Peak Estimation and Recovery with Occupation Measures

Peak Estimation aims to find the maximum value of a state function achieved by a dynamical system. This problem is non-convex when considering standard Barrier and Density methods for invariant sets, and has been treated heuristically by using auxiliary functions. A convex formulation based on occupation measures is proposed in this paper to solve peak estimation. This method is dual to the auxiliary function approach. Our method will converge to the optimal solution and can recover trajectories even from approximate solutions. This framework is extended to safety analysis by maximizing the minimum of a set of costs along trajectories.

[1]  Ali Jadbabaie,et al.  Safety Verification of Hybrid Systems Using Barrier Certificates , 2004, HSCC.

[2]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[3]  Matteo Tacchi,et al.  Transient Stability Analysis of Power Systems via Occupation Measures , 2018, 2019 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT).

[4]  Umberto Bortolozzo,et al.  Rogue waves and their generating mechanisms in different physical contexts , 2013 .

[5]  Colin Neil Jones,et al.  Convex Computation of the Maximum Controlled Invariant Set For Polynomial Control Systems , 2013, SIAM J. Control. Optim..

[6]  Didier Henrion,et al.  GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi , 2003, TOMS.

[7]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[8]  Graziano Chesi,et al.  On the Computation of the Peak of the Impulse Response of LTI Systems , 2019, Proceedings of the 2019 2nd International Conference on Information Science and Systems.

[9]  Thorsten Theobald,et al.  Exploiting Symmetries in SDP-Relaxations for Polynomial Optimization , 2011, Math. Oper. Res..

[10]  J. Lasserre,et al.  Detecting global optimality and extracting solutions in GloptiPoly , 2003 .

[11]  Giovanni Fantuzzi,et al.  Bounding Extreme Events in Nonlinear Dynamics Using Convex Optimization , 2019, SIAM J. Appl. Dyn. Syst..

[12]  Victor M. Preciado,et al.  Safety Verification of Nonlinear Polynomial System via Occupation Measures , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

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

[14]  A. Rantzer,et al.  On Analysis and Synthesis of Safe Control Laws , 2004 .

[15]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[16]  Emmanuel Trélat,et al.  Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations , 2007, SIAM J. Control. Optim..

[17]  Guy A. Dumont,et al.  How control theory can help us control Covid-19 , 2020, IEEE Spectrum.

[18]  Johan Löfberg,et al.  Pre- and Post-Processing Sum-of-Squares Programs in Practice , 2009, IEEE Transactions on Automatic Control.

[19]  Sanjoy K. Mitter,et al.  Convex Optimization in Infinite Dimensional Spaces , 2008, Recent Advances in Learning and Control.

[20]  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).