Adaptive Control Of Linear Time-varying Systems

This paper mainly studies the problems of adaptive control for linear time-varying systems with unmodeled dynamics. It is composed of two parts: 一, Adaptive control of time-varying systems with unmodeled dynamics and relative degree 1.Consider the following SISO linear time-varying systems with unmodeled dynamicsy(t) = (1 + μ₁â–³₁(s))A(s, t)^-1(B(s, t)u(t) + D(s)w(t)) + μ₂â–³₂(s)y(t).where u(t), y(t) is the input and output of the systems, respectively, w(t) is the disturbance of the systems, A(s,t) = sⁿ + s^n-1a_n-1(t) + ... + sa₁(t) +1, j = 0, ... , n - 1) are unknown time-varying parameter, â–³₁(t), â–³₂(t) denote input and output unmodeled dynamics, respectively, μ₁, μ2 ≥ 0 denote the amplitude of unmodeled dynamics, s denotes the differential operator, that is, for any differentiable function x(t), s(x(t)) (?) x(t).The control objective is to design an adaptive controller such that all the signals of the closed-loop systems are bounded, and the output tracks as close as possible a reference signal.We give the following assumptions for the systems:Assumption 1 B(s,t) is Hurwitz, the order n of the systems and the sign of the high-frequency gain b_n-1(t) are known, without loss of generality, we assume b_n-1(t) > 0, and bi(t) ∈ L2, i = 0,1, ... , n - 1.Assumption 2 â–³₁(s) and â–³₂(s) are stable and strictly proper.Assumption 3 are uniformly bounded with uniformly bounded derivatives, and there is σ > 0 such that |a_n-1(t)| ≤ σ.Assumption 4 for Vt > 0, w(t) is bounded.In this part, for relative-degree-one time-varying systems with unmodeled dynamics, we give the design of adaptive controller, and analyze stability of the closed-loop systems.^> Adaptive control of time-varying systems with unmodeled dynamics and relative degree n>1.In this part, by the basis of the above section, we further study adaptive control for the time-varying systems with relative degree n>1.Consider the following SISO linear time-varying systems with unmodeled dynamics:y(t) = (1 + μ1A1{s))A(s, t)^l(B(s, t)u{t) + D(s)w(t)) + μ2A2{s)y(t).where u(t), y(t) denote the input and output of the systems, respectively, w(t) is the disturbance of the systems, A(s, t) = sn + s""1^-!^) + ? ? ? + sai(t) + cbo(t), B(s, t) — smbm(t) + â–  â–  â–  + sbⁱ(t) + b^o(t), CLi(t), bj(t)(i — 0, â–  â–  ? ,n — l,j = 0, ? ? â–  ,m) are unknown time-varying parameter, Ai(s), A2(s) are input and output unmodeled dynamics, respectively, μl, μ-i > 0 denote the amplitude of unmodeled dynamics, s denotes the differential operator, that is, for any differentiable function x(t), s(x(t)) = x(t).The control objective is to design an adaptive controller such that all the signals of the closed-loop systems are bounded, and the output tracks as close as possible a reference signal.In this part, for time-varying systems with unmodeled dynamics and relative degree n > 1, we give the design of adaptive controller, and analyze stability of the closed-loop systems.