1. No category

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)

N
dH i
 1  cos  j ya  j   2 H i  H i 1
dt
j 1
H i represents the monopole strength of
the i th flame element averaged over the
cross-sectional area of the duct.
Nonlinearity introduced due to
combustion subsystem

... discretized front tracking equation
Acoustic Subsystem
N
acoustic velocity
u   cos  j y  j  t 
j 1
  M

p  
j
j 1 
N
acoustic pressure
• Governing equations:

 sin  j y  j  t 

fluctuating
heat release
d j
 
 j 

dt
j



P
d 

    j  j  2 j  j   2 sin  j y f   H i
dt  j 
i 1
 2 
  c  sin  j ya  u
 M 
...momentum
...energy
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

 
j 

j



d 

    j  j  2 j  j   2 sin  j y f
dt  j 
N
dH i
 1  cos  j ya  j   2 H i  H i 1
dt
j 1

d
dt
=
A
 H
P
i 1
i
 2 c

 M

 sin  j ya  u


Bu
Linear Quadratic (LQ) Regulator
u  K  such that the cost functional
2
N 

 j  j   P
2
2

J     j  t   
    Hi t  
 j   i 1
 j 1 

 T 
 lc u T u
is minimized.
Linear Quadratic (LQ) Regulator
u  K
Open loop plant :
(without control)
Closed loop plant :
(with control)
d
 A
dt
d
 A  Bu
dt
  A  BK  
Ac
LMI optimization problem
- Linear Matrix Inequalities (LMI): inequalities defined for matrix variables
min  c :
PAc  Ac H P  0
P  PH  0
I  P  cI
variables: P,  c
 c is the upper bound on the
energy   T   of the plant
controlled using the LQ
Regulator
d
 A  Bu
dt
  A  BK  
Ac
Actuator Placement
using LMI based Optimization Techniques
lc  1
* Actuator located closer to flame
results in lower bounds  c on Emax
* This feature is highlighted when
lc  10
the controller is aggressive ( lc  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
ya  0.3
ya  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.