Model Predictive Control

184 10 Constrained Optimal Control
J ∗ N−1 (xN−1) minµ,uN −1 µ
subj. to µ ≥ QxN−1p + RuN−1p + P(AxN−1 + BuN−1)p
xN−1 ∈ X, uN−1 ∈ U,
AxN−1 + BuN−1 ∈ Xf
(10.88)
By Theorem 6.4, J ∗ N−1 is a convex and piecewise affine function of xN−1, the
corresponding optimizer u∗ N−1 is piecewise affine and continuous, and the feasible
set XN−1 is a polyhedron. Without any loss of generality we assume J ∗ N−1 to be
described as follows: J ∗ N−1 (xN−1) = maxi=1,...,nN −1{cixN−1+di} (see Section 4.1.5
for convex PPWA functions representation) where nN−1 is the number of affine
components comprising the value function J ∗ N−1 . At step j = N − 2 of dynamic
programming (10.84)-(10.86) we have
J ∗ N−2 (xN−2) minuN −2 QxN−2p + RuN−2p + J ∗ N−1 (AxN−2 + BuN−2)
subj. to xN−2 ∈ X, uN−2 ∈ U,
AxN−2 + BuN−2 ∈ XN−1
(10.89)
Since J ∗ N−1 (x) is a convex and piecewise affine function of x, the problem (10.89)
can be recast as the following mp-LP (see Section 4.1.5 for details)
J ∗ N−2 (xN−2) minµ,uN −2 µ
subj. to µ ≥ QxN−2p + RuN−2p + ci(AxN−2 + BuN−2) + di,
i = 1, . . .,nN−1,
xN−2 ∈ X, uN−2 ∈ U,
AxN−2 + BuN−2 ∈ XN−1
(10.90)
J ∗ N−2 (xN−2), u∗ N−2 (xN−2) and XN−2 are computed by solving the mp-LP (10.90).
By Theorem 6.4, J ∗ N−2 is a convex and piecewise affine function of xN−2, the
corresponding optimizer u∗ N−2 is piecewise affine and continuous, and the feasible
set XN−2 is a convex polyhedron.
The convexity and piecewise linearity of J ∗ j and the polyhedra representation
of Xj still hold for j = N − 3, . . .,0 and the procedure can be iterated backwards
in time, proving the theorem. ✷
Consider the state feedback piecewise affine solution (10.83) of the CFTOC (10.73)
for p = 1 or p = ∞ and assume we are interested only in the optimal controller at
time 0. In this case, by using duality arguments we can solve the equations (10.84)-
(10.86) by using vertex enumerations and one mp-LP. This is proven in the next
theorem.
Theorem 10.8 The state feedback piecewise affine solution (10.83) at time k = 0
of the CFTOC (10.73) for p = 1 or p = ∞ is obtained by solving the equations
(10.84)-(10.86) via one mp-LP.