Analysis of Control Relevant Coupled Nonlinear Oscillatory Systems

Abstract

The paper proposes and analyzes two prototype structures of coupled generalized van der Pol equations able to describe self-excitation of simultaneous oscillations with distinct frequencies. These structures are relevant for describing oscillations phenomena which may be encountered on systems subject to control. These structures are analyzed using the Krylov-Bogoliubov averaging method. This analysis allows to establish conditions for the occurrence of the various operation regimes. The usefulness of the results is illustrated by their application to the straightforward analysis of the properties of a combustion instability model.

Self-excited Oscillations and Fuel Control of a Combustion Process in a Rijke Tube

“…In this paper, the phenomenon of instability of the combustion process due to the feedback with negative damping is described by means of the low order model, which consists of generalized 2-DOF autonomous van der Pol oscillator. The coupled van der Pol oscillator model is experimentally verified and it is possible to analyse the appearance of self-excited oscillations, which can evolve to limit cycles in accordance with experimental observations [20].…”

“…In the next section, the Extended Lindstedt-Poincare (EL-P) method with multiple time scales [28] is applied The instability problem of the combustion process has recently been analysed in the paper of Landau et al [20] by applying the Krilov and Bogoliubov method [22]. However, an alternative way to reach the same goal by using the EL-P method with multiple time scales [26], [28], [29] exists.…”

“…(1) can be represented by the control structure [20] shown in Fig. 2a, where the control plant consists of two harmonic oscillators in forward branches and the feedback consists from derivative of the flame heat release rate multiplied by ε.…”

“…The model of combustion instability is based on the approach, proposed by Culick [6,7], which is applied and further developed by Sterling [33], Murray et al [24], Peracchio and Proscia [27] and Dunstan [14], respectively. The use of van der Pol oscillator in modelling of combustion instabilities is first suggested by Awad and Culick [3] and then applied by Landau et al [20]. All these models, including the model of the control of the com bustion instabilities, presented in this paper, assume the linear acoustics and propagation of standing acoustic waves in the combustion chamber.…”

“…The physics of coupling between acoustic waves and unsteady heat release, concisely described through a generalized energy equation and followed by concepts of feedback control, the choice of sensors and actuators and the control of full-scale combustion systems, is presented by Dowling and Morgans [12] and the references therein. In order to obtain an efficient model of active control in this paper, the reduced-order model of combustion instability process is proposed in the form of generalized, coupled van der Pol differential equations [20]. This model does not rely on derivation of governing PDEs from conservation laws of mass, momentum and energy, respectively, but is built using a grey-box system approach [14].…”