The Successive Approximation Procedure For Finite-time Optimal Control Of Bilinear Systems

Most systems are nonlinear in industrial process, and all work with external disturbance. The traditional method is to linearization, however, when the input variable margin is bigger, the operating point move, the external disturbance fluctuation is obvious, especially when the object is complex industrial process. For instance, chemical production, process control, biological, ecological and sociological. This simple linear approximation model is can't satisfy those processes. In order to describe the actual object better, we can use the mathematical model with simple structure which can reflect or approximate the true characteristics of the original system model. So we found the bilinear systems which can well control the actual systems and improve the control effect.Based on the present situation on the optimal control problem of bilinear systems this thesis studies the approximate optimal disturbance rejection control for bilinear systems by using the successive approximation approach. The major results are shown as follows:Give a brief overview of bilinear systems, introduce the current domestic and foreign bilinear system situation, development. And overview of optimal control theory and its main research method. The latest research tendency of optimal control problem for bilinear systems and the main methods of disturbance disposal are presented. Then gives the theoretical basis: differential constraints, the Hamiltonian canonical equation. Based on the optimal control theory of linear systems we study the optimal control for bilinear systems. design simple optimal control law for linear systems, and then propose optimal control for general bilinear systems based on linear theory, describe bilinear system model and give the design of the successive approximation optimal control law. For a class of bilinear systems whose additive external disturbance is denoted by exosystem, we give the approximation design procedure of the optimal disturbance rejection control law. By introducing the design method of optimal linear systems, the bilinear systems with general external disturbance and sine disturbance optimal control is given. We analyze the bilinear system with disturbance, and the nonlinear two-point boundary value (TPBV) problem, which is derived from the original nonlinear optimal control problem, is rewritten into a new TPBV form whose state vector and adjoint vector are coupled, then we construct sequences make the new TPBV problem decoupled by using the successive approximation approach, and the new TPBV problem is transformed into a sequence of nonhomogeneous linear TPBV problems. We prove the sequence of the solutions uniformly converges to the optimal control law for the systems. the optimal disturbance rejection control law is obtained in finite-time domain and infinite-time domain by solving the adjoint vector sequence iteratively, which consists of an accurate linear term and a nonlinear compensating series.A successive approximation approach designing the optimal tracking control law is discussed for bilinear systems with general and sinusoidal (periodical) disturbances and respect to the quadratic performance index. Firstly, we study the MIMO linear system's optimal tracking control, give the model of linear system with sinusoidal (periodical) disturbances, design a feedforward - feedback optimal control law. Then we design output trackers for the bilinear system with general and periodical disturbances respectively. The optimal tracking control law which we get can makes the output meet the requirements of actual process.Overall thesis summarized as follows: First give the theoretical basis, based on optimal control of the linear optimal control, we study the bilinear systems, including general sinusoidal disturbance with the successive approximation approach. And we also give the optimal tracking control for bilinear systems with the general disturbance and sinusoidal disturbances. The approach we use is successive iterative approximation algorithm. This algorithm can broaden the study of the bilinear systems, improve the control performance of bilinear systems. At last concluded thesis work and innovation, and give the direction for future research...