NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

3.5 Successive Galerkin Approximations
The primary purpose for implementing the four previous designs was to pro-
vide a reasonably complete set of control laws with which to compare the successive
Galerkin approximation algorithm as developed in [5]. This control strategy seeks to
solve the non-linear H 2 and H1 problems by approximating the solutions to their
associated Hamilton-Jacobi equations. These equations are non-linear partial dif-
ferential equations that are impossible to solve analytically in the general case. To
accomplish the approximation the Hamilton-Jacobi equations are rst reduced to an
in nite sequence of linear partial di erential equations, named generalized Hamilton-
Jacobi equations. Second, Galerkin's method is used to approximate the solutions
of these linear equations, and the combination of these two steps yields a control
algorithm that converges to the optimal solution as the order of the approximation
and the number of iterations goes to in nity.
3.5.1 The H 2 Problem
The SGA technique was rst applied to the non-linear optimal H 2 problem,
where the goal is to minimize a cost functional V with respect to some u (x):
_x = f(x)+g(x)u
Z
V (x) =min l(x)+kuk
u
2
R dt
where l(x) is some cost function that depends on the state x, and R is the matrix
that weights the cost of the control. Throughout this development we shall assume
that f(0) = 0, that l(x) is a positive de nite function, and that f(x) is observable
through l(x). The solution to this minimization problem is given by the full-state
feedback control law
u (x) =; 1
2 R;1 T @V
g
@x
24
(3.22)