Page 805 - Mechanical Engineers' Handbook (Volume 2)

796 Neural Networks in Feedback Control Systems

M(q)¨q V (q,˙q)(˙q G(q) F(˙q) (1)
m
d
n
with q(t) R the joint variable vector, M(q) an inertia matrix, V a centripetal/Coriolis
m
matrix, G(q) a gravity vector, and F( ) representing friction terms. Bounded unknown dis-
turbances and modeling errors are denoted by and the control input torque is (t).
d
The sliding-mode control approach of Slotine 13,14 can be generalized to NN control
systems. Given a desired arm trajectory q (t) R ,define the tracking error e(t) q (t)
n
d d
T
q(t) and the sliding variable error r ˙e e, where 0. A sliding-mode manifold
is defined by r(t) 0. The NN tracking controller is designed using a feedback linearization
approach to guarantee that r(t) is forced into a neighborhood of this manifold. Define the
nonlinear robot function
ƒ(x) M(q)(¨q ˙e) V (q,˙q)(˙q e) G(q) F(˙q) (2)
m
d
d
with the known vector x(t) of measured signals suitably defined in terms of e(t), q (t). The
d
NN input vector x can be selected, for instance, as
T T
x [e T ˙ e T q T d ˙ q T d ¨ q ] (3)
d
3.1 Multilayer NN Controller
A NN controller may be designed based on the functional approximation properties of NNs,
as shown in Ref. 15. Thus, assume that ƒ(x) is unknown and given approximately as the
T
output of a NN with unknown ‘‘ideal’’ weights W, V so that ƒ(x) W (V x) with
T
an approximation error. The key is now to approximate ƒ(x) by the NN functional estimate
ˆ
ˆ
ˆ
ˆ
ˆ
T
T
ƒ(x) W (Vx) , with V, W the current (estimated) NN weights as provided by the tuning
algorithms. This is nonlinear in the tunable parameters V ˆ . Standard adaptive control ap-
proaches only allow LIP controllers.
Now select the control input
ˆ
ˆ
W (Vx) Kr v (4)
T
T
v
with K a symmetric positive-definite (PD) gain and v(t) a certain robustifying function
v
detailed in Ref. 15. This NN control structure is shown in Fig. 4. The outer PD tracking
loop guarantees robust behavior. The inner loop containing the NN is known as a feedback
linearization loop, 16 and the NN effectively learns the unknown dynamics online to cancel
the nonlinearities of the system.
Let the estimated sigmoid Jacobian be ˆ d (z)/dz z ˆVx . Note that this jacobian is
T
easily computed in terms of the current NN weights. Then, the next result is representative
of the sort of theorems that occur in NN feedback control design. It shows how to tune or
train the NN weights to obtain guaranteed closed-loop stability.
Theorem (NN Weight Tuning for Stability) Let the desired trajectory q (t) and its derivatives
d
be bounded. Take the control input for (1) as (4). Let NN weight tuning be provided by
˙ ˙ T ˆ
ˆ
ˆ
ˆ
ˆ ˆ
T
T
T
T
W Fˆ r F Vxr F r W V Gx(ˆ Wr) G r V ˆ (5)
with any constant matrices F F 0, G G 0, and scalar tuning parameter 0.
T
T
ˆ
ˆ
Initialize the weight estimates as W 0, V random . Then the sliding error r(t) and NN
ˆ
ˆ
weight estimates W, V are uniformly ultimately bounded.