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Local Stability Analysis Using Simulations and SumofSquares Programming
, 2008
"... The problem of computing bounds on the regionofattraction for systems with polynomial vector fields is considered. Invariant subsets of the regionofattraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for Lyapunov functions are used ..."
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Cited by 36 (15 self)
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The problem of computing bounds on the regionofattraction for systems with polynomial vector fields is considered. Invariant subsets of the regionofattraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates are assessed solving linear sumofsquares optimization problems. Qualified candidates are used to compute invariant subsets of the regionofattraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.
Stability region analysis using polynomial and composite polynomial Lyapunov functions and sumofsquares programming
 IEEE Transactions on Automatic Control
, 2008
"... We propose using (bilinear) sumofsquares programming for obtaining inner bounds of regionsofattraction for dynamical systems with polynomial vector fields. We search for polynomial as well as composite Lyapunov functions, comprised of pointwise maximums of polynomial functions. Results for sever ..."
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Cited by 22 (3 self)
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We propose using (bilinear) sumofsquares programming for obtaining inner bounds of regionsofattraction for dynamical systems with polynomial vector fields. We search for polynomial as well as composite Lyapunov functions, comprised of pointwise maximums of polynomial functions. Results for several examples from the literature are presented using the proposed methods and the PENBMI solver. I.
Stability region analysis using simulations and sumofsquares programming
 In Proceedings of the American Control Conference
, 2007
"... Abstract — The problem of computing bounds on the regionofattraction for systems with polynomial vector fields is considered. Invariant sets of the regionofattraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for the Lyapunov funct ..."
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Cited by 11 (6 self)
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Abstract — The problem of computing bounds on the regionofattraction for systems with polynomial vector fields is considered. Invariant sets of the regionofattraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for the Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of the Lyapunov function candidates are assessed solving linear sumofsquares optimization problems. Qualified candidates are used to compute provably invariant subsets of the regionofattraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on several small examples drawn from the literature. I.
An offline MPC strategy for nonlinear systems based on SOS programming
 Nonlinear Model Predictive Control, volume 384 of Lecture Notes in Control and Information Sciences
, 2009
"... A novel moving horizon control strategy for inputsaturated nonlinear polynomial systems is proposed. The control strategy makes use of the so called sumofsquares (SOS) decomposition, i.e. a convexification procedure able to give sufficient conditions on the positiveness of polynomials. The compl ..."
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Cited by 3 (0 self)
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A novel moving horizon control strategy for inputsaturated nonlinear polynomial systems is proposed. The control strategy makes use of the so called sumofsquares (SOS) decomposition, i.e. a convexification procedure able to give sufficient conditions on the positiveness of polynomials. The complexity of SOSbased numerical methods is polynomial in the problem size and, as a consequence, computationally attractive. SOS programming is used here to derive an “offline ” model predictive control (MPC) scheme and analyze in depth his properties. Such an approach may lead to less conservative MPC strategies than most existing methods based on the global linearization approach. An illustrative example is provided to show the benefits of the proposed SOSbased algorithm.
Complexity in automation of SOS proofs: An illustrative example
 In 45th IEEE Conf. on Decision and Control
, 2006
"... Abstract — We present a case study in proving invariance for a chaotic dynamical system, the logistic map, based on Positivstellensatz refutations, with the aim of studying the problems associated with developing a completely automated proof system. We derive the refutation using two different forms ..."
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Cited by 2 (1 self)
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Abstract — We present a case study in proving invariance for a chaotic dynamical system, the logistic map, based on Positivstellensatz refutations, with the aim of studying the problems associated with developing a completely automated proof system. We derive the refutation using two different forms of the Positivstellensatz and compare the results to illustrate the challenges in defining and classifying the ‘complexity ’ of such a proof. The results show the flexibility of the SOS framework in converting a dynamics problem into a semialgebraic one as well as in choosing the form of the proof. Yet it is this very flexibility that complicates the process of automating the proof system and classifying proof ‘complexity.’ I.
A robust control approach to understanding nonlinear mechanisms in shear flow turbulence
, 2010
"... ..."
Assessment of aircraft flight controllers using nonlinear robustness analysis techniques
"... Summary. The current practice to validate flight control laws relies on applying linear analysis tools to assess the closed loop stability and performance characteristics about many trim conditions. Nonlinear simulations are used to provide further confidence in the linear analyses and also to unco ..."
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Summary. The current practice to validate flight control laws relies on applying linear analysis tools to assess the closed loop stability and performance characteristics about many trim conditions. Nonlinear simulations are used to provide further confidence in the linear analyses and also to uncover dynamic characteristics, e.g. limit cycles, which are not revealed by the linear analysis. This chapter reviews nonlinear analysis techniques which can be applied to systems described by polynomial dynamic equations. The proposed approach is to approximate the aircraft dynamics using polynomial models. Nonlinear analyses can then be solved using sumofsquares optimization techniques. The applicability of these methods is demonstrated with nonlinear analyses of an F/A18 aircraft and NASA’s Generic Transport Model aircraft. These nonlinear analysis techniques can fill the gap between linear analysis and nonlinear simulations and hence used to provide additional confidence in the flight control law performance. 1
Acknowledgments
, 2010
"... I want to thank all those around me who supported me through the years. It has been a long journey that was crazy at times. I would never have been able to get any research together without your constant encouragement and cheer. For that I want to say thanks to my parents, my brother, Joe, and every ..."
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I want to thank all those around me who supported me through the years. It has been a long journey that was crazy at times. I would never have been able to get any research together without your constant encouragement and cheer. For that I want to say thanks to my parents, my brother, Joe, and everyone else in lab 15 for dealing with me. I also want to single out and thank Pete Seiler for his countless hours devoted to helping me iron out little odds and ends. I hope to one day explain to you something going on “under the hood”. Finally, none of this would have been possible without the guidance of my advisor Gary Balas. I have come a long way since undergrad thanks to you. No matter what, I will always remember to step back and first consider the results of the linear analysis.