Transmission Zeros in Structural Control with Collocated Multi-Input/Multi-Output Pairs

Abstract

I N ACTIVE vibration control of flexible structures with a singleinput/single-output (SISO) collocated (dual) actuator/sensor pair, the transmission zeros exhibit the well-known property of interlacing, that is, the poles and zeros alternate along the imaginary axis (strictly if the structure is undamped, slightly in the left-half plane for a lightly damped structure). This property is the origin of the guaranteed stability of the so-called low authority control (LAC) strategies for active damping [1,2]. Besides, the transmission zeros coincide with the poles (natural frequencies) of a modified system which depends on the sensor configuration. For a displacement (velocity) sensor, the modified system is the constrained system where the degree of freedom (DOF) alongwhich the control operates is blocked [3,4]. For a force sensor, the modified system is obtained by removing the contribution of the active member to the global stiffness matrix of the structure [2]. As the gain increases, the closedloop poles start from the open-loop poles and those which remain at finite distance move on loops going asymptotically to the transmission zeros. For displacement sensors, the extension of the SISO result to multi-input/multi-output (MIMO) collocated pairs has already been discussed in the literature [5] and, based on the fact that transmission zeros give identically zero output response, it has been inferred that, for MIMO systems, the transmission zeros are the eigenvalues of an associated constrained modes problem. To the authors’ knowledge, however, no formal proof is available. This Note provides such a proof for an undamped structure; the two cases (force actuator, displacement sensor and displacement actuator, force sensor) are discussed separately. II. Force Actuator, Displacement Sensor