An iterative procedure for optimal nonlinear tracking problem

It has been shown that for a class of nonlinear systems x=f(x)+g(x)u, the solution of the infinite horizon optimal regulation problem leads to a state dependent Ricatti equation. Under appropriate assumptions the optimal control may be obtained from the point-wise solution of an algebraic Ricatti equation during state evolution. This method cannot be used in finite horizon optimal regulation and finite or infinite horizon of optimal tracking problem. To solve the nonlinear quadratic regulation or tracking problems, two forward and backward equations corresponding to the state and co-state systems respectively must be solved and then the optimal control can be derived. Since the co-state equation is state-dependent and it develops backward and the state is not accessible in whole time, then the control law cannot be calculated. To overcome this problem, an iterative procedure is proposed. This method can be applied to both finite and infinite horizon optimal regulation and tracking problems. Simulation results are given for the nonlinear benchmark problem introduced in and Lorenz attractor as a chaotic system.