Bisimilar control affine systems

The notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation. These equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning. In systems theory, a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems. In this paper, we introduce two notions of bisimulation for nonlinear systems. We present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions. Furthermore, we also show that all bisimilar systems of different dimension are of this form.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

A simple graph theoretic characterization of reachability for positive linear systems

In this paper we consider discrete-time linear positive systems, that is systems defined by a pair (A,B) of non-negative matrices. We study the reachability of such systems which in this case amounts to the freedom of steering the state in the positive orthant by using non-negative control sequences. This problem was solved recently [Canonical forms for positive discrete-time linear control systems, Linear Algebra Appl., 310 (2000) 49]. However we derive here necessary and sufficient conditions for reachability in a simpler and more compact form. These conditions are expressed in terms of particular paths in the graph which is naturally associated with the system.     

Model order-reduction for discrete-time switched linear systems

This article describes the problem of model order-reduction for a class of hybrid discrete-time switched linear systems composed of linear discrete-time invariant subsystems with a switching rule. This article investigates two novel approaches to model order-reduction. The first approach consists in evaluating the error approximation performance; the problems are solved using the robust stability results of the switched systems. The second approach presents the reachability and observability Gramians of the switched systems, which allows a balanced truncation model reduction procedure. A numerical example shows the effectiveness of the proposed approaches.     

Linear and Branching System Metrics

We extend the classical system relations of trace inclusion, trace equivalence, simulation, and bisimulation to a quantitative setting in which propositions are interpreted not as boolean values, but as elements of arbitrary metric spaces. Trace inclusion and equivalence give rise to asymmetrical and symmetrical linear distances, while simulation and bisimulation give rise to asymmetrical and symmetrical branching distances. We study the relationships among these distances and we provide a full logical characterization of the distances in terms of quantitative versions of LTL and mu-calculus. We show that, while trace inclusion (respectively, equivalence) coincides with simulation (respectively, bisimulation) for deterministic boolean transition systems, linear and branching distances do not coincide for deterministic metric transition systems. Finally, we provide algorithms for computing the distances over finite systems, together with a matching lower complexity bound.     

Existence and Verification for Decentralized Nondeterministic Discrete‐Event Systems Under Bisimulation Equivalence

In this paper, we study the decentralized control problem for nondeterministic discrete‐event systems (DESs) under bisimulation equivalence. In order to exactly achieve the desired specification in the sense of bisimulation equivalence, we present a synchronous composition for the supervised system based on the simulation relation between the specification and the plant. After introducing the notions of simulation‐based controllability and simulation‐based coobservability, we present the necessary and sufficient condition for the existence of a decentralized supervisor such that the controlled system is bisimilar to the specification, and an algorithm for verifying the simulation‐based coobservability is proposed by constructing a computational tree.     

How non-zero initial conditions affect the minimality of linear discrete-time systems

From the state-space approach to linear systems, promoted by Kalman, we learned that minimality is equivalent with reachability together with observability. Our past research on optimal reduced-order LQG controller synthesis revealed that if the initial conditions are non-zero, minimality is no longer equivalent with reachability together with observability. In the behavioural approach to linear systems promoted by Willems, that consider systems as exclusion laws, minimality is equivalent with observability. This article describes and explains in detail these apparently fundamental differences. Out of the discussion, the system properties weak reachability or excitability, and the dual property weak observability emerge. Weak reachability is weaker than reachability and becomes identical only if the initial conditions are empty or zero. Weak reachability together with observability is equivalent with minimality. Taking the behavioural systems point of view, minimality becomes equivalent with observability when the linear system is time invariant. This article also reveals the precise influence of a possibly stochastic initial state on the dimension of a minimal realisation. The issues raised in this article become especially apparent if linear time-varying systems (controllers) with time-varying dimensions are considered. Systems with time-varying dimensions play a major role in the realisation theory of computer algorithms. Moreover, they provide minimal realisations with smaller dimensions. Therefore, the results of this article are of practical importance for the minimal realisation of discrete-time (digital) controllers and computer algorithms with non-zero initial conditions. Theoretically, the results of this article generalise the minimality property to linear systems with time-varying dimensions and non-zero initial conditions.     

Periodic stabilizability of switched linear control systems

Stabilizability via direct/observer-based state feedback control for discrete-time switched linear control systems (SLCSs) is investigated in this paper. For an SLCS, the control factors are not only the control input but also the switching signal, and they need to be designed in order to stabilize the system. As a result, stabilization design for SLCSs is more complicated than that for non-switched ones. Differently from the existing approaches, a periodic switching signal and piecewise constant linear state feedback control are adopted to achieve periodic stabilizability for such systems. It is highlighted that multiple feedback controllers need to be designed for one subsystem. For discrete-time SLCSs, it is proved that reachability implies periodic stabilizability via state feedback. A necessary and sufficient criterion for periodic stabilizability is also established. Two stabilization design algorithms are presented for real application. Moreover, it is proved that reachability and observability imply periodic stabilizability via observer-based feedback for discrete-time SLCSs. Periodic detectability, as the dual concept of periodic stabilizability, is discussed and the stabilization design algorithms via observer-based feedback are presented as well.     

A complexity reduction approach to detectability of switching systems

In this article, we address detectability for the class of linear switching systems. We focus on some hybrid state-space decompositions of the original switching system based on hybrid invariant subspaces, which yield a complexity reduction in checking detectability. We show that the reduced system extracted from the original system is the minimal bisimilar switching system associated with it. An example is also included which shows the applicability and benefits of the proposed approach.     

Filling Out the Gaps: A Padding Algorithm for Transforming Out Timing Leaks

It has been shown that secret information can be leaked to external observers through covert timing channels. In this paper we are concerned with a kind of timing attack that wants to differentiate two processes, presented as probabilistic transition systems, by observing their timing behaviour. Our goal is to make the processes indistinguishable i.e. bisimilar, by adding virtual (dummy) states and transitions to the original processes (padding). Instead of padding the processes with whole virtual copies of their counterparts - as done by some padding algorithms - we present an algorithm that uses the bisimulation equivalence relation - computed as a lumping partition - as the main criterion to optimise the padding procedure.     

A new perspective on criteria and algorithms for reachability of discrete-time switched linear systems

The paper presents a unified perspective on geometric and algebraic criteria for reachability and controllability of controlled switched linear discrete-time systems. Direct connections between geometric and algebraic criteria are established as well as that between the subspace based controllability/reachability algorithm and Kalman-type algebraic rank criteria. Also the existing geometric criteria is simplified and new algebraic conditions on controllability and reachability are given.     

Linear systems theory revisited

This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether or not the linear system has state dimensions that vary with time (2) bounds on the time axis of the linear system (3) whether or not the initial state is non-zero and (4) whether or not the system is time invariant. Also (5) time-reversibility of systems plays a role. Discrete-time linear strictly proper systems are considered. A recently published result is used to argue that all the results carry over to continuous time. Out of the investigation two types of definitions emerge. One type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions. This paper reveals that time-varying (state) dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and (weak) reachability together with observability at the systems level. Besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time-varying discrete-time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily.     

Reachability and observability of switched linear systems with continuous-time and discrete-time subsystems

In this paper, the reachability and observability criteria of switched linear systems with continuous-time and discrete-time subsystems are obtained. These criteria show that the reachable set may not be a subspace in the state space, because of the existence of discrete-time subsystems. Therefore, the definition of span reachability is proposed. Moreover, we demonstrate that the reachable set is equivalent to subspace if the discrete-time subsystems are reversible. The subspace algorithms for span reachability and unobservability are provided. One example is introduced to illustrate the effectiveness of the proposed criteria.     

State feedback stabilization of linear impulsive systems

We address the fundamental problem of state feedback stabilization for a class of linear impulsive systems featuring arbitrarily-spaced impulse times and possibly singular state transition matrices. Specifically, we show that a strong reachability property enables a state feedback law to be constructed that yields a uniformly exponentially stable closed-loop system. The approach adopts a receding horizon strategy involving a weighted reachability gramian in a manner reminiscent of well-known results for time-varying linear systems for both continuous and discrete-time cases.     

Bisimulation Minimization and Symbolic Model Checking

State space minimization techniques are crucial for combating state explosion. A variety of explicit-state verification tools use bisimulation minimization to check equivalence between systems, to minimize components before composition, or to reduce a state space prior to model checking. Experimental results on bisimulation minimization in symbolic model checking contexts, however, are mixed. This paper explores bisimulation minimization as an optimization in symbolic model checking of invariance properties. We consider three bisimulation minimization algorithms. From each, we produce a BDD-based model checker for invariant properties and compare this model checker to a conventional one based on backwards reachability. Our comparisons, both theoretical and experimental, suggest that bisimulation minimization is not viable in the context of invariance verification, because performing the minimization requires as many, if not more, computational resources as model checking the unminimized system through backwards reachability.     

生成诊断公式的有限状态进程等价验证

分析有限状态进程互模拟等价判定技术,探讨了诊断公式的生成问题.给出了将有限状态进程转化为带标号的迁移系统,修改了Paige和Trajan求解最粗划分的算法,使其适用于带标号的迁移系统.给出生成Hennessy-Milner逻辑描述的诊断公式的算法,当两个进程不能互模拟时,产生两个诊断公式.算法的时间复杂度为O(m log n),空间复杂度为O(m+n).     

Multi-core symbolic bisimulation minimisation

We introduce parallel symbolic algorithms for bisimulation minimisation, to combat the combinatorial state space explosion along three different paths. Bisimulation minimisation reduces a transition system to the smallest system with equivalent behaviour. We consider strong and branching bisimilarity for interactive Markov chains, which combine labelled transition systems and continuous-time Markov chains. Large state spaces can be represented concisely by symbolic techniques, based on binary decision diagrams. We present specialised BDD operations to compute the maximal bisimulation using signature-based partition refinement. We also study the symbolic representation of the quotient system and suggest an encoding based on representative states, rather than block numbers. Our implementation extends the parallel, shared memory, BDD library Sylvan, to obtain a significant speedup on multi-core machines. We propose the usage of partial signatures and of disjunctively partitioned transition relations, to increase the parallelisation opportunities. Also our new parallel data structure for block assignments increases scalability. We provide SigrefMC, a versatile tool that can be customised for bisimulation minimisation in various contexts. In particular, it supports models generated by the high-performance model checker LTSmin, providing access to specifications in multiple formalisms, including process algebra. The extensive experimental evaluation is based on various benchmarks from the literature. We demonstrate a speedup up to 95\(\times \) for computing the maximal bisimulation on one processor. In addition, we find parallel speedups on a 48-core machine of another 17\(\times \) for partition refinement and 24\(\times \) for quotient computation. Our new encoding of the reduced state space leads to smaller BDD representations, with up to a 5162-fold reduction.

     

REACHABILITY AND CONTROLLABILITY OF DISCRETE-TIME LINEAR DYNAMICAL SYSTEMS

The concepts of reachability and controllability do not coincide but reflect a fundamental difference between the dynamical characteristics of different classes of multi-variable systems. This difference is discussed in relation to discrete-time linear dynamical systems and is illustrated by a simple numerical example.     

混合系统的形式验证技术及其在化工过程控制中的应用

针对PLC等逻辑控制器控制连续对象的可靠性问题。给出了混合系统的形式验证的方法,即用混合矩形自动机建模,通过其商迁移的可达性分析,证明了控制程序的正确性,应用实例表明该方法是可行和有效的。