Yangjia Li

Validated Simulation-Based Verification of Delayed Differential Dynamics

Verification by simulation, based on covering the set of time-bounded trajectories of a dynamical system evolving from the initial state set by means of a finite sample of initial states plus a sensitivity argument, has recently attracted interest due to the availability of powerful simulators for rich classes of dynamical systems. System models addressed by such techniques involve ordinary differential equations (ODEs) and can readily be extended to delay differential equations (DDEs). In doing so, the lack of validated solvers for DDEs, however, enforces the use of numeric approximations such that the resulting verification procedures would have to resort to (rather strong) assumptions on numerical accuracy of the underlying simulators, which lack formal validation or proof. In this paper, we pursue a closer integration of the numeric solving and the sensitivity-related state bloating algorithms underlying verification by simulation, together yielding a safe enclosure algorithm for DDEs suitable for use in automated formal verification. The key ingredient is an on-the-fly computation of piecewise linear, local error bounds by nonlinear optimization, with the error bounds uniformly covering sensitivity information concerning initial states as well as integration error.

Computing reachable sets of linear vector fields revisited

The reachability problem is one of the most important issues in the verification of hybrid systems. But unfortunately the reachable sets for most of hybrid systems are not computable except for some special families. In our previous work, we identified a family of vector fields, whose state parts are linear with real eigenvalues, while input parts are exponential functions, and proved its reachability problem is decidable. In this paper, we investigate another family of vector fields, whose state parts are linear, but with pure imagine eigenvalues, while input parts are trigonometric functions, and prove its reachability problem is decidable also. To the best of our knowledge, the two families are the largest families of linear vector fields with a decidable reachability problem. In addition, we present an approach on how to abstract general linear dynamical systems to the first family. Comparing with existing abstractions for linear dynamical systems, experimental results indicate that our abstraction is more precise.

Safe Over- and Under-Approximation of Reachable Sets for Delay Differential Equations

Delays in feedback control loop, as induced by networked distributed control schemes, may have detrimental effects on control performance. This induces an interest in safety verification of delay differential equations (DDEs) used as a model of embedded control. This article explores reachable-set computation for a class of DDEs featuring a local homeomorphism property. This topological property facilitates construction of over- and under-approximations of their full reachable sets by performing reachability analysis on the boundaries of their initial sets, thereby permitting an efficient lifting of reach-set computation methods for ODEs to DDEs. Membership in this class of DDEs is determined by conducting sensitivity analysis of the solution mapping with respect to the initial states to impose a bound constraint on the time-lag term. We then generalize boundary-based reachability analysis to such DDEs. Our reachability algorithm is iterative along the time axis and the computations in each iteration are performed in two steps. The first step computes an enclosure of the set of states reachable from the boundary of the step’s initial state set. The second step derives an over- and under-approximations of the full reachable set by including (excluding, resp.) the obtained boundary enclosure from certain convex combinations of points in that boundary enclosure. Experiments on two illustrative examples demonstrate the efficacy of our algorithm.

Approximate Bisimulation and Discretization of Hybrid CSP

Hybrid Communicating Sequential Processes (HCSP) is a powerful formal modeling language for hybrid systems, which is an extension of CSP by introducing differential equations for modeling continuous evolution and interrupts for modeling interaction between continuous and discrete dynamics. In this paper, we investigate the semantic foundation for HCSP from an operational point of view by proposing notion of approximate bisimulation, which provides an appropriate criterion to characterize the equivalence between HCSP processes with continuous and discrete behaviour. We give an algorithm to determine whether two HCSP processes are approximately bisimilar. In addition, based on that, we propose an approach on how to discretize HCSP, i.e., given an HCSP process A, we construct another HCSP process B which does not contain any continuous dynamics such that A and B are approximately bisimilar with given precisions. This provides a rigorous way to transform a verified control model to a correct program model, which fills the gap in the design of embedded systems.

Generating semi-algebraic invariants for non-autonomous polynomial hybrid systems

Hybrid systems are dynamical systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template.