Model Reference Adaptive Control, Based On High Frequency Gain Matrix Factorization

This paper mainly deals with model reference adaptive control (MRAC) for multivariable systems using K_p = L₂D₂S₂ factorization. It is composed of two parts.一.Model reference adaptive control (MRAC) for multivariable systems using high frequency gain matrix K_p = L₂D₂S₂ factorizationConsider the following ideal MIMO systemsy = G(s)u, (2.2.1)where G{s) ∈ R^m×m[s] is the transfer function matrix, u,y ∈ R^m.Control objective is to design a control u such that all the signals in the closed-loop system are uniformly bounded, and the output y tracks the following reference model output y_m, wherey_m = W_m(s)r, (2.2.2)r ∈ R^m is any given piecewise continuous uniformly bounded input, and r ∈ l_∞. The following assumptions on the system and the reference model are needed.System assumptions:(A1) The transmission zeros of G(s) have negative real parts, and every lent of G(s) is analytic in Re[s] ≥ (δ₀)/2 for some δ₀ > 0.(A2) G(s) is strictly proper, has full rank and its modified left interactormatrixis diagnoal and known, where h_ij(s),j = 1, ....., m -1 i = 2, ......, m are some polynomials, d_i(s), i = 1, ........ m are arbitray monic Hurwitz polynomials with nonzero degree.(A3) The observability index v of G(s) is known, define the row degree Ai(s) of A(s) = (a_ij(s)) aswhere δ[p(s)] is the degree of a polynomial p(s), then the observability index v of transfer function matrix G(s) = A^-1(s)B(s) is defined by(A4) For the high frequency gain matrix , the signsof its leading principal minors are known.Reference model assumptions:(Ml) All of the transmission zeros of W_m(s) have negative real parts, and every element of W_m(s) is analytic in for the above δ₀ > 0.(M2) The zeros structure at infinity of W_m(s) is the same as that of G(s), i.e., is finite and nonsingular. Without loss of generality, wecan choose W_m(s) = ξ^-m(s).In this part, we give rigorously the stability analysis and the convergence proof of the tracking error for the closed-loop system.