Experimental Identification and Control of a Cantilever Beam Using ERA/OKID with a LQR Controller
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Experimental Identification and Control of a Cantilever Beam Using ERA/OKID with a LQR Controller L. A. Gagg F. · S. M. da Conceição · C. H. Vasques · G. L. C. M. de Abreu · V. Lopes Jr. · M. J. Brennan
Received: 31 August 2012 / Revised: 26 March 2013 / Accepted: 25 December 2013 / Published online: 11 March 2014 © Brazilian Society for Automatics–SBA 2014
Abstract This paper presents an experimental study of the system identification and vibration control of a cantilever beam. For system identification, a white noise was applied, and the response signal was measured. These signals were used to feed the eigensystem realization algorithm– Observer/Kalman Filter identification method. The identified system was reduced using the Hankel norm model. An linear quadratic regulator controller was projected to operate just on the first two natural frequencies of the structure. The damping ratio of the first mode was effectively increased from 0.009 to 0.046. Keywords Optimal control · Linear quadratic regulator · ERA · OKID
1 Introduction A relatively new area of engineering is that of smart structures, in which actuators and sensors are incorporated onto a structure to enable it to adapt to its environment. One application area of importance is the active control of vibrations (Gatti et al. 2007; Zhang et al. 2008). This is of concern in this paper, particularly, the active vibration control of a cantilever beam. Initially, to implement active control, a dynamic model of the mechanical system is required. A worthy model is L. A. Gagg F. (B) · S. M. da Conceição · C. H. Vasques · G. L. C. M. de Abreu · V. Lopes Jr. · M. J. Brennan GMSINT – Grupo de Materiais e Sistemas Inteligentes, UNESP/FEIS – Faculdade de Engenharia de Ilha Solteira, Av. Brasil, 56, Ilha Solteira, SP CEP 15385-000, Brazil e-mail: [email protected] URL: http://www.dem.feis.unesp.br/gmsint
that one whose mathematical representation is able to make a good prediction of the behavior of the system in time and space. The process of achieving this representation from observations and experimental data is called identification (Ljung 1987). In this field, the good intuition of the researcher is of the most value (Santo 2001; Deistler 2004). One of the first identification and modeling methods is called Least Squares Method and was developed by Gauss in 1795 (Monteiro 2006). (Juang and Pappa 1985) developed an identification method, which is convenient for this purpose. This method makes possible to represent the system in terms of its state variables. The algorithm used in this case, which was initially called the eigensystem realization algorithm (ERA), has had two extensions. One of these is the ERA/DC (Juang and Wright 1988), where DC abbreviation means “data correlation”, and the other is the ERA/OKID (Juang and Pappa 1985), where OKID means Observer/Kalman Filter. The latter method was designed to be used mainly in the identification of lightly damped structures. Originally, this technique was conceived for the identification of modal parameters
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