Optimal control for linear singular system using genetic programming

Abstract

In this paper, optimal control for linear singular system with quadratic performance is obtained using genetic programming (GP). The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the matrix Riccati differential equation (MRDE), obtained from well known traditional Runge–Kutta (RK) method and genetic programming method. To obtain the optimal control, the solution of MRDE is computed based on grammatical evolution. Accuracy of the solution of the GP approach to the problem is qualitatively better. An illustrative numerical example is presented for the proposed method.

Introduction

Genetic programming is an evolutionary algorithm that attempts to evolve solution to problem by using concepts taken from naturally occurring evolving process [14]. The technique is based on the evolution of a large number of candidate solutions through genetic operations such as reproduction, crossover and mutation. It is based upon the genetic algorithm (GA) [11], which exploits the process of natural selection based on a fitness measure to breed a population of trial solution that improves over time. While a GA usually operates on (coded) strings of numbers but GP uses the principles and ideas from biological evolution to guide the computer to acquire desired solution. The search space is much too large to attempt a brute force search, the method must be utilize to reduce the number of examined solutions. In this search we would often find that initially the population looks a bit like a cloud of randomly selected points, but that generation after generation it moves in the search space following a well defined trajectory. The generation is achieved with the help of grammatical evolution, because grammatical evolution can produce programmes in an arbitrary language, the genetic operation are faster and also because it is more convenient to symbolically differentiate mathematical expression. The code production is performed using a mapping process governed by grammar expressed in Backus NaurForm (BNF) [22]. In analogy to nature, the potential solution as an individual in some collection or population of potential solutions. The individuals who are stronger, meaning higher ranked according to fitness function, will be used to determine the next collection of potential solution. By employing analogs of reproduction and mutation a new generation will arise.

This means that GP has advantages over other algorithms as it can perform optimization at a structural level. This enabled Koza [14] to demonstrate the application of GP algorithm to a number of problem domains, including regression, control and classification. Research in this area has grown rapidly and encompassed a wide range of problems. GP techniques have been successfully applied in various engineering fields like signal processing [29], electrical circuit design [15], scheduling [21], process controller evolution [30] and modelling of both steady-state and dynamic processes [20].

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], [8], [18].

Many practical processes can be modelled as descriptor systems such as constrained control problems, electrical circuits, certain population growth models and singular perturbations. In the past years, stability and control problems of descriptor systems have been extensively studied due to the fact that the descriptor system better describes physical systems than the state space systems. Compared with state space systems, the descriptor system has a more complicated yet richer structure. Furthermore, the study of the dynamic performance of descriptor systems is much more difficult than that for state space systems since descriptor systems usually have three types of modes, namely, finite dynamic modes, impulsive modes and non dynamic modes [9], while the latter two do not appear in the state space systems.

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 [10] showed that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Solving the Matrix Riccati Differential Equation (MRDE) is the central issue in optimal control theory. The needs for solving such equations often arise in analysis and synthesis such as linear quadratic optimal control systems, robust control systems with H₂ and H_∞ control [31] performance criteria, stochastic filtering and control systems, model reduction, differential games, etc. One of the most intensely studied nonlinear matrix equations arising in mathematics and engineering is the Riccati equation. This equation, in one form or another, has an important role in optimal control problems, multivariable and large scale systems, scattering theory, estimation, detection, transportation and radiative transfer [12]. A variety of numerical algorithms have been developed for linear and nonlinear systems [4]. The solution of this equation is difficult to obtain from two points of view. One is nonlinear and the other is in matrix form. Most general methods to solve MRDE with a terminal boundary condition are obtained on transforming MRDE into an equivalent linear differential Hamiltonian system [13]. By using this approach, the solution of MRDE is obtained by partitioning the transition matrix of the associated Hamiltonian system [28]. Another class of method is based on transforming MRDE into a linear matrix differential equation and then solving MRDE analytically or computationally [19], [25], [27]. However, the method in [24] is restricted for cases when certain coefficients of MRDE are non-singular. In [13], an analytic procedure of solving the MRDE of the linear quadratic control problem for homing missile systems is presented. The solution K(t) of MRDE is obtained by using $K(t)=\frac{p(t)}{f(t)}$, where f(t) and p(t) are solutions of certain first order ordinary linear differential equations. However, the given technique is restricted to single input.

Although parallel algorithms can compute the solutions faster than sequential algorithms, there have been no report on solutions for MRDE using GP, that is compared with RK method solutions. Recently, Balasubramaniam et al. [1], [2] focused upon to find the optimal control for singular systems using neural networks. These papers motivated to study the optimal control of the singular system using GP. Many methods have been proposed for solving ordinary differential equations and partial differential equations such as RK, Predictor-Corrector method [16] and feed forward neural networks [17]. Recently methods based on GP have also been proposed in [3], [5]. This paper focuses upon the implementation of genetic programming approach for solving MRDE in order to get the optimal solution. The method forms generation of trail solution expressed in an analytical closed form. When the solution is not in closed form, it will produces approximate solution which is closer to analytical solution. An example is given which illustrates the advantage of the method and accurate solutions compared to RK method.

This paper is organized as follows. In Section 2, the statement of the problem is given. In Section 3, solution of the MRDE is presented. In Section 4, Genetic programming methodology is given In Section 5, numerical example is discussed. The final conclusion section demonstrates the efficiency of the method.