Dieky Adzkiya

Finite Abstractions of Max-Plus-Linear Systems

This work puts forward a novel technique to generate finite abstractions of autonomous and nonautonomous Max-Plus-Linear (MPL) models, a class of discrete-event systems used to characterize the dynamics of the timing related to successive events that synchronize autonomously. Nonautonomous versions of MPL models embed within their dynamics nondeterminism, namely a signal choice that is usually regarded as an exogenous control or schedule. In this paper, abstractions of MPL models are characterized as finite-state Labeled Transition Systems (LTS). LTS are obtained first by partitioning the state space (and, for the nonautonomous model, by covering the input space) of the MPL model and by associating states of the LTS to the introduced partitions, then by defining relations among the states of the LTS based on dynamical transitions between the corresponding partitions of the MPL state space, and finally by labeling the LTS edges according to the one-step timing properties of the events of the original MPL model. In order to establish formal equivalences, the finite abstractions are proven to either simulate or to bisimulate the original MPL model. This approach enables the study of general properties of the original MPL model by verifying (via model checking) equivalent logical specifications over the finite LTS abstraction. The computational aspects related to the abstraction procedure are thoroughly discussed and its performance is tested on a numerical benchmark.

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

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.

VeriSiMPL: verification via bisimulations of MPL models

VeriSiMPL (very simple) is a software tool to obtain finite abstractions of Max-Plus-Linear (MPL) models. MPL models (Sect. 2), specified in MATLAB, are abstracted to Labeled Transition Systems (LTS). The LTS abstraction is formally put in relationship with the concrete MPL model via a (bi)simulation relation. The abstraction procedure (Sect. 3) runs in MATLAB and leverages sparse representations, fast manipulations based on vector calculus, and optimized data structures such as Difference-Bound Matrices. LTS abstractions can be exported to structures defined in the PROMELA. This enables the verification of MPL models against temporal specifications within the SPIN model checker (Sect. 4). The toolbox is available at n n http://sourceforge.net/projects/verisimpl/

Abstraction and verification of autonomous Max-Plus-Linear systems

This work investigates the use of finite abstractions for the verification of autonomous Max-Plus-Linear (MPL) models. Abstractions are characterized as finite-state labeled transition systems (LTS) and are obtained by first partitioning the state space of the MPL and associating states of the LTS to the partitions, then by defining relations among the vertices of the LTS, corresponding to dynamical transitions between the MPL state partitions, and finally by labeling the LTS edges according to one-step time properties of the events of the MPL model. In order to establish formal equivalences, the finite LTS abstraction is proven to either simulate or to bisimulate the original MPL model, the difference depending on its determinism. The computational performance of the abstraction procedure is tested on a benchmark. The work then studies properties of the original MPL model by verifying equivalent specifications on the finite LTS abstraction.

Forward Reachability Computation for Autonomous Max-Plus-Linear Systems

This work discusses the computation of forward reachability for autonomous (that is, deterministic) Max-Plus-Linear (MPL) systems, a class of continuous-space discrete-event models that are relevant for applications dealing with synchronization and scheduling. Given an MPL model and a set of initial states, we characterize and compute its “reach tube,” namely the sequential collection of the sets of reachable states (these sets are regarded step-wise as “reach sets”). We show that the exact computation of the reach sets can be quickly and compactly performed by manipulations of difference-bound matrices, and derive explicit worst-case bounds for the complexity of these operations. The concepts and techniques are implemented within the toolbox VeriSiMPL, and are practically elucidated by a running example. We further display the computational performance of the approach by two concluding numerical benchmarks: the technique comfortably handles reachability computations over twenty-dimensional MPL models (i.e., models with twenty continuous variables), and it clearly outperforms an alternative state-of-the-art approach in the literature.

Finite Abstractions of Stochastic Max-Plus-Linear Systems

This work investigates the use of finite abstractions to study the finite-horizon probabilistic invariance problem over Stochastic Max-Plus-Linear (SMPL) systems. SMPL systems are probabilistic extensions of discrete-event MPL systems that are widely employed in the engineering practice for timing and synchronisation studies. We construct finite abstractions by re-formulating the SMPL system as a discrete-time Markov process, then tailoring formal abstraction techniques in the literature to generate a finite-state Markov Chain (MC), together with precise guarantees on the level of the introduced approximation. This finally allows to probabilistically model check the obtained MC against the finite-horizon probabilistic invariance specification. The approach is practically implemented via a dedicated software, and elucidated in this work over numerical examples.

Backward Reachability of Autonomous Max-Plus-Linear Systems

Abstract This work discusses the backward reachability of autonomous Max-Plus-Linear (MPL) systems, a class of continuous-space discrete-event models that are relevant for applications dealing with synchronization and scheduling. Given an MPL system and a continuous set of final states, we characterize and compute its “backward reach tube” and “backward reach sets, ” namely the set of states that can reach the final set within a given event interval or at a fixed event step, respectively. We show that, in both cases, the computation can be done exactly via manipulations of difference-bound matrices. Furthermore, we illustrate the application of the backward reachability computations over safety and transient analysis of MPL systems.

Symbolic abstractions for the scheduling of event-triggered control systems

In this paper, the problem of scheduling event-triggered networked control systems sharing a communication channel is addressed. Event-triggered control strategies effectively reduce the usage of resources in the implementation of control loops, in particular communication bandwidth. However, there is a lack of a well-established framework to analyze their corresponding communication load and synthesize schedulers. We focus on the case of linear-time-invariant plants and propose a procedure to build a timed automaton that captures the sampling behavior of each event-triggered controller. We show that these timed automata approximately simulate the controllers sampling behavior. Finally, a conflict-free scheduling policy is synthesized using timed game automata to guarantee reliable communication in the network.

Finite abstractions of nonautonomous Max-Plus-Linear systems

This work puts forward a technique to generate finite abstractions of nonautonomous Max-Plus-Linear (MPL) models, a known class of discrete-event systems characterizing the timing related to event counters. Nonautonomous models embed an external input (namely a nondeterministic choice, regarded as an exogenous control signal) in the dynamics. Abstractions are characterized as finite-state Labeled Transition Systems (LTS). LTS are obtained first by partitioning the state space of the MPL model and by associating states of the LTS to the introduced partitions, then by defining relations among the states of the LTS, corresponding to the dynamical (nonautonomous) transitions between the MPL state partitions, and finally by labeling the LTS edges according to the one-step timing properties related to the events of the original MPL model. In order to establish formal equivalences, the finite LTS abstraction is proven either to simulate or to bisimulate the original MPL model. The computational performance of the abstraction procedure is tested on a numerical benchmark. The approach enables the study of properties of the original MPL model by verifying equivalent specifications over the finite LTS abstraction.

Identification and estimation of state variables on reduced model using balanced truncation method

In this paper, we study the identification of variables on a model reduction process and estimation of variables on reduced system. We aim to relate variables on reduced and original system, so that we can compare the estimation accuracy of the original system and reduced system. As such, the objective of this paper is to discuss identification and estimation of variables on reduced model. First, model order reduction is done by using balanced truncation method. This process begins with the construction of balanced system. After that, we identify the relationship between variables of the balanced system and the original system. Then, we eliminate variables of the balanced system that have a small influence on the system. Furthermore, we estimate state variables on the original system and reduced system using a Kalman Filter algorithm. Finally, we compare the estimation result of the identified reduced and original system.