A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953-967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded -type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst. Control Lett. 49 (2003) 81-89; SIAM J. Control Optim. 42 (2003) 1671-1702].
Keywords: infinite-dimensional control systems, semigroups, Lyapunov functionals, circle criterion
@article{COCV_2006__12_1_169_0,
author = {Grabowski, Piotr and Callier, Frank M.},
title = {On the circle criterion for boundary control systems in factor form : {Lyapunov} stability and {Lur'e} equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {169--197},
year = {2006},
publisher = {EDP Sciences},
volume = {12},
number = {1},
doi = {10.1051/cocv:2005027},
zbl = {1105.93044},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv:2005027/}
}
TY - JOUR AU - Grabowski, Piotr AU - Callier, Frank M. TI - On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 169 EP - 197 VL - 12 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2005027/ DO - 10.1051/cocv:2005027 LA - en ID - COCV_2006__12_1_169_0 ER -
%0 Journal Article %A Grabowski, Piotr %A Callier, Frank M. %T On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 169-197 %V 12 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv:2005027/ %R 10.1051/cocv:2005027 %G en %F COCV_2006__12_1_169_0
Grabowski, Piotr; Callier, Frank M. On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur'e equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 169-197. doi: 10.1051/cocv:2005027
[1] and, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988) 837-852. | Zbl
[2] , On a generalization of the Kalman-Yacubovic lemma. Appl. Math. Optim. 31 (1995) 177-187. | Zbl
[3] , Frequency domain stability of nonlinear feedback systems with unbounded input operator. Dynam. Contin. Discrete Impuls. Syst. 7 (2000) 351-368. | Zbl
[4] and, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica 28 (1992) 757-770. | Zbl
[5] , Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Syst. 14 (2001) 299-337. | Zbl
[6] , Regular linear systems and their reciprocals: application to Riccati equations. Syst. Control Lett. 49 (2003) 81-89.
[7] , Riccati equations for stable well-posed linear systems: The generic case. SIAM J. Control Optim. 42 (2003) 1671-1702. | Zbl
[8] and, An Introduction to Infinite-Dimensional Linear Systems Theory. Heidelberg, Springer (1995). | Zbl | MR
[9] , and, Stability results of Popov-type for infinite - dimensional systems with applications to integral control. Proc. London Math. Soc. 86 (2003) 779-816. | Zbl
[10] ,, and, Analysis and Synthesis of Time-Delay Systems. Warsaw & Chichester: PWN and J. Wiley (1989). | Zbl
[11] , On the spectral - Lyapunov approach to parametric optimization of DPS. IMA J. Math. Control Inform. 7 (1990) 317-338. | Zbl
[12] , The LQ controller problem: an example. IMA J. Math. Control Inform. 11 (1994) 355-368. | Zbl
[13] , On the circle criterion for boundary control systems in factor form. Opuscula Math. 23 (2003) 1-25. | Zbl
[14] and, Admissible observation operators. Duality of observation and control using factorizations. Dynamics Continuous, Discrete Impulsive Systems 6 (1999) 87-119. | Zbl
[15] and, 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 07 (2000), FUNDP, Namur, Belgium.
[16] and, Boundary control systems in factor form: Transfer functions and input-output maps. Integral Equations Operator Theory 41 (2001) 1-37. | Zbl
[17] and, Circle criterion and boundary control systems in factor form: Input-output approach. Internat. J. Appl. Math. Comput. Sci. 11 (2001) 1387-1403. | Zbl
[18] and, On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations. Facultés Universitaires Notre-Dame de la Paix à Namur, Publications du Département de Mathématique, Research Report 05 (2002), FUNDP, Namur, Belgium.
[19] and, Toeplitz Forms and Their Application, Berkeley: University of California Press (1958). | Zbl | MR
[20] , Banach Spaces of Analytic Functions. Englewood Cliffs: Prentice-Hall (1962). | Zbl | MR
[21] and, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lect. Notes Control Inform. Sci. 164 (1991) 1-160. | Zbl
[22] and, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part I: Abstract Parabolic Systems, Cambridge: Cambridge University Press, Encyclopedia Math. Appl. 74 (2000). | Zbl | MR
[23] and, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Part II: Abstract Hyperbolic-Like Systems over a Finite Time Horizon, Cambridge: Cambridge University Press, Encyclopedia Math. Appl. 75 (2000). | Zbl | MR
[24] and, The frequency domain theorem for continuous one-parameter semigroups, IZVESTIJA ANSSSR. Seria matematicheskaya. 41 (1977) 895-911 (in Russian). | Zbl
[25] and, Absolute stability results for well-posed infinite-dimensional systems with low-gain integral control. ESAIM: COCV 5 (2000) 395-424. | Zbl | Numdam
[26] J.-Cl. Louis and D.Wexler, The Hilbert space regulator problem and operator Riccati equation under stabilizability. Annales de la Société Scientifique de Bruxelles 105 (1991) 137-165. | Zbl
[27] and Vû Quôc Phong, Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 37-41. | Zbl
[28] , On the stability of systems having several equilibrium points. Appl. Sci. Res. 21 (1969) 218-233. | Zbl
[29] , and, The computation of stability regions for systems with many singular points. Intern. J. Control 17 (1973) 641-652. | Zbl
[30] , New direct Lyapunov-type method for studying synchronization problems. Automatica 13 (1977) 139-151. | Zbl
[31] and, On the existence of solution of some operator inequalities. Sibirsk. Mat. Zh. 16 (1975) 563-571 (in Russian). | Zbl
[32] and, Riccati equations for strongly stabilizable bounded linear systems. Automatica 34 (1998) 953-967. | Zbl
[33] , Kalman-Popov-Yacubovich theorem: an overview and new results for hyperbolic control systems. Nonlinear Anal. Theor. Methods Appl. 30 (1997) 735-745. | Zbl
[34] , Dissipativity and Lur'e problem for parabolic boundary control system, Research Report, Dipartamento di Matematica, Politecnico di Torino 1 (1997) 1-27; SIAM J. Control Optim. 36 (1998) 2061-2081. | Zbl
[35] , The Kalman-Yacubovich-Popov theorem for stabilizable hyperbolic boundary control systems. Integral Equations Operator Theory 34 (1999) 478-493. | Zbl
[36] , Semigroups of Linear Operators and Applications to Partial Differential Equations. New York, Springer-Verlag (1983). | Zbl | MR
[37] , Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. | Zbl
[38] , Quadratic optimal control of stable well-posed linear systems through spectral factorization. Math. Control Signals Systems 8 (1995) 167-197. | Zbl
[39] and, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup. Trans. Amer. Math. Soc. 354 (2002) 3229-3262. | Zbl
[40] , Nonlinear Systems Analysis. 2nd Edition, Englewood Cliffs NJ, Prentice-Hall (1993). | Zbl
[41] , Transfer functions of regular linear systems. Part I: Characterization of regularity. Trans. AMS 342 (1994) 827-854. | Zbl
[42] , Riccati Equations in Hilbert Spaces: A Popov function approach. Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands (1994).
[43] and, Optimal control of stable weakly regular linear systems. Math. Control Signals Syst. 10 (1997) 287-330. | Zbl
[44] , An Introduction to Nonharmonic Fourier Series. New York, Academic Press (1980). | Zbl | MR
Cité par Sources :
