Model Predictive Control

12.5 State Feedback Solution, 1-Norm and ∞-Norm Case 255
12.5 State Feedback Solution, 1-Norm and ∞-Norm Case
In this chapter we show how to find a state feedback solution to CROC problems,
namely an explicit function u∗ k (xk) mapping the state xk to its corresponding optimal
input u∗ k , ∀k = 0, . . .,N − 1. The reader is assumed to be familiar with the
concept of multiparametric programming presented in Chapter 6.
Multiparametric Programming with Piecewise-Linear Cost
Consider the optimization problem
J ∗ (x) = min
z
J(z, x)
subj. to Gz ≤ W + Sx,
(12.70)
where z ∈ R s are the optimization variables, x ∈ R n is the vector of parameters,
J(z, x) is the objective function and G ∈ R m×s , W ∈ R m , and S ∈ R m×n . Problem
(12.70) with J(z, x) = c ′ z is a multiparametric linear program (mp-LP) (see
Chapter 6) .
For a given polyhedral set K ⊆ R n of parameters, solving (12.70) amounts to
determining the set K ∗ ⊆ K of parameters for which (12.70) is feasible, the value
function J ∗ : K ∗ → R, and the optimizer function 1 z ∗ : K ∗ → R s .
The properties of J ∗ (·) and z ∗ (·) have been analyzed in Chapter 6 and summarized
in Theorem 6.4. Below we give some results based on this theorem.
Lemma 8 Let J : R s × R n → R be a continuous convex piecewise affine function
of (z, x) of the form
J(z, x) = Liz + Hix + Ki for [ z x ] ∈ Ri
(12.71)
where {Ri} nJ
i=1 are polyhedral sets with disjoint interiors, R nJ i=1 Ri is a polyhedron
and Li, Hi and Ki are matrices of suitable dimensions. Then the multiparametric
optimization problem (12.70) is an mp-LP.
Proof: As J is a convex piecewise affine function, it follows that J(z, x) can be
rewritten as J(z, x) = maxi=1,...,s {Liz + Hix + Ki} (see Section 4.1.5). Then, it
is easy to show that (12.70) is equivalent to the following mp-LP: minz,ε ε subject
to Cz ≤ c + Sx, Liz + Hix + Ki ≤ ε, i = 1, . . .,s. ✷
Lemma 9 Let f : Rs × Rn × Rnw → R and g : Rs × Rn × Rnw → Rng be functions
of (z, x, w) convex in w for each (z, x). Assume that the variable w belongs to the
polyhedron W with vertices { ¯wi} NW
i=1 . Then the min-max multiparametric problem
J ∗ (x) = minz maxw∈W f(z, x, w)
subj. to g(z, x, w) ≤ 0 ∀w ∈ W
1 In case of multiple solutions, we define z ∗ (x) as one of the optimizers.
(12.72)