Arbitrarily fast and robust tracking by feedback

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Arbitrarily fast and robust tracking by feedbackMatei KelemenPublished online: 08 Nov 2010.

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Arbitrarily fast and robust tracking by feedback

MATEI KELEMEN{

This work is dedicated to Professor Yakar Kannai, on his 60th birthday

The output of a singe-input±single-output linear feedback system with more than one pole in excess over the zeros in theloop transmission cannot track arbitrarily fast its input (by the root locus). In this work we extend the linear feedback sothat some of the open loop poles may depend on the open loop gain; we call this new class quasi-linear feedback systems.We then derive time domain, pole-zero, and frequency domain conditions which ensure arbitrarily fast and robusttracking by quasi-linear feedback, for an arbitrary number of poles in excess over the zeros. We prove that in a particularcase these conditions are equivalent, and that the boundedness in frequency of the closed loop transfer function is nolonger necessary for achieving arbitrarily fast tracking. The robustness is to external disturbances and initial conditions,and the open loop has to be minimum phase. Some examples are presented which illustrate these results. They also showthat this good performance can be obtained with a reduced control e� ort, and that quasi-linear feedback can alleviate thelimitation on performance of non-minimum phase open loops.

1. Introduction

1.1. An example of feedback control

Let us consider the plant

P…s† ˆ1

s2

which is a double integrator. As simple as it appears thismathematical model was used, under some normaliza-tion, to the practical feedback control of a hard diskdrive servo system (see Remark 9 in } 4).

In this section we provide a frequency domainfeedback design to control this plant. Therefore webuild around it a feedback loop with compensatorG…s† ˆ k…s ‡ 1†=…s ‡ 2† (®gure 1). We raise the gaink > 0 to increase the performance of the closed loopand use the lead±lag compensation for ensuring somephase margin. Nevertheless, for k ˆ 1000 we obtain aquite slow and highly oscillatory response to a step input(see ®gure 2(a)). However, with the slightly di� erentcompensator Gk…s† ˆ k…s ‡ 1†=…s ‡ 2k0:6† the resultsare completely di� erent (®gure 2(b)). The simulationswere carried out with the MathWorks SimulinkTM soft-ware.

This great improvement of performance wasobtained without any change in the order of compensa-tion (which is already minimal), but by letting the poleof the compensator depend on the gain k itself: theexponent of k was f ˆ 0 in ®gure 2(a) and f ˆ 0:6 in®gure 2(b). The ®rst compensator is a linear one and we

call the second compensator `quasi-linear’ (the terminol-ogy will be clari®ed later).

The linear compensator cannot secure any phase

margin when the gain increases unboundedly becauseit has two poles more than zeros in the loop transmis-sion. This fact does not prevent the quasi-linear com-

pensator to achieve high performance because thedestabilizing lag e� ect is `pushed’ towards larger fre-quencies by the pole which wanders farther away withthe increase of the gain. The linear compensator might

achieve the same performance but only augmentinginde®nitely its chain of lead±lag compensation; andeven then a formal proof of convergence is needed.

Detailed explanations about what causes this changein performance is provided in } 4.2. However, ®gure 2(b)points intuitively towards our main result: for a well-

designed quasi-linear feedback system when the gainincreases unboundedly the tracking error shrinks tozero over a time domain which expands to the wholeopen positive semi-axis.

1.2. Path to generalization: statement of the problemand method of proof

The main objective of feedback control is to ensuregood tracking of the input to the closed loop by its out-put. And this despite the uncertainty in parameters, the

external disturbances and initial conditions which maya� ect the open loop, so the tracking is robust as well.

In some situations it is possible to obtain even an

arbitrarily fast and robust tracking by feedback(AFRTF). This means that over all the uncertainty set:the worst (tracking) settling time can be imposed arbi-trarily small; the closed loops are Hurwitz (exponen-

tially) stable and the poles are bounded away from the

International Journal of Control ISSN 0020±7179 print/ISSN 1366±5820 online # 2002 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/00207170110121899

INT. J. CONTROL, 2002, VOL. 75, NO. 6, 443 ±465

Received 15 February 2001. Revised 14 December 2001.{ Electrical Engineering Department (GREÂ PCI), EÂ cole de

Technologie SupeÂ rieure, UniversiteÂ du QueÂ bec, 1100, rueNotre-Dame Ouest, MontreÂ al, QueÂ bec H3C 1K3, Canada.e-mail: [email protected]

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imaginary axis by a ®xed margin; and the peaks (in time)

of the tracking errors satisfy a unique ®nite bound.

In this article, working with single-input±single-

output systems, we show that AFRTF is achievable

for minimum phase open loops, i.e. when the zeros of

the transfer functions are in the left half of the complex

plane. The robustness, however, is only for external dis-

turbances and initial conditions because for reducing by

feedback the uncertainty in the parameters of the plant a

very special analysis is needed, which is the topic of afuture work.

In ®gure 3 the feedback scheme to which we refer inthis work is represented: here u…t† is the reference input,y…t† is the output, w…t† is the external disturbance, c arethe initial conditions, kL…s† is the loop transmission, k isa positive gain, and U…s†, Y…s†, W…s† and C…s† are theLaplace transforms of the corresponding time domainfunctions.

Usually kL…s† contains the plant, independent of k,and the (feedback) compensator which is the onedepending on k. We do not distinguish between plantand compensator except for the examples of } 4. Thereason is twofold: (1) In this article our main purposeis to describe the new class of quasi-linear feedbacksystems and its properties, the systems being representedby their closed loop transfer functions. (2) A systematicdesign procedure for quasi-linear compensators (to bedeveloped in a future work) will dwell in the frequencydomain and will be based on the Nichols chart. Thischart is a logarithmic complex plane which provides adirect relation between the closed loop transfer functionand the loop transmission which is actually designed(see, for instance, Horowitz 1993). In the examples of} 4 the Nichols chart was implicitly present in the waythe excess of number of poles over zeros in the looptransmission, d , was treated.

To ®nd conditions for AFRTF let us consider theconvolution system

y…t† ˆ…t

0

h…t ¡ ½†u…½† d½; t > 0

If h…t† (the impulse response of this system) is the Diracdistribution ¯…t† we have formally

y…t† ˆ…t

0

¯…t ¡ ½†u…½† d½ ˆ u…t†; t > 0

This displays a perfect tracking by the output y…t† of theinput u…t†. Unfortunately the Dirac distribution is notthe impulse response of a dynamical system. Thereforewe have to look for approximations. An approximationby functions of ¯…t† is given in Schwartz (1966, Theorem13, p. 95 and note, p. 96). Thus if the sequence of func-tions hk…t† are de®ned for t ¶ 0 su� cient conditions forconvergence to ¯…t† are

444 M. Kelemen

Figure 1. Feedback control of the double integrator.

Figure 2. (a) Linear feedback design for the double integra-tor: f ˆ 0. (b) Quasi-Linear feedback design for thedouble integrator: f ˆ 0:6 > 0:

Figure 3. Feedback structure.

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limk ! 1

hk…t† ˆ 0 uniformly in any set

0 < T µ t µ 1=T < 1

limk ! 1

…T

0

hk…½† d½ ˆ 1 for any T > 0

…tf

0

jhk…½†j d½ µ M < 1; tf > 0 fixed;

k > 0 large; 0 < M

independent of k

9>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>;

…1†

It follows from this Theorem (and its proof) that

limk ! 1

yk…t† ˆ limk ! 1

…t

0

hk…t ¡ ½†u…½† d½ ˆ u…t†; t > 0

where u…t† is a test function, i.e. it is in®nitely di� erenti-able and with bounded support. We have to removeboth these constraints because we are interested in alarger class of inputs for which neither condition issatis®ed. Moreover, we wish to estimate the di� erencejy…t† ¡ u…t†j to measure directly (not in an averagingnorm) the tracking property of the system. Thereforewe have to particularize the choice of our hk…t† func-tions.

In this work we are interested in single-input±single-output linear time invariant feedback systems. With zeroinitial conditions and external disturbance the closedloop transfer function from ®gure 3 is

Y…s†U…s† ˆ T…s† ˆ kL…s†

1 ‡ kL…s† …2†

Then using the inverse Laplace transform L¡1, we de®ne

hk…t† ˆ L¡1…T…s††and

Hk…t† ˆ L¡1…T…s†=s† ˆ…t

0

hk…½† d½; t ¶ 0

They are in fact the impulse response and the unit stepresponse of T…s†.

A simple example but illustrative for our method ofproof is obtained for

L…s† ˆ …s ‡ z†=……s ‡ p1†…s ‡ p2†† Re z > 0

Thus for t ¶ 0

hk…t† ˆ e¡~zzt k¡~zz ‡ z

¡~zz ‡ ~pp1

‡ e¡~pp1t k¡~pp1 ‡ z

¡~pp1 ‡ ~zz

Hk…t† ˆk

~zz

…¡~zz ‡ z†…¡~zz ‡ ~pp1† ‡

k

~pp1

…¡~pp1 ‡ z†…¡~pp1 ‡ ~zz†

¡ 1

~zze¡~zzt k

¡~zz ‡ z

¡~zz ‡ ~pp1

¡ 1

~pp1

e¡~pp1 t k¡~pp1 ‡ z

¡~pp1 ‡ ~zz

9>>>>>>>>=

>>>>>>>>;

…3†

Here by the root locus method (Krall 1961) ~zz ! z and~pp1 ! k as k goes to in®nity. It can be checked directlythat hk…t† satis®es the conditions (1), and that Hk…t†

converges to 1 uniformly in t 2 ‰T ; 1† for any T > 0.Moreover, we can get a useful estimate of jyk…t† ¡ u…t†j.For this we integrate by parts in the previous convolu-tion equation and get

yk…t† ˆ Hk…t†u…0† ‡…t

0

Hk…t ¡ ½† _uu…½† d½

The class to which u…t† belongs will be speci®ed later.We can express

Hk…t† ˆ ¬k ¡ e¡t~zz k…t†

where limk ! 1

¬k ˆ 1; limk ! 1

j k…t†j ˆ 0 …4†

uniformly in t ¶ T > 0, T arbitrary positive, and j k…t†jis bounded from above for large k > 0 with a boundindependent of t and k. Then we get

yk…t† ˆ ¬ku…t† ¡ e¡t~zz k…t†u…0† ¡…t

0

e¡½ ~zz k…½† _uu…t ¡ ½† d½

so

jyk…t† ¡ u…t†j µ ju…t†jj¬k ¡ 1j ‡ j k…t†jju…0†j

‡ TB j _uuj ‡ j _uuj…t

T

e¡Re ~zz½ j k…½†j d½

where B is the upper bound on j k…t†j. Therefore withthe properties of ¬k and k…t† from above and since¡Re z < 0 we have for u…t† and _uu…t† bounded and forevery T > 0

limk ! 1

jyk…t† ¡ u…t†j µ B T j _uuj uniformly in t ¶ T > 0

Hence we obtained in the limit an arbitrarily fast track-ing for t > 0 by the output yk…t† of the input u…t†because T > 0 can be taken arbitrarily small.

The question is if such a result holds true for moregeneral systems. When there is an arbitrary but ®nitenumber of zeros with negative real parts in L…s† andonly one pole more than the zeros in L…s† the result isstill true, as above. If there are three or more poles inexcess over the zeros the result cannot be true becauseby the root locus the system will destabilize when k goesto in®nity. For two poles in excess the system mayremain stable but when k tends to in®nity any marginof stability (in frequency domain) is violated and thetracking result does not work either. All these state-ments are formally proved in } 3.3.

In the next section we extend the notion of linearfeedback by allowing some poles of L…s† to depend onk. Thus we expect to remove the restriction of the excessof number of poles over zeros in the open loop kL…s†, toperformance of the closed loop T…s† from (2). We callthis new class quasi-linear feedback systems.

Arbitrarily fast and robust tracking 445

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1.3. Previous work

The technical challenge which generated this workwas the concept of àrbitrarily large feedback bene®ts’

(ALFB) as formulated in Horowitz (1993). The con-ditions presented there were expressed in frequency

domain to ensure arbitrarily large bandwidth and a®xed peaking to the closed loop. The open question

which we wish to answer is what does this amount toin time domain. One su� cient condition for ALFB is

that the excess of poles over zeros in the (not uncertain)plant to be µ 1; this coincides with the condition found

above with the root locus.We now comment brie¯y on the other works

which provide a natural context for the results reportedhere. In frequency domain the use of high gain linear

feedback for achieving good performance despite un-certainty in environment can be traced back to

Horowitz (1963) and even to Bode (1945). The classicalfrequency domain methods based on Bode plots,

Nyquist stability, gain and phase margins produced

many useful feedback designs. However, only smallplant variations and external disturbances could behandled. The ®rst systematic feedback design method

to achieve high performance despite large variations in

plant parameters and in external disturbances appearsto be the one presented in Horowitz and Sidi (1972).

Since this method treated quantitatively both thenarrowness of the time domain speci®cations and the

extent of the plant and disturbance uncertainty itbecame known as the quantitative feedback theory

(QFT). This method has another useful feature: thedesign is done in frequency domain (easier to perform)

to achieve time domain speci®cations which are the onesrelevant in applications. Nevertheless, the translation of

speci®cations between time and frequency domains isstill unsolved theoretically. Later, with the advent of

H1 control (Zames 1981), the emphasis was on theoptimization (in the H1 norm) of the peak in frequency

of the closed loop response. This, in turn, ensured the

reduction of the sensitivity of the closed loop to externaldisturbances. All these feedback design methods arelinear, thus subjected to the limitations mentioned

before.

The perfect tracking problem by feedback with statespace methods also has a long history, among the ear-

liest important results being those obtained inKwakernaak and Sivan (1972). For a list of the signi®-

cant works in this area one may consult the references ofa recent monograph (Chen 2000). In this book a new

and e� ective method is also presented to solve therobust and perfect tracking with state and output feed-

back. The robustness is to initial conditions and externaldisturbances, and the tracking performance is measured

in integral norms. This method involves the parametri-

zation of some of the poles of the closed loop, as we alsodo here, but in a di� erent framework.

1.4. Present work

The main contribution of this article is the extensionof linear feedback systems to quasi-linear ones wheresome of the poles of the open loop depend on theopen loop gain. Thus we were able to prove that arbi-trarily fast and robust tracking by feedback is achievablefor the closed loop, for any number of poles in excessover the zeros of the open loop if this one is minimumphase. Working with quasi-linear feedback allowed us:

. to derive time domain, pole-zero and frequencydomain conditions for AFRTF, and to prove therelation and, in a particular case, the equivalenceamong them. Surprisingly, the condition ofboundedness in frequency of the closed loop (pres-ent in QFT and H1 linear designs) is no longernecessary for achieving AFRTF;

. to propose a method for the theoretical solution ofthe problem of feedback design in frequencydomain to achieve performance in time domain.Compatible with this goal the time domain quan-tities in our results are measured in absolute valuesnot in averaging norms;

. to reduce the control e� ort while achievingAFRTF, and to alleviate the non-minimumphase plant limitation to performance, in someexamples presented in this work.

1.5. Future work

Certain open problemsÐwhose solutions are usefulfor the design of quasi-linear feedback systemsÐarestated in some of the remarks, corollaries and notes tothe technical proofs. Besides, we point out three addi-tional ones. The ®rst is the robustness to plant uncertainpoles and zeros, which requires a detailed analysis whenthe gain k goes to in®nity, before generalizing the resultsobtained in this article. Another problem is the feedbackcontrol of non-linear plants. In view of the resultspresented here one avenue to solution seems to be totreat non-linearities as equivalent disturbances, as inHorowitz (1993) . Finally, for the multi-input-multi-output case the appropriate distribution of feedbackbene®ts among various channels has to be solved inour setting.

1.6. Notations and organization of the paper

Our notations are standard but we recall that t is thetime variable, s is the usual symbol for a complex vari-able, i is the imaginary unit, ! is the (real) frequencyvariable, o…1† is a function which goes to zero in

446 M. Kelemen

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absolute value when a corresponding variable goes toin®nity, O…1† is a function which remains bounded inabsolute value when a corresponding variable goes toin®nity, and L¡1 is the inverse Laplace transform.Some speci®c notations: the sign `~’ covers the poles(they all depend on k) of the closed loop transfer func-tions; the hypotheses and conclusions are denoted by …i†and …c† with appropriate indices.

Finally the paper is organized as follows. In } 2 wepresent our extended class of feedback together with itsproperties captured in three lemmas, } 3 contains ourthree Theorems concerning the arbitrarily fast androbust tracking by feedback from time domain, complexplane and frequency domain points of view. Section 4describes in detail three examples which illustrate ourresults and methods, and } 5 contains the proofs of ourtheorems and lemmas.

2. Quasi-linear Feedback

The purpose of this section is to extend the class offeedback systems (2) so as to remove the demandingconstraint imposed on the performance of the closedloop T…s† by the excess in the number of poles overzeros of the open loop kL…s†. We also point out someinherent properties of this new class of feedbacksystems.

2.1. Extended feedback class

Let N…s† and Dk…s† be two monic, coprime poly-nomials in the complex variable s, with degrees m andm ‡ d respectively, m ¶ 0, d ¶ 1 given integers andk > 0 a real number

N…s† ˆ …s ‡ z1† ¢ ¢ ¢ …s ‡ zm†

Dk…s† ˆ …s ‡ p1† ¢ ¢ ¢ …s ‡ pm‡d†

Here the complex numbers z1; . . . ; zm, are independentof k but p1; . . . ; pm‡d ; may depend on it.

We now embed the open loop, with L…s† replaced byLk…s† ˆ N…s†=Dk…s†, into the feedback structure from®gure 3 with zero initial conditions and external disturb-ances. Thus we obtain the rational transfer function ofthe closed loop

Yk…s†U…s† ˆ Tk…s† ˆ

kN…s†=Dk…s†1 ‡ kN…s†=Dk…s†

ˆ kN…s†Dk…s† ‡ kN…s†

N…s†Dk…s†

ˆ Lk…s†

9>>>>>>>>=

>>>>>>>>;

…5†

We have considered d ¶ 1 for two reasons.Mathematically both the open loop and the closedloop will result in actual (not distribution) time domain

solutions. From a feedback control point of view anexcess of (number of) poles over zeros is useful for sen-sor noise reduction (Horowitz and Sidi 1972). Whenm ˆ 0 it follows that N…s† ² 1 so Tk…s† has no zeros.In this case the statements and proofs presented in thisarticle become simpler. We work, however, in the gen-eral case.

De®nition 1: If Dk…s† is independent of k the feed-back loop is linear. If Dk…s† depends on k we call thefeedback quasi-linear.

We chose this terminology because the open loopremains linear in the s (and t) variable but its polesdepend non-linearly on the gain k.

Throughout this article we assume that this class oftransfer functions is subjected to the following require-ment whose justi®cation will become clear in the lemmasbelow:

…r1† k > 0 appears in Dk…s† only with rational expo-nents. Denote by f the maximal exponent and supposef < 1.

In the linear case f ˆ 0 or f ˆ ¡1.Since f is rational we can rewrite Tk…s† from (5) as

Tk…s† ˆ Tzk…s†Tdk…s† …6†

where

Tzk…s† ˆ …s ‡ z1†…s ‡ z2† ¢ ¢ ¢ …s ‡ zm†…s ‡ ~zz1†…s ‡ ~zz2† ¢ ¢ ¢ …s ‡ ~zzm† ;

Re z1 µ ¢ ¢ ¢ µ Re zm

Tdk…s† ˆ k

…s ‡ ~pp1† ¢ ¢ ¢ …s ‡ ~ppd† ; Re ~pp1 µ ¢ ¢ ¢ µ Re ~ppd

~pp` ˆ a`kn`R …1 ‡ o`R

…1†† ‡ ib`knÌ …1 ‡ oÌ

…1††;

k > 0 large

n`R; nÌ

are rational numbers, and denote n` ˆmax fn`R

; nÌg. By de®nition if a`0

ˆ 0 then n`0Rˆ ¡1

and if b`0ˆ 0 then n`0 I

ˆ ¡1. The dependence on kde®nes m ‡ d branches of poles.

In this article when a property is true for `large k’ itmeans that there is a ®nite k0 > 0 such that the propertyis veri®ed for all k ¶ k0. For instance, the stability ofTk…s† and the boundedness of the function k…t† from (4)can be assured only for large k > 0 because only forlarge k the loci of poles are known to stay in the lefthalf of the complex plane and also not to intersect.Moreover, since we have a ®nite number of poles andzeros there is a ®nite k0 for all the properties provedhere. Another situation for using `large k’ is when wereplace 1 ‡ o…1† by 1 as in the example (7) below.

The asymptotic behaviour of the zeros of equationDk…s† ‡ kN…s† ˆ 0 is obtained as follows. First we mul-


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tiply the equation by k to some exponent so that no

coe� cient of s has a negative power of k, and denotek1=g ˆ z where g is the least common multiple of thedenominators of the (new) exponents of k, see …r1†.Then extend z to a complex variable and apply(HoÈ rmander 1983, Lemma A.1.3, p. 363) to the poly-nomial in …s; z† thus obtained, which leads to the expan-

sion of its roots in a series of rational powers of z. Wereturn now to the k > 0 variable seen as a complexnumber , i.e. k e2ºi. By emphasizing the real and imagin-

ary parts we get the above result. A similar expansioncan be obtained also for large k < 0, from the sameformula in z but with a di� erent set faj; njR ; bj; njI

gbecause now the complex number is k eºi. In view ofLemma 1 below in the case of the expansions of

~zz1; . . . ; ~zzm the corresponding exponents of k cannot be

positive because the numbers z1; . . . ; zm are ®nite. Suchexpansions exist for the zeros of Dk…s† as well but theyare not needed explicitly here. For actual computation

of the expansions one can use the Puiseux series methoddescribed in Hille (1962, } 12.3).

Before proceeding we give two examples of type (2),

(3), but in the framework of quasi-linear feedback. Theywill be used in several places in this article. Both ex-amples display a ®nite stability margin: ¡a < 0 for the

®rst one and ¡~pp < 0 for the second. Therefore in bothcases the function k…t† will contain, for t > 0, an expo-nential with exponent going to ¡1 when k goes to in®-

nity.The ®rst example is

Tk…s† ˆk

…s ‡ ~pp1†…s ‡ ~pp2†…s ‡ ~pp3† ; ~pp1 ˆ k1=4;

~pp2;3 ˆ a § ik3=8; k > 0; a > 0 …7†

hk…t† ˆ L¡1…Tk…s†† ˆ k1=4 e¡k1=4t

1 ‡ ……a ¡ k1=4†2†=k3=4¡ …e¡at†

£ k1=4 cos …k3=8t† ‡ k1=8…1 ¡ a=k1=4† sin …k3=8t†1 ‡ ……a ¡ k1=4†2†=k3=4

;

t ¶ 0

Hk…t† ˆ L¡1…Tk…s†=s† ˆ…t

0

hk…½† d½ º 1 ¡ e¡k1=4 t

‡ e¡at‰…ak¡1=2 ‡ k¡1=4† cos …k3=8t†

‡ …ak¡5=8 ¡ k¡1=8† sin …k3=8t†Š

Hk…t† ˆ ¬k ¡ e¡at k…t†; ¬k º 1

k…t† º e¡…k1=4¡a†t ¡ ‰…ak¡1=2 ‡ k¡1=4† cos …k3=8t†

‡ …ak¡5=8 ¡ k¡1=8† sin …k3=8t†Š

Above where the approximation sign appears we re-placed 1 ‡ o…1† by 1 because we considered k > 0 large.

The second example is

Tk…s† ˆ k

…s ‡ k1=4 ‡ ik1=2†…s ‡ k1=4 ¡ ik1=2†; k > 0

…8†

hk…t† ˆ L¡1…Tk…s†† ˆ k1=2 e¡k1=4t sin…k1=2t†


0

hk…½† d½

ˆ k

k ‡ k1=2¡ e¡…k1=4t†

£ k1=4 sin …k1=2t† ‡ k1=2 cos …k1=2t†k ‡ k1=2

k1=2

Hk…t† ˆ ¬k ¡ e¡~ppt k…t†; ¬k ˆ k

k ‡ k1=2

k…t† ˆ e¡……k1=4¡~pp†t† k1=4 sin …k1=2t† ‡ k1=2 cos …k1=2t†k ‡ k1=2

k1=2

Here ~pp is an arbitrary real positive number.

2.2. Properties of the transfer function Tk…s†Next we present three lemmas which describe useful

properties of Tk…s† from (5) or, equivalently, (6); theywill be employed also in subsequent sections. These lem-mas bear witness to the power of the (seemingly) weakrequirement …r1†. Lemma 1 extends to the case of quasi-linear feedback two results of the classical root locusmethod (Krall 1961, Theorems 2 and 3). Lemma 2gives characterizations of some frequency domain prop-erties (high gain and boundedness of the closed loop). InLemma 3 some results necessary for the analysis ofquasi-linear feedback loops are proved. Conclusion…c1† of Lemma 3 is stated also in Lemma 1 …c2† due toits relevance to the root locus results.

Certain properties presented in these lemmas are truein the particular case of linear feedback as well (wheresome statements can be strengthened).

Lemma 1: (partial extension of the root locusmethod):

(i) The condition f < 1 from …r1† implies:

…c1† ~zz` 6ˆ z` for all k and limk ! 1 ~zz` ˆ z`, for all `;…c2† ~pp` is not identically 0 and, moreover, n` > 0, for

all `;limk !1 ¦d

`ˆ1~pp`=k ˆ 1

n1 ‡ ¢ ¢ ¢ ‡ nd ˆ 1 …9†

Because d ¶ 1, Tk…i!† cannot converge to 1 uni-formly in ! when k goes to in®nity. A simple exampleis T…ik† ˆ k=…ik ‡ k† which is 1=

��2

pin absolute value

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for all k 6ˆ 0. Therefore we shall use a weaker notion ofconvergence adapted from Rudin (1970, De®nition10.26, p. 214). Nevertheless, it will capture the highgain feedback property present in (5) when k goes toin®nity.

De®nition 2: We say that the sequence of functionsTk…i!† converges to 1 uniformly on compact subsets ofthe real numbers if for every such compact set Sand for every ° > 0 there is a number K ˆ K…S; °† sothat if k ¶ K…S; °† then jTk…i!† ¡ 1j µ ° for all ! 2 S.The abbreviation we shall use is: Tk…i!† converges…1 ¡ UCS†.

This de®nition implies that the zeros of N…s† cannotbe on the imaginary axis. Indeed, if they were Tk…i!†would be zero at the imaginary roots of N…s† thus notconverging to 1. We note that there are no poles of Tk…s†which could cancel the zeros because N…s† and Dk…s† arecoprime (no common zeros) by assumption, see alsoLemma 1 …c1†.

Therefore we shall include this constraint on N…s† inthe hypotheses of the following lemma, where it willensure also the convergence to 1 of Tzk…i!† when kgoes to in®nity, uniformly in !.

Lemma 2 (frequency domain properties): SupposeN…s† has no zeros on the imaginary axis. Assume alsothat

…i1† Tk…i!† converges …1 ¡ UCS†.Then…c11† a necessary and su� cient condition to satisfy …i1†

is f < 1;…c12† necessary and su� cient conditions for satisfying

…i1† are:~pp` is not identically 0 and, moreover, n` > 0 forall `;limk ! 1 ¦d

`ˆ1~pp`=k ˆ 1;~zz` 6ˆ z` for all k and limk ! 1 ~zz` ˆ z`, for all `.

…i2† Suppose all non-zero b` from …6† are distinct.Then the following statements are equivalent:…c2† jTk…i!†j is bounded in ! uniformly in large k, i.e.

there is a ®nite number B > 0 independent of !and k such that sup! jTk…i!†j µ B for largek > 0;

…c3† all a` 6ˆ 0, and for all `: n`R¶ nÌ

or

n`R¡ nÌ

‡ §dj 6ˆ` maxf…n` ¡ nj†; …njI

¡ nj†; …njR¡ nj†g ¶ 0

If all n` are equal the last condition reduces (also) to: alln`R

¶ nÌ.

Remark 1: It is worth emphasizing the followingproperty: if N…s† does not vanish on the imaginaryaxis Lemma 2 has proved that, with the feedbackstructure (5) under requirement …r1†, Tk…i!† alwaysconverges …1 ¡ UCS†. This is true in the linear case

too where f is ¡1 or 0 so < 1. It should be notedthat convergence …1 ¡ UCS† does not imply uniformboundedness as shown by T…s† ˆ k=…s2 ‡ k†: T…i!†converges …1 ¡ UCS† but T…§i

��k

p† ˆ 1 whether k > 0

is ®nite or goes to in®nity.

Remark 2: Here the uniform boundedness dependsonly on the exponents of the roots of Tk…s†. Whensome upper limit is imposed on jTk…i!†j this conditionmay depend on the coe� cients too, as in the QFT andsuboptimal H1 designs.

Let Pd…s† be the polynomial of degree d ¶ 1, withsolutions ¡~pp1; . . . ; ¡~ppd from the denominator of Tdk…s†in (6). Using (5) information on its coe� cients can beobtained from the identity Dk…s† ‡ k…s ‡ z1† £ ¢ ¢ ¢ £…s ‡ zm† ² Pd…s†…s ‡ ~zz1† £ ¢ ¢ ¢ £ …s ‡ ~zzm† by identi®ca-tion of the coe� cients of the monomials sj; j ¶ m.Thus the leading coe� cient of Pd…s† is 1, its last coe� -cient divided by k tends to 1 when k goes to in®nity, andthe other coe� cients have the exponent of k smallerthan 1. We used here the fact that the maximal exponentof k in Dk…s† is f < 1 owing to …r1†. Section 4.2 providesan actual computation of a Pd…s†.

We associate with Pd…s† the polynomial Qd…sk†obtained (much like in the case of homogeneouspolynomials) from it by division with k and with thechange of variable sk ˆ s=…k1=d†. Hence the roots of Pd

are those of Qd multiplied by k1=d . By construction theleading coe� cient of Qd is 1 and its last coe� cienttends to 1 when k goes to in®nity, so with the rel-ation among coe� cients and roots of a polynomialthe absolute value of the product of its roots tendsto 1.

Lemma 3 (prerequisites for the analysis of quasi-linearfeedback loops):

…c1† ~pp` is not identically 0 and, moreover, n` > 0 forall `;limk ! 1 ¦d

`ˆ1 ~pp`=k ˆ 1;n1 ‡ ¢ ¢ ¢ ‡ nd ˆ 1.

…c21† A necessary and su� cient condition for all n` tobe equal (to 1=d see the previous relation) is thatthe exponents of k in the coe� cients 2; . . . ; d ; ofthe polynomial Qd…sk† to be non-positive.

…c22† A su� cient condition for all the branches p` from…6† to be asymptotically uniformly distributed inthe complex plane is that the exponents of kin the coe� cients 2; . . . ; d; of the polynomialQd…sk† to be negative; in this case also all n`

are equal.…c3† If the coe� cients of the polynomial Dk…s†‡ kN…s†

are real a necessary condition for its roots to be inthe left half of the complex plane and boundedaway from the imaginary axis by a ®xed marginfor large k, is that the maximal exponent of k of


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every coe� cient to be non-negative; it followsf ¶ 0. (As in …6† if a coe� cient is zero we con-sider the corresponding exponent of k to be ¡1.)

For d ¶ 3 and general coe� cients a necessary con-dition for the roots of Dk…s† ‡ kN…s† to be in the openleft half of the complex plane for large k > 0, is f > 0.

If d ˆ 1 or d ˆ 2 the Hurwitz stability with a ®xedstability margin for large k > 0 can be achieved by linearfeedback as well.

3. The results

In this section we present our main results, Theorems1, 2 and 3. They all deal with the arbitrarily fast androbust tracking by feedback problem, from the timedomain, pole-zero and frequency domain points ofview. Both the quasi-linear feedback (De®nition 1) andthe linear one are treated. The Theorems are presentedin decreasing order of generality. This will be illustratedby (7)Ðapplicable to Theorem 1 but not Theorem 2Ðand (8) applicable to both. Linear feedback systems(Theorem 3) represent but a small fraction of the classof quasi-linear feedback ones (Theorems 1 and 2).

The high performance of the dynamical systems pre-sented in this section is based on the concept of highgain feedback. This notion is described formally inDe®nition 2 and in Lemma 2 …c11†, …c22†. The bounded-ness in frequency of feedback system, as represented inLemma 2 …c2†, …c3†, is used in Theorems 2 and 3. Ingeneral high gain feedback and stability are in con¯ict.Lemma 3 …c3† gives conditions which resolve (positively)this contradiction. Finally, the extended root locus ofLemma 1 (based on requirement …r1†) is used in allthree Theorems.

3.1. Arbitrarily fast and robust tracking by feedback:time domain properties

We now generalize the properties of function Hk…t†from (4) and consider also robustness to external dis-turbances. Intuitively we show that if the system repre-sented by Tk…s† has some `good’ properties for a stepinput 1=s then it will have certain `good’ properties for alarge class of input and disturbance functions.

Conditions for arbitrarily fast and robust tracking byfeedback (AFRTF):

Suppose it is given the transfer function Tk…s† from(5) or equivalently (6), which satis®es the requirement…r1† and, moreover, for large k > 0 all its poles are in a®xed left plane of the complex plane. The impulseresponse of this system is hk…t† ˆ L¡1…Tk…s††; t ¶ 0.The step response Hk…t† ˆ L¡1…Tk…s†=s† ˆ

„ t

0hk…½† d½ ,

t ¶ 0, can be expressed with the residue Theorem as

Hk…t† ˆ ¬k ¡ e¡t~pp k…t†; ¬k ˆ Tk…0† …10†

Here with the expansion (6) for the poles ¡~pp` and ¡~zz`

we de®ne ~pp as a ®nite real number such that

1 > ¡~pp > max`ˆ1;...;m‡d

limk ! 1

Re pole`fTk…s†g …11†

Two examples of the representation (10) can be found in(7) and (8). The ®niteness of ~pp is made possible by thelocation of poles mentioned above; this shows that forany unstable pole ¡~pp` (i.e. a` < 0) we have necessarilyn`R

µ 0.Now the following are properties required for

AFRTF:

…pr1† limk ! 1 ¬k ˆ 1; …¤†…pr2† for every T > 0 limk ! 1 j k…t†j ˆ 0 uniformly

in t 2 ‰T ; 1†;…pr3† supt¶0 j k…t†j µ B < 1, k > 0 large, B

independent of k;…pr4† there exists a number ® > 0 such that

¡~pp µ ¡® < 0.

For further use we de®ne for each T > 0 thesequence …k;T † ˆ supt¶T j k…t†j. Then

…pr2† () limk ! 1

…k;T † ˆ 0

for every T > 0.We see that …pr4†, even with ® ¶ 0, constraints the

zeros and poles of Tk…s† from (6) to have all Re z` > 0(with Lemma 1 …c1†) respectively all a` > 0, n`R

¶ 0,¡Re ~zz` < 0 and moreover all ~zz` to be bounded awayfrom the imaginary axis for large k; the latter is obtainedagain with Lemma 1 …c1†. At the same time if property…pr4† is true so is …pr1†, which follows from (10), (6) andLemma 1 …c1†, …c2†, hence the properties …¤† are notindependent.

This strong e� ect justi®es a few comments about thede®nition of ~pp. In view of …pr4† the number ¡~pp is a timedomain stability margin, that is why we chose it to be®nite (but it could be arbitrarily large in absolute value).Also, we did not use the sign `¶’ in (11) in order toprevent k…t† becoming unbounded in t and thus notsatisfying …pr2† and …pr3† (even for ® > 0). This unde-sired e� ect happens when the closest poles to the ima-ginary axis are multiple (Hirsch and Smale 1974,Theorem, p. 139), or converging toward each other(e.g. when there are multiple zeros closest to the imagin-ary axis). Moreover, the same use of the sign `>’ in (11)allows us to consider situations when the maximal limitis ®nite and is approached from the right as in the caseof some unstable open loops; for the equal sign k…t†would be again unbounded in t for all (®nite) k shortof k ˆ 1.

In the next Theorem the initial conditions are set tozero due to a certain technical consideration brie¯y men-tioned at the beginning of the proof of this theorem.Nevertheless, some of these issues do not apply to

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Theorems 2 and 3 where the initial conditions to theoutput of the open loop are taken into account (seealso the notes to the proof of this Theorem).

We can now state Theorem 1.

Theorem 1 (AFRTF in time domain): It is given theset of transfer functions …5†Ðor equivalently …6†Ðsatis-fying requirement …r1†. Assume that:

…i1† the properties …pr1† to …pr4† from …¤† are veri®ed;…i2† the reference input u…t† and the disturbance input

w…t† are bounded functions with support t ¶ 0.Moreover, for t > 0 they are continuous and continuouslydi� erentiable with bounded derivatives. The boundson the functions are denoted by Bu ˆ supt¶0 juj < 1,Bw ˆ supt¶0 jwj < 1, B _uu ˆ supt>0 j _uu…t†j < 1 and B _ww ˆsupt>0 j _ww…t†j < 1.

Then hypothesis …i1† is necessary and su� cient tosatisfy conclusion …c1†. For constant u and w from …i2†,…pr4† can be weakened to ® ¶ 0.

…c1† for every T > 0 and large k > 0 and for anyfunctions from …i2† there is a non-negative ®nitenumber C…u;w;®† independent of k, t and T suchthat the below estimates are true

jyk…t† ¡ u…t†j µ j¬k ¡ 1j…Bu ‡ Bw† ‡ …k;T†C…u;w;®†

‡ …B _uu ‡ B _ww†B T ; t 2 ‰T ; 1†

1 > C…u;w;®† ˆ

ju…0†j ‡ jw…0†j; u and w constant;

…® ¶ 0†ju…0†j ‡ jw…0†j ‡ …B _uu ‡ B _ww†

„ 10

e¡®½ d½;

otherwise …® > 0†

8>>><

>>>:

jyk…t† ¡ u…t†j µ j¬k ¡ 1j…Bu ‡ Bw† ‡ B …ju…0†j ‡ jw…0†j†

‡ …B _uu ‡ B _ww†B T ; t 2 ‰0; T†

these estimates lead to the following inequalities: for everyu; w, T > 0 and ° > 0 there is a number K…u;w†…T ; °† > 0so that if k ¶ K…u;w†…T ; °† then

jyk…t† ¡ u…t†j µ …B _uu ‡ B _ww†B T ‡ ° < 1; t 2 ‰T ; 1†jyk…t† ¡ u…t†j µ B …ju…0†j ‡ jw…0†j†

‡ …B _uu ‡ B _ww†B T ‡ ° < 1; t 2 ‰0; T†

…i3† Suppose the functions u…t† and w…t† belong to asubclass of …i2† such that the bounds on them andtheir derivatives are prespeci®ed.

Then…c2† the results in …c1† are independent of any particu-

lar functions u, w, when the bounds are replacedaccording to …i3†. No weakening of …pr4† is poss-ible here.

…c3† If the polynomial Dk…s† from …5† has a root atzero (the open loop contains an integrator) theconclusions …c1† and …c2† are true even with thebounds on u…t† and w…t† removed for t > 0. The

bounds on u…0† and w…0† have to be still taken

into account.

Corollary 1:(A) It follows from conclusion …c1† that the tracking

precision jyk…t† ¡ u…t†j can be made arbitrarilysmall, uniformly in t 2 ‰T ; 1†, by taking T > 0and ° > 0 small enough and k > 0 su� cientlylarge; at the same time the peak of the trackingerror is bounded independently of t ¶ 0 and largek. Hence the `tube’ in which yk…t† evolvesaround u…t† is shrinking to 0 as it gets closerto t ˆ 0; the rate of convergence depends on Tvia …k;T†. If u and w are constant after t ˆ 0 thetracking precision becomes zero when k goes toin®nity (uniformly in t 2 ‰T ; 1†, T > 0 arbitra-rily small). In view of …c2† we have AFRTF of the

feedback system Tk…s† or equivalently hk…t†, therobustness being to unknown external disturb-ances w; in addition there is also uniformity ofthe results in the functions to be tracked u.

(B) For any T > 0 the function Hk…t† de®ned at …10†converges to 1 uniformly in t 2 ‰T ; 1† when kgoes to in®nity, and it is bounded independentlyof large k and t ¶ 0, as a consequence of proper-ties …pr1† to …pr4† from …¤†. At the same timeTk…i!† converges uniformly to 1 on compact setsof frequencies, see Lemma 2. These two facts canbe used for the translation of speci®cationsbetween time and frequency domains, in feedbackdesign. For this a proof is needed that the com-posite operator of Fourier transform and time dif-

ferentiation is continuous together with its inversebetween the sets fHk…t†jk > 0g andfTk…i!†jk > 0g with their respective notions ofconvergence.

(C) In example …7† hk…t† does not converge to theDirac distribution in the sense of …1† while Hk…t†converges to 1 as above. (The ®rst requirement of…1† is not satis®ed, e.g. for t ˆ º=2 andk3=8 ˆ 1; 2; 3; . . . .) This suggests that the func-tion Hk…t† rather than hk…t† is suitable for thetranslation of speci®cations.

(C) The results presented in this article are true evenwith k…1 ‡ o…1†† replacing the gain k in ®gure 3and in …5†. In this case, however, k should bebigger to obtain the same level of accuracy asin the original situation, for o…1† to becomenegligible.

This extension makes possible to express theloop transmission Lk…s† in …7† and …8† as a prod-

uct of strictly proper plants (independent of k)and quasi-linear (k dependent) compensators . Inthose examples the plants had to be ®xed num-bers. Now for …7† the plant can be P…s† ˆ 1=s, the


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quasi-linear compensator Gk…s† ˆ …k ‡ a2k1=4†=…s2 ‡ …2a ‡ k1=4†s ‡ …a2 ‡ k3=4 ‡ 2ak1=4†† ando…1† ˆ a2k1=4=k. For …8† again P…s† ˆ 1=s andthen Gk…s† ˆ …k ‡ k1=2†=…s ‡ 2k1=4†† and o…1† ˆk1=2=k.

Remark 3:(a) The key result is conclusion …c1†. We emphasize

that: (1) We established there an equivalencebetween the conditions …¤† for AFRTF and theestimates and inequalities satis®ed by jyk…t†¡u…t†j. In this sense the conditions …¤† are uniquefor achieving AFRTF. (2) We showed after …¤†that ® ¶ 0 ensures the Hurwitz stability with a®xed stability margin of the poles of Tk…s†, for klarge. This stability property calls in …c1†, viaLemma 3 …c3† and requirement …r1†, for0 < f < 1 if d ¶ 3.

(b) An integrator in the open loop means that thesteady state error for a step input becomes zero,which is a large time performance. Conclusion…c3† shows that even for an unbounded input(like the ramp) the arbitrarily fast and robusttracking property begins at an arbitrarily smalltime.

(c) In example (7) we have limk !1 jTk…§ik3=8†j ˆ 1(compatible with Lemma 2 …c2†, …c3†). On theother hand hypothesis …i1† of Theorem 1 is ful-®lled. This shows that for quasi-linear feedbackAFRTF can be achieved without Tk…i!† to bebounded in the frequency domain. However,such a boundedness is necessary in the linearQFT and H1 designs.

Remark 4: One can relax to some extent the con-ditions the functions u and w have to satisfy. We men-tion two possibilities based on the estimates from …c1†.

(a) They may have a discrete set of points whereonly side derivatives exist, and which are uni-formly bounded over this set. (The ùsual’ deri-vatives are bounded as before.) Then the proofof the Theorem still works producing the sameresults.

(b) They may have a ®nite number of ®nite jumps andthe results in the Theorem still hold piecewiselyin time. (Hypothesis …i2† allows for one ®nitejump at t ˆ 0.) For an in®nite number of ®nitejumps care should be taken for the accumulationof tracking errors.

3.2. Pole-zero and frequency domain conditions for AFRTF

In this section, for a particular structure of poles, wederive complex plane (pole-zero) and frequency domain

conditions which ensure AFRTF in time domain of the

system with impulse response hk…t†. This behaviour wasdescribed in Theorem 1 but now the robustness is also tothe initial conditions on the output of the open loop (inaddition to external disturbances). At the same time wepoint out the relation among time domain, pole-zero

and frequency domain conditions for AFRTF. Finally,we prove also that the function hk…t† approximates theDirac distribution.

To the requirement …r1† on the transfer function (5),or equivalently (6), from } 2 we add now a new one:

…r2† All the zeros of N…s† and the poles of Tdk…s† from…6† are simple. Moreover, if two poles of Tdk…s†have the same exponent n` in the expansion …6†then their di� erence behaves as Akn` …1 ‡ o…1††too, for some positive constant A.

The second part of this requirement ensures thatthere is no loss in the power of k at the denominatorwhen the residue Theorem is applied to Tk…s†. For thisthe poles to be bounded away from each other is not

enough as shown by Tk…s† ˆ k=……s ‡ 1 ‡��k

p†…s ‡

��k

p††.

We have considered the poles and zeros to be distinct forconvenience, to simplify computations. Note that withLemma 1 …c1† the poles ~zzj are distinct for large k > 0 andwould satisfy automatically the second part of …r2† (they

are all ®nite). Also, because the poles ¡~pp` extend toin®nity when k goes to in®nity (Lemma 1) no ~pp` canbe equal to, or approach a ~zzj, so these two subsets ofpoles are naturally separated.

Theorem 2 (AFRTF: pole-zero and frequency domainconditions): Let the class of transfer functions …5†Ðorequivalently …6†Ðsubjected to the requirements …r1† and

…r2†. Suppose:

…i1† a` > 0 and n`R> 0 for all the poles ¡~pp` of Tdk…s†,

and Re z` > 0 for all the zeros ¡z` of Tzk…s†.Then

…c11† hypothesis …i1† is su� cient to satisfy …pr1† to…pr4† from …¤†.

…c12† all the conclusions of Theorem 1 are veri®ed.Moreover, if Co…t; k† denotes the free responsedue to the initial conditions on the output of the

open loop we have the following estimates: thereare two non-negative ®nite numbers Bop and Boz

independent of k, t, T, but depending on theinitial conditions on the output of the openloop such that for any T > 0 and large k

jCo…t; k†j µ Bop e¡Tak…nR †‡ Boz=k1¡f ; t 2 ‰T ; 1†;

0 < a ˆ min a` 0 < nR ˆ min n`R

jCo…t; k†j µ Bop ‡ Boz=k1¡f ; t 2 ‰0; T† …f < 1†

9>>>=

>>>;

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With these estimates conclusions …c1† and …c2† of Theorem

1 can be extended to robustness in initial conditions too,

by adding up the right-hand sides of the above estimates to

the right-hand sides of the estimates from …c1† of Theorem1. Only the second inequality from Theorem 1 …c1† is

modi®ed by adding to its right-hand side the number Bop.

…i2† Assume that all n` ˆ 1=d. (If all n` are equal they

have to be equal to 1=d, Lemma 1 …c2†.)Then

…c2† hypothesis …i1† is necessary to satisfy …pr2† and

…pr4† from …¤†. (See also the notes to the proof of

this Theorem.)…i3† Assume in addition to …i2† that all the non-zero

numbers b` from …6† are distinct. (This is hypoth-

esis …i2† of Lemma 2.)

Then

…c31† su� cient conditions to satisfy …i1† are: Hurwitz

stability with a ®xed stability margin for large

k > 0 of the poles of Tk…s† (this notion is inde-

pendent of …i2† and …i3†), and the boundedness in

! uniformly in large k of Tk…i!†. The ®rst con-

dition is necessary to satisfy …i1†, but not the

second one, see …8† (discussed in the proof);

…c32† The boundedness in ! uniformly in large k of

Tk…i!† is necessary for satisfying …i1† if and

only if all n`R¶ nÌ

.

…c4† Hypotheses …i1†, …i2†, …i3† and the boundedness

in ! uniformly in large k of Tk…i!† are su� cient

conditions for hk…t† to converge to the Dirac

distribution ¯…t† in the sense of …1†:

Remark 5: ConditionsÐbased on the coe� cients of

the polynomial Dk…s† ‡ kN…s†Ðfor satisfying hypoth-esis …i2† and the stability part of …i1† can be found in

Lemma 3. For the latter compare also Remark 3(a)(2).

Remark 6: From the second part of …c31† and (8) itfollows that it is possible for the quasi-linear feedback

Tk…s† to enjoy arbitrarily fast and robust tracking and,

at the same time, for jTk…i!†j to break any upper

bound when k goes to in®nity (compatible with Lem-ma 2 …c2†, …c3†). This represents a clear departure from

the linear feedback case, where unbounded T…i!† is as-sociated with the lack of AFRTF (Theorem 3 for

d > 1). In general if jTk…i!†j breaks any bounds when

k goes to in®nity this can occur only when ! goes also

to in®nity, see the note to the proof of Lemma 2 andthe fact that all a` 6ˆ 0 by …i1†.

As in (8), (7)Ðwhich `belongs’ to the more relaxed

hypotheses of Theorem 1, see also Remark 3(c)Ð

becomes unbounded in the frequency domain because

it has poles for which the imaginary parts run faster to

in®nity than the real parts. (In (8) the real parts them-

selves go to ¡1.)

3.3. Linear feedback

In this section we particularize our result on quasi-linear feedback to the case of a linear one. We recoverthus some results already known from the design oflinear feedback systems. At the same time we providea complete characterization of linear systems fromAFRTF point of view. To proceed we assume nowthat Dk…s† from (5) does not depend on k. Then f is¡1 or 0 so requirement …r1† is satis®ed. The closedloop becomes again (2)

T…s† ˆ kN…s†D…s† ‡ kN…s†

According to Lemma 1 …c1†

limk !1

~zzj ˆ zj; j ˆ 1; . . . ; m …12†

and with Lemma 3 …c22† and its proof

~pp` ˆ ¡k1=d e…2`¡1†ºi=d ‡ O`…1†; ` ˆ 1; . . . ; d …13†

Therefore requirement …r2† is satis®ed as well, see alsohypothesis …i1† below.

An early but di� erent proof of these relations can befound in Krall (1961, Theorems 2 and 3). For (13) it isgiven there also a rate of convergence to these `rays’. Wecan state now our

Theorem 3 (Linear feedback): Assume that:…i1† m, d are given and suppose all the zeros ¡z` are

simple (for convenience, as in Theorem 2) andwith Re z` > 0.

Then…c1† hypothesis …i2† of Theorem 2 is satis®ed for every

d ¶ 1 while hypothesis …i1† of that Theorem isveri®ed if and only if d ˆ 1. For d ˆ 2 the secondcondition from hypothesis …i1† of Theorem 2 isnot satis®ed and for d ¶ 3 the ®rst condition ofthat hypothesis is violated.

…c2† If d ˆ 1 all the conclusions of Theorems 2 and 1are true including the robustness to initial valuesof Theorem 2 and the su� cient conditions con-tained in …c31† and …c4† of the same Theorem.Moreover, the AFRTF requirements in timedomain (…i1† of Theorem 1), complex plane(…i1† of Theorem 2) and frequency domain (thesu� cient conditions from …c31† of Theorem 2) aresatis®ed and also equivalent if and only if d ˆ 1.

For d ˆ 2 either property …pr4† of hypothesis …i1† ofTheorem 1 is not satis®ed or if …pr4† is true then …pr2† isnot; for d ¶ 3 property …pr4† of that hypothesis is vio-lated. Therefore the conclusions of Theorem 1 are notapplicable for d ¶ 2.

…c3† The following classi®cation holds


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if d ˆ 1; limk ! 1 sup! jT…i!†j ˆ 1 ˆ limk ! 1 jT…i!†jfor every given !;

if d ˆ 2; limk ! 1 jT…§i��k

p…1 ‡ o…1††j ˆ 1;

if d ¶ 3; jT…i!0†j ˆ 1for some finite !0 and 0 < k0 < 1:

8>>>>>><

>>>>>>:

For d ˆ 1 we have recovered the boundedness in ! uni-formly in large k > 0 of T…i!†, from Lemma 2. The cased ˆ 2 is compatible with Lemma 2 …c2†, …c3†. If d ¶3 T…i!† becomes unbounded at a certain ®nite k and !so the present case complements Lemma 2 …c2†, …c3† wherethe boundedness is for large k only.

Remark 7: This theorem ensures, for d ˆ 1, AFRTFbehaviour in time domain in the sense of Theorem 1.If we wish to have arbitrarily fast tracking for d > 1while remaining within the realm of linear feedbacksystems one possibility is to relax the hypothesis of thisTheorem and let m depend on k. Then we get into aniterating procedure of loop shaping of L…i!† resem-bling the QFT design (Horowitz and Sidi 1972), tocombat perpetually the destabilizing e� ect of two ormore poles in excess of zeros of L…s†. The di� erencefrom the QFT design is that now k goes to in®nity,and so will the number of poles and zeros, therefore aproof of convergence of this method is necessary.

4. Examples

In this section we present two readily computableexamples which illustrate the e� ectiveness and ease ofquasi-linear feedback to achieve arbitrarily fast androbust tracking by feedback, in the sense of Theorem1 …c1†. We compute only the transfer function Tk…s†from (6) and analyse its properties. The robustnessto disturbances and initial conditions follows fromTheorems 2 or 3. The requirements …r1† and …r2† aresatis®ed in these examples.

At the same time we show the capability of quasi-linear feedback to reduce the control e� ort while ensur-ing the good performance mentioned before. The thirdexample presented shows that by quasi-linear feedbackit is possible to alleviate the limitation to performanceimposed by a non-minimum phase plant.

4.1. Plant: integrator

The transfer function is P…s† ˆ 1=s and the excess ofthe number of poles over zeros is d ˆ 1. ThereforeAFRTF can be obtained with a linear compensator(see Theorem 3). Indeed, with the simple compensatorG…s† ˆ k we obtain the closed loop

T…s† ˆ P…s†G…s†1 ‡ P…s†G…s† ˆ k

s ‡ k

It is straightforward to check that T…s† has all the (good)properties of Theorem 3. Because it may be a usefulillustration we compute

hk…t† ˆ L¡1…T…s†† ˆ k e¡kt

Hk…t† ˆ L¡1…T…s†=s† ˆ…t

0

hk…½† d½ ˆ 1 ¡ e¡kt

Hk…t† ˆ ¬k ¡ e¡~ppt k…t†

~pp arbitrary real positive number;

¬k ˆ 1; k…t† ˆ e…¡k‡~pp†t

which shows directly the validity of the AFRTF proper-ties …¤† from the beginning of } 3, and therefore ofTheorem 1 too. By Theorem 3 …c2† no strictly properlinear feedback can achieve AFRTF because d would bebigger than 1.

Nevertheless, AFRTF can be achieved with a strictlyproper but quasi-linear feedback. For instance Gk…s† ˆk=…s ‡ 2±

��k

p†; k > 0; 0 < ± µ 1. The closed loop now

becomes

Tk…s† ˆk

s2 ‡ 2±��k

ps ‡ k

For ®xed k this is the transfer function of a lineardamped oscillator with poles ¡~pp1;2 ˆ ¡±

��k

p§

i��k

p ��1 ¡ ±2

p. If k is allowed to increase unboundedly

we see that all the hypotheses , and therefore conclu-sions, of Theorem 2 are satis®ed so we have an arbitra-rily fast and robust tracking property. As before theAFRTF properties …¤† can be veri®ed directly

hk…t† ˆ L¡1…Tk…s†† ˆ k1=2 e¡k1=2±t

…1 ¡ ±2†1=2sin …k1=2…1 ¡ ±2†1=2t†


0

hk…½† d½

ˆ 1 ¡ e¡k1=2±t ±

…1 ¡ ±2†1=2sin ……1 ¡ ±2†1=2k1=2t†

"

‡ cos ……1 ¡ ±2†1=2k1=2t†#

Hk…t† ˆ ¬k ¡ e¡~ppt k…t†

~pp arbitrary real positive number; ¬k ˆ 1

k…t† ˆ e…¡k1=2±‡~pp†t ±

…1 ¡ ±2†1=2sin ……1 ¡ ±2†1=2k1=2t†

"

‡ cos ……1 ¡ ±2†1=2k1=2t†#

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Remark 8: One of the advantages of quasi-linearfeedback over the linear one is that it greatly reducesthe control e� ort (and the sensor noise) required to ob-tain a certain performance. The transfer function ofthe control e� ortÐcomputed from the input to theclosed loop to the input of the plantÐhas the expres-sion Ce ˆ G…s†=…1 ‡ P…s†G…s††. A `too high’ control ef-fort is not practical because it creates a large controlsignal at the plant input (particularly at high frequen-cies) which can saturate it.

Thus for the linear feedback we have for any given k

lim!!1

jCe…i!†j ˆ lim!!1

k!��!2 ‡ k2

p ˆ k

On the other hand for quasi-linear feedback, with itsgreater number d from (6),

lim!!1

jCe…i!†j ˆ lim!!1

k!��…¡!2 ‡ k†2 ‡ 4k±2!2

q ˆ 0

Even if we consider ! approaching in®nity as kx x > 0;the quasi-linear feedback retains its advantage. Indeedfor linear feedback the fastest growth with k of jCe…ikx†jis as k=

��1 ‡ …k=kx†2

q, obtained for x > 1. In the quasi-

linear case the fastest growth with k of jCe…ikx†j is as��k

p=…2±†, obtained for x ˆ 1=2.

4.2. Plant: double integrator

Its transfer function is

P…s† ˆ 1

s2…14†

and because d ˆ 2 > 1 arbitrarily fast tracking per-formance cannot be obtained with a linear feedback,Theorem 3 …c2†.

We employ, therefore, a quasi-linear feedback of theform

Gk…s† ˆk…s ‡ z†s ‡ ak f

; z > 0; a > 0; 0 < f < 1; k > 0

giving the closed loop

Tk…s† ˆ P…s†Gk…s†1 ‡ P…s†Gk…s† ˆ k…s ‡ z†

s3 ‡ ak f s2 ‡ ks ‡ kz

We check now hypothesis …i1† of Theorem 2. TheHurwitz stability conditions ak f > 0 k > 0 kz > 0and ak f k > kz are satis®ed for large k. (Without azero at the numerator the stability conditions cannotbe ful®lled (Gantmacher 1966, (82) p. 218).)

Since f < 1 it follows from Lemma 1 that thedenominator has a zero ¡~zz which goes to ¡z when kapproaches in®nity. To ®nd the behaviour of the poles¡~pp1;2 of the characteristic polynomial we divide it bys ‡ ~zz and get

s2 ‡ …ak f ¡ ~zz†s ‡ …k ¡ ~zz…ak f ¡ ~zz††

With the condition of perfect divisibility (remainderidentically zero)

kz ¡ …k ¡ ~zz…ak f ¡ ~zz††~zz ² 0

we obtain the equation

s2 ‡ …ak f ¡ ~zz†s ‡ kz=~zz ˆ 0 …15†

having solutions for large k (when ~zz º z)

¡~pp1;2 º ak f

2…¡1 §

��1 ¡ …4=a2†k…1¡2f †

q†

Explicitly

¡~pp1;2 º 12…¡ak f § i2k1=2†; if 2f < 1

¡~pp1;2 º ak1=2

2…¡1 §

��1 ¡ 4=a2

q†; if 2f ˆ 1

¡~pp1 º …¡1=a†k1¡f ; ¡~pp2 º ¡ak f ; if 2f > 1

9>>>>>=

>>>>>;

For the last case ¡~pp1 was obtained by expanding thesquare root around » ˆ 0, where » ˆ k1¡2f . Now sincea > 0 and z > 0 hypothesis …i1† of Theorem 2 is satis®edfor every 0 < f < 1. (If f ˆ 1=2 we consider for conve-nience a 6ˆ 2.) Hence, by Theorem 2 …c11† the doubleintegrator with a well-designed quasi-linear feedbackachieves AFRTF. Also, the left-hand side of (15) pro-vides an example of computing the polynomial Pd…s†from Lemma 3, and illustrates conclusions …c21† and…c22† of that Lemma as well.

Remark 9: If we consider the normalization s ˆ so=…44296000†1=2 the plant (14) in the original variableso, represents the simpli®ed model of a voice coilmotor (VCM) actuator (see Chen 2000, (14.3.1±2)p. 369). Therefore, our quasi-linear feedback can beused for the practical design of hard disk drive servosystems as it was successfully done in the aforemen-tioned reference (with state space methods). It is usefulto generalize the simple quasi-linear feedback designprocedure presented here, including a study of whenthe rationality of the exponents in …r1† can be relaxed(as in this example).

4.3. Plant: non-minimum phase

The notion of quasi-linear feedback is useful alsooutside the class of systems considered here, in particu-lar for non-minimum phase ones. They have zeros in theright half of the complex plane, which limit the raise ofthe gain k (due to stability) hence the achievable per-formance too.

Indeed, let us consider the linear feedback system


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T…s† ˆ kL…s†1 ‡ kL…s† ; L…s† ˆ …a ¡ s†

…s ‡ b†…s ‡ c†a > 0; b > 0; c > 0; k > 0

with the characteristic equation s2 ‡ …b ‡ c ¡ k†s ‡…bc ‡ ka† ˆ 0. For stability it is necessary and su� cientthat b ‡ c ¡ k > 0 and bc ‡ ka > 0, or k < b ‡ c.

On the other hand for the quasi-linear feedback

Tk…s† ˆ kLk…s†1 ‡ kLk…s† ; Lk…s† ˆ …a ¡ s†

…s ‡ b†…s ‡ ck f †a > 0; b > 0; c > 0; f < 1

the characteristic equation is s2 ‡ …b ‡ ck f ¡ k†s ‡…bck f ‡ ka† ˆ 0. The necessary and su� cient conditionfor stability is now k < b ‡ ck f .

We see that in the second case the constraint on k forstability is weaker than in the ®rst case and, moreover, itvanishes for f > 1 and large enough k. This considerableincrease available in k can be used to improve perform-ance.

5. Proofs

Proof of Lemma 1: …c1† The property that for all `,~zz` 6ˆ z` for any k follows from the assumption that Dk…s†and N…s† are coprime for all k.

The rest of the proof extends the (short) proof basedon the residue Theorem in Krall (1961, Theorem 3) to k-depending zeros of Dk…s†. With 0 denoting derivativewith respect to s we have with (5)

1

2ºi

I

js¡zj jˆ°

…Dk…s† ‡ kN…s††0

Dk…s† ‡ kN…s†ds

ˆ1

2ºi

I

js¡zj jˆ°

N 0…s†N…s† ds

‡1

2ºi

I

js¡zj jˆ°

1

k

D0k…s†N…s† ¡ Dk…s†N 0…s†

N…s†…Dk…s†=k ‡ N…s†† ds

…16†

In the above equation the left-hand side and the ®rstterm of the right-hand side represent the number ofzeros of Dk…s† ‡ kN…s† respectively of N…s† in the(open) disc with centre zj and radius ° chosen so thatno zero is on the boundary (see also Rudin 1970,Theorem 10.30, p. 215). Recall that the number ofzeros zj is ®nite. This choice of ° is possible for k su� -ciently large because, by the expansion in (6), the zerosof Dk…s† ‡ kN…s† either extend to in®nity or approachsome ®nite points when k goes to in®nity.

If ° is small enough for N…s† will be only zj, multi-plicity included, in that disc. On the other hand theintegrand of the last term of this relation can be madearbitrarily small for any such °. Indeed, on the boundaryof the disc s ˆ zj ‡ ° ei³, 0 µ ³ µ 2º, where all zj are

independent of k. Hence the increase in k of Dk…s†=k isas A°k

f ¡1 for large k, where f < 1 by assumption and A°

is a positive number not depending on k. Now N…s† isindependent of k. Thus for every small enough ° thedenominator is bounded away from zero for su� cientlylarge k while the numerator divided by k can be madearbitrarily small for such k. Hence the integral becomeszero for large enough k because it is small and also thedi� erence of two integers.

Therefore the number of zeros of Dk…s† ‡ kN…s† inthe disc with radius ° and centre zj coincides with theorder of zero zj, which proves this point if we considerarbitrarily small values of ° > 0 in (16).

…c2† is proved at Lemma 3 …c1†. &

Note to the proof: These arguments do not workdirectly to prove the converse implication. One reasonis that there may be cancellations in the numerator ofthe second integral from the right-hand side of (16),resulting in the loss of the maximal exponent of k, f ,there.

Proof of Lemma 2: …c11† Let be an arbitrary com-pact set of frequencies S. Then in view of (5) De®ni-tion 2Ðtherefore hypothesis …i1†Ðis equivalent to

limk ! 1

max!2S

Dk…i!†Dk…i!† ‡ kN…i!†

0

Now for su� ciency we assume that f < 1 and shouldprove that …i1† is true. (The necessary condition for …i1†to be true, given by the reasoning after De®nition 2, issatis®ed due to the hypothesis that N…s† has no imagin-ary zeros.) Thus because N…s† does not vanish on theimaginary axis and S is compact both by assumption,there is a positive ®nite number MS such that for k ¶ 1we have

max! 2 S

jDk…i!†=kN…i!†j µ kmaxf0;f g

kMS; MS < 1

This leads to the above limit and therefore to …i1†.For necessity we argue by contradiction showing

that if f ¶ 1 there are some frequencies ·!! (compactsets) for which the above limit is non-zero, i.e. …i1† isnot veri®ed. The frequency ·!! can be any point forwhich the polynomial ~DDk…i!† multiplying k f in Dk…i!†is non-zero. Such a point exists for any ®nite f ; if not thede®nition of f as maximal exponent of k is contradicted.( ~DDk…i!† ² 0 if and only if f ˆ ¡1, i.e. Dk…s† ˆ sm‡d ; inthis case however f < 1.) Now if f > 1 the limit is 1instead of 0. For f ˆ 1 the limit is non-zero but ®nitewhen ~DDk…i·!!† ‡ kN…i·!!† 6ˆ 0, and in®nite otherwise.Therefore we should have f < 1. Which proves thispoint.

…i1† ) …c12† Since …i1† is supposed true then by thenecessity part of the previous point we have f < 1. Thus

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the hypothesis of Lemma 1 is satis®ed so our conclusionis true because it coincides with the conclusions of thatlemma.

For the second and ®rst conditions we give addi-tional, direct proofs. For the second condition note byLemma 1 that Tzk…i!† from (6) tends to 1 uniformly in !when k goes to in®nity. Indeed, because N…s† has noimaginary zeros owing to …i1† assumed true, everyblock …i! ‡ z`†=…i! ‡ ~zz`† goes to 1 uniformly in !when k tends to in®nity, therefore their product toosince m is ®nite. So we have to study only Tdk…s† from(6). Thus the second property of …c12† follows by con-sidering …i1† and the compact set f! ˆ 0g.

For the ®rst property note, by contradiction, that if a~pp` is identically zero then from (6) and the fact thatTzk…0† approaches 1 (see above), we get jTk…0†j ˆ 1.Thus not 1 as expected from hypothesis …i1†. Supposenow, again by contradiction, there are `0 numbers suchthat n1 µ 0; . . . ; n`0

µ 0. It is useful to express Tdk…i!† as

Tdk…i!† ˆ k=¦d`ˆ1~pp`

…i!=~pp1 ‡ 1† ¢ ¢ ¢ …i!=~ppd ‡ 1†…17†

Then because ~pp1; . . . ; ~pp`0are bounded they cannot com-

pensate for the increase of the compact set of frequen-cies from the denominator of (17), increase which willforce Tdk…i!† towards zero. Recall that the numerator of(17) tends to 1 as before, as does Tzk…i!† because thereare no imaginary zeros (see above). This contradictsassumption …i1† and thus proves this point.

…c12† ) …i1† It is straightforward by using (17) andthe convergence to 1 uniformly in ! of Tzk…i!† from (6),proved as in the previous point. The latter property is sobecause N…s† has no imaginary zeros by assumption.

…c2†()…c3† We have to ®nd all the cases whichensure that Tdk…i!† is uniformly bounded becauseTzk…i!† approaches 1 uniformly in ! when k goes toin®nity. This last property was proved at the point…i1† ) …c12† using the fact that N…s† has no imaginaryzeros which is true here by assumption. With (6), (9) andby dropping the index of the o…1† functions we get

jTdk…i!†j ˆYd

`ˆ1

1

‰…!k¡n` ‡ b`knÌ

¡n` …1 ‡ o…1†††2 ‡ …a`kn`R

¡n` …1 ‡ o…1†††2Š1=2

We begin with the case when all b` ˆ 0 in (6). Then fromthe de®nition in (6) and Lemma 1 …c2† we have that allnÌ

ˆ ¡1, all a` 6ˆ 0 and all n`Rˆ n`. Hence both …c2†

and …c3† are veri®ed.Next we argue by contradiction so we have to ®nd

the conditions for which limk ! 1 sup! jTdk…i!†j ˆ 1.Thus we see that if an a` ˆ 0 then Tdk…i!† becomes in®-nity for the frequency !k ˆ ¡b`k

nÌ …1 ‡ o…1†† which can-cels the ®rst term of the same factor (no need of limithere). There is in fact a family indexed by k of ®nitecancelling frequencies, but we are concerned with k

going to in®nity according to the de®nition of bounded-ness of Tk…i!† from …c2†. The number b` is non-zerobecause n`R

ˆ ¡1 (a` ˆ 0) see the convention after(6), and since necessarily nÌ

ˆ n` > 0 by Lemma 1 …c2†.Therefore we consider now all a` 6ˆ 0. There is a

sequence !!k such that limk ! 1 sup! jTdk…i!!k†j ˆ 1 ifand only if at least one factor from the denominatorapproaches zero when k goes to in®nity, while the rateof increase with k of the other factors (combined) doesnot o� set the decrease of the considered factor. (Theactual limit is needed here because all a` 6ˆ 0.) This fac-tor tends to zero if and only if each term from the squarebracket goes to zero when k tends to in®nity. Assumethat such a factor of the denominator has the index `.Then necessarily n`R

< n` so nÌˆ n`, and !!k ˆ !k ˆ

¡b`kn` …1 ‡ o…1†† which again cancels the ®rst bracket;

b` 6ˆ 0 since otherwise nÌˆ ¡1 by de®nition, see (6),

thus forcing by Lemma 1 …c2† the contradictory choicen`R

ˆ n`…> 0†. (If we consider a !!k found by equatingthe ®rst bracket of the ` factor with some oo…1† the supre-mum of Tdk…i!† would not be achieved: the ®rstbracketÐtherefore the ` factor itselfÐwould not beminimal while the behaviour in large k of the otherfactors is not a� ected.) Hence this ` factor approacheszero as ja`jk…n`R

¡nÌ†…1 ‡ o…1††. For the frequency !!k each

remaining factor j 6ˆ ` has the value

‰…¡b`kn`¡nj …1 ‡ o…1†† ‡ bjk

njI¡nj …1 ‡ o…1†††2

‡…ajknjR

¡nj …1 ‡ o…1†††2Š1=2

By hypothesis …i2† there cannot be cancellations in the®rst bracket. Therefore, considering all the situations theabove factor of the denominator behaves as

C…j;`†kmax‰…n`¡nj †;…njI

¡nj†;…njR¡nj†Š

for some constant 0 < C…j; † µ max fjajj; jbjj; jb`j; jbj ¡ b`jg.Then for large k, recalling that Tzk…i!† approaches 1

uniformly in ! and that !!k º ¡b`kn` , we have approxi-

mately

jTk…¡ib`kn` †j ˆ 1

ja`jk…n`R¡nÌ

†¦dj 6ˆ`C…j;`†k

max ‰…n`¡nj†;…njI¡nj†…njR

¡nj†Š

Thus summarizing, to be unbounded it is necessary andsu� cient that: at least one a` ˆ 0, or (if all a` 6ˆ 0) for atleast one ` both n`R

< n`…ˆ nÌ† and

n`R¡ nÌ

‡Xd

j 6ˆ`

maxf…n` ¡ nj†; …njI¡ nj†; …njR

¡ nj†g < 0

This amounts, by negation, to …c3†. When all n` areequal the sum in …c3† becomes zero with the de®nitionof the exponents from (6), so the last condition in …c3†reduces to all n`R

¶ nÌ.

Which proves the lemma. &


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Note to the proof: The frequency !!k ˆ ¡b`kn` …1 ‡ o…1††

with which the unboundedness of Tk…i!† was obtainedwhen all a` 6ˆ 0, goes itself to in®nity. This is due tothe fact that k has to go to in®nity for the terma`k

…n`R¡n`† to vanish.

Proof of Lemma 3: …c1† By construction (the pream-ble to this lemma) Pd…s† cannot have identically zerosolutions because its last coe� cient 6ˆ 0 (tends to k).Here we used also the relation among the roots andcoe� cients of a polynomial.

Now by contradiction assume that there is only onesolution ¡~pp` of Pd…s† with n` µ 0. Then because theexponent k of the last (i.e. d ‡ 1) coe� cient is 1 thesum of the exponents nj of the rest of the solutionsshould be ¶ 1. Such a sum will appear in one of theterms of the d coe� cient of Pd…s† thus making its expo-nent of k to be ¶ 1 which is contrary to its a prioriproperties. (No cancellations can occur because theexponents of the other terms are a� ected by n` µ 0.)An analogue reasoning holds for more solutions withn` µ 0, but involving correspondingly lesser order coef-®cients. These contradictions with the construction ofPd…s† prove the ®rst part of this point.

The relation limk ! 1 ¦d`ˆ1~pp`=k ˆ 1 is precisely the

property of the last coe� cient of Pd…s† described beforethe statement of this lemma, the coe� cient being ¦d

`ˆ1~pp`.The fact that n1 ‡ ¢ ¢ ¢ ‡ nd ˆ 1 is a consequence of

the previous relation.…c21† Suppose all the exponents of k in the coe� -

cients 2; . . . ; d ; of Qd…sk† are non-positive. Then when kgoes to in®nity the solutions of Qd…sk† ˆ 0 will approachthe solutions of an algebraic equations with coe� cientsindependent of k, i.e. some numbers q1; . . . ; qd indepen-dent of k. (Recall that the solutions of an algebraicequation depend continuously on its coe� cients.)Returning to the original variable s ˆ sk…k1=d† we getthe approximate solutions for large k of Pd…s† ˆ 0 tobe q1k1=d ; . . . ; qdk1=d . The numbers q1; . . . ; qd are non-zero; more precisely jq1 £ ¢ ¢ ¢ £ qd j ˆ 1 by the relationamong zeros and coe� cients of Qd…sk† (preamble to thislemma).

Conversely, assume that n1 ˆ ¢ ¢ ¢ ˆ nd ˆ 1=d . ThenPd…s† approaches a homogeneous polynomial in …s; k1=d†when k goes to in®nity. Hence the exponents of k in thecoe� cients 2; . . . ; d, of the associated polynomial Qd…sk†cannot be bigger than zero. Which proves this point.

…c22† Suppose all the exponents of k of the coe� -cients 2; . . . ; d of Qd…sk† are now negative. Then thelimiting (in k) equation of Qd…sk† ˆ 0 becomessdk ‡ 1 ˆ 0 with roots q` ˆ e…2`¡1†ºi=d ; ` ˆ 1; . . . ; d .

Thus all the solutions are uniformly distributed on theunit circle in the complex plane. By construction theroots of Pd…s† have the stated properties for large kand, moreover, all n` ˆ 1=d see …c21†.

The linear case satis®es the conditions of this pointbecause Dk…s† is independent of k. Hence all the coe� -cients of Pd…s† are also independent of k except the lastone for which the exponent of k is 1 (see the preamble tothis lemma). This helps establish formulas of the sol-utions of Pd…s†; they will be used in Theorem 3.

Thus we write Qd…sk† ˆ …sdk ‡ 1† ‡ …1=k1=d†R…sk; k†

where R…sk; k† is a remainder polynomial and forwhich the maximal exponent of k of the non-leadingcoe� cients is negative. This representation is a directconsequence of Pd…s† from above. The 1=k1=d multiply-ing R…sk; k† is the smallest decay in k which can beextracted and corresponds to the coe� cient of s…d¡1†

k

assumed non-zero in the remainder polynomial, so themaximal exponent of k of this coe� cient is zero.Consider now the solutions q` of sd

k ‡ 1 ˆ 0 determinedbefore, which are independent of k. Recall that thelimiting equation (for k ˆ 1) of Qd…sk† is sd

k ‡ 1.Then by continuity the roots of Qd…sk† ˆ 0 areq`…1 ‡ o`…1††; ` ˆ 1; . . . ; d ; the functions o`…1† aresome series in negative rational powers of k see (6).

This type of formula was employed implicitly in thepoints …c21†, …c22†. Now, however, we need more speci®cinformation, i.e. concerning the maximal exponentse` < 0 of k in o`…1†. To be a solution the sum of all`monomials’ with the same non-positive power of khas to be identically zero, when (the solution) is replacedin Qd…sk†. Proceeding in this way we get the maximalpower of k in sd

k ‡ 1 to be 1=k¡e` (qd` is cancelled by 1),

and the maximal power of k in …1=k1=d†R…sk; k† to be1=k1=d (see the properties of R…sk; k† above). Thusnecessarily all e` ˆ ¡1=d . We return now to the s vari-able, i.e. we multiply by k1=d . Therefore the solutionsof Pd…s† are: ¡~pp` ˆ k1=dq` ¡ O`…1†; ` ˆ 1; . . . ; d , so~pp` ˆ ¡k1=d e…2`¡1†ºi=d ‡ O`…1†; ` ˆ 1; . . . ; d.

…c3† The proof of the ®rst part follows from aconsequence of the Hurwitz stability criterion (Gant-macher 1966, (82) p. 218). This shows that if the rootsare in the open left half of the complex plane the co-e� cients of the polynomial should be positive if theleading coe� cient is positive, as here. Now if at leastone coe� cient has its maximal exponent of k negativethen that coe� cient tends to 0 when k goes to in®nity.Thus in the limit at least one root will be on the ima-ginary axis (in the best case when there are no zeros inthe open right half plane). Therefore, by continuity, forarbitrary large k a root will be arbitrarily close to theimaginary axis. Hence every margin separating the zerosfrom the imaginary axis will be violated. This contra-diction with the hypothesis proves that all the maximalexponents of k of the coe� cients must be non-negative.

For the second part, if f µ 0 then the equationDk…s† ‡ kN…s† approaches a linear feedback one whenk goes to in®nity (k disappears from Dk…s†). ThereforeÐby the root locus Krall (1961, Theorem 2) and also by

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the last part of the proof of the above conclusion …c22†Ðat least one branch of zeros will cross into the open righthalf of the complex plane for large k, when d ¶ 3, againby continuity. This contradiction with the location ofthe roots proves the claim.

The last statement is a direct consequence of theexpression of the poles of linear feedback systemsfrom …c22†, for d ˆ 1. For d ˆ 2 we use Krall (1961,Theorem 2) which shows that the result can be obtainedwith a ®nite sequence of lead-lag compensations.Examples: T…s† ˆ k=…s ‡ k†, ¡~pp1 ˆ ¡k, O1…1† ˆ 0, re-spectively T…s† ˆ k=……s ‡ 1†2 ‡ k†; ¡~pp1;2 ˆ ¡1 § ik1=2;O1;2…1† ˆ 1 .

Which proves the lemma. &

Proof of Theorem 1: …c1† From ®gure 3 with L…s†replaced by Lk…s† ˆ N…s†=Dk…s† from (5) and i0, o0 re-presenting the vector of initial conditions to the inputrespectively the output of the open loop we have

Yk…s† ˆ kN…s†Dk…s† ¨…s† ¡ kC…s; i0†

Dk…s†‡ C…s; f ; o0†

Dk…s†‡ W…s†

¨…s† ˆ U…s† ¡ Yk…s†

Yk…s† ˆ kN…s†=Dk…s†1 ‡ kN…s†=Dk…s†

U…s† ‡ W…s†

¡ kN…s†=Dk…s†1 ‡ kN…s†=Dk…s†

W…s†

¡kC…s; i0†

Dk…s† ‡ kN…s† ‡C…s; f ; o0†

Dk…s† ‡ kN…s†

9>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>;

…18†

The functions C…s; i0† and C…s; f ; o0† are the polynomialsin s of degrees µ m ¡ 1 and µ m ‡ d ¡ 1 with coe� -cients depending on the initial conditions of the inputrespectively output of the open loop (DiStefano et al.1976, p. 63). Since the open loop has the formkN…s†=Dk…s† the maximum exponent of k in C…s; f ; o0†is f < 1, see the requirement …r1†.

We see now that we cannot expect the initial con-ditions of the input to the open loop i0, to be attenuatedby feedback because of the k at the numerator.Therefore we consider them to be zero which is thecase in most practical applications. For the initial con-ditions of the output to the open loop o0, before apply-ing the residue Theorem to the last term of (18), acharacterization of the closed loop poles allowed bythe properties …pr1† to …pr4† from …¤† is needed, and inparticular of those having bounded real parts butunbounded imaginary ones. Such a study will be per-formed in a future work, therefore for the time being weconsider the initial conditions to the output of the openloop to be zero as well. See, however, the notes to thisproof.

Now with zero initial conditions the time responseobtained from (18) with inverse Laplace transform is

yk…t† ˆ…t

0

hk…t ¡ ½†u…½† d½ ‡ w…t†

¡…t

0

hk…t ¡ ½†w…t† d½

hk…t† ˆ L¡1…Tk…s††; t ¶ 0

Su� ciency: We assume that hypothesis …i1† is true soproperties …pr1† to …pr4† from …¤† are veri®ed. We haveto prove the required estimates and that they lead to thestated inequalities. We provide the proof only for non-constant functions from …i2†. For constant ones theproof becomes simpler because below the integrals con-taining _uu and _ww vanish.

Integration by parts in the above equation yields

yk…t† ˆ Hk…t†u…0† ‡…t

0

Hk…t ¡ ½† _uu…½† d½ ‡ w…t†

¡ Hk…t†w…0† ‡…t

0

Hk…t ¡ ½† _ww…½† d½

³ ´

(Here u…0† and w…0† are in fact the limits when tapproaches 0 from the right.) Applying (10) we obtain

yk…t† ¡ u…t† ˆ …¬ku…t† ¡ u…t†† ¡ e¡t~pp k…t†u…0†

¡…t

0

k…½† e¡½ ~pp _uu…t ¡ ½† d½

‡ …w…t† ¡ ¬kw…t†† ‡ e¡t~pp k…t†w…0†

‡…t

0

k…½† e¡½ ~pp _ww…t ¡ ½† d½ …19†

The integrals from above can be estimated in the sameway. Working with the ®rst integral we split it in two so

Ik…t† ˆ I1k…t† ‡ I2k…t† ˆ…T

0

k…½† e¡½~pp _uu…t ¡ ½† d½

‡…t

T

k…½† e¡½ ~pp _uu…t ¡ ½† d½

With …pr4† and …pr3† and hypothesis …i2† we get for arbi-trary T > 0 and large k > 0

jI1k…t†j µ B _uuB T ; t 2 ‰0; T†

jI2k…t†j µ …k;T†B _uu

…1

0

e¡®½ d½ < 1; t 2 ‰T ; 1† …20†

where …k;T† was de®ned after the properties …pr1† to…pr4† from …¤†. An analogous result holds for the inte-gral involving _ww with B _uu replaced by B _ww.

Then with the functions from …i2†, with (19), (20)Ðand therefore with …pr3†, …pr4† tooÐwe obtain for everyT > 0 and large k > 0 the required estimates


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jyk…t† ¡ u…t†j µ j¬k ¡ 1j…Bu ‡ Bw† ‡ …k;T†C…u;w;®†

‡ …B _uu ‡ B _ww†B T ; t 2 ‰T ; 1†

1 > C…u;w;®† ˆ ju…0†j ‡ jw…0†j

‡ …B _uu ‡ B _ww†…1

0

e¡®½ d½

jyk…t† ¡ u…t†j µ j¬k ¡ 1j…Bu ‡ Bw†

‡ B …ju…0†j ‡ jw…0†j†

‡ …B _uu ‡ B _ww†B T ; t 2 ‰0; T†

9>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>;

…21†

We prove now the inequalities from …c1†. Thus with…pr1† the ®rst term of the right-hand side of the ®rstestimate can be made arbitrarily small when k is su� -ciently large. For the second term we use …pr2† and theequivalence: for any T > 0, limk ! 1 …k;T† ˆ 0 ()…pr2†. Thus we obtain from the de®nition of the limitthat for every T > 0 and every » > 0 there is aK…T ; »† > 0 such that if k ¶ K…T ; »† then j …k;T†j µ ».Now 0 µ C…u;w;®† from (21) is ®nite and independent ofk; t; T ; by …i2† and …pr4†. (C…u;w;®† ˆ 0 if and only if u ² 0and w ² 0, which is not relevant for tracking.) Thereforeif » is small enough and k > 0 is su� ciently large the®rst inequality is true. To prove the second inequality itsu� ces …pr1† so (only here) T does not appear in K . Forensuring the ®niteness of the right-hand side of theseinequalities we use property …pr3†. Note that K…T ; »†depends also on the functions u, w. Which proves thesu� ciency.

Necessity: Suppose conclusion …c1† is true, i.e. theestimates are veri®ed and they lead to the stated in-equalities. We have to prove that properties …pr1† to…pr4† from …¤† are then satis®ed. The functions u…t†and w…t† belong to the class described by hypothesis…i2†.

Property …pr3† is true owing to the ®niteness of theright-hand side of the inequalities from …c1†. Property…pr4† has to be true otherwise C…u;w;®† from (21) wouldnot be ®nite because the integral

„ 10

e¡®½ d½; ® µ 0would diverge. We see from (21) that formally C…u;w;®†does not depend on ® if and only if both u, w are con-stant. Nevertheless, ® does not become a free parameter.Indeed, remaking the estimates (21)Ðin the same waythey were obtained at the su� ciency partÐ startingfrom (19) with u and w constant, we see that now

® ¶ 0 is necessary because of e¡~ppt multiplying u…0† andw…0† in (19). This is a weaker form of …pr4†. The newlycomputed C…u;w;®† ˆ ju…0†j ‡ jw…0†j.

Next, since the estimates lead to the inequalities in…c1† and that ° > 0 can be taken arbitrarily small it

follows that each term in the ®rst estimate, except thethird one, should vanish when k tends to in®nity. Therecannot be cancellations because all the terms are non-negative. Thus limk ! 1 j¬k ¡ 1j ˆ 0, i.e. …pr1† is satis-®ed. From the second term we have that for everyT > 0 limk ! 1 …k;T† ˆ 0. This follows from the factthat C…u;w;®† is non-zero, ®nite and independent ofk; t; T ; by assumption, see also (21) with …pr4† alreadyproved. Note that if u and w are constant then C…u;w;®† ˆju…0†j ‡ jw…0†j < 1, as above. The case u ² 0 and w ² 0which renders C…u;w;®† ˆ 0 is not considered because itis not relevant for tracking. Then with the equivalence:for every T > 0 limk ! 1 …k;T† ˆ 0 () …pr2† we obtainthat property …pr2† is also veri®ed. Which proves thenecessity part and conclusion …c1†.

…c2† This is straightforward because now the esti-mates (21) are independent of any particular function aslong as it belongs to the speci®ed subclass of …i2†.

…c3† A root at zero of Dk…s† means that Tk…0† ˆ 1in (5). N…0† cannot spoil this property because it is non-zero since N…s† and Dk…s† are coprime by assumption.On the other hand Tk…0† ˆ ¬k see (10). Therefore ¬k ˆ 1and u…t† and w…t† cancel out in the expression ofyk…t† ¡ u…t† from (19). Thus Bu and Bw do not appearanymore in the estimates (21). However u…0† and w…0†do remain there.

Which proves this point and the Theorem. &

Notes to the proof: Even in the framework of the pres-ent theorem, equation (7) shows the same good behav-iour concerning the initial conditions on the output tothe open loop as in Theorems 2 and 3. For this we haveused the residue theorem for the last term of (18) withDk…s† ˆ s3 ‡ a2s

2 ‡ a1s1 ‡ a0; a2 ˆ 2a ‡ k1=4; a1 ˆ a2‡k3=4 ‡ 2ak1=4; a0 ˆ a2k1=4 and C…s; f ; o0† ˆ o0

0…s2‡a2s

1‡ a1s0†‡ o10…s1 ‡ a2s

0† ‡ o20…s0†. Here o`

0 is the initialcondition for the derivative ` of the output of the openloop. Thus the worst decay (corresponding to o0

0) for thepoles ¡a § ik3=8 was k¡1=8, obtained since k3=4 cancelledout at the numerator. The behaviour corresponding tothe pole ¡k1=4 is as in Theorem 2 …c12†. We considera > 0 to satisfy …pr4† but the same result is true witha ˆ 0.

The initial conditions on the output of the openloop can be viewed as the disturbance C…s; f ; o0†=Dk…s† (see also Horowitz (1993, p. 318). Then themethod of Theorem 1 may be applied but with theadditional assumption of stability of the open loopkN…s†=Dk…s†.

Proof of Theorem 2: To begin with we need the im-pulse response and unit step response of Tk…s† from(6). Thus we have to compute

hk…t† ˆ L¡1…Tk…s††

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and


0

hk…½† d½; t ¶ 0

For this we decompose Tk…s† into easily computableelements. Since the poles are simple by …r2† and thecomment after it the partial fraction expansion ofTk…s† has the following form (see DiStefano et al.1976, p. 65)

Tk…s† ˆXm

`ˆ1

·`

1

s ‡ ~zz`

‡Xd

`ˆ1

²`c` k1

s ‡ ~pp`

…22†

c` ˆ 1

…¡~pp` ‡ ~pp1† ¢ ¢ ¢ …¡~pp` ‡ ~pp`¡1†…¡~pp` ‡ ~pp`‡1† ¢ ¢ ¢ …¡~pp` ‡ ~ppd†

²` ˆ …¡~pp` ‡ z1† ¢ ¢ ¢ …¡~pp` ‡ zm†…¡~pp` ‡ ~zz1† ¢ ¢ ¢ …¡~pp` ‡ ~zzm†

·` ˆ …¡~zz` ‡ z1† ¢ ¢ ¢ …¡~zz` ‡ z`†z‚‚‚‚‚‚‚}|‚‚‚‚‚‚‚{

¢ ¢ ¢ …¡~zz` ‡ zm†…¡~zz` ‡ ~zz1† ¢ ¢ ¢ …¡~zz` ‡ ~zz`¡1†…¡~zz` ‡ ~zz`‡1† ¢ ¢ ¢ …¡~zz` ‡ ~zzm†

£ k

…¡~zz` ‡ ~pp1† ¢ ¢ ¢ …¡~zz` ‡ ~ppd†

With the residue Theorem we get

hk…t† ˆXd

`ˆ1

²`c` k e¡~pp`t ‡Xm

`ˆ1

·` e¡~zz`t …23†

and integration in time gives

Hk…t† ˆXd

`ˆ1

²`c` k

~pp`

¡Xd

`ˆ1

²`c`k

~pp`

e¡t~pp`

‡Xm

`ˆ1

·`

~zz`

¡Xm

`ˆ1

·`

~zz`

e¡t~zz`

By rearranging the terms above we put Hk…t† into theform (10) with ~pp de®ned in (11)

Hk…t† ˆ ¬k ¡ e¡t~pp k…t†; ¬k ˆXd

`ˆ1

²`c` k

~pp`

‡Xm

`ˆ1

·`

~zz`

k…t† ˆXd

`ˆ1

²`c`k

~pp`

e¡t…~pp`¡~pp† ‡Xm

`ˆ1

·`

~zz`

e¡t…~zz`¡~pp†

9>>>>>=

>>>>>;

…24†

…c11† To prove directly property …pr1† of …¤† weexpress ®rst ¬k in an appropriate way by recallingfrom (10) that ¬k ˆ Tk…0†. Then equating Tk…0† in therepresentations (6) and (22) we get

¬k ˆXm

`ˆ1

·`

~zz`

‡Xd

`ˆ1

²`c`k

~pp`

ˆ z1 ¢ ¢ ¢ zm

~zz1 ¢ ¢ ¢ ~zzm

k

¦d`ˆ1~pp`

This formula makes sense because all ~pp` and ~zz` arenon-zero for large k by Lemma 1 …c2† respectively by

hypothesis …i1† and Lemma 1 …c1†. Now …pr1† is trueby applying both conclusions of Lemma 1 to the right-hand side of the previous equality; no z` is zero by …i1†.Property …pr1† can be obtained also indirectly from …pr4†,see the consequence of …pr4† after …¤†.

Property …pr4† is satis®ed owing to hypothesis …i1†,and also to Lemma 1 …c1† which shows that all ~zz` are inthe left half of the complex plane and bounded awayfrom the imaginary axis for large k > 0.

For …pr2† we distinguish in the expression of k…t†from (24) between the terms with exponent containing ~zz`

and those containing ~pp`: The latter verify …pr2† becauseby hypothesis …i1† all n`R

> 0 and since

limk ! 1

kv e¡kx

ˆ 0; for any real v; x > 0 …25†

also, the coe� cients of the exponentials have at most apolynomial growth in k (all ~pp` are non-zero for large k asbefore, and distinct by …r2†). For the former terms of

k…t†, and large k, the exponentials are bounded bythe de®nition of ~pp from (11) but the coe� cients go tozero when k tends to in®nity. Indeed, all ~zz` are boundedaway from zero for large k by …i1† and Lemma 1 …c1†,and the ®rst limit from below concludes the argument

limk ! 1

·` ˆ 0; limk ! 1

²` ˆ 1 …26†

Both limits were obtained by using Lemma 1 …c1†, …c2† inthe expression of ·` respectively ²` from (22). For the®rst limit we used also the fact that all ~zz` are distinct forlarge k see the comment after requirement …r2†.

To prove …pr3† note that all the exponentials in k…t†from (24) are bounded in t ¶ 0 independently of large kby the de®nition of ~pp from (11). In view of (26) thecoe� cients containing ·` in (24) are bounded withrespect to large k (in fact we showed above that theygo to 0). For the remaining coe� cients, ²`c`k=~pp`, wehave ®rst that all ~pp` are bounded away from zero forlarge k by Lemma 1 …c2†. Next, by …r2† the exponent of kin the expression …~pp` ¡ ~ppj† is the maximum between theexponents of ~pp` and ~ppj for every ` and j 6ˆ `. Thus thesum of the exponents of k at the denominator of eachcoe� cient is at least 1 according to c` from (22) and (9)valid due to Lemma 1. Hence, together with the secondlimit of (26) also the coe� cients involving ²` arebounded with respect to large k, because the exponentof k at the numerator is 1. Therefore k…t† is bounded int ¶ 0 independently of large k. Which proves …pr3† andthis point.

…c12† The properties …pr1† to …pr4† of …¤† were ver-i®ed above. Therefore all the conclusions of Theorem 1are true for the class of functions …i2† from there.

To estimate the contribution of the initial conditionson the output of the open loop we have to compute from(18) the term (27) (see below).


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In (27) we have used the residue Theorem with thedenominator in the representation (6). We need also theexplicit expression of the function C…s; f ; o0† fromDiStefano et al. (1976, p. 63). Below the component o`

0

of the vector o0 is the initial condition on the ` derivativeof the output of the open loop. Thus if we write thepolynomial from (5) as

Dk…s† ˆ sm‡d ‡ ddm‡ d¡1sm‡ d¡1 ‡ ddm‡ d¡2sm‡ d¡2

‡ ¢ ¢ ¢ ‡ dd1s ‡ dd0 …28†

then

C…s; f ; o0† ˆ o00…sm‡ d¡1 ‡ ddm‡ d¡1s

m‡ d¡2

‡ ¢ ¢ ¢ ‡ dd2s ‡ dd1†

‡ o10…sm‡ d¡2 ‡ ddm‡ d¡1s

m‡ d¡3

‡ ¢ ¢ ¢ ‡ dd2† ‡ ¢ ¢ ¢ ‡ om‡ d¡10 …1†

Now for the second sum from (27) we see that the expo-nentials are bounded independently of t ¶ 0 and large kbecause by …i1† and Lemma 1 …c1† all ¡Re ~zzj < 0 forsuch k. However, the coe� cients of these exponentialsare decaying with k. Indeed, the power of k at thenumerator is at most f < 1 (requirement …r1†) because,we recall, ~zz` has no positive exponent of k in its expan-sion while C…s; f ; o0† itself has f < 1 as maximal expo-nent of k. On the other hand the denominator of thesecoe� cients has the exponent of k equal to 1 in view ofthe product ¦d

`ˆ1~pp` which can be extracted there, andbecause all ~zzj are distinct for large k (see the commentafter …r2††. This product tends to k by Lemma 1 …c2††.Therefore, the second sum decays with k as 1=k1¡f ;f < 1:

For the ®rst sum from (27) all the terms decay expo-nentially to zero when k goes to in®nity, uniformly int ¶ T > 0, for every T > 0. This is due to a` > 0 andn`R

> 0 in all the exponentials see …i1† and (25), while thecoe� cients have at most a polynomial increase in k (thepoles are distinct by …r2†).

We prove now that in fact all the coe� cients arebounded with respect to large k. The exponent of k atthe denominator of the coe� cient ` is at least1 ¡ n` ‡ mn`. Indeed, 1 ¡ n` is obtained as in the proofof property …pr3† of conclusion …c11† noting that now

~pp` is missing from the denominator, hence the ¡n`.

To evaluate the exponent of k at the numerator we

use the identity Dk…¡~pp`† ‡ kN…¡~pp`† ² 0 which is true

because ¡~pp` is a pole of Tk…s†. So the exponent of k in

Dk…¡~pp`† is exactly 1 ‡ mn` because the coe� cients ofN…s† are independent of k and n` > 0 (see Lemma 1

…c2††. Now by inspection, in the polynomial from (28)

multiplying o00, we see that for each coe� cient dd`; ` ¶ 1,

the power of s decreases with one compared to Dk…s†and dd0 does not even appear. Thus the maximal expo-nent of k of the polynomial multiplying o0

0 is at most

1 ‡ mn` ¡ n` (could be less if there are cancellations).

The exponents of k in the polynomials multiplying the

other initial conditions are even smaller. This proves theclaim of boundedness of the coe� cients because the

exponent of k at the denominator is at least as big as

that from the numerator.

Now by …i1† we have for the pole representation in

(6) that a ˆ min a` > 0 and nR ˆ min n`R> 0. Therefore,

by summarizing the previous arguments, we obtain theestimates from the statement of this conclusion. They

lead (directly) to the extension of conclusions …c1† and

…c2† of Theorem 1 to the robustness in the initial con-

ditions on the output of the open loop too. Which

proves this point.…c2† First we show by counterexample that hypoth-

esis …i2† is needed to prove this point. So let us consider

equation (7). Then the properties …pr1† to …pr4† of …¤† are

satis®ed but not …i1† because n2;3Rˆ 0. Hypothesis …i2†

will prevent the occurrence of this phenomenon, i.e. if aterm of k…t† from (24) with ~p`p` at the exponent decays to

zero with k it will decay due to the exponential, not the

coe� cient.

We return now to the proof where we have to showthat properties …pr2† and …pr4† from …¤† imply hypoth-

esis …i1†. Thus from …pr4†Ðsee also its consequence

after …¤†Ðit follows that all a` > 0, all n`R¶ 0, and all

Re z` > 0. Therefore to prove …i1† we have to show only

that all n`R> 0. In view of …pr2† we study the behaviour

of every term of k…t† when k goes to in®nity. We begin

with the terms which have ~zz` at the exponent. They go to

zero when k tends to in®nity as in the proof of …pr2†from conclusion …c11†: …pr4† is now assumed true

so all Re z` > 0 as in hypothesis …i1† which was used in…c11†.

462 M. Kelemen

Co…t; k† ˆ L¡1…C…s; f ; o0†=…Dk…s† ‡ kN…s†††

ˆXd

`ˆ1

C…¡~pp`; f ; o0†…¡~pp` ‡ ~pp1† ¢ ¢ ¢ …¡~pp` ‡ ~pp`¡1†…¡~pp` ‡ ~pp`‡1† ¢ ¢ ¢ …¡~pp` ‡ ~ppd†…¡~pp` ‡ ~zz1† ¢ ¢ ¢ …¡~pp` ‡ ~zzm† e¡~pp`t

‡Xm

jˆ1

C…¡~zzj; f ; o0†…¡~zzj ‡ ~zz1† ¢ ¢ ¢ …¡~zzj ‡ ~zzj¡1†…¡~zzj ‡ ~zzj‡1† ¢ ¢ ¢ …¡~zzj ‡ ~zzm†…¡~zzj ‡ ~pp1† ¢ ¢ ¢ …¡~zzj ‡ ~ppd†

e¡~zzj t …27†

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We continue by proving that the coe� cients ²`c`k=~pp`

in k…t† have a ®nite, di� erent from zero limit when kgoes to in®nity. Indeed with hypothesis …i2† all n` ˆ 1=d .On the other hand by …r2† the maximal exponent of k ofeach factor at the denominator is 1=d . Hence the sum ofthe exponents of k at the denominator of each suchcoe� cient is d=d ˆ 1, which equals the exponent of kat the numerator. Therefore, with the second limit from(26), all the coe� cients are bounded away from zero,and moreover are bounded (as in the proof of …c11†),and have a di� erent from 0 limit by the asymptoticexpansion of ~pp` from (6). Thus when limk! 1 k…t† ˆ 0,according to …pr2†, no exponential term from (24)(including coe� cient) goes to zero when k goes to in®-nity, uniformly (or not) in t ¶ T , T arbitrary positive,unless all n`R

> 0, i.e. all Re ~pp` ! 1.It remains the possibility that there are some n`R

ˆ 0but the sum of the corresponding terms in k…t† vanisheswhen k ! 1, uniformly in t ¶ T , T arbitrary positive.This cannot happen either except when all n`R

> 0.Arguing by contradiction assume that n1R

ˆ n2Rˆ

¢ ¢ ¢ n…`0¡1†Rˆ n`0R

ˆ 0. We show, for some t > 0 (in factany) and some unbounded sequence fknng, that this sumis not approaching zero, thus …pr2† is contradicted. Forthis we evaluate for large k > 0 the sum of the corre-sponding terms k…t† from (24) for t, 2t; . . . ; t…`0 ¡ 1†,t`0, t > 0 arbitrary. Then with (6) and …i2† we get

k…t†

k…t2†

¢ ¢ ¢

k…t…`0 ¡ 1††

k…t`0†

2666666664

3777777775

º

e¡…a1‡ib1k1=d ¡~pp†t ¢ ¢ ¢ e¡…a`0‡ib`0

k1=d ¡~pp†t

e¡…a1‡ib1k1=d ¡~pp†t2 ¢ ¢ ¢ e¡…a`0‡ib`0

k1=d ¡~pp†t2

¢ ¢ ¢

e¡…a1‡ib1k1=d ¡~pp†t…`0¡1† ¢ ¢ ¢ e¡…a`0‡ib`0

k1=d ¡~pp†t…`0¡1†

e¡…a1‡ib1k1=d¡~pp†t`0 ¢ ¢ ¢ e¡…a`0‡ib`0

k1=d ¡~pp†t`0

2666666664

3777777775

£

²1c1k=~pp1

²2c2k=~pp2

¢ ¢ ¢

²…`0¡1†c…`0¡1†k=~pp…`0¡1†

²`0c`0

k=~pp`0

2666666664

3777777775

Above we have neglected o…1† in the expressions1 ‡ o…1† from the matrix so we have used the approx-imation sign, which is the ®rst step of our proof.

Note that the determinant is a Vandermonde one(Bellman 1970, p. 193). It is bounded away from zeroif the entries of the ®rst row are bounded away fromzero (they are), together with all the di� erence functionsobtained by subtracting any two elements (from distinctlocations) of this row. It is our case for any given t > 0and for some sequence fknng which tends to in®nity. Thissequence is found as follows. Consider the worst situa-tion when all a` are equal. By requirement …r2† all b` aredistinct. Therefore the set of singular k for which the

di� erence functions mentioned above vanish is: k…j1;j2† ˆ…2ºL=…bj1

¡ bj2†t†d ; 1 µ j1 < j2 µ `0; L ˆ 0; 1; 2; ¢ ¢ ¢ : . It

is a discrete set of points of the real axis. We take now asmall open set around zero in the complex plane and getthe intersection of its inverse images through all thedi� erence functions, on the real k axis. This is an openset because the functions are continuous and their num-ber is ®nite. If the open set around zero is su� cientlysmall the inverse image will shrink towards the k…j1 ;j2†points in disjoint open subsets. (The open set in thecomplex plane has not to shrink to zero for gettingthese disjoint open subsets because all the functionsare uniformly continuous.) Thus on the complement ofthe union of these disjoint open sets all the di� erencefunctions are bounded away from zero. Hence fknng canbe selected from this complement which, by construc-tion, extends to in®nity. Therefore for any given t > 0and all knn the determinant is bounded away from zero.

The next step is to consider o…1† in the matrix sothe equality is restored above. For k large enough thediscrete sets indexed by o…1† where the determinantvanishes are close enough to the discrete set fk…j1 ;j2†gfound before (in the limit they coincide with it). Thusthe complement set established above still exists (dimin-ished to a small degree depending on how large k is).Therefore for any given t > 0 and all knn from this new(still unbounded) set the determinant is bounded awayfrom zero.

Finally we showed before that the components of thevector multiplying the matrix are bounded away fromzero for large k. Thus the right-hand side of the vectorequality is bounded away from zero for any given t > 0and large knn, otherwise at least one of the two precedingproperties would not be satis®ed. Therefore at least onecomponent of the vector from the left-hand side isbounded away from zero for a given t > 0 and largeknn. Hence …pr2† is violated so we must have all n`R

> 0.Then hypothesis …i1† of this Theorem is satis®ed, whichproves this point.

…c31† Suppose Tk…i!† is bounded in ! uniformly inlarge k and the poles of Tk…s† are Hurwitz stable with a®xed stability margin for large k. Then the conditions ona` and z` of …i1† are satis®ed in view of the Hurwitzstability mentioned above, and with Lemma 1 …c1†.From the uniform boundedness of Tk…i!† and hypoth-eses …i2†, …i3† we haveÐvia Lemma 2 …c2†, …c3†Ðthatn`R

¶ nÌfor all `. Now the ®rst property of Lemma 1

…c2† leads to all n`R> 0. Lemma 2 is applicable because

no zero of N…s† is imaginary which was just proved atthe beginning of this point. Hence hypothesis …i1† isveri®ed.

The converse implication is not true in general asshown by equation (8). All the provisions of hypotheses…i1†, …i2† and …i3† are satis®ed. However jTk…§ik1=2†j goesto in®nity together with k.


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Nevertheless, hypothesis …i1† impliesÐwith Lemma 1…c1†Ðthe Hurwitz stability with a ®xed stability marginfor large k, of the poles of Tk…s†.

…c32† This is a direct consequence of …c3† and …c2†from Lemma 2. We use here hypotheses …i2†, …i3†, and…i1† which ensures that all a` 6ˆ 0…> 0† and no z` is ima-ginary.

…c4† To prove that hk…t† from (23) converges to theDirac distribution we have to show that the threerequirements from (1) are satis®ed. The ®rst one followsby using …r2†, …i1†, (25), (26) for the terms with exponen-tial containing ~pp` in (23); we employ …r2†, (26), …i1† forthe other terms. The second property is a weaker versionof the result on Hk…t† from Corollary 1(B) (the unifor-mity in t is not required now). For the third property wehave to prove the boundedness independently of large kof two types of expressions

²`c` k

Re ~pp`

¡ ²`c`k

Re ~pp`

e¡tRe ~pp` ;·`

Re ~zz`

¡ ·`

Re ~zz`

e¡t Re ~zz`

obtained by upper bounding the absolute value of eachterm of hk…t† from (23) and then integrating in time. Thesecond expression and the second term of the ®rstexpression go to zero when k tends to in®nity as in theproof of property …pr2† from conclusion …c11†, whichinvolves …i1†. With …r2† we have proved in …pr3† of…c11† that ²`c` k=~pp` was bounded independently oflarge k. The same argument applies to the ®rst term ofthe ®rst expression if we show that all n` ˆ n`R

. Indeed,with …i2†, …i3† and Lemma 2 …c2†, …c3†Ðvalid in view of…i1† which ensures that no zero of N…s† is imaginaryÐwehave all n`R

¶ nÌ. Thus in the requirement …r2† all

n` ˆ n`Rand so the remaining expression is bounded

independently of large k. Hence the third property of(1) is satis®ed.

Which proves this point and the theorem. &

Notes to the proof: For proving …c2† we did not em-ploy explicitly …pr3† of …¤† but a stronger property: thecombination of requirement …r2† and hypothesis …i2†.This ensured not only the boundedness of the coe� -cients ²`c`k=~pp` from (24) but that they are boundedaway from zero as well. Property …pr3† is true as in theproof of point …c11† but with hypothesis …i1† replacedby …pr4†; recall that …i2† was not employed to prove…pr3†.

We did not use …pr1† either since it is not related to…i1†. However …pr1† is true as well as a consequence of…pr4†.

Proof of Theorem 3: …c1† From (13) we have forevery d that all n` ˆ 1=d so hypothesis …i2† of Theorem 2is satis®ed for each d.

Suppose now d ˆ 1. Then from (13) we have a1 ˆ 1,n1R

ˆ 1 and n1Iµ 0. Hence hypothesis …i1† of Theorem 2

is also satis®ed, see …i1† of this Theorem too. For d ˆ 2we have again from (13) that n…1;2†R

µ 0 andn…1;2†I

ˆ 1=2. Therefore hypothesis …i1† of Theorem 2 isnot veri®ed. For every d ¶ 3 and (13) it exists at leastone `0 for which a`0

ˆ ¡ cos ……2`0 ‡ 1†º=d† < 0. Thushypothesis …i1† of Theorem 2 is not satis®ed either.

…c2† The ®rst and second part of the conclusion arestraightforward to check. We use here …i1† and …c1† ofthis Theorem and all the hypotheses of Theorem 2together with (13); compare also the proof of the pre-vious point.

For d ˆ 2 hypothesis …i1† of Theorem 1 is not validbecause of property …pr2†: this property is not veri®ed inview of Theorem 2 …c2† with …i1† of that Theorem notsatis®ed and …pr4† assumed true. If d ¶ 3 hypothesis …i1†of Theorem 1 is not veri®ed due to property …pr4† see(13). So the last statement of this conclusion is true sinceproperties …pr1† to …pr4† from …¤† are necessary and suf-®cient for satisfying conclusion …c1† of Theorem 1.

…c3† For d ˆ 1 we use ®rst the fact that Tzk…i!†from (6) goes to 1 uniformly in ! when k approachesin®nity (the proof is the one from the beginning of theproof of Lemma 2 part …i1† )…c12†). This is due to (12)and …i1† which prevents imaginary zeros. Then togetherwith (13) we see that the limit is 1 for every given !,which proves the last equality. To prove the ®rst one,note that for every k > 0 there is a !k which maximizesjTk…i!†j. However, when k goes to in®nity jTzk…i!k†japproaches 1, as before. Therefore for large k thesupremum is increasingly determined by jTdk…i!k†j ˆk=

��!2

k ‡ k2…1 ‡ o…1††q

, which is 1 in the limit; !k hasto go to 0 otherwise the supremum is not achieved.Note that if N…s† ˆ 1 then !k ˆ 0 for all k.

If d ˆ 2 the required limit is obtained by using (13)and also the limiting property of Tzk…i!† from above.

For d ˆ 3 there is at least one branch of poles whichwill cross the imaginary axis and enter into the right orleft half plane (depending on where the `rays’ originate)for large enough k, see (13). Denote by k0 > 0 and !0

the value of k and ! at this crossing. Since the denomi-nator of Tk0

…i!0† from (6) vanishes by assumption andthe numerator Tk0

…i!0† 6ˆ 0Ðno z` is imaginary see…i1†Ðwe obtained the required property.

Which proves this point, and the Theorem. &

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DiStefano, J. J., Stubberud, A. R., and Williams, I. J.,1976, Feedback and Control Systems (New York: McGraw-Hill).

464 M. Kelemen

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Krall, A., 1961, An extension and proof of the root-locusmethod. Journal of the Society of Industrial AppliedMathematics, 9, 644±653.

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