Direct Adaptive Control for Infinite-dimensional Symmetric Hyperbolic Systems

Abstract Given a linear continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known and unknown waveform, we show that there exists a stabilizing direct model reference adaptive control law with certain disturbance rejection and robustness properties. The closed loop system is shown to be exponentially convergent to a neighborhood with radius proportional to bounds on the size of the disturbance. The plant is described by a closed densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. Symmetric Hyperbolic Systems of partial differential equations describe many physical phenomena such as wave behavior, electromagnetic fields, and quantum fields. To illustrate the utility of the adaptive control law, we apply the results to control of symmetric hyperbolic systems with coercive boundary conditions.

[1]  D. D’Alessandro.  Introduction to Quantum Control and Dynamics , 2007 .

[2]  Alberto Bressan,et al.  Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations , 2012 .

[3]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[4]  J. Wen.  Time domain and frequency domain conditions for strict positive realness , 1988 .

[5]  F. John.  Partial differential equations , 1967 .

[6]  Mark J. Balas,et al.  Robust adaptive model tracking for distributed parameter control of linear infinite-dimensional systems in Hilbert space , 2014, IEEE/CAA Journal of Automatica Sinica.

[7]  R. Curtain,et al.  Functional Analysis in Modern Applied Mathematics , 1977 .

[8]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[9]  Mark J. Balas,et al.  Adaptive Tracking Control of Linear Systems with Unknown Delays and Persistent Disturbances (or Who You Callin' Retarded?) , 2009 .

[10]  Mark J. Balas,et al.  Direct Adaptive Rejection of Persistent Disturbances , 2000 .

[11]  Silvan S. Schweber,et al.  An Introduction to Relativistic Quantum Field Theory , 1962 .

[12]  Mark J. Balas,et al.  ADAPTIVE CONTROL OF PERSISTENT DISTURBANCES FOR AEROSPACE STRUCTURES , 2000 .

[13]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

[14]  Michael E. Taylor,et al.  Partial Differential Equations , 1996 .

[15]  M. Balas,et al.  A non-orthogonal projection approach to characterization of almost positive real systems with an application to adaptive control , 2004, Proceedings of the 2004 American Control Conference.

[16]  Robert Fuentes,et al.  DISTURBANCE ACCOMMODATION FOR A CLASS OF TRACKING CONTROL SYSTEMS , 2000 .

[17]  Mark J. Balas,et al.  Trends in large space structure control theory: Fondest hopes, wildest dreams , 1982 .

[18]  M. J. Balas,et al.  Robust model reference adaptive control with disturbance rejection , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

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

[20]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[21]  F. Trèves.  Basic Linear Partial Differential Equations , 1975 .

[22]  Panos J. Antsaklis,et al.  A Linear Systems Primer , 2007 .