Generating Box Invariants

Tiwari, Ashish

doi:10.1007/978-3-540-78929-1_58

Cited by 19 publications

(20 citation statements)

References 21 publications

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“…Alternatively, SAT Modulo Theory decision procedures and polynomial systems [16,17,9,18,19] could also, eventually, lead to decision procedures for invariant generation. Nonetheless, despite significant progress over the years in static analysis and formal methods for algorithms and programs verification [8,20,21,9,11,[22][23][24][25]16,12,26], the problem of invariant generation for hybrid systems remains very challenging for non-linear discrete systems as well as for non-linear differential systems with non-abstracted local and initial conditions.…”

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confidence: 98%

Generating invariants for non-linear hybrid systems

Rebiha

Moura

Matringe

2015

Theoretical Computer Science

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We describe powerful computational techniques, relying on linear algebraic methods, for generating ideals of non-linear invariants of algebraic hybrid systems. We show that the preconditions for discrete transitions and the Lie-derivatives for continuous evolution can be viewed as morphisms, and so can be suitably represented by matrices. We reduce the non-trivial invariant generation problem to the computation of the associated eigenspaces or nullspaces by encoding the consecution requirements as specific morphisms represented by such matrices. Our methods are the first to establish very general sufficient conditions that show the existence and allow the computation of invariant ideals. Our approach also embodies a strategy to estimate certain degree bounds, leading to the discovery of rich classes of inductive invariants. By reducing the problem to related linear algebraic manipulations we are able to address various deficiencies of other state-of-the-art invariant generation methods, including the efficient treatment of non-linear hybrid systems. Our approach avoids first-order quantifier eliminations, Gröbner basis computations or direct system resolutions, thereby circumventing difficulties met by other recent techniques.

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confidence: 98%

Generating invariants for non-linear hybrid systems

Rebiha

Moura

Matringe

2015

Theoretical Computer Science

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“…Methods for automatic continuous invariant generation have been reported by numerous authors [49,59,18,53,52,25,63,16,30,54], but in practice often result in "coarse" invariants that cannot be used to prove the property of interest, or require an unreasonable amount of time due to their reliance on expensive real quantifier elimination algorithms. Stability analysis (involving a linearisation; see [56] for details) can be used to suggest a polynomial function V : R n → R, given by V (x) = 50599.6 − 14235.7x1 + 1234.22x for which we can reasonably conjecture that V (x) ≤ 1400 defines a positively invariant set under the flow of our non-linear system.…”

Section: Continuous Invariantmentioning

confidence: 99%

Verifying Safety and Persistence Properties of Hybrid Systems Using Flowpipes and Continuous Invariants

Sogokon

Jackson

Johnson

2017

Lecture Notes in Computer Science

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We propose a method for verifying persistence of nonlinear hybrid systems. Given some system and an initial set of states, the method can guarantee 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 flow-pipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study concerning 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 flow-pipes or just reasoning about invariants alone can be insufficient. The case study also nicely shows the richness of systems that the method can handle: the case study features a mode with non-polynomial (nonlinear) ODEs and we manage to prove the persistence property with the aid of an automatic prover specifically designed for handling transcendental functions.

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“…In the past few years, there has been an extensive push for extending formal verification approaches to also verify physical and cyber-physical systems. Broadly speaking, these techniques can be classified as follows: 1) reach-set methods that compute the set of all reachable states of the system, either exactly [20], or approximately [7], [35], [19], [12], [17], [15] 2) abstraction-based methods that first abstract the system and then analyze the abstraction [33], [1], [8] 3) certificate-based methods that directly search for certificates of correctness (such as inductive invariants and Lyapunov functions) of systems [30], [25], [27], [23], [16], [31], [4], [24] While all these techniques have had some success, the certificate-based methods are turning out to be particularly effective in proving deep properties of complex systems. Certificate-based methods work by fixing a template for the "certificate of correctness", and casting the verification problem as a problem of finding an appropriate instantiation of the template.…”

Section: Introductionmentioning

confidence: 99%

Compositionally analyzing a proportional-integral controller family

Tiwari

2011

IEEE Conference on Decision and Control and European Control Conference

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Abstract-We define always eventually region stability and then formulate the absolute always eventually region stability problem as the problem of finding a class of plants that a generic proportional-integral (PI) controller can always eventually stabilize. We use real quantifier elimination methods to solve the absolute always eventually region stability problem. The class of plants found in our solution includes nonlinear and switched plant models. Our analysis reveals that any PI controller that satisfies two assumptions of the form: (a) the proportional gain Kp and the integral gain Ki satisfy an inequality, and (b) the integral of the error term in the controller is saturated, then such a PI controller can be used to establish always eventually region stability for any plant that suitably responds to the control input. Such a result is useful for compositionally verifying a system consisting of a plant and a controller.

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Generating Box Invariants

Cited by 19 publications

References 21 publications

Generating invariants for non-linear hybrid systems

Generating invariants for non-linear hybrid systems

Verifying Safety and Persistence Properties of Hybrid Systems Using Flowpipes and Continuous Invariants

Compositionally analyzing a proportional-integral controller family

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