Frequency Weighted Model Order Reduction
Techniques
by
Wan Mariam binti Wan Muda
A thesis submitted to the School of Electrical, Electronic
and Computer Engineering in partial fulfilment of the
requirements for the degree of Doctor of Philosophy
Faculty of Engineering, Computing and Mathematics
University of Western Australia
2012
Statement of Originality
The contents of this thesis are the results of original research, and have not been
submitted for a higher degree at any other university or institution.
Much of the work in this thesis has been published for publication in refereed inter-
national conferences Chapters 3-5 of this thesis are based on the work described in
these papers.
Refereed International Conference Papers:
∙ Wan Muda W.M., Sreeram V. and Iu H.C., “Passivity-Preserving Frequency
Weighted Model Order Reduction Techniques for General Large-Scale RLC
Systems”, 11th International Conference on Control, Automation, Robotics and
Vision, Singapore, pp. 1310 - 1315, Dec 7-10, 2010 (Chapter 5).
∙ Wan Muda W.M., Sreeram V. and Iu H.C., “Frequency Weighted Balanced
Truncation with Special Weight”, 8th Asian Control Conference, Kaohsiung,
Taiwan, pp. 1443 - 1448, May 15-18, 2011 (Part of Chapter 3).
∙ Wan Muda W.M., Sreeram V. and Iu H.C., “An Improved Algorithm for Partial
Fraction Expansion Based Frequency Weighted Balanced Truncation”, Ameri-
can Control Conference, San Francisco, California, USA, pp. 5037 - 5042, June
29 - July 1, 2011 (Chapter 4).
∙ Wan Muda W.M., Sreeram V. and Iu H.C., “An Improved Algorithm for Fre-
quencyWeighted Balanced Truncation”, 50th IEEE Conference on Decision and
Control and European Control Conference, Orlando, Florida, USA, pp.7182 -
7187, December 12-15, 2011 (Part of Chapter 3).
ii
My doctoral studies were conducted under the guidance of Professor Victor Sreeram
as my supervisor.
The research described in this thesis is the result of a collaborative effort with my
Ph.D supervisor Professor Victor Sreeram and Professor Herbert Ho Ching Iu, and
professors from Carleton University Ottawa Canada, Professor Ramachandra Achar
and Professor Michel Nakhla, and Mr. Behzad Nouri. However, majority of the work
is my own.
Wan Mariam binti Wan Muda
School of Electrical, Electronic and Computer Engineering
The University of Western Australia
Crawley, W.A. 6009 Australia.
iii
Acknowledgements
I wish to thank, first and foremost, to my main supervisor, Professor Victor
Sreeram, and co-supervisor, Professor Herbert Ho-Ching Iu, for giving me oppor-
tunity to further my PhD studies under their supervision. Without their untiring
support, patience, guidance and persistent help, this thesis would not have been pos-
sible.
My sincere thanks also go to Government of Malaysia and University Malaysia
Terengganu (UMT) for giving me a chance to study abroad, and providing the finan-
cial support for me and my family.
I am deeply grateful to all my Malaysian friends whom I have come across during
my studies here in University of Western Australia (UWA), especially to Ms Wahiza
Wahi, Ms Rasheeda Mohd Zamin, and their family for their kindness and sincere
friendship.
To my former colleagues, Dr. Shafishuhaza Sahlan and Dr. Hamdan Daniyal,
thank you for inspiring me to not give up on my study with your hard work, patience,
and success. To my colleagues, Mr. Hammad Khan, Mr. Sachit Gopalan, and Ms.
Xiaoyuan Wang, thanks for helping me with my study. I have benefited a lot from
our discussions.
Last but not least, I owe my deepest gratitude to my lovely husband, Izham bin
Ashab, and my beautiful kids, Muhammad Iqbal and Marsya Irdina, for always being
there for me through my ups and downs, accepting me for who I am, and supporting
me spiritually throughout my life.
iv
Abstract
This thesis investigates the frequency weighted balanced model order reduction prob-
lem for linear time invariant systems.
First, two new frequency weighted balanced truncation techniques based on zero cross-
terms are proposed. Both methods are applicable for single-sided weighting, and are
based on modifications to Sreeram and Sahlan’s technique. The first method uses the
properties of all-pass function to transform the original frequency weighted model
order reduction problem into an equivalent unweighted model reduction problem,
while in the second method, the relationship between the final and the intermediate
reduced order model used in Sreeram and Sahlan’s technique is derived. Numerical
examples show that a significant error reduction can be achieved using both methods.
Second, we present an improvement to frequency weighted balanced truncation tech-
nique based on well-known partial fraction expansion idea. The method yields stable
reduced-order models for double-sided weightings. Two numerical examples includ-
ing a practical application example, show a significant improvement over the other
well-known techniques.
Lastly, we present passivity preserving frequency-weighted model order reduction
techniques for general large-scale RLC (resistor-inductor-capacitor) systems. Three
well-known frequency weighted balanced truncation techniques (Enns’, Wang et al.’s
and Lin and Chiu’s), which preserve only stability and not passivity are generalized
v
to include passivity. Conditions under which the passivity is preserved are also de-
rived. Four practical examples are given to show the validity and effectiveness of the
proposed algorithms using different weighting functions.
vi
Contents
Statement of Originality ii
Acknowledgements iv
Abstract v
List of Tables xii
List of Figures xiii
1 Introduction 1
1.1 Overview of Model Reduction Techniques . . . . . . . . . . . . . . . . 1
1.2 Organization and Contributions . . . . . . . . . . . . . . . . . . . . . 4
2 Frequency Weighted Balanced Model Reduction Techniques: A Re-
view 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Truncated Balanced Realization (TBR) . . . . . . . . . . . . . 12
2.2.2 Singular Perturbation Approximation (SPA) . . . . . . . . . . 14
vii
2.3 Motivation and Problem Formulation . . . . . . . . . . . . . . . . . . 16
2.3.1 Interconnect Network . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 The Importance of Passivity Preserving Model Reduction Tech-
niques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Passivity Preserving Model Reduction Techniques: A Review . 21
2.3.4 Modern Controller Reduction . . . . . . . . . . . . . . . . . . 23
2.3.5 Frequency Weighted Model Reduction . . . . . . . . . . . . . 25
2.4 Frequency Weighted Balanced Truncation Techniques . . . . . . . . . 27
2.4.1 Enns’ Technique . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Generalization of Lin and Chiu’s Technique . . . . . . . . . . 31
2.4.3 Varga and Anderson’s modification to Lin and Chiu’s Technique 34
2.4.4 Sreeram’s Technique . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.5 Wang et al.’s Technique . . . . . . . . . . . . . . . . . . . . . 35
2.4.6 Varga and Anderson’s modification to Wang et al.’s Technique 37
2.4.7 Influence of Cross-terms . . . . . . . . . . . . . . . . . . . . . 38
2.4.8 Sreeram and Sahlan’s Technique . . . . . . . . . . . . . . . . . 40
2.4.9 FrequencyWeighted Balanced Truncation based on Partial Frac-
tion Expansion Techniques . . . . . . . . . . . . . . . . . . . . 44
2.4.10 Ghafoor and Sreeram’s Technique . . . . . . . . . . . . . . . . 47
2.4.11 Sahlan and Sreeram’s Technique . . . . . . . . . . . . . . . . . 48
2.5 Passivity Preserving Balanced Truncation Techniques . . . . . . . . . 50
2.5.1 Phillips et al.’s Technique . . . . . . . . . . . . . . . . . . . . 51
2.5.2 Gugercin and Antoulas’s Technique . . . . . . . . . . . . . . . 53
2.5.3 Unneland et al.’s Technique . . . . . . . . . . . . . . . . . . . 56
viii
2.5.4 Boyuan et al.’s Technique . . . . . . . . . . . . . . . . . . . . 57
2.5.5 Heydari and Pedram’s Technique . . . . . . . . . . . . . . . . 58
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 Frequency Weighted Balanced Truncation based on Zero Cross-terms
Techniques 64
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Generalization of Lin and Chiu’s Technique . . . . . . . . . . 67
3.2.2 Influence of Cross-terms . . . . . . . . . . . . . . . . . . . . . 69
3.2.3 Sreeram and Sahlan’s Technique . . . . . . . . . . . . . . . . . 71
3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.1 New Method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.2 New Method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Frequency Weighted Balanced Truncation based on Partial Fraction
Expansion Techniques 99
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.1 Sreeram and Anderson’s Technique . . . . . . . . . . . . . . . 101
4.2.2 Sahlan and Sreeram’s Technique . . . . . . . . . . . . . . . . . 103
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
ix
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5 Passivity Preserving Frequency Weighted Balanced Truncation Tech-
niques 122
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 Phillips et al.’s Technique . . . . . . . . . . . . . . . . . . . . 126
5.2.2 Frequency Weighted Balanced Truncation Techniques . . . . . 128
5.2.2.1 Enns’ Technique . . . . . . . . . . . . . . . . . . . . 129
5.2.2.2 Wang et al.’s Technique . . . . . . . . . . . . . . . . 130
5.2.2.3 Lin and Chiu’s Technique . . . . . . . . . . . . . . . 131
5.2.3 Heydari and Pedram’s Technique . . . . . . . . . . . . . . . . 132
5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3.1 Positive-Real Enns’ Technique . . . . . . . . . . . . . . . . . . 137
5.3.2 Positive-Real Wang et al.’s Technique . . . . . . . . . . . . . . 139
5.3.3 Positive-Real Lin and Chiu’s Technique . . . . . . . . . . . . . 142
5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Conclusions 153
6.1 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Bibliography 157
x
Appendixes 170
Appendix A: Transforming Lur’e equation to ARE . . . . . . . . . . . . . 170
Appendix B: Solving Lur’e equation for D=0 . . . . . . . . . . . . . . . . 172
Appendix C: Factorization Formula for Co-inner Function (Single-sided) . 177
Appendix D: Factorization Formula for Inner/Co-inner Functions (Double-
sided) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Appendix E: The Modified Nodal Analysis (MNA) Formulation . . . . . . 182
xi
List of Tables
3.1 Weighted errors for Example 1 . . . . . . . . . . . . . . . . . . . . . . 94
3.2 Weighted errors for Example 2 using Algorithm 1 . . . . . . . . . . . 96
3.3 Weighted errors and error bounds for Example 2 using Algorithm 2 . 96
4.1 Weighted errors and error bounds for Example 3 (double-sided case) . 117
4.2 Weighted errors and error bounds for Example 3 (single-sided case) . 117
4.3 Weighted errors for Example 4 . . . . . . . . . . . . . . . . . . . . . . 120
xii
4.4 Frequency weighted error versus parameters (� and �) for Example 3 119
4.5 Frequency weighted error versus parameters (� and �) for Example 4 121
5.1 Interconnect circuit represented with RLC lumped segment model . . 124
5.2 Illustration of passivity behaviour via Nyquist plots . . . . . . . . . . 125
5.3 Illustration of passivity behaviour via Nyquist plots for Example 5 . . 147
5.4 itℎ section of a two port RLC network . . . . . . . . . . . . . . . . . . 148
5.5 Eigenvalues of Real(Y(s)) for Example 6 . . . . . . . . . . . . . . . . 148
5.6 Approximation errors for Example 6 . . . . . . . . . . . . . . . . . . 149
5.7 A lumped RLC circuit for Example 7 . . . . . . . . . . . . . . . . . . 150
5.8 Nyquist plots and approximation errors for Example 7 . . . . . . . . 150
5.9 Nyquist plots and approximation errors for Example 8 . . . . . . . . 152
1 A lumped RLC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 182
xiv
List of Figures
2.1 Classification of model order reduction techniques . . . . . . . . . . . 8
2.2 Extensions of truncated balanced realization (TBR) technique . . . . 11
2.3 High Speed Interconnect Effects [1] . . . . . . . . . . . . . . . . . . . 17
2.4 Time response of input and ouput signal of Zc(s) . . . . . . . . . . . 21
2.5 Closed Loop System Diagram . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Two-sided frequency weighted model reduction . . . . . . . . . . . . . 26
2.7 Input frequency weighted model reduction . . . . . . . . . . . . . . . 26
2.8 Output frequency weighted model reduction . . . . . . . . . . . . . . 27
2.9 Input-Output augmented system . . . . . . . . . . . . . . . . . . . . 29
2.10 The main property of zero cross-terms based technique . . . . . . . . 39
2.11 The main property of partial fraction expansion based technique . . . 45
3.1 Maximum singular value of input weight V (s) for Example 2 . . . . . 95
3.2 Maximum singular value of reduction error for Example 2 . . . . . . . 97
4.1 Summary of the new method based on partial fraction expansion method106
4.2 Maximum singular value of input weight V (s) for Example 3 . . . . . 116
4.3 Weighted model reduction error for Example 3 . . . . . . . . . . . . . 118
xiii
Elementary Notations and Terminologies
Transfer function G(s) = C(sI − A)−1B +D
⇔ State-space realization {A,B,C,D}
∥G(j!)∥∞
sup! �(G(j!)), if G(j!) is a transfer function (matrix)
where �(G(j!)) is a maximum singular value of G(j!)
P > 0 Positive definite matrix P
Symmetric matrix P with positive eigenvalues
P ≥ 0 Positive semidefinite matrix P
Symmetric matrix P with non-negative eigenvalues
XT or X ′ Transpose of matrix or vector X
X∗ Complex conjugate transpose of matrix or vector X
X−1 Inverse of matrix X
�i(X) Eigenvalues of X
�(X) Singular values of X
xv
Chapter 1
Introduction
1.1 Overview of Model Reduction Techniques
Simulation plays an important role in the analysis and design of a dynamic system.
For a realistic simulation, a mathematical model which is used to describe the char-
acteristics of a dynamic system should consider all behaviors of the system. As a
result, one can obtain a fairly complex and a high order model for the system. This
complexity makes it difficult to analyse and have a good understanding of the sys-
tem behaviour. Furthermore, simulation of a high order system is computationally
demanding and is therefore not recommended. This problem is considerably eased if
the system is replaced with a good approximation low order model. The process of
deriving a low order model from a high order system is known as model reduction.
The model reduction techniques have been used widely in different applications
such as in control systems [42, 19, 28, 48], electromagnetic systems (EM) [88, 10, 9, 54],
electro-thermal micro-electromechanical systems (MEMS) [13, 12, 7, 14, 8], large
1
lumped RLC networks [65, 72, 71, 61, 82, 73], and distributed transmission lines
[1, 73].
One of the important factors in model reduction is approximation error which
is computed from the difference between the output of the original system and the
reduced order model for a given input. In addition, the system properties such as
stability and passivity are very important to be preserved in model reduction. Nu-
merical properties such as accuracy of computation, computational speed and storage
requirements play a vital role in the computational efficiency of the model reduction
techniques. The error-bound formula [77, 67] for a model reduction technique gives
some idea of the approximation error one can expect while using the technique and is
useful for the designer for choosing the right technique for the application concerned.
Considering the above factors, balanced truncation or truncated balanced realiza-
tion (TBR) technique [42] which was inspired by balanced realization [47, 46], has
received considerable attention over the last few decades and many methods have
been proposed [19, 40, 77, 69, 76, 24, 58, 70].
Ideally, it is important that the approximation error is small for all frequencies.
However, sometimes the error in a certain frequency band is more important than
other frequencies. This is true when using the reduced order model in feedback control
design [3, 19]. This motivated the introduction of frequency weightings in the model
reduction procedure that gave rise to what is known as frequency weighted model
reduction problem [19, 18, 2, 3].
Frequency weighted model reduction technique was first proposed by Enns [19, 18]
in his Ph.D thesis. His method is an extension of the balanced truncation [42] to
include frequency weightings. The weights are utilized for shaping the frequency of
2
the model reduction error. The frequency weightings may include input weighting,
output weighting or both. With only one weighting present, stability of the reduced
order models is guaranteed. However, with both the weightings present, Enns’ method
may yield unstable models for stable original systems. To guarantee the stability of
the reduced order model in the case of double-sided weightings, several modifications
to Enns’ method have been proposed in the literature [40, 69, 77, 76].
Lin and Chiu’s technique [40] and its generalization [69] proposed a simple modifi-
cation to Enns’ technique provided that there are no pole-zero cancellations between
the original system and the weights [76]. The technique was then modified by Varga
and Anderson [76], and Sreeram [67]. Another modification to Enns’ technique was
proposed by Wang et al. [77] which not only guarantees stability in the case of double-
sided weightings but also yields simple and elegant error bounds. The method was
improved by Varga and Anderson [76].
The second approach in frequency weighted balanced truncation technique is based
on zero cross-terms. Gestel et al. have pointed out in [75] that the frequency weighted
balanced truncation technique has large frequency weighted error due to nonzero
cross-terms in the Gramians for augmented system realization. By transforming the
Gramians with nonzero cross-terms into new Gramians with zero cross-terms, the
reduction error can be improved. This is basically the property of the Lin and Chiu’s
technique. Recently, Sreeram and Sahlan [70] improved the Lin and Chiu’s technique
using the properties of inner/co-inner functions (which preserve the property of zero
cross-terms) discussed in [86] to reduce the reduction error.
Another approach in frequency weighted balanced truncation technique is based
on partial fraction expansion idea introduced by Al-Saggaf and Franklin [2, 3]. In
3
this technique, the weightings were introduced such that the reduction error has zeros
at the poles of the frequency weightings. Other frequency weighted model reduction
methods based on partial fraction expansion idea include [31, 68, 84, 85, 24, 58]. Error
bounds exist for some special types of weighting functions [2, 3, 68, 24, 58]. However
the approximation error obtained using these methods is generally larger compared
to Enns’ method except Zhou’s method [85] where an optimization is used to improve
the approximation error.
Besides frequency weighted balanced truncation, another extension of balanced
truncation [42] has been introduced by Phillips et al. [53]. Unlike the original balanced
truncation, the method proposed by Phillips et al. preserves passivity in addition to
stability of the original system. Heydari and Pedram [30] extended [53] to include
weighting functions such that the reduction error can be minimized in some frequency
range of interest, whilst passivity of the original system is preserved. However, the
method proposed by Heydari and Pedram is limited to strictly proper original system.
In this thesis, we focus on the frequency weighted model reduction problem mo-
tivated by the fact that the existing methods have significant deficiencies including
high approximation error. Several modifications to the existing techniques have been
proposed to overcome some of these shortcomings.
1.2 Organization and Contributions
In this thesis, several methods for the frequency weighted model order reduction
techniques have been proposed.
Chapter 2 presents an overview of model order reduction techniques. First, we
4
present the properties of balanced truncation technique and its closely related method,
singular perturbation approximation. Then the passivity preserving model order re-
duction problem and the frequency weighted model reduction problem are formulated.
Several existing techniques are also discussed in detail to provide a background infor-
mation required to understand the methods proposed in the subsequent chapters.
In Chapter 3, we propose two frequency weighted balanced truncation techniques
based on zero cross-terms. The methods are modification to Sreeram and Sahlan’s
technique [70] which is conceptually based on inner/co-inner properties [86]. First, we
explain the concept of zero cross-terms and inner/co-innner functions, then we review
the Sreeram and Sahlan’s technique. In the first method, we propose improvement
to Sreeram and Sahlan’s technique for special type of weighting functions. While
in the second method, we modify the Sreeram and Sahlan’s technique by using the
relationship between the intermediate reduced order model and the final reduced order
model. Both methods are applicable to single-sided case (input or output weighting)
only. Numerical examples show that both methods give a significant approximation
error reduction compared to the existing techniques.
Chapter 4 discusses the frequency weighted balanced truncation based on partial
fraction expansion technique. A new algorithm which is an improvement to Sahlan
and Sreeram’s technique [58] is developed. The method guarantees stability of the
original system in the case of double-sided weightings. Two numerical examples in-
cluding a practical application example are given to show the effectiveness of the
proposed method in reducing the approximation errors in the selected band of fre-
quencies.
In Chapter 5, we propose three new algorithms based on passivity preserving fre-
5
quency weighted balanced truncation. First, we review the passivity-preserving model
order reduction techniques [53, 30], and frequency weighted balanced truncation tech-
niques [19, 40, 69, 77]. Then, we propose three new algorithms which are extensions
of the frequency weighted balanced truncation techniques (Enns’ [19], Wang et al.’s
[77], and Lin and Chiu’s [40, 69]) to include passivity in addition to stability. Four
practical examples are presented to show the validity and effectiveness of the proposed
algorithms.
Chapter 6 summarizes the main contributions and presents some suggestions for
future research.
6
Chapter 2
Frequency Weighted Balanced
Model Reduction Techniques: A
Review
2.1 Introduction
Since 1980s several model order reduction (MOR) techniques with different proper-
ties have been proposed in the literature. They can be broadly classified into three
categories (see Figure 2.1):
1. Moment Matching Techniques (MMT)
2. Singular Value Decomposition (SVD) based Techniques
3. Combination of MMT and SVD based Techniques
7
MOR
SVDMMT MMT + SVD
Explicit Implicit SPATBR Hankel approximation
Figure 2.1: Classification of model order reduction techniques
8
Moment matching techniques (MMT) are based on retaining a certain parameters
(Markov parameters or time moments) of the original system in the reduced order
model. The main idea of the methods is to remove unnecessary poles because only
dominant poles have any significant effect on the overall system performance. The
method uses explicit or implicit moment matching to approximate an rtℎ order model
by matching its pole to the moments of a high order original system. The explicit
method [55] is numerically unstable, thus the implicit methods which are also known
as projection based methods have been extensively studied recently [21, 65, 50, 6,
66, 20, 32]. The main advantage of the moment matching methods is computational
efficiency, thus they are suitable to reduce very high order systems. However, the
techniques suffer from a few disadvantages. The methods can only reduce the system
down to a certain level. If the system is reduced too small compared to the original
order, the reduction errors will be large. In addition, the methods have no error
bounds.
The singular value decomposition (SVD) based methods approximate the reduced
order models based on their roots in the singular value decomposition and they in-
volve approximating matrices by means of matrices of lower rank. A well-known
technique in SVD based approach is the truncated balanced realization (TBR) pro-
posed by Moore [42]. In this approach, the given system is transformed to a balanced
realization in which each state is equally controllable and observable. The reduced
order models are obtained by directly truncating the least significant states. Two
other closely related model order reduction methods are balanced singular pertur-
bation approximation (SPA) [22, 41] and optimal Hankel norm approximation [25]
(see Figure 2.1). The SVD based model reduction methods are popular in model
9
reduction for control due to the guaranteed stability and the availability of easily
computable frequency response error bounds [19, 77, 67]. The other advantage of
SVD based methods is capability of obtaining models with a better error control
compared to the MMT based methods. Although the methods have error bounds,
they are computationally intensive as they require solving two large Lyapunov or
Lur’e equations.
Since MMT and SVD based methods have their own advantages as well as disad-
vantages, a new trend in model reduction techniques is to combine the good properties
of the two methods. Combination can be in the form of using an iterative method like
PRIMA [50] as the first level of reduction and SVD-based techniques as the second
stage of reduction so that the reduced order models preserve passivity while retaining
error bounds property as in [53, 80], or they also can be combined by solving the Lya-
punov equations (instead of solving the Lur’e equations which is more computational
infeasible), then using congruence transformation (instead of similarity transforma-
tion) to preserve the passivity of the original system as discussed in [81]. Another
way of combination involves the projection based methods (which is used in MMT
methods) and SVD based methods to increase the computational efficiency in solving
the Lyapunov or the Lur’e equations [80, 56].
Considering the continuous research effort has been made in reducing the compu-
tational cost to solve the Lyapunov and the Lur’e equations in SVD based methods
as discussed in [36, 57, 52, 39, 80, 79], and also, the methods can be applied as second
level reduction, we choose this group in our research. Thus, in the next section, we
discuss the SVD based methods in more detail.
10
TBR
FWBT (2.4) PPBT (2.5)
Figure 2.2: Extensions of truncated balanced realization (TBR) technique
In this chapter, we first review the balanced truncation method [42] and the
balanced singular perturbation approximation method [22, 41]. Then we formulate
the passivity preserving, and frequency weighted model reduction problems. Next we
discuss both of the extensions of truncated balanced realization (TBR) (see Figure
2.2) which are the frequency weighted balanced truncation (FWBT) [19, 40, 69, 77,
76, 68, 70, 58, 24], and passivity preserving balanced truncation (PPBT) [53, 29,
74, 30, 81] methods, in the sections shown in the round brackets (). Several critical
remarks about the different techniques are given.
2.2 Preliminaries
Since the proposed methods in this thesis are based on balanced realization, the
truncated balanced realization technique and its closely related singular perturbation
approximation method are first reviewed.
11
2.2.1 Truncated Balanced Realization (TBR)
Consider an ntℎ order, stable, and minimal original system G(s) represented in state
space below:
x(t) = Ax(t) + Bu(t) (2.1a)
y(t) = Cx(t) +Du(t), (2.1b)
where x(t) ∈ ℜn, u(t) ∈ ℜp, and y(t) ∈ ℜq. The corresponding transfer function is
G(s) = C(sI − A)B + D, and the state space also can be written in a matrix form
G(s) =
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦.
The system is controllable if it is possible to transfer its state from initial state
x(t0) to any desired state x(tf ) in specified finite time by input u(t), and it is observ-
able if every state x(t0) can be identified by measurement of the output y(t). The
corresponding controllability and observability Gramians, P and Q respectively, can
be obtained by solving the following Lyapunov equations:
AP + PAT +BBT = 0 (2.2a)
ATQ+QA+ CTC = 0. (2.2b)
Hankel singular values Σ(�i) which carry useful information about the input-
output behavior of the system can be obtained from√
�i(PQ) where �i(PQ) is the
eigenvalues of PQ. For a balance system
Pbal = Qbal = T TQT = T−1PT−T = Σ =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
�1 0 ⋅ ⋅ ⋅ 0
0 �2 ⋅ ⋅ ⋅ 0
......
. . ....
0 0 ⋅ ⋅ ⋅ �n
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (2.3)
12
the Hankel singular values �i are arranged from the strongest state, �1 (the most
controllable and observable) to the weakest state, �n (the least controllable and ob-
servable) in diagonal form.
To transform the system into a balanced form, P and Q which are positive defi-
nite for a controllable and observable system can be factorized into LcLTc and LoL
To
respectively using Cholesky factorization. Then, the singular value decomposition
UΣV T of LTo Lc can be obtained. The balancing transformation matrix T in (2.3) can
be computed from
T = LcV Σ1
2 T−1 = Σ1
2UTLTo .
Using the transformation matrix, the balanced realization can be computed as follows:
Abal = T−1AT =
⎡
⎢
⎣
A11 A12
A21 A22
⎤
⎥
⎦, Bbal = T−1B =
⎡
⎢
⎣
B1
B2
⎤
⎥
⎦,
Cbal = CT =
[
C1 C2
]
, Dbal = D (2.4)
where A11 ∈ ℜr×r (r < n). If the Hankel singular values in (2.3) is partitioned
into Σ =
⎡
⎢
⎣
Σ1 0
0 Σ2
⎤
⎥
⎦, where Σ1 = diag{�1, �2, . . . , �r}, Σ2 = diag{�r+1, . . . , �n},
�i ≥ �i+1, i = 1, 2, . . . , n− 1, �r > �r+1, an rtℎ reduced order model Gr(s) = C1(sI−
A11)−1B1 +D can be obtained by truncating the balanced realization corresponding
to the weak states Σ2.
The error bounds [19, 25] of the method can then be obtained from
∥G(s)−Gr(s)∥∞ ≤ 2n
∑
i=r+1
�i. (2.5)
The H∞ norm ∥G∥∞
of a system is defined as the maximum of the highest peak of
the frequency response of G.
13
Remark 1 Some important properties of the balanced realization are:
1. Balanced realization technique can only be applied to an asymptotically stable
system (A,B,C,D) for obtaining its balanced realization (Abal, Bbal, Cbal, Dbal).
2. The subsystem Aii is asymptotically stable if Σ1 and Σ2 have no common diag-
onal element. Furthermore, the subsystem (Aii, Bi, Ci) for (i = 1, 2) is control-
lable and observable.
Remark 2 The balanced truncation error ∥G(s)−Gr(s)∥∞ tends to zero at very high
frequencies.
Remark 3 It has been shown in [53] that for a positive real system in (2.1) with
A = AT , A ≤ 0, B = CT , D ≥ 0 such as in RC circuit, a reduced order model
obtained from the truncated balanced realization method is positive real.
2.2.2 Singular Perturbation Approximation (SPA)
Let the stable original system have the balanced realization (2.4), and the transfer
function G(s) be written in the form:
G(s) =
[
C1 C2
]
⎡
⎢
⎣
sIr − A11 −A12
−A21 sIn−rA22
⎤
⎥
⎦
⎡
⎢
⎣
B1
B2
⎤
⎥
⎦.
Decomposing the transfer function G(s) = G1(s) +G2(s) gives
G1(s) = Cspa(s)(sIr − Aspa(s))−1Bspa(s) +D
G2(s) = C2(sIn−r − A22)−1B2
14
where
Aspa(s) = A11 + A12(sIn−r − A22)−1A21 (2.6a)
Bspa(s) = B1 + A12(sIn−r − A22)−1B2 (2.6b)
Cspa(s) = C1 + C2(sIn−r − A22)−1A21. (2.6c)
If the subsystem G2(s) is stable and its states have very fast transient dynamics in the
neighbourhood of s = �0, then by ignoring G2(s), the reduced order model of G(s) can
be approximated by Gspa(�0) = Cspa(�0)(sI − Aspa(�0))−1Bspa(�0) +Dspa(�0) where
Dspa(�0) = D+C2(�0I −A22)−1B2, and Aspa(�0), Bspa(�0), Cspa(�0) are defined as in
(2.6) by substituting s with �0.
The two extreme cases of the generalized balanced singular perturbation approx-
imation are:
1. at �0 = 0, the reduced order model is
Gspa(0) = Cspa(0)(sI − Aspa(0))−1Bspa(0) +Dspa(0)
where
Aspa(0) = A11 + A12A−122 A21
Bspa(0) = B1 + A12A−122 B2
Cspa(0) = C1 + C2A−122 A21
Dspa(0) = D + C2A−122 B2
which is the balanced singular perturbation approximation [22, 41].
2. at �0 = ∞, the reduced order model corresponds to the balanced truncation
[42], as Aspa(∞) → A11, Bspa(∞) → B1, Cspa(∞) → C1 and Dspa(∞) → D.
15
Remark 4 The balanced singular perturbation approximation [22, 41] and balanced
truncation [42] are related via a frequency inversion s → 1/s, as follows:
1. Given G(s) in the balanced realization form, define H(s) = G(1/s).
2. Let Hr(s) be a reduced order model obtained via balanced truncation of H(s).
3. Set Gr(s) = Hr(1/s), where Gr(s) is the reduced order model obtained via bal-
anced truncation of G(s).
Remark 5 The reduced order models obtained via the balanced singular perturbation
approximation [22, 41] are also stable and balanced. Moreover, the error bounds for
the balanced truncation also holds for the balanced singular perturbation approxima-
tion [22, 41]. However, the reduced order models obtained via the balanced singular
perturbation approximation [22, 41] may be proper even for strictly proper original
systems.
Remark 6 The balanced singular perturbation approximation scheme [22, 41] yields
better approximation at low frequencies in contrast to the balanced truncation tech-
nique [42].
2.3 Motivation and Problem Formulation
Since the proposed methods consider both stable and passive systems, the passivity
preserving model order reduction problem is first discussed here. Then, frequency
weighted model order reduction problem is formulated.
16
2.3.1 Interconnect Network
Recent trends in very large-scale integration (VLSI) technology and computer-aided
design (CAD) techniques at both the chip and package levels are that the central
processor switching times are reaching the vicinity of sub-nano seconds and commu-
nication switches are being designed to transmit data that have bit rates at multiples
of Gb/s. At these higher data rates and operating frequencies, previously negligi-
ble effects of interconnects, such as ringing, distortion, reflections and cross-talk, as
shown in Figure 2.3, tend to become the major bottlenecks in the design as well as
validation of high-speed designs.
Figure 2.3: High Speed Interconnect Effects [1]
The interconnect network which is modeled by hundreds of thousands of RLC
elements [23, 87, 11, 17, 63] yield a very high order system n ≈ 105− 106. Simulation
tool like SPICE becomes inadequate for such complex circuits as they are very CPU
expensive. A standard practice to deal with this issue is to use model-order reduction
prior to performing transient analysis. It has also been high-lighted in the recent
17
literature [1, 53, 60] that, it is very important to preserve the passivity of reduced
order models as any loss of passivity in the model can lead to artificial oscillations
[1] during transient simulations (when connected and simulated with the rest of the
circuitry).
2.3.2 The Importance of Passivity Preserving Model Reduc-
tion Techniques
A system is passive if it cannot generate energy, and can only absorb energy supplied
from the sources connected to the system to excite it [5]. In terms of transfer function,
a system is passive if its transfer function is positive real. The transfer function G(s)
is said to be positive real if:
∙ All elements of G(s) are analytic in Re[s] > 0.
∙ The matrix GH(s) +G(s) ≥ 0 in {s : Re(s) > 0}.
In the above, GH denotes Hermitian (complex conjugate and transpose).
Consider an ntℎ order passive system of an m-port network in a state space form
(2.1) represented by {A,B,C,D} where p = q = m (the number of input and output
interconnect network are equal [50, 53, 30, 60, 56]) , is positive real if it satisfies the
following lemma:
Lemma 1 Positive Real Lemma [5, 57]
For a minimal and positive-real system represented by G(s) = {A,B,C,D}, there
exist real matrices Q > 0, Q ∈ ℜn×n, Ko ∈ ℜm×n, and Jo ∈ ℜm×m , such that the
following three equivalent statements hold:
18
1. Matrices Q, Ko and Jo satisfy
ATQ+QA = −KTo Ko (2.7a)
QB − CT = −KTo Jo (2.7b)
JTo Jo = D +DT . (2.7c)
2. The transfer function G(s) = C(sI − A)−1B +D satisfies:
G(s) +GT (−s) = RT (−s)R(s) (2.8)
where R(s) = Ko(sI − A)−1B + Jo is the Youla’s right spectral factor of G(s).
3. For all x and u in (2.1) satisfy
d
dt
1
2xTQx+
1
2yTr yr = yTu (2.9)
where yr(t) = Kox(t) + Jou(t) is the output equation of R(s).
The matrices Q, Ko, Jo in the above lemma can be obtained in many ways.
Equation (2.7) can be rewritten as the following algebraic Riccati equation (ARE)
(Appendix A shows how to obtain ARE from the Lur’e equation):
ATQ+QA+QB(D +DT )−1BTQ+ CT (D +DT )−1C = 0 (2.10)
where A = A − B(D + DT )−1C. Solving (2.10) for Q and (2.7c) for Jo, the matrix
Ko can then be obtained from (2.7b). For a low order system, (2.8) can be used to
compute Jo and Ko using comparison method as shown in [5], then, the matrix Q can
be obtained from solving the Lyapunov equation (2.7a).
Equation in (2.9) is an energy balance equation for a passive system. The rate
of change of the system’s energy, ddt
12xTQx, plus the power dissipation rate, 1
2yTr yr, is
equal to the input power supplied to a positive real system, yTu.
19
Remark 7 For D + DT is singular, (2.7) can be solved using Algorithm II of [57].
For a strictly proper system D = 0, Appendix B shows step-by-step guides to solve
the equation.
Passivity of a system can be checked using Nyquist plot [73]. For a passive system,
the Nyquist plot should lie entirely in the right half of the complex plane. In addition,
it also can be verified using the following theorem [60]:
Theorem 1 [60] The state space {A,B,C,D} is passive iff the following Hamiltonian
matrix (M) has no imaginary eigenvalues:
M =
⎡
⎢
⎣
A− B(D +DT )−1C B(D +DT )−1BT
−CT (D +DT )−1C −AT + CT (D +DT )−1BT
⎤
⎥
⎦.
Remark 8 The passivity checking Theorem 1 can only be applied for a system with
D +DT > 0.
Passivity is very important to be preserved in reduced order models. Stable but
not passive model may yield unstable system when the model is connected to other
passive system [35]. Let Gr(s) =s+4
s2+2s+5is a stable reduced order model with poles
at −1 ± 2i and a zero at −4. When the model is connected to transfer function
Yrc(s) = 0.06 + 0.056s which represents a capacitor and a resistor in parallel, the
overall impedance is Zc(s) = 1Gr(s)+Yrc(s)
= s2+2s+50.056s3+0.172s2+1.4s+4.3
. The poles for the
new impedance are −3.0714,±5i, indicate the system is unstable.
When the system is connected to a current source u(t) = sin !t, the signal from
both input u(t) and output y(t) of the system are connected to a scope as shown in
Figure 2.4. The figure shows that the input signal has been amplified to 3 × 104 at
t = 10s, indicating the system Zc(s) generates energy and thus is not passive.
20
The above example shows that the stable but not passive reduced order model may
yield unstable and nonpassive system when it is connected to other passive systems.
Since passivity implies stability, but not vice versa, passivity preserving techniques
are very important in model reduction.
0 2 4 6 8 10−6
−4
−2
0
2
4
6
Time(s)
Sig
nal u
(t)
and
y(t)
InputOutput
(a) Normal view
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3x 10
4
Time(s)
Sig
nal u
(t)
and
y(t)
InputOutput
(b) Zoom-out
Figure 2.4: Time response of input and ouput signal of Zc(s)
2.3.3 Passivity Preserving Model Reduction Techniques: A
Review
To overcome the aforementioned difficulties in interconnect simulations, the model
order reduction problem for interconnect network has been introduced in [55]. The
method is called an asymptotic waveform evaluation (AWE). The main idea of the
21
proposed method is to remove unnecessary poles, from the fact that among very large
number of poles in the interconnect network, only dominant poles have any significant
effect on the overall system performance. The method used explicit moment match-
ing (see Figure 2.1) via Pade approximation to approximate an rtℎ order model by
matching its pole to the first 2r moments of a high order original system. Although
the method is numerically unstable, it led to extensively research on model order
reduction problem in this area.
To overcome the stability problem in AWE [55], the first projection based method
(implicit moment matching) (see Figure 2.1) which is called the Pade via Lanczos
(PVL) was proposed by Feldmann and Freund [21]. In this method, a Lanczos process
is used to project the moment space onto an orthonormal Krylov subspace. The
projection process preserves the moment information and numerically more stable
compared to explicit moment matching technique. The Krylov subspace can also be
projected by the Arnoldi process [65]. Both methods (PVL and Arnoldi) guarantee
stability, but not passivity.
Inspired by [33] which introduced congruence transformation to preserve passivity
in reduced order models of RC circuits, Odabasioglu et al. [50] extended the Arnoldi
technique [65] to include guaranteed passivity. The method is known as PRIMA
(Passive Reduced order Interconnect Macromodeling Algorithm). Instead of using
Hessenberg matrix to compute the reduced order model as in [65], PRIMA used the
Krylov subspace vector to form the projector for the congruence transformation to
preserve the passivity of RLC system provided the circuit matrices are in a passive
form [50].
PRIMA method is computationally efficient, therefore it is suitable to reduce a
22
very high order system. The main disadvantage of PRIMA is the condition of the
original system matrices to be in a passive form. Such condition can only be found in
electromagnetic (EM) networks [88, 10, 9, 54]. Other passive systems obtained from
measured data [16, 59], from second-level reduction algorithm [53], and embedded
state-space system [60] (and references therein) which has no special structure, hence
cannot use PRIMA method as it may yield nonpassive models as demostrated in
[53, 60]. Moreover, PRIMA method has no error bounds.
Therefore, passivity preserving methods based on balanced truncation techniques
(one of the SVD based methods (see Figure 2.1)) have been studied intensively re-
cently in interconnect networks [72, 71, 53, 30, 80, 79, 81]. The balanced truncation
based methods are already well-developed in the control area [42, 19, 4, 49, 28, 86,
48, 51, 6, 62]. To get a better understanding of the balanced truncation based meth-
ods, we present the controller reduction problem in the next section, which leads to
the importance of frequency weighted balanced truncation techniques (FWBT) (see
Figure 2.2).
2.3.4 Modern Controller Reduction
Consider a feedback control system as shown in Figure 2.5(a) where P (s) is a linear
time-invariant plant with input w and output z, controlled by a full order controller
K(s), the reduced order model of the controller Kr(s) can be obtained as in Figure
2.5(b).
Partitioning P (s) =
⎡
⎢
⎣
P11(s) P12(s)
P21(s) P22(s)
⎤
⎥
⎦, K(s) and Kr(s) can then be expressed
23
P (s)
K(s)
z w
(a)
P (s)
Kr(s)
z w
(b)
Figure 2.5: Closed Loop System Diagram
in linear fractional transformation form as in [86]:
Tzw(s) = P11(s) + P12(s)K(s)(I − P22(s)K(s))−1P21(s)
Tzw(s) = P11(s) + P12(s)Kr(s)(I − P22(s)Kr(s))−1P21(s).
Suppose K(s) and Kr(s) have same number of right half plane poles, then the closed
loop system Tzw(s) is stable if either of the following sufficient conditions is satisfied
∥
∥(I − P22(s)K(s))−1P22(s)(K(s)−Kr(s))∥
∥
∞< 1
or
∥
∥(K(s)−Kr(s))(I − P22(s)K(s))−1P22(s)∥
∥
∞< 1.
By substracting Tzw(s) from Tzw(s) as follows:
Tzw(s)− Tzw(s) = P12(s)K(s)(I − P22(s)K(s))−1P21(s) (2.11)
−P12(s)Kr(s)(I − P22(s)Kr(s))−1P21(s)
≈ P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s),
(2.11) suggests the following approximation problem. Find the reduced order con-
troller Kr(s) such that the full order controller K(s) and the reduced order controller
24
Kr(s) has the same number of poles in the open right half plane, and the index
∥P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)∥∞ is minimized.
Minimizing ∥P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)∥∞
is the optimal solution which sometimes may not be found, so we seek a stabilizing
reduced order controller Kr(s) such that
∥
∥P12(s)(I −K(s)P22(s))−1 (K(s)−Kr(s)) (I − P22(s)K(s))−1P21(s)
∥
∥
∞<
where is a positive constant.
Note that, in a special case when P (s) =
⎡
⎢
⎣
0 P12(s)
P21(s) 0
⎤
⎥
⎦=
⎡
⎢
⎣
0 W (s)
V (s) 0
⎤
⎥
⎦,
then Tzw(s)− Tzw(s) = P12(s) (K(s)−Kr(s))P21(s).
2.3.5 Frequency Weighted Model Reduction
The above controller reduction problems can be summarized as frequency weighted
model reduction problem. Given the original full order stable system G(s) = C(sI −
A)−1B + D, the stable input weighting system V (s) = Cv(sI − Av)−1Bv + Dv
and the stable output weighting system W (s) = Cw(sI − Aw)−1Bw + Dw, where
{A,B,C,D}, {Av, Bv, Cv, Dv} and {Aw, Bw, Cw, Dw} are their ntℎ, ptℎ and qtℎ order
minimal realizations respectively. The objective is to find a lower order stable model
Gr(s) = Cr(sI − Ar)−1Br +Dr where {Ar, Br, Cr, Dr} is an rtℎ order (r < n) mini-
mal realization, such that ∥W (s) (G(s)−Gr(s))V (s)∥∞
is made as small as possible.
This is known as the two sided frequency weighted model reduction problem (see
Figure 2.6).
If one of the weights is identity, the problem is known as the one sided frequency
weighted model reduction (see Figure 2.7 and Figure 2.8), where the objective is to
25
V(s)
G(s)
-Gr(s)
+ W(s)u y
Figure 2.6: Two-sided frequency weighted model reduction
V(s)
G(s)
-Gr(s)
+u y
Figure 2.7: Input frequency weighted model reduction
find a stable lower order model Gr(s), such that ∥(G(s)−Gr(s))V (s)∥∞
(in case
of input weight) and ∥W (s) (G(s)−Gr(s))∥∞ (in case of output weight) is made as
small as possible. Enns [19] was the first, who formulated this problem by introducing
the frequency weights to balanced truncation [42] scheme.
26
G(s)
-Gr(s)
+ W(s)u y
Figure 2.8: Output frequency weighted model reduction
2.4 FrequencyWeighted Balanced Truncation Tech-
niques
Frequency weighted balanced truncation technique which was introduced by Enns
[19] is one of the extensions of balanced truncation technique [42] (see Figure 2.2).
Adding frequency weighting in the model order reduction problem was motivated by
the controller reduction problem explained in the previous section. In this technique,
instead of minimizing the reduction error over all frequencies, the weighted reduction
error can be minimized in a certain frequency band shaped by the chosen weighting
functions. The frequency weightings may include input weighting, output weighting
or both. The stability of reduced order models is guaranteed only when one weighting
is present.
The original Lin and Chiu’s technique [40] present a simple modification to Enns’
technique to overcome the potential drawback of instability when both weightings are
present. The method was later improved by Sreeram et al. [69] to include proper
27
weights. However, the method cannot be used in controller reduction applications due
to no pole-zero cancellation assumption required in the method [76]. The problem
was rectified by Varga and Anderson [76]. Another modification to Enns’ technique
was proposed by Wang et al. [77] which not only guarantees stability in the case
of double-sided weightings but also yields simple and elegant error bounds. The
approximation error in Wang et al.’s technique was later improved by Varga and
Anderson [76]. These two methods, [77] and [76] were later pointed out in Sreeram’s
paper [67] of being realization dependant. This basically means that for the same
original system, different models can be obtained from different realizations. Although
the stability of reduced order models is guaranteed, the reduction errors obtained from
[40, 69, 76, 67, 77] are at best slightly lower than Enns’ method and hence may be
considered still too large in most applications.
Another group of frequency weighted balanced truncation techniques are proposed
based on zero cross-terms. By transforming the original Gramians with nonzero cross-
terms into new Gramians with zero cross-terms, the reduction error can be improved
[75]. This is the main property of Lin and Chiu’s technique. Modifications of Lin and
Chiu’s technique proposed by Varga and Anderson [76], and Sreeram [67], may change
this property. Using the properties of inner/co-inner functions [86] (which preserve
the zero cross-terms in Lin and Chiu’s technique [40, 69]), Sreeram and Sahlan [70]
modified Lin and Chiu’s technique to reduce the approximation error.
The third group of frequency weighted balanced truncation techniques is partial
fraction expansion based methods which is originally proposed by Latham and Ander-
son [38]. Inspired by [38], Al-Saggaf and Franklin [3] proposed a method for frequency
weighted model reduction. The technique is then generalized by Sreeram and An-
28
V(s) G(s) W(s)input output
Figure 2.9: Input-Output augmented system
derson [68] to include double-sided weighting. However, the method can only handle
strictly proper weighting functions. The method was then improved by Ghafoor and
Sreeram [24] to include proper weights, but the method is ad hoc with no theoritical
justification. Improved technique was proposed by Sahlan and Sreeram [58] which is
conceptually simple and elegant.
This section reviews some of the well-known frequency weighted balanced trunca-
tion techniques as mentioned above. LetG(s) = {A,B,C,D}, V (s) = {Av, Bv, Cv, Dv}
and W (s) = {Aw, Bw, Cw, Dw} be the stable original system, and the stable input
and output weights respectively as shown in Figure 2.9. The augmented system
W (s)G(s)V (s) shown in the figure can be represented by the following realization:
W (s)G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwC BwDCv BwDDv
0 A BCv BDv
0 0 Av Bv
Cw DwC DwDCv DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦. (2.12)
The controllability and observability Gramians of the augmented realization{
A, B, C, D}
29
are given by:
P =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Pw P12 P13
P T12 PE P23
P T13 P T
23 Pv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Q =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Qw Q12 Q13
QT12 QE Q23
QT13 QT
23 Qv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(2.13)
where P and Q satisfy the following Lyapunov equations:
AP + P AT + BBT = 0 (2.14a)
AT Q+ QA+ CT C = 0. (2.14b)
Assuming that there are no pole-zero cancellations in W (s)G(s)V (s) the Gramians
P and Q are positive definite.
2.4.1 Enns’ Technique
Expanding (2, 2) blocks of (2.14) yield the following equations:
APE + PEAT +XE = 0
ATQE +QEA+ YE = 0
where
XE = BCvPT23 + P23C
Tv B
T + BDvDTv B
T (2.15a)
YE = CTBTwQ12 +QT
12BwC + CTDTwDwC. (2.15b)
Diagonalizing the weighted Gramians {PE, QE} yields
T−1E PET
−TE = T T
EQETE = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
30
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Transforming and partitioning the
original system realization we have
⎡
⎢
⎣
T−1E ATE T−1
E B
CTE D
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A11 A12 B1
A21 A22 B2
C1 C2 D
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Enns’ reduced-order model is then given by GE(s) = {A11, B1, C1, D}.
Essentially, Enns’ technique is based on diagonalizing simultaneously the solutions
of Lyapunov equations as given in (2.14). However, Enns’ technique cannot guarantee
the stability of reduced order models as XE and YE may not be positive semidefinite
(see (2.15)). Several modifications to Enns’ technique are proposed in the literature
to overcome the stability problem [40, 69, 77].
2.4.2 Generalization of Lin and Chiu’s Technique
In order to solve the stability problem in Enns’ technique, the generalization of Lin
and Chiu’s technique proposed in [69] first defines
X = P23P−1v and Y = Q−1
w Q12.
Using the following transformation matrix
T =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I −Y 0
0 I X
0 0 I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
the original Gramians of the augmented system{
P , Q}
in (2.13) are transformed
into P = T−1P T−T and Q = T T QT respectively, which have partial block diagonal
31
structures as shown below:
P =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Pw P12 P13
P T12 PLC 0
P T13 0 Pv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Q =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Qw 0 Q13
0 QLC Q23
QT13 QT
23 Qv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
where PLC = PE−P23P−1v P T
23 andQLC = QE−QT12Q
−1w Q12. Other matrices Pw, P12, Q13,
Q23 and Qv which are not important for our proposed algorithms are not given here.
The corresponding state-space realization has the following structure:
W (s)G(s)V (s) =
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦
=
⎡
⎢
⎣
T−1AT T−1B
CT D
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw X12 X13 X1
0 A X23 X2
0 0 Av Bv
Cw Y1 Y2 DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦(2.16)
32
where
X12 = Y A− AwY + BwC (2.17a)
X23 = AX −XAv + BCv (2.17b)
X13 = BwCX + Y AX + BwDCv + Y BCv − Y XAv (2.17c)
X1 = BwDDv + Y BDv − Y XBv (2.17d)
X2 = BDv −XBv (2.17e)
Y1 = DwC − CwY (2.17f)
Y2 = DwCX +DwDCv (2.17g)
D = DwDDv. (2.17h)
The new realization{
A, B, C}
now satisfies the following Lyapunov equations:
AP + P AT + BBT = 0
AT Q+ QA+ CT C = 0.
Diagonalizing the weighted Gramians {PLC , QLC} of the new system {A,X2, Y1}
which satisfy
APLC + PLCAT +X2X
T2 = 0
ATQLC +QLCA+ Y T1 Y1 = 0
yields
T−1LCPLCT
−TLC = T T
LCQLCTLC = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
33
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. The reduced order model is then
obtained by transforming, partitioning and truncating the original system realization.
Since the realization {A,X2, Y1} satisfies the Lyapunov equation, stability of the
models obtained from the technique is guaranteed for double-sided weightings.
2.4.3 Varga and Anderson’s modification to Lin and Chiu’s
Technique
In controller reduction applications, since the weights are of the form (I+G(s)K(s))−1
and (I + G(s)K(s)−1G(s)) where K(s) is the controller for the plant G(s), Lin and
Chiu’s requirement of no pole/zero cancellation between the weights and the controller
will not be satisfied.
To overcome this drawback, Varga and Anderson [76] proposed a method based
on diagonalizing simultaneously the Gramians PV A and QV A
T−1PV AT−T = T TQV AT = diag(�1, �2, . . . , �n)
where
PV A = PE − �2cP23P
−1V P T
23 (2.18)
QV A = QE − �2oQ
T12Q
−1W Q12, (2.19)
0 ≤ �c, �o ≤ 1, and �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Reduced order
models are then obtained by transforming, partitioning, and truncating the original
system realization.
Remark 9 When �c = �o = 0, it can be seen that this method is equal to Enns’
technique with no guaranteed stability.
34
Remark 10 When �c = �o = 1, this method is equal to Lin and Chiu’s technique
with guaranteed stability.
2.4.4 Sreeram’s Technique
Another method based on modification to Lin and Chiu’s technique was presented
in [67]. The method is based on balancing{
A, Bs, Cs
}
where Bs and Cs are given
below:
Bs =
[
�sB X2
]
, Cs =
⎡
⎢
⎣
�sC
Y1
⎤
⎥
⎦
whereX2 and Y1 are as defined in equation (2.17). By varying the user-defined param-
eters �s and �s, it was shown that the weighted approximation error can be reduced.
However, the reduction procedure proposed was adhoc without any theoretical justi-
fication. Similar technique based on partial fraction expansion can be found in [24].
However, an advantage of this scheme is the existence of a priori error bounds:
Theorem 2 Let G(s) be a stable transfer function of order n and V (s) and W (s) be
the weighting functions. If Gr(s) is a stable reduced order model, then the following
error bounds holds [67]:
∥W (s)(G(s)−Gr(S))V (s)∥∞
≤2
�s�s
∥W (s)∥∞∥V (s)∥
∞
n∑
i=r+1
�i
2.4.5 Wang et al.’s Technique
Wang et al.’s technique [77] used the property of symmetric matrices of XE and YE in
(2.15) to transform the matrices into positive semidefinite matrices using orthogonal
35
eigen decomposition of symmetric matrices as follows:
XE = UBSBUTB
YE = VCZCVTC
The matrices Bw and Cw which are fictitious input and output matrices introduced
in this technique are determined from:
Bw = UB ∣SB∣1
2
Cw = ∣ZC ∣1
2 V TC
New controllability and observability Gramians (Pw, Qw) are obtained as the so-
lutions to Lyapunov equations
APw + PwAT +BwB
Tw = 0
ATQw +QwA+ CTwCw = 0
are then diagonalized. Since
XE ≤ BwBTw ≥ 0
YE ≤ CTwCw ≥ 0
and {A,Bw, Cw} is minimal, stability of reduced order models in the case of double-
sided weighting is guaranteed.
Remark 11 If XE and YE are positive semidefinite, then Pw = PE and Qw = QE.
This implies Enns’ and Wang et al.’s techniques will be identical under this condition.
Remark 12 The following error bounds holds for this technique:
∣∣W (s) (G(s)−Gr(s))V (s)∥∞ ≤ k
n∑
i=r+1
�i
36
where k = 2∥W (s)L∥∞∥KV (s)∥∞ with
L = CVCdiag(∣z1∣−1/2 , ∣z2∣
−1/2 , . . . , ∣zni∣−1/2 , 0, . . . , 0)
K = diag(∣s1∣−1/2 , ∣s2∣
−1/2 , . . . , ∣sno∣−1/2 , 0, . . . , 0)UT
BB
wℎere ni = rank(X) and no = rank(Y )
Note that the above constant matrices depend only on the weights and the original
system.
This is the first a priori error bounds formula proposed for any double-sided model
reduction technique. Other error bounds [35], [69] proposed for Enns’, and Lin and
Chiu’s technique are not a priori error bounds.
2.4.6 Varga and Anderson’s modification to Wang et al.’s
Technique
Varga and Anderson’s [76] modification to Wang et al.’s [77] technique is aimed at
reducing the Gramians’ distance to Enns’ choice i.e. sizes of [Pw − PE] and [Qw −QE].
This is done by simultaneously diagonalizing the Gramians PV A and QV A as shown
below:
T T QV AT = T−1PV AT−T = diag(�1, �2, . . . , �n)
where the pair of Lyapunov equations are given as
APV A + PV AAT + BV AB
TV A = 0 (2.20)
AT QV A + QV AA+ CTV ACV A = 0 (2.21)
and �i ≥ �i+1, i = 1, 2, ⋅ ⋅ ⋅ , n − 1 and �r > �r+1. The new pseudo input and
output matrices BV A and CV A are defined as BV A = UV A1S1/2V A1
and CV A = R1/2V A1
V TV A1
37
respectively and UV A1, SV A1
, RV A1and VV A1
are obtained from the orthogonal eigen
decomposition of symmetric matrices
X =
[
UV A1UV A2
]
⎡
⎢
⎣
SV A10
0 SV A2
⎤
⎥
⎦
⎡
⎢
⎣
UTV A1
UTV A2
⎤
⎥
⎦
Y =
[
VV A1VV A2
]
⎡
⎢
⎣
RV A10
0 RV A2
⎤
⎥
⎦
⎡
⎢
⎣
V TV A1
V TV A2
⎤
⎥
⎦
where
⎡
⎢
⎣
SV A10
0 SV A2
⎤
⎥
⎦= diag {s1, s2, ⋅ ⋅ ⋅ , sn},
⎡
⎢
⎣
RV A10
0 RV A2
⎤
⎥
⎦= diag {r1, r2, ⋅ ⋅ ⋅ , rn}
and SV A1> 0, SV A2
≤ 0, RV A1> 0 and RV A2
≤ 0. Reduced order model is then
obtained by transforming and partitioning the original system. Since
X ≤ BV ABTV A ≤ BwB
Tw ≥ 0
Y ≤ CTV ACV A ≤ CT
wCw ≥ 0
and {A,BV A, CV A} is minimal, stability of the reduced order model for two-sided
frequency weighting is guaranteed.
2.4.7 Influence of Cross-terms
As pointed out in [75], frequency weighted balanced truncation technique has a large
frequency weighted error due to nonzero P23 and Q12 in (2.13).
Lemma 2 [75] The class of input weight V (s) = {Av, Bv, Cv, Dv} corresponding to
P23 = 0 has to satisfy the following two equations:
B(CvPv +DvBTv ) = 0
AvPv + PvATv +BvB
Tv = 0.
38
Original Gramians of augmented system with nonzero cross-terms
P =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
P11 P12 P13
P T12 P22 P23
P T13 P T
23 P33
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Q =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Q11 Q12 Q13
QT12 Q22 Q23
QT13 QT
23 Q33
⎤
⎥
⎥
⎥
⎥
⎥
⎦
New Gramians of augmented system with zero cross-terms
P =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
P11 P12 P13
P T12 P22 0
P T13 0 P33
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Q =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Q11 0 Q13
0 Q22 Q23
QT13 QT
23 Q33
⎤
⎥
⎥
⎥
⎥
⎥
⎦
are transformed into
Figure 2.10: The main property of zero cross-terms based technique
The class of output weight W (s) = {Aw, Bw, Cw, Dw} corresponding to Q12 = 0 has
to satisfy the following two equations:
CT (BTwQw +DT
wCw) = 0
ATwQw +QwAw + CT
wCw = 0.
To reduce the reduction error, the original Gramians (P , Q) in (2.13) with nonzero
cross-terms can be transformed into new Gramians with zero cross-terms (P23 =
Q12 = 0) (as in Lin and Chiu’s technique) as shown in Figure 2.10. The modifications
of Lin and Chiu’s technique discussed in [69, 70] which preserve the property of zero
cross-terms can be classified as frequency weighted model order reduction techniques
based on zero cross-terms.
39
2.4.8 Sreeram and Sahlan’s Technique
Inspired by inner/co-inner properties discussed in [86] as in the following lemma,
Lemma 3 [86] V (s) = {Av, Bv, Cv, Dv} is a co-inner function (V (s)V ∗(s) = I) if
and only if
AvPv + PvATv + BvB
Tv = 0
CvPv +DvBTv = 0
DvDTv = I
and W (s) = {Aw, Bw, Cw, Dw} is an inner function (W ∗(s)W (s) = I) if and only if
ATwQw +QwAw + CT
wCw = 0
BTwQw +DT
wCw = 0
DTwDw = I,
Sreeram and Sahlan [70] present an improved Lin and Chiu’s technique by decompos-
ing the transformed augmented systemW (s)G(s)V (s) in (2.16) into a new augmented
system W (s)G(s)V (s) where the new weigths (W (s) and V (s)) have inner/co-inner
properties.
The new original G(s), and the new weights V (s) and W (s) have new realizations
{
A,B,C,D}
, and{
Av, Bv, Cv, Dv
}
and{
Aw, Bw, Cw, Dw
}
respectively. In the new
weights{
Av, Bv, Cv, Dv
}
and{
Aw, Bw, Cw, Dw
}
, the cross-terms P23 = 0 andQ12 = 0
are preserved. The new parameters in the above equation are defined by
Bw =
[
Bw Aw I
]
(2.22a)
Dw =
[
Dw Cw 0
]
(2.22b)
40
B =
[
B −X AX
]
(2.22c)
C =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
C
−Y
Y A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(2.22d)
D =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
D 0 CX
0 0 0
Y B −Y X Y AX
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(2.22e)
Cv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(2.22f)
Dv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Dv
Bv
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (2.22g)
Using the above definition, the equations in (2.17) can now be expressed as:
X12 = BwC (2.23a)
X23 = B Cv (2.23b)
X13 = BwD Cv (2.23c)
X1 = BwD Dv (2.23d)
X2 = B Dv (2.23e)
Y1 = DwC (2.23f)
Y2 = DwD Cv (2.23g)
D = DwD Dv. (2.23h)
41
Diagonalizing the weighted Gramians{
P ,Q}
of the new systemG(s) ={
A,B,C,D}
which satisfy
AP + PAT + B BT
= 0
ATQ+QA+ CTC = 0
yielding
T−1SSPT−T
SS = T TSSQTSS = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Instead of reducing G(s), the
technique reduces the new original system G(s) using balanced truncation to obtain
an rtℎ order intermediate reduced order model Gr(s) ={
Ar, Br, Cr, Dr
}
. The final
reduced-order model Gr(s) = {Ar, Br, Cr, Dr} is then obtained by simply deleting the
extra rows in Cr, extra columns in Br, and both extra rows and columns in Dr. Since
the realization{
A,B,C}
is minimal and the weighted Gramians{
P ,Q}
satisfy the
above Lyapunov equations, the technique yields stable models in the case of double-
sided weightings. Although the method is simple and elegant, approximation error
reduction obtained from this technique is very small and is often negligible.
Remark 13 The reduction error in this method is small because the new weights that
are supposed to be inner/co-inner functions, are actually not, as one of the conditions
(DvDTv = I for input weight and DT
wDw = I for output weight) in the Lemma 3 are
missing in their lemma [70].
Remark 14 Note that Lemma 3 implies Lemma 2 and not vice versa.
Remark 15 Sreeram and Sahlan in [70] show that the parameters defined in (2.22)
are not unique. However, there are factorizations that can make the second condition
42
of the Lemma 3 to be not satisfied. For example, consider Cv and Dv defined in
(2.22). Substituting the matrices in the second equation of the Lemma 3 we get
CvPv +DvBTv = 0
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Pv +
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Dv
Bv
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
BTv = 0
⎡
⎢
⎢
⎢
⎢
⎢
⎣
CvPv
AvPv
Pv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎣
DvBTv
BvBTv
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
∕= 0.
From the last equation, we can see that, for whatever values of {Av, Bv, Cv, Dv}, the
summation of Pv + 0 ∕= 0, and for a stable input weighting, Pv ∕= 0. So, the Lemma
3 is not satisfied. In [70], they have proven that B(CvPv + DvBTv ) = 0, but from
Remark 14, the equation in Lemma 2 indicates that the cross-terms are zero, but not
necessarily inner/co-inner functions.
Remark 16 There are also factorizations which satisfy the second condition of the
Lemma 3. For example, consider parameters below which were defined in [70]:
B =
[
B −P23
]
Cv =
⎡
⎢
⎣
−DvBTv P
−1v
−P−1v BvB
Tv P
−1v
⎤
⎥
⎦Dv =
⎡
⎢
⎣
Dv
P−1v Bv
⎤
⎥
⎦.
43
Substituting the parameters in the second equation of the Lemma 3 we get
CvPv +DvBTv = 0
⎡
⎢
⎣
−DvBTv P
−1v
−P−1v BvB
Tv P
−1v
⎤
⎥
⎦Pv +
⎡
⎢
⎣
Dv
P−1v Bv
⎤
⎥
⎦BT
v = 0
⎡
⎢
⎣
−DvBTv
−P−1v BvB
Tv
⎤
⎥
⎦+
⎡
⎢
⎣
DvBTv
P−1v BvB
Tv
⎤
⎥
⎦= 0
Since the first and second equations in the Lemma 3 are satisfied (the first equation
has been proven in [70]), then DvDT
v = V (s)V ∗(s) are true. However DvDT
v ∕= I,
thus the new weight using the factorization above is not a co-inner function.
2.4.9 Frequency Weighted Balanced Truncation based on Par-
tial Fraction Expansion Techniques
Al-Saggaf and Franklin [2, 3] introduced a new version of frequency weighted model
reduction method based on partial faction expansion. But there are some limitation
which are (i) it can be used with single-sided weighting only, (ii) the ouput matrix
of input weight or input matrix of output weight have to be square and nonsingular
and (iii) the original system and weighting function need to be strictly proper.
Sreeram and Anderson [68] generalized [2, 3] to include double-sided weighting
functions. Although the method can only handle strictly proper weighting functions,
the derivation presented here is generalized to include proper weights as these equa-
tions will be required in the next section.
The technique first transforms the augmented system realization (2.12) into a
44
Original augmented system realization
W (s)G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwC BwDCv BwDDv
0 A BCv BDv
0 0 Av Bv
Cw DwC DwDCv DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
New augmented system realization
W (s) + G(s) + V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw 0 0 X1
0 A 0 X2
0 0 Av Bv
Cw Y1 Y2 DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Aw X1
Cw 0
⎤
⎥
⎦+
⎡
⎢
⎣
A X2
Y1 DwDDv
⎤
⎥
⎦+
⎡
⎢
⎣
Av Bv
Y2 0
⎤
⎥
⎦
are transformed into
Figure 2.11: The main property of partial fraction expansion based technique
45
block diagonal form (see Figure 2.11) by the following transformation matrix:
T =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I −Y R
0 I X
0 0 I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
The matrices in Figure 2.11 can be written as:
X12 = Y A− AwY + BwC = 0 (2.24a)
X23 = AX −XAv + BCv = 0 (2.24b)
X13 = AwR−RAv + BwCX + Y AX + BwDCv (2.24c)
+Y BCv − Y XAv = 0 (2.24d)
X1 = BwDDv + Y BDv − Y XBv −RBv (2.24e)
X2 = BDv −XBv (2.24f)
Y1 = DwC − CwY (2.24g)
Y2 = DwCX +DwDCv + CwR (2.24h)
D = DwDDv. (2.24i)
In this method, the following Gramians
Ppf = PE − P23XT −XP T
23 +XPvXT
Qpf = QE −Q12Y − Y TQT12 + Y TQwY
which satisfy the following Lyapunov equations
APpf + ATPpf +X2XT2 = 0 (2.25a)
ATQpf +QpfA+ Y T1 Y1 = 0 (2.25b)
46
are simultaneously diagonalized.
Since the realization {A,X2, Y1} is minimal and the Gramians diagonalized satisfy
the Lyapunov equations, the partial fraction expansion technique yields stable models
in the case of double-sided weightings.
Note that the frequency weighted error can be large with this method. However,
the error can be reduced for strictly proper original systems and the weights (D =
0, Dv = 0 and Dw = 0) if the reduction error is made to have zeros at the poles of
input weight or output weight as shown in [68].
2.4.10 Ghafoor and Sreeram’s Technique
Sreeram and Anderson’s [68] method was later generalized by Ghafoor and Sreeram
[24] to include proper weights. In this method, a new frequency weighted balanced
reduction technique is proposed which is based on parameterized combination of the
truncated balanced realization [42] and the partial fraction expansion technique [68].
Instead of simultaneously diagonalizing Ppf and Qpf in (2.25), the Gramians Pgs
and Qgs defined as follows
Pgs = P + �2gsPpf
Qgs = Q+ �2gsQpf
are simultaneously diagonalized. In the above equations, �gs and �gs are real con-
stants, while P and Q are the unweighted Gramians satisfying
AP + PAT +BBT = 0
ATQ+QA+ CTC = 0.
47
The Gramians Pgs and Qgs in the above equations satisfy
APgs + ATPgs +BgsBTgs = 0
ATQgs +QgsA+ CTgsCgs = 0
where
Bgs =
[
B �gsX2
]
Cgs =
⎡
⎢
⎣
C
�gsY1
⎤
⎥
⎦
are fictitious input and output matrices.
Remark 17 Note that when �gs = 0 and �gs = 0, the new input and output matrices
are equal to B and C respectively i.e.,
Bgs∣�gs=0 = B
Cgs∣�gs=0 = C
Remark 18 The realization {A,Bgs, Cgs} is stable and minimal.
Remark 19 Although, the method gives lower error, the method is adhoc with no
theoretical justification, for simultaneously diagonalizing Pgs and Qgs.
2.4.11 Sahlan and Sreeram’s Technique
As in the Sreeram and Anderson’s method [68], Sahlan and Sreeram’s method [58] also
involves decomposing the augmented system W (s)G(s)V (s) into W (s)+ G(s)+ V (s)
(see Figure 2.11)) using partial fraction expansion. These terms are then recombined
to obtain a new augmented system W (s)G(s)V (s) such that
W (s)G(s)V (s) = W (s) + G(s) + V (s) = W (s)G(s)V (s)
48
where G(s) ={
A,B,C,D}
is the new original system, and V (s) ={
Av, Bv, Cv, Dv
}
and W (s) ={
Aw, Bw, Cw, Dw
}
are the new input and output weights respectively.
The new parameters in the above equations are given by
Bw =
[
Bw Aw I
]
Dw =
[
Dw Cw 0
]
B =
[
B −X AX
]
C =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
C
−Y
Y A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
D =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
D 0 CX
0 0 R
Y B −R− Y X Y AX
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Cv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Dv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Dv
Bv
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Using the above definition, parameters in (2.24), can now be expressed as:
X12 = BwC
X23 = B Cv
X13 = BwD Cv
X1 = BwD Dv
X2 = B Dv
Y1 = DwC
Y2 = DwD Cv
D = DwD Dv.
Diagonalizing the weighted Gramians{
P ,Q}
of the new systemG(s) ={
A,B,C,D}
49
which satisfy
AP + PAT + B BT
= 0 (2.26a)
ATQ+QA+ CTC = 0 (2.26b)
yielding
T−1SSPT−T
SS = T TSSQTSS = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Instead of reducing G(s), the
technique reduces the new original system G(s) using balanced truncation to obtain
an rtℎ order intermediate reduced order model Gr(s) ={
Ar, Br, Cr, Dr
}
. The final
reduced-order model Gr(s) = {Ar, Br, Cr, Dr} is then obtained by simply deleting the
extra rows in Cr, extra columns in Br, and both extra rows and columns in Dr. Since
the realization{
A,B,C}
is minimal and the weighted Gramians{
P ,Q}
satisfy the
Lyapunov equations (2.26), the technique yields stable models in the case of double-
sided weightings. Although the method is simple and elegant, approximation error
reduction obtained from this technique is very small and is often negligible.
2.5 Passivity Preserving Balanced Truncation Tech-
niques
Besides the frequency weighted balanced truncation techniques discussed in the previ-
ous section, another extension of balanced truncation is passivity preserving balanced
truncation techniques (see Figure 2.2). Considering the factors that i) the passivity
preserving model order reduction technique is very important as discussed in Section
2.3.2, ii) the limitations of moment matching techniques as presented in Section 2.3.3,
50
iii) the continuous research effort has been made in reducing the computational de-
mand in SVD based methods [36, 57, 52, 39, 80, 79], and iv) the fact that the SVD
based methods can be applied as second level reduction [53, 80, 79, 30], a number
of passivity preserving model order reduction methods based on balanced truncation
have been proposed in the literature [53, 29, 80, 79, 81, 74, 56, 30]. This section re-
views some of the methods in this class, which include both unweighted and weighted
methods.
2.5.1 Phillips et al.’s Technique
Positive real truncated balanced realization (PR-TBR) proposed by Phillips et al.
[53] is an extension of truncated balanced realization (TBR) proposed by Moore
[42]. Unlike TBR (which solve two Lyapunov equations to obtain the Gramians P
and Q (see Section 2.2.1)), for a positive real system G(s) with minimal realization
{A,B,C,D}, the matrices P > 0 and Q > 0 satisfy the following Lur’e equations:
AP + PAT = −KcKTc (2.27a)
PCT −B = −KcJTc (2.27b)
JcJTc = D +DT (2.27c)
ATQ+QA = −KTo Ko (2.28a)
QB − CT = −KTo Jo (2.28b)
JTo Jo = D +DT . (2.28c)
51
Note that, the matrices P and Q in (2.27) and (2.28) are analogous to the controlla-
bility and observability Grammians. In fact, they are controllability and observability
Grammians of the system with input matrix Kc and output matrix Ko, respectively.
For D + DT > 0, the positive definite matrices P and Q of the above Lur’e
equations can be obtained by solving the dual pair algberaic Riccati equations (AREs)
as follows:
AP + PAT + (PCT − B)(D +DT )−1(CP −BT ) = 0 (2.29a)
ATQ+QA+ (QB − CT )(D +DT )−1(BTQ− C) = 0 (2.29b)
Similar to TBR method (see Section 2.2.1), by balancing the Gramians, the bal-
anced realization can be obtained using similarity transformation. The reduced order
models can then be obtained by partitioning and truncating the balanced realization.
Remark 20 The idea of computing the Gramians from the Lur’e equations is ob-
tained from the first item in the positive real lemma (see Lemma 1).
Remark 21 To preserve positive-realness of the original transfer function, instead
of solving Lyapunov equations as in the TBR method [42], the PR-TBR method [53]
computes controllability (P ) and observability (Q) Gramians from two sets of Lur’e
equations.
Remark 22 If D + DT is singular, the Lur’e equation can be solved using the al-
gorithm proposed by [57]. For a strictly proper system D = 0, Appendix B shows
step-by-step procedure for solving the equation.
52
Remark 23 Let L(s) = {A,Kc, C, Jc} and R(s) = {A,B,Ko, Jo} are left and right
power spectral factors of G(s) respectively, they satisfy
Φ = G(s) +GT (−s) = L(s)LT (−s) = RT (−s)R(s).
Remark 24 If input matrix of L(s), Kc, and output matrix of R(s), Ko, are known,
the Gramians obtained from solving the Lyapunov equations (2.27a) and (2.28a) pre-
serve the passivity of G(s).
2.5.2 Gugercin and Antoulas’s Technique
Gugercin and Antoulas [29] proposed a modification to balanced stochastic realization
[26] such that the error bounds can be written in terms of the original system G(s)
and its reduced order model Gr(s), not in terms of the power spectral factors L(s)
(see Remark 23). In addition, the modification guarantees passivity of the models.
In [26], by considering L(s) = {A,Kc, C, Jc} as the original system, the con-
trollability Gramian P is obtained from the Lyapunov equation (2.27a). Then, the
observability Gramian is computed from (2.29b), by first obtaining the input matrix
B of R(s) from (2.27b).
Remark 25 As mentioned in [27], the original balanced stochastic realization [26]
yields models with closeness in phase to the original system L(s), thus it is also called
a phase matching reduction technique. The method often gives passive models without
any guarantee.
Remark 26 For an asymptotically stable, minimal, square and nonsingular, L(s),
with det(Jc) ∕= 0, then the reduced order model Lr(s) obtained by balanced truncation
53
is asymptotically stable, minimal and satisfies
∥
∥L(s)−1(L(s)− Lr(s))∥
∥
∞≤
q∏
i=k+1
1 + �i
1− �i
− 1 (2.30a)
∥
∥Lr(s)−1(L(s)− Lr(s))
∥
∥
∞≤
q∏
i=k+1
1 + �i
1− �i
− 1 (2.30b)
Remark 27 [85] from (2.30), stochastic balancing is the same as frequency weighted
balanced truncation method with output weight = L(s)−1 or = Lr(s)−1.
Remark 28 The method can’t be used for a system with Jc = 0.
Gugercin and Antoulas [29] modified balanced stochastic realization [26] such that
the bound (2.30) can be written in terms of the original system and its reduced order
model, not in terms of the power spectral factors of G(s). Without loss of generality,
let {A,B,C,D} is a balanced realization, and R2 = (D +DT )−1, thus (2.29) can be
written as follows:
AΣ + ΣAT + (ΣCT −B)R2(CΣ−BT ) = 0
ATΣ + ΣA+ (ΣB − CT )R2(BTΣ− C) = 0
Expanding the above equations,
(A−BR2C)Σ + Σ(A− BR2C)T + ΣCTRRCΣ + BRRBT = 0 (2.31a)
(A−BR2C)TΣ + Σ(A− BR2C) + ΣBRRBTΣ + CTRRC = 0 (2.31b)
Equations in (2.31) can be seen as a new AREs for G(s) ={
A, B, C, 0}
where
A = A− BR2C, B = BR, C = RC.
54
The reduced order model of G is obtained by balanced truncation Gr(s) ={
A11, B1, C1, 0}
and the error bounds is obtained from
∥
∥
∥G(s)− Gr(s)
∥
∥
∥
∞
= 2n
∑
i=r+1
Πi
where Πi is singular values decomposition of G(s).
If we post and pre mulitply G(s) and Gr(s) by R and −R, we can have new
matrices functions
Θ(s) = RG(s)(−R) = RC(sI − A)−1B(−R)
=
⎡
⎢
⎣
A−BR2C −BR2
R2C 0
⎤
⎥
⎦
Θr(s) = RGr(s)(−R) = RC1(sI − A11)−1B1(−R)
=
⎡
⎢
⎣
A11 −B1R2C1 −B1R
2
R2C1 0
⎤
⎥
⎦
with new error bounds
2 ∥R∥2n
∑
i=r+1
Πi =∥
∥
∥(R)(G(s)− Gr(s))(−R)
∥
∥
∥
∞
= ∥Θ(s)−Θr(s)∥∞
=∥
∥(Θ(s) +R2)− (Θr(s) +R2)∥
∥
∞. (2.32)
Since Θ(s) +R2 is equal to⎡
⎢
⎣
A B
C R−2
⎤
⎥
⎦
−1
=
⎡
⎢
⎣
A− BR2C −BR2
R2C R2
⎤
⎥
⎦
then
Θ(s) +R2 = (C(sI − A)−1 + B +D +DT )−1 = (G(s) +DT )−1
Θr(s) +R2 = (C1(sI − A11)−1 +B1 +D +DT )−1 = (Gr(s) +DT )−1
55
Equation (2.32) can then be rewritten as
2 ∥R∥2n
∑
i=r+1
Πi =∥
∥(G(s) +DT )−1 − (Gr(s) +DT )−1∥
∥
∞
=∥
∥(DT +G(s))−1(G(s)−G(r)(s))(DT +Gr(s))−1∥
∥
∞(2.33)
Remark 29 Equation (2.33) can be seen as error bounds for frequency weighted
method with (DT +G(s))−1 is output weight and (DT +Gr(s))−1 is input weight.
Remark 30 The method can’t be used for a passive system with D = 0.
2.5.3 Unneland et al.’s Technique
Unneland et al. [74] proposed a method which is a combination of TBR proposed
by Moore [42] and PR-TBR by Phillips et al. [53]. Consider G(s) = {A,B,C,D}
as in (2.1), the method combines TBR and PR-TBR to reduce the computational
demand of PR-TBR, and at the same time, preserve the passivity of the original
system. Without loss of originality, assume G(s) is in a balanced form. Then, the
Gramians are obtained from combination of
AΣ + ΣAT +BBT = 0
ATΣ + ΣA+ (ΣB − CT )(D +DT )−1(BTΣ− C) = 0
or
ATΣ + ΣA+ CTC = 0
AΣ + ΣAT + (ΣCT −B)(D +DT )−1(CΣ−BT ) = 0.
Since Σ > 0 satisfies the AREs, the passivity is guaranteed.
56
2.5.4 Boyuan et al.’s Technique
In [81], Boyuan et al. proposed a method which combines the properties of MMT and
SVD based methods. Consider a positive real system in descriptor form as follows:
Ex = Ax+ Bu (2.34a)
y = Cx+Du (2.34b)
The controllability and observability Gramians are then obtained by solving the fol-
lowing generalized Lyapunov equations:
APET + EPAT + BBT = 0
ATQE + ETQA+ CTC = 0.
Similar to TBR method [42], for a stable system with P > 0 and Q > 0, the Gramians
can be factorized into Cholesky factors P = LPLTP and Q = LQL
TQ. Singular value
decomposition of LTPE
TLQ =
[
U1 U2
]
⎡
⎢
⎣
Σ1 0
0 Σ2
⎤
⎥
⎦
⎡
⎢
⎣
V T1
V T2
⎤
⎥
⎦can be computed. The
dominant basis T = LPU1Σ−1/21 are calculated to project the system onto orthogonal
Krylov subspaces (also known as reachability-observability subspaces). Similar to
PRIMA method, the orthonormal basis matrix X generated from the projection is
used to preserve passivity of the original system using congruence transformation as
follows:
Er = XTEX, Ar = XTAX, Br = XTB Cr = CX.
Remark 31 Similar to PRIMA, the method is applicable to a system in a passive
form.
57
Remark 32 Similar to SVD based method, it has error bounds.
Remark 33 Since the first step in the method requires solving for the Gramians
from the dual Lyapunov equations, it is computationally infeasible for a very high
order system. Thus, it is suitable for a second level reduction.
Remark 34 The method can be generalized to preserve structure and reciprocity [81].
2.5.5 Heydari and Pedram’s Technique
Heydari and Pedram [30] extended PR-TBR [53] method to include frequency weight-
ings. This is the first passivity preserving frequency weighted balanced trunca-
tion technique. Consider a state space in equation (2.1) with realization G(s) =
{A,B,C, 0}. The positive real input and output weighting functions can be written
as:
Wi(s) = Ci(sI − Ai)−1Bi +Di
Wo(s) = Co(sI − Ao)−1Bo +Do.
Input and ouput augmented systems can then be obtained as follows:
G(s)Wi(s) =
⎡
⎢
⎣
Ai Bi
C i Di
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A BCi BDi
0 Av Bi
C 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Wo(s)G(s) =
⎡
⎢
⎣
Ao Bo
Co Do
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A 0 B
BoC Ao 0
DoC Co 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
58
Substituting the above augmented realizations into the Lur’e equations (2.27) and
(2.28) we have
AiPL + PLAT
i = −Kc,iKT
c,i (2.35a)
PLCT
i −Bi = −Kc,iJTc,i (2.35b)
Jc,iJTc,i = Di +D
T
i (2.35c)
AT
o QL +QLAo = −KT
o,oKo,o (2.36a)
QLBo − CT
o = −KT
o,oJo,o (2.36b)
JTo,oJo,o = Do +D
T
o (2.36c)
Partitioning the matrices
PL =
⎡
⎢
⎣
P11 P12
P T12 P22
⎤
⎥
⎦Kc,1 =
[
Kc1 Kc2
]
QL =
⎡
⎢
⎣
Q11 Q12
QT12 Q22
⎤
⎥
⎦Ko,1 =
⎡
⎢
⎣
Ko1
Ko2
⎤
⎥
⎦
the (1,1) blocks of (2.35) and (2.36) can then be written as:
AP11 + P11AT = −X (2.37a)
P11CT −BDi = −Kc1J
Tc,i (2.37b)
Jc,iJTc,i = Di +D
T
i (2.37c)
59
ATQ11 +Q11A = −Y (2.38a)
Q11B − CTDTo = −KT
o1Jo,o (2.38b)
JTo,oJo,o = Do +D
T
o (2.38c)
where
X = BCiPT12 + P12C
Ti B
T +Kc1KTc1
Y = CTBTo Q
T12 +Q12BoC +KT
o1Ko1
Note that the equations for X and Y given in (23) and (24) of [30] have typographical
errors and hence are dimensionally incorrect. The correct equations are shown above.
Since X and Y are symmetric matrices, then there exist orthogonal matrices Ux
and Vy, and diagonal matrices Sx and Zy satisfy
X = UxSxUTx
Y = VyZyVTy .
The fictitious input and output matrices can be obtained from
B BT
= Ux ∣Sx∣UTx
CTC = Vy ∣Zy∣V
Ty .
Suppose that rank(X) = i and rank(Y ) = j, where 1 ≤ i, j ≤ n, we can write
B = Uxdiag(
∣sx,1∣1/2 , . . . ∣sx,i∣
1/2 , 0 . . . , 0)
C = diag(
∣zy,1∣1/2 , . . . ∣zy,j∣
1/2 , 0 . . . , 0)
V Ty .
60
Substituting X and Y with the new positive semidefinite input B BTand output
CTC matrices, the positive semidefinite Gramians P and Q can be obtained by solving
the following equations:
AP + PAT = −B BT
(2.39a)
PCT − B = −Kc1JTc,i (2.39b)
Jc,iJTc,i = Di +D
T
i (2.39c)
AT Q+ QA = −CTC (2.40a)
QB − CT = −KTo1Jo,o (2.40b)
JTo,oJo,o = Do +D
T
o (2.40c)
Similar to TBR method [42], reduced order models are then obtained by transforming,
partitioning, and truncating the original system realization.
Remark 35 Similar to Wang et al.’s method [77], the fictitious input and output
matrices B and C are used in this method, to find the new Gramians.
Remark 36 The method is limited to stricly proper (D = 0) original system.
Remark 37 In a standard Lur’e equation as in Lemma 1, the matrices Q, Ko and
Jo are unknown. But in (2.39) and (2.40), the matrices B BTand C
TC are known.
Thus, P and Q can be obtained by solving the Lyapunov equations (2.39(a)) and
(2.40(a)) respectively, see Remark 24.
61
Remark 38 Comparing (2.37) and (2.38) with (2.39) and (2.40), one can see that
by changing P11 and Q11 to P and Q, the matrices BDi and DoC are also changed to
new matrices B and C respectively where B = PCT +Kc1JTc,i and C = BT Q+JT
o,oKo1.
Remark 39 Although the mathematical derivation in Heydari and Pedram’s tech-
nique shows the single-sided case, similar to Wang et al.’s technique, it can be easily
extended to double-sided case.
2.6 Summary
The existing techniques discussed in the previous section can be summarized as fol-
lows.
The reduced order models obtained using Enns’ method [19], Varga and Anderson
with Lin and Chiu’s modification [76] are not guaranteed to be stable. Other methods
that include Lin and Chiu’s [40] and its generalization [69], Sreeram’s [67], Wang et
al.’s [77], Varga and Anderson with Wang et al.’s modification [76], Ghafoor and
Sreeram’s [24], Sreeram and Sahlan’s [70], and Sahlan and Sreeram’s [58] techniques,
produce stable reduced order models.
In controller reduction case, Enns’ method and Varga and Anderson’s modification
to Lin and Chiu’s method [76], yield the same reduced order controller model. If Enns’
method [19] yields unstable reduced order controller, so does Varga and Anderson’s
[76].
Lin and Chiu’s method [40], and Sreeram’s method [67] cannot be used for con-
troller reduction applications due to pole-zero cancellations of the controller with the
weight.
62
Frequency response error bounds are available for Sreeram’s method [67], Wang
et al.’s method [77], Varga and Anderson’s [76] modification to Wang et al.’s [77],
Ghafoor and Sreeram’s method [24], Sreeram and Sahlan’s [70], and Sahlan and
Sreeram’s method [58].
Phillips et al.’s [53], Unneland et al.’s [74], and Boyuan et al.’s [81] techniques pro-
duce passive reduced order models. Gugercin and Antoulas’ technique [29] guarantees
passivity of reduced order models but the approximation error obtained from the tech-
nique is in the form of frequency weighted model order reduction based method with
the weighting functions are in terms of the original system and the reduced order
model. Heydari and Pedram’s technique [30] was the only passivity preseving fre-
quency weighted model reduction technique available in the literature so far, but the
technique is limited to strictly proper original system.
2.7 Conclusion
Important properties of balanced realization/truncation and balanced singular pertur-
bation approximation have been presented. The passivity preserving and frequency
weighted model order reduction problems have been formulated. Several frequency
weighted model order reduction methods were reviewed and their important proper-
ties were outlined. In the literature, although Enns’ method [19] may yield unstable
reduced order models, they provide a better reduction error when compared to other
well-known methods [40, 69, 77, 76, 70, 58]. Therefore, better frequency weighted bal-
anced truncation techniques are required, that can guarantee both stability or/and
passivity, and low reduction error, as in the unweighted case.
63
Chapter 3
Frequency Weighted Balanced
Truncation based on Zero
Cross-terms Techniques
3.1 Introduction
Lin and Chiu’s technique [40] has been proposed to overcome the stability problem
in Enns’ technique [19]. The method has been generalized in [69] to include proper
weights. The main property of Lin and Chiu’s technique is the original Gramians with
nonzero cross-terms has been transformed into new Gramians with zero cross-terms
as shown in Figure 2.10 in the previous chapter. As pointed out in [75], frequency
weighted balanced truncation method with zero cross-terms may reduce the reduction
errors. However, Lin and Chiu’s technique can only guarantee stability for double
sided weightings when there are no pole-zero cancellations between the original system
64
and the weights [76]. Furthermore, in most cases, the technique still yield large
approximation error.
Sreeram and Sahlan [70] improved Lin and Chiu’s technique by decomposing
the transformed augmented system in [40] into a new augmented system and new
weights. Their method does not only guarantee stability of the original system in
case of double-sided weightings, but also simple, elegant and easily computable error
bounds. However, using their method there is only slight improvement in the ap-
proximation error reduction over Enns’ method by varying user-chosen parameters.
This is because only one of the conditions for the new weights to be inner/co-inner
functions is satisfied.
In this chapter, we first review the generalized Lin and Chiu’s technique, the prop-
erties of zero cross-terms, and inner/co-inner functions. Then, we discuss Sreeram
and Sahlan’s technique to provide some background information needed in the pro-
posed methods. Then, we propose two modifications to Sreeram and Sahlan’s method
[70].
In our first method [43], we modify Sreeram and Sahlan’s method [70] such that
the new weights are all-pass functions [86], which is a special case of inner/co-inner
function. In the second method [44], again we modify Sreeram and Sahlan’s method
[70] by implementing the relationship between the intermediate and the final reduced
order models. Both methods are applicable for single-sided case only (input or output
weighting). Numerical examples are presented to demonstrate the effectiveness of the
techniques.
65
3.2 Preliminaries
This section reviews some of the well-known frequency weighted balanced truncation
techniques. Let G(s), V (s) and W (s) be the stable original system and the sta-
ble input and output weights respectively. Let {A,B,C,D}, {Av, Bv, Cv, Dv} and
{Aw, Bw, Cw, Dw} be their corresponding minimal realizations respectively. Consider
the augmented system G(s)V (s) and W (s)G(s) represented by the following realiza-
tion:
G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A BCv BDv
0 Av Bv
C DCv DDv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Ai Bi
Ci Di
⎤
⎥
⎦(3.1a)
W (s)G(s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwC BwD
0 A B
Cw DwC DwD
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Ao Bo
Co Do
⎤
⎥
⎦. (3.1b)
The controllability and observability Gramians of the augmented realization are given
by:
Pi =
⎡
⎢
⎣
PE P12
P T12 Pv
⎤
⎥
⎦Qo =
⎡
⎢
⎣
Qw Q12
QT12 QE
⎤
⎥
⎦(3.2)
66
where Pi and Qo satisfy the following Lyapunov equations:
AiPi + PiATi + BiB
Ti = 0 (3.3a)
ATo Qo + QoAo + CT
o Co = 0. (3.3b)
Assuming that there are no pole-zero cancellations in G(s)V (s) and W (s)G(s), the
Gramians, Pi and Qo are positive definite.
3.2.1 Generalization of Lin and Chiu’s Technique
The generalization of Lin and Chiu’s technique [69] modified Enns’ technique [19]
using the following transformation matrices
Ti =
⎡
⎢
⎣
I X
0 I
⎤
⎥
⎦To =
⎡
⎢
⎣
I −Y
0 I
⎤
⎥
⎦
where
X = P12P−1v and Y = Q−1
w Q12, (3.4)
to transform the original Gramians of augmented system{
Pi, Qo
}
in (3.2) into block
diagonal matrices as shown below:
Pi = T−1i PiT
−Ti =
⎡
⎢
⎣
PLC 0
0 Pv
⎤
⎥
⎦Qo = T T
o QoTo =
⎡
⎢
⎣
Qw 0
0 QLC
⎤
⎥
⎦
where PLC = PE − P12P−1v P T
12 and QLC = QE −QT12Q
−1w Q12.
67
The corresponding state-space realizations have the following structures:
⎡
⎢
⎣
Ai Bi
Ci Di
⎤
⎥
⎦=
⎡
⎢
⎣
T−1i AiTi T−1
i Bi
CiTi Di
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A X23 X2
0 Av Bv
C Y2 DDv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.5a)
⎡
⎢
⎣
Ao Bo
Co Do
⎤
⎥
⎦=
⎡
⎢
⎣
T−1o AoTo T−1
o Bo
CoTo Do
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Aw X12 X1
0 A B
Cw Y1 DwD
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.5b)
where
X23 = AX −XAv +BCv (3.6a)
X2 = BDv −XBv (3.6b)
Y2 = CX +DCv (3.6c)
Di = DDv (3.6d)
X12 = Y A− AwY + BwC (3.6e)
Y1 = DwC − CwY (3.6f)
X1 = BwD + Y B (3.6g)
Do = DwD (3.6h)
The new realizations{
Ai, Bi
}
and{
Ao, Co
}
now satisfy the following Lyapunov
68
equations:
AiPi + PiATi + BiB
Ti = 0
ATo Qo + QoAo + CT
o Co = 0
Diagonalizing the weighted Gramians {PLC , QLC} of the new system {A,X2, Y1}
which satisfy
APLC + PLCAT +X2X
T2 = 0
ATQLC +QLCA+ Y T1 Y1 = 0
yields
T−1LCPLCT
−TLC = T T
LCQLCTLC = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Using the same similarity
transformation matrix TLC , the balanced realizations can be computed. Reduced
order model is then obtained by partitioning and truncating the balanced realizations.
3.2.2 Influence of Cross-terms
As pointed out in [75], frequency weighted balanced truncation technique has large
frequency weighted error due to nonzero P12 and Q12 in (3.2).
Lemma 4 [75] The class of input weight V (s) = {Av, Bv, Cv, Dv} corresponding to
P12 = 0 has to satisfy the following two equations:
B(CvPv +DvBTv ) = 0
AvPv + PvATv + BvB
Tv = 0
69
The class of output weight W (s) = {Aw, Bw, Cw, Dw} corresponding to Q12 = 0 has
to satisfy the following two equations:
CT (BTwQw +DT
wCw) = 0
ATwQw +QwAw + CT
wCw = 0
Lemma 5 [86] V (s) = {Av, Bv, Cv, Dv} is a co-inner function (V (s)V ∗(s) = I) if
and only if
AvPv + PvATv + BvB
Tv = 0
CvPv +DvBTv = 0
DvDTv = I
Similarly, W (s) = {Aw, Bw, Cw, Dw} is an inner function (W ∗(s)W (s) = I) if and
only if
ATwQw +QwAw + CT
wCw = 0
BTwQw +DT
wCw = 0
DTwDw = I
Note that V ∗(s) and W ∗(s) are used to denote the complex conjugate transpose of
V (s) and W (s) respectively.
Remark 40 The matrices functions V (s) and W (s) need not be square to be co-
inner/inner function. If the co-inner and inner matrices functions (V (s) and W (s))
are square then they satisfy the following:
V (s)V ∗(s) = V ∗(s)V (s) = W ∗(s)W (s) = W (s)W ∗(s) = I
which implies they are all-pass functions.
70
Remark 41 Note that Lemma 5 implies Lemma 4 and not vice versa.
3.2.3 Sreeram and Sahlan’s Technique
Sreeram and Sahlan’s technique [70] improved Lin and Chiu’s technique [69] using
the properties of zero cross-terms (see Lemma 4) and inner/co-inner function (see
Lemma 5).
The mathematical derivation of Sreeram and Sahlan’s technique [70] (see Section
2.4.8) is modified here to single-sided case as we consider this case in both methods
proposed in this chapter.
In [70], they present an improved Lin and Chiu’s technique by decomposing the
transformed augmented system G(s)V (s) and W (s)G(s) in (3.5) into new augmented
systems as follows:
G(s)V (s) = Gi(s)V (s) (3.7a)
W (s)G(s) = W (s)Go(s) (3.7b)
where Gi(s) ={
A,B,C,Di
}
and Go(s) ={
A,B,C,Do
}
are the new original systems
and the new weights V (s) ={
Av, Bv, Cv, Dv
}
and W (s) ={
Aw, Bw, Cw, Dw
}
. In the
new weights,{
Av, Bv, Cv, Dv
}
and{
Aw, Bw, Cw, Dw
}
the cross-terms P12 = 0 and
Q12 = 0 respectively.
71
The new parameters in the above equations are given by
B =
[
B −X AX
]
(3.8a)
Di =
[
D 0 CX
]
(3.8b)
Cv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.8c)
Dv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Dv
Bv
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.8d)
C =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
C
−Y
Y A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.8e)
Do =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
D
0
Y B
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.8f)
Bw =
[
Bw Aw I
]
(3.8g)
Dw =
[
Dw Cw 0
]
(3.8h)
72
Using the matrices defined in (3.8), the equations in (3.6) can now be expressed as:
X23 = B Cv (3.9a)
X2 = B Dv (3.9b)
Y2 = DiCv (3.9c)
Di = DiDv (3.9d)
X12 = BwC (3.9e)
Y1 = DwC (3.9f)
X1 = BwDo (3.9g)
Do = DwDo (3.9h)
Diagonalizing the weighted Gramians{
P ,Q}
of the new system{
A,B,C}
which
satisfy
AP + PAT + B BT
= 0 (3.10a)
ATQ+QA+ CTC = 0 (3.10b)
yielding
T−1SSPT−T
SS = T TSSQTSS = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Instead of reducing G(s), the
technique reduce the new original system Gx(s) by balanced truncation to obtain
an rtℎ order intermediate reduced-order model Gr,x(s) where x = i, o depending on
input or output weighting. The final reduced-order model Gr,x(s) is obtained by
simply deleting the extra rows in Cr,o and Dr,o, and extra columns in Br,i and Dr,i.
Although the method is simple and elegant, approximation error reduction obtained
from this technique is very small and is often negligible.
73
Remark 42 The new weight V (s) obtained in the decomposition (3.7a) satisfies only
the first two conditions of Lemma 5. The third condition is not satisfied i.e,
DvDT
v ∕= I.
Hence V (s) is not co-inner as claimed in [70]. Similarly
DT
wDw ∕= I
and hence W (s) in (3.7b) is not inner.
Remark 43 Appendix C and Appendix D show that there are conditions need to be
satisfied to decompose (3.7) into inner/co-inner function for both single and double-
sided cases.
3.3 Main Results
In this chapter, we improve Sreeram and Sahlan’s technique [70] in two different
ways to reduce the model reduction error. In the first method [43], instead of using
the properties of inner/co-inner function, we decompose the transformed augmented
system in Lin and Chiu’s technique [69] using the property of all-pass function (see
Remark 40). In the second method [44], the only difference between the proposed
method and Sreeram and Sahlan’s technique [70], is in the way the final reduced order
model is obtained. Instead of directly truncating the intermediate reduced order
model as in [70], we compute the final reduced order model from the relationship
between the final and the intermediate reduced order model.
74
3.3.1 New Method 1
In this section, we decompose the transformed augmented system in [69] into a new
augmented system where the new weights are square matrices and all-pass functions
(see Remark 40). The proposed frequency weighted model reduction method is appli-
cable when only one weight is present, i.e., either input or output weight. Furthermore
the weight has to be strictly proper and output matrix Cv of the input weight has
to be square and nonsingular or the input matrix Bw of the output weight has to be
square and nonsingular.
The proposed method can be described conceptually as follows: A single-sided
frequency weighted model reduction problem given by
∥W (s)(G(s)−Gr(s))∥∞ or ∥(G(s)−Gr(s))V (s)∥∞
is transformed into an equivalent problem:
∥W (s)(Go(s)−Gr,o(s))∥∞ or ∥(Gi(s)−Gr,i(s))V (s)∥∞
where W (s) and V (s) are all-pass functions and Gx(s) (x = i or x = o depending on
the input or output weight respectively) is the new original system. The intermediate
reduced order model, Gr,x(s) is first calculated by balanced truncation of Gx(s). Since
the property of all-pass functions ∥W (s)∥∞ = ∥V (s)∥∞ = 1, the final reduced order
model, Gr(s) is then obtained to satisfy either
∥W (s)(G(s)−Gr(s))∥∞ = ∥(Go(s)−Gr,o(s))∥∞
or
∥(G(s)−Gr(s))V (s)∥∞ = ∥(Gi(s)−Gr,i(s))∥∞,
75
depending on the type of original single-sided problem.
To decompose the transformed augmented realizations in (3.5) into new form
of augmented systems where the new weights are all-pass functions, we define new
parameters:
B = −XBv (3.11a)
C = −CwY (3.11b)
D = 0 (3.11c)
Dv = I (3.11d)
Cv = −BTv P
−1v (3.11e)
Dw = I (3.11f)
Bw = −Q−1w CT
w . (3.11g)
Theorem 3 Given the original system G(s) = {A,B,C,D} and the weight V (s) =
{Av, Bv, Cv, 0}, with nonzero cross-term (i.e. P12 ∕= 0), then the new weight V (s) =
{
Av, Bv, Cv, Dv
}
with zero cross-term (i.e P 12 = 0) is an all-pass function.
Proof 1 The first equation of Lemma 5 is easily proved from the (2,2) block of (3.3a).
The second and third equations of the lemma can be rewritten using the parameters
defined in (3.11) as follows:
CvPv +DvBTv =
(
−BTv P
−1v
)
Pv + (I)BTv = 0
DvDT
v = I.
Since Lemma 5 conditions are satisfied, V (s) ={
Av, Bv, Cv, Dv
}
is a co-inner func-
tion. Furthermore V (s) is a square matrix function satisfying V (s)V∗
(s) = V∗
(s)V (s) =
I. Therefore it is an all-pass function.
76
Remark 44 Similarly, we can show that for output weighting function W (s) =
{Aw, Bw, Cw, 0} with nonzero cross-term (i.e. Q12 ∕= 0), the new weight W (s) =
{
Aw, Bw, Cw, Dw
}
with zero cross-term (i.e Q12 = 0) is an all-pass function.
Theorem 4 Given the original system G(s) = {A,B,C,D} and the weight V (s) =
{Av, Bv, Cv, 0} with nonzero cross-term (i.e. P12 ∕= 0) and the output matrix Cv
square and nonsingular, then the new original system Gi(s) ={
A,B,C,D}
and the
new weight V (s) ={
Av, Bv, Cv, Dv
}
satisfy the following relationship:
∥(G(s)−Gr(s))V (s)∥∞
=∥
∥(Gi(s)−Gr,i(s))V (s)∥
∥
∞
=∥
∥Gi(s)−Gr,i(s)∥
∥
∞
where Gr,i(s) ={
Ar, Br, Cr, Dr
}
is an rtℎ order intermediate reduced order model of
Gi(s), and Gr(s) = {Ar, Br, Cr, Dr} is an rtℎ order final reduced order model of G(s).
Proof 2 From (3.5a) with Dv = 0, we have
G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A X23 X2
0 Av Bv
C Y2 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Equivalently, we can rewrite the equation as
G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A X23 X2
0 Av Bv
C 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎣
Av Bv
Y2 0
⎤
⎥
⎦.
Using the new parameters defined in (3.11), we can factorize the first two matrices
in (3.6) as follows:
X23 = B Cv
X2 = B Dv
77
and zero entries in a new augmented system can be written as:
0 = D Cv
0 = D Dv
which gives
G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A B Cv B Dv
0 Av Bv
C D Cv D Dv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎣
Av Bv
Y2 0
⎤
⎥
⎦
=
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦.
⎡
⎢
⎣
Av Bv
Cv Dv
⎤
⎥
⎦+
⎡
⎢
⎣
Av Bv
Y2 0
⎤
⎥
⎦
= Gi(s)V (s) + Y (s). (3.12)
Let Gr(s) = Cr(sIr − Ar)−1Br + Dr be an rtℎ order model of the original sys-
tem G(s), then the augmented system, Gr(s)V (s) can be represented by the following
realization:
Gr(s)V (s) =
⎡
⎢
⎣
Ar Br
Cr Dr
⎤
⎥
⎦.
⎡
⎢
⎣
Av Bv
Cv 0
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar BrCv 0
0 Av Bv
Cr DrCv 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Ai,r Bi,r
Ci,r Di,r
⎤
⎥
⎦.
The controllability Gramian of the augmented realization {Ai,r, Bi,r} is given by Pi,r =
78
⎡
⎢
⎣
P22,r P23,r
P T23,r Pv
⎤
⎥
⎦which satisfies the following Lyapunov equation:
Ai,rPi,r + Pi,rATi,r + Bi,rB
Ti,r = 0.
Similar to the original augmented systems (3.1) which have been transformed into new
augmented systems (3.5), the augmented system of reduced order model Gr(s)V (s) can
also be transformed using a transformation matrix
Ti,r =
⎡
⎢
⎣
I Xr
0 I
⎤
⎥
⎦
such that the Gramian of the transformed augmented system Pi,r is in a block diagonal
form. The transformed augmented realization can be written as:
Gr(s)V (s) =
⎡
⎢
⎣
T−1i,r Ai,rTi,r T−1
i,r Bi,r
Ci,rTi,r Di,r
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar X23r X2r
0 Av Bv
Cr Y2r 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar X23r X2r
0 Av Bv
Cr 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎣
Av Bv
Y2r 0
⎤
⎥
⎦(3.13)
where
X2r = −XrBv (3.14a)
X23r = ArXr + BrCv −XrAv (3.14b)
Y2r = CrXr +DrCv. (3.14c)
79
From the new augmented system Gi(s) ={
A,B,C,D}
with D = 0 obtained in
(3.12), an rtℎ order intermediate reduced order model Gr,i(s) ={
Ar, Br, Cr, Dr
}
with
Dr = 0 can be computed directly by using balanced truncation. Using the parameters
{
Cv, Dv
}
defined in (3.11), equations in (3.14a) and (3.14b) can then be factorized
as follows:
−XrBv = BrDv (3.15a)
ArXr + BrCv −XrAv = BrCv (3.15b)
and two zero blocks in the new augmented system (3.13) can be replaced by:
0 = DrCv
0 = DrDv
such that
Gr(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar BrCv BrDv
0 Av Bv
Cr DrCv DrDv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎣
Av Bv
Y2r 0
⎤
⎥
⎦
=
⎡
⎢
⎣
Ar Br
Cr Dr
⎤
⎥
⎦.
⎡
⎢
⎣
Av Bv
Cv Dv
⎤
⎥
⎦+
⎡
⎢
⎣
Av Bv
Y2r 0
⎤
⎥
⎦
= Gr,i(s)V (s) + Y r(s). (3.16)
If Gr(s) = Cr(sIr − Ar)−1Br +Dr is an rtℎ order final reduced order model of G(s),
the matrices Ar and Cr can be obtained directly from the intermediate reduced order
model Gr,i(s) ={
Ar, Br, Cr, Dr
}
. The unknown matrix Br can be obtained by solving
80
(3.15) while matrix Dr can be computed by equating (3.14c) to Y2 in (3.6)
Y2r = CrXr +DrCv
= Y2 (3.17)
such that Y (s) in (3.12) is equal to Y r(s) in (3.16). Note that, in solving (3.17) for
Dr, matrix Cv needs to be square and nonsingular.
The realizations Y (s) and Y r(s) can be cancelled out when substracting (3.16)
from (3.12) as follows:
(G(s)−Gr(s))V (s) = Gi(s)V (s) + Y (s)−Gr,i(s)V (s)− Y r(s)
=(
Gi(s)−Gr,i(s))
V (s).
Since all-pass function∥
∥V (s)∥
∥
∞= 1, we have
∥(G(s)−Gr(s))V (s)∥∞
=∥
∥Gi(s)−Gr,i(s)∥
∥
∞.
Remark 45 Similarly, given the original system G(s) = {A,B,C,D} and the weight
W (s) = {Aw, Bw, Cw, 0} with nonzero cross-term (i.e. Q12 ∕= 0) and the matrix Bw
square and nonsingular, then the new original system Go(s) ={
A,B,C,D}
and the
new weight W (s) ={
Aw, Bw, Cv, Dw
}
satisfy the following relationship:
∥W (s) (G(s)−Gr(s))∥∞ =∥
∥Go(s)−Gr,o(s)∥
∥
∞
where Gr(s) = {Ar, Br, Cr, Dr} and Gr,o(s) ={
Ar, Br, Cr, Dr
}
are rtℎ order models
of G(s) and Go(s) respectively.
Theorem 5 If {A,B,C,D} is stable and minimal, then the new realization{
A,B,C,D}
is also stable and minimal.
81
Proof 3 Follows immediately from the stability and minimality of {A,B,C,D}.
Theorem 6 If{
A,B,C,D}
is stable and minimal, then an rtℎ order model of the
new realization Gr,i(s) ={
Ar, Br, Cr, 0}
obtained using the balanced truncation tech-
nique is also stable and minimal.
Proof 4 Follows immediately from the stability and minimality of{
A,B,C,D}
.
Theorem 7 If a given original system G(s) is stable and minimal, then an rtℎ order
model Gr(s) obtained from the proposed method is also stable and minimal.
Proof 5 This follows immediately from the stability and minimality of Gr,i in Theo-
rem 6 as it has the same Ar as Gr,i.
A step-by-step algorithm for the proposed method can be given as follows:
Algorithm 1 New method 1
1. Given a stable and minimal G(s) and V (s), solve (3.3a) for the Gramian Pi.
2. Compute X from (3.4)
3. Compute B from (3.11).
4. Compute controllability Gramian P from{
A,B}
and observability Gramian Q
from {A,C}.
5. Find the transformation matrix T to diagonalize the Gramians:
T−1PT−T = T TQT = diag {�1, �2, . . . , �n} .
82
6. Compute the frequency weighted balanced realization
⎡
⎢
⎣
T−1AT T−1B
CT D
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar A12 Br
A21 A22 B2
Cr C2 Dr
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
7. An rtℎ order intermediate model of G(s) can be obtained as Gr(s) ={
Ar, Br, Cr, Dr
}
.
8. Solve (3.15) for Br and (3.17) for Dr. The final reduced order model is given
by Gr(s) = {Ar, Br, Cr, Dr}.
9. Calculate the weighted error = ∥(G(s)−Gr(s))V (s)∥∞.
There are some limitations of the proposed method which are:
1. The chosen weighting functions must be strictly proper (Dv = Dw = 0).
2. It can be applied for single-sided only.
3. The output matrix of input weight (Cv) and input matrix of output weight (Bw)
have to be square and nonsingular.
3.3.2 New Method 2
Refer to (3.7), let Gr,x(s) is an rtℎ order model of G(s), and Gr,x(s) is an rtℎ order
model of Gx(s) (x = i, x = o depending on the input or output weight). In [70],
the final reduced order model Gr(s) is obtained by deleting the extra rows and/or
columns of realization in the intermediate reduced order model Gr,x(s). In the new
83
method 2, [70] is modified such that the final reduced order model is obtained from the
relationship between the intermediate and the final reduced order models as follows:
Gr,i(s)V (s) = Gr,i(s)V (s) (3.18a)
W (s)Gr,o(s) = W (s)Gr,o(s) (3.18b)
Let Gr,x(s) = Cr,x(sI − Ar,x)−1Br,x +Dr,x where (x = i, x = o depending on the
input or output weight) and Dr,x = D, then the augmented systems Gr,i(s)V (s) and
W (s)Gr,o(s) are given by:
Gr,i(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar,i Br,iCv Br,iDv
0 Av Bv
DwCr,i DCv DDv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Ar,i Br,i
Cr,i Dr,i
⎤
⎥
⎦
W (s)Gr,o(s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwCr,o BwD
0 Ar,o Br,o
Cw DwCr,o DwD
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Ar,o Br,o
Cr,o Dr,o
⎤
⎥
⎦
where the Gramians
Pr,i =
⎡
⎢
⎣
P11,r P12,r
P T12,r Pv
⎤
⎥
⎦Qr,o =
⎡
⎢
⎣
Qw Q12,r
QT12,r Q22,r
⎤
⎥
⎦
satisfy the following Lyapunov equations:
Ar,iPr,i + Pr,iATr,i + Br,iB
Tr,i = 0
ATr,oQr,o + Qr,oAr,o + CT
r,oCr,o = 0
84
Similar to (3.7),W (s)Gr,o(s) andGr,i(s)V (s) can also be decomposed intoGr,i(s)V (s)
and W (s)Gr,o(s) where Gr,x(s) is an rtℎ order model of Gx(s).
Let Tr,i =
⎡
⎢
⎣
I Xr
0 I
⎤
⎥
⎦and Tr,o =
⎡
⎢
⎣
I −Yr
0 I
⎤
⎥
⎦be the transformation matrices
required to take the Gramians{
Pr,i, Qr,o
}
into block diagonal matrices as follows:
Pr,i = T−1r,i Pr,iT
−Tr,i =
⎡
⎢
⎣
P11,r 0
0 Pv
⎤
⎥
⎦Qr,o = T T
r,oQr,oTr,o =
⎡
⎢
⎣
Qw 0
0 Q22,r
⎤
⎥
⎦
then the corresponding state-space realizations can be written as:
Gr,i(s)V (s) =
⎡
⎢
⎣
T−1r,i Ar,iTr,i T−1
r,i Br,i
Cr,iTr,i Dr,i
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar,i X23,r X2,r
0 Av Bv
Cr,i Y2,r DDv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.19a)
W (s)Gr,o(s) =
⎡
⎢
⎣
T−1r,o Ar,oTr,o T−1
r,o Br,o
Cr,oTr,o Dr,o
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Aw X12,r X1,r
0 Ar,o Br,o
Cw Y1,r DwD
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.19b)
85
where
X23,r = Ar,iXr −XrAv + Br,iCv
X2,r = Br,iDv −XrBv
Y2,r = Cr,iXr +DCv
X12,r = YrAr,o − AwYr + BwCr,o
Y1,r = DwCr,o − CwYr
X1,r = BwD + YrBr,o
From (3.7), we can obtain W (s), V (s) and Gx(s). Similar to [70], an rtℎ order
intermediate reduced order model Gr,x(s) is computed here using balanced truncation
method. Let Gr,i(s) ={
Ar,i, Br,i, Cr,i, Dr,i
}
or Gr,o(s) ={
Ar,o, Br,o, Cr,o, Dr,o
}
be
the intermediate reduced order model obtained from Gx(s), then we can write the
augmented systems as
Gr,i(s)V (s) =
⎡
⎢
⎣
Ar,i Br,i
Cr,i Dr,i
⎤
⎥
⎦
⎡
⎢
⎣
Av Bv
Cv Dv
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar,i Br,iCv Br,iDv
0 Av Bv
Cr,i Dr,iCv Dr,iDv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.20a)
W (s)Gr,o(s) =
⎡
⎢
⎣
Aw Bw
Cw Dw
⎤
⎥
⎦
⎡
⎢
⎣
Ar,o Br,o
Cr,o Dr,o
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwCr,o BwDr,o
0 Ar,o Br,o
Cw DwCr,o DwDr,o
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(3.20b)
86
Equating equations (3.19) and (3.20) gives
X23,r = Br,iCv (3.21a)
X2,r = Br,iDv (3.21b)
X12,r = BwCr,o (3.21c)
Y1,r = DwCr,o (3.21d)
Y2,r = Dr,iCv (3.21e)
DDv = Dr,iDv (3.21f)
X1,r = BwDr,o (3.21g)
DwD = DwDr,o (3.21h)
Rewriting the first four of (3.21) we get
X23,r = Ar,iXr −XrAv +Br,iCv = Br,iCv (3.22a)
X2,r = Br,iDv −XrBv = Br,iDv (3.22b)
X12,r = YrAr,o − AwYr +BwCr,o = BwCr,o (3.22c)
Y1,r = DwCr,o − CwYr = DwCr,o (3.22d)
In (3.22), the matrices{
Ar,i, Br,i, Ar,o, Cr,o
}
are obtained from the intermediate re-
duced order model Gr,x(s). Solving (3.22) forXr, Br,i, Yr, Cr,o one can obtain Gr,i(s) =
{Ar,i, Br,i, Cr,i, D} or Gr,o(s) = {Ar,o, Br,o, Cr,o, D} depending on input or output
weighting.
Note that, since Dr,x = D, and to ensure that the last four equations of (3.21) are
87
satisfied, the matrices Dr,i and Dr,o are defined as:
Dr,i =
[
D 0 Cr,iXr
]
Dr,o =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
D
0
YrBr,o
⎤
⎥
⎥
⎥
⎥
⎥
⎦
using the matrices obtained from (3.22).
To solve the equations (3.22c) and (3.22d), we can rewrite them as
⎡
⎢
⎣
−I ⊗ Aw + ATr,o ⊗ I I ⊗ Bw
−I ⊗ Cw I ⊗Dw
⎤
⎥
⎦
⎡
⎢
⎣
V ec(Yr)
V ec(Cr,o)
⎤
⎥
⎦
=
⎡
⎢
⎣
V ec(BwCr,o)
V ec(DwCr,o)
⎤
⎥
⎦(3.23)
where V ec(X) denotes the vector formed by stacking the columns of X into one
long vector. The coefficient matrix on the left of the above equation has full rank,
guaranteeing solvability of the equation when
⎡
⎢
⎣
−Aw + �I Bw
−Cw Dw
⎤
⎥
⎦
has full rank for all � = �i(Ar,o), i = 1, . . . , r [85], where �(X) denotes the eigenvalues
of X. However, there is a unique solution if and only if the matrix on the left of (3.23)
is square. Similarly Xr and Br,i, provided they exist, are uniquely determined if and
only if V (s) is square.
Remark 46 The condition that⎡
⎢
⎣
−Aw + �I Bw
−Cw Dw
⎤
⎥
⎦
88
has full rank at some �i is effectively a condition that W (�i) has full rank there. This
observation follows immediately from the identity:
⎡
⎢
⎣
−Aw + �I Bw
−Cw Dw
⎤
⎥
⎦=
⎡
⎢
⎣
I 0
Cw(Aw − �iI)−1 I
⎤
⎥
⎦
⎡
⎢
⎣
−Aw + �I Bw
0 W (�i)
⎤
⎥
⎦
We say effectively, since there remains open the possibility that W (s) could have a
pole at �i. A similar remark applies to the input weight V (�i).
Remark 47 Note that if the weights W (s) and V (s) have full row and column rank
respectively, the requirement for them to have this property for the particular values
of � = �i(Ar,o) will be generally satisfied.
Theorem 8 If G(s) = {A,B,C,D} is stable and minimal then Gr,x(s) obtained from
the proposed method is also stable and minimal.
Proof: It has been proven in [70] that for a stable and minimal original systemG(s) =
{A,B,C,D}, the new realization Gx(s) is also stable and minimal. Since Gr,x(s) is
obtained by balanced truncation of Gx(s), stability of Gr,x(s) follows immediately.
As a result, Gr,x(s) which is the reduced order model obtained using the proposed
technique is also guaranteed to be stable for stable original systems as it has the same
Ar as Gr,x(s).
Theorem 9 If Gr,x(s) is an rtℎ order model of the given original system G(s) and
Gr,x(s) is an rtℎ order model of the new system Gx(s), then
∥(G(s)−Gr,i(s))V (s)∥∞
=∥
∥(Gi(s)−Gr,i(s))V (s)∥
∥
∞
∥W (s)(G(s)−Gr,o(s))∥∞ =∥
∥W (s)(Go(s)−Gr,o(s))∥
∥
∞
89
Proof: From (3.7), we have
G(s)V (s) = Gi(s)V (s) (3.24a)
W (s)G(s) = W (s)Go(s) (3.24b)
From (3.18) we also have
Gr,i(s)V (s) = Gr,i(s)V (s) (3.25a)
W (s)Gr,o(s) = W (s)Gr,o(s) (3.25b)
Substracting (3.25) from (3.24) we have
(G(s)−Gr,i(s))V (s) = (Gi(s)−Gr,i(s))V (s)
W (s)(G(s)−Gr,o(s)) = W (s)(Go(s)−Gr,o(s))
Corollary 1
∥(G(s)−Gr,i(s))V (s)∥∞
=∥
∥(Gi(s)−Gr,i(s))V (s)∥
∥
∞
≤ 2∥
∥V (s)∥
∥
∞
n∑
i=r+1
�i
∥W (s)(G(s)−Gr,o(s))∥∞ =∥
∥W (s)(Go(s)−Gr,o(s))∥
∥
∞
≤ 2∥
∥W (s)∥
∥
∞
n∑
i=r+1
�i
where �i are the singular values of Gx(s).
Remark 48 If the reduced order model Gr,x(s) is obtained without frequency weight-
ing, then V (s) = W (s) = I. The following result of [19, 25] can be obtained easily:
∥(G(s)−Gr(s))∥∞ ≤ 2n
∑
i=r+1
�i.
90
Algorithm 2 New Method 2
1. Given a stable and minimal G(s) and V (s) solve (3.3a) for the Gramians P .
2. Compute X from (3.4).
3. Compute the fictitious input and output matrices B from (3.8a).
4. Calculate the transformation matrix T which balance{
A,B,C}
to diagonalize
the Gramians:
T−1PT−T = T TQT = diag {�1, �2, . . . , �n}
5. Compute the frequency weighted balanced realization
⎡
⎢
⎣
T−1AT T−1B
CT Di
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar,i A12 Br,i
A21 A22 B2
Cr,i C2 Dr,i
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
6. Solve (3.22a) to (3.22b) for Br,i.
7. An rtℎ order model is given by Gr,i(s) = {Ar,i, Br,i, Cr,i, D}.
8. Calculate the weighted error =
∥(G(s)−Gr(s))V (s)∥∞
Remark 49 To reduce the approximation error, the matrices B and C used in the
proposed algorithm can be made to be functions of free parameter � as follows:
B =
[
B −�X AX
]
91
To ensure that equations in (3.9) are valid, we need to have
Cv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
�
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Dv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Dv
Bv
�
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Note that, � can be any scalar values other than zeros. By varying the scalar �, one
can easily reduce the weighted approximation errors.
Remark 50 Similar to [77, 70] the proposed method is realization dependent. For
different realization of input and output weights, different reduced order models and
weighted approximation errors are obtained.
Remark 51 Similar to [70], the factorization in (3.8) is not unique.
3.4 Simulation Results
In this section we compare the proposed methods with other well-known methods
such as Enns’ [19], Lin and Chiu’s (LC’s) [40, 69], Varga and Anderson’s (VA’s) [76],
Wang et al.’s (Wang’s) [77], and Sreeram and Sahlan’s (SS’s) [70] techniques using
two numerical examples.
92
Example 1 We consider a fourth-order system (Example 2 of [77]) as follows:
A =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−0.6503 −0.2734 0.0030 −0.1815
0.2883 −1.0171 0.0102 −1.2651
0.0377 0.1087 −0.0011 −3.2129
0.8699 −4.6643 16.1671 −18.3349
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
B =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
3.3317 3.2155
−1.9209 −0.0978
−4.5402 2.6599
−17.4882 6.0988
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
C =
[
31.5142 6.4374 −0.0750 4.3834
]
with the following input weight:
Av =
⎡
⎢
⎣
−8 0
1 −5
⎤
⎥
⎦Bv =
⎡
⎢
⎣
5
10
⎤
⎥
⎦Cv =
⎡
⎢
⎣
−2 2
3 1
⎤
⎥
⎦
Since the new method 1 (Algorithm 1) yields proper models (Gr(s) = Cr(sIr −
Ar)−1Br+Dr withDr ∕= 0) for strictly proper original systems (see Step 8 of Algorithm
1), to make a fair comparison, we use the singular perturbation approximation [22, 41]
(see Remark 5) of well known methods instead of direct truncation [42].
Simulation results are shown in Table 3.1. The figures in the last column give
the approximation error improvement (in percentage) of the proposed technique over
Enns’ technique. From the table, it can be seen that the proposed method gives the
lowest errors compared to other well-known techniques. For the 3rd order model, the
proposed method yields a very low error which is ≈ 77% less than Enns’ technique.
93
Table 3.1: Weighted errors for Example 1
Enns’ LC’s VA’s Wang SS’s Algorithm 1
Order Error Error Error Error �SS Error Error % improvement
1 29.0893 38.9031 30.6917 30.5901 9.9 28.2842 20.0975 30.91
2 2.9625 4.7611 3.1112 3.0063 3.1 2.9941 0.8728 70.54
3 0.2853 1.1361 0.2927 2.8247 3.2 4.7264 0.0667 76.62
Example 2 For comparison purposes, we consider the fourth-order system used in
[40, 69, 77, 70]
A =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−1 0 0 0
0 −2 0 0
0 0 −3 0
0 0 0 −4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
B =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 5
1/2 −3/2
1 −5
−1/2 1/6
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
C =
⎡
⎢
⎣
1 0 1 0
4/15 1 0 1
⎤
⎥
⎦
with the following input weight [40]:
V (s) =4.5
s+ 4.5I2
where I2 denotes a 2nd order of identity matrix.
Let �max[G] denote the maximum singular value of G. The maximum singular
value of input weight V (s) is given in Figure 3.1. From the figure, we can see that
the considered weighting function is a low pass filter with the passband frequency is
all frequencies lower than 4.5 rad/s.
The simulation results are shown in Table 3.2. From the table, we can see clearly
that the proposed method is well ahead of all the other methods in reducing the
94
10−2
10−1
100
101
102
0
0.2
0.4
0.6
0.8
1
Frequency ω (rad/sec)
σ max
[V(jω
)] (
abs)
Figure 3.1: Maximum singular value of input weight V (s) for Example 2
weighted approximation error.
Using the same example, we use Algorithm 2 and compare it with other tech-
niques. Table 3.3 shows the reduction errors ∥(G(s)−Gr(s))V (s)∥∞
obtained from
Algorithm 2 and the other existing techniques. Note that, since Algorithm 2 uses
direct truncation to reduce the balanced system, other techniques also use the same
method to get the models.
It is clear from the table that the proposed technique gives the lowest errors
compared to other well-known techniques. Figure 3.2 gives the plot of �max[(G(j!)−
Gr(j!))V (j!)] for r = 3 with various methods. The figure shows clearly that the
proposed method not only gives the lowest value of ∥(G(s)−Gr(s))V (s)∥∞
but also
a significant improvement to existing techniques in the selected band of frequencies.
95
Table 3.2: Weighted errors for Example 2 using Algorithm 1
Enns’ LC’s VA’s Wang SS’s Algorithm 1
Order Error Error Error Error �SS Error Error % improvement
1 0.3775 0.3993 0.3820 0.3780 0.7 0.5543 0.3476 7.92
2 0.0724 0.0877 0.0742 0.0678 0.1 0.0619 0.0599 17.27
3 0.0175 0.0183 0.0177 0.0180 0.1 0.0314 0.0164 6.29
Table 3.3: Weighted errors and error bounds for Example 2 using Algorithm 2
Enns’ LC’s VA’s Wang SS’s Algorithm 2
Order Error Error Error Error Error bound �SS Error Error bound � Error Error bound
1 0.5568 0.5605 0.5545 0.5556 1.3024 4 0.5545 1.5783 12 0.5487 1.2736
2 0.0620 0.0619 0.0616 0.0663 0.2806 54 0.0619 2.4475 10 0.0610 0.2126
3 0.0322 0.0314 0.0319 0.0326 0.0614 39 0.0314 0.3562 3 0.0281 0.0341
96
10−2
10−1
100
101
102
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Frequency ω [rad/s]
σ max
[(G
(jω)−
Gr(jω
))V
(jω)]
TBREnnsVarga and AndersonWang et al.Lin and ChiuSreeram and SahlanProposed
Figure 3.2: Maximum singular value of reduction error for Example 2
3.5 Conclusion
Two frequency weighted balanced truncation algorithms based on zero cross-terms
(the main property of Lin and Chiu’s technique [40, 69]) are proposed. Both meth-
ods are modifications to Sreeram and Sahlan’s technique [70]. The first method
decomposes the augmented systems obtained from Lin and Chiu’s technique into
new augmented systems and new weights using the properties of all-pass function.
Although the first method is limited to special weighting functions (single-sided and
strictly proper), the method doesn’t depend on the free parameter to reduce the re-
duction error. In addition, the approximation error reduction achievable is significant
as illustrated in the numerical examples. The second method improved Sreeram and
Sahlan’s method [70] using the relationship between the intermediate and the final
reduced order models. By varying user-chosen free parameter, the example indicates
97
that a significant improvement over the existing techniques [40, 19, 69, 77, 76, 70] can
be achieved.
98
Chapter 4
Frequency Weighted Balanced
Truncation based on Partial
Fraction Expansion Techniques
4.1 Introduction
An alternative method to guarantee stability of the reduced order models in the
two sided weighting case, where Enns’ method [19] cannot, is the partial fraction
expansion based technique [68]. The method was originally proposed by Latham and
Anderson [38], which was later modified by Al-Saggaf and Franklin [2, 3] to include
frequency weightings. Other frequency weighted balanced truncation based on partial
fraction expansion techniques include [2, 3, 68, 85, 24, 58] and related references.
Error bound exists for some special types of weighting function [68, 24, 58]. However,
the approximation error obtained using these methods is generally larger compared to
99
Enns’ method [19], with the exception of the method by Zhou [85] where optimization
is used to improve the approximation error.
In this chapter, we present further improvements to the partial fraction expansion
technique [45] which yields substantial approximation error reduction compared to
Enns’ technique. The method is also elegant with simple and easily computable error
bounds, and is illustrated by two examples.
4.2 Preliminaries
This section reviews some of the well-known frequency weighted balanced truncation
based on partial fraction expansion techniques.
Let G(s), V (s) and W (s) be the stable original system and the stable input and
output weights respectively. Let {A,B,C,D}, {Av, Bv, Cv, Dv} and {Aw, Bw, Cw, Dw}
be their corresponding minimal realizations respectively. Consider the augmented
system W (s)G(s)V (s) represented by the following realization:
W (s)G(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwC BwDCv BwDDv
0 A BCv BDv
0 0 Av Bv
Cw DwC DwDCv DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦. (4.1)
The controllability and observability Gramians of the augmented realization{
A, B, C, D}
100
are given by:
P =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Pw P12 P13
P T12 PE P23
P T13 P T
23 Pv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Q =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Qw Q12 Q13
QT12 QE Q23
QT13 QT
23 Qv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
where P and Q satisfy the following Lyapunov equations:
AP + P AT + BBT = 0 (4.2a)
AT Q+ QA+ CT C = 0. (4.2b)
4.2.1 Sreeram and Anderson’s Technique
Sreeram and Anderson generalized the partial fraction expansion based technique pro-
posed in [2, 3] to include double-sided weighting [68]. Although the method can only
handle strictly proper weighting functions, the derivation presented in this chapter is
generalized to include proper original systems as these equations will be required in
the main section of the chapter.
The technique first transforms the augmented system realization (4.1) into a block
diagonal form by the following transformation matrix:
T =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I −Y R
0 I X
0 0 I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
101
Transforming the augmented system realization (4.1), we have:
W (s)G(s)V (s) =
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦
=
⎡
⎢
⎣
T−1AT T−1B
CT D
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw X12 X13 X1
0 A X23 X2
0 0 Av Bv
Cw Y1 Y2 DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= W (s) + G(s) + V (s)
=
⎡
⎢
⎣
A B
C D
⎤
⎥
⎦(4.3)
where
X12 = Y A− AwY + BwC = 0 (4.4a)
X23 = AX −XAv + BCv = 0 (4.4b)
X13 = AwR−RAv + BwCX + Y AX + BwDCv
+Y BCv − Y XAv = 0 (4.4c)
X1 = BwDDv + Y BDv − Y XBv −RBv (4.4d)
X2 = BDv −XBv (4.4e)
Y1 = DwC − CwY (4.4f)
Y2 = DwCX +DwDCv + CwR (4.4g)
D = DwDDv. (4.4h)
102
Remark 52 Equation (4.4c) has a unique solution R if and only if �−� ∕= 0 for all
� ∈ �(Aw) and � ∈ �(Av) [37], where �(Z) denotes the spectrum of the matrix Z.
Instead of balancing and truncating the original system {A,B,C}, the method
balances and truncates the new system {A,X2, Y1} to obtain the reduced-order mod-
els.
Note that the frequency weighted error can be large with this method. However,
the error can be reduced for strictly proper original systems and the weights (D =
0, Dv = 0 and Dw = 0) if the reduction error is made to have zeros at the poles of
input weight or output weight as shown in [68].
4.2.2 Sahlan and Sreeram’s Technique
As in the previous method, Sahlan and Sreeram’s method [58] involves decomposing
the augmented system W (s)G(s)V (s) into W (s) + G(s) + V (s) (see equation (4.3))
using partial fraction expansion. These terms are then recombined to obtain a new
augmented system W (s)G(s)V (s) such that
W (s)G(s)V (s) = W (s) + G(s) + V (s) = W (s)G(s)V (s) (4.5)
where G(s) ={
A,B,C,D}
is the new original system, and V (s) ={
Av, Bv, Cv, Dv
}
and W (s) ={
Aw, Bw, Cw, Dw
}
are the new input and output weights respectively.
103
The new parameters in the above equations are given by
Bw =
[
Bw Aw I
]
(4.6a)
Dw =
[
Dw Cw 0
]
(4.6b)
B =
[
B −X AX
]
(4.6c)
C =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
C
−Y
Y A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(4.6d)
D =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
D 0 CX
0 0 R
Y B −R− Y X Y AX
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(4.6e)
Cv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(4.6f)
Dv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Dv
Bv
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (4.6g)
104
Using the matrices defined in (4.6), the equations in (4.4) can now be expressed as:
X12 = BwC (4.7a)
X23 = B Cv (4.7b)
X13 = BwD Cv (4.7c)
X1 = BwD Dv (4.7d)
X2 = B Dv (4.7e)
Y1 = DwC (4.7f)
Y2 = DwD Cv (4.7g)
D = DwD Dv. (4.7h)
Remark 53 From equation (4.5), we have
W (s)G(s)V (s) = W (s)G(s)V (s).
This relation is valid even if
W (s)G(s)V (s) ∕= W (s) + G(s) + V (s)
which is the case when R in (4.4c) does not exist (see Remark 52).
Diagonalizing the weighted Gramians{
P ,Q}
of the new system G(s) ={
A,B,C,D}
which satisfy
AP + PAT + B BT
= 0 (4.8a)
ATQ+QA+ CTC = 0 (4.8b)
yielding
T−1SSPT−T
SS = T TSSQTSS = diag(�1, �2, . . . , �r, �r+1, . . . , �n)
105
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Instead of reducing G(s), in this
technique the new original system G(s) is reduced using balanced truncation to obtain
Gr(s) ={
Ar, Br, Cr, Dr
}
. The final reduced-order model Gr(s) = {Ar, Br, Cr, Dr}
is obtained by simply deleting the extra rows in Cr, extra columns in Br and both
extra rows and columns in Dr. Since the realization{
A,B,C}
is minimal and the
weighted Gramians{
P ,Q}
satisfy the Lyapunov equations (4.8), the technique yields
stable models in the case of double-sided weightings. Although the method is simple
and elegant, approximation error reduction obtained from this technique is very small
and is often negligible.
In the next section we present an improvement to this technique to obtain a
significant weighted error reduction not reported so far with any technique.
4.3 Main Results
Step 1: Figure 2.11 as in [68, 24, 58]
Step 2: W (s) + G(s) + V (s) = W (s)G(s)V (s) as in [58]
Step 3: Find Gr(s) as in [58]
Step 4: Find Gr(s) from W (s)(G(s)− Gr(s))V (s) = W (s)(G(s)− Gr(s))V (s)
Figure 4.1: Summary of the new method based on partial fraction expansion method
106
As shown in Figure 4.1, the proposed method [45] can be explained using the
following steps.
Step 1 The augmented system W (s)G(s)V (s) is decomposed using partial frac-
tion expansion to obtain W (s)+ G(s)+ V (s) (see Figure 2.11). This step is the same
in all three partial fraction expansion techniques [68, 24, 58] and can be written as
follows
W (s)G(s)V (s) = W (s) + G(s) + V (s).
Step 2 The block diagonalized augmented system W (s) + G(s) + V (s) is recon-
structed to find a new augmented system W (s)G(s)V (s). This step is the same as in
[58] and is written as
W (s) + G(s) + V (s) = W (s)G(s)V (s).
Step 3 Intermediate reduced order model Gr(s) = Cr(sI − Ar)−1Br + Dr is
obtained from G(s) by using balanced truncation. This step is same as in [58].
Step 4 which is the final step, is different to the technique of [58]. In [58] the
final reduced order model is obtained by directly deleting the extra rows in Cr, extra
columns in Br and extra rows and columns in Dr. In the proposed method, if Gr(s) =
Cr(sI −Ar)−1Br +Dr is the final reduced-order model then the matrices Cr, Br and
Dr are chosen such that
W (s)Gr(s)V (s) = W (s)Gr(s)V (s).
To find the final reduced-order model Gr(s) in the proposed technique, let Gr(s) =
Cr(sI − Ar)−1Br +Dr with Dr = D, then the augmented system W (s)Gr(s)V (s) is
107
given by:
W (s)Gr(s)V (s) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwCr BwDCv BwDDv
0 Ar BrCv BrDv
0 0 Av Bv
Cw DwCr DwDCv DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎣
Ar Br
Cr Dr
⎤
⎥
⎦.
Let Tr =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
I −Yr Rr
0 I Xr
0 0 I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
be the transformation matrix required to take the aug-
mented system into a block diagonal form, then
W (s)Gr(s)V (s) =
⎡
⎢
⎣
T−1r ArTr T−1
r Br
CrTr Dr
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw X12,r X13,r X1,r
0 Ar X23,r X2,r
0 0 Av Bv
Cw Y1,r Y2,r DwDDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.9)
108
where
X12,r = YrAr − AwYr + BwCr = 0 (4.10a)
Y1,r = DwCr − CwYr (4.10b)
X23,r = ArXr −XrAv + BrCv = 0 (4.10c)
X2,r = BrDv −XrBv (4.10d)
X13,r = AwRr −RrAv + BwCrXr + YrArXr + BwDCv
+YrBrCv − YrXrAv = 0 (4.10e)
X1,r = BwDDv + YrBrDv − YrXrBv −RrBv
Y2,r = DwCrXr +DwDCv + CwRr (4.10f)
Dr = DwDDv. (4.10g)
Since we know Gr(s) from Step 3 and the new weights W (s) and V (s) from Step
2, we can write the augmented system as
W (s)Gr(s)V (s) =
⎡
⎢
⎣
Aw Bw
Cw Dw
⎤
⎥
⎦
⎡
⎢
⎣
Ar Br
Cr Dr
⎤
⎥
⎦
⎡
⎢
⎣
Av Bv
Cv Dv
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Aw BwCr BwDrCv BwDrDv
0 Ar BrCv BrDv
0 0 Av Bv
Cw DwCr DwDrCv DwDrDv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
. (4.11)
To find Gr(s) such that
W (s)Gr(s)V (s) = W (s)Gr(s)V (s)
109
we need to equate equations (4.9) and (4.11). This gives
X12,r = BwCr (4.12a)
Y1,r = DwCr (4.12b)
X23,r = BrCv (4.12c)
X2,r = BrDv (4.12d)
X13,r = BwDrCv (4.12e)
X1,r = BwDrDv (4.12f)
Y2,r = DwDrCv (4.12g)
Dr = DwDrDv. (4.12h)
Rewriting the first four equations of (4.10) and (4.12), we get
X12,r = YrAr − AwYr + BwCr = BwCr (4.13a)
Y1,r = DwCr − CwYr = DwCr (4.13b)
X23,r = ArXr −XrAv + BrCv = BrCv (4.13c)
X2,r = BrDv −XrBv = BrDv. (4.13d)
In (4.13), the matrices{
Br, Cr
}
are obtained from the intermediate reduced order
model Gr(s) = Cr(sI−Ar)−1Br+Dr in Step 3. Solving the equations for Yr, Cr, Xr
and Br, the final reduced order model Gr(s) = {Ar, Br, Cr, D} can be obtained.
Note that, since Dr = D, and to ensure that the last four equations of (4.12) are
satisfied, the matrix Dr is defined as:
Dr =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
D 0 CrXr
0 0 Rr
YrBr −Rr − YrXr YrArXr
⎤
⎥
⎥
⎥
⎥
⎥
⎦
110
using the matrices obtained from (4.13).
To solve the equations (4.13a) and (4.13b), we can rewrite them as
⎡
⎢
⎣
−I ⊗ Aw + ATr ⊗ I I ⊗ Bw
−I ⊗ Cw I ⊗Dw
⎤
⎥
⎦
⎡
⎢
⎣
V ec(Yr)
V ec(Cr)
⎤
⎥
⎦=
⎡
⎢
⎣
V ec(BwCr)
V ec(DwCr)
⎤
⎥
⎦(4.14)
where V ec(X) denotes the vector formed by stacking the columns of X into one
long vector. The coefficient matrix on the left of the above equation has full rank,
guaranteeing solvability of the equation when
⎡
⎢
⎣
−Aw + �I Bw
−Cw Dw
⎤
⎥
⎦
has full rank for all � = �i(Ar), i = 1, . . . , r [85, 86], where �(X) denotes the eigenval-
ues of X. However, there is a unique solution if and only if the matrix on the left of
(4.14) is square. Similarly Xr and Br, provided they exist, are uniquely determined
if and only if V (s) is square.
Remark 54 The condition that⎡
⎢
⎣
−Aw + �I Bw
−Cw Dw
⎤
⎥
⎦
has full rank at some �i is effectively a condition that W (�i) has full rank there. This
observation follows immediately from the identity
⎡
⎢
⎣
−Aw + �I Bw
−Cw Dw
⎤
⎥
⎦=
⎡
⎢
⎣
I 0
Cw(Aw − �iI)−1 I
⎤
⎥
⎦
⎡
⎢
⎣
−Aw + �I Bw
0 W (�i)
⎤
⎥
⎦.
We say ’effectively’, since the possibility remains open that W (s) could have a pole
at �i. A similar remark applies to the input weight V (�i).
111
Remark 55 Note that if the weights W (s) and V (s) have full row and column rank
respectively, the requirement for them to have this property for the particular values
of � = �i(Ar) will be generally satisfied.
Theorem 10 If G(s) = {A,B,C,D} is stable, then Gr(s) = {Ar, Br, Cr, D} ob-
tained from the proposed method is also stable.
Proof 6 It has been proven in [58] that for a stable original system G(s) = {A,B,C,D},
the new realization G(s) ={
A,B,C,D}
is also stable. The stability of Gr(s) =
Cr(sI −Ar)−1Br +Dr follows immediately as it is obtained by balanced truncation of
G(s). The stability of the reduced order model obtained by using the proposed tech-
nique Gr(s) = Cr(sI − Ar)−1Br + Dr is also guaranteed as it has the same Ar as
Gr(s).
Theorem 11 If Gr(s) = {Ar, Br, Cr, D} is an rtℎ order model of the given original
system G(s) and Gr(s) ={
Ar, Br, Cr, Dr
}
is an rtℎ order model of the new system
G(s), then
∥W (s)(G(s)−Gr(s))V (s)∥∞
=∥
∥W (s)(G(s)−Gr(s))V (s)∥
∥
∞.
Proof 7 From Step 1 and Step 2 of the proposed method
W (s)G(s)V (s) = W (s)G(s)V (s). (4.15)
From Step 4 of the proposed method
W (s)Gr(s)V (s) = W (s)Gr(s)V (s). (4.16)
Substracting (4.16) from (4.15) we have
W (s)(G(s)−Gr(s))V (s) = W (s)(G(s)−Gr(s))V (s).
112
Corollary 2
∥W (s)(G(s)−Gr(s))V (s)∥∞
=∥
∥W (s)(G(s)−Gr(s))V (s)∥
∥
∞
≤ 2∥
∥V (s)∥
∥
∞
∥
∥W (s)∥
∥
∞
n∑
i=r+1
�i
where �i are the singular values of G(s).
Remark 56 If the reduced order model Gr(s) is obtained without frequency weighting,
then V (s) = W (s) = I. The following result of [19, 25] can be obtained easily:
∥(G(s)−Gr(s))∥∞ ≤ 2n
∑
i=r+1
�i.
A step-by-step algorithm for the proposed method can be obtained as follows:
Algorithm 3 New method based on partial fraction expansion method
1. Given a stable and minimal G(s), V (s) and W (s), compute Y and X from
(4.4a) and (4.4b) respectively.
2. Compute the fictitious input and output matrices B and C from (4.6c) and
(4.6d) respectively.
3. Calculate the transformation matrix T which balances{
A,B,C}
to diagonalize
the Gramians:
T−1PT−T = T TQT = diag {�1, �2, . . . , �n} .
4. Compute the frequency weighted balanced realization
⎡
⎢
⎣
T−1AT T−1B
CT D
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Ar A12 Br
A21 A22 B2
Cr C2 D
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
113
5. Solve (4.13a) to (4.13d) for Xr, Yr, Br, Cr.
6. An rtℎ order model can be obtained as Gr(s) = {Ar, Br, Cr, D}.
7. Calculate the weighted error =
∥W (s)(G(s)−Gr(s))V (s)∥∞
Remark 57 In the above algorithm, the values of R and Rr do not have any affect on
the approximation errors. The matrices only determine the values of X13 and X13,r
respectively. In other words, the equation
∥W (s)(G(s)−Gr(s))V (s)∥∞
=∥
∥W (s)(G(s)−Gr(s))V (s)∥
∥
∞
is true due to Remark 53.
Remark 58 To reduce the approximation error, the matrices B and C used in the
proposed algorithm 3 can be made to be functions of free parameters � and � as
follows:
B =
[
B −�X AX
]
C =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
C
−�Y
Y A
⎤
⎥
⎥
⎥
⎥
⎥
⎦
To ensure that equations in (4.7) are valid, we need to have
Cv =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Cv
Av
�
I
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Bw =
[
BwAw
�I
]
.
Note that � and � can be any scalar values other than zeros. By varying the scalars
� and � one can easily reduce the weighted approximation errors.
114
Remark 59 Similar to [77, 58] the proposed method is realization dependent. For
different realization of input and output weights, different reduced order models and
weighted approximation errors are obtained.
4.4 Simulation Results
Example 3 For comparison purposes we consider the fourth-order system used in
[40, 69, 77, 24, 58].
A =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−1 0 0 0
0 −2 0 0
0 0 −3 0
0 0 0 −4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
B =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 5
1/2 −3/2
1 −5
−1/2 1/6
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
C =
⎡
⎢
⎣
1 0 1 0
4/15 1 0 1
⎤
⎥
⎦
with the following weighting functions where I2 denotes a 2nd order of identity matrix:
V (s) = W (s) =s+ 9
s+ 4.5I2 = {−4.5I2, 3I2, 1.5I2, I2} .
Note that the input and output weights used in this example are equal. Since the
solutions of (4.4c) and (4.10e) do not exist (see Remark 52), thus we set R = Rr = 0
(see Remark 57).
Let �max[G] denote the maximum singular value of G. The maximum singular
value of input weight V (s) is given in Figure 4.2. From the figure we can see clearly
that the considered weighting function is a low pass filter.
Simulation results for double-sided weights are shown in Table 4.1 while for the
single-sided case they are shown in Table 4.2. The figures in the last column give
approximation error improvement (in percentage) of the proposed technique over
115
10−2
10−1
100
101
102
0
1
2
3
4
5
6
7
Frequency ω (rad/sec)
σ max
[V(jω
)] (
dB)
Figure 4.2: Maximum singular value of input weight V (s) for Example 3
Enns’ technique. Due to space limitations, only three existing techniques (Enns’ [19],
Ghafoor and Sreeram’s (GS) [24]; and Sahlan and Sreeram’s (SS) [58] techniques) are
considered here for comparison. Results for other well-known techniques such as Lin
and Chiu’s [69], Varga and Anderson’s [76] and Wang et al.’s [77] techniques can be
obtained in Table 1 of [24]. Note that, user-chosen parameters � and � for Ghafoor
and Sreeram’s, and Sahlan and Sreeram’s techniques are taken directly from their
best results (see Table 1 of [24, 58]). It is clear from the tables that the proposed
technique gives the lowest errors compared to other well-known techniques for both
cases.
116
Table 4.1: Weighted errors and error bounds for Example 3 (double-sided case)
Enns’ [19] GS’s [24] SS’s [58] Proposed Method
Order Error �GS �GS Error Error bound �SS �SS Error Error bound � � Error Error bound %
1 2.1291 0.4 0.4 2.1126 3.4848 8.1 6.7 2.1158 13.922 8.4 1.5 1.9480 12.9628 8.50
2 0.2660 0.8 2.9 0.2709 2.3312 180 110 0.3149 0.5290 16.6 40.0 0.2398 38.2713 9.85
3 0.1131 0.8 0.5 0.1084 0.1786 103.9 35.8 0.1081 0.3104 2.3 3.2 0.1003 0.5238 11.31
Table 4.2: Weighted errors and error bounds for Example 3 (single-sided case)
Enns’ [19] GS’s [24] SS’s [58] Proposed Method
Order Error �GS Error Error bound �SS Error Error bound � Error Error bound %
1 1.1310 5 1.1221 11.5618 7.4 1.1188 1.4841 3.9 1.0293 2.3170 9.01
2 0.1342 10 0.1334 4.1179 500 0.1563 0.2470 20 0.1254 1.2296 6.56
3 0.0654 2 0.0647 0.0994 142.9 0.0676 0.0683 1.7 0.0544 0.0951 16.82
117
If we plot the input-output weighted model reduction error �max[W (j!)(G(j!)−
Gr(j!))V (j!)] for order 2, we obtain Figure 4.3(a) and for the input weighted model
reduction error �max[(G(j!) − Gr(j!))V (j!)] for order 3, we obtain Figure 4.3(b).
From both figures, we can see that the proposed method not only gives the lowest val-
ues of ∥W (s)(G(s)−Gr(s))V (s)∥∞for double-sided case and ∥(G(s)−Gr(s))V (s)∥
∞
for single-sided case, but also the lowest errors in the selected band of frequencies.
10−2
10−1
100
101
102
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Frequency ω [rad/s]
σ max
[W(jω
)(G
(jω)−
Gr(jω
))V
(jω)]
EnnsSSGSProposed
(a) Double-sided case
10−2
10−1
100
101
102
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Frequency ω [rad/s]
σ max
[(G
(jω)−
Gr(jω
))V
(jω)]
EnnsSSGSProposed
(b) Single-sided case
Figure 4.3: Weighted model reduction error for Example 3
In Figure 4.4, the frequency weighted model reduction errors are plotted versus
parameters (� and �) for order r = 1 and order r = 3 respectively. Generally for
order r = 1 the reduction errors are lower for higher values of � and �, whilst for
order r = 3 the reduction errors are lower for higher values of �.
118
02
46
810
0
5
101
2
3
4
5
beta (β)alpha (α)
Wei
ghte
d er
ror
(a) r = 1
01
23
45
0
2
4
60.1
0.15
0.2
0.25
beta (β)alpha (α)
Wei
ghte
d er
ror
(b) r = 3
Figure 4.4: Frequency weighted error versus parameters (� and �) for Example 3
Example 4 Consider the one-link flexible robot arm controller reduction problem [34]
as in Example 2 of [24]. The transfer function of the flexible robot arm from the motor
voltage signal to angular position of a load mass is given by
G(s) =4445.7
s4 + 28.3s3 + 364.1s2 + 2386.9s
A convex optimization based fifth-order controller transfer function is given by
K(s) =s5 + 3.1s4 + 4.4s3 + 3.2s2 + 1.3s+ 0.2
s5 + 3s4 + 4.3s3 + 3s2 + 1.2s+ 0.2
The input weight V (s) = (I+G(s)K(s))−1 and output weight W (s) = (I+G(s)K(s))−1G(s)
are used as the frequency weighted models.
Simulation result is shown in Table 4.3. The figures in the last column gives
approximation error improvement (in percentage) of the proposed technique over
119
Table 4.3: Weighted errors for Example 4
Enns’ [19] GS’s [24] SS’s [58] Proposed Method
Order Error �GS �GS Error �SS �SS Error � � Error %
(×10−4) (×10−4) (×10−4) (×10−4)
1 192.13 0.3 0.1 191.77 100 100 191.26 0.1 0.8 198.00 -3.06
2 178.91 2 2 149.71 8 10 154.75 0.7 0.9 141.27 21.04
3 219.23 2.5 0.25 151.15 10 10 158.17 0.3 0.2 143.02 34.76
4 3.0830 0.25 1.5 2.8624 100 100 3.2893 2.0 0.8 3.0582 0.80
Enns’ technique. From the table, even though the reduction error for the proposed
method is slightly higher for order 1, but it gives a significant improvement for order
2 and order 3.
Figure 4.5 shows the frequency weighted model reduction error versus parameters
(� and �) for order r = 1 and order r = 3 respectively. For order r = 1, the reduction
errors are generally lower for lower values of � and �. For order r = 3, the reduction
errors are smaller for higher values of �.
4.5 Conclusion
An improved frequency weighted balanced truncation based on partial fraction expan-
sion is presented. The method guarantees the stability of reduced order models for
double-sided weights. Furthermore, the approximation error can be reduced by vary-
ing user-chosen free parameters � and �. The simulation results indicate a significant
120
00.5
11.5
2
0
0.5
1
1.5
20.01
0.02
0.03
0.04
0.05
0.06
beta (β)alpha (α)
Wei
ghte
d er
ror
(a) r = 1
00.5
11.5
2
0
0.5
1
1.5
20.01
0.02
0.03
0.04
0.05
0.06
0.07
beta (β)alpha (α)
Wei
ghte
d er
ror
(b) r = 3
Figure 4.5: Frequency weighted error versus parameters (� and �) for Example 4
improvement over the existing techniques [19, 69, 77, 76, 24, 58].
121
Chapter 5
Passivity Preserving Frequency
Weighted Balanced Truncation
Techniques
5.1 Introduction
Recent trends in VLSI technology and computer-aided design (CAD) techniques at
both the chip and package levels are that the central processor switching times are
reaching the vicinity of sub-nano seconds and communication switches are being de-
signed to transmit data that have bit rates at multiples of Gb/s. At these higher data
rates and operating frequencies, previously negligible effects of interconnects, such as
ringing, distortion, reflections and cross-talk tend to become the major bottlenecks in
the design as well as validation of high-speed designs. Direct analysis of interconnect
networks is very expensive in terms of computational cost. This is because, inter-
122
connect networks from modern high-speed designs tend to contain large number of
RLC components and circuit nodes. A standard practice to deal with this issue is
to use model-order reduction prior to performing transient analysis. It has also been
high-lighted in the recent literature [1, 53, 60] that, it is very important to preserve
the passivity of reduced-order models as any loss of passivity in the model can lead to
artificial oscillations [1] during transient simulations (when connected and simulated
with the rest of the circuitry).
In the available literature, several techniques can be found for the reduction of
large interconnect networks [1, 60, 55, 21, 50, 83, 32, 64, 15, 53, 30, 80, 79]. They can
be broadly classified into two categories: 1) moment matching techniques ([55, 21, 50,
83, 32]) and 2) singular value decomposition (SVD) based methods ([64, 15, 53, 30,
80, 79]). A well-known technique in SVD based approach is the truncated balanced
realization (TBR), originally proposed for continuous systems by Moore [42]. It is a
very elegant method as it has error bounds and is capable of reducing approximation
error better than moment matching methods. To preserve the passivity of the original
system, positive-real truncated balanced realization (PR-TBR) was introduced in [53].
Similar to the standard TBR method, this method also has error bounds.
Enns [19] extended the TBR method to include frequency weightings. Using a
chosen weighting function, the reduction errors can be minimized in some frequency
range of interest. With only one weighting present, the stability of reduced-order
models is guaranteed. However, in case of double-sided weightings, Enns’ method
may yield unstable models for stable original systems. The original Lin and Chiu’s
technique [40] and its generalization [69] present a modification to Enns’ technique
to guarantee stability in the case of double-sided weightings provided that there are
123
no pole-zero cancellations between the original system and the weights [76]. Another
modification to Enns’ technique proposed by Wang et al. [77] not only guarantees the
stability of reduced-order models, but also provides easily computable error bounds.
A major shortcoming of all these based frequency-weighted model-order reduction
(FWMOR) techniques, when applied to passive systems, is that they only guarantee
stability but not passivity of the reduced-order models. This is illustrated through a
numerical example below. For this purpose, a single port RLC network [64] given in
Figure 5.1 is considered.
+−Vin(t)
Iin(t)
R R1 L1
C1 C2
R150 L150
Cload
Figure 5.1: Interconnect circuit represented with RLC lumped segment model
The values of the parameters used are R = 1Ω, Ri = 0.1Ω, Li = 1mH, Ci =
Cload = 1�F and order n = 301. The chosen input and output weighting functions
are:
Wi(s) = Wo(s) =s+ 1× 109
s+ 3× 109(5.1)
In the experiment, the original system was reduced to a system of order r = 5
using the Enns’ [19], the Lin and Chiu’s [40, 69] and the Wang et al.’s [77] techniques.
The passivity of the reduced-order models are verified using Hamiltonian Theorem
2 of [60] by finding the eigenvalues of the Hamiltonian matrices, which detected a pair
124
of imaginary eigenvalues, indicating that the reduced-models are non-passive.
For the purpose of illustration, passivity verification is also demonstrated using
the Nyquist plots, shown in Figure 5.2. It is well known that the Nyquist plot of
a positive-real transfer-function should lie entirely in the right-half of the complex
plane [73].
−1 −0.5 0 0.5 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Real Axis
Imag
inar
y A
xis
OriginalEnnsWang et al.Lin and Chiu
(a) Normal view
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
Real Axis
Imag
inar
y A
xis
OriginalEnnsWang et al.Lin and Chiu
(b) Enlarged view
Figure 5.2: Illustration of passivity behaviour via Nyquist plots
When the figure is zoomed in as in Figure 5.2(b), we can see clearly that the
reduced-order models obtained from the standard FWMOR methods are not passive
as the Nyquist plots extend to the left-half of the complex plane.
To address the above discussed difficulty, in this chapter, we present
1. positive-real Enns’ method
125
2. positive-real Lin and Chiu’s method
3. positive-real Wang et al.’s method
to ensure the passivity of the resulting reduced-order models. Passivity proofs and
computational steps of each of the methods are given. Also numerical examples
and comparisons are provided to validate the passivity and accuracy of the proposed
methods.
This chapter is organized as follows. Section 5.2 summarizes the relevant state-
space concepts required for understanding the proposed methods. In Section 5.3, we
present the three positive-real frequency weighted model order reduction techniques
followed by step-by-step details of the proposed algorithms. Section 5.4 discusses the
simulation results and Section 5.5 concluding remarks.
5.2 Preliminaries
In this section, we review some of the well-known techniques related to the passivity-
preserving model-order reduction technique and FWMOR methods.
5.2.1 Phillips et al.’s Technique
Consider an ntℎ order minimal positive-real transfer-function of an m-port network:
(see Appendix E to see a step-by-step modified nodal analysis (MNA) formulation to
obtain a state-space from an RLC system):
H(s) = C(sI − A)−1B +D (5.2)
126
Equation (5.2) can also be represented in a state-space form as below:
x = Ax+ Bu (5.3a)
y = Cx+Du (5.3b)
where A ∈ ℜn,n, B ∈ ℜn,m, C ∈ ℜm,n, D ∈ ℜm,m , x ∈ ℜn,1 and u, y ∈ ℜm,1. The
goal of model order reduction is to obtain an rtℎ order model Hr(s) = {Ar, Br, Cr, D}
with r << n while maintaining the accuracy as well as important characteristics of the
original system such as stability and passivity. Since passivity implies stability and not
vice versa, passivity-preserving property is very important in model order reduction
algorithms. If H(s) represents the Y (admittance) or Z (impedance) parameters
system, positive-realness of H(s) implies that the underlying state-space description
represents a passive system [53].
To preserve positive-realness of the reduced transfer-function, instead of solving
Lyapunov equations as in the TBR method [42], the PR-TBR method [53] computes
controllability (P ) and observability (Q) Gramians from two sets of Lur’e equations
as follows:
AP + PAT = −KcKTc (5.4a)
PCT −B = −KcJTc (5.4b)
JcJTc = D +DT (5.4c)
ATQ+QA = −KTo Ko (5.5a)
QB − CT = −KTo Jo (5.5b)
JTo Jo = D +DT (5.5c)
127
Passive reduced order models are then obtained by partitioning and truncating the
balanced realization similar to the TBR method.
5.2.2 Frequency Weighted Balanced Truncation Techniques
This section reviews some of the well-known frequency-weighted balanced truncation
techniques as relevant to the proposed work presented in this chapter. Consider a
transfer-function represented by (5.2) and let the minimal input and output weighting
functions (Wi(s),Wo(s)) be:
Wi(s) = Ci(sI − Ai)−1Bi +Di (5.6a)
Wo(s) = Co(sI − Ao)−1Bo +Do (5.6b)
where Ai ∈ ℜv,v, Bi ∈ ℜv,m, Ci ∈ ℜm,v, Di ∈ ℜm,m, Ao ∈ ℜw,w, Bo ∈ ℜw,m, Co ∈
ℜm,w, Do ∈ ℜm,m, v and w are the number of states of input and output weights,
respectively. The augmented systems can be expressed in state space form by:
H(s)Wi(s) =
⎡
⎢
⎣
Ai Bi
C i Di
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A BCi BDi
0 Ai Bi
C DCi DDi
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(5.7a)
Wo(s)H(s) =
⎡
⎢
⎣
Ao Bo
Co Do
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A 0 B
BoC Ao BoD
DoC Co DoD
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (5.7b)
Next, the controllability and observability Gramians of the augmented realizations
{
Ai, Bi
}
and{
Ao, Co
}
are given by:
P =
⎡
⎢
⎣
P11 P12
P T12 P22
⎤
⎥
⎦Q =
⎡
⎢
⎣
Q11 Q12
QT12 Q22
⎤
⎥
⎦(5.8)
128
where P ∈ ℜN,N and Q ∈ ℜN,N satisfy the following Lyapunov equations:
AiP + PAT
i + BiBT
i = 0 (5.9a)
AT
o Q+ QAo + CT
o Co = 0. (5.9b)
Assuming that there are no pole-zero cancellations in H(s)Wi(s) and Wo(s)H(s), the
resulting Gramians P and Q are positive definite.
5.2.2.1 Enns’ Technique
Expanding the (1, 1) sub-matrix blocks of (5.9) yields the following:
AP11 + P11AT +XE = 0 (5.10a)
ATQ11 +Q11A+ YE = 0 (5.10b)
where
XE = BCiPT12 + P12C
Ti B
T + BDiDTi B
T (5.11a)
YE = CTBTo Q
T12 +Q12BoC + CTDT
o DoC. (5.11b)
Diagonalizing the weighted Gramians {P11, Q11} yields
T−1E P11T
−TE = T T
EQ11TE = diag(�1, �2, . . . , �r, �r+1, . . . , �n) (5.12a)
where �1 ≥ �2 ≥ . . . ≥ �r > �r+1 ≥ . . . ≥ �n > 0. Transforming and partitioning the
original system realization, we have
⎡
⎢
⎣
T−1E ATE T−1
E B
CTE D
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A11 A12 B1
A21 A22 B2
C1 C2 D
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (5.13a)
129
Enns’ reduced order model is then given by GE(s) = {A11, B1, C1, D}.
Essentially, Enns’ technique is based on diagonalizing simultaneously the solutions
of Lyapunov equations as given in (5.10). However, Enns’ technique cannot guarantee
the stability of reduced order models as XE and YE may not be positive semidefinite.
Several modifications to Enns’ technique are proposed in the literature to overcome
the stability problem [40, 69, 77].
5.2.2.2 Wang et al.’s Technique
The stability problem in Enns’ technique is solved in Wang et al.’s technique [77]
by making the matrices XE and YE positive semidefinite. In this technique, new
controllability and observability Gramians are obtained as the solutions to Lyapunov
equations
APW + PWAT +BWBTW = 0 (5.14a)
ATQW +QWA+ CTWCW = 0 (5.14b)
are diagonalized. The matrices BW and CW in the above Lyapunov equations are
fictitious input and output matrices determined from:
BW = UB ∣SB∣1
2 (5.15a)
CW = ∣ZC ∣1
2 V TC (5.15b)
130
The terms on RHS of the above equations, UB, SB, ZC and VC are obtained from the
following orthogonal eigen decomposition of symmetric matrices:
XE = UBSBUTB (5.16a)
YE = VCZCVTC . (5.16b)
Since
XE ≤ BWBTW ≥ 0 (5.17a)
YE ≤ CTWCW ≥ 0 (5.17b)
and {A,BW , CW} is minimal, stability of reduced order models in the case of double-
sided weighting is also guaranteed.
5.2.2.3 Lin and Chiu’s Technique
Another modification to Enns’ technique were proposed by Lin and Chiu [40], and
its generalization in [69]. In this method [69], instead of diagonalizing Gramians P11
and Q11, the following Gramians
PLC = P11 − P12P−122 P T
12 (5.18a)
QLC = Q11 −Q12Q−122 Q
T12 (5.18b)
are simultaneously diagonalized.
Note that the Gramians P11 − P12P−122 P T
12 and Q11 − Q12Q−122 Q
T12 are the Schur
complements of the (1,1) blocks of the matrices P and Q and satisfy the following
131
Lyapunov equations:
APLC + PLCAT +BLCB
TLC = 0
ATQLC +QLCA+ CTLCCLC = 0
where
BLC = BDi − P12P−122 Bi (5.19a)
CLC = DoC − CoQ−122 Q
T12. (5.19b)
Since the realization {A,BLC , CLC} is minimal and the Gramians diagonalized satisfy
the Lyapunov equations, Lin and Chiu’s technique yields stable models in the case of
double-sided weighting.
5.2.3 Heydari and Pedram’s Technique
Recently, a passivity-preserving FWMOR technique (namely, spectrally weighted bal-
anced truncation (SBT)) has been proposed in [30]. The method is an extension of
PR-TBR [53] (as discussed in Section 5.2.1) to include frequency weightings.
By substituting the augmented systems (5.7a) and (5.7b) in the Lur’e equations
(5.4) and (5.5) respectively, controllability PL ∈ ℜN,N and observability QL ∈ ℜN,N
Gramians of the augmented systems are obtained by solving the following Lur’e equa-
tions:
AiPL + PLAT
i = −Kc,iKT
c,i (5.20a)
PLCT
i −Bi = −Kc,iJTc,i (5.20b)
Jc,iJTc,i = Di +D
T
i (5.20c)
132
AT
o QL +QLAo = −KT
o,oKo,o (5.21a)
QLBo − CT
o = −KT
o,oJo,o (5.21b)
JTo,oJo,o = Do +D
T
o (5.21c)
Note that, although the technique [30] was proposed for strictly proper original sys-
tem, the derivation presented here is generalized to include proper weights and the
original system.
In literature, there are a number of methods to solve the Lur’e equations [79, 80,
5, 57, 78]. When Ri = Di +DT
i > 0 and Ro = Do +DT
o > 0, the solutions of these
equations are generally obtained by solving algebraic Riccati equations (AREs) as
discussed in [57, 79, 80, 78]. Also, where the matrices Ri and Ro are singular or zeros,
the Gramians PL and QL can be computed by using the algorithm such as in [57].
Equations in (5.20) and (5.21) can be written as the following AREs respectively:
(Ai − BiR−1i C i)PL + PL(Ai −BiR
−1i C i)
T + PLCT
i R−1i C iPL + BiR
−1i B
T
i = 0
(Ao −BoR−1o Co)
TQL +QL(Ao − BoR−1o Co) +QLBoR
−1o B
T
o QL + CT
o R−1o Co = 0
Solving the above AREs, we obtain the frequency weighted Gramians PL and QL
for the augmented systems. Since Ri and Ro ∈ ℜm,m, then other matrices can be
obtained as follows:
Jc,i ∈ ℜm,i = R1
2
i V (5.22a)
Jo,o ∈ ℜj,m = UR1
2
o (5.22b)
Kc,i ∈ ℜN,i = (Bi − PLCT
i )R−
1
2
i V (5.22c)
Ko,o ∈ ℜj,N = UR−
1
2
o (Co −BT
o QL) (5.22d)
133
where V and U are arbitrary real orthogonal and not necessarily square matrices; i.e.,
V V T = UTU = Im. The number of columns i (for Jc,i and Kc,i), and number of rows
j (for Jo,o and Ko,o) must be ≥ m such that Jc,iJTc,i and JT
o,oJo,o, have rank equal to
m. Thus, all matrices in (5.22) are not unique.
The matrices PL, QL, Kc,i and Ko,o can be partitioned as follows:
PL =
⎡
⎢
⎣
P11 P12
P T12 P22
⎤
⎥
⎦QL =
⎡
⎢
⎣
Q11 Q12
QT12 Q22
⎤
⎥
⎦Kc,i =
⎡
⎢
⎣
Kc1
Kc2
⎤
⎥
⎦Ko,o =
[
Ko1 Ko2
]
(5.23)
Defining new variables
X = BCiPT12 + P12C
Ti B
T +Kc1KTc1 (5.24a)
Y = CTBTo Q
T12 +Q12BoC +KT
o1Ko1 (5.24b)
Expanding the (1, 1) blocks of (5.20) and (5.21) yield the following:
AP11 + P11AT = −X (5.25a)
P11CT − B = −Kc1J
Tc,i (5.25b)
Jc,iJTc,i = Di +D
T
i (5.25c)
ATQ11 +Q11A = −Y (5.26a)
Q11B − CT = −KTo1Jo,o (5.26b)
JTo,oJo,o = Do +D
T
o (5.26c)
where B = BDi − P12CTi D
T and C = DoC −DTBTo Q
T12.
134
To guarantee passivity of reduced-order models, [30] replaced the indefinite sym-
metric matrices X and Y , which can be decomposed into UxSxUTx and VyZyV
Ty re-
spectively, by fictitious input and output matrices obtained from
Kc,1KT
c,1 = Ux ∣Sx∣UTx
KT
o,1Ko,1 = Vy ∣Zy∣VTy
Suppose that rank(X) = i and rank(Y ) = j, where 1 ≤ i, j ≤ n, we can write
Kc,1 = Uxdiag(
∣sx,1∣1/2 , . . . ∣sx,i∣
1/2 , 0 . . . , 0)
(5.27a)
Ko,1 = diag(
∣zy,1∣1/2 , . . . ∣zy,j∣
1/2 , 0 . . . , 0)
V Ty (5.27b)
After the matrices Kc,1 and Ko,1 are obtained, Heydari and Pedram [30] didn’t
explain in details the implication of replacing the matrices X and Y with the positive
semidefinite Kc,1KT
c,1 and KT
o,1Ko,1 in solving the frequency weighted Gramians.
Using the matrices Kc,1 and Ko,1 defined in (5.27), we can rewrite (5.25) and
(5.26) as follows:
APSBT + PSBTAT = −Kc,1K
T
c,1 (5.28a)
PSBTCT − B = −Kc,1J
T
c,i (5.28b)
J c,iJT
c,i = Di +DT
i (5.28c)
ATQSBT +QSBTA = −KT
o,1Ko,1 (5.29a)
QSBTB − CT = −KT
o,1Jo,o (5.29b)
JT
o,oJo,o = Do +DT
o (5.29c)
135
Refer to (5.28) and (5.29), in standard Lur’e equations, the matrices Kc,1KT
c,1 and
KT
o,1Ko,1 are normally unknown but not in this case. Therefore using the computed
matrices Kc,1KT
c,1 and KT
o,1Ko,1, frequency weighted Gramians PSBT and QSBT can
be solved from (5.28a) and (5.29a) respectively, using any Lyapunov equation solver.
Comparing equations (5.25b) and (5.26b) with equations (5.28b) and (5.29b) respec-
tively, note that P11 and Q11 have been changed to PSBT and QSBT . To make sure
the Lur’e equations are satisfied, the matrices B and C have to be replaced by B and
C which can be computed from the following equations:
B = PSBTCT +Kc,1J
T
c,i
C = BTQSBT + JT
o,oKo,1
where matrices J c,i and Jo,o can be obtained as follows:
J c,i =
[
Jc,i 0
]
Jo,o =
⎡
⎢
⎣
Jo,o
0
⎤
⎥
⎦(5.30)
The zero blocks in the definition of (5.30) are chosen such that J c,i and Kc,1 have the
same number of columns, and Jo,o and Ko,1 have the same number of rows.
Partitioning Kc,1 and Ko,1, we have:
Kc,1 =
[
Kc11 Kc22
]
Ko,1 =
⎡
⎢
⎣
Ko11
Ko22
⎤
⎥
⎦
where Kc11 ∈ ℜn,m, Kc22 ∈ ℜn,i−m, Ko11 ∈ ℜm,n and Ko22 ∈ ℜj−m,n.
The corresponding AREs can be written as follows:
AiPSBT + PSBT ATi + PSBTC
TR−1i CPSBT + BR−1
i BT = −Kc22KTc22
ATo QSBT +QSBT Ao +QSBTBR−1
o BTQSBT + CTR−1o C = −KT
o22Ko22
where Ai = A− BR−1i C and Ao = A−BR−1
o C.
136
5.3 Main Results
In this section, the three FWMOR techniques (Enns’, Wang et al.’s and Lin and
Chiu’s) are generalized to include passivity in addition to stability. In the passivity-
preserving FWMOR algorithms presented here, they differ from the existing tech-
niques, in the way the frequency-weighted Gramians matrices (P,Q) are computed.
5.3.1 Positive-Real Enns’ Technique
Similar to standard Enns’ technique [19], the positive-real Enns’ method proposed
here to retain passivity also takes P11 and Q11 as the frequency weighted Gramians
for the original system. Define
Φ = P12CTi B
T + BCiPT12 (5.31a)
� = CTBTo Q
T12 +Q12BoC (5.31b)
Theorem 12 A reduced order model obtained from the positive-real Enns’ method
is guaranteed to be passive if and only if matrices Φ and � in (5.31) are positive
semidefinite.
Proof 8 Assuming Φ = ΦΦT and � = �T�, then equations (5.25) and (5.26) can be
rewritten as:
AP11 + P11AT = −(Kc1K
Tc1 + ΦΦT ) = −Kc,1K
T
c,1 (5.32a)
P11CT − B = −Kc,1J
T
c,i (5.32b)
J c,iJT
c,i = Di +DT
i (5.32c)
137
ATQ11 +Q11A = −(KTo1Ko1 + �T�) = −K
T
o,1Ko,1 (5.33a)
Q11B − CT = −KT
o,1Jo,o (5.33b)
JT
o,oJo,o = Do +DT
o (5.33c)
where
Kc,1 =
[
Kc1 Φ
]
Ko,1 =
⎡
⎢
⎣
Ko1
�
⎤
⎥
⎦(5.34)
Note that the second and third equations of (5.32) and (5.33) actually remain un-
changed from the corresponding equations in (5.25) and (5.26).
The corresponding AREs [5] of (5.32) and (5.33) can be written as follows:
AiP11 + P11ATi + P11C
TR−1i CP11 + BR−1
i BT = −ΦΦT (5.35a)
ATo Q11 +Q11Ao +Q11BR−1
o BTQ11 + CTR−1o C = −�T� (5.35b)
where Ai = A−BR−1i C and Ao = A−BR−1
o C. The positive semidefinite matrices P11
and Q11 satisfy (5.35) if and only if Φ and � are positive semidefinite. A step-by-step
algorithm of this technique is given below:
Algorithm 4 Positive-Real Enns’ Technique
1. Solve (5.20) for PL and (5.21) for QL.
2. Partition PL and QL as in (5.23), the frequency weighted Gramians of the
system are defined as P = P11 and Q = Q11.
3. Find the transformation matrix, T to diagonalize the Gramians:
T−1PT−T = T TQT = diag {�1, �2, . . . , �n} (5.36)
138
4. Compute the frequency weighted balanced realizations
⎡
⎢
⎣
T−1AT T−1B
CT D
⎤
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A11 A12 B1
A21 A22 B2
C1 C2 D
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
5. An rtℎ order model can be obtained as Hr(s) = {A11, B1, C1, D}.
6. Calculate the weighted error =
∥Wo(s)(H(s)−Hr(s))Wi(s)∥∞ (5.37)
Remark 60 If Di = Do = 0, the condition for passivity is the same as in Theorem
12.
Remark 61 If matrices Φ and � are indefinite, Lyapunov equations in (5.25a) and
(5.26a) and Lur’e equations (5.25) and (5.26) may not be satisfied, which may result
in instability and nonpassivity of the reduced order models, respectively.
5.3.2 Positive-Real Wang et al.’s Technique
Positive-real of Wang et al.’s method are proposed to rectify the stability and pas-
sivity problems in the positive-real Enns’ technique by guaranteeing the stability and
passivity of reduced order models. The methods (similar to the standard Wang et
al.’s method [77]) are based on replacing X and Y in (5.25a) and (5.26a) respectively
with guaranteed positive semidefinite matrices Kc,1KT
c,1 and KT
o,1Ko,1.
139
Note that, in SBT method [30] discussed in Section 5.2.3, the matrices X and
Y have been replaced by Kc,1KT
c,1 and KT
o,1Ko,1. If we refer to (5.24), the matrices
Kc1KTc1 and KT
o1Ko1 are already positive semidefinite, thus, instead of computing
eigenvalue decomposition of X and Y as in SBT, we compute the corresponding
eigenvalue decomposition for symmetric matrices Φ = UΦSΦUTΦ and � = V�Z�V
T� as
defined in (5.31). We define the posivite semidefinite matrices ΦwΦTw and �Tw�w as
shown below:
ΦwΦTw = UΦ ∣SΦ∣U
TΦ (5.38a)
�Tw�w = V� ∣Z�∣VT� (5.38b)
Suppose that rank(X) = i and rank(Y ) = j, where 1 ≤ i, j ≤ n, we can write
Φw = UΦdiag(
∣sΦ,1∣1/2 , . . . ∣sΦ,i∣
1/2 , 0 . . . , 0)
�w = diag(
∣z�,1∣1/2 , . . . ∣z�,j∣
1/2 , 0 . . . , 0)
V T�
The Lur’e equations for the new positive-real Wang et al.’s method can be written
as follows:
APw + PwAT = −Kc,1K
T
c,1 (5.39a)
PwCT − B = −Kc,1J
T
c,i (5.39b)
J c,iJT
c,i = Di +DT
i (5.39c)
ATQw +QwA = −KT
o,1Ko,1 (5.40a)
QwB − CT = −KT
o,1Jo,o (5.40b)
JT
o,oJo,o = Do +DT
o (5.40c)
140
Once we calculate the matrices Kc,1KT
c,1 and KT
o,1Ko,1, we can solve for Pw and
Qw from (5.39a) and (5.40a) using any Lyapunov equation solver as in SBT. The
corresponding AREs are:
(A− BR−1i C)Pw + Pw(A− BR−1
i C)T + PwCTR−1
i CPw + BR−1i BT = −ΦwΦ
Tw
(A−BR−1o C)TQw +Qw(A− BR−1
o C) +QwBR−1o BTQw + CTR−1
o C = −�Tw�w
Similar to SBT, the matrices B and C can be computed from the following equations:
B = PwCT +Kc,1J
T
c,i
C = BTQw + JT
o,oKo,1
where matrices J c,i and Jo,o can be obtained from (5.30).
Theorem 13 A reduced order model obtained from the positive-real Wang et al.’s
method is guaranteed to be passive.
Proof 9 The positive-real Wang et al.’s method satisfy the Lur’e equations and thus
is guaranteed to yield passive models.
Remark 62 For posivite semidefinite Φ and �, the positive-real Enns’, SBT and the
new positive-real Wang et al.’s methods are identical. The methods yield the same
pairs of P and Q and hence are guaranteed to yield passive models.
Remark 63 For Di = Do = 0, the reduced order model obtained is also guaranteed
to be passive.
A step-by-step algorithm of the positive-real of Wang et al.’s technique is as fol-
lows:
141
Algorithm 5 Positive-Real Wang et al.’s Method
1. Solve (5.20) for PL and (5.21) for QL.
2. Compute the matrices Kc,1 and Ko,1 from (5.34).
3. Solve for P = Pw and Q = Qw.
4. Follow Algorithm 4 (Steps 3-6)
5.3.3 Positive-Real Lin and Chiu’s Technique
The positive-real Lin and Chiu’s technique is based on block diagonalization of the
Gramians for the augmented systems to obtain the frequency weighted Gramians of
the original system.
Since the chosen weighting functionsWi(s) = {Ai, Bi, Ci, Di} andWo(s) = {Ao, Bo, Co, Do}
are assumed to be minimal and stable, P22 and Q22 will be positive definite due to
controllability and observability of {Ai, Bi} and {Ao, Co} respectively. Let transfor-
mation matrices required to block diagonalize the Gramians be:
Ti =
⎡
⎢
⎣
I P12P−122
0 I
⎤
⎥
⎦, To =
⎡
⎢
⎣
I 0
−Q−122 Q
T12 I
⎤
⎥
⎦
The block diagonalized Gramians are
Pi = T−1i PLT
−Ti =
⎡
⎢
⎣
P 0
0 P22
⎤
⎥
⎦, Qo = T T
o QLTo =
⎡
⎢
⎣
Q 0
0 Q22
⎤
⎥
⎦
where P = P11 − P12P−122 P T
12 and Q = Q11 − Q12Q−122 Q
T12. The corresponding state-
142
space realizations of the new augmented systems are:
Hi(s) =
⎡
⎢
⎣
Ai Bi
Ci Di
⎤
⎥
⎦=
⎡
⎢
⎣
T−1i AiTi T−1
i Bi
C iTi Di
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A Ai,12 B
0 Ai Bi
C C2 Di
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Ho(s) =
⎡
⎢
⎣
Ao Bo
Co Do
⎤
⎥
⎦=
⎡
⎢
⎣
T−1o AoTo T−1
o Bo
CoTo Do
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A 0 B
ATo,12 Ao B2
C Co Do
⎤
⎥
⎥
⎥
⎥
⎥
⎦
where
Ai,12 = AP12P−122 + BCi − P12P
−122 Ai
B = BDi − P12P−122 Bi
C2 = CP12P−122 +DCi
ATo,12 = Q−1
22 QT12A+ BoC − AoQ
−122 Q
T12
B2 = Q−122 Q
T12B + BoD
C = DoC − CoQ−122 Q
T12
The new realizations of augmented systems now satisfy the following Lur’e equations:
143
AiPi + PiATi = −Kc,iK
Tc,i (5.41a)
PiCTi − Bi = −Kc,iJ
Tc,i (5.41b)
Jc,iJTc,i = Di +D
T
i (5.41c)
ATo Qo +QoAo = −KT
o,oKo,o (5.42a)
QoBo − CTo = −KT
o,oJo,o (5.42b)
JTo,oJo,o = Do +D
T
o (5.42c)
where Kc,i = (Bi − PiCTi )R
−1
2
i V and Ko,o = UR−
1
2
o (Co − BTo Qo). Expanding (1,1)
entries of (5.41) and (5.42), we have:
AP + PAT = −KcKTc (5.43a)
PCT − B = −KcJTc,i (5.43b)
Jc,iJTc,i = Di +D
T
i (5.43c)
AT Q+ QA = −KTo Ko (5.44a)
QB − CT = −KTo Jo,o (5.44b)
JTo,oJo,o = Do +D
T
o (5.44c)
where Kc = (B − PCT )R−
1
2
i V , Ko = UR−
1
2
o (C − BT Q) and the matrices Jc,i, Jo,o
can be obtained from (5.22). The corresponding AREs for the Lur’e equations can
144
be written as follows:
(A− BR−1i C)P + P (A− BR−1
i C)T + PCTR−1i CP + BR−1
i BT = 0
(A−BR−1o C)T Q+ Q(A−BR−1
o C) + QBR−1o BT Q+ CTR−1
o C = 0
Theorem 14 A reduced order model obtained from the positive-real Lin and Chiu’s
method is guaranteed to be passive.
Proof 10 Since the Lur’e equations (5.43) and (5.44) are satisfied, passivity of the
models are guaranteed.
A step-by-step algorithm of this method is given below:
Algorithm 6 Positive-Real Lin and Chiu’s Method
1. Solve (5.20) for PL and (5.21) for QL.
2. Partition PL and QL as in (5.23), the frequency weighted Gramians of the
system can be computed from P = P11−P12P−122 P T
12 and Q = Q11−Q12Q−122 Q
T12.
3. Follow Algorithm 4 (Steps 3 - 6)
Remark 64 Positive-real Lin and Chiu’s method can also be easily shown to yield
passive models for augmented systems with Di = Do = 0.
Remark 65 Similar to standard frequency weighted balanced truncation techniques,
all the proposed positive-real frequency weighted methods can be extended to double-
sided easily.
145
5.4 Simulation Results
In this section, four practical examples are given to show validity and the effectiveness
of the proposed algorithms using different weighting functions.
Example 5 In this example, we consider the interconnect circuit described in Section
5.1 (Figure 5.1) while applying the proposed positive-real FWMOR techniques.
The passivity of the models from the proposed techniques are confirmed using
Hamiltonian Theorem 2 of [60] by finding the eigenvalues of the Hamiltonian matrices,
and no imaginary eigenvalues were found indicating that the reduced-order models
are passive.
For further illustration, Nyquist plots for the original system and reduced-order
models are shown Figure 5.3. From the figure, it can be seen that the proposed
reduced models are passive as the Nyquist plots lie entirely in the right-half of the
complex plane.
Example 6 In this example, a two port network from [79, 80] as shown in Figure 5.4
is analyzed. In this experiment, the system with order n = 2000 is reduced to models
of order r = 10 using the proposed techniques. The parameters used are R = 0.1Ω,
G = 1℧, C = 0.1F and L = 0.1H. A transfer-function as in (5.45) which is a high-
pass filter (HPF) is chosen as input and output weighting functions in this example.
Note that I2 denotes an identity matrix of order 2.
Wi(s) = Wo(s) =s+ 0.01
s+ 0.05I2 (5.45)
146
−1 −0.5 0 0.5 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Real Axis
Imag
inar
y A
xis
OriginalPR−EnnsPR−Wang et al.PR−Lin and Chiu
(a) Normal view
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
Real AxisIm
agin
ary
Axi
s
OriginalPR−EnnsPR−Wang et al.PR−Lin and Chiu
(b) Enlarged view
Figure 5.3: Illustration of passivity behaviour via Nyquist plots for Example 5
The passivity of the models are confirmed using Hamiltonian Theorem 2 of [60]
by finding the eigenvalues of the Hamiltonian matrices, and no imaginary eigenvalues
were found indicating that the models are passive.
For the purpose of illustration, passivity verification is also illustrated by plotting
eigenvalues of the Re(Y(s)) in Figure 5.5 versus frequency at discrete frequency points.
As seen from the graph, both the eigenvalues are above zero thus confirming the
passivity of the models.
Let �max[H] denote the maximum singular value of H. The approximation errors
for this example are plotted in Figure 5.6. The dashed line is the cutoff frequency at
f1 = 0.05 rad/s and the passband for the chosen HPF is all frequencies to the right
of the dashed line (> f1). The figure clearly demonstrates the merits of the proposed
147
−
C
L
G
−
I2+ Port 2
V2
R
I1Port 1 +
V1
Figure 5.4: itℎ section of a two port RLC network
0 200 400 600 800 10000
1
2
3
4
5
6
7
8
9
10
Frequency ω [rad/s]
1st E
igen
valu
e
PR−Wang et al.PR−EnnsPR−Lin and Chiu
(a) 1st Eigenvalue
0 200 400 600 800 10000
1
2
3
4
5
6
7
8
9
10
Frequency ω [rad/s]
1st E
igen
valu
e
PR−Wang et al.PR−EnnsPR−Lin and Chiu
(b) 2nd Eigenvalue
Figure 5.5: Eigenvalues of Real(Y(s)) for Example 6
148
positive-real FWMOR algorithms wherein the reduction errors from the proposed
methods are lower than the unweighted model-order reduction method (PR-TBR)
when the frequencies are in the region of interest f1 < f < 1 rad/s.
10−3
10−2
10−1
100
101
102
103
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Frequency ω [rad/s]
σ max
[ Wo (
jω)
(H (
jω)
− H
r (jω
)) W
i (jω
)]
PR−TBRPR−EnnsPR−Wang et al.PR−Lin and ChiuSBT
Figure 5.6: Approximation errors for Example 6
Example 7 We consider a single port lumped RLC network of order n = 2000 as
shown in Figure 5.7 with the following values for circuit parameters: Ri = 0.1Ω,
Gi = 10Ω, Ci = 0.01F , Li = 0.01H and i = 1, 2, 3 . . . 1000. The system is then
reduced to order r = 10. Here we use a low-pass filter (LPF) as the input weighting
function with a transfer-function given by:
Wi(s) =10
s+ 10(5.46)
Note that, in this example, Di = 0. The passivity of the reduced-order models
are confirmed using Hamiltonian Theorem 2 of [60] by finding the eigenvalues of the
149
+−Vin(t)
Iin(t)
R1
L1
G1C1
R1000
L1000
Vout
G1000C1000
Figure 5.7: A lumped RLC circuit for Example 7
Hamiltonian matrices, and no imaginary eigenvalues were found indicating that the
models are passive. For further illustration, Nyquist plots for the original system and
reduced-order models are shown in Figure 5.8(a). From the figure, it can be seen that
the proposed reduced-order models are passive as the Nyquist plots lie entirely in the
right-half of the complex plane.
−8 −6 −4 −2 0 2 4 6 8 10−6
−4
−2
0
2
4
6
Real Axis
Imag
inar
y A
xis
OriginalPR−TBRPR−EnnsPR−Wang et al.PR−Lin and ChiuSBT
(a) Nyquist plots
10−3
10−2
10−1
100
101
102
103
0
1
2
3
4
5
6
7
8x 10
−5
Frequency ω [rad/s]
σ [
(H (
jω)
− H
r (jω
)) W
i (jω
)
PR−TBRPR−EnnsPR−Wang et al.PR−Lin and ChiuSBT
(b) Approximation erros
Figure 5.8: Nyquist plots and approximation errors for Example 7
150
The results for approximation errors are shown in Figure 5.8(b). The cutoff fre-
quency for the chosen LPF is indicated as a dashed line which is f2 = 10 rad/s. Thus,
the passband is all frequencies to the left of the dashed line. It can be observed clearly
that all the proposed positive-real FWMOR techniques in this paper perform better
than the unweighted method (PR-TBR) in reducing the approximation errors in the
frequency band of interest.
Example 8 We consider again the circuit of Example 3 with the circuit parameters
are Ri = 0.1Ω, Gi = 1℧, Ci = 0.1F , and Li = 0.1H. In this example, we use a HPF
(5.45) (without I2) as the input weighting function and a LPF (5.46) as the output
weighting function. For the double-sided case, the combination of these weighting
functions yield a band-pass filter (BPF).
The passivity of the reduced-order models are confirmed using Hamiltonian Theo-
rem 2 of [60] by finding the eigenvalues of the Hamiltonian matrices, and no imaginary
eigenvalues were found indicating that the models are passive. For further illustra-
tion, Nyquist plots for the original system and reduced-order models are shown in
Figure 5.9(a). From the figure, it can be seen that the proposed reduced models are
passive as the Nyquist plot lie entirely in the right-half of the complex plane.
Approximation errors for this example are shown in Figure 5.9(b). The passband
frequency is 0.05 rad/s< f3 < 10 rad/s which is shown by the dashed lines in the
figure. The proposed methods provide better error rate compared to the unweighted
method (PR-TBR).
151
−8 −6 −4 −2 0 2 4 6 8 10−6
−4
−2
0
2
4
6
Real Axis
Imag
inar
y A
xis
OriginalPR−TBRPR−EnnsPR−Wang et al.PR−Lin and ChiuSBT
(a) Nyquist Plots
10−2
10−1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−4
Frequency ω [rad/s]
σ[ W
o (jω
) (H
(jω
) −
Hr (
jω))
Wi (
jω)]
PR−TBRPR−EnnsPR−Wang et al.PR−Lin and ChiuSBT
(b) Approximation errors
Figure 5.9: Nyquist plots and approximation errors for Example 8
5.5 Conclusion
Three frequency weighted model order reduction methods (Enns’, Wang et al.’s and
Lin and Chiu’s) are generalized to include passivity in addition to stability. The
derivations and simulations show that positive-real Lin and Chiu’s and positive-real
Wang et al.’s methods are guaranteed to yield passive models for passive original
systems. Necessary conditions are developed to ensure the passivity of reduced models
from the positive-real Enns’ method, the passivity of models are guaranteed only
under some conditions. From the simulations, it was observed that all the proposed
positive-real FWMOR methods yield significantly better error performance compared
to the PR-TBR method in reducing the reduction errors using all the chosen filters.
152
Chapter 6
Conclusions
6.1 Overview of the Thesis
This thesis has investigated the frequency weighted model reduction problem. Both
stable and passive systems were considered.
In Chapter 2, a review of several passivity preserving balanced truncation and
frequency weighted model reduction methods was presented. Problems involving pas-
sivity preserving model reduction techniques and frequency weights were considered.
Several critical remarks about the different methods were given.
In Chapter 3, two frequency weighted balanced truncation algorithms based on
zero cross-terms were proposed. The first method is applicable to special weighting
functions (single-sided and strictly proper). It does not depend on the free parameter
to reduce the reduction error. Furthermore, the approximation error reduction achiev-
able is significant as illustrated in the numerical examples. In the second method, a
modification to Sreeram and Sahlan’s method [70] yield a better error reduction over
153
the existing techniques [40, 19, 69, 77, 76, 70] by varying user-chosen free parameters.
In Chapter 4, a parameterized method which is based on partial fraction expansion
technique [68] is presented. This method has the following advantages: (i) guaranteed
stability of models in the case of double-sided weightings (unlike the well-know Enns’
technique [19]), (ii) simple, elegant and easily computable error bounds, (iii) appli-
cability to both continuous and discrete systems, (iv) choice of anti-stable (as well
as stable) weighting functions, (v) a choice of free parameters to reduce the weighted
error and error bounds, and (vi) easy applicability to controller reduction problems
(unlike the technique of Lin and Chiu [40, 69]).
In Chapter 5, the three frequency weighted balanced truncation techniques (Enns’,
Lin and Chiu’s, and Wang et al.’s) are generalized to include passivity in addition to
stability. The derivation and simulations show that positive-real Lin and Chiu’s, and
positive-real Wang et al.’s techniques are guaranteed to yield passive models for pas-
sive original systems. Necessary conditions are developed to ensure the passivity of
reduced order models from the Enns’ method. From the simulations, it was observed
that the proposed positive-real frequency weighted balanced truncation techniques
yield significantly better error performance compared to the passivity preserving un-
weighted method (PR-TBR [53]).
6.2 Future Work
In this section, we propose the following topics for future research that may further
enhance and extend the research results described in this thesis.
∙ Similar to methods proposed in [77, 58, 70], the new method 2 in Chapter 3 and
154
the method proposed in Chapter 4 are realization dependent. Which realization
of the new original system can produce lower approximation error remains an
open question.
∙ In the proposed parameterized frequency weighted model reduction schemes
(the new method 2 in Chapter 3, and Chapter 4), it is not clear what values of
� and � give the best results. Further research is necessary to find the optimum
values of parameters � and �.
∙ The new method 1 proposed in Chapter 3 is applicable only to special weight
(single-sided and strictly proper). Further investigation could remove the limi-
tations, while remaining its capability in reducing the reduction error.
∙ Similar to Sreeram and Sahlan’s technique [70], the new weights in the new
method 2 are not inner/co-inner functions. Factorization of the parameters in
Lin and Chiu’s technique [69] to satisfy all properties of inner/co-inner func-
tions, which theoretically may give good approximation errors, remains an open
problem.
∙ In Chapter 5, for different weighting functions yield different results. A system-
atic way of choosing weighting functions to minimize the approximation errors
needs further investigation.
∙ Derivation of error bounds for the passivity preserving frequency weighted bal-
anced truncation methods (Chapter 5) can be very useful. A further investi-
gation is needed to find whether the error bounds exist for all the proposed
algorithms.
155
∙ The RLC system considered in Chapter 5 is normally represented in descriptor
form. The methods proposed in the chapter are based on state-space formula-
tion. It is interesting to see whether the proposed techniques can be extended
to handle descriptor form representation.
156
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Appendixes
Appendix A: Transforming Lur’e equation to ARE
Consider a positive-real system represented by a state-space below:
x = Ax+ Bu (1a)
y = Cx+Du (1b)
where A ∈ ℜn,n, B ∈ ℜn,m, C ∈ ℜm,n and D ∈ ℜm,m. The system satisfies the
following Lur’e equation:
ATQ+QA = −KTo Ko (2a)
QB − CT = −KTo Jo (2b)
JTo Jo = D +DT . (2c)
For a positive-real system with D + DT > 0, the Lur’e equation in (2) can be
solved using the following algebraic Riccati equation (ARE):
ATQ+QA+QB(D +DT )−1BTQ+ CT (D +DT )−1C = 0 (3)
where A = A− B(D +DT )−1C.
170
The ARE in (3) can be obtained from the Lur’e equation (2) as follows: Let define
R = D +DT . Note that R is a symmetric matrix, then Jo = R1
2 . Now consider (2b)
which can be written as follows:
QB − CT = −KTo R
1
2
(QB − CT )R−1
2 = −KTo
R−1
2 (BTQ− C) = −Ko
Now consider (2a) which can be written as follows:
ATQ+QA+KTo Ko = 0
ATQ+QA+ (QB − CT )R−1
2R−1
2 (BTQ− C) = 0
ATQ+QA+ (QB − CT )R−1(BTQ− C) = 0
ATQ+QA+QBR−1BTQ+ CTR−1C −QBR−1C − CTR−1BTQ = 0
(AT − CTR−1BT )Q+Q(A−BR−1C) +QBR−1BTQ+ CTR−1C = 0
171
Appendix B: Solving Lur’e equation for D=0
This appendix shows that Lur’e equation can be solved for the case D = 0. Consider a
positive-real system represented by a state-space as in (1) of Appendix A with D = 0.
The system satisfies the following Positive Real Lemma [57]:
Lemma 6 ([57] Lemma 2.1) The transfer function G(s) with a minimal state space
description {A,B,C} is positive real if there exist real matrices Q ∈ ℜn,n > 0, Ko ∈
ℜm,n, such that the following three equivalent statements hold:
1. Matrices Q and Ko satisfy the following Lur’e eqn:
ATQ+QA = −KTo Ko (4a)
QB = CT (4b)
2. The transfer function H(s) = Ko(sI − A)−1B satisfies
G(s) +GT (−s) = HT (−s)H(s) (5)
3. For all x and u satisfying (1), the energy balanced for the system can be written
as:
d
dt
1
2xTQx+
1
2yT y = yTu, (6)
where y = Kox.
To compute Q and Ko in (4), first, the original system {A,B,C} are transformed
into{
A, B, C}
using the following transformation matrix T:
T ∈ ℜn,n =
[
B V
]
, T−1 =
⎡
⎢
⎣
X
V
⎤
⎥
⎦(7)
172
where V ∈ ℜn,n−m is chosen such that CV = 0 and V TV = I. Partitioning T−1,
the matrix X = (CB)−1C. Note that, it has been proven in [57] that T is always
invertible if CB is invertible. From equation (4b),
CB = BTQB > 0 (8)
it can be observed clearly that CB is a sysmmetric matrix and it is always invertible
for a positive real system.
Using similarity transformation matrix T, (1) can be transformed into:
˙x = Ax+ Bu (9a)
y = Cx (9b)
where
B =
⎡
⎢
⎣
Im
0
⎤
⎥
⎦C =
[
C1 0
]
and C1 = CB. The new realization{
A, B, C}
satisfies the following Lur’e equation:
AT Q+ QA = −KTo Ko (10a)
QB = CT (10b)
where Q = T TQT , Ko = KoT .
173
Expanding (10b)
QB = CT
⎡
⎢
⎣
Q11 Q12
QT12 Q22
⎤
⎥
⎦
⎡
⎢
⎣
I
0
⎤
⎥
⎦=
⎡
⎢
⎣
CT1
0
⎤
⎥
⎦
⎡
⎢
⎣
Q11
QT12
⎤
⎥
⎦=
⎡
⎢
⎣
CT1
0
⎤
⎥
⎦
where Q11 = CT1 = CB, and Q12 = 0. The matrices
{
Q, Ko
}
can be partitioned as
follows:
Q =
⎡
⎢
⎣
Q11 0
0 Q22
⎤
⎥
⎦Ko =
[
Ko1 Ko2
]
(11)
The corresponding energy balanced equation for the transformed realization{
A, B, C}
can be written as follows:
d
dt
1
2xT Qx+
1
2yTt yt = yTu; yt = Kox (12)
Expanding (9),⎡
⎢
⎣
˙x1
˙x2
⎤
⎥
⎦=
⎡
⎢
⎣
A11 A12
A21 A22
⎤
⎥
⎦
⎡
⎢
⎣
x1
x2
⎤
⎥
⎦+
⎡
⎢
⎣
I
0
⎤
⎥
⎦u
y =
[
CB 0
]
⎡
⎢
⎣
x1
x2
⎤
⎥
⎦
we can have:
˙x1 = A11x1 + A12x2 + u (13)
˙x2 = A21x1 + A22x2 (14)
y = CBx1 (15)
174
The input u in (13) can be expressed as:
u = ˙x1 − A11x1 − A12x2 (16)
Define y as follows:
y = CBu− y (17)
= CB( ˙x1 − A11x1 − A12x2)− y
= −CBA11x1 − CBA12x2 + CB ˙x1 − y
= −CBA11x1 − CBA12x2 + y − y
= −CBA11x1 − CBA12x2 (18)
From equations (14) and (18), a new system can be written as follows:
˙x2 = Ax2 + Bu (19a)
y = Cx2 + Du (19b)
where
A = A22 B = A21 C = −CBA12 D = −CBA11 u = x1 (20)
The new realization{
A, B, C, D}
is partial inverse of{
A, B, C, 0}
, with D+DT > 0
which has been proven in [57] is guaranteed for a positive real system. The new
realization satisfies:
AT Q+ QA = −KTo Ko (21a)
QB − CT = −KTo Jo (21b)
JoJo = D + DT (21c)
175
and its corresponding energy balanced equation is as follows:
d
dt
1
2xT2 Qx2 +
1
2yTℎ yℎ = uT y
d
dt
1
2xT2 Qx2 +
1
2
∥
∥
∥Kox2 + Jou
∥
∥
∥
2
= uT y (22)
Now, refer to (12) which can be written as follows:
d
dt
1
2xT Qx+
1
2
∥
∥
∥Kox
∥
∥
∥
2
= yTu
d
dt
1
2
[
xT1 xT
2
]
⎡
⎢
⎣
Q11 0
0 Q22
⎤
⎥
⎦
⎡
⎢
⎣
x1
x2
⎤
⎥
⎦+
1
2
∥
∥
∥
∥
∥
∥
∥
[
Ko1 Ko2
]
⎡
⎢
⎣
x1
x2
⎤
⎥
⎦
∥
∥
∥
∥
∥
∥
∥
2
= yTu
d
dt
1
2xT1 Q11x1 +
d
dt
1
2xT2 Q22x2 +
1
2
∥
∥
∥Ko1x1 + Ko2x2
∥
∥
∥
2
= yTu
xT1 Q11
˙x1 +d
dt
1
2xT2 Q22x2 +
1
2
∥
∥
∥Ko1x1 + Ko2x2
∥
∥
∥
2
= yTu
From (20), (15) and (17), we can rewrite the above equation as follows:
uTCB ˙u+d
dt
1
2xT2 Q22x2 +
1
2
∥
∥
∥Ko1u+ Ko2x2
∥
∥
∥
2
= (uTCB)((CB)−1y + (CB)−1y)
uTCB ˙u+d
dt
1
2xT2 Q22x2 +
1
2
∥
∥
∥Ko1u+ Ko2x2
∥
∥
∥
2
= uT y + uT y
uTCB ˙u+d
dt
1
2xT2 Q22x2 +
1
2
∥
∥
∥Ko1u+ Ko2x2
∥
∥
∥
2
= uT y + uTCB ˙u
d
dt
1
2xT2 Q22x2 +
1
2
∥
∥
∥Ko1u+ Ko2x2
∥
∥
∥
2
= uT y (23)
Comparing (23) and (22), we can see that:
Q = Q22 Ko = Ko2 Jo = Ko1
Solving (21) for Q, Ko, Jo will indirectly solve (10). Using the same similarity trans-
formation matrix T defined in (7), the matrices Q and Ko can be transformed back
to the original Q and Ko which are the solutions of (4).
176
Appendix C: Factorization Formula for Co-inner Func-
tion (Single-sided)
This appendix shows that the parameters in Lin and Chiu’s technique [40, 69] can be
factorized to be inner/co-inner functions for single-sided case with a condition. Con-
sider the following seven equations need to be satisfied by V (s) ={
Av, Bc, Cv, Dv
}
to be a co-inner function:
X23 = AX −XAv + BCv
= −(BDv −XBv)BTv P
−1v = B Cv (24a)
X2 = BDv −XBv = B Dv (24b)
Y2 = CX +DCv = D Cv (24c)
D = DDv = D Dv (24d)
0 = AvPv + PvATv + BvB
Tv (24e)
0 = CvPv +DvBTv (24f)
I = DvDT
v . (24g)
Remark 66 The first four of (24) can be rewritten as:⎡
⎢
⎣
X23 X2
Y2 D
⎤
⎥
⎦=
⎡
⎢
⎣
B
D
⎤
⎥
⎦
[
Cv Dv
]
Since all the parameters in the right hand side are unknown, the equation cannot be
solved directly.
Remark 67 To satisfy (24f), the matrices Cv and Dv are linearly dependent where
Cv = −DvBTv P
−1v .
177
Remark 68 Comparing X23 and X2, we also can see that these two matrices also
dependent each other by X23 = −X2BTv P
−1v .
Remark 69 Equation in Remark 66 can now be rewritten as follows:
⎡
⎢
⎣
−X2BTv P
−1v X2
Y2 D
⎤
⎥
⎦=
⎡
⎢
⎣
B
D
⎤
⎥
⎦
[
−DvBTv P
−1v Dv
]
=
⎡
⎢
⎣
−B DvBTv P
−1v B Dv
−D DvBTv P
−1v D Dv
⎤
⎥
⎦
Comparing both sides of the equation, the matrices Y2 and D need to be dependent by
Y2 = −DBTv P
−1v to ensure all the unknown matrices exist.
From Remark 69, we can say that the factorization of Lin and Chiu’s parameter
[69] can be factorized to be inner/co-inner funtion if Y2 = −DBTv P
−1v .
178
Appendix D: Factorization Formula for Inner/Co-
inner Functions (Double-sided)
This appendix shows that the parameters in Lin and Chiu’s technique [40, 69] can be
factorized to be inner/co-inner functions for double-sided case with conditions. Con-
sider the following fourteen equations need to be satisfied for V (s) ={
Av, Bv, Cv, Dv,}
and W (s) ={
Aw, Bw, Cw, Dw
}
to be inner/co-inner functions:
X12 = Y A− AwY + BwC
= −Q−1w CT
w (DwC − CwY ) = BwC (25a)
X23 = AX −XAv + BCv
= −(BDv −XBv)BTv P
−1v = B Cv (25b)
X2 = BDv −XBv = B Dv (25c)
Y1 = DwC − CwY = DwC (25d)
X13 = BwCX + Y AX + BwDCv + Y BCv − Y XAv = BwD Cv (25e)
X1 = BwDDv + Y BDv − Y XBv = BwD Dv (25f)
Y2 = DwCX +DwDCv = DwD Cv (25g)
D = DwDDv = DwD Dv (25h)
0 = AvPv + PvATv +BvB
Tv (25i)
0 = CvPv +DvBTv (25j)
I = DvDT
v (25k)
0 = ATwQw +QwAw + CT
wCw (25l)
179
0 = BT
wQw +DT
wCw (25m)
I = DT
wDw (25n)
Remark 70 The first eight of (25) can be rewritten as follows:
⎡
⎢
⎢
⎢
⎢
⎢
⎣
X12 X13 X1
Y1 Y2 D
0 X23 X2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Bw 0
Dw 0
0 B
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
C D Cv D Dv
0 Cv Dv
⎤
⎥
⎦
Since all the parameters in the right hand side are unknown, the equation cannot be
solved directly.
Remark 71 From (25j) and (25m), we can see that Cv and Bw are depend on Dv
and Dw respectively. Rewriting the equations, we have Cv = −DvBTv P
−1v and Bw =
−Q−1w CT
wDw.
Remark 72 The matrix X23 depends on X2, and X12 depends on Y1. Rewriting the
equations (25b) and (25a), we have X23 = −X2BTv P
−1v and X12 = −Q−1
w CTwY1.
Remark 73 Rewriting the equation in Remark 70 by substituting the correponding
180
matrices defined in the above remarks, we can have:
⎡
⎢
⎢
⎢
⎢
⎢
⎣
−Q−1w CT
wY1 X13 X1
Y1 Y2 D
0 −X2BTv P
−1v X2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
−Q−1w CT
wDw 0
Dw 0
0 B
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
C −D DvBTv P
−1v D Dv
0 −DvBTv P
−1v Dv
⎤
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
−Q−1w CT
wDwC Q−1w CT
wDwD DvBTv P
−1v −Q−1
w CTwDwD Dv
DwC −DwD DvBTv P
−1v DwD Dv
0 −B DvBTv P
−1v B Dv
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Comparing both sides of the equation, we can see that the matrices X13, X1, and Y2
must depend on D to guarantee the solutions exist.
From Remark 73, we can see that the parameters in (25) can be factorized into
inner/co-inner function if
X13 = Q−1w CT
wDBTv P
−1v (26)
X1 = −Q−1w CT
wD (27)
Y2 = −DBTv P
−1v (28)
181
Appendix E: The Modified Nodal Analysis (MNA)
Formulation
Refer to a two-port network (Fig. 3 of [60]) as shown below:
P1
R1 I1L1
C1
R2 L2
C2
R3 L3
C3
L4 R4
P2
Figure 1: A lumped RLC circuit
where R1 = 1Ω, L1 = 0.2nH, C1 = 10pF, R2 = 0.2Ω, L2 = 0.8nH, C2 = 2pF, R3 =
0.1Ω, L3 = 3nH, C3 = 1pF, L4 = 4nH, R4 = 25Ω.
To find Y11, voltage at P2 is zero. Then, we can have:
I1 = IL1
0 = I1 − IL1(29)
C1VC1= IL1
− IL2(30)
C2VC2= IL2
− IL3(31)
C3VC3= IL3
− IL4(32)
L1IL1= VP1
− VC1−R1IL1
(33)
L2IL2= VC1
− VC2−R2IL2
(34)
182
L3IL3= VC2
− VC3−R3IL3
(35)
L4IL4= VC3
−R4IL4(36)
0 = −VP1+ u (37)
We can arrange the above equations in a matrix form as follows:
183
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 0 0 0 0 0 0 0
0 C1 0 0 0 0 0 0 0
0 0 C2 0 0 0 0 0 0
0 0 0 C3 0 0 0 0 0
0 0 0 0 L1 0 0 0 0
0 0 0 0 0 L2 0 0 0
0 0 0 0 0 0 L3 0 0
0 0 0 0 0 0 0 L4 0
0 0 0 0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
VP1
VC1
VC2
VC3
IL1
IL2
IL3
IL4
I1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 0 0 −1 0 0 0 1
0 0 0 0 1 −1 0 0 0
0 0 0 0 0 1 −1 0 0
0 0 0 0 0 0 1 −1 0
1 −1 0 0 −R1 0 0 0 0
0 1 −1 0 0 −R2 0 0 0
0 0 1 −1 0 0 −R3 0 0
0 0 0 1 0 0 0 −R4 0
−1 0 0 0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
VP1
VC1
VC2
VC3
IL1
IL2
IL3
IL4
I1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
0
0
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
u
(38)
184
For Y11, y = I1. We can rewrite y in a matrix form as follows:
y =
[
0 0 0 0 0 0 0 0 1
]
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
VP1
VC1
VC2
VC3
IL1
IL2
IL3
IL4
I1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(39)
Equations (38) and (39) for admittance Y11 can be rewritten as:
C ˙x = −Gx+ Bu
y = Lx+ Du (40)
where D = 0.
For simplicity, consider the system in (40) as a single-input single-output (SISO)
system, and C ∈ ℜN,N with rank n, singular value decomposition of C can be written
as C = U
⎡
⎢
⎣
Σ 0
0 0
⎤
⎥
⎦V T . Define a matrix T =
⎡
⎢
⎣
Σ−1 0
0 0
⎤
⎥
⎦, (40) can be transformed
into
C ˙x = −Gx+ Bu
y = Lx+ Du (41)
185
where C = TUT CV, G = TUT GV, B = TUT B, L = LV, D = D.
Rewriting (41),
⎡
⎢
⎣
In 0
0 0
⎤
⎥
⎦
⎡
⎢
⎣
x
x2
⎤
⎥
⎦= −
⎡
⎢
⎣
G11 G12
G21 G22
⎤
⎥
⎦
⎡
⎢
⎣
x
x2
⎤
⎥
⎦+
⎡
⎢
⎣
B1
B2
⎤
⎥
⎦u
y =
[
L1 L2
]
⎡
⎢
⎣
x
x2
⎤
⎥
⎦+ Du (42)
we can have
x = −G11x−G12x2 + B1u (43)
0 = −G21x−G22x2 + B2u (44)
y = L1x+ L2x2 + Du (45)
Refer to [71], for an RLC circuit, the matrix G22 is nonsingular. Then, eliminating
the variable x2, we have
x = Ax+ Bu
y = Cx+Du (46)
where
A = G11 −G12G−122 G21 (47)
B = B1 −G12G−122 B2 (48)
C = L1 − L2G−122 G21 (49)
D = D + L2G−122 B2 (50)
Using steps from (40) to (46), we can transform (38) and (39) into its equivalent
186
{A,B,C,D} as follows:
A = 1× 1012
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−0.0062 0 0 0 0 0 0.0002
0 −0.0000 0 0 0 0.0003 −0.0003
0 0 −0.0003 0 0.0013 −0.0013 0
0 0 0 −0.0050 −0.0050 0 0
0 0 −0.1000 0.1000 0 0 0
0 −0.5000 0.5000 0 0 0 0
−1.0000 1.0000 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
B = 1× 109
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
5
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
C =
[
0 0 0 1 0 0 0
]
D = 0
Note that, for this particular case, D = 0. For other circuits, D can be nonzero.
For multi-input multi-output (MIMO) system as in Fig. 1, the process of obtaining
the state-space form is similar to SISO system. When voltage at P2 is zero, the
admittance matrix Y21 can be obtained if y = −IL4. Using similar process to compute
Y11 and Y21, the admittance matrices Y12 and Y22 can be obtained when voltage at P1
is zero. Then, the overall system can be written as Y =
⎡
⎢
⎣
Y11 Y12
Y21 Y22
⎤
⎥
⎦. From Y , the
realization in (40) can then be obtained. Using the same steps as for SISO system,
the state-space in (46) can be obtained.
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