Generalized Sampled-Data Stabilization of Well-Posed Linear Infinite-Dimensional Systems

We consider well-posed linear infinite-dimensional systems, the outputs of which are sampled in a generalized sense using a suitable weighting function. Under certain natural assump- tions on the system, the weighting function, and the sampling period, we show that there exists a generalized hold function such that unity sampled-data feedback renders the closed-loop system exponentially stable (in the state-space sense) as well as L 2 -stable (in the input-output sense). To illustrate our main result, we describe an application to a structurally damped Euler-Bernoulli beam.

[1]  Tzyh-Jong Tarn,et al.  Periodic output feedback stabilization of neutral systems , 1990, 29th IEEE Conference on Decision and Control.

[2]  George Weiss,et al.  The representation of regular linear systems on Hilbert spaces , 1989 .

[3]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[4]  George Weiss.  Admissibility in input elements for diagonal semigroups on l 2 , 1988 .

[5]  Stuart Townley,et al.  Discrete-time low-gain control of uncertain infinite-dimensional systems , 1997, IEEE Trans. Autom. Control..

[6]  Raymond M. Redheffer,et al.  Completeness of sets of complex exponentials , 1977 .

[7]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[8]  D. Salamon.  Infinite Dimensional Linear Systems with Unbounded Control and Observation: A Functional Analytic Approach. , 1987 .

[9]  Stuart Townley,et al.  Generalized sampled data feedback control of distributed parameter systems 1 1 This work was support , 1998 .

[10]  O. Staffans.  Quadratic optimal control of stable well-posed linear systems , 1997 .

[11]  Mituhiko Araki,et al.  Recent Developments in Digital Control Theory , 1993 .

[12]  M. F. Rubinstein,et al.  Dynamics of structures , 1964 .

[13]  D. L. Russell,et al.  Admissible Input Elements for Systems in Hilbert Space and a Carleson Measure Criterion , 1983 .

[14]  O. Staffans.  Well-Posed Linear Systems , 2005 .

[15]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[16]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[17]  R. Rebarber.  Spectral determination for a cantilever beam , 1989 .

[18]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .

[19]  Ruth F. Curtain,et al.  Well posedness of triples of operators (in the sense of linear systems theory) , 1989 .

[20]  Tzyh Jong Tarn,et al.  Stabilization of infinite-dimensional systems with periodic feedback gains and sampled output, , 1988, Autom..

[21]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[22]  D. Russell,et al.  A mathematical model for linear elastic systems with structural damping , 1982 .

[23]  Hartmut Logemann,et al.  Time-Varying and Adaptive Discrete-Time Low-Gain Control of Infinite-Dimensional Linear Systems with Input Nonlinearities , 2000, Math. Control. Signals Syst..

[24]  Olof J. Staffans,et al.  Transfer functions of regular linear systems. Part II: The system operator and the lax-phillips semigroup , 2002 .

[25]  Cornelius T. Leondes,et al.  On the design of linear time-invariant systems by-periodic output feedback Part II. Output feedback controllability† , 1978 .

[26]  Stuart Townley,et al.  Stabilization of distributed parameter systems by piecewise polynomial control , 1997, IEEE Trans. Autom. Control..

[27]  Stuart Townley,et al.  Stability of infinite-dimensional sampled-data systems , 2003 .

[28]  Dietmar A. Salamon,et al.  Realization theory in Hilbert space , 1988, Mathematical systems theory.

[29]  Hartmut Logemann,et al.  Discrete-Time and Sampled-Data Low-Gain Control of Infinite-Dimensional Linear Systems in the Presence of Input Hysteresis , 2002, SIAM J. Control. Optim..

[30]  R. Curtain.  Infinite-Dimensional Linear Systems Theory , 1978 .

[31]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[32]  P. Kabamba.  Control of Linear Systems Using Generalized Sampled-Data Hold Functions , 1987, 1987 American Control Conference.