=Paper=
{{Paper
|id=Vol-1256/poster2
|storemode=property
|title=Observer Design and Feedback Controller Synthesis with Observer in Idempotent Semiring
|pdfUrl=https://ceur-ws.org/Vol-1256/poster2.pdf
|volume=Vol-1256
|dblpUrl=https://dblp.org/rec/conf/vecos/AbdesselamKL14
}}
==Observer Design and Feedback Controller Synthesis with Observer in Idempotent Semiring==
https://ceur-ws.org/Vol-1256/poster2.pdf
120
Observer design and feedback controller
synthesis with observer in idempotent
semiring
Aldjia Nait Abdesselam Redouane Kara Jean-Jacques Loiseau
L2CSP laboratory L2CSP laboratory IRCCyN
Mouloud Mammeri UNiversity Mouloud Mammeri University UMR CNRS 6597
Route de Hasnaoua Route de Hasnaoua BP 92101 1 rue de la No 44321
BP 17, 15000, Tizi-Ouzou BP 17, 15000, Tizi-Ouzou Nantes Cedex 3
Algeria Algeria France
abdeslamaldja@yahoo.fr redouk@yahoo.fr Jean-Jacques.Loiseau@irccyn.ec-nantes.fr
In this paper, we present an observer design and a feedback controller with observer for a discrete
event system involving synchronization phenomena. These systems can be described by linear models
in the idempotent semiring. The approach follows the same principle as the Luenberger observer used
in continuous systems. Theoretical results are applied to an industrial process and simulation results are
reported to show the effectiveness of these methods in the estimation for min max plus linear systems using
Scilab.
idempotent semiring, observer, dioids, (max, +) linear systems, feedback controller
1. INTRODUCTION systems and provide a framework for analytical
techniques to meet the goals of design, control
A discrete event system (DES) is a dynamic system and performance evaluation. For about 30 years,
whose behavior can be described by means of a set a particular algebraic structure, called Dioids has
of time-consuming activities, performed according motivated the elaboration of a new linear system
to a prescribed ordering. Events correspond to theory (Baccelli (1992); Cuninghame-Green (1979);
starting or ending some activity (Cassandras (1999); Cohen (1984)). This theory offers a striking analogy
Cohen (1984)). These systems can represent with conventional linear system theory such as state
a great number of processes characterized as representation, transfer matrices, corrector synthesis
being concurrent, asynchronous, distributed or and identification theory (Cohen (1999); Lhommeau
parallel, such as flexible manufacturing systems, (2003); Cottenceau (1999)).
multiprocessor systems or transportation networks. In control theory, a state observer is a system
If the concerned systems are characterized by delay that provides an estimate of the internal state of
and synchronization phenomena, the Timed Event a given real system from measurements of the
Graphs (TEG) constitute interesting models. Timed input and output of the real system. the observer
Event Graphs are a subclass of timed Petri Net in was first proposed and developed in (Luenberger
which all places have a single transition upstream (1964, 1966)). Since these early papers, which
(A single upstream transition means that there is concentrated on observers for purely deterministic
no competition in either consumption or supply of continuous linear systems, observer theory has been
token in TEG) and a single one downstream (means extended by several researchers to include discrete
that all potential conflicts in using tokens in places event dynamic systems, in particular Timed Event
have been already arbitrated). This class of system Graph.
plays an important role because of its deterministic The observer design problem of Timed Event Graph
temporal behavior. has received much attention over the last few years.
In opposition to continuous systems, Timed Event A first problem considered is to estimate state in
Graphs are not modeled through differential or presence of disturbances for max-min plus linear
difference equations. An appropriate model is system initially developed by (Laurent (2010)). Here,
developed to describe the behavior of these
121
the main approach is based on the dioid of series equal to 0. We adopt the usual notation, so that
Maxin [γ, δ]. A second objective is to use an observer the symbol ⊕ stands for the max operation,
to feedback controller in order to obtain a desired and ⊗ stands for the addition. Notice that
behavior. ⊗ (+∞) = (−∞) + (+∞) = = (−∞) in
In this paper, the approaches are applied to an Rmax .
industrial process. Simulation results using Scilab
are reported. 2.1.2. Example 2.
The article is organized as follows. In section 2 we ((min, +)algebra).Rmin = (R ∪ {−∞} ∪
recall basic notions and results about idempotent {+∞}, min, +) is also a commutative dioid, for
semiring and residuation theory (Cohen (1998a)). which equals to +∞, and e equals to 0. We shall
A brief description of the industrial plant is given, denote ⊕ the min operation in the sequel, and the
and we then introduce the modeling of Timed Event symbol ⊗ will stand for the addition. Notice that
Graph in the dioid of formal series Max in [γ, δ] in ⊗ (−∞) = (+∞) + (−∞) = = (+∞) in Rmin .
section 3. In section 4, the observer is designed by
analogy with the classical Luenberger observer for
linear systems and controller synthesis with observer 2.1.3. Remark.
is obtained by considering residuation theory which Most of the time the symbol ⊗ will be omitted as in
allows the inversion of mapping in section 5. Finally, conventional algebra, moreover ai = a ⊗ ai−1 and
an example of production process is given in section a0 = e.
6.
Definition 4 (Order relation). A set D is said to be
2. PRELIMINARIES ordered if there exisists a binary relation such that
the following conditions are satisfied for all a, b and c
In this section we give the notations and some in D:
algebraic tools concerning the dioid and residuation
theories. • Reflexive: every element is in relation with itself
(a a);
2.1. Definitions
• Antysymmetric: if a b and b a ⇒ a = b.
Definition 1 (Monoid). A Monoid is a set D,
endowed with an internal law noted ⊕, which is • Transitive: if a b and b c ⇒ a c.
associative and has a neutral element, denoted ,
∀a ∈ D, a ⊕ = ⊕ a = a. In a dioid, the relation associated with max
application is an oreder relation which correspond to
Definition 2 (Dioid or idempotent semiring). A dioid the usual order ≤, a b ⇔ b = a ⊕ b ⇔ a ≤ b.
(D, ⊕, ⊗) is an algebraic structure, endowed with The relation associated with min application is an
two internal operations, denoted by ⊕ and ⊗. oreder relation which correspond to the reverse of
The operation ⊕ is associative, commutative and the usual order ≥, a b ⇔ b = a ⊕ b ⇔ a ≥ b.
idempotent, that is a ⊕ a = a. The operation ⊗ is
associative (but not necessarily commutative), and Definition 5 (Majorant and minorant). Let (D, D )
distributive at left with respect to ⊕: ∀ a, b, c ∈ D, (a ⊕ be an ordered set, C ⊂ D a non-empty subset of D,
b) ⊗ c = (a ⊗ c) ⊕ (b ⊗ c), and at right: ∀ a, b, c ∈ and a, b ∈ C.
D, a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c). The neutral elements
• An element x ∈ D satisfying ∀b ∈ C, b x is
of ⊕ and ⊗ are represented by and e respectively,
called majorant of set C.
and is absorbing for ⊗: ∀ a ∈ D, a ⊗ = ⊗ a =
• An element y ∈ D satisfying ∀b ∈ C, y b is
One says that the dioid is commutative provided that called minorant of set C.
the law ⊗ is commutative.
In particular, if the upper bound (i.e. the least
majorant) or/and lower bound (i.e. the greatest
Definition 3 (Complete dioid). A dioid D is said to minorant) of set a, b exist, we denote them by a ∨ b
be complete if it is closed for infinte for infinite sums and a ∧ b, respectively.
and if the product disitributes over infinte sums. A
dioid is said complet 2.2. Matrix dioid
2.1.1. Example 1. Let (D, ⊕, ⊗) be a given dioid, and denote Dn×n the
((max, +)algebra).Rmax = (R ∪ {−∞} ∪ set of square n × n matrices with entries over D.
{+∞}, max, +) is a commutative dioid with zero The sum and the product over D extend as usually
element equal to −∞, and the unit element e over Dn×n as follows:
122
denoted respectively by L]a (x) = a ◦ x and Ra] (x) =
(A ⊕ B)ij = Aij ⊕ Bij x ø a, were ◦ and ø are the left and right residuation
n
M respectively.
(A ⊗ B)ij = (Aik ⊗ Bkj ).
k=1 Theorem 2 The mappings x 7→ a ◦ x and x 7→ xøa
satisfy the following properties:
One can see that (Dn×n , ⊕, ⊗) is a dioid. The neutral
matrix for the law ⊕ is the matrix with entries equal to a ◦ a = (a ◦ a)∗ , aøa = (aøa)∗ , (4)
, the identity matrix for the law ⊗ is the matrix with
entries equal to e on the diagonal and elsewhere.
Notice that the products of matrices in Rmax and in
Rmin are not equal, and do not equal the usual sum a(a ◦ (ax)) = ax, ((xa)øa)a = xa, (5)
of matrices. b ◦ a ◦ x = (ab) ◦ x, xøaøb = xø(ba), (6)
∗ ∗ ∗ ∗ ∗ ∗
a ◦ (a x) = a x, (a x)øa = a x, (7)
Theorem 1 (Kleene star operator). Over a complete
dioid D, the implicit equation x = ax L ⊕ b admits (a ◦ x) ∧ (a ◦ y) = a ◦ (x ∧ y), (8)
x = a∗ b as least solution. where a∗ = i∈N ai
(xøa) ∧ (yøa) = (x ∧ y)øa. (9)
In the following this operator will sometimes be
represented by the mapping K : D → D,x 7→ x∗ .
Furthermore, letting a, b ∈ D, Kleene star operator The sum, the product and the residuation of
satisfies the following well known properties : : matrices are defined after the sum, product and the
residuation of scalars in D.
(a∗ )∗ = a∗ , a∗ a = aa∗ , a(ba)∗ = (ab)∗ a (1)
(a ⊕ b)∗ = (a∗ ⊕ b)∗ = (a ⊕ b∗ )∗ = (a∗ ⊕ b∗ )∗ (2) 3. TIMED EVENT GRAPH
∗
Thereafter, the operator a+ = i
L
i∈N + a = aa = A Timed Event Graph (TEG) is a subclass of timed
∗
a a is also considered, it satisfies the following Petri Net where each place has a single input
properties: transition and a single output transition. For more
details about Petri net see David and Alla (1997).
a+ a∗ , (a+ )∗ = a∗ , (ab∗ )+ = a(a ⊕ b)∗ (3)
Inversion of mappings is an important issue in
many control applications. Unfortunately, in general
manner, mappings defined over idempotent semiring
do not admit inverse. However the residuation
theory allows to characterize the solution set of an
inequality such as f (x) b. The reader may consult
Cohen (1998a) to obtain a complete presentation of
this theory.
Definition 6 (Isotone mapping). f is an isotone
mapping if it preserves order, that is, a b ⇒ f (a) Figure 1: Production process
f (b).
Definition 7 (Residuated mapping). An isotone 3.1. Plant description
mapping f : D → C, where D and C are ordered
sets, is a residuated mapping if for all b ∈ C The process we study here (see Martinez (2003);
there exists a greatest element x that satisfies the Amari (2004)) is composed of three conveyor belts
inequality f (x) b. This greatest element is denoted connected by loops. the parts are made on an
by f ] (b) and mapping f ] is called the residual of f . extruding machine in loop 3. Loop 1 and loop 2 are
Dually, if there exists a least element x for the both similar one to each other. they are dedicated to
inequality f (x) b, it is denoted by f [ (b). Mapping a thermal processing of the parts. Loop 3 processes
f [ is called the dual residual of f . parts that are conveyed on pallets to one of the other
loops. we study loop 2 Figure 2 (identical process
Corollary 1 The mappings La : x 7→ a ⊗ x for loop1). Parts arrive from loop 3 at point A and
and Ra : x 7→ x ⊗ a defined over a complete an operator fixes them to point I. Here they enter
idempotent semiring D are both residuated Cohen inside the furnace. This element is a channel divided
(1998a). Their residuals are isotone mappings into two sections. Inside the former section parts are
123
respectively. whereas r is a monomial γ ν δ τ which
reproduces the pattern q along the slope τν .
Considering the Timed Event Graph in Figure2. The
dynamic behavior of this system can be expressed
as follow: x1 (k) = max(x7 (k−5)+4, x2 (k−1), u1 (k)),
x2 (k) = max(x1 (k) + 1, x3 (k − 2)), x3 (k) =
max(x2 (k)+3, x4 (k−2), w1 (k)), x4 (k) = max(x3 (k)+
10, x5 (k − 2)), x5 (k) = max(x4 (k) + 10, x6 (k − 3)),
x6 (k) = max(x5 (k) + 3, x7 (k − 1)), x7 (k) =
max(x6 (k) + 2, x1 (k − 2), u2 (k), w2 (k)), y(k) = x7 (k).
In terms of Max Plus notation, we obtain the
following linear equations: x1 (k) = 4 ⊗ x7 (k − 5) ⊕
x2 (k − 1) ⊕ u1 (k), x2 (k) = 1 ⊗ x1 (k) ⊕ x3 (k − 2),
x3 (k) = 3 ⊗ x2 (k) ⊕ x4 (k − 2) ⊕ w1 (k),
x4 (k) = 10 ⊗ x3 (k) ⊕ x5 (k − 2), x5 (k) =
10 ⊗ x4 (k) ⊕ x6 (k − 3)), x6 (k) = 3 ⊗ x5 (k) ⊕ x7 (k − 1),
Figure 2: Petri Net of the loop2
x7 (k) = 2 ⊗ x6 (k) ⊕ x1 (k − 2) ⊕ u2 (k) ⊕ w2 (k)),
y(k) = x7 (k).
heated and they are next cooled down inside the consequently, The transitions are related as follows
latter. Once, pallets come outside the furnace (point over Max in [γ, δ]:
O), they are transferred to a second operator who x1 = γ 5 δ 4 x7 ⊕ γx2 ⊕ u1 , x2 = δx1 ⊕ γ 2 x3 ,
3 2
removes parts from the pallets. Thus, parts are taken x3 = δ x2 ⊕ γ x4 ⊕ w 1 , x4 = δ 10 x3 ⊕ γ 2 x5 ,
10 3
away at point E according to the external resources. x5 = δ x4 ⊕ γ x6 , x6 = δ 3 x5 ⊕ γx7 ,
2 2
Finally, the free pallets are released and transfer to x7 = δ x6 ⊕ γ x1 ⊕ u2 ⊕ w2 , y = x7 .
point A. We obtain the following state space representation
The main problem is to achieve the thermal over Max in [γ, δ]:
treatment on loop 1 or loop 2 without major failures.
In figure 1, d and l are assumed to be the durations of x = Ax ⊕ Bu ⊕ Rw = A∗ Bu ⊕ A∗ Rw (10)
operations and the conveyor capacities respectively. ∗
y = Cx = CA Bu ⊕ CA Rw ∗
(11)
This physical process (loop2) is modelled thanks
γ 5 δ4
to a TEG. Transition u1 models parts arrivals from γ
δ 2
loop3, u2 models the necessity of a resource to carry γ
the terminated part and Transition y represents the δ3 γ 2
departure of an achieved part. Figure 2 shows a With: A = δ 10 γ2 ,
10 3
model of the plant. δ γ
3
δ γ
γ 2 δ 2
e x1
3.2. Timed Event Graph description in dioids x2
e x3
Timed Event Graph can be expressed by linear
B= , R = , x = x4 ,
relations over some dioids Cohen (1999). By x5
associating with each transition x a dater function,
x6
in which x(k) is equal to the date when which the
e e x7
firing numbered k occurs, it is possible to obtain a
linear state representation in Rmax . there is another C = e, , ut = u1 u2 ,
representation of TEG in Rmin , a function of time wt = w1 w2 .
t, corresponding to the cumulated number of firings where u, y and x are respectively the input, output
of the transition at time t. such a function is called and state vector. w represents uncontrollable inputs,
a counter. A two-dimensional representation of each entry of w corresponds to a transition which
input-output maps called Max disable the firing of internal transition of the graph,
in [γ, δ] is considered
here Cohen (1998b), where γ is an indeterminate and then decreases the performance of the system.
which may also be considered as the backward A, B, C and R are given matrices over Max in [γ, δ].
shift operator in the event domain, and δ is the CA∗ B is the input/output transfer matrix and CA∗ R
backward shift operator in the time domain. this is the disturbance/output transfer matrix.
property means that each entry can be written as
an expression of the form s = p ⊕ qr∗ in which
p and q are polynomials in (γ, δ) which represent
the transient behavior and the repeated pattern
124
1966), we present an observer design (inspired from
the work of (Laurent (2010)) for Timed Event Graph
modeled in Max in [γ, δ]. Figure3 depicts the observer
structure whose equations are:
x̂ = Ax̂⊕Bu⊕L(ŷ⊕y) = Ax̂⊕Bu⊕LC x̂⊕LCx (12)
ŷ = C x̂ (13)
Where L is an observer gain matrix.
Using equation (10), we obtain the following
structure:
x̂ = Ax̂ ⊕ Bu ⊕ LC x̂ ⊕ LC(A∗ Bu ⊕ A∗ Rw)
x̂ = (A ⊕ LC)x̂ ⊕ Bu ⊕ LCA∗ Bu ⊕ LCA∗ Rw
x̂ = (A ⊕ LC)∗ Bu ⊕ (A ⊕ LC)∗ LCA∗ Bu ⊕
(A ⊕ LC)∗ LCA∗ Rw
Figure 3: observer structure By applying Kleene star properties we are:
(A ⊕ LC)∗ = A∗ (LCA∗ )∗ (14)
3.3. Periodicity, causality and asymptotic slope
Replacing (14) in previous equation, we obtain:
Definition 8 (Periodicity) A series s ∈ Max in [γ, δ] is x̂ = A∗ (LCA∗ )∗ Bu ⊕ A∗ (LCA∗ )∗ LCA∗ Bu ⊕
said to be periodic if it can be written as s = p ⊕ A∗ (LCA∗ )∗ LCA∗ Rw
q(γ ν δ τ ) with p and q two polynomials and ν, τ ∈ N . and by recalling that (LCA∗ )∗ LCA∗ = (LCA∗ )+ ,
A matrix is said to be periodic if all its entries are this equation may be written as follows:
periodic. x̂ = A∗ (LCA∗ )∗ Bu ⊕ A∗ (LCA∗ )+ Bu ⊕
∗ ∗ +
A (LCA ) Rw
Definition 9 (Causality) A series s ∈ Max in [γ, δ] is Equation (3) yields: (LCA∗ )+ ≤ (LCA∗ )∗
causal if s = . The set of causal elements of Then the observer equation may be written as
Max
in [γ, δ] has a complete dioid structure denoted by follows:
Max+
in [γ, δ]
x̂ = A∗ (LCA∗ )∗ Bu ⊕ A∗ (LCA∗ )+ Rw
Definition 10 (asymptotic slope) The asymptotic
slope of a periodic series s = p ⊕ q(γ ν δ τ )∗ denoted x̂ = (A ⊕ LC)∗ Bu ⊕ (A ⊕ LC)∗ LCA∗ Rw (15)
σ∞ (s) is defined as the ratio σ∞ (s) = ν/τ The objective, is to calculate the greatest observa-
tion matrix L such that the estimated vector x̂ be
Let s1 and s2 be two periodic series such that ν1 , ν2 as close as possible to state x, under the constraint
6= 0 et τ1 , τ2 6= 0), then x̂ ≤ x , formally it can be written :
σ∞ (s1 ⊕ s2 ) = min(σ∞ (s1 ), σ∞ (s2 )), (A ⊕ LC)∗ Bu ⊕ (A ⊕ LC)∗ LCA∗ Rw ≤ A∗ Bu ⊕ A∗ Rw
σ∞ (s1 ⊗ s2 ) = min(σ∞ (s1 ), σ∞ (s2 )), Or equivalently:
If σ∞ (s1 ) ≤ σ∞ (s2 ) then
(A ⊕ LC)∗ B ≤ A∗ B (16)
σ∞ (s2 ◦ s1 ) = σ∞ (s1 ) ∗ ∗ ∗
(A ⊕ LC) LCA R ≤ A R (17)
otherwise s2 ◦ s1 = .
Lemma 1 The greatest matrix L such that (A ⊕
LC)∗ B ≤ A∗ B is given by :
L1 = (A∗ B)ø(CA∗ B) (18)
4. OBSERVER DESIGN
Lemma 2 The greatest matrix L such that (A ⊕
The system evolves according to its state vector LC)∗ LCA∗ R ≤ A∗ R is given by :
equations. Or, In many systems of practical
importance, the entire state vector is not available L2 = (A∗ R)ø(CA∗ R) (19)
for measurement. When faced with this difficulty , a
solution is to provide an estimate of the internal state
The greatest observer matrix such that x̂ ≤ x is:
of the given plant from measurements of the input
and output of the real system. By analogy with the L = L1 ∧ L2
classical Luenberger observer Luenberger (1964,
125
Using the residuation properties, the following
equivalences are given :
CA∗ B(K(A ⊕ Lopt C)∗ B)∗ Gref
⇔ (K(A ⊕ Lopt C)∗ B)∗ CA∗ B ◦ Gref
⇔ K(A ⊕ Lopt C)∗ B CA∗ B ◦ Gref
⇔ K CA∗ B ◦ Gref ø(K(A ⊕ Lopt C)∗ B) = Kopt
6. ILLUSTRATION
In order to illustrate results presented previously, we
consider a Timed Event Graph depicted in figure,
2. Transitions w1 and w2 represent uncontrollable
Figure 4: Controller using observer
inputs. each one corresponds to a transition which
delays or disables the firing of internal transition
5. FEEDBACK CONTROLLER SYNTHESIS WITH of the graph. In our example, w1 corresponds to a
OBSERVER failure at entry inside the furnace, the beginning of
the thermal process, is modeled by x3 , the firing of
Now, we consider a problem of controller synthesis this transition is reported to time 25 instead of time
with an observer, for TEG, in an objective of 15. w2 means that the operator removes parts from
reference model matching. This problem can be the pallets at time 45 instead of time 39. Then, the
described in the following way: Taking a TEG of state are delayed by disturbances whose trajectories
which one knows the transfer matrix, we estimate a are as follows :
state and we compute a controller which leads to a 3 25
w1 γ δ
closed loop system whose behavior is as close as =
w2 γ 2 δ 45
possible to the given reference model Gref .
The input output transfet function is expressed by: Using minmaxgb Toolbox for Scilab (see
http://www.maxplus.org/), the observer matrix is
y = Hu, such that H = CA∗ B given by:
The observer equation is given by:
x̂ = (A ⊕ LC)∗ Bu L11 = γ 5 δ 4 ⊕γ 6 δ 6 ⊕γ 7 δ 8 ⊕γ 8 δ 10 ⊕γ 9 δ 12 ⊕(γ 10 δ 37 ⊕γ 11 δ 39
Acontroller denoted K is added between x̂ and u, ⊕γ 12 δ 47 ⊕ γ 13 δ 49 ⊕ γ 14 δ 57 )[γ 5 δ 33 ]∗
the input is described by:
L21 = γ 5 δ 5 ⊕γ 6 δ 7 ⊕γ 7 δ 9 ⊕γ 8 δ 11 ⊕γ 9 δ 18 ⊕(γ 10 δ 38 ⊕γ 11 δ 40
∗ ∗
u = K x̂ ⊕ v = (K(A ⊕ LC) B) v (20)
⊕γ 12 δ 48 ⊕ γ 13 δ 50 ⊕ γ 14 δ 58 )[γ 5 δ 33 ]∗
Then, the system equation is given by: L31 = ⊕(γ 5 δ 8 ⊕γ 6 δ 10 ⊕γ 7 δ 18 ⊕γ 8 δ 20 ⊕γ 9 δ 28 )[γ 5 δ 33 ]∗
x = A∗ Bu = A∗ B(K(A ⊕ LC)∗ B)∗ v L41 = ⊕(γ 5 δ 18 ⊕γ 6 δ 20 ⊕γ 7 δ 28 ⊕γ 8 δ 30 ⊕γ 9 δ 38 )[γ 5 δ 33 ]∗
L51 = γ 4 ⊕(γ 5 δ 28 ⊕γ 6 δ 30 ⊕γ 7 δ 38 ⊕γ 8 δ 40 ⊕γ 9 δ 48 )[γ 5 δ 33 ]∗
y = Cx = CA∗ B(K(A ⊕ LC)∗ B)∗ v (21)
L61 = γ⊕γ 2 δ 2 ⊕γ 3 δ 4 ⊕γ 4 δ 6 ⊕(γ 5 δ 31 ⊕γ 6 δ 33 ⊕γ 7 δ 41 ⊕γ 8 δ 43
Firstly, we calculate the observer matrix Lopt =
(A∗ B)ø(CA∗ B). Then, within the framework of ⊕γ 9 δ 51 )[γ 5 δ 33 ]∗
feedback controller synthesis, we have to find for
L71 = e⊕γδ 2 ⊕γ 2 δ 4 ⊕γ 3 δ 6 ⊕γ 4 δ 8 ⊕(γ 5 δ 33 ⊕γ 6 δ 35 ⊕γ 7 δ 43
a given Gref , a greatest Kopt with respect to the
residuation theory. ⊕γ 8 δ 45 ⊕ γ 9 δ 53 )[γ 5 δ 33 ]∗
and ŷ is equal to y:
Kopt = H ◦ Gref ø((A ⊕ Lopt C)∗ B) (22)
ŷ = δ 29 ⊕ γδ 31 ⊕ γ 2 δ 45 ⊕ γ 3 δ 50 ⊕ γ 4 δ 52 ⊕ γ 5 δ 62 ⊕ γ 6 δ 64
Proof 1 We calculate the greatest solution Kopt in
order that the controlled system (with estimate state) ⊕(γ 7 δ 78 ⊕ γ 8 δ 83 ⊕ γ 9 δ 88 ⊕ γ 10 δ 95 ⊕ γ 11 δ 98 )[γ 5 δ 33 ]
will behave as close as possible to a given reference
model. The simulation results are Shown in figure (Fig. 5,
Fig. 6). The estimated states of Timed Event Graph
CA∗ B(K(A ⊕ Lopt C)∗ B)∗ Gref x̂1 , x̂2 , x̂3 , x̂4 , x̂5 , x̂6 , x̂7 are compared to the actual
126
state. x1 , x2 , x3 , x4 , x5 , x6 , x7 . We remark a little
difference between some state and corresponding
estimated state : x̂3 and x3 , x̂4 and x4 , x̂5 and x5 ,
x̂6 and x6 . it implies that the disturbances applied
for transition x3 reduce the system performances,
but these disturbances are not acknowledge by
the observer. it is assumed that the model and
the initial state correspond to the fastest behavior,
ie. ŷ and y are equivalent, and there are equality
between asymptotic slope of the state and the one
of estimated state, σ∞ (ŷ) = σ∞ (y) = 5/33.
Consider a problem of controller synthesis with
observer, we propose to compute a greatest K so
that the system has a transfer relation close to a
given reference transfer Gref .
The objective of the reference model is to impose a
desired behavior Gref to a given system H, then in Figure 5: Estimated state of x1 , x2 , x3 and x4
our example, we consider Gref = H.Using Scilab,
we calculate H:
H11 = δ 29 ⊕γ 1 δ 31 ⊕γ 2 δ 39 ⊕γ 3 δ 41 ⊕γ 4 δ 49 ⊕(γ 5 δ 37 ⊕γ 11 δ 39 [γ 5 δ 33 ]∗
H12 = γ 0 δ 0 ⊕γ 1 δ 2 ⊕γ 2 δ 4 ⊕γ 3 δ 6 ⊕γ 4 δ 8 ⊕(γ 5 δ 33 ⊕γ 6 δ 35 ⊕γ 7 δ 43
⊕γ 8 δ 45 ⊕ γ 9 δ 53 [γ 5 δ 33 ]∗
For Lopt obtained, we compute the greatest
realizable feedback Kopt using the expression:
Kopt = H ◦ Hø((A ⊕ Lopt C)∗ B)
therefore, we can compute the controller and the
output of the system:
u = (K(A ⊕ LC)∗ B)∗ v
Figure 6: Estimated state of x5 , x6 , x7 and y
∗ ∗ ∗
y = CA B(K(A ⊕ LC) B) v
Assume that: Gkopt = CA∗ B(K(A ⊕ LC)∗ B)∗ 7. CONCLUSION
In this paper, we have applied to an industrial
Gkopt11 = δ 29 ⊕ γ 1 δ 31 ⊕ γ 2 δ 39 ⊕ γ 3 δ 41 process an observer design and the synthesis of
controller with observer. this system is modeled by
⊕γ 4 δ 49 ⊕ (γ 5 δ 37 ⊕ γ 11 δ 39 [γ 5 δ 33 ]∗ . Timed Event Graph which can be described by
linear equations in idempotent semiring. Firstly, The
estimation state in presence of perturbation based
Gkopt12 = γ 0 δ 0 ⊕ γ 1 δ 2 ⊕ γ 2 δ 4 ⊕ on Luenberger observer, is considered, and the
effectiveness of this method is shown by simulation
γ 3 δ 6 ⊕ γ 4 δ 8 ⊕ (γ 5 δ 33 ⊕ γ 6 δ 35 ⊕ γ 7 δ 43 results given by the use of Scilab. Afterwards, A
⊕γ 8 δ 45 ⊕ γ 9 δ 53 [γ 5 δ 33 ]∗ . synthesis of controller with observer for TEG in a
model reference is presented, we have shown that
it is possible to preserve its own transfer with a
For any TEG, it is possible to preserve its own
greatest realizable feedback control.
transfer with either a greatest realizable feedback
control. We have: Gkopt Gref = H
REFERENCES
Amari, S., Demongodin, I. and Loiseau, J.J. (2004)
Sizing, cycle time and plant control using dioid
algebra. Springer, pp.77-85.
127
Baccelli, F., Cohen, G., Olsder, G. and Quadrat, J. Luenberger, D G. (1964) Observers for multivariable
(1992) Synchronization and Linearity: An Algebra Systems, IEEE Trans. on Automatic Control , Vol.
for Discrete Event Systems. Wiley and Sons. AC-II,NO.2, pp.190-197, April 1966.
Cassandras, C.G. and Lafortune, S. (1992) Introduc- Martinez, C. and Castagna, P. (2003) Sizing of an
tion to discrete event systems. Kluwer Academic. industrial plant using tight time constraints using
complementary approaches: (max, +) algebra
Cohen, G., Moller, P., Quadrat, J.P. and viot, M.
and computer simulation. Simulation Modelling
(1984)Linear system theory for discrete event
Practice and Theory, Elsevier, Vol. 11,pp.75-88.
systems. In 23rd IEEE Conf. on Decision and
Control, Las Vegas, Nevada.
Cohen, G. (1998a) Residuation and applica-
tions,In Algébres Max-plus et applications en in-
formatique, Ecole de printemps d’informatique
thèorique, Noirmoutier, France.
Cohen, G. (1998b)Two-dimentional domain rep-
resentation of timed event graphs,In algébres
Max-Plus et applications en informatique et au-
tomatique, Ecole de printemps d’informatique
théorique, Noirmoutier, France.
Cohen, G., Gaubert, S. and Quadrat, J. (1999)
Max-plus algebra and system theory,Where we
are and where to go now, Annuel Reviews in
control,(23),pp. 207-219.
Cottenceau, B. Contribution á la commande des
systèmes á événements discrets: Synthése
de correcteurs pour les graphes d’événements
temporisés dans les dioı̈des. Phd thesis, Angers
University.
Cuninghame-Green, R.A. Minimax Algebra. (1979)
Number 166 in Lecture notes in Economics and
Mathematical Systems, Springer.
David, R. and Alla, H. (1997) Du Grafcet aux
réseaux de Petri. Ed. Hermes, 2e édition revue et
augmentée.
Houssin, L. (2003) Contribution á l’étude des
systèmes événements discrets dans l’algébre des
dioı̈des. Applications aux systémes de transport.
Rapport de stage de DEA, DEA Automatique et
informatique Appliquée, Nantes-Angers, France.
Laurent, H., Carlos, A.M., Bertrand, C. and Mehdi,
L. (2010) observer design for (max, +)linear
systems. IEEE Transactions on Automatic Control,
VOL.55.
Lhommeau, M. (2003) Etude de systèmes á
événements discrets dans l’Algébre (max, +):
Synthèse de correcteurs robustes dans un
dioı̈de d’intervalles, Synthèse de correcteurs en
présence de perturbation.Phd Thesis, Angers
University.
Luenberger, D G. (1964) Observing the State of a
Linear System , IEEE Trans. on Military Electronics
, Vol. MIL-8,pp.74-80, April 1964.