Simulation and Bisimulation of Nonlinear Control Systems with Admissible Classes of Inputs and Disturbances

Grasse, Kevin A.

doi:10.1137/050638072

Cited by 19 publications

(29 citation statements)

References 13 publications

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“…Some steps in this direction have been taken; see for example [2], [10], and [12]. The problem of achievability for general systems has been addressed in [9].…”

Section: Discussionmentioning

confidence: 99%

“…Let and be two systems of the form given in (2). A subspace is a bisimulation relation with respect to and if and only if the following hold true: (4) There is also a one-sided notion of bisimulation called simulation.…”

Section: Proposition 2: [12 Theorem 210]mentioning

confidence: 99%

“…We first define a bisimulation relation as introduced in [6] and [12]. Consider two dynamical systems described by the following equations: (2) where , , , , and . Here, is the state of the system ; , are inputs, while and are outputs that take values in the finite dimensional real vector spaces , , , and , respectively.…”

Section: Definitionsmentioning

confidence: 99%

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The Achievable Dynamics via Control by Interconnection

Vinjamoor¹,

Schaft²

2011

IEEE Trans. Automat. Contr.

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Abstract-We consider the problem of finding a controller such that, when interconnected to the plant, we obtain a system that is equivalent to a desired system. Here, "equivalence" is formalized as "bisimilarity." We give necessary and sufficient conditions for the existence of such a controller. The systems we consider are linear input-state-output systems. A comparison is made to previously obtained results about achievable/implementable behaviors in the behavioral approach to systems theory. Among the advantages of using the notion of bisimilarity is the fact that it directly applies to state-space systems, while the computations involved are operations on constant matrices.Index Terms-Achievability, bisimulations, canonical controller, interconnection, linear systems. I. MOTIVATIONA BASIC question in systems and control theory is the following: Given a plant system, by constructing another dynamical system (called a controller) and interconnecting this to the plant, what are the possibilities of modifying its input-output behavior? Before we begin addressing this question, we first explain and motivate the setting we shall work with.We consider plant systems with two types of inputs and and two types of outputs and ; see Fig. 1. The first type of input together with the first type of output describes the interaction of the system with its environment and can be used for performance specifications (for example, as in control). We call the pair the manifest variables of the system. The second type of input and the second type of output are variables that are to be connected to the controller system. Hence, we call the pair the control variables of the system as depicted in Fig. 1. Now suppose we are given a plant together with another system called the desired system (or specification) , having the same set of manifest variables . The aim is to design a controller , if it exists, so that when we attach the controller to the plant , this controlled system behaves exactly like the desired system . A. More General InterconnectionsThe allowed controller interconnections that we consider are more general than the ones usually seen in controller design Manuscript received July 23, 2009; revised March 05, 2010; June 29, 2010; July 19, 2010; and July 29, 2010; accepted August 17, 2010. Date of publication September 02, 2010; date of current version May 11, 2011. Recommended by Associate Editor A. Astolfi.The authors are with the Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen 9700 AK, The Netherlands (e-mail: h.g.vinjamoor@rug.nl; a.j.van.der.schaft@rug.nl).Digital Object Identifier 10.1109/TAC.2010.2072650 techniques and depicted in Fig. 1. We first motivate our more general controller interconnection and then state precisely what type of interconnections we allow in the next section. Usually, control theory deals with feedback controllers, i.e., controllers that accept the output of the plant as their input and produce an output that acts as an input to the plant...

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“…Some steps in this direction have been taken; see for example [2], [10], and [12]. The problem of achievability for general systems has been addressed in [9].…”

Section: Discussionmentioning

confidence: 99%

Section: Proposition 2: [12 Theorem 210]mentioning

confidence: 99%

See 1 more Smart Citation

The Achievable Dynamics via Control by Interconnection

Vinjamoor¹,

Schaft²

2011

IEEE Trans. Automat. Contr.

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“…Recently, the same notion was shown to be relevant for continuous [vdS04,TP04,Gra07], switched [PvdSdB06], hybrid [HTP05] and abstract state systems [PvdSB05]. What makes bisimulation appealing it the possibility of rendering systems of different "sizes" equivalent.…”

Section: Introductionmentioning

confidence: 99%

Controller synthesis for bisimulation equivalence

Tabuada¹

2008

Systems & Control Letters

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Abstract. The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discrete-event systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that the arguments underlying proofs of existence and methods for the construction of controllers are extraneous to the particular class of systems being considered and thus can be presented in greater generality.

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“…Reduction of continuous control systems to continuous control systems with lower dimensional state space has been addressed in [vdS04,Gra07,TP04,PvdSB06]; (ii) Quantized control systems: finite abstractions have been studied in [BMP02,BMP06] for certain classes of control systems with quantized inputs;…”

Section: Introductionmentioning

confidence: 99%

Approximately bisimilar symbolic models for nonlinear control systems

2008

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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.

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Simulation and Bisimulation of Nonlinear Control Systems with Admissible Classes of Inputs and Disturbances

Cited by 19 publications

References 13 publications

The Achievable Dynamics via Control by Interconnection

The Achievable Dynamics via Control by Interconnection

Controller synthesis for bisimulation equivalence

Approximately bisimilar symbolic models for nonlinear control systems

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