Mixed H₂/H_? Control Problem Based On Stackelberg Game Approach

Optimal control is one of the important research area of modern control theory,the main idea is to select the optimal controller from a class of allowable controller-s according to the mathematical model of the controlled object,so as to make the controlled object run from an initial state to a specified target state,and optimize a given performance index.The research on H2 optimal control and H? optimiza-tion control has great significance to the development of modern control theory.H2 optimal control can make the system have fast response speed and other excellent characteristics,but it lacks robustness to uncertainties caused by some extra distur-bances or modeling errors.H? control has good robustness to the uncertainty of the system,but it sacrifices some systems' performance.Based on this,combining the advantages of the two design methods,mixed H2/H?,control has been intro-duced to minimize a desired index function while guarantee the robust performance of the system.The problem is very difficult to solve.So in most of the solutions,mixed H2/H? optimization problem has been simplified,deformed or translated in-to optimal control problem with Nash game approach,or converted into a problem solving Riccati equation problem,the LMI problem and convex optimization prob-lem by adding some limited conditions and assumptions.In this paper,the mixed H2/H? control problem is firstly converted into Stackergerg optimal control prob-lem,then the optimal solution of the problem is obtained,and the results are extended to time delay systems and stochastic systems.The closed-loop solution of the mixed H2/H? control problem is obtained under the condition that the control weighting matrix of the performance index is positive semi-definite.The main chapter arrangements and the innovation achievements are as follows.First,the background and research status of this article are given.In this chapter,the importance of mixed H2/H? control in theory and practice is described,at the same time,the existing research achievements and existing problems are pointed out.Second,the open-loop Stackelberg game approach on mixed H2/H? control problem is considered.The mixed H2/H? control problem is equivalently described as Leader-followers optimal control,namely the Stackelberg control,wherein,the control input is treated as leader and the disturbance is treated as follower.The op-timal control theory is applied to study Stackelberg game control.By introducing a new co-state to capture the future information of the leader,sufficient and necessary condition for the unique solution of H2/H? open-loop control is given.Under stan-dard assumption,the analytical solution of the optimal control strategy is given in terms of three decoupled and symmetric Riccati equations.Third,we apply the open-loop Stackelberg game approach on mixed H2/H?control problem with input-delay.Necessary and sufficient condition for the unique-ness and existence of the problem is given,moreover,the analytical expression of the optimal control strategy is given.By introducing two costate state variable to get future information and introducing a new kind of state to obtain the influence of the past,the non-causality caused by the input delay in the system is overcome.Thus,sufficient and necessary condition is presented to guarantee the existence of the unique solution of the mixed H2/H? control problem with input-delay and the optimal controllers are designed by solving the three decoupled symmetric Riccati equations.Fourth,we study the linear discrete-time stochastic mixed H2/H? control with(x,u,v)-dependent noise.Based on the results of the certain system,by applying stochastic control maximum principle,the solution of the problem is equivalently transformed as the solution of the forward and backward stochastic difference equa-tions(FBSDEs).And then we introduce a new co-states to obtain the FBSDEs.The solution of the FBSDEs is given by solving symmetric and decoupled Riccati equa-tions,spontaneously,we get the open-loop solution of the leader-follower stochastic game problem.Then,the multiplicative noise stochastic mixed H2/H? open loop control problem is solved.Fifth,for mixed H2/H? control problem with positive semi-definite control weighting matrix,the closed loop solution of H2/H? mixed control is obtained by using the regular Riccati equation.The key technology is based on a two-step optimization process.The first step is to obtain the optimal control strategy(u*,w*).Due to the positive semi-definite property of the control weighting matrix,arbitrarily item appear on u*.In the second step,by matrix conversation,the arbitrarily item is reformulated as strategy which ia waiting to be solved in H2 optimization procedure.Based on these two optimization procedures,under regular assumption,the closed-loop solution of mixed H2/H? positive semi-definite control problem is given.