Problem 1.10

Regular feedback implementability of linear differential behaviors

H. L. Trentelman

Mathematics Institute

University of Groningen

P.O. Box 800, 9700 AV Groningen

The Netherlands

[email protected]

1 INTRODUCTION

In this short paper, we want to discuss an open problem that appears in the context of interconnection of systems in a behavioral framework. Given a system behavior, playing the role of plant to be controlled, the problem is to characterize all system behaviors that can be achieved by interconnecting the plant behavior with a controller behavior, where the interconnection should be a regular feedback interconnection.

More specifically, we will deal with linear time-invariant differential systems, i.e., dynamical systems given as a triple where is the time-axis, and where called the behavior of the system is equal to the set of all solutions of a set of higher order, linear, constant coefficient, differential equations. More precisely,

The first three of these conditions state that, in the interconnection of an along the terminals of the interconnected system one can identify a signal flow that is compatible with the signal flow diagram of a feedback configuration with proper transfer matrices. The fourth condition states that this feedback interconnection is “well-posed.” The equivalence of the property of being a regular feedback interconnection with these four conditions was studied for the “full interconnection case” in [8] and [2].

2 STATEMENT OF THE PROBLEM

Effectively, a characterization of all such behaviors gives a characterization of the “limits of performance” of the given plant under regular feedback control.

3 BACKGROUND

Our open problem is to find conditions for a given to be implementable by regular feedback. An obvious necessary condition for this is that is implementable, i.e., it can be achieved by interconnecting the plant with a controller by (just any) interconnection through the interconnection variable c. Necessary and sufficient conditions for implementability have been obtained in [7]. These conditions are formulated in terms of two behaviors derived from the full plant behavior

P and N are both in and are called the manifest plant behavior and hidden behavior associated with the full plant behavior respectively. In [7] it has been shown that is implementable if and only if

i.e., contains , and is contained in This elegant characterization of the set of implementable behaviors still holds true if, instead of (ordinary) linear differential system behaviors, we deal with nD linear system behaviors, which are system behaviors that can be represented by partial differential equations of the form

with a polynomial matrix in n indeterminates. Recently, in [6] a variation of condition (2) was shown to be sufficient for implementability of system behaviors in a more general (including nonlinear) context.

For a system behavior to be implementable by regular feedback, another necessary condition is of course that is regularly implementable, i.e., it can be achieved by interconnecting the plant with a controller by regular interconnection through the interconnection variable c. Also for regular implementability necessary and sufficient conditions can already be found in the literature. In [1] it has been shown that a given is regularly implementable if and only if, in addition to condition (2), the following condition holds:

case = 0 (which is equivalent to the “full interconnection case”), conditions (2) and (3) for regular implementability in the context of nD system behaviors can also be found in [4]. In the same context, results on regular implementability can also be found in [9].

We finally note that, again for the full interconnection case, the open problem stated in this paper has recently been studied in [3], using a somewhat different notion of linear system behavior, in discrete time. Up to now, however, the general problem has remained unsolved.

BIBLIOGRAPHY

[1] M. N. Belur and H. L. Trentelman, “Stabilization, pole placement and regular implementability, ” IEEE Transactions on Automatic Control, May 2002.

[2] M. Kuijper, “Why do stabilizing controllers stabilize ?” Automatica, vol. 31, pp. 621-625, 1995.

[3] V. Lomadze, On interconnections and control, manuscript, 2001.

[4] P. Rocha and J. Wood, “Trajectory control and interconnection of nD systems, ” SIAM Journal on Contr. and Opt., vol. 40, no 1, pp. 107-134, 2001.

[5] J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory: A Behavioral Approach, Springer Verlag, 1997.

[6] A. J. van der Schaft, Achievable behavior of general systems, manuscript, submitted for publication, 2002.

[7] J. C. Willems and H. L. Trentelman, “Synthesis of dissipative systems using quadratic differential forms, Part 1, ” IEEE Transactions on Automatic Control, vol. 47, no. 1, pp. 53-69, 2002.

[8] J. C. Willems, “On Interconnections, Control and Feedback, ” IEEE Transactions on Automatic Control, vol. 42, pp. 326-337, 1997.

[9] E. Zerz and V. Lomadze, “A constructive solution to interconnection and decomposition problems with multidimensional behaviors, ” SIAM Journal on Contr. and Opt., vol. 40, no 4, pp. 1072-1086, 2001.