Abstract
In this paper we discuss certain optimization problems which have their origin in infinite-dimensional control problems. The optimization problems are of the form:
be factored as Φ= ψ̄W, ψ an inner function and W a proper rational funtion. Problems of sensitivity minimization for plants with a rational function with a single delay in the input lead to an optimization problem of the above type. Existence of a solution to such problems is easy to prove but in general there is no unique solution. We discuss the question of uniqueness based on a criterion due to Adamjan, Arov and Krein and recent work of Sarason. For some special cases, we give a parametrization of all solutions when there is no uniqueness. These problems are equivalent ot extension problems for Hankel operators.
We discuss how the ideas and methods discussed in this paper can be used to solve problems of sensitivity minimization for certain distributed parameter systems.
This research has been supported by the Air Force Office of Scientific Research under Grant AFOSR 85–0227, Army Research Office under Contract DAAG29-84-0005
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adamjan, VM, Arov, DZ and Krein, MG [1968] Infinite Hankel Matrices and Generalized Problems of Caratheodory, Fejer and I. Schur, Functional Anal. and Applns. 2, pp. 269–281.
Desoer, CA and Vidyasagar [1975] Feedback Systems: Input-Output Properties, Academic Press, New York.
Fagnani, F [1986] Problemi di Minimizzazione in H∞ Per Sistemi Dinamici in Dimensione Infinita, Thesis, Laurea, University of Pisa.
Flamm, DS [1985] H∞-Optimal Sensitivity for Delay Systems, Ph.D. Thesis Proposal, Dept. of Elec. Eng. and Computer Science, M.I.T., Cambridge, MA.
Flamm, DS [1986] Control of Delay Systems for Minimax Sensitivity, Ph.D. Thesis LIDS-TH-1560, M.I.T., Cambridge, MA 02139
Flamm, DS and Mitter, SK [1986] H∞-Sensitivity Minimization for Delay Systems: Part I. Technical Report LIDS-P-1605, M.I.T., Cambridge, MA, Abridged version to appear in Systems Control Letters, 1987.
Foias, A, Tannenbaum, A and Zames, G [1986] Weighted Sensitivity Minimization for Delay Systems, IEEE Trans, on Automatic Control, AC-31 (81), pp. 763–766.
Foures, Y and Segal, IE [1955] Causality and Analyticity, Trans. Am. Math. Soc., 78, pp. 385–405.
Fuhrmann, P [1981] Linear Systems and Operators in Hilbert Space, McGraw Hill, New York.
Garnett, JB [1981] Bounded Analytic Functions, Academic Press, New York.
Glover, K [1984] All Optimal Hankel-norm Approximations of Linear Multivariable Systems and their L∞-Bounds, International Journal of Control, 39, pp. 1115–1193.
Hoffman, K [1962] Banach Space of Analytic Functions, Prentice Hall, Englewood Cliffs, N.J.
Sarason, D [1967] Generalized Interpolation in H∞, Trans. Am. Math. Soc., 127, pp. 179–203.
Sarason, D [1985] Operator-theoretic aspects of the Nevanlina-Pick Interpolation Problems, in Operator and Function Theory, ed. S.C. Power, pp. 279–314.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fagnani, F., Flamm, D., Mitter, S.K. (1987). Some Min-Max Optimization Problems in Infinite-Dimensional Control Systems. In: Curtain, R.F. (eds) Modelling, Robustness and Sensitivity Reduction in Control Systems. NATO ASI Series, vol 34. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-87516-8_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-87516-8_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-87518-2
Online ISBN: 978-3-642-87516-8
eBook Packages: Springer Book Archive