Adaptive nonlinear control of self-excited oscillations in Rijke-type thermoacoustic systems is considered. To demonstrate the methodology, a well-accepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopole-like actuators. To facilitate the derivation of the adaptive control law, the dynamic model is recast as a set of nonlinear ordinary differential equations, which are amenable to control design. The control-oriented nonlinear model includes unknown, unmeasurable, nonvanishing disturbances in addition to parametric uncertainty in both the thermoacoustic dynamic model and the actuator dynamic model. To compensate for the unmodeled disturbances in the dynamic model, a robust nonlinear feedback term is included in the control law. One of the primary challenges in the control design is the presence of input-multiplicative parametric uncertainty in the dynamic model for the control actuator. This challenge is mitigated through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunov-based adaptive control law. To address practical implementation considerations, where sensor measurements of the complete state are not available for feedback, a detailed analysis is provided to demonstrate that system observability can be ensured through judicious placement of pressure (and/or velocity) sensors. Based on this observability condition, a sliding-mode observer design is presented, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunov-based stability analysis is provided to prove that the proposed closed-loop active thermoacoustic control system achieves asymptotic (zero steady-state error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop control system under various sets of operating conditions.
Part of the book: Adaptive Robust Control Systems