Machine-Made Synthesis of Stabilization System by Modified Cartesian Genetic Programming

Abstract

A numerical solution of the problem of the general synthesis of a stabilization system by a symbolic regression method is considered. The goal is to automatically find a feedback control function using a computer so that the control object can reach a given terminal position from anywhere in a given region of the initial conditions with an optimal value of the quality criterion. Usually, the control synthesis problem is solved analytically or technically taking into account the specific properties of the mathematical model. We suppose that modern numerical approaches of symbolic regression can be applied to find a solution without reference to specific model equations. It is proposed to use the numerical method of Cartesian genetic programming (CGP). It was developed for automatic writing of programs but has never been used to solve the synthesis problem. In the present work, the method was modified with the principle of small variations in order to reduce the search area and increase the rate of convergence. To apply the general principle of small variations to CGP, we developed special types of variations and coding. The modified CGP searches for the mathematical expression of the feedback control function in the form of a code and, at the same time, the optimal value of the parametric vector which is also a new feature-simultaneous tuning of the parameters inside the search process. This approach enables working with objects and functions of any type, which is not always possible with analytical methods. The need to use the received solution on the onboard processor of the control object imposes certain restrictions on the used basic set of elementary functions. This article proposes the theoretical foundations of the study of these functions, and the concept of the space of machine-made functions is introduced. The capabilities of the approach are demonstrated on the numerical solution of the control system synthesis problems for a mobile robot and a Duffing model.