Performance bounds for feedback control ofnon-minimum phase MIMO systems with arbitrarydelay structure

E.I. Silva and M.E. Salgado

Abstract: The paper derives 2-norm performance bounds for the feedback control of discrete-timeMIMO non-minimum phase (NMP) systems with arbitrary delay structure. Also, the associatedoptimal controller, in Youla-parameterised form, is explicitly obtained. The derivation of theseresults uses a special interactor matrix to extract the delays. It is shown that this interactor is unique,and a building algorithm is also proposed.

1 Introduction

Fundamental limitations and performance bounds in controlsystems have been important research topics since Bode’spioneering work [1–4]. The relevance of those topics lies intheir ability to provide answers to some fundamentalquestions:

. What limitations are inescapable on all possible linearcontrol system designs for a given linear model?. What is the best achievable performance consideringsome specific class of controllers and peformance indexes?

The significance of these questions is manyfold. One not soevident issue is that the answer to them provides abenchmark against which, for instance, non-linear controlperformance can be compared and measured.

Answers to the first question have been sought for morethan sixty years and were first introduced by Bode’ssensitivity integrals [1]. Those results have been extendedto the unstable and NMP SISO case [5] and to the discretetime case [6]. They rely entirely upon the analytic propertiesof the sensitivity and complementary sensitivity functionsand are expressed using Poisson or Cauchy type integrals.Extensions of limitations to the MIMO case can be found in[3] and [7]. Those results state that delays, unstable polesand NMP zeros are sources of limitations and that, in theMIMO case, their directions also play an important role.In later work [4] those results were complemented with thecomputation of bounds for the sensitivity and complemen-tary sensitivity peaks, which are specially relevant forrobust control design.

In the above mentioned work the negative effect thatunstable poles and NMP zeros have on the performance ofthe control system is recurrent. However, those resultscannot be used to compute the best achievable performancefor the control of a given process model. In other words,

they define what cannot be achieved, but they cannot beused to determine the best controller (in some appropriatesense) for a given plant.

Results related to the second question (performancebounds) are more recent and the research has followed twodifferent, although connected, paths: the deterministic andthe stochastic approaches.

Many of the results available in the deterministic case usea 2-norm based performance index and, therefore, theyadmit time domain interpretation using Parseval’s relations.Abundant references can be found in [1] and [8] for theSISO case. An advantage of this approach is that it can beconnected to optimal synthesis techniques and hence, can beused to obtain a controller capable of achieving that bestperformance [2].

Results for the MIMO case have been presented in [9],where the best achievable performance for a continuoustime MIMO process is evaluated using a quadraticperformance index. The case of one- and two-degrees-of-freedom controllers are considered there. Again, thebest performance is limited by the unstable poles, NMPzeros and delays of the system. It is worth noting that in[9] only measurement delays are considered. Thisparticular structure is not flexible enough to encompassmultivariable models with delayed interactions, whichare frequent in industrial processes. Extensions to two-degree-of-freedom control architectures are considered in[10] and [11].

Performance limitations for discrete time MIMO plantshave been considered in [12], where a natural extension of[9] is presented. Again, only restrictive delay structures areexplicitly considered.

Comprehensive surveys of the research using an stochas-tic setting can be found in [13] and [14]. Most of the workreviewed in those references is based upon the researchreported in [15]. In the latter work, focused on SISOsystems, minimum variance control (MVC) [16] is used as abenchmark, and closed-loop data is used to evaluate loopperformance. For the SISO case, several indexes have beenproposed to compare the performance of a given controlloop to the ideal minimum variance strategy [17–20].

Classical MVC [16] cannot stabilise NMP plants [13, 14,21] and therefore, the ideal minimum variance performancemeasure is an unachievable lower bound for the control ofNMP plants. This feature is highly significant since the

q IEE, 2005

IEE Proceedings online no. 20045056

doi: 10.1049/ip-cta:20045056

The authors are with the Department of Electronic Engineering,Universidad Tecnica Federico Santa Marıa, Valparaso, Chile

E-mail: eduardo.silva@elo.utfsm.cl

Paper received 24th May 2004. Originally published online 7th February2005

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 211

difference between the ideal minimum variance performanceand the achievable variance may be very large for processeswith NMP zeros near (1, 0) [22].

In the MIMO case, minimum variance performancebounds require knowledge of an interactor matrix of theprocess [13, 14] which captures the system delay structure.Once this matrix has been computed, one can extend SISOperformance measures to the MIMO case as proposed, forinstance, in [13], in [14] and in the references therein. It isworth pointing out that the interactor matrix is non-unique,as discussed in [23]. Therefore, the selected structure for theinteractor may have impact on the computation of theperformance bounds depending on the metrics being used.In particular, the unitary interactor introduced in [24] isuseful when considering quadratic performance measures,since it allows one to compute the best performance bound[25, 26].

In the MVC-based approach, significant work is devotedto identification of the interactor matrix and estimation ofthe performance bound using closed-loop data. However, noachievable bound, in closed form, is computed for systemswith NMP zeros. Also, to the best of the authors’knowledge, the MVC based strategy to assess and tomonitor performance does not deliver a controller in closedform. This feature limits the designer’s ability toanalytically assess the effects of plant features such as thelocation of NMP zeros, delay structure and signaldirectionality.

At this stage of the discussion it should be said that thereexists a connection between the two approaches mentioneddealing with the performance limitation issue. As a matterof fact, the SISO minimum variance control problemconsidering an ARIMA(n, 1, p) model for the disturbancecan be formulated as the problem of finding a stabilisingcontroller that achieves the minimum 2-norm of the(weighted) error due to a step change in the reference, andtherefore the solution of both problems are closely related.

In this paper we follow a deterministic approach and themain contributions are, first, the derivation of a closed formexpression for the best achievable performance in the linearcontrol of a stable NMP discrete time MIMO plant with anarbitrary delay structure, i.e. where delays appear not only inthe measurements but also in the channel interactions.A second contribution is the construction of a closed formexpression for the controller which achieves that perform-ance bound. Both results are described in a way thathighlights the interplay between performance, NMP zeros,delays and multivariable directionality.

The results presented in this paper are based upon aspecial case of the interactor matrix [2, 24, 27, 28] althoughthe proofs proceed along different paths to those in thereferred works.

2 Definitions

This section introduces basic definitions and the notationused throughout the paper. For any complex number z, �zzrepresents its conjugate. Given M 2 C

m�n; MH denotes itshermitian (conjugate transpose) and for a rational transfer

matrix MðzÞ 2 Cm�n we define the operation ð�Þ� as

M�ðzÞ ¼ MH�

1�zz

�which reduces to M�ðzÞ ¼ MT ðz�1Þ in

the real rational case, i.e. when MðzÞ 2 Rn�n; 8z 2 R: Note

that in either case, ð�Þ� reduces to ð�ÞH when z ¼ e jo:We say that a rational transfer matrix MðzÞ 2 C

n�n isunitary if and only if M�ðzÞMðzÞ ¼ I: Note that M(z)unitary implies that Mðe joÞ is unitary in the traditional sensefor all o 2 R; i.e. MHðe joÞMðe joÞ ¼ I.

L2 is defined as the Hilbert space of all matrix functionsmeasurable over the unit circle, i.e. for jzj ¼ 1;with the usualinner product [12]. The norm induced by this inner product iscalled 2-norm and will be denoted by k � k2:H2 � L2, the setof all matrix functions that are analytic outside the unit circleðjzj>1Þ, andH?

2 � L2; the set of all matrix functions that areanalytic inside the unit circle ðjzj< 1Þ; form an orthogonalsubspace pair [12].

3 Zero factorisations in discrete-time MIMOsystems

A MIMO system with transfer function GoðzÞ has a zero atz ¼ zo if an only if GoðzoÞ is singular. A key issue regardingthis definition is that the set of zeros of a system includesfinite zeros as well as zeros at infinity. The zeros at infinitydescribe the relative degree and, hence, they capture thedelay structure of discrete time systems.

In this section we consider the factorisation of NMP zerosin discrete time MIMO systems. Zeros in MIMO systemscan be factorised in several ways using, for example, the socalled interactor matrices and z-interactors [2, 29]. Theinteractor matrices defined in [28] and [29], for example,have the important property of being unique and triangular,but they are not necessarily unitary (see Example 25.2 in[2]). This feature makes such interactors unsuitable for thederivations in Section 4.

To compute the performance bounds we need two keyresults. The first one is given in the following lemma.

Lemma 1: Consider a real rational transfer matrix GðzÞ 2C

n�n: Assume that G(z) has a zero at z ¼ c with multiplicitymc associated to the unitary direction hc (there may be morezeros located somewhere else). Define

GGðzÞ ¼ LcðzÞGðzÞ ð1Þwhere

LcðzÞ ¼1 � c

1 � �cc

1 � �ccz

z � chch

Hc þ UcUH

c

¼ hc Uc½ �1 � c

1 � �cc

1 � �ccz

z � c0

0 In�1

264

375 hH

c

UHc

24

35 ð2Þ

and Uc is chosen so that hc Uc½ � is unitary. Then,

1. GGðzÞ has mc � 1 zeros at z ¼ c associated to the directionhc and, possibly, one additional zero at z ¼ 1

�ccassociated

to hc.2. Except for the previous fact, GGðzÞ shares its poles andzeros with G(z), but not necessarily the associateddirections.3. LcðzÞ is unitary and has unity DC gain, i.e. Lcð1Þ ¼ I:4. LcðzÞ is biproper.

Proof: See [12]. A

Note that from lemma 1

GðzÞ ¼ L�1c ðzÞGGðzÞ;

with L�1c ðzÞ ¼ 1 � �cc

1 � c

z � c

1 � �cczhch

Hc þ UcUH

c ð3Þ

which means that LcðzÞ extracts one of the zeros of G(z) inz ¼ c associated to hc: Furthermore, provided that hH

c GðzÞis analytical at z ¼ 1=�cc; the effect of LcðzÞ is to replace oneNMP zero at z ¼ c in G(z) by its stable reflection.

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005212

A second key result is given in lemma 2. This lemmacaptures the idea that the delay structure can be visualised asthe structure of system zeros at infinity.

Lemma 2: Assume that G(z) has a zero at infinity withmultiplicity m1 associated to the unitary direction h1(there may be more zeros located somewhere else). Define

~GGðzÞ ¼ L1ðzÞGðzÞ ð4Þ

L1ðzÞ ¼ zh1hH1 þ U1UH

1

¼ h1 U1½ �z 0

0 In�1

24

35 hH

1

UH1

24

35 ð5Þ

and U1 is chosen so that h1 U1½ � is unitary. Then,

1. ~GGðzÞ has m1 � 1 zeros at infinity associated to thedirection h1 and, possibly, one zero in z ¼ 0 associatedto h1:2. Except by the previous fact, ~GGðzÞ shares its poles andzeros with G(z), but not necessarily their directions.3. L1ðzÞ is unitary and has unity DC gain, i.e. L1ð1Þ ¼ I:4. L1ðzÞ is a polynomial matrix.

Proof: We proceed by parts:

1. G(z) has a zero with multiplicity m1 at infinity associatedto the direction h1 iff

hH1GðzÞ ¼

Xn

i¼1

�1iGi�ðzÞ ¼

1Qm1i¼1 ðz � piÞ

DðzÞ ð6Þ

where Gi�ðzÞ denotes the ith row of G(z), �1ithe ith

component of h1 and fpigi¼1;...;m; are poles of hH1GðzÞ.

DðzÞ is a row vector such that Dð1Þ 6¼ 0 and Dð1Þ has

finite entries.Also, from (4) it follows that

hH1 ~GGðzÞ ¼ hH

1L1ðzÞGðzÞ ¼ zhH1GðzÞ ð7Þ

which shows that ~GGðzÞ has a zero in z ¼ 0 with left directionh1, if and only if the row vector hH

1GðzÞ is analytical atz ¼ 0: Therefore, from (6) and (7)

hH1 ~GGðzÞ ¼ 1Qm1�1

i¼1 ðz � piÞ� z

z � pm1

DðzÞ

¼ 1Qm1�1i¼1 ðz � piÞ

�DDðzÞ ð8Þ

which shows that ~GGðzÞ has m1 � 1 zeros at infinity with leftdirection h1.2. From the Smith form of L1ðzÞ [30], it is clear that L1ðzÞhas no poles and only one zero at the origin. Therefore, atmost one zero will be added to G(z) to form ~GGðzÞ. It wasshown above that the effect of L1ðzÞ is to cancel one zero atinfinity of G(z) and, possibly, to substitute it by a zero atz ¼ 0. Therefore, the rest of the poles and zeros of G(z) arealso poles and zeros of ~GGðzÞ, although not necessarily withthe same directions.3. To prove this, it suffices to note that the fact that h1 andh1 U1½ � are unitary implies

L�1ðzÞL1ðzÞ ¼ z�1h1hH

1 þ U1UH1

� �zh1hH

1 þ U1UH1

� �¼ I ð9Þ

and L1ð1Þ ¼ I:4. Straightforward from (5). A

From lemma 2 we have that

GðzÞ ¼ L�11 ðzÞ ~GGðzÞ;

with L�11 ðzÞ ¼ 1

zh1hH

1 þ UcUHc ð10Þ

which says that L1ðzÞ extracts one of the zeros of G(z) inz ¼ 1; i.e. the relative degree is reduced by one.

3.1 Unitary interactors

In this section, lemma 2 is used to find a unitary interactormatrix with unity DC gain. It is also shown that an interactorwith those properties is unique.

Definition 1: Given any real, proper and non-singulartransfer matrix GoðzÞ 2 C

n�n, then any polynomial transfermatrix, jgðzÞ, that satisfies

limz!1

jgðzÞGoðzÞ ¼ K ð11Þ

where K is a real non-singular matrix (and therefore onlywith finite entries), will be called a (left) interactor matrix ofGoðzÞ:The above definition is equivalent to saying that apolynomial transfer matrix, jgðzÞ, is a (left) interactormatrix for another transfer matrix, GoðzÞ, if their productjgðzÞGoðzÞ is biproper. Also note that, in general, theinteractor matrix is non-unique [23], but it can be madeunique if some special constraints are imposed [2, 29, 31].Note that the previous definition allows the interactor to beNMP, but as we will see in the following sections, it ispreferable to choose an interactor matrix with all its zerosin the stability region.

The following result is extracted from [24]:

Theorem 1: Given any proper and non-singular (almosteverywhere) transfer matrix GoðzÞ 2 C

n�n, then there existsa unitary interactor, jgUðzÞ, which satisfies

detfjgUðzÞg ¼ kzm ð12Þ

where m is the relative degree of GoðzÞ and k is anappropriated real number. In the sequel, jgUðzÞ will becalled a unitary interactor.

jgUðzÞ is non-unique, but if jgU1ðzÞ and jgU2ðzÞ areunitary interactors, then the only matrix T(z) satisfying

jgU1ðzÞ ¼ TðzÞjgU2ðzÞ ð13Þ

is a unitary constant one.

Proof: See [24] and [27] where an explicit algorithm tobuild unitary interactors is given. A

It must be also noted that the unitary interactor definedabove has no especial structure (compare with Wolovichand Falb’s triangular interactor [28]). It has, however, thekey property of being unitary.

The significance of a unitary interactor L(z) in theframework of this paper is that

kLðzÞFðzÞk2 ¼ kFðzÞk2 ð14Þfor any FðzÞ 2 L2. This fact will be used in the followingSection.

From theorem 1 we have the following corollary:

Corollary 1: Given any real, proper and non-singular(almost everywhere) transfer matrix GoðzÞ 2 C

n�n, thenthere exists a unique unitary interactor with unity DC gain.

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 213

Proof: If there exist two distinct unitary interactors withunitary DC gain, then upon considering (13) it followsthat T ¼ I: This contradicts our supposition and provesthe result. A

The unitary interactor with unity DC gain described bycorollary 1 can be built as follows.

Lemma 3: Consider any real, proper and non-singular(almost everywhere) transfer matrix GoðzÞ 2 C

n�n withrelative degree equal to m (that is, with m zeros atinfinity [30]). The unitary interactor with unity DC gainis given by

jj gUðzÞ ¼ LmðzÞ . . .L1ðzÞ ¼Ymi¼1

Lm�iþ1ðzÞ ð15Þ

where we consider the auxiliary definition

GiðzÞ ¼ LiðzÞGi�1ðzÞ; i ¼ 1; . . . ;m ð16Þ

with GoðzÞ ¼ GoðzÞ and where LiðzÞ is defined in (5) (seelemma 2) making h1 ¼ hi, where hi is the directionassociated to the first zero of Gi�1ðzÞ at infinity (theordering is arbitrary).

Proof: To prove our assertion it suffices to prove that thematrix given by (15) corresponds to the interactor matrixdefined in definition 1 with the properties specified intheorem 1 and in corollary 1.

. From lemma 2 it follows thatQm

i¼1 Lm�iþ1ðzÞ is unitary,has unitary DC gain and is a polynomial matrix. Moreover,it follows that det

Qmi¼1 Lm�iþ1ðzÞf g ¼ zm, which shows that

(12) is verified with k ¼ 1:. It remains to prove that

Qmi¼1 Lm�iþ1ðzÞ satisfies (11).

We first write

G1ðzÞ ¼ L1ðzÞGoðzÞ ð17Þ

and, in agreement with lemma 2, we have that G1ðzÞ sharespoles and zeros with GoðzÞ except one zero at infinity thathas been replaced (or cancelled) by a zero at the origin. Thisimplies that the relative degree (number of zeros at infinity)of G1ðzÞ is smaller, in one unit, than the relative degree ofGoðsÞ.

If we write now

G2ðzÞ ¼ L2ðzÞG1ðzÞ ¼ L2ðzÞL1ðzÞGoðzÞ ð18Þ

and follow the previous argument, it turns out that therelative degree of G2ðzÞ is smaller than the relative degree ofGoðzÞ in two units. Therefore, generalising the analysis wehave that

GmðzÞ ¼ LmðzÞGm�1ðzÞ ¼ LmðzÞ � � �L1ðzÞGoðzÞ ð19Þ

is a biproper matrix (i.e. without zeros at infinity) and,therefore, it is non-singular for z ! 1.

The discussion above allows us to conclude that (15)defines the unique unitary interactor with unity DC gain ofGoðzÞ.

The previous ideas are next illustrated with anexample.

Example 1: Consider a continuous-time plant having amodel given by

GocðsÞ ¼

1:116e�7s

s þ 0:22310

0:8926e�5s

s þ 0:2231

0:5579e�6s

s þ 0:2231

1:116

s þ 0:2231

1:339e�4s

s þ 0:2231

00:5579

s þ 0:2231

0:6694e�4s

s þ 0:2231

2666666664

3777777775

ð20Þ

Assume a sampling interval of 1[s ] and a zero-order hold atthe input of the plant, then the corresponding discrete timemodel is found to be

GoðzÞ ¼

5

ð5z � 4Þz70

4

ð5z � 4Þz5

2:5

ð5z � 4Þz6

5

5z � 4

6

ð5z � 4Þz4

02:5

5z � 4

3

ð5z � 4Þz4

266666664

377777775

ð21Þ

Thus

detfGoðzÞg ¼ 25

z11ð5z � 4Þ3ð22Þ

which implies that GoðzÞ has m ¼ 14 zeros at infinity, i.e. ithas relative degree equal to 14, and no NMP zeros. Note thatthere are 11 poles at the origin which take the delays of theplant model into account.

The unitary interactor with unity DC gain will beconstructed following the procedure outlined in the proofof lemma 3. The first factor L1ðzÞ is given by

L1ðzÞ ¼1 0 0

0 z 0

0 0 1

24

35 ð23Þ

and therefore,

G1ðzÞ ¼ L1ðzÞGoðzÞ ¼

5

ð5z � 4Þz70

4

ð5z � 4Þz5

2:5

ð5z � 4Þz5

5z

5z � 4

6

ð5z � 4Þz3

02:5

5z � 4

3

ð5z � 4Þz4

266666664

377777775

ð24ÞNote that

detfG1ðzÞg ¼ 25

ð5z � 4Þ3z10ð25Þ

what implies that G1ðzÞ has 13 zeros at infinity, i.e. it hasrelative degree equal to 13. Note that this verifies that L1ðzÞcancels one of the zeros at infinity of GoðzÞ with its zero atthe origin.

Proceeding in a similar fashion, one can construct the 14factors that form the unitary interactor of GoðzÞ. Theprocedure is easily implemented automatically using asuitable software, such as Matlab. Note that G14ðzÞ is suchthat

detfG14ðzÞg ¼ 25z3

ð5z � 4Þ3ð26Þ

which implies that the relative degree of G14ðzÞ is zero and,therefore, G14ðzÞ is biproper and the procedure stops.

From (15) the unitary interactor with unity DC gain isthen given by

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005214

jj gUðzÞ ¼Y14

i¼1

L14�iþ1ðzÞ

¼z6 0

0 0:2zðz6 þ 4Þ0 �2=5zðz � 1Þðz þ 1Þðz2 þ 1 þ zÞðz2 � z þ 1Þ

264

0

�2=5zðz � 1Þðz þ 1Þðz2 þ 1 þ zÞðz2 � z þ 1Þ0:2zð4z6 þ 1Þ

375

ð27Þ

3.2 Unitary zero-interactors

This section provides a precise expression for a unitaryzero-interactor for any square transfer matrix.

Definition 2: Consider a proper and non-singular (almosteverywhere) transfer matrix GoðzÞ 2 C

n�n, with no zeros onthe unit circle, but having q NMP finite zeros, denotedjointly by fcigi¼1;...;q; jcij>1, then any minimum phasebiproper matrix, jcðzÞ; will be called a zero-interactor ofGoðzÞ if it satisfies

limz!ci

jcðzÞGoðzÞ ¼ Ki 8i 2 f1; . . . ; qg ð28Þ

where Ki is a non-singular real matrix.

The previous definition is equivalent to saying that anymatrix, jcðzÞ, is a (left) zero-interactor of a transfer matrix,GoðzÞ, if jcðzÞGoðzÞ has all its zeros inside the unit circle.Note that the definition can be applied to minimum phasezeros as well, but only the NMP zero-interactor will proveuseful in what follows.

Expressions for certain classes of zero-interactors can befound in [2] and in [29].

The following lemma provides an explicit expressionfor the unitary zero-interactor of a given transfer matrix:

Lemma 4: Consider a proper and non-singular (almosteverywhere) transfer matrix GoðzÞ 2 C

n�n with q finiteNMP zeros and no zeros on the unit circle. One unitaryzero-interactor with unity DC gain is given by

jj cUðzÞ ¼ LcqðzÞ � � �Lc1ðzÞ ¼Yq

i¼1

Lcq�iþ1ðzÞ ð29Þ

where the following auxiliary definition is considered

GiðzÞ ¼ LciðzÞGi�1ðzÞ; i ¼ 1; . . . ; q ð30Þand LciðzÞ is defined in (2) (see lemma 1) with hc ¼ hi,where hi is the direction of the NMP zero of Gi�1ðzÞlocated at z ¼ ci.

Proof: To prove the result we will proceed as in the proof oflemma 3. It suffices to prove that the matrix given by (29)satisfies all the properties of the interactor defined indefinition 2, that it is unitary and that it has unity DC gain.

. From lemma 1 we have thatQq

i¼1 Lcq�iþ1ðzÞ is unitary,has unitary DC gain and is a biproper minimum phasetransfer matrix.. It only remains to prove that

Q1i¼q Lcq�iþ1ðzÞ satisfies

(28). To that end we write

G1ðzÞ ¼ Lc1ðzÞGoðzÞ ð31ÞFrom Lemma 1, we have that G1ðzÞ shares the poles andzeros of GoðzÞ except for one zero at z ¼ c1 of GoðzÞ that has

been replaced by a zero at z ¼ 1�cc1

in G1ðzÞ; provided thathc1

HGoðzÞ is analytical at z ¼ 1�cc1

.Writing now

G2ðzÞ ¼ Lc2ðzÞG1ðzÞ ¼ Lc2ðzÞLc1ðzÞGoðzÞ ð32Þand repeating the previous argument, it is clear that G2ðzÞhas the same poles and zeros of GoðzÞ except for two zeros,originally at z ¼ c1 y z ¼ c2 in GoðzÞ, that were replaced by

zeros at z ¼ 1�cc1

and z ¼ 1�cc2

in G2ðzÞ, provided that �c1HGoðzÞ

is analytical at z ¼ 1�cc1

and �c2HG1ðzÞ is analytical at z ¼ 1

�cc2.

Therefore,

GqðzÞ ¼ LcqðzÞGq�1ðzÞ ¼ LcqðzÞ . . .Lc1ðzÞGoðzÞ ð33Þ

is a matrix without NMP zeros. Hence jj cUðzÞGoðzÞ is non-singular at z ¼ ci; 8i.

The previous discussion allows one to conclude that (29)defines a unitary zero-interactor with unity DC gain. A

The relevance of a unitary zero-interactor LcðzÞ lies in theproperty (14). This fact will be used in the following sectionto derive performance bounds in MIMO control systems.

4 Performance bounds in MIMO systems

In this section we compute the minimum value of the costfunctional

J ¼X1k¼0

eTðkÞeðkÞ ¼ kEðzÞk22 ð34Þ

where e(k) denotes the tracking error of a one-degree-of-freedom control loop, for a step reference signal, r(k),applied at k ¼ 0. This means that rðkÞ ¼ vmðkÞ withv 2 R

n�1, and where mðkÞ denotes the unit step function.Using the Youla parameterisation of all stabilisingcontrollers (see, for example, [2]) it is possible to re-writeJ as

J ¼ ðI � GoðzÞQðzÞÞ vz

z � 1

��������2

2

ð35Þ

where Q(z) is the Youla parameter.

4.1 Case I: stable, minimum phase plant

Theorem 2: Consider a proper stable minimum-phasetransfer matrix GoðzÞ 2 C

n�n with relative degree (numberof zeros at infinity) equal to m and such that Goð1Þ is non-singular.

1. The optimal causal Youla parameter that stabilises theplant and that minimises J, as defined in (35), is given by

QoptðzÞ ¼ arg minQðzÞ2S

J ¼ ~GG�1o ðzÞ ð36Þ

where S denotes the set of all stable and proper rationaltransfer matrices KðzÞ 2 C

n�n and ~GGoðzÞ satisfies

~GGoðzÞ ¼ jj gUðzÞGoðzÞ ð37Þ

where jjgUðzÞ is the unitary interactor matrix with unity DCgain for GoðzÞ.2. The minimum cost J is given by

Jopt ¼ minQðzÞ2S

J ¼Xm

i¼1

jhHi vj2 ð38Þ

where hi corresponds to the direction of the first zero atinfinity of Gi�1ðzÞ, with Gi�1ðzÞ defined in lemma 3.

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 215

Proof: We will proceed by parts:

1. Using (37) in (35) we have that

J ¼ zjj�1

gUðzÞðjj gUðzÞ � ~GGoðzÞQðzÞÞ v

z � 1

��������2

2

ð39Þ

Then, using the fact that jj gUðzÞ and zIn are unitary transfermatrices and using elementary 2-norm properties, (39) canbe written as

J ¼ ðjj gUðzÞ � ~GGoðzÞQðzÞÞ v

z � 1

��������2

2

ð40Þ

Note that since jj gUðzÞ has unity DC gain it is necessary thatQð1Þ ¼ ~GG

�1o ð1Þ ¼ G�1

o ðzÞ (the last equality follows from(37)) in order to J to be finite. Equation (40) leads to

J ¼ ðjj gUðzÞ � InÞv

z � 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}AðzÞ

þ ðIn � ~GGoðzÞQðzÞÞ v

z � 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}BðzÞ

���������

���������

2

2

ð41Þ

From theorem 1, jj gUðzÞ is a polynomial matrix and,therefore, has not finite poles. Moreover, jj gUð1Þ ¼ I) jjgUðzÞ � In ¼ ðz � 1Þ ~NNðzÞ. Therefore, A(z) is analyticalfor all jzj � 1, which means that AðzÞ 2 H?

2 [12, 32]. On theother hand, ~GGoðzÞ is stable since GoðzÞ is itself stable andjjgUðzÞ does not add poles to Go when constructing ~GGoðzÞ(see (37)). Moreover, Q(z) must be stable in order to havea stable control loop (since Go is stable, this suffices to

guarantee stability) and since Qð1Þ ¼ ~GG�1o ð1Þ ¼ G�1

o ð1Þ )ðIn � ~GGoðzÞQðzÞÞ ¼ ðz � 1Þ ~~NN~NNðzÞ; it is clear that B(z) has allits poles inside the unit circle and, therefore, BðzÞ 2 H2.Therefore,

J ¼ ðjj gUðzÞ � InÞv

z � 1

��������2

2

þ ðIn � ~GGoðzÞQðzÞÞ v

z � 1

��������2

2

ð42Þ

From (42) it is clear that

QoptðzÞ ¼ arg minQðzÞ2S

J ¼ ~GG�1o ðzÞ ð43Þ

Note that given (37) and (11), ~GGoðzÞ is biproper.Furthermore, since GoðzÞ is minimum phase and the zerosof ~GGoðzÞ are the zeros of GoðzÞ plus up to m zeros at theorigin (see (12)), it is clear that ~GGoðzÞ is minimum phase.Hence, ~GG

�1o ðzÞ is stable and biproper. Note that it is verified

that Qoptð1Þ ¼ ~GG�1o ð1Þ what validates our previous assump-

tion regarding the DC gain of QoptðzÞ.This discussion allows us to conclude that (43) is the

optimal Youla parameter.2. Replacing (36) in (42) the optimal cost can beexpressed as

Jopt ¼ minQðzÞ2S

J

¼ vH 1

2p

Z p

�p

jj gUðe joÞ � In

e jo � 1

!Hjj gUðe joÞ � In

e jo � 1do

" #v

ð44Þ

using the definition z ¼ e jo and Cauchy’s residue theorem(see, for example, [1]), (44) can be written as

Jopt ¼ vHXk

i¼1

Resz¼zi

1

z

jj ~gUðzÞ � In

z�1 � 1

!jj gUðzÞ � In

z � 1

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

OðzÞ

266664

377775v

ð45Þwhere fzigi¼1...k denotes the set of all poles of O(z) inside theunit circle. It is straightforward to prove, using the fact thatthe unitary interactor jj gUðzÞ is a polynomial matrix and hasunity DC gain, that the only poles of O(z) inside the unit circleare at the origin. This implies that it suffices to calculate theresidues of O(z) at the origin to evaluate (45). Also, using thefact that the interactor is unitary, it follows that

OðzÞ ¼ �2In

ðz � 1Þ2þ

jj gUðzÞðz � 1Þ2

þjj ~gUðzÞðz � 1Þ2

ð46Þ

However, given the explicit expression of the interactorgiven by (15) and the properties of each of its factors (seelemma 2), it is clear that the first two summands in (46) haveno poles at the origin and, consequently, their residues atz ¼ 0 are zero. It only remains to evaluate the residues of thethird term in (46).

Using (15), (2) and the results mentioned in the lastparagraphs, one has that

Xk

i¼1

Resz¼ziOðzÞ ¼ Resz¼0

jj ~ggUðzÞðz � 1Þ2

¼ Resz¼0

Qmi¼1 hi Ui½ �

1z

0

0 In�1

� �hH

i

UHi

" #

ðz � 1Þ2

¼ Resz¼0

1

zmðz � 1Þ2

�Ymi¼1

hihHi þ z In � hih

Hi

� �� �ð47Þ

Note that the right side of (47) has at least relative degreeequal to 2. This implies, in accordance with lemma 1 in [8],that the sum of residues at each singularity of thatexpression equals zero.

Therefore,

Resz¼0

1

zmðz � 1Þ2

Ymi¼1

hihHi þ z In � hih

Hi

� �� �¼ �Resz¼1

1

zmðz � 1Þ2

Ymi¼1

hihHi þ z In � hih

Hi

� �� �ð48Þ

But if we define LiðzÞ ¼ hihHi þ z In � hih

Hi

� �� �, then

Resz¼1

1

zmðz � 1Þ2

Ymi¼1

LiðzÞ ¼ limz!1

d z�mQm

i¼1 LiðzÞ½ �dz

ð49Þ

It can be verified that LiðzÞjz¼1 ¼ In anddLiðzÞ

dz

���z¼1

¼ In � hihHi . Moreover, applying elementary

derivative properties it follows that

d z�mQm

i¼1LiðzÞ½ �dz

¼�mz�m�1Ymi¼1

LiðzÞ

þz�mXm

j¼1

Yj�1

i¼1

LiðzÞ !

dLjðzÞdz

Ymi¼jþ1

LiðzÞ !( )

ð50Þ

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005216

This result, when combined with the properties of LiðzÞmentioned above, allows us to conclude that the limitingoperation in (49) can be written as

limz!1

d z�mQm

i¼1 LiðzÞðzÞ½ �dz

¼ �Xm

j¼1

hjhHj ð51Þ

Equation (51), together with (49), (48), (47) and (45)lead to

Jopt ¼ vHXm

j¼1

hjhHj

" #v ð52Þ

which proves the result. A

The previous result cannot be derived using Wolovichand Falb interactor matrices [2, 28] since that interactor isnot unitary. This confirms that the Wolovich and Falbinteractor is not as suitable as other versions of the interactormatrix, which is in agreement with the findings in [26], inthe framework of minimum variance control.

The result of theorem 2 can be particularised to thecase when a step change is applied to the reference inthe ith channel. In this case, the 2-norm of the error (see(35)) is given by

Ji ¼ ðI � GoðzÞQðzÞÞ eiz

z � 1

��������2

2

ð53Þ

where ei is the ith canonical basis vector of Cn�1. If one

is interested in considering the sum of the 2-norm of theerrors resulting of the application of successive stepchanges in each channel, the corresponding cost can bewritten as

JF ¼Xn

i¼1

Ji ð54Þ

In the case of a stable, minimum phase and propertransfer function GoðzÞ, the minimum value of JF isgiven by the relative degree (number of zeros at infinity)of GoðzÞ, as follows from a straightforward calculationbased upon theorem 2.

Example 2: Consider the plant of example 1 given by (21)and its unitary interactor with unity DC gain given by (27).According to theorem 2 the optimum Youla parameter isgiven by QoptðzÞ ¼ ~GG

�1o ðzÞ ¼ G�1

14 ðzÞ and the minimumcost by

Jopt ¼ vHX14

i¼1

hihHi v ¼ vH

6 0 0

0 11=5 �12=5

0 �12=5 29=5

24

35v ð55Þ

Figure 1 shows the temporal evolution of the errorconsidering v ¼ ½ 1 �1 0:5 �H . The cost obtained upondirect evaluation of the left expression in (34) using thesimulation results, is given by Jsim ¼ 12:05, which is equalto the cost obtained evaluating (55) in this case. A

4.2 Case II: stable, NMP plant

In this subsection we will extend the result of theorem 2to the NMP case.

Theorem 3: Consider a proper stable NMP transfer matrixGoðzÞ 2 C

n�n with relative degree (number of zeros atinfinity) equal to m, q NMP zeros denoted by fcigi¼1���q andsuch that Goð1Þ is nonsingular. Then,

1. The optimal causal Youla parameter that stabilisesthe plant and achieves a finite cost J, as defined in (35), isgiven by

QoptðzÞ ¼ arg minQðzÞ2S

J ¼ ~GG�1o ðzÞ ð56Þ

where S denotes the set of all stable and proper rationaltransfer matrices KðzÞ 2 C

n�n and ~GGoðzÞ satisfies

~GGoðzÞ ¼ jj cUðzÞjj gUðzÞGoðzÞ ð57Þ

where jj gUðzÞ is the unitary interactor matrix with unity DC

gain of GoðzÞ and jj cUðzÞ is the unitary zeros-interactor

defined in lemma 4 for jj gUðzÞGoðzÞ.2. The minimum cost J is given by

Jopt ¼ minQ2S

J ¼Xm

i¼1

hgHi v

�� ��2 þXq

i¼1

jcij2 � 1

j1 � cij2hcH

i v�� ��2 ð58Þ

where hgi corresponds to the direction of the first zero at

infinity of Gi�1ðzÞ, with Gi�1ðzÞ defined in lemma 3considering GoðzÞ as the plant whose zeros at infinityshould be removed. hc

i corresponds to the direction ofthe NMP zero of Gi�1ðzÞ, located at z ¼ ci, with Gi�1ðzÞdefined in lemma 4, considering jj gUðzÞGoðzÞ as the plantwhose finite NMP zeros should be removed.

Proof: The proof goes along the same lines as the proof oftheorem 2.

1. Substituting (57) in (35) one gets

J ¼ ðjj cUðzÞjj gUðzÞ � InÞv

z � 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}AðzÞ

þ ðI � ~GGoðzÞQðzÞÞ v

z � 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}BðzÞ

���������

���������

2

2

ð59ÞGiven the properties of the interactor matrices involved in(59), A(z) has finite DC gain since the factor z � 1 issimplified. Therefore, all the poles of A(z) are outside theunit circle. Note that some of them are at infinity. Thisimplies that AðzÞ 2 H?

2 . On the other hand, B(z) is stablewith finite DC gain because Q(z) must be chosen with theinverse DC gain of the plant and this is the DC gain of ~GGoðzÞ.Moreover, B(z) is strictly proper and stable so thatBðzÞ 2 H2.

Fig. 1 Error evolution using optimal controller and r(k) ¼[ 1 �1 0:5 ]Hm(k)

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005 217

Therefore,

J ¼ ðjj cUðzÞjj gUðzÞ� InÞv

z� 1

��������2

2

þ ðI � ~GGoðzÞQðzÞÞ v

z� 1

��������2

2

ð60Þwhich clearly implies that

QoptðzÞ arg minQðzÞ2S

J ¼ ~GG�1o ðzÞ ð61Þ

2. From (61) and (60),

Jopt ¼ minQðzÞ2S

J ¼ ðjj cUðzÞjj gUðzÞ � InÞv

z � 1

��������2

2

¼ vHXk

i¼1

Resz¼zi

�2In

ðz � 1Þ2þjj cUðzÞjj gUðzÞ

ðz � 1Þ2

("

þðjj cUðzÞjj gUðzÞÞ�

ðz � 1Þ2

)#v ð62Þ

where we have followed the same procedure we used toprove theorem 2 and fzigi¼1...;k corresponds to the set ofstable poles of each of the quantities whose residues must beevaluated in (62). Given the properties of the unitaryinteractor matrices involved, it is clear that the first andsecond summand in (62) have no poles inside the unit circleand therefore, their residues are zero. This implies that

Jopt ¼ vHXk

i¼1

Resz¼zi

ðjj cUðzÞjj gUðzÞÞðz � 1Þ2

�" #v ð63Þ

Using (15) and (29) one can write

ðjj cUðzÞjj gUðzÞÞ�

ðz�1Þ2

¼ Ym

j¼1

hgj Ug

j

� � 1�zz

0

0 0

� �hgH

j

UgHj

" #þ In

( )

�Yg

j¼1

hcj Uc

j

� � 1��cci

1�ci

z�ci

1��cciz�1 0

0 0

" #hcH

j

UcHj

" #þ In

( )!,ðz�1Þ2

ð64Þwhere it becomes apparent that the first factor of the rightside numerator of (64) has all its poles at the origin and,since jcij>1, the second factor has all its poles inside theunit circle. Proceeding as is the proof of theorem 2, it can beverified that the relative degree of (64) is equal to two andtherefore, one can use the same ideas that showed useful inthat proof. Proceeding in this way, it is straightforward toobtain

Jopt ¼ vHXm

i¼1

hgi h

gHi þ

Xq

i¼1

jcij2 � 1

j1 � cij2hc

ihcHi

" #v ð65Þ

which completes the proof. A

5 Discussion of the results

Expressions (38) and (58) can be interpreted as a measure ofhow far Q(z) is from the perfect inverse of the plant modelGoðzÞ, in a direction v. Thus, zero cost is equivalent toperfect inversion. It is known that the process characteristicsthat prevent perfect (stable and feasible) inversion are theprocess delays (zeros at infinity) and its finite NMP zeros.In the first case, it is impossible to build a physicallyrealizable inverse and in the second, the resulting inversewould be unstable. The results of theorem 3 are then

sensible: the optimal cost, as measured by (34), is boundedfrom below by the presence of finite and non-finite NMPzeros.

The analysis of the closed form expressions of theminimal costs (38) and (58) also shows that there is a strongdependence upon the direction of the reference. This meansthat not only the presence of some non-invertible charac-teristic will increase the functional value. Indeed, there is aninterplay between directionality, NMP zeros and delays.This is a key issue, and it implies that experiments poorlydesigned might obscure some of the process main features.Consider for instance a process with only one finite NMPzero. If one selects a reference with a direction that isorthogonal to the NMP zero direction, the particular costwill be equal to the cost achieved in the case of a processthat shares the delay structure of the first plant, but that isminimum phase. This is not surprising since directionality isa key issue in MIMO systems [2].

Finally it is interesting to examine more carefully the effectof the location of the finite NMP zeros on the value of thefunctional (34). According to theorem 3 (see (58)) the generalterm that weights the product between the zero direction andthe reference (or output disturbance) direction is

PðciÞ ¼jcij2 � 1

j1 � cij2ð66Þ

This function tends to infinity as ci ! 1 and to zero ifci ! �1. This means that only zeros near z ¼ 1 will cause ahigh rising of the minimum functional value, but other zeros,although having magnitude near one, won’t have animportant effect. This can be best viewed in Fig. 2 wherePðciÞ is plotted considering ci ¼ re jy with r 2 ð1; 3� andy 2 ½�p; p�. The result shown in Fig. 2 is in completeagreement with [22] and other classical results [2]: NMPzeros near one are very hard to deal with, in particular theymakes a process very hard to control.

6 Conclusions

This paper presents the computation of an achievableperformance bound for the feedback control of stableMIMO systems with arbitrary delay structure and NMPzeros. The result also includes a closed form expression forthe Youla controller, which allows one to achieve thatperformance bound. The expression for the bound explicitlyincludes the key plant features that determine and limit thecontrol performance.

The main results rely entirely on the unitary property of theproposed interactors and zero-interactors. As a byproduct,we have proven the uniqueness of the unitary interactor with

Fig. 2 P(ci) as a function of r and y, where ci ¼ re jy

IEE Proc.-Control Theory Appl., Vol. 152, No. 2, March 2005218

unity DC gain, which in some way completes the availableresults on that topic.

Future work in the areas covered by the paper shouldinclude the discussion of performance limits in the fulldecentralised case in order to evaluate, for example, theconsequential performance deterioration.

7 Acknowledgments

The authors gratefully acknowledge the support receivedthrough grants Fondecyt 1040313 and UTFSM 230425. Theyare also grateful for helpful discussions with Juan I. Yuz.

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