Circle criterion and boundary control systems in factor form: input-output approach

• Autorzy: Grabowski P. , Callier F. M.
• Języki publikacji: EN

Abstrakty

EN

A circle criterion is obtained for a SISO Lur'e feedback control system consisting of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the transfer function belongs to H^∞(Π⁺) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be well-posed, i.e. for any initial state x₀ Є H the truncated input and output signals u_T, y_T belong to L²(0, T) for any T > 0. The technique of the proof adapts Desoer-Vidyasagar's circle criterion method (Desoer and Vidyasagar, 1975, Ch. 3, Secs. 1 and 2, pp. 37-43, Ch. 5, Sec. 2, pp. 139-142 and Ch. 6, Secs. 3 and 4, pp. 172-174), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two transmission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.

Słowa kluczowe

EN

infinite dimensional control systems; semigroups; input-output relations

PL

system nieskończenie wymiarowy; system sterowania; półgrupa

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2001
• Tom: Vol. 11, no. 6
• Strony: 1387--1403
• Opis fizyczny: Bibliogr. 14 poz., rys.

Twórcy

autor

Grabowski P.

• Institute of Automatics, Academy of Mining and Metallurgy, al. Mickiewicza 30/B1, 30-059 Kraków, Poland

autor

Callier F. M.

• University of Namur (FUNDP), Department of Mathematics, Rempart de la Vierge 8, B-5000 Namur, Belgium

Bibliografia

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• [2] Bucci F. (2000): Frequency domain stability of nonlinear feedback systems with unbounded input operator. - Dyn. Cont. Discr. Impuls. Syst., Vol. 7, No. 3, pp. 351-368.
• [3] Desoer C.A. and Vidyasagar M. (1975): Feedback Systems: Input-Output Properties. - New York: Academic Press.
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• [7] Grabowski P. (1994): The LQ controller problem: An example. - IMA J. Math. Contr. Inf., Vol. 11, No. 4, pp. 355-368.
• [8] Grabowski P. and Callier F.M. (1999): Admissible observation operators. Duality of observation and control using factorizations. - Dyn. Cont., Discr. Impuls. Syst., Vol. 6, pp. 87-119.
• [9] Grabowski P. and Callier F.M. (2000): On the circle criterion for boundary control systems in factor form: Lyapunov approach - Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 00-07, Namur, Belgium: FUNDP. Submitted to Int. Eqns. Oper. Theory.
• [10] Grabowski P. and Callier F.M. (2001): Boundary control systems in factor form: Transfer functions and input-output maps - Int. Eqns. Oper. Theory, Vol. 41, pp. 1-37.
• [11] Logemann H. (1991): Circle criterion, small-gain conditions and internal stability for infinite-dimensional systems. - Automatica, Vol. 27, No. 4, pp. 677-690.
• [12] Logemann H. and Curtain R.F. (2000): Absolute stability results for well-posed infinite-dimensional systems with low-gain integral control. - ESAIM: Contr. Optim. Calc. Var., Vol. 5, pp. 395-424.
• [13] Pazy A. (1993): Semigroups of Linear Operators and Applications to PDEs. - Berlin: Springer.
• [14] Vidyasagar M. (1993): Nonlinear Systems Analysis, 2nd Ed. - Englewood Cliffs NJ: Prentice-Hall.