/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

This paper presents algorithms for the automatic synthesis of real-time controllers by finding a winning strategy for certain games defined by the timed-automata of Alur and Dill. In such games, the outcome depends on the players' actions as well as on their timing. We believe that these results will pave the way for the application of program synthesis techniques to the construction of real-time embedded systems from their specifications.

On the synthesis of discrete controllers for timed systems

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.

Feedback Refinement Relations for the Synthesis of Symbolic Controllers

We introduce SCOTS a software tool for the automatic controller synthesis for nonlinear control systems based on symbolic models, also known as discrete abstractions. The tool accepts a differential equation as the description of a nonlinear control system. It uses a Lipschitz type estimate on the right-hand-side of the differential equation together with a number of discretization parameters to compute a symbolic model that is related with the original control system via a feedback refinement relation. The tool supports the computation of minimal and maximal fixed points and thus natively provides algorithms to synthesize controllers with respect to invariance and reachability specifications. The atomic propositions, which are used to formulate the specifications, are allowed to be defined in terms of finite unions and intersections of polytopes as well as ellipsoids. While the main computations are done in C++, the tool contains a Matlab interface to simulate the closed loop system and to visualize the abstract state space together with the atomic propositions. We illustrate the performance of the tool with two examples from the literature. The tool and all conducted experiments are available at www.hcs.ei.tum.de.

SCOTS: A Tool for the Synthesis of Symbolic Controllers

Mathematical Control Theory

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.

Optimized State Space Grids for Abstractions

Local characterization of strongly convex sets

Necessary and sufficient conditions for convexity and strong convexity, respectively, of sublevel sets that are defined by finitely many real-valued $C^{1,1}$-maps are presented. A novel characterization of strongly convex sets in terms of the so-called local quadratic support is proved. The results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous-time nonlinear systems to be strongly convex. An application of these conditions is a novel method to over-approximate attainable sets when strong convexity is present.

Classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems

We prove that strong structural controllability of a pair of structural matrices $(\mathcal{A},\mathcal{B})$ can be verified in time linear in $n + r + \nu$, where $\mathcal{A}$ is square, $n$ and $r$ denote the number of columns of $\mathcal{A}$ and $\mathcal{B}$, respectively, and $\nu$ is the number of non-zero entries in $(\mathcal{A},\mathcal{B})$ We also present an algorithm realizing this bound, which depends on a recent, high-level method to verify strong structural controllability and uses sparse matrix data structures Linear time complexity is actually achieved by separately storing both the structural matrix $(\mathcal{A},\mathcal{B})$ and its transpose, linking the two data structures through a third one, and a novel, efficient scheme to update all the data during the computations We illustrate the performance of our algorithm using systems of various sizes and sparsity

Alexander Weber

Papers

Feedback Refinement Relations for the Synthesis of Symbolic Controllers

Optimized State Space Grids for Abstractions

Local characterization of strongly convex sets

Classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems

A linear time algorithm to verify strong structural controllability