By Edited by Magdi S. Mahmoud

This quantity brings concerning the modern leads to the sphere of discrete-time platforms. It covers papers written at the subject matters of strong keep watch over, nonlinear structures and up to date functions. even if the technical perspectives are diverse, all of them geared in the direction of targeting the updated wisdom achieve by means of the researchers and offering potent advancements alongside the structures and regulate area. every one subject has an in depth discussions and recommendations for destiny perusal via investigators.

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E satisfies rankE = r ≤ n. Unforced descriptor system (1) with u(k) = 0 is denoted by the pair ( E, A). 1. (Dai [4]) (i) The pair ( E, A) is said to be regular if det(zE − A) is not identically zero. (ii) The pair ( E, A) is said to be causal if it is regular and deg(det(zE − A)) = rank( E). (iii) Define the generalized spectral radius as ρ( E, A) = max |λ|. The pair ( E, A) is λ∈{det(zE− A)=0} said to be stable if ρ( E, A) < 1. (iv) The pair ( E, A) is said to be admissible if it is regular, causal and stable.

Zou, Y. Improved conditions for delay-dependent robust stability and stabilization of uncertain discrete-time systems, Asian Journal of Control, 2005;7 344-348. , Lam, J. & Zou, Y. New results on delay-dependent robust H∞ control for systems with time-varying delays, Automatica, 2006;42 343-348. [35] Ye, D. & Yang, G. H. Adaptive robust H∞ state feedback control for linear uncertain systems with time-varying delay, International Journal of Adaptive Control and Signal Processing, 2008;22 845-858.

5772/ 51538 41 Robust Control Design of Uncertain Discrete-Time Descriptor Systems with Delays 13 In this case, a feedback gain in the controller (4) is given by K = − Λ −1 Ψ T . (30) Proof: (Sufficiency) The closed-loop system (17) with the feedback gain (30) is given by Ex (k + 1) = ( A − B2 Λ−1 Ψ T + Hc Fc (k) Gc ) x (k) + B1 w(k), z(k) = (C − DΛ−1 Ψ T − αDΦ(k)Λ−1 Ψ T ) x (k) where Gc = G T −ΨΛ−1 T (31) . Then, by some mathematical manipulation, we have Q1 S T ( A − B2 Λ−1 Ψ T ) + ( A − B2 Λ−1 Ψ T ) T SQ1T + ε 1 GcT Gc + ε 2 ΨΛ−1 Λ−1 Ψ T = Q1 S T A − Q1 S T B2 Λ−1 Ψ T + A T SQ1T − ΨΛ−1 B2T SQ1T + ε 1 G T G +(ε 1 + ε 2 )ΨΛ−1 Λ−1 Ψ T , (C − DΛ−1 Ψ T ) T Z −1 (C − DΛ−1 Ψ T ) = C T Z −1 C − ΨΛ−1 DZ −1 C − C T Z −1 DΛ−1 Ψ T + ΨΛ−1 DZ −1 DΛ−1 Ψ T , −1 1 −1 T T T T T T −1 ( A − B2 Λ−1 Ψ T + ε− 1 Hc Hc SQ1 ) Γ ( A − B2 Λ Ψ + ε 1 Hc Hc SQ1 ) T − 1 − 1 T − 1 T − 1 − 1 T − 1 T − 1 − = Θ A Γ Θ A − ΨΛ B2 Γ Θ A − Θ A Γ B2 Λ Ψ + ΨΛ B2 Γ B2 Λ 1 Ψ T , 1 −1 T T −1 T [( A − B2 Λ−1 Ψ T ) T SQ2T + ( Q1 S T + ( A − B2 Λ−1 Ψ T + ε− 1 Hc Hc SQ1 ) Γ ) Θ B ]W 1 T SQ T ) T Γ−1 ) Θ )] T H H ×[( A − B2 Λ−1 Ψ T ) T SQ2T + ( Q1 S T + ( A − B2 Λ−1 Ψ T + ε− c c B 1 1 = YW −1 Y T − ΨΛ−1 B2T (SQ2T + Γ−1 Θ B ) T W −1 Y T − YW −1 (SQ2T + Γ−1 Θ B ) T B2 Λ−1 Ψ T +ΨΛ−1 B2T (SQ2T + Γ−1 Θ B )W −1 (SQ2T + Γ−1 Θ B ) T B2 Λ−1 Ψ T .

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