Optimizing the performance of the feedback controller for state‐based switching bilinear systems

Abstract

Summary This article is concerned with the design and performance optimization of feedback controllers for state‐based switching bilinear systems (SBLSs), where subsystems take the form of bilinear systems in different state space polyhedra. First, by further dividing the subregions into smaller regions and designing region‐dependent feedback controllers in the resulting regions, the SBLSs can be transformed into corresponding switching linear systems (SLSs). Then, for these SLSs, by imposing contractility conditions on the Lyapunov functions, an upper bound on the infinite horizon quadratic cost can be obtained. Optimizing this upper bound yields the controller design. The optimization problem is formulated as a linear matrix inequalities optimization problem, which can be solved efficiently. Finally, the stability of the close‐loop system under the proposed controller is established step by step through a decreasing overall Lyapunov function.

Convergence of the optimization method for feedback stabilization of linear delay systems

“…Here, K * is the optimal solution for the problem (12) with Equations ( 5) and ( 6), F[K * , t] is the solution to the differential Equation ( 5) when K = K * . Furthermore, K (n) * is the minimizer of the problem (14) with Equations ( 15) and ( 16) dividing T into n segments, F n (K (n) * ) is the numerical solution of Equation ( 15) when K = K (n) * . To prove this theorem, the following lemma is proved firstly.…”

“…For the stabilization of time‐delay systems, the early method is to transform the analysis of a related Riccati equation and give the controller design method by solving the Riccati equation. In Reference 14, according to Lyapunov theory, the designs of state feedback controllers for iterative algorithms based on linear matrix inequalities (LMIs) have been extensively studied. For the stabilization of time‐delay systems, based on the Riccati equation, a feedback low‐gain controller is proposed by solving a minimization problem 15 .…”

Hierarchical maximum likelihood generalized extended stochastic gradient algorithms for bilinear‐in‐parameter systems

“…The ML‐MISG algorithm can combine the filtering technique and the hierarchical identification methods 61‐66 to study the identification problem of other linear and nonlinear systems 67‐70 …”