Transfer functions of regular linear systems Part II: The system operator and the Lax-Phillips semigroup
Authors:
Olof Staffans and George Weiss
Journal:
Trans. Amer. Math. Soc. 354 (2002), 3229-3262
MSC (2000):
Primary 93C25; Secondary 34L25, 37L99, 47D06
DOI:
https://doi.org/10.1090/S0002-9947-02-02976-8
Published electronically:
April 3, 2002
MathSciNet review:
1897398
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as ``Part I''. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by ,
would be the
-dependent matrix
. In the general case,
is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks
and
, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of
where the right lower block is the feedthrough operator of the system. Using
, we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the ``initial time'' is
. We also introduce the Lax-Phillips semigroup
induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index
which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of
and also the points where
is not invertible, in terms of the spectrum of the generator of
(for various values of
). The system
is dissipative if and only if
(with index zero) is a contraction semigroup.
- 1.
D.Z. Arov and M.A. Nudelman.
Passive linear stationary dynamical scattering systems with continuous time.
Integral Equations and Operator Theory, 24:1-45, 1996. MR 96k:47016 - 2.
G. Avalos, I. Lasiecka, and R. Rebarber.
Lack of time-delay robustness for stabilization of a structural acoustics model.
SIAM J. Control and Optimization, 37:1394-1418, 2000. MR 2000e:93064 - 3.
G. Doetsch.
Handbuch der Laplace Transformation, Band I.
Birkhäuser Verlag, Basel, 1950. MR 13:230f - 4.
K.-J. Engel.
On the characterization of admissible control- and observation operators.
Systems and Control Letters, 34:225-227, 1998. MR 99d:93007 - 5.
P. Grabowski and F.M. Callier.
Admissible observation operators. Semigroup criteria of admissibility.
Integral Equat. & Operator Theory, 25:182-198, 1996. MR 97d:93011 - 6.
J.W. Helton.
Systems with infinite-dimensional state space: the Hilbert space approach.
Proceedings of the IEEE, 64:145-160, 1976. MR 54:4764 - 7.
E. Hille and R.S. Phillips.
Functional Analysis and Semi-Groups.
American Mathematical Society, Providence, Rhode Island, revised edition, 1957. MR 19:664d - 8.
D. Hinrichsen and A. J. Pritchard.
Robust stability of bilinear evolution operators on Banach spaces.
SIAM J. Control Optim., 32:1503-1541, 1994. MR 95i:93109 - 9.
B. Jacob and J.R. Partington.
The Weiss conjecture on admissibility of observation operators for contraction semigroups.
Integral Equations and Operator Theory, 40:231-243, 2001. - 10.
B. Jacob and H. Zwart.
Realization of inner functions.
Preprint, Twente, 1998. - 11.
V. Katsnelson and G. Weiss.
A counterexample in Hardy spaces with an application to systems theory.
Zeitschrift für Analysis und ihre Anwendungen, 14:705-730, 1995. MR 96m:47124 - 12.
P.D. Lax and R.S. Phillips.
Scattering Theory.
Academic Press, New York, 1967. MR 36:530 - 13.
P.D. Lax and R.S. Phillips.
Scattering theory for dissipative hyperbolic systems.
J. Functional Analysis, 14:172-235, 1973. MR 50:5502 - 14.
H. Logemann, R. Rebarber, and G. Weiss.
Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop.
SIAM J. Control and Optim., 34:572-600, 1996. MR 97c:93073 - 15.
H. Logemann and E.P. Ryan.
Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities.
SIAM J. Control and Optimization, 38:1120-1144, 2000. MR 2002c:93137 - 16.
H. Logemann, E.P. Ryan, and S. Townley.
Integral control of infinite-dimensional linear systems subject to input saturation.
SIAM J. Control and Optimization, 36:1940-1961, 1998. MR 99f:93078 - 17.
H. Logemann, E.P. Ryan, and S. Townley.
Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain.
IEEE Trans. Autom. Control, 44:1315-1319, 1999. CMP 99:14 - 18.
H. Logemann and S. Townley.
Discrete-time low-gain control of uncertain infinite-dimensional systems.
IEEE Trans. Autom. Control, 42:22-37, 1997. MR 98a:93034 - 19.
H. Logemann and S. Townley.
Low gain control of uncertain regular linear systems.
SIAM J. Control and Optimization, 35:78-116, 1997. MR 97m:93048 - 20.
K.A. Morris.
Justification of input-output methods for systems with unbounded control and observation.
IEEE Trans. Autom. Control, 44:81-84, 1999. MR 99j:93083 - 21.
R. Ober and S. Montgomery-Smith.
Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions.
SIAM J. Control and Optimization, 28:438-465, 1990. MR 91d:93019 - 22.
R. Ober and Y. Wu.
Infinite-dimensional continuous-time linear systems: stability and structure analysis.
SIAM J. Control and Optim., 34:757-812, 1996. MR 97d:93077 - 23.
R.E.A.C. Paley and N. Wiener.
Fourier Transforms in the Complex Domain.
American Mathematical Society, Providence, Rhode Island, 1934. MR 98a:01023 (latest reprint) - 24.
A. Pazy.
Semi-Groups of Linear Operators and Applications to Partial Differential Equations.
Springer-Verlag, Berlin, 1983. MR 85g:47061 - 25.
R. Rebarber.
Conditions for the equivalence of internal and external stability for distributed parameter systems.
IEEE Trans. Autom. Control, 38:994-998, 1993. MR 94b:93100 - 26.
R. Rebarber.
Exponential stability of coupled beams with dissipative joints: a frequency domain approach.
SIAM J. Control Optim., 33:1-28, 1995. MR 95i:93103 - 27.
R. Rebarber and S. Townley.
Robustness and continuity of the spectrum for uncertain distributed parameter systems.
Automatica, 31:1533-1546, 1995. MR 96f:93045 - 28.
D. Salamon.
Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach.
Trans. Amer. Math. Soc., 300:383-431, 1987. MR 88d:93024 - 29.
D. Salamon.
Realization theory in Hilbert space.
Math. Systems Theory, 21:147-164, 1989. MR 89k:93038 - 30.
O.J. Staffans.
Quadratic optimal control of stable well-posed linear systems.
Trans. Amer. Math. Soc., 349:3679-3715, 1997. MR 97k:49011 - 31.
O.J. Staffans.
Coprime factorizations and well-posed linear systems.
SIAM J. Control Optim., 36:1268-1292, 1998. MR 99g:93049 - 32.
O.J. Staffans.
Quadratic optimal control of well-posed linear systems.
SIAM J. Control Optim., 37:131-164, 1998. MR 2000i:93046 - 33.
O.J. Staffans.
Feedback representations of critical controls for well-posed linear systems.
Internat. J. Robust and Nonlinear Control, 8:1189-1217, 1998.MR 99m:93043 - 34.
O.J. Staffans.
On the distributed stable full informationminimax problem.
Internat. J. Robust and Nonlinear Control, 8:1255-1305, 1998. MR 99m:93034 - 35.
O.J. Staffans.
Lax-Phillips scattering and well-posed linear systems.
In Proceedings of the 7th IEEE Mediterranean Conference on Control and Systems, CD-ROM, Haifa, Israel, July 28-30, 1999. - 36.
O.J. Staffans.
Well-Posed Linear Systems.
Book manuscript, 2002. - 37.
E.G.F. Thomas: Vector valued integration with applications to the operator valued
space, IMA J. on Math. Control and Inform., 14:109-136, 1997. MR 99d:28016
- 38.
G. Weiss.
Admissibility of unbounded control operators.
SIAM J. Control Optim., 27:527-545, 1989. MR 90c:93060 - 39.
G. Weiss.
Admissible observation operators for linear semigroups.
Israel J. Math., 65:17-43, 1989. MR 90g:47082 - 40.
G. Weiss.
The representation of regular linear systems on Hilbert spaces.
In Control and Estimation of Distributed Parameter Systems, vol. 91 of ISNM, eds. F. Kappel, K. Kunisch, W. Schappacher, pp. 401-416, Birkhäuser-Verlag, Basel, 1989. MR 91d:93026 - 41.
G. Weiss.
Transfer functions of regular linear systems. Part I: Characterizations of regularity.
Trans. Amer. Math. Soc., 342:827-854, 1994. MR 91f:93074 - 42.
G. Weiss.
Regular linear systems with feedback.
Math. Control, Signals and Systems, 7:23-57, 1994. MR 96i:93046 - 43.
G. Weiss.
A powerful generalization of the Carleson measure theorem?
In Open Problems in Math. Systems and Control Theory, eds. V. Blondel, E. Sontag, M. Vidyasagar, J. Willems, pp. 267-272, Springer-Verlag, London, 1999. MR 2000g:93003 - 44.
G. Weiss and R.F. Curtain.
Dynamic stabilization of regular linear systems.
IEEE Trans. Autom. Control, 42:4-21, 1997. MR 98d:93084 - 45.
M. Weiss and G. Weiss.
Optimal control of stable weakly regular linear systems.
Math. Control, Signals and Systems, 10:287-330, 1997. MR 99h:49037 - 46.
G. Weiss and M. Häfele.
Repetitive control of MIMO systems usingdesign.
Automatica, 35:1185-1199, 1999. CMP 2001:12 - 47.
Y. Yamamoto.
Realization theory of infinite-dimensional linear systems, parts I and II.
Math. Systems Theory, 15:55-77,169-190, 1981. MR 83j:93031a;MR 83j:93031b - 48.
K. Zhou, J. Doyle and K. Glover.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, 1996.
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Additional Information
Olof Staffans
Affiliation:
Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland
Email:
Olof.Staffans@abo.fi
George Weiss
Affiliation:
Department of Electrical & Electronic Engineering, Imperial College of Science & Technology, Exhibition Road, London SW7 2BT, United Kingdom
Email:
G.Weiss@ic.ac.uk
DOI:
https://doi.org/10.1090/S0002-9947-02-02976-8
Keywords:
Well-posed linear system,
(weakly) regular linear system,
operator semigroup,
system operator,
generating operators,
well-posed transfer function,
scattering theory,
Lax-Phillips semigroup,
dissipative system
Received by editor(s):
February 23, 2001
Received by editor(s) in revised form:
November 16, 2001
Published electronically:
April 3, 2002
Article copyright:
© Copyright 2002
American Mathematical Society