Robust controller synthesis of switched systems using counterexample guided framework

Abstract

We investigate the problem of synthesizing robust controllers that ensure that the closed loop satisfies an input reach-while-stay specification, wherein all trajectories starting from some initial set I, eventually reach a specified goal set G, while staying inside a safe set S. Our plant model consists of a continuous-time switched system controlled by an external switching signal and plant disturbance inputs. The controller uses a state feedback law to control the switching signal in order to ensure that the desired correctness properties hold, regardless of the disturbance actions. Our approach uses a proof certificate in the form of a robust control Lyapunov-like function (RCLF) whose existence guarantees the reach-while-stay specification. A counterexample guided inductive synthesis (CEGIS) framework is used to find a RCLF by solving a ∃∀∃∀ formula iteratively using quantifier free SMT solvers. We compare our synthesis scheme against a common approach that fixes disturbances to nominal values and synthesizes the controller, ignoring the disturbance. We demonstrate that the latter approach fails to yield a robust controller over some benchmark examples, whereas our approach does. Finally, we consider the problem of translating the RCLF synthesized by our approach into a control implementation. We outline the series of offline and real-time computation steps needed. The synthesized controller is implemented and simulated using the Matlab(tm)/Simulink(tm) model-based design framework, and illustrated on some examples.

Learning control lyapunov functions from counterexamples and demonstrations

“…In this problem, the goal is to reach a target set T from an initial set I, while staying in a safe set S, wherein I ⊆ S. The set S is assumed to be compact. By combining the local Lyapunov function and a barrier certificate, one can define a smooth, Lyapunov-like function V , that satisfies the following conditions (see [79]):…”

“…We briefly sketch the argument as to why such a Lyapunov-like function satisfies the reach-while-stay, referring the reader to our earlier work on control certificates for a detailed proof [79]. Suppose we have found a function V satisfying (27).…”

“…The CEGIS procedure has been used for the synthesis of CLFs recently by authors [77,79], combining it with SDP solvers for verifying CLFs. The key difference here lies in the use of the demonstrator module that simplifies the learning module.…”

Formal Synthesis of Analytic Controllers for Sampled-Data Systems via Genetic Programming

“…Other related work is found in [16], in which robust CLFs for switched systems with reach-while-stay (RWS) specifications are synthesized using a counterexampleguided synthesis. However, in [16] the controller modes are pre-specified, while in our approach these modes can also be discovered automatically, eliminating the need for prior input space discretization. Furthermore, this paper extends the set of specifications to include invariance of the goal set.…”

“…Furthermore, this paper extends the set of specifications to include invariance of the goal set. Finally, similar to [15], the theoretical lower bounds on the minimum dwell-times reported in [16] are often very conservative.…”