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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Abstract-We consider here the problem of finding a controller such that when interconnected to the plant, we obtain a system which is asymptotically equivalent to a desired system. Here 'asymptotic equivalence' is formalized as 'asymptotic bisimilarity'. Intuitively speaking, two systems are asymptotically bisimilar if the difference between their outputs decays to zero with time. We give necessary and sufficient conditions for the existence of such a controller. These conditions can be verified computationally using standard algorithms in linear geometric control. The systems we consider are linear time invariant input-state-output systems.