Active control of a smart beam with time delay by Legendre wavelets

Abstract Active vibration control in a flexible beam is investigated by taking into account the time-delay which may occur in voltage controlled piezoelectric actuators. The voltage applied to a piezoelectric patch actuator to suppress the dynamic responses of the beam at a specified terminal time is computed by solving an optimal control problem. The quadratic performance index function consists of related potential energy and kinetic energy at the terminal time as well as control effort as a penalty term. The solution method is based on a combination of spatial modal expansion and direct approximation approaches. The spatial modal expansion approach is used to transfer the optimal control problem of distributed parameter system (DPS) into the optimal control problem of a linear time-invariant lumped parameter system (LPS). A direct state-control parametrization approach is formulated where Legendre wavelets are employed to solve time delayed LPS. The operational matrices of integration and delay are utilized to reduce the solution of linear time-varying delayed systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the applicability and the efficiency of the proposed method and the numerical results indicate the effectiveness of the proposed control mechanism.

[1]  Nastaran Vasegh,et al.  Suboptimal control of linear systems with delays in state and input by orthonormal basis , 2011, Int. J. Comput. Math..

[2]  Alan J. Laub,et al.  Matrix analysis - for scientists and engineers , 2004 .

[3]  Arch W. Naylor,et al.  Linear Operator Theory in Engineering and Science , 1971 .

[4]  Simon X. Yang,et al.  A Discrete Optimal Control Method for a Flexible Cantilever Beam with Time Delay , 2006 .

[5]  Xing Tao Wang,et al.  Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials , 2007, Appl. Math. Comput..

[6]  Michael J. Chajes,et al.  STABILITY OF ACTIVE-TENDON STRUCTURAL CONTROL WITH TIME DELAY , 1993 .

[7]  P. Wang,et al.  Optimum control of distributed parameter systems with time delays , 1964 .

[8]  Ibrahim Sadek.  Necessary and sufficient conditions for the optimal control of distributed parameter systems subject to integral constraints , 1988 .

[9]  Taher Abualrub,et al.  A computational method for solving optimal control of a system of parallel beams using Legendre wavelets , 2007, Math. Comput. Model..

[10]  I. Sadek.  Optimal control of time-delay systems with distributed parameters , 1990 .

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

[12]  J. C. Bruch,et al.  A Maximum Principle for Nonconservative Self-Adjoint Systems , 1989 .

[13]  Xiaofan Yang,et al.  Techniques for solving integral and differential equations by Legendre wavelets , 2009, Int. J. Syst. Sci..

[14]  J. C. Bruch,et al.  Optimal piezo-actuator locations/lengths and applied voltage for shape control of beams , 2000 .

[15]  S. A. Yousefi,et al.  The linear Legendre mother wavelets operational matrix of integration and its application , 2006, J. Frankl. Inst..

[16]  I. Sadek,et al.  Optimal Control of a Class of Time-delayed Distributed Systems by Orthogonal Functions , 1998 .