Global and Semi-Global Stabilization of Linear Systems With Multiple Delays and Saturations in the Input

• Published in: SIAM journal on control and optimization Vol. 48; no. 8; pp. 5294 - 5332
• Main Authors: Zhou, Bin, Lin, Zongli, Duan, Guang-Ren
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Applied sciences; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control theory. Systems; Delay; Eigenvalues; Exact sciences and technology; Linear equations; Linear systems; Mathematical analysis; Mathematical models; Mathematics; Matrix; Ordinary differential equations; Output feedback; Saturation; Sciences and techniques of general use; Stabilization; Studies; System theory; State feedback; delayed input; stabilization; Differential equation; Feedback regulation; Multiple delay; Linear control; multiple time delays; Time varying system; Linear equation; Eigenvalue problem; Delay time; actuator saturation; Matrix equation; time-varying delays; Actuator; Stability; 34D23; Operator equation; Functional equation; Input saturation; Delay system; Lyapunov equation; 93D15; Global stabilization; 34K06; Open loop; Riccati equation; 93C30; parametric Lyapunov matrix equation; Output feedback
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/090771673

This paper studies stabilization problems for linear systems with multiple delays in the input. Two types of delays are considered. The first type of delays is constant delays, which can be arbitrarily large, while the second type is time-varying with an arbitrarily large bound. With the first type of delays, under the condition that the open loop system is absolutely controllable with all its eigenvalues on the imaginary axis, (globally) stabilizing state and output feedback laws are constructed based on the solution to a family of parametric Riccati equations, which can be obtained explicitly through the solution of a parametric linear matrix equation. With the second type of delays, under the condition that the open-loop system is absolutely controllable with all its eigenvalues on the imaginary axis being zero, (global) state and output feedback laws are explicitly constructed based on the solution to a similar family of parametric Riccati equations. When the input is also subject to magnitude saturation, it is shown that semiglobal stabilization, instead of global stabilization, can still be achieved. Numerical examples illustrate the effectiveness of the proposed approach. [PUBLICATION ABSTRACT]