Impact of Convection and Diffusion Processes in

Fundamental Limitations of Combustion Control

Prashant G. Mehta∗, Andrzej Banaszuk∗,

Marios C. Soteriou∗

United Technologies Research Center, East Hartford, CT 06040, U.S.A

This paper is aimed at understanding the role of physical processes of convection anddiffusion on the fundamental limitations arising in fuel control of thermoacoustic instabil-ities. For this purpose, we develop a physics based reduced order “heat release” modelthat is analytical and incorporates a) distributed flame dynamics, b) exothermicity relateddilatation effects, and c) distributed fuel dynamics including diffusion. Results show thatconvection leads to the presence of distributed delay, while diffusion, depending upon dif-fusivity and exothermicity constants, may lead to the presence of right half plane zeroes inthe response of heat release to the fuel control command. We isolate these non-minimumphase aspects of dynamics and apply the theory of fundamental limitations to obtain con-clusions on the impact of convection and diffusion on ones ability to use fuel based feedbackcontrol in controlling heat release.

I. Introduction

Thermoacoustic instabilities arise in power generation devices such as gas turbines and rockets whenacoustic modes couple with unsteady heat released due to combustion in a positive feedback loop; see recentreview articles.1, 2 Active control using fuel modulation is a viable means for controlling these instabilitiesby suppressing the pressure oscillations.3–5 However, there is no unanimous consensus on the obtainableperformance using feedback control and experiments have reported a wide range of performance as measuredby dB-reduction in pressure amplitude; see the introduction in6 for a summary of many of the experimentalconclusions. In some of our earlier papers,7–9 we have applied the control theory of fundamental limitationsand showed, with experimentally obtained input-output models, how factors such as delay, actuator controlbandwidth, and unstable poles can limit the obtainable performance irrespective of control design.

From our earlier work on combustion control,9 and from related work on cavity flow control of,10, 11

it is by now well known that delays that arise on account of the physical process of convection adverselyaffect obtainable performance and in particular, can lead to the phenomenon of peak splitting. Performancelimitations due to delay arise because one can not stably invert the effect of delay in controller.12, 13 Anotherirreversible process that is universal in many flow applications (including combustion) is that of turbulentmixing. To the best of our knowledge, implications of this irreversibility on obtainable performance withflow or fuel control has not been studied before.

In the spirit of flow models presented in our recent work,14, 15 we model convection using potential flows,that incorporate the effect of exothermicity but neglect the vortical effect of bluffbody wake dynamics. Wemodel the effect of turbulent mixing of fuel using a linear diffusion equation with appropriate boundaryconditions and (eddy) diffusivity constant. Our goal is to understand the implications of these two physicalprocesses, convection and diffusion, on the fundamental limitations in controlling the heat release (using fuelmodulation) in a distributed environment. We do so by 1) obtaining control-oriented linear models from thedistributed fuel input to the scalar heat release output, 2) understanding the non-minimum phase aspectsof these dynamics (zeroes of the system), and 3) applying the theory of fundamental limitations to obtainconclusions on the impact of convection and diffusion on ones ability to use feedback control in controllingheat release.

∗Research Engineer, UTRC, 411 Silver Lane, East Hartford, CT 06040.

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2nd AIAA Flow Control Conference28 June - 1 July 2004, Portland, Oregon

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Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

The outline of this paper is as follows. In Section II, we summarize the physical problem, includingequations of combustion and fuel transport (convection & diffusion). In Section III, we present the control-oriented linear models for these equations, highlighting the input-output responses of interest. In Section IV,we present an analysis of these models to obtain results on fundamental limitations arising in the presenceof convection and diffusion. Finally, we draw some conclusions in Section V.

II. Physical Problem

We consider the physical problem of premixed combustion stabilized by a single rectangular bluff bodyflameholder of height h in a channel of height H (see Figure 1 for a schematic). The size of both the bluff

Figure 1. Schematic of the physical problem.

body and the channel in the third dimension (z) is large so that two-dimensionality is assumed to apply.The Mach number is low and both reactants (of density ρu) and products (of density ρb) are assumed tobehave as ideal gases. On account of the low Mach number, we assume that the density jump arises entirelybecause of the temperature difference between the reactants and the products. Premixed combustion ismodeled using two flamesheets anchored to the two lips of the flameholder. For fluid dynamics, we retain theexothermic effects of burning but neglect vortical effects and the effect of geometrical expansion downstreamof the bluffbody assuming a constant inflow of reactants with velocity U0.

The details of the equations corresponding to the physical problem appear in14 and these are onlysummarized here. We model the flame motion by a version of the G-equation

−∂ξ

∂t+ (U0 + ue + ua) − ve

∂ξ

∂y= ST

[

1 +

(

∂ξ

∂y

)2]

1

2

, (1)

where the flamesheet is described by the connected locus of the points xf = (ξ(y), y). The flame motionarises because of the local (at the flame front) fluid velocity (U0 + ue + ua, ve), where subscripts e and a

represent the expansion and acoustic velocities respectively, and the flame speed (defined with respect to thereactants)

ST (xf ) = ST [Yf (xf )] (2)

is a function of the local fuel mass fraction; hats in this paper denote perturbation quantities, while baron a variable denotes mean quantities, the underline notation is reserved for vector quantities. Figure 2plots a typical ST [Yf ] as a function of Yf . We reserve the square bracket notation ST [·] for the function todistinguish it from the flame speed ST . The expansion component (ue, ve) of the fluid velocity arises as aconsequence of burning and is deduced from the solution of the continuity equation

∇ · ue = −1

ρ

Dρ

Dt, (3)

where DDt

is the material (substantial) derivative and ρ is the density. For this paper, we make an additionalsimplifying assumption of shallow flame angle (see14, 16, 17) by replacing the nonlinearity

[

1 +

(

∂ξ

∂y

)2]

1

2

≈∂ξ

∂y, (4)

in Eq. (1).The local fuel mass fraction seen at the flame front arises due to the convection and turbulent diffusion of

the fuel-air mixture in the duct. We make a simplifying assumption that the convection process is dominant

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only along the axial x direction and that the diffusion process is dominant only along the transverse y

direction to obtain a model for fuel transport as

∂Yf

∂t+ (U0 + ue)

∂Yf

∂x= β0

∂2Yf

∂y2, (5)

with boundary conditions

∂Yf

∂y

∣

∣

∣

∣

y= H2

= 0, (6)

∂Yf

∂n

∣

∣

∣

∣

(x=ξ−(y),y)

= 0, (7)

where the first boundary condition models the assumption that no fuel is exchanged across the duct wallat y = H

2 and the second boundary condition models the assumption of completely premixed flame (no fueltransport on account of diffusion at the flame boundary (x = ξ−(y), y), where the superscript “−” denotesthe fact that the boundary condition is applied just ahead of the flame, in the region of reactants), and n

denotes the unit vector normal to the flame. In light of the shallow flame angle assumption, we replace theboundary condition (7) by

∂Yf

∂y

∣

∣

∣

∣

(x=ξ−(y),y)

= 0. (8)

Finally, a model of the initial upstream fuel profile

Yf (x = 0, y, t) = Y 0f + y0

f (y, t), (9)

provides the third boundary condition at the entrance of each of the half channels upstream (on either side)of the bluff body - Y 0

F is a spatially uniform mean that provides a uniformly premixed gaseous mixture andy0

f (y, t) denotes the possibly non-uniform time-varying gaseous fuel control input signal. Here, we assumethat the fuel distribution is such that a lean condition (where the fuel mass fraction YF is less than thestoichiometric value - see Figure 2) always applies for fuel at the flame front.

Figure 2. Plot of a typical flame speed function ST [Yf ] : the peak corresponds to stoichiometric conditionwhere the flame burns with maximal flame speed S0.

We are interested in controlling, using upstream gaseous fuel control y0f (y, t), the integral heat release

perturbation response

q(t) = (γ − 1)ρu∆H

∫

Ω

ST (xf )Yf (xf )δ(x − xf )dA − Q (10)

downstream in the duct (see also15); ρu is the density of the unburnt mixture, ∆H is the heat of reaction,δ(·) denotes the delta distribution as defined in,14 Ω

.= [0, L]× (−H

2 , H2 ) is the 2D combustor domain, and Q

is the mean heat release (corresponding to the uniform mean fuel Y 0F ). We are interested in controlling the

heat release perturbation q(t) in Eq. (10) because it can “drive” the thermoacoustic pressure oscillations - forphysical models of acoustics, see.18–20 Here, we assume an empirical second order model for the acoustics.We present this model along with the control-oriented linear discretized models of flame and fuel dynamicsin the following section.

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III. Control-oriented modeling

A. Mean flame

In,14 we derived a reduced order model for the expansion velocity field that is used to construct the reducedorder model for flame dynamics (expressed here for the upper flame) as

∂ξ

∂t+ (ST + 2α(

H

2− y))

∂ξ

∂y= U0 + ua + 2αξ, (11)

where the boundary condition ξ(h2 , t) = 0, and α = µST

His a frequency scale arising as a consequence of

exothermicity and confinement effects; µ.= (ρu

ρb− 1). In the absence of any unsteadiness due to acoustics or

fuel, we obtain the steady mean flame equation

(y − b)dξ

dy+ ξ = −

U0

2α, (12)

where the boundary condition ξ(h2 ) = 0 and b

.= H

2

[

1 + 1µ

]

- the bar in ξ is used to distinguish the mean

solution from its unsteady counterpart ξ. The mean flame solution is

ξ(y) =U0

2α

(y − h2 )

(b − y), y ∈ [

h

2,H

2), (13)

and the flame saturates on the channel wall at y = H2 . On substituting y = H

2 into Eq. (13) yields

L = U0

ST

(

H2 − h

2

)

, the value of x along the duct at which the flame hits the channel wall - L being independentof the heat release parameter µ. However, for realistic density ratios across the flame, a curved mean flameanalytically described by Eq. (13) is obtained. In the limit µ → 0, however, this curved flame approaches thelinear V flame solution well-known in literature.17, 19 Physically, the curved solution arises because of theexpansion effects : a) higher value of effective streamwise flow velocity U0 + ue downstream of the bluffbodyleads to a shallowing of the flame there, and b) the motion of the flame in the cross-stream direction due toa higher value of ve near the lip leads to a bulging of the flame there.

B. Linearized flame dynamics

Linearizing the Eq. (11) about the mean in Eq. (13) leads to a linear flame model

∂ξ

∂t+ 2α(b − y)

∂ξ

∂y= ua(ξ(y), y, t) −

U0

ST

S′T yf (ξ(y), y, t) + 2αξ, (14)

that captures the effect of infinitesimal longitudinal acoustic velocity perturbation ua and that of infinitesimal

fuel perturbation yf ; S′T

.=

dST [Yf ]dYf

∣

∣

∣

Yf=Y 0

f

. Notationally, the complete flame solution is a function ξ(y, t) =

ξ(y) + ξ(y, t) and the complete fuel mass faction solution is a field Yf (x, y, t) = Y 0f + yf (x, y, t); yf (ξ(y), y, t)

is then the fuel mass fraction perturbation yf (x, y, t) as evaluated at the mean flame x = ξ(y). Taking theLaplace transform and multiplying by an integrating factor leads to

d

dy

[

(y − b)1−s2α Ξ(y, s)

]

=1

2α

[

Ua −U0

ST

S′T Yf

]

(ξ(y), y, s)(y − b)−s2α , (15)

where the capitalized symbols denote the Laplace transforms. As we are interested in analyzing the effectof distributed acoustic perturbations, it is convenient to make a change of co-ordinate from the independentco-ordinate y → x, where x denotes the mean flame solution ξ in Eq. (13). On making such a co-ordinatechange, we to obtain an equivalent model

d

dx

[

a(x)s2α

−1Ξ(x, s)]

=

[

Ua

U0−

S′T

S0T

Yf

]

(x, s)a(x)s2α

−2, (16)

where a(x).= 1 + 2α

U0x and explicit integration gives

Ξ(x, s) =1

a(x)s2α

−1

∫ x

0

a(z)s2α

−2

[

Ua(z, s)

U0−

S′T

S0T

Yf (z, s)

]

dz. (17)

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C. Fuel dynamics

We next use the convection-diffusion Eq. (5) to obtain a model from the control input y0f (y, t) (upstream fuel

perturbation at x = 0) to yf (ξ(y), y, t) (fuel perturbation felt at the mean flame x = ξ(y)). On linearizingabout its mean Y 0

f premixed value, we obtain the equation in perturbation yf as

∂yf

∂t+ U0a(x)

∂yf

∂x= β0

∂2yf

∂y2. (18)

Taking the Laplace transform, denoting β.= β0

U0, and multiplying by an integrator factor gives

d

dx

[

a(x)s2α Yf

]

(x, y, s) =β

a(x)

∂2

∂y2

[

a(x)s2α Yf

]

(x, y, s), (19)

with boundary conditions (9), (6), and (8)

Yf (x = 0, y, s) = Y 0f (y, s), (20)

∂Yf

∂y

∣

∣

∣

∣

∣

y= H2

=∂Yf

∂y

∣

∣

∣

∣

∣

(x=ξ−(y),y)

= 0. (21)

In the characteristic co-ordinates, the solution of interest Yf (ξ(y), y, s) - Laplace transform of the fuel massfraction perturbation felt at the flame - can then easily be obtained by applying the semigroup Tx (corre-

sponding to the right hand side of Eq. (19) - linear operator βa(x)

∂2

∂y2 with Neumann boundary conditions)

to the “initial”-condition Y 0f (y, s) as

Yf (x = ξ(y), y, s) = a(x)−s2α TxY 0

f (x = 0, y, s). (22)

Indeed, use the co-ordinateZf

.= a(x)

s2α Yf (23)

to transform the heat equation in to its standard form, albeit with x replacing the more standard t co-ordinate, with Neumann boundary conditions and initial condition Zf (x = 0, y, s) = a(0)

s2α Y 0

f (y, s) =

Y 0f (y, s); s arises simply as a parameter.

D. Discretization

xxxx

x x

y0

x0

y1

x1

y2

y3

x2 x3

Dis

cret

izat

ion

alon

g y

Discretization along x

Mean Flame

xxx x x

Figure 3. Schematic of the physical problem.

For the purpose of control analysis and design, we discretize the distributed problem (as shown in theFigure 3) to obtain an input-output map from the discrete control input vector

Y0

f [i](s).= Y 0

f (x = 0, y = yi, s), (24)

to the fuel mass fraction (computed at the flame) vector

Y f [i](s).= Yf (x = xi, yi, s) (25)

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and the flame perturbation vectorΞ[i](s)

.= Ξ(y = yi, s), (26)

where yiNi=1 is a discretization of the interval [h

2 , H2 ] for the upper flame and xi

.= ξ(y = yi). Using this

discretization, we construct the semigroup (matrix) X - solution of the heat equation - such that

Zf [i](s) =

N∑

j=1

Xij Y0

f [j](s), (27)

where the vector Zf [i](s).= Zf (x = xi, y = yi, s). Using Eqs. (27) and (23), we obtain

Y f [i](s) = a(xi)− s

2α

N∑

j=1

Xij Y0

f [j](s), (28)

which represents the discretized solution of the convection-diffusion Eq. (19).

Remark 1.

1. If the diffusivity β = 0, the matrix X is an identity, and the solution

Y f [i](s) = a(xi)− s

2α Y0

f [i](s), (29)

arises as a distributed delay and represents the fact that the individual “straight line” fuel trajectoriesconvect with a finite (x-dependent) delay 1

2αlog a(x) to the individual flame elements.

2. The matrix X is a doubly stochastic matrix,21 i.e., it has all positive entries (for β > 0), it preservesthe mass of the fuel, i.e,

N∑

j=1

Xij = 1, ∀ i;N

∑

i=1

Xij = 1, ∀ j, (30)

and its largest eigenvalue is 1 with a uniform fuel mass fraction φ0

.= [1, ..., 1] as the corresponding

(left and right) eigenvector.

3. In the limit as diffusivity β → ∞,

limβ→∞

Xij =1

N, (31)

and the flame sees a uniform mixture regardless of the control input’s spatial distribution y0f (y, t) - any

variations (along y) in the control input is instantaneously mixed to become uniform.

Finally, the discrete flame solution is obtained by integrating the Eq. (17) as

Ξ[i](s) = a(xi)1− s

2α

i∑

j=1

a(zj)s2α

−2 Ua

U0dzj − a(zj)

−2 S′T

S0T

Zf [j]dzj . (32)

E. Heat release

For a shallow flame, using Eq. (10), we have a formula for the integral heat release

Q(s) = g1Ξ[N ](s) + g2

N∑

i=1

Y f [i](s)dxi, (33)

where g1 = and g2 =(ST Yf )′

S0

TY 0

f

g1.20 On account of fuel perturbation alone (with acoustic perturbation Ua = 0

in Eq. (32)), the heat release response thus decomposes into two parts: 1) arising because of flame response,and 2) arising because of upstream fuel perturbation as it is felt downstream at the (nominally fixed) flame.

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Below, we analyze the two responses separately. We show that the part (1) due to the flame response alonearises as a delay, while the part (2) due to fuel perturbation alone arises as a weighted sum of parallel delays.

We begin by writing the heat release response on account of flame dynamics alone as

Q1(s).= g1Ξ[N ](s)

= g1(−S′

T

S0T

)a(L)−s2α

a(L)

N∑

j=1

N∑

k=1

a(zj)−2dzjXjkY

0

f [k](s)

(34)

= g1L(−S′

T

S0T

)e−sτ(µ) < φ(µ, β), Y0

f >, (35)

where τ(µ).= 1

2αlog(a(L)) is the delay and φ(µ, β)[k] = a(L)

L

∑Nj=1 a(zj)

−2dzjXjk. The expression in Eq. (35)shows that flame dynamics arise as a delay and in the general case, the flame does not respond to the fuelperturbations orthogonal to the direction φ(µ, β).

Remark 2.

1. For µ = 0,

τ(0) = limµ→0

1

2αlog a(L) =

L

U0, (36)

and φ(µ = 0, β = ·)[k] = 1N

phi0, the uniform vector. The proof follows by noting that for µ = 0, we

have a(·) = 1, dzj = LN

, and the term in square bracket in Eq. (34) can be simplified as

a(L)N

∑

j=1

N∑

k=1

a(zj)−2dzjXjkY

0

f [k](s)

=L

N

N∑

j=1

N∑

k=1

XjkY0

f [k](s)

=L

N< φ

0, Y

0

f > (37)

by the fuel conservation property of the doubly stochastic matrix X.

2. For very large diffusion limit (β → ∞), Xij → 1N

and

N∑

k=1

XjkY0

f [k] → y0, ∀ j ∈ [1, ..., N ]. (38)

where y0 = 1N

∑Nk=1 Y

0

f [k] corresponds to the total fuel. We have

a(L)N

∑

j=1

N∑

k=1

a(zj)−2dzjXjkY

0

f [k](s)

= a(L)N

∑

j=1

a(zj)−2dzjy0

= y0a(L)

∫ L

0

a(z)−2dz

= Ly0 =L

N

N∑

k=1

Y0

f [k] =L

N< φ

0, Y

0

f > . (39)

As a result, φ(µ = ·, β → ∞) = φ0

regardless of the value of µ.

3. For the general case, numerical evidence suggests that φ(µ, β) ≈ φ0.

We thus obtain

Q1(s) ≈ g1L

N(−

S′T

S0T

)e−sτ(µ) < φ0, Y

0

f (s) >= g1L(−S′

T

S0T

)e−sτ(µ)y0(s), (40)

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with equality either for the µ = 0 or β = ∞ limit. We next analyze the heat release perturbation on accountof fuel perturbation alone (nominally fixed flame)

Q2 = g2

N∑

i=1

a(xi)−

s2α

N∑

j=1

Xij Y0

f [j]dxi =< A(s), Y0

f > . (41)

where A(s)[j].= g2

∑Ni=1 e−sq(µ)[i]dxiXij is the transfer function from Y

0

f [j] → Q2. The expression simplifiessomewhat for the two extreme cases

Remark 3.

1. For β = 0 limit (no diffusion case), we have sum of parallel delays

Q2(s) = g2

N∑

i=1

e−sq(µ)[i]dxiY0

f [i] =< A0(s), Y0

f >, (42)

where q(µ)[i].= 1

2αlog(a(xi)) is a delay vector, and in the limit q(0)[i] = xi

U0. A0(s) is the resulting

heat release transfer function vector (“wide”) that highlights the rank 1 nature of the response.

2. For β = ∞ (diffusion dominant) limit, Xij = 1N

and we have control only along one direction (that ofuniform density). In particular, for this case,

Q2 = g2

N∑

i=1

a(xi)−

s2α dxi

1

N

N∑

j=1

Y0

f [j](s),

= g2

N∑

i=1

a(xi)− s

2α dxiy0(s), (43)

and on integration, we obtain the SISO response as

Q2 = g2U0(1 − e−τ(µ)(s−2α))

s − 2αy0(s) = A∞y0(s), (44)

where A∞(s) =< A(s), φ0

>. We note that identical transfer function also results (for arbitrary β) if

one were to restrict the control input Y0

f to the subspace of uniform (in y) functions.

F. Acoustics

For the purpose of this study, we assume a particularly simple, but relevant, empirical model of the acousticsas a lightly damped oscillator. Such a model structure arises naturally (see1, 9) if one is interested onlyin a single (typically bulk) acoustic mode. We go a step further and assume the oscillator model to infact describe the thermoacoustic model for premixed combustion. Indeed the heat release dynamics for thethermoacoustic problem arise because of the presence of Ua on the right hand side of the flame equation,that provides a feedback coupling to the acoustics. We incorporate this feedback in to an oscillator modelthat we write as

(s2 + 2ζω0s + ω20)Pa = sQc(s) + N(s), (45)

where Qc(s) is the scalar heat release perturbation entirely because of fuel control (obtained from equa-tions (32)-(33) after substituting Ua = 0 in Eq. (32) and N(s) models noise as motivated in.8, 9

IV. Fundamental limitations study

A. Control problem

Figure 4 depicts the block diagram of the feedback control problem. In the absence of fuel control (Qc(s) = 0in Eq. (45)), the assumed second order dynamics of the thermoacoustic model may be lightly damped or

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Thermoacoustics

aPN

Flame/Fuel cQ

][ˆ 0 ⋅

fY

Controller

Figure 4. Feedback control problem: design controller to stabilize the thermoacoustic plant and reject thedisturbance N (bold arrows denote vector signals).

even unstable. In the former case, with broadband noise N , large oscillations will result with spectral powerconcentrated near the frequency ω0. In the later case, depending upon the noise power and nature of limitingnonlinearities, large oscillations can once again arise as either a large noise driven limit cycle, or simply noisedriven oscillations (without a limit cycle) - see22 for a theoretical framework and23 for detailed analysis ofa specific system. In order to suppress the oscillations, one designs a feedback controller C(s) that uses

the pressure signal as its input, and generates the fuel distribution Y0

f [·](s) at its output. The disturbancerejection control objective for designing C is such that a) the block diagram depicted in Figure 4 is stable,and b) the sensitivity of the loop is suitably small at and near ω0, and not too large at other frequencies.The performance objective for the sensitivity function is illustrated in Figure 5.

ω0

∆ω

ε

S(jω)1

ω

α+<∞

1S

attenuation level

(perf. bandwidth)

Peaking not too large

Figure 5. Performance objective: Keep the Sensitivity function inside the shaded area.

We have shown in Eq. (35) that the heat release response due to flame dynamics alone arises muchlike a delay. The fundamental limitations arising for combustion systems with delay has been studied in7, 9

and here we restrict our attention to the study where the heat release response arises only due to the Q2

component (physical assumption where g2 >> g1). In the following sections, we are interested in establishingfundamental limitations on the control of these oscillations, due to the presence of physical processes ofconvection and diffusion. We assume that the thermoacoustic model (Eq. (45)), without control, is unstable(ζ < 0).

B. Effect of distributed delay

In order to study the fundamental limitation on account of the distributed delay, we consider the limit whereβ = 0. Noting,

Q2 =< A0(s), Y0

f (s) >= g2L

N< e−sq(µ), Y

0

f (s) > . (46)

Instead of analyzing this problem directly, we consider fundamental limitations for a problem where a portionof the controller is apriori fixed. Indeed the limitations in obtainable performance can be no worse thanwhen

Y0

f [i](s) = e−s(τ(µ)−q(µ)[i])Y1

f [i](s), (47)

where Y1

f (s) now represents the new control input vector. With such a control structure, we now obtain afixed delay τ(µ) for the dynamics, i.e,

Q2 = g2L

Ne−sτ(µ) < φ

0, Y

1

f (s) >= g2Le−sτ(µ)y1(s), (48)

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where y1(s).= 1

N

∑Nj=1 Y

1

f [j](s) is the new control input. We thus have a single-input-single-output (SISO)system with a constant delay together with an unstable pole, a situation that has been studied earlier in thecontext of fundamental limitations. As one of many typical results, we have for ζ < 0,

log ‖ S ‖∞≥ τ(µ)p, (49)

where p.= −ζω0, the real part of the unstable pole.12 Additional results, using area formula, appear in the

work of Freudenberg,24 and similar results for combustion control appear in7–9 and for cavity control appearin.10

Remark 4. In the above analysis, we have substituted the effect of distributed delay by a more convenientlargest delay τ(µ). One of the reasons for apriori assuming the control structure in Eq. (47) is that thedisturbance rejection problem, as it is stated, is somewhat ill suited for the β = 0 case. For instance,one solution for the stated problem is where control merely fuels the lip of the flame, where the delay isnegligible. Such a solution entirely circumvents the limitation on account of the distributed delay. However,this solution is not practical, as we would ideally like to distribute uniformly, over flame, the action of control.The assumed control architecture allows us to accomplish that (over all frequencies).

C. Effect of diffusion

In order to study the fundamental limitation where diffusion is dominant, we consider the limit whereβ → ∞. The results of such a study are also relevant where the control input is apriori restricted to the one-

dimensional uniform subspace (Y0

f [i] = Y0

f [j] = y0). Denoting y0(s).= 1

N

∑Nj=1 Y

0

f [j](s), the optimal control

analysis and design then is for single-input-single-output system, with input y0 and output Pa. Furthermore,using Eq. (44), the heat release response

Q2(s)

y0(s)= A∞(s) = g2U0

(1 − e−τ(µ)(s−2α))

s − 2α(50)

has right half plane zeroes z∞n ∞n=1 at

z∞n = 2α + iωn, (51)

where ωn.= 2nπ

τ(µ) . Applying the theory of fundamental limitations, we show that when ωn is near ω0, the

disturbance rejection problem becomes rather impossible. The unstable pole pair in Eq. (45) appears as anunstable zero pair of the sensitivity function. The sensitivity function can then be factorized as an all-passtransfer function Sap in series with a minimum phase transfer function Smp,

25 i.e.,

S(s) = Smp(s)Sap(s), (52)

where

Sap(s) =s2 + 2ζpω0s + ω2

0

s2 − 2ζ0ω0s + ω20

(53)

At each of the zeros z∞n , we have

1 = S(z∞n ) = Smp(z∞n )Sap(z

∞n ) (54)

andSmp(z

∞n ) = S−1

ap (z∞n ) (55)

If S is stable then Smp is analytic in CRHP and using Maximum Modulus theorem,26

‖ S ‖∞≥| S−1ap (z∞n ) |> 1. (56)

This shows that as the unstable pole of the thermoacoustic model approaches one of the unstable zeros (sayz∞1 ) of the fuel-flame path, we necessarily get peaking, so called waterbed effect, in the sensitivity function.Noting,

| S−ap1(z∞1 ) |≈|

z∞1 − p∗1z∞1 − p1

|.= log M, (57)

where p1.= −ζω0+iω0, and p∗1

.= ζω0 +iω0, we have the result that the peaking in the sensitivity explodes as

the right half plane pole gets close to the right half plane zero - just for stabilization. With the requirements of

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performance, sensitivity reduction in a frequency band (see Figure 5), the peaking in the sensitivity functionworsens. In order to compute a bound on ‖ S ‖∞, we use the Poisson integral relationship evaluated at thezero z1 (assumed here to be closest to the unstable pole),

log M = log | S−1ap (z∞1 ) |=

1

π

∫ ∞

−∞

log | S(ω) |2α

(2α)2 + (ω − ω1)2 dω, (58)

and it follows that in order to get performance, expressed in terms of attenuation-bandwidth product, of∆ω log( 1

ε), the peaking in sensitivity

log ‖ S ‖∞≥ C∆ω log(1

ε) + log M, (59)

where

C.=

1

π∆ω

∫ ω0−∆ω2

ω0−∆ω2

2α

(2α)2 + (ω − ω1)2 dω ≈

1

2πα, (60)

last approximation being valid, where ω0 ≈ ω1 and ∆ω << ω0.

D. Effect of diffusion + delay

In this section, we numerically investigate the transfer function from Y0

f → Q2 as diffusivity β increases.We have

Q2 =< A(s), Y0

f (s) >, (61)

where A(s)[j] = g2

∑Ni=1 e−sq(µ)[i]dxiXij is the transfer function from Y

0

f [j] → Q2. Indeed, this transfer

function is simply proportional to the delay e−sq(µ)[i]s in β = 0 limit and is given by A∞(s) =< A(s), φ0

>

(see Eq. (50)) in the β = ∞ limit. Note that while the delay has no finite zeros, the transfer function A∞(s)has a sequence of right half plane zeroes z∞

i . For a fixed µ, diffusion leads to two effects:

Appearance of finite and right half plane zeroes. As β increases from zero, the individual transfer

yi

βc(yi)µ=1

µ=2

µ=5

Boundary between right half and left half zeroes

Figure 6. Effect of diffusion on βc[yj ], the critical diffusivity value beyond which fuel injection at yj leads to a

right half plane zero in the heat release response. Non-dimensional parameter values - STU0

= 0.1, hH

= 0.25.

function A(s)[j] now has a sequence of finite zeros. An individual such zero (say z1[j]) is damped (stable) forextremely low values of β, crosses the imaginary axis for a critical value of βc[j], and z1[j] → z∞1 as β → ∞.Moreover, for any positive value of µ, we have

βc[N ] < βc[N − 1] < ... < βc[1] < ∞, (62)

where Figure 6 shows the dependence of βc on exothermicity (parameter µ). For values of β > βc[1], diffusionensures that right half plane zeros always arise irrespective of what shape (in y) one chooses for the controlinput. Figure 7 compares the transfer function magnitude and phases for fuel injection at yc = H+h

2 for afixed value of µ = 1 but for three values of diffusivity constant: β = 0, where the heat release response arisesas a delay, for β = β−

c [yc] just below the critical value, which shows a lightly damped left half plane zero,and for β = β+

c [yc] just above the critical value which shows a right half plane zero.

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American Institute of Aeronautics and Astronautics

Nor

mal

ized

mag

nitu

dePh

ase

-deg

rees

(solid); (dashed); (dash-dot) 0=β −= cββ += cββ

−= cββ

+= cββ0=β

0=β

Figure 7. Heat release response for fuel injection at yc = H+h2

as the diffusivity β is raised. Non-dimensional

parameter values - µ = 1, ST

U0= 0.1, h

H= 0.25.

Diminishing of control authority. In addition to the appearance of the right half plane zeros (forsufficiently large β), diffusion has the effect of decreasing the control gain in directions orthogonal to theuniform direction. Appearance of zeros (both LHP and RHP) in individual transfer functions is one man-ifestation of this effect. As we have already shown, in the direction of uniform fuel, the transfer functionis given by A∞(s) =< A(s), φ

0> irrespective of the value of parameter β. Figure 8 compares the A∞(iω)

against

A∞⊥(iω).= max

Y0

f∈φ⊥

0

< A(iω), Y0

f >

‖ Y0

f ‖, (63)

for µ = 0 and three different values of β. For β = 0, the transfer function A(s)[j] is proportional to a delay

with a (normalized) gain of unity. For such a case, with control input Y0

f restricted to the subspace spannedby φ

0, marginally damped (for µ = 0) zeroes arise and lead to performance limitations as shown in the

preceding section. The figure, however, shows that by suitably choosing control input in an orthogonal (toφ

0) direction, one can “recover” the control gain and hence bypass the limitation arising due to right half

plane zeroes. However, as β increases, the gain in the orthogonal direction decreases (because of mixing dueto diffusion) and for values of β ≥ 0.1, the limitations discussed for β = ∞ limit in the preceding Sectionapply for any practical control solution.

Frequency (ω)

Nor

mal

ized

mag

nitu

de

0

)(φ

ωiA∞

)( ωiA ⊥∞

)( ωiA ⊥∞0/)( φωiA∞ (solid) and (dashed)

0=β

01.0=β

1.0=β

Figure 8. Effect of diffusion on the control gain (A∞⊥(iω)) orthogonal to the uniform direction. Non-

dimensional parameter values - µ = 0, ST

U0= 0.1, h

H= 0.25.

V. Conclusion

In this paper, we have presented a reduced order heat release model that captures the physical effects ofconvection and diffusion in fuel transport. Our primary objective in doing so was to isolate non-minimumphase dynamics with the view of understanding fundamental limitations using fuel-based feedback control ofthermoacoustic instabilities. Results show that convection leads to the presence of distributed delay, whilediffusion leads to the presence of finite, and depending upon diffusivity and exothermicity constants, righthalf plane zeroes in the response.

We applied the theory of fundamental limitations to analyze the impact of non-minimum phase dynamics,in particular the right half plane zeroes, on stabilization and disturbance rejection of thermoacoustic insta-

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American Institute of Aeronautics and Astronautics

bilities. We showed that the presence of (sufficiently large) diffusion can lead to both poor control authority(on account of lightly damped zeroes, as well as low gain in directions orthogonal to uniform direction) andlarge peaking in sensitivity function (on account of right half plane zeroes).

VI. Acknowledgements

Supported by AFOSR grant FA9550-04-C-00442.

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