128 research outputs found
Convex Programs for Temporal Verification of Nonlinear Dynamical Systems
A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates
Verifying safety and persistence in hybrid systems using flowpipes and continuous invariants
We describe a method for verifying the temporal property of persistence in non-linear hybrid systems. Given some system and an initial set of states, the method establishes that system trajectories always eventually evolve into some specified target subset of the states of one of the discrete modes of the system, and always remain within this target region. The method also computes a time-bound within which the target region is always reached. The approach combines flowpipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study showing that potentially destructive stick-slip oscillations of an oil-well drill eventually die away for a certain choice of drill control parameters. The case study demonstrates how just using flowpipes or just reasoning about invariants alone can be insufficient and shows the richness of systems that one can handle with the proposed method, since the systems features modes with non-polynomial ODEs. We also propose an alternative method for proving persistence that relies solely on flowpipe computation
A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates
This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method
Direct methods for deductive verification of temporal properties in continuous dynamical systems
This thesis is concerned with the problem of formal verification of correctness
specifications for continuous and hybrid dynamical systems. Our main focus will
be on developing and automating general proof principles for temporal properties
of systems described by non-linear ordinary differential equations (ODEs) under
evolution constraints. The proof methods we consider will work directly with the
differential equations and will not rely on the explicit knowledge of solutions,
which are in practice rarely available. Our ultimate goal is to increase the
scope of formal deductive verification tools for hybrid system designs. We
give a comprehensive survey and comparison of available methods for checking
set invariance in continuous systems, which provides a foundation for safety
verification using inductive invariants. Building on this, we present a technique
for constructing discrete abstractions of continuous systems in which spurious
transitions between discrete states are entirely eliminated, thereby extending
previous work. We develop a method for automatically generating inductive
invariants for continuous systems by efficiently extracting reachable sets from
their discrete abstractions. To reason about liveness properties in ODEs, we
introduce a new proof principle that extends and generalizes methods that have
been reported previously and is highly amenable to use as a rule of inference
in a deductive verification calculus for hybrid systems. We will conclude with a
summary of our contributions and directions for future work
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