Linear matrix inequality based model predictive controller

Abstract

A model predictive controller based on linear matrix inequalities (LMIs) is presented. As in standard model predictive control (MPC) algorithms, at each (sampling) time, a convex optimisation problem is solved to compute the control law. The optimisation involves constraints written as LMIs, including those normally associated with MPC problems, such as input and output limits. Even though a state-space representation is used, only the measurable output and the extreme values of the unmeasurable states are used to determine the controller, hence, it is an output feedback control design method. Stability of the closed-loop system is demonstrated. Based on this MPC, a Lyapunov matrix is built and the controller computation is set in a more standard MPC framework. The design techniques are illustrated with numerical examples

Full text

LINEAR MATRIX INEQUALITIES
Linear matrix inequality based model predictive
controller
E. Granado, W. Colmenares, J. Bernussou and G. Garcı
´a
Abstract: A model predictive controller based on linear matrix inequalities (LMIs) is presented. As
in standard model predictive control (MPC) algorithms, at each (sampling) time, a convex
optimisation problem is solved to compute the control law. The optimisation involves constraints
written as LMIs, including those normally associated with MPC problems, such as input and output
limits. Even though a state-space representation is used, only the measurable output and the
extreme values of the unmeasurable states are used to determine the controller, hence, it is an output
feedback control design method. Stability of the closed-loop system is demonstrated. Based on this
MPC, a Lyapunov matrix is built and the controller computation is set in a more standard MPC
framework. The design techniques are illustrated with numerical examples.
1 Introduction
Model predictive control (MPC) is the most popular
industrial MIMO control strategy [1]. From among the
reasons for this popularity we highlight that all real systems
are subjected to physical constraints, such as an actuator’s
operation limits, and they may be explicitly considered, in
the MPC formulation. It is mainly a control technique for
systems with slow dynamics but its application to more
demanding systems is an area of current interest [2].
In the MPC scheme, the control law is obtained from an
optimisation problem whose objective function weights
the control efforts and the deviations from the set point. The
optimisation problem normally includes constraints on the
input (hard constraints), output and state (soft constraints).
The optimisation is performed over a (prediction) horizon
which is continuously moved forward in time, since only the
first control law is applied (out of those calculated over the
horizon) to the system [3, 4].
MPC was introduced in the 1970s [5], and considerable
research has been performed in the area to ensure, the
stability and feasibility of the problem [6 – 10]. MPC is a
methodology that, working always in the time domain, lets
the operator easily, handle physical performance require-
ments, such as upper and lower bounds on the process
variables, and tuning of the closed-loop. At the same time,
very little knowledge of the theory involved is required.
An extension to the output feedback case, of the
methodology proposed in [11] is now presented.
The unconstrained case as well as input and output
restrictions are considered. To ensure stability, an infinite
horizon is included in the objective quadratic function. The
optimisation problem is formulated in terms of linear matrix
inequalities (LMIs). Dedicated powerful LMI algorithms
already exist that allow problem solution in polynomial time,
and in many cases in times comparable to those necessary to
obtain an analytical solution to a similar problem [12].An
‘on-line’ and in-time solution is fundamental to MPC.
In order to only use the measurable output, a dynamic
output feedback controller (not an observer) is calculated at
each iteration. Since not all states are available, either the
extreme values or some of the statistical properties of the
unmeasured states are used. This approach provides, not
only a controller but also a Lyapunov matrix. This is used to
set the problem in a more standard framework. An observer
is required and stability is no longer guaranteed.
2 Problem statement
Consider the discrete linear time invariant system rep-
resented by:
xðkþ1Þ¼AxðkÞþBuðkÞ
yðkÞ¼CxðkÞð1Þ
where xðkÞ2<
n,uðkÞ2<
mand yðkÞ2<
qare respectively,
the state, input and output of the system. A2<
nn,B2
<nmand C2<
qnare constant matrices.
We want to find, at each sampling time, k, a control law in
the MPC framework, that stabilises (1), with the following
representation (dynamic controller):
xcðkþ1Þ¼AcxcðkÞþBcyðkÞ
uðkÞ¼CcxcðkÞð2Þ
where xcðkÞ2<
nand Ac;Bc;Ccare matrices of appropriate
dimensions. The problem reduces, for each sampling time,
to the determination of the matrices Ac;Bc;Cc, so that the
closed-loop system is stable.
qIEE, 2003
IEE Proceedings online no. 20030703
doi: 10.1049/ip-cta:20030703
E. Granado and W. Colmenares are with the Universidad Simo
´n Bolı
´var,
Dpto. Procesos y Sistemas, Apartado 89000, Caracas 1080, Venezuela
J. Bernussou and G. Garcı
´a are with the LAAS-CNRS 7, Av. Du Colonel
Roche 31077, Toulouse, France
Paper first received 1st October 2001 and in revised form 21st March 2003
IEE Proc.-Control Theory Appl., Vol. 150, No. 5, September 2003528

The closed-loop system (1) and (2) may be represented
by:
^
xxðkþ1Þ¼ ^
AA^
xxðkÞþ ^
BBuðkÞ
uðkÞ¼K^
xxðkÞ
yðkÞ¼ ^
CC^
xxðkÞð3Þ
where
^
AA ¼A0
BcCA
c
"#
2<
2n2n^
BB ¼B
0
"#
2<
2nm;
K¼0Cc

2<
m2n^
CC ¼C0

2<
q2n
ð4Þ
and
^
xxðkÞ¼ xðkÞ
xcðkÞ

2<
2n
ð5Þ
As mentioned above, we consider an infinite horizon
objective function to ensure stability. In terms of the
variables of the closed-loop system, the function is given by:
J1ðkÞ¼X
1
i¼0^
xxðkþi=kÞT^
QQ^
xxðkþi=kÞþuðkþi=kÞTRuðkþi=kÞ
ð6Þ
where
^
QQ¼Q0
00
 ð7Þ
Q50;R>0, and ^
xxðkþi=kÞrepresents the prediction of ^
xx at
instant kþi, given ^
xxðk=kÞ. Evidently, ^
xxðk=kÞ¼^
xxðkÞ.
Let us introduce the quadratic function:
Vðxaðk=kÞÞ ¼ ^
xxðk=kÞTP^
xxðk=kÞ;P>0ð8Þ
If we have that ^
xxð1=kÞ¼0 then Vð^
xxð1=kÞÞ ¼ 0.
Let us suppose for the moment, that for any instant kand
i50, the following condition is satisfied:
Vð^
xxðkþiþ1=kÞÞ  Vð^
xxðkþi=kÞÞ4
^
xxðkþi=kÞT^
QQ^
xxðkþi=kÞþuðkþi=kÞTRuðkþi=kÞð9Þ
If we sum inequality (9) from i¼0uptoi¼1 with
^
xxð1=kÞ¼0, we obtain:
Vð^
xxðk=kÞÞ4J1ðkÞð10Þ
that is, (8) is an upper bound for the objective function (6).
The algorithm proposed, much as in [11], is that of
minimising (8) subject to conditions that ensure (9), at
each sampling time.
Some, but not all of the states are measurable. As for the
rest, even if unknown, their extreme values or some
statistics must be available in order to define their condition
in probabilistic terms.
It is possible then to partition the state vector xðkÞas:
xðkÞ¼ xmðkÞ
xrðkÞ

2<
n
ð11Þ
where xmðkÞ2<
prepresents the measurable (known) states
and xrðkÞ2<
npthe non-measurable states, p4q. In this
work, the unknown states will be characterised by:
xmin
ri4xriðkÞ4xmax
rii¼1;...;npð12Þ
Let fxr1;xr2;...;xrlg, be the vertices of the polyhedra in
<npobtained by convex combination of the extreme values
of the unknown states. Note that l¼2np. Also, for any
measured value xmðkÞ, let:
x
jðkÞ¼ xmðkÞ
xrj

and ^
xx
jðkÞ¼
xmðkÞ
xrj
xcðkÞ
2
43
5j¼1;...;l
ð13Þ
Observe that
x
ðkÞ2<
nand ^
xx
ðkÞ2<
2n, also xcðkÞis the
initial (known) condition of the controller, normally set to
zero. Note also that other characterisations based on
probabilistic parameters such as probability density, mean
Ehxriand correlation matrix EhxrxT
rimay be also used [13].
As mentioned, all real processes have constraints in their
variables, those restrictions may be included in the
algorithm as ‘sufficient’ conditions expressed as LMIs.
The input constraints uðkÞnormally represent physical
limits (such as valve saturation). They are usually
considered to be ‘hard’ constraints since they have to be
satisfied. We will consider this constraint through the
Euclidean norm, given by:
uðkþi=kÞ


24umax i50
Output constraints are less restrictive, since they normally
represent performance requirements. We will consider
them, similarly, through the Euclidean norm, given in this
case by:
yðkþi=kÞ


24ymax i51
The vector yðkþi=kÞ, represents a system’s predicted
output at time kþi, based on the output at time k,yðk=kÞ.
The output constraints are imposed on future values (i50),
since it does not make any sense to apply it to the actual
value (i¼0).
3 Main result
The following theorem gives conditions for the existence
of a stabilising controller:
Theorem 1: Given the known limit vectors ymax and umax,
system (1) is stabilised by a controller of the form (2) if
there exist symmetric positive definite matrices X;Y2<
n,
and matrices F2<
nm;L2<
qnand Z2<
nn, that are
solutions of the following optimisation problem:
min ð14Þ
subject to
YIY
x
jðkÞ
IX
x
jðkÞ
Y
x
jðkÞ

T
x
jðkÞTI
2
6
6
43
7
7
5
>0j¼1;2;...;l
ð15Þ
YIYAþFC Z 00
IXAAXþBL 00
ðYAþFCÞTATYI0Q1=2
ZTðAXþBLÞTIXL
TR1=2XQ1=2
000R1=2LI0
00Q1=2Q1=2X0I
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
>0
ð16Þ
IEE Proc.-Control Theory Appl., Vol. 150, No. 5, September 2003 529

YI 0
IX L
T
0Lu
2
maxI
2
6
43
7
5>0
ð17Þ
YI ðCAÞT
IXðCAX þCBLÞT
CA CAX þCBL y2
maxI
2
6
43
7
5>0
ð18Þ
Proof: Since (8) is an upper bound for the objective function
(6), we may try to minimise such a bound by solving the
following problem:
min ð19Þ
subject to
^
xxðk=kÞTP^
xxðk=kÞ< ð20Þ
or equivalently:
^
xxðk=kÞT^
PP1^
xxðk=kÞ<1ð21Þ
where ^
PP ¼P1and using the Schur complement, inequal-
ity (20) maybe written as:
^
PP ^
xxðk=kÞ
^
xxðk=kÞTI
"#
>0
ð22Þ
Let us partition matrices ^
PP and ^
PP1in the form:
^
PP ¼XU
UT^
XX
"#
^
PP1¼YV
VT^
YY
"#
ð23Þ
and define the matrix:
T¼YI
VT0
"# ð24Þ
Without loss of generality, we may suppose that matrix Vis
a non-singular matrix [14] and therefore Tis also regular.
Pre-multiplying (22) by TT
1and post-multiplying it by T1;
with:
T1¼T0
0I

yields:
YIYxðk=kÞ
IXxðk=kÞ
Yxðk=kÞ

Txðk=kÞTI
2
6
43
7
5>0
ð25Þ
but (25) results from certain convex combinations of (15),
for any possible xðkÞ:Hence, (14) and (15) follow.
To obtain (16), we substitute (3) and (8) into (9) to obtain:
ð^
AA þ^
BBKÞTPð^
AA þ^
BBKÞPþ^
QQ þKTRK <0ð26Þ
Therefore, if inequality (16) is satisfied, the closed-loop
system is stable. Assuming a perfect model, at the next
iteration at least one feasible solution exists (the one
associated with the actual optimal solution but starting at
kþ1). The optimal cost at time kþ1ðJ1ðkþ1ÞÞ is smaller
than that at kðJ1ðkÞÞ since matrix Ris definite positive.
Hence, J1is a Lyapunov function to the MPC problem and
stability is guaranteed.
By Schur’s complement, (26) is equivalent to:
^
PP ð^
AA þ^
BBKÞ^
PP 00
^
PPð^
AA þ^
BBKÞT^
PP ^
PPKTR1=2^
PPQ1=2
0R1=2K^
PP I0
0Q1=2^
PP 0I
2
6
6
6
6
6
4
3
7
7
7
7
7
5
>0
ð27Þ
Pre-multiplying (27) by TT
2and post-multiplying by T2;
where:
T2¼
T000
0T00
00I0
000I
2
6
6
6
6
4
3
7
7
7
7
5
yields:
TT^
PPT TTð^
AA þ^
BBKÞ^
PPT 00
TT^
PPð^
AA þ^
BBKÞTTT
T^
PPT TT^
PPKTR1=2TT^
PPQ1=2
0R1=2K^
PPT I0
0Q1=2^
PPT 0I
2
6
6
6
6
6
4
3
7
7
7
7
7
5
>0
ð28Þ
Substituting (4), (23) and (24) into (28) and by defining:
F¼VBc
L¼CcUT
Z¼YAX þFCX þYBL þVAcUTð29Þ
we obtain (16).
Regarding the input and output constraints. Observe that
if conditions (9) and (15) are satisfied then:
^
xxðkþ1=kÞTP^
xxðkþ1=kÞ<; i51ð30Þ
that is, the ellipsoid:
"¼z=zTPz <

is an invariant ellipsoid for the predicted values of the states.
Then we may, at instant k;impose the following
Euclidean constraint to all future controls (even if we will
only use the next one):
uðkþi=kÞ


24umax i50ð31Þ
then it follows that:
max
i50

uðkþi=kÞ


2
2¼max
i50

K^
xxðkþi=kÞ


2
2
4max
z2"

Kz


2
2
¼maxð^
PP1=2KTK^
PP1=2Þð32Þ
From (32), by using Schur’s complement, we can write:
^
PP ^
PPKT
K^
PPu
2
maxI
"#
>0ð33Þ
pre-multiplying (33) by TT
1and post-multiplying by T1;and
substituting (4), (23) and (24) into the resulting inequality
and by definitions (29), (17) is obtained.
Regarding the output, and again using the Euclidean
norm, we want to ensure that yðkþi=kÞ


24ymax;i51:We
have that:
IEE Proc.-Control Theory Appl., Vol. 150, No. 5, September 2003530

max
i50yðkþiþ1=kÞ



2
2¼max
i50
^
CCð^
AA þ^
BBKÞ^
xxðkþi=kÞ





2
2;i50
4max
z2"
^
CCð^
AA þ^
BBKÞz





2
2
¼maxð^
PP1=2ð^
AA þ^
BBKÞT
^
CCT^
CCð^
AA þ^
BBKÞ^
PP1=2Þ
ð34Þ
then yðkþi=kÞ


24ymax;i51 if:
^
PP ^
PPð^
AA þ^
BBKÞT^
CCT
^
CCð^
AA þ^
BBKÞ^
PPy
2
maxI
"#
>0
ð35Þ
pre-multiplying (35) by TT
1and post-multiplying by T1and
by substituting (4), (23), (24) and (29) into the resulting
inequality, yields (18).
3.1 Controller construction
Once a solution to (14) is obtained, a controller might be
readily built by assuming any regular matrix U(on which no
restraint has been imposed, other than regularity [14]) and
the controller is given by:
.V¼ðIYXÞðUTÞ1
.Cc¼LðUTÞ1
.Bc¼V1F
.Ac¼V1ZðUTÞ1
3.2 Peak bound constraints
In theorem 1 we have restrictions on the input and output
variables through the Euclidean norm. Other time-domain
specifications may be included. For instance, peak bounds
on each component of the control variable, may be
accounted for by:
ujðkþi=kÞ
4uj;max;i50;j¼1;2;...;mð36Þ
Now:
max
i50jujðkþi=kÞj2¼max
i50jK^
xxðkþi=kÞjj2
4max
z2"Kz
jj
2
ðK^
PP1=2Þj





2
2(Cauchy-Schawarz inequality)
¼ðK^
PPKTÞjj
ð37Þ
Thus, the existence of a matrix Ssuch that:
~
PP ~
PPKT
K~
PPS
"#
>0
ð38Þ
with Sjj4u2
j;max;i50;j¼1;2;...;m;guarantees that
ujðkþi=kÞ4uj;max, for all predicted values and all entries
ði;jÞ:
Pre-multiplication of (38) by TT
1and post-multiplication
by T1;will lead to a LMI in the same variables as in
theorem 1.
4 Numerical example
Next, we present a numerical example to feature the
algorithm. A model taken from [8], slightly modified
(a measurable output is included in the model, i.e. not all
states are available for feedback), is used. The model is:
xðkþ1Þ¼ 10:1
00:99
"#
xðkÞþ 0
0:0787
"#
uðkÞ
yðkÞ¼ 10

xðkÞ
The sampling time is 0.1 s, and the state initial condition is
xð0Þ¼½0:05 0 T. Also Q¼Iand R¼0:000 02. The
second state ðx2Þwill be unmeasured, and its known bounds
are:
0:14x240:1
The control bound was set to: umax ¼1;obtaining the
profiles for the output yand control ushown in Figs. 1 and 2
respectively.
5 A more classical approach
The preceeding Section depicts a controller that assures
stability of the closed-loop and input and output constraints
satisfaction if some LMI conditions are verified.
Another important result is that a Lyapunov matrix (either
Por ^
PPÞis obtained. This second result may be used to set the
approach in a more standard framework of the MPC.
In this Section such an approach is formulated and it is
compared, using the same numerical example, with the
LMI-based method previously presented. The results
obtained are better than the previous ones.
To transform the infinite horizon problem into a finite
horizon problem with a terminal cost, we will follow the
ideas of [6]. Let us start with the objective function that we
Fig. 2 The control performance profile of the LMI-based
constrained MPC
Fig. 1 The output performance profile of the LMI-based
constrained MPC
IEE Proc.-Control Theory Appl., Vol. 150, No. 5, September 2003 531

partition at a finite horizon N:
J1ðkÞ¼X
N1
i¼0ð^
xxðkþi=kÞT^
QQ^
xxðkþi=kÞþuðkþi=kÞTRuðkþi=kÞ
þX
1
i¼N
ð^
xxðkþi=kÞT^
QQ^
xxðkþi=kÞþuðkþi=kÞTRuðkþi=kÞÞ
ð39Þ
The idea behind this more classical approach, is to
compute at time kthe optimal control sequence
ðu0ðk=kÞ;...;u0ðkþN1=kÞ;u0ðkþN=kÞ;...;u0ð1=kÞ;that
minimises (39) subject to the system dynamics (1) and the
extreme absolute values ymax and umax :Starting at t¼kþN;
the system will be a closed-loop with the same controller
computed in Sections 3.1 and (3). Minimising (39) yields
the sequence: u0ðk=kÞ;...;u0ðkþN1=kÞ;K^
xxðkþN=kÞ;...;
K^
xxðkþi=kÞ;...
A few comments are now in order. The first one is: in the
more classical approach, based on the perfect model
assumption, if the first optimisation at time kis feasible,
so will all the others. The optimal solution of the previous
iteration is a feasible solution to the next iteration. Also,
if problem (14) has a solution, then the more classical
approach is always feasible, since a solution of (14) is a
feasible solution to the more classical approach.
Second, if an optimal sequence up to t¼kþNis found
ðu0ðk=kÞ;K;u0ðkþN1=kÞÞ then no constraint will be
violated from there on. It is a consequence of the control law
that is used starting at t¼kþN:
An immediate consequence of the feasibility assurance is
that J1from (41), is in fact a Lyapunov function and hence,
the stability of this MPC scheme.
Let us now concentrate on the reformulation to a finite
horizon problem.
The second term in (39) is equivalent to [6]:
X
1
i¼N
ð^
xxðkþi=kÞT^
QQ^
xxðkþi=kÞþ^
uuðkþi=kÞTR^
uuðkþi=kÞÞ ¼
^
xxðkþN=kÞTX
1
i¼0
½ð ^
AATþ^
KKT^
BBTÞið^
QQ þ^
KKTR^
KKÞð ^
AA þ^
BB ^
KKÞi
^
xxðkþN=kÞ
ð40Þ
Since the matrix ð^
AA þ^
BBKÞis stable then the second term in
(40) converges to a matrix:
P¼X
1
i¼0
ðð ^
AATþKT^
BBTÞið^
QQ þKTRKÞð ^
AA þ^
BBKÞiÞ
which satisfies:
ð^
AA þ^
BBKÞTPð^
AA þ^
BBKÞ¼Pð^
QQ þKTRKÞ
that is, the Lyapunov matrix computed in the preceeding
Section (see (23) and the definition of ^
PPÞ:
Consequently, the objective function (40) may be written
as:
J1¼^
xxðkþN=kÞT^
PP ^
xxðkþN=kÞ
þX
N1
i¼0
ð^
xxðkþi=kÞT^
QQ^
xxðkþi=kÞþuðkþi=kÞTRuðkþi=kÞÞ
ð41Þ
As a consequence, the minimisation of (41) subject to the
system dynamics (1) and the feasible set determine by ymax
and umax renders a stable MPC. It is a finite horizon MPC
problem, and the constraints are to be verified only up to the
horizon N:
In (41) the initial state xðkÞis required to predict the
future values. Given that this is not always available, an
observer has to be used to estimate it. The introduction of
the observer precludes us from invoking condition (9) and
hence, the stability guarantee is lost. However, if the
observer is stable, i.e. the estimate converges asymptotically
to the real value, it can be expected to recover the stability
condition as t!1[2]. Other prediction scenarios, free from
the initial conditions, are currently under evaluation.
The observer structure used is [2]:
z
ðk=kÞ¼
z
ðk=k1Þþ ^
KK0ðyðkÞC
z
ðk=k1ÞÞ
z
ðkþ1=kÞ¼A
z
ðk=kÞþBuðkÞð42Þ
where
z
ðkÞis the estimate of xðkÞ:The observer (42) is stable
if ðAK0CÞis stable [2],K0¼A^
KK0:
6 Numerical example. Classical approach
In this Section, the previous numerical example is run
within the finite horizon framework. The horizon was set to
N¼10 and the observer’s eigenvalues was chosen to be
(0.25, 0.25). The results are highlighted in Fig. 3.
7 Conclusions
An output feedback control design method has been
presented for MPC based on LMI constraints. The controller
ensures stability of the closed-loop system. Input and output
constraints were included (as sufficient conditions) as
additional LMIs. Other performance requirements normally
formulated in the frequency domain such a H1and H2may
equally be incorporated. Since some of the states are
unmeasurable, they have been characterised by their
extreme values, much as in a robust control scheme. A
Lyapunov matrix that allows us to set the problem in a more
standard framework and with apparently, better numerical
results has also been presented.
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