Optimal Control Of Continuous Chaotic Systems

The optimal control theory which developed from the late 1950s to the early 1960s is an important branch of modern control theory; its formation and development have laid the foundation of modern control theory. Chaotic motion is a complex motion, whose equation is certain but the trajectory of the orbits is stochastic. There are lots of chaotic phenomena in real world, the chaotic systems which have a high application value are a kind of the special nonlinear system, and the optimal control of chaotic systems gets widespread application in the control of systems.In this paper, we present some techniques for the optimal control of chaotic systems based on the basic characteristics of chaotic systems and the basic theory of optimal control.(1) The dynamic behaviors of a new chaotic system are described. Then a quadratic performance is given and a simple linear state feedback controller is designed based on Pontryagin Minimum Principle and the resulted control law is proved to be optimal control. The system orbit can be controlled to its originally unstable zero equilibrium point by the designed controller. The structure of the controller is simple and the controller is easy to attain. Based on Lyapunov functional method, the stability of system is proved. Simulation results of the transient process of the states of the closed control system are provided to demonstrate the effectiveness of the suggested scheme.(2)A method of optimal control which takes the orbit of system to any desired point is provided based on Lorenz system. Based on Hamilton-Jacobi-Bellman equation, the problem of constructing the optimal controller comes down to the problem of solving the partial differential equations. The optimal controller is obtained through constructing Lyapunov function. The result of simulation shows the effectiveness of the method.(3)The paper is devoted to discuss the problem of optimal control of unified chaotic systems with complete unknown parameters. The equilibrium points of unified chaotic systems were given and stability of the equilibrium points was analyzed. Then the state feedback controllers are designed based on Pontryagin Minimum Principle. Theorize analysis show that the equilibrium points of systems, which are essentially unstable, can be stabilized by the optimal controllers. The effectiveness of the method is verified according to numerical simulations of unified chaotic systems with complete unknown parameters.(4) A nonlinear state feedback controller is designed for controlling the error system of synchronization of Rossler using time minimization control approach. Based on the Lyapunov stability theory, the designed controller is proved enable to globally stabilize asymptotically the controlled system to its zero point and minimize the proposed cost functional. The numerical simulation shows the effectiveness and readiness of the controller.