Model Predictive Control

264 12 Constrained Robust Optimal Control
x2
10
5
0
-5
-10
-10 -5 0 5 10
x1
(a) CROC-OL
x2
10
5
0
-5
-10
-10 -5 0 5 10
x1
(b) CROC-CL
Figure 12.7 Polyhedral partition of the state-space corresponding to the
explicit solution of CROC-OL and CROC-CL
1 1
with N = 5, P = Q = [ 0 1 ], R = 1.8 and we set Xf = X.
CROC-OL. The min-max problem is formulated as in (12.41)–(12.45). The resulting
polyhedral partition consists of 22 regions and it is depicted in Figure 12.7(a). We
remark that the CROC-OL problem (12.41)–(12.45) is infeasible for horizon N greater
than five.
CROC-CL. The min-max problem is formulated as in (12.47)–(12.49) and solved
using the approach of Theorem 12.3. The resulting polyhedral partition consists of
64 regions and is depicted in Figure 12.7(b).
12.8 Robust Receding Horizon Control
A robust receding horizon controller for system (12.38)-(12.40) which enforces the
constraints (12.39) at each time t in spite of additive and parametric uncertainties
can be obtained immediately by setting
u(t) = f ∗ 0 (x(t)), (12.104)
where f ∗ 0 (x0) : R n → R m is the solution to the CROC-OL or CROC-CL problems
discussed in the previous sections. In this way we obtain a state feedback strategy
defined at all time steps t = 0, 1, . . ., from the associated finite time CROC
problems.
If f0 is computed by solving CROC-CL (12.47)–(12.51) (CROC-OL (12.41)–
(12.45)), then the RHC law (12.104) is called a robust receding horizon controller
with (open-loop) closed-loop predictions. The closed-loop system obtained by controlling
(12.38)-(12.40) with the RHC (12.104) is
x(k+1) = A(w p )x(k)+B(w p )f0(x(k))+Ew a fcl(x(k), w p , w a ), k ≥ 0 (12.105)