Self-tuning control of a nonlinear model of combustion instabilities

We present a self-tuning scheme for adapting the parameters of a proportional integral (PI) controller proposed by Fung and Yang 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. The adaptation mechanism is designed to stabilize both modes and prevent the phenomenon observed by Candel and coworkers 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. 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 linear ID procedure with input excitation, we propose a simple nonlinear ID approach based on limit cycles (internal excitation) which exploits the quadratic character of the nonlinearities. Simulations illustrate the scheme’s capability to attenuate limit cycles and its robustness to magnitudeand rate-saturation of the actuator.