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Linear Relaxations of Polynomial Positivity for Polynomial Lyapunov Function Synthesis [article]

Mohamed Amin Ben Sassi and Sriram Sankaranarayanan and Xin Chen and Erika Abraham
2014 arXiv   pre-print
In this paper, we examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs.  ...  A common approach to Lyapunov function synthesis starts from a desired parametric polynomial form of the polynomial Lyapunov function.  ...  In this paper, we focus on the synthesis of polynomial Lyapunov functions for proving the stability of autonomous systems with polynomial dynamics using linear programming (LP) relaxations.  ... 
arXiv:1407.2952v2 fatcat:43rnag74lrb6zovft6uai4buym

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Mohamed Amin Ben Sassi, Sriram Sankaranarayanan, Xin Chen, Erika Ábrahám
2015 IMA Journal of Mathematical Control and Information  
We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs.  ...  Subsequently, we show how these LP relaxations can be used to search for Lyapunov functions for polynomial ODEs by formulating LP instances.  ...  In this paper, we focus on the synthesis of polynomial Lyapunov functions for proving the stability of autonomous systems with polynomial dynamics using linear programming (LP) relaxations.  ... 
doi:10.1093/imamci/dnv003 fatcat:p65om6rrbnewpg5pexcu4nm63i

Lyapunov Function Synthesis using Handelman Representations

Sriram Sankaranarayanan, Xin Chen, Erika ábrahám
2013 IFAC Proceedings Volumes  
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 branch-and-relax scheme for discovering Lyapunov functions.  ...  ACKNOWLEDGMENTS We are grateful to the anonymous reviewers for their helpful comments.  ... 
doi:10.3182/20130904-3-fr-2041.00198 fatcat:ywrwmmkew5a5pjhk55yohxqcre

Lyapunov Function Synthesis - Infeasibility and Farkas' Lemma * *This work is supported by the Danish Council for Independent Research under grant number DFF - 4005-00452 in the project CodeMe

Tobias Leth, Christoffer Sloth, Rafał Wisniewski, Sriram Sankaranarayanan
2017 IFAC-PapersOnLine  
In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector fields defined on simplices.  ...  Abstract: In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector fields defined on simplices.  ...  Andersen from MOSEK for bringing Farkas' Lemma to our attention, fruitful discussions, and technical support.  ... 
doi:10.1016/j.ifacol.2017.08.339 fatcat:25l34y2zmnedzizy2x3yceioku

Stabilization of polynomial dynamical systems using linear programming based on Bernstein polynomials [article]

Mohamed Amin Ben Sassi, Sriram Sankaranarayanan
2015 arXiv   pre-print
Our approach uses Bernstein polynomials to build a linear relaxation of polynomial optimization problems, and the use of a so-called "policy iteration" approach to deal with bilinear optimization problems  ...  In this paper, we deal with the problem of synthesizing static output feedback controllers for stabilizing polynomial systems.  ...  Linear and bilinear feasibility programs for existence of Lyapunov function (Problem 1) Let V (x, c) be the assumed polynomial form for the Lyapunov function with unknowns c.  ... 
arXiv:1501.04578v1 fatcat:5rcehcgwbbgercprphsnry2wgm

A convex data-driven approach for nonlinear control synthesis [article]

Hyungjin Choi, Umesh Vaidya, Yongxin Chen
2020 arXiv   pre-print
Unlike Lyapunov based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate.  ...  Our method is founded on the density function based almost everywhere stability certificate that is dual to the Lyapunov function for dynamic systems.  ...  Sum of squares SOS optimization [15] - [18] is a relaxation of positive polynomial constraints appearing in polynomial optimization problems which are generally difficult to solve.  ... 
arXiv:2006.15477v1 fatcat:n3e4ae6xh5bw3phz3qc7vy4vbu

Formal Synthesis of Lyapunov Neural Networks [article]

Alessandro Abate, Daniele Ahmed, Mirco Giacobbe, Andrea Peruffo
2020 arXiv   pre-print
We propose an automatic and formally sound method for synthesising Lyapunov functions for the asymptotic stability of autonomous non-linear systems.  ...  Our method synthesises Lyapunov functions faster and over wider spatial domains than the alternatives, yet providing stronger or equal guarantees.  ...  the problem as a linear program (LP), or restrict V to be a sum-of-squares (SOS) function and relax the synthesis problem into a linear matrix inequalities (LMI) program [13] .  ... 
arXiv:2003.08910v2 fatcat:7a7mzu3t2nckpkbauhgyq67knm

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Amir Ali Ahmadi, Pablo A. Parrilo
2014 53rd IEEE Conference on Decision and Control  
, and controller synthesis can be addressed by a synergy of classical tools from Lyapunov theory and modern computational techniques from algebraic optimization.  ...  In this paper, we give a brief overview of our recent research efforts (with various coauthors) to (i) enhance the scalability of the algorithms in this field, and (ii) understand their worst case performance  ...  Lyapunov functions for their example is not robust to arbitrarily small perturbations. • On the topic of existence of sos Lyapunov functions: Does existence of a polynomial Lyapunov function for a polynomial  ... 
doi:10.1109/cdc.2014.7039734 dblp:conf/cdc/AhmadiP14 fatcat:kaahigfyavfsta47vwldaiht64

An improved nonlinear H∞ synthesis for parameter-dependent polynomial nonlinear systems using SOS programming

Dan Zhao, JianLiang Wang
2009 2009 American Control Conference  
State feedback control synthesis problems for a class of polynomial nonlinear systems are investigated in this paper.  ...  Hence the proposed methodology can be extended to the synthesis for the parameter-dependent polynomial systems.  ...  In this paper we revisit the Lyapunov-based state feedback synthesis for a class of nonlinear systems whose dynamics are described by polynomials.  ... 
doi:10.1109/acc.2009.5160334 dblp:conf/amcc/ZhaoW09 fatcat:jioklyp65recrljzkesrpq3y4u

Model-based control and stability analysis of discrete-time polynomial fuzzy systems with time delay and positivity constraints

Xiaomiao Li, Kamyar Merhan
2019 IEEE transactions on fuzzy systems  
To relax the conservativeness of the obtained stability results, two main methods are proposed in this paper: first, the piecewise linear membership functions (PLMFs) are used to introduce the approximate  ...  This paper proposes a novel Lyapunov stabilization analysis of discrete-time polynomial-fuzzy-model-based control systems with time delay under positivity constraint.  ...  Therefore, linear copositive function is proposed and widely employed to consider the innate features of positivity for the stability and positivity analyses of positive systems [16] - [19] .  ... 
doi:10.1109/tfuzz.2019.2893344 fatcat:lxmqa6at7fb7pf4vjo3pzljh7a

Bernstein Polynomial Relaxations for Polynomial Optimization Problems [article]

Mohamed Amin Ben Sassi, Sriram Sankaranarayanan
2015 arXiv   pre-print
In this paper, we examine linear programming (LP) relaxations based on Bernstein polynomials for polynomial optimization problems (POPs).  ...  We present a progression of increasingly more precise LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials.  ...  Lyapunov Stability Proofs A standard approach to prove stability for polynomial dynamical systems is to find a polynomial Lyapunov certificate which consists on a positive definite function decreasing  ... 
arXiv:1509.01156v1 fatcat:rngoh4ydovgwboykxybfcnjtzm

A Convex Data-Driven Approach for Nonlinear Control Synthesis

Hyungjin Choi, Umesh Vaidya, Yongxin Chen
2021 Mathematics  
Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems.  ...  Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate.  ...  ., for the U.S.  ... 
doi:10.3390/math9192445 fatcat:c4qaj3qjorgondtwn74v3del2m

Lyapunov redesign of adaptive controllers for polynomial nonlinear systems

Qian Zheng, Fen Wu
2009 2009 American Control Conference  
To achieve better controlled performance, the Lyapunov functions will be relaxed from quadratic to higher order and the resulting controller gain is generalized from constant to parameter dependent.  ...  The synthesis conditions of adaptive control will be formulated as polynomial matrix inequalities and are solvable by recast the resulting conditions into a Sum of Squares (SOS) optimization problem, from  ...  All of the synthesis conditions are formulated in the frame work of polynomial/constant linear matrix inequalities and solvable using available SOS programming software package.  ... 
doi:10.1109/acc.2009.5160128 dblp:conf/amcc/ZhengW09 fatcat:rpxq435cmrhynogwscusrvocbu

Suboptimal stabilizing controllers for linearly solvable system

Yoke Peng Leong, Matanya B. Horowitz, Joel W. Burdick
2015 2015 54th IEEE Conference on Decision and Control (CDC)  
It is shown that this linear partial differential equation can be relaxed to a linear differential inclusion, allowing for approximating polynomial solutions to be generated using sum of squares programming  ...  This paper presents a novel method to synthesize stochastic control Lyapunov functions for a class of nonlinear, stochastic control systems.  ...  Lyapunov theory may be generalized from analysis to synthesis of control systems using Control Lyapunov Function (CLF) [1] .  ... 
doi:10.1109/cdc.2015.7403348 dblp:conf/cdc/LeongHB15 fatcat:6uyvfxrstvczjoidtl6zethmfe

Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations

Hadi Ravanbakhsh, Sriram Sankaranarayanan
2017 Robotics: Science and Systems XIII  
We present a technique for learning control Lyapunov (potential) functions, which are used in turn to synthesize controllers for nonlinear dynamical systems.  ...  Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.  ...  All opinions expressed are those of the authors and not necessarily of the NSF.  ... 
doi:10.15607/rss.2017.xiii.049 dblp:conf/rss/RavanbakhshS17 fatcat:o5h32scirnbzneqyra2tclvwk4
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