Self-tuning control of a nonlinear model of combustion instabilities

We present a self-tuning scheme for adapting the parameters of a PI controller proposed by Fung and Yang (1992) for stabilization of a Culick-type model of nonlinear acoustic oscillations in combustion chambers. Our adaptation criterion is Lyapunov-based and its objective is the regulation of nonlinear pressure oscillations to zero. We focus on a two-mode model and first develop a design based on an assumption that the amplitudes of the two modes are available for measurement. Instead of estimating the damping coefficients of each mode (which are dependent on the unknown air/fuel ratio), the Lyapunov approach allows us to adapt the parameters of the controller directly. The adaptation mechanism is designed to stabilize both modes and prevent the phenomenon observed by Billoud, Galland, Huu, and Candel (1992), whose adaptive controller stabilizes the first but (under some conditions) apparently destabilizes the second mode. We also prove that the adaptation mechanism is robust to a time delay inherent to the actuation approach via heat release. In order to avoid requirements for sophisticated sensing of the mode amplitudes needed for feedback, we also develop an adaptation scheme which employs only one pressure sensor. Our approach is based on the idea to use the squares of the pressure and its derivative instead of the squares of the amplitudes of the modes to drive the adaptation. In order for the adaptation scheme to be implementable, it is also necessary to know the control input matrix of the system. Rather than performing a complicated linear ID procedure, we propose a simple nonlinear ID approach that exploits the quadratic character of the nonlinearities and identifies the input matrix from steady-state limit cycle data. Simulations illustrate the capability of the scheme to attenuate limit cycles without the knowledge of the growth coefficients.