Page 274 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Figure 18.1 Hammerstein system.

In this chapter, we investigate the identification and composite control
design of a Hammerstein system with linear dynamics and a static hysteresis
non-linearity described by Preisach model. For the identification of linear
dynamics, a Hankel matrix is calculated to determine the order n of linear
transfer function. Then, the blind identification method is used to obtain
the coefficients of the transfer function by using the measurable output
y(t) only, and the unmeasured variable x(t) can be further computed by
using the output y(t) and the identified transfer function together. Then
a novel deterministic identification of the Preisach model is proposed by
using the calculated x(t). Finally, we use the identified system dynamics to
design a composite control to achieve output tracking. The presented con-
trol consists of two controllers: 1) a discrete inverse model-based controller
(DIMBC) based on the identified models to compensate for the unde-
sired dynamics (e.g., hysteresis); 2) a discrete adaptive sliding mode control
(DASMC) to retain tracking performance. Numerical simulations are given
to validate the proposed identification and control methods.

18.2 PROBLEM FORMULATION

Considering a Hammerstein system as shown in Fig. 18.1, which includes
a static hysteresis non-linearity followed by a linear transfer function as

y(z) b 1z −1 + b 2z −2 + ... + b nz −n
G(z) = = (18.1)
x(z) 1 + a 1z −1 + a 2z −2 + ... + a nz −n
where n is the order of transfer function G(z) and a i , b i, i = 1,2,...n are the
unknown coefficients of G(z).
The dynamics of hysteresis can be described by the following non-linear
function

x(t) = f (u(t)) (18.2)
In this chapter, the Preisach operator [19]willbeused. Forbrevity,
some basis of Preisach operator will be provided again. Considering a pair
of thresholds (α,β)with α β, then the Preisach operator γ α,β [·,·] is de-