A factorization approach for solving the Hamilton-Jacobi equations in nonlinear optimal control
The Hamilton-Jacobi equation (HJE) arose early in the last century in the study of the calculus of variation, classical mechanics and Hamiltonian systems. Recently, there has been a renewed interest in HJEs arising in various analysis and synthesis problems in systems theory. The HJE despite providing a necessary and sufficient condition for an optimal control, is very difficult to solve for general nonlinear systems, and therefore its application remained limited to linear systems. Yet, the HJE has been studied extensively in the literature from diverse areas of science and engineering, varying from mathematical physics, to mechanics, control theory, and to partial differential equations. In this dissertation, some analytical approaches for solving the HJEs arising in H∞, mixed H2/H∞ and H2 control problems for nonlinear systems are developed. Two major approaches are presented. The first approach is essentially an inversion or factorization method, and involves solving the HJE like a scalar quadratic algebraic equation with the gradient of the smooth scalar function as unknown. Since the HJE is a quadratic equation in the gradient of the unknown scalar function, we obtain two parameterized solutions which represent a parameterization of all solutions to the HJE. Thus, the problem is reduced to that of factorization of a scalar algebraic equation which we call the discriminant equation (or inequality). The main difficulties with this approach however are: (i) even after obtaining a solution to the discriminant equation, there is no guarantee that the gradient vector obtained subsequently represents a scalar function (i.e. represents a symmetric solution to the HJE); and (ii) there is no guarantee that the resulting solution is positive-definite. However, these difficulties can still be overcome by some additional constraints to the problem. Computational procedures for determining symmetric elementary solutions are then presented. The second approach involves converting the first-order HJ partial differential equation (PDE) to a second-order PDE. Then using a suitable parameterization, this second-order PDE is converted to a coupled system of higher-order nonlinear PDEs which can be solved using some available SYMBOLIC manipulation packages or by other methods. In general, there are no systematic procedures for solving the resulting system of higher-order PDEs, but various ad-hoc procedures can be used. This presents the most serious limitations of the approach. Both the time-varying and time-invariant systems are considered.
nonlinear systems, riccati equations, nonlinear h-infinity control, optimal control, hamilton-jacobi equations
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