Model Predictive Control

202 11 Receding Horizon Control
with the input constraints
and the state constraints
−10
≤ x(k) ≤
−10
− 1 ≤ u(k) ≤ 1, k = 0, . . . , N − 1 (11.18)
10
, k = 0, . . . , N − 1. (11.19)
10
In the following, we study the receding horizon control problem (11.7) with
p(xN) = x ′ NPxN, q(xk, uk) = x ′ kQxk + u ′ kRuk for different horizons N and
weights R. We set Q = I and omit both the terminal set constraint and the
terminal weight, i.e. Xf = R 2 , P = 0.
Fig. 11.4 shows closed-loop trajectories for receding horizon control loops that
were obtained with the following parameter settings
Setting 1: N = 2, R = 10
Setting 2: N = 3, R = 2
Setting 3: N = 4, R = 1
For Setting 1 (Fig. 11.4(a)) there is evidently no initial state that can be steered
to the origin. Indeed, it turns out, that all non-zero initial states x(0) ∈ R 2
diverge from the origin and eventually become infeasible. Different from that,
Setting 2 leads to a receding horizon controller, that manages to get some of the
initial states converge to the origin, as seen in Fig. 11.4(b). Finally, Fig. 11.4(c)
shows that Setting 3 can expand the set of those initial states that can be brought
to the origin.
Note the behavior of particular initial states:
1. Closed-loop trajectories starting at state x(0) = [−4,7] behave differently
depending on the chosen setting. Both Setting 1 and Setting 2 cannot bring
this state to the origin, but the controller with Setting 3 succeeds.
2. There are initial states, e.g. x(0) = [−4,8.5], that always lead to infeasible
trajectories independent of the chosen settings. It turns out, that no setting
can be found that brings those states to the origin.
These results illustrate that the choice of parameters for receding horizon control
influences the behavior of the resulting closed-loop trajectories in a complex
manner. A better understanding of the effect of parameter changes can be gained
from an inspection of maximal positive invariant sets O∞ for the different settings,
and the maximal control invariant set C∞ as depicted in Fig. 11.5.
The maximal positive invariant set stemming from Setting 1 only contains the
origin (O∞ = {0}) which explains why all non-zero initial states diverge from the
origin. For Setting 2 the maximal positive invariant set has grown considerably,
but does not contain the initial state x(0) = [−4,7], thus leading to infeasibility
eventually. Setting 3 leads to a maximal positive invariant set that contains this
state and thus keeps the closed-loop trajectory inside this set for all future time
steps.
From Fig. 11.5 we also see that a trajectory starting at x(0) = [−4, 8.5] cannot
be kept inside any bounded set by any setting (indeed, by any controller) since
it is outside the maximal control invariant set C∞.