Frequency domainsolutionto delay-type Nehari problem

Transcription

1 Available online at Automatica 39 (2003) Brief paper Frequency domainsolutionto delay-type Nehari problem Qing-Chang Zhong Department of Electrical and Electronic Engineering, mperial College, Exhibition Road, ndon, SW7 2BT, UK Received 17 November 2001; received inrevised form 30 September 2002; accepted 24 October 2002 Abstract This paper generalizes the frequency-domainresults onthe delay-type Nehari problem inthe stable case to the unstable case. The solvability condition of the delay-type Nehari problem is formulated in terms of the nonsingularity of three matrices. The optimal value opt is the maximal (0; ) such that one of the three matrices becomes singular. All sub-optimal compensators are parameterized in a transparent structure incorporating a modied Smith predictor.? 2002 Elsevier Science Ltd. All rights reserved. Keywords H control; Smith predictor; Nehari problem; Time delay systems; Riccati equation; L 2 [0;h-induced norm 1. ntroduction The H control of processes with delay(s) has been an active research area since the mid-1980s. There are mainly three kinds of methods operator-theoretic methods (Foias, Ozbay, & Tannenbaum, 1996; Dym, Georgiou, & Smith, 1995; Zhou & Khargonekar, 1987), state-space methods (Nagpal & Ravi, 1997; Tadmor, 1997a, 2000; Basar & Bernhard, 1995) and frequency-domain methods (Mirkin, 2000; Meinsma & Zwart, 2000; Meinsma, Mirkin, & Zhong, 2002; Zhong, 2003). t is well known that a large class of H control problems can be reduced to Nehari problems (Francis, 1987). This is still true inthe case for systems with delay(s), where the simplied problem is a delay-type Nehari problem. Some papers, e.g. (Zhou & Khargonekar, 1987; Flamm & Mitter, 1987), were devoted to calculate the inmum of the delay-type Nehari problem inthe stable case. t was shownin(zhou & Khargonekar, 1987) that this problem in the stable case is equivalent to calculating an L 2 [0;h-induced norm. However, for the unstable case, it becomes much more involved. Tadmor (1997b) presented a state space solutionto this problem inthe unstable case, inwhich a dierential/algebraic matrix Riccati equation-based method was used. The optimal value relies on the solution of a dierential Riccati equation. Although the expositions are very elegant, the suboptimal solution is too complicated and the structure is not transparent. Motivated by the idea of Meinsma and Zwart (2000), this paper presents a frequency domain solution to the delay-type Nehari problem inthe unstable case. The maintool used here is the J -spectral factorizationof generic para-hermitian matrices (Kwakernaak, 2000; Zhong, 2001). The optimal value opt of the delay-type Nehari problem is formulated ina simple, clear way it is the maximal such that one of three matrices becomes singular when decreases from + to 0. Hence, one need no longer solve a dierential Riccati equation. A prominent advantage is that the suboptimal solutions have a transparent structure incorporating a modied Smith predictor. This paper was presented at the 15th FAC World Congress, Barcelona, Spain, July 21 26, This paper was recommended for publication in revised form by Associate Editor Patrizio Calaneri under the direction of Editor Roberto Tempo. Tel ; fax address zhongqc@ic.ac.uk (Q.-C. Zhong). URL http//members.fortunecity.com/zhongqc /03/$ - see front matter? 2002 Elsevier Science Ltd. All rights reserved. P S (02)

2 500 Q.-C. Zhong / Automatica 39 (2003) Notation Givena matrix A, A T and A denote its transpose and complex conjugate transpose, respectively, and A stands for (A 1 ) whenthe inverse A 1 exists. A rational transfer matrix G(s)=D + C(s A) 1 B is frequently denoted as A B C D and its conjugate is dened as [ A G (s)=[g( s ) C = B D A completion operator h is dened as h {G} = A B A B Ce Ah e sh = Ĝ(s) e sh G(s); 0 C D where h 0. This follows (Mirkin, 2000), except for a small adjustment in the notation. This operator maps any rational transfer matrix G into an nite impulse response (FR) block. Let N11 N 12 N = N 21 N 22 bea2 2 block transfer matrix. The upper linear fractional transformation is dened as F u (N; Q) =N 22 + N 21 Q( N 11 Q) 1 N 12 provided that ( N 11 Q) 1 exists. Another two less-used linear fractional transformations, called homographic transformations (Kimura, 1996; Delsarte, Genin, & Komp, 1979), are dened as follows provided that the corresponding inverse exists H r (N; Q)=(N 11 Q + N 12 )(N 21 Q + N 22 ) 1 ; H l (N; Q)= (N 11 QN 21 ) 1 (N 12 QN 22 ); where the subscript l stands for left and r for right. These transformations correspond to the matrix multiplication, or chain scattering (Kimura, 1996), from the left side and from the right side, respectively H r (N 1 ; H r (N 2 ;Q)) = H r (N 1 N 2 ;Q); H l (N 1 ; H l (N 2 ;Q)) = H l (N 2 N 1 ;Q); where N 1 and N 2 are 2 2 block transfer matrices with appropriate dimensions. A signature matrix is dened as p 0 J p;q = 0 q and is reduced to J whenthe indices p and q are obvious or irrelevant. A -related signature matrix is dened as 0 J = 0 2 The following Hamiltonian matrices are frequently used in this paper [ A 2 BB A 0 H c = ; H o = ; 0 A C C A [ A 2 BB H = C C A H c relates to the control matrix B and H o relates to the observation matrix C. The Hamiltonian-matrix exponential with respect to H is dened as = =e Hh ; 21 22

3 Q.-C. Zhong / Automatica 39 (2003) where h 0 inthis paper. This matrix plays quite animportant role inh -control of dead-time systems. This matrix holds some nice properties, see the technical-report version (Zhong, 2001) of this paper. 2. Problem formulation The Delay-type Nehari Problem (NP h ) is described as follows Givena minimal state-space realization G (s) = A B ; (1) C 0 which is not necessarily stable and h 0, characterize the optimal value opt =inf{ G (s)+e sh K(s) L K(s) H } and for a given opt, parameterize the suboptimal set of proper and causal K(s) H such that G (s)+e sh K(s) L (2) A similar Nehari problem with distributed delays, which canbe reformulated as e sh G (s)+k(s) inthe single delay case, was studied in(tadmor, 1995). Although this questionlooks similar to (2) intheir forms, they are entirely dierent in essence and the problem studied here and in (Tadmor, 1997b) is much more complicated. t is well known (Gohberg, Goldberg, & Kaashoek, 1993) that this problem is solvable i opt = e sh G ; where denotes the Hankel operator. nspecting the transfer matrix e sh G (s), one can see that opt is not less than the L 2 [0;h-induced norm of G (s) (Foias et al., 1996; Zhou & Khargonekar, 1987; Gu, Chen, & Toker, 1996), i.e., opt h = G (s) L2[0;h Hence, it is assumed that h in the sequel. Under this condition, matrix 22, i.e. the (2; 2)-block of, is always nonsingular. A simple formula to compute the L 2 [0;h-induced norm h was givenin(zhou & Khargonekar, 1987) as h = max { det 22 =0}; (3) i.e., the maximal that makes 22 singular or the maximal root of det 22 =0. 3. Main result and the proof Theorem 1. For a given minimally-realized transfer matrix A B G (s)= C 0 having neither j!-axis zero nor j!-axis pole, the following algebraic Riccati equations [ Lc H c =0 and Ho =0 L c always have unique solutions L c 6 0 and L o 6 0 such that A + 2 BB L c and A + L o C C are stable, respectively. The optimal value opt of the delay-type Nehari problem (2) is opt = max { h ; 1 ; 2 }; where { ( ) } 0 h = max det 0 =0 ; { ( ) } 1 = max det [0 =0 ; { ( ) } 2 = max det [ L c =0

4 502 Q.-C. Zhong / Automatica 39 (2003) Fig. 1. Representation of the problem as a block diagram. Furthermore, for a given opt, all K(s) H satisfying (2) can be parameterized as ( ) 0 K(s)=H r W 1 (s);q(s) ; (4) (s) where { ( )} G (s) (s)= h F u ; 2 G (s) ; (5) 0 A + 2 BB L c ( L oh L c ) 1 L oh C ( L oh L c ) 1 (L oh )B W 1 (s)= C 0 (6) 2 B (21 11L c ) 0 with L oh = H r (; L o ), and Q(s) H is a free parameter. Remark 2. Here, h ensures the nonsingularity of 22 ; 1 ensures the existence of L oh and 2 ensures the existence of the J -spectral factorizationand the stability of K(s). The structure of K(s) is showninfig. 1 (the dashed box). t consists of an innite-dimensional block (s), which is a FR block (i.e. a modied Smith predictor), and a nite-dimensional block W 1 (s). The right-upper tag means W 1 (s) maps the right variables to the left variables while a left upper tag, if any, means the matrix maps the left variables to the right variables. Proof. Using the chain-scattering representation (Kimura, 1996), G (s)+e sh K(s)=H r (G(s);K(s)); where G(s) = e sh G (s) 0 t has already beenwell known(kimura, 1996; Meinsma & Zwart, 2000; Green, Glover, Limebeer, & Doyle, 1990) that the H control problem H r (G(s);K(s)) is equivalent to that G (s)j G(s) has a J -spectral factor V (s) such that the (2; 2)-block of G(s)V 1 (s) is bistable. Hence, in this proof, we characterize the conditions to meet these requirements. Here, [ e sh G (s) G (s)j G(s)= e sh G (s) G (s)g (s) 2 The main idea underlying here is to nd a unimodular matrix to equivalently rationalize the system and then to nd the J -spectral factorizationof the rationalized system. This idea was rstly proposed in(meinsma & Zwart, 2000) for a two-block problem. The result there was obtained for a stable G(s) and cannot be directly used here because G(s) here is

5 Q.-C. Zhong / Automatica 39 (2003) not necessarily stable. We borrowed some ideas from there but we use an elementary tool (the similarity transformation) to nd the realization of the rationalized system. (s) givenin(5) canbe decomposed into two parts (s)=f 0 (s)+f 1 (s)e sh with ( ) G (s) F 1 (s)=f u ; 2 G (s) (7) 0 Since (s) is an FR block, nding the J -spectral factorizationof G (s)j G(s) is equivalent to nding the J -spectral factorizationof the following matrix (Kimura, 1996; Meinsma & Zwart, 2000) (s) = (s) 0 G (s)j G(s) ; 0 (s) which is rationalized to be [ 2 ( 2 G G ) 1 + F0 (G G 2 )F 0 F0 (G G 2 ) (s)= (8) (G G 2 )F 0 G G 2 Substitute (1) into (7) and use the star product formula (Zhou & Doyle, 1997), thenthe realizationof F 1 (s) canbe obtained as A 2 BB 0 F 1 (s)= C C A C (9) 0 2 B 0 Accordingly, F 0 (s) has the following realization according to the denition of the operator h A 2 BB 0 F 0 (s)= C C A C (10) 2 B 21 2 B 11 0 Substitute (1), (9) and(10) into (8), thenthe realizationof 1 (s) can be obtained after tedious matrix manipulations, see Appendix, as A 2 BB B 1 0 A C 2 21 B (s)= C B 21 2 B Using the J -spectral factorizationtheory of generic para-hermitianmatrices (Zhong, 2001), 1 (s) has a J -spectral co-factorization if and only if the two Riccati equations [ Lc H c [ L c = 0 and h h Ho 1 =0 have unique symmetric stabilizing solutions L c 6 0andL oh, respectively, and det( L oh L c ) 0. Since the Hamiltonian matrix inthe last Riccati equationis similar to H o, the unique stabilizing solution L oh canalso be obtained as L oh =( 11 L o + 12 )( 21 L o + 22 ) 1 = H r (; L o ) provided that 21 L o + 22 is nonsingular, where L o 6 0 is the unique stabilizing solution of Ho =0

6 504 Q.-C. Zhong / Automatica 39 (2003) The J -spectral co-factor of 1 (s) is A + 2 BB L c ( L oh L c ) 1 L oh C ( L oh L c ) 1 (L oh ) 1 B W 1 0 (s)= C 0 2 B (21 11L c ) 0 1 Hence, the following equality is obtained (s) 0 G (s)j G(s)= W0 (s)jw 0 (s) 0 (s) Since W 0 (s) and 0 (s) are all bistable, 0 W 0 (s) (s) is a J -spectral factor of G (s)j G(s). This means that any K(s) inthe form of ( ) 0 K(s)=H r W 1 (s);q(s) ; (s) where W 1 (s) is as givenin(6) and Q(s) H is a free parameter, satises G (s)+e sh K(s) L norder to guarantee K(s) H, the bistability of the (2; 2)-block of the matrix 11 (s) 12 (s) 0 (s)= = G(s) W 1 (s) 21 (s) 22 (s) (s) is required. Using similar arguments of Meinsma and Zwart (2000), the bistability of 22 (s) is equivalent to the existence of L oh (equivalently, the nonsingularity of 21 L o + 22 ) and the nonsingularity of L oh L c (equivalently, L c L oh )not only for but also for any number larger than. n summary, the solvability conditions are (i) h so that 22 = [ 0 0 is always nonsingular; (ii) There exists a 1 0 such that 21 L o + 22 = [ 0 is always nonsingular for 1 ; (iii) There exists a 2 0 such that L c L oh is always nonsingular for 2. Whencondition(ii) is satised, the nonsingularity of L c L oh is equivalent to that of ( L c L oh )( 21 L o + 22 )= 21 L o + 22 L c ( 11 L o + 12 )= [ Lc. The minimal satisfying these conditions is the optimal value opt = max{ h ; 1 ; 2 }, as givenintheorem 1. This completes the proof. 4. Special cases Case 1 fa is stable, then L o =0, L c = 0 and L oh = Condition (iii) is always satised and condition (ii) becomes the same as condition (i), i.e., only the nonsingularity of 22 is required. nthis case, opt = h. Corollary 3. For a given minimally realized stable transfer matrix A B G (s)= C 0

7 Q.-C. Zhong / Automatica 39 (2003) having neither j!-axis zero nor j!-axis pole, the delay-type Nehari problem (2) is solvable i h (or equivalently, 22 is nonsingular not only for but also for any number larger than ). Furthermore, if this condition holds, then K(s) is parameterized as (4) but A W 1 C 22 B (s)= C 0 2 B 21 0 Remark 4. The cost to be paid for the instability of G (s) is the nonsingularity of the matrices [ 0 and [ Lc, inadditionto the matrix [ 0 0 for the stable case, for and any number larger than. Case 2 f delay h = 0, then = and L oh = L o. Conditions (i) and (ii) are always satised and L oh always exists. Hence, the conditions are reduced to the nonsingularity of L o L c for any 2. nthis case, opt = 2 = max{ det ( L o L c )=0}. Corollary 5. For a given minimally-realized transfer matrix A B G (s)= C 0 having neither j!-axis zero nor j!-axis pole, the delay-free Nehari problem (to nd K(s) H such that G (s) + K(s) L ) is solvable i max{ det( L o L c )=0}. Furthermore, if this condition holds, then K(s) is parameterized as K(s)=H r (W 1 (s);q(s)); where A + 2 BB L c ( L o L c ) 1 L o C ( L o L c ) 1 B W 1 (s)= C 0 2 B L c 0 and Q(s) H is a free parameter. Remark 6. This is analternative solutionto the well-knownnehari problem which has beenaddressed extensively, e.g. in(francis, 1987; Greenet al., 1990). The A-matrix A is not split here. n common situations, it was handled by modal decomposition, see, e.g. (Greenet al., 1990), and the A-matrix A was split into a stable part and an anti-stable part. Case 3 f delay h = 0 and A is stable, then L o =0, L c =0, L oh =0, and =. The conditions are always satised for any 0. K(s) is thenparameterized as K(s)=H r (W 1 (s);q(s)), where G W 1 (s)= 0 This is obvious. For the stable delay-free Nehari problem, the solutionis denitely K(s)= G (s)+q(s) for any 0, where Q(s) H is a free parameter. nthis case, opt =0. [ 5. Conclusion The solution to the delay-type Nehari problem in the stable case is extended to the unstable case. The additional cost paid for the instability is the nonsingularity of two additional matrices. All sub-optimal compensators are parameterized in a transparent structure incorporating a modied Smith predictor, which is the only innite-dimensional part in the controller.

8 506 Q.-C. Zhong / Automatica 39 (2003) Acknowledgements This research was supported initially by the srael Science Foundation (Grant No. 384/99) and then by the EPSRC (Grant No. GR/N38190/1). Q.-C. Zhong thanks the Editors and the anonymous Reviewers for their constructive comments and appreciates the help from Dr. Gjerrit Meinsma, Dr. Leonid Mirkin, Dr. George Weiss and Dr. Kemin Zhou. Appendix. Realizations of 1 (s) and (s) [ F Although 0 (s) is unstable, (s) can be calculated as follows without changing the result after a series of 0 simplication (only for calculation, not for implementation) [ F [ (s)= 0 2 ( 2 G G ) G G 2 0 (11) F 0 Substituting the realizations of G (s) andf 0 (s) into (11), (s) is A 2 BB C C A C A BB 21 2 BB 11 0 B 0 0 C C A (s)= BB A C C 2 21BB BB B BB 2 BB A 2 11BB BB B A 2 BB C C A C 0 C C B 0 0 B 21 B = = A 2 BB C C A C A BB 21 2 BB 11 0 B 0 0 C C A BB A C C 2 21BB BB 11 C 21B BB 2 BB A 2 11BB BB B A 2 BB C C A C C B 0 0 B 21 B A BB 21 2 BB 11 0 B C C A BB A C C 2 21BB BB 11 C 21B BB 2 BB A 2 11BB BB B A 2 BB C C A C C B 0 0 B 21 B norder to further simplify (s), invert it A 2 BB B C C A A C 2 21 B 1 (s)= BB A B A 2 BB C C C C A C C B B 21 2 B

9 Q.-C. Zhong / Automatica 39 (2003) A 2 BB B C C A A C 2 21 B BB A B = A 2 BB B C C A B C B 2 B 11 2 B 21 2 B 21 2 B A 2 BB 0 2 B = C C A B 0 0 A 0 C 2 21 B + 2 BB A B 0 C 0 2 B 11 2 B A 2 BB B = 0 A C 2 21 B C B 21 2 B A 2 BB B C C A B B 21 2 B (s) is now obtained, by inverting 1 (s) and then using a similarity transformation 1,as A 0 12C B C (s)= C A 11C 0 C 11 C B 0 2 References Basar, T., & Bernhard, P. (1995). H -optimal control and related minimax design problems A dynamic game approach (2nd ed.). Boston Birkhauser. Delsarte, P. H., Genin, Y., & Kamp, Y. (1979). The Nevanlinna-Pick problem for matrix-valued functions. SAM Journal on Applied Mathematics, 36, Dym, H., Georgiou, T., & Smith, M. C. (1995). Explicit formulas for optimally robust controllers for delay systems. EEE Transactions on Automatic Control, 40(4), Flamm, D. S., & Mitter, S. K. (1987). H sensitivity minimization for delay systems. Systems & Control Letters, 9, Foias, C., Ozbay, H., & Tannenbaum, A. (1996). Robust control of innite dimensional systems Frequency domain methods. n Lecture Notes in Control and nformatic Sciences, Vol ndon Springer. Francis, B. A. (1987). A course in H control theory. n Lecture Notes in Control and nformatic Sciences, Vol. 88. New York Springer. Gohberg,., Goldberg, S., & Kaashoek, M. A. (1993). Classes of linear operators, Vol.. Birkhauser. Basel. Green, M., Glover, K., Limebeer, D., & Doyle, J. (1990). A J-spectral factorization approach to H control. SAM Journal of Control and Optimization, 28(6), Gu, G., Chen, J., & Toker, O. (1996). Computation of L 2 [0;h induced norms. n Proceedings of the 35th EEE conference on decision & control, Kobe, Japan(pp ). Kimura, H. (1996). Chain-scattering approach to H control. Boston Birkhauser. Kwakernaak, H. (2000). A descriptor algorithm for the spectral factorization of polynominal matrices. n FAC symposium on robust control design, Prague, Czech Republic. Meinsma, G., & Zwart, H. (2000). On H control for dead-time systems. EEE Transactions on Automatic Control, 45(2), Meinsma, G., Mirkin, L., & Zhong, Q.-C. (2002). Control of systems with /O delay via reduction to a one-block problem. EEE Transactions on Automatic Control, 47(11). Mirkin, L. (2000). On the extraction of dead-time controllers from delay-free parametrizations. n Proceedings of the second FAC workshop on linear time delay systems, Ancona, taly (pp ). Nagpal, K. M., & Ravi, R. (1997). H control and estimation problems with delayed measurements State-space solutions. SAM Journal on Control and Optimization, 35(4), Tadmor, G. (1995). The Nehari problem in systems with distributed input delays is inherently nite dimensional. Systems & Control Letters, 26(1),

10 508 Q.-C. Zhong / Automatica 39 (2003) Tadmor, G. (1997a). Robust control in the gap A state space solution in the presence of a single input delay. EEE Transactions on Automatic Control, 42(9), Tadmor, G. (1997b). Weighted sensitivity minimization in systems with a single input delay A state space solution. SAM Journal on Control and Optimization, 35(5), Tadmor, G. (2000). The standard H problem in systems with a single input delay. EEE Transactions on Automatic Control, 45(3), Zhong, Q.-C. (2001). Frequency domain solution to the delay-type Nehari problem using J -spectral factorization. Technical report, Department of E.E. Engineering, mperial College of Science, Technology and Medicine. Available at http// Zhong, Q.-C. (2003). On standard H control of processes with a single delay. EEE Transactions on Automatic Control, 48. Zhou, K., & Doyle, J. C. (1997). Essentials of robust control. Upper Saddle River, NJ Prentice-Hall. Zhou, K., & Khargonekar, P. P. (1987). On the weighted sensitivity minimization problem for delay systems. Systems & Control Letters, 8, Qing-Chang Zhong was born in 1970 in Sichuan, China. He obtained the M.S. degree in Electrical Engineering in 1997 from Hunan University, China and the Ph.D. degree in 2000 in Control Theory and Engineering from Shanghai Jiao Tong University, China. He held a postdoctoral position at Technion-srael nstitute of Technology, srael. He is currently a Research Associate at the Department of Electrical and Electronic Engineering of mperial College of Science, Technology and Medicine, ndon, UK. His current research focuses on H-innity control of time-delay systems, control in power electronics, process control, control using delay elements and control in communication networks.