NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

the new control, then update the control again and so on. Thus we actually perform
two simultaneous iterations, one for w (x) and one for u (x):
@V (ij)T
(f + gu
@x
(i) + kw (ij) )
(i)
2
+ l + u
R ; 2 (ij) 2
w
P
=0 (3.29)
w (ij+1) (x) = 1
2 2 P ;1 k T (ij)
@V
(x) (x) (3.30)
@x
u (i+1) (x) =; 1
2 R;1g T (i1)
@V
(x) (x) (3.31)
@x
where i and j range from zero to 1. If u (0) (x) asymptotically stabilizes the system
_x = f + gu (0) on IR n , then Equations (3.29), (3.30) and (3.31) converge to
the solution of the HJI equation pointwise on as shown in [5]. V (ij) (x) is again
approximated via a global Galerkin approximation scheme where the coe cients are
found by solving the algebraic Galerkin equation:
Z @V (ij)T
@x
(f + gu + kw)+l + kuk 2
R
!
k dx =0
k = 1:::N and is found via a bisection search algorithm. The le hji.m is a
straightforward implementation of this algorithm in Matlab code, where the integrals
are computed numerically using Matlab's quad function. This software package again
only requires three things: the set , the basis elements f jg N
j=1, and an initial
stabilizing control u (0) . To implement this control in hardware we used the following:
=[;:2:2] [;2:5 2:5] [; 2 2 ] [;20 20]
f jg = fx 2
1x 2
2x 2
3x 2
4x 1x 2x 1x 3x 1x 4x 2x 3x 2x 4x 3x 4g
u (0) (x) =41y ; 1:5_y ; 2:6 ; :19 _ :
We used the same basis functions, , and initial control as in the HJB case. This
control strategy has all of the strengths of the HJB law, and it additionally improves
the robustness of the control.
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