Reduced-order model feedback control design: numerical implementation in a thin shell model

Reduced-order models employing the Lagrange and popular proper orthogonal decomposition (POD) reduced-basis methods in numerical approximation and feedback control of systems are presented and numerically tested. The system under consideration is a thin cylindrical shell with surface-mounted piezoceramic actuators. Donnell-Mushtari equations, modified to include Kelvin-Voigt damping, are used to model the system dynamics. Basis functions constructed from Fourier polynomials tensored with cubic splines are employed in the Galerkin expansion of the full-order model. Reduced-basis elements are then formed from full order approximations of the exogenously excited shell taken at different time instances. Numerical examples illustrating the features of the reduced-basis methods are presented. As a first step toward investigating the behavior of the methods when implemented in physical systems, the use of reduced-order model feedback control gains in the full order model is considered and numerical examples are presented.

[1]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[2]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[3]  Harvey Thomas Banks,et al.  Reduced order model feedback control design: computational studies for thin cylindrical shells , 1998 .

[4]  Kazufumi Ito,et al.  Approximation in LQR problems for infinite dimensional systems with unbounded input operators , 1994 .

[5]  Harvey Thomas Banks,et al.  Parameter estimation in a structural acoustic system with fully nonlinear coupling conditions , 1996 .

[6]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[7]  Ricardo C.H. Rosario,et al.  Computational Methods for Feedback Control in Structural Systems , 1998 .

[8]  Kazufumi Ito,et al.  Well posedness for damped second-order systems with unbounded input operators , 1995, Differential and Integral Equations.

[9]  Ralph C. Smith,et al.  Spline Approximation of Thin Shell Dynamics , 1997 .

[10]  Michael A. Demetriou,et al.  UTILIZATION OF COUPLING EFFECTS IN COMPENSATOR DESIGN FOR STRUCTURAL ACOUSTIC SYSTEMS , 1996 .

[11]  Janet S. Peterson,et al.  The Reduced Basis Method for Incompressible Viscous Flow Calculations , 1989 .

[12]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1979 .

[13]  Ahmed K. Noor,et al.  Reduced basis technique for collapse analysis of shells , 1981 .

[14]  Harvey Thomas Banks,et al.  Smart material structures: Modeling, estimation, and control , 1996 .

[15]  A. Noor.  Recent advances in reduction methods for nonlinear problems. [in structural mechanics , 1981 .

[16]  Harvey Thomas Banks,et al.  Experimental Confirmation of a PDE-Based Approach to Design of Feedback Controls , 1997 .

[17]  Dennis A. Nagy,et al.  Modal representation of geometrically nonlinear behavior by the finite element method , 1979 .

[18]  Harvey Thomas Banks,et al.  Convergence of approximations in feedback control of structures , 2001 .

[19]  R. C. Smith,et al.  LQR Control of Thin Shell Dynamics: Formulation and Numerical Implementation , 1998 .

[20]  H. Tran,et al.  Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor , 2002 .

[21]  Ralph C. Smith,et al.  LQR Control of Shell Vibrations via Piezoceramic Actuators , 1998 .

[22]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[23]  H. T. Banks,et al.  Well-Posedness of a Model for Structural Acoustic Coupling in a Cavity Enclosed by a Thin Cylindrical Shell , 1995 .