Decomposition of Reachable Sets and Tubes for a Class of Nonlinear Systems

• Published in: IEEE transactions on automatic control Vol. 63; no. 11; pp. 3675 - 3688
• Main Authors: Chen, Mo, Herbert, Sylvia L., Vashishtha, Mahesh S., Bansal, Somil, Tomlin, Claire J.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Autonomous systems; Computation; computational efficiency; Control systems; control theory; Decomposition; Disturbances; Dynamical systems; Electron tubes; Nonlinear analysis; Nonlinear control; Nonlinear control systems; Nonlinear systems; Polytopes; reachability analysis; Safety; scalability; System dynamics; system verification; Trajectory; Tubes; Vehicle dynamics
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2797194

Reachability analysis provides formal guarantees for performance and safety properties of nonlinear control systems. Here, one aims to compute the backward reachable set (BRS) or tube (BRT)-the set of states from which the system can be driven into a target set at a particular time or within a time interval, respectively. The computational complexity of current approaches scales poorly, making application to high-dimensional systems intractable. We propose a technique that decomposes the dynamics of a general class of nonlinear systems into subsystems which may be coupled through common states, controls, and disturbances. Despite this coupling, BRSs and BRTs can be computed efficiently using our technique without incurring additional approximation errors and without the need for linearizing dynamics or approximating sets as polytopes. Computations of BRSs and BRTs now become orders of magnitude faster, and for the first time BRSs and BRTs for many high-dimensional nonlinear control systems can be computed using the Hamilton-Jacobi formulation. In situations involving bounded adversarial disturbances, our proposed method can obtain slightly conservative results. We demonstrate our theory by numerically computing BRSs and BRTs for several systems, including the six-dimensional Acrobatic Quadrotor using the HJ formulation.