Design of SPR Systems with Dynamic Compensators and Output Variable Structure Control

This paper presents necessary and sufficient conditions for the following problem: given a linear time invariant plant G(s) = N(s)D(s) ^-1 = C(sI -A)^-1B, with m inputs, p outputs, p > m, rank(C) = p, rank(B) = rank(CB) = m, find a tandem dynamic controller G_c(s) = D_c(s)^-1N_c(s) = C_c(sI - A_c)^-1B_c B_c + D_c, with p inputs and m outputs and a constant output feedback matrix K_o isin Ropf^mtimesp such that the feedback system is strictly positive real (SPR). It is shown that this problem has solution if and only if all transmission zeros of the plant have negative real parts. When there exists solution, the proposed method firstly obtains G_c (s) in order to all transmission zeros of G_c(s)G(s) present negative real parts and then K_o is found as the solution of some linear matrix inequalities (LMIs). Then, taking into account this result, a new LMI based design for output variable structure control (VSC) of uncertain dynamic plants is presented. The method can consider the following design specifications: matched disturbances or nonlinearities of the plant, output constraints, decay rate and matched and nonmatched plant uncertainties