Page 224 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Non-singular Terminal Sliding Mode Funnel Control of Servo Systems With Input Saturation 223

T
Hence, for any control gain μ>P N +
˜ W φ(X)
,Eq. (14.29)can be
rewritten as
1 (p+q)/q
˙ V 1 ≤− |s 2 |
k m β
1 (p+q)/(2q) 1 2 (p+q)/(2q)
≤− (2k m ) ( s )
k m β 2k m 2 (14.30)
1 (p+q)/(2q) (p−q)/(2q)
=− 2 k m V (p+q)/(2q)
β 1
=−k 2V 1 k 3
where k 2 = 1/β2 (p+q)/(2q) (p−q)/(2q) ,and 0 < k 3 = (p+q)/(2q)< 1 are all pos-
k m
itive constants.
Then, we can obtain
˙ V 1 + k 2V k 3 ≤ 0 (14.31)
1
According to Lemma 14.1, it can be concluded that the terminal sliding
manifold s 2 can converge to the equilibrium point within a finite time t 1
V 1−k 3 (t 0 )
given by t 1 = .
k 2 (1−k 3 )
3) Once the sliding surface s 2 = 0 is reached, the states of system (14.23)
will remain on it and the system has the invariant properties. On the sliding
surface s 2 = 0, we can obtain from (14.16)that

s ˙ 1 =−αs 1 (14.32)

Select the following Lyapunov function
1 2
V 2 = s (14.33)
2 1
and differentiating V 2 along (14.32)yields

2
˙ V 2 =−αs ≤ 0 (14.34)
1
Then, we can conclude that the tracking error s 1 will converge to zero
exponentially. Then based on (14.20), we can claim that s 2 also converges
to zero.Hence,from(14.13), the tracking error e will be retained within
the prescribed bound. This completes the proof.

14.4 SIMULATIONS

In this section, simulations are conducted to verify the proposed neural net-
work based non-singular terminal sliding mode funnel control (NTSMFC),
and show its superior performance in comparison to the following three
control approaches.