Gunther Reissig

Gunther Reissig, Alexander Weber, Matthias Rungger

We present an abstraction and refinement methodology for the automated controller synthesis to enforce general predefined specifications. The designed controllers require quantized (or symbolic) state information only and can be interfaced with the system via a static quantizer. Both features are particularly important with regard to any practical implementation of the designed controllers and, as we prove, are characterized by the existence of a feedback refinement relation between plant and abstraction. Feedback refinement relations are a novel concept introduced in this paper. Our work builds on a general notion of system with set-valued dynamics and possibly non-deterministic quantizers to permit the synthesis of controllers that robustly, and provably, enforce the specification in the presence of various types of uncertainties and disturbances. We identify a class of abstractions that is canonical in a well-defined sense, and provide a method to efficiently compute canonical abstractions. We demonstrate the practicality of our approach on two examples.

Alexander Weber, Matthias Rungger, Gunther Reissig

The practical impact of abstraction-based controller synthesis methods is currently limited by the immense computational effort for obtaining abstractions. In this note we focus on a recently proposed method to compute abstractions whose state space is a cover of the state space of the plant by congruent hyper-intervals. The problem of how to choose the size of the hyper-intervals so as to obtain computable and useful abstractions is unsolved. This note provides a twofold contribution towards a solution. Firstly, we present a functional to predict the computational effort for the abstraction to be computed. Secondly, we propose a method for choosing the aspect ratio of the hyper-intervals when their volume is fixed. More precisely, we propose to choose the aspect ratio so as to minimize a predicted number of transitions of the abstraction to be computed, in order to reduce the computational effort. To this end, we derive a functional to predict the number of transitions in dependence of the aspect ratio. The functional is to be minimized subject to suitable constraints. We characterize the unique solvability of the respective optimization problem and prove that it transforms, under appropriate assumptions, into an equivalent convex problem with strictly convex objective. The latter problem can then be globally solved using standard numerical methods. We demonstrate our approach on an example.

Gunther Reissig, Matthias Rungger

We present novel results on the solution of a class of leavable, undiscounted optimal control problems in the minimax sense for nonlinear, continuous-state, discrete-time plants. The problem class includes entry-(exit-)time problems as well as minimum time, pursuit-evasion and reach-avoid games as special cases. We utilize auxiliary optimal control problems (`abstractions') to compute both upper bounds of the value function, i.e., of the achievable closed-loop performance, and symbolic feedback controllers realizing those bounds. The abstractions are obtained from discretizing the problem data, and we prove that the computed bounds and the performance of the symbolic controllers converge to the value function as the discretization parameters approach zero. In particular, if the optimal control problem is solvable on some compact subset of the state space, and if the discretization parameters are sufficiently small, then we obtain a symbolic feedback controller solving the problem on that subset. These results do not assume the continuity of the value function or any problem data, and they fully apply in the presence of hard state and control constraints.