Model Predictive Control

338 15 Optimal Control of Hybrid Systems
x 1 and x 2
Example 15.7 Consider the problem of regulating the hybrid spring-mass system (15.57)
described in Examples 14.2 and 15.5 to the origin. The finite time constrained op-
1 0
timal control problem (15.7) with cost (15.6) is solved with N = 3, P = Q = [ 0 1 ],
R = [
0.2 0
0 1 ]. Its state feedback solution (15.9) u∗ (x(0)) = f ∗ 0 (x(0)) at time 0 is
implemented in a receding horizon fashion, i.e. u(x(k)) = f ∗ 0 (x(k)).
The state-feedback solution was determined in Example 15.5 for the case of no terminal
constraint (Figure 15.5(a)). Figure 15.9 depicts the corresponding closed-loop
trajectories starting from the initial state x(0) = [3 4] ′ .
The state-feedback solution was determined in Example 15.5 for terminal constraint
Xf = [−0.01, 0.01] × [−0.01, 0.01] (Figure 15.5(b)). Figure 15.10 depicts the corresponding
closed-loop trajectories starting from the initial state x(0) = [3 4] ′ .
4
3
2
1
0
−1
0 5 10
time
(a) State Trajectories (x1 dashed line
and x2 solid line)
u 1 and u 2
3
2
1
0
−1
−2
−3
0 5 10
time
(b) Optimal Inputs (u1 solid line and
the binary input u2 with a starred line)
Figure 15.9 Example 15.7: MPC control of system (15.57) without terminal
constraint