Page 822 - Mechanical Engineers' Handbook (Volume 2)

9 Optimal Control Using NNs 813

Moreover, if the initial NN weights are selected to yield an admissible control, then the
control is admissible (which implies stability) at each iteration.
The control given by this approach is shown in Fig. 17. It is a feedback control in terms
of a nonlinear NN. This approach has also been given for constrained input systems, such
as industrial and aircraft actuator systems.

9.2 NN H Control Using the Hamilton–Jacobi–Isaacs Equation

Many systems contain unknown disturbances, and the optimal control approach just given
may not be effective. In this case, one may use the H design procedure.

Consider the dynamical system in Fig. 18, where u(t) is an action or control input, d(t)
is a disturbance or opponent, y(t) is the measured output, and z(t) is a performance output
2
T
2
with z h h u . Here we take full state feedback y x and desire to determine the
action or control u(t) u(x(t)) such that, under the worst disturbance, one has the L gain
2
bounded by a prescribed so that

T
z(t) dt (hh u ) dt
2
2
0 0 2
2 2
d(t) dt d(t) dt
0
0
This is a differential game with two players 41,42 and can be confronted by defining the utility
T
2
2
r(x,u,d) h (x)h(x) u(t) d(t) 2
and the long-term value (cost-to-go)

V(x(t)) r(x,u,d) dt (h (x)h(x) u(t) d(t) ) dt (11)
2
2
2
T
t t
The optimal value is given by

V*(x(t)) min max r(x,u,d) dt
u(t) d(t) t
The optimal control and worst-case disturbance are given by the stationarity conditions as

V (x)
L

L

Figure 17 Nearly optimal NN feedback control for constrained input nonlinear systems.