14 Mixed H2∕H∞ Control with Actuator Failures: the Finite-Horizon Case This chapter deals with the fault-tolerant control problem for a class of nonlinear stochastic time-varying systems with actuator failures. Both H2 and H∞ performance requirements are taken into consideration. The proposed actuator failure model is quite general and could cover several frequently seen actuator failure phenomena as special cases. The stochastic nonlinearities are quite general and could represent several types of nonlinear systems. It is the purpose of this chapter to find equilibrium strategies of a two-player Nash game, and meanwhile both H2 and H∞ performances are achieved via a proposed state feedback control scheme, which is characterized by the solution to a set of coupled matrix equations. The feedback gains can be solved recursively backward in k. A numerical computing algorithm is presented and then a simulation example is given to illustrate the effectiveness and applicability of the proposed algorithm. The rest of the chapter is arranged as follows: Section 14.1 formulates the mixed H2 ∕H∞ control problem for the nonlinear stochastic time-varying systems with actuator failures. In Section 14.2, the H∞ performance is analyzed in terms of the Riccati equation method. Section 14.3 gives the methodology to solve the addressed multi-objective problem and outlines the computational algorithm to recursively obtain the required parameters. A numerical example is presented in Section 14.4 to show the effectiveness and applicability of the proposed algorithm. Section 14.5 draws the summary. Variance-Constrained Multi-Objective Stochastic Control and Filtering, First Edition. Lifeng Ma, Zidong Wang and Yuming Bo. © 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd. 272 Variance-Constrained Multi-Objective Stochastic Control and Filtering 14.1 Problem Formulation Consider the following nonlinear stochastic time-varying system defined on k ∈ [0, N − 1]: ⎧ x(k + 1) = A(k)x(k) + (B(k)u(k)) + D(k)(k) ⎪ + f (x(k), u(k), k), (14.1) ⎨ ⎪ y(k) = C(k)x(k) + E(k)u(k) + g(x(k), u(k), k) ⎩ where x(k), y(k), and u(k) stand for the system state, control output, and control input respectively, (k) represents the external disturbance belonging to l2 , and A(k), B(k), C(k), D(k), and E(k) are known time-varying matrices with appropriate dimensions. For notation simplicity, we denote f (k) ≜ f (x(k), u(k), k), g(k) ≜ g(x(k), u(k), k), (k) ≜ [f T (k) gT (k)]T . For all x(k) and u(k), (k) is assumed to satisfy {(k)|x(k)} = 0, (14.2) {(k)T (j)|x(k)} = 0, k ≠ j, ] [ q 11 12 ∑ (k) Π (k) Π i i {(k)T (k)|x(k)} = 12 22 Π (k) Π (k) i=0 i i { T × x (k)Γi (k)x(k) } +uT (k)Ξi (k)u(k) , (14.3) (14.4) where Π11 (k) ≥ 0, Π22 (k) ≥ 0, Γi (k) ≥ 0, Ξi (k) ≥ 0, and Π12 (k) (i = 1, 2, … , q) are i i i known matrices with compatible dimensions. The nonlinear function (⋅) ∶ ℝm → ℝm is defined as follows: (B(k)u(k)) = R(k)B(k)u(k), (14.5) where R(k) = diag{r1 (k), r2 (k), … , rm (k)}, 0 ≤ ril (k) ≤ ri (k) ≤ riu (k) < ∞, (14.6) with ril (k) ≤ 1 and riu (k) ≥ 1 being lower and upper bounds on ri (k). Remark 14.1 The time-varying equation (14.5) is used to interpret the occurrence of incomplete information phenomenon due to the possible actuator failures. In such a model, when ri (k) = 1, it means that the actuator is in good condition; otherwise Mixed H2 ∕H∞ Control with Actuator Failures: the Finite-Horizon Case 273 there might be partial or complete actuator failure, which leads to incomplete information. The model proposed above could explain several types of failures occurring in engineering practice, including output quantization studied in Ref. [89] and sensor saturation studied in Ref. [132]. Let R(k) = H(k) + H(k)N(k), where H(k) = diag{h1 (k), h2 (k), … , hm (k)}, ril (k) + riu (k) , 2 N(k) = diag{n1 (k), n2 (k), … , nm (k)}, hi (k) = ni (k) = (14.7) ri (k) − hi (k) . hi (k) Denoting L(k) = diag{l1 (k), l2 (k), … , lm (k)} with li (k) = (riu (k) − ril (k))∕(riu (k) + ril (k)), we can easily obtain N T (k)N(k) ≤ LT (k)L(k) ≤ I. (14.8) So far, we have successfully converted the influence caused by the actuator failures on the system output sampling into certain norm-bounded parameter uncertainties. Then the original system (14.1) can be reformulated as follows: ̄ + D(k)(k) ⎧ x(k + 1) = A(k)x(k) + B(k)u(k) ⎪ + H(k)N(k)B(k)u(k) + f (k), ⎨ ⎪ y(k) = C(k)x(k) + E(k)u(k) + g(k), ⎩ (14.9) ̄ where B(k) = H(k)B(k). In the following, we give the design objective of this chapter. Firstly, we consider the following H∞ performance index for system (14.9): {N−1 } ∑ J∞ = (‖y(k)‖2 − 2 ‖(k)‖2 ) − 2 ‖x(0)‖2W , (14.10) k=0 where W > 0 is a known weighting matrix. In this chapter, it is our objective to design a state feedback controller of the form u(k) = K(k)x(k) such that: (R1) For a pre-specified H∞ disturbance attenuation level > 0, J∞ ≤ 0 holds for all nonzero (k) and x(0). (R2) When there exists a worst-case disturbance such that the H∞ disturbance attenuation ∑N−1 level is 2 maximized, the controlled output energy defined by J2 = k=0 {‖y(k)‖ } is minimized. 274 Variance-Constrained Multi-Objective Stochastic Control and Filtering Noticing that there exist parameter uncertainties in (14.9), we shall further reconstruct the system so as to obtain a system without N(k) as follows: { ̄ ̄ x(k + 1) = A(k)x(k) + B(k)u(k) + D(k)(k) + f (k), (14.11) y(k) = C(k)x(k) + E(k)u(k) + g(k), where [ ] (k) , −1 (k)N(k)B(k)u(k) ] [ ̄ D(k) = D(k) −1 (k)H(k) , (k) = with (k) ≠ 0 being any known real-valued constant. We now consider the following performance index: {N−1 ∑( J̄ ∞ = ‖y(k)‖2 + ‖(k)L(k)B(k)u(k)‖2 k=0 ) − 2 ‖(k)‖2 ) } (14.12) − 2 ‖x(0)‖2W . Next, a lemma is given to reveal the relationship between J∞ and J̄ ∞ . Lemma 14.1.1 Given a positive scalar . The performance indices J∞ and J̄ ∞ are defined by (14.10) and (14.12) respectively; then J∞ ≤ 0 is implied by J̄ ∞ ≤ 0. Proof. If J̄ ∞ ≤ 0 holds, it is found that ∑ N−1 {‖y(k)‖2 + ‖(k)L(k)B(k)u(k)‖2 } k=0 ≤ 2 (N−1 ∑ ) {‖(k)‖2 } + ‖x(0)‖2W (14.13) . k=0 Expanding both sides of the inequality shown above, we have ∑ N−1 2 ̄ {‖y(k)‖2 + ‖(k)L(k)B(k)u(k)‖ } k=0 ≤ 2 (N−1 ∑ ) {‖(k)‖2 } + ‖x(0)‖2W k=0 ̄ where L(k) = √ LT (k)L(k) − N T (k)N(k). (14.14) , Mixed H2 ∕H∞ Control with Actuator Failures: the Finite-Horizon Case This indicates that ∑ N−1 {‖y(k)‖2 } ≤ k=0 (N−1 ∑ 275 ) {‖(k)‖2 } + ‖x(0)‖2W . (14.15) k=0 It follows directly that J∞ ≤ 0. Moreover, it is worth pointing out that N(k) can be any arbitrary matrix corresponding to any arbitrary actuator failure. When there is no actuator failure occurring, that is, ril (k) = ri (k) = riu (k) = 1, then we have N(k) = L(k) = 0, and therefore J∞ = J̄ ∞ . As a result, we can see that J̄ ∞ is a tight upper bound of J∞ . The proof is complete. ◽ Based on the above discussions, the purpose of this chapter could be restated as follows. In this chapter, it is our objective to design a state feedback controller of the form u(k) = K(k)x(k) such that: (G1) For a pre-specified H∞ disturbance attenuation level > 0, J̄ ∞ ≤ 0 holds for all nonzero (k) and x(0). (G2) When there exists a worst-case disturbance ∗ (k) such that the H∞ disturbance ∑N−1attenuation level is maximized, the output energy defined by J2 = k=0 {‖y(k)‖2 } is minimized. In other words, we aim at finding equilibrium strategies u∗ (k) = K(k)x(k) that satisfy the Nash equilibria defined by J̄ ∞ (u∗ (k), (k)) ≤ J̄ ∞ (u∗ (k), ∗ (k)) ≤ 0, (14.16) J2 (u(k), ∗ (k)) ≥ J2 (u∗ (k), ∗ (k)). (14.17) Now we are in a situation to deal with the mixed H2 ∕H∞ control problem for the nonlinear stochastic system with possible actuator failures. 14.2 H∞ Performance In this section, we shall firstly analyze the system disturbance attenuation level. A theorem that plays a vital role in the controller design stage will be presented to give a necessary and sufficient condition guaranteeing the pre-specified H∞ specification. To this end, we obtain the unforced system by setting u(k) = 0 in system (14.11) as follows: { ̄ x(k + 1) = A(k)x(k) + D(k)(k) + f (k), (14.18) y(k) = C(k)x(k) + g(k). We now consider the following backward recursion: P(k) = AT (k)P(k + 1)A(k) + CT (k)C(k) −1 ̄ ̄ T (k)P(k + 1)A(k) (k)D + AT (k)P(k + 1)D(k)Φ + q ∑ i=0 22 Γi (k)(tr[P(k + 1)Π11 i (k)] + tr[Πi (k)]), (14.19) Variance-Constrained Multi-Objective Stochastic Control and Filtering 276 where ̄ T (k)P(k + 1)D(k). ̄ Φ(k) = 2 I − D (14.20) The following theorem gives a necessary and sufficient condition for system (14.18) capable of satisfying the pre-specified H∞ requirement. Theorem 14.2.1 Given a positive scalar > 0 and a positive definite matrix W. The pre-specified H∞ disturbance attenuation level defined in (G1) can be achieved for all nonzero x(0) and (k) if and only if, with the final condition P(N) = 0, there exist solutions P(k) (0 ≤ k < N) to (14.19) such that Φ(k) > 0 and P(0) ≤ 2 W. Proof Sufficiency By defining Jk = xT (k + 1)P(k + 1)x(k + 1) − xT (k)P(k)x(k), we have (14.21) {( )T ̄ A(k)x(k) + D(k)(k) + f (k) P(k + 1) ( )} ̄ × A(k)x(k) + D(k)(k) + f (k) − xT (k)P(k)x(k) ( {Jk } = = xT (k) AT (k)P(k + 1)A(k) − P(k) ∑ q + ) (14.22) ( ) x(k) Γi (k) tr[P(k + 1)Π11 i (k)] i=0 ̄ + 2xT (k)AT (k)P(k + 1)D(k)(k) ̄ T (k)P(k + 1)D(k)(k). ̄ + T (k)D Adding the following zero term ‖y(k)‖2 − 2 ‖(k)‖2 − (‖y(k)‖2 − 2 ‖(k)‖2 ) (14.23) to both sides of (14.22) and then taking the mathematical expectation results in {Jk } ( = xT (k) AT (k)P(k + 1)A(k) − P(k) + CT (k)C(k) + q ∑ i=0 T ( ) 22 Γi (k) tr[P(k + 1)Π11 i (k)] + tr[Πi (k)] x(k) ̄ + 2x (k)A (k)P(k + 1)D(k)(k) ( ) ̄ T (k)P(k + 1)D(k) ̄ (k) + T (k) − 2 I + D {( )} 2 2 2 − ‖y(k)‖ − ‖(k)‖ . T (14.24) Mixed H2 ∕H∞ Control with Actuator Failures: the Finite-Horizon Case 277 Completing the squares of (k), we have {Jk } = − ((k) − ∗ (k))T Φ(k)((k) − ∗ (k)) − {‖y(k)‖2 − 2 ‖(k)‖2 }, where ̄ T (k)P(k + 1)A(k)x(k). ∗ (k) = Φ−1 (k)D (14.25) (14.26) Now, summing (14.25) from 0 to N − 1 with respect to k leads to {N−1 } ∑ Jk = {xT (N)P(N)x(N)} − xT (0)P(0)x(0) k=0 ∑( N−1 =− ((k) − ∗ (k))T Φ(k) ((k) − ∗ (k)) (14.27) k=0 ) +{‖y(k)‖2 − 2 ‖(k)‖2 } . Noticing P(N) = 0, we can find from (14.27) that ∑( N−1 ) {‖y(k)‖2 − 2 ‖(k)‖2 } − 2 ‖x(0)‖2W k=0 T = x (0)(P(0) − 2 W)x(0) ∑ (14.28) N−1 − ((k) − ∗ (k))T Φ(k)((k) − ∗ (k)). k=0 Therefore, since Φ(k) > 0 and P(0) ≤ 2 W, we arrive at J̄ ∞ ≤ 0, (14.29) sup J∞ = sup J̄ ∞ = 0. {x(0),(k)} {x(0),(k)} (14.30) which implies J∞ ≤ 0. Moreover, Necessity See the proof of Ref. [66]. The proof is now complete. ◽ In this section, we have analyzed the H∞ performance of the unforced system and obtained an important necessary and sufficient condition that will play a key role in the controller design procedure. This will be discussed in the following section. 14.3 Multi-Objective Controller Design In this section, we aim to find the desired Nash equilibrium strategies with the prescribed H∞ constraint. We shall present a theorem to give a necessary and sufficient condition of the existence of such a control scheme. Then a computational algorithm will be proposed to obtain the numerical values of the feedback gain at each sampling instant k. Variance-Constrained Multi-Objective Stochastic Control and Filtering 278 14.3.1 Controller Design Firstly, for notational convenience, we denote ̃ ̄ A(k) = A(k) + B(k)K(k), ̄ C(k) = C(k) + E(k)K(k). (14.31) Implementing u(k) = K(k)x(k) to system (14.11), we obtain the following closed-loop nonlinear stochastic system: { ̃ ̄ x(k + 1) = A(k)x(k) + D(k)(k) + f (k), (14.32) ̄ y(k) = C(k)x(k) + g(k). The following theorem gives a necessary and sufficient condition for the existence of the desired multi-objective controller in terms of certain coupled matrix equations. Theorem 14.3.1 Given an H∞ performance index > 0. With the final condition P(N) = 0 and Q(N) = 0, there exists a state feedback controller u(k) = K(k)x(k) for system (14.1) such that both the design requirements (G1) and (G2) can be satisfied simultaneously, if and only if there exist solutions P(k) and Q(k) to the following coupled matrix equations: q ∑ ( ̃ + C̄ T (k)C(k) ̄ + P(k) = Ã T (k)P(k + 1)A(k) ( (Γi (k) + K T (k)Ξi (k)K(k)) i=0 )) + 2 (k)K T (k)BT (k)LT (k)L(k)B(k)K(k) × tr[P(k + ) ( T ̃ ̃ ̄ (k)P(k + 1)A(k) Φ (k) D (k)P(k + 1)A(k) , (14.33) + D 1)Π11 i (k)] + tr[Π22 i (k)] )T −1 ( T ̄ P(0) ≤ 2 W, (14.34) ̄ + CT (k)C(k) Q(k) = Ā T (k)Q(k + 1)A(k) ∑ q ( ) 22 Γi (k) tr[P(k + 1)Π11 i (k)] + tr[Πi (k)] i=0 ( ) ( ̄ + ET (k)C(k) T Ψ−1 (k) B̄ T (k)Q(k + 1)A(k) ̄ − B̄ T (k)Q(k + 1)A(k) ) + ET (k)C(k) , ̄ T (k)P(k + 1)D(k) ̄ > 0, Φ(k) = 2 I − D (14.35) (14.36) ̄ + E (k)E(k) Ψ(k) = B (k)Q(k + 1)B(k) ̄T + q ∑ T ( ) 22 Ξi (k) tr[P(k + 1)Π11 i (k)] + tr[Πi (k)] > 0, (14.37) i=0 ( ) ̄ + ET (k)C(k) , K(k) = −Ψ−1 (k) B̄ T (k)Q(k + 1)A(k) (14.38) ̄ T (k)P(k + 1)A(k), ̃ T(k) = Φ−1 (k)D (14.39) where ̄ ̄ A(k) = A(k) + D(k)T(k). (14.40) Mixed H2 ∕H∞ Control with Actuator Failures: the Finite-Horizon Case 279 Proof Sufficiency Firstly, according to Theorem 14.2.1, if there exist solutions P(k) to (14.33) such that Φ(k) > 0, system (14.11) will meet the pre-specified H∞ requirement and the worst-case disturbance can be obtained as ∗ (k) = T(k)x(k). Now we are in a situation to find the feedback controller gain K(k) such that the output energy can be minimized with the worst-case disturbance. To this end, when the worst-case disturbance happens, by applying ∗ (k) = T(k)x(k), system (14.11) can be rewritten as { ̄ ̄ x(k + 1) = (A(k) + D(k)T(k))x(k) + B(k)u(k) + f (k), (14.41) y(k) = C(k)x(k) + E(k)u(k) + g(k). By defining the following quadratic index J2k = xT (k + 1)Q(k + 1)x(k + 1) − xT (k)Q(k)x(k), we could have (14.42) {( )T ̄ ̄ A(k)x(k) + B(k)u(k) + f (k)) Q(k + 1) ( )} ̄ ̄ × A(k)x(k) + B(k)u(k) + f (k) {J2k } = − xT (k)Q(k)x(k) ( ̄ − Q(k) = xT (k) Ā T (k)Q(k + 1)A(k) + q ∑ i=0 ) (14.43) Γi (k)tr[P(k + 1)Π11 i (k)] x(k) ( ̄ + u (k) B̄ T (k)Q(k + 1)B(k) T + q ∑ ( Ξi (k) tr[P(k + ) ) 1)Π11 i (k)] u(k) i=0 ̄ + 2xT (k)Ā T (k)Q(k + 1)B(k)u(k). It then follows that {J2k + ‖y(k)‖2 − ‖y(k)‖2 } { ( = + xT (k) AT (k)Q(k + 1)A(k) − Q(k) + CT (k)C(k) q ∑ i=0 ( Γi (k) tr[P(k + 1)Π11 i (k)] + ) tr[Π22 i (k)] ) x(k) Variance-Constrained Multi-Objective Stochastic Control and Filtering 280 ̄ + CT (k)E(k))u(k) + 2xT (k)(AT (k)Q(k + 1)B(k) ( ̄ + ET (k)E(k) + uT (k) B̄ T (k)Q(k + 1)B(k) + q ∑ ( Ξi (k) tr[P(k + i=0 1)Π11 i (k)] + ) tr[Π22 i (k)] ) u(k) } − ‖y(k)‖ 2 . (14.44) Completing the squares of u(k), we obtain {J2k } = {(u(k) − u∗ (k))T Ψ(k)(u(k) − u∗ (k)) − ‖y(k)‖2 }, (14.45) where Ψ(k) is defined in (14.37) and u∗ (k) = K(k)x(k) with K(k) being defined in (14.38). Moreover, we have } {N−1 ∑ 2 J2 = ‖y(k)‖ k=0 = {N−1 ∑ } (u(k) − u∗ (k))T Ψ(k)(u(k) − u∗ (k)) (14.46) k=0 T + x (0)Q(0)x(0) − xT (N)Q(N)x(N). Noting Q(N) = 0, we can see that, when u(k) = u∗ (k), the output energy J2 is minimized at J2min = xT (0)Q(0)x(0), which means that the desired Nash game strategies expressed by (14.16) and (14.17) are achieved, and the design requirements (G1) and (G2) are satisfied simultaneously. Necessity If u∗ (k) = K(k)x(k) and ∗ (k) = T(k)x(k) are the desired equilibrium strategies satisfying J̄ ∞ (u∗ (k), (k)) ≤ J̄ ∞ (u∗ (k), ∗ (k)), J2 (u(k), ∗ (k)) ≥ J2 (u∗ (k), ∗ (k)), (14.47) and by implementing u∗ (k) = K(k)x(k) to system (14.11), the closed-loop system (14.32) can be obtained. Therefore, according to Theorem 14.2.1, the matrix equation (14.33) admits solutions P(k) > 0 (0 ≤ k < N). Furthermore, the worst-case disturbance, if it exists, can be represented by ∗ (k) = T(k)x(k). Substituting ∗ (k) = T(k)x(k) to system (14.11), we have system (14.41). Then the existence of the strategy u∗ (k) = K(k)x(k) indicates the existence of K(k) over the finite horizon Mixed H2 ∕H∞ Control with Actuator Failures: the Finite-Horizon Case 281 [0, N − 1]. Therefore, we can see from (14.38) that Ψ(k) should be nonsingular for all 0 ≤ k < N. Observing equation (14.35), given the final condition Q(N) = 0, we could learn that Q(k) can always be obtained by solving (14.35) with nonsingular Ψ(k) and backward in time. The proof is now complete. ◽ Remark 14.2 The proposed theorem gives a necessary and sufficient condition for the existence of the required multi-objective controller. By means of the Nash game approach, we have successfully converted the controller existence problem into the feasibility of certain coupled matrix equations. We should point out that the actuator failure model considered in this chapter is quite general and therefore can be applied in many branches of reliable control and signal processing problems, such as control with actuator saturation, filtering with missing measurements, and so on. 14.3.2 Computational Algorithm As mentioned above, the state feedback controller gains K(k) and T(k) can be obtained by solving the presented coupled matrix equations. In the next stage, it is our aim to propose an algorithm to get the numerical values of these desired parameters at each time point k recursively. Mixed H2 ∕H∞ Reliable Controller Design Algorithm Step 1. Set the finite time N and k = N − 1. Set the pre-specified H∞ index > 0. Set P(N) = Q(N) = 0. Select the positive definite matrices W > 0 and (k) > 0 properly. Step 2. With all the pre-set parameters and the available P(k + 1) and Q(k + 1), solve the set of coupled matrix equations (14.36) to (14.38) to get {Φ(k), Ψ(k), K(k), T(k)}. Step 3. With the obtained {Φ(k), Ψ(k), K(k), T(k)}, solve (14.33) and (14.35) for P(k) and Q(k) respectively. Step 4. If k = 0, go to Step 5. Else, set k = k − 1 and go to Step 2. Step 5. Stop. Remark 14.3 The Mixed H2 ∕H∞ Reliable Controller Design Algorithm gives a recursive way to obtain the numerical values of the feedback gain at each time point k. It should be noticed that the existence of the controller is expressed by the feasibility of coupled Riccati equations; therefore, the presented algorithm has a much lower computing complexity than the traditional LMI method. However, since the Riccati equations are backward in time, this algorithm is only suitable for the offline design. The possible research topic in the future is to develop the multi-objective controller design method, which could be expressed forward in time. Variance-Constrained Multi-Objective Stochastic Control and Filtering 282 14.4 Numerical Example In this section, numerical examples are presented to demonstrate the effectiveness of the Mixed H2 ∕H∞ Reliable Controller Design Algorithm proposed in this chapter. Set N = 5, = 1.3, W = 1.4, (k) = 1 and the final condition P(5) = Q(5) = 0. Consider a one-dimensional system with the following time-varying parameters over the finite horizon [0, 5]: A(k) = 0.3 + 0.1 cos(5k), B(k) = 0.6 − 0.2 sin(8k), C(k) = 0.4 + 0.1 sin(10k), D(k) = 0.3 − 0.1e0.2k , E(k) = 0.2, Π11 i (k) = 0.25, Π12 i (k) = 0.4, Π22 i (k) = 0.64, Γi (k) = 0.25, Ξi (k) = 0.36. We now consider the following two different actuator failure cases. Case 1: Let the actuator failure matrix be R(k) = 1.4sin2 (x(k)). Therefore, it can be found that 0 ≤ R(k) ≤ 1.4 and H(k) = 0.7, L(k) = 1. Using the developed computational algorithm, we can check the feasibility of the coupled Riccati-like equations and then calculate the desired controller parameters, which are shown below. • Set k = 4. • Then we obtain [ ] 1.6900 0 Φ(4) = > 0, 0 1.6900 Ψ(4) = 0.2704 > 0, [ ] 0 T(4) = , 0 K(4) = −0.3510. • With the obtained {Φ(4), Ψ(4), K(4), T(4)}, we have P(4) = 0.3519, Q(4) = 0.3519. By similar recursions, it can be found that: • k = 3: [ ] 1.6851 −0.0377 Φ(3) = > 0, −0.0377 1.3986 Ψ(3) = 0.4073 > 0, P(3) = 0.2647, • k = 2: K(3) = −0.2619, Q(3) = 0.2653. [ ] 1.6840 −0.0363 Φ(2) = > 0, −0.0363 1.4708 Ψ(2) = 0.3504 > 0, P(2) = 0.3861, [ ] 0.0024 T(3) = , 0.0186 [ ] 0.0014 T(2) = , 0.0084 K(2) = −0.3584, Q(2) = 0.3862. Mixed H2 ∕H∞ Control with Actuator Failures: the Finite-Horizon Case • k = 1: [ ] 1.6778 −0.0625 Φ(1) = > 0, −0.0625 1.3703 Ψ(1) = 0.3358 > 0, P(1) = 0.3178, • k = 0: [ ] 0.0119 T(1) = , 0.0609 K(1) = −3.3308, Q(1) = 0.3243. [ ] 1.6773 −0.0578 Φ(0) = > 0, −0.0578 1.4268 Ψ(0) = 0.3562 > 0, P(0) = 0.3441, 283 [ ] 0.0105 T(0) = , 0.0479 K(0) = −0.3950, Q(0) = 0.3487. It can be verified that P(0) ≤ 2 W, which means that {Φ(k), Ψ(k), K(k), T(k), P(k), Q(k)} are the solutions of the set of coupled matrix equations (14.33) to (14.38). Therefore, the Nash equilibrium strategies u∗ (k) = K(k)x(k) and ∗ (k) = T(k)x(k) have been found. Case 2: Let the actuator failure matrix be R(k) = sign((x(k)) + 1.2. Therefore, it can be found that 0.2 ≤ R(k) ≤ 2.2 and H(k) = 1.2, L(k) = 0.8333. By a similar simulation, it can be found that: • k = 4: [ ] 1.6900 0 Φ(4) = > 0, 0 1.6900 Ψ(4) = 0.2704 > 0, P(4) = 0.3519, • k = 3: Q(4) = 0.3519. ] 1.6851 −0.0647 Φ(3) = > 0, −0.0647 0.8336 P(3) = 0.2610, • k = 2: K(4) = −0.3510, [ Ψ(3) = 0.6113 > 0, [ ] 0.0006 T(3) = , 0.0080 K(3) = −0.2262, Q(3) = 0.2611. [ ] 1.6841 −0.0614 Φ(2) = > 0, −0.0614 1.0548 Ψ(2) = 0.3504 > 0, P(2) = 0.3861, [ ] 0 T(4) = , 0 [ ] −0.0009 T(2) = , −0.0098 K(2) = −0.3584, Q(2) = 0.3862. 284 • k = 1: Variance-Constrained Multi-Objective Stochastic Control and Filtering [ ] 1.6778 −0.1069 Φ(1) = > 0, −0.1069 0.7523 Ψ(1) = 0.3948 > 0, P(1) = 0.3131, • k = 0: K(1) = −0.4090, Q(1) = 0.3325. [ ] 1.6775 −0.0977 Φ(0) = > 0, −0.0977 0.9280 Ψ(0) = 0.4709 > 0, P(0) = 0.3305, [ ] 0.0121 T(1) = , 0.1064 ] 0.0069 , T(0) = 0.0535 [ K(0) = −0.4163, Q(0) = 0.3361. It can be verified that P(0) ≤ 2 W, which means that {Φ(k), Ψ(k), K(k), T(k), P(k), Q(k)} are the solutions of the set of coupled matrix equations (14.33) to (14.38). Therefore, the Nash equilibrium strategies u∗ (k) = K(k)x(k) and ∗ (k) = T(k)x(k) have been found. 14.5 Summary In this chapter, the mixed H2 ∕H∞ control problem has been studied for a type of nonlinear stochastic systems against actuator failures. The actuator failure model is quite general and could stand for several common failures as special cases. A state feedback control scheme has been proposed for the stochastic time-varying systems satisfying both H2 and H∞ performance indices simultaneously. The solvability of the addressed multi-objective control problem is expressed by the feasibility of certain coupled matrix equations. The numerical values of the controller gains can be obtained by the developed computing algorithm. An illustrative example is given to show the effectiveness and applicability of the proposed design strategy.

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