Feedback Linearisation for Nonlinear Vibration Problems

Abstract

Feedback linearisation is a well-known technique in the controls community but has not been widely taken up in the vibrations community. It has the advantage of linearising nonlinear system models, thereby enabling the avoidance of the complicated mathematics associated with nonlinear problems. A particular and common class of problems is considered, where the nonlinearity is present in a system parameter and a formulation in terms of the usual second-order matrix differential equation is presented. The classical texts all cast the feedback linearisation problem in first-order form, requiring repeated differentiation of the output, usually presented in the Lie algebra notation. This becomes unnecessary when using second-order matrix equations of the problem class considered herein. Analysis is presented for the general multidegree of freedom system for those cases when a full set of sensors and actuators is available at every degree of freedom and when the number of sensors and actuators is fewer than the number of degrees of freedom. Adaptive feedback linearisation is used to address the problem of nonlinearity that is not known precisely. The theory is illustrated by means of a three-degree-of-freedom nonlinear aeroelastic model, with results demonstrating the effectiveness of the method in suppressing flutter.

Magnetorheological elastomer-based variable stiffness flexible coupling for vibration isolation

“…Nonlinear isolators have a wider frequency band of vibration attenuation and can effectively reduce vibration with lighter auxiliary mass, since the mass of the system is not the main objective of this study, the linear model assumption showed satisfying results in terms of shifting the natural frequency away from the fundamental frequency. This is to avoid the non-linear modeling complexity and errors (Jiffri et al, 2014).…”

Experimental Nonlinear Control for Flutter Suppression in a Nonlinear Aeroelastic System

“…The method is well documented in texts such as [19,20]. A recent publication [21] attempts to illustrate the application of input-output linearisation in second-order elasto-mechanical systems, such as the present one. A brief explanation of the underlying theory of SISO input-output linearisation is now presented.…”

On the Use of Linear and Nonlinear Controls for Mechanical Systems Subjected to Friction-Induced Vibration

“…The principle is to define a control vector which allows to completely neutralize the nonlinear phenomena. The feedback linearization method using the Lie algebra [8,9] can be simplified when the system dynamics is defined by second-order differential equations as explained by Jiffri et al [11]. The dynamic equation of the system studied is defined including the controller by…”

“…First preliminary studies which are based on notions of passive or active control were initiated on simple phenomenological models subject to mono-instability [5][6][7]. In the field of nonlinear active control, linearization state feedback [8][9][10][11] is one of the possible solutions to control such a system. The principle of this method is to apply an active control vector which makes it possible to compensate exactly the nonlinear behavior of the system.…”