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
Thermoacoustic Instabilities

Acoustics
Occur due to positive feedback
mechanism between
combustion and acoustic
subsystems
Heat Release
Representative system:
ducted premixed flame
Schuller (2003)
Model
of the ducted premixed flame
Control Framework
•
•
LQ Regulator
Actuator Placement
Kalman filter
•LMI based techniques
•based on Hankel singular values
Conclusions
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
Combustion Subsystem
• Governing equation (linear)
dH i
dt
N

  1  cos  j y a  j   2 H i  H  i 1 
j 1
H i represents the m onopole strength of
the i
th
flam e elem ent averaged over the
cross-sectional area of the duct.
N onlinearity introduced due to
com bustion subsystem

... discretized front tracking equ ation
Acoustic Subsystem
N
u 
acoustic velocity
 cos  j y    t 
j
j 1
  M 
  j  sin  j y   j  t 
j 1 

N
p 
acoustic pressure
• Governing equations:
fluctuating
heat release
d j
  
 j 

dt
j



d   

    j   j  2 
dt  j 
...m om entum
j
 j  
2 sin  j y f
P
 H
i
...energy
i 1
 2 c

 M

 sin  j y a  u

contribution
from controller
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
State-Space Representation
d j

dt
d   


dt  j 
dH i

dt
  j   j  2 
2 sin  j y f
j  
j 
P
 H
i 1
N

dt
d
  
j 

j



j 1
=

 1  cos  j y a  j   2 H i  H  i 1 
A
i
 2 c

 M

 sin  j y a  u


B u
Linear Quadratic (LQ) Regulator
u  K  such that the cost functional
 N

J   
 j 1
  
T
2

 j  j  
2
 j  t   
 
j  





is minimized.
T
lc u u
P
 H t 
i
i 1
2




Linear Quadratic (LQ) Regulator
u  K
Open loop plant :
(without control)
d
dt
d
Closed loop plant :
(with control)
 A
 A  Bu
dt
  A  BK  
Ac
LMI optimization problem
- Linear Matrix Inequalities (LMI): inequalities defined for matrix variables
m in  c :
P Ac  Ac P  0
H
P  P
H
0
I  P   cI
d
 A  Bu
dt
variables: P ,  c
 c is the upper bound on the
energy

T
  of the plant
controlled using the LQ
R egulator
  A  BK  
Ac
Actuator Placement
using LMI based Optimization Techniques
* A ctuator located closer to flam e
lc  1
results in low er bounds  c on E m ax
* T his feature is highlighted w hen
l c  10
the controller is aggressive ( l c  1)
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
Controllability–Observability Measures
•
Measure of controllabilityobservability 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
Locations
closer to the
flame
LMI based techniques
Antinodes of
the least
stable modes
Measures based on HSVs.
The techniques give contradictory results
Actuator Placement
Numerical Validation
open loop
y a  0.3
y a  0.5
In the presence of transient growth, actuators placed according to LMI
techniques give better performance than when placed based on HSV
measures
Actuator Placement
Numerical Validation
0.5
0.3
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.
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.