Let Gc(s) be rational function

Gc(s) = c(s− z1)(s− z2) · · · (s− zm)(s− p1)(s− p2) · · · (s− pn)

,

where r := n−m > 0. It is known that when a continuous-time plant Gc(s)is discretized under sampling time T to Gd(z) then the poles pk (k = 1, . . . , n)are transformed as

pk → λk := epkT , (3.64)

in which the stability is preserved. There is unfortunately no simple transfor-mation which shows how the zeros of continuous-time plant zi (i = 1, . . . , m)are mapped by sampling. Generically, the transfer function Gd(z) has n − 1zeros. For particular values of T , some zeros may go to infinity, or they maybe canceled by poles. However, for limiting cases where T is sufficientlysmall, there is a useful characterization [2]. As sampling time T → 0, m zerosof Gd(z) go to 1 as

zi → ηi := eziT (3.65)

and the remaining r − 1 zeros of Gd(z) go to the zeros of polynomial

Br(z) = br1z

n−1 + br2z

n−2 + . . . + brr, (3.66)

where

brk :=

k∑

l=1

(−1)k−llr(

r + 1k − 1

), k = 1, . . . , n.

It is important to note that the number of non-minimum phase zeros of Br(z)is r−1

2 for odd r and r2 or r

2 − 1 for even r. This fact means that in many casesnon-minimum phase zeros will appear as the sampling time is decreasedeven though the continuous-time plant may be minimum phase.

Summarizing, if the continuous-time plant Gc(s) has unstable poles pk(k =1, . . . , n) and non-minimum phase zeros zi (i = 1, . . . , m), then if T is suffi-ciently small the corresponding discrete-time plant Gd(z) has unstable polesλk = epkT (k = 1, . . . , n) and non-minimum phase zeros ηi = eziT (i =1, . . . , m) and η?

i (i = 1, . . . ,m?), where m? is either r−12 , r

2 , or r2−1. Note that

the limiting zeros η?i are all finite. As a direct implication, the corresponding

delta domain plant GT (δ) has unstable poles ρk := epkT−1T (k = 1, . . . , n) and

non-minimum phase zeros ζi = eziT−1T (i = 1, . . . ,m) and ζ?

i = η?i−1T (i =

1, . . . , m?).

Convergence of the Expression

Here we will show the continuity properties of the delta domain expressions,i.e., we demonstrate that delta domain expressions in Theorem 3.8 converge


to their continuous-time counterparts in Theorem 3.3 when the samplingtime T tends to zero.

To avoid ambiguity, we denote by

Pc(s) =(Pc1(s), Pc2(s), . . . , Pcm(s)

)T,

the respecting continuous-time plant. Suppose that Pc(s) has unstable polespk(k = 1, . . . , np) and Pci(s) has non-minimum phase zeros zij(i = 1, . . . , m, j =1, . . . , ni). Under the zero-order hold operations we obtain the correspondingdelta domain plant

PT (δ) =(PT1(δ), PT2(δ), . . . , PTm(δ)

)T,

where PT (δ) has unstable poles ρk (k = 1, . . . , nρ) with nρ = np, and PTi(δ)has non-minimum phase zeros ζij (i = 1, . . . ,m, j = 1, . . . , ni) and ζ?

ij (i =1, . . . , m, j = 1, . . . , n?

i ).Now we are ready to show the convergence. Recall the index sets Kz, Kp,

and Kpi introduced in Subsection 3.2.2. From the previous part we know thefollowing pole-zero relationships:

ρk =epkT − 1

T, k = 1, . . . , nρ,

ζij =ezijT − 1

T, i = 1, . . . , m, j = 1, . . . , ni,

ζ?ij =

η?ij − 1T

, i = 1, . . . , m, j = 1, . . . , n?i ,

where η?ij are the zeros of polynomial Br(z) defined in (3.66). Since ζij → zij

and ζ?ij →∞ as T tends to zero, obviously we have

limT→0

Jδs1 =∑

i∈Kz

ν2i

ni∑

j=1

2Re zij

|zij |2 =: Jcs1. (3.67)

Immediately by fact that ρk → pk as T tends to zero we also have

limT→0

Jδu1 =∑

i∈Kz

ν2i

∑

k∈Kpi

2Re pk

|pk|2 =: Jcu1. (3.68)

Next, since PTi(δ) → Pci(s) and WTu(δ) → Wcu(s) as T tends to zero, and

limT→0

T 2

2(1− cos ωT )=

1ω2

, limT→0

ejωT − 1T

= jω

then we obtain

limT→0

Jδs2 =1π

∑

i∈Kz

ν2i

∫ ∞

0

log[ |Pci(0)|2‖Pc(0)‖2

‖Pc(jω)‖2 + |Wcu(jω)|2|Pci(jω)|2

]dω

ω2=: Jcs2.

(3.69)


We show the convergence of Jδu2 part by part. Since

limT→0

qk =∏

`∈Kp,` 6=k

p` − pk

p` + pk=: σk,

we readily have

limT→0

(T |ρk|2 + 2 Re ρk)(T |ρ`|2 + 2Re ρ`)qkq`ρkρ`(T ρkρ` + ρk + ρ`)

=4Re pk Re p`

σkσ`pkp`(pk + p`).

Now we deal with terms that contain the inner factor ΘT i(ρk). First note thatin delta domain we define Θ∼

T i(ρk) := ΘTT i(

−ρk

Tρk+1 ). Hence, it is clear that−ρk

Tρk+1 → −pk as T tends to zero. Recall the inner-outer factorization suchthat [

MT (δ)NT (δ)

]= ΘT i(δ)ΘTo(δ).

Here we set Wcu(s) = 1 without loss of generality. Note that we have thefollowing representation:

[MT (δ)NT (δ)

]=

AT + BT FT BT

FT ICc + DcFT Dc

,

where matrix FT is chosen such that (AT + BT FT ) is stable. Since all thematrices converge to the continuous-time counterparts as T tends to zero,then it is sufficient to conclude that ΘT i(δ) → Θci(s) as T → 0. Therefore,

limT→0

Jδu2 =∑

k,`∈Kp

4 Re pk Re p`(1−Θ∼ci (pk)Θci(0))(1−Θ∼

ci (p`)Θci(0))(pk + p`)pkp`σkσ`

=: Jcu2,

(3.70)where we define Θ∼

ci (pk) := ΘTci(−pk).

Overall, (3.67)–(3.70) show the continuity property:

limT→0

J∗δ = J∗c , (3.71)

where J∗c is given in Theorem 3.3. In other words, we completely recover thecontinuous-time expression from the delta domain expression stand point bymaking the sampling time approaches zero.

3.5 Delay-time Case

It is well-known that the time delays generally degrade the optimal trackingperformance in much the same manner as non-minimum phase zeros [14,58]. In this section, we exploit the delta domain expression to re-derive the

3.5 Delay-time Case 53

analytical closed-form expression of the optimal tracking performance fordelay-time systems.

We consider the unity feedback control system depicted in Fig. 3.1, whereP is the SIMO plant to be controlled with delay in the input port:

P (s) =

P1(s)P2(s)

...Pm(s)

e−τs, (3.72)

where Pi (i = 1, . . . , m) are scalar transfer functions and τ ≥ 0 is the time-delay. We minimize the performance index given in (3.9), that is

Jτ :=∫ ∞

0

(‖e(t)‖2 + |uw(t)|2)dt.

Under the zero-order hold operations of sampling time T , we can obtainthe delta domain counterpart of the plant (3.72) as follows

P (δ) =1

(Tδ + 1)τ/T+1

P1(δ)P2(δ)

...Pm(δ)

. (3.73)

Note that the relative degree τ/T is resulted from the discretization of thedelay part e−τs. We will also receive additional relative degree 1 contributedby Pi(s) (i = 1, . . . , m) provided that T is sufficiently small. We denote byρk ∈ Dc

T (k = 1, . . . , nρ) the unstable poles of P (δ) and by ζij ∈ DcT (i =

1, . . . , m, j = 1, . . . , ni) the non-minimum phase zeros of Pi(δ).

Theorem 3.9 Suppose that the delay-time plant P (s) given in (3.72) has unstablepoles pk ∈ C+ (k = 1, . . . , np) and Pi(s) has non-minimum phase zeros zij ∈C+ (i = 1, . . . , m, j = 1, . . . , ni). Then, the optimal tracking performance J∗τ isgiven by

J∗τ = Jτs1 + Jτs2 + Jτu1 + Jτu2, (3.74)

where

Jτs1 := τ +∑

i∈Izν2

i

ni∑

j=1

2Re zij

|zij |2 , Jτs2 := Jcs2, Jτu1 := Jcu1, Jτu2 := Jcu2

with Jcs2, Jcu1, and Jcu2 have the same expressions as those given by Theorem 3.3.

Proof. Since P (δ) has extra non-minimum phase zeros at infinity with mul-tiplicity τ/T + 1, then based on the continuity property of Theorem 3.8 weobtain


Jτs1 = limT→0

∑

i∈Izν2

i

ni+τ/T+1∑

j=1

(2Re ζij

|ζij |2 + T

)

= limT→0

∑

i∈Izν2

i

τ/T+1∑

j=1

T + limT→0

∑

i∈Izν2

i

ni∑

j=1

(2Re ζij

|ζij |2 + T

)

= τ +∑

i∈Izν2

i

ni∑

j=1

2 Re zij

|zij |2 .

Note that ν is a unitary vector. The proof for other terms follow the continuityproperty of Theorem 3.8. ¥

3.6 Sampled-data Case

Modern control systems are almost always implemented in a digital com-puter. It is thus important to have an appreciation of the impact of imple-menting a particular control law in digital form. Studies in this subject in-clude the frequency domain analysis for sampled-data systems, in which oneof the work introduced the use of lifting techniques. By lifting, a signal val-ued in a finite dimensional space is bijectively mapped onto a signal valuedin infinite dimensional spaces. Attractively, by this transformation it has be-come possible to view sampled-data system as an LTI discrete-time systemwith built-in inter-sample behavior.

Working with sampled-data system is naturally harder than that withcontinuous-time system or discrete-time system. In sampled-data feedbackcontrol problems we have a continuous-time plant and want to design a sta-bilizing discrete-time controller.

Problems concerning on the fundamental design limitations in sampled-data control systems have been widely investigated since last decade [26, 29,47] and one pertains to the tracking performance limitations of stable plantis recently studied in [15], which gives the analytical closed-form expressionof the optimal tracking performance. In this paper, the problem of track-ing a step reference signal using sampled-data control systems is studied byadopting a frequency domain lifting technique. Consequently, the problem isthen reduced to one of tracking in equivalent discrete-time system, in whichthe solution of the latter problem is readily available [58].

For scalar stable case, the optimal tracking performance is governed byfollowing expression [15]:

J∗sd =nz∑

k=1

2zk

+ T

nς∑

k=1

ςk + 1ςk − 1

+

T 2

2π

∫ π/T

0

log

[T 2

∑∞k=−∞ |Pk(jω)Hk(jω)|2

∣∣∑∞k=−∞ Pk(jω)Hk(jω)

∣∣2]

dω

1− cosωT, (3.75)

3.6 Sampled-data Case 55

Fig. 3.10. The sampled-data feedback control system.

where zk and ςk are respectively the non-minimum phase zeros of Pc(s)and a product of ZOH equivalent discretized transfer functions defined byPd(z) := zZPm(s)H(s)ZF (s). Here, H is the zero-order hold of sam-pling time T , F is anti-aliasing filter, Pm is minimum phase part of Pc, andPk is the harmonics of Pc, i.e., Pk(s) = Pc(s + 2πjk/T ).

It is shown in (3.75) that the optimal performance is explicitly determinedby the sampling time, the non-minimum phase zeros of the continuous-timeplant, the non-minimum phase zeros of the discretized plant, and the aliasingeffect incurred by the sampling-hold operations.

Our approach presented in this section is completely different. Insteadof compute the exact value of the optimal tracking performance as did in[15], we employ an approximation approach by implementing fast samplingtechnique. We benefit our preceding result on tracking performance problemof SIMO discrete-time system. By this approach, we can extend the trackingerror problem to unstable case and possibly with control input penalty.

We consider the generic setup of a single-input single-output (SISO)sampled-data feedback control system depicted in Fig. 3.10, where Pc(s)represents the continuous-time plant and Kd(z) the discrete-time stabiliz-ing controller. The signal r is the reference input and considered to be a unitstep function. Signals u, y, e := r − y denote the control input, measurementoutput, and error response, respectively. While, ek and uk represent digitalsignals relate to e and u conduced by the sampler S and the zero-order holdH of sampling time T . The set of all stabilizing controllers is then defined by

Ksd := Kd(z) : Kd(z) stabilizes Pc(s). (3.76)

We want to minimize the following tracking measure with respect to allstabilizing controllers:

Jsd =∫ ∞

0

(|e(t)|2 + |uw(t)|2) dt, (3.77)

where uw(t) = WuL−1u(s). Here we consider a real constant weightingfunction Wu for simplicity since it will also give a constant under sampling.We implement Assumptions 3.1 and 3.3, i.e., we assume that Nc(0) 6= 0 andPc(s) has a pole at s = 0.

3.6.1 Fast Sampling

Under the fast sampling technique, we embed a fast sampler Sf with sam-pling time T/N at the reference input and the plant output, from which


Fig. 3.11. Approximation of the sampled-data feedback control system.

Fig. 3.12. Equivalence of Fig. 3.11.

we subdivide the k-th sampling interval [kT, (k + 1)T ) into N subintervals[kT + i

N T, kT + i+1N T ), i = 0, 1, . . . , N − 1. By this process, the feedback con-

trol setup of Fig. 3.10 can be approximated by that of Fig. 3.11. Equivalently,Fig. 3.11 can be re-arranged such that it becomes Fig. 3.12. In these figureswe denote

rk :=

rk

rk

...rk

, yk :=

yk0

yk1

...ykN−1

,

where rk is a discrete-time unit step function and yki = y(kT + iN T ), for

i = 0, 1, . . . , N − 1.Suppose that the transfer functions of the continuous-time plant Pc(s)

and its discretized plant Pd(z) are determined by

Pc(s) =[

A BC D

], Pd(z) =

[Ad Bd

Cd Dd

],

where

Ad = eAT , Bd =∫ T

0

eAtB dt, Cd = C, Dd = D.

It is not difficult to verify that

yki = CdeA iN T x(kT + i

N T ) +(Cd

∫ iN T

0eAtB dt + Dd

)u(kT + i

N T )

for i = 0, 1, . . . , N − 1. The transfer functions from uk to yki , denoted by Pfi ,are then determined by

Pfi(z) =[

Ad Bd

Cdi Ddi

], (3.78)

where


Cdi= CdeA i

N T , Ddi= Cd

∫ iN T

0

eAtB dt + Dd.

Obviously, Pf0(z) = Pd(z) and we can see that Pfi (i = 0, 1, . . . , N − 1) havethe same set of unstable poles. Furthermore, we define

Pf(z) =(Pf0(z), Pf1(z), . . . , PfN−1(z)

)T. (3.79)

To solve the problem we implement the plant augmentation strategy ini-tially introduced in [31]. Let define the augmented plant

Pa(s) :=[

Wu

Pc(s)

],

from which by fast sampling procedure we can approximate the performanceindex (3.77) by

Jf :=T

N

∞∑

k=0

‖rak − yak‖2, (3.80)

where

rak =[

0N

rk

], yak =

[uwk

yk

]

with 0N := ( 0, . . . , 0 )T ∈ RN and uwk := ( uwk0 , . . . , uwkN−1 )T. We put afactor of T

N as implication of the sampling and hold operations. Furthermore,by Parseval’s identity we obtain

Jf =T

N

∥∥∥∥[

01N

]rk −

[√NWPf

]uk

∥∥∥∥2

2

, (3.81)

where 1N := ( 1, . . . , 1 )T ∈ RN . Note that originally we fast-sample thesignal uw such that we obtain the transfer functions (W, . . . , W )T of N -tuple.Since the sampling points are all constant we represent them only by singlepoint

√NW .

3.6.2 Approximation of the Optimal Performance

Let the coprime factorization of Pf0 is given by

Pf0(z) = Nf0(z)M−1f0

(z),

where

Mf0(z) =nλ∏

k=1

z − λk

1− λkz

with λk (k = 1, . . . , nλ) are the unstable poles of Pf0(z). Since Pfi (i =0, . . . , N − 1) have only common unstable poles then the coprime factoriza-tion of Pf is given by


Pf(z) = Nf(z)M−1f0

(z),

where Nf = ( Nf0 , Nf1 , . . . , NfN−1 )T. Youla parameterization (2.8) tells thatthe stabilizing digital controller is parameterized by

Kd =Yf0 −Mf0Qf

Nf0Qf −Xf0

,

where Qf ∈ RH∞ is a scalar free parameter. Since ek = Mf0(Xf0−Nf0Qf)rk, ityields uk = −Mf0(Yf0 −Nf0Qf)rk and furthermore yk = −Nf(Yf0 −Mf0Qf)rk.Consequently the minimum value of (3.81) is given by

J∗f =T

Ninf

Qf∈RH∞

∥∥∥∥νf + Nfa(Yf0 −QfMf0)

z − 1

∥∥∥∥2

2

, (3.82)

where νf = ( 0, 1, . . . , 1 )T ∈ RN+1 and

Nfa(z) =[√

NWuMf0(z)Nf(z)

].

Furthermore, we can write (3.82) as

J∗f =T

Ninf

Qfa∈RH∞

∥∥∥∥[I + Nfa(Yf0a −QfaMf0)]νf

z − 1

∥∥∥∥2

2

,

where Qfa = (0TN , Qf) and Yf0a = (0T

N , Yf0).Fortunately, the last expression of J∗f is coincident with that of the opti-

mal tracking performance for SIMO discrete-time case J∗d in (3.49) wheneverνa = νf . The only difference is that Mf0 is scalar, but Ma is square. Hence,by defining an inner-outer factorization such that Nfa = ΘiΘo we can invokethe further derivation of (3.49) to produce

J∗f =T

N

∥∥∥∥(Θ∼

i −Θ∼i (1)) νf

z − 1

∥∥∥∥2

2

+∥∥∥∥

(I −ΘiΘ∼i ) νf

z − 1

∥∥∥∥2

2

+

infQfa∈RH∞

∥∥∥∥[Θ∼i (1) + Θo (Yf0a −QfaMf0)] νf

z − 1

∥∥∥∥2

2

.

Theorem 3.10 Consider the sampled-data system depicted in Fig. 3.10 with anSISO plant Pc(s). Let ηij (i = 0, . . . , N − 1, j = 1, . . . , ni) be the non-minimumphase zeros of Pfi(z) and λk (k = 1, . . . , nλ) be the unstable poles of Pf(z). Thenthe approximation value of the optimal tracking error performance is given by

J∗f = Jfs1 + Jfs2 + Jfu2, (3.83)

where


Jfs1 :=T

N

N−1∑

i=0

ni∑

j=1

|ηij |2 − 1|ηij − 1|2 ,

Jfs2 :=T

2πN

N−1∑

i=0

∫ π

0

log[ |Pfi(1)|2‖Pf(1)‖2

‖Pf(ejθ)‖2 + NW 2u

|Pfi(ejθ)|2]

dθ

1− cos θ,

Jfu2 :=T

N

nλ∑

k,`=1

(|λk|2 − 1)(|λ`|2 − 1)(1−Θ∼i (λk)Θi(1))(1−Θ∼

i (λ`)Θi(1))hkh`(λk − 1)(λ` − 1)(λkλ` − 1)

,

with

hk :=

1 ; nλ = 1,∏

j 6=k

λk − λ`

1− λ`λk; nλ ≥ 2.

Proof. If Pc(s) has unstable poles pk (k = 1, . . . , np) then the discretized plantPfi will have only common unstable poles λk (k = 1, . . . , nλ), where λk =epkT and np = nλ. Consequently, Jfu2 is non-negative since Mf0(λk)νf = 0,but Jfu1 = 0. Note that if Pc(s) is marginally stable then Jfu2 = 0, and hencewe can compute J∗f without using Θ∼i . ¥

Example 3.6 Having a sampled-data feedback control system in Fig. 3.10, we con-sider the following SISO continuous-time plant:

Pc(s) =s− x

s(s + 1), x > 0.

Note that Pc(s) is marginally stable and has a non-minimum phase zero at s = x. Itis not difficult to verify that

Ad =[

e−T 01− e−T 1

],

Bd =[

1− e−T

T + e−T − 1

],

Cdi =((1 + x)e−

iN T − x, −x

),

Ddi = 1 + x(1− iT/N)− (1 + x)e−iN T .

Suppose that nfi(z) is the numerator of Pfi(z). Then, nfi(1) = x(1 − e−T )(2 −2e−T − T ) for all i = 0, . . . , N − 1. Consequently,

|Pfi(1)|2‖Pf(1)‖2 =

|nfi(1)|2∑N−1i=0 |nfi(1)|2

=1N

,

from which we simplify

Jfs2 =T

2πN

N−1∑

i=0

∫ π

0

log[

1N

‖Pf(ejθ)‖2 + NW 2u

|Pfi(ejθ)|2]

dθ

1− cos θ.


1 1.5 2 2.5 3

0.8

1

1.2

1.4

1.6

1.8

2

2.2

x

J* sd ,

J* f

ExactApproximation

T = 0.1sec.,N = 30, Wu = 0

T = 0.01sec.,N = 3, Wu = 0

Fig. 3.13. The exact and approximation values of the optimal tracking performancefor sampled-data systems with Wu = 0.

1 1.5 2 2.5 30.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

x

J* f

Wu = 5 x 10−5

Wu = 3 x 10−5

Wu = 1 x 10−5

Fig. 3.14. The approximation values of the optimal tracking performance for sampled-data systems with Wu 6= 0.

First we consider a case without input penalty, i.e., Wu = 0. We compute the optimaltracking performance for different pairs of T, N: 0.1sec., 30 and 0.01sec., 3by using Theorem 3.10. We also compute the exact value by using [15, Theorem1]. Fig. 3.13, which plots the optimal performance for x from 1 to 3, shows that weapproximate the exact results well. Particularly if the sampling time T is small, Ncan be made small. Second we consider nonzero Wu. We select Wu = 5, 3, 1 ×10−5 and compute the optimal performance for T = 0.01sec. and N = 3. Fig. 3.14shows that the results converge to those of the first case as Wu gets smaller.

3.7 Summary 61

3.7 Summary

We have examined the H2 tracking performance problem in SIMO LTI feed-back control system. We have formulated and solved the tracking problem inminimizing the tracking error under control input penalty with respect to astep reference input. We characterize and quantify the tracking performancelimitations may arise in terms of the dynamics and structure of the plant. Inother words we derive the analytical closed-form expression of the optimalperformance.

We have made few corrections toward the existing results of continuous-time case and have presented a new result for discrete-time counterpart. Ap-plication of the bilinear transformation enables us to derive the discrete-timeresults straightforwardly. We have also reformulated and resolved the prob-lem in the delta domain, from which we can recover the continuous-timeresults by approaching the sampling time to zero. Additionally, we have pro-vided the approximation of the optimal performance in sampled-data feed-back control system by applying the fast sampling technique.

In general, our results show that the optimal tracking performance isproperly characterized by the plant’s non-minimum phase zeros and unsta-ble poles, plant gain, and the direction of the reference input. The derivedexpressions are not complete yet in the sense that they include the inner fac-tor.

4

Regulation Performance Limitations

In this chapter, we investigate the regulation properties pertaining to single-input multiple-output (SIMO) linear time-invariant (LTI) systems. We pro-vide analytical closed-form expressions of the best achievable H2 optimalregulation performances against impulsive disturbance inputs for unstableand non-minimum phase continuous-time and discrete-time systems. Wealso modify the latter results by means the delta operator and show the con-tinuity property. In this step, we can also solve the regulation problem ofdelay-time systems.

We here mainly focus on the two-kind of optimal regulation problem,namely the energy regulation problem and the output regulation problem. Inthe former problem, the regulation performance is measured by minimizingthe energy of control input, while in the latter problem, by minimizing theenergy of control input jointly with the energy of measurement output. Re-sults onH2 energy regulation problem of continuous-time case can be foundin [31]. Equivalent results in SISO systems but articulated in term of signal-to-noise ratio constrained channels are found in [4, 42]. Meanwhile, resultson H2 output regulation problem of minimum phase SISO/MIMO systemsare presented in [14]. In the present work we complete the result on outputregulation problem for continuous-time case by considering non-minimumphase system. While in the discrete-time case we provide new results.

Applications of our results on the regulation problem of three-disk tor-sional system and magnetic bearing system can be found in Chapter 5.

4.1 Regulation Performance Problem

This section is devoted to the formulation of theH2 optimal regulation prob-lem. We consider the feedback control setup depicted by Fig. 4.1. In this setupthe functions Ws and Wy respectively denote the weighting for the sensitiv-ity reduction and that for the disturbance attenuation, which are stable andminimum phase. We note that in regulation problem the output responses

64 4 Regulation Performance Limitations

Fig. 4.1. The regulation setup.

are solely resulted from the disturbance input signal d, which is an impulsefunction. In some sense we can say that the regulation problem is the dualof the tracking problem. In regulation problem the disturbance signal entersthe system through the plant input, while in tracking problem the referencesignal commands the system through the plant output.

Our primary objective in this work is not on how to find an optimal con-troller K which stabilizes the feedback and regulate the system output tozero. Rather, we want to relate the optimal regulation performance with thecharacteristics of the plant P .

4.1.1 Integral Formulae

We have a number of integral formulae which play important roles in oursubsequent derivation.

Lemma 4.1 (Poisson-Jensen Formula) Let f is analytic in Dc and di (i =1, . . . , nd) be the zeros of f in Dc, counting their multiplicities. If z ∈ Dc andf(z) 6= 0, then

log |f(z)| = 1π

∫ π

0

Re(

zejθ + 1zejθ − 1

)log |f(ejθ)| dθ −

nd∑

i=1

log∣∣∣∣1− diz

z − di

∣∣∣∣ . (4.1)

Proof. The Poisson-Jensen formula can be found in many standard books oncomplex analysis. See for instance [20]. ¥

Lemma 4.2 Let g is analytic in C+ and ci (i = 1, . . . , nc) be the zeros of g in C+,counting their multiplicities. If s ∈ C+ and g(s) 6= 0, then

log |g(s)| = 2π

∫ ∞

0

Re(

1 + jωs

s + jω

)log |g(jω)|

1 + ω2dω −

nc∑

i=1

log∣∣∣∣ci + s

ci − s

∣∣∣∣ . (4.2)

Proof. This is the continuous-time version of the Poisson-Jensen formula.Perform the bilinear transformation over Lemma 4.1 to prove it. ¥

Lemma 4.3 (Bode’s Attenuation Integral Formula) Let f(s) ∈ F and denotef(jω) = f1(ω) + jf2(ω). Suppose that f(s) is conjugate symmetric, i.e., f(s) =f(s). Then whenever f(∞) exists,

lims→∞

s[f(s)− f(∞)] =1π

∫ ∞

−∞[f1(ω)− f1(∞)] dω. (4.3)

4.1 Regulation Performance Problem 65

Proof. See [53, pp. 49]. ¥

Lemma 4.4 Consider a conjugate symmetric function f(s). Suppose that f(s) isanalytic and has no zero in C+. Then provided that f(∞) 6= 0,

lims→∞

s logf(s)f(∞)

=1π

∫ ∞

−∞log

∣∣∣∣f(jω)f(∞)

∣∣∣∣ dω. (4.4)

Proof. An immediate consequence of Lemma 4.3. ¥

Lemma 4.5 If f(z) ∈ RH∞, then

1π

∫ π

0

Re f(ejθ) dθ = f(∞). (4.5)

Proof. Suppose that

f(z) =anzn + an−1z

n−1 + . . . + a1z + a0

bnzn + bn−1zn−1 + . . . + b1z + b0,

is stable transfer function, where ak, bk ∈ R (k = 0, 1, . . . , n) and bn 6= 0. Wemay write

f(z) =an

bn+ f(z),

where

f(z) =cn−1z

n−1 + . . . + c1z + c0

(z − pn)(z − pn−1) . . . (z − p1)(z − p0),

withck = ak − anbk

bn, k = 0, 1, . . . , n− 1,

and pk are the poles of f which laid inside the unit disk, i.e., pk ∈ D, fork = 0, 1, . . . , n. Hence,

1π

∫ π

−π

Re f(ejθ) dθ =2an

bn+

1π

∫ π

−π

Re f(ejθ) dθ.

By Cauchy integral,

1π

∫ π

−π

f(ejθ) dθ = − jπ

∫

Dg(z) dz,

where g(z) = f(z)/z. Residue theorem gives

− jπ

∫

Dg(z) dz = 2

(Res g(0) +

n∑

k=0

Res g(pk)

)= −2πRes g(∞) = 0.

The proof is complete by fact that f(∞) = an/bn. ¥


4.1.2 Energy Regulation Problem

We here consider the energy regulation problem, in which the regulation per-formance is measured by the energy of the control input u. For a given im-pulse signal d we define the regulation performance as

Ec :=∫ ∞

0

|u(t)|2 dt, (4.6)

Ed :=∞∑

k=0

|u(k)|2, (4.7)

for continuous-time and discrete-time cases, respectively. From Parsevalidentity we can deduce that the best achievable regulation performances byall stabilizing controllers in set K are given by

E∗c = inf

K∈K‖K(s)So(s)P (s)d(s)‖22,

E∗d = inf

K∈K‖K(z)So(z)P (z)d(z)‖22,

where So is the output sensitivity function defined in (3.2). By consideringBezout identity (2.5) and Youla parameterization (2.8) and by noting thatSo = XM −NQM , d = 1, the optimal performance can further be expressedas

E∗ = infQ∈RH∞

‖Y N −MQN‖22, (4.8)

where E∗ stands either E∗c for continuous-time case or E∗

d for discrete-timecase.

4.1.3 Output Regulation Problem

An extension of the preceding problem can be made by considering a mini-mization problem of the energy of the measurement output simultaneouslywith that of the control input. In the present work we consider a more generalproblem, i.e., we consider the following performance index

Ec :=∫ ∞

0

(‖yw(t)‖2 + |u(t)|2) dt, (4.9)

Ed :=∞∑

k=0

(‖yw(k)‖2 + |u(k)|2) , (4.10)

where the signal yw is the weighted regulation output, i.e.,

yw(t) =[

yw1(t)yw2(t)

]=

[L−1Ws(s)[u(s) + d(s)]

L−1Wy(s)y(s)]

,

yw(k) =[

yw1(k)yw2(k)

]=

[Z−1Ws(z)[u(z) + d(z)]

Z−1Wy(z)y(z)]

.


Note that the first and second elements of yw account the sensitivity reduc-tion and the disturbance attenuation, respectively. If Ws = 0 and Wy = 0 thenthe problem reduces to an energy regulation one, which is discussed in theprevious subsection.

It follows from the well-known Parseval identity that

E = ‖yw1‖22 + ‖yw2‖22 + ‖u‖22holds, where E stands either Ec for the continuous-time case or Ed for thediscrete-time case. Then, it is immediate to obtain that

E = ‖WsSid‖22 + ‖WyPSid‖22 + ‖KPSid‖22, (4.11)

= ‖Ws(1−KSoP )d‖22 + ‖WySoP d‖22 + ‖KSoP d‖22, (4.12)

where Si and So are the input and output sensitivity functions defined in (3.1)and (3.2), respectively. From (4.11) it is clear that we consider a more gen-eral case of regulation problem, where we minimized the regulation outputunder sensitivity and complementarity sensitivity constraints. Furthermore,Bezout identity (2.5) and Youla parameterization (2.8) provide

E = ‖Ws(1 + Y N −MQN)‖22 + ‖Wy(XN −NQN)‖22 + ‖Y N −MQN‖22,from which we then want to determine the best achievable regulation perfor-mance with respect to all stabilizing controllers expressed in the followingthree-block optimal control problem:

E∗ = infQ∈RH∞

∥∥∥∥∥∥

Ws(1 + Y N −MQN)Wy(XN −NQN)

Y N −MQN

∥∥∥∥∥∥

2

2

. (4.13)

4.2 Continuous-time Case

In this section we provide the analytical closed-form expressions of the opti-mal regulation performances for continuous-time system.

Recall the coprime factorizations of the plant P (s) given in (2.4). Withoutloss of generality we may fix the scalar transfer function M(s) as

M(s) :=np∏

k=1

s− pk

s + pk, (4.14)

where pk (k = 1, . . . , np) are the unstable poles of P (s). It is useful to point-out that M(∞) = 1. Next, let introduce the following index set:

Nz := i : N(zi) = 0, zi ∈ C+. (4.15)

Note thatNz contains the index set of all common non-minimum phase zerosof P (s), counting their multiplicities.



Result on the energy regulation problem of SIMO continuous-time systemcan be found in [31]. We present the result here and establish its proof. Thefollowing theorem gives the minimal value of (4.6), i.e.,

E∗c = inf

K∈K

∫ ∞

0

|u(t)|2 dt,

which is further expressed by (4.8).

Theorem 4.1 ( [31]) Suppose that the plant P (s) given in (2.4) has common non-minimum phase zeros zi (i ∈ Nz) and unstable poles pk (k = 1, . . . , np). Then theoptimal energy regulation performance is given by

E∗c = Ecm + Ecn, (4.16)

where

Ecm := 2np∑

k=1

pk,

Ecn :=∑

i,j∈Nz

4Re zi Re zj

aiaj(zi + zj)αiαj

with

ai :=

1 ; #Nz = 1∏

j∈Nz,j 6=i

zj − zi

zj + zi, ; #Nz ≥ 2 ,

αi := 1−np∏

k=1

zi + pk

zi − pk.

Proof. See Appendix A.2. ¥Theorem 4.1, which is valid for SISO and SIMO systems, shows that un-

stable poles and non-minimum phase zeros of the plant completely charac-terize the optimal energy regulation performance. Note that only the com-mon non-minimum phase zero gives an effect, otherwise Ecn = 0. This theo-rem, however, reveals that in SIMO systems non-minimum phase zero doesnot always give a contribution on the optimal performance.

The following corollary shows that how the unstable pole closes to non-minimum phase zero can degrade the optimal performance.

Corollary 4.1 Suppose that the plant P (s) has single unstable pole p and commonnon-minimum phase zeros zi (i = 1, . . . , nz). Then,

E∗c = 2p

[nz∏

i=1

zi + p

zi − p

]2

.


−1 −0.5 0 0.5 1 1.5 2 2.5 30

100

200

300

400

500

600

p

E* c


Fig. 4.2. The energy regulation performance for continuous-time system (Example4.1).

We consider two examples to confirm the validity the expression givenby Theorem 4.1.

Example 4.1 Consider an SISO plant given by

P (s) =2s− 3s− p

,

in which P (s) has a non-minimum phase zero at s = 32 and possibly an unstable pole

at s = p. We compute the optimal energy regulation performance by using Matlabtoolbox and expression of Theorem 4.1. Fig. 4.2 shows that two computations matchwell. Particulary, whenever P (s) is stable then E∗

c = 0 and if p approaches thenon-minimum phase zero then the performance becomes very large. In general, theoptimal performance of a SISO plant with one non-minimum phase zero z and oneunstable pole p is determined by Corollary 4.1 as

E∗c = 2p

(z + p

z − p

)2

.

Example 4.2 We consider an SISO plant

P (s) =(4s− 3)(s− z)

s2 − 4,

which possesses non-minimum phase zeros at s = 34 and possibly at s = z and

unstable pole at s = 2. We compute the optimal energy regulation for z from −1 to4. Fig. 4.3 plots the results.


Result on the output regulation problem of MIMO system is provided in [14].We extend the result in two senses: we consider a problem for non-minimum


−1 −0.5 0 0.5 10

20

40

60

80

100

120

140

160

180

z

E* c

by Toolboxby expression

1 2 30

1000

2000

3000

4000

5000

6000

7000

8000

9000

z

E* c

Fig. 4.3. The energy regulation performance for continuous-time system (Example4.2).

phase system as well as that under a sensitivity constraint, even our result isonly valid for SIMO system.

In order for Ec in (4.9) to be finite, it is necessary that P (s)d(s) ∈ L2 andWs(s)d(s) ∈ L2. Since d(t) is an impulse function so that d(s) = 1, we needthe following assumption.

Assumption 4.1 P (s) and Ws(s) are strictly proper.

We are now ready to provide the best achievable output regulation per-formance. We give the gives the minimal value of (4.9), i.e.,

E∗c = inf

K∈K

∫ ∞

0

(‖yw(t)‖2 + |u(t)|2) dt.

In other words, we provide the analytical closed-form expression of (4.13).

Theorem 4.2 Suppose that the plant P (s) given in (2.4) has common non-minimumphase zeros zi(i ∈ Nz) and unstable poles pk (k = 1, . . . , np). Define the inner-outerfactorization such that

Ws(s)Wy(s)N(s)

−1

= Λi(s)Λo(s).

Then, under Assumption 4.1, the optimal output regulation performance is given by

E∗c = Ecm + Ecn, (4.17)

where


Ecm := 2np∑

k=1

pk +1π

∫ ∞

0

log(1 + ‖Ws(jω)‖2 + ‖Wy(jω)P (jω)‖2) dω,

Ecn :=∑

i,j∈Nz

4Re zi Re zj

aiaj(zi + zj)αiαj

with

ai :=

1 ; #Nz = 1∏

j∈Nz,j 6=i

zj − zi

zj + zi, ; #Nz ≥ 2 ,

αi := 1− Λo(zi)np∏

k=1

zi + pk

zi − pk.


Theorem 4.2 shows that if we simultaneously optimize the energy of thecontrol input and the system output as well as the sensitivity reduction, thenwe have an additional integral term imposed by the gain of the plant and thatof the weighting functions. Meanwhile, the plant unstable poles and non-minimum phase zeros contribute their effects in a similar manner as do inthe energy regulation problem, except there is an effect caused by Λo(s). Insome cases we may obtain an explicit formula for Λo(s) if Wy = 0, sinceΛo(s) does not depend on the plant P (s). In general, since Λo is stable andminimum phase, its absolute value at zi can be obtained from Lemma 4.2,that is

|Λo(zi)| = exp

1π

∫ ∞

0

Re(

1 + jωzi

zi + jω

)log |Λo(jω)|2

1 + ω2dω

,

where|Λo(jω)|2 = 1 + ‖Ws(jω)‖2 + ‖Wy(jω)P (jω)‖2.

Particularly, if Ws(s) = 0 and Wy(s) = 0, which imply the integral termis zero and Λo(s) = 1, then we can recover the results of energy regulationproblem obtained in Theorem 4.1.Remark: The complexity of Ecn-term can be reduced by considering simplecases. For stable case we may obtain

Ecn =∑

i,j∈Nz

4Re zi Re zj

aiaj(zi + zj)[1− Λo(zi)][1− Λo(zj)].

And whenever the plant has single unstable pole p and single non-minimumphase zero z, we have

Ecn = 2z

[1− Λo(z)

z + p

z − p

]2

,


−1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

900

1000

p

E* c


Fig. 4.4. The output regulation performance for continuous-time system (Example4.3).

which shows that unstable pole closes to non-minimum phase zero will de-teriorate the performance.

Now we pick one example to confirm the effectiveness of the derivedexpression in Theorem 4.2.

Example 4.3 Consider an SISO plant described by

P (s) =s− 3

(4s− 1)(s− p).

Clearly, P (s) has one non-minimum phase zero at s = 3 and possibly two unstablepoles at s = 1

4 and s = p. We compute the optimal output regulation performanceE∗

c obtained by Theorem 4.2 (circled-line) and numerically calculated by Matlabtoolbox (starred-line) for p from −1 to 6. Here we take the weighting functions asfollow:

Ws(s) =1

3s + 1, Wy(s) =

2s + 13s + 1

.

Fig. 4.4 shows that two computations match rather well. Particularly, when p closesto 3, the performance will be unbounded since it happens almost unstable pole-zerocancellation.

4.3 Discrete-time Case

Now we provide the analytical closed-form expression of the optimal regu-lation performance for discrete-time systems. It is worth to point out that inthe regulation problem, we have to exploit a certain function evaluated at in-finity, which is laid on the jω-axis (boundary of s-domain) but not on the unit


circle (boundary of z-domain). It means that derivation process in discrete-time case is not parallel with that of continuous-time case. This is contrastwith the tracking problem, where the derivations for the discrete-time arealmost parallel to those for the continuous-time case, see the derivation ofTheorems 3.1, 3.2, 3.3, and 3.4, 3.5, 3.6.

Recall the coprime factorizations of P (z) in (2.4). Without loss of general-ity, it is possible to set

M(z) = B(z) :=nλ∏

k=1

z − λk

λkz − 1(4.18)

where λk (k = 1, . . . , nλ) are the unstable poles of P (z). It is important to notethat B(z) is inner function and

B(∞) =nλ∏

k=1

1λk

.

To facilitate our derivation we define

N(z) = zN(z) (4.19)

and introduce the following index set:

Nη := i : N(ηi) = 0, ηi ∈ Dc. (4.20)

The definition of N(z) indicates that we decrease by one the relative degreeof P (z) since we may allow the implementation of the biproper controllers.While, Nη contains the index set of all common non-minimum phase zeros ofP (z) with counting multiplicities except one zero at infinity, i.e., the numberof zeros at infinity in Nη is equal to the relative degree of P (z) minus one.


Here we provide the minimal value of (4.7), i.e.,

E∗d = inf

K∈K

∞∑

k=0

|u(k)|2.

In other words, we derive the analytical closed-form expression of the op-timal energy regulation performance. Note that the optimal performance isdetermine by (4.8).

Theorem 4.3 Suppose that the SIMO plant P (z) is given in (2.4) has commonnon-minimum phase zeros ηi (i ∈ Nη) and unstable poles λk (k = 1, . . . , nλ). Thenthe optimal energy regulation performance is given by


E∗d = Edm + Edn, (4.21)

where

Edm :=nλ∏

k=1

|λk|2 − 1,

Edn :=∑

i,j∈Nη

(|ηi|2 − 1)(|ηj |2 − 1)bibj(ηiηj − 1)

βiβj

with

bi :=

1 ; #Nη = 1∏

j∈Nη,j 6=i

ηi − ηj

ηjηi − 1; #Nη ≥ 2

βi :=nλ∏

k=1

λk −nλ∏

k=1

λkηi − 1ηi − λk

.


The most important fact revealed by Theorem 4.3 is that the contribu-tion of unstable poles is given in product way instead of in summation likein continuous-time case, see Theorem 4.1. We may explain this differenceas follows. Suppose that the continuous-time plant P (s) has unstable polespk (k = 1, . . . , np). Then the discretization process with sampling time Tseconds will produce a discrete-time plant P (z), which has unstable polesλk (k = 1, . . . , nλ), where λk = epkT . It is obvious that

nλ∏

k=1

λk = exp

T

np∑

k=1

pk

.

This, however, gives an insight that in discrete-time case unstable poles con-tribute more detrimental effect than those in continuous-time case. For SIMOcase, condition ηi (i ∈ Nη) means that only the common non-minimum phasezeros give effect. Hence, if P (z) has no common non-minimum phase zerothen E∗

dn = 0.

Example 4.4 We consider the following SISO plant

P (z) =1

z − λ,

in which P (z) has relative degree 1 and possibly an unstable pole at z = λ. Ob-viously, if |λ| ≤ 1 then E∗

d = 0. For |λ| > 1 we consider two cases. First, weimplement a biproper controller. The optimal performance is then given by

E∗d = λ2 − 1.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

2

4

6

8

10

12

λ

E* d


strictly propercontroller

bipropercontroller

Fig. 4.5. The energy regulation performance for discrete-time system (Example 4.4).

Second, we apply a strictly proper controller. By this assumption, plant P (z) hasone non-minimum phase zero at z = ∞. Hence,

E∗d = (λ2 − 1) + lim

η→∞(λ2 − 1)2

η2 − 1(η − λ)2

= (λ2 − 1)λ2.

We also numerically compute the optimal performance by using Matlab toolbox.Fig. 4.5 depicts the computation results for λ from −2 to 2.

Now we provide two direct implications of Theorem 4.3 in the followingcorollaries.

Corollary 4.2 Suppose that the SIMO plant P (z) given in (2.4) is minimum phase,has relative degree 2 and unstable poles λk (k = 1, . . . , nλ). Then,

E∗d = Edm + Edn, (4.22)

where

Edm :=nλ∏

k=1

|λk|2 − 1,

Edn :=

[nλ∏

k=1

|λk|2] ∣∣∣∣∣

nλ∑

k=1

|λk|2 − 1λk

∣∣∣∣∣

2

.

Proof. The proof of Edm is obvious. If P (z) has only one unstable pole λ thenfrom the expression in Theorem 4.3 we obtain

Edn = (λ2 − 1)2 = λ2

(λ− 1

λ

)2

= |λ|2∣∣∣∣|λ|2 − 1

λ

∣∣∣∣2

.

If P (z) has two unstable poles λ1, λ2 then


Edn = (λ21λ2 + λ1λ

22 − λ1 − λ2)2 = |λ1|2|λ2|2

∣∣∣∣|λ1|2 − 1

λ1+|λ2|2 − 1

λ2

∣∣∣∣2

.

Subsequently, if P (z) has three unstable poles λ1, λ2, λ3 then

Edn = |λ1|2|λ2|2|λ3|2∣∣∣∣|λ1|2 − 1

λ1+|λ2|2 − 1

λ2+|λ3|2 − 1

λ3

∣∣∣∣2

.

In general, if P (z) has nλ unstable poles λk (k = 1, . . . , nλ) then

Edn =

[nλ∏

k=1

|λk|2] ∣∣∣∣∣

nλ∑

k=1

|λk|2 − 1λk

∣∣∣∣∣

2

.

It is proved. Note that the expressions in this corollary are similar to those inProposition 4.3 of [42]. ¥Corollary 4.3 Suppose that the SIMO plant P (z) given in (2.4) has relative degreev, common non-minimum phase zeros ηi (i ∈ Nη), and only one unstable pole λ.Then,

E∗d = λ2(v−1)(λ2 − 1)

∏

i∈Nη

ληi − 1ηi − λ

2

. (4.23)

Proof. Let the plant P (z) has only one unstable pole λ. In addition, if P (z) hasrelative degree 1 and one common non-minimum phase zero η, then from theexpressions in Theorem 4.3 we obtain

E∗d = (λ2 − 1)

(λη − 1η − λ

)2

.

If P (z) has relative degree 2 and two common non-minimum phase zerosη1, η2, then

E∗d = λ2(λ2 − 1)

(λη1 − 1η1 − λ

λη2 − 1η2 − λ

)2

.

Furthermore, if P (z) has relative degree 3 and three common non-minimumphase zeros η1, η2, η3, then

E∗d = λ4(λ2 − 1)

(λη1 − 1η1 − λ

λη2 − 1η2 − λ

λη3 − 1η3 − λ

)2

.

In general, if P (z) has relative degree v and common non-minimum phasezeros ηi (i ∈ Nη), then

E∗d = λ2(v−1)(λ2 − 1)

∏

i∈Nη

ληi − 1ηi − λ

2

.

It is proved. ¥



Here we provide the minimal value of (4.10), i.e.,

E∗d = inf

K∈K

∞∑

k=0

(‖yw(k)‖2 + |u(k)|2) ,

which further expressed by (4.13). Note that the plant P (z) and the weightingfunction Ws(z) are not necessary to be strictly proper to regulate a discrete-time system.

Theorem 4.4 Suppose that the SIMO plant P (z) is given in (2.4) has commonnon-minimum phase zeros ηi (i ∈ Nη) and unstable poles λk (k = 1, . . . , nλ).Define the inner-outer factorization such that

Ws(z)Wy(z)N(z)

−1

= Λi(z)Λo(z).

Then the optimal output regulation performance is given by

E∗d = Edm + Edn, (4.24)

where

Edm :=exp

1π

∫ π

0

log(1 + ‖Ws(ejθ)‖2 + ‖Wy(ejθ)P (ejθ)‖2)dθ

nλ∏

k=1

|λk|2 − 1,

Edn :=∑

i,j∈Nη

(|ηi|2 − 1)(|ηj |2 − 1)bibj(ηiηj − 1)

βiβj

with

bi :=

1 ; #Nη = 1∏

j∈Nη,j 6=i

ηj − ηi

ηiηj − 1; #Nη ≥ 2 ,

βi := Λo(∞)nλ∏

k=1

λk − Λo(ηi)nλ∏

k=1


.


Theorem 4.4 points-out that when we minimized the energy of controlinput simultaneously with that of system output then an additional termcaused by the plant gain influences the optimal performance. Differ fromits continuous-time counterpart in Theorem 4.2, the effect of the plant gainis expresses in exponential way rather than in linear manner. This fact sug-gests that the plant gain contributes more detrimental effects than those in


continuous-time case. The remainders are almost similar with those of theenergy regulation problem, i.e., Theorem 4.3, except the existence of the outerfunction Λo(z). The expression for Λo(∞) and |Λo(ηi)| can be derived fromPoisson-Jensen formula in Lemma 4.1 as follow:

Λo(∞) = exp

12π

∫ π

0

log |Λo(ejθ)|2 dθ

,

|Λo(ηi)| = exp

12π

∫ π

0

Re(

ηiejθ + 1ηiejθ − 1

)log |Λo(ejθ)|2 dθ

,

where|Λo(ejθ)|2 = 1 + ‖Ws(ejθ)‖2 + ‖Wy(ejθ)P (ejθ)‖2.

Again, if Ws(z) = 0 and Wy(z) = 0 then the expression reduces to thatof Theorem 4.3. We can also consider some special cases relate to the typeof the plant. Let introduce the following mild definition. The plant P (z) iscalled non-minimum phase if P (z) has common non-minimum phase zero.Otherwise, minimum phase.

Corollary 4.4 Suppose that the SIMO plant P (z) is given in (2.4). The followingsare direct implications of Theorem 4.4:

1. If P (z) is unstable and minimum phase (possibly with relative degree 1), then

E∗d = exp

1π

∫ π

0

log(1 + ‖Ws(ejθ)‖2 + ‖Wy(ejθ)P (ejθ)‖2) dθ

nλ∏

k=1

|λk|2−1.

2. If P (z) is stable and non-minimum phase then E∗d = Edm + Edn, where

Edm := exp

1π

∫ π

0

log(1 + ‖Ws(ejθ)‖2 + ‖Wy(ejθ)P (ejθ)‖2)dθ

− 1,

Edn :=∑

i,j∈Nη

(|ηi|2 − 1)(|ηj |2 − 1)bibj(ηiηj − 1)

(Λo(∞)− Λo(ηi))(Λo(∞)− Λo(ηj)).

3. If P (z) is stable and minimum phase then

E∗d = exp

1π

∫ π

0


− 1.

The following illustrative example confirms the validity of the result inTheorem 4.4.

Example 4.5 We consider an SISO plant given by

P (z) =4z2 − 9

z(3z + 4)(z − λ),


−3 −2.5 −2 −1.50

2

4

6x 10

4

λ

E* d

by Toolboxby expression

−1.5 −1 −0.5 00

2

4

6x 10

4

λ

E* d

0 0.5 1 1.5200

300

400

500

600

λ

E* d

1.5 2 2.5 3100

200

300

400

500

λ

E* d

Fig. 4.6. The output regulation performance for discrete-time system (Example 4.5).

which has non-minimum phase zeros at z = 32 and z = − 3

2 , and unstable polesat z = − 4

3 and possibly at z = λ. Fig. 4.6 plots Theorem 4.4 based computation(circled-line) and MATLAB toolbox-based computation (starred-line) for λ from −3to 3, where we set Ws(z) = 1 and Wy(z) = 1. The figure clearly shows that theexpression in Theorem 4.4 is correct and that E∗

d becomes larger when λ approachesto one of the non-minimum phase zeros.

4.3.3 Output Regulation Problem with Noise

In the recent years, there is a growing attention related to the research activityin feedback control with communication constraints [4, 42]. In this part, weconsider a feedback control system in which there exists a communicationlink, see Fig. 4.7. In digital communication, the link consists of some pre andpost processing equipments for the signal that are sent through the commu-nication channel, which might be in the form of filter, A-D converter, coder,modulator, decoder, demodulator, and D-A converter.

In this study, an signal-to-noise ratio constrained channel will be consid-ered and all pre and post signal processing are restricted to LTI filtering andD-A/A-D type operations. Thus, the communication link simplifies to the

Fig. 4.7. Feedback control systems with a communication link.


Fig. 4.8. Feedback control systems with a noisy channel.

noisy channel, Fig. 4.8. Here, y ∈ Rm is the measurement output and n ∈ Rm

is a zero mean additive white Gaussian noise with intensity Ω, i.e.,

E [n(k)] = 0, E [n(k)nT(κ)] = Ωδ(k − κ),

where E [·] represents the expectation operator and δ is the unitary impulsefunction:

δ(k − κ) =

1, k = κ0, k 6= κ.

We define‖y‖ =

√E [y′y].

and assume a given energy constraint Y , such that it is required

‖y‖2 < Y,

for some predetermined value Y > 0. The stabilization problem addressedin this section is then can be stated as a problem of finding the smallest valueof ‖y‖. Further we may write

‖y‖2 = ‖T (z)‖22 Ω, (4.25)

where T = PK(I + PK)−1. Note that by taking Ω as an identity, we thenhave a the problem of stabilization which minimize

Ed :=∞∑

k=0

‖y(k)‖2. (4.26)

By Parseval identity we can write

Ed = ‖T (z)n(z)‖22 = ‖T (z)1‖22 = ‖(I − So(z))1‖22,

where So(z) is the sensitivity function defined in (3.2). Then we can write theoptimal performance as

E∗d = inf

Q∈RH∞‖[I − (X −NQ)M ]1‖22. (4.27)

Without loss of generality we may set M(z) = B(z)I , where B(z) is given by(4.18). Hence,


E∗d = inf

Q∈RH∞‖(B−1I −X + NQ)1‖22

= infQ∈RH∞

‖(B−1 −B−1(∞))I + (B−1(∞)I −X + NQ)1‖22= ‖(B−1 −B−1(∞))I‖22 + inf

Q∈RH∞‖(B−1(∞)I −X + NQ)1‖22.

Now we are ready to provide the analytical closed-form expression of E∗d .

Theorem 4.5 Consider the feedback control setup given by Fig. 4.8, where P (z) isSIMO plant given in (2.3), which has common non-minimum phase zeros ηi (i ∈Nη) and unstable poles λk (k = 1, . . . , nλ). Then the optimal output regulationperformance is given by

E∗d = Edm + Edn, (4.28)

where

Edm := m

[nλ∏

k=1

|λk|2 − 1

],

Edn := m∑

i,j∈Nη

(|ηi|2 − 1)(|ηj |2 − 1)bibj(ηiηj − 1)

βiβj

with

bi :=

1 ; #Nη = 1∏

j∈Nη,j 6=i

ηi − ηj

ηjηi − 1; #Nη ≥ 2 ,

βi :=nλ∏

k=1

λk −nλ∏

k=1


.

Proof. Follow the proof of Theorem 4.3. ¥

Note that if m = 1, i.e., we consider an SISO plant P (z), then the expres-sions of this theorem is similar with those of Theorem 4.3. It can be under-stood since ‖y‖2 is completely determined by the complementary sensitivityfunction T (z), then the problem can be similarly treated as an energy regula-tion problem.

4.4 Delta Domain Case

In this section we reformulate and solve the optimal regulation problem interm of the delta operator in order to link the continuous-time and discrete-time results derived in the previous sections.

Consider the coprime factorizations of P (δ) given in (2.4), in which with-out loss of generality we may set M = H , where


H(δ) := B(Tδ + 1) =nλ∏

k=1

(Tδ + 1)− λk

λk(Tδ + 1)− 1, (4.29)

with λk are the unstable poles of P (z). It is easy to show that H is inner indelta domain, i.e., H( e−jωT−1

T )H( ejωT−1T ) = 1 and H(∞) = B(∞). We remark

that H possesses non-minimum phase zeros ρk ∈ DcT at ρk = (λk−1)/T (k =

1, 2, . . . , nρ), in which they also act as the unstable poles of P (δ). Note thatnρ = nλ.

As the disturbance input signal, we consider an impulse function in thefollowing form:

d(k) =

1T , for k = 00, for k 6= 0 , (4.30)

where its D-transform is dT (δ) = 1. We define

N(δ) = δN(δ), (4.31)

and introduce the following index set:

Nζ := i : N(ζi) = 0, ζi ∈ DcT . (4.32)

Note thatNζ contains the index set of all common non-minimum phase zerosof P (δ) with counting multiplicities except one zero at infinity.


Here we reformulate and solve the energy regulation problem, which is pre-viously discussed in Subsection 4.3.1, in terms of delta operator. In otherwords, we minimize the following performance index

Eδ := T

∞∑

k=0

|u(k)|2. (4.33)

By Parseval’s identity we may write (4.33) as

Eδ = ‖K(δ)So(δ)P (δ)dT (δ)‖22.

The optimal performance is then deduced as

E∗δ = inf

Q∈RH∞‖Y N −MQN‖22, (4.34)

The analytical closed-form expression of the optimal performance (4.34) isprovided in the following theorem.


Theorem 4.6 Suppose that the plant P (δ) given in (2.4) has common non-minimumphase zeros ζi (i ∈ Nζ) and unstable poles ρk (k = 1, . . . , nρ). Then the optimal en-ergy regulation performance is given by

E∗δ = Eδm + Eδn, (4.35)

where

Eδm :=1T

[nρ∏

k=1

|Tρk + 1|2 − 1

],

Eδn :=∑

i,j∈Nζ

(T |ζi|2 + 2 Re ζi)(T |ζj |2 + 2 Re ζj)gigj(T ζiζj + ζi + ζj)

γiγj

with

gi :=

1 ; #Nζ = 1∏

j∈Nζ ,j 6=i

ζi − ζj

T ζjζi + ζj + ζi; #Nζ ≥ 2 ,

γi :=nρ∏

k=1

(T ρk + 1)−nρ∏

k=1

T ρkζi + ρk + ζi

ζi − ρk.

Proof. By invoking the proof of Theorem 4.3 we can immediately write

E∗δ = E1 + E2,

where

E1 := ‖H−1(∞)−H−1‖22,E2 := inf

Q∈RH∞‖H−1(∞)− X + QN‖22.

Note that since H is inner,

E1 = ‖H−1(∞)H(δ)− 1‖22 = 1T ‖B−1(∞)B(z)− 1‖22.

The last holds from the norm relation (2.20). We then show that E1 = Eδm byfact that

‖B−1(∞)B(z)− 1‖22 =nλ∏

k=1

|λk|2 − 1 =nρ∏

k=1

|Tρk + 1|2 − 1.

We can show that E2 = Eδn by performing the partial fraction expansion asdid in the proof of Theorem 4.3. The factor 1

T appears from the fact that∥∥∥∥

1(Tδ + 1)− ηi

∥∥∥∥2

2

=1T

∥∥∥∥1

z − ηi

∥∥∥∥2

2

=1T

1|ηi|2 − 1

,

where ηi = Tζi + 1. We complete the proof. ¥



We consider the following performance index

Eδ := T

∞∑

k=0

(‖yw(k)‖2 + |u(k)|2), (4.36)

where yw(k) is given by

yw(k) =[

yw1(k)yw2(k)

]=

[D−1Ws(δ)[u(δ) + d(δ)]

D−1Wy(δ)y(δ)]

.

The optimal output regulation performance is also be provided by (4.13).Now we are ready to provide the analytical expressions of the optimal per-formance in delta domain. First we reformulate Lemmas 4.1 and 4.5 in deltadomain, respectively as follow.

Lemma 4.6 Let h is analytic in DcT and σi (i = 1, . . . , nσ) be the zeros of h in Dc

T ,counting their multiplicities. If δ ∈ Dc

T and h(δ) 6= 0, then

log |h(δ)| = T

π

∫ π/T

0

Re[(Tδ + 1)ejωT + 1(Tδ + 1)ejωT − 1

]log

∣∣∣∣h(

ejωT − 1T

)∣∣∣∣ dω

−nσ∑

i=1

log∣∣∣∣Tδσi + σi + δ

δ − σi

∣∣∣∣ . (4.37)

Lemma 4.7 If h ∈ RH∞, then

T

π

∫ π/T

0

Re h

(ejωT − 1

T

)dω = h(∞). (4.38)

Theorem 4.7 Suppose that the plant P (δ) given in (2.4) has common non-minimumphase zeros ζi(i ∈ Nζ) and unstable poles ρk (k = 1, . . . , nρ). Define the inner-outerfactorization such that

Ws(δ)Wy(δ)N(δ)

−1

= Λi(δ)Λo(δ).

Then the optimal output regulation performance is given by

E∗δ = Eδm + Eδn, (4.39)

where

Eδm :=1T

[|Λo(∞)|2

nρ∏

k=1

|Tρk + 1|2 − 1

],

Eδn :=∑

i,j∈Nζ


γiγj


with

|Λo(∞)|2 = exp

Tπ

∫ π/T

0log(1 + ‖Ws( ejωT−1

T )‖2+

‖Wy( ejωT−1T )P ( ejωT−1

T )‖2) dω

,

gi :=

1 ; #Nζ = 1∏

j∈Nζ ,j 6=i

ζi − ζj

T ζjζi + ζj + ζi; #Nζ ≥ 2 ,

γi := Λo(∞)nρ∏

k=1

(T ρk + 1)− Λo(ζi)nρ∏

k=1


ζi − ρk.

Proof. The proof is almost parallel to the discrete-time case, i.e., the proof ofTheorem 4.4. Use Lemma 4.7. ¥

Theorem 4.7 shows that the expressions of the optimal performance arenot only explicitly characterized by the unstable poles and non-minimumphase zeros of the plant but also by the sampling time. Actually, it is notdifficult to deduce that E∗

δ = E∗d/T , where we impose the relations λk =

Tρ+1 and ηi = Tζi+1 and note that the expression of |Λo(ζi)| can be derivedby Lemma 4.6.

In the next section we will inspect the convergence of the expressions ofTheorem 4.7 when we let the sampling time approach zero.

4.4.3 Continuity Property

Now we will demonstrate that E∗δ converges to E∗

c when the sampling timeT tends to zero. We deal with the output regulation case. We follow the wayof the tracking performance case in Subsection 3.4.4.

We denote by

Pc(s) =(Pc1(s), Pc2(s), . . . , Pcm(s)

)T,

the respecting continuous-time plant. Suppose that Pc(s) has unstable polespk (k = 1, . . . , np) and common non-minimum phase zeros zi (i ∈ Nz). Un-der the zero-order hold operations with sufficiently small sampling time weobtain the corresponding delta domain plant

PT (δ) =(PT1(δ), PT2(δ), . . . , PTm(δ)

)T,

where PT (δ) has unstable poles ρk (k = 1, . . . , nρ) with nρ = np, and commonnon-minimum phase zeros ζi (i ∈ Nζ) and ζ?

i (i ∈ N?ζ). Note that the latter are

the limiting zeros of PT (δ).Recall the following pole-zero relationships:


ρk =epkT − 1

T,

ζi =eziT − 1

T,

ζ?i =

η?i − 1T

,

where η?i are the zeros of polynomial Br(z) defined in (3.66). It is easy to

verify that Eδm can be written as Eδm = EH + ER, where

EH :=1T

[nρ∏

k=1

|Tρk + 1|2 − 1

]

ER :=|Λo(∞)|2 − 1

T

nρ∏

k=1

|Tρk + 1|2.

Since

EH ≈ 2nρ∑

k=1

ρk

and ρk → pk as T tends to zero, then we get

limT→0

EH = 2np∑

k=1

pk.

Next, since ejωT−1T → jω and |Tρk + 1|2 → 1 as T tends to zero, and also

WT s(δ), WTy(δ), PT (δ) converge to the continuous-time counterparts Wcs(s),Wcy(s), Pc(s), we have

limT→0

ER =1π

∫ ∞

0

log(1 + ‖Wcs(jω)‖2 + ‖Wcy(jω)Pc(jω)‖2)dω.

These two facts then show that

limT→0

Eδm = Ecm. (4.40)

We show the convergence of Eδn part by part. First we only consider a casewhere the delta domain plant has no limiting zero. Since ζi → zi as T → 0,we have

limT→0

gi =∏

j∈Nz,j 6=i

zj − zi

zj + zi=: ai.

Subsequently we obtain

limT→0


=4Re zi Re zj

aiaj(zi + zj).


To inspect the convergence of γi, we know that limT→0 |Λo(∞)|2 = 1. Thus,limT→0 Λo(∞) = 1. We also know that

limT→0

nρ∏

k=1

(T ρk + 1) = 1

and

limT→0

nρ∏

k=1


ζi − ρk=

np∏

k=1

zi + pk

zi − pk.

Now we only need to show the convergence of ΛTo(ζi). From Lemma 4.6 wehave

|ΛTo(ζi)| = exp

T

2π

∫ πT

0

Re[(Tζi + 1)ejωT + 1(Tζi + 1)ejωT − 1

]log

∣∣∣∣ΛTo

(ejωT − 1

T

)∣∣∣∣2

dω

.

Since

limT→0

T Re[(Tζi + 1)ejωT + 1(Tζi + 1)ejωT − 1

]= 2Re

[1

zi + jω

]=

21 + ω2

Re[1 + jωzi

zi + jω

]

implies limT→0 ΛTo(ζi) = Λco(zi), then we achieve limT→0 γi = αi. Commu-nicating all the above facts yields

limT→0

Eδn = Ecn. (4.41)

Therefore, (4.40) and (4.41) conclude

limT→0

E∗δ = E∗

c ,

i.e., the δ-domain solution converges to the corresponding s-domain solutionas sampling time T approaches to zero.

Showing the convergence of Eδn where there exist some i, j ∈ N?ζ is more

complicated. Thus, for simplicity we only consider three simplest cases.Case 1: The plant PT (δ) has only one common non-minimum phase zero

ζ?, i.e., the only zero is a limiting zero. The continuous-time counterpart ofthis case is that the plant Pc(s) is minimum phase with relative degree 2, 3,or 4. In this case we have

Eδn := (Tζ?2 + 2ζ?)γ2,

where

γ := ΛTo(∞)nρ∏

k=1

(T ρk + 1)− ΛTo(ζ?)nρ∏

k=1

T ρkζ? + ρk + ζ?

ζ? − ρk.

Then the following holds:


limT→0

Eδn = limζ?→∞

(η? + 1)ζ?

[Λco(∞)− Λco(ζ?)

np∏

k=1

ζ? + pk

ζ? − pk

]2

= 0,

which means that the limiting zero ζ? does not give any contribution. Hence,we maintain (4.41).

Case 2: The plant PT (δ) has one common non-minimum phase ’usual’zero ζ1 := ζ and one common non-minimum phase limiting zero ζ2 := ζ?.The continuous-time counterpart of this case is that the plant Pc(s) has onlyone minimum phase z with relative degree 2, 3, or 4. In this case we maywrite Eδn = E1

δn + E2δn + E3

δn, where

E1δn :=

(Tζ21 + 2ζ1)γ2

1

g21

,

E2δn :=

(Tζ22 + 2ζ2)γ2

2

g22

,

E3δn :=

2(Tζ21 + 2ζ1)(Tζ2

2 + 2ζ2)γ1γ2

g1g2(Tζ1ζ2 + ζ1 + ζ2).

The last two terms amount the effects caused by the limiting zero ζ?. Since

limT→0

g1 = limT→0

ζ − ζ?

Tζζ? + ζ + ζ?= lim

ζ?→∞z − ζ?

zη? + ζ?= −1,

limT→0

g2 = 1

hold, we obtain

limT→0

E1δn = 2z

[1− Λco(z)

np∏

k=1

z + pk

z − pk

]2

,

which shows that it converges to the corresponding continuous-time term.Furthermore, we see that limT→0 E2

δn = 0 from Case 1. Lastly, we have

limT→0

E3δn = 0,

since

limT→0

−2(Tζ2 + 2ζ)(Tζ? + 2ζ?)(Tζζ? + ζ + ζ?)

= limζ?→∞

4z(η? + 1)ζ?

ζ? + zη?= 4z(η? + 1),

limT→0

γ1 = 1− Λco(z)np∏

k=1

z + pk

z − pk,

limT→0

γ2 = limζ?→∞

[1− Λco(ζ?)

np∏

k=1

pk(η? + 1) + ζ? + pk

ζ? − pk

]= 0.


Therefore, limT→0 E2δn = 0 and limT→0 E3

δn = 0 reveal that the limiting zeroζ? does not give any effect when the sampling time tends to zero.

Case 3: The only non-minimum phase zeros of PT (δ) are limiting zerosζ?i ∈ N?

ζ . The continuous-time counterpart of this case is that the plant Pc(s)is minimum phase with relative degree more than 4. Immediately we have

limT→0

gi =∏

j∈N?ζ ,j 6=i

η?i − η?

j

η?j η?

i − 1=: c?

i ,

limT→0

γi = limζ?

i →∞

[1− Λco(ζ?

i )np∏

k=1

pk(η?i + 1) + ζ?

i + pk

ζ?i − pk

]= 0,

limT→0

(T |ζi|2 + 2Re ζi)(T |ζj |2 + 2Re ζj)gigj(T ζiζj + ζi + ζj)

= limζ?

j→∞(η?2

i − 1)(η?j + 1)ζ?

j

c?i c

?j (η

?i η?

j − 1).

Consequently,limT→0

Eδn = 0

under L’Hopital’s rule.Cases 1–3 represent the all situation may occur in general, where there

exist interaction within usual zeros and limiting zeros, and those among lim-iting zeros. We have shown that the limiting zeros do not give effects on theoptimal performance.

Now we pick one example for illustrating the delta domain result and itsconvergence.

Example 4.6 Reconsider the SISO continuous-time plant given in Example 4.3.Implementation of zero-order hold operation yields the corresponding delta domainplant

P (δ) =c(δ − ζ)

(δ − ρ1)(δ − ρ2),

which has also one non-minimum phase zero at δ = ζ and possibly two unstablepoles at δ = ρ1 and δ = ρ2. Under the corresponding weighting functions Ws(δ)and Wy(δ), Fig. 4.9 shows the computation of E∗

c by using Theorem 4.2 (solid line)and that of E∗

δ by using Theorem 4.7 for sampling time T = 0.10, 0.05 seconds(dashed and dash-dotted lines, respectively). This confirms our result that E∗

δ con-verges to E∗

c as T approaches zero.

4.5 Delay-time Case

In this section we provide a tiny contribution on the regulation problem ofpure delay-time systems. First we present an implication of our precedingresult on energy regulation problem of delta domain case.


−1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800

900

1000

p

E* c ,

E* δ

Continuous−timeDelta−time (T = 0.10)Delta−time (T = 0.05)

Fig. 4.9. The convergence of the delta domain solution to its continuous-time coun-terpart (Example 4.6).

Corollary 4.5 Suppose that P (δ) has relative degree v, common non-minimumphase zeros ζi (i ∈ Nζ), and has only one unstable pole ρ. Then, the optimal en-ergy regulation performance is given by

E∗δ = (Tρ + 1)2(v−1)(Tρ2 + 2ρ)

∏

i∈Nζ

Tζiρ + ζi + ρ

ζi − ρ

2

.

Proof. This is the delta domain counterpart of Corollary 4.3. Use the relationλ = Tρ + 1, η = Tζ + 1, and note that E∗

δ = E∗d/T . ¥

From now on, we study the energy regulation problem of continuous-time delay systems. We consider the regulation setup depicted by Fig. 4.10,where the SISO plant P (s) has a pure delay-time in the input port:

P (s) =P0(s)s− p

e−τs (4.42)

with τ ≥ 0 is the delay time. Note that we consider a simple plant P (s) whichhas only one unstable pole p ∈ C+. P0(s) is stable and has non-minimumphase zeros zi (i = 1, . . . , nz).

We formulate and minimize the following performance index

Fig. 4.10. The regulation setup for delay systems.


Ec :=∫ ∞

0

|u(t)|2 dt (4.43)

with respect to an impulse disturbance input d(t). We provide the analyticalclosed-form expression of the minimal energy regulation performance E∗

c inthe following proposition.

Proposition 4.1 Let the plant P (s) is given in (4.42). Then, the optimal energyregulation performance is given by

E∗c = 2p e2pτ

[nz∏

i=1

zi + p

zi − p

]2

. (4.44)

Proof. We follow an indirect way to prove Proposition 4.1, i.e., by using con-tinuity property of delta domain expression. It is known that the zero-orderhold operation with sampling time T will convert the continuous-time delayplant P (s) given in (4.42) onto its delta domain counterpart P (δ) as follows:

P (δ) =P0(δ)δ − ρ

(Tδ + 1)−τ/T ,

where P0(δ) is stable and has non-minimum phase zeros ζi (i = 1, . . . , nζ),and ρ is the unstable pole of P (δ). Note that τ/T relative degrees are con-tributed by the discretization of the delay part, while 1 relative degree is fromthat of P0(s). The optimal performance is then can be obtained by applicationof Corollary 4.5, that is

E∗δ = (Tρ + 1)2τ/T (Tρ2 + 2ρ)

[ nζ∏

i=1

Tζiρ + ζi + ρ

ζi − ρ

]2

.

By facts that ρ = (epT − 1)/T and ζi = (eziT − 1)/T , then we immediatelyhave

limT→0

E∗δ = 2p e2pτ

[nz∏

i=1

zi + p

zi − p

]2

.

Hence, by the continuity property we can derive the energy regulation per-formance for delay-time system (4.42). ¥

Remark: Proposition 4.1 tells that the time-delay gives its effects in expo-nential way as well as the unstable pole. It also admits that unstable poleand non-minimum phase zero which close each other generally worsen theregulation performance. Furthermore, if zi = ∞ then E∗ = 2pe2pτ , whichconfirms the result in [5]. Additionally if τ = 0, i.e., we consider an LTI case,then

E∗ = 2p

[nz∏

i=1

zi + p

zi − p

]2

,


which can be confirmed by Corollary 4.1.

It is well-known that Pade approximation can be used to approximate thedelay-time part. The first-order approximation provides

e−τs ≈ 2/τ − s

2/τ + s. (4.45)

Hence, for minimum phase case of P0(s), we obtain

P (s) ≈ PPade(s) =P0(s)s− p

2/τ − s

2/τ + s.

Note that PPade(s) has not only one unstable pole p but also one extra non-minimum phase zero 2/τ . It can be calculated by Theorem 4.1 that the opti-mal energy regulation performance of PPade(s) is given by

E∗Pade = 2p

[2/τ + p

2/τ − p

]2

. (4.46)

Furthermore, Taylor expansion gives

E∗c = 2p e2pτ ≈ 2p + 4τp2 + 2τ2p3 + · · · ,

E∗Pade = 2p

[2/τ + p

2/τ − p

]2

≈ 2p + 4τp2 + 4τ2p3 + · · · ,

which show that Pade approximation works well only for the small valuesof p and τ . We confirm this fact by example as follows.

Example 4.7 We consider the following delay-time plant

P (s) =e−0.1s

s− p, p > 0.

With delay time τ = 0.1sec., the Pade approximation plant PPade(s) has one non-minimum phase zero at s = 20. Hence,

E∗c = 2p e0.2p,

E∗Pade = 2p

[20 + p

20− p

]2

.

Figs. 4.11 and 4.12 plot these two optimal performances with respect to unstablepole location p. It is shown that for bigger p, E∗

Pade will be unbounded whenever pis getting closer to 20 since it happens almost unstable pole-zero cancellation. Twocalculations are very close each other but only for small p.

4.6 Summary 93

0 5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

p

Opt

imal

Per

form

ance

E*c

E*Pade

Fig. 4.11. Optimal energy regulation performance for delay-time systems via deltaapproach and Pade approximation.

0 1 2 3 4 50

5

10

15

20

25

30

p

Opt

imal

Per

form

ance

E*c

E*Pade

Fig. 4.12. Delta approach and Pade approximation match well for small p.

4.6 Summary

In this chapter we have examined the H2 regulation performance problemin SIMO LTI feedback control system. We have formulated and solved theproblem in minimizing the energy of the system output under the control in-put and sensitivity constraints against the impulsive disturbance input. Weprovide the analytical closed-form expression of the optimal regulation per-formance in terms of the dynamics and structure of the plant. Toward theexisting result of continuous-time case, we have extended the problem to anon-minimum phase system. While in discrete-time case, we have providednew results.

Our result shows that, in discrete-time case, the contribution of unsta-ble poles and plant gain are given in product and exponential ways. This


differs from the continuous-time case, where those are given in summationand plain ways. The difference is caused by a fact that the derivation of theexpressions for both two cases are completely different. In this process wehave to exploit certain functions evaluated at infinity, which is laid on thejω-axis (boundary of s-domain) but not on the unit circle (boundary of z-domain). It means that derivation process in discrete-time case is not parallelwith that of continuous-time case. This is contrast with the tracking problem,where the derivations for the discrete-time are almost parallel to those for thecontinuous-time case.

Additionally, we have reformulated and resolved the regulation problemin delta domain by means the delta operator. Analysis on continuity propertyshows that we can recover the continuous-time expression from the deltadomain expression stand point by taking the sampling time to zero. Based onthe delta domain expression, we have also derived the analytical closed-formexpression of the optimal energy regulation performance for pure delay-timesystem.

In general, we show that the optimal regulation performance is prop-erly characterized by the plant’s unstable poles and common non-minimumphase zeros, plant gain, and the outer factor. The last can be fairly explainedby Poisson-Jensen formula.

5

Applications

This chapter is devoted to application issues. We demonstrate how to ap-ply the analytical closed-form expressions to some physical systems, namelythree-disk torsional system, inverted pendulum system, and magnetic bear-ing system.

5.1 Three-disk Torsional System

5.1.1 Problem Setting

Fig. 5.1 depicts the three degree of freedom torsional apparatus, where thesystem has three disks of inertia J1, J2, J3, respectively, due to the disksthemselves and the masses we affix to the disks, and damping coefficientsc1, c2, c3 due to friction in the bearings and other components supporting thedisks. A torsional rod connecting all three disks with torsional spring con-stants k1 and k2 in between the disks. The actuation device, a high torquebrushless DC servo motor, applies torque to the bottom disk. The control in-put u is the voltage applied to the control motor. As feedback sensors, highresolution optical encoders are attached at each disk to measure the angulardisplacements θ1, θ3, and θ3.

The equations of motion for the three-disk torsional system are as follow:

Fig. 5.1. Three-disk torsional system.

96 5 Applications

J1θ1 + c1θ1 + k1θ1 − k1θ2 = u,

J2θ2 + c2θ2 − k1θ1 + (k1 + k2)θ2 − k2θ3 = 0,

J3θ3 + c3θ3 − k2θ2 + k2θ3 = 0.

We denote by P1(s), P2(s), and P3(s), the transfer functions from appliedvoltage u to regulated outputs θ1, θ3, and θ3, respectively. Then they are givenby 6th order models

P1(s) =N1(s)sD(s)

, P2(s) =N2(s)sD(s)

, P3(s) =N3(s)sD(s)

,

where

N1(s) = J2J3s4 + (J2c3 + J3c2)s3 + (J2k2 + c2c3 + J3k1 + J3k2)s2 +

(c2k2 + c3k1 + c3k2)s + k1k2,

N2(s) = k1(J3s2 + c3s + k2),

N3(s) = k1k2,

and

D(s) = J1J2J3s5 + (J1J2c3 + J1J3c2 + J2J3c1)s4 +

[J1(J2k2 + J3k1 + J3k2 + c2c3) + J2(J3k1 + c1c3) + J3c1c2]s3 +[J1(c2k2 + c3k1 + c3k2) + J2(c1k2 + c3k1) +J3(c1k1 + c1k2 + c2k1) + c1c2c3]s2 +[(J1 + J2 + J3)k1k2 + c1(c2k2 + c3k1 + c3k2) + c2c3k1]s +(c1 + c1 + c1)k1k2.

Expression for D(s) requires the measurement or estimation of eight dy-namic parameters for its coefficients. The more practical forms expressed interms of measurable frequencies, damping, and gains, are given by

P1(s) =K1(s2 + 2ζz1ωz1s + ω2

z1)(s2 + 2ζz2ωz2s + ω2

z2)

s(s + c)(s2 + 2ζp2ωp2s + ω2p2

)(s2 + 2ζp3ωp3s + ω2p3

),

P2(s) =K2(s2 + 2ζzωzs + ω2

z)s(s + c)(s2 + 2ζp2ωp2s + ω2

p2)(s2 + 2ζp3ωp3s + ω2

p3),

P3(s) =K3

s(s + c)(s2 + 2ζp2ωp2s + ω2p2

)(s2 + 2ζp3ωp3s + ω2p3

).

Suppose that we fix the parameters J1 = 0.002713, J2 = 0.001802, andJ3 = J2 (all in kg m2), c1 = 0.008778, c2 = 0.001834, and c3 = c2 (all inkg m/s2), and k1 = 3.0568 and k2 = 2.6570 (all in kg m/s). Then the transferfunctions are all (marginally) stable and minimum phase. The location of thepoles and zeros is depicted by Fig. 5.2. Note that in this case all the plantshave the same set of poles.

5.1 Three-disk Torsional System 97

−2 −1.5 −1 −0.5 0−80

−60

−40

−20

0

20

40

60

80

Real Axis

Imag

inar

y A

xis

PolesZeros of Disk 1Zeros of Disk 2

Fig. 5.2. The location of poles and zeros for torsional system.

5.1.2 Tracking Performance Limitation

In this part we apply our analytical expressions of the optimal tracking per-formance to the torsional system. We consider the tracking error problemunder control input penalty (See Subsection 3.2.3), i.e., we compute

J∗c := infK∈K

∫ ∞

0

(|e(t)|2 + |uw(t)|2) dt,

where uw(t) = L−1Wu(s)u(s). Note that the plants all have an integrator,thus the required assumptions are satisfied.

Since the plants are (marginally) stable and minimum phase, then theoptimal tracking performances are given by Corollary 3.2 as follow:

J∗c,i =1π

∫ ∞

0

log[1 +

|Wu(jω)|2|Pi(jω)|2

]dω

ω2, (5.1)

for i = 1, 2, 3. Fig. 5.3 plots the optimal performance J∗c,i for Wu from 0 to 1. Itis shown that disk 1 provides the best tracking performance and disk 3 givesthe worst.

5.1.3 Regulation Performance Limitation

In this subsection we apply our analytical expressions of the optimal regula-tion performance to the torsional system. We consider the output regulationproblem, where we minimize the energy of control input jointly with that ofsystem output (See Subsection 4.2.2), i.e., we compute

E∗c := inf

K∈K

∫ ∞

0

(|yw(t)|2 + |u(t)|2) dt,

98 5 Applications

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Wu

J* c

Disk 1Disk 2Disk 3

Fig. 5.3. The optimal tracking performance of three-disk torsional system.

0.03 0.035 0.04 0.045 0.05 0.0551.5

2

2.5

3

E* c

Disk 1Disk 2Disk 3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

E* c

10 15 20 25 30 35 4050

100

150

200

Wy

E* c

Fig. 5.4. The optimal regulation performance of three-disk torsional system.

where yw(t) = L−1Wy(s)y(s). Note that the plants are all strictly proper,thus the required assumption is satisfied.

Since the plants are (marginally) stable and minimum phase, then theoptimal regulation performances are given by Theorem 4.2 as follow:

E∗c,i =

1π

∫ ∞

0

log(1 + |Wy(jω)Pi(jω)|2) dω, (5.2)

for i = 1, 2, 3. Fig. 5.4 plots the optimal performance E∗c,i for different ranges

of Wy. Differ to tracking performance case, regulation performance problem

5.2 Inverted Pendulum System 99

Fig. 5.5. The inverted pendulum system.

of three-disk torsional system reveals interesting facts. Disk 1 provides thebest performance if we take the output weighting function Wy very small,i.e., Wy < 0.05, but the worst performance for very big Wy , i.e., Wy > 27.

5.2 Inverted Pendulum System


For an illustration of our results on tracking performance limitations we con-sider the inverted pendulum system shown in Fig. 5.5, where an invertedpendulum is mounted on a motor driven-cart. We assume that the pendu-lum moves only in the vertical plane, i.e., two dimensional control problem.We here assume that M , m, and 2l respectively denote the mass of the cart,the mass of the pendulum, and the length of the pendulum. We also assumethat the friction between the track and the cart is µt and that between thependulum and the cart is µp. We consider an uniform pendulum so that itsinertia is given by I = 1

3ml2.The equations of motion between the control input u, which is the force

to the cart, the position of the cart x, the angle of the pendulum θ are repre-sented by

43ml2θ + µpθ −mglθ = −mlx,

(M + m)x + µtx−mlθ = u,

under the assumption that the angle θ is small. Taking the Laplace transformof the system equations yields

43ml2Θ(s)s2 + µpΘ(s)s−mglΘ(s) = −mlX(s)s2,(M + m)X(s)s2 + µtX(s)s−mlΘ(s)s2 = U(s).

Then, the transfer functions from u to x (denoted by Px), from u to θ (denotedby Pθ), and that from u to the position of the edge of the pendulum x + 2lθ(denoted by Pxθ) are, respectively, given by

100 5 Applications

Px(s) =43ml2s2 + µps−mgl

sD(s),

Pθ(s) =−mls

D(s),

Pxθ(s) =− 2

3ml2s2 + µps−mgl

sD(s),

whereD(s) = a3s

3 + a2s2 + a1s + a0,

with a3 := 13 (4M +m)ml2, a2 := (M +m)µp + 4

3µtml2, a1 := −(M +m)mgl+µpµt, and a0 := −µtmgl. Note that all of the plants have a common unstablepole at p. Additionally, Px(s) has a non-minimum phase zero at

zx =−3µp +

√9µ2

p + 48m2gl3

8ml2, (5.3)

and Pxθ(s) has two non-minimum phase zeros at

zxθ =3µp ±

√9µ2

p − 24m2gl3

4ml2. (5.4)

To study the tracking performance of inverted pendulum system, we con-sider three different cases:

C1: The control objective in this case is to control the cart position x(t), i.e.,we consider an SISO plant P (s) = Px(s).

C2: We shall control the cart position x(t) and the pendulum angle θ(t),i.e., we consider an SITO plant

P (s) =[

Px(s)Pθ(s)

].

We select ν = (1, 0)T as the step input direction.C3: In this case we control the cart position x(t) and the position of the

edge of the pendulum x + 2lθ, i.e., we consider an SITO plant

P (s) =[

Px(s)Pxθ(s)

].

We choose ν = (1, 1)T as the step input direction.


We here give the analytical closed-form expression for the best achievabletracking error performance. We assume µp = µt = 0 for simplicity, i.e., thereare no frictions imposed, from which the unstable pole is determined by


1 1.5 2 2.5 315.5

15.6

15.7

15.8

15.9

16

16.1

16.2

16.3

16.4

l (m)

J* c1

Case C1

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

l (m)

J* c2 ,

J* c3

Cases C2 and C3

Case C2

Case C3

Case C1

Fig. 5.6. Tracking error performance of inverted pendulum system.

p =

√3(M + m)g(4M + m)l

. (5.5)

We denote by J∗c1, J∗c2, and J∗c3, the optimal tracking performances for CasesC1, C2, and C3, respectively. According to Theorem 3.2 we obtain

J∗c1 = 4

√l

3g

√

M + m +√

M + 14m

√M + m−

√M + 1

4m

2

,

J∗c2 = 4

√l

3g+

1π

∫ ∞

0

log[1 +

9ω4

(4lω2 + 3g)2

]dω

ω2,

J∗c3 = 4

√l

3g+

1π

∫ ∞

0

log[12

+12

(2lω2 − 3g)2

(4lω2 + 3g)2

]dω

ω2

+1π

∫ ∞

0

log[12

+12

(4lω2 + 3g)2

(2lω2 − 3g)2

]dω

ω2.

We remark that in Case C1, the optimal tracking performance is imposedby both non-minimum phase zero zx and unstable pole p. Hence, J∗c1 de-pends on the mass of the cart M , the mass and the length of the pendulum,m and l. While in Cases C2 and C3 the unstable pole p does not give an ef-fect since its direction is not coincident with that of step input signal ν, i.e.,M(p)ν 6= 0. Consequently, the optimal performances J∗c2 and J∗c3 dependonly on l but are independent of M and m.

Furthermore, if we assume that the ratio between the mass and the lengthof the pendulum is constant, i.e., m

l = ψ for a real constant ψ, then the length

102 5 Applications

which gives the lowest possible performance in Case C1 is

l∗ =(√

265− 5)M2ψ

. (5.6)

It can be readily seen that we can make l∗ small by reducing the mass of thecart M or by increasing the pendulum parameter ψ. We may also rewrite(5.6) as

m∗

M=√

265− 52

≈ 5.6394, (5.7)

which suggests that the lowest possible value of the optimal tracking per-formance can be achieved as long as the ratio between the the mass of thependulum and that of the cart satisfies (5.7), regardless the type of materialwe use for the pendulum.

Fig. 5.6 illustrates the relationship between the length of the pendulumand the optimal performance for ψ = 2.145πkg/m and M = 2kg. In CaseC1, l ≈ 1.7m gives the lowest possible performance. While in Cases C2 andC3, shorter pendulum is preferred since it provides smaller tracking perfor-mance.

5.2.3 Tracking Error Problem under Control Penalty

This part considers the tracking error problem under control input penalty ofinverted pendulum system. We here assume µp 6= 0, µt 6= 0, and Wu(s) = 1.We define the following functions:

f1(ω) := ω2[(a0 − a2ω2)2 + (a1ω − a3ω

3)2],f2(ω) := m2l2( 4

3 lω2 + g)2 + µ2pω2,

f3(ω) := m2l2( 23 lω2 − g)2 + µ2

pω2,

and

Jz :=16ml2

−3µp +√

9µ2p + 48m2gl3

,

which denotes the effect caused by the non-minimum phase zero of Px(s),i.e., zx. According to Theorem 3.3, the optimal performances for Cases C1,C2, and C3, are respectively given by

J∗c1 = Jz +1π

∫ ∞

0

log[1 +

f1(ω)f2(ω)

]dω

ω2+

2[1−Θ∼i (p)Θi(0)]2

p,

J∗c2 = Jz +1π

∫ ∞

0

log[1 +

f1(ω) + m2l2ω4

f2(ω)

]dω

ω2,

J∗c3 = Jz +2µp

mgl+

1π

∫ ∞

0

log[[f1(ω) + f2(ω) + f3(ω)]2

f2(ω)f3(ω)

]dω

ω2,


1 2 3 40.9

0.95

1

1.05

1.1

1.15

1.2x 10

5

l (m)

J* c1

Case C1

0 0.5 1 1.5 23

3.5

4

4.5

5

5.5

6

l (m)

J* c2

Case C2

Fig. 5.7. Tracking error performance under control input penalty of inverted pendu-lum system.

where Θi(s) in J∗c1 is determined from the inner-outer factorization[

Mx(s)Nx(s)

]= Θi(s)Θo(s)

with Px(s) = Nx(s)M−1x (s). The existence of Θi(s) makes the analytical for-

mula incomplete since we don’t know the closed-form expression of thistransfer function.

Fig. 5.7 depicts the relationship between the length of the pendulum andthe optimal performance for ψ = 2.145πkg/m, M = 2kg, µt = 1

4 , and µp = 13 .

We can see from the figure that the values of J∗c1 are quite large which arecaused by the third term of J∗c1, and that the optimal length of the pendulumwhich gives the lowest possible performance is around 2.8m. The values ofoptimal costs J∗c2 for Case C2 is quite small in comparison with J∗c1. The mainreason is that the unstable pole p does not give any effect since p is a commonpole of Px(s) and Pθ(s), and its direction is not coincident with that of stepinput signal, i.e., M(p)ν 6= 0. In this case, the best length of the pendulumis about 0.2m. Note that in Case C3, the second term is the effect caused bynon-minimum zeros of Pxθ(s), i.e., zxθ, but J∗c3 = ∞ since the integral term isinfinity. The integral term accounts the total variation of the plant directionwith frequency. It is understood that a rapid change of plant direction atlow frequency will impose a more deteriorate constraint upon the achievabletracking performance.

104 5 Applications

Fig. 5.8. Active magnetic bearing system.

5.3 Magnetic Bearing System


In this section we implement our results on regulation performance limi-tation provided in Chapter 4 to study the performance limitations in a mag-netic bearing system, which has been widely investigated, see e.g., [39,56,57].We consider a simple active magnetic bearing (AMB) depicted in Fig. 5.8.AMBs suspend the levitated object (generally, a rotor) of mass M by forces oftwo opposing magnetic attractions which are supplied by power switchingamplifiers of voltages V1, V2 and currents I1, I2. AMBs use actively controlledelectromagnetic forces to control the position of the rotor or other ferromag-netic body in air which has nominal air gap g0.

If we assume that the state variable can be forced to track some con-stant trajectory Φ0 by appropriate choice of control input u, then a linearizingmodel may be realized as follows [39]:

ddt

xvφ

=

0 1 00 0 Φ0

µΦ0 0 −µ

xvφ

+

001

u,

yc =[−Φ0 0 1

]

xvφ

, yp = x,

where x and v respectively denote the normalized position of the rotor andits derivative, and φ, u, and yc are respectively normalized differences of thefluxes (Φ1, Φ2), input voltages (V1, V2), and output currents (I1, I2) of left andright magnetics, which are given by

φ :=Φ1 − Φ2

AgBsat, u :=

V1 − V2

V0, yc :=

I1 − I2

Isat

with appropriate constant Ag, Bsat, V0, and Isat.We define by Pc the transfer function from the control input to the current

sensor, i.e., from u to yc, by Pp the transfer function from the control input tothe position sensor, i.e., from u to yp, and by Pcp the transfer function from

5.3 Magnetic Bearing System 105

the control input to the both of current and position sensors, i.e., from u to yc

and yp. In other words, we define

Pc(s) =s2 − Φ2

0

s3 + µs2 − µΦ20

,

Pp(s) =Φ0

s3 + µs2 − µΦ20

,

Pcp(s) =[

Pc(s)Pp(s)

].

It is clear that Pc(s) has one non-minimum phase zero at Φ0 and one un-stable pole p lying between 0 and Φ0, depending on the value of Φ0 and µ.It is known that physically reasonable value of σ := Φ0/µ ranges betweenabout 0.3 and 3 and that, the unstable pole p ranges from about 0.6Φ0 and0.9Φ0 [39]. Further, Pp(s) has no non-minimum phase zero but one unstablepole at p. Meanwhile, Pcp(s) has no common non-minimum phase zero butone unstable pole, also at p.

We here examine the regulation performance limitation of the AMB asmeasured by the H2 norm. In other words, we minimize the following per-formance measure

Ec =∫ ∞

0

(|yw1(t)|2 + ρc|yc(t)|2 + ρp|yp(t)|2 + |u(t)|2)dt, (5.8)

where ρc and ρp are some constants. Indeed, the above performance index re-veals the most natural setting in the control application, where yw1 accountsthe sensitivity measure, yc and yp quantify the regulation outputs, and con-trol input u represents the disturbance attenuation. In practical situation, thecurrent should be bounded. Thus, include the current sensor yc into the per-formance index to be minimized is a more realistic setup.

We examine the following three cases relate to the choice of the weightingfunction Wy :

C1: For P (s) = Pc(s),

Wy,c(s) =

[ρc

ρpΦ0(s+Φ0)2

].

C2: For P (s) = Pp(s),

Wy,p(s) =

[ρc(s+Φ0)

2

Φ0

ρp

].

C3: For P (s) = Pcp(s),

Wy,cp(s) =[

ρc 00 ρp

].

106 5 Applications

Note that the definitions Wy(s) above assure that the performance indexesfor the three cases are completely the same, i.e.,

Wy,c(s)Pc(s) = Wy,p(s)Pp(s) = Wy,cp(s)Pcp(s).

By letting the weighting function Wy depend on parameters ρp and ρc wegain more degree of freedom. We immediately can assign ρc = 0 such thatwe regulate only the position yp. For all the cases we choose the weightingfunction Ws as

Ws(s) =w

1 + τs,

where τ is the bandwidth and w is a real constant.To facilitate our analysis, we define by E∗

c,c, E∗c,p, E∗

c,cp, the optimal per-formances correspond to Pc(s), Pp(s), and Pcp(s), respectively.

5.3.2 Regulation Problem: Continuous-time Case

We first note that the first terms of (4.17), Ecm, are the same for all the threecases. Since Pp(s) has no non-minimum phase zero and Pcp(s) has no com-mon one (note that only common non-minimum phase zero gives limita-tion), we can see that Ecn = 0 and hence the optimal performance of thesetwo cases are equal. On the other hand, Pc(s) has one non-minimum phasezero at Φ0, and hence Ecn is always positive. This observation implies thatthe following relations generally hold:

E∗c,c > E∗

c,p = E∗c,cp, (5.9)

where

E∗c,c − E∗

c,p = 2Φ0

[1− Λo(Φ0)

Φ0 + p

Φ0 − p

]2

.

This can be confirmed by the following further investigation. First we con-sider a case where ρc = ρp = 0, i.e., Wy(s) = 0. For this special case, clearlywe obtain

Λo(s) =√

1 + w2 + τs

1 + τs.

Then, the closed-form expression of the optimal performances can be ex-pressed as

E∗c,p = E∗

c,cp = 2p +1π

∫ ∞

0

log[1 +

w2

1 + ω2τ2

]dω = 2p +

√1 + w2 − 1|τ |

and

E∗c,c − E∗

c,p = 2Φ0

[1−

√1 + w2 + τΦ0

1 + τΦ0

Φ0 + p

Φ0 − p

]2

> 0.

5.3 Magnetic Bearing System 107

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

20

40

60

80

100

120

140

160

180

200

220

µ

E* c,

c


(a) Current sensor E∗c,c

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

µ

E* c,

p


(b) Position sensor E∗c,p

Fig. 5.9. The optimal regulation performances with respect to µ (ρc = ρp = 0).

We now compute the optimal performances with the following physicalparameters [39]: Φ0 = 0.288, µ = 0.582, from which we get p = 0.242. For theweighting function Ws(s), we take w = 1 and τ = 1. The computation resultsgive E∗

c,c = 117.7013 and E∗c,p = E∗

c,cp = 0.8983. The complete calculationsfor different value of µ are depicted by Figs. 5.9(a) and 5.9(b). We also providethe numerical calculation by using Matlab toolbox. The calculations confirmthe equality and inequality relations (5.9).

Next, we consider a case where ρc = ρp = 1. Note that in computationE∗

c,c, Λo(s) is determined from the inner-outer factorization

Ws(s)Wy,c(s)Nc(s)

−1

= Λi(s)Λo(s),

where Nc(s) is the coprime factor of Pc(s), i.e.,

Pc(s) = Nc(s)M−1c (s).

The computation results provide E∗c,c = 716.5626 and E∗

c,p = E∗c,cp = 1.5821.

One physical interpretation can be gained from relations (5.9) is that puttingextra sensor does not not always conduce an performance improvement.

5.3.3 Regulation Problem: Discrete-time Case

We here discuss the discrete-time case. We assume that the correspondingdiscrete-time transfer functions Pc(z), Pp(z), Pcp(z) are obtained from thezero-order hold operations of Pc(s), Pp(z), Pcp(s), respectively. By these op-erations, we know that Pc(z) has two non-minimum phase zeros at 1.3351and at infinity. Pp(z) has also two non-minimum phase zeros at −3.2498 andat infinity, while Pcp(z) has no common non-minimum phase zero except at

108 5 Applications

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

50

100

150

200

250

300

µ

E* d,

c


(a) Current sensor E∗d,c

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

µ

E* d,

p


(b) Position sensor E∗d,p

0.35 0.4 0.45 0.5 0.55 0.6 0.651.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

µ

E* d,

cp


(c) Current-position sensor E∗d,cp

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

E* d,

p − E

* d,cp

T

(d) Differences between E∗d,p and E∗

d,cp

Fig. 5.10. The optimal regulation performances for the discretized plant with respectto µ (ρc = ρp = 0).

infinity. All the plants have one unstable pole at 1.2738. The optimal perfor-mance then can be computed based on Theorem 4.4, and we can show thefollowing relations:

E∗d,c > E∗

d,p > E∗d,cp. (5.10)

For example, for ρc = ρp = 0 with the sampling time T = 1 second, wehave E∗

d,c = 150.3615, E∗d,p = 1.7506, and E∗

d,cp = 1.3383, which impliesthat using multiple sensors has an advantage for the discrete-time system.For the complete calculations of different value of µ, see Figs. 5.10(a), 5.10(b),and 5.10(c). The calculations confirm the inequality relations (5.10). However,the differences between E∗

d,p and E∗d,cp become smaller when the sampling

time T is reduced, as shown in Fig. 5.10(d).

6

Conclusion

We have examined the H2 optimal tracking and regulation problems insingle-input multiple-output (SIMO) linear time-invariant (LTI) feedbackcontrol systems. Instead of seeking the optimal controllers, we have been de-riving the analytical closed-form expressions of the optimal performances interms of the plant dynamics and structures. In other words, we have quanti-fied and characterized the fundamental performance limitations arise in theH2 optimal control problem and provided guidelines for plant design fromthe view point of control. We have provided the comprehensive and unifiedsolutions to the problem since we derive the analytical expressions for threedomains: continuous-time, discrete-time, delta-time systems, and show theirunification.

We rely our derivation on the factorization approach such as the coprimefactorizations of the plant which led to the Bezout identity and the Youlaparameterization of the all stabilizing controllers, and the inner-outer factor-ization of transfer function. By deeply exploiting this approach,

(i) We have provided the analytical closed-form expressions of the optimalperformance in tracking step input signal. The tracking ability itself ismeasured by the error between the input reference signal and the mea-surement output, possibly under control input constraint. In H2 optimalcontrol setting, the tracking performance J is given by

J = ‖e(·)‖22,J = ‖e(·)‖22 + ‖Wu(·)u(·)‖22,

where Wu is the stable and minimum phase weighting function. Briefly,we found that• Continuous-time Case: The available continuous-time results [12]

and [31] contain small mistake. We correct the mistake by explicitlyaccounting an additional effect Jcu1 caused by the plant’s unstablepoles pk. However, the additional term Jcu1 is equal to zero when-

110 6 Conclusion

ever the plant P is either scalar or SIMO with the set of all unstablepoles of Pi (i = 1, . . . , m) are completely the same.

• Discrete-time Case: The derivation process of the discrete-time caseis almost similar to that of the continuous-time case. The key ideaproposed in this way is the application of the bilinear transforma-tion, from which we enable to derive two key lemmas from thecontinuous-time counterparts.

• Delta Domain Case: The derivation of the delta-time expressions in-cluding two key lemmas can be obtained in straightforward mannerby variables substitution. Differ to the discrete-time case, the expres-sions in the delta-time case contain the sampling time T . We demon-strated that the continuous-time expressions can be completely re-covered by approaching T to zero. In this regard, we show that thelimiting zeros of the discretized plant do not give any effects to theperformance provided the sampling time is sufficiently small.

• Delay-time Case: By exploiting the delta domain expressions we con-firm that the time delay degrades the tracking performance in muchthe same manner as non-minimum phase zeros.

• Sampled-data Case: We have employed an approximation approachby implementing fast sampling technique to derive the similar resultof sampled-data feedback control systems. Fast sampling process ofsampling time T/N enables us to approximate the sampled data feed-back control system to one of discrete-time system. Immediately wecan derive the approximation of the optimal performance by invok-ing the discrete-time result. We have shown by example that we canapproximate the exact result quite well. Particularly if the samplingtime T is small, N can also be made small.

In general, the optimal tracking performance of SIMO system is explicitlycharacterized by the plant’s non-minimum phase zeros ηij and unstablepoles λk, the plant direction which mostly determined by the plant gain,and the reference input direction ν. Furthermore, problem of minimizingthe tracking error under control input penalty provides additional limitsimposed by Wu, which appears in the logarithmic term and inner factor.If we set Wu = 0 then we can easily obtain the non-penalty result.

(ii) We have provided the analytical closed-form expressions of the optimalregulation performance against an impulsive disturbance input d. In thisproblem the regulation performance is measured by the energy of thecontrol input u, possibly under system output yw := ( yT

w1, yTw2 )T and

sensitivity constraints. In H2 optimal control setting, the regulation per-formance E is given by

E = ‖u(·)‖22,E = ‖yw1(·)‖22 + ‖yw2(·)‖22 + ‖u(·)‖22,

where yw1 measures the sensitivity reduction and yw2 accounts the regu-lated output. We briefly summarize as follows.

6 Conclusion 111

• Continuous-time Case: We have completed the results in continuous-time system by deriving the analytical closed-form expression fornon-minimum phase system and have extended it by considering aproblem with sensitivity penalty.

• Discrete-time Case: We have provided the first result on the regu-lation performance limitation of the discrete-time systems. Differ tothe tracking problem, the derivation process of the optimal regula-tion performance for discrete-time case is not parallel with that ofcontinuous-time case since we have to exploit a certain function eval-uated at infinity which is laid on the jω-axis (boundary of s-domain)but not on the unit circle (boundary of z-domain). Consequently, theunstable poles of discrete-time system give their effects in exponentialway, while those of continuous-time systems contribute their effect inlinear manner.

• Delta Domain Case: We have demonstrated that the continuous-timeexpressions can be completely recovered from the delta-time expres-sions stand-point by approaching the sampling time T to zero. Wealso show that the limiting zeros do not contribute any effects on theoptimal regulation performance.

• Delay-time Case: We have provided the analytical closed-form ex-pression of the optimal energy regulation performance for simpleSIMO delay-time system, where the plant is non-minimum phase andhas only single unstable pole. The derivation has been carried-out byinvoking discrete-time and delta-time expressions of the correspond-ing problem. This tiny contribution shows that the time delay as wellas the unstable pole give effects in exponential manner.

Generally speaking, the optimal regulation performance of SIMO sys-tem is explicitly characterized by the plant’s non-minimum phase zerosηij and unstable poles λk as well as the plant gain. It is important topoint-out that only the common non-minimum phase zeros influence theperformance. Furthermore, if we set Ws = Wy = 0 then we obtain the op-timal energy regulation performance, i.e., the problem without penalty.

(iii) We have confirmed the effectiveness of the derived expressions by sev-eral illustrative examples. We have also shown how to apply the analyt-ical expressions to practical applications including the control of three-disk torsional system, the determination of the optimal parameters ininverted pendulum system, and the selection of the sensor strategy inmagnetic bearing system. It has been demonstrated that the analyticalclosed-form expressions of the optimal performance are quite useful indetermining the easily controllable plant.

A

Some Proofs

A.1 Proof of Theorem 3.5

Given the inner-outer factorization (3.44), we define a norm preserving func-tion

Ψ(z) :=[

Θ∼i (z)

I −Θi(z)Θ∼i (z)

],

i.e., Ψ∼(ejθ)Ψ(ejθ) = I . By pre-multiplying Ψ(z) to (3.45), we obtain


∥∥∥∥∥[Θ∼i + Θo(Y −QM)]ν

z − 1

∥∥∥∥∥

2

2

+∥∥∥∥

(I −ΘiΘ∼i )ν

z − 1

∥∥∥∥2

2

=∥∥∥∥

(Θ∼i −Θ∼i (1))νz − 1

∥∥∥∥2

2

+ infQ∈RH∞

∥∥∥∥∥[Θ∼i (1) + Θo(Y −QM)]ν

z − 1

∥∥∥∥∥

2

2

+

∥∥∥∥(I −ΘiΘ

∼i )ν

z − 1

∥∥∥∥2

2

= J∗1 + J∗2 ,

where

J∗1 :=∥∥∥∥

(Θ∼i −Θ∼i (1))νz − 1

∥∥∥∥2

2

+∥∥∥∥

(I −ΘiΘ∼i )ν

z − 1

∥∥∥∥2

2

,

J∗2 := infQ∈RH∞

∥∥∥∥∥[Θ∼i (1) + Θo(Y −QM)]ν

z − 1

∥∥∥∥∥

2

2

.

Note that J∗1 is the optimal performance corresponding to the stable part ofthe plant, that is N(z). Since N = ΘiΘo, where Θi = ( w1, . . . , wm )T, thenwi(z) (i = 1, . . . ,m) has the same set of non-minimum phase zeros as Ni(z).It is immediate from Lemma 3.4 that

114 A Some Proofs

w′i(1)wi(1)

= −ni∑

j=1

|ηij |2 − 1|ηij − 1|2 −

∑

k∈Jzi

|λk|2 − 1|λk − 1|2 +

12π

∫ π

−π


∣∣∣∣dθ

1− cos θ.

The second term of above equation appears since the set of non-minimumphase zeros of Ni(z) contains the unstable poles of Pj(z), i 6= j. And bynoting that |wi(ejθ)| = |Ni(ejθ)|/‖N(ejθ)‖ = |Pi(ejθ)|/‖P (ejθ)‖ we show that

J∗1 = Jds1 + Jds2 + Jdu1.

Now we carry on the derivation of J∗2 . We define g(z) := M(z)ν. Sinceg(λk) = 0 for all k ∈ Jp, and since g(z) is left invertible, we can factorize it inthe form of g(z) = m(z)h(z), where m(z) is left invertible in RH∞ and h(z)is inner function defined by

h(z) :=∏

k∈Jp

z − λk

1− λkz.

Consequently,

J∗2 = infQ∈RH∞

∥∥∥∥∥

[Θ∼

i (1) + ΘoY

hν −ΘoQm

]h

z − 1

∥∥∥∥∥

2

2

= infQ∈RH∞

∥∥∥∥∥

[Θ∼

i (1) + ΘoY

hν −ΘoQm

]1

z − 1

∥∥∥∥∥

2

2

.

Let denote R1(z) := Θo(z)Y (z). Based on a partial fraction expansion proce-dure used in [12, 17, 58], it is possible to write

Θ∼i (1) + R1(z)h(z)

=∑

k∈Jp

1vk

Θ∼i (1) + R1(λk)

hk+ R2(z),

where

vk(z) :=z − λk

1− λkz,

hk :=∏

`∈Jp,` 6=k

λk − λ`

1− λ`λk,

and R2(z) is in RH∞. Then,

J∗2= infQ∈RH∞

∥∥∥∥∥∥

∑

k∈Jp

1vk

Θ∼i (1) + R1(λk)hk

ν + R2ν −ΘoQm

1

z − 1

∥∥∥∥∥∥

2

2

= infQ∈RH∞

∥∥∥∥∥∥

∑

k∈Jp

[1vk− 1− λk

1− λk

]Θ∼i (1) + R1(λk)

hkν + R3ν −ΘoQm

1

z − 1

∥∥∥∥∥∥

2

2

,

A.1 Proof of Theorem 3.5 115

where

R3(z) = R2(z) +∑

k∈Jp

Θ∼i (1) + R1(λk)

hk.

Since [1vk− 1− λk

1− λk

]1

z − 1∈ H⊥2 ,

and(R3(z)ν −Θo(z)Q(z)m(z))

1z − 1

∈ H2,

then we have

J∗2 =

∥∥∥∥∥∥

∑

k∈Jp

[1vk− 1

]Θ∼i (1) + R1(λk)

hkν

1

z − 1

∥∥∥∥∥∥

2

2

+

infQ∈RH∞

∥∥∥∥(R3ν −ΘoQm)1

z − 1

∥∥∥∥2

2

.

Direct calculation yields[

1vk− 1− λk

1− λk

]1

z − 1=

|λk|2 − 1(z − λk)(1− λk)

.

Moreover, since Θo is right-invertible and m is left-invertible we may prop-erly select a Q such that

infQ∈RH∞

∥∥∥∥(R3ν − CQm)1

z − 1

∥∥∥∥2

2

= 0.

Hence, by choosing ν = Θi(1) we obtain

J∗2 =

∥∥∥∥∥∥∑

k∈Jp

1z − λk

(|λk|2 − 1)(1−R1(λk)Θi(1))hk(1− λk)

∥∥∥∥∥∥

2

2

.

Since XM = I + ΘiR1 which implies Θ∼i XM = Θ∼i + R1, then we obtain

R1(λk) = −Θ∼i (λk) for all k ∈ Jp. Therefore,

J∗2 =∑

k,`∈Jp

(|λk|2 − 1)(|λ`|2 − 1)hkh`(1− λk)(1− λ`)

(1−Θ∼i (λk)Θi(1))(1−Θ∼i (λ`)Θi(1))×⟨

1z − λk

,1

z − λ`

⟩.

Now we calculate the inner product term. Note that for a given stableplant

116 A Some Proofs

P (z) =[

A BC D

],

itsH2 norm is determined by ‖P (z)‖22 = D2 +CLCT, where L is the solutionof L = ALAT + BBT. Since for λ ∈ Dc,

∥∥∥∥1

z − λ

∥∥∥∥2

2

=∥∥∥∥

1λz − 1

∥∥∥∥2

2

,

and1

λz − 1=

[1/λ 11/λ 0

]

is stable, then we obtain∥∥∥∥

1z − λ

∥∥∥∥2

2

=1

|λ|2 − 1.

Consequently, ⟨1

z − λk,

1z − λ`

⟩=

1λkλ` − 1

,

and then we show that J∗2 = Jdu2. The proof is now complete.


Since M(s) defined in (4.14) is an inner, we can write (4.8) as

E∗c = ‖M−1Y N −QN‖22.

From Bezout identity (2.6) we have MX − Y N = I , and then M−1Y N =X −M−1. This enables us to write

E∗c = ‖ −M−1 + X −QN‖22

= ‖(1−M−1)− (1− X + QN)‖22.

For any Q(s) ∈ RH∞ such that (1− X + QN) ∈ H2 then

E∗c = ‖1−M−1‖22 + inf

Q∈RH∞‖1− X + QN‖22.

Let denote by E1 the first term of above equation and by E2 the second term.By noting that

1− s + pk

s− pk=−2Re pk

s− pk,

then


E1 =

∥∥∥∥∥np∑

k=1

−2Re pk

s− pk

∥∥∥∥∥

2

2

=np∑

k=1

4 Re2pk

∥∥∥∥1

s− pk

∥∥∥∥2

2

.

Since ∥∥∥∥1

s− pk

∥∥∥∥2

2

=1

2 Re pk,

we conclude that

E1 = 2np∑

k=1

Re pk = 2np∑

k=1

pk.

We show that E1 = Ecm. To obtain the closed-form expression of E2, we em-ploy the standard partial fraction expansion technique. First, since N(zi) = 0for all i ∈ Nz and N(s) is left-invertible, we can perform the factorization

N(s) = n(s)a(s),

where n(s) is left-invertible in RH∞ and a(s) is defined by

a(s) :=∏

i∈Nz

s− zi

s + zi.

Since a(s) is inner, we then obtain

E2 = infQ∈RH∞

∥∥∥∥∥1− X

a+ Qn

∥∥∥∥∥

2

2

.

An expansion procedure provides

E2 = infQ∈RH∞

∥∥∥∥∥∑

i∈Nz

s + zi

s− zi

1− X(zi)ai

+ R1 + Qn

∥∥∥∥∥

2

2

= infQ∈RH∞

∥∥∥∥∥∑

i∈Nz

[s + zi

s− zi− 1

]1− X(zi)

ai+ R2 + Qn

∥∥∥∥∥

2

2

,

where

ai :=∏

j∈Nz,j 6=i

zj − zi

zj + zi,

R1(s) :=1− X(s)

a(s)−

∑

i∈Nz

s + zi

s− zi

1− X(zi)ai

,

R2(s) := R1(s) +∑

i∈Nz

1− X(zi)ai

.

118 A Some Proofs

Splitting into orthogonal subspaces yields

E2 =

∥∥∥∥∥∑

i∈Nz

[s + zi

s− zi− 1

]1− X(zi)

ai

∥∥∥∥∥

2

2

+ infQ∈RH∞

‖R2 + Qn‖22 .

Since n(s) is left invertible, we may select a Q such that infQ∈RH∞ ‖R2 +Qn‖22 = 0. This provides

E2 =

∥∥∥∥∥∑

i∈Nz

2Re(zi)(1− X(zi))ai

1s− zi

∥∥∥∥∥

2

2

.

By noting that X(zi) = M−1(zi) for all i ∈ Nz we conclude

E2 =∑

i,j∈Nz

4Re(zi)Re(zj)aiaj(zi + zj)

(1−M−1(zi))(1−M−1(zj)).

We show that E2 = Ecn by defining αi := 1 − M−1(zi). The proof is nowcomplete.


The proof of the minimum phase part of this theorem mainly follows thatof [14, Theorem 3]. Since M(s) in (4.14) is inner then we may write (4.13) as

E∗c = inf

Q∈RH∞

∥∥∥∥∥∥

Ws(M−1 + M−1Y N −QN)Wy(XN −NQN)M−1Y N −QN

∥∥∥∥∥∥

2

2

.

From Bezout identity (2.6) we obtain identities X = M−1 + M−1Y N andNX = XN , such that we may write

E∗c = inf

Q∈RH∞

∥∥∥∥∥∥

Ws(X −QN)Wy(NX −NQN)X −M−1 −QN

∥∥∥∥∥∥

2

2

= infQ∈RH∞

∥∥∥∥[

WX −WQN

X −M−1 −QN

]∥∥∥∥2

2

,

where

W :=[

Ws

WyN

].

Based on the orthogonal subspaces splitting we write

E∗c = EM + EQ,

where


EM :=∥∥∥∥[

01−M−1

]∥∥∥∥2

2

,

EQ := infQ∈RH∞

∥∥∥∥[

WX −WQN

1− X + QN

]∥∥∥∥2

2

.

The proof of Theorem 4.1 shows that

EM = 2np∑

k=1

pk.

Next, we write

EQ = infQ∈RH∞

∥∥∥∥[

WX

1− X

]−

[W−1

]QN

∥∥∥∥2

2

and perform an inner-outer factorization such that[

W (s)−1

]= Λi(s)Λo(s),

from which we have

|Λo(jω)|2 = 1 + ‖Ws(jω)‖2 + ‖Wy(jω)N(jω)‖2= 1 + ‖Ws(jω)‖2 + ‖Wy(jω)P (jω)‖2

and Λo(∞) = 1. Let define the following norm preserving function

Γ (z) :=[

Λ∼i (s)I − Λi(s)Λ∼i (s)

],

i.e., Γ∼(jω)Γ (jω) = I . By pre-multiplying Γ we obtain

EQ = infQ∈RH∞

∥∥∥∥Γ

[WX

1− X

]−

[W−1

]QN

∥∥∥∥2

2

= infQ∈RH∞

∥∥∥R1 − ΛoQN∥∥∥

2

2+ ‖R2‖22,

where

R1 := ΛoX − Λ−Ho ,

R2 :=[

W (ΛHo Λo)−1

1− (ΛHo Λo)−1

].

Further,

EQ = infQ∈RH∞

∥∥∥ΛoX − 1− ΛoQN∥∥∥

2

2+

∥∥Λ−Ho − 1

∥∥2

2+ ‖R2‖22.

120 A Some Proofs

Let denote

EQ1 := infQ∈RH∞

∥∥∥ΛoX − 1− ΛoQN∥∥∥

2

2,

EQ2 :=∥∥Λ−H

o − 1∥∥2

2+ ‖R2‖22.

We can similarly follow the partial fraction expansion used in the proof ofTheorem 4.1 to show that EQ1 = Ecn. Direct calculation by using H2 normdefinition yields

EQ2 = − 1π

∫ ∞

−∞

(Re Λ−1

o (jω)− 1)

dω.

Since Λ−1o (∞) = 1 we can apply Lemma 4.3, which gives

EQ2 = − lims→∞

s[Λ−1o (s)− 1]

= lims→∞

s2[Λ−1o (s)]′

= − lims→∞

s log Λ−1o (s).

The last two equalities hold under L’Hopital rule. Application of Lemma 4.4yields

− lims→∞

s log Λ−1o (s) =

−1π

∫ ∞

−∞log |Λ−1

o (jω)| dω.

Therefore,

EQ2 =1π

∫ ∞

0

log |Λo(jω)|2 dω

=1π

∫ ∞

0

log(1 + ‖Ws(jω)‖2 + ||Wy(jω)P (jω)‖2) dω.

The proof is complete by fact that EM + EQ2 = Ecm.


Since we set M(z) = B(z) and B(z) is an inner function then (4.8) becomes

E∗d = inf

Q∈RH∞‖B−1Y N −QN‖22.

From Bezout identity (2.6) we have MX − Y N = I , or equivalently BX =Y N + I , and thus B−1Y N + B−1 = X ∈ RH∞. This enables us to write

E∗d = inf

Q∈RH∞‖ −B−1 + X −QN‖22.


For any Q ∈ RH∞ such that (B−1(∞) − X + QN) ∈ H2 and together withthe fact that (B−1(∞)−B−1) ∈ H⊥2 , then we may write E∗

d = E1 +E2, where

E1 := ‖B−1(∞)−B−1‖22,E2 := inf

Q∈RH∞‖B−1(∞)− X + QN‖22.

We shall show that E1 = Edm and E2 = Edn. Since B(z) is inner then

E1 = ‖B−1(∞)B(z)− 1‖22 =

∥∥∥∥∥nλ∏

k=1

λkz − |λk|2λkz − 1

− 1

∥∥∥∥∥

2

2

.

Let define

E1,N :=

∥∥∥∥∥N∏

k=1


− 1

∥∥∥∥∥

2

2

, (A.1)

and claim that

E1,N =N∏

k=1

|λk|2 − 1.

We rely on the mathematical induction to prove our claim. For N = 1, it istrue that

E1,1 =∥∥∥∥

1− |λ1|2λ1z − 1

∥∥∥∥2

2

= |λ1|2 − 1.

Further, let define

χ(z) :=λNz − 1

λNz − |λN |2λN − |λN |2

λN − 1.

It is not difficult to show that χ(z) is an inner function. Then by pre-multiplyingχ(z) to (A.1), we may obtain

E1,N =

∥∥∥∥∥λN − |λN |2

λN − 1

[N−1∏

k=1


− λNz − 1λNz − |λN |2

]∥∥∥∥∥

2

2

= |λN |2∥∥∥∥∥

[N−1∏

k=1


− 1

]−

[λNz − 1

λNz − |λN |2− 1

]∥∥∥∥∥

2

2

= |λN |2[E1,N−1 +

∥∥∥∥λNz − 1

λNz − |λN |2− 1

∥∥∥∥2

2

].

A direct calculation shows that∥∥∥∥

λNz − 1λNz − |λN |2

− 1∥∥∥∥

2

2

=1

|λN |2∥∥∥∥|λN |2 − 1z − λN

∥∥∥∥2

2

=|λN |2 − 1|λN |2 .

122 A Some Proofs

Hence, we may write E1,N as a recursive expression

E1,N = |λN |2E1,N−1 + |λN |2 − 1.

Suppose it is true that

E1,N−1 =N−1∏

k=1

|λk|2 − 1.

Then we get

E1,N = |λN |2[

N−1∏

k=1

|λk|2 − 1

]+ |λN |2 − 1 =

N∏

k=1

|λk|2 − 1.

And this proves our claim and shows that E1 = Edm. Now we take care ofE2, which can be further written as

E2 = infQ∈RH∞

∥∥∥z[B−1(∞)− X] + QN∥∥∥

2

2,

where N(z) = zN(z). Since N(ηi) = 0 for all i ∈ Nη and since N(z) isleft-invertible, it can be factorized as N(z) = n(z)b(z), where n(z) is left-invertible in RH∞ and b(z) is defined by

b(z) :=∏

i∈Nη

z − ηi

ηiz − 1.

Therefore, we obtain

E2 = infQ∈RH∞

∥∥∥∥∥z[B−1(∞)− X]

b+ Qn

∥∥∥∥∥

2

2

.

Based on the standard partial fraction expansion procedure, we may write

z[B−1(∞)− X(z)]b(z)

=∑

i∈Nη

1vi

z[B−1(∞)− X(ηi)]bi

+ R1,

where

vi(z) :=z − ηi

ηiz − 1,

bi :=∏

j∈Nη,j 6=i

ηi − ηj

ηjηi − 1,

and R1(z) is in RH∞. Then,


E2 = infQ∈RH∞

∥∥∥∥∥∥∑

i∈Nη

1vi

z[B−1(∞)− X(ηi)]bi

+ R1 + Qn

∥∥∥∥∥∥

2

2

= infQ∈RH∞

∥∥∥∥∥∥∑

i∈Nη

[1vi− ηi

]z[B−1(∞)− X(ηi)]

bi+ R2 + Qn

∥∥∥∥∥∥

2

2

,

where

R2(z) = R1(z) +∑

i∈Nη

ηiz[B−1(∞)− X(ηi)]bi

.

Since [1vi− ηi

]∈ H⊥2 ,

and(R2(z) + Q(z)n(z)) ∈ H2,

then we have

E2 =

∥∥∥∥∥∥∑

i∈Nη

[1vi− ηi

]z[B−1(∞)− X(ηi)]

bi

∥∥∥∥∥∥

2

2

+ infQ∈RH∞

‖R2 + Qn‖22 .

Since n(z) is left invertible we may select a Q such that

infQ∈RH∞

‖R2 + Qn‖22 = 0.

By fact that [1vi− ηi

]=|ηi|2 − 1z − ηi

,

and X = B−1Y N + B−1 thus X(ηi) = B−1(ηi) for all i ∈ Nη , we get

E2 =

∥∥∥∥∥∥∑

i∈Nη

(|ηi|2 − 1)(B−1(∞)−B−1(ηi))bi(z − ηi)

∥∥∥∥∥∥

2

2

.

Further, since ∥∥∥∥1

z − ηi

∥∥∥∥2

2

=1

|ηi|2 − 1,

we then show that E2 = Edn by defining

βi :=nλ∏

k=1

λk −nλ∏

k=1


.

124 A Some Proofs


Since M(z) = B(z) is an inner function then we may write (4.13) as

E∗d = inf

Q∈RH∞

∥∥∥∥∥∥

Ws(B−1 + B−1Y N −QN)Wy(XN −NQN)

B−1Y N −QN

∥∥∥∥∥∥

2

2

.

From Bezout identity (2.6) we obtain identities X = B−1 + B−1Y N andNX = XN , such that we may write

E∗d = inf

Q∈RH∞

∥∥∥∥∥∥

Ws(X −QN)Wy(NX −NQN)−B−1 + X −QN

∥∥∥∥∥∥

2

2

= infQ∈RH∞

∥∥∥∥[

WX −WQN

−B−1 + X −QN

]∥∥∥∥2

2

,

where

W :=[

Ws

WyN

].

Based on the orthogonal subspaces splitting we write

E∗d = EB + EQ,

where

EB :=∥∥∥∥[

0B−1(∞)−B−1

]∥∥∥∥2

2

,

EQ := infQ∈RH∞

∥∥∥∥[

WX −WQN

B−1(∞)− X + QN

]∥∥∥∥2

2

.

The proof of Theorem 4.3 shows that

EB =nλ∏

k=1

|λk|2 − 1. (A.2)

Next, we write

EQ = infQ∈RH∞

∥∥∥∥[

WX

B−1(∞)− X

]−

[W−1

]QN

∥∥∥∥2

2

and perform an inner-outer factorization such that[

W (z)−1

]= Λi(z)Λo(z),

where the inner factor Λi(z) is a stable factor and the outer factor Λo(z) rep-resents the minimum phase part. Note that Λi(z) is a column vector transferfunction, Λo(z) is a scalar transfer function, and


|Λo(ejθ)|2 = 1 + ‖Ws(ejθ)‖2 + ‖Wy(ejθ)N(ejθ)‖2= 1 + ‖Ws(ejθ)‖2 + ‖Wy(ejθ)P (ejθ)‖2.

Note that |Λo(∞)|2 6= 1 since infinity does not lie in the unit circle and wehave no properness assumption on Ws(z) and P (z). Let define a norm pre-serving function

Γ (z) :=[

Λ∼i (z)I − Λi(z)Λ∼i (z)

],

i.e., Γ∼(ejθ)Γ (ejθ) = I . By pre-multiplying Γ we obtain

EQ = infQ∈RH∞

∥∥∥∥Γ

[WX

B−1(∞)− X

]−

[W−1

]QN

∥∥∥∥2

2

= infQ∈RH∞

‖C1 − ΛoQN‖22 + ‖C2‖22,

where

C1 := ΛoX − Λ−Ho B−1(∞),

C2 :=[

W (ΛHo Λo)−1

1− (ΛHo Λo)−1

]B−1(∞).

Further, based on the orthonormal subspace splitting we may write

EQ = EQ1 + EQ2 + EQ3 ,

where

EQ1 := infQ∈RH∞

‖ΛoX − Λo(∞)B−1(∞)− ΛoQN‖22,

EQ2 := |B−1(∞)|2‖Λ−Ho − Λo(∞)‖22,

EQ3 := ‖C2‖22.

By following the partial fraction expansion as did in the proof of Theorem4.3 we provide EQ1 = Edn. Next, direct calculation yields

EQ2 =|Λo(∞)|2|B(∞)|2 +

12π|B(∞)|2

∫ π

−π

(|Λ−1o (ejθ)|2 − 2ReΛ−1

o (ejθ)Λo(∞))dθ,

and similarly,

EQ3 =1

|B(∞)|2 −1

2π|B(∞)|2∫ π

−π

|Λ−1o (ejθ)|2dθ.

Therefore,

EQ2 + EQ3 =|Λo(∞)|2 + 1|B(∞)|2 − 2Λo(∞)

π|B(∞)|2∫ π

0

Re Λ−1o (ejθ) dθ.

126 A Some Proofs

Since Λo is an outer factor, then Λ−1o is in RH∞. Invoking Lemma 4.5 yields

EQ2 + EQ3 =|Λo(∞)|2 + 1|B(∞)|2 − 2

|B(∞)|2 =|Λo(∞)|2 − 1|B(∞)|2 . (A.3)

Since

|B(∞)|2 =nλ∏

k=1

1|λk|2 ,

then (A.2) together with (A.3) produce

EB + EQ2 + EQ3 = |Λo(∞)|2nλ∏

k=1

|λk|2 − 1.

We then show that EB + EQ2 + EQ3 = Edm by application of Poisson-Jensenformula in Lemma 4.1, by fact that Λo(z) is a stable and minimum phasefunction, i.e.,

|Λo(∞)|2 = exp

1π

∫ π

0

log |Λo(ejθ)|2 dθ

= exp

1π

∫ π

0


.

We complete the proof of Theorem 4.4.

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List of Publications

All the work presented in this thesis have been carried out at the Departmentof Information Physics and Computing, Graduate School of Information Sci-ence and Technology, the University of Tokyo Japan, during the Ph.D. studyin the period of 2003–2006 supported and funded by Ministry of Education,Culture, Sports, Science and Technology of Japan. This thesis is based on thefollowing publications and manuscripts.

• Journal Papers(1) S. Hara and T. Bakhtiar, ”H2 control performance limitations for SIMO

linear discrete-time systems.” To appear in Trans. ISCIE, Oct. 2006. (inJapanese)

(2) T. Bakhtiar and S. Hara, ”H2 regulation performance limitations forSIMO linear time-invariant feedback control systems.” Automatica.(provisionally accepted)

(3) T. Bakhtiar and S. Hara, ”H2 performance limitations for single-input/single-output linear systems with time delay.” Trans. ISCIE. (accepted,in Japanese)

(4) S. Hara, T. Bakhtiar, and M. Kanno, ”The best achievable H2 track-ing performances for SIMO feedback control systems.” Submitted toJournal of Control Science and Engineering.

• International Conference Papers(5) T. Bakhtiar and S. Hara, ”Tracking performance limits for SIMO dis-

crete-time feedback control systems,” in Proc. SICE Annual Conference2004, Sapporo, Japan, Aug. 2004, pp. 1825– 1830.1

(6) T. Bakhtiar and S. Hara, ”H2 control performance limitations for SIMOsystems: a unified approach.” In Proc. 6th Asian Control Conference(ASCC2006), Bali, Indonesia, July 2006, pp. 547–555.

(7) T. Bakhtiar and S. Hara, ”H2 regulation performance limits for SIMOfeedback control systems.” In Proc. 17th International Symposium on

1 The first author won the Young Author Award when presenting this paper.

134 References

Mathematical Theory of Networks and Systems (MTNS2006), Kyoto, Japan,July 2006, pp. 1966–1973.

(8) S. Hara, M. Kanno, and T. Bakhtiar, ”Optimal tracking performancefor SIMO feedback control systems: analytical closed-form expres-sions and guaranteed accuracy computation.” Accepted for presen-tation at the 45th IEEE Conference on Decision and Control (CDC2006),San Diego, USA, Dec. 2006.

• Domestic Conference Papers(9) T. Bakhtiar and S. Hara, ”H2 regulation performance limitations for

unstable/non-minimum phase SIMO systems,” in Proc. 34th SICESymposium on Control Theory, Osaka, Japan, Oct.–Nov. 2005, pp. 489–492.

(10) S. Hara and T. Bakhtiar, ”H2 tracking and regulation performance lim-its for SIMO feedback control systems,” in Proc. 33rd SICE Symposiumon Control Theory, Hamamatsu, Japan, Nov. 2004, pp. 139–142.

(11) S. Hara, M. Kanno, and T. Bakhtiar, ”H2 optimal tracking performanceand its guaranteed accuracy computation: SIMO plant case,” in Proc.6th SICE Annual Conference on Control Systems, Nagoya, Japan, May–June 2006, pp. 229–234. (in Japanese)

• Technical Reports(12) T. Bakhtiar and S. Hara, ”H2 regulation performance limitations for

SIMO linear time-invariant feedback control systems: a unified ap-proach.” Technical Reports. Department of Mathematical Informatics,the University of Tokyo. METR2006-11, Feb. 2006.

(13) S. Hara and T. Bakhtiar, ”H2 tracking performance limitations forSIMO feedback systems: a unified approach to control input penaltycase.” Technical Reports. Department of Mathematical Informatics,the University of Tokyo. METR2006-33, May 2006.

Biography

Toni Bakhtiar was born in June 27, 1972 in Tuban, East Java, Indonesia, wherehe spent most of his childhood. He finished his elementary school in 1985 atSD Negeri Kutorejo 3 Tuban, junior high school in 1988 at SMP Negeri 3Tuban, and senior high school in 1991 at SMA Negeri 1 Tuban.

In 1991, just after completing his high school, he went to Bogor, WestJava, for pursuing the undergraduate study at Department of Mathematics,Bogor Agricultural University, Indonesia. He received a Sarjana Sains de-gree in mathematics in 1996. Subsequently, he received a Master of Sciencedegree in technical mathematics in 2000 from Department of Control, Op-timization, and Stochastic, Delft University of Technology, the Netherlands.Since 2003 he has been working as a Ph.D. student at Department of Infor-mation Physics and Computing, the University of Tokyo, Japan. He is therecipient of the Young Author’s Award in the 2004 SICE Annual Conferenceheld in Sapporo, Japan.

Toni Bakhtiar is with Department of Mathematics, Bogor AgriculturalUniversity since 1997.

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