Stability Analysis For Constrained Switched Systems And A Solution To Its Optimal Control Problem

Constrained problems have been studied by more and more researchers, as its widely use in practical problem, which mainly contain state constrained problem and controller constrained problem. The stability of switched linear positive system subject to actuator saturation is been studied firstly, whose state and controller are both limited, designing switched law to ensure the stability and positivity of system. However, in addition to stability, the optimal performance also tend to be considered in actual life. Therefore, this paper analyzes the optimal control problem of switched nonlinear system whose controller is constrained, and gives the necessary condition to optimal control in the second part. The main contributions are as follows:In this section, stabilization for a class of positive switched linear systems subject to actuator saturations is investigated. Two state feedback control law was devised, one by exploiting minimum dwell time method which guarantees that the closed-loop system is positive and locally exponential stable, another by state switching which guarantees positivity and locally asymptotical stability. Several sufficient conditions for stabilization and positivity are derived via the multiple linear copositive Lyapunov functions method for the above two cases respectively. The results obtained are formulated in terms of inequalities which are solved by nonlinear programming. Then convex analysis on enlarging the attractive domain is investigated. Finally, two numerical examples are given to illustrate the effectiveness of the two proposed approach respectively.In this section, optimal control problem of switched nonlinear system whose controller is constrained is analyzed. Firstly, some approaches to general optimal control problem are introduced, which are variational method for unconstrained control and Pontryagin’s minimum principle method for constrained control. Then it’s extended to switched system. Optimal control problem of switched nonlinear system with fixed switching sequence whose controller is constrained is analyzed and a two-stage algorithm is proposed for this problem. In the first stage, fix switching time, and then obtain the optimal solution to switched system in this condition; in the second stage, change the switching time and then derive the optimal control of switched system. It should be noted that in the first stage variational method and minimum principles method are extended to switched system, and in the second stage, the optimal control result is obtained by using genetic algorithm when changing the switching time, without regard to solving the partial derivatives of the objective function which choosing switching time as its independent variables.