Proceedings of the
46th IEEE Conference on Decision and Control
New Orleans, LA, USA, Dec. 12-14, 2007
FrB03.5
Offset-Free Reference Tracking for Predictive Controllers
Urban Maeder, and Manfred Morari
Automatic Control Laboratory, ETH Zurich, Physikstrasse 3, ETL K13.1, CH – 8092 Zurich, Switzerland
maeder | morari @control.ee.ethz.ch
Abstract— We discuss the offset-free reference tracking problem for linear constrained systems in the presence of disturbances and plant-model mismatch. Introducing disturbance
models, we show how to construct a controller / observer
combination, such that zero offset is achieved. Contrary to
other approaches, the plant model is augmented only by as
many disturbance states as there are tracked variables, thus
yielding the controller with minimal complexity according to
the internal model principle.
when explicit Model Predictive Control is used [2], where
the complexity increases quickly with the number of state
variables.
II. P RELIMINARIES
Consider the feedback system in Figure 1. Let the nominal
model be defined by
x(k + 1) = Ax(k) + Bu(k)
y(k) = Cx(k)
I. I NTRODUCTION
In Model Predictive Control (MPC), a model of the plant
is used to predict the future evolution of the system state
[6], [7]. At each time step an optimization problem is solved
over the sequence of future input moves, possibly subject to
constraints. The first optimal control move obtained by the
optimization is then applied to the plant. At the next time
step, the prediction horizon of the optimization problem is
shifted forward in time and the procedure is repeated.
By the internal model principle [4], there have to be as
many integrators in the controller as there are variables for
offset-free control. Most traditional methods for LQG-style
state-feedback controllers suggest adding the integral of the
tracking error as additional state variable [3], [5]. Such a
scheme is inherently offset-free, but limited in the choice of
the disturbance model. In a constrained setup, these methods
suffer from windup effects [1], [8].
While MPC directly considers constraints, applying these
methods to MPC has some disadvantages. For instance,
the choice of an invariant terminal set which guarantees
feasibility is not clear. Some traditional methods do not
allow freedom in designing the disturbance model, since they
only consider the tracking error. For MPC, a two-degreeof-freedom method, where reference and output signals are
considered independently, is therefore preferable.
For these reasons, interest in disturbance model and observer based approaches has increased [10], [9]. Both teams
of authors independently derived conditions for offset-free
reference tracking. They both prove that if the plant is
augmented with as many disturbance states as there are
measured variables, there will be no offset. In the case where
only a subset of the measured variables is actually to be
controlled with zero offset, this method yields too complex
models. By the internal model principle, it should be possible
to add only as many disturbance states as there are outputs
to control with zero offset. Adding more variables than
necessary will lead to a unnecessarily complex optimization
problem in MPC. This effect is particularly bothersome
1-4244-1498-9/07/$25.00 ©2007 IEEE.
(1)
where x(k) ∈ Rnx , u(k) ∈ Rnu and y(k) ∈ Rny . We assume
(A, B) stabilizable and (C, A) detectable with C full row
rank. The controlled variables are defined as
z(k) = Hy(k)
(2)
with z(k) ∈ Rnz , H full row rank and nu ≥ nz . Input and
states are constrained
x(k) ∈ X ,
u(k) ∈ U,
(3)
where Y and U are convex sets with the origin in their
interior. The goal is to asymptotically eliminate the control
error given a constant reference signal r∞ , that is
z(k) → r∞ ,
k → ∞,
(4)
in the presence of constant, non-decaying disturbances and
plant-model mismatch, assuming stability of the closed loop.
The receding-horizon optimal control problem with
quadratic performance cost is considered
∗
JN
(x(k)) :=
min
u0 ,...,uN −1
N
−1
X
(uTi Rui + xTi Qxi )
i=0
+ xTN QN xN
s. t. xi ∈ X, ui−1 ∈ U, ∀ i ∈ {1, . . . , N },
xN ∈ T ,
xi+1 = Axi + Bui ,
R ≻ 0, Q 0, QN 0.
(5a)
(5b)
(5c)
(5d)
(5e)
The optimization problem is defined by the system model
(A, B), the prediction horizon N , the cost matrices Q and R,
the terminal weight QN , the terminal set T and the current
state x0 .
The problem is repeatedly solved at every time step k for
x0 = x(k) the current the state vector. The optimal input
u(k) = u∗o is applied to the plant, the rest of the optimal
open-loop control sequence u∗1 . . . u∗N −1 is discarded. We
note that the resulting control law is linear in some region
5252
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
u
y
P
H
FrB03.5
Assume that both the system and controller converge to
steady-state and the reference signal is given by r(k) = rss .
If ∃E ∈ Rnx̄ ×nz and E full column rank such that
z
E T (Ak − I)
T
E Bk
E T Br
K
r
Fig. 1.
Control System Structure
containing the origin, where none of the constraints are
active.
The terminal weight and the terminal set can be chosen
such that the nominal closed-loop system is guaranteed to be
stable [6]. For example, if QN is the solution to the discrete
algebraic Riccati equation
0 = Q + AT QN A − QN
− (AT QN B)(B T QN B + R)−1 (B T QN A),
(6)
(7)
If T is such that it is invariant and feasible under this local
controller, then both closed-loop stability and feasibility hold.
For reference tracking, it is clear that zero offset can only
be achieved when the controller is locally unconstrained,
around the requested set point. Thus, we will assume there
exists a set T which contains the requested set point and
which is rendered invariant by the linear controller Kx . Also,
Kx must be feasible within T . The system state must enter
the set T , otherwise the reference cannot be feasibly attained.
In the following, it will be assumed the state enters T and
the local control law is linear. The analysis of offset can thus
be performed for the linear control law Kx .
The state vector is assumed not to be directly measurable;
Hence, a linear state observer is employed.
x̂(k + 1|k) = (A − LC)x̂(k|k − 1) + Bu(k) + Ly(k). (8)
The control signal is
u(k) = −Kx x̂(k|k − 1).
(9)
Introducing the variables x̄(k) = x̂(k|k−1), Ak = A−LC −
BK, Bk = L and Ck = −Kx , the controller dynamics are
defined by
x̄(k + 1) =
u(k) =
Ak x̄(k) + Bk y(k) + Br r(k)
Ck x̄(k) + Dr r(k)
(11a)
= H,
= −I,
(11b)
(11c)
then there is zero offset in the controlled variable z(k), i.e.
z(k) → zss = rss for k → ∞.
Proof. Let uss , yss , x̄ss and rss be the signals at steady-state
and the controller’s state vector, respectively. For the system
to be in steady-state, (Ak − I)x̄ss + Bk yss + Br rss = 0 must
hold. Left multiplying by E T yields E T Bk yss +E T Br rss =
0 and by (11b) and (11c) Hyss = rss and hence zss = rss .
It is observed that E spans the left eigenspace of the
integrator modes (λ = 1) of Ak . Hence, Ak must contain
at least nz integrator modes for E to exist. Equations (11b)
and (11c) impose conditions on the input directions of the
integrators of K in the signals y(k) and r(k) [11].
III. D ISTURBANCE M ODEL
then the local unconstrained control law is
u(k) = −Kx x(k),
Kx = (B T QN B + R)−1 (B T PN A).
= 0,
(10)
with x̄(k) ∈ Rn̄x , u(k) ∈ Rnu and y(k) ∈ Rny
We first state a fundamental, sufficient condition for zero
offset.
Theorem 1 (Zero Offset): Consider the feedback system
shown in Figure 1, let the controller K be given by (10).
To account for the disturbances and plant-model mismatch,
we augment the plant model by a disturbance with integrator
B
A Bd
, Bm =
Am =
, Cm = [C Cd ] , (12)
0 I
0
with Bd and Cd of appropriate dimension. This model
assumes the disturbance is constant over time. This is a
common assumption in practice, when offset-free control at
steady-state is needed. The choice of the matrices Bd and
Cd may be motivated by a real, physical disturbance acting
on the system, but they may also be chosen freely, as design
parameters. The augmented model (Cm , Am ) is assumed to
be detectable.
A. Equivalence of State- / Output Disturbance Models
In the following, the equivalence of state and output disturbance models will be established. A similarity transform
is introduced
I −T12
T =
.
(13)
0
I
Thus, the transformed system matrices are
A (I − A)T12 − Bd
−1
T Am T
=
0
I
Cm T −1
= [C
(14)
CT12 + Cd ] .
It is easy to see that any disturbance model with Cd 6= 0 and
Bd = 0 (output disturbance) can be transformed into a pure
state disturbance model by solving
CT12 = −Cd .
(15)
for T12 . Since C has full row rank, a solution always exists.
5253
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
Conversely, a state disturbance model (Bd 6= 0, Cd = 0)
can be transformed into a pure output disturbance model by
solving
(I − A)T12 = Bd
(16)
for T12 . A solution exists iff span(Bd ) ⊆ span(I − A),
which is trivially true for plants with no eigenvalues at 1.
For simplicity, we will assume Cd = 0 from now on.
Since output disturbance models can always be transformed
to state disturbance models by similarity transform, this
is completely general. The proofs and methods are also
straightforward to extend to the case Cd 6= 0.
IV. C ONTROLLER
In this section, we develop a general method for tracking
constant reference signals in the presence of constant disturbances. The approach is well known and essentially similar
to [9] and [10]. The idea is to first compute the steady-state
target xss (rss , dss ) and uss (rss , dss ) for a given disturbance
and reference, and then devise a stabilizing controller of the
form
u(k) = uss − k(x̂(k|k − 1) − xss ).
(17)
The target variables xss and uss are determined by solving
a steady-state equation
I −A
HC
−B
0
xss
uss
=
Bd
0
dss +
0
I
A solution exists for any rss , dss if
I − A −B
rank
= nx + nz .
HC
0
rss . (18)
(19)
Assume nu ≥ nz . Then, the rank condition (19) can be stated
equivalently in the frequency domain. Let
G(z) = HC(zI − A)−1 B.
=
=
Xd dss + Xr rss ,
Ud dss + Ur rss .
(21a)
(21b)
The MPC problem (5) is modified
∗
JN
(x(k)) :=
min
u0 ,...,uN −1
N
−1
X
((ui − uss )T R(ui − uss )
i=0
+ (xi − xss )T Q(xi − xss )
(22a)
T
+ (xN − xss ) QN (xN − xss )
s. t. xN ∈ T (xss , uss ),
(22b)
(5b), (5d), (5e).
(22c)
The terminal set T now depends on the target variables.
If (22) is feasible, then the optimal control law is linear in
x(k) in some region around the target. Let this unconstrained
controller be given by
u(k) = uss − Kx (x̂(k|k − 1) − xss ).
Introducing the feedback gain matrices
Kd = −Kx Xd − Ud ,
(23)
Kr = −Kx Xr − Ur ,
(24)
Equation (23) is rewritten as linear feedback law of the
ˆ
−
augmented state estimate x̄(k) = [x̂(k|k − 1)T d(k|k
T T
1) ] , where dss is replaced by the current estimate of
the disturbance. Similarly, rss is replaced by the current
reference value. The feedback law is
u(k) = −[Kx Kd ]x̄(k) − Kr r(k).
(25)
This shows the degrees of freedom of the control structure
in both the reference and control error.
V. E STIMATOR
A state estimator is constructed for the augmented model
(Am , Cm ). The estimator gain matrix is
Lx
.
(26)
L=
Ld
The controller dynamics can be written as
x̄(k + 1) =
u(k) =
Ak x̄(k) + Bk y(k) + Br r(k)
Ck x̄(k) + Dr r(k)
(27)
Inserting the augmented model description (12), the control
law (25) and the estimator gain (26) yields the controller
dynamics
A − BKx − Lx C Bd − BKd
,
Ak =
−Ld C
I
BKr
Lx
(28)
,
, Br =
Bk =
0
Ld
Ck = −[Kx Kd ],
Dr = −[Kr ].
A. Conditions
(20)
Then, (19) holds if and only if z = 1 is not a transmission
zero for G(z), i.e. the steady-state gain matrix G(1) has full
row rank [11]. Let the solution to (18) exist and be given by
xss
uss
FrB03.5
For brevity, the following notation is introduced
Φ = I − A + BKx ,
Ψ = Φ + Lx C.
(29)
Consider the nominal system (12) subject to the control
law (25). The steady-state gains are easily computed and
they are given by zss = HCΦ−1 (Bd − BKd )dss for the
disturbance and zss = HCΦ−1 BKr rss for the reference.
Nominal tracking (disturbances are measurable, no plantmodel mismatch) can thus be achieved by choosing Kd and
Kr correctly. For offset-free control, the estimator has to be
considered additionally.
In the following, conditions are given for the matrices
Kx , Kd , Kr , Lx and Ld such that (11a-c) hold for the
dynamic system (28). These results are mostly equivalent
to the conditions given in [9] and [10].
Lemma 1 (Conditions for zero offset): Consider the controller defined by (27) and (28). Assume the closed loop
reaches steady state, the reference r(k) = rss constant and
the following holds
ker(Ld ) ⊆ ker H(I + CΦ−1 Lx )
(30a)
HCΦ−1 BKr = I
(30b)
HCΦ−1 (Bd − BKd ) = 0.
(30c)
5254
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
Then, there is zero offset in the controlled variable, i.e.
z(k) → zss = rss .
Proof. Equation (30a) can be rewritten as ker(Ld C) ⊆
ker(HCΦ−1 Ψ). If it holds, there exist ∆ ∈ Rnz ×nd which
chooses a linear combination of the rows of Ld C such that
∆Ld C = HCΦ−1 Ψ.
(31)
Consider E T = [−HCΦ−1 ∆] as candidate left eigenspace
of Ak to the eigenvalue λ = 1. By using (30), we check if
the conditions of Theorem 1 hold.
i) Using (11a) yields E T (Ak − I) = HCΦ−1 Ψ −
∆Ld C − HCΦ−1 (Bd − BKd ) . The first element
is zero by using (31), the second element vanishes
by (30c), which proves the presence of the integrator
modes.
ii) Checking for the integrator’s input directions using
(11b), we check if E T Bk = H and E T Br = −H.
For the former, we get E T Bk = −HCΦ−1 Lx + ∆Ld ,
and (31)
−HCΦ−1 Lx C + ∆Ld C
−1
= HC
FrB03.5
the Hautus observability condition for the system (Am −
[LTx 0]T Cm , Ĉ) at λ = 1


Ĉ
0
rank  I − A + Lx C −Bd  < nx + nd ,
(35)
0
0
it is clear that detectability does not hold. Therefore, L̂d
cannot stabilize all nd integrator modes, nor can Ld if it is
rank deficient.
Lemma 3 (Estimator): Consider the controller defined by
(27) and (28). Assume A − Lx C is stable and nd = nz .
Consider the estimator gain
Lx + L̄x H̄
L̄ =
(36)
L̄d H̄
where H̄ = H(I + CΦ−1 Lx ). Assume L̄x and L̄d can be
chosen such that Am − L̄Cm stable. Then, (30a) holds for
the estimator gain L̄.
Proof. Inserting (36) into (31) yields ∆L¯d H̄C =
HCΦ−1 (I − A + BKx + (Lx + L̄x H̄)C). For the theorem
to hold, ∆ must exist. Substituting H̄ yields
−1
−HCΦ Lx C + HCΦ Ψ = HC
HC Φ−1 (−Lx C + Ψ) = HC
{z
}
|
∆L̄d H(I + CΦ−1 Lx )C
HCΦ−1 Ψ +
L̄x H(I + CΦ−1 Lx )C
I
For the reference direction, we immediately get
E T Br = −HCΦ−1 BKr = −I by (30b).
Remark. When Kd and Kr are computed as in (21) and
(24), it is easy to check that (30b) and (30c) hold. The
construction of the estimator gains Lx and Ld given Kx
such that (30a) also holds will be discussed next.
B. Properties
In the following, the problem of constructing the estimator
gains such that (30a) holds is discussed. In principle, given
Kx , and an Lx which stabilizes A−Lx C, the set of candidate
Ld is defined by
∆Ld = H(I + CΦ−1 Lx ).
=
(32)
In order to derive a construction method for Ld , we will
state some properties first.
Lemma 2 (Rank of Ld ): Consider the the linear system
defined by the matrices
A
Bd
, Cm = [C 0] ,
(33)
Am =
0 Ind ×nd
where A ∈ Rnx ×nx , Cm ∈ Rny ×nx and Bd ∈ Rnx ×nd . Assume (Am , Cm ) detectable. Consider the class of estimator
T
gains L = LTx LTd
such that Am − LCm is stable. Then
rank(Ld ) = nd .
(34)
Proof. Choose an arbitrary Lx such that A− Lx Cm is stable.
Suppose there exists a Ld such that rank(Ld ) < nd and
Am −LCm is stable. Since rank(Ld C) < nd , we can choose
Ĉ, L̂d such that L̂d Ĉ = Ld C and rank(Ĉ) < nd . Applying
∆L̄d HCΦ−1 Ψ =
∆L̄d
=
HCΦ−1 (I + L̄x HCΦ−1 )Ψ
I + HCΦ−1 L̄x .
From the last equation, it is immediately clear that ∆ exists,
since L̄d has full rank by Lemma 2.
C. Construction Method
The previous sections suggest a direct construction method
for L.
Algorithm 1: Consider the linear system as in (1) and
(12). Assume Cd = 0, (Cm , Am ) detectable and nd = nz .
Suppose Kd and Kr are known and (30c), (30b) hold.
1) Compute Lx such that A − Lx C is stable and
(H̄Cm , Ā) detectable, where H̄ = H(I + C(I − A +
BKx )−1 Lx ) and Ā = Am − [LTx 0]T Cm .
2) Compute L̄ such that Ā − L̄H̄Cm is stable.
3) The final estimator gain
Lx + L̄x H̄
L2 =
(37)
L̄d H̄
stabilizes Am −L2 Cm and - by Lemma 3 - (30a) holds.
A better way to construct the estimator gain is described
in the following, modified algorithm, which allows to move
the poles associated to the integrator modes independently
from the other modes.
Algorithm 2: 1) Compute Lx such that A − Lx C is
stable and (H̄Cm , Ā) detectable, where H̄ = H(I +
C(I − A + BKx )−1 Lx ) and Ā = Am − [LTx 0]T Cm .
2) Apply the linear transform
I −(I − A + Lx C)−1 Bd
T =
(38)
0
I
5255
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
which brings the system to block-diagonal form
A − Lx C 0
Āt = T ĀT −1 =
(39)
0
I
C̄t
= H̄Cm T −1
(40)
−1
= H̄ C C(I − A + Lx C) Bd . (41)
3) Compute L̄d such that
(I − L̄d H̄C(I − A + Lx C)−1 Bd )
(42)
is stable.
4) Recover the estimator gain for the original system
0
Lx
−1
(43)
+T
L2 =
L̄d H̄
0
Algorithm 2 does not move the eigenvalues already established by A − Lx C. Hence, the integrator eigenvalues can
be placed independently.
D. Summary
An algorithm was proposed for constructing the estimator
such that offset-free control is achieved. The approach uses
a minimal order disturbance model. For every controlled
variable, one disturbance state is added, whereas existing
methods suggest adding as many disturbance states as there
are measured variables. In MPC, this improvement leads to
simpler optimization problems and thus a better performance
of the optimization algorithm.
Assuming stabilizability and detectability of the system,
necessary conditions for the existence of a tracking controller
are that the plant has no transmission zero at 1, that is
I − A −B
rank
= nx + nz ,
(44)
HC
0
and that the chosen disturbance model is detectable. Algorithm 1 and 2 require detectability of (Ā, H̄C) for the
construction of the estimator. It is an open question whether
there always exists an estimator gain Lx for a given Kx
such that detectability holds. Given the local linear control
law Kx , the resulting estimator will remove offset of the
closed loop in the controlled variables. Algorithm 2 describes
a method to construct the estimator in two steps. The closedloop estimator poles for the disturbances can be moved
independently of the poles of A − Lx C.
VI. E XAMPLE
As an example, we consider the double integrator
1 1
0.5
A=
, B=
, C = I.
0 1
1
(46)
The actuator is constrained by
|u| ≤ 0.2.
The dynamics of the physical plant are slightly different and
given by
1
1
Ar =
.
(48)
−0.1 0.9
Furthermore, the system shall be subject to disturbances. The
disturbance model is given by
1
Bd =
.
(49)
.5
Solving the target problem for this disturbance model, we
get Xd = [0 −3/4]T , Ud = −1/2, Xr = [1 0]T and Ur = 0.
For the cost function, we choose
Q = I, R = I,
(47)
(50)
which yields the LQ feedback gains Kx = [0.435 1.03],
Kd = 1.271 and Kr = −0.435. Closed loop performance
is analyzed for two estimators, defined by the gain matrices
L1 and L2 . L1 is the standard Kalman filter gain, while L2
is computed employing Algorithm 2.


1.26 0.87
L1 =  0.29 0.68  ,
0.33 0.13


1.51 1.25
L2 =  0.38 0.83  .
0.47 0.37
(51)
It is easy to check that L1 does not satisfy (30a), while L2
does. To account for the input constraint, we set up the MPC
problem, slightly modifying (5)
∗
JN
(x(k)) :=
min
u0 ,...,uN −1
N
−1
X
(ui − ut )T R(ui − ut )
i=0
+ (xi − xt )T Q(xi − xt )
+ (xN − xt )T QN (xNp − xt )
s. t. |ui−1 | < 0.2, ∀ i ∈ {1, . . . , N },
xi+1 = Axi + Bui + Bd d0
(52)
xt = Xd d0 + Xr r0 , ut = Ud d0 + Ur r0 .
Qn is set to the solution to the DARE which was obtained
while computing Kx . To close the control loop, we set x0 =
ˆ
x̂(k|k − 1), d0 = d(k|k
− 1) and r0 = r(k) and solve (52)
for every time step.
The results are plotted in Figures 2 and 3 for a prediction
horizon of N = 1. For the standard Kalman estimator gain
L1 , the closed loop shows offset. The second plot shows no
offset, but virtually the same performance. This suggests the
proposed method does fit nicely into the classical framework
of controller - observer design.
VII. C ONCLUSION
(45)
The goal is to track x1 , that is
H = [1 0].
FrB03.5
We have seen that robust reference tracking is not straightforward in Model Predictive Control. First we noted the
classic integral control schemes proposed in the literature
[3], [5] cannot be used for MPC straightforwardly. A twodegree-of-freedom approach employing a disturbance model
which may reflect knowledge of the physical structure of the
system is better suited.
5256
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
FrB03.5
x1(k)
2
x (k)
2
x (k)
2
1.5
1
x (k)
2
1.5
r(k)
1
1
0.5
0.5
0
r(k)
0
0
5
10
15
20
time
25
30
35
40
0
1
5
10
15
20
time
25
30
35
40
1
u(k)
u(k)
0.5
0.5
0
0
0
5
10
15
20
time
25
30
35
1
40
0
x (k)
5
10
15
20
time
25
30
35
x (k)
1
1
1
x (k)
x (k)
2
0
40
2
0
d(k)
−1
d(k)
−1
−2
0
5
10
15
20
time
25
30
35
1
−2
40
u(k)
0
5
10
15
20
time
25
30
35
40
u(k)
1
0.8
0.6
0.5
0.4
0.2
0
0
0
5
10
15
20
time
25
30
35
40
0
Fig. 2. Reference step (top plots) and disturbance steps (bottom plots) for
L1 estimator
5
10
15
20
time
25
30
35
40
Fig. 3. Reference step (top plots) and disturbance steps (bottom plots) for
L2 estimator
R EFERENCES
However, the disturbance modeling approach is difficult:
The integrators which are required for robustness do not appear automatically in the controller; they are tightly coupled
to the observer used. The solution proposed in the literature
was to add as many disturbance states to the model as there
are measured variables. This may lead to an unnecessary
increase in the dimension of the state space.
In this paper, we investigated the nature of the problem
further. We showed that the number of disturbance states can
in fact be reduced to the number of controlled variables. The
construction method for the observer gain starts with a linear
state-feedback controller and then constructs an observer
gain which yields zero offset. It is simple and based on
the standard state-space algorithms. By using a predictive
controller with a linear gain around the origin, the results
can be used directly for MPC. In the nominal case (no plant
- model mismatch), stability, feasibility and optimality of the
MPC apply.
It is clear from the construction method that the integrators
only appear for a matching controller / estimator pair. As
soon as the MPC is not unconstrained and the feedback law
changes, the integrators disappear. This avoids windup of the
integrators.
[1] Karl Johan Aström and Lars Rundqwist. Integrator windup and how
to avoid it. In American Control Conference, volume 2, pages 1693–
1698, 1989.
[2] Alberto Bemporad, Manfred Morari, V. Dua, and E. N. Pistikopoulos.
The explicit solution of model predictive control via multiparametric
quadratic programming. In Proceedings of the American Control
Conference, 2000.
[3] Edward J. Davison. Robust Servomechanism Problem, chapter 46,
pages 731–747. IEEE Press, 1996.
[4] B. A. Francis and W. M. Wonham. The internal model principle of
control theory. Automatica, 12:457–465, 1976.
[5] Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. Feedback Control of Dynamic Systems. Addison Welsey, 1994.
[6] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica,
36:789–814, 2000.
[7] M. Morari and J.H. Lee. Model predictive control: past, present and
future. Computers & Chemical Engineering, 23(4–5):667–682, 1999.
[8] E. F. Mulder, M. V. Kothare, and Manfred Morari. Multivariable antiwindup controller synthesis using linear matrix inequalities. Automatica, 37:1407–1416, 2001.
[9] Kenneth R. Muske and Thomas A Badgwell. Disturbance models for
offset-free linear model predictive control. Journal of Process Control,
12:617–632, 2002.
[10] Gabriele Pannocchia and James B. Rawlings. Disturbance models for
offset-free model predictive control. AIChE Journal, 49(2):426–437,
2003.
[11] Sigurd Skogestad and Ian Postlethwaite. Multivariable Feedback
Control. John Wiley & Sons, 1996.
5257