Approximately symbolic models for a class of continuous-time nonlinear systems

Discrete abstractions have become a standard approach to assist control synthesis under complex specifications. Most techniques for the construction of discrete abstractions are based on sampling of both the state and time spaces, which may not be able to guarantee safety for continuous-time systems. In this work, we aim at addressing this problem by considering only state-space abstraction. Firstly, we connect the continuous-time concrete system with its discrete (state-space) abstraction with a control interface. Then, a novel stability notion called controlled globally asymptotic/practical stability with respect to a set is proposed. It is shown that every system, under the condition that there exists an admissible control interface such that the augmented system (composed of the concrete system and its abstraction) can be made controlled globally practically stable with respect to the given set, is approximately simulated by its discrete abstraction. The effectiveness of the proposed results is illustrated by a simulation example.Comment: Accepted by the 58th IEEE Conference on Decision and Control, Nic

Approximately bisimilar symbolic models for nonlinear control systems

Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suitable to describe software and hardware interfacing the physical world. For this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each "symbol" corresponds to an "aggregate" of states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provide a unified language to study problems of control in which software and hardware interact with the physical world. Furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to a symbolic model. The approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired. Furthermore if the state space of the control system is bounded the obtained symbolic model is finite. For digital control systems, and under the stronger assumption of incremental input-to-state stability, symbolic models can be constructed through a suitable quantization of the inputs.Comment: Corrected typo

Symbolic models for nonlinear control systems without stability assumptions

Finite-state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications. Most techniques for the construction of such symbolic models have two main drawbacks: either they can only be applied to restrictive classes of systems, or they require the exact computation of reachable sets. In this paper, we propose a new abstraction technique that is applicable to any smooth control system as long as we are only interested in its behavior in a compact set. Moreover, the exact computation of reachable sets is not required. The effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle.Comment: 11 pages, 2 figures, journa