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CONVEX RELAXATIONS OF INTEGRAL VARIATIONAL PROBLEMS: POINTWISE DUAL RELAXATION AND sum-OF-squares OPTIMIZATION

SIAM JOURNAL ON OPTIMIZATION 2023年第2期33卷 481-512页

作者： Chernyavsky, Alexander Bramburger, Jason J. Fantuzzi, Giovanni Goluskin, David SUNY Buffalo Dept Math Buffalo NY 14260 USA Concordia Univ Math & Stat Montreal PQ H3G 1M8 Canada Imperial Coll London Dept Aeronaut London SW7 2AZ England Univ Victoria Dept Math & Stat Victoria BC V8P 5C2 Canada

We present a method for finding lower bounds on the global infima of integral variational problems, wherein integral(Omega) f(x, u(x), del u(x))dx is minimized over functions u: Omega subset of R-n -> R-m satisfying given equality or inequality constraints. Each constraint may be imposed over Omega or its boundary, either pointwise or in an integral sense. These global minimizations are generally nonconvex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the original problem. The PDR can be derived by dualizing and relaxing the original problem;its constraints are pointwise equalities or inequalities over finite-dimensional sets rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of (x, u, del u), the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions m, n and the number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou [IEEE Trans. Automat. Control, 61 (2016), pp. 1649-1654]. We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincare inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several examples.

关键词： calculus of variations convex relaxation polynomial optimization sum-of-squares programming

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Impulse-based controller synthesis for a class of polynomial systems

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INTERNATIONAL JOURNAL OF MODELLING IDENTIFICATION AND CONTROL 2023年第2期42卷 153-162页

作者： Ye, Qian Wuxi Inst Technol Coll IoT Technol Wuxi 214121 Peoples R China

This paper develops a time-triggered impulsive control strategy for stabilisation problem of a class of polynomial systems. The proposed hybrid control method combines time-triggered impulsive control and state-feedback control. To obtain the control feedback gain, the matrix sum-of-squares programming is employed to solve the feasible solutions of constructed positive polynomial problems. Compared with a single time-triggered impulsive control or the pure state-feedback control, the proposed control method can effectively combine the advantages of the two control methods. In particular, the design of the controller is more flexible and the control effect is greatly improved. Finally, two simulation examples including the Chua's chaotic system are provided to illustrate the effectiveness and superiority of the proposed method.

关键词： polynomial system impulse-based control state-feedback control sum-of-squares programming

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Data-driven Control of Polynomial Systems Using Dissipativity-Based Inequalities and Petersen's Lemma 12

Data-driven Control of Polynomial Systems Using Dissipativit...

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12th International Conference on Control Mechatronics and Automation

作者： Silva, Joao Gabriel N. Madeira, Diego de S. Correia, Wilkley B. Univ Fed Ceara Dept Elect Engn Fortaleza Ceara Brazil

ISBN: (纸本)9798331517519;9798331517526

In engineering science, the strategies of data-based control synthesis are valuable tools that allow to obtain a control law without the need to identify or to model the system through its related physics. However, the inherent presence of noise when acquiring data represents an obstacle in the validity of these data to represent accurately the system. In this paper, Lyapunov stability theory, dissipativity theory and Petersen's lemma are utilized to develop an algorithm. The proposed algorithm uses sum-of-squares programming to design a control law that guarantees asymptotic stability of polynomial systems with noisy data. In addition, the develop method is tested in comparison to a D-K iteration method in order to evaluate its performance in terms of computational costs.

关键词： dissipativity theory sum-of-squares programming data-driven nonlinear control

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Safe-by-design planner-tracker synthesis with a hierarchy of system models

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ANNUAL REVIEWS IN CONTROL 2022年 53卷 138-146页

作者： Schweidel, Katherine S. Yin, He Smith, Stanley W. Arcak, Murat Univ Calif Berkeley Dept Mech Engn Berkeley CA 94720 USA Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA USA

We present a safe-by-design trajectory planning and tracking framework for nonlinear dynamical systemsusing a hierarchy of system models. The planning layer uses a low-fidelity model to plan a feasible trajectorysatisfying the planning constraints, and the tracking layer utilizes the high-fidelity model to design a controllerthat restricts the error states between the low- and high-fidelity models to a bounded set. The simplicity ofthe low-fidelity model enables the planning to be performed online (e.g. using Model Predictive Control) andthe tracking controller and error bound are derived offline (e.g. using sum-of-squares programming). Thiserror bound is then used by the planner to ensure safety for the combined planner-tracker system To providefreedom in the choice of the low-fidelity model, we allow the tracking error to depend on both the states andinputs of the planner. The goal of this article is to provide a tutorial review of this hierarchical frameworkand to illustrate it with examples, including a design for vehicle obstacle avoidances

关键词： Motion planning Hierarchical control sum-of-squares programming Model Predictive Control

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sum-of-squares Collision Detection for Curved Shapes and Paths 23

Sum-of-Squares Collision Detection for Curved Shapes and Pat...

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ACM SIGGRAPH Conference

作者： Zhang, Paul Marschner, Zoe Solomon, Justin Tamstorf, Rasmus MIT 77 Massachusetts Ave Cambridge MA 02139 USA Walt Disney Animat Studios Burbank CA 91521 USA

ISBN: (纸本)9798400701597

sum-of-squares programming (SOSP) has recently been introduced to graphics as a unified way to address a large set of difficult problems involving higher order primitives. Unfortunately, a challenging aspect of this approach is the computational cost-especially for problems involving multiple geometries like collision detection. In this paper, we present techniques to reduce the cost of SOSP significantly. We use these improvements to speed up difficult problems like collision detection between Bezier triangles by as much as 300x. In addition, motivated by hair bundle simulation, we present SOSP based collision detection on the tapered cubic cylinder. We also present an algebraic formulation of rigid body motion enabling SOSP based collision detection for curved geometries and trajectories simultaneously. While these new formulations are complex, our speedups make them feasible. These advances improve the applicability of SOSP based collision detection and enable the continued progress of higher-order geometry processing.

关键词： collision detection sum-of-squares programming parametric shape analysis

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Safety Verification and Controller Synthesis for Systems with Input Constraints 22

Safety Verification and Controller Synthesis for Systems wit...

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22nd World Congress of the International Federation of Automatic Control (IFAC)

作者： Wang, Han Margellos, Kostas Papachristodoulou, Antonis Univ Oxford Dept Engn Sci Oxford England

ISBN: (纸本)9781713872344

In this paper we consider the safety verification and safe controller synthesis for nonlinear control systems. The Control Barrier Certificates (CBC) approach is proposed as an extension to the barrier certificates approach. Our approach can be used to characterize control invariance of a given set in terms of safety of a general nonlinear control system subject to input constraints. From the point of view of controller design, the proposed method provides an approach to synthesize a safe control law that guarantees that the trajectories of the system starting from a given initial set do not enter an unsafe set. Unlike the related control barrier functions formulations, our formulation only considers the vector field within the tangent cone of the zero level set defined by the certificates, and is shown to be less conservative by means of numerical evidence. For polynomial systems with semi-algebraic initial and safe sets, CBCs and safe control laws can be synthesized using sum-of-squares decomposition and semi-definite programming. Numerical examples demonstrate the efficacy of our approach. Copyright (c) 2023 The Authors.

关键词： safety verification nonlinear system sum-of-squares programming

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Designing Safety Certificates for H∞ Control of Unknown Linear Systems

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IEEE CONTROL SYSTEMS LETTERS 2022年 6卷 2990-2995页

作者： Tooranjipour, Pouria Kiumarsi, Bahare Michigan State Univ Dept Elect & Comp Engn E Lansing MI 48824 USA

This letter synthesizes safety barrier certificates for H-infinity control of unknown linear systems. Inspired by the idea of expanding the domain of attraction (DoA) using control barrier functions (CBFs), we construct a region, called safe optimal DoA, in which both optimality and safety are simultaneously respected. The effect of unknown disturbances is attenuated by unifying the robustness property of CBFs and the H-infinity control methods. To construct the control barrier certificates, a feasible optimization problem is developed using the relaxed game algebraic Riccati equation. A sum-of-squares (SOS)-based safe policy iteration algorithm is then proposed to solve the optimization problem iteratively. The resulting optimal policy minimizes a predefined cost function over an obtained maximum barrier-certified region for which there is no conflict between safety and optimality. Furthermore, to remedy the requirement of having the system dynamics, an online data-driven approach is presented using off-policy reinforcement learning. Finally, to demonstrate the efficacy of the proposed controller, a numerical example is given.

关键词： Safety-critical control control barrier functions safe reinforcement learning sum-of-squares programming

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On Estimating the Probabilistic Region of Attraction for Partially Unknown Nonlinear Systems: An sum-of-squares Approach 34

On Estimating the Probabilistic Region of Attraction for Par...

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34th Chinese Control and Decision Conference (CCDC)

作者： Huang, Hejun Hang, Dongkun Chinese Univ Hong Kong Dept Mech & Automat Engn Hong Kong Peoples R China

ISBN: (纸本)9781665478960

Estimating the region of attraction for partially unknown nonlinear systems is a challenging issue. In this paper, we propose a tractable method to generate an estimated region of attraction with probability bounds, by searching an optimal polynomial barrier function. Chebyshev interpolants, Gaussian processes and sum-of-squares programmings are used in this paper. To approximate the unknown non-polynomial dynamics, a polynomial mean function of Gaussian processes model is computed to represent the exact dynamics based on the Chebyshev interpolants. Furthermore, probabilistic conditions are given such that all the estimates are located in certain probability bounds. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

关键词： Region of attraction sum-of-squares programming Chebyshev interpolants Gaussian processes

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SOStab: A Matlab toolbox for transient stability analysis

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ELECTRIC POWER SYSTEMS RESEARCH 2024年 235卷

作者： Drobot, Stephane Tacchi, Matteo Cardozo, Carmen Jones, Colin Reseau Transport Elect R&D Div La Def France Univ Grenoble Alpes Grenoble INP GIPSA Lab CNRS Grenoble France Ecole Polytech Fed Lausanne Automat Control Lab Lausanne Switzerland

This paper presents a new Matlab toolbox, aimed at facilitating the use of polynomial optimization for stability analysis of nonlinear systems. In the past decade several decisive contributions made it possible to recast this type of problems as convex optimization ones that are tractable in modest dimensions. However, available software requires their user to be fluent in sum-of-squares programming, preventing them from being more widely explored by practitioners. To address this issue, SOStab entirely automates the writing and solving of optimization problems, and directly outputs relevant data for the user, while requiring minimal input. In particular, no specific knowledge of optimization is needed for implementation. The toolbox allows a user to obtain outer and inner approximates of the Region of Attraction (RoA) of the operating point of different grid connected devices such as synchronous machines and power converters.

关键词： AC power systems Software sum-of-squares programming Transient stability analysis

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sum-of-squares certificates for Vizing's conjecture via determining Grobner bases

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JOURNAL OF SYMBOLIC COMPUTATION 2024年 120卷

作者： Gaar, Elisabeth Siebenhofer, Melanie Johannes Kepler Univ Linz Inst Prod & Logist Management Linz Austria Alpen Adria Univ Klagenfurt Klagenfurt Austria

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class g of all graphs with ng vertices and domination number kg and reformulated Vizing's conjecture as the problem that for all graph classes g and 7-t the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of kg, ng, kn, and nn. In this paper, we consider their approach for kg = kn= 1. For this case we are able to derive the unique reduced Grobner basis of the Vizing ideal. Based on this, we deduce the minimum degree (ng +nn -1)/2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes g and 7-t with ng +nn -1 = d for general d, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS certificates for all graph classes g and 7-t with kg = kn= 1 and ng +nn & LE;15.& COPY;2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

关键词： Vizing's conjecture Grobner basis Algebraic model sum-of-squares programming Semidefinite programming

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