Model Predictive Control

11.7 Offset-Free Reference Tracking 225
Delta Input (δu) Formulation.
In the δu formulation, the MPC scheme uses the following linear time-invariant
system model of (11.46):
⎧
⎨
⎩
x(t + 1) = Ax(t) + Bu(t)
u(t) = u(t − 1) + δu(t)
y(t) = Cx(t)
(11.69)
System (11.69) is controllable if (A, B) is controllable. The δu formulation often
arises naturally in practice when the actuator is subject to uncertainty, e.g. the
exact gain is unknown or is subject to drift. In these cases, it can be advantageous
to consider changes in the control value as input to the plant. The absolute control
value is estimated by the observer, which is expressed as follows
ˆx(t + 1) A B ˆx(t) B Lx
=
+ δu(t) +
û(t + 1) 0 I û(t) I
The MPC problem is readily modified
Lu
minδu0,...,δuN −1 yk − rk2 Q + δuk2 R
subj. to xk+1 = Axk + Buk, k ≥ 0
yk = Cxk k ≥ 0
xk ∈ X, uk ∈ U, k = 0, . . .,N − 1
The control input applied to the system is
xN ∈ Xf
uk = uk−1 + δuk, k ≥ 0
u−1 = û(t)
x0 = ˆx(t)
(−ym(t) + Cˆx(t))
(11.70)
(11.71)
u(t) = δu ∗ 0 + u(t − 1). (11.72)
The input estimate û(t) is not necessarily equal to the actual input u(t). This
scheme inherently achieves offset-free control, there is no need to add a disturbance
model. To see this, we first note that δu ∗ 0 = 0 in steady-state. Hence our analysis
applies as the δu formulation is equivalent to a disturbance model in steady-state.
This is due to the fact that any plant/model mismatch is lumped into û(t). Indeed
this approach is equivalent to an input disturbance model (Bd = B, Cd = 0). If
in (11.71) the measured u(t) were substituted for its estimate, i.e. u−1 = u(t − 1),
then the algorithm would show offset.
In this formulation the computation of a target input ūt and state ¯xt is not
required. A disadvantage of the formulation is that it is not applicable when there
is an excess of manipulated variables u compared to measured variables y, since
detectability of the augmented system (11.69) is then lost.
Minimum-Time Controller
In minimum-time control, the cost function minimizes the predicted number of
steps necessary to reach a target region, usually the invariant set associated to