Page 51 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Figure 3.3 Profile of prescribed performance function.

where −δ and δ i are design parameters. An example of PPF and error
i
constraint (3.13)isshownin Fig. 3.3.
From (3.12)and (3.13), one can see that −δ μ i0 defines the lower bound
i
of the undershoot and δ i μ i0 defines the upper bound of the maximum
overshoot. The decreasing rate κ i denotes the required convergence speed
of tracking error [22]. Hence, the transient and steady-state performance
can be designed in a priori by tuning the parameters −δ , δ i , κ i , μ i0,and
i
μ i∞. To design control with prescribed performance, an error transform is
used to transform the original tracking error system with the constrained
tracking error bound (3.13) into an equivalent “unconstrained” one [29].
With this purpose, we define a smooth, strictly increasing function S i (z i )
of the transformed error z i, which fulfills the following properties:
1) −δ < S i (z i )< δ i , ∀z i ∈ L ∞.
¯
i
2) lim S i (z i ) = δ i ,and lim S i (z i ) =−δ .
i
z i →+∞ z i →−
Based on these properties of S i (z i ),Eq. (3.13) equals
e i (t) = μ i (t)S i (z i ). (3.14)

Then, the transformed error z i can be calculated by

e i (t)
−1
z i = S . (3.15)
i
μ i (t)
For any initial condition e i (0), if parameters μ i (0), δ i,and δ are selected
i
that −δ μ i (0)< e i (0)< δ i μ i (0) and z i can be controlled to be bounded, then
i
−δ < S i (z i )< δ i holds, such that the condition −δ μ i (t)< e i (t)< δ i μ i (t) is
i i
guaranteed. In this chapter, the following function is used in the control