Automatica 38 (2002) 467–475www.elsevier.com/locate/automatica

Brief Paper

Robust Nyquist array analysis based on uncertainty descriptionsfrom system identi'cation�

Dan Chen, Dale E. Seborg ∗

Department of Chemical Engineering, University of California, Santa Barbara, CA 93106, USA

Received 25 April 2000; revised 18 July 2001; received in 'nal form 21 August 2001

Abstract

A robust Nyquist array analysis for MIMO systems is proposed based on uncertainty descriptions obtained from systemidenti'cation. Two types of statistical-based uncertainty error bounds for the frequency response are obtained: element boundsand column bounds. Gershgorin’s theorem and the concepts of diagonal dominance and Gershgorin bands are extended to includemodel uncertainty. Robust stability theorems are developed based on ellipsoidal uncertainty descriptions obtained from systemidenti'cation. An example is given to illustrate the robust Nyquist array analysis. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Statistical model uncertainty; Gershgorin theorem; Robust Nyquist array analysis

1. Introduction

Robust control based on frequency domain analysissuch as H∞ methods and �-synthesis has been an ac-tive research 'eld during the past 20 years (Skogestad& Postlethwaite, 1996; Zhou, Doyle, & Glover, 1996).These techniques provide very powerful, complex ap-proaches, but the resulting control systems usually havea complicated structure and are high order. Thus, thesedesign techniques are not directly applicable to the de-centralized PI=PID control systems that are widely usedin the process industries. Furthermore, a typical startingpoint for contemporary robust control algorithms is thathard bounds are used to characterize model uncertainties.This formulation disregards the statistical model uncer-tainty descriptions that are readily available from systemidenti'cation (Goodwin, 1999). EBorts to obtain a bettermatch between robust control and system identi'cationhave received attention recently. For example, Braatzand Crisalle (1998) developed a robustness analysis forSISO systems with ellipsoidal parametric uncertainty

� This paper was not presented at any IFAC meeting. This paperwas recommended for publication in revised form by Associate EditorTor Arne Johansen under the direction of Editor Sigurd Skogestad.

∗ Corresponding author. Tel.: + 1-805-893-3352; fax: +1-805-893-4731.E-mail address: seborg@engineering.ucsb.edu (D.E. Seborg).

that is naturally obtained from system identi'cation. Arobust stability analysis directly based on the con'denceellipses of system frequency response from identi'cationhas been developed by Cooley and Lee (1998). Somegeneral conditions for the robustness of MIMO systemswith structured uncertainty such as ellipsoidal uncertaintyhave been developed in the framework of �-synthesis(Khatri & Parrilo, 1998; Chellaboina, Haddad, &Bernstein, 1998), but just as for other �-synthesis meth-ods these robustness analyses cannot be applied for 'xedstructure control system design. In the present paper, theresearch objective is to develop a robust control tech-nique based on statistical uncertainty information ob-tained from system identi'cation, which can be appliednot only as a robust stability analysis method for MIMOsystems but also as a simple robust design approach fordecentralized control systems.Frequency domain techniques, such as Nyquist array

analysis (Rosenbrock, 1974) and quantitative feedbacktheory (QFT) (Horowitz, 1982), provide powerful toolsfor the analysis and design of decentralized control sys-tems. QFT is a generalized loop-shaping method based onNichols plots that addresses model uncertainty for singleloop and decentralized control systems (Horowitz, 1992);thus, it is primarily a robust design methodology. Modi'-cations of QFT have been developed that address robust-ness to parametric and unstructured uncertainties (Braatz,1994; Jayasuriya & Zhao, 1994; Chait, Chen, & Hollot,

0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0005-1098(01)00207-2

468 D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475

1999; Lee, Chait, & Steinbuch, 2000). A disadvantageof the QFT design procedure is that it is rather compli-cated for MIMO systems. Design approaches based oncombinations of the inverse Nyquist array (INA) analy-sis and QFT have been proposed (Nwokah, Nordgren, &Grewal, 1995; Chen, Wang, & Wang, 1991), but do notconsider ellipsoidal uncertainty descriptions.The Nyquist array method for MIMO systems is

much more straightforward than the QFT design pro-cedure and is easier to implement, especially for 'xedstructure controllers such as PID control (Maciejowski,1989). Nyquist array analysis provides a useful exten-sion of the SISO Nyquist method to MIMO systems.Both Gershgorin bands, which address loop interactions,and Nyquist loci have been used to design controllers.Unlike the QFT method, model uncertainty is not con-sidered directly in the original Nyquist array method.Based on a large amount of numerical calculation, Gaoand Zhang (1993) derived empirical robust stabilityconditions for the inverse Nyquist array method and2 × 2 and 3 × 3 systems. More general and theoreticalrobust Nyquist array methods have been developed toaccommodate system uncertainty that can be describedby deterministic error bounds on individual transferfunctions (Arkun, Manousiouthakis, & Putz, 1984), orbe expressed as parametric uncertainty in the rationalpolynomial functions that describe transfer functions (DeSantis & Vicino, 1996; Kontogiannis & Munro, 1997).Since these methods are restricted to transfer functionmodels with rational forms, systems with time delayscannot be analyzed without approximating each time de-lay as a rational function (e.g., a PadMe approximation).Furthermore, these approaches are not general enough toutilize the statistical uncertainty information, especiallycorrelation information that is available from systemidenti'cation. Thus, unnecessary conservativeness canbe introduced by these approaches.In the present paper, Nyquist array analysis is extended

to more general system descriptions and to more gen-eral model uncertainty descriptions, based on statisticaluncertainty bounds obtained from system identi'cation.Two types of uncertainty bounds are obtained for the fre-quency response matrix: bounds for individual elementsand column bounds for each column which are structurederror bounds and include correlation information betweentransfer functions. A new robust Gershgorin theorem isderived for these error bounds descriptions.The new results provide a less conservative robust sta-

bility analysis and a simple robust design approach fordecentralized control systems.

2. Uncertainty description

Consider a MIMO linear system with nu inputs andny outputs. A discrete-time dynamic model can be

written as

y(t)=G(q−1;�)u(t) +H(q−1;�)e(t); (1)

where y(t)∈Rny , e(t)∈Rny , u(t)∈Rnu and

�=[�1; : : : ; �p]T; (2)

G(q−1;�)=∞∑

k=1

Gk(�)q−k ; (3)

H(q−1;�)= I+∞∑

k=1

Hk(�)q−k : (4)

Vector � denotes the model parameters to be identi'ed.In Eqs. (1)–(4), q−1 denotes the backward shift opera-tor and e(t) is a sequence of independent random vectorswith zero mean values and covariance matrices �e. Thesubsequent analysis is not restricted to the model formgiven in Eqs. (1)–(4). In fact, if the system can be de-scribed by any linear model, the subsequent analysis isstill valid.Assume that �=[�1; : : : ; �p]

T is an asymptoticallynormal and unbiased estimate of �. The estimated co-variance matrix of the parameter estimation error �� isassumed to be obtained during system identi'cation.Therefore, an asymptotic limit exists,

(�−�)T�−1� (�−�)→ �2p; (5)

where p is the number of model parameters and �2p is thechi-square distribution with p degrees of freedom. Thisresult enables the construction of con'dence bounds

P[(�−�)T�−1� (�−�)6 �2�;p]= 1− �; (6)

where 1− � is the probability that the true parameters liewithin the speci'ed ellipsoidal con'dence region about� and �2�;p is the upper 100� percentage point of the �2

probability distribution with p degrees of freedom.Let gkl(e j!;�) be the frequency response from input

l to output k. The corresponding estimated frequencyresponse is gkl(e

j!), gkl(e j!; �). Express the frequencyresponse in terms of real and imaginary parts:

gkl(e j!;�)= akl(!;�) + jbkl(!;�);

gkl(ej!)= akl(!) + jbkl(!); (7)

where akl(!) and bkl(!) denote the correspondingestimates.The gradient of akl(!;�) with respect to the param-

eters � is represented by

akl;�(!)=@

@�T akl(!;�) (8)

D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475 469

which is a 1 × p row vector. Denote the gradient ofbkl(!;�) as

bkl;�(!)=@

@�T bkl(!;�): (9)

De'ne

∇gkl(!),

[akl;�(!)

bkl;�(!)

]�

; (10)

a 2 × p matrix which is evaluated at �= �. Therefore,the estimation error for akl(!;�) and bkl(!;�) can beapproximated by a truncated Taylor series expansion,

Rgkl(!),

[akl(!)− akl(!;�)

bkl(!)− bkl(!;�)

]

≈ ∇gkl(!)(�−�): (11)

For the remainder of the paper, we use the simpler nota-tion, akl(!)= akl(!;�) and bkl(!)= bkl(!;�). The sub-sequent discussion utilizes the following lemma.

Lemma 1 (Wahlberg & Ljung; 1992). Let x∈Rn; �∈Rn×n be positive de1nite; and assume that

xT�−1x6 1: (12)

Consider y=Ax; where y∈Rp; p6 n; A∈Rp×n and Ahas full row rank. Then

yT(A�AT)−1y6 1: (13)

2.1. Element bounds for the frequency response

Con'dence bounds for each frequency response, gkl,can be derived using Lemma 1 with Eqs. (6) and (11)

Rgkl(!)T[∇gkl(!)��∇gkl(!)T]−1Rgkl(!)6 �2�;p:

(14)

Let �kl(!)=∇gkl(!)��∇gkl(!)T, which represents the2 × 2 covariance matrix for akl(!) and bkl(!) (Ljung,1985). The singular value decomposition (SVD) for thesymmetric �kl(!) matrix is

�kl(!)=Ukl(!)

[�1; kl(!) 0

0 �2; kl(!)

]Ukl(!)T; (15)

where �1; kl(!)¿�2; kl(!) and Ukl(!) is a 2 × 2 unitarymatrix.The following theorem and proof is a formal statement

of a result in Cooley and Lee (1998) that was statedwithout proof.

Theorem 1 (Element bounds). An error bound for eachfrequency response; gkl(e j!;�); is given by

|gkl(e j!;�)− gkl(ej!)|= ‖Rgkl(!)‖26

√�2�;p�1; kl(!):

(16)

Proof. Substituting Eq. (15) into Eq. (14) gives,

Rgkl(!)TUkl(!)

1�1; kl(!)

0

01

�2; kl(!)

Ukl(!)TRgkl(!)6 �2�;p: (17)

Let y=[y1; y2]T =Ukl(!)TRgkl(!), therefore the aboveequation can be written as

y21�1; kl(!)

+y22

�2; kl(!)6 �2�;p: (18)

This implies that

y21�1; kl(!)

+y22

�1; kl(!)6 �2�;p (19)

because �1; kl(!)¿�2; kl(!). Hence,

‖y‖226 �2�;p�1; kl(!): (20)

Because Ukl(!) is a unitary matrix, ‖y‖22 = ‖Rgkl(!)‖22,which implies that inequality (16) holds.

Theorem 1 provides a circular bound for each fre-quency response element, gkl(e j!; ;�). This theorem canalso be proven by using the Rayleigh–Ritz inequal-ity (Rugh, 1996). The uncertainty bounds given byTheorem 1 include a degree of conservatism that isintroduced by Lemma 1 and Eq. (19).

2.2. Column bounds for the frequency response

The error bounds in Eq. (16) provide independentbounds on each element of the frequency responsematrix. However, they ignore the covariance informa-tion between diBerent elements. In multivariable designproblems, the interaction between elements is an im-portant factor. In Nyquist array analysis, the radius ofeach Gershgorin circle depends on the entire columnor row of the transfer function matrix. Therefore, ig-noring the uncertainty interaction between diBerent el-ements can produce very conservative stability results.In this section, a new formulation of the error boundfor each column of the frequency response matrix isderived from statistical information provided by systemidenti'cation.

470 D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475

De'ne

∇gl(!)= col([∇g1l(!); : : : ;∇gnyl(!)]); (21)

where col(·) is an operator that stacks the columns ontop of each other and yields a column vector. Thus,

Rgl(!), col([Rg1l(!); : : : ;Rgnyl(!)])

≈ ∇gl(!)(�−�): (22)

From Lemma 1,

Rgl(!)T[∇gl(!)��∇gl(!)T]−1Rgl(!)6 �2�;p: (23)

Let �l(!)=∇gl(!)��∇gl(!)T. Therefore, �l(!) is a2ny × 2ny symmetric matrix representing the covariancematrix of [a1l(!); b1l(!); : : : ; anyl(!); bnyl(!)]

T (Ljung,1985). The SVD for this matrix is

�l(!)=Ul(!)diag[�1; l(!); : : : ; �2ny;l(!)]Ul(!)T; (24)

where �1; l(!)¿�2; l(!)¿ · · ·¿�2ny;l(!), Ul(!) is a2ny ×2ny unitary matrix, and diag[ · ] denotes a diagonalmatrix with elements �1; l(!); : : : ; �2ny;l(!).Therefore, a new bound for the estimation error of each

column of the system frequency response matrix can bederived based on covariance information and the largestsingular value.

Theorem 2 (Column bounds). The bound for ‖Rgl(!)‖2;the error for the lth column of frequency responsematrix G(e j!;�); is given by

‖Rgl(!)‖2 =( ny∑

k=1

|gkl(ej!;�)− gkl(ej!)|2

)1=2

6√

�2�;p�1; l(!): (25)

Proof. Similar to Theorem 1.

In analogy with Theorem 1, the uncertainty boundsgiven by Theorem 2 may also be conservative due toLemma 1 and Eq. (19). Eq. (25) gives the error boundfor a column of the estimated frequency response matrixwhile Eq. (16) gives the error bound for each element.Thus, it is inappropriate to compare these two boundsdirectly. But if the error bound for an entire column ofthe estimated frequency response matrix is obtained byadding the error bounds in Eq. (16) for each element inthis column, this error bound will be larger than the errorbound obtained directly from Eq. (25), unless this columnhas only one nonzero element. The element bound (16)is more conservative because the correlation between dif-ferent elements in the column is ignored while the co-variance information of the entire column is included inthe error bound in Eq. (25).

3. Robust Gershgorin bands

The key to Nyquist array analysis is Gershgorin’stheorem. Next, a new robustness theorem is developedfor uncertain complex matrices based on Gershgorin’stheorem.

Theorem 3 (Robust Gershgorin′s theorem). For an n× nuncertain complex matrix Z =[zkl]n×n=[akl + jbkl]n×n

with the nominal value Z =[zkl]n×n=[akl + jbkl]n×n; theeigenvalues of Z lie in the union of the n circles

|#(Z)− zll|6Rl; l=1; 2; : : : ; n; (26)

where Rl= Rl +RRl and

Rl=min

n∑k=1; k �=l

|zkl|;√

n − 1 n∑

k=1; k �=l

|zkl|21=2 ;

(27)

RRl=min

n∑k=1

|zkl − zkl|;√

n

(n∑

k=1

|zkl − zkl|2)1=2

:

(28)

The eigenvalues of Z also lie in the union of the circles;each centered on zkk with radius Rk = Rk +RRk; where

Rk =min

n∑l=1;l �=k

|zkl|;√

n − 1 n∑

l=1;l �=k

|zkl|21=2 ;

(29)

RRk =min

n∑l=1

|zkl − zkl|;√

n

(n∑

l=1

|zkl − zkl|2)1=2

:

(30)

Proof. See the appendix.

Theorem 3 is a robust version of Gershgorin’s theo-rem for uncertain matrices. According to this theorem,the eigenvalues of an uncertain matrix must be locatedinside the union of circles de'ned by the estimated val-ues and the uncertainty bounds. Thus, for an MIMO con-trol problem, the eigenvalues of the frequency responsematrix lie within the union of circles centered on the esti-mated frequency response. Therefore, Theorem 3 can beused for a Nyquist array analysis of uncertain models.

De#nition (Diagonal dominance). For an n×n uncertaincomplex matrix Z which has a nominal value Z , if

|zll|¿ Rl= Rl +RRl for l=1; 2; : : : ; n (31)

then, Z is said to have robust column diagonal dominancefor Rl and RRl in Eqs. (27) and (28). If Rk and RRk are

D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475 471

de'ned as in Eqs. (29) and (20), Z is said to have robustrow diagonal dominance.

De#nition (Robust Gershgorin bands). Consider ann × n transfer function matrix Z(s) with the nominalvalue Z(s). At each zll(j!), superimpose a circle ofradius Rl(!) + RRl(!) de'ned as Eqs. (27), (28) or(29), (30) (making the same choice for all the diagonalelements at each frequency). The ‘bands’ obtained inthis way are called robust Gershgorin bands; each iscomposed of robust Gershgorin circles.

4. Robust Nyquist array analysis

Theorem 3 and robust diagonal dominance provide thebasis for a robust Nyquist stability criterion for uncertainprocess models. First, a preliminary result is presented inLemma 2.

Lemma 2. Suppose that Z(s) is an n× n uncertain ma-trix with the nominal value Z(s) and that Z(s) is robustdiagonally dominant for l=1; 2; : : : ; n and for all s on theNyquist contour. If the lth robust Gershgorin band ofZ(s) encircles the origin Nl times counterclockwise anddet[Z(s)] encircles the origin N times counterclockwise;then

N =n∑

l=1

Nl: (32)

Proof. The robust diagonal dominance of Z(s) ensuresthat each of its robust Gershgorin bands does not coverthe origin. According to Theorem 3, the union of thesebands contains the characteristic loci of Z(s) which aredenoted by #l(s) (l=1; : : : ; n). Therefore by the principleof the argument (Maciejowski, 1989),

2'N =Rarg det[Z(s)]=n∑

l=1

Rarg #l(s)

=n∑

l=1

2'Nl; (33)

where Rarg denotes the change in the argument as straverses the Nyquist contour. Hence, Eq. (32) holds.

Based on Theorem 3 and Lemma 2, a new robuststability theorem can be obtained for the closed-loopsystem consisting of a square transfer function G(s)connected with a diagonal controller gain matrixK=diag{k1; : : : ; kn}. Each ki is a real, nonzero constantin a negative-feedback loop.

Theorem 4 (Robust Nyquist stability). Suppose thatG(s) is an n × n system and has an estimatedfrequency response matrix G(s)= [gkl(s)]n×n; that

K=diag[k1; : : : ; kn] and that the matrix; K−1 + G(s);has robust column diagonal dominance on the Nyquistcontour; i.e.;∣∣∣∣ 1kl

+ gll(s)∣∣∣∣¿ Rl(s)= Rl(s) +RRl(s); (34)

where

Rl(s)=min

n∑k=1; k �=l

|gkl(s)|;

√n − 1

n∑

k=1; k �=l

|gkl(s)|21=2 ; (35)

RRl(s) =min

n∑k=1

|gkl(s)− gkl(s)|;

√n

(n∑

k=1

|gkl(s)− gkl(s)|2)1=2

(36)

for each l and for all s on the Nyquist contour. Let thelth robust Gershgorin band of G(s); which is composedof circles centered at gll(s) with radius Rl(s); encirclethe critical point (−1=kl; 0); Nl times counterclockwise.Then the negative feedback system with return ratio,−G(s)K, is stable if and only if

n∑l=1

Nl=P0; (37)

where P0 is the number of unstable poles of G(s).

Proof. Let N and N ′ be the number of counterclockwiseencirclements of the origin made by det[I+G(s)K] andby det[K−1 +G(s)], respectively. We must have N =N ′

(Maciejowski, 1989). Because the lth robust Gershgorinband of G(s) encircles the point (−1=kl; 0); Nl timescounterclockwise, this means that the lth robust Gersh-gorin band of K−1 + G(s) encircles the origin Nl timescounterclockwise. Condition (34) ensures that K−1 +G(s) is robust diagonally dominant, so that the assump-tions of Lemma 2 are satis'ed for Z(s)=K−1 + G(s).Therefore,

N ′=n∑

l=1

Nl ⇒ N =n∑

l=1

Nl: (38)

From the corollary of Theorem 4:1 on page 141 in Rosen-brock (1974), a necessary and suTcient condition for sta-bility is that N =P0. Therefore, Eq. (37) is the necessaryand suTcient condition for stability.

Remark. For a continuous-time system, G(s) gives thefrequency response of the system as s traverses theNyquist contour. Correspondingly, for a discrete-timesystem, the matrix G(e j!) for all !∈ [ − '; '] provides

472 D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475

the same system frequency response information. There-fore, the results in Lemma 2 and Theorem 4 are also ap-plicable to discrete-time systems if G(s) is replaced byG(e j!).

Corollary. Consider an n × n system G(e j!) withthe estimated frequency response matrix G(e j!)=[gkl(e

j!)]n×n and covariance information as described inSection 2. Thus; the robust Gershgorin bands at the(1−�)100% con1dence level are composed of the circlescentered at gll(e

j!) and having radii; Rl(!)= Rl(!) +RRl(!); de1ned as

Rl(!) =min

n∑k=1; k �=l

|gkl(ej!)|;

√n − 1

n∑

k=1; k �=l

|gkl(ej!)|2

1=2 ; (39)

RRl(!)=min{RRel ;RRc

l}; (40)

RRel(!)=

n∑k=1

√�2�;p�1; kl(!); (41)

RRcl(!)=

√n√

�2�;p�1; l(!): (42)

Let P0 be the number of unstable open-loop poles whichis assumed to remain constant for G(e j!) and G(e j!).A decentralized controller; K=diag[k1; : : : ; kn]; can bedesigned so that the robust Gershgorin bands excludethe critical point (−1=kl; 0) but encircle it counterclock-wise Nl times for all l=1; 2; : : : ; n. Then the closed-loopsystem is stable at the (1 − �)100% con1dence level ifEq. (37) holds.

Proof. Substituting the uncertainty descriptions ofEqs. (16) and (25) in Section 2 into Eqs. (35) and (36),then the radius of each robust Gershgorin circle at the(1− �)100% con'dence level is obtained as Eqs. (39)–(42). Therefore, according to Theorem 4 the closed-loopsystem is stable at the (1 − �)100% con'dence level ifEq. (37) holds.

Theorem 4 and the corollary provide a simple methodfor analyzing robust stability for a multivariable sys-tem which has an estimated model and statistical un-certainty description obtained from system identi'cation.The upper bound on the magnitude of the controller gainkl for robust stability can be easily obtained from thistheorem. Furthermore, robust design methods for decen-tralized control systems can be developed based on theproposed robust stability theorems.

5. Example

Wood and Berry (1973) developed the following em-pirical model of a pilot-scale distillation column that isused to separate a methanol–water mixture,

[XD(s)

XB(s)

]=

12:8e−s

16:7s+ 1−18:9e−3s21s+ 1

6:6e−7s

10:9s+ 1−19:4e−3s14:4s+ 1

[

R(s)

S(s)

]; (43)

where XD and XB are the overhead and bottom composi-tions of methanol, respectively, R is the reVux Vow rate,and S is the steam Vow rate to the reboiler.A MIMO PRBS signal was designed and used as the

excitation signal for process identi'cation. Gaussian mea-surement noise was added to provide a signal-to-noiseratio of three. The sampling period was chosen asRt=1 min. A subspace identi'cation technique, canon-ical variate analysis (CVA) (Larimore, 1996), andthe ADAPTX software were used to identify a statespace model from a 1200-point data set. The frequencyresponse and the corresponding covariance for eachelement are also calculated by the software. Fig. 1 com-pares the estimated model with the true model and 95%con'dence intervals.Based on the estimated frequency response and covari-

ance information, the uncertainty bounds for the radii ofthe Gershgorin bands were calculated. The 95% con'-dence bounds for the Gershgorin band radii are shown inFig. 2. Notice that the element bounds are more conser-vative than the column bounds.By applying Theorem 4, a proportional-only, decentral-

ized controller, K=diag[0:3;−0:16], has been obtainedthat guarantees robust stability for all the possible sys-tems that are within the 95% con'dence bands of the esti-mated model. Fig. 3 shows that the Gershgorin bands forthis controller do not quite cover the critical point. Thus,it can be concluded that any controller with smaller gains(in absolute value) can guarantee robust stability, if thiscontrol structure is used.

6. Conclusions

A robust Nyquist array analysis for MIMO systems hasbeen developed based on uncertainty descriptions that arereadily available from system identi'cation techniques.Two types of uncertainty bounds on the system frequencyresponse are derived: an uncertainty bound for each el-ement of the system frequency response and an uncer-tainty bound for each column of the system frequencyresponse. Gershgorin’s theorem is extended to includemodel uncertainty. Robust diagonal dominance and ro-bust Gershgorin bands are de'ned for uncertain system

D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475 473

Fig. 1. The frequency response with 95% con'dence bands.

Fig. 2. The uncertainty bounds for the Gershgorin band radii.

Fig. 3. Gershgorin bands with proportional-only, decentralized control.

models. Based on these results, new robust stabilitytheorems are developed for systems with model uncer-tainty. The robust Nyquist array analysis is illustrated inan application to the widely used, Wood–Berry distilla-tion column model.

Acknowledgements

The authors acknowledge the UCSB Process ControlConsortium for 'nancial support and Dr. Wallace Lari-more (Adaptics, Inc.) for his assistance and for providing

474 D. Chen, D.E. Seborg / Automatica 38 (2002) 467–475

the ADAPTx software. The reviewers suggested a num-ber of useful modi'cations and references which havebeen included.

Appendix

Proof of Theorem 3. Let # be any eigenvalue of Z . Thenthere is at least one nonnull x such that

xTZ = #xT: (A.1)

Suppose the lth component of x has the largest modulus,so x can be normalized as [x1; : : : ; xl−1; 1; xl+1; : : : ; xn]

T

with |xk |6 1 for k �= l. Equating the lth element on eachside of (A.1) gives

n∑k=1

xkzkl= #xl= # ⇔ # − zll=n∑

k=1; k �=l

xkzkl: (A.2)

Hence

# − zll − (zll − zll)

=n∑

k=1; k �=l

zklxk +n∑

k=1; k �=l

(zkl − zkl)xk (A.3)

⇒ |# − zll|6∣∣∣∣∣∣

n∑k=1; k �=l

zklxk

∣∣∣∣∣∣+∣∣∣∣∣

n∑k=1

(zkl − zkl)xk

∣∣∣∣∣ :(A.4)

By using the triangle inequality and the Cauchy–Schwarzinequality (Anton, 1987), two inequalities can be ob-tained for the 'rst term on the right-hand side,∣∣∣∣∣∣

n∑k=1; k �=l

zklxk

∣∣∣∣∣∣6n∑

k=1; k �=l

|zklxk |=n∑

k=1; k �=l

|zkl| |xk |

6n∑

k=1; k �=l

|zkl|; (A.5)

∣∣∣∣∣∣n∑

k=1; k �=l

zklxk

∣∣∣∣∣∣6 n∑

k=1; k �=l

|zkl|2n∑

k=1; k �=l

|xk |21=2

6√

n − 1 n∑

k=1; k �=l

|zkl|21=2

: (A.6)

Therefore, the following inequality must be held:∣∣∣∣∣∣n∑

k=1; k �=l

zklxk

∣∣∣∣∣∣

6min

n∑k=1; k �=l

|zkl|;√

n − 1 n∑

k=1; k �=l

|zkl|21=2 :

(A.7)

Similarly, the following inequality can be obtained forthe second term on the right-hand side:∣∣∣∣∣

n∑k=1

(zkl − zkl)xk

∣∣∣∣∣6min

n∑k=1

|zkl − zkl|;√

n

(n∑

k=1

|zkl − zkl|2)1=2

:

(A.8)

By combining Eqs. (A.4), (A.7) and (A.8), Eq. (26) canbe obtained. Thus Theorem 3 is proved.

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Dale E. Seborg received his B.S. degreefrom the University of Wisconsin and hisPh.D. degree from Princeton University,both in chemical engineering. From 1968to 1977, he was a faculty member atthe University of Alberta in Edmonton,Canada. He joined UCSB in 1977 andserved as the department chair for threeyears. Dr. Seborg’s research interestsare in the areas of process control

and monitoring. He is the co-author of a widely used textbook, Pro-cess Dynamics and Control (1989), with Prof. Duncan Mellichamp(UCSB) and Prof. Tom Edgar (UT-Austin). This textbook has beentranslated into Korean and Japanese. Dr. Seborg is the co-editor ofthree books and has taught a variety of short courses around the world.He is an active industrial consultant. Dr. Seborg is the recipient, orco-recipient, of several national awards that include the AmericanStatistical Association’s Statistics in Chemistry Award (1994), theAmerican Automatic Control Council’s Education Award (1993), theAmerican Society of Engineering Education’s Meriam-Wiley Dis-tinguished Author Award (1990), and the Joint Automatic ControlConference Best Paper Award (1973). Within IFAC, he was the NOCCo-Chair for SYSID-2000 and the IPC Chair for ADCHEM-1988.He also served as the General Chair for the 1992 American Con-trol Conference and co-organized the 1981 Chemical Process Control(CPC-2) Conference. He has been a director of the American Auto-matic Control Council and the AIChE CAST Division. He currentlyserves on the editorial board of the IEE Proc. on Control Theoryand Applications. Previously, he was an Associate Editor at Largefor the IEEE Trans. on Automatic Control, and a member of theeditorial boards for the Int. J. of HVAC & R Research and the Int.J. of Adaptive Control & Signal Processing.

Dan Chen received her B.S. degree inChemical Engineering and her M.S. de-gree in Industrial Process Control fromZhejiang University, China, in 1993 and1996. She is currently a Ph.D. student inDepartment of Chemical Engineering atthe University of California, Santa Bar-bara. Her research interests include ro-bustness analysis by utilizing system iden-ti'cation and statistical uncertainty de-scriptions, decentralized control systemdesign and analysis.