8. Conclusion

A robust and adaptive nonlinear control method is presented, which asymptotically regulates thermoacoustic oscillations in a Rijke-type system in the presence of dynamic model uncertainty and unknown disturbances. 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. 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. A sliding-mode observer design is developed, 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 20 different sets of operating conditions.
