Approximately bisimilar symbolic models for nonlinear control systems

Abstract

Abstract. 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.

Symbolic models for nonlinear control systems affected by disturbances

“…In this paper we face the problem of deriving symbolic models for nonlinear control systems affected by disturbances. The presence of disturbances requires us to replace the notion of approximate bisimulation employed in [PGT08,GPT10,PPDT10] with the notion of alternating approximate bisimulation introduced in [PT09] and inspired by Alur and coworkers' alternating bisimulation [AHKV98]. As discussed in [PT09,Tab09] this notion is a key ingredient when constructing symbolic models of systems affected by disturbances because it guarantees that control strategies synthesized on the symbolic models can be readily transferred to the original model.…”

Symbolic Design of Networked Control Systems with State Prediction

“…The stability of the controlled system with the approximate simulation is investigated with a Lyapunov-like function [5]. The approximate (bi)simulation-based abstraction has been studied for nonlinear systems [6], [7], switched linear systems [8], and time-delay systems [9].…”

Parametrization of completeness in symbolic abstraction of bounded input linear systems

“…Finite symbolic abstractions of control systems are used in algorithmic controller synthesis [7,8,9]. Since digital implementations of continuous control systems [4] have quantized and bounded input space, we consider the setting of bounded and quantized-input control systems.…”