SUFFICIENT CONDITION OF MAX-PLUS ELLIPSOIDAL
INVARIANT SET AND COMPUTATION OF FEEDBACK CONTROL
OF DISCRETE EVENT SYSTEMS
Mourad Ahmane and Laurent Truffet
Institut de Recherche en Communications et Cybern
´
etique de Nantes
IRCCyN, UMR-CNRS 6597,
´
Ecole Centrale Nantes
1 rue de la No
´
e, BP 92101, 44321 Nantes Cedex 3, France
and
´
Ecole des Mines de Nantes
4 rue Alfred Kastler, BP 20722, 44307 Nantes Cedex 3 France
Keywords:
Max-plus algebra, geometric approach, set invariance, residuation.
Abstract:
Haar’s Lemma (1918) provides the algebraic characterization of the inclusion of polyhedral sets. This Lemma
has been involved many times in automatic control of linear dynamical systems when the constraint domains
(state and/or control) are polyhedrons. More recently, this Lemma has been used to characterize stochastic
comparison w.r.t linear orderings of Markov chains with different state spaces. Stochastic comparison is
involved in the simplification of complex stochastic systems in order to control the approximation error made.
In this paper we study the positive invariance of a max-plus ellipsoid by a max-plus linear dynamical system.
We remark that positive invariance of max-plus ellipsoid is a particular case of polyhedron inclusion and we
use Haar’s Lemma to derive sufficient condition for the positive invariance. As an application we propose a
method to compute a static state feedback control.
1 INTRODUCTION
In 1918, Haar demonstrated the following result
which provides an algebraic characterization of the
inclusion of two polyhedral sets. For more details see
e.g. (Haar, 1918). This result can be found in (Hen-
net, 1989) where it is called extended Farkas’Lemma.
Let
n
denotes the component-wise order on R
n
(i.e.
x
n
y i, x
i
y
i
).
Result 1 (Haar’s Lemma) Let Φ (resp. Ψ) be a m×
d (resp. m
× d) matrix. Let p (resp. q) be a m
(resp. m
) dimensional column vector. The following
assertion
6= P, p) P, q), (1)
which is can be written as
6= {x R
d
|Φx
m
p} {x R
d
|Ψx
m
q},
is true iff
H R
m
×m
, (a).H 0
m
×m
,
(b).Ψ = HΦ,
(c).Hp
d
q.
In the case where p and q are equal to the null vector
(i.e. the homogeneous case), Haar’s Lemma reduces
to Farkas’ Lemma (Farkas, 1902). A recent reference
for such material is ((Urruty and al., 2001), pp. 58-
61).
Max-plus algebraic approach to modelling Discrete
Event Systems (DES) such as manufacturing systems,
communication protocols (TCP), computer networks,
dynamic programming, transportation networks (see
e.g. (Baccelli and al., 1992)) is now almost classical.
The geometric approach for the control of DES is one
of the main topics of research already mentioned by
(Cohen and al.,1999) which has not yet been so de-
veloped than in linear algebra.
In linear algebra, the geometric approach for the
positive invariance has been mainly developed by
(Wonham, 1985) in the case of vectorial subspaces
(see also (Basile and al., 1992)). The positive invari-
ance of a subset of the state space of a given dynami-
cal system is characterized by the following property:
if at some time a positively invariant set contains the
state system, then it will contains it also in the future.
(See e.g. the survey paper (Blanchini, 1999) on this
subject and references therein). It is known at least
since (Kalman and al., 1960) that stability in the sense
of Lyapuonv of classical linear systems has links with
the existence of a positively invariant ellipsoid
{x| x
T
P x c}.
99
Ahmane M. and Truffet L. (2006).
SUFFICIENT CONDITION OF MAX-PLUS ELLIPSOIDAL INVARIANT SET AND COMPUTATION OF FEEDBACK CONTROL OF DISCRETE EVENT
SYSTEMS.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 99-106
DOI: 10.5220/0001219600990106
Copyright
c
SciTePress
At the end of the 1980’s, (Bitsoris, 1988), (Hennet,
1989), and also more recently (Farina and al., 1998)
have explored and characterized other kind of posi-
tively invariant sets.
In (Ahmane and al., 2004) it has been shown that
the algebraic characterization of the inclusion of two
polyhedral sets provided by Haar’s Lemma is the fun-
damental notion for the characterization of the posi-
tive invariance of discrete-time Markov chains in par-
ticular and linear systems in general. This idea is
adapted here in the context of the control of Dis-
crete Event Systems (DES) using the properties of A-
invariance and (A B F )-invariance.
Let (S, , , ε, e) = (R
{−∞, +∞}, max, +, −∞, 0) be a complete idem-
potent semiring (see definitions in section 2.3). Let us
now consider the linear systems over (S, , , ε, e)
specified by (d, , A) and defined by:
(d, , A) :
x(0) S
d
,
x(n) = A x(n 1), n 1,
(2)
with d N, x(n) is the state vector at time instant
n and A S
d×d
is the state matrix. The vectorial
equation (2) means that
i, x
i
(n) = max
j=1,...,d
(A
i,j
+ x
j
(n 1)).
Define now the Max-Plus ellipsoidal set as follows:
E(P, w, α) = {x S
d
| x
T
P x
(α)
w}, (3)
where matrix P S
d×d
, w S \ {ε}, (·)
T
denotes
the transpose operator and
x S
d
, x
(α)
def
= (x
α
1
, . . . , x
α
d
)
T
= (x
α
j
)
T
j=1,...,d
,
with
a S, α R, a
α
def
= αa,
with αa denotes the usual multiplication α × a.
In linear algebra, E(P, w, α) coincides with the notion
of ellipsoid.
The main results of this paper are as follows. First,
Using Haar’s Lemma (see Result 1), we obtain suffi-
cient condition under which the following assertion is
true (cf. Theorem 1):
x(0) S
d
, [x(0) E(P, w, α) n, x(n) E(P, w, α)].
(4)
We will then introduce the concept of the positive in-
variance by the max-plus linear mapping x 7→ A x
or the A-invariance of the set E(P, w, α). It means
that if the state of the system at time instant 0 is on
the set E(P, w, α), one is sure that this state will re-
main within the set E(P, w, α) at every time instant
n. Second, we show that this sufficient condition for
the positive invariance of such set allow us to deter-
mine a max-plus linear state feedback control law. It
means to find a feedback F S
m×d
such that the
following linear system over a complete idempotent
semiring (S, , , ε, e) defined by:
(
x(0) S
d
, A S
d×d
, B S
d×m
x(n) = A x(n 1) B u(n),
u(n) = F x(n 1), n = 1, 2, . . . ,
verifies the assertion (4). We will also then introduce
the concept of the (A B F )-invariance of the set
E(P, w, α). It means that if the state of the system at
time instant 0 is in the set E(P, w, α), one is sure with
applying an adequate control u that this state will re-
main within the set E(P, w, α) at time instant n. Oth-
erwise says, the state x(n) can be forced to remain
inside E(P, w, α) by an adequate choice of the con-
trol u. To obtain such feedback on the state system,
the residuation theory is one of the main tool.
The paper is organized as follows. In Section 2, we
introduce the main notations used in this paper and
we present the main definitions of max-plus algebra.
The main references are e.g. ((Baccelli and al., 1992),
(Blyth and al. 1972), (Golan, 1992)). In Section 3,
we identify and characterize the positive invariance of
a max-plus ellipsoidal set. Sufficient conditions for
A-invariance and (A B F )-invariance are pro-
vided using Haar’s Lemma (Result 1). As an applica-
tion, Subsection 3.1 and 3.2 are respectively devoted
to the existence of a max-plus linear state feedback
controller and to a method for computing this max-
plus linear state feedback control law. Finally, some
conclusions are given in Section 4.
2 BACKGROUND
2.1 Notations and Definitions
(·)
T
denotes the transpose operator.
m
denotes the component-wise ordering of S
m
,
for all m N.
All vectors are column vectors.
If A, B S
m×n
then A B denotes the entry-
wise comparison of the matrices A and B.
2.2 Ordered Sets and Elements of
Residuation Theory
Let (Ω, ) be a partially ordered set. (Ω, ) is a
sup semilattice (resp. inf semilattice) iff any set
{ω
1
, ω
2
} has a supremum
W
{ω
1
, ω
2
} (resp. an
infimum
V
{ω
1
, ω
2
}). (Ω, ) is a lattice iff (Ω, )
is a sup and inf semilattice. (Ω, ) is complete iff
any set A has a supremum
W
A. A com-
plete ordered set is also a complete lattice because
ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
100
V
A
def
=
W
{ω : a A, ω a}. A lattice
is distributive iff and are distributive with respect
to (w.r.t) one another.
A map f : (Ω, ) (Ω
, ), where (Ω, ) and
(Ω
, ) are two ordered sets, is (, )-monotone if
it is a compatible morphism with respect to and
. The map f : (Ω, ) (Ω
, ) is residuated iff
there exists a map f
: (Ω
, ) (Ω, ) such that:
ω , ω
, f (ω) ω
ω f
(ω
).
This relation can be defined as follows:
f
(·)
def
=
_
{ω : f (ω) · }.
A monotone map f : (Ω, ) (Ω
, ), where
(Ω, ) and (Ω
, ) are complete sets, is said to be
continuous iff A , f(
W
A) =
W
f(A),
W
(resp.
W
) denotes the supremum w.r.t (resp. );
f(A)
def
= {f (a) : a A}.
the following result (see e.g. (Blyth and al. 1972,
Th 5.2) or (Baccelli and al., 1992, Th 4.50)) provides
a characterization for a residuable function over two
complete ordered sets.
Result 2 Let (Ω, ) and (Ω
, ) two complete sets.
A map f : (Ω, ) (Ω
, ) is residuated iff f is
continuous and f(
V
Ω) =
V
.
2.3 Basic Algebraic Structures
For any set S, (S, , , ε, e) is a semiring if (S, , ε)
is a commutative monoid, (S, , e) is a monoid,
distributes over , the neutral element ε for is also
absorbing element for , i.e. a S, εa = aε =
ε, and e is the neutral element for .
(S, , , ε, e) is an idempotent semiring (called
also dioid) if (S, , , ε, e) is a semiring, the inter-
nal law is idempotent, i.e. a S, a a = a. If
(S, , e) is a commutative monoid, then the idempo-
tent semiring (S, , , ε, e) is said commutative.
(S, , , ε, e) is a an idempotent semifield
if (S, , , ε, e) is an idempotent semiring and
(S\{ε}, , e) is a group, i.e. (S\{ε}, , e) is a
monoid such that all its elements are invertible
(a S\{ε}, a
1
: a a
1
= a
1
a = e).
Also if (S\{ε}, , e) is a commutative monoid,
then the idempotent semifield (S, , , ε, e) is said
commutative.
Let (S, , , ε, e) be an idempotent semiring. Each
element of S
n
is a n-dimensional column vector. We
equip S
n
with the two laws and · as follows:
x, y S
n
, (x y)
i
= x
i
y
i
, s S, (s x)
i
def
=
s x
i
, i = 1, . . . , n. The addition and the mul-
tiplication are naturally extended to matrices with
compatible dimension. Any n × p matrix A is asso-
ciated with a (, )-linear map A : S
p
S
n
. The
(i, j) entry, the l
th
row-vector and the k
th
column-
vector of matrix A, are respectively denoted a
i,j
, a
l,·
and a
·,k
. Let (S, , , ε, e) be an idempotent semi-
ring or an idempotent semifield, then (S, , ε) is an
idempotent monoid, which can be equiped with the
natural order relation defined by:
a, b S, a b
def
a b = b. (5)
We say that (S, , , ε, e) is complete if it is com-
plete as a naturally ordered set and if the respec-
tive left and right multiplications, λ
a
, ρ
a
: S S,
λ
a
(x) = a x, ρ
a
(x) = x a are continuous for all
a S. In such case we adopt the following notations
for a, b S:
λ
a
(b)
not.
= a\b
def
=
{x S : x a b},
ρ
a
(b)
not.
= b/a
def
=
{x S : a x b}.
A typical example of complete dioid is the top com-
pletion of an idempotent semifield. Let us note that
if a S is invertible then: a\b = a
1
b and
b/a = b a
1
. Let us note also that as S is com-
plete it possesses a top element
W
S
not.
= = +.
We have by convention the following identities:
ε = ε = ε,
a S, a = , a = a = a.
(6)
We suppose besides that (for a discussion to this topic,
see e.g. ((Baccelli and al., 1992, p. 163-164)):
a 6= ε, a = a = . (7)
By definition of / (idem for \) and properties of , ε
and of [7] we have for all a S:
a/ε = , /a = , (8a)
a/ =
ε if a 6=
if a =
ε/a =
ε si a 6= ε
if a = ε
.
(8b)
The operations ·/·, ·\· are extended to matrices and
vectors with compatible dimensions assuming that all
the elements of these matrices and vectors are in a
complete set S:
(A\y)
i
=
j
(a
j,i
\y
j
); (9a)
(A\B)
i,j
def
= (
{X : A X B})
i,j
=
k
(a
k,i
\b
k,j
);
(9b)
(D/C)
i,j
def
= (
{X : X C D})
i,j
=
l
(d
i,l
/c
j,l
).
(9c)
3 APPLICATIONS FOR THE
CONTROL OF DISCRETE
EVENT SYSTEMS
In this section, we identify and characterize the pos-
itive invariance of the max-plus ellipsoidal set pre-
viously defined by (3). Using Haar’s Lemma (cf.
SUFFICIENT CONDITION OF MAX-PLUS ELLIPSOIDAL INVARIANT SET AND COMPUTATION OF
FEEDBACK CONTROL OF DISCRETE EVENT SYSTEMS
101
Result 1), we give sufficient conditions for the A-
invariance (cf. Theorem 1) and (A B F )-
invariance (cf. Theorem 2). (Truffet, 2004) treated
the case where the variable α is equal to 1. He gave
necessary and sufficient conditions for the positive in-
variance of the max-plus ellipsoidal set previously de-
fined in the case where α = 1. As an application we
show that this sufficient characterization for the pos-
itive invariance of such sets allow us to determine a
max-plus linear state feedback control law ( F to be
determined).
3.1 A-invariance
Let us consider the following linear system (d, , A)
over a complete idempotent semiring (S, , , ε, e)
defined as follows:
(d, , A) :
x(0) S
d
,
x(n) = A x(n 1), n 1,
where x(n) is the state vector at time instant n and
A S
d×d
is the state matrix.
Recall also the max-plus ellipsoidal set defined by
(3):
E(P, w, α) = {x S
d
| x
T
P x
(α)
w},
P S
d×d
, w S \ {ε}.
(10)
where
x S
d
, x
(α)
def
= (x
α
1
, . . . , x
α
d
)
T
= (x
α
j
)
T
j=1,...,d
,
We are interested now in conditions under which
the following sets inclusion is true:
A E(P, w, α) E(P, w, α), (11)
where, by definition,
A E(P, w, α)
def
= {A x : x E(P, w, α)}.
We will then introduce the concept of the A-
invariance of the set E(P, w, α).
Definition 1 (A-invariant set) The set E(P, w, α) is
positively invariant by the linear mapping x 7→ A x
or A-invaraint if the last assertion defined by (11) is
verified.
This last definition means that, if the state vector of
the system (d, , A) at time instant 0 is in the set
E(P, w, α), one is sure that this state vector will re-
main it at time instant n. Otherwise says, the equation
(11) means that:
x(0) S
d
, [x(0) E(P, w, α)
n N, x(n) E(P, w, α)]
. (12)
The last assertion (12) can be rewritten at each time
instant by:
x S
d
, [x E(P, w, α) A x E(P, w, α)].
Using the definition of E(P, w, α) given by (10),
this last implication is equivalent to:
x S
d
, [x
T
P x
(α)
w
(A x)
T
P (A x)
(α)
w]
.
(13)
Lemma 1 For any matrix P S
d×d
we have:
x S
d
, (x
T
P x
(α)
w) (x
(α)
x
T
w/P )
(14)
Proof.
We have:
x
T
P x
(α)
=
M
i,j
(x
i
P
i,j
x
α
j
).
Since is commutative and = , the following
inequality
x
T
P x
(α)
w,
is logically equivalent to:
i, j = 1, ..., d, x
α
j
x
i
P
i,j
w.
what implies, by residuation,
i, j = 1, ..., d, x
α
j
x
i
w/P
i,j
,
which ends the proof.
Notice that α R, then it can be take some positive
or negative values, therefore in what follows we treat
the two cases: α R
+
and α R
.
α is a negative real number (α R
)
Proposition 1 The assertion
x S
d
, [x
T
P x
(α)
w
x
T
A
T
P A
(α)
x
(α)
w],
(15)
implies the assertion defined above by (13).
Proof.
It is sufficient to remark that :
x S
d
, α R
; (Ax)
(α)
A
(α)
x
(α)
,
where the matrix A
(α)
is defined by:
A
(α)
=
A
α
1,1
. . . A
α
1,d
.
.
.
.
.
.
.
.
.
A
α
d,1
. . . A
α
d,d
.
and
(A x)
T
= x
T
A
T
.
The result is now achieved by noticing that and
are non-decreasing w.r.t. .
In the following theorem, using Haar’s Lemma (see
Result 1) we give sufficient condition for the A-
invariance of the set E(P, w, α).
ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
102
Theorem 1 Let us assume that the set E(P, w, α)
is not empty. Let also A S
d×d
. The set
E(P, w, α) defined by (10) is positively invariant
by the linear mapping x 7→ A x or A-invariant
if there exists a d
2
× d
2
-matrix H such that the fol-
lowing conditions hold:
(a). H 0
d
2
×d
2
(elementwise) ,
(b). Q = HQ (linear algebra) ,
(16)
(c). i, j = 1, . . . , d;
N
k,l∈{1,...,d}
(w/P
k,l
)
(H
(i1)d+j,(k1)d+l
)
w/(A
T
P A
(α)
)
i,j
(17)
where Q denotes the d
2
× d-matrix defined by:
i, j = 1, . . . d, Q
(i1)d+j,·
= b
T
i
+ αb
T
j
, (18)
with (b
i
)
i=1,...,d
senotes the canonical basis of R
d
:
b
i
= (δ
{k=i}
)
k=1,...,n
; δ
{·}
= 1 if assertion {·} is
true and 0 otherwise.
Proof.
Based on Lemma 1, The assertion (15) given in
Proposition 1 can be rewritten as the following sets
inclusion:
P(Q, p) P(Q, q),
where, by definition,
P(Q, ·) = {x R
d
|Qx ·}.
The matrix Q is defined by (18), and the d
2
-
dimensional vectors p and q are respectively de-
fined by:
i, j = 1, . . . , d.
p
(i1)d+j
= w/P
i,j
, (19)
q
(i1)d+j
= w/(A
T
P A
(α)
)
i,j
. (20)
Using Haar’s Lemma (cf. Result 1), with in our
case Φ = Ψ = Q in assertion (1), we obtain the
conditions (a), (b) and (c), and the proof is now
achieved.
α is a positive real number (α R
+
)
Remark 1 In the case where α is a non-negative
real number, the sufficient condition of Theorem 1
is also necessary because :
x S
d
, α R
+
; (Ax)
(α)
= A
(α)
x
(α)
.
3.2 (A B F )-invariance
Let us consider now the following linear system over
a complete idempotent semiring (S, , , ε, e) de-
fined by:
x(0) S
d
,
x(n) = A x(n 1) B u(n), n 1
u(n) = F x(n 1),
(21)
with A S
d×d
, B S
d×m
and F S
m×d
to be
determined ( Any F is called a feedback) such that
the following assertion to be true:
x(0) S
d
, [x(0) E(P, w, α)
n N, x(n) E(P, w, α)]
, (22)
where E(P, w, α) is the set previously defined by
(10). It means that if the state of the system at time
instant 0 is in the set E(P, w, α), one is sure with ap-
plying an adequate control u that this state will remain
within the set E(P, w, α) at time instant n.
Notice that (21) can be rewritten as follows:
x(0) S
d
,
x(n) = (A B F ) x(n 1),
(23)
If the system (23) verifies the condition (22), we
will then introduce the concept of the (A B F )-
invariance of the set E(P, w, α). For more details, see
e.g. (Dorea, 1997), (Lhommeau, 2003), (Castelan and
al., 1993).
3.2.1 Existence Condition of Max-plus Linear
State Feedback Control
By definition of the system (21), the condition (22)
is equivalent to say that there exists a feedback F
S
m×d
such that the set E(P, w, α) is (A B F )-
invariant. In the following theorem, we give a suffi-
cient condition for the (A B F )-invariance of the
set E(P, w, α).
Theorem 2 Let us assume that the set E(P, w, α) is
not empty. The set E(P, w, α) is (A B F )-
invariant if there exists a d
2
× d
2
-matrix H such that
the following conditions hold:
(a). H 0
d
2
×d
2
(elementwise) ,
(b). Q = HQ (linear algebra) ,
(24)
(c). i, j = 1, . . . , d;
k,l∈{1,...,d}
(w/P
k,l
)
(H
(i1)d+j,(k1)d+l
)
w/((A B F )
T
P (A B F )
(α)
)
i,j
(25)
where matrix Q is defined by (18).
Proof.
We just have to apply the result of Theorem 1 with
matrix A replaced by matrix (A B F ).
From this sufficient condition of the existence of
static state feedback control law we can elaborate a
methodology based on linear programming to com-
pute F .
3.2.2 Computation of the Max-plus Linear State
Feedback Control
In this subsection, we give a method divided in three
main steps to compute all possible F such that the
following condition holds true:
SUFFICIENT CONDITION OF MAX-PLUS ELLIPSOIDAL INVARIANT SET AND COMPUTATION OF
FEEDBACK CONTROL OF DISCRETE EVENT SYSTEMS
103
x(0) S
d
, [x(0) E(P, w, α) n,
(A B F ) x(n) E(P, w, α)].
Step I:
We first compute the d
2
-dimensional vector ˆq by solv-
ing (see Theorem 1):
i, j = 1, . . . , d; ˆq
(i1)d+j
(H
(i1)d+j,·
) p
under:
(H
(i1)d+j,·
) Q = Q
(i1)d+j,·
,
(H
(i1)d+j,·
0 (elementwise),
(26)
because the left term of inequality (25):
O
k,l∈{1,...,d}
(w/P
k,l
)
(H
(i1)d+j,(k1)d+l
)
can be rewritten in linear algebra as:
i, j = 1, . . . , d; (H
(i1)d+j,·
) p,
with p and Q are respectively defined by (19) and
(18).
The resolution of the system (26) can be done us-
ing for example the simplex algorithm proposed by
Dantzig in 1947 (see e.g. (Schrijver, 1986)).
Step II:
Let us consider now an unknown d × d-dimensional
matrix G. We denote q(G) the d
2
-dimensional vec-
tor defined by (20, with matrix A replaced by matrix
G). Then, we have to find the solutions in G of the
following inequality:
ˆq q(G),
where ˆq is defined by the systems of inequalities (26).
This is equivalent to find the solutions of the follow-
ing system of d
4
-inequalities:
(i). i, j, k, l = 1, . . . , d,
G
k,i
G
α
l,j
(w/P
k,l
)/ˆq
(i1)d+j
.
This last system of inequalities can be rewritten in lin-
ear algebra as:
Q
g ψ (27)
with ψ is the d
4
-dimensional vector defined by:
(ii). i, j, k, l = 1, . . . d,
ψ
(i1)d
3
+(j1)d
2
+(k1)d+l
= (w/P
k,l
)/ˆq
(i1)d+j
,
(28)
and g is the d
2
-dimensional vector associated to the
unknown matrix G and defined by:
(iii). i, j = 1, . . . d, g
(i1)d+j
= G
i,j
, (29)
and Q
is the d
4
× d
2
-dimensional matrix defined by:
(iv). i, j, k, l = 1, . . . d,
Q
(i1)d
3
+(j1)d
2
+(k1)d+l,·
= b
T
(k1)d+i
+ αb
T
(l1)d+j
,
(30)
where (b
m
)
m=1,...,d
2
denotes the canonical basis of
R
d
2
.
The problem is now to enumerate all possible so-
lutions in g of the system of inequalities (27). This,
can be done by using the Γ-algorithm (Castillo and
al., 1999) pp. 161-162.
Step III:
The final step is to find all possible feedback F such
that:
A B F = G, (31)
where A, B are given matrices. The matrix G S
d×d
is computed by using of the formulæ(29) from a solu-
tion g of the system of inequality (27).
In the case of the assertion (31), using elements of
residuation theory (see for example (Baccelli and al.,
1992)), it can be shown that F is a solution of (31) iff
A G
B (B\G) = G
.
Example 1 Let us consider the following linear sys-
tem given over a complete idempotent semiring
(S, , , ε, e) = (R {−∞, +∞}, max, +, −∞, 0):
x(0) S
2
,
x(n) = A x(n 1)
0
1
|
{z}
B
u(n),
u(n) = F x(n 1),
Let us consider also the following set:
E(P, w, α) = {x S
d
| x
T
0 8
0 0
P
x
(α)
w},
with α = 1 and w = 9.
Now we try to find the feedback F such the set
E(P, w, α) is (A B F )-invariant.
STEP I:
Using (19) and (18) we obtain:
Q =
1 + α 0
1 α
α 1
0 1 + α
=
0 0
1 2
2 1
0 0
p =
9
1
9
9
.
Using condition (b) of Theorem 1, we obtain for
example:
H =
0 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
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104
and by using (26) we obtain:
ˆq =
0
1
9
0
.
STEP II:
Using (30):
Q
=
1 + α 0 0 0
1 0 α 0
α 0 1 0
0 0 1 + α 0
1 α 0 0
1 0 0 α
0 α 1 0
0 0 1 α
α 1 0 0
0 1 α 0
α 0 0 1
0 0 α 1
0 1 + α 0 0
0 1 0 α
0 α 0 1
0 0 0 1 + α
=
0 0 0 0
1 0 1 0
1 0 1 0
0 0 0 0
1 1 0 0
1 0 0 1
0 1 1 0
0 0 1 1
1 1 0 0
0 1 1 0
1 0 0 1
0 0 1 1
0 0 0 0
0 1 0 1
0 1 0 1
0 0 0 0
,
Using (28), the column vector ψ is
ψ =
9
1
9
9
8
0
8
8
0
8
0
0
9
1
9
9
.
Using (27):
The set of all solutions in g is:
g =
g
1
g
2
g
3
g
4
=
3
1
9
3
,
6
5
13
6
,
8
4
12
8
, ...
For example, taking
g =
3
1
9
3
,
and then by (29):
G =
3 1
9 3
.
STEP III:
If the matrix A of the system satisfied the condition
A G, we can compute the matrix F using residua-
tion theory, then the equation (31) for example admits
the solution
F = B\G = [
3 1
] .
4 CONCLUSION
In this paper we identified and characterized the pos-
itive invariance of max-plus ellipsoidal set. Based on
Haar’s lemma (cf. Result 1)(also called an extension
of Farkas’ Lemma due to (Hennet, 1989)), sufficient
condition for the A-invariance (cf. Theorem 1) and
the (A B F )-invariance (cf. Theorem 2) are pro-
vided. This sufficient condition leads us to provide a
methodology for computing and enumerating all pos-
sible max-plus linear state feedback control laws.
ACKNOWLEDGMENTS
This work is partially supported by the CNRS project
”MathSTIC 2004-05” entitled ’Control of Discrete
Events Systems via techniques of Comparison and
Aggregation of Stochastic Systems’ and by the Pro-
gramme ”Optimisation des Processus Industriels:
Optimisation-Ordonnancement et Pilotage” of the
grant between the Govt of France and the region Pays-
De-La-Loire 2000-06.
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