NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

where V satis es the well known Hamilton-Jacobi-Bellman (HJB) equation, written
as follows:
@V T
@x
1 @V
f + l ;
4 @x gR;1 T
T @V
g
@x
=0: (3.23)
To implement the rst step of the SGA algorithm we write the HJB equation as
@V T
@x
(f + gu)+l + kuk2 R =0 (3.24)
u (x) =; 1
2 R;1 T @V
g
@x
: (3.25)
Equation (3.24), a linear partial di erential equation, is the Generalized Hamilton-
Jacobi Bellman equation (GHJB). The usefulness of writing the HJB equation in
this form is that now V and u are decoupled. Assuming that we start with some
stabilizing control, u (0) (x), we can perform an in nite sequence of iterations to nd
the optimal control, u (x):
@V (i)T
@x (f + gu(i) )+l(x)+
(i)
u
2
=0 R (3.26)
u (i+1) (x) =; 1
2 R;1g T (i) @V
(x) (x)
@x
(3.27)
where i ranges from 0 to 1. If u (0) (x) asymptotically stabilizes the system on IR n ,
then Equations (3.26) and (3.27) describe a sequence of iterations, which was shown
in [16] to converge to the solution of the HJB equation point wiseon . Thus instead
of computing V and u simultaneously as in the HJB equation, we compute them
iteratively. The only problem left is to solve the GHJB equation at each step of the
iteration.
The solution V (i) of Equation (3.26) on can be approximated via a global
Galerkin approximation scheme as follows. Let V (i)
N (x) = PN j=1 c(i)
j
j(x), where the
set f j(x)g 1 j=1 is a complete basis for L 2( ) and j(0) = 0. The coe cients c (i)
j are
found by solving the algebraic Galerkin equation
Z
@V (i)T
@x (f + gu(i) )+l +
(i)
u
2
R
k =1:::N.
25
k dx =0