Computational techniques for reachability analysis of Max-Plus-Linear systems

Abstract

This work discusses a computational approach to reachability analysis of Max-Plus-Linear (MPL) systems, a class of discrete-event systems widely used in synchronization and scheduling applications. Given a set of initial states, we characterize and compute its “reach tube,” namely the collection of set of reachable states (regarded step-wise as “reach sets”). By an alternative characterization of the MPL dynamics, we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference-bound matrices, and further derive worst-case bounds on the complexity of these operations. The approach is also extended to backward reachability analysis. The concepts and results are elucidated by a running example, and we further illustrate the performance of the approach by a numerical benchmark: the technique comfortably handles twenty-dimensional MPL systems (i.e. with twenty continuous state variables), and as such it outperforms the state-of-the-art alternative approaches in the literature.

Introduction

Reachability analysis is a fundamental problem in the area of formal methods, systems theory, and performance and dependability analysis. It is concerned with assessing whether a certain state of a system is attainable from given initial states of the system. The problem is particularly interesting and compelling over models with continuous components—either in time or in (state) space. Over the first class of models, reachability has been widely investigated over discrete-space systems, such as timed automata (Alur and Dill, 1994, Behrmann et al., 2004), Petri nets (Kloetzer et al., 2010, Murata, 1989), or hybrid automata (Henzinger & Rusu, 1998). On the other hand, much research has been directed to computationally push the envelope for reachability analysis of continuous-space models. Among the many approaches for deterministic dynamical systems, we report here the use of face lifting (Dang & Maler, 1998), the computation of flow-pipes via polyhedral approximations (Chutinan & Krogh, 2003), later implemented in Silva, Richeson, Krogh, and Chutinan (2000), the formulation as solution of Hamilton–Jacobi equations (Mitchell, Bayen, & Tomlin, 2005) (related to the study of forward and backward reachability Mitchell, 2007), the use of ellipsoidal techniques (Kurzhanskiy & Varaiya, 2007), later implemented in Kurzhanskiy and Varaiya (2006), and the use of differential inclusions (Asarin, Schneider, & Yovine, 2007). Techniques that have displayed scalability features (albeit at the expense of precision due to the use of over-approximations) are the use of low-dimensional polytopes (Han & Krogh, 2006) and the computation of reachability using support functions (Le Guernic & Girard, 2009).

Max-Plus-Linear (MPL) systems are discrete-event systems (Baccelli et al., 1992, Hillion and Proth, 1989) with continuous variables that express the timing of the underlying sequential events. MPL systems are used to describe the timing synchronization between interleaved processes, and as such are widely employed in the analysis and scheduling of infrastructure networks, such as communication and railway systems (Heidergott, Olsder, & van der Woude, 2006) or production and manufacturing lines (Roset, Nijmeijer, van Eekelen, Lefeber, & Rooda, 2005). They are related to a subclass of timed Petri nets, namely timed-event graphs (Baccelli et al., 1992). MPL systems are classically analyzed over properties such as transient and periodic regimes (Baccelli et al., 1992), or ultimate dynamical behavior (De Schutter, 2000). They can be simulated (though not verified) via the max-plus toolbox for Scilab (Plus, 1998).

Reachability analysis of MPL systems from a single initial condition has been investigated in Cohen, Gaubert, and Quadrat (1999), Gaubert and Katz (2003) and Gazarik and Kamen (1999) by leveraging the computation of the reachability matrix, which leads to a parallel with reachability for discrete-time linear dynamical systems. It has been shown in Gaubert and Katz (2006, Sec. 4.13) that the reachability problem for autonomous MPL systems with a single initial condition is decidable—this result does not hold for a general, uncountable set of initial conditions. Furthermore, the existing literature does not deal with backward reachability analysis.

Under the requirement that the set of initial conditions is expressed as a max-plus polyhedron (Gaubert and Katz, 2007, Zimmermann, 1977), forward reachability analysis can be performed over the max-plus algebra. Similarly for backward reachability analysis of autonomous MPL systems, where in addition the system matrix has to be max-plus invertible. A matrix is max-plus invertible if and only if there is a single finite element (not equal to $-\infty$) in each row and in each column. Despite the requirements, computationally the approach based on max-plus polyhedra can be advantageous since its time complexity is polynomial. To the authors’ best knowledge, there are no approaches for solving the backward reachability problem of nonautonomous MPL systems in the max-plus algebra. Let us also mention that reachability analysis has been used to determine a static max-plus linear feedback controller for a nonautonomous MPL system such that the trajectories lie within a given target tube (Ahmane & Truffet, 2007, Sec. 4.3). In each event step, the target tube is defined as a max-plus polyhedron (Ahmane & Truffet, 2007, Eqs. (8) and (11)).

In this work, we generalize the results for reachability analysis of MPL systems. We extend the results for forward reachability by considering an arbitrary set of initial conditions. Additionally for backward reachability analysis, we are able to handle nonautonomous MPL systems and state matrices that are not max-plus invertible. The approach is as follows. We first alternatively characterize MPL dynamics by Piece-wise Affine (PWA) systems, and show that they can be fully represented by Difference-Bound Matrices (DBM) (Dill, 1990, Sec. 4.1), which are quite simple to manipulate computationally. We further claim that DBM are closed over PWA dynamics, which leads to being able to map DBM-sets through MPL systems. Then given a set of initial states, we characterize and compute its “reach tube”, namely the union of sets of reachable states (aggregated step-wise as “reach sets”). The set of initial conditions is assumed to be a union of finitely many DBM, which contains the class of max-plus polyhedra (cf. Section  2.3) and of max-plus cones as a special case. The approach is also applied to backward reachability analysis. Due to the computational emphasis of this work, we provide a quantification of the worst-case complexity of the algorithms and of the operations that we discuss throughout the work. Interestingly, DBM and max-plus polyhedra have been used for reachability analysis of timed automata (Bengtsson, 2002, Lu et al., 2012) and implemented in UPPAAL (Behrmann et al., 2004) and in opaal (Dalsgaard et al., 2011), respectively. Although related scheduling problems can be solved via timed automata (Abdeddaïm, Asarin, & Maler, 2006), this does not imply that we can employ related techniques for reachability analysis of MPL systems since the two modeling frameworks are not comparable.

Computationally, the present contribution leverages a related, recent work in Adzkiya, De Schutter, and Abate (2012) and Adzkiya, De Schutter, and Abate (2013b), which has explored an approach to analysis of MPL systems that is based on finite-state abstractions. In particular, the technique for reachability computation on MPL systems discussed in this work is implemented in the VeriSiMPL (“very simple”) software toolbox, which is freely available at Adzkiya and Abate (2013). To the best of our knowledge, there does not exist any computational toolbox for general reachability analysis of MPL systems, nor is it possible to leverage current software for related timed-event graphs or timed Petri nets. As further elaborated later, reachability computation for MPL systems can be alternatively tackled using the Multi-Parametric Toolbox (MPT) (Kvasnica, Grieder, & Baotić, 2004). In a numerical case study, we display the scalability of the tool versus model dimension, and benchmark its computation of forward reachability sets against the alternative numerical approach based on the MPT software (Kvasnica et al., 2004).

This manuscript represents an extension of the results in Adzkiya et al., 2014a, Adzkiya et al., 2014b to forward and backward reachability of non-autonomous MPL systems. Further, this article provides a more thorough connection and indeed an extension of existing literature: we have proved that a given max-plus polyhedron can be expressed as a union of finitely many DBM (cf. Proposition 7). Moreover, we explicitly show that, under some assumptions, the number of PWA regions generated by $A^{\otimes N}$ is higher than that generated by $A$ (cf.  Proposition 12). This is an important result that allows elucidating the computational complexity of the discussed batch vs. one-shot procedures, which are later on implemented in the computational benchmark.

The article is structured as follows. Section  2 introduces models and preliminary notions. The procedure for forward and backward reachability analysis is discussed in Sections  3 Forward reachability analysis, 4 Backward reachability analysis, respectively. Section  5 tests the developed approach over a computational benchmark, whereas a running case study is discussed throughout the manuscript. Finally, Section  6 concludes the work.