Composite quadratic Lyapunov functions for constrained control systems

• Published in: IEEE transactions on automatic control Vol. 48; no. 3; pp. 440 - 450
• Main Authors: Hu, Tingshu, Lin, Zongli
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2003; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuators; Applied sciences; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control systems; Control theory; Control theory. Systems; Ellipsoids; Exact sciences and technology; Feedback; Hulls; Hulls (structures); Invariants; Level set; Linear matrix inequalities; Linear systems; Lyapunov functions; Lyapunov method; Mathematical models; Stability analysis; Studies; Sufficient conditions; Invariant; Convex set; Stability; Saturation; Feedback regulation; Continuous time; Linear time; Constrained optimization; Control constraint; Convex hull; Linear system; Ellipsoid; Actuator; State constraint; Lyapunov function; Quadratic function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2003.809149

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.