For the velocity subsystem of AHV (17), considering the saturation characteristic of [PHI] (11), and adopting the control law (26) and the adaptive law (37) under the premise of Theorem 1, the closed-loop control system is semiglobally uniformly ultimately bounded
Then all of the signals in [V.sub.L] are uniformly ultimately bounded
. This completes the proof.
Then, by selecting the design parameters appropriately, the designed control scheme can guarantee that all signals in the closed-loop system are bounded; for example, the signals [mathematical expression not reproducible] are uniformly ultimately bounded
By combining backstepping design and two independent Lyapunov functions, a novel adaptive neural control scheme was presented to guarantee all the signals in the closed-loop system are uniformly ultimately bounded
, while this control scheme achieves predefined transient and steady-state tracking control performances concerning the link angular position and velocity tracking errors.
Therefore, synchronization error e and estimation error [PHI] - [??] are uniformly ultimately bounded
. The size of compact set [parallel][PHI] - [??][parallel] [less than or equal to] [[PHI].sub.m]/2 + [square root of ([[PHI].sup.2.sub.m]/4 + p[[zeta].sub.m]/[k.sub.c])] can be minimized by selecting a large value of [k.sub.c] for a given value of p.
Considering the identification model (6), the identification error (8) will be uniformly ultimately bounded
(UUB) if the weights updating laws are as follows:
Thus, [s.sub.2] and [[??].sub.2] are uniformly ultimately bounded
. Assume that [absolute value of [[??].sup.*.sub.2]] [less than or equal to] [[xi].sub.2], where [[xi].sub.2] is positive.
Given any constant [mu] > 0, for any bounded initial conditions satisfying the prescribed performance (8) and V(0) < [mu], there exists design parameters [C.sub.1], [C.sub.2], [C.sub.3], [C.sub.4], [[GAMMA].sub.2], [[sigma].sub.2], [[GAMMA].sub.4], [[sigma].sub.4], [[tau].sub.1], [[tau].sub.2], and [[tau].sub.3], such that the proposed control scheme guarantees that (1) all the signals in the closed-loop system are uniformly ultimately bounded
and (2) the tracking error converges to a small neighborhood around zero with the prescribed performance (8) in a finite time [T.sub.1].
From proposition 2 in , [t.sub.f] = [infinity], it can be concluded that all error signals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the closed-loop system are semiglobally uniformly ultimately bounded
Then all the signals involved are semiglobally uniformly ultimately bounded
Then, the estimate error x is uniformly ultimately bounded
Moreover, it has been proven that the designed controller is able to guarantee that all signals are semiglobally uniformly ultimately bounded
. Finally, simulation verifies the feasibility and effectiveness of the obtained theoretical results.