[en] This paper studies the possibility of extending the already proved link between the pole-zero distance and the maximum reachable damping ratio in single input single output (SISO) systems to multiple inputs multiple outputs (MIMO) ones. This extension is shown to be possible when the considered system presents specific properties: (i) it is equipped with collocated transducers with small authority, (ii) the system has a small modal density in the frequency band of interest and (iii) a low authority control law is used. It is indeed demonstrated that when these three conditions are satisfied, the analytical development of the closed-loop poles convergence is equivalent to the one observed with SISO cases, except that the anti-resonances are replaced by the transmission zeros (TZs). Consequently, it is concluded that the maximum reachable damping ratio is directly proportional to the pole-transmission zero distance for such MIMO systems. This conclusion is demonstrated with two numerical examples (a cantilever beam and a simply supported plate) and experimentally validated on a cantilever beam where all the studied systems are equipped with two collocated pairs of piezoelectric patches.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others
Collette, Christophe ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Active aerospace structures and advanced mechanical systems
Language :
English
Title :
On the link between pole-zero distance and maximum reachable damping in MIMO systems
Gripp, J.A.B., Rade, D.A., Vibration and noise control using shunted piezoelectric transducers: A review. Mech. Syst. Signal Process. 112 (2018), 359–383.
Zhao, Guoying, Raze, Ghislain, Paknejad, Ahmad, Deraemaeker, Arnaud, Kerschen, Gaëtan, Collette, Christophe, Active nonlinear inerter damper for vibration mitigation of duffing oscillators. J. Sound Vib., 473, 2020, 115236.
Macfarlane, A.G.J., Karcanias, N., Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex-variable theory. Internat. J. Control 24:1 (1976), 33–74.
Williams, T., Constrained modes in control theory: Transmission zeros of uniform beams. J. Sound Vib. 156:1 (1992), 170–177.
Miu, D.K., Physical interpretation of transfer function zeros for simple control systems with mechanical flexibilities. J. Dyn. Syst. Meas. Control 113:3 (1991), 419–424.
Trevor Williams, Transmission zeros of non-collocated flexible structures-Finite-dimensional effects, in: Dynamics Specialists Conference, 1992, p. 2116.
Davison, E.J., Wang, S.H., Properties and calculation of transmission zeros of linear multivariable systems. Automatica 10:6 (1974), 643–658.
Preumont, André, De Marneffe, Bruno, Krenk, Steen, Transmission zeros in structural control with collocated multi-input/multi-output pairs. J. Guid. Control Dyn. 31:2 (2008), 428–432.
Preumont, Andre, Voltan, Matteo, Sangiovanni, Andrea, Mokrani, Bilal, Alaluf, David, Active tendon control of suspension bridges. Smart Struct. Syst. 18:1 (2016), 31–52.
Lu, Yifan, Amabili, Marco, Wang, Jian, Yang, Fei, Yue, Honghao, Xu, Ye, Tzou, Hornsen, Active vibration control of a polyvinylidene fluoride laminated membrane plate mirror. J. Vib. Control 25:19–20 (2019), 2611–2626.
Balmes, Etienne, Structural dynamics toolbox 7.1 (for use with MATLAB). 1991.
Pathak, Shashank, Piron, Dimitri, Paknejad, Ahmad, Collette, Christophe, Deraemaeker, Arnaud, On transmission–zeros of piezoelectric structures. J. Intell. Mater. Syst. Struct., 2021.
Deraemaeker, Arnaud, Tondreau, Gilles, Bourgeois, Frédéric, Equivalent loads for two-dimensional distributed anisotropic piezoelectric transducers with arbitrary shapes attached to thin plate structures. J. Acoust. Soc. Am. 129:2 (2011), 681–690.
Fanson, J.L., Caughey, T.K., Positive position feedback control for large space structures. AIAA J. 28:4 (1990), 717–724.
Moheimani, S.O. Reza, Fleming, Andrew J., Piezoelectric Transducers for Vibration Control and Damping, Vol. 1. 2006, Springer.
Bavafa-Toosi, Yazdan, Introduction to Linear Control Systems. 2017, Academic Press.