Abstract
In this paper, new conditions of the stability, stabilization and L 2 -gain performance of periodic piecewise systems are proposed. Both the continuous and discontinuous Lyapunov functions with dwell-time related time-varying Lyapunov matrix polynomial are adopted, and methods guaranteeing the positive and negative definiteness of a matrix polynomial are introduced. Exponential stability conditions are derived based on continuous and discontinuous Lyapunov functions, respectively. A stabilizing controller with time-varying controller gain is designed with continuous Lyapunov function and the weighted L 2 -gain performance based on the discontinuous Lyapunov matrix polynomial is studied as well. Numerical examples are used to verify the effectiveness of the proposed methods.
Highlights
The periodic system gives a framework for modeling systems with periodic features in engineering, ecological, and economic fields, such as rotor-blade systems [1], satellite attitude control systems [2], communication systems [3]
Periodic piecewise system is a special kind of periodic system because it consists of several subsystems in one period and the switching signal between subsystems is fixed and periodic
The weighted L2-gain performance analysis is carried out based on the discontinuous Lyapunov function with timevarying Lyapunov matrix polynomial, which has not been reported in previous works
Summary
The periodic system gives a framework for modeling systems with periodic features in engineering, ecological, and economic fields, such as rotor-blade systems [1], satellite attitude control systems [2], communication systems [3]. Since control problems of continuoustime periodic systems are more challenging than the discretetime periodic systems [8], analysis and synthesis results of periodic piecewise systems can help study continuous-time periodic systems. P. Li et al.: New Conditions of Analysis and Synthesis for Periodic Piecewise Linear Systems matrix formats. Nonconservative results are obtained for switched systems with guaranteed dwell time and arbitrary switching [26], [27] These methods are adopted for periodic piecewise systems in [15] to handle the time-varying Lyapunov matrix polynomial. Motivated by the above works, new conditions of the stability, stabilizing and L2-gain performance are proposed in this paper based on the matrix polynomial approach. The continuous Lyapunov function could result in a directly solvable periodic time-varying controller. 2. The weighted L2-gain performance analysis is carried out based on the discontinuous Lyapunov function with timevarying Lyapunov matrix polynomial, which has not been reported in previous works.
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