169 Discrete Time Mixed LQR/H∞ Control Problems − T − T −γ −2 AT X∞ B1U1 1B1 X∞ A + ATU B2U 1B2 U A −1 ˆ −1 ˆ T −1 −1 ˆ −1 ⎤ ⎡ −1 ˆ −U + U B1U B1 U1 U1 B1U ⎥ BT X A ˆ = AT X∞ B ⎢ ∞ − ˆT − − U 1B1 U1 U2 ⎥ ⎢ ⎣ ⎦ ˆ ˆ ˆ ˆ ˆ = AT X B( BT X B + R )−1 BT X A ∞ ∞ ∞ We have T T − T T AK X∞ AK − X∞ + γ −2 AK X∞ BKU 1BK X∞ AK + C KC K + Q + K T RK ˆ ˆ ˆ ˆ ˆ = AT X A − X − AT X B( BT X B + R )−1 BT X A + C T C + Q ∞ ∞ ∞ ∞ ∞ 1 − T − T + (K + U 1B2 U A)T U ( K + U 1B2 U A) Also, noting that the discrete time Riccati equation (13) and substituting the above equality for (15), we get ∞ ∞ k =0 k =0 2 − T ˆ ˆ J = ∑ xT ( k )Ωx( k ) = ∑ { −ΔV ( x( k )) − z + γ w − γ U1 ( w − γ −2U 1BK X∞ AK x ) + U2 (K − T + U 1B2 U A)x (16) } − Based on the above, it is clear that if K = −U 1BTU A , then we get 2 T − T inf{ J } = x0 X∞ x0 − z + γ w − γ U ( w − γ −2U1 1BK X∞ AK x ) K (17) − T ˆk ˆ By letting w( k ) = γ −2U1 1BK X∞ AK x( k ) for all k ≥ , we get that x( k ) = AK x0 with AK which T −1 ˆ T ˆ ˆ ˆ ˆ ˆ belongs to L2 [0, +∞ ) since AK = A − B( B X∞ B + R ) B X∞ A is stable Also, we have 2 w( k ) = γ −4 xT X w x0 , z( k ) = xT X z x0 0 Then it follows from (17) that sup inf{ J } = xT ( X∞ + γ −2 X w − X z )x0 w∈L2 + K Thus we conclude that there exists an admissible state feedback controller such that T sup inf{ J } = x0 ( X∞ + γ −2 X w − X z )x0 subject to Tzw w∈L2 + K ∞ and any admissible controller uncertainty, determine an admissible non-fragile controller F∞ such that 170 New Trends in Technologies ˆ sup { J } subject to Tzw ( z) ∞ < γ w∈L2 + where, the controller uncertainty ΔF( k ) considered here is assumed to be of the following structure: ΔF( k ) = H K F( k )EK where, H K and EK are known matrices of appropriate dimensions F( k ) is an uncertain matrix satisfying F T ( k )F( k ) ≤ I with the elements of F( k ) being Lebesgue measurable If this controller exists, it is said to be a non-fragile discrete time state feedback mixed LQR/ H ∞ controller In order to solve the problem defined in the above, we first connect the its design criteria with the inequality (11) Lemma 4.1 Suppose that γ > , then there exists an admissible non-fragile controller F∞ that achieves ˆ sup { J } = xT X∞ x0 subject to Tzw w∈L2 + ∞ 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 U BF X∞ AF is stable Then it follows from ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ Lemma 2.3 that Tzw ∞ < γ Using the same argument as in the proof of Lemma 2.3, we get that AF is stable and J can be rewritten as follows: ˆ ∞ ∞ − T J = ∑ { −ΔV ( x( k )) − z + γ w − γ U ( w − γ −2U1 1BF X∞ AF x ) ˆ ˆ ∞ k =0 T +x T ( AF ˆ ∞ X∞ AF − X∞ + γ ˆ −2 ∞ − T X∞ BF U1 1BF X∞ AF ˆ ˆ ˆ ∞ ∞ ∞ ∞ T AF ˆ T + CF ˆ ∞ (18) ∞ ˆT ˆ C F + Q + F∞ RF∞ )x} ˆ ∞ Substituting (11) for (18) to get 2 − T J < xT X∞ x0 − z + γ w − γ U1 ( w − γ −2U1 1BF X∞ AF x ) ˆ ˆ ∞ ∞ (19a) or T − T ˆ J < x0 X∞ x0 − z − γ U ( w − γ −2U 1BF X∞ AF x ) ˆ ˆ ∞ By letting − T w = γ −2U1 1BF X∞ AF x ˆ ˆ ∞ ∞ − T ˆ AF = AF + γ −2 BF U 1BF X∞ AF ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ for all k≥0, we ∞ get (19b) that ˆ which belongs to L2 [0, +∞ ) since AF ˆ ∞ ˆk x( k ) = AF x0 ˆ ∞ with is stable It follows 171 Discrete Time Mixed LQR/H∞ Control Problems ˆ from (19b) that sup{ J } w∈L = xT X∞ x0 Thus, we conclude that there exists an admissible 2+ ˆ non-fragile controller such that sup{ J } w∈L = xT X∞ x0 subject to Tzw ∞ < γ Q E D 2+ − T Remark 4.1 In the proof of Lemma 4.1, we let w = γ −2U1 1BF X∞ AF x for all k ≥ to get that ˆ ˆ ∞ − T ˆk ˆ x( k ) = AF x0 with AF = AF + γ −2 BF U 1BF X∞ AF ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ˆ which belongs to L2 [0, +∞ ) since AF ˆ ∞ is stable Also, we have w 2 = γ −4 xT X w x0 , z 2 = xT X z x0 Then it follows from (19a) that J < xT ( X∞ + γ −2 X w − X z )x0 ∞ (20) ∞ T − T T ˆk ˆk ˆk ˆk where, X w = ∑ {( AF )T AF X∞ B1U1 B1 X∞ AF AF } , and X z = ∑ {( AF )T C F C F AF } ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ k =0 ∞ ∞ ∞ ∞ k =0 ∞ ∞ ∞ ∞ ˆ Note that AF depends on the controller uncertainty ΔF( k ) , thus it is difficult to find an ˆ ∞ upper bound of either of X w and X z This implies that the existence of controller uncertainty ΔF( k ) makes it difficult to find sup w∈L2+ { J } by using (20) Thus, it is clear that the existence of the controller uncertainty makes the performance of the designed system become bad In order to give necessary and sufficient conditions for the existence of an admissible nonfragile controller for solving the non-fragile discrete-time state feedback mixed LQR/ H ∞ control problem, we define the following parameter-dependent discrete time Riccati equation: T T ˆ ˆ ˆ ˆ ˆ AT X∞ A − X∞ − AT X∞ B( BT X∞ B + R )−1 BT X∞ A + ρ EK EK + C C + Qδ = ˆ where, B = ⎡γ −1B1 ⎣ ˆ ⎡−I B2 ⎤ , R = ⎢ ⎦ ⎣0 ⎤ , Qδ = Q + δ I with δ > being a sufficiently small I + R⎥ ⎦ T T constant, ρ is a given number satisfying ρ I − H KU H K > , U = I − γ −2 B1 X∞ B1 U2 = T B2 U B2 (21) + I + R and U = X∞ + γ −2 − T X∞ B1U1 1B1 X∞ >0, If A is invertible, the parameter- dependent discrete time Riccati equation (21) can be solved by using the following symplectic matrix T T ˆˆ ˆ ˆˆ ˆ ⎡ A + BR −1BT A−T ( ρ EK EK + C C + Qδ ) −BR −1BT A−T ⎤ ˆ S∞ := ⎢ ⎥ T T A −T − A−T ( ρ EK EK + C C + Qδ ) ⎢ ⎥ ⎣ ⎦ The following theorem gives the solution to non-fragile discrete time state feedback mixed LQR/ H ∞ control problem Theorem 4.1 There exists a non-fragile discrete time state feedback mixed LQR/ H ∞ controller iff for a given number ρ and a sufficiently small constant δ > , there exists a stabilizing solution X∞ ≥ to the parameter-dependent discrete time Riccati equation (21) T T such that U = I − γ −2 B1 X∞ B1 > and ρ I − H KU H K > 172 New Trends in Technologies Moreover, this non-fragile discrete time state feedback mixed LQR/ H ∞ controller is − F∞ = −U 1BTU A ˆ and achieves sup{ J } w∈L2+ = xT X∞ x0 subject to Tzw ∞ < γ Proof: Sufficiency: Suppose that for a given number ρ and a sufficiently small constant δ > , there exists a stabilizing solution X∞ ≥ to the parameter-dependent Riccati T T equation (21) such that U = I − γ −2 B1 X∞ B1 > and ρ I − H KU H K > This implies that the ˆ ( BT X B + R )−1 BT X A is stable Define respectively the ˆ ˆ ˆ ˆ solution X∞ ≥ is such that A − B ∞ ∞ state matrix and controlled output matrix of closed-loop system − T AF = A + B2 ( −U 1B2 U A + H K F( k )EK ) ˆ ∞ − C F = C + D12 ( −U 1BTU A + H K F( k )EK ) ˆ ∞ − − T and let AF∞ = A − B2U 1BTU A and F∞ = −U 1B2 U A + H K F( k )EK , then it follows from the square completion that 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 ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ − T T T T T = AT X∞ A − X∞ + γ −2 AT X∞ B1U1 1B1 X∞ A + C C + Q + F∞ B2 U A + ATU B2 F∞ + F∞ U F∞ (22) − T − T T = AT X∞ A − X∞ + γ −2 AT X∞ B1U1 1B1 X∞ A + C C + Q − ATU B2U 1B2 U A + ΔN T T where, ΔN = EK F T ( k )H KU H K F( k )EK T Noting that ρ I − H KU H K > , we have T T T T ΔN = −EK F T ( k )( ρ I − H KU H K )F( k )EK + ρ EK F T ( k )F( k )EK ≤ ρ EK EK (23) Considering (22) and (23) to get 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 ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ T T − T − T ≤ AT X∞ A − X∞ + γ −2 AT X∞ B1U 1B1 X∞ A + C C + Q + ρ EK EK − ATU B2U 1B2 U A (24) Also, 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 − 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 This implies that (21) can be rewritten as − T T − T T AT X∞ A − X∞ + γ −2 AT X∞ B1U 1B1 X∞ A + C C + Qδ − ATU 3B2U 1B2 U A + ρ EK EK = (25) Thus, it follows from (24) and (25) that there exists a non-negative-definite solution to the inequality 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 ˆ ˆ ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 173 Discrete Time Mixed LQR/H∞ Control Problems − T ˆ ˆ ˆ ˆ ˆ Note that A − B( BT X∞ B + R )−1 BT X∞ A = AF∞ + γ −2 B1U 1B1 X∞ AF∞ is stable and ΔF( k ) is an − T admissible uncertainty, we get that AF + γ −2 BF U1 1BF X∞ AF is stable By Lemma 4.1, there ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ exists a non- fragile discrete time state feedback mixed LQR/ H ∞ controller Necessity: Suppose that there exists a non-fragile discrete time state feedback mixed LQR/ H ∞ controller By Lemma 4.1, there exists a stabilizing solution X∞ ≥ to the T inequality (11) such that U = I − γ −2 BF X∞ BF > , i.e., there exists a symmetric nonˆ ˆ ∞ ∞ − T negative-definite solution X∞ to the inequality (11) such that AF + γ −2 BF U1 1BF X∞ AF ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ is T stable and U = I − γ −2 BF X∞ BF > for any admissible uncertainty ΔF( k ) ˆ ˆ ∞ ∞ Rewriting (11) to get T T − T T T ˆ AF∞ X∞ AF∞ − X∞ + γ −2 AF∞ X∞ B1U1 1B1 X∞ AF∞ + C F∞ C F∞ + Q + F∞ RF∞ + ΔN < T T ˆ ΔN = ( ATU B2 + F∞ U )ΔF( k ) + ΔF T ( k )( B2 U A + U F∞ ) + ΔF T ( k )U ΔF( k ) (26) T Note that ρ I − H KU H K > and T T T T ˆ ΔN = ρ EK F T ( k )F( k )EK + ( ATU B2 + F∞ U )H K ( ρ I − H KU H K )−1 H K T T T T × ( B2 U A + U F∞ ) − (( ATU B2 + F∞ U )H K ( ρ I − H KU H K )−1 − EK FT ( k )) T T T T × ( ρ I − H KU H K )(( ρ I − H KU H K )−1 H K ( B2 U A + U F∞ ) − F( k )EK ) (27) T T T T T ≤ ρ EK EK + ( ATU B2 + F∞ U )H K ( ρ I − H KU H K )−1 H K ( B2 U A + U F∞ ) It follows from (26) and (27) that T T − T T T T AF∞ X∞ AF∞ − X∞ + γ −2 AF∞ X∞ B1U1 1B1 X∞ AF∞ + C F∞ C F∞ + Q + F∞ RF∞ + ρ EK EK T T T T +( ATU B2 + F∞ U )H K ( ρ I − H KU H K )−1 H K ( B2 U A + U F∞ ) < (28) Using the argument of completion of squares as in the proof of Theorem 3.1 in Furuta & − Phoojaruenchanachai (1990), we get from (28) that F∞ = −U 1BTU A , where X∞ is a symmetric non- negative-definite solution to the inequality − T T − T T AT X∞ A − X∞ + γ −2 AT X∞ B1U 1B1 X∞ A + C C + Q − ATU B2U 1B2 U A + ρ EK EK < or equivalently, X∞ is a symmetric non-negative-definite solution to the parameterdependent discrete time Riccati equation − T T − T T AT X∞ A − X∞ + γ −2 AT X∞ B1U 1B1 X∞ A + C C + Qδ − ATU B2U 1B2 U A + ρ EK EK = (29) Also, we can rewrite that Riccati equation (29) can be rewritten as T T ˆ ˆ ˆ ˆ ˆ AT X∞ A − X∞ − AT X∞ B( BT X∞ B + R )−1 BT X∞ A + ρ EK EK + C C + Qδ = (30) 174 New Trends in Technologies by using the similar standard matrix manipulations as in the proof of Theorem 3.1 in Souza − T ˆ ˆ ˆ ˆ ˆ & Xie (1992) Note that A − B( BT X∞ B + R )−1 BT X∞ A = AF∞ + γ −2 B1U 1B1 X∞ AF∞ and ΔF( k ) is − T an admissible uncertainty, the assumption that AF + γ −2 BF U1 1BF X∞ AF ˆ ˆ ˆ ˆ ∞ ∞ ∞ ∞ is stable implies ˆ ˆ ˆ ˆ ˆ that A − B( BT X∞ B + R )−1 BT X∞ A is stable Thus, we conclude that for a given number ρ and a sufficiently small number δ > , the parameter-dependent discrete time Riccati equation T T (30) has a stabilizing solution X∞ and U = I − γ −2 B1 X∞ B1 > and ρ I − H KU H K > Q E D Numerical examples In this section, we present two examples to illustrate the design method given by Section and 4, respectively Example Consider the following discrete-time system in Peres and Geromel (1993) x( k + 1) = Ax( k ) + B1 w( k ) + B2 u( k ) z( k ) = C 1x( k ) + D12 u( k ) where, ⎡ 0.2113 0.0087 0.4524 ⎤ ⎡ 0.6135 0.6538 ⎤ ⎢ ⎥ ⎢ ⎥ A = ⎢0.0824 0.8096 0.8075 ⎥ , B2 = ⎢ 0.2749 0.4899 ⎥ , ⎢0.7599 0.8474 0.4832 ⎥ ⎢0.8807 0.7741 ⎥ ⎣ ⎦ ⎣ ⎦ ⎡1 ⎢0 ⎢ C1 = ⎢0 ⎢ ⎢0 ⎢0 ⎣ 0 0⎤ ⎡0 ⎤ ⎥ ⎢0 ⎥ 0⎥ ⎢ ⎥ ⎥ , D12 = ⎢0 ⎥ and B1 = I ⎥ ⎢ ⎥ 0⎥ ⎢1 0⎥ ⎢0 1⎥ 0⎥ ⎦ ⎣ ⎦ In this example, we will design the above system under the influence of state feedback of the form (3) by using the discrete-times state feedback mixed LQR/ H ∞ control method displayed in Theorem 3.1 All results will be computed by using MATLAB The above system is stabilizable and observable, and satisfies Assumption 3, and the eigenvalues of matrix A are p1 = 1.6133 , p2 = 0.3827 , p3 = −0.4919 ;thus it is open-loop unstable ⎡1 0 ⎤ ⎡1 0⎤ ⎢ ⎥ Let γ = 2.89 , R = ⎢ ⎥ , Q = ⎢0 ⎥ , we solve the discrete-time Riccati equation (13) to get ⎣0 ⎦ ⎢0 ⎥ ⎣ ⎦ ⎡ 2.9683 1.1296 0.1359 ⎤ ⎢ ⎥ X∞ = ⎢1.1296 6.0983 2.4073 ⎥ > , ⎢0.1359 2.4073 4.4882 ⎦ ⎥ ⎣ 175 Discrete Time Mixed LQR/H∞ Control Problems ⎡ 0.6446 −0.1352 −0.0163 ⎤ ⎢ ⎥ T U = I − γ −2 B1 X∞ B1 = ⎢ −0.1352 0.2698 −0.2882 ⎥ > ⎢ −0.0163 −0.2882 0.4626 ⎥ ⎣ ⎦ Thus the discrete-time state feedback mixed LQR/ H ∞ controller is ⎡ −0.3640 −0.5138 −0.3715 ⎤ K=⎢ ⎥ ⎣ −0.2363 −0.7176 −0.7217 ⎦ Example Consider the following discrete-time system in Peres and Geromel (1993) x( k + 1) = Ax( k ) + B1 w( k ) + B2 u( k ) z( k ) = C 1x( k ) + D12 u( k ) under the influences of state feedback with controller unceratinty of the form (4), where, A , B1 , B2 , C and D12 are the same as ones in Example 1; the controller uncertainty ΔF( k ) satisfies ΔF( k ) = EK F( k )EK , F T ( k )F( k ) ≤ I ⎡1 0⎤ 0⎤ ⎡0.0100 where, EK = ⎢0 ⎥ , H K = ⎢ ⎢ ⎥ 0.0100 ⎥ ⎣ ⎦ ⎢0 1 ⎥ ⎣ ⎦ In this example, we illustrate the proposed method by Theorem 4.1 by using MATLAB As stated in example 1, the system is stabilizable and observable, and satisfies Assumption 3, and is open-loop unstable ⎡1 0⎤ ⎡1 0⎤ ⎢ ⎥ Let γ = 8.27 , R = ⎢ ⎥ , Q = ⎢0 ⎥ , ρ = 3.7800 , and δ = 0.0010 , then we solve the ⎣0 ⎦ ⎢0 ⎥ ⎣ ⎦ parameter-dependent discrete-time Riccati equation (21) to get ⎡18.5238 3.8295 0.1664 ⎤ ⎢ ⎥ X∞ = ⎢ 3.8295 51.3212 23.3226 ⎥ > , ⎢ 0.1664 23.3226 22.7354 ⎥ ⎣ ⎦ ⎡ 0.7292 −0.0560 −0.0024 ⎤ ⎡609.6441 723.0571 ⎤ T U = I − γ B1 X∞ B1 = ⎢ −0.0560 0.2496 −0.3410 ⎥ > , U = ⎢ ⎥, ⎢ ⎥ ⎣723.0571 863.5683 ⎦ ⎢ −0.0024 −0.3410 0.6676 ⎥ ⎣ ⎦ ⎡ 14.2274 −0.0723 ⎢ T ρ I − H KU H K = ⎢ −0.0723 14.2020 ⎢ ⎣ 0 ⎤ ⎥ ⎥>0 14.2884 ⎥ ⎦ 176 New Trends in Technologies Based on this, the non-fragile discrete-time state feedback mixed LQR/ H ∞ controller is ⎡ −0.4453 −0.1789 −0.0682 ⎤ F∞ = ⎢ ⎥ ⎣ −0.1613 −1.1458 −1.0756 ⎦ Conclusion In this chapter, we first study the discrete time state feedback mixed LQR/ H ∞ control problem In order to solve this problem, we present an extension of the discrete time bounded real lemma In terms of the stabilizing solution to a discrete time Riccati equation, we derive the simple approach to discrete time state feedback mixed LQR/ H ∞ control problem by combining the Lyapunov method for proving the discrete time optimal LQR control problem with the above extension of the discrete time bounded real lemma, the argument of completion of squares of Furuta & Phoojaruenchanachi (1990) and standard inverse matrix manipulation of Souza & Xie (1992).A related problem is the standard H ∞ control problem (Doyle et al., 1989a; Iglesias & Glover, 1991; Furuta & Phoojaruenchanachai, 1990; Souza & Xie, 1992; Zhou et al 1996), another related problem is the H ∞ optimal control problem arisen from Basar & Bernhard (1991) The relations among the two related problem and mixed LQR/ H ∞ control problem can be clearly explained by based on the discrete time reference system (9)(3) The standard H ∞ control problem is to find an admissible controller K such that the H ∞ -norm of closed-loop transfer matrix from disturbance input w to the controlled output z is less than a given number γ > while the H ∞ optimal control roblem arisen from Basar & Bernhard (1991) is to find an admissible controller such that the H ∞ -norm of closed-loop transfer matrix from disturbance input w to the controlled output z0 is less than a given number γ > for the discre time reference system (9)(3) Since the latter is equivalent to the problem that is to find an admissible ˆ controller K such that sup w∈L2+ infK { J } , we may recognize that the mixed LQR/ H ∞ control problem is a combination of the standard H ∞ control problem and H ∞ optimal control problem arisen from Basar & Bernhard (1991) The second problem considered by this chapter is the non-fragile discrete-time state feedback mixed LQR/ H ∞ control problem with controller uncertainty This problem is to extend the results of discrete-time state feedback mixed LQR/ H ∞ control problem to the system (2)(4) with controller uncertainty In terms of the stabilizing solution to a parameter-dependent discrete time 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Banda S S and Chang B (1992) Necessary and sufficient conditions for mixed H and H ∞ optimal control, IEEE Trans Aut Control, 37 (3), PP 355-358 K Zhou, Glover K., Bodenheimer B and Doyle J C (1994) Mixed H and H ∞ performance objectives I: robust performance analysis, IEEE Trans Aut Control, 39 (8), PP 15641574 K Zhou, Doyle J C and Glover K (1996) Robust and optimal control, Prentice-Hall, INC., 1996 184 Discrete Time Systems and a scalar λ > satisfying Π= ¯ ¯ ¯ Φ + λET E H T ¯ H − λI < 0, (11) where Φ is given in Theorem 3.1, and T T T T T ¯ H = − H T T1 − H T T2 − H T T3 − H T T4 − H T T5 , and ¯ E = E + E1 K Ed + Eb K 0 Proof: Replacing A, Ad , B and Bd in (7) with A + HF (k) E, Ad + HF (k) Ed , B + HF (k) E1 and B + HF (k) Eb , respectively, we obtain a robust stability condition for the system (5) ¯ ¯ ¯ ¯ Φ + H T F (k) E + E T F T (k) H < (12) By Lemma 2.2, a necessary and sufficient condition that guarantees (12) is that there exists a scalar λ > such that ¯ ¯ ¯ ¯ Φ + λET E + H T H < (13) λ Applying Schur complement formula, we can show that (13) is equivalent to (11) State feedback sabilization This section proposes a state feedback stabilization method for the uncertain discrete-time delay system (1) First, stabilization of nominal system is considered in Section 4.1 Then, robust stabilization is proposed in Section 4.2 4.1 Stabilization First, we consider stabilization for the nominal system (3) Our problem is to find a control gain K such that the closed-loop system (6) is asymptotically stable Unfortunately, Theorem 3.1 does not give LMI conditions to find K Hence, we must look for another method Theorem 4.1 Given integers dm and d M , and scalars ρi , i = 1, · · · , Then, the controller (4) ¯ ¯ ¯ asymptotically stabilizes the time-delay system (3) if there exist matrices P1 > 0, P2 > 0, Q1 > 0, ¯ ¯ ¯ Q2 > 0, S > 0, M > 0, G, Y ⎡ ⎤ ⎡ ⎤ ¯ ¯ L1 N1 ⎢ N2 ⎥ ¯ ¯ ⎢ L2 ⎥ ⎢ ⎥ ⎢ ⎥ ¯ ¯ ¯ ¯ L = ⎢ L3 ⎥ , N = ⎢ N3 ⎥ , ⎢ ⎥ ⎢ ⎥ ⎣ N4 ⎦ ⎣ L4 ⎦ ¯ ¯ ¯ ¯ L5 N5 satisfying Ψ= Ψ1 + Θ L + Θ T + Θ N + Θ T + Θ T + Θ T L √ N T ¯ dM ZT √ ¯ dM Z 0, P2 > 0, Q1 > 0, Q2 > 0, ¯ ¯ S > 0, M > 0, G, Y ⎡ ⎤ ⎡ ⎤ ¯ ¯ L1 N1 ¯ ¯ ⎢ L2 ⎥ ⎢ N2 ⎥ ⎢ ⎥ ⎢ ⎥ ¯ ¯ ¯ ¯ L = ⎢ L3 ⎥ , N = ⎢ N3 ⎥ , ⎢ ⎥ ⎢ ⎥ ¯ ¯ ⎣ L4 ⎦ ⎣ N4 ⎦ ¯5 ¯ L N5 186 Discrete Time Systems and a scalar λ > satisfying Λ= ˆ ˆ ˆ Ψ + λ H T H ET ˆ E − λI < 0, (16) where ˆ H = − ρ1 H T − ρ2 H T − ρ3 H T − ρ4 H T − ρ5 H T , and ˆ E = EY T + E1 G Ed Y T + Eb G 0 In this case, a controller gain in the controller (4) is given by (15) Proof: Replacing A, Ad , B and Bd in (14) with A + HF (k) E, Ad + HF (k) Ed , B + HF (k) E1 and B + HF (k) Eb , respectively, we obtain robust stability conditions for the system (1): ¯ ¯ ¯ ¯ Ψ + H T F (k) E + E T F T (k) H < (17) By Lemma 2.2, a necessary and sufficient condition that guarantees (17) is that there exists a scalar λ > such that ¯ ¯ ¯ ¯ (18) Ψ + λ H T H + ET E < λ Applying Schur complement formula, we can show that (18) is equivalent to (16) State estimation All the information on the state variables of the system is not always available in a physical situation In this case, we need to estimate the values of the state variables from all the available information on the output and input In the following, we make analysis of the existence of observers Section 5.1 analyzes the observer of a nominal system, and Section 5.2 considers the robust observer analysis of an uncertain system 5.1 Observer analysis Using the results in the previous sections, we consider an observer design for the system (1), which estimates the state variables of the system using measurement outputs x (k + 1) = ( A + ΔA) x (k) + ( Ad + ΔAd ) x (k − dk ), y(k) = (C + ΔC ) x (k) + (Cd + ΔCd ) x (k − dk ) (19) (20) where uncertain matrices are of the form: H ΔA ΔAd F ( k ) E Ed = ΔC ΔCd H2 where F (k) ∈ l × j is an unknown time-varying matrix satisfying F T (k) F (k) ≤ I and H, H2 , E and Ed are constant matrices of appropriate dimensions We consider the following system to estimate the state variables: ¯ ˆ ˆ ˆ x (k + 1) = A x(k) + K (y(k) − C x(k)) (21) 187 Robust Control Design of Uncertain Discrete-Time Systems with Delays ¯ ˆ where x is the estimated state and K is an observer gain to be determined It follows from (19), (20) and (21) that ˜ ˜ ˜ ˜ ˜ ˜ xc (k + 1) = ( A + HF (k) E ) xc (k) + ( Ad + HF (k) Ed ) xc (k − dk ) where T xc = [xT eT ]T , (22) ˆ e(k) = x (k) − x (k) and A Ad ˜ ˜ , Ad = , A= ¯ ¯ A − KC Ad − KCd H ˜ ˜ , E = E , Ed = Ed ¯ H − KH2 We shall find conditions for (22) to be robustly stable In this case, the system (21) becomes an observer for the system (19) and (20) For nominal case, we have ˜ H= ˜ ˜ xc (k + 1) = Axc (k) + Ad xc (k − dk ) (23) We first consider the asymptotic stability of the system (23) The following theorem gives conditions for the system (23) to be asymptotically stable ¯ Theorem 5.1 Given integers dm and d M , and observer gain K Then, the system (23) is ˜ ˜1 ∈ 2n×2n , < P2 ∈ 2n×2n , < Q1 ∈ 2n×2n , ˜ asymptotically stable if there exist matrices < P ˜ ˜ ˜ < Q2 ∈ 2n×2n , < S ∈ 2n×2n , < M ∈ 2n×2n , ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ˜ ˜ ˜ T1 L1 N1 ˜ ⎢ L2 ⎥ ⎢ N2 ⎥ ⎢ T2 ⎥ ˜ ˜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ˜ ˜ ˜ ˜ ˜ ˜ L = ⎢ L3 ⎥ ∈ 10n×2n , N = ⎢ N3 ⎥ ∈ 10n×2n , T = ⎢ T3 ⎥ ∈ 10n×2n ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ N4 ⎦ ⎣ T4 ⎦ ⎣ L4 ⎦ ˜ ˜ ˜ ˜ ˜ ˜ T5 N5 L5 satisfying ˜ Φ= where ˜ ˜ ˜ ˜ ˜ ˜ ˜ Φ1 + Ξ L + Ξ T + Ξ N + Ξ T + Ξ T + Ξ T L √ N T ˜ dM ZT ⎡ √ ˜ dM Z satisfying ˜ Π= ˆ ˆ ˆ ˜ Φ + λET E H T ˆ H − λI are given weighting matrices Definition 1: For system (1), if there exists state-feedback controller, such that the faulty closed-loop system (6) will meet the following indices constraints simultaneously, a The closed-loop system is quadratic D stabilizable with constraint Φ(q , r ) , Φ(q , r ) denotes the disc with centre q + j0 and the radius r , where r and q are known constants with |q|+ r < b The H∞ norm of the closed-loop transfer function is strictly less than a given positive scalar γ , c The closed-loop value of the cost function (7) exists an upper bound satisfying J ≤ J ∗ , then for all admissible uncertainties and possible faults, the given indices, quadratic D stabilizability index Φ(q , r ) , H∞ norm bound γ > and cost function performance J * > are said to be consistent, state-feedback controller u( k ) = Kx( k ) is said to be satisfactory fault-tolerant controller Now, the satisfactory fault-tolerant control problem considered in this paper is stated in the following Problem: For the system (1) with actuator failure, given the quadratic D stabilizability index Φ(q , r ) , H∞ norm bound γ > and the cost function (7), determine a control law u( k ) = Kx( k ) so that the closed-loop system satisfies criteria (a), (b) and (c) simultaneously Main results Lemma 1: Consider the actuator fault model (3), for any matrix R = RT > and scalar ε > , if R −1 − ε I > then ( MRM ≤ M0 R −1 − ε I ) −1 M + ε −1M0 JJM0 (8) Lemma 2: Consider the system (1) subject to faults, given index Φ(q , r ) , if there exists gain matrix K and symmetric positive matrix P such that the following matrix inequality ⎡ −P −1 ⎢ ⎢ A C − qI ⎣ ( ) T A C − qI ⎤ ⎥ 0(i = ~ 3) such that the following linear matrix inequality ⎡Σ11 ⎢ ⎢ * ⎢ * ⎢ ⎢ * ⎢ ⎢ * ⎣ AX + BY − qX −r X * * * X ε 3BJ Y −ε 1I * −ε 2I + ε 3J * ⎤ ⎥ ⎥ ⎥ , U = ⎢ ⎥, ⎢ ⎥ ? ?72 3.0 571 863.5683 ⎦ ⎢ −0.0024 −0.3410 0.6 676 ⎥ ⎣ ⎦ ⎡ 14.2 274 ... continuous -time delay systems In addition, most results have focused on state feedback stabilization of discrete- time systems with time- varying delays Only a few results on observer design of discrete- time