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72
Analysis of Nonpolynomial Systems Using the Sum of Squares Decomposition
 Positive Polynomials in Control, volume 312 of Lecture Notes in Control and Information Sciences
, 2005
"... Summary. Recent advances in semidefinite programming along with use of the sum of squares decomposition to check nonnegativity have paved the way for efficient and algorithmic analysis of systems with polynomial vector fields. In this paper we present a systematic methodology for analyzing the more ..."
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Cited by 20 (6 self)
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Summary. Recent advances in semidefinite programming along with use of the sum of squares decomposition to check nonnegativity have paved the way for efficient and algorithmic analysis of systems with polynomial vector fields. In this paper we present a systematic methodology for analyzing the more general class of nonpolynomial vector fields, by recasting them into rational vector fields. The sum of squares decomposition techniques can then be applied in conjunction with an extension of the Lyapunov stability theorem to investigate the stability and other properties of the recasted systems, from which properties of the original, nonpolynomial systems can be inferred. This will be illustrated by some examples from the mechanical and chemical engineering domains. 1
Analysis of Nonlinear TimeDelay Systems Using the Sum of Squares Decomposition
, 2004
"... The use of the sum of squares decomposition and semidefinite programming have provided an efficient methodology for analysis of nonlinear systems described by ODEs by algorithmically constructing Lyapunov functions. Based on the same methodology we present an algorithmic procedure for constructing L ..."
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Cited by 17 (7 self)
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The use of the sum of squares decomposition and semidefinite programming have provided an efficient methodology for analysis of nonlinear systems described by ODEs by algorithmically constructing Lyapunov functions. Based on the same methodology we present an algorithmic procedure for constructing LyapunovKrasovskii functionals for nonlinear time delay systems described by Functional Differential Equations (FDEs) both for delaydependent and delayindependent stability analysis. Robust stability analysis of these systems under parametric uncertainty can be treated in a unified way. We illustrate the results with an example from population dynamics.
Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using SumofSquares Optimization
, 2003
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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.
Maximum block improvement and polynomial optimization
 SIAM Journal on Optimization
"... Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with t ..."
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Cited by 15 (5 self)
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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.
Convex programs for temporal verification of nonlinear dynamical systems
 SIAM J. Control Optim
"... Abstract. A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the se ..."
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Cited by 12 (1 self)
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Abstract. A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates.
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
Globally Optimal Affine and Metric Upgrades in Stratified Autocalibration
"... We present a practical, stratified autocalibration algorithm with theoretical guarantees of global optimality. Given a projective reconstruction, the first stage of the algorithm upgrades it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally ..."
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Cited by 6 (3 self)
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We present a practical, stratified autocalibration algorithm with theoretical guarantees of global optimality. Given a projective reconstruction, the first stage of the algorithm upgrades it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, the algorithm upgrades this affine reconstruction to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a postprocessing step. For each stage, we construct and minimize tight convex relaxations of the highly nonconvex objective functions in a branch and bound optimization framework. We exploit the problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. The convex relaxation techniques presented here are general enough that we expect them to be of use to computer vision researchers solving optimization problems in multiview geometry and elsewhere. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data. MATLAB code for the implementation is made available to the community for facilitating further research. 1
Periodically Controlled Hybrid Systems Verifying A Controller for An Autonomous Vehicle
"... Abstract. This paper introduces Periodically Controlled Hybrid Automata (PCHA) for describing a class of hybrid control systems. In a PCHA, control actions occur roughly periodically while internal and input actions may occur in the interim changing the discretestate or the setpoint. Based on perio ..."
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Cited by 6 (3 self)
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Abstract. This paper introduces Periodically Controlled Hybrid Automata (PCHA) for describing a class of hybrid control systems. In a PCHA, control actions occur roughly periodically while internal and input actions may occur in the interim changing the discretestate or the setpoint. Based on periodicity and subtangential conditions, a new sufficient condition for verifying invariance of PCHAs is presented. This technique is used in verifying safety of the plannercontroller subsystem of an autonomous ground vehicle, and in deriving geometric properties of planner generated paths that can be followed safely by the controller under environmental uncertainties.
Control and verification of the safetyfactor profile in tokamaks using sumofsquares polynomials
 in 18th IFAC World Congress
, 2011
"... Abstract: In this paper, we propose a method of using the sumofsquares methodology to synthesize controllers for plasma stabilization in Tokamak reactors. We use a partial differential model of the poloidal magnetic flux gradient and attempt to stabilize a reference safetyfactor profile. Our meth ..."
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Cited by 5 (5 self)
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Abstract: In this paper, we propose a method of using the sumofsquares methodology to synthesize controllers for plasma stabilization in Tokamak reactors. We use a partial differential model of the poloidal magnetic flux gradient and attempt to stabilize a reference safetyfactor profile. Our methods utilize fullstate feedback control and are based on solving a dual version of the Lyapunov operator inequality. In addition, we implement the controller insilico using experimental conditions inferred from the Tore Supra tokamak. 1.