Quantitative analysis of continuous weighted marked graphs

This paper deals with the computation of the asymptotic firing rate vector and the stationary marking of continuous weighted marked graphs under infinite servers semantics. The continuous weighted marked graphs are a particular class of continuous Petri net where each place has exactly one input and one output transition. First, we give an explicit formula to compute the asymptotic firing rate vector of transitions using the structure of the given net. Then, under the assumption that there exists only one critical circuit in the strongly connected continuous neutral weighted marked graphs, an original approach to compute the vector of stationary marking is presented. Finally, an application to a flexible manufacturing system is given.     

Analysis of strong and strong periodic detectability of bounded labeled Petri nets

Detectability describes the property of a system to uniquely determine, after a finite number of observations, the current and the subsequent states. Different notions of detectability have been proposed in the literature. In this paper, we formalize and analyze strong detectability and strong periodic detectability for systems that are modeled as labeled Petri nets with partial observation on their transitions. We provide three new approaches for the verification of such detectability properties using three different structures. The computational complexity of the proposed approaches is analyzed and the three methods are compared. The main feature of all the three approaches is that they do not require the calculation of the entire reachability space or the construction of an observer. As a result, they have lower computational complexity than other methods in the literature.     

ON/OFF strategy based minimum-time control of continuous Petri nets

This paper focuses on the target marking control problem of timed continuous Petri nets (TCPN), aiming to drive the system from an initial state to a desired final one. This problem is similar to the set-point control problem in a general continuous-state system. In a previous work, a simple and efficient ON/OFF controller was proposed for Choice-Free nets, and it was proved to be minimum-time (Wang, 2010). However, for general TCPN the ON/OFF controller may bring the system to “blocking” situations due to its “greedy” firing strategy, and the convergence to the final state is not ensured. In this work the ON/OFF controller is extended to general TCPN by adding more “fair” strategies to solve conflicts in the system: the ON/OFF+ controller is obtained by forcing proportional firings of conflicting transitions. Nevertheless, such kind of controller might highly slow down the system when transitions have flows of different orders of magnitude, therefore a balancing process is introduced, leading to the B-ON/OFF controller. A third approach introduced here is the MPC-ON/OFF controller, a combination of Model Predictive Control (MPC) and the ON/OFF strategy; it may achieve a smaller number of time steps for reaching the final states, but usually requires more CPU time for computing the control laws. All the proposed extensions are heuristic methods for the minimum-time control and their convergences are proved. Finally, an application example of a manufacturing cell is considered to illustrate the methods. It is shown that by using the proposed controllers, reasonable numbers of time steps for reaching the final state can be obtained with low computational complexity.     

Homothecy,bifurcations, continuity and monotonicity in timed continuous Petri nets under infinite server semantics

This work studies how equilibrium markings and throughputs change in Timed Continuous Petri Net (TCPN) systems as transition firing rates vary. In particular, it analyzes the bifurcations of the former, and the discontinuities and non-monotonicities of the latter; specifically, using structural objects of the net, such as P-semiflows, T-semiflows, and configurations, among others, the following properties can be obtained. For Join Free TCPN systems, a sufficient structural condition guaranteeing that the equilibrium markings do not bifurcate when firing rates vary, is derived. A dual result is obtained for Choice Free TCPN systems. For Mono-T-Semiflow TCPN systems, the equilibrium throughput is investigated; using a time-scale (a homothetic) property it is proven that a discontinuity of the equilibrium throughput implies its non-monotonicity, even if not evident at first glance. This is a connection of two timed behavioral properties of the equilibrium throughput. Moreover, a sufficient structural condition, parametrized by the equilibrium markings, ensures its continuity under firing rate variations. It is also proven that the monotonicity of the equilibrium throughput can be characterized by the previous structural condition. The convergence of the marking evolution of TCPN systems to its equilibrium markings is also discussed.     

On the fluidization of Petri nets and marking homothecy

The analysis of Discrete Event Dynamic Systems suffers from the well known state explosion problem. A classical technique to overcome it is to relax the behavior by partially removing the integrality constraints and thus to deal with hybrid or continuous systems. In the Petri nets framework, continuous net systems (technically hybrid systems) are the result of removing the integrality constraint in the firing of transitions. This relaxation may highly reduce the complexity of analysis techniques but may not preserve important properties of the original system. This paper deals with the basic operation of fluidization. More precisely, it aims at establishing conditions that a discrete system must satisfy so that a given property is preserved by the continuous relaxation. These conditions will be mainly based on the marking homothetic behavior of the system. The focus will be on logical properties as boundedness, B-fairness, deadlock-freeness, liveness and reversibility. Furthermore, testing homothetic monotonicity of some properties in the discrete systems is also studied, as well as techniques to improve the quality of the fluid relaxation by removing spurious solutions.     

Sequence of decisions on discrete event systems modeled by Petri nets with structural alternative configurations

The management of certain systems, such as manufacturing facilities, supply chains, or communication networks implies assessing the consequences of decisions, aimed for the most efficient operation. This kind of systems usually shows complex behaviors where subsystems present parallel evolutions and synchronizations. Furthermore, the existence of global objectives for the operation of the systems and the changes that experience the systems or their environment during their evolution imply a more or less strong dependence between decisions made at different time points of the life cycle. This paper addresses a complex problem that is scarcely present in the scientific literature: the sequences of decisions aimed for achieving several objectives simultaneously and with strong influence from one decision to the rest of them. In this case, the formal statement of the decision problem should take into account the whole decision sequence, making impractical the solving paradigm of “divide and conquer”. Only an integrated methodology may afford a realistic solution of such a type of decision problem. In this paper, an approach based on the formalism of the Petri nets is described, several considerations related to this problem are presented, a solving methodology based on the previous work of the authors, as well as a case-study to illustrate the main concepts.     

A linear characterization of the switching dynamic behavior of timed continuous Petri nets with structural conflicts

The behavior of timed continuous Petri nets (TCPN) can be ruled by linear equations during certain time elapses (IB-states), but changes in the marking and conflict solving policies make nonlinear the complete computation of the behavior. In this paper a global characterization of the switching behavior of TCPN through Mixed Linear Integer Programming (MLIP) is presented. The contribution is an analytical technique to compute the evolution graph of a TCPN, which allows deriving MLIP problems from TCPN models including cycles and structural conflicts; conflict resolution policies by priorities and sharing are considered.     

Hybrid and Hybrid Adaptive Petri Nets: On the computation of a Reachability Graph

Petri Nets (PNs) constitute a well known family of formalisms for the modelling and analysis of Discrete Event Dynamic Systems (DEDS). As general formalisms for DEDS, PNs suffer from the state explosion problem. A way to alleviate this difficulty is to relax the original discrete model and deal with a fully or partially continuous model. In Hybrid Petri Nets (HPNs), transitions can be either discrete or continuous, but not both. In Hybrid Adaptive Petri Nets (HAPNs), each transition commutes between discrete and continuous behaviour depending on a threshold: if its load is higher than its threshold, it behaves as continuous; otherwise, it behaves as discrete. This way, transitions adapt their behaviour dynamically to their load. This paper proposes a method to compute the Reachability Graph (RG) of HPNs and HAPNs.     

基于观测器的一类时滞不确定非线性离散系统的鲁棒模糊控制

讨论一类基于Takagi-Sugeno(T-S)模糊模型的具有时滞的不确定非线性离散系统的模糊鲁棒控制问题。在系统状态不可测的情况下。设计模糊观测器。利用LMI方法,得到模糊控制系统鲁棒稳定的充分条件。     

Stability analysis and control synthesis for a class of switched neutral systems

In this paper, the problems of asymptotical stability and stabilization of a class of switched neutral control systems are investigated. A delay-dependent stability criterion is formulated in term of linear matrix inequalities (LMIs) by using quadratic Lyapunov functions and inequality analysis technique. The corresponding switching rule is obtained through dividing the state space properly. Also, the synthesis of stabilizing state-feedback controllers are done such that the close-loop system is asymptotically stable. Two numerical examples are given to show the proposed method.     

Robust stability analysis of uncertain systems with two additive time-varying delay components

This paper is concerned with stability analysis for uncertain systems. The systems are based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. The relationship between the time-varying delay and its upper bound is taken into account when estimating the upper bound of the derivative of Lyapunov functional. As a result, some less conservative stability criteria are established for systems with two successive delay components and parameter uncertainties. Numerical examples show that the proposed criteria are effective and are an improvement over some existing results in the literature.     

Properties of the Optimal Ellipsoids Approximating the Reachable Sets of Uncertain Systems

The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In the paper, the optimal outer-ellipsoidal approximation of reachable sets is considered, and attention is paid to the criterion associated with the projection of the approximating ellipsoid onto a given direction. The nonlinear differential equations governing the evolution of ellipsoids are analyzed and simplified. The asymptotic behavior of the ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets.     

Strongly absolute stability of Lur'e type differential-algebraic systems

In this paper, we consider Lur'e type differential-algebraic systems (LDS) and introduce the concept of strongly absolute stability. Such a notion is a generalization of absolute stability for Lur'e type standard state-space systems (LSS). By a Lur'e type Lyapunov function, we derive an LMI based stability criterion for LDS to be strongly absolutely stable. Using extended strictly positive realness (ESPR), we present the frequency-domain interpretation of the obtained criterion, by which we simplify the criterion and show that the criterion is a generalization of the well-known Popov criterion. Finally, we illustrate the effectiveness of the main results by a numerical example.     

Absolute stability criteria for a class of nonlinear singular systems with time delay

This paper deals with the problem of absolute stability for a class of time-delay singular systems with sector-bounded nonlinearity. Both delay-independent and delay-dependent criteria are presented and formulated in the form of linear matrix inequalities (LMIs). Neither model transformation nor a bounding technique for cross-terms, nor a slack variable method is involved in obtaining the stability criteria. Numerical examples are given to show the effectiveness and improvements over some existing results.     

On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations

The issue of robustly exponential stability for uncertain neutral-type systems is considered in this paper. The uncertainties are nonlinear and the delays are time-varying. In terms of a linear matrix inequality (LMI), the new sufficient stability condition with delay dependence is presented. The model transformation and bounding techniques for cross terms are avoided based on an integral inequality. Two illustrative examples are proposed to show the effectiveness of our method.     

Global asymptotic stability analysis of discrete-time Cohen–Grossberg neural networks based on interval systems

The global asymptotic stability of discrete-time Cohen–Grossberg neural networks (CGNNs) with or without time delays is studied in this paper. The CGNNs are transformed into discrete-time interval systems, and several sufficient conditions for asymptotic stability for these interval systems are derived by constructing some suitable Lyapunov functionals. The conditions obtained are given in the form of linear matrix inequalities that can be checked numerically and very efficiently by using the MATLAB LMI Control Toolbox. Finally, some illustrative numerical examples are provided to demonstrate the effectiveness of the results obtained.     

Robust fault tolerant control of continuous-time switched systems: An LMI approach

The present paper proposes a new robust fault tolerant control (RFTC) design for continuous-time switched systems. The main objective is to design in an integrated way the couple (controller, observer) that allows to stabilize switched systems even in the presence of actuator faults. A state/fault estimation observer is designed to simultaneously estimate system state and actuator faults. Based on this observer, a fault tolerant controller is developed to stabilize the system and accommodate the actuator faults automatically. The RFTC problem is formalized in the form of linear matrix inequalities (LMI) rather than bilinear matrix inequalities (BMI), to avoid the difficulty of solving BMIs. Finally, a numerical example composed of unstable subsystems is studied to show the applicability and efficiency of the obtained results.     

Semidefinite representation of convex sets

Let be a semialgebraic set defined by multivariate polynomials g _i (x). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S _i ) be the boundary of S (resp. S _i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of , does there exist an LMI representable set Ŝ in some higher dimensional space whose projection down onto equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g _i (x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ ^T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g _i (x) is either sos-concave ( − ∇² g _i (x) = W(x) ^T W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S _i is either sos-convex or poscurv-convex (S _i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g _i (x) ≠ 0 on ∂ S _i ∩ S), then S is SDP representable. This also holds for S _i for which ∂ S _i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).