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. 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