By Khalid Abidi, Jian-Xin Xu
This e-book covers a large spectrum of structures similar to linear and nonlinear multivariable structures in addition to keep watch over difficulties comparable to disturbance, uncertainty and time-delays. the aim of this publication is to supply researchers and practitioners a guide for the layout and alertness of complex discrete-time controllers. The ebook offers six diversified keep watch over ways counting on the kind of procedure and keep watch over challenge. the 1st and moment methods are according to Sliding Mode keep an eye on (SMC) idea and are meant for linear structures with exogenous disturbances. The 3rd and fourth ways are in line with adaptive regulate concept and are geared toward linear/nonlinear platforms with periodically various parametric uncertainty or platforms with enter hold up. The 5th process is predicated on Iterative studying keep watch over (ILC) conception and is aimed toward doubtful linear/nonlinear structures with repeatable projects and the ultimate method relies on fuzzy good judgment keep watch over (FLC) and is meant for hugely doubtful platforms with heuristic regulate wisdom. distinctive numerical examples are supplied in each one bankruptcy to demonstrate the layout approach for every keep watch over process. a couple of useful keep an eye on functions also are provided to teach the matter fixing approach and effectiveness with the complicated discrete-time keep an eye on methods brought during this book.
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Extra resources for Advanced Discrete-Time Control: Designs and Applications
1, ek is O T 2 if δ k = O T 3 . When the system relative degree is 2, C B = 0, and CΓ = C BT + 1 1 ABT 2 + A2 BT 3 + · · · 2! 3! = 1 1 C ABT 2 + C A2 BT 3 + · · · = O T 2 . 2! 3! 130) Similarly CΦΓ = C I + AT + 1 2 2 A T + ··· 2! Γ = C (I + O (T )) Γ = CΓ + O T 2 = O T 2 . 131) 42 2 Discrete-Time Sliding Mode Control Now rewrite δ k = C dˆ k − dk + dk−1 − dˆ k−1 − CΦ dk − dˆ k = CΓ ηˆ k − ηk + ηk−1 − ηˆ k−1 − CΦΓ ηk − ηˆ k + O T 3 . 3, is O (T ). 132) δ k = O T 2 · (O (T ) + O (T )) − O T 2 · O (T ) + O T 3 = O T 3 and consequently ek is ultimately O T 2 .
4 Discrete-Time Integral Sliding Mode Control 23 guarantees the desired closed-loop performance. Thus, we have xk+1 = (Φ − Γ K ) xk + dk − Γ (DΓ )−1 Ddk−1 − Γ (DΓ )−1 D (dk−1 − dk−2 ) . 56), the disturbance estimate dˆ k has been replaced by dk−1 . 57) ζ k = dk − 2Γ (DΓ )−1 Ddk−1 + Γ (DΓ )−1 Ddk−2 . 58) where The magnitude of ζ k can be evaluated as below. 58) yield ζ k = (dk − 2dk−1 + dk−2 ) + I − Γ (DΓ )−1 D (2dk−1 − dk−2 ) . 1, it has been shown that (dk − 2dk−1 + dk−2 ) ∈ O T 3 . 4) we have I − Γ (DΓ )−1 D (2dk−1 − dk−2 ) = I − Γ (DΓ )−1 D Γ (2fk−1 − fk−2 ) + T Γ (2vk−1 − vk−2 ) + O T 3 2 Note that I − Γ (DΓ )−1 D Γ = 0, thus I − Γ (DΓ )−1 D Γ (2fk−1 − fk−2 ) + Furthermore, I − Γ (DΓ )−1 D O T 3 .
46) is O T 3 , and lim k→∞ xk ≤ O T 2 . 47) leads to 22 2 Discrete-Time Sliding Mode Control σ k+1 = (DΦ + E) xk + D (Γ uk + dk ) + ε k − Dx0 . 48) The equivalent control is found by solving for σ k+1 = 0 uk = (DΓ )−1 Dx0 − (DΓ )−1 ((DΦ + E) xk + Ddk + ε k ) . 49) would require a priori knowledge of the disturbance dk . 2) leads to the closed-loop equation in the sliding mode xk+1 = Φ − Γ (DΓ )−1 (DΦ + E) xk − Γ (DΓ )−1 ε k + Γ (DΓ )−1 Dx0 + dk − Γ (DΓ )−1 D dˆ k . 51) Let us derive the sliding dynamics.