Publication e-STA e-STA 2004-4
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        <identifier identifierType="DOI">10.23723/545:2004-4/20035</identifier><creators><creator><creatorName>M. Lhommeau</creatorName></creator><creator><creatorName>L. Hardouin</creatorName></creator><creator><creatorName>C. A. Maia</creatorName></creator><creator><creatorName>R. Santos-Mendes</creatorName></creator></creators><titles>
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	    <date dateType="Created">Sat 30 Sep 2017</date>
	    <date dateType="Updated">Sat 30 Sep 2017</date>
            <date dateType="Submitted">Sat 17 Feb 2018</date>
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COMMANDE ET ANALYSE DE ROBUSTESSE DE SYSTÈMES (MAX, +)-LINÉAIRES CONTROL AND ROBUSTNESS ANALYSIS FOR (MAX, +)-LINEAR SYSTEMS M. Lhommeau ∗ L. Hardouin ∗ C.-A. Maia ∗∗ R. Santos-Mendes ∗∗∗ ∗ Laboratoire d’Ingénierie des Systèmes Automatisés 62, avenue Notre-Dame du lac 49000 ANGERS, FRANCE ∗∗ Depto. de Engenharia Elétrica - UFMG Av. Antônio Carlos 6627 - Pampulha 31270-010 Belo Horizonte - MG - BRAZIL ∗∗∗ Faculdade de Engenharia Elétrica e de Computação, Universidade Estadual de Campinas P.O. Box 6101, 13083-970 Campinas, SP, BRAZIL Résumé: Ce papier s’intéresse à la synthèse de correcteurs pour les systèmes (max, +)-linéaires. L’objectif est de comparer les performances et la robustesse de deux stratégies de commande introduites respectivement dans (Cottenceau et al., 2001) et (Maia et al., 2003). Pour chacune des stratégies, l’influence possible des dérives du système par rapport à son modèle sont analysées. Nous montrons, que la structure de commande mettant en oeuvre un précompensateur et un retour de sortie donne de meilleures performances. Abstract: This paper deals with controller synthesis for (max,+) linear system. It aims at comparing the performances and the robustness of two control strategies introduced in (Cottenceau et al., 2001) and (Maia et al., 2003) respectively. In both strategies, the influences of the possible mismatches between the system and its model are analyzed. This work shows that the control strategy using simultaneously a precompensator and a feedback controller (introduced in (Maia et al., 2003)) gives better performances. Keywords: Discrete event dynamic systems, Timed Petri nets, (max, +) algebra, Timed event Graphs, dioid, idempotent semiring, Control Systems. 1. INTRODUCTION Timed Event Graphs (TEG) constitute a subclass of timed Petri nets in which each place has ex- actly one upstream and one downstream transi- tion. It is well known that the timed/event be- havior of a TEG, under earliest functioning rule 1 , can be expressed by linear relations over some dioids, namely idempotent semiring (Baccelli et al., 1992). Strong analogies then appear be- tween the classical linear system theory and the 1 i.e. a transition is fired as soon as it is enabled (max, +)-linear system theory. In particular, the concept of control is well defined in the context of TEG study. It refers to the firing-control of the TEG input transitions in order to reach the desired performance. In the literature, an optimal control for TEG exists and is proposed in (Cohen et al., 1989). For a given reference input, this open-loop structure control yields the latest input firing date in order to obtain the output before the desired date. One possible approach for the control of TEG is the model reference technique in which a given model describes the desired per- formance and the design goal is achieved through the calculation of a precompensator or a feed- back controller (Cottenceau et al., 2001; Luders and Santos-Mendes, 2002). The control strategies based on feedback control, although favoring sta- bility, are limited in the sense that the reference model must satisfy certain restrictive conditions. Lately, a new technique for the design of con- trollers in which a precompensator and a feedback controllers are calculated simultaneously was in- troduced by (Maia et al., 2003). This paper aims at comparing the performances and robustness of the above mentioned control methods. More precisely, we will compare performances regarding the just-in-time criterion and we will compare ro- bustness, regarding possible mismatches between the system and its model. The paper is organized as follows. Section 2 introduces some algebraic tools concerning the dioid and residuation theo- ries. Section 3 is devoted to recall some elements of DES representation over particular dioids and this section presents three control strategies. Section 4 is dedicated to the analysis of the performances and the robustness of these control strategies. 2. ALGEBRAIC PRELIMINARIES A dioid D is an idempotent semiring, that is an algebraic structure with two internal operations denoted by ⊕ and ⊗. The neutral elements of ⊕ and ⊗ are represented by ε and e respectively. In a dioid, a partial order relation is defined by a º b iff a = a ⊕ b and x ∧ y denotes the greatest lower bound between x and y. A dioid D is said to be complete if it is closed for infinite ⊕-sums and if ⊗ distributes over infinite ⊕-sums. Most of the time the symbol ⊗ will be omitted as in conventional algebra. Theorem 1. ((Baccelli et al., 1992), th. 4.75). The implicit equation x = ax ⊕ b defined over a com- plete dioid D, admits x = a∗ b as least solution, where a∗ = L i∈N ai (Kleene star operator). It will be sometimes represented by the following mapping : K : D → D, x 7→ L i∈N xi . TEG control problems (Cohen et al., 1989), stated in a just-in-time context, usually involves the in- version of isotone mappings 2 , that is, one must find x such that f(x) = y (where f is isotone). Residuation Theory (Blyth and Janowitz, 1972) deals with such problems stated in partially or- dered sets. Definition 2. (Residual and residuated mapping). A mapping f : D → E between two ordered sets is residuated if it is isotone, and if, for all y ∈ E, the subset {x ∈ D | f(x) ¹ y} admits a maximal element, denoted f] (y). The isotone mapping f] : E → D is called the residual of f. The residual f] is the only isotone mapping satisfying the following properties : f ◦ f] ¹ Id and f] ◦ f º Id, (1) where Id is the identity mapping respectively on D and E. Lemma 3. ((Cohen, 1998)). • If f : D → E and g : E → F are residuated mappings, then f ◦ g is also residuated and (f ◦ g)] = g] ◦ f] . • If f is a residuated mapping from D → E, then f ◦ f] ◦ f = f. The mappings La : x 7→ a ⊗ x and Ra : x 7→ x ⊗ a defined over a complete dioid D are both residu- ated ((Baccelli et al., 1992), p. 181). Their residu- als are isotone mappings denoted respectively by L] a(x) = a◦ \x and R] a(x) = x◦ /a. Some useful dioid formulæ involving these residuals are given below. a(a◦ \x) ¹ x and (x◦ /a)a ¹ x (2) a(a◦ \(ax)) = ax (3) a◦ \a = (a◦ \a)∗ (4) (a∗ ) 2 = a∗ (5) x◦ \ (a∗ x) = (a∗ x) ◦ \ (a∗ x) (6) Definition 4. (Restricted mapping). Let f : D → E be a mapping and B ⊂ E with f(D) ⊂ B. We will denote B|f : D → B the mapping defined by f = iB ◦B| f, where iB : B → E, x 7→ x is the canonical injection. Definition 5. (Closure mapping). An isotone map- ping f : D → D defined on an ordered set D is a closure mapping if f º Id and f ◦ f = f. Remark 6. According to (5), the Kleene star op- erator is a closure mapping since a∗ º a and (a∗ )∗ = a∗ . 2 f is an isotone mapping if it preserves order, that is, a ¹ b =⇒ f(a) ¹ f(b). Theorem 7. ((Cottenceau et al., 2001)). Let f : D → D be a closure mapping. Then, Imf|f is a residuated mapping whose residual is the canoni- cal injection iImf : Imf → D, x 7→ x. Example 8. Mapping ImK|K : D → ImK is a residuated mapping whose residual is ¡ ImK|K ¢] = iImK. This means that x = a∗ is the greatest solution to inequality x∗ ¹ a∗ . Actually, this greatest solution achieves equality. Theorem 9. ((Gaubert, 1992)). Let f : D → D be a residuated closure mapping, we have f = f] ◦ f and f = f ◦ f] . 3. CONTROL METHOD Firstly, let us consider the following (max, +)- linear system x(k) = Ax(k − 1) ⊕ Bu(k), y(k) = Cx(k), (7) where x(k) ∈ Z n×1 max, u(k) ∈ Z p×1 max and y(k) ∈ Z m×1 max are respectively the state, input and output vectors of the system. The matrices A, B and C are of proper sizes and have entries ranging over Zmax. We know from (Baccelli et al., 1992) that (7) represents the behavior of a class of dis- crete event systems called Timed Event Graphs (TEG). In the case of a TEG, x (resp. u and y) is a vector associated to the internal (resp. input and output) transitions, and xi(k) represents the kth firing dates of the internal transitions which are labelled xi. Following the conventional ap- proach, it is possible to define the transformation x(γ) = L k∈Z x(k)γk where γ is a backward shift operator in event domain (that is y(γ) = γx(γ) ⇔ {y(k)} = {x(k − 1), ∀k} , see (Baccelli et al., 1992), p. 228). This transformation is analogous to the Z-transform used in discrete-time classical con- trol theory and the formal series x(γ) is a syn- thetic representation of the trajectory x(k). The set of the formal series in γ is a dioid denoted by Zmax[[γ]]. By using γ-transform, we obtain the following representation of (7) : X(γ) = AγX(γ) ⊕ BU(γ), Y (γ) = CX(γ), where U(γ), X(γ) and Y (γ) are the γ-transform of u, x and y respectively. The implicit equation for the vector X, namely X = AγX ⊕ BU which is solved (thanks to theorem 1) by X = (Aγ)∗ BU. Finally, we obtain the input-output representation (transfer matrix) Y = HU with H(γ) = C(Aγ)∗ B. (8) Herein, three control strategies for the systems are presented, and their performances are compared in section 4. They are based on the Just-in-Time criterion and on the model reference approach (Cottenceau et al., 2001). They can be described as follows : let H ∈ (Zmax[[γ]])m×p be the transfer matrix of the plant, given by (8), and Gref ∈ (Zmax[[γ]])m×p be the reference model, i.e., the desired transfer matrix for the controlled system, what are the controllers leading to the greatest controlled system lower than the reference model. The precompensation problem is depicted Fig. 1.(a). It is an open-loop strategy. The relation between the input V ∈ Zmax[[γ]]p , and the output Y is denoted Gc and the relation between V and U is denoted Guv. They are given by Y