Self-tuning control of a nonlinear model of combustion instabilities

We present a self-tuning scheme for adapting the parameters of a proportion al integral (PI) controller proposed by Fung and Yang for stabilization of a Culick-type! model of nonlinear acoustic oscillations in combustion chamb ers, Our adaptation criterion is Lyapunov-based and its objective is the re gulation of nonlinear pressure oscillations to zero. We focus on a two-mode model and first develop a design based on an assumption that the amplitude s of the two modes are available for measurement. The adaptation mechanism is designed to stabilize both modes and prevent the phenomenon observed bg Candel and coworkers whose adaptive controller stabilizes the first but (un der some conditions) apparently destabilizes the second mode, We also prove that the adaptation mechanism is robust to a time delay inherent to the ac tuation approach via heat release. In order to avoid requirements for sophi sticated sensing of the mode amplitudes needed for feedback, we also develo p an adaptation scheme which employs only one pressure sensor. In order for the adaptation scheme to be implementable, it is also necessary to know th e control input matrix: of the system, Rather than performing a linear ID p rocedure with input excitation, we propose a simple nonlinear ID approach b ased on limit cycles (internal excitation) which exploits the quadratic cha racter of the nonlinearities, Simulations illustrate the scheme's capabilit y to attenuate limit cycles and its robustness to magnitude- and rate-satur ation of the actuator.