Permissive Barrier Certificates for Safe Stabilization Using Sum-of-squares

Motivated by the need to simultaneously guarantee safety and stability of safety-critical dynamical systems, we construct permissive barrier certificates in this paper that explicitly maximize the region where the system can be stabilized without violating safety constraints. An iterative search algorithm is developed to search for the maximum volume barrier certified region of safe stabilization. The barrier certified region, which is allowed to take any arbitrary shape, is proved to be strictly larger than safe regions generated with Lyapunov sublevel set based methods. The proposed approach effectively unites a Lyapunov function with multiple barrier functions that might not be compatible with each other. Simulation results of the iterative search algorithm demonstrate the effectiveness of the proposed method.

[1]  Francis Eng Hock Tay,et al.  Barrier Lyapunov Functions for the control of output-constrained nonlinear systems , 2009, Autom..

[2]  James Anderson,et al.  Region of attraction analysis via invariant sets , 2014, 2014 American Control Conference.

[3]  A. Papachristodoulou,et al.  On the construction of Lyapunov functions using the sum of squares decomposition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[4]  Bayu Jayawardhana,et al.  Uniting Control Lyapunov and Control Barrier Functions , 2014, 53rd IEEE Conference on Decision and Control.

[5]  Bayu Jayawardhana,et al.  Stabilization with guaranteed safety using Control Lyapunov-Barrier Function , 2016, Autom..

[6]  Li Wang,et al.  Safe Learning of Quadrotor Dynamics Using Barrier Certificates , 2017, 2018 IEEE International Conference on Robotics and Automation (ICRA).

[7]  Magnus Egerstedt,et al.  Safe certificate-based maneuvers for teams of quadrotors using differential flatness , 2017, 2017 IEEE International Conference on Robotics and Automation (ICRA).

[8]  Matthias Althoff,et al.  Estimating the domain of attraction based on the invariance principle , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[9]  Paulo Tabuada,et al.  Correctness Guarantees for the Composition of Lane Keeping and Adaptive Cruise Control , 2016, IEEE Transactions on Automation Science and Engineering.

[10]  Koushil Sreenath,et al.  Exponential Control Barrier Functions for enforcing high relative-degree safety-critical constraints , 2016, 2016 American Control Conference (ACC).

[11]  Amir Ali Ahmadi,et al.  Control design along trajectories with sums of squares programming , 2012, 2013 IEEE International Conference on Robotics and Automation.

[12]  Xiangru Xu.  Control sharing barrier functions with application to constrained control , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[13]  Yeung Sam Hung,et al.  Analysis and Synthesis of Nonlinear Systems With Uncertain Initial Conditions , 2008, IEEE Transactions on Automatic Control.

[14]  George J. Pappas,et al.  A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates , 2007, IEEE Transactions on Automatic Control.

[15]  Didier Henrion,et al.  Convex Computation of the Region of Attraction of Polynomial Control Systems , 2012, IEEE Transactions on Automatic Control.

[16]  G. Chesi.  Domain of Attraction: Analysis and Control via SOS Programming , 2011 .

[17]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[18]  Rafael Wisniewski,et al.  Compositional safety analysis using barrier certificates , 2012, HSCC '12.

[19]  Li Wang,et al.  Multi-objective compositions for collision-free connectivity maintenance in teams of mobile robots , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[20]  Graziano Chesi,et al.  Rational Lyapunov functions for estimating and controlling the robust domain of attraction , 2013, Autom..

[21]  B. Tibken.  Estimation of the domain of attraction for polynomial systems via LMIs , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[22]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[23]  Aaron D. Ames,et al.  Control barrier function based quadratic programs with application to bipedal robotic walking , 2015, 2015 American Control Conference (ACC).

[24]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[25]  Paulo Tabuada,et al.  Control barrier function based quadratic programs with application to adaptive cruise control , 2014, 53rd IEEE Conference on Decision and Control.