Functional Method And Application Of Optimal Control For Switching Systems

Switched systems is an important kind of hybrid systems with research significance,which is widely used in many research fields.The key of the switching systems is that it has a switching strategy,so that the system runs in a given switching rule,and ultimately achieve the desired results,such as stability,optimal or suboptimal performance index.This paper studies the following three innovative conclusions:Firstly,we abstract the system state equation and performance index functional in the engineering optimal control problem.The definition of the second variation of the dual functional and the second-order part of the operator are given.Based on the known first variation of the abstract dual functional and the linear part of the operator,we get the second variation expression of the new functional of the composite operator and the sufficient condition of the functional to reach the extreme value.Finally,we get a general conclusion of linear(or nonlinear)switched system when its performance index reaches the optimum.Then,we consider that in specific case,the performance index of the system is often expressed by the functional in the form of abstract integral.When the motion state of the system satisfies a set of differential equations,for this kind of problem,we obtain the sufficient conditions when the performance index reaches the extreme value.Secondly,we consider the linear switching systems with quadratic performance index function,and design the controller with limited initial state and limited terminal state.Finally,we study the switching rules of two-dimensional linear switching systems.We classified the two-dimensional linear switching system,and the optimal switching problem when the system state motion is moved to a particular angle is analyzed.