Optimal multivariable controller design using an ITSE performance index

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Optimal multivariable controller design using an ITSEperformance indexD.S. Carrasco a & M.E. Salgado aa Department of Electronic Engineering , Universidad Técnica Federico Santa María , Casilla110-V, Valparaíso, ChilePublished online: 13 Oct 2010.

To cite this article: D.S. Carrasco & M.E. Salgado (2010) Optimal multivariable controller design using an ITSE performanceindex, International Journal of Control, 83:11, 2340-2353, DOI: 10.1080/00207179.2010.520033

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International Journal of ControlVol. 83, No. 11, November 2010, 2340–2353

Optimal multivariable controller design using an ITSE performance index

D.S. Carrasco and M.E. Salgado*

Department of Electronic Engineering, Universidad Tecnica Federico Santa Marıa, Casilla 110-V, Valparaıso, Chile

(Received 5 March 2010; final version received 24 August 2010)

This article deals with the design of optimal controllers and, as an embedded problem, with the computation ofachievable performance bounds in the control of non-minimum phase, unstable multi-input multi-outputsystems. The cost function comprises a time-weighted measure of the tracking error (ITSE) and an incrementalquadratic penalisation of the control effort. The proposed methodology relies on the properties of a frequencydomain generalisation of the ITSE index. The solution to the optimisation problem can be computed usingorthonormal basis function expansions, and a closed-form expression is found for the optimal coefficients of theexpansion. Numerical examples are presented to illustrate the results.

Keywords: optimal control; multivariable systems; performance bounds; ITSE index

1. Introduction

The design of optimal controllers is primarily the resultof minimising a mathematical criterion that captures arequired control objective. This problem is usually castas the minimisation of a mixed index that penalises acombination of a measure of the tracking error and ameasure of the control effort, for a given referencesignal (Anderson and Moore 1971; Zhou, Doyle, andGlover 1996; Goodwin, Graebe, and Salgado 2001).On the other hand, performance bounds define the bestperformance that can be achieved when using a linearfeedback control loop, thus creating a benchmarkagainst which different design methodologies can becompared (see Chen and Middleton (2003) and thereferences therein). Then it is not rare that bothconcepts can be addressed under the same framework,since the performance bound of a system represents theideal optimal control goal, which is quantified onlyby the tracking error component of a cost function,without any other constraint or penalisation.

Most of the known results in optimal control andperformance bounds use standard quadratic measuresof the tracking error (ISE) (Chen, Hara, and Chen 2003;Silva and Salgado 2005), to benefit from the knownanalytic properties of its frequency domain counterpart,the 2-norm. In this article, instead of using the tradi-tional ISE/2-norm index, we focus on establishingresults with the Integral of Time Squared Error(ITSE) index (Ogata 1970), combined with a measureof the incremental control effort. In this way, thesolution to the performance bound problem is

embedded in the solution to the more general optimal

control problem.Time-weighted quadratic cost functions have been

used limitedly in the literature: algorithms have been

proposed to evaluate time weighted quadratic perfor-

mance indexes (Fukata and Tamura 1984) to design

SISO controllers with a basic structure (Ogata 1970),

as well as to design feedback gains in the control of

continuous and sampled-data systems (Fukata, Mohri,

and Takata 1981, 1983). The benefits of using this kind

of index come from the fact that it assigns small weight

to unavoidable initial errors, favouring a quick settling

of the controlled plant output.Early proposals to minimise a time-weighted index

are described in Ramani and Atherton (1974) (and the

references therein) and Dan-Isa and Atherton (1997).

Nevertheless, those approaches are restricted either to

static state/output feedback regulators or to the PID

controller structure. Here we focus instead on the more

general optimal design of unstructured, multivariable,

discrete-time controllers that minimise an ITSE crite-

rion for unstable, non-minimum phase plant models.To the best of the authors’ knowledge, a procedure

to design unstructured, multivariable, dynamic optimal

controllers, using a time-weighted (ITSE) performance

index, has not been proposed yet. Carrasco and

Salgado (2009) give a first approach to this issue, but

it is limited to non-minimum phase, stable, SISO plant

models. The contribution of this work is a methodol-

ogy to design optimal stabilising controllers and to

compute achievable performance bounds in the control

*Corresponding author. Email: [email protected]

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2010 Taylor & Francis

DOI: 10.1080/00207179.2010.520033

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of non-minimum phase, unstable and multivariablesystems, subject to a unit step reference or outputdisturbance vector, with zero steady-state trackingerror. The performance is measured using the ITSEindex of the tracking error, combined with a quadraticmeasure of the incremental control energy.

The layout of this article is as follows. Section 2defines notation and background material. Section 3defines the cost function representing the problem ofinterest and gives preliminary results regarding itsdecomposition. Sections 4 and 5 present the mainresults on performance bounds and optimal control,respectively, while Sections 6 and 7 present numericalexamples for such results. In Section 8 final conclu-sions are drawn.

2. Preliminaries

This section provides definitions, describes back-ground material and reviews some results that will beused throughout this article.

2.1 Basic notation and function spaces

For any complex number z, z represents its conjugate,and for M2C

m�n, MH denotes its Hermitian (conju-gate transpose). For a rational transfer matrixM[z]2C

m�n the operation (�)� is defined as M�½z� ¼MH½1z�, which in the real rational case reduces toM�[z]¼MT[z�1]. Note that (�)� reduces to (�)H whenz¼ ej!.L2 is defined as the Hilbert space of all matrix

functions measurable over the unit circle, with theinner product

hF,Gi ¼1

2�

Z �

��

trace F ½e j!�HG ½e j!�

� �� d!: ð1Þ

The norm induced by this inner product is knownas the 2-norm and it is denoted by k�k2. H2 and H

?2 are

subspaces of L2 containing analytic functions for jzj41and jzj51, respectively. Both spaces are orthogonalcomplements in L2, therefore F [z]2L2 admits thedecomposition

F ½z� ¼ Fs½z� þ Fu½z�, ð2Þ

where Fs[z]2H2 and Fu½z� 2 H?2 . RL2(RH1) is the

class of real rational proper (stable) transfer functions,bounded on the unit circle.

A rational matrix transfer function V [z]2Cn�n is

said to be unitary if and only if V�[z]V [z]¼ I, andtherefore V ½z�F ½z�

�� 22¼ F ½z�V ½z�� 2

2¼ F ½z�� 2

2.

2.2 Vectorisation and the Kronecker product

Denote the vectorisation of a matrix transfer functionF ½z� 2 Lm�n2 by Fv[z]¼ vec(F [z])}, where Fv½z� 2 L

mn�12

is the vector obtained by stacking the columns of F [z]into a single column. An important property of thevec(�) operator is F ½z�

�� 22¼ vec F ½z�ð Þ�� 2

2(Rotkowitz

and Lall 2005).Given two matrix transfer functions F ½z� 2 Lm�n2

and G ½z� 2 Ls�q2 , then F ½z� � G ½z� 2 L

ms�nq2 denotes the

Kronecker product of F and G. A useful propertyrelating the vec(�) operator and the Kroneckerproduct is

vec ABCð Þ ¼ ðCT� AÞvec Bð Þ ð3Þ

where A, B, C are matrices of appropriate dimensions.

2.3 Spectral factorisation

Let F [z]2L2 be a transfer function matrix that satisfiesF�[z]¼F [z] and F [1]� 0, then F [z] admits thefactorisation

F ½z� ¼ Op½z� �O�p ½z� ¼ O�c ½z� �Oc½z�, ð4Þ

where Op[z], Oc[z]2H2 and, since they are not unique,they can be chosen to have only minimum-phase zeros.An algorithm to compute this factorisation can befound in Denham (1975).

2.4 Orthonormal basis functions

Consider a SISO stable transfer function G[z]2H2. Let{Bi[z]}i¼1,2, . . . be a sequence of orthonormal functions(Heuberger, Van den Hof, and Bosgra 1995) that forma complete set in H2, so that

1

2�

Z �

��

Bk½ej!�Bl ½e

�j!�d! ¼1, ðk ¼ l Þ

0, ðk 6¼ l Þ:

�ð5Þ

Then there always exists a set of coefficients{�1,�2, . . .} such that G[z] can be written as

G½z� ¼X1i¼1

�i �Bi ð6Þ

Particular choices for this basis are the pulse functions(FIR) and the discrete Laguerre functions Heubergeret al. (1995).

2.5 D-product

Let f [k] and g[k] be discrete-time real functions ofdimensions n� n. Let F [z] and G[z] be their corre-sponding Z-transforms, analytical for jzj ¼ 1. Then it ispossible to define the product

F ½z�,G ½z��

D¼�1

2�j

IC

trace F�½z� �dG ½z�

dz

� �dz, ð7Þ

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where the integral travels counterclockwise along theunit circle C.

In the sequel, we will refer to this product as theD-product.

Lemma 2.1: The D-product has the followingproperties:

P1.

F�½z�,G�½z��

D¼�1

2�j

IC

trace F ½z� �dG�½z�

dz

� �dz: ð8Þ

P2.

F ½z�,G ½z��

D¼ � F�½z�,G�½z�

� �D: ð9Þ

P3.

F ½z�,G ½z��

D¼ G ½z�,F ½z��

D: ð10Þ

P4. If F [z]2H2 and G ½z� 2 H?2 , then

F ½z�,G ½z��

D¼ 0: ð11Þ

P5. For every constant matrix c of appropriatedimensions:

c,G ½z��

D¼ G ½z�, c� �

D¼ 0: ð12Þ

P6. Invariance to shifting

F ½z� þ c1,G ½z� þ c2� �

D¼ F ½z�,G ½z��

Dð13Þ

for every constant matrices c1, c2 of appropriatedimensions.

Proof: See the Appendix. œ

2.6 T-product

When the sumP1

k¼0 trace k f ½k�g½k�T� �

converges, it ispossible to define a time-domain product related to theD-product. We define the T-product as

f ½k�, g½k��

T¼X1k¼0

trace k f ½k�g½k�T� �

: ð14Þ

The relationship between the two products comesfrom

f ½k�, g½k��

T¼ F ½z�,G ½z��

D, ð15Þ

which is a straightforward fact to prove.

2.7 The D-product as a pseudo norm

Consider the D-product of F [z] with itself, which wedenote by D{F [z]}¼hF,F iD2R. When F [z]¼Z{f [k]}2H2, then we have

D F ½z��

¼X1k¼0

trace k f ½k� f ½k�T� �

4 0: ð16Þ

Although this particular case coincides with theHilbert–Schimdt–Hankel norm, in strict sense, theD-product does not induce a norm in L2 since it lacksthe positive definiteness property. However, a partic-ular and important property can be proved.

Lemma 2.2: Orthogonality Let F [z]2H2 andG ½z� 2 H?2 , then

D F ½z� þ G ½z��

¼ D F ½z��

þD G ½z��

, ð17Þ

where D{F [z]}40 and D{G[z]}50.

Proof: Straightforward from the definition and thelisted D-product properties. œ

Also, the main results in this article require theproperty stated in the following lemma.

Lemma 2.3: Given a matrix transfer functionF ½z� 2 Lm�n2 , then

D F ½z��

¼ D vec F ½z�ð Þ� �

: ð18Þ

Proof: Direct upon considering: (i) the vectorisationof a matrix is a linear transformation; (ii) the vec(�)operator is unitary (Khargonekar and Rotea 1991),i.e. the usual inner product is preserved and (iii) theD-product can always be written in terms of the usualinner product. œ

3. The cost function

The definition of the multivariable cost function is firstintroduced. We then present a set of decompositionswhich are necessary to ensure the existence of anoptimal solution.

3.1 Definition

We first recall that the main objective is to obtain aminimum achievable value for the ITSE index of thetracking error, in a one-degree-of-freedom control loopsubject to a unit step vector reference with a givendirection, penalising at the same time the controleffort. A characterisation of this problem is given by

Jv ¼X1k¼0

k � e½k�Te½k�

þ � �X1k¼0

u½kþ 1� � u½k�ð ÞT u½kþ 1� � u½k�ð Þ, ð19Þ

where e[k]2Rn denotes the tracking error vector and

u[k]2Rn denotes the controller output vector.

As usual, in similar optimal control problems, thepenalty on the control effort indirectly imposes asoft constraint on the speed of convergence of the

2342 D.S. Carrasco and M.E. Salgado

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closed-loop response; this penalisation depends directly

on the value chosen for the parameter �. Contrary to

the tracking error measure in the cost function, the

control signal variations are equally weighted through-

out time. This choice will damp excessive initial

variations in the actuation.Now, if we apply the definitions of the D-product

and the 2-norm to (19), then the cost function can be

expressed as

Jv ¼ D E ½z��

þ � � ðz� 1ÞU ½z�� 2

2, ð20Þ

where E[z] and U[z] are the Z-transforms of e[k] and

u[k], respectively.It is then possible to use the relationships derived

by the closed-loop sensitivity functions, i.e.

E ½z� ¼ So½z�R½z� ð21Þ

U ½z� ¼ Suo½z�R½z�, ð22Þ

where So[z] is the nominal sensitivity function, Suo[z] is

the nominal control sensitivity (Goodwin et al. 2001)

and R[z] is the Z-transform of the reference vector.An important issue is that the reference vector is a

unit step vector with direction l,

R½z� ¼1

1� z�1� m, ð23Þ

and that the optimal cost and the solution to the

optimisation problem will depend on the specific value

of this vector. A more general approach is to consider

the direction vector l as a zero-mean unit-variance

random variable, that is,

E mf g ¼ 0 ð24Þ

E mmT� �

¼ I: ð25Þ

These assumptions on the reference imply that, if

we take the average of all possible directions and

redefine the cost function to be J¼E{Jv}, then

J ¼ E D So½z�R½z��

þ � � E ðz� 1ÞSuo½z�R½z�� 2

2

n o,

ð26Þ

which leads to

J ¼ D So½z� �1

1� z�1

� �þ � � Suo½z�

�� 22: ð27Þ

For this cost function to be well defined in the time

domain and in the frequency domain, it must be

assumed that the error e[k] converges to zero expo-

nentially fast. This is always the case for a stable

control loop with integral action.

Using the results presented in Salgado and Silva

(2006), we can parameterise the sensitivity functions as

So½z� ¼ Soo½z� � Vc½z�X½z�Vp½z� ð28Þ

Suo½z� ¼ G�10 ½z�Too½z� þ G�10 ½z�Vc½z�X½z�Vp½z�, ð29Þ

where G0[z] is the nominal plant model, unstable and

non-minimum phase that always admits a coprime

matrix fraction description in RH1 given by

G0½z� ¼ DI�1½z�NI½z� ¼ ND½z�DD

�1½z� (Maciejowski1989); Soo[z], Too[z] are admissible closed-loop sensi-

tivity functions for G0[z]; X [z] is a stable and proper

transfer function matrix; Vc[z] (Vp[z]) is the inverse ofthe generalised left (right) unitary interactor (Silva and

Salgado 2005) for ND[z] (DI[z]), which by definition has

unit DC-gain, i.e. Vc[1]¼ I (Vp[1]¼ I).In this context, the closed loop with sensitivity (28)

has integral action if and only if there existseX ½z� 2 RH1 (Salgado and Silva 2006) such that

X½z� ¼ Soo½1� þ ð1� z�1ÞeX ½z�: ð30Þ

With all these relations in mind, the cost function(27) can always be written as

J ¼ JeðeXÞ þ � � JuðeXÞ, ð31Þ

where

JeðeXÞ ¼ D We½z� � Vc½z�eX ½z�Vp½z�n o

ð32Þ

JuðeXÞ ¼ Wu½z� þ G�10 ½z�Vc½z� 1� z�1 eX ½z�Vp½z�

�� 22

ð33Þ

with We[z]¼ (Soo[z]�Vc[z]Soo[1]Vp[z])/(1� z�1)

and Wu½z� ¼ G�10 ½z�ðToo½z� þ Vc½z�Soo½1�Vp½z�Þ.

3.2 Decomposition

We now present a decomposition of JeðeXÞ. The next

two lemmas contain a key result in the development ofthis work. The following two lemmas use this result to

perform the decomposition.

Lemma 3.1: Let Vc[z], Vp[z]2RS1 be the inverse of

generalised left and right unitary interactors (respec-

tively) of dimensions n� n. Then, the functions

Hc½z� ¼ �z � V�c ½z� �

dVc½z�

dz, ð34Þ

Hp½z� ¼ �z �dVp½z�

dz� V�p ½z�: ð35Þ

admit a spectral factorisation.


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Proof: We shall only prove the result for Hc[z], since

the proof for Hp[z] is completely analogous. We then

need to prove the two necessary and sufficient condi-

tions for the existence of a spectral factorisation, i.e.

Hc½z� ¼ H�c ½z� andHc[1]� 0. Since Vc[z] is unitary, then

dV�c ½z�Vc½z�

dz¼

dV�c ½z�

dzVc½z� þ V �c ½z�

dVc½z�

dz¼ 0 ð36Þ

so that

V �c ½z�dVc½z�

dz¼ �

dV�c ½z�

dzVc½z� ð37Þ

¼ �dV�c ½z�

dz�1dz�1

dzVc½z� ð38Þ

¼ �dV�c ½z�

dz�1ð�z�2ÞVc½z�: ð39Þ

Multiplying by �z on both sides, we have

�z � V �c ½z�dVc½z�

dz¼ �z � V �c ½z�

dVc½z�

dz

� ��, ð40Þ

which means Hc½z� ¼ H�c ½z�. On the other hand, to

prove Hc[1]� 0 we just need to prove that

dVc½z�

dz

z¼1

0 ð41Þ

since Vc[1]¼ I by definition. From Silva and Salgado

(2005) we know that

Vc½z� ¼ n½z��1 ¼Yni¼1

Ln�iþ1½z�

!�1¼Yni¼1

Li½z��1 ð42Þ

from which, knowing that Li[1]¼ I by the definition in

Silva and Salgado (2005), we can write

dVc½z�

dz

z¼1

¼XNk¼1

dLk½z��1

dz�YN

i¼1,i6¼k

Li½z��1

( ) z¼1

ð43Þ

¼XNk¼1

dLk½z��1

dz

z¼1

: ð44Þ

Now, also from Silva and Salgado (2005),

we know that

Lk½z��1¼

1� ck1� ck

z� ck1� ckz

� 1

� �gkgk

H þ I, ð45Þ

where ck is the location of the k-th NMP zero, and �k isthe associated direction. Then,

dLk½z��1

dz

z¼1

¼1� ck1� ck

1� jckj2

1� ckzð Þ2

� �gkgk

H

z¼1

ð46Þ

¼1� jckj

2

j1� ckj2

� �gkgk

H: ð47Þ

Finally, since gkgkH � 0 by definition and jckj

241,

inequality (41) holds and the proof is complete. œ

Lemma 3.2: Let F [z]2L2 be an n� n matrix transfer

function. Consider Vc[z], Vp[z] as in Lemma 3.1. Then,

D Vc½z�F ½z��

¼ D F ½z��

þ Oc½z�F ½z�� 2

2ð48Þ

D F ½z�Vp½z��

¼ D F ½z��

þ F ½z�Op½z�� 2

2, ð49Þ

where Oc[z], Op[z]2RH2 come from the spectral

factorisation Hc½z� ¼ O�c ½z�Oc½z�, Hp½z� ¼ Op½z�O�p ½z�,

with Hc[z], Hp[z] defined as in Lemma 3.1.

Proof: Using the D-product definition and expanding

the derivative


¼�1

2�j

IC

trace Vc½z�F ½z�ð Þ�d Vc½z�F ½z�ð Þ

dz

� �dz

¼�1

2�j

IC

trace F�½z�V�c ½z�dVc½z�

dzF ½z�

� �dz

þ�1

2�j

IC

trace F�½z�V�c ½z�Vc½z�dF ½z�

dz

� �dz

and since V�c ½z�dVc½z�dz ¼ �

1zO�c ½z�Oc½z� and

Vc�½z�Vc½z� ¼ I, then we have


¼ D F ½z��

þ Oc½z�F ½z�� 2

2: ð50Þ

On the other hand,


¼�1

2�j

IC

trace F ½z�Vp½z� �d F ½z�Vp½z�

dz

� �dz

¼�1

2�j

IC

trace V�p ½z�F�½z�

dF ½z�

dzVp½z�

� �dz

þ�1

2�j

IC

trace V�p ½z�F�½z�F ½z�

dVp½z�

dz

� �dz,

where, using the trace commutative property, the fact

thatdVp½z�

dz V�p ½z� ¼ �1zOp½z�O

�p ½z� and noticing that,

since Vp[z] is unitary and square, Vp½z�V�p ½z� ¼ I,

we get


¼ D F ½z��

þ F ½z�Op½z�� 2

2, ð51Þ

which completes the proof. œ

Lemma 3.3: Consider the cost function JeðeXÞ definedin (32). Define A½z� ¼ Vc

�1½z�We½z�Vp�1½z� � eX ½z�, with

Oc[z], Op[z] as in Lemma 3.2, then

JeðeXÞ ¼ D A½z��

þ Oc½z�A½z�� 2

2þ A½z�Op½z�� 2

2: ð52Þ

Proof: Proof is straightforward upon considering

Lemma 3.2. First, define F [z]¼A[z]Vp[z] and


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use (48), so that

JeðeXÞ ¼ D Vc½z�A½z�Vp½z��

¼ D A½z�Vp½z��

þ Oc½z�A½z�Vp½z�� 2

2

¼ D A½z�Vp½z��

þ Oc½z�A½z�� 2

2: ð53Þ

Then, applying (49) completes the proof. œ

Lemma 3.4: Define Me½z� 2 H2,Mi½z� 2 H?2 as the

additive expansion of Vc�1½z�We½z�Vp

�1½z� 2 L, so that

A½z� ¼Mi½z� þMe½z� � eX ½z�, with eX ½z� 2 RH1. Then,the cost function (52) can be written as

JeðeXÞ ¼ D Mi½z��

þD Me½z� � eX ½z�n oþ Nci½z�� 2

2þ Nce½z� þOc½z�ðMe½z� � eX ½z�Þ�� 2

2

þ Npi½z�� 2

2þ Npe½z� þ ðMe½z� � eX ½z�ÞOp½z�� 2

2,

ð54Þ

where Nce[z], Npe[z]2H2 and Nci½z�,Npi½z� 2 H?2 come

from the additive expansions

Oc½z�Mi½z� ¼ Nci½z� þNce½z� ð55Þ

Mi½z�Op½z� ¼ Npi½z� þNpe½z�: ð56Þ

Proof: Proof is straightforward from the orthogonal-

ity property of both the D-product and the 2-norm.œ

Remark 1: The expression found in (54) admits the

decomposition

JeðeXÞ ¼ Je1 þ Je2ðeXÞ ð57Þ

Je1 ¼ D Mi½z��

þ Nci½z�� 2

2þ Npi½z�� 2

2ð58Þ

Je2ðeXÞ ¼ D Me½z� � eX ½z�n oþ Nce½z� þOc½z�ðMe½z� � eX ½z�Þ�� 2

2

þ Npe½z� þ ðMe½z� � eX ½z�ÞOp½z�� 2

2: ð59Þ

This implies that Je1 can be interpreted as a base value

for the cost function that depends on Vc[z] and Vp[z],

that is, it can be considered as an inherent minimal cost

based on the location and direction of the non-mini-mum phase zeros and the unstable poles of the plant.

4. Achievable performance bounds

In this section we present the methodology to solve theoptimisation problem for the case �¼ 0. The relevanceof this case is that it yields an achievable performancebound, and also, it introduces the technique to be usedfor the general case, � 6¼ 0. This performance bound isa benchmark for any linear control design strategy,provided it is measured with the ITSE criterion of thetracking error for a random step vector referencesatisfying (24) and (25).

4.1 Optimal solution

Assume that the optimal solution for eX ½z� exists and isdefined as

eXopt¼ arg mineX2RH1 J ¼ arg mineX2RH1 Je2, ð60Þ

where eXoptis a stable and proper matrix transfer

function.As known from Carrasco and Salgado (2009), for

the SISO stable case, when the plant has only one non-minimum phase zero at z¼ a with jaj41, the optimalsolution is a constant number (eXopt ¼Mi½1=a�). Forthe general SISO case (two or more NMP zeros) theoptimal solution would be instead a non-constantstable transfer function. We now extend the SISOsolution to the case at hand, where we can alwayschoose the following structure for the optimal solution,eXopt

½z� ¼Me½z� þ eF ½z� ð61Þ

where eF ½z� is a stable matrix transfer function whoseelements can be parameterised using a linear combi-nation of orthonormal basis. Although a stable func-tion can be represented exactly by an infinite expansionin a complete orthonormal basis, for practical reasonsonly an N-term truncated expansion is used. Hence,if we consider eF ½z� 2 C

n�n, then, an particularexpansion is

eF ½z� ¼eF11½z� eF12½z� . . . eF1n½z�eF21½z� eF22½z� . . . eF2n½z�

..

. ... . .

. ...

eFn1½z� eFn2½z� . . . eFnn½z�

266664377775

n�n

¼

�111 B1 þ � � � þ �11N BN �121 B1 þ � � � þ �

12N BN . . . �1n1 B1 þ � � � þ �

1nN BN

�211 B1 þ � � � þ �21N BN �221 B1 þ � � � þ �

22N BN . . . �2n1 B1 þ � � � þ �

2nN BN

..

. ... . .

. ...

�n11 B1 þ � � � þ �n1N BN �n21 B1 þ � � � þ �

n2N BN . . . �nn1 B1 þ � � � þ �

nnN BN

266664377775,


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where the terms �i and Bi are those defined in

Section 2.4. The latter expression means that eF ½z� canbe represented by eF ½z� ¼ bT½z� � h, ð62Þ

where b is the matrix containing the N terms of the

basis for each element of eF ½z� and h is a matrix of real

coefficients:

bT½z� ¼

B1 . . . BN 01 . . . 0N . . . 01 . . . 0N

01 . . . 0N B1 . . . BN . . . 01 . . . 0N

..

. ... . .

. ...

01 . . . 0N 01 . . . 0N . . . B1 . . . BN

266664377775

n�nN

ð63Þ

hT ¼

�111 . . .�11N �211 . . .�21N . . . �n11 . . .�n1N�121 . . .�12N �221 . . .�22N . . . �n21 . . .�n2N

..

. ... . .

. ...

�1n1 . . .�1nN �2n1 . . .�2nN . . . �nn1 . . .�nnN

266664377775

n�nN

:

ð64Þ

Thus, the problem of minimising Je2ðeX ½z�Þ is now

transformed into a problem of minimising Je2ðeXðhÞÞ.However, this will yield a suboptimal solution, since

bT[z]h is only an approximation of eF ½z�.Now, in order to ensure a closed form expression

for the optimal coefficients hopt, we make use of

Lemma 2.3. First, we use (62) in (61) and replace it in

(59)), obtaining

Je2ðeXðhÞÞ ¼ D bT½z�h� �

þ Nce½z� �Oc½z�bT½z�h

�� 22

þ Npe½z� � bT½z�hOp½z�� 2

2: ð65Þ

Then, vectorising the arguments of (65) and apply-

ing property (3), we have

Je2ðeXðhvÞÞ ¼ D BvIhvf g þ Avc � Bvchvk k22

þ Avp � Bvphv�� 2

2, ð66Þ

where

hv ¼ vec hð Þ ð67Þ

BvI ¼ ðIn�n � b½z�TÞ ð68Þ

Avc ¼ vec Nce½z�ð Þ ð69Þ

Bvc ¼ ðIn�n �Oc½z�b½z�TÞ ð70Þ

Avp ¼ vec Npe½z�

ð71Þ

Bvp ¼ ðOp½z�T� b½z�TÞ ð72Þ

with In�n defined as an n� n identity matrix.

Lemma 4.1: Consider Je2 as defined in (66), then

hoptv ¼ argmin Je2ðeXðhvÞÞ ¼ L�1e Ke, ð73Þ

where

Ke ¼1

2�j

IC

B�vcAvcdz

zþ

1

2�j

IC

B�vpAvpdz

zð74Þ

Le ¼1

2�j

IC

B�vcBvcdz

zþ

1

2�j

IC

B�vpBvpdz

z

þ�1

2�j

IC

B�vIdBvI

dzdz: ð75Þ

Proof: Using integral definition of the D-product and

the 2-norm in (66), we have

Je2 ¼1

2�j

IC

trace ðAvc � BvchvÞ�ðAvc � BvchvÞ

� � dzz

þ1

2�j

IC

trace ðAvp � BvphvÞ�ðAvp � BvphvÞ

� � dzz

þ�1

2�j

IC

trace ðBvIhvÞ� d

dzðBvIhvÞ

� �dz: ð76Þ

Expanding each term, reordering the matrices and

using trace properties, we can write

Je2 ¼ trace Pef g � trace Ke1hvf g � trace hTvKe2

� �þ trace hTv Lehv

� �, ð77Þ

where

Pe ¼1

2�j

IC

A�vcAvcdz

zþ

1

2�j

IC

A�vpAvpdz

zð78Þ

Ke1 ¼1

2�j

IC

A�vcBvcdz

zþ

1

2�j

IC

A�vpBvpdz

zð79Þ

Ke2 ¼1

2�j

IC

B�vcAvcdz

zþ

1

2�j

IC

B�vpAvpdz

zð80Þ

Le ¼1

2�j

IC

B�vcBvcdz

zþ

1

2�j

IC

B�vpBvpdz

z

þ�1

2�j

IC

B�vIdBvI

dzdz: ð81Þ

Now, using trace derivative properties (Petersen

and Pedersen 2008), we have

@Je2@hv¼ �KT

e1 � Ke2 þ ðLe þ LTe Þh

optv ¼ 0: ð82Þ

Using Ke¼ (Ke1)T¼Ke2 and Le ¼ LT

e , it follows

that

hoptv ¼ L�1e Ke, ð83Þ

from where the result follows. œ


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This result allows to have an explicit expression forthe performance bound of a given plant:

Je2ðeXðhoptv ÞÞ ¼ D BvIhoptv

� �þ Avc � Bvch

optv

�� 22

þ Avp � Bvphoptv

�� 22, ð84Þ

where hoptv is the coefficient vector defined inLemma 4.1.

5. Optimal controller design

Since we already have a methodology to minimise thecase �¼ 0, we now extend our results to a more generalframework using a similar approach.

In order to find an expression that minimises thecase � 6¼ 0, as in the previous case, we first replace (61)and (62) in (33), obtaining

JuðeXðhÞÞ ¼ ��G�1o ½z�ðToo½z� þ Vc½z�ðSoo½1�

þ ð1� z�1ÞMe½z�ÞVp½z�Þ

þ G�1o ½z�Vc½z�ð1� z�1Þb½z�ThVp½z��22: ð85Þ

Again, vectorising the argument of (85) and con-veniently applying (3), we have

Ju ¼ Avu þ Bvuhvk k22, ð86Þ

where hv¼ vec(h) and

Avu ¼ vec G�1o ½z�ðToo½z� þ Vc½z�ðSoo½1�

þ ð1� z�1ÞMe½z�ÞVp½z�Þ

ð87Þ

Bvu ¼ ðVp½z�T� G�1o ½z�Vc½z�ð1� z�1Þb½z�TÞ: ð88Þ

Lemma 5.1: Consider eXopt½z� 2 RH1

n�n as definedin (61) and (62). Also consider the cost functiondefined in (31), with � 6¼ 0, Je as in (57), (58) and (66),Ju as in (86), then

hoptv ¼ argmin JðeXðhvÞÞ ¼ Le þ �Luð Þ�1 Ke � �Kuð Þ,

ð89Þ

where Le and Ke are defined as in Lemma 4.1, and

Ku ¼1

2�j

IC

B�vuAvudz

zð90Þ

Lu ¼1

2�j

IC

B�vuBvudz

z: ð91Þ

Proof: Consider Ju defined in (86). Then, using theintegral definition of the 2-norm, we have

Ju ¼1

2�j

IC

trace ðAvu þ BvuhvÞ�ðAvu þ BvuhvÞ

� � dzz:

ð92Þ

Expanding the argument and using trace properties, wecan write

Ju ¼ trace Puf g � trace Ku1hvf g � trace hTvKu2

� �,

þ trace hTv Luhv� �

ð93Þ

where

Pu ¼1

2�j

IC

A�vuAvudz

zð94Þ

Ku1 ¼1

2�j

IC

A�vuBvudz

zð95Þ

Ku2 ¼1

2�j

IC

B�vuAvudz

zð96Þ

Lu ¼1

2�j

IC

B�vuBvudz

z: ð97Þ

This means, along with Lemma 4.1, the total costfunction J can be written as

J ¼ Je1 þ trace Pe þ �Puf g � trace ðKe1 � �Ku1Þhv� �

� trace hTv ðKe2 � �Ku2Þ� �

þ trace hTv ðLe þ �LuÞhv� �

:

ð98Þ

Taking the derivative, and using trace derivativeproperties (Petersen and Pedersen 2008), we have

@J

@hv¼ �ðKe1 � �Ku1Þ

T� ðKe2 � �Ku2Þ

þ ðLe þ �Lu þ ðLe þ �LuÞTÞhoptv ¼ 0:

ð99Þ

Now, note that Ku¼ (Ku1)T¼Ku2 and Lu ¼ LT

u , andsince Ke¼ (Ke1)

T¼Ke2 and Le ¼ LT

e , we have

hoptv ¼ Le þ �Luð Þ�1 Ke � �Kuð Þ: ð100Þ

œ6. MIMO 2\ 2 Example (j^ 0)

In this section we present a numerical example thatillustrates the results obtained for the case (�¼ 0), thatis, the performance bound is computed for a givenplant model. This specific example deals with the caseof a plant having an unstable pole, two zeros at infinityand a distributed zero (a non-existent concept in SISOsystems).

6.1 Definitions

First of all, we define the nominal plant model as

G0½z� ¼

0:1

ðz� 1:1Þ

1

z2

0:1

ðz� 1:1Þ

0:8

z

26643775: ð101Þ


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Remark 2: The solution to the associated minimisa-

tion problem is the same as that for any plant having

the form

G0½z� ¼

0:1

ðz� 1:1Þ

1

z2

0:1

ðz� 1:1Þ

0:8

z

26643775Gad½z�, ð102Þ

where Gad[z] is stable, biproper and minimum phase.

For the model defined in (101), the determinant

shows a non-minimum phase distributed zero at

c¼ 1.25 with left direction g¼ [�0.7071 0.7071]T, two

zeros at infinity and one unstable pole:

detfG0½z�g ¼0:08ðz� 1:25Þ

z2ðz� 1:1Þð103Þ

Then, the inverse of the zero (pole) interactor

matrix Vc[z] (Vp[z]) are given by

Vc½z� ¼

0:1ðzþ 1Þ

zðz� 0:8Þ

0:9ðz� 1Þ

zðz� 0:8Þ

0:9ðz� 1Þ

zðz� 0:8Þ

0:1ðzþ 1Þ

zðz� 0:8Þ

26643775 ð104Þ

Vp½z� ¼

0:045455ðzþ 1Þ

ðz� 0:9091Þ

�0:95455ðz� 1Þ

ðz� 0:9091Þ

�0:95455ðz� 1Þ

ðz� 0:9091Þ

0:045455ðzþ 1Þ

ðz� 0:9091Þ

26643775:ð105Þ

Also, a choice for admissible sensitivity and com-

plementary sensitivity closed-loop functions is

Too½z� ¼

0:15ðzþ 0:4Þ

zðz� 0:8Þ

0:95ðz� 0:9895Þ

zðz� 0:8Þ

0:95ðz� 0:9895Þ

zðz� 0:8Þ

0:15ðzþ 0:4Þ

zðz� 0:8Þ

26643775ð106Þ

Soo½z�

¼

ðzþ0:05944Þðz�1:009Þ

zðz�0:8Þ

�0:95ðz�0:9895Þ

zðz�0:8Þ�0:95ðz�0:9895Þ

zðz�0:8Þ

ðzþ0:05944Þðz�1:009Þ

zðz�0:8Þ

26643775:

ð107Þ

The following are the necessary matrix transfer

functions to apply the minimisation methodology:

From which we can obtain the decomposition

Vc[z]�1We[z]Vp[z]

�1¼Me[z]þMi[z]

Me½z� ¼0 0

0 0

� �ð110Þ

Mi½z� ¼

�1:175zðz�0:05319Þ

ðz�1:25Þ

0:075zðzþ17:5Þ

ðz�1:25Þ0:075zðzþ17:5Þ

ðz�1:25Þ

�1:175zðz�0:05319Þ

ðz�1:25Þ

26643775:ð111Þ

Then, the spectral factor that comes from

O�c ½z�Oc½z� ¼ �zV�c ½z�V

0c½z� and the one that comes

from Op½z�Op�½z� ¼ �zVp

0½z�Vp�½z� are given by

Oc½z� ¼

1:1325ðz�0:6325Þ

zðz�0:8Þ

�0:13246ðzþ0:6325Þ

zðz�0:8Þ�0:13246ðzþ0:6325Þ

zðz�0:8Þ

1:1325ðz�0:6325Þ

zðz�0:8Þ

26643775

ð112Þ

Op½z� ¼

0:2083

ðz� 0:9091Þ

0:2083

ðz� 0:9091Þ0:2083

ðz� 0:9091Þ

0:2083

ðz� 0:9091Þ

26643775: ð113Þ

With this we can decompose now Oc[z]Mi[z]¼

Nce[z]þNci[z], where

Nce½z� ¼

0:94868

ðz� 0:8Þ

�0:94868

ðz� 0:8Þ�0:94868

ðz� 0:8Þ

0:94868

ðz� 0:8Þ

26643775 ð114Þ

Nci½z� ¼

�1:3406ðzþ0:9615Þ

ðz�1:25Þ

0:24057ðzþ11:07Þ

ðz�1:25Þ

0:24057ðzþ11:07Þ

ðz�1:25Þ

�1:3406ðzþ0:9615Þ

ðz�1:25Þ

26643775,ð115Þ

and at the same time Mi[z]Op[z]¼Npe[z]þNpi[z], where

Npe½z� ¼

�0:2083

ðz� 0:9091Þ

�0:2083

ðz� 0:9091Þ�0:2083

ðz� 0:9091Þ

�0:2083

ðz� 0:9091Þ

26643775 ð116Þ

Npi½z� ¼�0:22913 �0:22913

�0:22913 �0:22913

� �: ð117Þ

We½z� ¼Soo½z� � Vc½z�Soo½1�Vp½z�

1� z�1ð108Þ

¼

ðz� 0:8707Þðzþ 0:05708Þðz2 � 2:091zþ 1:098Þ

zðz� 0:8Þðz� 0:9091Þðz� 1Þ

�0:99545ðz� 1:057Þðz2 � 1:796zþ 0:8124Þ

zðz� 0:8Þðz� 0:9091Þðz� 1Þ

�0:99545ðz� 1:057Þðz2 � 1:796zþ 0:8124Þ

zðz� 0:8Þðz� 0:9091Þðz� 1Þ

ðz� 0:8707Þðzþ 0:05708Þðz2 � 2:091zþ 1:098Þ

zðz� 0:8Þðz� 0:9091Þðz� 1Þ

2666437775 ð109Þ


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6.2 Minimum values

For this example, the base value of the cost function

turns out to be

Je1 ¼ Nci½z�� 2

2þ Npi½z�� 2

2þD Mi½z�

� �¼ 15, 21:

ð118Þ

Table 1 shows the obtained values of Je2ðeXðhoptÞÞand the corresponding coefficients hopt for different

choices of the length of the expansion N, where

Laguerre orthonormal basis are used with Laguerre

pole p¼ 0.4.If we consider the minimum value obtained with

N¼ 5, then

JeðeXðhoptÞÞ ¼ 16:2325: ð119Þ

This value is the minimum achievable performance

cost in a closed-loop control for the plant model (101),

quantified by the ITSE index of the tracking error. We

obtain a convergent series of parameters for each of the

elements of the MIMO system. Also, although the

defined plant model is not symmetric, the optimal

coefficients that represent the off-diagonal elements are

the same. This latter fact can be checked by computing

the matrix eXðhoptÞ, given byeX11ðhoptÞ

¼0:65176zðz2�0:9883zþ0:2488Þðz2�0:7212zþ0:1735Þ

ðz�0:4Þ5

ð120Þ

eX21ðhoptÞ

¼�1:0135zðz2�1:148zþ0:366Þðz2�0:5308zþ0:1303Þ

ðz�0:4Þ5

ð121Þ

eX12ðhoptÞ

¼�1:0135zðz2�1:148zþ0:366Þðz2�0:5308zþ0:1303Þ

ðz�0:4Þ5

ð122Þ

eX22ðhoptÞ

¼0:65176zðz2�0:9883zþ0:2488Þðz2�0:7212zþ0:1735Þ

ðz�0:4Þ5:

ð123Þ

Similarly, we can compute the optimal nominal

sensibility So[z]

So11 ½z� ¼

ðz�0:7815Þðz�0:9123Þðz�1Þðzþ0:9519Þ

� ðz2�1:01zþ0:276Þðz2�0:6506zþ0:1432Þ

� �zðz�0:8Þðz�0:9091Þðz�0:4Þ5

ð124Þ

So21 ½z� ¼

�1:826ðz� 0:7853Þðz� 0:9062Þ

�ðz� 1Þðz2 � 1:097zþ 0:3416Þ

�ðz2 � 0:5132zþ 0:1171Þ

8><>:9>=>;

zðz� 0:8Þðz� 0:9091Þðz� 0:4Þ5

ð125Þ

So12 ½z� ¼

�1:826ðz� 0:7853Þðz� 0:9062Þðz� 1Þ

�ðz2 � 1:097zþ 0:3416Þ

�ðz2 � 0:5132zþ 0:1171Þ

8><>:9>=>;

zðz� 0:8Þðz� 0:9091Þðz� 0:4Þ5

ð126Þ

So22 ½z� ¼

ðz�0:7815Þðz�0:9123Þðz�1Þðzþ0:9519Þ

�ðz2�1:01zþ0:276Þðz2�0:6506zþ0:1432Þ

� �zðz�0:8Þðz�0:9091Þðz�0:4Þ5

:

ð127Þ

Both matrices show that the optimal transfer

functions matrix are symmetrical.

Table 1. Values of Je2ðeXðhoptÞÞ.N Je2ðh

optL Þ h

optL

1 1.0964 0:6447 �1:0768

�1:0768 0:6447

� �3 1.0256 0:6788 �1:0784

�0:0609 0:0432

0:0300 �0:0741

�1:0784 0:6788

0:0432 �0:0609

�0:0741 0:0300

2666666664

37777777755 1.0225 0:6806 �1:0767

�0:0615 0:0437

0:0342 �0:0714

�0:0075 �0:0016

0:0008 �0:0116

�1:0767 0:6806

0:0437 �0:0615

�0:0714 0:0342

�0:0016 �0:0075

�0:0116 0:0008

26666666666666666664

37777777777777777775


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7. MIMO 2\ 2 example (j 6 0)

In this section we present a numerical example thatillustrates the results obtained for the general case(� 6¼ 0), considering the plant model defined in (101).In particular, we analyse the effect of the parameter �on the optimal cost of Je and Ju. In addition, wecompare the step responses of the optimal nominalsensibility and control sensibility for different values of�.

7.1 Effect of the parameter j

Table 2 shows the optimal values of cost function fordifferent choices of the parameter �, as well as theoptimal parameter matrix hopt. The Laguerre pole isp¼ 0.4 and the expansion length is N¼ 5.

From Table 2 we establish two facts: first, as themagnitude of � increases, so does the obtained value ofJe, meaning that the achieved performance of thetracking error deteriorates. Second, as � increases, theperformance index of the control effort (Ju) diminishes,which is expected since it has a larger weight in the costfunction.

7.2 Closed-loop dynamics

The resulting closed-loop dynamics shown in Figures 1and 2 illustrate the cases �¼ 0, �¼ 0.1 and �¼ 10 fora step reference vector with direction l¼ [�0.70710.7071]T. From those figures it is direct that as thevalue of the parameter � increases, the closed-loopdynamics become slower, which corroborates the ideathat the effect of Ju in the cost function is to moderatethe speed of convergence of the resulting closed-loopsignals, specifically the control signals.

8. Conclusions

We have presented a methodology for the optimisationof a mixed performance multivariable cost function.In this setting, we propose a methodology to computeachievable performance bounds, by minimising theITSE index of the tracking error in a one-degree-of-freedom closed loop. In addition, we propose ageneralisation of this methodology for optimal con-troller design, by adding a measure of the controlenergy.

The usage of an orthonormal basis expansion of theoptimal modified Youla parameter and the vectorisa-tion of some matrix expressions are key tools to ensurea closed solution for the optimisation problem. Inparticular, with the vectorisation, the choice of thefactorisation of the expansion does not need a specificstructure.

The solution to the cheap control case (�¼ 0)depends only on the system interactors. However, thesolution for the general case � 6¼ 0 depends on the fullknowledge of the plant model.

Finally, throughout this article we presented someexamples illustrating the proposed methodology. Theinfluence of the expansion length and the parameter �were the factors considered. As in the SISO case, thetime-weighted nature of the ITSE index and its

Table 2. Values of the cost function J(hopt).

� Je Ju J hopt

0 16.2325 948.1867 16.2325 0:6806 �1:0767

�0:0615 0:0437

0:0342 �0:0714

�0:0075 �0:0016

0:0008 �0:0116

�1:0767 0:6806

0:0437 �0:0615

�0:0714 0:0342

�0:0016 �0:0075

�0:0116 0:0008

26666666666666666664

377777777777777777750.1 25.5852 91.5906 34.7443

0:1105 �0:3505

0:1735 �0:2978

0:0970 �0:0827

0:0196 �0:0673

0:0057 �0:0051

0:4460 0:3630

�0:1846 �0:5997

�0:1441 0:2623

�0:0893 �0:0518

�0:0301 0:0275

26666666666666666664

3777777777777777777510 78.7288 11.6202 194.9305 �0:2401 0:5946

�0:4052 0:0114

�0:3056 0:0298

�0:1717 0:0344

�0:0938 0:0433

1:3738 0:0750

0:7316 �0:6687

0:4413 �0:5160

0:2778 �0:2789

0:1249 �0:1298

26666666666666666664

37777777777777777775


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frequency domain equivalent (the D-product) isreflected on the achieved closed-loop dynamics andthe associated time response.

Further research directions should include theproblem of orthogonal basis selection to attain aparsimonious description of the functions involved.Also, additional light can be shed on the proposeddesign methodology by studying its application to

control problems which have been tackled using

different control strategies.

Acknowledgements

The authors gratefully acknowledge the support of UTFSMand FONDECYT through grant 1080274.

−10

0

10

20

30

Channel 1 u[k] l= 10

l= 0.1l= 0

0 5 10 15 20 25 30 35 40−8

−6

−4

−2

0

Sample number k

Channel 2 u[k] l= 10

l= 0.1l= 0

Figure 2. Step response of Suo[z].

−2.5

−2

−1.5

−1

−0.5

0

Channel 1 error l= 10

l= 0.1l= 0

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

Sample number k

Channel 2 error l= 10

l= 0.1l= 0

Figure 1. Step response of So[z].


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Control System Design, Englewood Cliffs, NJ: Prentice

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Appendix: Proof of D-product properties

P1. This proof is straightforward from the definition.P2. From the definition, considering z¼ u�1 and using

trace properties, we have

F ½z�,G ½z��

D¼�1

2�j

IC


dz

� �dz

¼�1

2�j

IC�

trace F ½u�T �dG ½u�1�

du

� �du

¼�1

2�j

IC�

tracedG ½u�1�T

du� F ½u�

� �du

¼�1

2�j

IC�

trace F ½u� �dG ½u�1�T

du

� �du

¼1

2�j

IC

trace F ½u� �dG ½u��

du

� �du

¼ � F�½z�,G�½z��

D, ðA1Þ

where the contour C� is the unit circle in the complexplane (clockwise).

P3. Consider the following integral:IC

traced F�½z�G ½z��

dz

� �dz ¼ 0: ðA2Þ

Expanding the derivative and applying trace properties(Petersen and Pedersen 2008), we haveI

C

trace G ½z�dF�½z�

dz

� �dzþ

IC

trace F�½z�dG ½z�

dz

� �dz ¼ 0:

ðA3Þ

Thus, multiplying by the right factor leads to

G�½z�,F�½z��

Dþ F ½z�,G ½z��

D¼ 0: ðA4Þ

Now, using P2 we know that

G�½z�,F�½z��

D¼ � G ½z�,F ½z�

� �D, ðA5Þ

which completes the proof.P4. If F [z]2H2 and G ½z� 2 H?2 , then

F ½z�,G ½z��

D¼�1

2�j

IC


dz

� �dz, ðA6Þ

where F�½z� � dG ½z�dz has all its poles outside the unit disc,which means this function is analytic inside the disc


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and therefore, by Cauchy’s Integral Theorem(Churchill and Brown 1984), the integral is zero.

P5. From the definition

c,G ½z��

D¼�1

2�j

IC

trace c �dG ½z�

dz

� �dz ðA7Þ

¼ trace c ��1

2�j

IC

dG ½z�

dz

� �dz: ðA8Þ

If we define Gij as the (i, j)-th element of G[z], then

�1

2�j

IC

dGij

dzdz ¼

�1

2�j

IC

dGij ¼ 0 ðA9Þ

since it represents a closed integral on Gij. This meansthe integral (A8) is also zero. The counterparthG[z], ciD¼ 0 comes from using P3.

P6. From the definition we know that

F ½z� þ c1,G ½z� þ c2� �

D¼�1

2�j

IC

trace F ½z� þ c1ð Þ��dG ½z�

dz

� �dz:

ðA10Þ

Now, note that

�1

2�j

IC

trace F ½z�þ c1ð Þ��dG ½z�

dz

� �dz¼ F ½z�,G ½z�

� �Dþ c1,G ½z��

D:

ðA11Þ

Finally, using P5 the result is obtained.


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Optimal multivariable controller design using an ITSE performance index