Towards nonconservative conditions for equilibrium stability. Applications to switching systems with control delay

Abstract

The objective of the paper is to give a unitary approach to nonconservative conditions of the equilibrium stability. The focus is on a real world system defined by a switched linear mathematical model having control delay. A control synthesis based on predictive state feedback is first applied to compensate the actuator delay. This results in a closed loop switching system free of delay. Then, a theorem giving conservative sufficient conditions of global uniform exponential stability for a switched system is proved. Next, a theorem expressing necessary and sufficient stability conditions is cited, in which the differences compared to the proved theorem are highlighted, but whose mechanism could not be applied in practice. Finally, it is shown how the problem of nonconservativeness can be solved starting from an important theorem given in the literature that uses the idea of homogeneous polynomial Lyapunov functions originating in the Hilbert’s famous 17th Problem. The paper investigates the complexity of the method in terms of numerical computation volume, and offers a way to circumvent the conflict between the method and the mathematical model. In this manner older and newer results are presented in a unitary concept and in a selfcontained approach. Numerical simulations are applied to a real-world model: an active vibration control for a physical model of an intelligent airplane wing in a wind tunnel. The substantial result of the simulations is the minimum guaranteed dwell time that characterizes the equilibrium stability of the switching system with control delay.