Model Predictive Control

314 15 Optimal Control of Hybrid Systems
are the continuous inputs and uℓ ∈ R mℓ are the binary inputs (Section 14.2.2). We
will require the following assumption.
Assumption 15.1 For the discrete-time PWA system (15.2) is, the mapping (xc(t), uc(t)) ↦→
xc(t + 1) is continuous.
Assumption 15.1 requires that the PWA function that defines the update of the
continuous states is continuous on the boundaries of contiguous polyhedral cells,
and therefore allows one to work with the closure of the sets ˜ C i without the need
of introducing multi-valued state update equations. With abuse of notation in the
next sections ˜ C i will always denote the closure of ˜ C i . Discontinuous PWA systems
will be discussed in Section 15.7.
15.2 Properties of the State Feedback Solution, 2-Norm
Case
Theorem 15.1 Consider the optimal control problem (15.7) with cost (15.6) and
let Assumption 15.1 hold. Then, there exists a solution in the form of a PWA
state-feedback control law
u ∗ k(x(k)) = F i kx(k) + G i k if x(k) ∈ R i k, (15.9)
where Ri k , i = 1, . . .,Nk is a partition of the set Xk of feasible states x(k), and
the closure ¯ Ri k of the sets Ri k has the following form:
and
¯R i k x : x(k) ′ Li k (j)x(k) + Mi k (j)x(k) ≤ Ni k (j) ,
, k = 0, . . .,N − 1,
j = 1, . . .,n i k
x(k + 1) = Aix(k) + Biu∗ k
x(k)
if u ∗ k (x(k))
(x(k)) + fi
∈ ˜ C i , i = {1, . . .,s}.
(15.10)
(15.11)
Proof: The piecewise linearity of the solution was first mentioned by Sontag
in [232]. In [177] Mayne sketched a proof. In the following we will give the proof
for u∗ 0 (x(0)), the same arguments can be repeated for u∗1 (x(1)), . . . , u∗N−1 (x(N −1)).
• Case 1: (ml = nl = 0) no binary inputs and states
Depending on the initial state x(0) and on the input sequence U = [u ′ 0,. . .,
u ′ k ], the state xk is either infeasible or it belongs to a certain polyhedron
˜C i , k = 0, . . . , N − 1. The number of all possible locations of the state
sequence x0, . . .,xN−1 is equal to sN . Denote by {vi} sN
i=1 the set of all possible
switching sequences over the horizon N , and by vk i the k-th element of the
sequence vi, i.e., vk i = j if xk ∈ ˜ Cj .