Bisimilar control ane systems

The fundamental notion of bisimulation equivalence for concurrent processes, has escaped the world of continuous, and subsequently, hybrid systems. Inspired by the categorical framework of Joyal, Nielsen and Winskel, we develop novel notions of bisimulation equivalence for dynamical systems as well as control systems. We prove that these notions can be captured by the abstract notion of bisimulation as developed by Joyal, Nielsen and Winskel. This is the first unified notion of system equivalence that transcends discrete and continuous systems. Furthermore, this enables the development of a novel and natural notion of bisimulation for hybrid systems, which is the final goal of this paper.

Electronic Notes in Theoretical …, 2003

In this paper we propose a new equivalence relation for dynamical and control systems called bisimulation. As the name implies this definition is inspired by the fundamental notion of bisimulation introduced by R. Milner for labeled transition systems. It is however, more subtle than its namesake in concurrency theory, mainly due to the fact that here, one deals with relations on manifolds. We further show that the bisimulation relations for dynamical and control systems defined in this paper are captured by the notion of abstract bisimulation of Joyal, Nielsen and Winskel (JNW). This result not only shows that our equivalence notion is on the right track, but also confirms that the abstract bisimulation of JNW is general enough to capture equivalence notions in the domain of continuous systems. We believe that the unification of the bisimulation relation for labeled transition systems and dynamical systems under the umbrella of abstract bisimulation, as achieved in this work, is a first step towards a unified approach to modeling of and reasoning about the dynamics of discrete and continuous structures in computer science and control theory.

Theoretical Computer Science, 2005

The fundamental notion of bisimulation equivalence for concurrent processes, has escaped the world of continuous, and subsequently, hybrid systems. Inspired by the categorical framework of Joyal, Nielsen and Winskel, we develop novel notions of bisimulation equivalence for dynamical systems as well as control systems. We prove that this notion can be captured by the abstract notion of bisimulation as developed by Joyal, Nielsen and Winskel. This is the first unified notion of system equivalence that transcends discrete and continuous systems. Furthermore, this enables the development of a novel and natural notion of bisimulation for hybrid systems, which is the final goal of this paper. This completes our program of unifying bisimulation notions for discrete, continuous and hybrid systems.

Automatica, 2003

The notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems. Bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics, and, in particular, reachability. In this paper, the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems. Given a discrete-time or continuous-time linear system, and a ÿnite observation map, we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system. Interestingly, the characterizations for discrete-time systems are more restrictive than for continuous-time systems, due to the existence of an atomic time step. We show that computing the coarsest bisimulation, which results in maximum complexity reduction, corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map. These results establish strong connections between complexity reduction concepts in control theory and computer science. ?

arXiv: Dynamical Systems, 2015

In this paper the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterization of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialized to the case where the matrix pencil $sE - A$ is regular. Furthermore, by developing a one-sided version of bisimulation, characterizations of simulation and abstraction are obtained.

2005

The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunov-like inequalities, which for deterministic systems can be computed using recent sum-of-squares techniques. Our framework is illustrated on a safety verification example.

2005

The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunov-like inequalities, which for deterministic systems can be computed using recent sum-of-squares techniques. Our framework is illustrated on a safety verification example.

Proceedings of the 44th IEEE Conference on Decision and Control, 2005

Given a plant system and a desired system, we study conditions for which there exists a controller that interconnected with the plant, yields a system that is bisimilar to the desired system. Some sufficient and some necessary conditions are provided in the general case of (non-deterministic) abstract state systems and stronger results are obtained for the special classes of autonomous abstract state systems, finite abstract state systems, and non-deterministic linear dynamical systems.

2004

A general notion of hybrid bisimulation is proposed and related to the notions of algebraic, state-space and input-output equivalences for the class of switching linear systems. An algebraic characterization of hybrid bisimulations and a procedure converging in a finite number of steps to the maximal hybrid bisimulation are derived. Bisimulation-based reduction and simulation-based abstraction are defined and characterized. Connections with observability are investigated.

Automatica, 2008

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.