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Absolute stability results for well-posed infinite-dimensionalsystems with applications to low-gain integral control

Published online by Cambridge University Press:  15 August 2002

Hartmut Logemann  and
Ruth F. Curtain
Hartmut Logemann
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K.; hl@maths.bath.ac.uk.
Ruth F. Curtain
Affiliation:
Mathematics Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands; R.F.Curtain@math.rug.nl.
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Abstract

We derive absolute stability results for well-posed infinite-dimensional systems which, in a sense, extend the well-known circle criterion to the case that the underlying linear system is the series interconnection of an exponentially stable well-posed infinite-dimensional system and an integrator and the nonlinearity ϕ satisfies a sector condition of the form (ϕ(u),ϕ(u) - au) ≤ 0 for some constant a>0. These results are used to prove convergence and stability properties of low-gain integral feedback control applied to exponentially stable, linear, well-posed systems subject to actuator nonlinearities. The class of actuator nonlinearities under consideration contains standard nonlinearities which are important in control engineering such as saturation and deadzone.

Keywords

Absolute stabilityactuator nonlinearitiescircle criterionintegral controlpositive realrobust trackingwell-posed infinite-dimensional systems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 395 - 424
Copyright
© EDP Sciences, SMAI, 2000

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References

M.A. Aizerman and F.R. Gantmacher, Absolute Stability of Regulator Systems. Holden-Day, San Francisco (1964).
B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach. Prentice Hall, Englewood-Cliffs, NJ (1973).
V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems. Academic Press, Boston (1993).
F. Bucci, Frequency-domain stability of nonlinear feedback systems with unbounded input operator. Preprint. Dipartimento de Matematica Applicata ``G. Sansone'', Università degli Studi di Firenze (1997) (to appear in Dynamics of Continuous, Discrete and Impulsive Systems).
F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York (1983).
F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998).
C. Corduneanu, Integral Equations and Stability of Feedback Systems. Academic Press, New York (1973).
C. Corduneanu, Almost Periodic Functions. Wiley, New York (1968).
R.F. Curtain, H. Logemann, S. Townley and H. Zwart, Well-posedness, stabilizability and admissibility for Pritchard-Salamon systems. Math. Systems, Estimation and Control 7 (1997) 439-476.
R.F. Curtain and G. Weiss, Well-posedness of triples of operators in the sense of linear systems theory, in Control and Estimation of Distributed Parameter System, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser Verlag, Basel (1989) 41-59.
G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990).
P.R. Halmos, Finite-Dimensional Vector Spaces. Springer-Verlag, New York (1987).
H.K. Khalil, Nonlinear Systems, 2nd Edition. Prentice-Hall, Upper Saddle River, NJ (1996).
S. Lefschetz, Stability of Nonlinear Control Systems. Academic Press, New York (1965).
G.A. Leonov, D.V. Ponomarenko and V.B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis. World Scientific, Singapore (1996).
B.A.M. van Keulen, H∞Control for Infinite-Dimensional Systems: A State-Space Approach. Birkhäuser Verlag, Boston (1993).
Logemann, H., Circle criteria, small-gain conditions and internal stability for infinite-dimensional systems. Automatica 27 (1991) 677-690. CrossRef
Logemann, H. and Ryan, E.P., Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities. SIAM J. Control Optim. 38 (2000) 1120-1144. CrossRef
Logemann, H., Ryan, E.P. and Townley, S., Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain. IEEE Trans. Auto. Control 44 (1999) 1315-1319. CrossRef
Logemann, H., Ryan, E.P. and Townley, S., Integral control of infinite-dimensional linear systems subject to input saturation. SIAM J. Control Optim. 36 (1998) 1940-1961. CrossRef
Logemann, H. and Townley, S., Low-gain control of uncertain regular linear systems. SIAM J. Control Optim. 35 (1997) 78-116. CrossRef
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
W. Rudin, Functional Analysis. McGraw-Hill, New York (1973).
Salamon, D., Realization theory in Hilbert space. Math. Systems Theory 21 (1989) 147-164. CrossRef
Salamon, D., Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach. Trans. Amer. Math. Soc. 300 (1987) 383-431.
O.J. Staffans, Well-Posed Linear Systems, monograph in preparation (preprint available at http://www.abo.fi/ staffans/).
Staffans, O.J., Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc. 349 (1997) 3679-3715. CrossRef
M. Vidyasagar, Nonlinear Systems Analysis, 2nd Edition. Prentice Hall, Englewood Cliffs, NJ (1993).
Weiss, G., Transfer functions of regular linear systems, Part I: Characterization of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854.
Weiss, G., Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527-545. CrossRef
Weiss, G., Admissible observation operators for linear semigroups. Israel J. Math. 65 (1989) 17-43. CrossRef
G. Weiss, The representation of regular linear systems on Hilbert spaces, in Control and Estimation of Distributed Parameter System, edited by F. Kappel, K. Kunisch and W. Schappacher. Birkhäuser Verlag, Basel (1989) 401-416.
Wexler, D., On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation. SIAM J. Math. Analysis 11 (1980) 969-983. CrossRef
Wexler, D., Frequency domain stability for a class of equations arising in reactor dynamics. SIAM J. Math. Analysis 10 (1979) 118-138. CrossRef