1. Control of Thermoacoustic Instabilities: Actuator-Sensor Placement Pushkarini Agharkar, Priya Subramanian, Prof. R. I. Sujith Department of Aerospace Engineering Prof. Niket Kaisare Department of Chemical Engineering Indian Institute of Technology, Madras Acknowledgements: Boeing Travel Grant, IIT Madras Alumni Affairs Association, IIT Madras

2. Thermoacoustic Instabilities Occur due to positive feedback mechanism between combustion and acoustic subsystems Representative system: ducted premixed flame Schuller (2003) Acoustics Heat Release

3. Model of the ducted premixed flame Control Framework LQ Regulator Kalman filter Actuator Placement LMI based techniques based on Hankel singular values Conclusions

4. Model of the ducted premixed flame acoustic subsystem combustion subsystem single actuator and sensor pair actuator adds energy to the system sensor measures acoustic pressure

5. Governing equation (linear) Combustion Subsystem

6. Acoustic Subsystem Governing equations: contribution from controller fluctuating heat release

7. Properties of the Model –Non-normality: due to coupling between combustion and acoustic subsystems –Nonlinearity: due to the equations of evolution of the flame front –Motivation: Reducing the transient growth and avoiding triggering

8. State-Space Representation

9. is minimized. such that the cost functional Linear Quadratic (LQ) Regulator

10. Open loop plant : (without control) Closed loop plant : (with control)

11. LMI optimization problem - Linear Matrix Inequalities (LMI): inequalities defined for matrix variables

12. Actuator Placement using LMI based Optimization Techniques

13. Controllability–Observability Measures Other ways to determine optimal placement of actuators and sensors Controllability-Observability measure based on Hankel singular values (HSVs). –measure = – Hankel singular value

14. Controllability–Observability Measures Measure of controllability- observability based on HSVs calculated for various actuator and sensor locations Locations of the antinodes of the third acoustic pressure mode give highest measure From numerical simulations, the third acoustic mode is also the highest energy state

15. The techniques give contradictory results Antinodes of the least stable modes Measures based on HSVs. Locations closer to the flame LMI based techniques

16. Actuator Placement Numerical Validation In the presence of transient growth, actuators placed according to LMI techniques give better performance than when placed based on HSV measures

17. Actuator Placement Numerical Validation 0.3 0.5 0.833 In the absence of transient growth, actuators placed according to HSV measures give better performance than in the presence of transient growth, but still not better than LMI techniques.

18. Conclusions Actuator-Sensor placement of non-normal systems requires different approaches than the ones used conventionally. For the ducted premixed flame model, actuators placed nearer to the flame give better overall performance. Controllers based on these actuators results in low transient growth as well as less settling time.