Page 154 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

150 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

1) Select the feedback gains k i and filter parameters μ i with respect to
(9.49), and the initial parameters ˆ (0) ≥ 0, ˆε i (0) ≥ 0 for all i = 1,··· ,n;
θ i
2) Derive the tracking error z 1 = x 1 − y d and the corresponding virtual
control α 1 based on (9.9)–(9.11);
3) For i = 2,··· ,n − 1, calculate s i−1 according to (9.6), and then the
virtual error z i = x i − s i−1 and the control α i via (9.29)–(9.31);

4) Calculate s n−1 according to (9.6), and the virtual error z n = x n − s n−1
and the real control v from (9.36)–(9.38);
5) Go back to Step 2) for next sampling interval.
ˆ
The initial condition of adaptive parameters θ i , ˆε i should be non-
ˆ
negative based on Theorem 9.1, i.e. θ i (0) ≥ 0, ˆε i (0) ≥ 0. To facilitate the
design, they can be simply specified as θ i (0) =ˆε i (0) = 0. Moreover, as stated
ˆ
in each step, γ i should be tuned to be large enough and ϑ i, c i1, c i2, c i3 should
be small to achieve small tracking errors. Based on the formulation of ϑ i,
γ i, it is desirable to choose the feedback gain k i and learning parameters
i,

ai as large as possible, which also benefits to justify the condition (9.49).
However, practical systems do not allow too large gains, which may result
in a high-gain control and divergence in the adaptation. Thus, these learn-
ing coefficients have to be chosen online, not excessively large, to guarantee
the stability and performance, i.e., by gradually increasing them from small
initial values until the tracking error and control signal started to exhibit
nervous behaviors.

9.4 SIMULATIONS

A numerical example is included to show the satisfactory performance of
the proposed ANDSC. Consider the following non-linear system:

⎧ 2(1−e −x 1 ) 2 2 2
⎪ ˙ x 1 = x 1e −0.5x 1 + x (t − τ 1 ) + (2 + x + sin(x (t − τ 1 )))x 2
1+e −x 1 1 1 1
⎪
⎪
˙ x 2 = x 1x + 2cos(x 1 )sin(x 2 )x 1 (t − τ 1 )x (t − τ 2 ) + (3 + x x ) (9.52)
⎨ 2 2 2 2
2
2
1 2
+cos(x 1 (t − τ 1 )x 2 (t − τ 2 )))u
⎪
⎪
⎪
⎩
y = x 1
The perturbed non-linear dead-zone is specified as
⎧
⎪ (1 − 0.3sin(v))(v − 2.5) if v ≥ 2.5
⎨
u(t) = D(v(t)) = 0 if − 1.5<v < 2.5 (9.53)
⎪
⎩ (0.8 − 0.2cos(v))(v + 1.5) if v ≤−1.5