Then according to Lyapunov theorem, error [??], NN weight error [??], predictor error e, and filtered error r are all UUB
. The control error [delta] is thus bounded based on (24) and Assumption 1.
then the closed-loop system is UUB
and the [H.sub.[infinity]] control performance (18) is guaranteed as prescribed [[rho].sup.2].
Following the same procedure as that of Theorem 11, we get that the local tracking errors are cooperatively UUB
For bounded initial conditions, if there exist a [C.sup.1] continuous and positive definite Lyapunov function V(x) satisfying a([parallel]x[parallel]) [less than or equal to] V(x) [less than or equal to] b([parallel]x[parallel]) such that [??](x) [less than or equal to] -[delta]V(x) + [pi], where a, b : [R.sup.n] [right arrow] R are class K functions and [delta], [pi] are positive constants, then the solution x(t) is uniformly ultimately bounded (UUB
Considering the identification model (6), the identification error (8) will be uniformly ultimately bounded (UUB
) if the weights updating laws are as follows:
where [??] is the estimation of a and is updated by (33a), then the system is ensured to be UUB
The stability analysis demonstrates a Uniformly Ultimately Bounded (UUB
) control method.
and [??] is updated by (29a), then the closed-loop system (11) is ensured to UUB
If the state tracking errors (17), the desired states (15), the auxiliary dynamic system (25), the surge control law [[tau].sub.uc] (26), the yaw control law [[tau].sub.rc] (27), and the adaptive laws (28) are applied to the hovercraft system represented by (6) and, for any bounded initial condition, the closed-loop control system signals [s.sub.u], [s.sub.r], [[xi].sub.u], [[xi].sub.r], [e.sub.u], [e.sub.[beta]], [x.sub.e], [y.sub.e] and [[??].sub.iu] and [[??].sub.ir], i = 1,2, ..., n, are uniformly ultimately bounded (UUB
Then, the estimate error x is uniformly ultimately bounded (UUB