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20
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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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.
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.
Analysis of nonlinear delay differential equation models of TCP/AQM protocols using sums of squares
 43rd IEEE Conference on Decision and Control
, 2004
"... Abstract — The simplest adequate models for congestion control for the Internet are in the form of deterministic nonlinear delay differential equations. However the absence of efficient, algorithmic methodologies to analyze them at this modelling level usually results in the investigation of their l ..."
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Cited by 8 (2 self)
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Abstract — The simplest adequate models for congestion control for the Internet are in the form of deterministic nonlinear delay differential equations. However the absence of efficient, algorithmic methodologies to analyze them at this modelling level usually results in the investigation of their linearizations including delays; or in the analysis of nonlinear yet undelayed models. In this paper we present an algorithmic methodology for efficient stability analysis of network congestion control schemes at the nonlinear delaydifferential equation model level, using the Sum of Squares decomposition and SOSTOOLS. I.
H.: Automatically discovering relaxed Lyapunov functions for polynomial dynamical systems
 Mathematics in Computer Science
, 2012
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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.
Simulationguided Lyapunov Analysis for Hybrid Dynamical Systems
"... Lyapunov functions are used to prove stability and to obtain performance bounds on system behaviors for nonlinear and hybrid dynamical systems, but discovering Lyapunov functions is a difficult task in general. We present a technique for discovering Lyapunov functions and barrier certificates for n ..."
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Lyapunov functions are used to prove stability and to obtain performance bounds on system behaviors for nonlinear and hybrid dynamical systems, but discovering Lyapunov functions is a difficult task in general. We present a technique for discovering Lyapunov functions and barrier certificates for nonlinear and hybrid dynamical systems using a searchbased approach. Our approach uses concrete executions, such as those obtained through simulation, to formulate a series of linear programming (LP) optimization problems; the solution to each LP creates a candidate Lyapunov function. Intermediate candidates are iteratively improved using a global optimizer guided by the Lie derivative of the candidate Lyapunov function. The analysis is refined using counterexamples from a Satisfiability Modulo Theories (SMT) solver. When no counterexamples are found, the soundness of the analysis is verified using an arithmetic solver. The technique can be applied to a broad class of nonlinear dynamical systems, including hybrid systems and systems with polynomial and even transcendental dynamics. We present several examples illustrating the efficacy of the technique, including two automotive powertrain control examples.
Domain of attraction: estimates for nonpolynomial systems via LMIs
 in Proc. of 16th IFAC World Congress on Automatic Control
, 2005
"... Abstract: Estimating the Domain of Attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been developed for computing the Largest Estimate of the DA (LEDA) corresponding to a Lyapunov function in the case of polynomial systems. I ..."
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Abstract: Estimating the Domain of Attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been developed for computing the Largest Estimate of the DA (LEDA) corresponding to a Lyapunov function in the case of polynomial systems. In the case of nonpolynomial systems, the computation of the LEDA is still an open problem. In this paper, an LMI technique is proposed to deal with such a problem for a class of nonpolynomial systems. The key point consists of using sum of squares relaxations for taking into account the worstcase remainders corresponding to truncated Taylor expansions of the nonpolynomial terms. As shown by some examples, low degree remainders may be sufficient to obtain almost tight estimates. Copyright c©2005 IFAC
Abstraction of Elementary Hybrid Systems by Variable Transformation
"... Elementary hybrid systems (EHSs) are those hybrid systems (HSs) containing elementary functions such as exp, ln, sin, cos, etc. EHSs are very common in practice, especially in safetycritical domains. Due to the nonpolynomial expressions which lead to undecidable arithmetic, verification of EHSs i ..."
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Elementary hybrid systems (EHSs) are those hybrid systems (HSs) containing elementary functions such as exp, ln, sin, cos, etc. EHSs are very common in practice, especially in safetycritical domains. Due to the nonpolynomial expressions which lead to undecidable arithmetic, verification of EHSs is very hard. Existing approaches based on partition of the state space or overapproximation of reachable sets suffer from state space explosion or inflation of numerical errors. In this paper, we propose a symbolic abstraction approach that reduces EHSs to polynomial hybrid systems (PHSs), by replacing all nonpolynomial terms with newly introduced variables. Thus the verification of EHSs is reduced to the one of PHSs, enabling us to apply all the wellestablished verification techniques and tools for PHSs to EHSs. In this way, it is possible to avoid the limitations of many existing methods. We illustrate the abstraction approach and its application in safety verification of EHSs by several real world examples.