טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentGrosman Benyamin
SubjectStability Analysis of Nonlinear Control Systems Using
Genetic Programming
DepartmentDepartment of Chemical Engineering
Supervisor Professor Daniel Lewin
Full Thesis textFull thesis text - English Version


Abstract

This thesis describes the use of genetic programming in stability analysis and control synthesis for nonlinear autonomous dynamic systems. The main ideas are associated with the Lyapunov direct method and optimal control synthesis driven by the solution of  the Hamilton-Jacobi-Bellman (HJB) equation.

            A novel genetic programming code was written for the purpose of disclosing non-trivial Lyapunov functions. These functions were used initially for stability analysis, and subsequently for the synthesis of nonlinear optimal controllers. The work required the transformation of abstract mathematical concepts into a computer language format. This included satisfying the general Lyapunov conditions for stability, the identification of connected sets, the detection of their boundaries and other related topics. In addition it was necessary to address optimal control issues, through the near-solution of the Hamilton-Jacobi-Bellman (HJB) equation.

            The GP has the capacity to discover non-trivial Lyapunov functions that achieve good approximations to the domains of attraction for a variety of nonlinear dynamic systems. Moreover, the task of finding an approximation to the solution of the HJB equation around a working point was demonstrated on a number of autonomous control systems. In cases where the results included non-polynomial terms that are difficult to solve analytically, this obstacle was overcome by using high-order Taylor series expansions. These expansions were shown to be proper Lyapunov functions, which were analyzed using a positivity test for multivariable polynomials.

            Numerous case-studies were examined, including a comparison of the method with the well-known work of Vennelli and Vidyasagar on detecting domains of attraction. Moreover, the control synthesis was compared with well-established control techniques such as feedback linearization as well as other related works on optimal control.

            The methodology demonstrated in this work represents a viable attractive alternative analysis method for the investigation of nonlinear dynamic systems, both in open and closed loop, which can be harnessed in numerous fields of research where a guideline for disclosing unknown Lyapunov functions is lacking.