On the Stabilization of a Network of a Class of SISO Coupled Hybrid Linear Subsystems via Static Linear Output Feedback

• Published in: Mathematics (Basel) Vol. 10; no. 7; p. 1066
• Main Author: De la Sen, Manuel
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.04.2022
• Subjects: Algebra; Control theory; Controllers; Couplings; Decentralized control; Design techniques; Discrete time systems; Dynamics; Food science; hybrid dynamic systems; Hybrid systems; Linear algebra; Linear systems; Mathematical analysis; Matrices (mathematics); Matrix algebra; Output feedback; SISO (control systems); Stabilization; static output feedback controller; Synthesis
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10071066

This paper deals with the closed-loop stabilization of a network which consists of a set of coupled hybrid single-input single-output (SISO) subsystems. Each hybrid subsystem involves a continuous-time subsystem together with a digital (or, eventually, discrete-time) one being subject to eventual mutual couplings of dynamics and also to discrete delayed dynamics. The stabilizing controller is static and based on linear output feedback. The controller synthesis method is of algebraic type and based on the use of a linear algebraic system, whose unknown is a vector equivalent form of the controller gain matrix, which is obtained from a previous algebraic problem version which is based on the ad hoc use of the matrix Kronecker product of matrices. As a first step of the stabilization, an extended discrete-time system is built by discretizing the continuous parts of the hybrid system and to unify them together with its digital/discrete-time ones. The stabilization study via static linear output feedback contains several parts as follows: (a) stabilizing controller existence and controller synthesis for a predefined targeted closed-loop dynamics, (b) stabilizing controller existence and its synthesis under necessary and sufficient conditions based on the statement of an ad hoc algebraic matrix equation for this problem, (c) achievement of the stabilization objective under either partial or total decentralized control so that the whole controller has only a partial or null information about couplings between the various subsystems and (d) achievement of the objective under small coupling dynamics between subsystems.