A Class of Control Certificates to Ensure Reach-While-Stay for Switched Systems

Abstract

In this article, we consider the problem of synthesizing switching controllers for temporal properties through the composition of simple primitive reach-while-stay (RWS) properties. Reach-while-stay properties specify that the system states starting from an initial set I, must reach a goal (target) set G in finite time, while remaining inside a safe set S. Our approach synthesizes switched controllers that select between finitely many modes to satisfy the given RWS specification. To do so, we consider control certificates, which are Lyapunov-like functions that represent control strategies to achieve the desired specification. However, for RWS problems, a control Lyapunov-like function is often hard to synthesize in a simple polynomial form. Therefore, we combine control barrier and Lyapunov functions with an additional compatibility condition between them. Using this approach, the controller synthesis problem reduces to one of solving quantified nonlinear constrained problems that are handled using a combination of SMT solvers. The synthesis of controllers is demonstrated through a set of interesting numerical examples drawn from the related work, and compared with the state-of-theart tool SCOTS. Our evaluation suggests that our approach is computationally feasible, and adds to the growing body of formal approaches to controller synthesis.

Formal Controller Synthesis via Genetic Programming

“…memory usage. Compared to the CEGIS methods in [33,49], our method is overall slower, but is able to discover the solution structures itself, whereas e.g. for the DC-DC boost converter the authors of [33] had to iteratively add barrier functions by hand before a solution was found.…”

“…We synthesize a controller κ for specification CS 1 with the safe, initial and goal set as in [33] 55, 5.75]. Given that the initial and goal sets are relatively close to the safe set, a second-order polynomial is likely not to suffice, and therefore we bias our solutions by including a pre-specified barrier function of the form:…”

“…Taking inspiration from synthesis of switching controllers based on a CLFB (see e.g. [33], [38]), the controller is based on the candidate LBF V , In other words, a mode q is selected so that it minimizes ∇V (s), f q (s) . This is achieved by the following controller:…”

“…[28][29][30][31][32]) and inspired by the idea of counterexample-guided inductive synthesis (CEGIS) (see e.g. [33,34]), we propose a framework combining genetic programming (GP) [35] and Satisfiability Modulo Theories (SMT) solvers [36] to automatically synthesize correct-by-design controllers for hybrid systems. This is done by codesigning a controller and Lyapunov barrier-like function for temporal logic specifications combining simple safety and reachability requirements.…”

“…Other related work includes [48], which finds the controller structure using a variant of GP and verifies it using quadratic Lyapunov functions for continuous-time systems. Several studies have proposed CEGIS based on SMT solvers for the synthesis of (control) certificate functions: in [34] Lyapunov and barrier functions are synthesized for the verification of continuous-time switched systems; in [49] and [33], control Lyapunov barrier-like functions for switched nonlinear systems are synthesized; in [50], hybrid effects caused by digital implementations are considered for linear open-loop systems; and in [26] controllers are synthesized for nonlinear systems with signal temporal logic specifications. The current proposal extends the literature by considering general hybrid systems and automatically evolving the solutions structures.…”