Page 242 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Figure 15.1 Spring mass and damper system.

which can represent a class of widely-used second-order electro-mechanical
servo systems, such as hydraulic systems, rigid robots, and so on [20,13,21].
Then the dynamics of the studied system (Fig. 15.1) are described as [10]

˙ x 1 = x 2
(15.61)
c
k
1
˙ x 2 =− x 1 − x 2 + v (u) + E (t)
m m m
where y = x 1 and x 2 are the position and velocity, respectively, m is the mass,
k is the stiffness constant of the spring and c is the damping coefficient,
E(t) = sin(2π t) denotes the external disturbances and uncertainties.
According to the proposed coordinate transform and the saturation ap-
proximation, system (15.61) can be transformed into

˙ x 1 = x 2
(15.62)
˙ x 2 = a(¯ x 2 ) + b(¯ x 2 ,v) · u
c
k
1
T
where ¯x 2 = [x 1 ,x 2 ] , a(¯ x 2 ) =− x 1 − x 2 + sin(2π t), b(¯ x 2 ,v) = .
m m m
In the simulation, the sinusoidal wave y d =−0.2cos(3πt) + 0.2is
adopted as the desired reference signal. The initial states and parameters of
T
T
the system are [x 1 ,x 2 ] = [0,0] , m = 1kg, c = 2Ns/m, and k = 8N/m.
The parameters of adaptive law are set as δ = 0.5, σ = 0.01, ˆε N = 0.01,
ε = 1. The time constant of filter is τ = 0.01. The NN parameters are
r 1 = 1, r 2 = 5, r 3 = 5, r 4 =−0.1, = 5. The feedback control gains are given
by k 1 = 10, k 2 = 8, λ = 5. The input saturation bound is υ max = 14 Nm.
Comparative tracking performances, tracking errors, and control inputs
are shown in Figs. 15.2–15.4, respectively. From Fig. 15.2 and Fig. 15.3,we
can see that compared with the proposed S1 method, S2 has larger over-
shoot, and S2, S3 have lager tracking errors than S1. From Fig. 15.4A, it is
found that compared with S2 and S3, the control input of S1 is smoother.
In particular, the compensation of input saturation by using the proposed
control in comparison to S2 is shown in Fig. 15.4B.