Page 283 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Control of Hammerstein Systems With Hysteresis Non-linearity 285

Figure 18.4 Structure of composite control system.

3) Estimate the unknown coefficients of G(z) by (18.10)and (18.16).
4) Calculate the variable x(t) based on the transfer function G(z) and
output measurements y(t).
5) Compute ˆ U and ˆ X based on (18.18) for one piecewise monotonic
section and calculate the Preisach density function μ from (18.19)with
ˆ X, ˆ U, and Preisach operator ˆω.
6) Go back to step 1, and conduct identification for next piecewise mono-
tonic section.

18.4 COMPOSITE CONTROL DESIGN AND ANALYSIS
After obtaining the model of Hammerstein system by using the previ-
ously presented system identification, we will further design a composite
control for system shown in Fig. 18.1 to achieve output tracking. The pro-
posed control consists of a feedforward controller and a feedback controller,
whichcan beshowninFig. 18.4. The feedforward controller is called dis-
crete inverse model-based controller (DIMBC), which includes the inverse
Preisach model (static non-linearity) and the inverse of the non-hysteretic
dynamics (linear dynamics). The feedback controller is a discrete adaptive
sliding mode control (DASMC).
ˆ −1
As showninFig. 18.4, DIMBC consists of two parts: G and f ˆ−1 ,
ˆ −1
where G denotes the estimation of the inverse linear dynamics G −1 and
ˆ−1 denotes the estimation of the inverse hysteresis non-linearity, respec-
f
tively. After the Hammerstein system is identified, the estimated model-
ˆ −1
based inversion f ˆ−1 and G can be implemented.
It is known that DIMBC provides a feedforward compensation for the
Hammerstein system, which can be taken as an open-loop control. Con-
sequently, the robustness may be a problem in the presence of modeling
uncertainties. In order to accommodate this problem, a feedback control
(DASMC) is further developed.