Study On Regular/Irregular Linear Quadratic Optimal Control Problem And Its Applications

Linear quadratic(LQ)optimal control is one of the core foundation and basic problems of modern control theory,which has important applications in engineering,economy and other fields.In recent decades,LQ control has been widely investigated,but there are still some basic problems facing challenges.The singular LQ control problem(regular/irregular control)studied in this paper is one of these challenges.Specifically,singular LQ control means that the control weighting matrix in quadratic performance index is singular,in which case partial controllers are not solvable,that is,the Riccati equation is irregular.In this paper,a novel decoupling method for forward and backward differential equations(FBDEs)is proposed.The necessary and sufficient conditions and explicit analytical solutions for irregular LQ control are obtained,which reveals the essential difference between regular and irregular control,that is,the state terminal needs to satisfy the constraints.Based on these,the results are applied to the study of finite-time stabilization,and the problem of regular/irregular finite-time optimal stabilization is solved.Meanwhile,the stochastic irregular LQ control problem with additive noise and multiplicative noise is studied,and the explicit solution of optimal control is obtained.Another contribution of this paper is that we have applied the design of regular controller(singular control)to consensus control and tracking control.For consensus control,the optimal consensus control strategy is designed,which not only makes the system achieve the mean-square consensus,but also minimizes the performance index.For tracking control,the analytical form of the finite-time horizon optimal tracking controller and the expression of the infinite-time horizon stable optimal controller are obtained.In this paper,the more general singular case is considered,where the control weighted matrix is singular,which is different from the existing results.The key technology is to make full use of the free term of regular control to solve the optimal consensus and tracking control problems.The specific research contents and achievements include the following aspects according to the order of chapters:1.The problem of determinate irregular LQ control and its application to finitetime stabilization are investigated.Based on the results of regular/irregular optimal control of deterministic systems,the conditions for ?-finite-time stabilization and ?finite-time optimal and stabilization with open-loop/closed-loop solvability are given,respectively.The open-loop solvability is equivalent to the range of two matrices satisfying the corresponding inclusion relation.The closed-loop solution is given in the form of state feedback.In particular,?-finite-time stabilizability of the system will generate the solvability of the irregular optimal control problem.2.The stochastic irregular LQ control problem is considered.The irregular optimal control problem is not solvable for the standard quadratic performance index when additive noise and multiplicative noise exist at the same time.By defining the terminal quadratic index of the expected product of state terminal,the problem can be transformed into a special kind of irregular mean-field control problem.Using the maximum principle,the solvability of the optimization problem is transformed into solving FBDEs.According to the Riccati equation,the analytical solution of FBDEs is obtained innovatively,that is,the non-homogeneous relationship between the costate,state and the expectation of the state is established,from which the solvable conditions of stochastic irregular optimal control and the form of optimal controller are presented.3.The application of regular optimal control in multi-agent consensus control is discussed.For Ito stochastic systems with multiplicative noise dependent on state and control input,under the consensus protocol of relative state feedback,the solvable critical parameters of parameterized algebraic Riccati equation are proposed innovatively.By designing consensus gain,the conditions of mean-square consensus of multi-agent systems are derived.Using the free term of the regular controller,the optimal consensus protocol is designed,and the optimal consensus control conditions of Ito stochastic multi-agent systems are given in regular case.4.The application of regular optimal control in tracking control is studied.For Ito stochastic systems with input delay and multiplicative noise,the solvability of optimal tracking control problem is equivalent to solving FBDEs by using maximum principle.Applying a proposed coupled Riccati like equation,the non-homogeneous relationship between backward and forward stochastic processes is creatively established,and the solvable conditions of tracking control and the analytic form of optimal controller are given in regular case.