Development of optimization Alghorithm for uncertain non-linear dynamical system

Abstract

Nonlinear optimal control problems are problems involving real world situations where the objectives are the maximization of the return from, or the minimization of the cost of, the operation of physical, social, and economic processes. Algorithms used to solve these problems are expected to satisfy the objectives consistently and since time translates into cost, must also be fast. An algorithm that definitely can satisfy the objectives is the Dynamic Integrated Systems Optimization and Parameter Estimation (DISOPE) algorithm. However, this algorithm has an inherent problem of slow convergence due to its gradient descent type updating mechanism. Hence, the purpose of this study is to overcome this convergence problem by modifying the mechanism. Two approaches were chosen for this purpose. The first is the use of momentum terms and the second is the parallel tangent method. Two new algorithms named DISOPE-MOMENTUM and DISOPE-PARTAN sprouted from these modifications and extensive simulations were performed to observe their performances. To strengthen the findings, theoretical analyses were done on each algorithm. These include optimality, stability, convergence, and the rate of convergence analyses. Based on the results of these simulations, we compared the number of iterations needed by each algorithm to arrive at the optimal solution and the CPU time taken for each algorithm to execute the search. From the theoretical analyses, comparisons were done on the speeds of contraction of the algorithms. Both new algorithms managed to arrive at the optimum in fewer numbers of iterations and in shorter CPU times than DISOPE without compromising on the accuracy of the solutions. The new algorithms also boast faster contractions. Both new algorithms performed better than DISOPE. This study succeeded in overcoming the problem of slow convergence and with the modifications, the new algorithms become more efficient in solving the optimal control problems.