Research On Improved Successive Approximation Approach For Optimal Control Of Nonlinear Systems

In recent years, the optimal control problem of nonlinear systems has been one of the most challenging problems in system and control. For nonlinear systems, its optimal control problem always gives rise to nonlinear Hamilton-Jacobi-Bellman equation or nonlinear two-point boundary value problem, both of which are hard to solve in general. Therefore many approximation approaches was introduced to solve this problem. The successive approximation approach is a familiar one. By transforming the nonlinear two-point boundary value problem into a sequence of linear two-point boundary value problem, the nonlinear two-point boundary value problem is solved iteratively by using this approach. And an approximate optimal control law consisting of a linear state feedback term and an open-loop approximate compensator term is designed. Comparing with the other approximate approaches, the successive approximation approach takes lower computational loads, and still with a good convergence. Therefore it has a good prospect of application. This dissertation presents an improved successive approximation approach by improving on the successive approximation approach in two aspects. Firstly, for affine nonlinear systems, the transformation from nonlinear two-point boundary value problem to the linear sequence of it is rebuilt to design the approximate optimal control law purely in close-loop linear state feedback form. As a result, the close-loop system under this control law becomes more robust. Secondly, the state sequence of the linear two-point boundary value problem sequence approaches the optimal state trajectory more rapidly during the iteration procedure by introducing a close-loop system sequence. Thus the astringency of the iteration procedure is increased.In this dissertation, the history of the development in the optimal control of nonlinear systems is firstly reviewed. And the latest research tendency and the main methods are summarized. By working on the mathematic origin of the successive approximation approach and its application to the nonlinear optimal control problem, a more robust and convergent approach named improved successive approximation approach is proposed. The major results of this dissertation are summarized as follows.1. Based on the improved successive approximation approach, the optimal control for a class of affine nonlinear systems is studied. By using the improved approach, the nonlinear two-point boundary value problem, which is the necessary condition of the optimal control problem, is transformed into a linear two-point boundary value problem sequence that is easy to solve. An approximate solution is obtained by truncating the sequence to a finite iteration, and then an approximate optimal control law with purely close-loop linear state feedback form is designed. Meanwhile, the convergence of the iteration is ensured by proving that the constructed sequence is uniformly convergent to the nonlinear two-point boundary value problem. Also a simulation example is employed to test the validity of the iteration algorithm proposed and its superiority upon the original approach.2. Based on the improved successive approximation approach, the optimal control for a class of Lipschitz-continues nonlinear systems is studied. By separating the affine nonlinear part with the other non-affine nonlinear part, the improved approach together with the original approach is employed to solve this problem. An approximate optimal control law with both a linear state feedback term and an open-loop nonlinear compensator is designed. And the convergence is also proved. Its superiority upon using the original approach alone is shown by a simulation example.3. The optimal control of the nonlinear similar composite system is studied. By using some decoupling methodology, the nonlinear similar composite system is transformed into an affine nonlinear system. Then optimal control of the affine nonlinear system, which is equivalent to the nonlinear similar composite system, is familiar with the problem studied in chapter 1. A close-loop state feedback optimal control is designed, and the convergence also is proved. A simulation example is employed to test the validity and efficiency of the algorithm proposed.4. The optimal control of the nonlinear interconnected large-scale system is studied. By using the improved successive approximation approach, the problem is solved iteratively. An approximate optimal control law with both a linear state feedback term and a nonlinear compensator is designed, and the convergence is also proved. A simulation example is employed to test the validity and efficiency of the algorithm proposed.5. The optimal tracking control of a class of affine nonlinear systems is studied. With respect to the error-based quadratic performance index, the optimal control problem is still come down to a nonlinear two-point boundary value problem. By introducing the improved approach, the problem is solved iteratively, and an approximate control law with both a linear state feedback term and a reference model state feedforward term is designed.6. The optimal sliding mode control of a class of affine nonlinear systems is studied. By introducing the improved approach, a virtual approximate optimal control of the nonlinear system is designed. Then, according to the theory of linear switching manifold design methodology, a linear optimal switching manifold is designed. And a sliding mode control law is obtained based on the designed linear optimal switching manifold.7. The conclusions are made. And the direction for the future study is indicated.