SMT-based and fixed-point approaches for state estimation in max-plus linear systems

Symbolic Reachability Analysis of High Dimensional Max-Plus Linear Systems

Max-Plus Linear (MPL) systems are an algebraic formalism with practical applications in transportation networks, manufacturing and biological systems. MPL systems can be naturally modeled as infinite-state transition systems, and exhibit interesting structural properties (e.g. periodicity or steady state), for which analysis methods have been recently proposed. In this paper, we tackle the open problem of specifying and analyzing user-defined temporal properties for MPL systems. We propose Time-Difference LTL (TDLTL), a logic that encompasses the delays between the discrete-time events governed by an MPL system, and characterize the problem of model checking TDLTL over MPL. We propose a family of specialized algorithms leveraging the periodic behaviour of an MPL system. We prove soundness and completeness, showing that the transient and cyclicity of the MPL system induce a completeness threshold for the verification problem. The algorithms are cast in the setting of SMT-based verification of infinite-state transition systems over the reals, with variants depending on the (incremental vs upfront) computation of the bound, and on the (explicit vs implicit) unrolling of the transition relation. Our comprehensive experiments show that the proposed techniques can be applied to MPL systems of large dimensions and on general TDLTL formulae, with remarkable performance gains against a dedicated abstraction-based technique and a translation to the nuXmv symbolic model checker.

Discrete Event Dynamic Systems (DEDS) are discrete-state systems whose dynamics areentirely driven by the occurrence of asynchronous events over time. Linear equations in themax-plus algebra can be used to describe DEDS subjected to synchronization and time delayphenomena. The reachability analysis concerns the computation of all states that can bereached by a dynamical system from an initial set of states. The reachability analysis problemof Max Plus Linear (MPL) systems has been properly solved by characterizing the MPLsystems as a combination of Piece-Wise Affine (PWA) systems and then representing eachcomponent of the PWA system as Difference-Bound Matrices (DBM). The main contributionof this thesis is to present a similar procedure to solve the reachability analysis problemof MPL systems subjected to bounded noise, disturbances and/or modeling errors, calleduncertain MPL (uMPL) systems. First, we present a procedure to partition the state spaceof an uMPL system into components that can be completely represented by DBM. Then weextend the reachability analysis of MPL systems to uMPL systems. Moreover, the results onreachability analysis of uMPL systems are used to solve the conditional reachability problem,which is closely related to the support calculation of the probability density function involvedin the stochastic filtering problem.

We examine the possibility of applying the concepts of the feeding and the project buffers of critical chain project management (CCPM) on max-plus linear discrete event systems, using a simple model and numerical examples in order to control the occurrence of an undesirable state change in discrete event systems. As a remarkable outcome, we found that the application of a CCPM based framework on max-plus linear system was proven to be effective.

This PhD thesis considers the development of optimization and model-based control techniques for max-plus linear (MPL) and continuous piecewise affine (PWA) systems. The three main topics investigated in this thesis are as follows: 1. Optimistic optimization and planning for model-based control of MPL systems; 2. Optimistic optimization for MPC of continuous PWA systems; 3. MPC for stochastic MPL systems with chance constraints

This paper introduces the abstraction of max-plus linear (MPL) systems via predicates. Predicates are automatically selected from system matrix, as well as from the specifications under consideration. We focus on verifying time-difference specifications, which encompass the relation between successive events in MPL systems. We implement a bounded model checking (BMC) procedure over a predicate abstraction of the given MPL system, to verify the satisfaction of time-difference specifications. Our predicate abstractions are experimentally shown to improve on existing MPL abstractions algorithms. Furthermore, with focus on the BMC algorithm, we can provide an explicit upper bound on the completeness threshold by means of the transient and the cyclicity of the underlying MPL system.

Book Abstract: Max-plus linear systems theory was inspired by and originated from classical linear systems theory more than three decades ago, with the purpose of dealing with nonlinear synchronization and delay phenomena in timed discrete event systems in a linear manner. Timed discrete event systems describe many problems in diverse areas such as manufacturing, communication, or transportation networks. This monograph provides a thorough survey of current research work in max-plus linear systems. It summarizes the main mathematical concepts required for a theory of max-plus linear systems, including idempotent semirings, residuation theory, fixed point equations in the max-plus algebra, formal power series, and timed-event graphs. The authors also review some recent major achievements in control and state estimation of max-plus linear systems. These include max-plus observer design, max-plus model matching by output or state feedback and observer-based control synthesis. Control and State Estimation for Max-Plus Linear Systems offers students, practitioners, and researchers an accessible and comprehensive overview of the most important recent work in max-plus Linear Systems.

Notice of Violation of IEEE Publication Principles<br><br>"The Application of Critical Chain Project Management Based on Max-plus Linear System"<br>by Wenjun Wang, Xiuming Li, Xiong Zeng<br>in the Proceedings of the 2010 2nd Conference on Environmental Science and Information Application Technology (ESIAT 2010), pp. 640-643<br><br>After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE's Publication Principles.<br><br>This paper contains significant portions of original text from the papers cited below. The original text was copied without attribution (including appropriate references to the original author(s) and/or paper title) and without permission.<br><br>Due to the nature of this violation, reasonable effort should be made to remove all past references to this paper, and future references should be made to the following articles:<br><br>"Toward the Application of a Critical-Chain-Project-Management-based framework on Max-plus Linear Systems"<br>by H. Takahashi, H.Goto, M. Kasahara<br>in Industrial Engineering and Management Systems, vol.8, no. 3, September 2009, pp. 155-161<br><br>"Application of a Critical Chain Project Management Based Framework on Max-plus Linear Systems"<br>by H. Takahashi, H.Goto, M. Kasahara<br>in International Journal of Computational Science, vol. 3, no. 2, April 2009, pp. 117-132<br><br> <br/> We have proposed a method for inserting time buffers without installing new processes in the MPL-CCPM representation. To achieve this, we changed the definition of the representation matrices of the target system. We partly reformulated the previous method. Moreover, we redefined the matrices after the CCPM has been applied for representing the structure of the modified system. If we regard the transportation time between processes or between a process and an external output as a size of the time buffer, we can insert the feeding and project buffers without installing new processes. In this paper, the effect of the resource conflict has not been considered for simplicity. Thus, we should develop a method for resolving the effect of resource conflicts in future work.

Max-plus linear systems are often used to model timed discrete-event systems, which represent system operations as discrete sequences of events in time. This paper presents the observer-based controller to solve the disturbance decoupling problem for max-plus linear systems where only estimations of system states are available for the controller. This observer-based controller leads to a greater control input than the one obtained with the output feedback strategy based on just-in-time criterion. A high throughput screening system in drug discovery illustrates this main result by showing that the scheduling obtained from the observer-based controller solving the disturbance decoupling problem is better than the scheduling obtained from the output feedback controller.

The objective of this paper is to provide a concise introduction to the max-plus algebra and to max-plus linear discrete-event systems. We present the basic concepts of the max-plus algebra and explain how it can be used to model a specific class of discrete-event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete-event systems because they can be described by a model that is “linear” in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete-event systems. Next, some control approaches for max-plus linear discrete-event systems, including residuation-based control and model predictive control, are presented briefly. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.

We provide an introduction to the max-plus algebra and explain how it can be used to model a specific class of discrete event systems with synchronization but no concurrency. Such systems are called max-plus linear discrete event systems because they can be described by a model that is ldquolinearrdquo in the max-plus algebra. We discuss some key properties of the max-plus algebra and indicate how these properties can be used to analyze the behavior of max-plus linear discrete event systems. We also briefly present some control approaches for max-plus linear discrete event systems, including model predictive control. Finally, we discuss some extensions of the max-plus algebra and of max-plus linear systems.

The max-plus linear systems have been studied for almost three decades, however, a well-established system theory on such specific systems is still an on-going research. The geometric control theory in particular was proposed as the future direction for max-plus linear systems by Cohen et al. [Cohen, G., Gaubert, S. and Quadrat, J.P. (1999), ‘Max-plus Algebra and System Theory: Where we are and Where to Go Now’, Annual Reviews in Control, 23, 207--219]. This article generalises R.E. Kalman's abstract realisation theory for traditional linear systems over fields to max-plus linear systems. The new generalised version of Kalman's abstract realisation theory not only provides a more concrete state space representation other than just a ‘set-theoretic’ representation for the canonical realisation of a transfer function, but also leads to the computational methods for the controlled invariant semimodules in the kernel and the equivalence kernel of the output map. These controlled invariant semimodules play key roles in the standard geometric control problems, such as disturbance decoupling problem and block decoupling problem. A queueing network is used to illustrate the main results in this article.

Conditional reachability of uncertain Max Plus Linear systems

This paper investigates the dimension reduction and feedback stabilization of max-plus linear systems and applies them to control and optimize the very large scale integration (VLSI) array processors. We introduce the weakly similar relation between max-plus matrices and the pseudoequivalent relation between autonomous max-plus linear systems, and point out that two systems are pseudoequivalent if and only if their state matrices are weakly similar. The reduced system is defined by using the pseudoequivalence, whose dimension is determined by the row rank of the original state matrix. We focus on obtaining a reduced system which maintains the stability and retains the steady-state period. An algorithm of polynomial complexity is developed to find such a reduced system. The reduced system is then used to design a state feedback controller to stabilize a max-plus linear system. Finally, we use the VLSI array processors as an example to demonstrate how the presented methods work in practical applications.

This paper discusses the reachability analysis (RA) of Interval Max-Plus Linear (IMPL) systems, a subclass of continuous-space, discrete-event systems defined over the max-plus algebra. Unlike standard Max-Plus Linear (MPL) systems, where the transition matrix is fixed at each discrete step, IMPL systems allow for uncertainty on state matrices. Given an initial and a target set, we develop algorithms to verify the existence of IMPL system trajectories that, starting from the initial set, eventually reach the target set. We show that RA can be solved by encoding the IMPL system, as well as initial and target sets, into linear real arithmetic expressions, and then checking the satisfaction of a resulting logical formula via a satisfiability modulo theory (SMT) solver. The performance and scalability of the developed SMT-based algorithms are shown to drastically outperform state-of-the-art RA algorithms applied to IMPL systems, which promises to usher their use in practical, industrial-sized IMPL models.