# Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

Hartmut Logemann; Ruth F. Curtain

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 5, page 395-424
- ISSN: 1292-8119

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topLogemann, Hartmut, and Curtain, Ruth F.. "Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 395-424. <http://eudml.org/doc/116566>.

@article{Logemann2010,

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.
},

author = {Logemann, Hartmut, Curtain, Ruth F.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Absolute stability; actuator nonlinearities;
circle criterion; integral control; positive real; robust tracking;
well-posed infinite-dimensional systems.; well-posed linear systems; absolute stability; circle criterion},

language = {eng},

month = {3},

pages = {395-424},

publisher = {EDP Sciences},

title = {Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control},

url = {http://eudml.org/doc/116566},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Logemann, Hartmut

AU - Curtain, Ruth F.

TI - Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 395

EP - 424

AB -
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.

LA - eng

KW - Absolute stability; actuator nonlinearities;
circle criterion; integral control; positive real; robust tracking;
well-posed infinite-dimensional systems.; well-posed linear systems; absolute stability; circle criterion

UR - http://eudml.org/doc/116566

ER -

## References

top- 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 Control7 (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).
- H. Logemann, Circle criteria, small-gain conditions and internal stability for infinite-dimensional systems. Automatica27 (1991) 677-690.
- H. Logemann and E.P. Ryan, Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities. SIAM J. Control Optim.38 (2000) 1120-1144.
- 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. Auto. Control44 (1999) 1315-1319.
- H. Logemann, E.P. Ryan and S. Townley, Integral control of infinite-dimensional linear systems subject to input saturation. SIAM J. Control Optim.36 (1998) 1940-1961.
- H. Logemann and S. Townley, Low-gain control of uncertain regular linear systems. SIAM J. Control Optim.35 (1997) 78-116.
- 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).
- D. Salamon, Realization theory in Hilbert space. Math. Systems Theory21 (1989) 147-164.
- D. Salamon, 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/).
- O.J. Staffans, Quadratic optimal control of stable well-posed linear systems. Trans. Amer. Math. Soc.349 (1997) 3679-3715.
- M. Vidyasagar, Nonlinear Systems Analysis, 2nd Edition. Prentice Hall, Englewood Cliffs, NJ (1993).
- G. Weiss, Transfer functions of regular linear systems, Part I: Characterization of regularity. Trans. Amer. Math. Soc.342 (1994) 827-854.
- G. Weiss, Admissibility of unbounded control operators. SIAM J. Control Optim.27 (1989) 527-545.
- G. Weiss, Admissible observation operators for linear semigroups. Israel J. Math.65 (1989) 17-43.
- 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.
- D. Wexler, On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation. SIAM J. Math. Analysis11 (1980) 969-983.
- D. Wexler, Frequency domain stability for a class of equations arising in reactor dynamics. SIAM J. Math. Analysis10 (1979) 118-138.

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