Reachability Analysis Of Hybrid Systems Research Articles

Safe & robust reachability analysis of hybrid systems

Hybrid systems—more precisely, their mathematical models—can exhibit behaviors, like Zeno behaviors, that are absent in purely discrete or purely continuous systems. First, we observe that, in this context, the usual definition of reachability—namely, the reflexive and transitive closure of a transition relation—can be unsafe, i.e., it may compute a proper subset of the set of states reachable in finite time from a set of initial states. Therefore, we propose safe reachability, which always computes a superset of the set of reachable states.Second, in safety analysis of hybrid and continuous systems, it is important to ensure that a reachability analysis is also robust w.r.t. small perturbations to the set of initial states and to the system itself, since discrepancies between a system and its mathematical models are unavoidable. We show that, under certain conditions, the best Scott continuous approximation of an analysis A is also its best robust approximation. Finally, we exemplify the gap between the set of reachable states and the supersets computed by safe reachability and its best robust approximation.

Parallel reachability analysis of hybrid systems in XSpeed

Reachability analysis techniques are at the core of the current state-of-the-art technology for verifying safety properties of cyber-physical systems (CPS). The current limitation of such techniques is their inability to scale their analysis by exploiting the powerful parallel multi-core architectures now available in modern CPUs. Here, we address this limitation by presenting for the first time a suite of parallel state-space exploration algorithms that, leveraging multi-core CPUs, enable to scale the reachability analysis for linear continuous and hybrid automaton models of CPS. To demonstrate the achieved performance speedup on multi-core processors, we provide an empirical evaluation of the proposed parallel algorithms on several benchmarks comparing their key performance indicators. This enables also to identify which is the ideal algorithm and the parameters to choose that would maximize the performances for a given benchmark.

Compile-Time Extensions to Hybrid ODEs

Reachability analysis for hybrid systems is an active area of development and has resulted in many promising prototype tools. Most of these tools allow users to express hybrid system as automata with a set of ordinary differential equations (ODEs) associated with each state, as well as rules for transitions between states. Significant effort goes into developing and verifying and correctly implementing those tools. As such, it is desirable to expand the scope of applicability tools of such as far as possible. With this goal, we show how compile-time transformations can be used to extend the basic hybrid ODE formalism traditionally supported in hybrid reachability tools such as SpaceEx or Flow*. The extension supports certain types of partial derivatives and equational constraints. These extensions allow users to express, among other things, the Euler-Lagrangian equation, and to capture practically relevant constraints that arise naturally in mechanical systems. Achieving this level of expressiveness requires using a binding time-analysis (BTA), program differentiation, symbolic Gaussian elimination, and abstract interpretation using interval analysis. Except for BTA, the other components are either readily available or can be easily added to most reachability tools. The paper therefore focuses on presenting both the declarative and algorithmic specifications for the BTA phase, and establishes the soundness of the algorithmic specifications with respect to the declarative one.

Efficient Geometric Operations on Convex Polyhedra, with an Application to Reachability Analysis of Hybrid Systems

We present a novel representation class for closed convex polyhedra, where a closed convex polyhedron is represented in terms of an orthogonal projection of a higher dimensional \(\mathcal{H}\)-polyhedron. The idea is to treat the pair of the projection and the polyhedron symbolically rather than to compute the actual described polyhedron. We call this representation a symbolic orthogonal projection, or a sop, for short. We show that fundamental geometrical operations, like affine transformations, intersections, Minkowski sums, and convex hulls, can be performed by simple block matrix operations on the representation. Due to the underlying \(\mathcal{H}\)-polyhedron, linear programs can be used to evaluate sops, e.g. the usage of template matrices easily yields tight over-approximations. Beyond this, we will also show how linear programming can be used to improve these over-approximations by adding additional supporting half-spaces, or even facet-defining half-spaces, to the over-approximation. On the other hand, we will also discuss some drawbacks of this representation, e.g. the lack of an efficient method to decide whether one sop is a subset of another sop, or the monotonic growth of the representation size under geometric operations. The second part deals with reachability analysis of hybrid systems with continuous dynamics described by linear differential inclusions and arbitrary affine maps for discrete updates. The invariants, guards, and sets of reachable states are given as convex polyhedra. First, we present a purely sop-based reachability algorithm where various geometric operations are performed exactly. Due to the monotonic growth of the representation size of the sops, this algorithm is not suited for practical applications. Then we propose a combination of the sop-based algorithm with a support function based reachability algorithm. Accompanied by some simple examples, we show that this combination results in an efficient reachability algorithm of better accuracy than pure support function based algorithm. We also discuss the current limitations of the proposed techniques and chart a path forward which might help us to solve linear programs over huge sops.

Level-Set Approach for Reachability Analysis of Hybrid Systems under Lag Constraints

This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists of the introduction of an adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite difference scheme for solving the system of inequalities is proposed, and the convergence of the numerical solution toward the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application. We are interested in the maximum range of hybrid vehicles.

Computing the Evolution of Hybrid Systems using Rigorous Function Calculus

Hybrid systems exhibit all the complexities of finite automata, nonlinear dynamic systems and differential equations, and are extremely difficult to analyze. A rigorous mathematical approach is needed to achieve provable approximation bounds along the computations. In this paper we describe a rigorous numerical calculus for working with functions that can be used for computing the evolution of nonlinear hybrid systems, and the implementation in the tool Ariadne for reachability analysis of hybrid systems. The method is based around expressing the sets attained during the evolution in terms of functions, and computing approximations to these functions, and allows highly accurate approximations for the evolved sets to be computed. An example of the control of the water level in a tank is presented.

A Hybrid System Approach to the Analysis and Design of Power Grid Dynamic Performance

In this paper, we describe an approach to the analysis and design of power grid dynamic performance based on hybrid systems theory. Power grid is a large-scale cyber-physical system for transmission of electrical energy. The joint dynamics of physical processes and cyber elements in power grids are typical of a mixture of continuous and discrete behaviors, that is, hybrid dynamics. We address problems on stability that are basic concerns in the performance of current and future power grids with the hybrid dynamics. Measures for stability of power grids are interpreted as safety specifications in hybrid system models and are translated into restrictions on the systems' reachable sets of states. Algorithmic reachability analysis of hybrid systems enables analysis of safe initial states and hence quantitative estimation of stability regions. Also it contributes to synthesis of safe initial states as well as switching conditions in order to satisfy safety specifications in a power grid. We demonstrate the approach for two problems on transient stability of the single machine/infinite bus (SMIB) system and on fault release control of a multimachine power grid.

Symbolic Reachability Analysis of Hybrid Systems

定义了一种称作混合区域的形式化结构表示矩形混合系统的状态集,它实际上是由一组特殊形式的线性不等式联立表示的多面体空间.证明了混合区域对于矩形混合系统的可达性操作的封闭性.此外,用矩形混合系统近似模拟非线性混合系统,相应地解决了非线性混合系统的可达性问题.使用混合区域,可以直接计算由某个正则的混合区域开始的可达集,这样,混合系统的可达性问题主要是求解混合区域的正则型问题,而这问题是一种线性规划问题,可以使用经典的线性规划算法加以解决.;A restricted constraint system called hybrid zone is formalized for the representation and manipulation of rectangular automata state-spaces. Hybrid zones are proved to be closed over reachability operations of rectangular hybrid systems. In addition, rectangular hybrid systems are used to simulate nonlinear hybrid systems, which enables us to use hybrid zones for reachability analysis of nonlinear hybrid systems. After the hybrid zone has been converted to the canonical form, reachability operations for hybrid systems can be implemented straightforwardly. Hence, the main computation is the operation for obtaining the canonical form of hybrid zones. Finding the canonical form can be automated by an algorithm for linear programming.

Safety verification and reachability analysis for hybrid systems

Safety verification and reachability analysis for hybrid systems is a very active research domain. Many approaches that seem quite different, have been proposed to solve this complex problem. This paper presents an overview of various approaches for autonomous, continuous-time hybrid systems and presents them with respect to basic problems related to verification.

Safety verification and reachability analysis for hybrid systems

Safety verification and reachability analysis for hybrid systems is a very active research domain. Many approaches that seem quite different, have been proposed to solve this complex problem. This paper presents an overview of various approaches for autonomous, continuous time hybrid systems and present them with respect to basic problems related to verification.

Analysis of lactose metabolism in E.coli using reachability analysis of hybrid systems

We propose an abstraction method for medium-scale biomolecular networks, based on hybrid dynamical systems with continuous multi-affine dynamics. This abstraction method follows naturally from the notion of approximating nonlinear rate laws with continuous piecewise linear functions and can be easily automated. An efficient reachability algorithm is possible for the resulting class of hybrid systems. An efficient reachability algorithm is possible for the resulting class of hybrid systems. An approximation for an ordinary differential equation model of the lac operon is constructed, and it is shown that the abstraction passes the same experimental tests as were used to validate the original model. The well studied biological system exhibits bistability and switching behaviour, arising from positive feedback in the expression mechanism of the lac operon. The switching property of the lac system is an example of the major qualitative features that are the building blocks of higher level, more coarse-grained descriptions. The present approach is useful in helping to correctly identify such properties and in connecting them to the underlying molecular dynamical details. Reachability analysis together with the knowledge of the steady-state structure are used to identify ranges of parameter values for which the system maintains the bistable switching property.

Predicate abstraction for reachability analysis of hybrid systems

Embedded systems are increasingly finding their way into a growing range of physical devices. These embedded systems often consist of a collection of software threads interacting concurrently with each other and with a physical, continuous environment. While continuous dynamics have been well studied in control theory, and discrete and distributed systems have been investigated in computer science, the combination of the two complexities leads us to the recent research on hybrid systems . This paper addresses the formal analysis of such hybrid systems. Predicate abstraction has emerged to be a powerful technique for extracting finite-state models from infinite-state discrete programs. This paper presents algorithms and tools for reachability analysis of hybrid systems by combining the notion of predicate abstraction with recent techniques for approximating the set of reachable states of linear systems using polyhedra. Given a hybrid system and a set of predicates, we consider the finite discrete quotient whose states correspond to all possible truth assignments to the input predicates. The tool performs an on-the-fly exploration of the abstract system. We present the basic techniques for guided search in the abstract state-space, optimizations of these techniques, implementation of these in our verifier, and case studies demonstrating the promise of the approach. We also address the completeness of our abstraction-based verification strategy by showing that predicate abstraction of hybrid systems can be used to prove bounded safety.

PREDICTING TRANSIENT INSTABILITY OF POWER SYSTEMS BASED ON HYBRID SYSTEM REACHABILITY ANALYSIS

This paper shows a novel method for predicting transient instability of power systems based on reachability analysis of hybrid systems. The analysis is performed by computing reachable sets of unsafe sets in nonlinear hybrid automata that represent both continuous electro-mechanical dynamics of generators and discrete operations by relay devices. The unsafe sets here are subsets of system states in which power systems show unacceptable operations such as stepping-out of generators. Then the transient instability of power systems can be estimated by investigating whether a system state exists in the reachable sets or not. The estimation is possible at any onset of accidental faults such as line and plant trips. This paper demonstrates the proposed method through an analysis of single-machine infinite bus system.

Symbolic Reachability Analysis of Hybrid Systems

A hybrid system consists of discrete dynamics mixed with continuous dynamics. Typical examples arise in mechanical systems such as robot controllers, engine controllers, and chemical batch processing systems. This tutorial paper outlines a methodology for verifying safety properties of a hybrid system by analyzing an overapproximation of its set of reachable states. A controller for the fluid level of a tank is studied as an example.