2 S. Evesque, AP Dow ling and AM Aweb.mit.edu/aaclab/pdfs/rsguide_final.pdf · Activ econ trol pro vides a w a y of extending their stable op erating range b y in terrupting the damaging

Self-tuning regulators for combustion

oscillations

By St�ephanie Evesque1y, Ann P. Dowling1 and Anuradha M.

Annaswamy2

1Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK2Department of Mechanical Engineering, Massachusetts Institute of Technology,

Cambridge MA 02139-4307, USA

Self-excited combustion oscillations arise from a coupling between unsteady com-bustion and acoustic waves, and can cause structural damage to many combustionsystems. Active control provides a way of extending their stable operating range byinterrupting the damaging thermo-acoustic interaction. The active controller con-sidered injects some fuel unsteadily into the burning region, thereby altering theheat release rate, in response to an input signal detecting the oscillation. Althoughthe feasibility of such control con�guration has been demonstrated on laboratory-scale experiments over �fteen years ago, the triple requirement for full-scale appli-cations is to adapt the controller response to varying operating conditions and toguarantee that the controller will cause no harm, while relying as little as possible ona particular combustion model. As a �rst step towards meeting these requirements,a Self-Tuning Regulator (STR) is proposed in this paper for a class of combustionsystems that satisfy some fairly non-restrictive assumptions. An open-loop transferfunction from the controller output voltage driving the actuator to the pressuresignal detecting the oscillation is derived, and is shown to be the product of a timedelay �tot and a transfer function that satis�es certain structural properties. Usingthe open-loop transfer function, it is shown that the stabilizing STR structure isthat of a simple phase lead compensator if �tot = 0, and the same compensatorcombined with a Smith Controller if �tot 6= 0. The adaptive rule for the adjustment

of the STR parameters is derived from a Lyapunov stability analysis. Promisingresults are obtained on a simulation based on a nonlinear premixed ame modeland on an experimental set-up which consists of a laminar ame burning in a Rijketube.

Keywords: active-adaptive control, self-tuning regulators, combustion

instability, Lyapunov stability, Rijke tube

1. Introduction

In order to meet stringent emission requirements, combustors are increasingly be-ing designed to operate in a lean premixed mode. Although this reduces the NOx

emissions, it has the disadvantage that premixed ames are particularly suscepti-ble to self-excited oscillations. These oscillations create large-scale pressure waves

y Present address: Lehrstuhl f�ur Thermodynamik, Technische Universit�at M�unchen, 85747

Garching, Germany

Article submitted to Royal Society TEX Paper

2 S. Evesque, AP Dowling and AM Annaswamy

which can cause structural damage to the combustion system. These self-excitedcombustion oscillations result from an interaction between unsteady combustionand acoustic waves: unsteady combustion generates sound, while acoustic waves re- ected from the boundaries of the combustor perturb the combustion still further.This problem is currently being experienced by all manufacturers of industrial gasturbines and is leading to long and costly development and commissioning times.The next generation of aeroengine combustors is also required to have low NOx

levels and damaging self-excited oscillations are also being experienced on devel-opment engines. Passive dampers are currently used within the industry but theyrequire substantial tuning and have a limited operational range in which they canstabilise the combustion system.

An alternative solution is to use active control to extend the stable operatingrange of a combustion system, by interrupting the damaging coupling between un-steady heat release and acoustic waves. The most practical active controller injectsunsteadily some fuel into the burning region, modifying thereby the unsteady heatrelease, in response to a sensor signal detecting the oscillation. However, to be use-ful in practice, an active controller needs to be e�ective across a range of operatingconditions. An eÆcient approach is to use an active-adaptive controller in which thecontroller transfer function is continually altered as the engine condition changes.One of the most popular adaptive schemes proposed for active control of combus-tion instability is the Least Mean Squares (LMS) algorithm applied to a FIR (FiniteImpulse Response) or IIR (In�nite Impulse Response) �lter. The LMS is very at-tractive because it does not require any theoretical model: the combustion processis considered as a `black box' and is learnt during a system identi�cation procedure,performed o�-line (Billoud et al. 1992) or on-line (Kemal & Bowman 1996; Evesque& Dowling 2001). However, the major drawback of the LMS controller, as well asother minimization schemes proposed for adaptive active combustion control (Blon-bou et al. 2000; Padmanabhan et al. 1996), is that no theoretical guarantee on thelong-term stability of the controller can be provided.

An eÆcient way to prevent any divergence of the adaptive control scheme is to

use systematic methods for designing stable adaptive systems. The adaptive con-troller, called STR (Self-Tuning Regulator) by Annaswamy et al.(1998), is designedbased on a Lyapunov stability analysis and is therefore guaranteed to be stable forany operating conditions. Furthermore, the STR has the advantage of avoiding asystem identi�cation procedure (which is one of the main diÆculties in implement-ing a LMS controller as discussed in details in (Evesque & Dowling 2001)), since ituses little information about the physical process. However, so far, only a speci�csimple combustor has been shown to have the structure required for the design ofa STR (Annaswamy et al. 1998).

Our purpose in this paper is to develop an active controller that is adaptableto varying conditions, and also guaranteed to cause no harm to the system, whilerelying as little as possible on a particular combustion system. We design there-fore an active-adaptive controller that can be applied to a class of combustors andactuation systems. We do this by determining the general features of a class ofself-excited combustion systems, rather than investigate a particular combustor indetail. This involves deriving a transfer function representing the open-loop process,which is the combination of the combustion system, the sensor and the actuator.The features of the open-loop process are then exploited to design a novel adaptive

Article submitted to Royal Society

Self-tuning regulators for combustion oscillations 3

controller, denoted STR, that is guaranteed to stabilise this class of combustionsystems, the major challenge being to guarantee stabilisation when there is a sig-ni�cant time delay between actuation and detection. The control design involves�rstly the choice of a �xed controller structure that is suÆcient to stabilise the sys-tem, and secondly the determination of adaptive laws for the controller parametersguaranteed to converge to stabilising values.

Hence, the paper is divided as follows: in x2, a class of self-excited combustionsystems is described, and some open-loop properties useful for control design arehighlighted. Section 3 describes the STR design. In x4, the performance of the STRis tested on a simulation based on Dowling's (1999) premixed ame model. In x5,the STR is implemented on an experimental set-up which consists of a laminar ame burning in a Rijke tube, i.e. a vertical cylindrical tube open at both ends.

2. General features of self-excited combustion systems with

actuation

(a) Derivation of the open-loop transfer function

A class of combustion systems, including lean premixed prevapourised (LPP)combustors and aeroengine afterburners, can be modelled as a combustion sectionembedded within a network of pipes, as shown in �gure 1. We will investigate linearlow frequency perturbations to the ow in such a pipework system.

The ow at inlet to the combustor is assumed to be isentropic, the frequenciesof interest are low and the combustion zone is short compared with the wavelength.In practice, the most challenging combustion instabilities occur at lean premixedconditions where the combustion is concentrated near the inlet of the combustor.Moreover, since only plane waves transport acoustic energy, it is suÆcient to con-sider one-dimensional disturbances. The pressure and velocity upstream the amecan therefore be written as a linear combination of the waves g and f , and down-stream the ame as a linear combination of the waves h and j:in �xu < x < 0,

p(x; t) = p1 + f

�t�

x

c1 + u1

�+ g

�t+

x

c1 � u1

�

u(x; t) = u1 +1

�1c1

�f

�t�

x

c1 + u1

�� g

�t+

x

c1 � u1

��(2.1)

in 0 < x < xd,

p(x; t) = p2+h

�t�

x

c2+u2

�+j

�t+

x

c2�u2

�

u(x; t) = u2+1

�2c2

�h

�t�

x

c2+u2

��j

�t+

x

c2�u2

��(2.2)

p,u,� and c are respectively the pressure, velocity, density and speed of sound. Theoverbar denotes a mean value, while 0 denotes a uctuation. The suÆces 1 and 2denote ow quantities just upstream and downstream of the combustion zone. Thedistances xu and xd are the lengths upstream and downstream of the combustionzone.



The boundary conditions of the combustor are characterized by upstream anddownstream pressure re ection coeÆcientsRu andRd respectively. Since this bound-ary condition neglects the conversion of combustion generated entropy waves intosound at any downstream nozzle, we expect it to be a good approximation whenthe time taken for entropy waves to convect through the straight duct, xd=�u2, ex-ceeds their di�usion time. We consider linear disturbances with time dependenceest, where s = i! is the Laplace variable, and ! is the complex frequency. For gen-eral boundary conditions of the combustor, Ru and Rd are frequency dependent,and therefore the re ected waves f and j are easily obtained from g and h usingthe following relationships:

f(t) = Ru(s)[g(t� �u)] at x = �xu

j(t) = Rd(s)[h(t� �d)] at x = xd; (2.3)

where �u = 2xu= �c1(1 � �M12) and �d = 2xd= �c2(1 � �M2

2) are respectively the

upstream and downstream propagation time delays, �M is the mean ow Machnumber, and Ru(s)[:] denotes an operator of the variable s = d

dt: for example,

x(t) = 1s+1

[y(t)] means that dxdt(t) + x(t) = y(t).

The equations of conservation of mass, momentum and energy across the short ame zone at x = 0 can be written in a form that is independent of downstreamdensity and temperature (Dowling 1997):

p2 � p1 + �1u1(u2 � u1) = 0

� 1(p2u2 � p1u1) +

1

2�1u1(u

22 � u

21) =

Q

Acomb

: (2.4)

Q is the instantaneous rate of heat release, Acomb is the combustor cross-sectionalarea and is the ratio of speci�c heat capacities. Substitution from (2.1), (2.2)and (2.3) into (2.4), making use of the isentropic condition p1=�

1 = p2=�

2 , andlinearising in the ow perturbations give the time evolution of the outgoing wavesg and h generated by the unsteady heat release Q(t):�

X11 X12

X21 X22

��g(t)

h(t)

�=

�Y11Ru Y12Rd

Y21Ru Y22Rd

��[g(t� �u)]

[h(t� �d)]

�+

0

Q(t)�Q

Acombc1

!(2.5)

where Xij and Yij are constant coeÆcients depending on the mean ow only, andare given in Appendix A.

After taking the Laplace transform of the system (2.5) (the same notation isused for a temporal signal and its Laplace transform, for example u(t) and u(s))and using (2.1) and the boundary condition (2.3), one obtains the transfer function

G(s) =u1(s)

Q(s)=

(RdY12e�s�d �X12)(Rue

�s�u � 1)

Acomb��1�c21 det(S)

(2.6)

where

S =

�X11 �RuY11e

�s�u X12 �RdY12e�s�d

X21 �RuY21e�s�u X22 �RdY22e

�s�d

�(2.7)

G(s) describes the acoustic response of the duct, ie the unsteady velocity u1(t) atthe ame due to the unsteady heat release Q(t).



Since the self-excited oscillation results from a coupling between unsteady heatrelease and acoustic waves, the forcing of unsteady heat release due to incoming owdisturbances at the ame must be also described. In many applications, the com-bustion responds most strongly to velocity uctuations. This is because in acousticwaves the fractional change in ow velocity is order �M�1 larger than the fractionalchange in pressure, ie a large factor at the low Mach numbers at which combus-tion can be sustained. This dependence on ow velocity can either be seen directlythrough its in uence on ame kinematics and shape in perfectly premixed systems(Flei�l et al. 1996; Dowling 1999) or indirectly through its in uence on fuel-air ra-tio and hence on the rate of combustion in partially premixed systems (Richard &Janus 1998; Venkataraman et al. 1999; Lieuwen et al. 1998). A transfer function

H(s) =Q(s)

u1(s)(2.8)

is introduced to describe this combustion response. In many circumstances, H(s)will include substantial time delays. Models for the ame transfer function H(s)have been published in the literature for di�erent combustors. However, the subjectof the present work is not to study the ame dynamics in details. Therefore, we willkeep the general form H(s) in this paper. The eigenfrequencies of the self-excitedcombustion system alone can be determined by combining equations (2.6) and (2.8).They satisfy

1�G(s)H(s) = 0: (2.9)

In order to apply active control to the self-excited combustion system shown in�gure 1, an actuator is used to inject an additional perturbation and hence inter-rupt the damaging coupling between unsteady combustion and acoustic waves. Thetwo most commonly used active control inputs are loudspeaker forcing and fuelforcing. E�ectiveness of the fuel injection technique has been extensively shownexperimentally (Langhorne et al. 1990; Sivasegaram et al. 1995; Banaszuk et al.

1999; Paschereit et al. 1999) and fuel forcing has been used in most full-scale de-montrations of active combustion control (Hantschk et al. 1996; Moran et al. 2000;Seume et al. 1998). Therefore we will concentrate on fuel forcing which is the mostrelevant method of actuation for practical applications. An actuator is driven toprovide extra fuel (sometimes mixed with air) which in turn produces additionalheat release.

We assume that the fuel injection is arranged so that the voltage Vc sent to theactuator results in an additional uctuating heat release Qc through the followingtransfer function

Qc(s)

Vc(s)=Wac(s)e

�s�ac ; (2.10)

where Wac(s) represents the actuator dynamics. Typically the actuator will be avalve with the characteristics of a mass-spring-damper system, whose dynamics aredescribed by the transfer function Wac(s). Wac(s) is therefore assumed to be of lowrelative degree and to have no unstable zeros (i.e. no zeros in Re(s) � 0). If a fuel-air mixture is injected directly into the combustion zone, the combustion response



will be instantaneous (�ac = 0). However, if only fuel is added, there will be a smallmixing time delay before it is burnt (�ac > 0). Often it is hazardous to inject fueldirectly into the ame. If the additional fuel is introduced some distance upstream ofthe combustion zone, there will be a convection time delay �ac between injection andcombustion. In a LPP system, it is convenient to modulate the main fuel suppliedin the premix ducts, in which case �ac may be a signi�cant proportion of the periodof the self-excited oscillations. Note that �ac is assumed to be independent of the ame radial position, which means that the same time delay occurs between all fuelinjection and its combustion. When the combustion zone is short this is triviallysatis�ed. If the combustion zone is extensive, it may be necessary just to injectthe fuel for control action in a localised region to meet this constraint. Althoughequation (2.10) and the condition that Wac(s) is of low relative degree with stablezeros are restrictive, they can be considered as design requirements of the actuationsystem.

In order to describe the impact of this input on the combustion system charac-teristics, we study the relationship between Vc, a voltage sent to the actuator, andPref , the uctuating pressure measured at a location xref (see �gure 1). That is,our goal is to characterize the transfer function

W (s) =Pref (s)

Vc(s); (2.11)

which represents the actuated open-loop combustion process, and which is repre-sented schematically in �gure 2. We derive this transfer function in the following.

In addition to the heat release Qc due to the actuation described in (2.10), thereis also unsteady heat release driven by the ow uctuations. We will denote thisnaturally occurring unsteady rate of heat release by Qn. It is related to the velocity

uctuations by the ame model in (2.8):

H(s) =Qn(s)

u1(s): (2.12)

For linear uctuations, we can superimpose the uctuating heat release due toexternal actuation Qc and the naturally occuring heat release Qn to give the total uctuating rate of heat release Q:

Q(s) = Qc(s) +Qn(s): (2.13)

The acoustic waves generated by Q(s) are described by (2.6), ie

G(s) =u1(s)

Q(s): (2.14)

From equations (2.10)-(2.13), one obtains that:

u1(s)

Vc(s)=G(s)Wac(s)e

�s�ac

1�G(s)H(s): (2.15)

If the unsteady pressure Pref is measured upstream the ame (xref � 0), thenPref is a linear combination of the upstream waves f and g. Using (2.1) and the



boundary condition (2.3) at x = �xu, one easily obtains:

Pref (s)

u1(s)= ��1 �c1

1 +Rue

�2s(xu+xref )

�c1(1��M21)

Rue�s�u � 1

e

sxref

�c1��u1 = F (s)esxref

�c1��u1 (2.16)

Therefore, it follows from equations (2.15) and (2.16) that the open-loop transferfunction of the actuated system given in (2.11) can be written in the form:

W (s) =Pref (s)

Vc(s)=W0(s)e

�s�tot ; (2.17)

where

�tot = �det + �ac (2.18)

is the total time delay in the actuated system, �ac is the time delay due to theactuation and �det = �xref=(�c1��u1) is the detection time delay due to the pressure

measurement location, and

W0(s) =F (s)G(s)Wac(s)

1�G(s)H(s); (2.19)

with G(s), H(s) and F (s) given in equations (2.6), (2.8) and (2.16) respectively. IfPref is measured downstream the ame (xref > 0), then Pref is a linear combinationof the downstream waves h and j. A simple calculation, using the wave structure inequations (2.1) and (2.2) and the continuity condition across the combustion zonein equation (2.4)a, shows that the transfer function Pref (s)=Vc(s) again has theform given in equation (2.17), but this time we have �det =

xref

�c2+�u2, and

W0(s) =F (s)Pdu(s)G(s)Wac(s)

1�G(s)H(s)(2.20)

where

Pdu(s) =�X11+RuY11e

�s�u

X12�RdY12e�s�D

1+Rde�2s

xd�x

ref

�c2(1��M22)

1+Rue�2s

xu+xref

�c1(1��M21)

(2.21)

(b) Properties of the open-loop system, useful for control design

The open-loop transfer function W (s) = Pref (s)=Vc(s) which represents theself-excited combustion system with actuator and sensor has been derived. It isthe product of a time delay �tot and a transfer function W0(s). We show now thatunder four fairly non-restrictive assumptions, a �nite dimensional approximationto W0(s) satis�es three properties that are very useful for control design. The fourassumptions made are:(i) The pressure re ection coeÆcients Ru(s) and Rd(s) at the upstream and down-stream boundaries of the combustor are rational and satisfy the following conditions



(given here for Ru(s) only):

jRu(s)j < 1 in Re(s) � 0

n� (Ru(s)) = 0;

jk0(Ru(s))j < 1; (2.22)

where the notation n�(r(s)) indicates the relative degree of a rational transfer

function r(s), which is equal to the number of poles of r(s) minus the number ofzeros of r(s); k0(r(s)) denotes the high-frequency gain of a rational transfer functionr(s) which is de�ned as

k0(r(s)) = lims!1

sn�(r(s))

r(s):

Physically, the conditions in (2.22) mean that the amplitude of the re ected wave ata combustor boundary is smaller than the amplitude of the incoming wave. Simpleduct terminations like open and choked ends trivially satisfy these conditions, pro-vided appropriate energy loss mechanisms are included. Conditions (2.22) are alsosatis�ed by re ection from general pipework con�gurations with negligible mean ow (see proof in Appendix B).(ii) The ame is stable when there is no driving velocity u1, i.e., the operator H(s),de�ned in equation (2.8), is stable.(iii) The ame response has a limited bandwidth, ie H(s)! 0 when s!1.(iv) The actuator dynamics Wac(s) is of low relative degree and has no unstablezeros. This can be achieved by a simple valve modulating the fuel supply.The assumptions (ii) and (iii) on the structure of the ame transfer function H(s)�t many ame models given in the literature, including premixed ames (Flei�let al. 1996; Dowling 1999) and LPP systems (Schuermans et al. 1990; Lieuwen &Zinn 1998; Peracchio & Proscia 1998; Hubbard & Dowling 1998). Physically, (ii)means that any hydrodynamic instabilities of the ame are neglected. It is only theinteraction with the acoustic waves that leads to instability.

The control design proposed in this paper assumes a linear �nite dimensionalexpression of the open-loop transfer function W0(s). Therefore, the `in�nite dimen-sional' expression of W0(s) given in equation (2.19) and which contains many timedelays needs to be expanded into a rational form. This is achieved by applying thePad�e approximation technique (Baker & Graves-Morris 1996) for each exponentialterm of W0(s). This technique has been widely used in handling systems with timedelays. We will use the notation [L=M ]f(s) to denote the (L;M)th order Pad�e ap-

proximant of a function f(s), which is a rational function whose numerator hasorder L and denominator order M . This rational function is chosen such that the�rst L+M + 1 terms in its power series will match those of f(s), ie

f(s) =

�L

M

�f(s)

+O(sL+M+1): (2.23)

Equation (2.23) shows that the frequency range over which the Pad�e approximationis valid increases with the value of L and M . The rational approximator, which weshall denote as W0r (s), is obtained from W0(s) using such a Pad�e expansion, byapproximating all terms of the form of e�s� by a rational function [M=M ]e�s� in



W0(s). The ame response function can be approximated by a rational functionHr(s), where assumptions (ii) and (iii) imply that (ii0) Hr(s) has stable poles, and(iii0) n�(Hr(s)) > 0, respectively. The resulting W0r (s) can be written as

W0r (s) = k0Z0(s)

R0(s)(2.24)

where k0 is the high frequency gain, and Z0(s) andR0(s) are two polynomials whoseleading coeÆcient is equal to one (the polynomials are then denoted as 'monic'),and are `coprime', i.e., have no common factors. In Appendix C, it is proved that,under the assumptions (i)-(iv),W0r (s) in (2.24) satis�es three useful properties thatare given below.Property 1. The zeros of Z0(s) are stable (ie are situated in the s-half-plane

Re(s) < 0).

Property 2.

n� (W0r (s)) = n

� (Wac(s)) :

Property 3.

sign (k0 (W0r (s))) = sign (k0(Wac(s))) :

Since the control design discussions (carried out in section 3) are predicatedsolely on W0r (s) while the main goal is to control the actual system which is char-acterized by W0(s), it is important to ensure that W0r (s) is a good approximationof W0(s). Eq. (2.23) implies that for a given order of the transfer function W0r (s),W0(s) behaves like the rational approximationW0r (s) over a �nite frequency range[0; !0]. It is therefore important to ensure in a given application that over the fre-quency range of interest, W0r (s) is indeed a valid approximation. This is discussedin section 4 where the STR is tested on a simulation in which fuel actuation isused. In section 5, the STR is tested on an experimental set-up with loudspeakeractuation, and there we investigate the in�nite dimensional plant W0(s) and �ndby inspection that it meets the conditions for the use of a STR over the frequencyrange of interest.

There is a straightforward reason why the open-loop transfer function W0r (s)satis�es properties 1 and 2. For Re(s) � 0, the amplitudes of the oscillations do notdecrease with time and assumption (i) regarding the combustor boundary condi-tions ensure that the largest contribution to Pref is from the acoustic wave leavingthe combustion zone, rather than the waves subsequently re ected from the bound-aries. Under these circumstances, the main structure of W0(s) in equations (2.19)and (2.20) is dominated by the properties of Wac(s), the actuator dynamics, theother multiplying factors do not introduce unstable zeros nor a�ect the relativedegree.

It is interesting to note that this argument remains true if the form of actua-tion is a loudspeaker, provided the loudspeaker is located within the combustionzone. However, this situation is more complicated when the loudspeaker is at ageneral axial position in the combustor. Then, since the combustion zone is an ac-tive component, it can re ect a wave of greater amplitude than the incident wave:if Rf denotes the re ection coeÆcient at the ame, jRf j > 1 is possible even inRe(s) � 0 (Poinsot et al. 1986). Under these circumstances, Pref=Vc can have un-stable zeros for some positions xref . Property 1 is therefore not true for general



loudspeaker axial positions. This is consistent with the observations of Annaswamyet al. (1997) who calculated Pref=Vc for a particular idealised combustor with loud-speaker actuation. They found that no simple relationship could be derived betweenthe locations of the sensor, actuator and ame, and the zeros stability. For instance,for the particular case of sensor and actuator collocated at the ame, their open-loop plant Pref=Vc had no unstable zeros. However, for the particular case of fuelforcing, the open-loop transfer function satis�es the properties 1, 2 and 3, whichgreatly help in the control design, as shown in the following section.

3. STR design

We have shown that, for a class of combustion systems with appropriate fuel actua-tion, the open-loop transfer function from the voltage Vc driving the fuel injector tothe pressure measurement Pref is the product of a pure time delay �tot and a trans-fer function W0(s). After W0(s) has been made rational through Pad�e expansions,the open-loop transfer function is described by

Pref (s)

Vc(s)=W0(s)e

�s�tot ' k0Z0(s)

R0(s)e�s�tot ; (3.1)

with k0 a constant, Z0 and R0 two coprime and monic polynomials, and �tot aknown time delay. R0 has degree n (possibly very large), and Z0 has degree n�n

�,with n

� equal to the relative degree of the actuator (property 2). We assume asimple actuation system, for example a valve modelled by a mass-spring-damperand take n� = 1 or 2. Further, Z0 is a stable polynomial (property 1), and the signof k0 is known, since the sign of k0(Wac) is known in a given application (property3). For ease of exposition, we assume in the following that k0 is positive.

When �tot is zero, it is easy to show (see details in Appendix D) that the followingadaptive controllers stabilize the system when n

� = 1 and n� = 2, respectively:

Vc(t) = kT (t):d(t)

_k(t) = �Pref (t)d(t)

9=; (n� = 1) (3.2)

Vc(t) = kT (t):d(t) + _k(t)T :da(t)

_k(t) = �sign(k0)Pref (t)da(t)

9=; (n� = 2) (3.3)

where

da(t) =1

s+ a[d(t)]; d(t)T = [Pref (t); V (t)];

V (t) =1

s+ zc[Vc(t)]; k(t)T = [k1(t); k2(t)]:

Both controllers are �rst order compensators whose gain and phase are modulatedby the parameters k1(t) and k2(t), respectively. Such a �rst order compensator isschematically represented on �gure 3.

With a delay, these controllers become inadequate, especially when �tot is ofthe order of the period of the unstable mode. Control of systems in the presence



of time delays has been extensively studied (Smith 1959; Manitius & Olbrot 1979;Ichikawa 1985). A popular approach to accommodate large delays is due to Smith(1959). The Smith Controller (SC) attempts to estimate future outputs Pref ofthe system using a known model, and provides an appropriate stabilization action.In (Manitius & Olbrot 1979), the SC was modi�ed to control systems that are open-loop unstable by using �nite-time integrals of inputs Vc to estimate future outputsPref (the goal of this modi�cation was to avoid unstable pole-zero cancellations).Ichikawa (1985) developed a SC controller with �nite-time integrals to serve asa pole-placement controller. In (Ortega & Lozano 1988; Ichikawa 1986), adaptiveversions of the pole-placement controller have been derived and proved to be stable.The approach from Ichikawa (1985) has been recently applied to a model-based�xed pole placement controller for combustion instabilities (Hathout et al. 2000).Inspired by these results, we shall design a STR controller for the class of systemsin (3.1), by proceeding in two stages. In the �rst stage, a controller structure ischosen that is guaranteed to stabilize the system for a certain choice of the controlparameters. In the second stage, an adaptive rule for adjusting these parameters isdetermined.

(a) Controller structure

The controller structure for time-delay systems that was suggested by Ichikawa(1985) has the following form:

Vc(t)=�C(s)

E(s)[Vc(t��tot)]�

B(s)

E(s)[Pref (t)]+

0Z��tot

�(�)Vc(t+�) d� (3.4)

where the polynomials C(s), B(s) and E(s) are of degree n � 2, n � 1, and n,respectively, and are chosen such that the denominator of the closed-loop transferfunction matches a chosen stable polynomial. This means that even for a plant oflow relative degree (n� = 1 or 2), if the plant order n is large, then the controllerdynamics C=E and B=E will be of high order. Since in (3.1), n is bound to be verylarge after a Pad�e approximation, our goal is to design a low-order controller ratherthan a pole placement controller as in (3.4). Therefore, the controller structure ischosen as

Vc(t) = �k1Pref (t)�k2

s+ zc[Vc(t)] + VSC(t) (3.5)

where the �rst two terms are similar to (3.2), and VSC(t) is an additional termused to accommodate the presence of the delay. Since the combustion system canbe unstable without control, VSC(t) is implemented using a �nite-time integral(Manitius & Olbrot (1979), Ichikawa (1985)) as

VSC(t) =

0Z��tot

�(�)Vc(t+ �) d�; (3.6)

where �(�) is a weighting function. We note that for a choice of �(�)

�(�) =

nXi=1

�ie��i�; (3.7)



where the �i are the n zeros of the polynomial R0(s), we obtain that

VSC(t) =

�n1(s)

R0(s)�

n2(s)

R0(s)e�s�tot

�[Vc(t)]; (3.8)

where

n1(s)

R0(s)=

nXi=1

�i

s� �i

n2(s)

R0(s)=

nXi=1

�i

s� �ie�i�tot : (3.9)

Using the controller structure given in equations (3.5) and (3.8) and shown in�gure 5(a), the closed-loop transfer function is given by

Wcl(s) =k0(s+ zc)Z0(s)e

�s�tot

Rcl(s)=Wcl0(s)e

�s�tot ; (3.10)

where

Rcl(s) = A(s) +B(s)e�s�tot ; (3.11)

with

A(s) = (s+ zc)(R0(s)� n1(s)) + k2R0(s) (3.12)

B(s) = (s+ zc) (n2(s) + k1k0Z0(s)) : (3.13)

We prove in the following that Rcl(s) has stable roots for a small �tot.If �tot = 0,

Rcl(s) = (s+ zc + k2)R0(s) + k1k0(s+ zc)Z0(s) (3.14)

and can be shown to be stable for certain choices of k1 and k2 by using simpleroot locus arguments and properties 1-2 (Evesque 2000). If �tot 6= 0, continuityarguments can be used to establish that Rcl(s) is still stable, as follows. Since thepolynomial n2(s) is of degree n� 1, its coeÆcients can be chosen such that

n2(s) = �k1k0Z0(s); (3.15)

which implies, from equation (3.13), that B(s) = 0 and therefore Rcl(s) = A(s).Furthermore, it is clear from equation (3.9) that the polynomials n1 and n2 of theSC are linked. More precisely, the choice of n2 in equation (3.15) imposes restrictionson n1: it means that the coeÆcients �i, and hence n1, are proportional to k0k1.Further, when �tot = 0, n2 = n1. Therefore, we will emphasise this scaling bywriting:

n2(s) = n1(s) + �totk0k1n3(s); (3.16)

where n3 is a polynomial of degree n � 1, with �nite coeÆcients. Using (3.15)and (3.16) in (3.11), the closed loop poles are the roots of

Rcl(s) = (s+zc+k2)R0(s)+k1k0(s+zc) [Z0(s)+�totn3(s)] : (3.17)



For `small' �tot, i.e. for �tot!z <O(1) where !z is the highest frequency amongthe zeros of Z0, the zeros of Z0(s) + �totn3(s) are close to the zeros of Z0(s), andhence are stable. Therefore, for jk1j large, n� 1 zeros of Rcl(s) are stabilised. The2 remaining zeros of Rcl(s) are obtained at large s: indeed, k�11 (s+ zc+k2)R0(s) isnot negligible compared to (s+ zc)(Z0(s) + �totn3(s)) when s �O(k

�11 ), and this is

where we �nd the 2 remaining zeros of Rcl(s). Looking at the large jsj asymptoticform of Rcl(s) shows that the 3 highest coeÆcients of Rcl(s) after division by sn�1

are

s2 + (k2 + k1k0�totn3;1)s+ (k1k0 + k1k0�totn3;1zc) (3.18)

where n3;1 is the highest coeÆcient in n3(s). The 2 remaining roots of Rcl(s) arestable if the polynomial (3.18) is stable, i.e. if

k2 + k1k0�totn3;1 > 0

k1k0(1 + �totn3;1zc) > 0: (3.19)

In the range of values of �tot considered (i.e. �tot!z<O(1)), one easily checks thatn3;1 �+1 and that equation (3.19) is satis�ed for some k1>0 and k2>0. Therefore,we proved that for `small' �tot, all the closed-loop poles, ie the roots of Rcl(s), arestabilised for some k1>0 and k2>0. That is, the plant as in (3.1) of relative degreen� � 2 with a time delay �tot not too large can be stabilized by the controller in

(3.5) and (3.6) (shown in �gure 5(a)).In practice, the constraint on the size of �tot is not overly conservative: in the

simulation in section 4, control is obtained for values of �tot up to 3 periods ofoscillations (see details in section 4), while in the experiment described in section5, control is obtained for �tot up to 5 cycles of oscillations.

(b) Derivation of the adaptive rule for the control parameters

The controller structure given in �gure 5(a) includes �xed controller parametersk1, k2, �i and �i which need to be chosen based on the plant parameters. Undervariations in the operating conditions, and also to ensure that the STR design is in-dependent of the details of the modelling, it is more appropriate to determine thosecontrol parameters adaptively. Hence, we suggest the adaptive controller shown in�gure 5(b), which is of the form

Vc(t) = �k1(t)Pref (t)� k2(t)V (t) + VSC(t) (3.20)

where V (t) = 1s+zc

[Vc(t)] and VSC(t), in lieu of (3.6), is approximated as follows:

VSC(t) =

NXi=1

�i(t)Vc(t� idt): (3.21)

For ease of exposition, we denote the controller parameters and data vectors k andd, respectively, as

� k(t)T = [�k1(t);�k2(t); �N (t); :::; �1(t)], and its error vector ~k = k�k�, and

� d(t)T = [Pref (t); V (t); Vc(t�Ndt); :::; Vc(t� dt)],



so that Vc(t) = kT (t):d(t). The discussions in x4(a) imply that for a constant valuek�, if k = k�, the closed-loop system is stable for a small �tot. Here, our goal isto determine an update-rule for k so that it stabilizes the self-excited combustionsystem.

General case : n�(W0r ) � 2. From x4(a), it follows that for a constant valuek�, the closed-loop system is stable for a small �tot. Denoting the correspondingtransfer function Wcl0 which is de�ned in equation (3.10) as Wcl�0

, it follows thatWcl�0

has stable poles and stable zeros with n�(Wcl�0

) = n�(W0r ) = 2. W �

cl0can be

made to e�ectively have relative degree 1 by using a modi�ed control signal Vc, asillustrated in �gure 6. This leads to the following underlying error model :

Pref (s) =Wm(s)e�s�tot [~kT (t):da(t)]; (3.22)

where da(t) =1

s+a[d(t)] and Wm(s) = (s+ a)W �

cl0(s) has relative degree 1 and is

SPR (Strictly Positive Real).However due to the presence of the time-delay �tot, we note that there can be

a signi�cant lag between the parameter error ~k and the output error Pref . As aresult, one cannot adopt the same approach as in Appendix D (case �tot = 0) toestablish stability. We note that it is possible to rewrite (3.22) as

Pref (s) =Wm(s)[~kT (t� �tot):da(t� �tot)]: (3.23)

If k(t) were to be adjusted as

_k(t� �tot) = �Pref (t)da(t� �tot) (3.24)

then the same approach as in Appendix D could be used to establish stability.However, equation (3.24) is a non-causal algorithm which obviously cannot be im-plemented. Suppose we choose, instead, an algorithm of the form

_k(t) = �Pref (t)da(t� �tot) (3.25)

even though equation (3.24) is the desired algorithm. Comparison of these twoforms suggests that provided that suÆcient bounds are maintained on ~k and itsderivatives over the interval [t � �tot; t], then it should be possible to establishstability of the update algorithm (3.25). Based on such a reasoning, Niculescu &

Annaswamy(2002) added a term0R

��tot

tRt+�

k_~k(�)k2d�d� to the Lyapunov function.

We use the same approach in Appendix E to establish stability of the closed-loopsystem. In summary, the adaptive controller in the presence of a delay is given by

Vc(t) = kT (t):d(t) + _k(t)T :da(t)

_k(t) = �Pref (t)da(t� �tot): (3.26)

The adaptive law in (3.26) can be generalised for an arbitrary sign of the highfrequency gain k0 as follows:

_k(t) = �sign(k0)Pref (t)da(t� �tot): (3.27)

It should be noted that as the time delay �tot is increased, N in (3.21) must beincreased and hence, more controller parameters are required.



Particular case : n�(W0r ) = 1. In this case, the manipulation on Vc shownin �gure 6 is not required, and therefore the following simpler algorithm can beimplemented:

Vc(t) = kT (t):d(t)

_k(t) = �Pref (t)d(t� �tot): (3.28)

4. Simulation results

(a) Open-loop plant

The STR will be now tested on a simulation based on Dowling's (1999) premixed ame model. This model describes the nonlinear oscillations of a premixed ameburning in a duct. Essentially, the constant ame speed model of Flei�l et al. (1996)has been extended by Dowling to include a ame holder at the centre of the ductand nonlinear e�ects. The level of background noise can be freely set. For detailson the model, the reader is asked to refer to (Dowling 1999).

For active instability control, the actuator chosen is a fuel injection systemrepresented in the frequency domain by the following second order transfer functionrelating the controller output voltage Vc to the additional heat release Qc producedfor control:

Wac(s) =Qc(s)

Vc(s)/

e�s�ac

s2=!2c + 2�s=!c + 1

; (4.1)

where the cut-o� frequency !c is chosen to be 1100 rad/s and the damping coeÆ-cient � = 1. The value of the time delay �ac is varied.

As it was done in section 2, the open-loop transfer function Pref=Vc =W0(s)e�s�tot ,

where Pref is the uctuating pressure, can be derived analytically. In order to ap-ply the proposed adaptive controller, W0(s) is required to satisfy properties 1, 2,and 3. As shown in Appendix C, these properties are satis�ed by the rational ap-proximator W0r (s) of W0(s) under assumptions (i)-(iv) which in turn hold for thecombustion system simulated in this section. The question that arises is if indeedthe actual transfer function W0(s) satis�es the relevant properties. Since W0(s) isin�nite-dimensional, and properties 1, 2, and 3 are for a rational operator, a di�er-ent tool for checking the applicability of the proposed approach toW0(s) is needed.An examination of the results of section 3 reveals that the main implication of theseproperties and the essential condition that is required for the proposed controllerto be applicable is that the underlying transfer function Wm(s) in (3.22) is SPR.The central property of a SPR operator is that its phase lies between ��=2 and�=2. Keeping in mind that an unstable and a stable pair of poles adds a phase of+� and ��, respectively (see Dorf and Bishop, 1995), we note (see �gure 7) thatover the frequency range of interest, the phase of W0 varies between 0 and 2.5�.Since W0 includes a pair of unstable poles at 380 rad/s, it follows that a transferfunction that has a pair of stable poles at 380 rad/s instead of the unstable polesof W0(s) and all of the remaining poles and zeros at almost the same locations as



W0(s), will have a phase that remains between ��=2 and �=2. This veri�es thatindeed the proposed controller is applicable to the problem considered in this sec-tion. As a cross-check, we have also included in �gure 7, the phase-plot of Wm(s),the closed-loop transfer function in (3.22) obtained with zc = 300 rad/s, a = 1000rad/s, and k

�

1 and k�

2 as control parameters in the closed-loop that correspond totypical values that stabilizeW0(s). This plot shows that indeed the phase ofWm(s)lies between ��=2 and �=2. Figure 8 shows that over the frequency range of inter-est, the rational approximation W0r (s) to W0(s) coincides with W0(s), which alsovalidates the applicability of the proposed approach.

(b) Closed-loop testing

(i) No time delay in the open-loop plant (�tot = 0)

The model parameters can be chosen so that the open-loop time delay �tot

equals zero. Hence the STR design given in equation (3.3) can be tested. In thesimulation, a convergence coeÆcient � > 0 has been added to the adaptive rule forthe control parameters. This does not modify the results of the Lyapunov stabilityanalysis (as can be easily checked by replacing ~kT ~k by (1=�)~kT ~k in the Lyapunovfunction (D 4)), but allows the speed of adaption to be varied.

Figure 9 shows the time evolution of the uctuating pressure Pref and thecontrol signal Vc. The controller is switched on at t = 0:15 s, when the limit cycleis already established (operating conditions: � = 0:7, �M1 = 0:08). The oscillationstend to zero within 0.5 s, but note that the settling time depends on the convergencecoeÆcient � chosen. Essentially, increasing � leads to a faster control but alsoproduces a larger maximum amplitude of Vc. In order to test the adaptabilityof the controller, the mean upstream Mach number �M1 is increased linearly (theunstable mode is then shifted from 58 to 63 Hz) after the oscillations have beendamped. Figure 9 shows that the STR maintains Pref to zero. Changes in the fuelair ratio � have also been successfully tested.

The controller action on the combustion instability can be physically interpretedusing the Rayleigh criterion (Rayleigh 1896). Rayleigh noted that acoustic wavesgain energy from their interaction with unsteady combustion if the mean value ofp0

fQ0 is positive, where p0f is the pressure uctuation at the ame, and Q

0 is therate of heat release. The Rayleigh criterion is mathematically expressed as follows:

p0

fQ0 =

ZT

p0

f (t)Q0(t)dt

�> 0 acoustic energy gain

< 0 acoustic energy loss(4.2)

where T is a period of oscillation. In �gure 10(a) is plotted the time evolution of thequantities p0f (t):Qn(t) and p

0

f (t):Qc(t), where Qc is the unsteady rate of heat releasedirectly produced from the controller (equation (2.10)), while Qn is the `natural'unsteady heat release response to incoming velocity uctuations (equation (2.12)).Before control is switched on, p0f (t) and Qn(t) are in phase (their product is posi-tive as shown in �gure 10(a)), indicative of a combustion instability. Once controlis switched on, an extra uctuating heat release Qc(t) is produced, which is adap-tively set out of phase with p

0

f (t) (the product p0

f (t):Qc(t) is negative), leading toa reduction of Qn(t) down to zero. Note that it is not necessary for Qc(t) to makep0

f(t)Q0(t) = p

0

f(t)(Qn(t) +Qc(t)) negative, only to reduce it suÆciently to ensure



that the rate at which acoustic energy is radiated from the end of the duct is greaterthan the rate of gain from the combustion. Oscillations then slowly decay.

(ii) Time delay in the open-loop plant (�tot 6= 0)

By changing the pressure measurement location and the time delay in the fuelinjection system, the overall time delay �tot in the open-loop plant can be freely setin the simulation. In the presence of a signi�cant time delay �tot = 23 ms, the STRdescribed by equation (3.26) is able to successfully control the plant, even if themean upstream Mach number �M1 is varied (see �gure 11). A similar performancewas achieved under variation of the fuel air ratio �.

Again, the controller action can be interpreted using the Rayleigh criterion, asillustrated in �gure 10(b) where all signals are measured at the ame location. Asfor the case �tot = 0, p0f (t) and Qn(t) are in phase before control is switched on(the product p0f (t):Qn(t) is positive), indicating strong driving towards instability.Control is switched on at t0 = 0:15 s. As seen from the adaptive law (3.25), Vcwill start to grow only at time t0 + �tot. Then, due to the time delay �ac in thefuel injection system (equation (4.1)), the heat release Qc will start to grow only attime t0+ �tot+ �ac = t0+2�tot, as observed on �gure 10(b) (note that the pressure uctuation p

0

f is taken at the ame, therefore from equation (2.18) we have that�tot = �ac). Rayleigh's criterion involves the pressure uctuation p

0

f (t) and theglobal uctuating heat release Q0(t) = Qn(t) + Qc(t). We observe in �gure 10(b)that the STR learns on line to make the product p0f (t):Qc(t) negative, and achievescontrol.

The robustness of the STR in the presence of noise has been tested by addingto the system broadband noise from an incoming sound wave having a rms velocity uctuation of 12% of the mean velocity. Figure 12 shows the resulting pressurefrequency spectrum before and after control. The fundamental mode of oscillationand its harmonics are completely removed by the controller, showing the STRrobustness to a signi�cant level of background noise.

We tested this simulation for increasing values of the time delay �tot. The resultsare shown in �gure 13(b). For �tot!u � 1 where !u is the main unstable mode,control is achieved with k1 > 0 and k2 > 0, in agreement with the theory presentedin section 3. Furthermore, for higher values of �tot, up to �tot!u � 3� (ie �tot �3 cycles of oscillation), we observe some periodic stability bands according to thevalues of !u�tot: a `stability band' corresponds to values of !u�tot for which controlis obtained after a �nite settling time. Between two consecutive stability bands,there are a few values of !u�tot for which control is not obtained (the settling timeis then in�nite). It was also observed that the sign of the �rst order compensatorgain k1 achieving control changes between two consecutive stability bands (see�gure 13(a)). These observations on sign(k1) and on the stability bands patterncan be interpreted as follows: for frequencies close to !u, the open-loop transferfunction W0(s) can be approximated by a second order system

k01

(s� �u)2 + !2u

(4.3)

where k0 is a constant. It is shown in Appendix F that a plant of order 2 and timedelay �tot, whose transfer function W0(s) is given by equation (4.3), is stable for



any delay �tot which satis�es

k0k1 sin(!u�tot) < 0

1= tan(!u�tot)�zc

!u< 0: (4.4)

Hence, such a plant is characterized by a pattern of periodic stability bands accord-ing to the values of !u�tot, as shown in �gure 13(a). We see also from equation (4.4)that the sign of k1 required for control satis�es

sign(k0k1) = �sign(sin(!u�tot)); (4.5)

in agreement with the simulation results (�gure 13(a)).

5. Experimental results

(a) Description of the experimental set-up

A vertical cylindrical tube of 75 cm length and 5 cm diameter, open at bothends, is considered and shown in �gure 14. A laminar ame, fed by a propanecylinder and stabilised by a grid, burns in the lower half of the tube. The couplingbetween the tube acoustics and the unsteady heat released by this ame leads toa self-excited combustion oscillation at the organ pipe fundamental frequency. Inorder to vary the frequency of the unstability, the upper portion of the tube has a`trombone-like' arrangement shown in �gure 14, so that the e�ective length L of thetube can be varied from 75 to 105 cm. The join between the main tube and the extratube length is carefully sealed. Without the extra tube length (L = 75 cm), themeasured natural mode of instability is 244 Hz. With the 30cm extra tube length(L = 105 cm), the natural mode is measured at 180Hz and the correspondingunsteady pressure amplitude is increased by a factor of 1.5. For active stabilitycontrol, the actuator chosen is a 50 W low frequency loudspeaker which is situatedclose to the lower end of the tube. The loudspeaker acts to modify the acousticboundary condition at the lower end of the organ pipe. A Br�uel & Kj�r type 4135microphone is ush-mounted in the tube wall to measure the unsteady pressure Prefin the tube. In a closed-loop con�guration (indicated in dotted line in �gure 14), thepressure signal Pref is sent to a 32-bit M62 digital signal processing (DSP) boarddeveloped by Innovative Integration, on which the control algorithm is implemented.The control signal Vc generated by the DSP board in response to the pressuremeasurement Pref is used to drive the loudspeaker.

Although this simple Rijke tube experimental set-up is very far from industrialcombustors, we thought the results obtained with the STR had a real interest forthis paper for the following reasons:

� The mechanism of actuation is here completely di�erent to the one studiedso far in the paper (i.e. fuel actuation). Therefore, we can test the robustnessof the STR to a new type of actuation.

� A drastic change in the frequency of the instability can be easily implementedthanks to the trombone-like arrangement of the tube, and the adaptability ofthe STR can be tested.



� The inherent open-loop time delay in the Rijke tube with loudspeaker actu-ation is very small, but an extra arti�cial time delay can be implemented onthe DSP board. Hence, the performance of the STR in the presence of largetime delays can be properly tested and used as a benchmark for future testson larger experimental rigs.

As mentioned above, the actuator used in this experiment di�ers from the fuelinjection system studied so far. Therefore, we need �rst to check if our experimentalset-up satis�es the open-loop properties required so that the STR is theoreticallyguaranteed to achieve control. This is done in the following section.

(b) Characterisation of the open-loop transfer function

The open-loop transfer function Pref=Vc from the voltage signal Vc driving theloudspeaker to the pressure measurement Pref can be measured experimentally:a white noise Vc is sent to the loudspeaker, while the pressure response Pref ismeasured, with or without a ame. The corresponding gain and phase of the transferfunction Pref (s)=Vc(s) =W0(s)e

�s�tot obtained are plotted in �gure 13(a) for a tubelength L equal to 75 cm. Without a ame (see �gure 13(a-1)), the phase decreases by� radians at 230, 450, and 680 Hz, indicating that there are complex-pair of stablepoles at these locations, while the phase increases by � radians at 420 and 840 Hzindicating that there are complex-pairs of stable zeros at these locations (Dorf &Bishop 1995). With a ame, the locations of the poles and zeros are modi�ed asshown in �gure 13(a-2), but we note that all of the zeros remain stable. Further, thetotal time delay �tot in our experimental open-loop con�guration is easily obtainedfrom the average slope of the phase of Pref (s)=Vc(s), and is found to be typicallyaround 1 to 1.5 ms.

The open-loop transfer functionW0(s) can also be obtained analytically, using asimple time lag model for the ame (see details in (Evesque 2000)). Gain and phaseof W0(s) are plotted in �gure 13(b). Note that �gure 13(b) is the Bode plot ofthe theoretical transfer function W0(s) without time delay �tot, which explains thatthere is no average slope in the predicted phase. There is a satisfying agreementbetween the analytical and the measured transfer function, as far as the locationof poles and zeros, as well as their stability, are concerned. The only noticeableanomaly can be found for the measured phase around the unstable pole at 244Hz. Indeed, the phase close to an unstable pole cannot be measured accurately(this is because the pressure measurement Pref has a very strong amplitude at thefrequency of unstability, and therefore does not `respond' coherently to the whitenoise sent by the loudspeaker). Therefore, the value of the measured phase around244 Hz should not be relied upon.

We deduce from these experimental and analytical results that the open-looptransfer function has only stable zeros, a positive gain at high frequencies and thatthe number of poles minus the number of zeros over the frequency range of interest[0� 1000] Hz is not larger than 2. Hence, the properties required for application ofthe STR are satis�ed.



(c) Small �tot

Since the measured overall time delay �tot from the voltage Vc to the pres-sure measurement Pref is within half a period of oscillation Tosc (�tot � 1:5ms �Tosc=2), it is expected that control can be achieved using a simple �rst order com-pensator (equation 3.3). This �rst order compensator is implemented, with �xedvalues for the control parameters k1 and k2. With L = 75 cm, an adequate choice ofk1, k2 and zc can be made to achieve control, as shown in �gure 16(a). Then, keep-ing the same �xed control parameters, the tube length is increased by 30 cm: theoscillation grows again but this time at 180Hz, and the �xed compensator, despitea signi�cant control e�ort Vc, is unable to regain control. The adaptive version ofthe �rst order compensator (equation (3.3)), is tested in a similar way, as indicatedin �gure 15(b). Starting from control parameters k1 and k2 equal to zero, controlis achieved within 0.4 second for L = 0:75 cm. Then L is suddendly increased to105 cm: after a short destabilization period during which the pressure oscillationis growing again, control is regained very quickly. Note that if the transition phase(from L = 75 to L = 105 cm) is performed at a slower rate (typically over 3 sec),then the pressure amplitude remains at a very low level all the time. This is a niceillustration of the greater eÆciency of adaptive controllers over �xed controllers tomaintain control under varying operating conditions.

Finally, the STR given in equation (3.26), i.e. a �rst order compensator associ-ated with a Smith Controller, is implemented for comparison. Control is achievedwith a faster settling time of typically 0.2 s.

(d) Large �tot

We would like to test the performance of the STR (equation (3.26)) in thepresence of a signi�cant time delay �tot in the open-loop plant. Since the Rijke tubehas an overall time delay of 1.5 ms only, an extra time delay �ext has been arti�ciallyintroduced on the DSP board, so that the control algorithm is not applied to thecurrent pressure input Pref (t), but to the pressure signal Pref (t � �ext) measured�ext seconds before. Therefore, the overall time delay �tot in the open-loop plant isnow �tot = 1:5ms + �ext. In the presence of a time delay �ext, the simple �rst ordercompensator of equation (3.3) is able to control the Rijke tube for �tot taking valuesup to 3Tosc=4 only, where Tosc is the period of oscillation. In contrast, �gure 17shows the STR given in equation (3.26), i.e. a �rst order compensator associatedwith a Smith Controller, controlling the Rijke tube in the presence of a knowntotal time delay �tot up to 5Tosc. Further, similarly to what we observed in section4 (see �gure 13), control is obtained during periodic `stability bands' according tothe value of !u�tot where !u is the dominant unstable mode at 244 Hz. Betweentwo consecutive stability bands, there is a small range of values of !u�tot for whichcontrol is not reached. Further, the sign of the controller gain k1 achieving controlalternates between two consecutive stability bands. This stability band patternand the sign of k1 obtained experimentally are in very good agreement with thestability behaviour of a second order plant of time delay �tot controlled by the STR(Appendix F). These experimental results provide further demonstration of thepotential of the STR to control plants having a time delay equal to several periodsof oscillation.



6. Conclusions

An adaptive control strategy for unstable combustion has been theoretically investi-gated and numerically and experimentally tested. The adaptability of the controlleris essential in practical combustion systems to allow a wide range of stable opera-tion. Further, a fundamental challenge in the adaptive control design that needs tobe addressed is to rely as little as possible on a particular combustion model (sincefull-scale models are still very inaccurate, and also to make the adaptive control de-sign generally applicable), while providing some guarantees that the controller willnot go unstable and cause any harm (this is a critical requirement in particular foraeroengines). Therefore, in this paper we focused on a class of combustion systemswhose general features can be simply described, and propose an adaptive controldesign, denoted as a Self-Tuning Regulator (STR), that is guaranteed, theoretically,to control such systems safely.

A class of self-excited combustion systems has been studied, and an analyticalexpression of the open-loop transfer function W (s) from the controller output volt-age Vc to the pressure Pref measured by a sensor has been derived. The actuatorconsidered is a fuel injection system, since it seems to be the most practical actua-tor and has been used in all full-scale demonstrations of active combustion control.When the control actuation is chosen appropriately, W (s) was shown to have avery simple form: it is the product of a time delay �tot (whose two contributions arethe propagating time delay �det from the ame to the sensor location, and the timedelay �ac between fuel injection all its combustion), and a transfer function W0(s).Under four fairly general assumptions given in x2 b , it was demonstrated that arational approximation to W0(s) has only stable zeros, the same relative degree asthe actuator transfer function, and that its high frequency gain has the same signas the high frequency gain of the actuator transfer function.

These three structural properties of the open-loop process have been exploitedto design a STR which is guaranteed to stabilise the combustion system representedby the rational approximation to W0(s). The adaptive control design in the case�tot = 0 corresponds to the one proposed by Annaswamy et al.(1998) for a particularcombustion model. In this paper it was shown that this STR design could be usedon a wider class of combustion systems. The controller consists of a simple phaselead compensator whose gain and phase are updated according to a rule derivedfrom a Lyapunov stability analysis.

For the more realistic case of a nonzero time delay �tot between controller outputvoltage Vc and pressure measurement Pref , a novel STR design has been developed,which guarantees the global stability of the system. In this case, the controller con-sists of a phase-lead compensator similar to the one used in the case of �tot = 0,combined with a Smith Controller whose coeÆcients `compensate' for the time de-lay �tot. A formal proof has been derived, showing that this controller structure isguaranteed to stabilise the system if �tot is `small', but simulation and experimentalresults indicate that control can be achieved for �tot up to a few periods of oscilla-tion Tosc. The adaptive rule for the Smith Controller and phase lead compensatorcoeÆcients is again derived from a Lyapunov stability analysis. The practical im-plementation of the STR for the case �tot 6= 0 remains very easy, requiring only thepressure signal Pref and past values of Vc over a time interval of length �tot.

The STR has been implemented on a simulation based on Dowling's (1999) non-



linear premixed ame model. For active stability control, the actuator considered isa fuel injection system and the sensor a pressure transducer. The STR achieved con-trol in the presence of a signi�cant time delay �tot in the open-loop plant (typicallyfor �tot up to 3Tosc) and also with a strong background noise. The way the STRproduced an extra unsteady heat release out of phase with the unsteady pressurecould be interpreted using the Rayleigh criterion.

Finally, the STR has been tested experimentally. The experimental set-up con-sists of a laminar ame burning in the lower half of a Rijke tube. The system isstabilized by an active control feedback loop, in which the actuator is a loudspeaker,the sensor a microphone and the control algorithm is implemented on a DSP board.The tube length L can be varied, leading to a shift in the frequency of the unstablemode. The STR demonstrated its ability to achieve and maintain control even whena 30% variation of the frequency of the instability was imposed. Further, when anextra time delay was added in the open-loop plant, a simple adaptive �rst ordercompensator failed to control the oscillation when the time delay exceeded threequarters of Tosc, whereas the STR (Smith controller associated with a �rst ordercompensator) could silence the Rijke tube for delays up to 5Tosc. These experimen-tal results will be used as benchmarks for further experimental tests of the STR ona 0.5 MW premixed rig.

Part of the work described in x4 was carried out while APD was the Jerome C Hunsaker

Visiting Professor and SE a Visiting Engineer at the Massachusetts Institute of Technology.

The support of MIT in making this collaboration possible is gratefully acknowledged. AMA

is sponsored by the OÆce of Naval Research, under grant N00014-99-1-0448, and SE by

Trinity College, Cambridge.

Appendix A.

The full form of the coeÆcients in equation (2.6) is:

X =

0@ �1 + �M1

�2� �u2

�u1

�� M

21

�1� �u2

�u1

�1 + �M1

��1 �c1��2 �c2

1� �M1

�1+ �M

21 �

�M

21 (1�

�M1)12

��u22�u21� 1�

�c2�c1

1+ �M2

�1+ �M1

�M2��1��2

1A (A 1)

Y =

0@ 1 + �M1

�2� �u2

�u1

�+ �M2

1

�1� �u2

�u1

��M1

��1 �c1��2 �c2

� 1

1+ �M1

�1+ �M

21 �

�M

21 (1 +

�M1)12

��u22�u21� 1�

�c2�c1

1� �M2

�1+ �M1

�M2��1��2

1A (A 2)

Appendix B. Pressure re ection coeÆcients in a pipework

system

We demonstrate here that assumption (i) is valid for a pipework sytem upstream(see �gure 18a) and downstream (see �gure 18b) of the combustion zone. Moreprecisely, we show by induction that if the boundary at x0 (see �gure 18a) and theboundary at z0 (see �gure 18b) satisfy the conditions of equation (2.22), then the



boundaries at x = �xu and x = xd just upstream and downstream the combus-tion zone (see �gure 1) are represented by re ection coeÆcients Ru(s) and Rd(s)satisfying equation (2.22).

We consider linear disturbances with time dependence est and assume there isno mean ow in the pipework system. The pressure re ection coeÆcient at an axialposition xi is denoted Rui(s). The cross-sectional area of the duct i between x = xi

and x = xi+1 is Ai. The mean speed of sound is �c and the mean density is ��. At x0,we assume that the pressure re ection coeÆcient Ru0(s) is rational and satis�es

jRu0(s)j < 1 in Re(s) � 0

n� (Ru0(s)) = 0;

jh0j < 1: (B 1)

where n� denotes the relative degree of Ru0(s) and h0 its high frequency gain. Theseassumptions are true for a choked end at x = x0 for instance. In the following, weshow by induction that for all i, the re ection coeÆcient Rui(s) satis�es jRui(s)j < 1in Re(s) � 0, has for relative degree n�(Rui(s)) = 0 and for high frequency gain hisuch that jhij < 1.

At x = xi, the re ected wave fi is related to the incoming wave gi as follows:

fi(t) = e�2s

xi+1�xi�c Rui(s) [gi(t)] : (B 2)

Writing the continuity of pressure and mass ux across x = xi+1, and using theboundary condition (B 2) leads to the following relationship between the re ectedwave fi+1 and the incoming wave gi+1 in the duct i+ 1:

fi+1(t) = e�2s

xi+2�xi+1

�c Rui+1(s) [gi+1(t)] ; (B 3)

where

Rui+1(s) =

Ki +Rui(s)e�s�i

1 +KiRui(s)e�s�i

(B 4)

Ki =Ai+1 �Ai

Ai+1 +Ai

(B 5)

�i =�2(xi+1 � xi)

�c: (B 6)

By inspection jKij < 1. Furthermore, from (B 4), one deduces that n�(Rui(s)) = 0implies that n�(Rui+1

(s)) = 0 after a Pad�e expansion [M=M ] for e�s�i is made(see de�nition of Pad�e expansion in x2 b ). Let us write Rui in the form Rui(s) =jRui(s)je

�s�i in equation (B 4). Then one easily obtains

jRui+1(s)j2=

K2i +jRui j

2e�2Æi+2KijRui(s)je

�Æi cos(�i)

1+K2i jRui j

2e�2Æi+2KijRui(s)je�Æi cos (�i)

(B 7)

where �i = Imag(s)�i+�i and Æi = Re(s)�i. The di�erence P between the numer-ator and the denominator of jRui+1

(s)j2 is equal to

P = (1�K2i )(jRui j

2e�2Re(s)�i � 1) (B 8)



Therefore, in Re(s) � 0, jRui(s)j < 1 implies that P < 0 and hence jRui+1j < 0.

From (B4), after a Pad�e expansion [M=M ] for e�s�i is made, one deduces that

hi+1 =Ki + hi(�1)

M

1 +Kihi(�1)M: (B 9)

and hence

jh2i+1j =K

2i + h

2i � 2(�1)MKihi

1 + 2Kihi(�1)M +K2i h

2i

: (B 10)

The di�erence Q between the numerator and the denominator of hi+1 is

Q = (1�K2i )(h

2i � 1): (B 11)

Hence, jhij < 1 implies that Q < 0 and therefore jhi+1j < 1.Similarly, for a pipework system downstream the ame, ended by a nozzle (see

�gure 18b), we assume that at z = z0, the pressure re ection coeÆcient Rd0 satis�esthe assumptions in (B 1). this is the case if the nozzle is compact (Marble & Candel1977). By induction, it can be shown that at any position z = zj , the downstreampressure coeÆcient Rdj and its high frequency gain hj satisfy

n�(Rdj ) = 0

jRdj (s)j < 1 in Re(s) � 0

jhj j < 1: (B 12)

Appendix C. Proofs of Properties 1, 2, and 3:

Proof of Property 1: We note that

W0r (s) =Fr(s)Gr(s)Wac(s)

1�Gr(s)Hr(s)(C 1)

where Ar(s) denotes a rational approximator of an operator A(s) obtained byapproximating all terms of the form of e�s� by a rational function [M=M ]e�s� inA(s). To show that the zeros of W0r (s) are stable, we examine the zeros of each ofthe factors in W0r (s).

It is evident from equation (2.19) that the poles of the ame transfer functionHr(s) become zeros of W0r (s). As noted in assumption (ii0), the poles of Hr(s)are stable and therefore Hr(s) does not introduce unstable zeros for W0r . Gr(s)

appears at the numerator and the denominator ofW0r (s), and hence its poles do notin uence the zeros of W0(s). Wac(s) has no unstable zeros (assumption (iv)). Fromequations (2.6) and (2.16), it follows that both the numerator and denominator ofFr(s)Gr(s) is the product of a constant with terms of the form k + R(s)

�M

M

�e�s�

where � is a time delay, k is a constant and R(s) is a rational transfer function. Toshow that Fr(s)Gr(s) has no zeros in Re(s) � 0, we note that at a zero,�

M

M

�e�s�

=�k

R(s): (C 2)



On Re(s) = 0, j [M=M ]e�s�

j = 1 since the numerator and denominator of the Pad�eapproximant are complex conjugates of each other, therefore equation (C 2) has noroots on Re(s) = 0. Further, from the properties of Pad�e approximants (Baker &Graves-Morris(1996)), it follows that

j

�M

M

�e�s�

j < 1 for Re(s) < 0: (C 3)

Since jRu(s)j < 1 and jRd(s) < 1j for Re(s) � 0 (assumption (i)), and sincejX12j > jY12j (see appendix A), we have that j � k=R(s)j > 1 for Re(s) � 0.Therefore, equation (C 2) has no roots on Re(s) > 0.

Proof of Property 2:

Since Fr(s) is of the form

Fr(s) = kF

1 +Ru [e�s� ]

pade

Ru [e�s�u ]pade � 1

where kF is a constant and � =2(xu+xref )

�c1(1� �M1), and since k0(Ru) < 1 (assumption (i)),

it follows that n�(Fr(s)) = 0. Noting that

Gr(s) = kG

(RdY12 [e�s�d ]

pade�X12)(Ru [e

�s�u ]pade

� 1)

det(S)

where kG is a constant, we consider both the numerator and denominator of Gr(s)separately. Since the numerator of Gr is similar to Fr, jX12j > jY12j, k0(Ru) < 1and k0(Rd) < 1, it follows that the relative degree of the numerator of Gr(s) is zero.Since X11Y21 � X21Y11 6= 0 for low Mach number ows, k0(Ru) < 1, k0(Rd) < 1and jX22j > jY22j, it follows that the relative degree of the denominator of Gr(s) iszero. From assumption (ii0), it follows that

n�

�Gr(s)

1�Gr(s)Hr(s)

�= n

�(Gr(s)):

Therefore, it follows that

n�

�Fr(s)Gr(s)

1�Gr(s)Hr(s)

�= 0

which proves Property 2.Proof of Property 3: Since n�(Gr) = 1 and n

�(Hr) > 0, it follows that

k0

�Fr(s)Gr(s)

1�Gr(s)Hr(s)

�= lim

s!1Fr(s)Gr(s):

Since lims!1 Fr(s) < 0 and lims!1Gr(s) < 0, property 3 holds.

Appendix D.

Simple root locus arguments show that a �rst order compensator as shown in �gure3(a) is suÆcient to stabilize the plant (3.1) when �tot = 0 (see details in (Evesque,



2000)). The control parameters k1 and k2 can be found adaptively using the adap-tive rule (3.3) if n� = 1 or (3.2) if n� = 2. Such a rule is derived from a Lyapunovstability analysis: it ensures the decay in time of an energy or Lyapunov function ofthe system, i.e it is guaranteed to lead to values k�1 and k�2 that stabilize the plant.The proof is brie y reproduced here for the case n� = 1; for more details on the casen� = 2, the reader is asked to refer to (Evesque, 2000) or (Annaswamy et al. 1998).

The deviation of the control parameters from values achieving control is denoted~k(t)T = [k1(t) � k

�

1 ; k2(t) � k�

2 ]. The adaptive closed-loop system shown in �gure3(b) can be represented by a so-called error model which relates the parametererror ~k to the observed error in Pref , as follows:

_x = Ax(t) + b

�~kT (t):d(t)

�(D 1)

Pref (t) = hT :x(t); (D 2)

where Wm(s) = hT (sI � A)�1b is a strictly positive real (SPR) operator (thiscomes from the fact that the rational approximationW0r (s) to the open-loop planthas only stable zeros and relative degree n� = 1), I is the identity matrix and x isthe state vector. We deduce using lemma 2.4 in (Narendra & Annaswamy 1989) thatgiven a matrix Qm symmetric strictly positive, there exists a matrix Pm symmetricstrictly positive, such that

ATPm + P

TmA = �Qm

Pmb = h: (D 3)

The Lyapunov function candidate is

Vl = xTPmx+ ~kT ~k (D 4)

which is positive de�nite. Using equations (D 1), (D 2) and (D 3), the time derivativeof Vl is obtained:

_Vl = �xTQmx+ 2~k ��Prefd+

_~k�

(D 5)

Choosing the adaptive rule for k as indicated in equation (3.2) ensures that_Vl = �xTQmx and therefore is negative. This proves that ~k and x are bounded.Then equation (D 1) implies that _x is bounded. Finally, lemma 2.12 (Narendra &Annaswamy 1989) shows that x, hence Pref , is guaranteed to converge asymptoti-cally to zero.

Appendix E.

We start from equation (3.22):

Pref (t) =Wm(s)e�s�tot [~kT (t):da(t)]; (E 1)

where Wm(s) =W�

cl0(s)(s+ a) is SPR and da(t) =

1s+a

[d(t)].The adaptive law is chosen as

_k(t) = �Pref (t)da(t� �tot); (E 2)



where a positive sign for k0 is assumed. Equation (E 1) can be expressed in atime-domain representation as

_x = Ax(t) + b

�~kT (t� �tot):da(t� �tot)

�Pref (t) = hT :x(t): (E 3)

x is the `state vector' of the system and (A;b;h) is the `state representation' ofWm(s), which means that Wm(s) = hT (sI � A)�1b. We note that equation (E 3)can be rewritten as

_x(t) = Ax+ b

�~kT (t):da(t� �tot)

�� bda

T (t� �tot)

0@ 0Z��tot

_~k(t+ �)d�

1A : (E 4)

Using equation (E2), this leads to

_x(t)=Ax+ b

�~kT (t):da(t��tot)

�+bda

T (t��tot)

0@ 0Z��tot

Pref (t+�)da(t+��tot)d�

1A(E 5)

Since Wm(s) is SPR, lemma 2.4 of Narendra & Annaswamy(1989) can be used:given a symmetric positive de�nite matrix Qm, there exists a symmetric positivede�nite matrix Pm, such that

ATPm + P

TmA = �Qm; Pmb = h; (E 6)

As in (Burton 1985) and (Niculescu et al. 1997; Niculescu & Annaswamy 2002),the Lyapunov function candidate is chosen as

Vl = xT (t)Pmx(t) + ~kT (t)~k(t) +

0Z��tot

tZt+�

k_~k(�)k2d�d� (E 7)

which is positive de�nite. Using equations (E 2) and (E5), (E 7) leads to the time-derivative

_Vl = xT (ATPm+P T

mA)x+2xT (t)Pmb�~kT (t):da(t��tot)

�

+2xT (t)Pmbda(t��tot)

0@ 0Z��tot


1A

�2Pref (t)~kT (t):da(t��tot)+

d

dt

0@ 0Z��tot

tZt+�

(Pref (�))2kda(��tot)k

2d�d�

1A(E 8)

Using equation (E6), we obtain

_Vl = �xTQmx+2Pref (t)da(t��tot)

0@ 0Z��tot


1A

+

0Z��tot

�kPref (t)da(t��tot)k

2�kPref (t+�)da(t+��tot)k2�d�: (E 9)



Denoting

y = Pref (t)da(t� �tot); w = Pref (t+ �)da(t+ � � �tot); (E 10)

equation (E9) can be rewritten as

_Vl = �xTQmx�

0Z��tot

�kw� yk2 � 2yTy

�d�

� �xT�Qm � 2�totkda(t� �tot)k

2hhT�x (E 11)

Hence, _Vl is negative if da(t � �tot) satis�es the matrix inequality (see (Bellman1953) for de�nition of a matrix inequality)

Qm > 2�totkda(t� �tot)k2hhT (E 12)

Condition (E12) is not easy to check at any time t. We show below that it can bereplaced by bounds on da over the interval [t0 � �tot; t0), where t0 is the time atwhich control is switched on.

Suppose that over [t0 � �tot; t0), da satis�es

supt2[t0��tot;t0)

kda(t)k2 � (E 13)

where is a positive real constant. Further, assume that a �xed delay value �1 issuch that

2�1 hhT< Qm (E 14)

Then, combining equations (E 13) and (E 14) leads to

Qm > 2�totkda(t)k2hhT (E 15)

for all t 2 [t0; t0 + �tot) and for all �tot smaller than �1. Equation (E 15) impliesthat Vl(t) is not increasing for t 2 [t0; t0 + �tot), as long as �tot � �1. Therefore, wehave

x(t)TPmx(t) � Vl(t) � Vl(t0) (E 16)

for all t 2 [t0; t0 + �tot), which means that x is bounded over [t0; t0 + �tot) and thecorresponding bound is given by

supt2[t0;t0+�tot)

kx(t)k2 �Vl(t0)

�min�Pm

(E 17)

where �min�Pm is the smallest eigen value of the positive symmetric matrix Pm.A constant matrix C exists such that da = Cx (this essentially means that da is apart of the state of the system). Therefore we have

supt2[t0;t0+�tot)

kda(t)k2 �

kCk2Vl(t0)

�min�Pm

(E 18)



which does not depend on .Now let us consider _Vl over the second delay interval [t0 + �tot; t0 + 2�tot). _Vl is

negative over this interval if �tot � �2 with �2 satisfying the inequality

2�2kCk2Vl(t0)

�min�Pm

hhT < Qm (E 19)

Hence Vl is not increasing at time [t0 + �tot; t0 + 2�tot) if �tot � �2.By repeating the process, it can be shown that the constructions above also hold

on the next delay intervals [t0+k�tot; t0+(k+1)�tot) for any positive integer k > 2.We conclude that for a time delay �tot smaller than �3 = min(�1; �2) and for dasatisfying the initial condition (E13), equation (E 12) is satis�ed at any time t > t0,ie _Vl(t) is negative at any time t > t0. Then, application of lemma 2.12 (Narendra &Annaswamy 1989) guarantees that Pref tends asymptotically to zero as long as thedelay �tot is smaller than �3 and for da satisfying the initial condition (E13). Notethat this initial condition is satis�ed in a practical self-excited combustion systemwhen control is switched on while the pressure limit cycle is already established.

Appendix F.

Consider an open-loop plant of order 2, de�ned by Z0(s) = 1 and R0(s) = (s ��u)

2+!2u where �u < 0 since the plant is open-loop unstable. The total time delay

in the plant is �tot and its high frequency gain is k0. Using the controller de�ned in�gure 5(a), the denominator of the closed-loop transfer function is easily calculatedanalytically and is found to be:

Rcl(s) = s3 + as

2 + bs+ c (F 1)

where

a = �2�u+zc+k2�k0k1e��u�tot

�sin(!u�tot)

!u

�

b = �2u+!

2u�2�u(zc+k2)+k0k1e

��u�tot

��u�zc

!usin(!u�tot)+cos(!u�tot)

�

c = (�2u+!2u)(zc+k2)+k0k1zce

��u�tot

��u

!usin(!u�tot)+cos(!u�tot)

�(F 2)

For very small �tot (�tot!u <O(1)), stability is guaranteed for some k1 > 0 andk2 > 0 (continuity of no time delay case). For larger �tot, assuming that j�uj << !u

(this is a realistic assumption: the growth rate is much smaller than the frequencyof the unstable mode), the Routh criteria applied to Rcl(s) shows that Rcl(s) is

stable for some k2 > 0 if

k0k1 sin(!u�tot) < 0

1= tan(!u�tot)�zc

!u< 0: (F 3)

Consider the angle � de�ned by 1tan(�)

= zc!u

and sin(�) > 0. We have:

� for � < !u�tot < � and k1 < 0, Rcl(s) is stable.



� for � + � < !u�tot < 2� and k1 > 0, Rcl(s) is stable.

Hence we deduce the periodic stability bands shown in �gure 13(a). Also note theresult derived from equation (F 3) that k1 required for control satis�es sign(k0k1) =�sign(sin(!u�tot)).

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Captions to Figures

Figure 1 : Open-loop self-excited combustion process with actuation.

Figure 2 : Schematic representation of the open-loop process (solid line), which can be

controlled by a feedback controller (dashed line).

Figure 3 : Low-order controller for �tot = 0, n� � 2. (a) �xed structure, (b) adaptive

version.

Figure 4 : A nth order controller structure for �tot 6= 0, proposed by Ichikawa (1985).

Figure 5 : Low-order controller for �tot 6= 0, n� � 2. (a) �xed structure, (b) adaptive

version.

Figure 6 : Modi�cation of the control input Vc.

Figure 7: Bode plot of open-loop transfer function W0(s) and phase of closed-loop transfer

function Wm(s).

Figure 8: Validation of rational approximation to W0(s) over the frequency range of in-

terest.

Figure 9: STR (�tot = 0) under varying operating conditions. M1 is varied linearly from

0.08 at t = 1:95 s to 0.095 at t = 2:1 s. : : : : controller OFF, | : controller ON after

t = 0:15 s.

Figure 10: Interpretation of the controller action using the Rayleigh criterion. (a) �tot = 0,

(b) �tot = 23 ms.

Figure 11: STR for �tot = 23 ms, xref = �xu=2,M1 is varied linearly from 0.08 at t = 1:95

s to 0.095 at t = 2:1 s. : : : : controller OFF, | : controller ON after t = 0:15 s.

Figure 12: STR performance in a noisy environment: additional white noise having a rms

velocity uctuation of 12 % of the mean velocity (� = 0:7, �M1 = 0:08).

Figure 13 STR performance in a simulation based on Dowling's (1999) premixed ame

model, compared with a theoretical study of a second order plant having the same time

delay �tot and stabilised by the STR (Appendix F). (a) k1 obtained in simulation and

sign(k0k1) required for stability for a second order plant. (b) simulation: control obtained

periodically up to !u�tot � 3 cycles of oscillation.

Figure 14 : Schematic representation of the experimental set-up.

Figure 15 : Comparison of measured and theoretically derived open-loop transfer function.

Figure 16 : Control of Rijke tube under varying tube length L. L has been varied between

the two vertical lines. (a) �xed �rst order compensator, (b) adaptive �rst order compen-

sator (equation (3.3)), (c) STR (equation (3.26)).

Figure 17 : EÆciency of STR at controlling the Rijke tube in the presence of a substantial

added time delay �ext.

Figure 18 : Upstream and downstream pipework systems.



: closed loopcontrol

actuator

Vc

controller

xxref xd-xu

1

g

zonecombustion

2

f

Ru Rd

h

j

+ entropy waves

0

Prefmeasured

Figure 1. Open-loop self-excited combustion process with actuation

e-s τdet

H(s)

G(s)

u1

acoustic waves

Qn

Qc Q

acoustic waves

Vc

Pref

CONTROLLER

ACTUATOR

SENSOR

COMBUSTION SYSTEM

flame dynamics

F(s)

+

+

from control output

til fuel combustion

Wac(s)e-s τac

Figure 2. Schematic representation of the open-loop process (solid line), which can be

controlled by a feedback controller (dashed line).



Vc Pref

k1

ko(Zo/Ro)

k2/(s+zc)

- -

+i

Vc Pref

k1

ko(Zo/Ro)

1/(s+zc)

k2

+

- -

i

(a)

(b)

Figure 3. Low-order controller for �tot = 0, n� � 2. (a) �xed structure, (b) adaptive

version.

SCV

C/E

B/E

SC

ko(Zo/Ro)

e-s τ tot

e-s τ totPrefVc

+

- -

i

Figure 4. A nth order controller structure for �tot 6= 0, proposed by Ichikawa (1985).



SCV

SC

k1

ko(Zo/Ro) e-s τ totVc Pref

k2/(s+zc)

+

- -

+i

SCV

SC

k1

ko(Zo/Ro) e-s τ totVc Pref

1/(s+zc)

k2

+

--

i

(a)

(b)

Figure 5. Low-order controller for �tot 6= 0, n� � 2. (a) �xed structure, (b) adaptive

version.

da

s+a1/(s+a)

dk

Vc

Figure 6. Modi�cation of the control input Vc.


36

S.Evesqu

e,APDowlin

gandAM

Annasw

amy

0200

400600

8001000

12001400

−50

−40

−30

−20

−10 00200

400600

8001000

12001400

−0.5 0

0.5 1

1.5 2

2.5

W0 (s)

Wm

(s)

frequ

ency (rad

/s)

gain (dB)phase (π rad)

Figure

7.Bodeplotofopen-looptra

nsfer

functio

nW

0 (s)andphase

ofclo

sed-loop

transfer

functio

nW

m(s).

0200

400600

8001000

12001400

−50

−40

−30

−20

−10 00200

400600

8001000

12001400

0

0.5 1

1.5 2

2.5

ration

al app

roxim

ation

to W

0 (s)

W0 (s)

frequ

ency (rad

/s)

phase (π rad) gain (dB)

Figure

8.Validatio

nofratio

nalapproximatio

nto

W0 (s)over

thefreq

uency

rangeof

interest.

Artic

lesubmitted

toRoyalSociety


0 0.5 1 1.5 2 2.5 3−0.4

−0.2

0

0.2

0.4

Vc

time (s)

0 0.5 1 1.5 2 2.5 3−0.2

−0.1

0

0.1

0.2

Pre

f

M1 increased from 0.08 to 0.095

Figure 9. STR (�tot = 0) under varying operating conditions. M1 is varied linearly from

0.08 at t = 1:95 s to 0.095 at t = 2:1 s. : : : : controller OFF, | : controller ON after

t = 0:15 s.



(a)

(b)

-

-

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time (s)

p’f(t)*Qc(t)

p’f(t)*Qn(t)

control switched ON

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

time (s)

p’f(t)*Qn(t)

p’f(t)*Qc(t)

control switched ON

2ttot

Figure 10. Interpretation of the controller action using the Rayleigh criterion. (a)

�tot = 0, (b) �tot = 23 ms.



0 0 . 5 1 1 . 5 2 2 . 5 3−0 . 2

−0 . 1

0

0 . 1

0 . 2

Pr

ef

0 0 . 5 1 1 . 5 2 2 . 5 3−0 . 4

−0 . 2

0

0 . 2

0 . 4

t i m e ( s )

Vc

M1 increased from 0.08 to 0.095

Figure 11. STR for �tot = 23 ms, xref = �xu=2, M1 is varied linearly from 0.08 at

t = 1:95 s to 0.095 at t = 2:1 s. : : : : controller OFF, | : controller ON after t = 0:15 s.

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0−8 0

−7 0

−6 0

−5 0

−4 0

−3 0

−2 0

−1 0

0

1 0

2 0

F r e q u e n c y , ( H z )

Sp

ec

tra

l P

ow

er

de

ns

ity

, (

dB

)

C o n t r o l l e r O N C o n t r o l l e r O F F

Figure 12. STR performance in a noisy environment: additional white noise having a rms

velocity uctuation of 12 % of the mean velocity (� = 0:7, �M1 = 0:08).



w u t tot

0 50 100 150 200 250

2

0

2

4

6

ttot

/ dt

sin(w1 t

tot)

k1

0 50 100 150 200 2500

10

20

30

40

50

60

ttot

/ dt

settling time (arbitrary units)

in simulation

(a)

(b)

control control controlcontrolcontrol obtained

in simulation

1

k0k1<0

k0k1>0

k0k1<0

k0k1>0

k0k1<0

k0k1>0

k0k1<0

k0k1>0

k0k1>0

theoretical stability bands and

w u t tot

k0k1>0sign(k0k1) for 2nd order plant

2p 6p4p0

u

2p 6p4p0

Figure 13. STR performance in a simulation based on Dowling's (1999) premixed ame

model, compared with a theoretical study of a second order plant having the same time

delay �tot and stabilised by the STR (Appendix F). (a) k1 obtained in simulation and

sign(k0k1) required for stability for a second order plant. (b) simulation: control obtained

periodically up to !u�tot � 3 cycles of oscillation.



75 cm

Vc

Pref

34 cmDSP board

amplifier propane

22 cm

5 cm

xref

-xu

0

x

xd

extra tubelength

30 cm

Figure 14. Schematic representation of the experimental set-up.



0 100 200 300 400 500 600 700 800 900 100080

60

40

20

0

Frequency (Hz)

Gai

n (d

B)

0 100 200 300 400 500 600 700 800 900 10002

1

0

1

2

3

Frequency (Hz)

Pha

se (

x pi

rad

ian)

0 100 200 300 400 500 600 700 800 900 1000

60

40

20

0

20

Frequency (Hz)

Gai

n (d

B)

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

Frequency (Hz)

Pha

se (

x pi

rad

ian)

(a-1) without flame (a-2) with flame

(a) open-loop transfer function measured experimentally

0 100 200 300 400 500 600 700 800 900 100060

40

20

0

20

40

Frequency (Hz)

Gai

n (d

B)

0 100 200 300 400 500 600 700 800 900 10002.5

2

1.5

1

0.5

0

0.5

Frequency (Hz)

Pha

se (

x pi

rad

ian)

0 100 200 300 400 500 600 700 800 900 100060

40

20

0

20

40

Frequency (Hz)

Gai

n (d

B)

th=2 ms

0 100 200 300 400 500 600 700 800 900 10000.5

0

0.5

1

1.5

2

2.5

3

3.5

Frequency (Hz)

Pha

se (

x pi

rad

ian)

(b-1) without flame (b-2) with flame

(b) open-loop transfer function derived theoretically

Figure 15. Comparison of measured and theoretically derived open-loop transfer function.



0 5 10 151000

500

0

500

1000

Time (sec)

Pre

f (P

a)

0 5 10 154

2

0

2

4

Time (sec)

Vc

(Vol

ts)

2 2.05 2.1 2.15 2.2500

0

500

Time (sec)

Pre

f (P

a)

2 2.05 2.1 2.15 2.24

2

0

2

4

Time (sec)

Vc

(Vol

ts)

L varied from 75 to 105 cmcontrol ON control ON

0

500

500

0

(a)

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2500

0

500

Time (sec)

Pre

f (P

a)

2.5 2.6 2.7 2.8 2.9 3 3.1 3.21

0.5

0

0.5

1

Time (sec)

Vc

(Vol

ts)

0 5 10 151000

500

0

500

1000

Time (sec)

Pre

f (Pa

)

0 5 10 154

2

0

2

4

Time (sec)

Vc

(Vol

ts)

control ONL varied from 75 to 105 cm

control ON

0

500

500

(b)

2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6500

0

500

Time (sec)

Pre

f (P

a)

2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6

1

0.5

0

0.5

1

Time (sec)

Vc

(Vol

ts)

0 5 10 151000

500

0

500

1000

Time (sec)

Pre

f (Pa

)

0 5 10 154

2

0

2

4

Time (sec)

Vc

(Vol

ts)

L varied from 75 to 105 cmcontrol ON control ON

0

500

500

(c)

zoom in

zoom in

zoom in

Figure 16. Control of Rijke tube under varying tube length L. L has been varied be-

tween the two vertical lines. (a) �xed �rst order compensator, (b) adaptive �rst order

compensator (equation (3.3)), (c) STR (equation (3.26)).



0 Tosc 2Tosc 3Tosc 4Tosc 5Tosc1.5 ms

0 5 10 15 20 25 30 353

2

1

0

1

2

3

4

wuttot

(rad/s)

final k1 when flame is controlled

sin(wuttot

)

k0k1>0 k0k1>0 k0k1>0

k0k1<0k0k1<0 k0k1<0 k0k1<0

k0k1>0

k0k1>0

k0k1<0 k0k1<0

k0k1>0

theoretical stability bands andsign(k0k1) for 2nd order plant

CONTROL of Rijke tube obtained

= 1.5ms + ttot ext

t

Figure 17. EÆciency of STR at controlling the Rijke tube in the presence of a

substantial added time delay �ext.

f i f i+1

g i+1g i

x0 x1 x i x i+1

Ruo

z1z jz j+1

a ja j+1

b j+1b j

z0

(a) upstream pipework system

(b) downstream pipework system

Figure 18. Upstream and downstream pipework systems.


Documents

2 S. Evesque, AP Dow ling and AM Aweb.mit.edu/aaclab/pdfs/rsguide_final.pdf · Activ econ trol pro vides a w a y of extending their stable op erating range b y in terrupting the damaging