Model Predictive Control

14.2 Piecewise Affine Systems 289
k(x1) b(u2)
M
x1
(a) System with low viscous friction
(u2 = 1)
u1
k(x1) b(u2)
M
x1
(b) System with high viscous friction
(u2 = 0)
Figure 14.4 Spring mass system of Example 14.2
40
35
30
25
k(x1)
20
15
10
5
0
-5
-10
-10 -8 -6 -4 -2 0 2 4 6 8 10
x1
Figure 14.5 Nonlinear spring force k(x1)
Example 14.2 Consider the spring-mass system depicted in Figure 14.4(a), where the
spring has the nonlinear characteristics described in Figure 14.5. The viscous friction
coefficient can be instantaneously switched from one value b1 to another different
value b2 by instantaneously changing the geometrical shape of the mass through a
binary input u2 (see Figure 14.4(b)).
The system dynamics can be described in continuous-time as:
M ˙x2 = u1 − k(x1) − b(u2)x2
where x1 and x2 = ˙x1 denote the position and the speed of the mass, respectively, u1
a continuous force input, and u2 binary input switching the friction coefficient. The
spring coefficient is
k(x1) =
and the viscous friction coefficient is
k1x1 + d1 if x1 ≤ xm
k2x1 + d2 if x1 > xm,
b(u2) =
b1 if u2 = 1
b2 if u2 = 0.
Assume the system description is valid for −5 ≤ x1, x2 ≤ 5, and for −10 ≤ u2 ≤ 10
(all units will be omitted here for simplicity).
u1