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Safety Verification and Controller Synthesis for Systems with Input Constraints

IFAC-PapersOnLine 2023年第2期56卷 1698-1703页

作者： Han Wang Kostas Margellos Antonis Papachristodoulou

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.

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

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Determining input-to-state and incremental input-to-state stability of nonpolynomial systems

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 2020年第12期30卷 4676-4689页

作者： Vosswinkel, Rick Roebenack, Klaus Tech Univ Dresden Inst Control Theory Fac Elect & Comp Engn D-01062 Dresden Germany Leipzig Univ Appl Sci HTWK Fac Elect Engn & Informat Technol Leipzig Germany

In this study, we propose constructive ways to determine input-to-state stability (ISS) as well as incremental ISS (delta ISS) of nonpolynomial dynamical systems. The developed procedures are based on sums-of-square decomposition. This tool is only applicable to polynomial systems. Thus, a rational recast of the nonpolynomial system description is used. This recast generally leads to an increased system order and additional constraints. These constraints must be respected in the resulting formulations. The proposed approach gives a unique and constructive procedure to determine the ISS and the delta ISS property, which is normally nontrivial and needs a good understanding of the system's dynamics. The proposed approaches are illustrated on several examples.

关键词： nonlinear feedback control input-to-state stability Lyapunov methods 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

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

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第34届中国控制与决策会议

作者： Hejun Huang Dongkun Han Department of Mechanical and Automation Engineering The Chinese University of Hong Kong

Estimating the region of attraction for partially unknown nonlinear systems is a challenging *** this paper,we propose a tractable method to generate an estimated region of attraction with probability bounds,by searching an optimal polynomial barrier *** interpolants,Gaussian processes and sum-of-squares programmings are used in this *** 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 ***,probabilistic conditions are given such that all the estimates are located in certain probability *** 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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Safety verification for distributed parameter systems using barrier functionals

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SYSTEMS & CONTROL LETTERS 2017年 108卷 33-39页

作者： Ahmadi, Mohamadreza Valmorbida, Giorgio Papachristodoulou, Antonis Univ Texas Austin Inst Computat Engn & Sci ICES Peter ODonnel Jr Bldg201 E 24th St Austin TX 78712 USA Univ Paris Saclay Univ Paris Sud CNRS Lab Signaux & SystCentraleSupelec 3 Rue Joliot Curie F-91192 Gif Sur Yvette France Univ Oxford Dept Engn Sci Parks Rd Oxford OX1 3PJ England

We study the safety verification problem for a class of distributed parameter systems described by partial differential equations (PDEs), i.e., the problem of checking whether the solutions of the PDE satisfy a set of constraints at a particular point in time. The proposed method is based on an extension of barrier certificates to infinite-dimensional systems. In this respect, we introduce barrier functionals, which are functionals of the dependent and independent variables. Given a set of initial conditions and an unsafe set, we demonstrate that if such a functional exists satisfying two (integral) inequalities, then the solutions of the system do not enter the unsafe set. Therefore, the proposed method does not require finite dimensional approximations of the distributed parameter system. Furthermore, for PDEs with polynomial data, we solve the associated integral inequalities using semi-definite programming (SDP) based on a method that relies on a quadratic representation of the integrands of integral inequalities. The proposed method is illustrated through examples. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Safety verification Barrier certificates sum-of-squares programming Distributed parameter systems

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NONLINEAR ANALYSIS OF VEHICLE CONTROL ACTUATIONS BASED ON CONTROLLED INVARIANT SETS

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INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND COMPUTER SCIENCE 2016年第1期26卷 31-43页

作者： Nemeth, Balazs Gaspar, Peter Peni, Tamas Hungarian Acad Sci Inst Comp Sci & Control Syst & Control Lab Kende Utca 13-17 H-1111 Budapest Hungary

In the paper, an analysis method is applied to the lateral stabilization problem of vehicle systems. The aim is to find the largest state-space region in which the lateral stability of the vehicle can be guaranteed by the peak-bounded control input. In the analysis, the nonlinear polynomial sum-of-squares programming method is applied. A practical computation technique is developed to calculate the maximum controlled invariant set of the system. The method calculates the maximum controlled invariant sets of the steering and braking control systems at various velocities and road conditions. Illustration examples show that, depending on the environments, different vehicle dynamic regions can be reached and stabilized by these controllers. The results can be applied to the theoretical basis of their interventions into the vehicle control system.

关键词： vehicle dynamics sum-of-squares programming Lyapunov method

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On the performance of nonlinear dynamical systems under parameter perturbation

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AUTOMATICA 2016年 63卷 265-273页

作者： Raman, Dhruva V. Anderson, James Papachristodoulou, Antonis Univ Oxford Dept Engn Sci Oxford OX1 3PJ England

We present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear ODEs, as initial conditions and parameters are varied within compact sets over which stability is guaranteed. This deviation is taken to be the integral over time of a user-specified, positive definite function of the difference between the trajectories, for instance the 2 norm. We use sum-of-squares programming to obtain two polynomials, which take as inputs the (possibly differing) initial conditions and parameters of the two families of ODEs, and output upper and lower bounds to this transient deviation. Equality can be achieved using symbolic methods in a special case involving Linear Time Invariant Parameter Dependent systems. We demonstrate the utility of the proposed methods in the problems of model discrimination, and location of worst case parameter perturbation for a single parameterised family of ODE models. (C) 2015 The Authors. Published by Elsevier Ltd.

关键词： Analysis of systems with uncertainties Application of nonlinear analysis and design sum-of-squares programming

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Dissipation inequalities for the analysis of a class of PDEs

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AUTOMATICA 2016年 66卷 163-171页

作者： Ahmadi, Mohamadreza Valmorbida, Giorgio Papachristodoulou, Antonis Univ Oxford Dept Engn Sci Parks Rd Oxford OX1 3PJ England

In this paper, we develop dissipation inequalities for a class of well-posed systems described by partial differential equations (PDEs). We study passivity, reachability, induced input-output norm boundedness, and input-to-state stability (ISS). We consider both cases of in-domain and boundary inputs and outputs. We study the interconnection of PDE-PDE systems and formulate small gain conditions for stability. For PDEs polynomial in dependent and independent variables, we demonstrate that sum-of-squares (SOS) programming can be used to compute certificates for each property. Therefore, the solution to the proposed dissipation inequalities can be obtained via semi-definite programming. The results are illustrated with examples. (C) 2016 The Authors. Published by Elsevier Ltd.

关键词： Distributed parameter systems Convex optimization sum-of-squares programming Dissipation inequalities Interconnected systems

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Deductive Proofs of Almost Sure Persistence and Recurrence Properties 22nd

Deductive Proofs of Almost Sure Persistence and Recurrence P...

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22nd International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS) held as Part of the European Joint Conferences on Theory and Practice of Software (ETAPS)

作者： Chakarov, Aleksandar Voronin, Yuen-Lam Sankaranarayanan, Sriram Univ Colorado Boulder Boulder CO 80303 USA

ISBN: (纸本)9783662496749;9783662496732

Martingale theory yields a powerful set of tools that have recently been used to prove quantitative properties of stochastic systems such as stochastic safety and qualitative properties such as almost sure termination. In this paper, we examine proof techniques for establishing almost sure persistence and recurrence properties of infinite-state discrete time stochastic systems. A persistence property lozenge square (P) specifies that almost all executions of the stochastic system eventually reach P and stay there forever. Likewise, a recurrence property square lozenge(Q) specifies that a target set Q is visited infinitely often by almost all executions of the stochastic system. Our approach extends classic ideas on the use of Lyapunov-like functions to establish qualitative persistence and recurrence properties. Next, we extend known constraint-based invariant synthesis techniques to deduce the necessary supermartingale expressions to partly mechanize such proofs. We illustrate our techniques on a set of interesting examples.

关键词： Temporal logic Stochastic systems Markov processes Stochastic control sum-of-squares programming

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A Dissipativity Approach to Safety Verification for Interconnected Systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2015年第6期60卷 1722-1727页

作者： Coogan, Samuel Arcak, Murat Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA

We propose a computational method for verifying a state-space safety constraint of a network of interconnected dynamical systems satisfying a dissipativity property. We construct an invariant set as the sublevel set of a Lyapunov function comprised of local storage functions for each subsystem. This approach requires only knowledge of a local dissipativity property for each subsystem and the static interconnection matrix for the network, and we pose the safety verification as a sum-of-squares feasibility problem. In addition to reducing the computational burden of system design, we allow the safety constraint and initial conditions to depend on an unknown equilibrium, thus offering increased flexibility over existing techniques.

关键词： Lyapunov methods safety verification sum-of-squares programming

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Local stability analysis using simulations and sum-of-squares programming

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AUTOMATICA 2008年第10期44卷 2669-2675页

作者： Topcu, Ufuk Packard, Andrew Seiler, Peter Univ Calif Berkeley Dept Mech Engn Berkeley CA 94720 USA Honeywell Technol Ctr Minneapolis MN 55418 USA

The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations 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 is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction 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. (c) 2008 Elsevier Ltd. All rights reserved,

关键词： Local stability Region-of attraction Nonlinear dynamics sum-of-squares programming Simulations

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