Page 820 - Mechanical Engineers' Handbook (Volume 2)

9 Optimal Control Using NNs 811

that as the system dynamical structure becomes more complex or the performance require-
ments become more stringent, it is necessary to add more feedback loops. Rigorous neuro-
controller design algorithms may be given in terms of Lyapunov energy-based techniques,
passivity, and so on.
Nonlinear optimal control design provides a very powerful theory that is applicable for
systems in any form. Solution of the so-called Hamilton–Jacobi (HJ) equations will directly
yield a controller with guaranteed properties in terms of stability and performance for any
sort of nonlinear system. Unfortunately, the HJ equations are difficult to solve and may not
even have analytic solutions for general nonlinear systems. In the special case of linear
optimal control, 37 solution techniques are available based on Riccati equation techniques,
and that theory provides a cornerstone of control design for aerospace systems, vehicles, and
industrial plants. It would be very valuable to have tractable controller design techniques for
general nonlinear systems. In fact, it has been shown that NNs afford computationally ef-
fective techniques for solving general HJ equations, and so for designing closed-loop con-
trollers for general nonlinear systems.

9.1 NN H Control Using the Hamilton–Jacobi–Bellman Equation
2
In work by Abu-Khalaf and Lewis it has been shown how to solve the Hamilton–Jacobi–
38
Bellman (HJB) equation that appears in optimal control for general nonlinear systems by a
successive approximation (SA) technique based on NNs. Rigorous results have been proven,
and a computationally effective scheme for nearly optimal controller design was provided
based on NNs.
This technique allows one to consider general affine nonlinear systems of the form
˙ x ƒ(x) g(x)u(x) (6)
To give internal stability and good closed-loop performance, one may select the L norm
2
performance index

V(x(0)) [Q(x) uRu] dt (7)
T
0
with matrix R positive definite and Q(x) generally selected as a norm. It is desired to select
the control input u(t) to minimize the cost V(x). Under suitable assumptions of detectability,
this guarantees that the states and controls are bounded and hence that the closed-loop
systems is stable.
An infinitesimal equivalent to the cost is given by
V
V T (ƒ gu) Q uRu Hx,
0 T ,u (8)
x x
which defines the Hamiltonian function H( ) and the costate as the cost gradient V/ x. This
is a nonlinear Lyapunov equation. It has been called a generalized HJB equation by Saridis
and Lee. 39
Differentiating with respect to the control input u(t)to find a minimum yields the control
in the form
1 V(x)
1T
u(x) Rg (x) (9)
2 x
Substituting this into the previous equation yields the HJB equation of optimal control