Model Predictive Control

10.2 Constrained Optimal Control Problem Formulation 165
where Px,u is a polyhedron in R n+m or mixed input and state constraints over a finite
time
[x(0) ′ , . . . , x(N − 1) ′ , u(0) ′ , . . . , u(N − 1) ′ ] ∈ Px,u,N
(10.27)
where Px,u,N is a polyhedron in R N(n+m) . Note that constraints of the type (10.27)
can arise, for example, from constraints on the input rate ∆u(t) u(t) −u(t −1). In
this chapter, for the sake of simplicity, we will use the less general form (10.25).
Define the cost function
N−1
J0(x(0), U0) p(xN) + q(xk, uk) (10.28)
where xk denotes the state vector at time k obtained by starting from the state
x0 = x(0) and applying to the system model
k=0
xk+1 = Axk + Buk
(10.29)
the input sequence U0 [u ′ 0, . . . , u ′ N−1 ]′ .
If the 1-norm or ∞-norm is used in the cost function (10.28), then we set
p(xN) = PxNp and q(xk, uk) = Qxkp + Rukp with p = 1 or p = ∞ and P,
Q, R full column rank matrices. Cost (10.28) is rewritten as
N−1
J0(x(0), U0) PxN p + Qxkp + Rukp
k=0
(10.30)
If the squared euclidian norm is used in the cost function (10.28), then we set
p(xN) = x ′ N PxN and q(xk, uk) = x ′ k Qxk + u ′ k Ruk with P 0, Q 0 and R ≻ 0.
Cost (10.28) is rewritten as
J0(x(0), U0) x ′ N−1
NPxN +
k=0
x ′ kQxk + u ′ kRuk
Consider the constrained finite time optimal control problem (CFTOC)
J ∗ 0 (x(0)) = minU0 J0(x(0), U0)
subj. to xk+1 = Axk + Buk, k = 0, . . .,N − 1
xk ∈ X, uk ∈ U, k = 0, . . .,N − 1
xN ∈ Xf
x0 = x(0)
(10.31)
(10.32)
where N is the time horizon and Xf ⊆ R n is a terminal polyhedral region. In (10.28)–
(10.32) U0 = [u ′ 0 , . . . , u′ N−1 ]′ ∈ R s , s mN is the optimization vector. We denote
with X0 ⊆ X the set of initial states x(0) for which the optimal control problem
(10.28)–(10.32) is feasible, i.e.,
X0 = {x0 ∈ R n : ∃(u0, . . . , uN−1) such that xk ∈ X, uk ∈ U, k = 0, . . .,N − 1, xN ∈ Xf
where xk+1 = Axk + Buk, k = 0, . . .,N − 1},
(10.33)