Certifying Safety for Dynamical Systems: Sufficiency, Necessity, and Regularity

Abstract: Beyond robustness of asymptotic stability, safety is one of the most important properties to guarantee for a dynamical system. A dynamical system is considered to be safe when trajectories starting from a given set of initial conditions avoid a set of points deemed unsafe. In applications, unsafe sets correspond to values of the state associated to potential system damage, lack of controllability, or are simply values of the state that do not satisfy given specifications. A powerful approach to solve the safety problem consists of finding a function of the state that, without loss of generality, is positive on the unsafe set and nonpositive on the initial set, and is nonincreasing when evaluated along trajectories to the system. A function with such a property is said to be a safety certificate (a.k.a. barrier function). In this talk, we introduce the general safety problem and present conditions guaranteeing safety. The emphasis is on infinitesimal conditions that are both necessary and sufficient, and involve the safety certificate, the dynamics of the system, and the unsafe set. We establish connections between these conditions and forward invariance of a set. We also point out that the class of functions that are continuous and depend only on the state of the system are not rich enough to certify safety. Based on this observation, we present a converse theorem for safety using (lower semicontinuous) time-varying safety certificates. Examples illustrate the ideas and results.

Bio: Ricardo G. Sanfelice is Professor and Department Chair of Electrical and Computer Engineering, University of California at Santa Cruz, CA, USA. He received his M.S. and Ph.D. degrees in 2004 and 2007, respectively, from the University of California, Santa Barbara. During 2007 and 2008, he was a Postdoctoral Associate at the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technology and visited the Centre Automatique et Systemes at the Ecole de Mines de Paris for four months. Prof. Sanfelice is the recipient of the 2013 SIAM Control and Systems Theory Prize, the National Science Foundation CAREER award, the Air Force Young Investigator Research Award, the 2010 IEEE Control Systems Magazine Outstanding Paper Award, the 2012 STAR Higher Education Award for his contributions to STEM education, and the 2020 ACM Test-of-Time Award from the HSCC. He is Associate Editor for Automatica, a Fellow of the IEEE, and has served as Chair of the Hybrid Systems Technical Committee from the IEEE Control Systems Society. He coauthored articles selected as finalists for the Best Student Paper Award (2014, 2019, and 2022) at the American Control Conference (ACC) and the International Conference on Automation Science and Engineering (CASE). He is Director of the Cyber-Physical Systems Research Center at UCSC and Director of the Center for Information Technology Research in the Interest of Society and the Banatao Institute (CITRIS) Aviation Initiative. His research interests are in modeling, stability, robust control, observer design, and simulation of nonlinear and hybrid systems with applications to robotics, power systems, aerospace, and biology.

https://cuboulder.zoom.us/j/190280621

Certifying Safety for Dynamical Systems: Sufficiency, Necessity, and Regularity

Abstract: Beyond robustness of asymptotic stability, safety is one of the most important properties to guarantee for a dynamical system. A dynamical system is considered to be safe when trajectories starting from a given set of initial conditions avoid a set of points deemed unsafe. In applications, unsafe sets correspond to values of the state associated to potential system damage, lack of controllability, or are simply values of the state that do not satisfy given specifications. A powerful approach to solve the safety problem consists of finding a function of the state that, without loss of generality, is positive on the unsafe set and nonpositive on the initial set, and is nonincreasing when evaluated along trajectories to the system. A function with such a property is said to be a safety certificate (a.k.a. barrier function). In this talk, we introduce the general safety problem and present conditions guaranteeing safety. The emphasis is on infinitesimal conditions that are both necessary and sufficient, and involve the safety certificate, the dynamics of the system, and the unsafe set. We establish connections between these conditions and forward invariance of a set. We also point out that the class of functions that are continuous and depend only on the state of the system are not rich enough to certify safety. Based on this observation, we present a converse theorem for safety using (lower semicontinuous) time-varying safety certificates. Examples illustrate the ideas and results.

Bio: Ricardo G. Sanfelice is Professor and Department Chair of Electrical and Computer Engineering, University of California at Santa Cruz, CA, USA. He received his M.S. and Ph.D. degrees in 2004 and 2007, respectively, from the University of California, Santa Barbara. During 2007 and 2008, he was a Postdoctoral Associate at the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technology and visited the Centre Automatique et Systemes at the Ecole de Mines de Paris for four months. Prof. Sanfelice is the recipient of the 2013 SIAM Control and Systems Theory Prize, the National Science Foundation CAREER award, the Air Force Young Investigator Research Award, the 2010 IEEE Control Systems Magazine Outstanding Paper Award, the 2012 STAR Higher Education Award for his contributions to STEM education, and the 2020 ACM Test-of-Time Award from the HSCC. He is Associate Editor for Automatica, a Fellow of the IEEE, and has served as Chair of the Hybrid Systems Technical Committee from the IEEE Control Systems Society. He coauthored articles selected as finalists for the Best Student Paper Award (2014, 2019, and 2022) at the American Control Conference (ACC) and the International Conference on Automation Science and Engineering (CASE). He is Director of the Cyber-Physical Systems Research Center at UCSC and Director of the Center for Information Technology Research in the Interest of Society and the Banatao Institute (CITRIS) Aviation Initiative. His research interests are in modeling, stability, robust control, observer design, and simulation of nonlinear and hybrid systems with applications to robotics, power systems, aerospace, and biology.

https://cuboulder.zoom.us/j/190280621