Control of Thermoacoustic Instabilities: Actuator Placement

Transcription

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

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

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

4 Model of the ducted premixed flame acoustic subsystem combustion subsystem 4

5 Model of the ducted premixed flame single actuator and sensor pair actuator adds energy to the system sensor measures acoustic pressure 5

6 Combustion Subsystem Governing equation (linear) dh dt i N j y a j Hi Hi j1 cos... discretized front tracking equation The combustion subsystem introduces nonlinearity 6

7 Acoustic Subsystem 7 acoustic velocity acoustic pressure N j1 u cos j y t N M p sin j y j t j1 j j Governing equations: d dt d dt j j j...momentum P j j 2 j j 2 sin j y f Hi...energy j i1 2 c M fluctuating heat release sin j y a u contribution from controller

8 8 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 Control objective is to reduce transient growth and avoid triggering of instability

9 State-Space Representation d dt j j j P d 2 c j j 2 j j 2 sin j y f Hi sin j ya u dt j i1 M N dh cos i j ya j Hi Hi dt j1 d dt = A Bu 9

10 Linear Quadratic (LQ) Regulator u K such that the cost functional j J t H t T is minimized. 2 N 2 P j 2 j i j1 j i1 T lcu u 10

11 Linear Quadratic (LQ) Regulator u K Open loop plant : (without control) d dt A Closed loop plant : (with control) d dt A Bu A BK A 11 c

12 LMI optimization problem - Linear Matrix Inequalities (LMI): inequalities defined for matrix variables min : H H PA A P 0 P P 0 variables: P, c c c c is the upper bound on the energy c I P I c of the plant controlled using the LQ Regulator T d dt A Bu A BK A c 12

13 Actuator close to the flame location gives the lowest γ c l c =10 l c =1 13

14 Controllability Observability Measures Other ways to determine optimal placement of actuators and sensors Controllability-Observability measure based on Hankel singular values (HSVs). measure = 2 i i Hankel singular value 14

15 Controllability Observability Measures Measure based on HSVs Locations of the antinodes of the third acoustic pressure mode give highest measure Third acoustic mode also least stable 15

16 Locations closer to the flame LMI based techniques Antinodes of the least stable modes Measures based on HSVs. The techniques give contradictory results 16

17 Actuator Placement Numerical Validation open loop y y a a In the presence of transient growth, actuators placed according to LMI based measures and not HSV measures give better performance 17

18 Actuator Placement Numerical Validation 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 measures. 18

19 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. 19