Pavithra Prabhakar

STORMED Hybrid Systems

We introduce STORMED hybrid systems, a decidable class of hybrid systems which is similar to o-minimal hybrid automata in that the continuous dynamics and constraints are described in an o-minimal theory. However, unlike o-minimal hybrid automata, the variables are not initialized in a memoryless fashion at discrete steps. STORMED hybrid systems require flows which are monotonic with respect to some vector in the continuous space and can be characterised as bounded-horizon systems in terms of their discrete transitions. We demonstrate that such systems admit a finite bisimulation, which can be effectively constructed provided the o-minimal theory used to describe the system is decidable. As a consequence, many verification problems for such systems have effective decision algorithms.

On the expressiveness of MTL in the pointwise and continuous semantics

We show that the pointwise version of the logic MTL is strictly less expressive than the continuous version, over finitewords. The proof is constructive in that we exhibit a timed language, which is definable in the continuous semantics but is not definable in the pointwise semantics.

Computing augmented finite transition systems to synthesize switching protocols for polynomial switched systems

This work is motivated by the problem of synthesizing mode sequences for continuous-time polynomial switched systems in order to guarantee that the trajectories of the system satisfy certain high-level specifications expressed in linear temporal logic. We use augmented finite transition systems as abstract models of continuous switched systems. Augmented finite transition systems are equipped with liveness properties that can be used to enforce progress in accordance with the underlying dynamics. We then introduce abstraction and refinement relations that induce a preorder on this class of finite transition systems. By construction, the resulting preorder respects the feasibility (i.e., realizability) of the synthesis problem. Hence, existence of a discrete switching strategy for one of these abstract finite transition systems guarantees the existence of a mode sequence for the continuous system such that all of its trajectories satisfy the specification. We also present an algorithm, which can be implemented using sum-of-squares based relaxations, to compute such high fidelity abstract models in a computationally tractable way. Finally, these ideas are illustrated on an example.

On the expressiveness of MTL with past operators

We compare the expressiveness of variants of Metric Temporal Logic (MTL) obtained by adding the past operators ‘S’ and ‘SI’. We consider these variants under the “pointwise” and “continuous” interpretations over both finite and infinite models. Among other results, we show that for each of these variants the continuous version is strictly more expressive than the pointwise version. We also prove a counter-freeness result for MTL which helps to carry over some results from [3] for the case of infinite models to the case of finite models.

Hybrid automata-based CEGAR for rectangular hybrid systems

In this paper, we present a counterexample guided abstraction refinement (CEGAR) framework for systems modelled as rectangular hybrid automata. The main difference, between our approach and previous proposals for CEGAR for hybrid automata, is that we consider the abstractions to be hybrid automata as well, as opposed to finite state systems. We show that the CEGAR scheme is semi-complete for the class of rectangular hybrid automata and complete for the subclass of initialized rectangular automata. We have implemented the CEGAR based algorithm in a tool called Hare, that makes calls to HyTech to analyze the abstract models and validate the counterexamples. The experimental evaluations demonstrate the merits of the approach.

A dynamic algorithm for approximate flow computations

In this paper we consider the problem of approximating the set of states reachable within a time bound T in a linear dynamical system, to within a given error bound ε. Fixing a degree d, our algorithm divides the interval [0,T] into sub-intervals of not necessarily equal size, such that a polynomial of degree d approximates the actual flow to within an error bound of ε, and approximates the reach set within each sub-interval by the polynomial tube. Our experimental evaluation of the algorithm when the degree d is fixed to be either 1 or 2, shows that the approach is promising, as it scales to large dimensional dynamical systems, and performs better than previous approaches that divided the interval [0,T] evenly into sub-intervals.

Hybrid Automata-Based CEGAR for Rectangular Hybrid Systems

In this paper we present a framework for carrying out counterexample guided abstraction-refinement CEGAR for systems modelled as rectangular hybrid automata. The main difference, between our approach and previous proposals for CEGAR for hybrid automata, is that we consider the abstractions to be hybrid automata as well. We show that the CEGAR scheme is semi-complete for the class of rectangular hybrid automata and complete for the subclass of initialized rectangular automata. We have implemented the CEGAR based algorithm in a tool called Hare, that makes calls to HyTech to analyze the abstract models and validate the counterexamples. Our experiments demonstrate the usefulness of the approach.

Abstraction Based Model-Checking of Stability of Hybrid Systems

In this paper, we present a novel abstraction technique and a model-checking algorithm for verifying Lyapunov and asymptotic stability of a class of hybrid systems called piecewise constant derivatives. We propose a new abstract data structure, namely, finite weighted graphs, and a modification of the predicate abstraction based on the faces in the system description. The weights on the edges trace the distance of the executions from the origin, and are computed by using linear programming. Model-checking consists of analyzing the finite weighted graph for the absence of certain kinds of cycles which can be solved by dynamic programming. We show that the abstraction is sound in that a positive result on the analysis of the graph implies that the original system is stable. Finally, we present our experiments with a prototype implementation of the abstraction and verification procedures which demonstrate the feasibility of the approach.

Pre-orders for reasoning about stability

Pre-orders between processes, like simulation, have played a central role in the verification and analysis of discrete-state systems. Logical characterization of such pre-orders have allowed one to verify the correctness of a system by analyzing an abstraction of the system. In this paper, we investigate whether this approach can be feasibly applied to reason about stability properties of a system. Stability is an important property of systems that have a continuous component in their state space; it stipulates that when a system is started somewhere close to its ideal starting state, its behavior is close to its ideal, desired behavior. In [Cuijpers 07], it was shown that stability with respect to equilibrium states is not preserved by bisimulation and hence additional continuity constraints were imposed on the bisimulation relation to ensure preservation of Lyapunov stability. We first show that stability of trajectories is not invariant even under the notion of bisimulation with continuity conditions introduced in [Cuijpers 07]. We then present the notion of uniformly continuous simulations --- namely, simulation with some additional uniform continuity conditions on the relation --- that can be used to reason about stability of trajectories. Finally, we show that uniformly continuous simulations are widely prevalent, by recasting many classical results on proving stability of dynamical and hybrid systems as establishing the existence of a simple, obviously stable system that simulates the desired system through uniformly continuous simulations.

Patching task-level robot controllers based on a local μ-calculus formula

We present a method for mending strategies for GR(1) specifications. Given the addition or removal of edges from the game graph describing a problem (essentially transition rules in a GR(1) specification), we apply a μ-calculus formula to a neighborhood of states to obtain a “local strategy” that navigates around the invalidated parts of an original synthesized strategy. Our method may thus avoid global resynthesis while recovering correctness with respect to the new specification. We illustrate the results both in simulation and on physical hardware for a planar robot surveillance task.