Thermoacoustic Instability: Model-based OptimalControl Designs and Experimental Validation �

A. M. Annaswamyy, M. Fleifily, J.W. Rumseyz, R. Prasanthx, J.P. Hathouty, and A.F. Ghoniemy

Abstract

Active control of thermoacoustic instability has been increasingly sought after in the pasttwo decades to suppress pressure oscillations while maintaining other performance objectivessuch as low NOx emission, high efficiency and power density. Recently, we have developed afeedback model of a premixed laminar combustor which captures several dominant features inthe combustion process such as heat release dynamics, multiple acoustic modes, and actuatoreffects [1]. In this paper, we study the performance of optimal control designs using themodel in [1] with additional effects of mean heat and mean flow, actuator dynamics, andinput saturation. These designs are verified experimentally using a 1kW bench-top combustorrig and a 0.2W loudspeaker over a range of flow rates and equivalence ratios. Our resultsshow that the proposed controllers, which are designed using a two-mode finite dimensionalmodel, suppress the thermoacoustic instability significantly faster than those obtained usingempirical approaches in similar experimental set-ups without creating secondary resonances,and guarantee stability robustness.

1 Introduction

In several applications such as propulsion, power generation, and heating, processes that involve

continuous combustion are encountered. One of the main characteristics of these processes is a

dynamic behavior denoted as thermoacoustic instability. In most cases, the instability occurs due

to a coupling between the unsteady components of pressure and heat release rate and manifests in

the form of growing pressure oscillations. Often these pressure oscillations become more severe

as the operating condition of the combustors change to meet specific performance criteria such as

�The work reported here was supported in part by the National Science Foundation under grant No. ECS-9296070.The third author was supported in part by an NSF Graduate Fellowship.

yDepartment of Mechanical Engineering, MIT, Cambridge, MA 02139zArthur D. Little Inc., Cambridge, MA 02140xScientific Systems Inc., Woburn, MA

1

operating lean to reduce NOx formation, increasing the thermal output, or reducing the size of the

combustor. Pressure oscillations are undesirable since they lead to excessive vibrations resulting in

mechanical failures, high levels of acoustic noise, high localized burn rates, and possibly component

melting. While passive approaches such as changing the flame anchoring point, installing baffles

and acoustic dampers, etc., have been sought to counter the instability, a desire to operate over

a wide range of conditions without running the risk of self-destruction, and maintaining various

performance measures at desirable levels, has led to exploring active control as a possible strategy

for achieving the desired performance.

Several experimental results have been reported over the past decade for controlling thermally

driven acoustic oscillations using active methods [2]- [12]. Active control has also been attempted

for realizing other objectives such as efficiency (complete combustion),high performance (increased

thermal output), and low NOx formation [13, 14]. In much of these efforts, the experimental

controllers are implemented using analog electronic circuitry whose components are designed so

as to provide the functionalities of a filter, phase-shifter, and an amplifier, and their parameters are

determined by trial-and-error so as to add the requisite phase. Typically, the results from these

experiments have demonstrated that the dominant thermoacoustic instability can be suppressed. In

many cases, however, secondary peaks at frequencies which were not excited in the uncontrolled

combustor appear (for example, [4, 7, 9]). Also, as operating conditions such as equivalence ratio

and the flow rate change, the controller would fail in suppressing the primary instability as well

[7]. Active-adaptive control strategies have been attempted in [9, 12] so as to expand the range

of operating conditions. Typically, these investigations have employed an adaptive filter and an

LMS-algorithm [15] for adjusting its coefficients. Studies have also been reported in [16] using an

observer-based approach wherein a real-time identification of the unstable modes is proposed to

cope with uncertainties in a combustion process.

An alternative prescriptive approach for designing active controllers is to use a model of the

combustion process by employing the conservation equations and constitutive relations that govern

the acoustics and combustion dynamics. This not only allows one to obtain fundamental insights

into the underlying mechanisms and quantify the system properties in relation to various physical

parameters but also allows the development of a systematic controls methodology that incorporates

features of optimization and robustness, and enables an enhanced range of operation. Attempts

have been made in this direction in [4],[17]-[19], where the effect of acoustics is characterized,

but the combustion dynamics is not modeled. In Ref. [4] a model-based control design is carried

out using a model with a single acoustic mode at which the combustor is unstable. In [17]-[19],

multiple acoustic modes are included but the coupling between acoustic modes is neglected, and

simulation studies are reported using data obtained from a solid rocket motor [20]. In [21] an active

controller is proposed using system identification at a stable operating point and the �-synthesis

2

control procedure.

We have recently developed a physically-based finite-dimensional model of a continuous com-

bustion process [22, 23] and a model-based control methodology [1, 24]. The model includes flame

kinematics derived assuming that the the flame is laminar and anchored on a perforated plate [25],

acoustics with longitudinal modes, and a loudspeaker as an actuator. In [22] and [23], various

properties of the model are derived, including the effect of coupling between acoustic modes when

a heat source and an active control source are present, and the cause of secondary peaks that occur

in the experimental investigations of active control. In [1] and [24], model-based control designs

based on the LQG-LTR approach and adaptation are proposed, and their advantages over empirical

control designs are discussed and validated using numerical studies of finite-dimensional models.

The main goal of this paper is to carry out an extensive study of model-based optimal control

designs and their experimental validation. The underlying model is an extension of that in [1] and

includes the effects of mean flow and mean heat additions, actuator dynamics, and input saturation

all of which have a significant impact on the efficacy of the control design. The optimal control

designs are based on the LQG/LTR and H1 approaches. The closed-loop performance using

both these controllers is compared through experimental studies and their robustness properties are

characterized. The experimental investigations are carried out using a bench-top rig which exhibits

several features that are commonly encountered in combustion processes such as limit-cycles,

bifurcation, and hysteresis, a condenser microphone as a sensor, and a loudspeaker as an actuator.

The control designs are also validated using a PDE model of the acoustics.

Section 2 presents the input-output model of the combustor starting from the conservation

equations and the flame surface kinematics. Section 3 presents the active control designs which are

then verified experimentally using a bench-top combustor rig and numerically using a PDE model

of the combustion acoustics.

2 A Physics-based Dynamic Model of a Premixed Combustor

Thermoacoustic instability is generated due to the feedback between combustion and acoustics.

That is, the heat release source responds dynamically to acoustic perturbations, and the acoustic

oscillations are excited by the unsteady heat release rate. In [25, 22], a dynamic model of

an one-dimensional rig was developed which captures the dominant interactions between these

two subsystems, starting from the conservation equations of mass, momentum, and energy, and

kinematics of a laminar flame. The acoustics was modeled primarily by considering longitudinal

modes, and linear dynamics. The kinematics of a premixed laminar flame was modeled in [25]

assuming that it was stabilized in a low velocity region, such as behind a perforated disk, and

3

flameholder

loudspeaker

ProductsReactants

xfxa�� @@@@ ��

xs

microphone

?

- -

Figure 1: Schematic of the combustor with a side-mounted loudspeaker.

representing the flame as a thin sheet moving with the local convective velocity plus a small

constant burning velocity in a normal direction to its surface relative to the reactants.

The following assumptions were made in the derivation of the model (see Fig. 1 for a schematic):

(A1) Effects of viscosity and heat conduction are negligible.

(A2) Acoustic effects are one-dimensional.

(A3) the flame zone is spatially localized, at x = xf (see Figure 1) with a heat release rate per unit

area, q0f .

(A4) Perturbations about the mean are small.

(A5) The premixed flame has a conical mean flame surface with a small apex angle.

(A6) A loudspeaker is side-mounted with�r as the ratio of cross-sectional areas of the loudspeaker

and the combustor, xa is the location of the loudspeaker and xl is the displacement of its

diaphragm.

(A7) The pressure perturbation p0 can be approximated as

p0 (x; t) = pnXi=1

i (x) �i (t) (1)

where p is the mean pressure, using basis functions

i (x) = sin (kix + �i0) ; (2)

where ki and �i0 are determined by the boundary conditions, and correspond to the wave

numbers and spatial mode shapes, respectively, of the unforced wave equation [26, 27, 28].

(A8) Effects of mean-flow and mean-heat addition are negligible.

4

(A9) The loudspeaker dynamics can be neglected so that xl = kai, where i is the input current into

the loudspeaker.

Using the above assumptions, the conservation equations can be used to obtain the following

equations governing the underlying acoustic and heat-release dynamics:

pnXi=1

i (x) �i � c2pnXi=1

i (x) �i (t) = ( � 1)� :q0f

�+ p�rxl (3)

u0f (t) =1

nXi=1

1k2i

:�i (t)

d i

dx

�xf�+ �a0q

0f (t) + �rxlH

�xf � xa

�(4)

:q0f = �!f

�q0f � gfu

0f

�(5)

where is the specific heat ratio, p is the mean pressure, c is the mean speed of sound, ao =

( � 1)= p, � represents the effect of the flow velocity before and ahead of the flame at xf , i.e.,

uf(t) = (1� �)u�f + �u+f 0 < � < 1;

df is the diameter of the flame1,

!f =

4Su�dp

!; gf =

�dpD

!2

nf�u∆qr;

∆qr is the heat release rate per unit mass of the mixture, D is the diameter of the flameholder, �u is

the density of the premixed reactants, and H(�) is Heaviside function.

We now extend the model in Eqs. (3)–(5) when both assumptions (A8) and (A9) are not valid,

both of which are restrictive and affect the performance of the active controller. Assumption (A8)

is unrealistic since both mean-flow and mean-heat effects are always present in a combustor. More

importantly, the basis functions (mode shapes) i(�) depend on the mean temperature field in the

combustor. This effect is quantified in Section 2.1. That the actuator dynamics can be neglected

is not realistic either, since the natural frequencies of a speaker are often of the same order of

magnitude as the unstable frequencies of the combustor. We include the dynamics of the actuator

in this paper as well.

2.1 Relaxation of Assumptions (A8) and (A9)

2.1.1 Effect of Mean Flow and Mean-Heat Additions

In a typical combustion system, the mean flow velocity is nonzero and there is a non-negligible

amount of mean heat release, which causes a significant change in the velocity as well as density

1For a laminar flame stabilized on a perforated disc with nf holes, the flame base corresponding to each holetends to increase due to the entrainment from the neighboring locations. We incorporate this effect by assuming thatdf = �dp, where dp is the diameter of the perforation of the flame holder, and � 2(1.2,2).

5

and temperature of the hot gases. It can be shown that due to the mean heat, c(x), u(x), and �(x)

experience a step increase at x = xf , and are constants otherwise [22]. Denoting the speed of

sound c(x) = c1 for x � xf and c(x) = c2 for x > xf , the following change of coordinates

z = x��

1�c1

c2

� �x� xf

�H�x� xf

�(6)

is needed. This implies that points with x > xf shrink in the transformed cordinate z, since c2 > c1.

This implies that the equivalent length decreases from L to Le, where

Le = L�xfL

+c1

c2

�1�

xfL

��:

We now discuss the effect of the mean heat addition on the mode shapes. In the absence of

mean heat addition, if the fundamental mode is (x) = sinfkx+�g, using the relation in (6), one

can show that with mean heat addition, it becomes

(x) = sinfkex+ �eg

whereke =

2��Le

4= kc; �e = � for 0 � x � xf

ke =c1c2: 2��Le

4= kh; �e = �+ ke

�1� c1

c2

�xf for xf < x � L

and � is a constant that depends on the boundary conditions of the combustor. For example, for an

open-open combustor, � = 2 and for a closed-open combustor, � = 4. Similar comparisons can

be derived for higher modes with and without mean heat addition. Figure 2 shows a comparison

between the fundamental mode shape with and without mean heat addition, for a combustor that is

open at both ends.

Since the wave number with no mean heat addition is given by k = 2��L

, it follows that the

effective wave number in the cold section is larger since kc > k and the wave number is smaller in

the hot section since kh < k. It is interesting to note that the average wave number kav is unchanged

from k since

kav =1L

Z L

0kedx = k:

The change in the wave number, in turn, affects the acoustic frequency as well as the relative

location of the flame. In both the cold and hot sections, the acoustic frequency is increased due to

heat addition. If ! is the frequency without heat addition,

!e = kec > !

since Le < L.

On the other hand, the relative flame location is moved toward the downstream end, since

�xf�= sin

�khxf + �

�= sin

�kexf + �

�6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless distance x/L

ψ (

x) "−" With mean heat

"−." Without mean heat

Figure 2: Fundamental wave with and without mean heat addition for a combustor that is open atboth ends.

where

exf = xf + �xf ; � =

�1�

xfL

��1�

c1

c2

�xfL

+c1

c2

�1�

xfL

�

The most important effect of heat addition on the instability properties is due to the change

from xf to exf . If xf is close to a node or an antinode of the mode shape, heat addition can cause a

transition from a stable mode to an unstable mode or vice versa. When it comes to control design

therefore, a robust performance with respect to c2 may be nontrivial to establish since perturbations

in c2 may change the number of unstable poles.

An additional point needs to be made regarding the mode shapes and their dependence on

boundary conditions. Since the perturbation in the heat release does not affect the mode shapes

substantially, we adopt the unperturbed mode shapes (defined in Eq. (2)) as our basis functions

for the pressure. ki and �i0 can be calculated in a straight forward manner for ideal boundary

conditions where the ends are “fully open” or “fully closed”. For non-ideal boundary conditions,

the mode shapes can still be derived assuming that the impedance between the velocity and the

pressure is jZ where Z is real, as shown in Appendix A. The case where Z is complex addresses

the effect of dissipation from the boundaries [29]. Since we have neglected the internal dissipation

in the combustor due to viscous effects in our model, it is reasonable to neglect the dissipation from

7

the boundaries as well.

2.1.2 Loudspeaker Dynamics

In order to ensure an accurate control design, the effect of the loudspeaker dynamics governing the

relationship between ua, the voltage applied to the loudspeaker, and the diaphragm acceleration xl,

must be taken into account. Neglecting effects of magnetic inductance, for small input amplitudes,

this model can be derived as

Gl (s) =k1s

2

mls2 + bls+ kl(7)

where ml, bl, and kl represent the mass, friction, and stiffness properties, respectively, of the

loudspeaker, and k1 is a calibration gain. Additional dynamics can arise from the housing used to

focus the acoustics of the loudspeaker onto to the combustor, such as a funnel or a waveguide [8].

This housing typically encloses some volume and can act as a Helmholtz resonator with a certain

damping and resonant frequency which could overlap with the acoustic range of the combustor,

making the task of designing a controller more difficult. It may be important to design this housing

so as to ensure minimal attenuation and minimize the introduction of additional dynamics. Also,

in order to ensure that the model in (7) is valid, care needs to be taken such that the input voltage

does not saturate the speaker.

2.2 An Extended Dynamic Model of the Combustor

The discussions in Section 2.1, together with the change in the mode-shape, effective acoustic

length, and the actuator dynamics, lead us to the following dynamic model:

�i + !2i �i +

nXj=1

dij:�j = bi

:q0f +bMi

q0f + bci xl (8)

y =nXi=1

cci�i; (9)

:q0f +bfq

0f = !fgf eu0f ; (10)

eu0f =nXi=1

�ci:�i +cMi

�i�+ kao�rxl (11)

mlxl + blxl + klxl = k1ua (12)

where y = p0(xs; t)=p, the normalized unsteady pressure component, is the output, ua is the input

voltage to the speaker, xa, xs, and xf are the locations of the actuator, the sensor, and the flame

respectively, kao = 0 if xa > xf and unity otherwise, xl is the position of the speaker diaphragm,

8

M is the Mach number of the mean-flow, and the parameters

bi = a0

E i

�xf�; ci =

1 k2

i

d i

dx

�xf�; bMi

= � a0

EMc1

d i

dx

�xf�; bci =

�rE

i (xa) ;

E =Z Le

0 2i dx; cci = i (xs) ; dij =

Mc1

E

Z Le

0 id j; cMi

= �Mc1

i

�xf�;

!i = cki; bf = !f�1� �a0gf

�; � 2 (0; 0:5); � 2 (1:2; 2):

In the above model, the model for heat-release dynamics was derived starting from the flame

kinematics relations and making simplifications based on the flame geometry and the structure

of the perforated disk as a flame stabilization mechanism [25]. The model indicates that q0f is

yet another state variable of the combustor system and must be taken into account in the analysis

and control synthesis. Since for premixed flames, Su << c, it follows that q0f typically exhibits

low frequency compared to acoustics. Other stabilization mechanisms such as rings, a bluff-body,

sudden expansion, or swirlers could lead to different frequency characteristics and parametric

dependencies.

Another point to note is that the control parameter bci in Eq. (8) can become zero if i(xa) = 0

for some actuator locations xa. One such example is an end-mounted loudspeaker with a combustor

that is open at the upstream and downstream ends. This in turn can affect the relative degree of the

open-loop transfer function, though it still remains controllable due to the action of the controller

on the flow velocity (see Eq. (11)).

2.3 Model Validation: Experimental Results

A bench-top combustor rig was constructed to evaluate the model-based approach to control design

(See Figures 3 and 4 for the set-up and its schematic). The test rig consists of an air supply through

a low-noise blower, a settling chamber, a rotameter for adjusting and measuring the air flow rate,

a fuel (propane) supply through a pressure regulator, a rotameter for adjusting and measuring the

fuel flow rate, and a nozzle for enhancing mixing between fuel and air. The combustion chamber

is a 5.3-cm diameter, 47-cm long tube closed at upstream end and open at downstream end. The

flame was anchored on a perforated disc with 80 holes fixed 26 cm from upstream end with

several ports included for mounting actuators and sensors. Pressure is measured using a calibrated

capacitance microphone, and a 0.2 W Radio Shack loudspeaker is used as an actuator. Due to

design limitations, we restricted our experimental investigations to the case when the loudspeaker

was side-mounted. Measurements on the test rig were recorded using a Keithley MetraByte DAS-

1801AO data acquisition and control board, with a maximum sampling frequency of 300 KHz.

The board was hosted in a Pentium PC. The sensors are connected to the board through appropriate

signal conditioning circuits. Most experiments were conducted with an equivalence ratio between

9

Figure 3: The Bench-top combustor rig.

0.69 and 0.74 and an air flow rate of 333 mL/s (0.38 g/s), which corresponded to an unstable

operating condition without control (Equivalence ratios of less than 0.69 corresponded to a stable

operating point). The flow rate was varied between 267 mL/s and 400 mL/s and the power of

the combustor was 0.831 kW. A sampling rate of 10 KHz was found to be more than sufficient to

prevent aliasing. The unstable frequency of the combustion process was found to be 470 Hz.

Using the information from the bench-top combustor rig, the combustor model was simulated as

in Eqs. (8) - (12) with the following parameters: L = 0:62m 2, = 1:4, p = 1atm, c1 = 347m=s,

M = 3:612 � 10�4, �u = 1:163kg=m3, ∆qr = 2:26 � 106J=kg (for � = 0:74), Su = 0:3m=s,

� = 0:5, � = 2:0, dp = 1:5 � 10�3m, D = 0:053m, and nf = 80. The choice of these values

follows directly from the geometry and fuel properties. For example, Su was chosen based on the

burning velocity for propane and accounting for heat losses at the walls of the combustor. The heat

of reaction ∆qr was found using the following equation:

∆qr = Cv�

�+ 15:6; (13)

2L corresponds to the acoustic length, which was determined by locating the pressure null in the combustor. It was

found that the length of the air/fuel feed tube as well as an end-correction at the downstream end contributed to thisacoustic length.

10

Air from blower

Propane

Hot water out

Microphoneports Cold water in

Loudspeaker

Products

DAS 1801AO

From

DAS 1801 AO

Flame

ToPentium - PC

@@@

���

��

@@

� ��

�

�

- ?

6

�� -

@@R

@@R

@@R �

XXXXXy

Figure 4: Schematic of the combustor test rig, data-acquisition, and control.

0 50 100 150 200 250

−100

−50

0

50

100

Time (msec)

Pres

sure

(Pa)

(ii) Experiment

<−−−−−−−−−−−Linear Region−−−−−−−−−><−−−−−−−−−−−Nonlinear Region−−−−−−>

0 50 100 150 200 250

−100

−50

0

50

100

Time (msec)

Pres

sure

(Pa)

(i) Simulation

Figure 5: Pressure oscillations for uncontrolled combustor (i) Simulation results using the twomode model and (ii) Experimental results.

11

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Combustor length (m)

Non

dim

ensi

onal

mod

e am

plitu

de

D B C AXf

Closed−Open Combustor

stable

unstable

Figure 6: Mode shapes for closed-open combustor boundary conditions and actuator-sensor-flamelocations.

whereCv is the enthalpy of reaction of propane, and 15.6 is the stoichiometric ratio between air and

fuel. � and �, which also affect the flame parameters, were lumped approximations, as mentioned

earlier. A closed-open boundary condition was chosen due to the structure of the flow conditions.

The effect of mean heat, as mentioned earlier, contributed to a reduction in the effective length,

Le = 0:535m. A damping ratio � = :0033 was added at all frequencies to account for passive

damping in the system, the effects of which were not included in the model. The choice of � was

therefore arbitrary, and was selected so as to match the experimental growth rates over as wide a

range of equivalence ratios as possible. The corresponding mode shapes, ki, were computed as

shown in Figure 6 and !i to be 162 Hz and 488 Hz for i = 1; 2. Denoting WA=B (s) as a transfer

function with the actuator at B and sensor at A, the resulting plant transfer functions are of the form

WD=D (s) = 2:15� 105(s+ 440)

�s2 � 83s+ 7:70� 106

�(s+ 14:9)

�s2 + 407s+ 1:03� 106

� �s2 � 63s+ 9:41� 106

� ;WC=D (s) = 8:38� 104

(s+ 589)�s2 � 26s+ 1:47� 107

�(s+ 14:9)

�s2 + 407s+ 1:03� 106

� �s2 � 63s+ 9:41� 106

� ;assuming that only the first two modes are present (see Figure 6 for locations of C and D). We

note that in this closed-open case, WD=D has unstable zeros even though the actuator-sensor pair

is collocated. The performance of the uncontrolled combustor for both the simulation and the

experiment is shown in Figure 5, which shows that over the first 70 milliseconds the simulation

12

and experimental growth rates match closely. Beyond this point, the pressure level continues to

grow in the linear model, as expected, while nonlinearities begin to dominate in the experimental

combustor and a limit cycle is reached. The experimental and predicted behavior of the combustor

differ more drastically for � < 0:65. The former led to a stable system while the latter yielded an

unstable system with a smaller growth rate. This may be due to the modeling error in the passive

damping mechanism, which may in fact be nonlinear and depend on �.

2.3.1 Actuation and Sensing

The efficacy of the active control designs were evaluated using a 0.25W loudspeaker as an actuator

and a microphone as a sensor. To determine the loudspeaker dynamics, using a function generator

and a photo sensor for measuring the displacement of the loudspeaker diaphragm, a frequency

analysis was carried out. This was used to determine the transfer function relating the voltage into

the loudspeaker to the acceleration of the loudspeaker diaphragm which resulted in

Gl(s) =35:5s2

s2 + 364s+ 3:320� 106(14)

The natural frequency of the loudspeaker, which was at 290 Hz, is on the order of the first acoustic

mode, indicating the necessity of including the actuator dynamics in the control design process.

Over the entire range of available input voltage from the data acquisition board, the speaker

dynamics was observed to be linear. To complete the model of the experimental system, a sensor

gain of 45:3Pa=V olt was included in the simulation.

For the 0.2W loudspeaker that we used, the housing dynamics was not important. This was due

to the location of the loudspeaker in the funnel which was used to mount the speaker to the side

of the combustor. Since the loudspeaker diameter was only slightly larger than the hole leading

to the combustor, the size of the cavity between the speaker and combustor was small, preventing

housing dynamics from having an effect.

In addition to the combustion dynamics, loudspeaker dynamics, and sensor gain, the power

limitations of the instrumentation were considered when designing a controller. The data acquisition

board was limited to an output of �10 volts, from which we computed the maximum diaphragm

acceleration that our experimental system could provide with the 0.2 W loudspeaker at the unstable

frequency, which was 600m=s2. This limitation on the maximum control effort was taken into

account when designing the controllers, the details of which are described in Section 3.

13

3 Control

In general, a host of control methods can be applied to stabilize the combustor whether the actuator

is side-mounted or end-mounted. The dominant features of the combustor that should be kept in

mind while carrying out the control design are (i) the order of the system including the actuator

dynamics is 2n + 3 , where n is the number of acoustic modes, (ii) the system has two complex

unstable poles, (iii) the system can have unstable zeros for a number of actuator-sensor locations

even when they are collocated, (iv) the different modes of the system are coupled [22], (v) all

states are not accessible, (vi) the system is controllable and observable for a number of actuator-

sensor positions, (vii) the actuator output is constrained to lie within specified bounds and (viii)

nonlinearities are present whose effect is a stabilizing one leading to limit-cycles.

Two approaches are presented in this section for suppressing the pressure oscillations, (i) LQG-

LTR control and (ii)H1 control, which are two of the most commonly used control methodologies.

The fundamental objectives of any model-based control design are (a) the realization of a desired

closed loop performance and (b) stability robustness with respect to modeling uncertainties. These

competing objectives are realized to varying degrees by the above control methods depending

upon the practical application under consideration. In the context of thermoacoustic instability, the

questions are how these two methods perform, and what their relative advantages and disadvantages

are. These are addressed in sections 3.1 and 3.2.

3.1 LQG-LTR control

The LQG-LTR control procedure consists of a combined estimator-state feedback design, with the

former assuming a fictitious Gaussian noise and a quadratic cost in the estimation error and the

latter based on a quadratic cost in the system response as well as the control effort. The controller

has the form:

:bx = Abx +Bu+H (y � C bx)

u = �K bxwhere the estimator gain H and the state feedback gain K are to be designed. The matrices A, B,

and C are from the plant state space model. An optimal control strategy proposed in [30] leads to

a natural specification of K and H . K is determined using the cost function

J =Z 1

0

�yTQy + uTRu

�dt Q = I; R = �I (15)

where � is a scaling factor that determines the trade-off between fast transients and magnitude of

the control input. In the LQG-LTR design, H is determined by posing the problem as the design

14

G

++

∆ P

yG

u

P

y∆

l

1z

c

Figure 7: Block diagram of the controlled combustion system.

of a Kalman filter which ensures that bx converges to x as efficiently as possible, by introducing a

fictitious input noise with a variance I and an output noise with a variance Rf = �I . One can use

the Matlab control toolkit to compute K and H by fine-tuning � and �.

As mentioned earlier, a dominant feature of the actuator is input saturation. In order to include

this effect explicitly in the control design, as well as to obtain stability robustness measures, we

provide another interpretation of the LQG methodology. The underlying control problem(see

Figure 7) is to ensure: (1) robust stability (i.e., the closed loop system is stable for all admissible

additive uncertainties ∆P ); (2) nominal performance (i.e., the nominal closed loop pressure signal y

is small); and (3) nominal actuator activity (i.e., kz1k1 = supt�0 jz1(t)j is within a specified bound).

In Figure 7, P denotes the nominal combustor model (WC=D orWD=D),Gl is the loudspeaker model,

Gc is the controller to be designed and ∆P represents the model uncertainty. The effect of ∆P can be

equivalently represented by a noise w of bounded energy with an unknown spectrum using a stable

minimum phase weighting function W1 so that y∆ = W1w. With this interpretation, the selection

of � in the LQG-LTR design procedure as the output noise variance implies that the control design

supposes the presence of a modeling uncertainty with an equivalent effect of a weighting function

W1 = �.

We can now formulate the combustor problem as the following energy-to-peak orL2-L1 control

problem [31, 32]: find a controller Gc such that the nominal closed loop is internally stable, and

for all w such that kwk2 � 1, y and z1 must be such that the peak value kz1k1 < � where � is a

specified positive number, and kyk1 is minimized. It can be shown [31, 32] that the above problem

is solved by an LQG controller with a particular choice of �. As a result, the amplitude constraint

on z1 is naturally accommodated in the control design by appropriately choosing �. Thus, the

LQG-LTR approach with � as a design parameter is essentially an L2-L1 control problem and

guarantees robust L1 performance in the presence of bounded energy disturbances.

15

0 50 100−1000

−500

0

500

1000

Time (msec)

Contr

ol eff

ort (m

/s^2)

D/D configuration

0 50 100−100

−50

0

50

100

Time (msec)

Pressu

re (Pa

)D/D configuration

0 50 100−100

−50

0

50

100

Time (msec)

Pressu

re (Pa

)

C/D configuration

0 50 100−1000

−500

0

500

1000

Time (msec)

Contr

ol eff

ort (m

/s^2)

C/D configuration

Figure 8: Pressure response and control input with D/D and C/D configurations, and LQG-LTRcontrol: Experimental results for � = 0:7.

3.1.1 Controller design and experimental verification

The above discussions indicate that the parameter � can be chosen so as to accommodate the

amplitude constraints in the input xl to the combustor and � determines the amount of stability

robustness. For the system model given by WD=D(s)Gl(s), a choice of � = 0:1 ensured that xlremained within a maximum limit of 600m=s2. A nominal value of � = 0:01 was chosen. The

resulting LQG controller was found to be

GD=D (s) =5:05� 103 (s� 2071)

�s2 � 418s+ 5:06� 106

� �s2 + 459s+ 1:54� 106

��s2 + 1378s+ 6:06� 106

� �s2 + 766s+ 1:05� 107

� �s2 + 750s+ 7:37� 105

�(16)

For the second configuration given by WC=D(s)Gl(s), the controller is:

GC=D (s) =5:05� 103 (s� 2071)

�s2 � 418s+ 5:06� 106

� �s2 + 459s+ 1:54� 106

��s2 + 1378s+ 6:06� 106

� �s2 + 766s+ 1:05� 107

� �s2 + 750s+ 7:37� 105

�(17)

The performance obtained from the benchtop rig using the above controllers with the actuator-

sensor pair at D/D and C/D are shown in Figure 8 which resulted in a settling time of 59 msec and

45 msec, respectively. It can be seen that these match the simulation results of the controlled

combustor using (16) and (17) shown in Figure 9 quite closely. The improvement in the performance

at C/D is possibly due to the fact that the sensor is closer to the anti-node when placed at C and

therefore results in a larger system gain. We also observed that the LQG-LTR control designs

based on the unstable mode alone or on the model where the speaker dynamics is neglected failed

16

0 20 40 60 80 100−100

−50

0

50

100

Time (msec)

Pressu

re (Pa

)D/D configuration

0 20 40 60 80 100−1000

−500

0

500

1000

Time (msec)

Contr

ol Eff

ort (m

/sec2 )

D/D configuration

0 20 40 60 80 100−100

−50

0

50

100

Time (msec)

Pressu

re (Pa

)

C/D configuration

0 20 40 60 80 100−1000

−500

0

500

1000

Pressu

re (Pa

)

Time (msec)

C/D configuration

Figure 9: Pressure response and control input with D/D and C/D configurations, and LQG-LTRcontrol: Simulation results for � = 0:7.

to stabilize the oscillations, indicating that for the configuration in the experiment, the inclusion of

the system dynamics at all frequencies lower than the unstable value at around 500 Hz is necessary.

The performance of the LQG controller was tested for flow rates between 267 mL/s and 400

mL/s and equivalence ratios between 0.69 and 0.74 and in all cases, the thermoacoustic instability

was successfully suppressed. Changes in the flow rate while maintaining the same equivalence

ratio did not affect the ability of the controller to stabilize the thermoacoustic instability, in contrast

to Ref. [7]. As � increased, the settling time increased (see Fig. 10) which may be due to the

fact that the pressure levels and therefore the required control effort increase with � whereas the

loudspeaker has limited control authority (see the control effort in Figure 8). For a smaller � which

is such that the control effort required was significantly larger than that which could be achieved

by the experimental system, the simulations indicated a fast settling time, but the experimental

controller was not be able to achieve the same performance due to loudspeaker saturation.

Using the LQG-LTR controller, we were able to suppress the pressure level from 250 Pa (at A)

to an ambient noise level of 1.5 Pa, which corresponds to a reduction of 45 dB. The residual noise

is mostly due to the blower which accounts for the small amplitude of the pressure oscillations

in steady-state. A power spectrum of the combustor with and without control is shown in Figure

11 along with the power spectrum of the system with no combustion for reference. The figure

demonstrates that no secondary peaks were observed since the LQG-LTR controller provides

appropriate phase compensation at all the modeled frequencies unlike phase-shift control designs

based on the behavior at the unstable frequency alone [23, 1]. We evaluated the performance

17

0.69 0.695 0.7 0.705 0.71 0.715 0.72 0.725 0.73 0.735 0.740

20

40

60

80

100

120

Settlin

g Tim

e (ms

ec)

Equivalence Ratio φ

D/D configuration

C/D configuration

Figure 10: 5%-settling time achieved using the LQG-LTR controller as a function of the equivalenceratio for the D/D and C/D configurations, respectively.

0 1000 2000 3000 400010

−8

10−6

10−4

10−2

100

102

104

106

(b) Without Combustion − Freq. (Hz)

Powe

r Den

sity

0 1000 2000 3000 400010

−8

10−6

10−4

10−2

100

102

104

106

(a) With Combustion − Freq. (Hz)

Powe

r Den

sity

Without Control −−

With Control _

Without Control −−

With Control _

Without Control −−

With Control _

Figure 11: Power spectrum of the pressure response (a) with and without control and (b) withblower noise without combustion.

18

robustness of the LQG design by perturbing many of the parameters in the model. We found that

a 20% change in L destabilized the closed-loop system, while the controller was successful in

stabilizing the combustor in the presence of 20% perturbations in �, Su, �, and �. In the latter case,

robustness was evaluated by perturbing Su, �, and � in the model and by changing � on-line in the

experiment by varying the fuel flow rate. The controller provided a robust performance over all

values of � 2 [0:55; 0:74] even though the uncontrolled model and the experiment differed in the

stability behavior for � < 0:69. This could be attributed to the fact that the former set of parameters

does not affect the model response at the unstable frequency whereas changes in L directly affect

the system poles and zeros, thereby changing the gain and phase at the unstable frequency.

Another important point to be noted about the control design is its inability to suppress instabil-

ities for equivalence ratios greater than 0.74, which resulted in pressure levels of over 250 Pa. We

observed that the loudspeaker reached saturation levels fairly quickly indicating that larger power

authority is required in the actuator. The requisite actuator selection in such a case needs to be

made such that the acoustic energy of the actuator can be appropriately focused.

3.1.2 Performance and stability robustness analysis

In order to analyze the stability robustness of the LQG-LTR controller, the Bode plot of the closed

loop transfer function Tz1w fromw to the controlled output z1 obtained usingWD=D(s) andGD=D(s)

as in (16) is shown in Figure 12. For comparison, the Bode magnitude plot of the nominal plant

model is also shown. The transfer function Tz1w has the following interpretations: (i) Tz1w(j!) is

the factor by which the control signal is amplified during closed loop operation and, (ii) since

y(j!) = y0Tyw

1� ∆PTz1w

(j!);

where y0 is the initial condition, 1=Tz1w(j!) is the amount of uncertainty that can be tolerated at

each frequency. From Figure 12, we observe that, as expected, the control effort is the largest at the

open loop unstable frequency of � 488 Hz. The figure also illustrates that the stability robustness

is rather large, especially in the [200, 400] Hz. The third plot included in Figure 12 corresponds to

the H1 controller which is discussed in Section 3.2.

As is well known, the drawback of the LQG-LTR method is the lack of precomputable stability

robustness bounds on the allowable uncertainty ∆P . It should be noted that the choice of �

provides some freedom in increasing the stability robustness. More flexibility in enhancing stability

robustness is obviously attainable by choosing W to be frequency dependent.

19

101

102

103

10−2

10−1

100

101

Frequency (Hz)

Magn

itude

Open loop plant

LQG closed loop

Hinf closed loop

Figure 12: The Bode plot of the open-loop transfer functionGlP of the combustor and closed looptransfer functions Tz1w obtained using the LQG-LTR controller and the H1 controller.

3.2 H1 control

The philosophy behind H1 control procedure is to ensure that desired measures of stability

robustness and performance specified in the frequency domain are achieved. The block diagram

for controller design model is as in Figure 13 where the effect of uncertainty on stability robustness

is represented using W1. We have also added a performance weighting function W2. Given

weighting functions W1 and W2, the H1 control problem is to minimize the performance measure

kTz2wk1 subject to the stability robustness constraint kTz3wk1 < 1. The H1 procedure [33]

guarantees that the closed loop system of Figure 7 achieves robust stability for all ∆P satisfying

k∆P (j!)k < jW1(j!)j

for all ! 2 [0;1). The standard approach for solving the above H1 problem is to use the

-iteration of Safonov and Chiang [34], included in the MATLAB robust control toolbox.

The block diagram in Figure 13 indicates that in order to generate a H1 controller for ther-

20

G

P

++ y

G

u

W22

l

w

z

c

Wl

z3

z1

Figure 13: Block diagram of the controlled combustor using a H1 controller.

moacoustic pressure suppression, one needs to specify W1 and W2. A point to note is that the H1

control procedure does not provide a natural way to include time domain specifications such as

actuator saturation limits. One needs to iterate the weights W1 and W2 in order to meet such time

domain requirements on u.

3.2.1 Controller design and experimental verification

In the context of thermoacoustic pressure suppression, the following guidelines can be used to arrive

at an initial set of weights. For example, since the model accuracy at low frequencies is typically

high, the weight W1, which represents modeling uncertainty, must have low Bode magnitudes at

low frequencies (recall the inequality k∆P (j!)k < jW1(j!)j). Similarly, since one of the H1design objectives is to have a sensitivity function with low magnitude at low frequencies, the weight

W2 must be high at low frequencies. Thus, we can start the design iteration with weights of the

form:

W1 = k1s+ �1p1

s+ p1and W2 = k2

s+ �2p2

s+ p2

where �1 < 1 and �2 > 1. For the combustor with dominant (unstable) frequency at 488 Hz, we

took the initial pole locations at 600 Hz.

The final set of weights are:

W1 =s+ 100�s+ 1200�

; W2 =5(s+ 1600�)4(s+ 200�)

The H1 controller obtained with these weights resulted in:

GD=D(s) =�337:32 (s+14:9)(s�666:33)(s+3770)(s�8689)[(s+203:5)2+994:32][(s+182)2+18132]

(s+11:6)(s+360:73)(s+628:32)[(s+592:1)2+888:72][(s+303)2+1997:62][(s+146:91)2+3050:82]

21

It can be easily seen that this controller exhibits certain desirable features such as adding gain

and phase at the unstable combustor frequency, large magnitudes at low frequencies. The Bode

plot of the control loop transfer function in Figure 12 shows smaller magnitudes indicating that a

large stability robustness bound. In experiments, it was found that (see Figure 14) this controller is

rather sluggish and settling time for pressure oscillations were of the order of 120 msec.

1.45 1.5 1.55 1.6 1.65−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Time (sec)

Loudsp

eaker

Accl.

(m/se

c^2)

1.45 1.5 1.55 1.6 1.65−100

−80

−60

−40

−20

0

20

40

60

80

100

Time (sec)

Pressu

re (Pa

)

Figure 14: Pressure response and control input with D/D configuration, and H1 control: Experi-mental results for � = 0:7. Controller was turned on at 1500 msec.

A more aggressive H1 controller was obtained by choosing W1 and W2 using closed-loop

transfer functions obtained from the LQG-LTR controller. The order of the controller was 21,

which after balanced truncation, was computed to be

�11616(s+ 537:3337)[(s+ 289:5)2 + 10212](s� 1598)[(s� 28:522 + 20522]

(s+ 342:4)[(s+ 316)2 + 7192][(s+ 259:6)2 + 24652][(s+ 639:9)2 + 31472];

and resulted in a settling time of 65msec.

3.3 Discussion

The main contribution of the model-based approach used in the control designs discussed in this

section is the optimization framework that it provides. In the LQG-LTR design, the cost function

J in (15) is minimized, whereas in the H1 method, the peak value of kTz2wk1 is minimized. A

22

direct consequence of such a systematic design procedure is the fast suppression of pressure in a

1 kW combustor using a 0.2W speaker with a minimal control effort (for example, 3 mW peak

electrical power in the LQG-LTR control design) and a guaranteed margin of stability robustness

(for example, in the H1 design). The model-based approach enabled pressure suppression over

a range of equivalence ratios (0.69-0.74) and flow rates (267mL/s-400mL/s) without resulting in

any secondary peaks. Beyond these ranges, the linearity of the heat release dynamics, and more

importantly, that of the actuator dynamics failed, thereby making the control design inadequate.

Combustor rigs of comparable power densities have been experimentally investigated in [5] and

[7], both of which used an empirical phase-shift controller. In [5], pressure suppression is achieved

in 80 msec. using a 10W speaker and a peak electrical power of 16mW. In [7], a 30W speaker is

used to suppress the pressure where the closed-loop system exhibits secondary peaks at 240Hz and

550Hz.

The results in sections 3.1 and 3.2 also show that various properties of the actuator can be

naturally accommodated in the design procedure. Given the open-loop instability of the combustion

system, a dominant constraint in the control design is the input amplitude. As mentioned earlier

the LQG-LTR procedure accommodates this in a straight forward manner, and as indicated by

Figures 8 and 14, results in a better performance than the H1 controller. It may be possible to

realize a similar performance with a H1 approach as well through successive iterations of W1 and

W2. But these may be large in number, and hence result in a significant computational burden.

The model-based control designs also provide quantitative measures of the robustness of the

controlled combustor. As seen above, both the LQG-LTR and H1 controllers (a) are successful

in pressure stabilization, and (b) provide a certain amount of stability robustness. While the H1controller guarantees a bound a priori, the robustness achievable from an LQG-LTR cannot be

precomputed. One can, however, iterate on the selection of a suitable W1 to enhance stability

robustness.

3.4 Simulation results using the PDE model

In this section, we verify the LQG-LTR control design using the PDE model of the combustor

acoustics given by @2p0

@t2� c2

1@2p0

@z2

!+ 2Mc1

@2p0

@z@t= ( � 1)

@q0f@t

+Mc1@q0f@z

!+ p�r

@va@t

!(18)

@p0

@t+Mc1

@p0

@z+m

c1

M

@u0

@z= ( � 1)q0f + p�rva (19)

23

where �(x)u(x) = m, and the flame model in (5), for various actuator-sensor configurations.

The PDEs were simulated using the Split-Coefficient-Matrix method [35] with a Courant number

CN = 0:85, which is defined as

CN =∆tc∆x

where ∆x and ∆t denote the step sizes in t and x, respectively. CN needs to be chosen to be less

than unity to ensure convergence of the numerical solution of the PDE when it has feedback control

inputs. The effect of adding CN is equivalent to adding damping to the system. To simulate a

similar effect in the two-mode model, a damping term was added to both modes with � = 0:0033.

This value was chosen so that the uncontrolled combustor exhibits similar responses using the PDE

model as well as the two-mode model. One can view the addition of the damping term in both these

models to represent any passive damping that may be present in the combustor. The output equation

and the actuator dynamics as in (9) and (7) were simulated with the same values for the parameters

as in Section 2.3. We assumed that the loudspeaker was side-mounted and that the actuator-sensor

location was D/D. We observed that both controllers are effective in stabilizing the system at D/D.

We also evaluated the performances of both the controllers as the sensor location was moved away

from the actuator. We found that for all sensor-actuator locations, the LQG controller resulted in

almost the same performance when evaluated using the PDE model or the two-mode model. The

corresponding pressure responses for A/D location, where the deviation between the PDE model

and the two-mode model is a maximum, are shown in Figure 15. We also observed that at the D/D

location, the pressure response obtained using the PDE model and the LQG controller resulted in

a similar performance to that obtained in the experiment, with a settling time of 40 msec. The

pressure response using the H1 controller led to a similar performance as in Figure 14 as well.

4 Summary

In this paper, we studied the performance of model-based controllers for suppressing thermoacoustic

instability in a premixed laminar combustor with a loudspeaker as an actuator and a microphone

as a sensor. Two control designs were discussed to suppress the pressure oscillations using (i) the

LQG-LTR method and (ii) the H1 approach. Their performance was validated experimentally

using a 1kW benchtop rig, and their robustness properties were discussed. The results indicated

that the LQG-LTR provides a better performance, while the H1 provides pre-computable stability

robustness bounds. Both control designs were effective though they were based on a linear model,

indicating that the limit-cycle behavior of the nonlinearity did not affect the performance of the

linear design.

The experimental results reported here represent the first of its kind where a predictive model-

based control design was used for combustion control. The results show that a systematic optimal

24

0 10 20 30 40 50 60 70 80 90 100−150

−100

−50

0

50

100

150(a) With the PDE model and LQG controller

Time (msec)P

ress

ure

(Pa)

0 10 20 30 40 50 60 70 80 90 100−150

−100

−50

0

50

100

150

Time (msec)

Pre

ssur

e (P

a)

(b) With the Two−mode model and LQG controller

Figure 15: Pressure responses obtained using both the PDE model and the two-mode model forLQG control for A/D location.

control design can be carried out for suppression of thermoacoustic instability and can lead to

a faster settling time and a reduced controlled effort, both of which are attractive features for

purposes of commercial implementation. Various features of the available actuator technology

such as control bandwidth and input saturation can be readily incorporated into the design to

ensure efficient utilization of the control energy. Measures of robustness of closed-loop system

stability can be derived, quantifying the requisite model accuracy. Bounds on the operating range

over which satisfactory performance can be realized can be ascertained using the proposed model-

based approach. We are currently evaluating nonlinear control strategies that take into account the

structure of the nonlinearities in the system to expand the scope of operation.

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[35] R.J. LeVeque. Numerical methods for conservation laws. Birkhauser Verlag, Basel, 1992.

Appendix A

We show that the solution p0(x; t) to the unperturbed wave equation

@2p0

@t2� c2@

2p0

@z2= 0 (20)

27

is of the form

p0 (x; t) = Reh (x) ej!t

iwhere i (x) = sin (kx+ �) ; (21)

in the presence of non-ideal boundary conditions given by

u0 (zn) =p0

jZn (!)(zn) ; n = 1; 2 (22)

where z1 and z2 denote the inlet and outlet, respectively, and Z1 and Z2 are real functions of !.

Assuming that p0 = Re(p(z)ej!t), (20) can be simplified as

d2p

dz2+ k2p = 0 (23)

where k = !=c, whose solution is of the form

p (z) = aejkz + be�jkz (24)

where a and b are two unknown parameters to be determined. The boundary conditions in (22) can

be expressed in terms of the pressure as follows: The unperturbed momentum equation

�@u0

@t+@p0

@z= 0 (25)

can be integrated to obtain

u0 =j

�!

dp

dzej!t: (26)

Combining Eq.(26) with Eq. (22), we obtain that, for n = 1; 2,

dp

dz(zn) = �

�!

Zn (!)p (zn) : (27)

Applying (27) to Eq.(24), a linear system of equations can be obtained as

jkejkzna� jke�jkznb = ��!

Zn (!)

hejkzna + e�jkznb

i; n = 1; 2: (28)

Nonzero solutions for a and b can be found from (28) if 1 +

�2c2

Z1 (!)Z2 (!)

!sin

�!L

c

�+

�c

z1 (!)�

�c

z2 (!)

!cos

�!L

c

�= 0 (29)

which can be solved to obtain admissible values of !. Using (28) to get a relation between a and

b, we obtain that

bp (z) =1vuutk2 +

�!

Z1 (!)

!2

"kcos (kz) +

�!

Z1 (!)sin (kz)

#(30)

28

Eq. (30) shows that bp(z) is real, and hence, it follows that (z) = bp(z) and that (21) is the solution

of (20) where

k =!

c; tan� =

k

�!Z1(!): (31)

The above derivation shows that for general boundary conditions determined by Z1 and Z2 in

Eq. (22), the mode shape is of the form (21) where k and � are given by Eq. (31). Eq. (31) also

shows that when Z1 =1 the mode shape is cos(kz) which corresponds to the case when the inlet

is “fully closed”, and when Z1 = 0 the mode shape is sin(kz) when the inlet is “fully open”. It

should also be noted that Z2(!) affects the solution of Eq. (29) and therefore the wave number k

of the mode shape.

29