Research On Adaptive And Optimal Control For Nonlinear Systems Based On Neural Networks

The optimal control problem of nonlinear systems has always been the chal-lenging problem in the control field. The existed optimal control methods, including variational calculus, maximum principle and dynamic programming, all have their respective limitations and the analytical optimal solution is hard to gain. Therefore, adaptive dynamic programming, as a new algorithm to approximately solve the op-timal control problem, has gained much attention from a lot of researchers. The adaptive dynamic programming algorithm can overcome the "curse of dimensional-ity", and meanwhile obtain the approximate optimal closed-loop feedback control law. However, the most existed results on adaptive dynamic programming focus on the stabilization control of unconstrained nonlinear systems. The optimal stabiliza-tion and optimal tracking problems for the constrained nonlinear systems are still unsolved. Also, the improvement and perfection of the adaptive dynamic program-ming algorithm, such as the stability analysis, the convergence analysis, have been a hot topic. Therefore, in this dissertation, the adaptive dynamic programming algorithm is proposed based on neural networks to solve the optimal stabilization control and adaptive optimal tracking control problems of constrained nonlinear sys-tems with rigorous convergence analysis, which provides new ideas and new results for the analysis and control of complex nonlinear systems. The main contents of the dissertation can be briefly described as follows:1. Propose a novel neural network controller design scheme for a class of afflne nonlinear system with unknown actuator dead-zone. First a neural network is intro-duced to estimate the partial unknown nonlinear dynamic, and then another static network is constructed as a novel compensator to overcome the unknown nonsym-metric dead-zone behavior of the actuator based on the implicit function theorem. Lyapunov theory is used to present the smooth control law and prove the uniform ultimate boundedness of the close-loop tracking error and the networks weights. Furthermore, the tracking error is able to converge to a small neighborhood around zero by adjusting the design parameters.2. Based on the RBF neural networks, solve the approximate optimal control problem of a class of constrained discrete-time nonlinear systems. First propose a novel nonquadratic functional to deal with the control constraints of nonlinear discrete-time systems and derive the corresponding discrete-time HJB equation. Then prove that the iterative cost function sequence converges to the optimal cost function, i.e., the infimum of all the cost functions obtained by all admissible control law sequences, and show that this optimal cost function satisfies the HJB equation. Furthermore, implement the iterative adaptive dynamic programming algorithm by introducing the costate function, which gets rid of the computations of a derivative term and a integral term appearing in solving the optimal control law. Meanwhile, the RBF neural networks utilized to approximate the costate function and the cor-responding optimal control law. Specifically, use a model network to approximate the nonlinear system dynamics, which renders the iterative adaptive dynamic pro-gramming algorithm suitable to plants whose mathematical models are unknown.3. Based on the GI-GDHP algorithm, the approximate optimal tracking control problem is studied for a class of discrete-time non-affine nonlinear systems. First, propose an infinite-time optimal tracking control scheme which is composed of a feedforward control term and a feedback control term for a class of discrete-time non-affine systems. Then solve the optimal control problem of the error system with time-varying parameters by converting the error system into a augmented system with the dimension augmentation technique. Furthermore, develop a new iteration algorithm named GI-GDHP algorithm for discrete-time non-affine nonlinear systems with unknown mathematic models.4. Propose two approximate optimal control schemes for a class of nonlinear descriptor systems with control constraint. The first scheme transforms the descrip-tor system into a common state space form, and then introduce the greedy iterative DHP(GI-DHP) algorithm to solve the approximate optimal control problem. The second scheme directly implements the costate function iteration and the optimal control law iteration to compute the approximate optimal control law for the orig-inal descriptor system, and provides the corresponding convergence analysis of the iteration process.5. The near-optimal control problem for nonlinear constrained discrete-time systems is solved by single network greedy iterative DHP(GI-DHP) algorithm. First, a nonquadratic functional is developed to change the constrained problem into a un-constrained problem so that the derived control policy is smooth. Then in order to overcome the difficulty of solving the HJB equation analytically, the GI-DHP algo-rithm is proposed to solve the optimal costate function and optimal control policy with associated convergence analysis. Moreover, a new implementation scheme for the iterative algorithm is presented, where the action network which is needed in the usual adaptive dynamic programming(ADP) scheme is removed. In the new scheme, only a network is used to approximate the costate function, and then the optimal control policy can be directly computed based on the costate function. In this way, the implementation process of the iterative algorithm is greatly simplified, and the computing burden can be effectively decreased.Finally, concluding remarks are given. Some unsolved problems and develop-ment direction for adaptive dynamic programming method are proposed. Further, the prospects of the future study are given.