Page 67 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

mance function (PPF) that characterizes the convergence rate, maximum
overshoot, and steady-state error is proposed and incorporated into the
control design. An output error transform is derived by applying the PPF
on the original system, such that the stabilization of the transformed sys-
tem can guarantee that the tracking error of the original system is strictly
retained within the set prescribed by PPF. The effect of frictions is ex-
plicitly considered by using a newly developed non-linear continuously
differentiable friction model [15]and [16]. This model can capture vari-
ous friction dynamics (e.g., Coulomb, Viscous, and Stribeck effects) and
has continuous characteristic functions, which is suitable for control design
and analysis. Then the friction model is lumped into the neural network
used for approximating other non-linear dynamics (e.g., resonances, dis-
turbances) and thus the associated primary parameters are online updated
together with NN weights. As a result, the costly offline identification of
friction is avoided without sacrificing tracking performance. Moreover,
only a scalar parameter, independent of the number of hidden nodes in
the neural network, is online updated to reduce the computational costs.
Practical experiments are carried out based on a laboratory turntable servo
platform, which reveal that the proposed adaptive prescribed performance
control (APPC) outperforms several other controllers.

4.2 PROBLEM FORMULATION AND PRELIMINARIES
4.2.1 Dynamic Model of Servo System

The turntable servo mechanism driven by a DC torque motor can be de-
scribed as [10]:

⎧
⎪ J¨q + f (q, ˙q) + T f + T l + T d = T m
⎨
dI a
K E ˙q + L a + R aI a = u (4.1)
dt
⎪
⎩
T m = K T I a
where q, ˙q are the angular position (rad) and velocity (rad/s), J is the in-
2
ertia (kg/m ), f (q, ˙q) is the unknown resonances and uncertainties; T d , T l ,
T f ,and T m are the unknown disturbance, load, friction, and the generated
torque, respectively; u is the control input voltage, I a, R a,and L a are the ar-
mature current, resistance, and inductance; K T is the electrical-mechanical
conversion constant and K E is the back electromotive force coefficient.
In practical servo systems, the electrical constant L a /R a is small, and the
electrical transients L adI a /dt is close to zero [6]. We define the system states