Active Control of Thermoacoustic Instabilities in Combustion Systems

1
Active Control of Thermoacoustic Instabilities in
Combustion Systems

• Dr N Ananthkrishnan Himani Jain
• Department of Aerospace Engineering
• Indian Institute of Technology Bombay

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Thermoacoustic Instability

• THERMAL source, e.g., flame, heater
• ACOUSTIC resonator, e.g., open-open tube
• Favorable conditions Rayleigh criterion
• E.g., Combustion chambers, Rijke tube
• Thermal source (flame) creating sound

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Early History - Thermoacoustics

• Heat-driven oscillations
• Higgins (1777) Singing flame
• Sondhauss (1850) Closed-open tube
• Rijke (1859) Open-open vertical tube
• Observations
• Frequency depends on length of tube
• Heat source must be in bottom half of tube
• Blocking convection through tube stops sound
• Turning tube horizontal stops sound
• Explanation Rayleigh (1878)

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Instabilities in Combustion Systems

• 1940 Solid, Liquid Rocket Engines (High
  frequency) - Passive control (baffles, resonance
  rods)
  
• 1950 Afterburners, Ramjets (High frequency)
• 1960 Apollo Liquid-Propellant Engine, Viking
  engine
  
• 1980 Ramjets, High BPR Turbofans (Low frequency)
  Passive control difficult
  
• Solid rockets in large ICBMs, Strap-on Booster
  Rockets (Low frequency)

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Modeling Acoustic Resonators

• Spring-mass-damper system (oscillator)
• X2??X?2Xf
• Multiple modes modeshapes depend on
• Geometry
• Boundary conditions
• Set of S-M-D systems
• Coupled due to nonlinear gas dynamics

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Thermal-Acoustic Feedback

• Acoustic modes driven by fluctuating heat
  release, which in turn depends on acoustic
  pressure/velocity
  
• Xi 2??Xi ?2Xi f(q)
• q g(p,v)
• Different physics ('g') for SRM/LRM/GTC/AB

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Rayleigh Criterion

• If heat be given to air at the moment of
  greatest condensation, or be taken from it at the
  moment of greatest rarefaction, the vibration is
  encouraged. On the other hand, if the heat be
  given at the moment of greatest rarefaction, or
  abstracted at the moment of greatest
  condensation, the vibration is discouraged.''
• Rayleigh Index

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Control Thermoacoustic Instability

• High frequency
• Passive control (rods, baffles, etc.)
• SRM Particle damping, e.g., add Al oxide
• Low frequency
• Passive control ineffective
• Active control Promising
• Advantages
• Reduced noise (important for GTE)
• Reduced NOx emission (low pollution)
• More compact combustors (higher heat
  release/volume)

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Active Control Basic Ideas

• Operate at high W and PHI0
• Passive Decrease 'r' to operate in stable
  region
  
• Active Stabilize at PHI0 during operation to
  right of stable region

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Active Control Requirements

• Physics, mechanisms, model
• Conflicts, system parameters, stability analysis
• Sensing, actuation, receptivity
• Active control law
• System nonlinearities Modal coupling
• Uncertainty in system parameters
• Noise Unmodeled dynamics
• Sensor/Actuator limits, bandwidths
• Robustness Reliability

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Adaptive Feedback Linearization

• Feedback linearizing controller
• Cancel nonlinearities
• Stability, frequency response, handle
  disturbances/noise
  
• Adaptation law
• Adapt controller gains to changes in system
  parameters
  
• Online estimate of unknown system parameters
• State estimator
• Estimate states for feedback to controller
• Periodic forcing
• Drive the parameter error (adaptation law) to
  convergence

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Ongoing work...

• Experiments on Rijke-tube configurations
• Dr SD Sharma, Dr N Ananthkrishnan, Dr DN Manik
  (funded by DRDO)
  
• Nonlinear modeling, stability analysis, control
• Dr N Ananthkrishnan (with Dr FEC Culick, Caltech)

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Model Reference Adaptive Controller (MRAC)

• Plant An experimental or mathematical model of
  the nonlinear system dynamics.
  
• Controller to make the plant-controller closed
  loop linear and identical to the reference
  model.
  
• Reference Model of the same order as plant
  model.
  
• Adaptation Law to evolve the controller
  parameters ?.

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Problems to be addressed

• Control regulating the plant state variables
  to the
  
• unstable equilibrium state
• Tracking making the plant states follow
  the
  
• reference model states
• Parameter convergence having the controller
• parameters converge to that special value
• which makes the closed-loop plant-controller
  system
  
• identical to the reference model.

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Example of Nonlinear Oscillator
Nonlinear Plant dynamics can be written as
Using experimental data for the involved
parameters, the open loop simulation is
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Control Law and Error Dynamics

• Feedback linearizing control is chosen to be of
  the form
  
• Now, the combined plant-controller system can be
  written as
  
• Reference model is taken as

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Control Law and Errror Dynamics (Contd.)

• The combined plant-controller dynamics will be
  exactly
  
• identical to the reference model if
• the tracking error dynamics can be written as
• or
• where,
  and ? is the parameter error vector

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Adaptation Law

• An adaptation law is required which derives
  output error e, plant
  
• states xp and parameter error ?, all to zero.
• To this end, the 9-dimensional dynamic system
  consisting of
  
• ,
  and is considered.
  
• To determine the stability of the equilibrium
  point ,
  
• a time-invariant Lyapunov function is chosen as
  follows
  
• where, Pe, Px are symmetric positive definite
  matrices and is a
  
• diagonal positive definite matrix.
• When the adaptation law is selected as
• the derivative of v along the trajectory of
  output error and parameter
  
• error dynamics reduces to

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Adaptation Law (Contd.)

• where, Qe,Qx are positive definite symmetric
  matrices and related
  
• to Pe,Px as follows
• If is chosen, then elements of
  can be obtained as
  
• and Qx is chosen to be , then elements
  of Px are obtained
  
• as times the corresponding element of Pe.

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Stability Analysis

• Unforced case
• In this case hence results of Lyapunov
  analysis are
  
• valid ? are bounded.
• From the equations, and are bounded as
  well.
  
• It follows is bounded also, so is
  uniformly continuous.
  
• Barbalats Lemma then guarantees the global
  asymptotic
  
• convergence of output error and plant states to
  zero.
  
• But equilibrium point
  is not stable
  
• asymptotically as convergence of ? to zero could
  not be
  
• guaranteed.

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Stability Analysis (Contd.)

• Forced case r(t)
• The forcing function should satisfy the following
  properties (i) its
  
• a smooth function, (ii) r(t) and all its time
  derivatives are bounded,
  
• (iii) it satisfies the finite energy condition,
  i.e.
  
• Lets choose the forcing function
  . As
  
• previously, .
• Hence, the conclusion once again is the same
  and as before, the
  
• parameter error is only bounded.
• An analysis that explicitly accounts for
  persistently exciting
  
• forcing may provide parameter error
  convergence.
  
• However, above analysis suggests (i) a suitable
  choice of forcing
  
• will result in faster convergence of parameters,
  (ii) a suitable
  
• choice for r(t) would be a sinusoidal signal
  modulated by a
  
• function that is exponentially decaying.

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Simulation Results
Figures above show the asymptotic convergence of
plant states and the 2-norm of output error to
zero using the forcing function
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Simulation Results (Contd.)
Figures below show the convergence of linear
parameter errors to zero.
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Simulation Results (Contd.)
Nonconvergence of non-linear parameters
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Comparison with Unforced Case
Non-convergence of linear parameters in the
absence of forcing
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Comparison with Unforced Case (Contd.)
Lower value of lnV is achieved with the
incorporation of forcing function leading to more
negative in comparison to the unforced
case.
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Effect of Measurement Noise

• In practice, the adpative feedback linearizatio
  scheme
  
• would have to perform when the plant outputs
  are
  
• corrupted by measurement noise.
• For this purpose, the following noise signal is
  considered
  
• in the plant output
• The baseline study of the closed-loop system is
  repeated
  
• with this noise signal and simulation is done.
• 2-norm of the output error shows that tracking is
  
  
• substantially achieved.
• Linear parameters do converge in a statistical
  sense.
  

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Effect of Measurement Noise (Contd.)
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Effect of Measurement Noise (Contd.)
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Conclusions

• A novel adaptation law has been derived using
• Lyapunov technique, including a forcing
  function.
  
• The stability analysis itself reveals the desired
  form of
  
• the forcing function.
• Baseline simulations show successful control,
  tracking
  
• and estimation of linear parameters.
• Results were compared with unforced case and
  effect
  
• of varying the constants in the forcing function
  was
  
• studied.
• Baseline results were successfully reproduced in
  the
  
• presence of measurement noise in the plant
  output
  
• signal.
• The same procedure can be applied to any
  nonlinear
  
• system of similar nature and Lyapunov analysis
  can be
  
• carried out on similar lines.