Solution of matrix Riccati differential equation for nonlinear singular system using neural networks

ABSTRACT In this paper, the solution of the matrix Riccati differential equation(MRDE) for nonlinear singular system is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the MRDE obtained from well known traditional Runge Kutta(RK)method and nontraditional neural network method. Accuracy of the neural solution to the problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method. Key words: Matrix Riccati differential equation, Nonlinear, Optimal control, Singular system, Runge Kutta method and Neural networks. 1. INTRODUCTION Neural networks or simply neural nets are computing systems, which can be trained to learn a complex relationship between two or many variables or data sets. Having the structures similar to their biological counterparts, neural networks are representational and computational models processing information in a parallel distributed fashion composed of interconnecting simple processing nodes [36]. Neural net techniques have been successfully applied in various fields such as function approximation, signal processing and adaptive (or) learning control for nonlinear systems. Using neural networks, a variety of off-line learning control algorithms have been developed for nonlinear systems [17, 25]. A variety of numerical algorithms have been developed for solving the algebraic Riccati equation. In recent years, neural network problems have attracted considerable attention of many researchers for numerical aspects for algebraic Riccati equations see [13, 14, 37, 3]. Singular systems contain a mixture of algebraic and differential equations. In that sense, the algebraic equations represent the constraints to the solution of the differential part. These systems are also known as degenerate, descriptor or semi-state and generalized state-space systems. The complex nature of singular system causes many difficulties in the analytical and numerical treatment of such systems, particularly when there is a need for their control. The system arises naturally as a linear approximation of system models or linear system models in many applications such as electrical networks, aircraft dynamics, neutral delay systems, chemical, thermal and diffusion processes, large-scale systems, robotics, biology, etc., see [6, 7, 11, 19]. Most of the research on nonlinear singular systems has focused primarily on issues related to solvability and numerical solutions for such systems [5, 9]. The literature on feedback control of nonlinear singular systems is sparse. The feedback stabilization problem for nonlinear singular systems is addressed by McClamroch [22]. In this paper, we make use of a result that generalizes the LQ theory to nonlinear systems to provide a nonlinear design method [4, 21]. This nonlinear quadratic (NLQ) method applies to systems having a broad class of nonlinear dynamics with state dependent weighting matrices. In brief, it turns out that the infinite time horizon LQ regulator problem, when solved afresh at every point on the state trajectory, leads to an asymptotically optimal control policy. The LQ regulator problem converges to the optimal control close to the origin. As the theory of optimal control of linear systems with quadratic performance criteria is well developed, the results are most complete and close to use in many practical designing problems. The theory of the quadratic cost control problem has been treated as a more interesting problem and the optimal feedback with minimum cost control has been characterized by the solution of a Riccati equation. Da Prato and Ichikawa [12] showed that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Solving the MRDE is the central issue in optimal control theory. The needs for solving such equations often arise in analysis and synthesis such as