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H.: Automatically discovering relaxed Lyapunov functions for polynomial dynamical systems
 Mathematics in Computer Science
, 2012
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Estimating the region of attraction of ordinary differential equations by quantified constraint solving
 In Proceedings of the 3rd WSEAS International Conference on DYNAMICAL SYSTEMS and CONTROL (CONTROL’07
, 2007
"... We formulate the problem of estimating the region of attraction using quantified constraints and show how the resulting constraints can be solved using existing software packages. We discuss the advantages of the resulting method in detail. 1 ..."
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We formulate the problem of estimating the region of attraction using quantified constraints and show how the resulting constraints can be solved using existing software packages. We discuss the advantages of the resulting method in detail. 1
A SEMIALGEBRAIC APPROACH FOR THE COMPUTATION OF LYAPUNOV FUNCTIONS
"... In this paper we deal with the problem of computing Lyapunov functions for stability verification of differential systems. We concern on symbolic methods and start the discussion with a classical quantifier elimination model for computing Lyapunov functions in a given polynomial form, especially in ..."
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In this paper we deal with the problem of computing Lyapunov functions for stability verification of differential systems. We concern on symbolic methods and start the discussion with a classical quantifier elimination model for computing Lyapunov functions in a given polynomial form, especially in quadratic forms. Then we propose a new semialgebraic method by making advantage of the local property of the Lyapunov function as well as its derivative. This is done by first using real solution classification to construct a semialgebraic system and then solving this semialgebraic system. Our semialgebraic approach is more efficient in practice, especially for loworder systems. This efficiency will be evaluated empirically.
Benchmark Examples Stability of Nonlinear ODE’s
, 2007
"... In this section, eight examples will be presented, for which we computed set Lyapunov functions using the method described in this paper. Here the target region TR is the set {x ∈ B: xi − ¯xi  < δ, 1 ≤ i ≤ n}, where B is a given box containing the equilibrium ¯x, and δ> 0 is a arbitrarily gi ..."
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In this section, eight examples will be presented, for which we computed set Lyapunov functions using the method described in this paper. Here the target region TR is the set {x ∈ B: xi − ¯xi  < δ, 1 ≤ i ≤ n}, where B is a given box containing the equilibrium ¯x, and δ> 0 is a arbitrarily given constant. Example 1 A simplified model of a chemical oscillator [2]. ˙x1 = 0.5 − x1 + x 2 1x2
Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems
"... Your article is protected by copyright and all rights are held exclusively by Springer Basel. This eoffprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to selfarchive your work, please use the accepted author’s version for posting to your own we ..."
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Your article is protected by copyright and all rights are held exclusively by Springer Basel. This eoffprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to selfarchive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication. Math.Comput.Sci.
Automatic Verification of Stability and Safety for Delay Differential Equations
"... Abstract. Delay differential equations (DDEs) arise naturally as models of, e.g., networked control systems, where the communication delay in the feedback loop cannot always be ignored. Such delays can prompt oscillations and may cause deterioration of control performance, invalidating both stabil ..."
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Abstract. Delay differential equations (DDEs) arise naturally as models of, e.g., networked control systems, where the communication delay in the feedback loop cannot always be ignored. Such delays can prompt oscillations and may cause deterioration of control performance, invalidating both stability and safety properties. Nevertheless, stateexploratory automatic verification methods have until now concentrated on ordinary differential equations (and their piecewise extensions to hybrid state) only, failing to address the effects of delays on system dynamics. We overcome this problem by iterating bounded degree intervalbased Taylor overapproximations of the timewise segments of the solution to a DDE, thereby identifying and automatically analyzing the operator that yields the parameters of the Taylor overapproximation for the next temporal segment from the current one. By using constraint solving for analyzing the properties of this operator, we obtain a procedure able to provide stability and safety certificates for a simple class of DDEs. 1
RSolver User Manual
"... Taketheexpressionx 2 +y 2 +z 2 ≤ 1, wherethevariablesx, y andz rangeover the real numbers. This expression represents a set of values—its solution set, which is a ball. We call such an expression a constraint. Solving a constraint means to find some interesting information about its solution ..."
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Taketheexpressionx 2 +y 2 +z 2 ≤ 1, wherethevariablesx, y andz rangeover the real numbers. This expression represents a set of values—its solution set, which is a ball. We call such an expression a constraint. Solving a constraint means to find some interesting information about its solution
Lyapunov Function Synthesis using Handelman Representations.
"... Abstract: We investigate linear programming relaxations to synthesize Lyapunov functions that establish the stability of a given system over a bounded region. In particular, we attempt to discover functions that are more readily useful inside symbolic verification tools for proving the correctness ..."
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Abstract: We investigate linear programming relaxations to synthesize Lyapunov functions that establish the stability of a given system over a bounded region. In particular, we attempt to discover functions that are more readily useful inside symbolic verification tools for proving the correctness of control systems. Our approach searches for a Lyapunov function, given a parametric form with unknown coefficients, by constructing a system of linear inequality constraints over the unknown parameters. We examine two complementary ideas for the linear programming relaxation, including interval evaluation of the polynomial form and “Handelman representations ” for positive polynomials over polyhedral sets. Our approach is implemented as part of a branchandrelax scheme for discovering Lyapunov functions. We evaluate our approach using a prototype implementation, comparing it with techniques based on SumofSquares (SOS) programming. A comparison with SOSTOOLS is carried out over a set of benchmarks gathered from the related work. The evaluation suggests that our approach using Simplex is generally fast, and discovers Lyapunov functions that are simpler and easy to check. They are suitable for use inside symbolic formal verification tools for reasoning about continuous systems.
ATTRACTION AND STABILITY OF NONLINEAR ODE’S USING CONTINUOUS PIECEWISE LINEAR APPROXIMATIONS
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