Page 85 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

RISE Based Asymptotic PPC of Servo Systems With Continuously Differentiable Friction Model 77

The problem to be addressed is to find a continuous control u such
that: 1) the angular position q can track a given time-varying trajectory q d
asymptotically; 2) the transient tracking error e 1 (t) = q − q d can be retained
within a prescribed bound for all t > 0; 3) all signals in the closed-loop
system are bounded.
Without loss of generality, the uncertainties d and its first two time
derivatives are bounded such that |d|≤ δ 1, |d|≤ δ 2 hold for positive con-
¨
˙
stants δ 1 and δ 2 [9]. Moreover, the reference trajectory q d to be tracked and
its first two time derivatives ˙q d , ¨q d are bounded. In this chapter, the friction
model (4.3) is employed to describe the friction dynamics.

5.2.2 Function Approximation Using ESN
As explained in Chapter 3, ESNs have been used to model and control
non-linear dynamical systems [20,21] due to its simple training procedure.
Hence, ESN will be used as the function approximator in this chapter.
For more details concerning the property and structure of ESN, we re-
fer to Chapter 3 and [20,21]. For the completeness of this chapter, we
briefly show the universal approximation of any continuous function f (·):
n
n
R − R on a sufficiently large compact set ⊂ R ,which is givenby
∗T
f (x) = X(x) + ε ∗ ∀x ∈ ⊂ R n (5.2)
0
∗
∗
where ε is the approximation error of the ESN bounded by |ε |≤ ε m,
is the ideal value of ESN weight. Because the ideal NN weight is
∗
∗
0 0
unknown, we can only use the estimated value 0 of in the control
∗
ˆ
0
design, which can be online updated.
5.2.3 Prescribed Performance Function and Error Transform
For the position tracking control, we define the tracking error as
(5.3)
e 1 (t) = q − q d
Then, to quantitatively study the transient response of the tracking error
e 1 (t), we will incorporate the PPF into the control designs. The readers can
refer to Chapter 3 and Chapter 4 for more explanations on the PPF and the
associated error transform. Again, for the completeness of this chapter, we
briefly introduce the PPF function to be used in this chapter, which is given
by the following positive decreasing smooth function μ(t) : R → R +
+
μ(t) = (μ 0 − μ ∞ )e −κt + μ ∞ (5.4)