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.
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