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Logics of Dynamical Systems
"... We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded ..."
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Cited by 18 (17 self)
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We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyberphysical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. Distributed hybrid systems combine distributed systems with hybrid systems, i.e., they are multiagent hybrid systems that interact through remote communication or physical interaction. Stochastic hybrid systems combine stochastic
Generating Polynomial Invariants for Hybrid Systems
, 2005
"... We present a powerful computational method for automatically generating polynomial invariants of hybrid systems with linear continuous dynamics. When restricted to linear continuous dynamical systems, our method generates a set of polynomial equations (algebraic set) that is the best such overappro ..."
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Cited by 17 (1 self)
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We present a powerful computational method for automatically generating polynomial invariants of hybrid systems with linear continuous dynamics. When restricted to linear continuous dynamical systems, our method generates a set of polynomial equations (algebraic set) that is the best such overapproximation of the reach set. This shows that the set of algebraic invariants of a linear system is computable. The extension to hybrid systems is achieved using the abstract interpretation framework over the lattice defined by algebraic sets. Algebraic sets are represented using canonical Gröbner bases and the lattice operations are effectively computed via appropriate Gr"obner basis manipulations.
Automatic invariant generation for hybrid systems using ideal fixed points
 In Hybrid Systems: Computation and Control
, 2010
"... We present computational techniques for automatically generating algebraic (polynomial equality) invariants for algebraic hybrid systems. Such systems involve ordinary differential equations with multivariate polynomial righthand sides. Our approach casts the problem of generating invariants for di ..."
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Cited by 13 (4 self)
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We present computational techniques for automatically generating algebraic (polynomial equality) invariants for algebraic hybrid systems. Such systems involve ordinary differential equations with multivariate polynomial righthand sides. Our approach casts the problem of generating invariants for differential equations as the greatest fixed point of a monotone operator over the lattice of ideals in a polynomial ring. We provide an algorithm to compute this monotone operator using basic ideas from commutative algebraic geometry. However, the resulting iteration sequence does not always converge to a fixed point, since the lattice of ideals over a polynomial ring does not satisfy the descending chain condition. We then present a boundeddegree relaxation based on the concept of “pseudo ideals”, due to Colón, that restricts ideal membership using multipliers with bounded degrees. We show that the monotone operator on bounded degree pseudo ideals is convergent and generates fixed points that can be used to generate useful algebraic invariants for nonlinear systems. The technique for continuous systems is then extended to consider hybrid systems with multiple modes and discrete transitions between modes. We have implemented the exact, nonconvergent iteration over ideals in combination with the bounded degree iteration over pseudo ideals to guarantee convergence. This has been applied to automatically infer useful and interesting polynomial invariants for some benchmark nonlinear systems.
Verification and synthesis using real quantifier elimination
, 2011
"... We present the application of real quantifier elimination to formal verification and synthesis of continuous and switched dynamical systems. Through a series of case studies, we show how firstorder formulas over the reals arise when formally analyzing models of complex control systems. Existing off ..."
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Cited by 13 (3 self)
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We present the application of real quantifier elimination to formal verification and synthesis of continuous and switched dynamical systems. Through a series of case studies, we show how firstorder formulas over the reals arise when formally analyzing models of complex control systems. Existing offtheshelf quantifier elimination procedures are not successful in eliminating quantifiers from many of our benchmarks. We therefore automatically combine three established software components: virtual subtitution based quantifier elimination in Reduce/Redlog, cylindrical algebraic decomposition implemented in Qepcad, and the simplifier Slfq implemented on top of Qepcad. We use this combination to successfully analyze various models of systems including adaptive cruise control in automobiles, adaptive flight control system, and the classical inverted pendulum problem studied in control theory.
Switching Logic Synthesis for Reachability
, 2010
"... We consider the problem of driving a system from some initial configuration to a desired configuration while avoiding some unsafe configurations. The system to be controlled is a dynamical system that can operate in different modes. The goal is to synthesize the logic for switching between the modes ..."
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Cited by 13 (6 self)
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We consider the problem of driving a system from some initial configuration to a desired configuration while avoiding some unsafe configurations. The system to be controlled is a dynamical system that can operate in different modes. The goal is to synthesize the logic for switching between the modes so that the desired reachability property holds. In this paper, we first present a sound and complete inference rule for proving reachability properties of single mode continuous dynamical systems. Next, we present an inference rule for proving controlled reachability in multimodal continuous dynamical systems. From a constructive proof of controlled reachability, we show how to synthesize the desired switching logic. We show that our synthesis procedure is sound and produces only nonzeno hybrid systems. In practice, we perform a constructive proof of controlled reachability by solving an ExistsForall formula in the theory of reals. We present an approach for solving such formulas that combines symbolic and numeric solvers. We demonstrate our approach on some examples. All results extend naturally to the case when, instead of reachability, interest is in until properties.
The Structure of Differential Invariants and Differential Cut Elimination
, 2011
"... not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: Proof theory, differential equations, differential cut elimination, logics of programs, The biggest challenge in hybrid systems verification is the handling o ..."
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Cited by 13 (12 self)
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not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Keywords: Proof theory, differential equations, differential cut elimination, logics of programs, The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closedform solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather adhoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which can be checked for invariance along a differential equation just by using their differential structure and without having to solve the differential equation. We study the structural properties of differential invariants. To analyze tradeoffs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that
Convex programs for temporal verification of nonlinear dynamical systems
 SIAM J. Control Optim
"... Abstract. 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 se ..."
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Cited by 12 (1 self)
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Abstract. 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.
Providing a basin of attraction to a target region by computation of Lyapunovlike functions
 In IEEE Int. Conf. on Computational Cybernetics
, 2006
"... Abstract — In this paper, we present a method for computing a basin of attraction to a target region for nonlinear ordinary differential equations. This basin of attraction is ensured by a Lyapunovlike polynomial function that we compute using an interval based branchandrelax algorithm. This alg ..."
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Cited by 11 (5 self)
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Abstract — In this paper, we present a method for computing a basin of attraction to a target region for nonlinear ordinary differential equations. This basin of attraction is ensured by a Lyapunovlike polynomial function that we compute using an interval based branchandrelax algorithm. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunovlike function to a system of linear interval inequalities that can then be solved exactly, and iteratively reduces the relaxation error by recursively decomposing the state space into hyperrectangles. Tests on an implementation are promising. I.
Deductive Verification of Continuous Dynamical Systems
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... We define the notion of inductive invariants for continuous dynamical systems and use it to present inference rules for safety verification of polynomial continuous dynamical systems. We present two different sound and complete inference rules, but neither of these rules can be effectively applied. ..."
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Cited by 11 (4 self)
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We define the notion of inductive invariants for continuous dynamical systems and use it to present inference rules for safety verification of polynomial continuous dynamical systems. We present two different sound and complete inference rules, but neither of these rules can be effectively applied. We then present several simpler and practical inference rules that are sound and relatively complete for different classes of inductive invariants. The simpler inference rules can be effectively checked when all involved sets are semialgebraic.
Verification constraint problems with strengthening
 In ICTAC, volume 3722 of LNCS
, 2006
"... Abstract. The deductive method reduces verification of safety properties of programs to, first, proposing inductive assertions and, second, proving the validity of the resulting set of firstorder verification conditions. We discuss the transition from verification conditions to verification constra ..."
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Cited by 8 (4 self)
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Abstract. The deductive method reduces verification of safety properties of programs to, first, proposing inductive assertions and, second, proving the validity of the resulting set of firstorder verification conditions. We discuss the transition from verification conditions to verification constraints that occurs when the deductive method is applied to parameterized assertions instead of fixed expressions (e.g., p0 +p1j +p2k> = 0, for parameters p0, p1, and p2, instead of 3+jk> = 0) in order to discover inductive assertions. We then introduce two new verification constraint forms that enable the incremental and propertydirected construction of inductive assertions. We describe an iterative method for solving the resulting constraint problems. The main advantage of this approach is that it uses offtheshelf constraint solvers and thus directly benefits from progress in constraint solving. 1 Introduction The deductive method of program verification reduces the verification of safetyand progress properties to proving the validity of a set of firstorder verification conditions [13]. In the safety case, the verification conditions assert thatthe given property is inductive: it holds initially ( initiation), and it is preservedby taking any transition ( consecution). Such an assertion is an invariant of theprogram. In the progress case, the verification conditions assert that a given