Discrete Time Systems Part 6 pptx

The Design of a Discrete Time Model Following Control System for Nonlinear Descriptor System 139 Both the controlled object and the reference model are controllable and observable N r ≠ Zeros of C [ zE − A]−1 B are stable γ f ( v( k )) ≤ α + β v( k ) ,(α ≥ 0, β ≥ 0,0 ≤ γ < 1) Existing condition of v( k ) is I + C v Bs ∂f ( v( k )) ≠ ∂vT ( k ) zE − A ≡ and rankE = deg zE − A = r ≤ n / Numerical simulation An example is given as follows: ⎤ ⎡ 1⎤ ⎡ ⎡0 ⎤ ⎡1⎤ ⎡0 ⎤ ⎢0 1⎥ x( k + 1) = ⎢ 0 ⎥ x( k ) + ⎢ ⎥ u( k ) + ⎢0 ⎥ f ( v( k )) + ⎢ ⎥ d( k ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1⎥ ⎢0.2 −0.5 0.6 ⎥ ⎢0 1⎥ ⎢1⎥ ⎢1⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ v( k ) = [ 1 1] x( k ), ⎡ 0.1 ⎤ ⎡ 1⎤ y( k ) = ⎢ x( k ) + ⎢ ⎥ 0.1 0.1⎥ ⎣ ⎦ ⎣ 1⎦ f ( v( k )) = (57) 3v ( k ) + v( k ) + + v4 (k) Reference model is given by ⎤ ⎡ ⎡0 ⎤ xm ( k + 1) = ⎢ x ( k ) + ⎢ ⎥ rm ( k ) −0.12 0.7 ⎥ m ⎣ ⎦ ⎣1⎦ y m ( k ) = [ ] xm ( k ) (58) rm ( k ) = sin( kπ / 16) In this example, disturbances d( k ) and d0 ( k ) are ramp and step disturbances respectively Then d( k ) and d0 ( k ) are given as d( k ) = 0.05( k − 85),(85 ≤ k ≤ 100) d0 ( k ) = 1.2,(20 ≤ k ≤ 50) (59) We show a result of simulation in Fig It can be concluded that the output signal follows the reference even if disturbances exit in the system Conclusion In the responses (Fig 1) of the discrete time model following control system for nonlinear descriptor system, the output signal follows the references even though disturbances exit in the system The effectiveness of this method has thus been verified The future topic is that the case of nonlinear system for γ ≥ will be proved and analysed 140 Discrete Time Systems Fig Responses of the system for nonlinear descriptor system in discrete time References Wu,S.; Okubo,S.; Wang,D (2008) Design of a Model Following Control System for Nonlinear Descriptor System in Discrete Time, Kybernetika, vol.44,no.4,pp.546-556 Byrnes,C.I; Isidori,A (1991) Asymptotic stabilization of minimum phase nonlinear system, IEEE Transactions on Automatic Control, vol.36,no.10,pp.1122-1137 Casti,J.L (1985) Nonlinear Systems Theory, Academic Press, London Furuta,K (1989) Digital Control (in Japanese), Corona Publishing Company, Tokyo Ishidori,A (1995) Nonlinear Control Systems, Third edition, Springer-Verlag, New York Khalil,H.K (1992) Nonlinear Systems, MacMillan Publishing Company, New York Mita,T (1984) Digital Control Theory (in Japanese), Shokodo Company, Tokyo Mori,Y (2001) Control Engineering (in Japanese), Corona Publishing Company, Tokyo Okubo,S (1985) A design of nonlinear model following control system with disturbances (in Japanese), Transactions on Instrument and Control Engineers, vol.21,no.8,pp.792799 Okubo,S (1986) A nonlinear model following control system with containing inputs in nonlinear parts (in Japanese), Transactions on Instrument and Control Engineers, vol.22,no.6,pp.714-716 Okubo,S (1988) Nonlinear model following control system with unstable zero points of the linear part (in Japanese), Transactions on Instrument and Control Engineers, vol.24,no.9,pp.920-926 Okubo,S (1992) Nonlinear model following control system using stable zero assignment (in Japanese), Transactions on Instrument and Control Engineers, vol.28, no.8, pp.939-946 Takahashi,Y (1985) Digital Control (in Japanese), Iwanami Shoten,Tokyo Zhang,Y; Okubo,S (1997) A design of discrete time nonlinear model following control system with disturbances (in Japanese), Transactions on The Institute of Electrical Engineers of Japan, vol.117-C,no.8,pp.1113-1118 Output Feedback Control of Discrete-time LTI Systems: Scaling LMI Approaches Jun Xu National University of Singapore Singapore Introduction Most physical systems have only limited states to be measured and fed back for system controls Although sometimes, a reduced-order observer can be designed to meet the requirements of full-state feedback, it does introduce extra dynamics, which increases the complexity of the design This naturally motivates the employment of output feedback, which only use measurable output in its feedback design From implementation point of view, static feedback is more cost effective, more reliable and easier to implement than dynamic feedback (Khalil, 2002; Kuˇ era & Souza, 1995; Syrmos et al., 1997) Moreover, many other problems are c reducible to some variation of it Simply stated, the static output feedback problem is to find a static output feedback so that the closed-loop system has some desirable characteristics, or determine the nonexistence of such a feedback (Syrmos et al., 1997) This problem, however, still marked as one important open question even for LTI systems in control engineering Although this problem is also known NP-hard (Syrmos et al., 1997), the curious fact to note here is that these early negative results have not prevented researchers from studying output feedback problems In fact, there are a lot of existing works addressing this problem using different approaches, say, for example, Riccati equation approach, rank-constrained conditions, approach based on structural properties, bilinear matrix inequality (BMI) approaches and min-max optimization techniques (e.g., Bara & Boutayeb (2005; 2006); Benton (Jr.); Gadewadikar et al (2006); Geromel, de Oliveira & Hsu (1998); Geromel et al (1996); Ghaoui et al (2001); Henrion et al (2005); Kuˇ era & Souza (1995); Syrmos et al (1997) and the c references therein) Nevertheless, the LMI approaches for this problem remain popular (Bara & Boutayeb, 2005; 2006; Cao & Sun, 1998; Geromel, de Oliveira & Hsu, 1998; Geromel et al., 1996; Prempain & Postlethwaite, 2001; Yu, 2004; Zeˇ evi´ & Šiljak, 2004) due to simplicity and c c efficiency Motivated by the recent work (Bara & Boutayeb, 2005; 2006; Geromel et al., 1996; Xu & Xie, 2005a;b; 2006), this paper proposes several scaling linear matrix inequality (LMI) approaches to static output feedback control of discrete-time linear time invariant (LTI) plants Based on whether a similarity matrix transformation is applied, we divide these approaches into two parts Some approaches with similarity transformation are concerned with the dimension and rank of system input and output Several different methods with respect to the system state dimension, output dimension and input dimension are given based on whether the distribution matrix of input B or the distribution matrix of output C is full-rank The other 142 Discrete Time Systems approaches apply Finsler’s Lemma to deal with the Lyapunov matrix and controller gain directly without similarity transformation Compared with the BMI approach (e.g., Henrion et al (2005)) or VK-like iterative approach (e.g.,Yu (2004)), the scaling LMI approaches are much more efficient and convergence properties are generally guaranteed Meanwhile, they can significantly reduce the conservatism of non-scaling method, (e.g.,Bara & Boutayeb (2005; 2006)) Hence, we show that our approaches actually can be treated as alternative and complemental methods for existing works The remainder of this paper is organized as follows In Section 2, we state the system and problem In Section 3, several approaches based on similarity transformation are given In Subection 3.1, we present the methods for the case that B is full column rank Based on the relationship between the system state dimension and input dimension, we discuss it in three parts In Subsection 3.2, we consider the case that C is full row rank in the similar way In Subsection 3.3, we propose another formulations based on the connection between state feedback and output feedback In Section 4, we present the methods based on Finsler’s lemma In Section 5, we compare our methods with some existing works and give a brief statistical analysis In Section 6, we extend the latter result to H∞ control Finally, a conclusion is given in the last section The notation in this paper is standard Rn denotes the n dimensional real space Matrix A > (A ≥ 0) means A is positive definite (semi-definite) Problem formulation Consider the following discrete-time linear time-invariant (LTI) system: x ( t + ) = A o x ( t ) + Bo u ( t ) (1) y ( t ) = Co x ( t ) (2) where x ∈ Rn , u ∈ Rm and y ∈ Rl All the matrices mentioned in this paper are appropriately dimensioned m < n and l < n We want to stabilize the system (1)-(2) by static output feedback u (t) = Ky(t) (3) ˜ x (t + 1) = Ax (t) = ( Ao + Bo KCo ) x (t) (4) The closed-loop system is The following lemma is well-known Lemma (Boyd et al., 1994) The closed-loop system (4) is (Schur) stable if and only if either one of the following conditions is satisfied: ˜ ˜ P > 0, A T P A − P < (5) ˜ ˜ Q > 0, AQ A T − Q < (6) Scaling LMIs with similarity transformation This section is motivated by the recent LMI formulation of output feedback control (Bara & Boutayeb, 2005; 2006; Geromel, de Souze & Skelton, 1998) and dilated LMI formulation (de Oliveira et al., 1999; Xu et al., 2004) Output Feedback Control of Discrete-time LTI Systems: Scaling LMI Approaches 143 3.1 Bo with full column-rank We assume that Bo is of full column-rank, which means we can always find a non-singular I matrix Tb such that Tb Bo = m In fact, using singular value decomposition (SVD), we can obtain such Tb Hence the new state-space representation of this system is given by A11 A12 − , B = Tb Bo , C = Co Tb A21 A22 − A = Tb Ao Tb = (7) The closed-loop system (4) is stable if and only if ˜ Ab = A + BKC is stable In this case, we divide it into situations: m = n − m, m < n − m, and m > n − m Let P= P11 P12 T P12 P22 ∈ Rn×n , P11 ∈ Rm×m , P12 ∈ Rm×( n−m) (8) For the third situation, let ( 1) ( 2) P12 = [ P12 P12 ], P11 = ( 1) ( 1) P11 ( 2) T P11 ( 2) P11 (9) ( 3) P11 ( 1) where P12 ∈ R( n−m)×( n−m) and P11 ∈ R( n−m)×( n−m) Theorem The discrete-time system (1)-(2) is stabilized by (3) if there exist P > defined in (8) and R, such that ⎧ ⎨ Φ (Θ1 ) < 0, m = n − m Φ (Θ2 ) < 0, m < n − m (10) ⎩ Φ (Θ3 ) < 0, m > n − m where ε ∈ R, Φ ( Θ1 ) = Θ2 = ⎣ (11) (12) ⎤ 0 P22 − ε Θ3 = and P2 > since P > Based on (17), we have ⎤ ⎡ 0 A − Λ0 ∗ ⎦ AT ⎣ P2 0, P2 > 0, P12 and R with P defined in (27), such that ⎧ ⎨ Υ(Λ1 ) < 0, m = n − m (28) Υ(Λ2 ) < 0, m < n − m ⎩ Υ(Λ3 ) < 0, m > n − m where ε ∈ R, ⎡ ⎤ 0 A − Λi ∗ ⎦ P2 Υ(Λi ) = ⎣ , RC + [ P11 P12 ] A − P11 AT Λ1 = P11 P12 T T , P12 P2 − ε2 P11 + εP12 + εP12 146 Discrete Time Systems ⎡ Λ2 = ⎣ ⎤ P11 T P12 P2 + ε Λ3 = P11 P12 P12 P ⎦, T + ε[ P12 0] − ε2 11 I ( 1) T T P12 P2 + εP12 P12 ( 1) ( 1) + εP12 − ε2 P11 Furthermore, a static output controller gain is given by (13) Proof: We only consider the first case Replacing P2 and R by P22 and K using (25) and (13), we can derive that (28) is a sufficient condition for (5) with the P defined in (8) 3.2 Co with full row-rank − When Co is full row rank, there exists a nonsingular matrix Tc such that Co To = [ Il 0] Applying a similarity transformation to the system (1)-(2), the closed-loop system (4) is stable if and only if ˜ Ac = A + BKC is stable − − where A = Tc Ao Tc , B = Tc Bo and C = Co Tc = [ Il 0] Similarly to Section 3.1, we can also divide this problem into three situations: l = n − l, Q11 Q12 l < n − l and l > n − l We use the condition (6) here and partition Q as Q = , T Q12 Q22 where Q11 ∈ Rl ×l Theorem The discrete-time system (1)-(2) is stabilized by (3) if there exist Q > and R, such that ⎧ ¯ ⎨ Γ (Θ1 ) < 0, l = n − l ¯ Γ (Θ ) < 0, l < n − l (29) ⎩ ¯2 Γ (Θ3 ) < 0, l > n − l where ε ∈ R, ¯ Γ (Θi ) = 0 , T Q22 + ε2 Q11 − εQ12 − εQ12 ¯ Θ1 = ⎡ ¯ Θ2 = ⎣ ( 1) ⎤ 0 Q22 + ε2 ¯ Θ3 = ( 1) ¯ AΘi A T − Q ∗ , ( A[ Q11 Q12 ] T + BR) T − Q11 0 ⎦, Q11 Q12 T −ε − ε[ Q12 0] 0 ( 1) Q22 + ε2 Q11 ( 1) T − εQ12 ( 1) − εQ12 , Q11 and Q12 are properly dimensioned partitions of Q11 and Q12 Furthermore, a static output feedback controller gain is given by − (30) K = RQ111 Output Feedback Control of Discrete-time LTI Systems: Scaling LMI Approaches 147 Proof: We only prove the first case l = n − l, since the others are similar Noting that ( BKC ) Q( BKC ) T = BKQ11 K T B and BKCQ = BK [ Q11 Q12 ], (6) is equivalent to − ( A[ Q11 Q12 ] T + BKQ11 ) Q111 ( A[ Q11 Q12 ] T + BKQ11 ) T Q11 − −A Q111 [ Q11 Q12 ] A T + AQA T − Q < T Q12 (31) Using the fact that Q− Q11 − Q111 [ Q11 Q12 ] = T Q12 0 T − Q12 Q111 Q12 we infer that stability of the close-loop system is equivalent to the existing of a Q > such that ¯ A Θ0 A T − Q ∗ 0, Ψc < 0} (39) −Wc1 AWc1 + BWc2 T Wc1 A T + Wc2 B T −Wc1 − ˜ Ko = {Ko = Wo11 Wo2 ∈ Rn×l : (Wo1, Wo2 ) ∈ Wo } and where Ψo = (40) Wo = {Wo1 ∈ Rn×n , Wo2 ∈ Rn×l : Wo1 > 0, Ψo < 0} (41) −W1o Wo1 A + Wo2 C T A T Wo1 + C T Wo2 −W1o Lemma L = ∅ if and only if ¯ Kc = Kc {Kc : Kc Yc = 0, Yc = N (C )} = ∅; or ¯ Ko = Ko {Kc : Yo Ko = 0, Yo = N ( B )} = ∅ In the affirmative case, any K ∈ L can be rewritten as K = Kc QC T (CQC T )−1 ; or K = ( B T PB )−1 B T PKo where Q > and P > are arbitrarily chosen Proof: The first statement has been proved in Geromel et al (1996) For complement, we give the proof of the second statement The necessity is obvious since Ko = BK Now we ¯ prove the sufficiency, i.e., given Ko ∈ Ko , there exists a K, such that the constraint Ko = BK BT P is solvable Note that for ∀ P > 0, Θ o = is full rank, where Yo = N ( B T ) In fact, T Yo B T PYo rank(Θ o Yo ) = rank( ) ≥ n − m Multiplying Θo at the both side of Ko = BK we have In − m B T PKo T Yo Ko = B T PBL Since B T PB is invertible, we have K = ( B T PB )−1 B T PK0 Hence, we can derive the result Lemma L = ∅ if and only if there exists Ec ∈ Rn×( n−l ) or Eo ∈ Rn×( n−m), such that one of the following conditions holds: rank( Tc = C T ) = n and C( Ec ) = ∅; or Ec rank( To = B Eo ) = n and O( Eo ) = ∅ where C( Ec ) = Wc O( Eo ) = Wo {(Wc1, Wc2 ) : CWc1 Ec = 0, Wc2 Ec = 0} T {(Wo1, Wo2 ) : B T Wo1 Eo = 0, Eo Wo2 = 0} In the affirmative case, any K ∈ L can be rewritten as K = Wc2 C T (CWc1 C T )−1 ; or K = ( B T Wo1 B )−1 B T Wo2 154 Discrete Time Systems Example Consider the unstable system as follows ⎡ 0.82 0.0576 0.2212 ⎢ 0.0574 0.0634 0.6254 ⎢ Ao = ⎢ 0.0901 0.7228 0.5133 ⎢ ⎣ 0.6967 0.0337 0.5757 0.1471 0.6957 0.2872 ⎡ 0.9505 ⎢ 0.3182 ⎢ Bo = ⎢ 0.2659 ⎢ ⎣ 0.0611 0.3328 Co = 0.8927 0.0926 0.2925 0.8219 0.994 ⎤ 0.0678 0.9731 ⎥ ⎥ 0.9228 ⎥ ⎥ 0.9587 ⎦ 0.5632 ⎤ 0.2924 0.4025 ⎥ ⎥ 0.0341 ⎥ ⎥ 0.2875 ⎦ 0.2196 0.5659 0.255 0.5227 0.0038 0.3608 0.8701 0.5918 0.1291 0.3258 0.994 This example is borrowed from (Bara & Boutayeb, 2006), where output feedback controllers have been designed For A22 from A, it has stable eigenvalue In this paper, we compare the design problem with the maximum decay rate, i.e., ˜ ˜ max ρ s.t A T P A − P < − ρP Note that in this example, m < n − m With ε = 0, i.e., using the method in (Bara & Boutayeb, 2006), we obtain the maximum ρ = 0.16, while Theorem gives ρ = 0.18 with ε = −0.09 ˆ However, Theorem only obtains a maximum ρ = 0.03 with a choice of Z = [ I2 I2 0] T Note that the solvability heavily depends on the choice of ε For example, when ε = 0.09 for Theorem 1, the LMI is not feasible Now we consider a case that A22 has an unstable eigenvalue Consider the above example with slight changes on Ao ⎡ 0.9495 ⎢ 0.8656 ⎢ Ao = ⎢ 0.5038 ⎢ ⎣ 0.13009 0.34529 0.12048 0.28816 0.46371 0.76443 0.61187 0.14297 0.67152 0.9712 0.47657 0.15809 0.19192 0.01136 0.93839 0.54837 0.46639 ⎤ 0.019139 0.38651 ⎥ ⎥ 0.42246 ⎥ ⎥ 0.4089 ⎦ 0.53536 We can easily verify that A22 from A has one unstable eigenvalue 1.004 Hence, the method in (Bara & Boutayeb, 2006) cannot solve it However, Theorem generates a solution as −0.233763 −0.31506 Meanwhile, Theorem also can get a feasible solution for K = −3.61207 0.376493 0.9373 −0.4008 ε = −0.1879 and K = Theorem via a standard SVD without scaling 1.5244 −0.7974 1.4813 0.5720 −0.3914 −0.3603 can also obtain K = using (43) or K = using (44) −2.3604 −1.1034 −3.7203 −1.8693 Example We randomly generate 5000 stabilizable and detectable systems of dimension n = 4(6, 6, 6, 7, 7), m = 2(3, 1, 5, 4, 3) and l = 2(3, 5, 1, 3, 4) Output Feedback Control of Discrete-time LTI Systems: Scaling LMI Approaches T1 155 T3 SeDuMi 5000 4982 SDPT3 4975 5000 Table Different solvability of different solvers T 1α T 4.2.2β 6.3.3 6.1.5 6.5.1 7.4.3 7.3.4 Y Y N N Y 4999 4999 4994 4996 4998 4998 N 1 Y 1 N 0 0 0 Superscriptγ: Y (N) means that the problem can (not) be solved by the corresponding theorems For example, the value of third row and third column means that in the random 5000 examples, there are cases that cannot be solved by Theorem while can be solved by Theorem Table Comparison of Theorem and Theorem Hence we can use Theorem and Theorem with ε = to solve this problem Note that different solvers may give different solvability For example, given n = 6, m = and l = 3, in a one-time simulation, the result is given in Table Thus in order to partially eliminate the effect of the solvers, we choose the combined solvability result from two solvers in this section Table shows the comparison of Theorem and Theorem Some phenomenons (the solvability of Theorem and Theorem depends on the l and m When m > l, Theorem tends to have a higher solvability than Theorem And vise verse.) was observed from these results obtained using LMITOOLS provided by Matlab is not shown here Extension to H∞ synthesis The aforementioned results can contribute to other problems, such as robust control In this section, we extend it to H∞ output feedback control problem Consider the following system: x (t + 1) = Ax (t) + B2 u (t) + B1 w (67) y(t) = Cx (t) + Dw (68) z(t) = Ex (t) + Fw (69) We only consider the case that B2 is with full rank and assume that the system has been transferred into the form like (7) Using the controller as (3), the closed-loop system is ˆ ˆ x (t + 1) = Ax (t) + Bw = ( A + B2 KC ) x (t) + ( B1 + B2 KD )w (70) z We attempt to design the controller, such that the L2 gain sup w ≤ γ It should be noted that all the aforementioned scaling LMI approaches can be applied here However, we only choose one similar to Theorem 156 Discrete Time Systems Theorem The discrete-time system (67)-(69) is stabilized by (3) and satisfies H∞ , if there exist a matrix P > defined in (8) and R, such that ⎧ ⎨ (Θ1 ) < 0, m = n − m (71) (Θ2 ) < 0, m < n − m ⎩ (Θ3 ) < 0, m > n − m where ε ∈ R, Θ i is defined in Theorem 1, (Θi ) = ⎤ − P11 RC + [ P11 P12 ] A RD + [ P11 P12 ] B1 ⎢ ∗ A T Θi A − P A T Θ i B1 ET ⎥ ⎥ ⎢ T ⎣ ∗ ∗ B1 Θ i B − γI FT ⎦ ∗ ∗ ∗ − γI ⎡ Proof: Following the arguments in Theorem 1, we can see that (71) implies ⎡ T ⎤ ˆ ˆ ˆ ˆ AT PB ET A PA − P ˆ ˆ (Θi ) = ⎣ B T P B − γI F T ⎦ < ∗ ∗ ∗ − γI (72) (73) Using bounded real lemma (Boyd et al., 1994), we can complete the proof Conclusion In this paper, we have presented some sufficient conditions for static output feedback control of discrete-time LTI systems Some approaches require a similarity transformation to convert B or C to a special form such that we can formulate the design problem into a scaling LMI problem with a conservative relaxation Based on whether B or C is full rank, we consider several cases with respect to the system state dimension, output dimension and input dimension These methods are better than these introduced in (Bara & Boutayeb, 2006) and might achieve statistical advantages over other existing results (Bara & Boutayeb, 2005; Crusius & Trofino, 1999; Garcia et al., 2001) The other approaches apply Finsler’s lemma directly such that the Lyapunov matrix and the controller gain can be separated, and hence gain benefits for the design All the presented approaches can be extended to some other problems Note that we cannot conclude that the approaches presented in this paper is definitely superior to all the existing approaches, but introduce some alternative conditions which may achieve better performance than others in some circumstances References Bara, G I & Boutayeb, M (2005) static output feedback stabilization with h∞ performance for linear discrete-time systems, IEEE Trans on Automatic Control 50(2): 250–254 Bara, G I & Boutayeb, M (2006) A new sufficient condition for the output feedback 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semidefinite-quadratic-linear programming, version 4.0, Manual, National University of Singapore, Singapore Xu, J & Xie, L (2005a) H∞ state feedback control of discrete-time piecewise affine systems, IFAC World Congress, Prague, Czech Xu, J & Xie, L (2005b) Non-synchronized H∞ estimation of discrete-time piecewise linear systems, IFAC World Congress, Prague, Czech Xu, J & Xie, L (2006) Dilated LMI characterization and a new stability criterion for polytopic uncertain systems, IEEE World Congress on Intelligent Control and Automation, Dalian, China, pp 243–247 158 Discrete Time Systems Xu, J., Xie, L & Soh, Y C (2004) H∞ and generalized H2 estimation of continuous-time piecewise linear systems, the 5th Asian Control Conference, IEEE, Melbourne, Australia Yu, J (2004) A new static output feedback approach to the suboptimal mixed H2 \ H∞ problem, Int J Robust Nonlinear Control 14: 1023–1034 Zeˇ evi´ , A I & Šiljak, D D (2004) Design of robust static output feedback for large-scale c c systems, IEEE Trans on Automatic Control 49(11): 2040–2044 10 Discrete Time Mixed LQR/H∞ Control Problems Xiaojie Xu School of Electrical Engineering, Wuhan University Wuhan, 430072, P R China Introduction This chapter will consider two discrete time mixed LQR/ H ∞ control problems One is the discrete time state feedback mixed LQR/ H ∞ control problem, another is the non-fragile discrete time state feedback mixed LQR/ H ∞ control problem Motivation for mixed LQR/ H ∞ control problem is to combine the LQR and suboptimal H ∞ controller design theories, and achieve simultaneously the performance of the two problems As is well known, the performance measure in optimal LQR control theory is the quadratic performance index, defined in the time-domain as ∞ J := ∑ ( xT ( k )Qx( k ) + uT ( k )Ru( k )) (1) k =0 while the performance measure in H ∞ control theory is H ∞ norm, defined in the frequency-domain for a stable transfer matrix Tzw ( z ) as Tzw ( z) ∞ := sup σ max [Tzw ( e jw )] w∈[ 0,2π ] where, Q ≥ , R > , σ max [•] denotes the largest singular value The linear discrete time system corresponding to the discrete time state feedback mixed LQR/ H ∞ control problem is x( k + 1) = Ax( k ) + B1 w( k ) + B2 u( k ) (2.a) z( k ) = C 1x( k ) + D12 u( k ) (2.b) with state feedback of the form u( k ) = Kx( k ) (3) where, x( k ) ∈ Rn is the state, u( k ) ∈ Rm is the control input, w( k ) ∈ Rq is the disturbance input that belongs to L2 [0, ∞ ) , z( k ) ∈ R p is the controlled output A , B1 , B2 , C and D12 are known matrices of appropriate dimensions Let x(0) = x0 The closed loop transfer matrix from the disturbance input w to the controlled output z is 160 New Trends in Technologies ⎡A Tzw ( z) = ⎢ K ⎣C K BK ⎤ := C K ( zI − AK )−1 BK 0⎥ ⎦ where, AK := A + B2 K , BK := B1 , C K := C + D12 K Recall that the discrete time state feedback optimal LQR control problem is to find an admissible controller that minimizes the quadratic performance index (1) subject to the systems (2) (3) with w = , while the discrete time state feedback H ∞ control problem is to find an admissible controller such that Tzw ( z) ∞ < γ subject to the systems (2)(3) for a given number γ > While we combine the two problems for the systems (2)(3) with w ∈ L2 [0,∞ ) , the quadratic performance index (1) is a function of the control input u( k ) and disturbance input w( k ) in the case of x(0) being given and γ being fixed Thus, it is not possible to pose a mixed LQR/ H ∞ control problem that is to find an admissible controller that achieves the minimization of quadratic performance index (1) subject to Tzw ( z) ∞ < γ for the systems (2)(3) with w ∈ L2 [0,∞ ) because the quadratic performance index (1) is an uncertain function depending on the uncertain disturbance input w( k ) In order to eliminate this difficulty, the design criteria of state feedback mixed LQR/ H ∞ control problem should be replaced by the design criteria sup inf{ J } subject to Tzw ( z) ∞ < γ w∈L2 + K because for all w ∈ L2 [0,∞ ) , the following inequality always exists inf{ J } ≤ sup inf{ J } K w∈L2 + K The stochastic problem corresponding to this problem is the combined LQG/ H ∞ control problem that was first presented by Bernstein & Haddad (1989) This problem is to find an admissible fixed order dynamic compensator that minimizes the expected cost function of the form J = lim Ε( xT Qx + uT Ru) subject to Tzw t →∞ ∞ is a given number ˆ ˆ ˆ ∞ ∞ ∞ ˆ Note that the feedback matrix F∞ of the considered closed-loop system is a function of the controller uncertainty ΔF( k ) , this results in that the quadratic performance index (1) is not only a function of the controller F∞ and disturbance input w( k ) but also a function of the controller uncertainty ΔF( k ) in the case of x(0) being given and γ being fixed We can easily know that the existence of disturbance input w( k ) and controller uncertainty ΔF( k ) makes it impossible to find sup w∈L2+ infK { J } , while the existence of controller uncertainty ΔF( k ) also makes it difficult to find sup w∈L2+ { J } In order to eliminate these difficulties, the design criteria of non-fragile discrete time state feedback mixed LQR/ H ∞ control problem should be replaced by the design criteria ˆ sup w∈L2+ { J } subject to Tzw ( z) ∞ < γ Motivation for non-fragile problem came from Keel & Bhattacharyya (1997) Keel & Bhattacharyya (1997) showed by examples that optimum and robust controllers, designed by using the H , H ∞ , l , and μ formulations, can produce extremely fragile controllers, in the sense that vanishingly small perturbations of the coefficients of the designed controller destabilize the closed-loop system; while the controller gain variations could not be avoided in most applications.This is because many factors, such as the limitations in available computer memory and word-length capabilities of digital processor and the A/D and D/A converters,result in the variation of the controller parameters in controller implementation Also, the controller gain variations might come about because of external effects such as temperature changes.Thus, any controller must be insensitive to the above-mentioned contoller gain variation The question arised from this is how to design a controller that is insensitive, or non-fragile to error/uncertainty in controller parameters for a given plant This 162 New Trends in Technologies problem is said to be a non-fragile control problem Recently, the non-fragile controller approach has been used to a very large class of control problems (Famularo et al 2000, Haddad et al 2000, Yang et al 2000, Yang et al 2001 and Xu 2007) The second aim of this chapter is to, based on the results of Xu (2007), present a non-fragile controller approach to the discrete-time state feedback mixed LQR/ H ∞ control problem with controller uncertainty This chapter is organized as follows In Section 2, we review several preliminary results, and present two extensions of the well known discrete time bounded real lamma In Section 3, we define the discrete time state feedback mixed LQR/ H ∞ control problem Based on this definition, we present the both Riccati equation approach and state space approach to the discrete time state feedback mixed LQR/ H ∞ control problem In Section 4, we intro-duce the definition of non-fragile discrete time state feedback mixed LQR/ H ∞ control problem, give the design method of a non-fragile discrete time state feedback mixed LQR / H ∞ controller, and derive the necessary and sufficient conditions for the existence of this controller In Section 5, we give two examples to illustrate the design procedures and their effectiveness, respectively Section gives some conclusions Throughout this chapter, AT denotes the transpose of A , A−1 denotes the inverse of A , A−T is the shorthand for ( A−1 )T , G~ ( z) denotes the conjugate system of G( z ) and is the shorthand for GT ( z −1 ) , L2 ( −∞ , +∞ ) denotes the time domain Lebesgue space, L2 [0, +∞ ) denotes the subspace of L2 ( −∞ , +∞ ) , L2 ( −∞ ,0] denotes the subspace of L2 ( −∞ , +∞ ) , L2 + is the shorthand for L2 [0, +∞ ) and L2 − is the shorthand for L2 ( −∞ ,0] Preliminaries This section reviews several preliminary results First, we consider the discerete time Riccati equation and discrete time Riccati inequality, respectively X = AT X( I + RX )−1 A + Q (5) AT X ( I + RX )−1 A + Q − X < (6) and with Q = QT ≥ and R = RT > We are particularly interested in solution s X of (5) and (6) such that ( I + RX )−1 A is stable A symmetric matrix X is said to the stabilizing solution of discrete time Riccati equation (5) if it satisfies (5) and is such that ( I + RX )−1 A is stable Moreover, for a sufficiently small constant δ > , the discrete time Riccati inequality (6) can be rewritten as X = AT X( I + RX )−1 A + Q + δ I (7) Based on the above relation, we can say that if a symmetric matrix X is a stabilizing solution to the discrete time Riccati equation (7), then it also is a stabilizing solution to the discrete time Riccati inequality (6) According to the concept of stabilizing solution of discrete time Riccati equation, we can define the stabilizing solution X to the discrete time Riccati inequality (6) as follow: if there exists a symmetric solution X to the discrete time Riccati inequality (6) such that ( I + RX )−1 A is stable, then it is said to be a stabilizing solution to the discrete time Riccati inequality (6) 163 Discrete Time Mixed LQR/H∞ Control Problems If A is invertible, the stabilizing solution to the discerete time Riccati equation (5) can be obtained through the following simplectic matrix ⎡ A + RA−T Q − RA−T ⎤ S := ⎢ ⎥ A −T ⎥ ⎢ − A−T Q ⎣ ⎦ (8) Assume that S has no eigenvalues on the unit circle, then it must have n eigenvalues in λi < and n in λi > ( i = 1, 2, , n , n + 1, , 2n ) If n eigenvectors corresponding to n eigenvalues in λi < of the simplectic matrix (8) is computed as ⎡ui ⎤ ⎢v ⎥ ⎣ i⎦ then a stabilizing solution to the discerete time Riccati equation (5) is given by X = [ v1 ][ u1 un ] −1 Secondly, we will introduce the well known discrete time bounded real lemma (see Zhou et al , 1996; Iglesias & Glover, 1991; Souza & Xie, 1992) Lemma 2.1 (Discrete Time Bounded Real Lemma) ⎡A B⎤ Suppose that γ > , M ( z) = ⎢ ⎥ ∈ RH ∞ , then the following two statements are ⎣C D ⎦ equivalent: i M( z) ∞ < γ ii There exists a stabilizing solution X ≥ ( X > if (C , A) is observable ) to the discrete time Riccati equation − AT XA − X + γ −2 ( AT XB + C T D)U1 ( BT XA + DT C ) + C T C = such that U = I − γ −2 (DT D + BT XB) > In order to solve the two discrete time state feedback mixed LQR/ H ∞ control problems considered by this chapter, we introduce the following reference system ⎡ C1 ⎤ ⎡ D12 ⎤ ⎢ ⎥ ⎢ ⎥ ˆ x( k + 1) = Ax( k ) + B1 w( k ) + B2 u( k ) z( k ) = ⎢ ⎡ I ⎤ ⎥ x( k ) + ⎢ ⎡0 ⎤ ⎥ u( k ) Ω ⎢ ⎥ Ω ⎢ ⎥ ⎢ ⎢ ⎣0 ⎦ ⎥ ⎣I ⎦⎥ ⎣ ⎦ ⎣ ⎦ (9) ⎡Q ⎤ ⎡ z( k ) ⎤ ˆ and z( k ) = ⎢ where, Ω = ⎢ ⎥ R⎥ ⎣ ⎦ ⎣ z0 ( k )⎦ The following lemma is an extension of the discrete time bounded real lemma Lemma 2.2 Given the system (2) under the influence of the state feedback (3), and suppose that γ > , Tzw ( z) ∈ RH ∞ ; then there exists an admissible controller K such that Tzw ( z) ∞ < γ if there exists a stabilizing solution X∞ ≥ to the discrete time Riccati equation T T − T T AK X∞ AK − X∞ + γ −2 AK X∞ BKU1 1BK X∞ AK + C KC K + Q + K T RK = (10) 164 New Trends in Technologies T such that U = I − γ −2 BK X∞ BK > Proof: Consider the reference system (9) under the influence of the state feedback (3), and define T0 as BK ⎤ ⎡ AK ⎢ ⎥ T0 ( z) := ⎢ ⎡ I ⎤ Ω ⎢ ⎥ 0⎥ ⎢ ⎥ ⎣K ⎦ ⎣ ⎦ ˆ then the closed-loop transfer matrix from disturbance input w to the controlled output z is ⎡Tzw ( z)⎤ ~ Tzw ( z) = ⎢ ˆ ˆ ˆ ⎥ Note that γ I − TzwTzw > is equivalent to ⎣ T0 ( z) ⎦ ~ γ I − TzwTzw > T0~T0 > for all w ∈ L2 [0, ∞ ) , and Tzw ( z) ∈ RH ∞ is equivalent to Tzw ( z) ∈ RH ∞ , so Tzw ( z) ∞ < γ implies Tzw ( z) ∞ < γ ˆ ˆ Hence, it follows from Lemma 2.1 Q.E.D To prove the result of non-fragile discrete time state feedback mixed LQR/ H ∞ control problem, we define the inequality T T − T T ˆT ˆ AF X∞ AF − X∞ + γ −2 AF X∞ BF U1 1BF X∞ AF + C F C F + Q + F∞ RF∞ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ (11) T where, U = I − γ −2 BF X∞ BF > ˆ ˆ ∞ ∞ In terms of the inequality (11), we have the following lemma: Lemma 2.3 Consider the system (2) under the influence of state feedback (4) with controler uncertainty, and suppose that γ > is a given number, then there exists an admissible nonfragile controller F∞ such that Tzw ∞ < γ if for any admissible uncertainty ΔF( k ) , there exists a stabilizing solution X∞ ≥ to the inequality (11) such that U = T I − γ −2 BF X∞ BF > ˆ ˆ ∞ ∞ Proof: Suppose that for any admissible uncertainty ΔF( k ) , there exists a stabilizing solution T X∞ ≥ to the inequality (11) such that U = I − γ −2 BF X∞ BF > This implies that the ˆ ˆ ∞ ∞ −2 −1 T solution X∞ ≥ is such that AF + γ BF U1 BF X∞ AF is stable ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ Let AF∞ = A + B2 F∞ and C F∞ = C + D12 F∞ ; then we can rewrite (11) as T T − T T AF∞ X∞ AF∞ − X∞ + γ −2 AF∞ X∞ BF U1 1BF X∞ AF∞ + C F∞ C F∞ + Q ˆ ˆ ∞ ∞ T T − T + F∞ RF∞ − ( ATU B2 + F∞ U )U ( B2 U A + U F∞ ) + ΔN F < − T T where, U = B2 U B2 + I + R , U = γ −2 X∞ BF U 1BF X∞ + X∞ , ˆ ˆ ∞ ∞ T T T −1 T ΔN F = ( A U B2 + F∞ U + ΔF ( k )U )U ( B2 U A + U F∞ + U ΔF( k )) Since ΔF( k ) is an admissible norm-bounded time- varying uncertainty, there exists a timevarying uncertain number δ ( k ) > satisfying T T − T T T AF∞ X∞ AF∞ − X∞ + γ −2 AF∞ X∞ BF U1 1BF X∞ AF∞ + C F∞ C F∞ + Q + F∞ RF∞ ˆ ˆ ∞ ∞ − T T − ( ATU B2 + F∞ U )U ( B2 U A + U F∞ ) + ΔN F + δ ( k )I = (12) 165 Discrete Time Mixed LQR/H∞ Control Problems − T Note that AF + γ −2 BF U 1BF X∞ AF is stable for any admissible uncertainty ΔF( k ) This ˆ ˆ ˆ ˆ ∞ ∞ −2 ∞ − ∞T implies that AF∞ + γ BF U1 1BF X∞ AF∞ is stable ˆ ˆ ∞ ∞ − T Hence, (U1 1BF X∞ AF∞ , AF∞ ) is detectable Then it follows from standard results on ˆ ∞ Lyapunov equations (see Lemma 2.7 a), Iglesias & Glover 1991) and the equation (12) that AF∞ is stable Thus, AF = AF∞ + B2 ΔF( k ) is stable for any admissible uncertainty ΔF( k ) ˆ ∞ Define V ( x( k )) := xT ( k )X∞ x( k ) , where, x is the solution to the plant equations for a given input w , then it can be easily established that ∞ = ∑ { −ΔV ( x( k )) + xT ( k + 1)X∞ x( k + 1) − xT ( k )X∞ x( k )} k =0 ∞ − T = ∑ { −ΔV ( x( k )) − z + γ w − γ U ( w − γ −2U1 1BF X∞ AF x ) ˆ ˆ 2 ∞ k =0 T ∞ T T T − T + x ( AF X∞ AF − X∞ + γ −2 AF X∞ BF U1 1BF X∞ AF + C F C F )x} ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ Add the above zero equality to J to get ∞ − T J = ∑ { −ΔV ( x( k )) − z + γ w − γ U ( w − γ −2U1 1BF X∞ AF x ) ˆ ˆ 2 ∞ k =0 T ∞ T T T − T ˆT ˆ + x ( AF X∞ AF − X∞ + γ −2 AF X∞ BF U1 1BF X∞ AF + C F C F + Q + F∞ RF∞ )x} ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ Substituting (11) for the above formula,we get that for any u( k ) and w( k ) and x(0) = , 2 − T J < − z + γ w − γ U ( w − γ −2U1 1BF X∞ AF x ) ˆ ˆ ∞ Note that z0 2 ∞ 2 ∞ − T ˆ ˆ = ∑ xT ( k )Ωx( k ) , and define that r := w − γ −2U 1BF X∞ AF x , we get ˆ ˆ ∞ k =0 ˆ2 z −γ w 2 < −γ U1 r ∞ 2 Suppose that Γ is the operator with realization ˆ x( k + 1) = ( A + B2 F∞ )x( k ) + BF w( k ) ˆ ∞ − T r ( k ) = −γ −2U1 1BF X∞ AF x( k ) + w( k ) ˆ ˆ ∞ ∞ which maps w to r − T ˆ Since Γ −1 exists ( and is given by x( k + 1) = ( A + B2 F∞ + γ −2 BF U1 1BF X∞ AF )x( k ) + BF r ( k ) , ˆ ˆ ˆ ˆ ∞ w( k ) = γ −2 − T U1 1BF X∞ AF x( k ) + r ( k ) ˆ ˆ ∞ ∞ ∞ ∞ ), we can write ˆ2 z − γ w < −γ U1 r 2 = −γ Γw ≤ κ w 2 ∞ 166 New Trends in Technologies for some positive κ This implies that there exists an admissible non-fragile controller such ~ that Tzw ∞ < γ Note that γ I − TzwTzw > is equivalent to ˆ ˆ ˆ ~ γ I − TzwTzw > T0~T0 > for all w ∈ L2 [0, ∞ ) so Tzw ˆ ∞ < γ implies Tzw controller such that Tzw ∞ ∞ < γ , and we conclude that there exists an admissible non-fragile < γ Q E D State Feedback In this section, we will consider the discrete time state feedback mixed LQR/ H ∞ control problem This problem is defined as follows: Given the linear discrete-time systems (2)(3) with w ∈ L2 [0,∞ ) and x(0) = x0 and the quadratic performance index (1), for a given number γ > 0, determine an admissible controller K that achieves sup inf{ J } subject to Tzw ( z) ∞ < γ w∈L2 + K If this controller K exists, it is said to be a discrete time state feedback mixed LQR/ H ∞ controller Here, we will discuss the simplified versions of the problem defined in the above In order to this, the following assumptions are imposed on the system Assumption (C , A) is detectable Assumption ( A, B2 ) is stabilizable T Assumption D12 [C D12 ] = [ I ] The solution to the problem defined in the above involves the discrete time Riccati equation T ˆ ˆ ˆ ˆ ˆ AT X∞ A − X∞ − AT X∞ B( BT X∞ B + R )−1 BT X∞ A + C C + Q = (13) ⎤ ˆ ⎡−I B2 ⎤ , R = ⎢ If A is invertible, the stabilizing solution to the ⎦ R + I⎥ ⎣ ⎦ discrete time Riccati equation (13) can be obtained through the following simplectic matrix ˆ where, B = ⎡γ −1B1 ⎣ T ˆˆ ˆ ˆˆ ˆ ⎡ A + BR −1BT A−T (C C + Q ) −BR −1BT A−T ⎤ S∞ := ⎢ ⎥ −T T − A (C C + Q ) A −T ⎢ ⎥ ⎣ ⎦ In the following theorem, we provide the solution to discrete time state feedback mixed LQR/ H ∞ control problem Theorem 3.1 There exists a state feedback mixed LQR/ H ∞ controller if the discrete time T Riccati equation (13) has a stabilizing solution X∞ ≥ and U = I − γ −2 B1 X∞ B1 > Moreover, this state feedback mixed LQR/ H ∞ controller is given by − T K = −U 1B2 U A T − T where, U = R + I + B2 U 3B2 , and U = X∞ + γ −2 X∞ B1U1 1B1 X∞ 167 Discrete Time Mixed LQR/H∞ Control Problems In this case, the state feedback mixed LQR/ H ∞ controller will achieve T sup inf{ J } = x0 ( X∞ + γ −2 X w − X z )x0 subject to Tzw w∈L2 + K − T ˆ AK = AK + γ −2 BKU1 1BK X∞ AK , where, ∞ , and let AK = A + B2 K and K = −U 1BTU A ; then AK is stable Proof: Suppose that the discrete time Riccati equation (13) has a stabilizing solution X∞ ≥ T and U = I − γ −2 B1 X∞ B1 > Observe that T ⎡γ −1B1 ⎤ ˆ ˆ ˆ BT X∞ B + R = ⎢ ⎥ X∞ ⎡γ −1B1 ⎣ T ⎢ B2 ⎥ ⎣ ⎦ ⎡−I B2 ⎤ + ⎢ ⎦ ⎣0 −U1 ⎤ ⎡ =⎢ T R + I ⎥ ⎢γ −1B2 X∞ B1 ⎦ ⎣ T γ −1B1 X∞ B2 ⎤ ⎥ T B2 X∞ B2 + R + I ⎥ ⎦ T − T Also, note that U = I − γ −2 B1 X∞ B1 > , U = X∞ + γ −2 X∞ B1U1 1B1 X∞ , and U = R + I T + B2 U B2 ; then it can be easily shown by using the similar standard matrix manipulations as in the proof of Theorem 3.1 in Souza & Xie (1992) that − ˆ − ˆT − − ˆ − ⎡ −U −1 + U1 1B1U 1B1 U 1 U 1B1U ⎤ ˆ ˆ ˆ ( BT X∞ B + R )−1 = ⎢ ⎥ − ˆT − − U 1B1 U 1 U2 ⎥ ⎢ ⎣ ⎦ T ˆ where, B1 = γ −1B1 X∞ B2 Thus, we have − T − T ˆ ˆ ˆ ˆ ˆ AT X∞ B( BT X∞ B + R )−1 BT X∞ A = −γ −2 AT X∞ B1U1 1B1 X∞ A + ATU B2U 1B2 U A Rearraging the discrete time Riccati equation (13), we get − T − T T X∞ = AT X∞ A + γ −2 AT X∞ B1U 1B1 X∞ A − ATU B2U 1B2 U A + C C + Q T − T − T − T = AT X∞ A + γ −2 AT X∞ B1U 1B1 X∞ A + C C + Q − ATU B2U 1B2 ( X∞ + γ −2 X∞ B1U 1B1 X∞ ) A − T − T − AT ( X∞ + γ −2 X∞ B1U 1B1 X∞ )B2U 1B2 U A T − T − − T + ATU B2U 1[ R + I + B2 ( X∞ + γ −2 X∞ B1U1 1B1 X∞ )B2 ]U 1B2 U A − T − T − T − T = ( AT X∞ A − ATU B2U 1B2 X∞ A − AT X∞ B2U 1B2 U A + ATU B2U 1B2 X∞ B2U 1B2 U A) T − − T − − T + (C C + ATU B2U 1U 1B2 U A) + ATU B2U RU 1B2 U A + Q − T − T − T + (γ −2 AT X∞ B1U 1B1 X∞ A − γ −2 ATU B2U 1B2 X∞ B1U1 1B1 X∞ A − T − T − T − T − T − γ −2 AT X∞ B1U 1B1 X∞ B2U 1B2 U A + γ −2 ATU B2U 1B2 X∞ B1U 1B1 X∞ B2U 1B2 U A) − − T − T − T = ( A − B2U 1BTU A)T X∞ ( A − B2U 1B2 U A) + (C − D12U 1B2 U A)T (C − D12U 1B2 U A) − T − T − T + K T RK + Q + γ −2 ( A − B2U 1B2 U A)T X∞ B1U1 1B1 X∞ ( A − B2U 1B2 U A) 168 New Trends in Technologies that is, T T − T T AK X∞ AK − X∞ + γ −2 AK X∞ BKU1 1BK X∞ AK + C KC K + Q + K T RK = (14) Since the discrete time Riccati equation (13) has a stabilizing solution X∞ ≥ , the discrete time Riccati equation (14) also has a stabilizing solution X∞ ≥ This implies that − T − T ˆ AK = AK + γ −2 BKU1 1BK X∞ AK is stable Hence (U1 1BK X∞ AK , AK ) is detectable Based on this, it follows from standard results on Lyapunov equations (see Lemma 2.7 a), Iglesias & Glover 1991) that AK is stable.Q E D Proof of Theorem 3.1: Suppose that the discrete time Riccati equation (13) has a stabilizing T solution X∞ ≥ and U = I − γ −2 B1 X∞ B1 > Then, it follows from Lemma 3.1 that AK is stable This implies that Tzw ( z) ∈ RH ∞ By using the same standard matrix manipulations as in the proof of Lemma 3.1, we can rewrite the discrete time Riccati equation (13) as follows: − T − T T AT X∞ A − X∞ + γ −2 AT X∞ B1U 1B1 X∞ A − ATU B2U 1B2 U A + C C + Q = or equivalently, T T − T T AK X∞ AK − X∞ + γ −2 AK X∞ BKU1 1BK X∞ AK + C KC K + Q + K T RK = Thus, it follows from Lemma 2.2 that Tzw ( z) ∞ < γ Define V ( x( k )) = xT ( k )X∞ x( k ) , where X∞ is the solution to the discrete time Riccati equation (13), then taking the difference ΔV ( x( k )) and completing the squares we get ΔV ( x( k )) = xT ( k + 1)X∞ x( k + 1) − xT ( k )X∞ x( k ) T T = xT ( k )( AK X∞ AK − X∞ )x( k ) + xT ( k ) AK X∞ BK w( k ) T T + wT ( k )BK X∞ AK x( k ) + wT ( k )BK X∞ BK w( k ) 2 − T = − z + γ w − γ U1 ( w − γ −2U1 1BK X∞ AK x ) − T T T T + xT ( AK X∞ AK − X∞ + γ −2 AK X∞ BKU 1BK X∞ AK + C KC K )x Based on the above, the cost function J can be rewritten as: ∞ ∞ k =0 k =0 T − T ˆ ˆ J = ∑ xT ( k )Ωx( k ) = ∑ { −ΔV ( x( k )) − z + γ w − γ U1 ( w − γ −2U 1BK X∞ AK x ) +x T ( AK X∞ AK − X∞ + γ −2 − T T AK X∞ BKU1 1BK X∞ AK T + C KC K (15) T + Q + K RK )x} On the other hand, it follows from the similar argumrnts as in the proof of Theorem 3.1 in Furuta & Phoojaruenchanachai (1990) that T T − T T AK X∞ AK − X∞ + γ −2 AK X∞ BKU 1BK X∞ AK + C KC K + Q + K T RK − T − T T = AT X∞ A − X∞ + γ −2 AT X∞ B1U 1B1 X∞ A − ATU B2U 1B2 U A + C C + Q − T − T + (K + U 1B2 U A)T U (K + U 1B2 U A) At the same time note that ... 0. 865 6 ⎢ Ao = ⎢ 0.5038 ⎢ ⎣ 0.13009 0.34529 0.12048 0.288 16 0. 463 71 0. 764 43 0 .61 187 0.14297 0 .67 152 0.9712 0.4 765 7 0.15809 0.19192 0.011 36 0.93839 0.54837 0. 466 39 ⎤ 0.019139 0.3 865 1 ⎥ ⎥ 0.422 46. .. solvers 154 Discrete Time Systems Example Consider the unstable system as follows ⎡ 0.82 0.05 76 0.2212 ⎢ 0.0574 0. 063 4 0 .62 54 ⎢ Ao = ⎢ 0.0901 0.7228 0.5133 ⎢ ⎣ 0 .69 67 0.0337 0.5757 0.1471 0 .69 57 0.2872... 0. 265 9 ⎢ ⎣ 0. 061 1 0.3328 Co = 0.8927 0.09 26 0.2925 0.8219 0.994 ⎤ 0. 067 8 0.9731 ⎥ ⎥ 0.9228 ⎥ ⎥ 0.9587 ⎦ 0. 563 2 ⎤ 0.2924 0.4025 ⎥ ⎥ 0.0341 ⎥ ⎥ 0.2875 ⎦ 0.21 96 0. 565 9 0.255 0.5227 0.0038 0. 360 8