Robust Nyquist array analysis based on uncertainty descriptions from system identication

Automatica 38 (2002) 467 475 Brief Paper www.elsevier.com/locate/automatica Robust Nyquist array analysis based on uncertainty descriptions from system identication 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 system identication. Two types of statistical-based uncertainty error bounds for the frequency response are obtained: element bounds and column bounds. Gershgorin s theorem and the concepts of diagonal dominance and Gershgorin bands are extended to include model uncertainty. Robust stability theorems are developed based on ellipsoidal uncertainty descriptions obtained from system identication. 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 analysis such as H methods and -synthesis has been an active research eld during the past 20 years (Skogestad & Postlethwaite, 1996; Zhou, Doyle, & Glover, 1996). These techniques provide very powerful, complex approaches, but the resulting control systems usually have a complicated structure and are high order. Thus, these design techniques are not directly applicable to the decentralized PI=PID control systems that are widely used in the process industries. Furthermore, a typical starting point for contemporary robust control algorithms is that hard bounds are used to characterize model uncertainties. This formulation disregards the statistical model uncertainty descriptions that are readily available from system identication (Goodwin, 1999). Eorts to obtain a better match between robust control and system identication have received attention recently. For example, Braatz and Crisalle (1998) developed a robustness analysis for SISO systems with ellipsoidal parametric uncertainty This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tor 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 identication. A robust stability analysis directly based on the condence ellipses of system frequency response from identication has been developed by Cooley and Lee (1998). Some general conditions for the robustness of MIMO systems with structured uncertainty such as ellipsoidal uncertainty have been developed in the framework of -synthesis (Khatri & Parrilo, 1998; Chellaboina, Haddad, & Bernstein, 1998), but just as for other -synthesis methods these robustness analyses cannot be applied for xed structure control system design. In the present paper, the research objective is to develop a robust control technique based on statistical uncertainty information obtained from system identication, which can be applied not only as a robust stability analysis method for MIMO systems but also as a simple robust design approach for decentralized control systems. Frequency domain techniques, such as Nyquist array analysis (Rosenbrock, 1974) and quantitative feedback theory (QFT) (Horowitz, 1982), provide powerful tools for the analysis and design of decentralized control systems. QFT is a generalized loop-shaping method based on Nichols plots that addresses model uncertainty for single loop and decentralized control systems (Horowitz, 1992); thus, it is primarily a robust design methodology. Modications of QFT have been developed that address robustness 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 disadvantage of the QFT design procedure is that it is rather complicated for MIMO systems. Design approaches based on combinations of the inverse Nyquist array (INA) analysis and QFT have been proposed (Nwokah, Nordgren, & Grewal, 1995; Chen, Wang, & Wang, 1991), but do not consider ellipsoidal uncertainty descriptions. The Nyquist array method for MIMO systems is much more straightforward than the QFT design procedure and is easier to implement, especially for xed structure controllers such as PID control (Maciejowski, 1989). Nyquist array analysis provides a useful extension 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 considered directly in the original Nyquist array method. Based on a large amount of numerical calculation, Gao and Zhang (1993) derived empirical robust stability conditions for the inverse Nyquist array method and 2 2 and 3 3 systems. More general and theoretical robust Nyquist array methods have been developed to accommodate system uncertainty that can be described by deterministic error bounds on individual transfer functions (Arkun, Manousiouthakis, & Putz, 1984), or be expressed as parametric uncertainty in the rational polynomial functions that describe transfer functions (De Santis & Vicino, 1996; Kontogiannis & Munro, 1997). Since these methods are restricted to transfer function models with rational forms, systems with time delays cannot be analyzed without approximating each time delay as a rational function (e.g., a Pade approximation). Furthermore, these approaches are not general enough to utilize the statistical uncertainty information, especially correlation information that is available from system identication. Thus, unnecessary conservativeness can be introduced by these approaches. In the present paper, Nyquist array analysis is extended to more general system descriptions and to more general model uncertainty descriptions, based on statistical uncertainty bounds obtained from system identication. Two types of uncertainty bounds are obtained for the frequency response matrix: bounds for individual elements and column bounds for each column which are structured error bounds and include correlation information between transfer functions. A new robust Gershgorin theorem is derived for these error bounds descriptions. The new results provide a less conservative robust stability analysis and a simple robust design approach for decentralized control systems. 2. Uncertainty description n y Consider a MIMO linear system with n u inputs and 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) R ny, e(t) R ny, u(t) R nu and =[ 1 ;:::; p ] T ; (2) G(q 1 ; )= G k ()q k ; (3) H(q 1 ; )=I + H k ()q k : (4) Vector denotes the model parameters to be identied. In Eqs. (1) (4), q 1 denotes the backward shift operator and e(t) is a sequence of independent random vectors with zero mean values and covariance matrices e. The subsequent analysis is not restricted to the model form given in Eqs. (1) (4). In fact, if the system can be described by any linear model, the subsequent analysis is still valid. Assume that ˆ =[ˆ 1 ;:::; ˆ p ] T is an asymptotically normal and unbiased estimate of. The estimated covariance matrix of the parameter estimation error is assumed to be obtained during system identication. Therefore, an asymptotic limit exists, ( ˆ ) T 1 ( ˆ ) 2 p; (5) where p is the number of model parameters and 2 p is the chi-square distribution with p degrees of freedom. This result enables the construction of condence bounds P[( ˆ ) T 1 ( ˆ ) 6 2 ;p]=1 ; (6) where 1 is the probability that the true parameters lie within the specied ellipsoidal condence region about ˆ and 2 ;p is the upper 100 percentage point of the 2 probability distribution with p degrees of freedom. Let g kl (e j! ; ) be the frequency response from input l to output k. The corresponding estimated frequency response is ĝ kl (e j! ), g kl (e j! ; ˆ). Express the frequency response in terms of real and imaginary parts: g kl (e j! ; )=a kl (!; )+jb kl (!; ); ĝ kl (e j! )=â kl (!)+jˆb kl (!); (7) where â kl (!) and ˆb kl (!) denote the corresponding estimates. The gradient of a kl (!; ) with respect to the parameters is represented by a kl; (!)= @ @ T a kl(!; ) (8)

D. Chen, D.E. Seborg / Automatica 38 (2002) 467 475 469 which is a 1 p row vector. Denote the gradient of b kl (!; ) as b kl; (!)= @ @ b kl(!; ): T (9) Dene akl; (!) g kl (!), b kl; (!) ; (10) ˆ a2 p matrix which is evaluated at = ˆ. Therefore, the estimation error for a kl (!; ) and b kl (!; ) can be approximated by a truncated Taylor series expansion, âkl (!) a kl (!; ) g kl (!), ˆb kl (!) b kl (!; ) g kl (!)( ˆ ): (11) For the remainder of the paper, we use the simpler notation, a kl (!)=a kl (!; ) and b kl (!)=b kl (!; ). The subsequent discussion utilizes the following lemma. Lemma 1 (Wahlberg & Ljung; 1992). Let x R n ; R n n be positive denite; and assume that x T 1 x 6 1: (12) Consider y = Ax; where y R p ;p6 n; A R p n and A has full row rank. Then y T (AA T ) 1 y 6 1: (13) 2.1. Element bounds for the frequency response Condence bounds for each frequency response, g kl, can be derived using Lemma 1 with Eqs. (6) and (11) g kl (!) T [ g kl (!) g kl (!) T ] 1 g kl (!) 6 2 ;p: (14) Let kl (!)= g kl (!) g kl (!) T, which represents the 2 2 covariance matrix for â kl (!) and ˆb kl (!) (Ljung, 1985). The singular value decomposition (SVD) for the symmetric kl (!) matrix is 1;kl (!) 0 kl (!)=U kl (!) U kl (!) T ; (15) 0 2;kl (!) where 1;kl (!) 2;kl (!) and U kl (!) isa2 2 unitary matrix. The following theorem and proof is a formal statement of a result in Cooley and Lee (1998) that was stated without proof. Theorem 1 (Element bounds). An error bound for each frequency response; g kl (e j! ; ); is given by g kl (e j! ; ) ĝ kl (e j! ) = g kl (!) 2 6 ;p 2 1;kl (!): (16) Proof. Substituting Eq. (15) into Eq. (14) gives, 1 0 g kl (!) T U kl (!) 1;kl (!) 1 0 2;kl (!) U kl (!) T g kl (!) 6 2 ;p: (17) Let y =[y 1 ;y 2 ] T = U kl (!) T g kl (!), therefore the above equation can be written as y 2 1 1;kl (!) + y2 2 2;kl (!) 6 2 ;p: (18) This implies that y 2 1 1;kl (!) + y2 2 1;kl (!) 6 2 ;p (19) because 1;kl (!) 2;kl (!). Hence, y 2 2 6 2 ;p 1;kl (!): (20) Because U kl (!) is a unitary matrix, y 2 2 = g kl(!) 2 2, which implies that inequality (16) holds. Theorem 1 provides a circular bound for each frequency response element, g kl (e j!; ; ). This theorem can also be proven by using the Rayleigh Ritz inequality (Rugh, 1996). The uncertainty bounds given by Theorem 1 include a degree of conservatism that is introduced by Lemma 1 and Eq. (19). 2.2. Column bounds for the frequency response The error bounds in Eq. (16) provide independent bounds on each element of the frequency response matrix. However, they ignore the covariance information between dierent elements. In multivariable design problems, the interaction between elements is an important factor. In Nyquist array analysis, the radius of each Gershgorin circle depends on the entire column or row of the transfer function matrix. Therefore, ignoring the uncertainty interaction between dierent elements can produce very conservative stability results. In this section, a new formulation of the error bound for each column of the frequency response matrix is derived from statistical information provided by system identication.

470 D. Chen, D.E. Seborg / Automatica 38 (2002) 467 475 Dene g l (!)=col([ g 1l (!);:::; g nyl(!)]); (21) where col( ) is an operator that stacks the columns on top of each other and yields a column vector. Thus, g l (!), col([g 1l (!);:::;g nyl(!)]) From Lemma 1, g l (!)( ˆ ): (22) g l (!) T [ g l (!) g l (!) T ] 1 g l (!) 6 2 ;p: (23) Let l (!)= g l (!) g l (!) T. Therefore, l (!) isa 2n y 2n y symmetric matrix representing the covariance matrix of [â 1l (!); ˆb 1l (!);:::;â nyl(!); ˆb nyl(!)] T (Ljung, 1985). The SVD for this matrix is l (!)=U l (!) diag[ 1;l (!);:::; 2ny;l(!)]U l (!) T ; (24) where 1;l (!) 2;l (!) 2ny;l(!), U l (!) is a 2n y 2n y unitary matrix, and diag[ ] denotes a diagonal matrix with elements 1;l (!);:::; 2ny;l(!). Therefore, a new bound for the estimation error of each column of the system frequency response matrix can be derived based on covariance information and the largest singular value. Theorem 2 (Column bounds). The bound for g l (!) 2 ; the error for the lth column of frequency response matrix G(e j! ; ); is given by g l (!) 2 = 6 ( ny ) 1=2 g kl (e j! ; ) ĝ kl (e j! ) 2 2 ;p 1;l (!): (25) Proof. Similar to Theorem 1. In analogy with Theorem 1, the uncertainty bounds given by Theorem 2 may also be conservative due to Lemma 1 and Eq. (19). Eq. (25) gives the error bound for a column of the estimated frequency response matrix while Eq. (16) gives the error bound for each element. Thus, it is inappropriate to compare these two bounds directly. But if the error bound for an entire column of the estimated frequency response matrix is obtained by adding the error bounds in Eq. (16) for each element in this column, this error bound will be larger than the error bound obtained directly from Eq. (25), unless this column has only one nonzero element. The element bound (16) is more conservative because the correlation between different elements in the column is ignored while the covariance information of the entire column is included in the error bound in Eq. (25). 3. Robust Gershgorin bands The key to Nyquist array analysis is Gershgorin s theorem. Next, a new robustness theorem is developed for uncertain complex matrices based on Gershgorin s theorem. Theorem 3 (Robust Gershgorin s theorem). For an n n uncertain complex matrix Z =[z kl ] n n =[a kl + jb kl ] n n with the nominal value Ẑ =[ẑ kl ] n n =[â kl + j ˆb kl ] n n ; the eigenvalues of Z lie in the union of the n circles (Z) ẑ ll 6 R l ; ; 2;:::;n; (26) where R l = ˆR l +R l and ˆR l = min ẑ kl ; 1=2 n 1 ẑ kl 2 ; (27) R l = min z kl ẑ kl ; ( ) 1=2 n z kl ẑ kl 2 : (28) The eigenvalues of Z also lie in the union of the circles; each centered on ẑ kk with radius R k = ˆR k +R k ; where ˆR k = min ẑ kl ; 1=2 n 1 ẑ kl 2 ;l k ;l k ; (29) R k = min z kl ẑ kl ; ( ) 1=2 n z kl ẑ kl 2 : (30) Proof. See the appendix. Theorem 3 is a robust version of Gershgorin s theorem for uncertain matrices. According to this theorem, the eigenvalues of an uncertain matrix must be located inside the union of circles dened by the estimated values and the uncertainty bounds. Thus, for an MIMO control problem, the eigenvalues of the frequency response matrix lie within the union of circles centered on the estimated frequency response. Therefore, Theorem 3 can be used for a Nyquist array analysis of uncertain models. Denition (Diagonal dominance). For an n n uncertain complex matrix Z which has a nominal value Ẑ, if ẑ ll R l = ˆR l +R l for l =1; 2;:::;n (31) then, Z is said to have robust column diagonal dominance for ˆR l and R l in Eqs. (27) and (28). If ˆR k and R k are

D. Chen, D.E. Seborg / Automatica 38 (2002) 467 475 471 dened as in Eqs. (29) and (20), Z is said to have robust row diagonal dominance. Denition (Robust Gershgorin bands). Consider an n n transfer function matrix Z(s) with the nominal value Ẑ(s). At each ẑ ll (j!), superimpose a circle of radius ˆR l (!) +R l (!) dened as Eqs. (27), (28) or (29), (30) (making the same choice for all the diagonal elements at each frequency). The bands obtained in this way are called robust Gershgorin bands; each is composed of robust Gershgorin circles. 4. Robust Nyquist array analysis Theorem 3 and robust diagonal dominance provide the basis for a robust Nyquist stability criterion for uncertain process models. First, a preliminary result is presented in Lemma 2. Lemma 2. Suppose that Z(s) is an n n uncertain matrix with the nominal value Ẑ(s) and that Z(s) is robust diagonally dominant for l =1; 2;:::;nand for all s on the Nyquist contour. If the lth robust Gershgorin band of Z(s) encircles the origin N l times counterclockwise and det[z(s)] encircles the origin N times counterclockwise; then N = N l : (32) Proof. The robust diagonal dominance of Z(s) ensures that each of its robust Gershgorin bands does not cover the origin. According to Theorem 3, the union of these bands contains the characteristic loci of Z(s) which are denoted by l (s)(l =1;:::;n). Therefore by the principle of the argument (Maciejowski, 1989), 2N =arg det[z(s)] = arg l (s) = 2N l ; (33) where arg denotes the change in the argument as s traverses the Nyquist contour. Hence, Eq. (32) holds. Based on Theorem 3 and Lemma 2, a new robust stability theorem can be obtained for the closed-loop system consisting of a square transfer function G(s) connected with a diagonal controller gain matrix K = diag{k 1 ;:::;k n }. Each k i is a real, nonzero constant in a negative-feedback loop. Theorem 4 (Robust Nyquist stability). Suppose that G(s) is an n n system and has an estimated frequency response matrix Ĝ(s)=[ĝ kl (s)] n n ; that K = diag[k 1 ;:::;k n ] and that the matrix; K 1 + G(s); has robust column diagonal dominance on the Nyquist contour; i.e.; 1 +ĝ k ll (s) l R l(s)= ˆR l (s)+r l (s); (34) where ˆR l (s)=min ĝ kl (s) ; 1=2 n 1 ĝ kl (s) 2 ; (35) R l (s)=min g kl (s) ĝ kl (s) ; n ( ) 1=2 g kl (s) ĝ kl (s) 2 (36) for each l and for all s on the Nyquist contour. Let the lth robust Gershgorin band of G(s); which is composed of circles centered at ĝ ll (s) with radius R l (s); encircle the critical point ( 1=k l ; 0); N l times counterclockwise. Then the negative feedback system with return ratio, G(s)K, is stable if and only if N l = P 0 ; (37) where P 0 is the number of unstable poles of G(s). Proof. Let N and N be the number of counterclockwise encirclements of the origin made by det[i + G(s)K] and by det[k 1 + G(s)], respectively. We must have N = N (Maciejowski, 1989). Because the lth robust Gershgorin band of G(s) encircles the point ( 1=k l ; 0); N l times counterclockwise, this means that the lth robust Gershgorin band of K 1 + G(s) encircles the origin N l times counterclockwise. Condition (34) ensures that K 1 + G(s) is robust diagonally dominant, so that the assumptions of Lemma 2 are satised for Z(s)=K 1 + G(s). Therefore, N = N l N = N l : (38) From the corollary of Theorem 4:1 on page 141 in Rosenbrock (1974), a necessary and sucient condition for stability is that N = P 0. Therefore, Eq. (37) is the necessary and sucient condition for stability. Remark. For a continuous-time system, G(s) gives the frequency response of the system as s traverses the Nyquist contour. Correspondingly, for a discrete-time system, 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. Therefore, the results in Lemma 2 and Theorem 4 are also applicable to discrete-time systems if G(s) is replaced by G(e j! ). Corollary. Consider an n n system G(e j! ) with the estimated frequency response matrix Ĝ(e j! )= [ĝ kl (e j! )] n n and covariance information as described in Section 2. Thus; the robust Gershgorin bands at the (1 )100% condence level are composed of the circles centered at ĝ ll (e j! ) and having radii; R l (!)= ˆR l (!)+ R l (!); dened as ˆR l (!)=min ĝ kl (e j! ) ; n 1 1=2 ĝ kl (e j! ) 2 ; (39) R l (!)=min{r e l; R c l}; (40) R e l(!)= 2 ;p 1;kl (!); (41) R c l(!)= n ;p 2 1;l (!): (42) Let P 0 be the number of unstable open-loop poles which is assumed to remain constant for G(e j! ) and Ĝ(e j! ). A decentralized controller; K = diag[k 1 ;:::;k n ]; can be designed so that the robust Gershgorin bands exclude the critical point ( 1=k l ; 0) but encircle it counterclockwise N l times for all l =1; 2;:::;n. Then the closed-loop system is stable at the (1 )100% condence level if Eq. (37) holds. Proof. Substituting the uncertainty descriptions of Eqs. (16) and (25) in Section 2 into Eqs. (35) and (36), then the radius of each robust Gershgorin circle at the (1 )100% condence level is obtained as Eqs. (39) (42). Therefore, according to Theorem 4 the closed-loop system is stable at the (1 )100% condence level if Eq. (37) holds. Theorem 4 and the corollary provide a simple method for analyzing robust stability for a multivariable system which has an estimated model and statistical uncertainty description obtained from system identication. The upper bound on the magnitude of the controller gain k l for robust stability can be easily obtained from this theorem. Furthermore, robust design methods for decentralized control systems can be developed based on the proposed robust stability theorems. 5. Example Wood and Berry (1973) developed the following empirical model of a pilot-scale distillation column that is used to separate a methanol water mixture, XD (s) = X B (s) 12:8e s 16:7s +1 6:6e 7s 10:9s +1 18:9e 3s 21s +1 19:4e 3s 14:4s +1 R(s) ; (43) S(s) where X D and X B are the overhead and bottom compositions of methanol, respectively, R is the reux ow rate, and S is the steam ow rate to the reboiler. A MIMO PRBS signal was designed and used as the excitation signal for process identication. Gaussian measurement noise was added to provide a signal-to-noise ratio of three. The sampling period was chosen as t =1 min. A subspace identication technique, canonical variate analysis (CVA) (Larimore, 1996), and the ADAPTX software were used to identify a state space model from a 1200-point data set. The frequency response and the corresponding covariance for each element are also calculated by the software. Fig. 1 compares the estimated model with the true model and 95% condence intervals. Based on the estimated frequency response and covariance information, the uncertainty bounds for the radii of the Gershgorin bands were calculated. The 95% condence bounds for the Gershgorin band radii are shown in Fig. 2. Notice that the element bounds are more conservative than the column bounds. By applying Theorem 4, a proportional-only, decentralized controller, K = diag[0:3; 0:16], has been obtained that guarantees robust stability for all the possible systems that are within the 95% condence bands of the estimated model. Fig. 3 shows that the Gershgorin bands for this 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 this control structure is used. 6. Conclusions A robust Nyquist array analysis for MIMO systems has been developed based on uncertainty descriptions that are readily available from system identication techniques. Two types of uncertainty bounds on the system frequency response are derived: an uncertainty bound for each element of the system frequency response and an uncertainty bound for each column of the system frequency response. Gershgorin s theorem is extended to include model uncertainty. Robust diagonal dominance and robust Gershgorin bands are dened for uncertain system

D. Chen, D.E. Seborg / Automatica 38 (2002) 467 475 473 Fig. 1. The frequency response with 95% condence 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 stability theorems are developed for systems with model uncertainty. The robust Nyquist array analysis is illustrated in an application to the widely used, Wood Berry distillation column model. Acknowledgements The authors acknowledge the UCSB Process Control Consortium for nancial support and Dr. Wallace Larimore (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 number of useful modications and references which have been included. Appendix Proof of Theorem 3. Let be any eigenvalue of Z. Then there is at least one nonnull x such that x T Z = x T : (A.1) Suppose the lth component of x has the largest modulus, so x can be normalized as [x 1 ;:::;x l 1 ; 1;x l+1 ;:::;x n ] T with x k 6 1 for k l. Equating the lth element on each side of (A.1) gives x k z kl = x l = z ll = x k z kl : (A.2) Hence ẑ ll (z ll ẑ ll ) = ẑ kl x k + ẑ ll 6 (z kl ẑ kl )x k (A.3) ẑ kl x k + (z kl ẑ kl )x k : (A.4) By using the triangle inequality and the Cauchy Schwarz inequality (Anton, 1987), two inequalities can be obtained for the rst term on the right-hand side, ẑ kl x k 6 ẑ kl x k = ẑ kl x k 6 ẑ kl x k 6 6 n 1 ẑ kl ; (A.5) ẑ kl 2 x k 2 ẑ kl 2 1=2 1=2 : (A.6) Therefore, the following inequality must be held: ẑ kl x k 6 min ẑ kl ; n 1 1=2 ẑ kl 2 : (A.7) Similarly, the following inequality can be obtained for the second term on the right-hand side: (z kl ẑ kl )x k ( ) 1=2 6 min z kl ẑ kl ; n z kl ẑ kl 2 : (A.8) By combining Eqs. (A.4), (A.7) and (A.8), Eq. (26) can be obtained. Thus Theorem 3 is proved. References Anton, H. (1987). Elementary linear algebra (5th ed.). New York: Wiley. Arkun, Y., Manousiouthakis, B., & Putz, P. (1984). Robust Nyquist array methodology: A new theoretical framework for analysis and design of robust multivariable feedback systems. International Journal of Control, 40, 602 629. Braatz, R. D. (1994). A reconciliation between quantitative feedback theory and robust multivariable control. Proceedings of 1994 American control conference, Baltimore, MD (pp. 3374 3378). Braatz, R. D., & Crisalle, O. D. (1998). Robustness analysis for systems with ellipsoidal uncertainty. International Journal of Robust and Nonlinear Control, 8, 1113 1117. Chait, Y., Chen, Q., & Hollot, C. V. (1999). Automatic loop-shaping of QFT controllers via linear programming. Journal of Dynamic Systems, Measurement and Control-Transactions of the ASME, 121, 351 357. Chellaboina, V.-S., Haddad, W. M., & Bernstein, D. S. (1998). Structured matrix norms for robust stability and performance with block-structured uncertainty. International Journal of Control, 71, 535 557. Chen, C. W., Wang, G. G., & Wang, S. H. (1991). New interpretation of a MIMO quantitative feedback theory. Proceedings of 1991 American control conference, Boston, MA (pp. 3125 3130). Cooley, B. L., & Lee, J. H. (1998). Integrated identication and robust control. Journal of Process Control, 8, 431 440. De Santis, E., & Vicino, A. (1996). Diagonal dominance for robust stability of MIMO interval systems. IEEE Transactions on Automatic Control, 41, 871 875. Gao, D., & Zhang, L. (1993). The robust inverse Nyquist array (RINA) method for the design of multivariable control systems. Proceedings of IEEE TENCON 93, Beijing, China (pp. 234 237). Goodwin, G. C. (1999). Identication and robust control: Bridging the gap. Seventh Mediterranean conference, Israel (pp. 1 22). Horowitz, I. M. (1982). Quantitative feedback theory. IEE Proceedings D, 129, 215 226. Horowitz, I. M. (1992). Quantitative feedback design theory (QFT). Boulder, CO: QFT Publications. Jayasuriya, S., & Zhao, Y. (1994). Robust stability of plants with mixed uncertainties and quantitative feedback theory. Journal of Dynamic Systems, Measurement and Control-Transactions of the ASME, 116, 10 16. Khatri, S., & Parrilo, P. A. (1998). Spherical. Proceedings of 1998 American control conference, Philadelphia, PA (pp. 2314 2318). Kontogiannis, E., & Munro, N. (1997). Extreme point solutions to the diagonal dominance problem and stability analysis of uncertain systems. Proceedings of 1997 American control conference, Albuquerque, NM (pp. 3936 3940).

D. Chen, D.E. Seborg / Automatica 38 (2002) 467 475 475 Larimore, W. E. (1996). ADAPTX users manual. Mclean, VA: Adaptics Inc. Lee, J. W., Chait, Y., & Steinbuch, M. (2000). On QFT tuning of multivariable controllers. Automatica, 36, 1701 1708. Ljung, L. (1985). Asymptotic variance expressions for identied black-box transfer function models. IEEE Transactions on Automatic Control, AC-30, 834 844. Maciejowski, J. M. (1989). Multivariable feedback design. Reading, MA: Addison-Wesley. Nwokah, O. D. I., Nordgren, R. E., & Grewal, G. S. (1995). Inverse Nyquist array: A quantitative theory. IEE Proceedings D, 142, 23 30. Rosenbrock, H. H. (1974). Computer-aided control system design. New York: Academic Press. Rugh, W. J. (1996). Linear system theory (2nd ed.). Upper Saddle River, NJ: Prentice-Hall. Skogestad, S., & Postlethwaite, I. (1996). Multivariable feedback control: Analysis and design. New York: Wiley. Wahlberg, B., & Ljung, L. (1992). Hard frequency-domain model error bounds from least-squares like identication techniques. IEEE Transactions on Automatic Control, 37, 910 912. Wood, R. K., & Berry, M. W. (1973). Terminal composition control of a binary distillation column. Chemical Engineering Science, 28, 1707 1717. Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control. Upper Saddle River, NJ: Prentice-Hall. Dale E. Seborg received his B.S. degree from the University of Wisconsin and his Ph.D. degree from Princeton University, both in chemical engineering. From 1968 to 1977, he was a faculty member at the University of Alberta in Edmonton, Canada. He joined UCSB in 1977 and served as the department chair for three years. Dr. Seborg s research interests are in the areas of process control and monitoring. He is the co-author of a widely used textbook, Process Dynamics and Control (1989), with Prof. Duncan Mellichamp (UCSB) and Prof. Tom Edgar (UT-Austin). This textbook has been translated into Korean and Japanese. Dr. Seborg is the co-editor of three books and has taught a variety of short courses around the world. He is an active industrial consultant. Dr. Seborg is the recipient, or co-recipient, of several national awards that include the American Statistical Association s Statistics in Chemistry Award (1994), the American Automatic Control Council s Education Award (1993), the American Society of Engineering Education s Meriam-Wiley Distinguished Author Award (1990), and the Joint Automatic Control Conference Best Paper Award (1973). Within IFAC, he was the NOC Co-Chair for SYSID-2000 and the IPC Chair for ADCHEM-1988. He also served as the General Chair for the 1992 American Control Conference and co-organized the 1981 Chemical Process Control (CPC-2) Conference. He has been a director of the American Automatic Control Council and the AIChE CAST Division. He currently serves on the editorial board of the IEE Proc. on Control Theory and Applications. Previously, he was an Associate Editor at Large for the IEEE Trans. on Automatic Control, and a member of the editorial 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 in Chemical Engineering and her M.S. degree in Industrial Process Control from Zhejiang University, China, in 1993 and 1996. She is currently a Ph.D. student in Department of Chemical Engineering at the University of California, Santa Barbara. Her research interests include robustness analysis by utilizing system identication and statistical uncertainty descriptions, decentralized control system design and analysis.