The Application Research Of Optimal Control Numerical Algorithms For Nonlinear Systems Based On The Switched System Method

Switched systems,as a special kind of hybrid systems,are formed by a series of continuous-time or discrete-time subsystems and a switching rule that orchestrates the switching among them at all time.In practical engineering,switched system methods have become an effective tool to solve the optimal control problem of the complex nonlinear system,and are widely applied in fields of mechanical systems,manufacturing,medical applications and many other fields.Therefore,the optimal control numerical algorithm for nonlinear systems based on the switched system method,are of great important theoretical significance and practical appliations.Main contributions in this paper are summarized as follows:1.In cancer chemotherapy,in order to reach the terminal target of minimizing the number of cancer cells,we propose an optimal control problem with the state and control input constraint.For the reasons that modes switch under state-dependent,and the existing optimal control approaches of switched systems with time-dependent switching can not directly used to solve this problem,so we need to develop a new numerical computation method to solve this problem.The problem is formulated as an optimal control problem of switched systems.Firstly,by introducing a binary function and relax the binary function and contain a penalty term on relaxation,we derive an equivalent optimal control problem of constrained nonlinear systems.With the help of the control variable parameterization,the time-scaling transformation,the smoothing technique,and the idea of ₁l penalty function method,the problem is converted into an optimal parameter selection problem.Then,a gradient-based continuous filled function algorithm is proposed to solve this problem.Finally,the simulation example shows the effectiveness of the algorithm.2.In parafoil system of considering obstacle avoidance,for achieving the minimization of the total energy loss in parafoil system,we research the optimal control problem with the state and control input constraint.Consider the open loop control method is often not robust in the practical engineering application,by introducing a novel piecewise state feedback controller,the original problem can be transformed into an optimal control problem of the closed-loop switched system with state inequality constraints.Firstly,with the help of the time-scaling transformation,the smoothing technique and the idea of ₁l penalty function method,the optimal control problem of switched systems is converted to an unconstrained optimal parameter selection problem.Then,an improved conjugate gradient algorithm is proposed to solve this problem and the convergence analysis is proved.Finally,the simulation example demonstrates the effectiveness of algorithm.