Infinite-dimensional Sylvester equations: basic theory and application to observer design

• Autorzy: Emirsajłow Z.
• Języki publikacji: EN

Abstrakty

EN

This paper develops a mathematical framework for the infinite-dimensional Sylvester equation both in the differential and the algebraic form. It uses the implemented semigroup concept as the main mathematical tool. This concept may be found in the literature on evolution equations occurring in mathematics and physics and is rather unknown in systems and control theories. But it is just systems and control theory where Sylvester equations widely appear, and for this reason we intend to give a mathematically rigorous introduction to the subject which is tailored to researchers and postgraduate students working on systems and control. This goal motivates the assumptions under which the results are developed. As an important example of applications we study the problem of designing an asymptotic state observer for a linear infinite dimensional control system with a bounded input operator and an unbounded output operator.

Słowa kluczowe

EN

infinite dimensional Sylvester equation; implemented semigroup; state observer design

PL

równanie nieskończenie wymiarowe; półgrupa złożona; obserwator stanu

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2012
• Tom: Vol. 22, no. 2
• Strony: 245--257
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Emirsajłow Z.

• Department of Control and Measurements, West Pomeranian University of Technology, Gen. Sikorskiego 37, 70-313 Szczecin, Poland,

Bibliografia

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• [2] Arendt, W., Raebiger, F. and Sourour, A. (1994). Spectral properties of the operator equation ax + xb = y, Quarterly Journal of Mathematics 45(2): 133-149.
• [3] Chen, C.-T. (1984). Linear Systems Theory and Design, Holt, Rinehart and Winston, New York, NY.
• [4] Emirsajłow, Z. (2005). Implemented Semigroup in Analysis of Infinite-Dimensional Sylvester and Lyapunov equations, Technical University of Szczecin Press, Szczecin, (in Polish).
• [5] Emirsajłow, Z. and Townley, S. (2000). From PDEs with a boundary control to the abstract state equation with an unbounded input operator, European Journal of Control 7(1): 1-23.
• [6] Emirsajłow, Z. and Townley, S. (2005). On application of the implemented semigroup to a problem arising in optimal control, International Journal of Control 78(4): 298-310.
• [7] Engel, J. and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, NY.
• [8] Gajic, Z. and Qureshi, M. (2000). Lyapunov Matrix Equation in Systems Stability and Control, Academic Press, San Diego, CA.
• [9] Kuehnemund, F. (2001). Bi-continuous Semigroups on Spaces with Two Topologies: Theory and Applications, Ph.D. thesis, Eberhard-Karls-Univerität Tuebingen, Tuebingen.
• [10] Phong, V. Q. (1991). The operator equation ax − xb = c with unbounded operators a and b and related Cauchy problems, Mathematische Zeitschrift 208(7): 567-588.