Research On Optimal Control Theory And Its Several Applications For Nonlinear Systems

The analysis and synthesize of optimal control is one of the hottest research topics in nonlinear control field. The main idea of optimal control theory is that it tries to determine an optimal control strategy for controlled dynamic systems such that the states are forced to the desired ones and a certain optimality is achieved. Recently, the optimal control topic has gradually attracted considerable attention and plentiful results have been published. For example, the minimum fuel problem and the smallest tracking error problem for spacecraft systems. However, there still exist many unsolved problems in optimal control field. In this paper, several optimal control problems are investigated, and the main contents are as follows:optimal control strategy design for systems with disturbance effects; finite-time based optimal control strategy design for nonlinear systems; applications of various optimal control strategies to various typical nonlinear systems. The results and contributions can be divided into two parts, i.e., the theory and the practical application. Specifically,(1) Optimal control strategy design for a class of nonlinear systems with disturbance. Firstly,θ—D method is applied to design a suboptimal control law with nominal model in the absence of disturbance. Then, disturbance is estimated by disturbance observer, and the disturbance estimation is integrated into the previous suboptimal controller to form a new optimal controller with disturbance attenuation. Finally, semi-global stability analysis of nonlinear systems with the composite suboptimal controller is proved.(2) Suboptimal sliding mode control design for a class of nonlinear affine systems with disturbance. First, a general form of integral sliding mode is presented. An extended θ— D method is developed for the optimal control problems characterized by a quadratic cost function with a cross term. Then the extended θ — D method is employed to determine a suboptimal integral sliding mode. Finally, the suboptimal integral sliding mode controller is obtained. Rigorous proof shows that the controller guarantees semi-global asymptotical stability of affine systems.(3) Optimal sliding mode control design for a class of affine systems with disturbance and constraints of states and control. First, a general form of integral sliding mode is presented. Pseudospectral method has a high convergence speed and performs well in solving optimal control problems with non-standard per-formance index, endpoint conditions and path constraints. In consideration of these advantages, an optimal integral sliding mode controller is determined by pseudospectral method. Finally, the stability analysis of optimal pseudospectral sliding mode method is discussed.(4) Finite-time optimal state feedback control design for a class of nonlinear systems. Firstly, a finite-time controller by adding a power integrator technique is presented. Then, the undetermined parameters of this finite-time controller are selected by using the inverse optimal method. Rigorous proof shows that the finite time controller based on inverse optimal method can guarantee finite-time stability of the closed-loop system.(5) Optimal output feedback control design for a class of nonlinear systems with partial measurable states. Firstly, an observer and a linear output feedback controller are proposed. Then, the parameters of this output feedback controller are selected appropriately by using inverse optimal method with respect to a meaningful cost function. Rigorous proof shows that the optimal output feedback controller based on inverse optimal method can guarantee the global stability of closed-loop system.(6) Optimal control design for overhead crane systems control problem with states and control con-straints. Here, the optimal controller is designed to minimize both the transport time and the pendulum angle because states and control inputs have many constraints in practice, for example, the absolute value of pen-dulum angle range should be less than 12° and control input must be in tolerance domain. Moreover, we consider the overhead crane optimal control problem with and without disturbance.(7) Optimal trajectory control design for power-descent phase of Mars landing. Firstly,θ — D method is applied to design a suboptimal control law with nominal powered-descent model for the case of quadratic cost function. Then, the disturbance is estimated by disturbance observer. A nonlinear optimal controller is skillfully constructed by optimal control and the disturbance estimation. Rigorous proof is also given to show the semi-global stability of power-descent system of Mars landing with the proposed controller.(8) Optimal control design for multi-input multi-output (MIMO) nonlinear rigid robotic manipulators system with constraints and disturbance. Firstly, a conventional integral sliding mode is designed. In consid-eration of the advantages of pseudospectral method, integral sliding mode control method and pseudospectral method are skillfully integrated to design an optimal integral sliding mode controller.(9) Finite-time optimal control design for a class of rigid spacecraft control problem. Firstly, a finite-time state feedback controller is designed, and the related stability analysis is also provided. Then, the undeter-mined parameters of this finite time controller are selected by using the inverse optimal method.