Time-varying Interconnected Systems Robust Decentralized Adaptive Stabilization

This paper mainly deals with a theoretical problem of robust adaptive control: Robust decentralized adaptive stabilization problem for time-varying interconnected systems with unmodelled dynamics, static interconnections and dynamic interconnections.Consider a class of interconnected systems consisting of N interconnected subsystems, and the ith subsystem is modelled bywhere u_i(t), y_i(t) ∈ R are the input and output of the ith subsystem, respectively. A_i0(s, t) = sⁿⁱ + s(n_i-1)a_i,ni-1(t)........ sa₍a_i1(t)+ a_i0(t), B_i0(s, t) =s^mib_i,mi(t) + ...... + sb_i1(t) + b_i0(t), a_ij(t) and b_ik(t) (j = 0, ........ ,n_i - 1, k = 0,......, m_i) are unknown time-varying parameters, D_i0(s) = (s^n_i-1,....., s, 1). f_ij(t>y_j) ∈ R^n_i and Δ_ij(s)y_j(i ≠ j) denote the static interconnections and dynamic interconnections from the jth subsystem to the ith subsystem, respectively. Δ_ii(s) is the unmodelled dynamics in the tth subsystem, and μ_ij, μ_ii > 0 (i, j' = 1, ....., N) specify the magnitudes of dynamic interconnections and unmodelled dynamics.The control objective is to design a robust decentralized adaptive output feedback controller for each subsystem so that all the signals in the closed-loop are bounded, and all the output y_i are regulated to zeros.To achieve the objective, we need the following assumptions for the ith subsystem.Assumption 1(1) B_i0(s,t) is Hurwitz polynomial, i.e. for any t ≥ 0, all zeros h_ij(t) of B_i0(s, t) satisfy the inequality. Re(h_ij(t)) < 0, j =1 , ....., m_i.