Approximate Equivalence of the Hybrid Automata with Taylor Theory

Hybrid automaton is a formal model for precisely describing a hybrid system in which the computational processes interact with the physical ones. The reachability analysis of the polynomial hybrid automaton is decidable, which makes the Taylor approximation of a hybrid automaton applicable and valuable. In this paper, we studied the simulation relation among the hybrid automaton and its Taylor approximation, as well as the approximate equivalence relation. We also proved that the Taylor approximation simulates its original hybrid automaton, and similar hybrid automata could be compared quantitatively, for example, the approximate equivalence we proposed in the paper

Automata-Based Analysis of Stage Suspended Boom Systems

A stage suspended boom system is an automatic steeve system orchestrated by the PLC (programmable logic controller). Security and fault-recovering are two important properties. In this paper, we analyze and verify the boom system formally. We adopt the hybrid automaton to model the boom system. The forward reachability is used to verify the properties with the reachable states. We also present a case study to illustrate the feasibility of the proposed verification

A Hybrid Denotational Semantics for Hybrid Systems -- Extended Version

27 pagesIn this article, we present a model and a denotational semantics for hybrid systems made of a continuous and a discrete subsystem. Our model is designed so that it may be easily used for modeling large, existing, critical embedded applications, which is a first step toward their validation. The discrete subsystem is modeled by a program written in an extension of an imperative language and the continuous subsystem is modeled by differential equations. We give to both subsystems a denotational semantics inspired by what is usually done for the semantics of computer programs and then we show how the semantics of the whole system is deduced from the semantics of its two components. The semantics of the continuous system is computed as the fix-point of a modified Picard operator which increases the information content at each step. This fix-point is computed as the supremum of a sequence of approximations and we show that this supremum exists and is the solution of a differential equation using Keye Martin's measurement theory. The semantics of the discrete system is given as a classical denotational semantics, except that special denotations are given for the actions of sensors and/or actuators

Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point

Even simple hybrid automata like the classic bouncing ball can exhibit Zeno behavior. The existence of this type of behavior has so far forced a large class of simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad-hoc restrictions to circumvent Zeno behavior or to abandon hybrid automata. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independent of the occurrence of a given event. Such an event can then even occur an unbounded number of times. This insight makes it possible to handle some types of Zeno behavior. If the post-Zeno state is defined explicitly in the given model of the hybrid automaton, the computed enclosure covers the corresponding trajectory that starts from the Zeno point through a restarted evolution

Smooth approximation of Lipschitz maps and their subgradients

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the map where it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology—that is, the weakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional—if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C∞ maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C∞ maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green’s theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction

Semantics and computation of the evolution of hybrid systems with ariadne

In this talk we will present material on the semantics, computability, and algorithms for the evolution of hybrid dynamical systems, and an overview of the tool Ariadne for verification of hybrid systems [1]. Hybrid systems are characterised by undergoing continuous evolution interspersed by discrete jumps. They exhibit all the complexities of finite automata, nonlinear dynamic systems and differential equations, and are extremely difficult to analyze. We will consider hybrid systems in which the continuous dynamics is given by a differential equation x = f(x), with discrete jumps x' = ri(x) which occur as soon as a guard condition gi(x) = 0 is activated. It is clear that the evolution of a hybrid system undergoes discontinuities, but since only continuous functions are computable, it is not clear to what extent, if any, it is possible to perform a rigorous analysis of a hybrid system. We will first show that we can define lower and upper semantics of evolution under which it is possible to compute reachable sets, and that away from discontinuity points (such as grazing or corner collision points), these semantics agree [2]. In order to perform reachability analysis, it is necessary to define the evolution over bounded initial sets of states. We show that this can be done using the operations of range, compose, flow and solve operations on functions. We will see that constrained image sets of the form {f(x) | x ? D | g(x) ? C}, are sufficient to express the evolution exactly, except for the case of degenerate (non-transverse) cross- ings [3]. The flow operation is the most computationally demanding, and we will give some details of the implementation and efficiency considerations [4]. We will give examples of reachability analysis in Ariadne, including electrical power converters and heating systems. Finally, we will outline some areas of active research, including differential inclusions [5] and modular reasoning

Recursive Solution of Initial Value Problems with Temporal Discretization

We construct a continuous domain for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.Comment: 50 pages, 6 figure