Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design

Abstract

The paper deals with the LPV stabilizability problem for linear plants whose parameters vary with time in a compact set. First, necessary and sufficient conditions for the existence of a linear gain-scheduled stabilizing compensator are given. Next, it is shown that, if these conditions are satisfied, any compensator transfer function depending on the plant parameters and internally stabilizing the closed-loop control system when the plant parameters are constant, can be realized in such a way that the closed-loop asymptotic stability is guaranteed under arbitrary parameter variations. To find one such realization, a reasonably simple and general algorithm based on Lyapunov equations and Cholesky factorization is provided. The result can profitably be used to achieve both pointwise optimality (or pole placement) and LPV stability. Some potential applications in adaptive control and online tuning are pointed out

Set-Theoretic Methods in Control

“…The following result establishes a separation principle for gain-scheduling design [BCMV10]. Consider the LPV systeṁ…”

“…7.4. It is also known that state feedback control and observer design are dual problems [BM03,BCMV10].…”

Mixed‐Objective Robust Dynamic Output Feedback Controller Synthesis for Continuous‐Time Polytopic Lpv Systems

“…In order to derive a solvable condition and reduce the conservativeness meanwhile, we develop a new parameterized linear matrix inequalities (PLMI) condition in the following. Similar to the introduction of the convexification method used in and , a finite dimensional PLMI condition that depends on the vertices of the polytope is presented in the following theorem, which can be efficiently solved by using standard numerical software. □Theorem Considering the system described in (1)–(2).…”

Gain‐Scheduled Control via Switching of LTI Controllers and State Reset

“…, (k, l) ∈  andR k andQ k are given by (16). Then S k = Y k − X −1 k > 0 and with the following coefficient matrices the closed-loop system (7)-(8) is exponentially stable and its L 2 -gain from w to z less than :…”

“…An alternating algorithm that solves a nonconvex inequality is proposed for synthesis of state-feedback gain-scheduled controllers [18]. Blanchini et al [16] showed a parameterization of stabilizing LPV controllers for LPV systems and applied it to gain-scheduled control with pole location. There are several papers that provide a method of interpolation of given or previously synthesized LTI controllers for finite points in the scheduling parameter [14,15,20].…”