Page 84 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

within a small bound, while the asymptotic convergence property cannot
be claimed.
In this chapter, we will develop a new adaptive control method for servo
mechanisms with unknown dynamics and frictions. This control scheme
can address both the transient response and the asymptotic convergence of
the tracking error simultaneously [19]. For this purpose, an improved PPF
is incorporated into the control design such that the tracking error can
be strictly retained within a prescribed set in the transient stage. To fur-
ther achieve asymptotic error convergence to zero, a smooth robust control
term based on the idea of RISE is introduced to compensate for the NN
approximation error and bounded disturbances. Finally, an adaptive law is
proposed to update the NN weight and friction model coefficients online.
It is shown that the use of a continuously differentiable friction model and
a smooth RISE term can lead to smooth compensation actions. Moreover,
the asymptotic control error convergence can be theoretically proved in
comparison to other PPF based controllers. Also, the transient response can
be prescribed even in the presence of unknown disturbances and NN ap-
proximation error. Experimental results based on the previously introduced
turntable servo system show that satisfactory transient and steady-state per-
formance of the proposed control system are achieved.

5.2 PROBLEM FORMULATION AND PRELIMINARIES

5.2.1 Dynamic Model of Servo System
In this chapter, we consider the position motion tracking control of the
same non-linear servo mechanism as used in Chapter 4. The overall
schematic diagram of the constructed position control system is given in
Figs. 4.1 and 4.2. Thus, according to Chapter 4, the dynamic model of
such systems can be described as

K T K T K E 1
¨ q = u − ˙ q − (f (q, ˙q) − T f (˙q)) + d (5.1)
JR a JR a J
where K E , d = (−T l − T d )/J denote the effect of loads and external distur-
bances. R a, L are the stator resistance and stator inductance; T f , T l ,and T d
represent the friction torque, load torque, and disturbance torque, respec-
tively; f (q, ˙q) denotes the unknown resonances and uncertainties. K T is the
torque constant, q, ˙q are the angular position and velocity of PMSM, and J
is the inertia, and d is the modeling uncertainties.