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26
Advanced methods and algorithms for biological networks analysis
 Proceedings of the IEEE
, 2006
"... Modeling and analysis of complex biological networks presents a number of mathematical challenges. For the models to be useful from a biological standpoint, they must be systematically compared with data. Robustness is a key to biological understanding and proper feedback to guide experiments, incl ..."
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Cited by 22 (5 self)
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Modeling and analysis of complex biological networks presents a number of mathematical challenges. For the models to be useful from a biological standpoint, they must be systematically compared with data. Robustness is a key to biological understanding and proper feedback to guide experiments, including both the deterministic stability and performance properties of models in the presence of parametric uncertainties and their stochastic behavior in the presence of noise. In this paper, we present mathematical and algorithmic tools to address such questions for models that may be nonlinear, hybrid, and stochastic. These tools are rooted in solid mathematical theories, primarily from robust control and dynamical systems, but with important recent developments. They also have the potential for great practical relevance, which we explore through a series of biologically motivated examples. Keywords—Biological networks, model invalidation, robust stability, sum of squares based software tools (SOSTOOLS), stochastic analysis. I.
Symbolic model checking of hybrid systems using template polyhedra
 In TACAS’08  Tools and Algorithms for
, 2008
"... Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs o ..."
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Cited by 20 (7 self)
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Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs overapproximations of the reachable states using template polyhedra. Therefore, operations used in symbolic model checking such as intersection, union and postcondition across discrete transitions over template polyhedra can be computed efficiently using template polyhedra without requiring expensive vertex enumeration. Additionally, the verification of hybrid systems requires techniques to handle the continuous dynamics inside discrete modes. We propose a new flowpipe construction algorithm using template polyhedra. Our technique uses higherorder Taylor series expansion to approximate the time trajectories. The terms occurring in the Taylor series expansion are bounded using repeated optimization queries. The location invariant is used to enclose the remainder term of the Taylor series, and thus truncate the expansion. Finally, we have implemented our technique as a part of the tool TimePass for the analysis of affine hybrid automata. 1
Reachability analysis for controlled discrete time stochastic hybrid systems
 in Hybrid Systems: Computation and Control  HSCC 2006
, 2006
"... Abstract. A model for discrete time stochastic hybrid systems whose evolution can be influenced by some control input is proposed in this paper. With reference to the introduced class of systems, a methodology for probabilistic reachability analysis is developed that is relevant to safety verificati ..."
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Cited by 17 (8 self)
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Abstract. A model for discrete time stochastic hybrid systems whose evolution can be influenced by some control input is proposed in this paper. With reference to the introduced class of systems, a methodology for probabilistic reachability analysis is developed that is relevant to safety verification. This methodology is based on the interpretation of the safety verification problem as an optimal control problem for a certain controlled Markov process. In particular, this allows to characterize through some optimal cost function the set of initial conditions for the system such that safety is guaranteed with sufficiently high probability. The proposed methodology is applied to the problem of regulating the average temperature in a room by a thermostat controlling a heater. 1
A tutorial on sum of squares techniques for system analysis
 In Proceedings of the American control conference, ASCC
, 2005
"... Abstract — This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We will present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential e ..."
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Cited by 17 (1 self)
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Abstract — This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We will present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential equations or differential algebraic equations, hybrid systems with nonlinear subsystems and/or nonlinear switching surfaces, and timedelay systems described by nonlinear functional differential equations. We will also discuss how different analysis questions such as model validation and safety verification can be answered for uncertain nonlinear and hybrid systems. I.
Computational Methods for Verification of Stochastic Hybrid Systems
 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS  PART A
, 2008
"... Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is ch ..."
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Cited by 14 (5 self)
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Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is challenging because of the interaction between the discrete and the continuous stochastic dynamics. In this paper, we propose a probabilistic method for verification of SHSs based on discrete approximations focusing on reachability and safety problems. We show that reachability and safety can be characterized as a viscosity solution of a system of coupled Hamilton–Jacobi–Bellman equations. We present a numerical algorithm for computing the solution based on discrete approximations that are derived using finitedifference methods. An advantage of the method is that the solution converges to the one for the original system as the discretization becomes finer. We also prove that the algorithm is polynomial in the number of states of the discrete approximation. Finally, we illustrate the approach with two benchmarks: a navigation and a room heater example, which have been proposed for hybrid system verification.
Relational Abstractions For Continuous and Hybrid Systems
"... Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any s ..."
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Cited by 13 (5 self)
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Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any state of the system to a state that can potentially be reached at some future time instant. Such relations are then interpreted as discrete transitions that model the continuous evolution of states over time. We adapt templatebased invariant generation techniques for continuous dynamics to derive relational abstractions for continuous systems with linear as well as nonlinear dynamics. Once a relational abstraction hasbeen derived,theresultingsystemis apurelydiscrete, infinitestatesystem. Therefore, techniquessuchas kinductioncan be directly applied to this abstraction to prove properties, and bounded modelchecking techniques applied to find potential falsifications. We present the basic underpinnings of our approach and demonstrate its use on many benchmark systems to derive simple and usable abstractions. 1
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.
On analysis and synthesis of safe control laws
 in Proceedings of the 42nd Allerton Conference on Communication, Control, and Computing
, 2004
"... Controller synthesis for nonlinear systems is considered with the following objective: no trajectory starting from a given set of initial states is allowed to enter into a given set of forbidden (unsafe) states. A methodology for safety verification using barrier certificates has recently been prop ..."
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Cited by 9 (2 self)
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Controller synthesis for nonlinear systems is considered with the following objective: no trajectory starting from a given set of initial states is allowed to enter into a given set of forbidden (unsafe) states. A methodology for safety verification using barrier certificates has recently been proposed. Here it is shown how a safe control law together with a corresponding certificate can be computed by means of convex optimization. A basic tool is the theory for density functions in analysis of nonlinear systems. Computational examples are considered. 1
Finitetime Regional Verification of Stochastic Nonlinear Systems
"... Abstract—Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the pro ..."
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Cited by 7 (4 self)
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Abstract—Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the probability of failure (defined as leaving a finite region of state space) over a finite time for stochastic nonlinear systems with continuous state. Our approach searches for exponential barrier functions that provide bounds using a variant of the classical supermartingale result. We provide a relaxation of this search to a semidefinite program, yielding an efficient algorithm that provides rigorous upper bounds on the probability of failure for the original nonlinear system. We give a number of numerical examples in both discrete and continuous time that demonstrate the effectiveness of the approach. I.
Timed relational abstractions for sampled data control systems. Submitted, Under review. 6 Supplementary Material Proof. (Proof sketch for Proposition 1) First, let p(x) be the linear expression c T y + d T z + e discovered in Step (6). Then, dp dt = cT (
 in Step (11). Let p1, p2 be as defined in Step (10). Then, d(p 2 1 + p 2 2) dt = 2p1(αp1 − βp2) + 2p2(βp1 + αp2) = 2α(p 2 1 + p 2 2) Hence, p1(x(t)) 2 + p2(x(t)) 2 = (p1(x(0)) 2 + p2(x(0)) 2 )e 2αt , and therefore, the relation
"... Abstract. In this paper, we define timed relational abstractions for verifying sampled data control systems. Sampled data control systems consist of a plant, modeled as a hybrid system and a synchronous controller, modeled as a discrete transition system. The controller performs control actions on t ..."
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Abstract. In this paper, we define timed relational abstractions for verifying sampled data control systems. Sampled data control systems consist of a plant, modeled as a hybrid system and a synchronous controller, modeled as a discrete transition system. The controller performs control actions on the plant by periodically sampling the state of the plant. The correctness of the system depends on the controller design as well as an appropriate choice of its sampling period. Our approach constructs a timed relational abstraction of the hybrid plant by replacing the continuous plant dynamics by relations. These relations map a state of the plant to states reachable within the sampling time period. We present techniques for building timed relational abstractions, while taking care of discrete transitions that can be taken by the plant between samples. The resulting abstractions are better suited for the verification of sampled data control systems. The abstractions focus on the states that can be observed by the controller at the sample times, while abstracting away behaviors between sample times conservatively. As the abstractions are discrete, infinitestate transition systems, conventional verification tools can be used. We use kinduction to prove safety properties and bounded model checking (BMC) to find potential falsifications. We present our idea, its implementation and results on many benchmark examples. 1