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In this case, a feedback gain in the controller (4) is given K = − Λ −1 Ψ T . (14) Proof: (Sufficiency) The closed-loop system (6) with the feedback gain (14) is given by Ex (k + 1) = ( A − BΛ−1 Γ T + Hc Fc (k) Gc ) x (k). where Gc = G T −ΨΛ−1 verified that T . Then, by some mathematical manipulation and (13), it can be 36 8 Advances in Discrete Time Systems Advances in Discrete Time Systems QS T ( A − BΛ−1 Ψ T ) + ( A − BΛ−1 Ψ T ) T SQ T − E T PE + εGcT Gc + ε−1 QS T Hc HcT SQ T +( A − BΛ−1 Ψ T + ε−1 Hc HcT SQ T ) T Γ−1 ( A − BΛ−1 Ψ T + ε−1 Hc HcT SQ T ) = QS T A + A T SQ T − E T PE + ( A + ε−1 Hc HcT SQ T ) T Γ−1 ( A + ε−1 Hc HcT SQ T ) +εG T G − QS T BΛ−1 Ψ T − ( A + ε−1 Hc HcT SQ T ) T Γ−1 BΛ−1 Ψ T − ΨΛ−1 B T SQ T −ΨΛ−1 B T Γ−1 ( A + ε−1 Hc HcT SQ T ) + ΨΛ−1 B T Γ−1 BΛ−1 Γ T + εΨΛ−1 Λ−1 Ψ T +ε−1 QS T Hc HcT SQ T = QS T A + A T SQ T − E T PE + ( A + ε−1 Hc HcT SQ T ) T Γ−1 ( A + ε−1 Hc HcT SQ T ) +εG T G − [ QS T + ( A + ε−1 Hc HcT SQ T ) T Γ−1 ] BΛ−1 Ψ T −ΨΛ−1 B T [SQ T + Γ−1 ( A + ε−1 Hc HcT SQ T )] + ΨΛ−1 Ψ T + ε−1 QS T Hc HcT SQ T = QS T A + A T SQ T − E T PE + εG T G + ε−1 QS T Hc HcT SQ T + Θ T Γ−1 Θ − ΨΛ−1 Ψ T < 0.
21] Tse, E. (1971). On the optimal control of stochastic linear systems. IEEE Trans. Aut. Control, 16(6), 776-785.  Xu, X. (1996). A study on robust control for discrete-time systems with uncertainty. A Master Thesis of 1995, Kobe university, Kobe, Japan, January, 1996.  Xu, X. (2008). Characterization of all static state feedback mixed LQR/H∞ controllers for linear continuous-time systems. Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, China, 678-682, July 16-18, 2008.
3(ii), we obtain the following result. 6. Given K, the descriptor system (6) is robustly admissible if and only if there exist matrix P and scalar ε > 0 such that − E T PE + ε( G T G + K T K ) ( A + BK ) T P P( A + BK ) −P 0 HT P 0 αB T P E T PE ≥ 0, (10) 0 0 PH αPB < 0. 3. Robust control design In the previous section, we have obtained robust admissibility conditions of the closed-loop system (6). Based on those conditions, we now seek how to calculate a feedback gain K in the controller (4).