Numerical methods for nonlinear optimal control problems using haar wavelet operational matrices / Waleeda Swaidan Ali - Waleeda Swaidan, Ali (2015)
Several computational methods have been proposed to solve optimal control problems. These methods are classified either as direct or indirect methods. This thesis is based on solving optimal control problems by using both direct and indirect methods. Orthogonal functions have been used to solve various problems of dynamic systems. A typical example is the Haar wavelet function, which is used in this work to convert the underlying differential equations in an optimal control problem into a system of linear algebraic equations. To establish an indirect method, we propose a novel feedback control method that uses only linear systems to solve affine nonlinear control system with quadratic cost function and infinite time horizon. This method is a numerical technique that is based on the combination of Haar wavelets operational matrices and successive Generalized Hamilton-Jacobi-Bellman (GHJB) equation. This method improves the closed-loop performance of stabilizing controls and reduces the problem of solving a nonlinear Hamilton-Jacobi-Bellman (HJB) equation to solve the corresponding GHJB equation. An interesting fact is that when the process of improving the controls and solving GHJB equation is iterated, the solution to the GHJB equation converges uniformly to the solution of the HJB equation which is in the form of the gradient of the Lyapunov functionV(x) . The Lyapunov function V(x) is the measure of the performance index, which can be determined by integrating V(x) parallel to the axes. In the process of establishing this novel feedback control method, we have to define new operational matrices of integration for a chosen stabilizing domain [ , ) and a new operational matrix for the product of two dimensional Haar wavelet functions. To establish a direct method, an efficient new algorithm is proposed to solve nonlinear optimal control problems with a finite time horizon under inequality constraints. In this technique, we parameterize both the states and the controls by using Haar wavelet functions and Haar wavelet operational matrix. The nonlinear optimal control problem is converted into a quadratic programming (QP) problem through the quasilinearization iterative technique. The inequality constraints for trajectory variables are transformed into quadratic programming constraints by using the Haar wavelet collocation method. The quadratic programming problem with linear inequality constraints is then solved by using standard QP solver. Both proposed numerical methods have been applied to several examples. The proposed methods obtain better or comparable results compared with other established methods. Moreover, the methods are attractive, stable, convergent and easily coded. The direct method has been applied in this thesis to solve a practical optimal control problem. This problem is the multi-item production-inventory model with stock-dependent deterioration rates and deterioration due to self-contact and the presence of the other stock. The problem is addressed by using four different types of demand rates namely, constant, linear, logistic and periodic demand rates. The solution to the model is discussed numerically and displayed graphically. By enhancing the resolution of the Haar wavelet, we can improve the accuracy of the states, controls and cost. Simulation results were also compared with those obtained by other researchers.