Optimal control of a beam with Kelvin–Voigt damping subject to forced vibrations using a piezoelectric patch actuator

A maximum principle is derived for the optimal control of a beam with Kelvin–Voigt damping subject to an external excitation with the control exercised by means of piezoelectric patch actuators. The objective functional is defined as a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time. A penalty term is also part of the objective functional defined as the control voltage used in the control process. The maximum principle makes use of a Hamiltonian defined in terms of an adjoint variable and the control function. The optimal control problem is expressed as a coupled system of partial differential equations in terms of state, adjoint and control variables subject to boundary, initial and terminal conditions. The solution is obtained by expanding the state and adjoint variables in terms of eigenfunctions and determining the optimal control using the maximum principle. Numerical examples are given to demonstrate the applicability and the efficiency of the proposed method.

[1]  Yalcin Yilmaz,et al.  Optimal control of a distributed parameter system with applications to beam vibrations using piezoelectric actuators , 2014, J. Frankl. Inst..

[2]  A. Ng,et al.  Actuator Placement Optimization and Adaptive Vibration Control of Plate Smart Structures , 2005 .

[3]  Sarp Adali,et al.  Active Open-Loop Control of Plates with Multiple Piezoelectric Patches via the Maximum Principle , 2014 .

[4]  M. Naushad Alam,et al.  Finite element analysis of actively controlled smart plate with patched actuators and sensors , 2010 .

[5]  I. Kucuk,et al.  Optimal vibration control of piezolaminated smart beams by the maximum principle , 2011 .

[6]  Juntao Fei,et al.  The Comparative Study of Vibration Control of Flexible Structure Using Smart Materials , 2010 .

[7]  Laura Menini,et al.  Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators , 2003, Simul. Model. Pract. Theory.

[8]  E. Roxin.  The existence of optimal controls. , 1962 .

[9]  Sarp Adali,et al.  A maximum principle approach to optimal control for one-dimensional hyperbolic systems with several state variables , 2006, J. Frankl. Inst..

[10]  Feng Chen,et al.  Optimal control of a beam with discontinuously distributed piezoelectric sensors and actuators , 2012 .

[11]  Abdolreza Ohadi,et al.  Bending vibration and buckling of non-uniform plate with time-dependent boundary conditions , 2011 .

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

[13]  Gangbing Song,et al.  Vibration control of civil structures using piezoceramic smart materials: A review , 2006 .

[14]  J. C. Bruch,et al.  Placement of Multiple Piezo Patch Sensors and Actuators for a Cantilever Beam to Maximize Frequencies and Frequency Gaps , 2009 .

[15]  K. Liew,et al.  Active control of FGM plates with integrated piezoelectric sensors and actuators , 2001 .

[16]  K. Ramesh Kumar,et al.  The optimal location of piezoelectric actuators and sensors for vibration control of plates , 2007 .

[17]  S. Narayanan,et al.  Active vibration control of beams with optimal placement of piezoelectric sensor/actuator pairs , 2008 .

[18]  J. C. Bruch,et al.  Optimal Control of Structural Dynamic Systems in One Space Dimension Using a Maximum Principle , 2005 .

[19]  J. Bruch,et al.  Integral equation approach for piezo patch vibration control of beams with various types of damping , 2008 .

[20]  Devendra P. Garg,et al.  Structural Damping and Vibration Control via Smart Sensors and Actuators , 2003 .

[21]  Carlos A. Mota Soares,et al.  Active control of adaptive laminated structures with bonded piezoelectric sensors and actuators , 2004 .

[22]  Martin Levesley,et al.  Vibration control of a flexible beam with integrated actuators and sensors , 2000 .

[23]  I. Bruant,et al.  Optimal Location of Actuators and Sensors in Active Vibration Control , 2005 .

[24]  Stephen J. Elliott,et al.  Modal response of a beam with a sensor–actuator pair for the implementation of velocity feedback control , 2005 .

[25]  Firooz Bakhtiari-Nejad,et al.  Vibration Optimal Control of a Smart Plate with Input Voltage Constraint of Piezoelectric Actuators , 2004 .

[26]  Xianmin Zhang,et al.  Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate , 2007 .

[27]  J. C. Bruch,et al.  Analytic and Finite Element Solutions for Active Displacement Feedback Control using PZT Patches , 2010 .

[28]  J. Dias Rodrigues,et al.  Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies , 2006 .

[29]  Carlos A. Mota Soares,et al.  Active control of forced vibrations in adaptive structures using a higher order model , 2005 .

[30]  L. Gallimard,et al.  Optimal piezoelectric actuator and sensor location for active vibration control, using genetic algorithm , 2010 .