Model Predictive Control

11.7 Offset-Free Reference Tracking 223
by (11.63). Left multiplying the second row of (11.62) by H and subtracting (11.63)
from the result, we obtain
(A − I)δx + Bδu = 0
HCδx = ε.
(11.64)
Next we prove that δx =0 and thus ε = 0.
Consider the MPC problem (11.58)-(11.59) and the following change of variables
δxk = xk − ¯xt, δuk = uk − ūt. Notice that Hyk − r(t) = HCxk + HCddk −
r(t) = HCδxk + HC¯xt + HCddk − r(t) = HCδxk from condition (11.59) with
ˆd(t) = dk. Similarly, one can show that δxk+1 = Aδxk + Bδuk. Then, the MPC
problem (11.58) becomes:
minδu0,...,δuN −1
δx ′ N−1
NPδxN +
k=0
δx ′ kQδxk + δu ′ kRδuk
subj. to δxk+1 = Aδxk + Bδuk, 0 ≤ k ≤ N
xk ∈ X, uk ∈ U, k = 0, . . .,N − 1
xN ∈ Xf
δx0 = δx(t),
δx(t) = ˆx(t) − ¯xt.
(11.65)
Denote by KMPC the unconstrained MPC controller (11.65), i.e., δu∗ 0 = KMPCδx(t).
At steady state δu∗ 0 → u∞ − ū∞ = δu and δx(t) → ˆx∞ − ¯x∞ = δx. Therefore, at
steady state, δu = KMPCδx. From (11.64)
(A − I + BKMPC)δx = 0. (11.66)
By assumption the unconstrained system with the MPC controller converges. Thus
KMPC is a stabilizing control law, which implies that (A − I + BKMPC) is nonsingular
and hence δx = 0. ✷
Remark 11.6 Theorem 11.4 was proven in [197] by using a different approach.
Remark 11.7 Theorem 11.4 can be extended to prove local Lyapunov stability of
the closed-loop system (11.61) under standard regularity assumptions on the state
update function f in (11.61) [178].
Remark 11.8 The proof of Theorem 11.4 assumes only that the models used for the
control design (11.1) and the observer design (11.47) are identical in steady state in
the sense that they give rise to the same relation z = z(u, d, r). It does not make
any assumptions about the behavior of the real plant (11.46), i.e. the model-plant
mismatch, with the exception that the closed-loop system (11.61) must converge to
a fixed point. The models used in the controller and the observer could even be
different as long as they satisfy the same steady state relation.
Remark 11.9 If condition (11.59) does not specify ¯xt and ūt uniquely, it is customary
to determine ¯xt and ūt through an optimization problem, for example, minimizing
the magnitude of ūt subject to the constraint (11.59) [197].