Adaptive Control of a Combustion Acoustics Model

with Input Saturation Using Control Hedging

Rashi Bansal∗ and N. Ananthkrishnan†

Indian Institute of Technology – IIT Bombay, Mumbai 400076, India

The presence of acoustics in combustion chambers is a major but unavoidable irritant. Apositive coupling between the acoustics field and the heat release source leads to thermoa-coustic instability, manifested with severe pressure oscillations. An effort has been made tosuppress the limit cycle oscillations using an adaptive feedback linearisation scheme. How-ever, the problem is made difficult by the fact that the control effort required is normallynot available due to actuator saturation limits. In this paper, a technique known as controlhedging is used to combat the effect of input saturation on the system dynamics. Also,an observer has been used to estimate the unmeasured states and a band-limited differen-tiator is used to estimate the acoustic velocity. Simulation results have been obtained toobserve the effect of variation in the parameters of higher modes of the instability model,and unmodeled dynamics.

Nomenclature

Cni, Dni gasdynamics terms

P symmetric 2 × 2 positive definite matrix

Xp, Xm plant and reference model state vectors, respectively

Xmod modified reference model state vector

c1 coupling constant

e error vector between plant and reference model

ep error vector between plant and modified reference model

e modified tracking error vector

r(t) forcing function

r1, r2 uniformly distributed random numbers

u control input

uc control input commanded

Γ diagonal weighting matrix

αn, θn parameters of linear combustion

δ1, δ2 controller parameters

δ∗1 , δ∗2 ideal controller parameters

δh hedge signal

ε1, ε2 error in controller parameters

ζn, ζn observer states

ηn amplitude of nth acoustic mode

ξ damping ratio of reference model

ωn frequency of reference model

∗Masters Student, Department of Aerospace Engineering; rashi@aero.iitb.ac.in.†Associate Professor, Department of Aerospace Engineering; akn@aero.iitb.ac.in. Senior Member AIAA.

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12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference)8 - 10 May 2006, Cambridge, Massachusetts

AIAA 2006-2611

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Combustion chambers, unfortunately, function as very effective acoustic resonators. A small fractionof the energy released in the combustion process is adequate to set up an acoustic wave system in the

combustion chamber – a phenomenon called combustion instability.1 Typically, this instability involves acoupling between the unsteady components of pressure and heat release rate, resulting in growing amplitudeof acoustic pressure oscillations. The pressure oscillations become more severe as the operating conditions ofcombustors change to meet specific performance criteria, and usually take the form of nonlinear oscillationscalled limit cycles.2,3 These oscillations are undesirable since they lead to excessive vibrations resulting inmechanical failures, high levels of acoustic noise, high burn rates of the fuel, and component melting.4 Toprevent degradation of system performance due to combustion instabilities, it becomes essential to providesome control to stabilize the oscillations. Passive approaches towards this end, like changing the location offuel injection, and changing the design of chamber to increase damping, have been shown to be unsystematicand inadequate in the face of changes in operating conditions and aging, and have given way to active andadaptive control for achieving the desired performance.5

A nonlinear acoustic wave model in a combustion environment has been developed by Culick.6 Fungand Yang,7 and Fung et al.,8 developed control-oriented extensions of Culick-type models and studied theeffect of PI compensators to achieve stability. However, the selection of gains required the knowledge of themodel parameters. If these parameters are not known or change with operating conditions, the mistunedcontroller may make one or more modes unstable. Hence, a need for adaptive control was felt. Krstic et al.9

focussed on the two-mode Culick model and developed a technique for self-tuning of the parameters of a PIcontroller to ensure stabilization of both the modes. When the actuator is saturated, the adaptive controllerin their case does not entirely suppress but only reduces the amplitude of the limit cycle. In another workby Mettenleiter et al.,10 the controller consists of an adaptive finite impulse response filter whose coefficientsare updated using an LMS algorithm. Recently, Evesque et al.11 developed an adaptive control design,called self-tuning regulator (STR), which attempts to rely as little as possible on a particular combustionmodel. In case of amplitude saturation, the STR design is modified, and it is seen that when the amplitudeconstraint becomes too severe, only small-amplitude initial oscillations can be controlled. A detailed reviewof these and other advances in active control of combustion acoustics has been presented by Dowling andMorgans.12

In the present work, an adaptive feedback linearizing control scheme has been used to suppress limitcycle oscillations in a four-mode Culick-type combustion acoustics model. The adaptive scheme is ableto simultaneously estimate the unknown parameters due to the linear combustion model in the first-modeacoustic dynamics. The control action may be thought to consist of two parts: i) The nonlinear terms drivingthe first-mode acoustics are canceled by a linearizing controller, and ii) The linear terms in the control lawmake the closed-loop plant-controller dynamics track a desired linear reference model. While the nonlinearterms in the control law, corresponding to second-order gasdynamic terms in the acoustic model, are takento be perfectly known, the linear terms are assumed unknown due to lack of knowledge of the correspondinglinear combustion terms in the model. Hence, an adaptive law is required to update the linear controllerparameters. Use of an adaptive feedback linearization scheme for the control of limit cycling motion has beenattempted in the past, but for a single degree-of-freedom model representing the rolling motion of a deltawing.13,14 Extension of those works to include simultaneous control and parameter estimation for the deltawing rolling problem has been recently presented by Jain et al.15 In contrast, the present problem considersa multi-degree-of-freedom model with four acoustic modes, which is the least number of modes that oughtto be considered for a first acoustic mode instability, following the recent work of Ananthkrishnan et al.16

The control design is based on full state feedback. The states corresponding to the first mode are availablebut those of higher acoustic modes are not. Estimates of the unavailable states are therefore obtained from anovel synchronization-based nonlinear observer, which uses only the first-mode acoustic pressure signal as themeasured variable. It may be noted that our observer design is fundamentally different from previous workin this direction,17 where the measured signal contained traces of all the acoustic modes, and the observeressentially decomposed the measured signal to extract the individual modal amplitudes and frequencies.

In practice, it is often the case that the required control (as computed by a controller of the feedbacklinearization or dynamic inversion type) is not available, and there is a limit to which control can be applied.It, thus, becomes essential to study the effect of input saturation on the control action and on parameterconvergence in such adaptive schemes, as well. With upper and lower bounds on the applied input inplace, the error driving the adaptive law also includes the error due to saturation and, hence, may result

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in incorrect adaptation. There has been a fair amount of work on adaptive control design in the presenceof input saturation. Monopoli18 suggested a modification of the adaptive control to account for saturationeffects, but no formal proof of stability was provided. Another concept of modifying the error, proportionalto the control deficiency, was suggested by Karason and Annaswamy.19 A fixed gain adjustment to thereference model, proportional to the control deficiency, called pseudo control hedging, was introduced byJohnson and Calise.20 In the pseudo control hedging scheme, the difference between the calculated controland the applied control due to input saturation, is called the hedge signal. The hedge signal results in alack of acceleration in the plant as compared to the reference acceleration. When the hedge is removedfrom the reference, the modified reference can be tracked within saturation limits. The tracking error willthen not contain the component due to saturation and, hence, the controller will adapt correctly.21 Anotherrecent approach, called positive µ-modification, has been proposed by Lavretsky and Hovakimyan,22 whichguarantees that the control never incurs saturation.23

In this paper, we incorporate the idea of control hedging,24 which is conceptually similar to pseudo controlhedging, into an adaptive feedback linearization scheme, with an observer for estimating the unknown states,to suppress the acoustic modes and obtain parameter convergence for the first mode of the four-mode Culickcombustion acoustics model, including effects of input saturation.

II. Nonlinear Model and Control Strategy

The nonlinear dynamics of acoustic waves in a combustion chamber as modeled by Culick6 is a setof linearly uncoupled second-order oscillators, one for each acoustic mode. The oscillations are coupleddue to nonlinear (second-order) gasdynamic effects. The modal natural frequencies have been assumed to beintegral multiples of the primary acoustic mode frequency, which is fairly accurate for the case of longitudinalmodes.16 The unforced coupled oscillator equations, with time nondimensionalized by the primary modefrequency, then appear as follows:

ηn − 2αnηn + n(n− 2θn)ηn = −n−1∑

i=1

(C(1)ni ηiηn−i + D

(1)ni ηiηn−i)−

∞∑

i=1

(C(2)ni ηiηn+i + D

(2)ni ηiηn+i) (1)

where ηn is the amplitude (nondimensional pressure) of the nth acoustic mode. The coefficients C and Dcan be exactly obtained from knowledge of the gasdynamics, and are given as:

C(1)ni =

−12γi(n− i) [n2 + i(n− i)(γ − 1)]

C(2)ni =

1γi(n+ i)

[n2 − i(n+ i)(γ − 1)]

D(1)ni =

γ − 14γ

[n2 − 2i(n− i)]

D(2)ni =

γ − 12γ

[n2 + 2i(n+ i)] (2)

The parameters αn and θn are defined as

αn = αn/ω1, θn = θn/ω1 (3)

where ω1 is the natural frequency of the first acoustic mode, and equals 5660 rad/s for the problem at hand.6

The parameters αn and θn are considered to be unknown for n = 1 and will be considered to be uncertainlater for higher n, and an adaptive law is required to estimate these parameters for n = 1. Nominal values ofthese parameters are listed in Table 1. Bifurcation analysis16,25 shows that the zero-amplitude equilibriumsolution for this model is linearly stable for α1 < 0 and becomes unstable for α1 > 0. Referring to Table 1, thenominal value of α1 is positive, hence the first mode is in unstable operation. Simulation of the uncontrolledplant for α1 = 0.005 shows limit cycle oscillations in all the acoustic modes. These are shown as projectionsin the ηi − ηi plane in Fig. 1. It is known16 that an unstable first mode drives the higher modes into limitcycle oscillations by a coupling mechanism due to the nonlinear gasdynamic terms in (1).

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Table 1. Data for parameters αn and θn.

Mode Number One Two Three Fourαn, 1/s 28.3 -324.8 -583.6 -889.4θn, rad/s 12.9 46.8 -29.3 -131.0

A. Control Strategy

The nonlinear plant dynamics is truncated to four modes, and the first mode dynamics is given as:

η1 = −(1− 2θ1)η1 + 2α1η1 − C(2)11 η1η2 − D(2)

11 η1η2 − C(2)12 η2η3 − D(2)

12 η2η3 − C(2)13 η3η4 − D(2)

13 η3η4 + u (4)

where u is the control input. The feedback linearizing control is taken to have the following form:

uc = (1− 2δ1)η1 − δ2η1 + C(2)11 η1η2 + D

(2)11 η1η2 + C

(2)12 η2η3 + D

(2)12 η2η3 + C

(2)13 η3η4 + D

(2)13 η3η4 + r(t) (5)

where uc is the control commanded by the controller, δ1 and δ2 are the controller parameters that need tobe adapted as they cannot be determined if the plant parameters, α1 and θ1, are unknown, and r(t) is theexternal forcing function. The control input u may, for example, arise from a secondary fuel injection, andneeds to be characterized based on the physics of the problem. This issue is not dealt with in the presentwork where it is assumed that an appropriate control is physically possible.

The block diagram of the adaptive feedback linearizing control scheme employed is shown in Fig. 2. Thereference model is a stable linear model that is to be tracked by the closed-loop plant-controller dynamics.The closed loop includes a saturation block that limits the control input u fed into the plant. The first-modeplant states, η1 and η1, are assumed measurable and are fed back to the controller, and are also used inthe adaptive law. The error e between the first-mode plant states and the reference model states drives theadaptive law, which evolves the controller parameters δ, which are then updated in the controller block. Asuitably weighted (by the constant c1) error between the first mode amplitude of the plant and the observeris used to drive the observer states to synchronize with the plant states. Estimates of the higher-mode plantstates obtained from the observer are used as feedback to the controller.

The adaptive law has been derived assuming that the commanded control is available to the plant, thatis, u = uc, and the effect of saturation has not been incorporated in the adaptive law, though simulationshave been done with control saturation included. The closed-loop plant-controller system for the first modehas been obtained by combining equations (4) and (5), assuming u = uc, as follows:

xp − (2α1 − δ2)xp − 2(θ1 − δ1)xp = r(t) (6)

where xp = η1 and xp = η1. The reference model is taken to be the following second-order linear system:

xm + 2ξωnxm + ω2nxm = r(t) (7)

where the constants ξ and ωn are chosen to have the following values: ξ = 0.1 and ωn = 1.The ideal values of the controller parameters, δ∗1 and δ∗2 , which provide for model matching between the

first mode of the closed-loop plant-controller system and the reference model, are obtained from (6) and (7),as follows:

2(θ1 − δ∗1) = −ω2n

2α1 − δ∗2 = −2ξωn (8)

The controller parameter errors are then defined as εi = δi − δ∗i . The adaptation scheme to tune thecontroller parameters δi can be derived by a Lyapunov analysis, as outlined by Singh et al.,13 which givesthe following parameter update law:

δ = Γ−1h(Xp)(eTPb) (9)

where, δT = [δ1, δ2], bT = [0, 1], XTp = [xp, xp], hT (Xp) = [2xp, xp], Γ = diag(0.1, 0.01), XT

m = [xm, xm],and e is the state error vector defined as e = [xp − xm, xp − xm]T = Xp − Xm. P is a 2 × 2 positive

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definite matrix. The form of the forcing function r(t) has been derived by Jain et al.15 and is given asr(t) = 0.1 exp(−t/300) sin(2 ∗ t).

Since it has been assumed that not all the states are measurable, a novel nonlinear observer based onsynchronization theory, first proposed by Ananthkrishnan,26 has been developed to estimate the unknownstates. The acoustic wave dynamics of (1) are resonantly coupled due to the nonlinear gasdynamic terms,and a single measured variable, such as η1, is sufficient to excite all the modes of the observer to trackthe states of the system. Here, it is assumed that only the first mode plant states, that is, η1 and η1, aremeasurable. The observer dynamical equations are identical to those of the plant except that the unstablefirst mode, whose parameters α1, θ1 are unknown, is replaced by the equation for the stable reference model.This equation is driven by a function proportional to the difference between the first mode amplitude ofthe plant and the observer, that is, c1(η1 − ζ1), where ζn and ζn are the observer states. c1 is the couplingconstant and it has been shown using bifurcation analysis that synchronization of the observer and plantstates is possible for any c1 > 0.26 Here, c1 has been chosen to be 0.4. The estimates of the states ζ2 throughζ4, and ζ2 through ζ4, are taken from the observer and fed to the controller.

The error between the plant states, Xp, and the reference model states, Xm, drives the adaptive law.But, this error also includes a component due to control saturation. Since the effect of saturation has notbeen taken into account in the derivation of the adaptive law, the adaptation may be incorrect. To accountfor this, control hedging has been found to be an effective tool which makes the plant track the referencemodel by removing the hedge signal from the reference model. The modification in the reference model isshown as:

xmod + 2ξωnxmod + ω2nxmod = r(t)− δh (10)

where δh = uc − u, and XTmod = [xmod, xmod] are the states of the modified reference model. The use of

control hedging also results in the convergence of controller parameters to their ideal values, even for fairlylow values of input saturation, though this was apparently not a factor considered by the proposers20 of thismethod.

The control going into the plant with the saturation block in place is given by:

u = (1− 2δ1)η1 − δ2η1 + C(2)11 η1η2 + D

(2)11 η1η2 + C

(2)12 η2η3 + D

(2)12 η2η3 + C

(2)13 η3η4 + D

(2)13 η3η4 + r(t)− δh (11)

To account for the input saturation, the error dynamics can be given as:

¨e1 + 2ξωn˙e1 + ω2

ne1 − (2α1 − δ2 + 2ξωn)xp − (2θ1 − 2δ1 + ω2n)xp = −δh (12)

where e = [e1, ˙e1]T = Xp −Xm.Let the modified tracking errors be ep = [ep1 , ep1 ]T = Xp −Xmod and em = [em1 , em1 ]T = Xmod −Xm,

such that e = ep + em. The dynamics of the error between the modified reference model and the originalreference model is obtained as:

em1 + 2ξωnem1 + ω2nem1 = −δh (13)

which is a stable second-order dynamics. The hedge signal, δh, acts as a disturbance input and deviates themodified reference trajectory from the original reference. The modified reference trajectory will track theoriginal reference if the control signal does not remain saturated for the entire interval. Whenever δh is zero,the modified reference tracks the original reference.21

The error dynamics between the plant states and the modified reference is given as:

ep1 + 2ξωnep1 + ω2nep1 − (2α1 − δ2 + 2ξωn)xp − (2θ1 − 2δ1 + ω2

n)xp = 0 (14)

which shows that the tracking error in the plant states does not include the effect of saturation.

III. Results

Simulation results are now presented to show the effectiveness of control hedging in obtaining parameterconvergence and tracking, in the presence of input saturation. Simulations are first carried out for a highvalue of input saturation of ±0.05, for which the parameter errors are seen to converge to zero, without the useof control hedging (see Fig. 3). Figure 3(a) shows that the first-mode plant states are stabilized, that is, thefirst-mode acoustic dynamics are suppressed. Similarly the higher-mode acoustic waves are also controlled(not shown in figure). Figure 3(b) plots the norm of state error which goes to zero, indicating good tracking

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performance, that is, the damping and natural frequency of the first acoustic mode match well with therespective values specified for the reference model. Figures 3(c) and 3(d) show that the commanded controlinput significantly exceeds the saturation limit initially; nevertheless, control and tracking are successfullyachieved. Additionally, Figs. 3(e) and 3(f) show that the controller parameters indeed converge to their idealvalues as defined in (8). Thus, it is also possible to estimate the linear combustion model terms θ1 and α1

in (4).When the simulations are repeated for a lower value of saturation limit of ±0.005, the results are not so

successful (see Fig. 4). For this case, as seen from Fig. 4(a), the plant states do fall to zero; hence, the acousticsare effectively suppressed. However, the error norm as seen in Fig. 4(b) does not readily fall to zero, suggestingthat the plant states do not track the reference model states well. The control commanded in Fig. 4(c) is seento be an order of magnitude larger than that in Fig. 3(c), whereas the control input available in Fig. 4(d)is, in comparison, insignificant. The large control command is on account of the incorrect adaptation whichdrives the parameter errors to large values. Effectively, the error signal in Fig. 4(b) contains the influence ofinput saturation, and this error drives the adaptive law into wrongly evolving the controller gain parameters,δ1, δ2. The controller parameter errors in Figs. 4(e) and 4(f), therefore, do not converge to zero. In fact,successful parameter error convergence is an indicator of effectiveness of the adaptive control scheme.

The situation is remedied when control hedging is used with the saturation limit of ±0.005 (see Fig. 5).Figures 5(a) and 5(b) show first-mode acoustic suppression and error dynamics which readily converges tozero. The control commanded for this case and the control available are shown in Figs. 5(c) and 5(d),respectively. The hedge signal shown in Fig. 6 is removed from the input signal to the reference model, andthe error dynamics now does not include the effect of saturation. The adaptive law is now able to successfullydrive the parameter errors to zero as seen from Figs. 5(e) and 5(f).

To verify the effectiveness of the observer in estimating unknown plant states, the dynamics of the com-bined plant-observer has been described by plotting the state trajectory on the appropriate projection plane.Figure 7 shows the projections on the ηn− ζn plane for the first and second mode variables. The projectionsare seen to tend towards a line of slope 1, which is the projection of the synchronization manifold.26 Thisimplies that the observer states first synchronize with the plant states before they collectively damp out tozero due to the control action. Similar plots for the third and fourth modes have not been shown since theiramplitudes are much smaller than that of the first and second modes.

IV. Estimation of acoustic velocity

In the above analysis, it has been assumed here that η1 is measurable, but in practice acoustic velocitymeasurements may not always be easy. Hence, it may be necessary to replace the η1 feedback signal by anestimate. Since the first mode of the observer is the stable reference model, ζ1 will not give the true acousticvelocity. Therefore, the estimate has been obtained by using a band-limited differentiator of the form sL

s+L ,which approaches an ideal differentiator as the bandwidth parameter, L, becomes large. Simulations havebeen carried out for a varying range of L and it has been observed that the parameter error reduces as Lincreases in the range 1 < L < 50. However, large values for L will lead to possibilities of high frequencynoise getting amplified. Moreover, simulations take longer time as L increases. Figure 8 shows the projectionof the closed-loop trajectories on the plane of actual and estimated acoustic velocity. The plot shows theprojection at a slope of 1, which suggests a very good estimate. The parameter errors, ε1 and ε2 also convergenearly to zero when the estimate of η1 is used in the adaptive and control laws, as shown in Figs. 9(a) and9(b).

V. Parameter variation and unmodeled dynamics

To verify the parameter convergence, both with and without saturation, simulations have been done withplant parameters varying by a reasonable amount. The parameter variation is modeled as follows

α′i = αi(1 + 0.5ri)

θ′i = θi(1 + 0.5ri) (15)

where ri are uniformly distributed random numbers such that ri ∈ [−1, 1] and the parameters are variedwithin 50% of their nominal values, αi and θi. It is seen that the parameter errors still converge to zero, and

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since the figures look identical to those obtained without parameter variation, they have not been shownhere.

To account for unmodeled dynamics, the fifth and the sixth modes of the instability have also beenincluded in the reduced-order instability model developed by Culick.16 These modes have not been cancelledby the controller, and these two modes constitute the unmodeled dynamics. The simulation results show thatthe controller parameters converge to their ideal values, showing that the higher order unmodeled modes donot effect the dynamics of the system. It has also been observed that even if the controller does not cancelthe nonlinear terms of the plant with four modes, the parameter errors still converge to zero. This happensbecause the nonlinear terms are very small as compared to the first mode plant states. Figure 10 shows thesum of nonlinear terms during the simulation interval. By comparing, it is seen that the magnitude of thissum is very small as compared to the magnitude of the first mode plant states. This means that even if allother higher mode states are not measured, the adaptive feedback linearization scheme results in parameterconvergence, hence, making the system stable.

VI. Conclusions

Active control strategies are necessary to effectively combat the growing heat release and limit cyclepressure oscillations set up in combustion chambers due to combustion instabilities. An adaptive feedbacklinearizing control strategy has been described in this work that suppresses the acoustic wave dynamics ina four-mode, nonlinear combustion acoustics model of the Culick type. The adaptive law has been shownto be capable of simultaneously estimating the unknown linear combustion model parameters for the firstacoustic mode. However, the adaptive law has been derived without considering input saturation constraints.Nevertheless, the control strategy is seen to function effectively for reasonably large saturation limits, butfails for low saturation levels. Under severe saturation limits, the adaptive control scheme is modified byusing a control hedging algorithm to stabilize the acoustic mode dynamics, and to correctly estimate thesystem parameters. The control law is based on a full state feedback scheme with the higher-mode statevariables obtained from a novel synchronization-based nonlinear observer design. The estimate of the acousticvelocity has been obtained using a band-limited differentiator, which gives best results for large values of L.The adaptive feedback linearizing scheme is seen to work well with high level of uncertainties in the linearcombustion model parameters. The effect of unmodeled dynamics on the performance of this scheme hasbeen observed by adding the 5th and the 6th modes in the four-mode Culick model. These modes, when notcancelled by the controller, do not effect the system dynamics because the magnitude of these modes is verysmall compared to the first mode.

VII. Acknowledgments

The second author would like to thank Dr. F.E.C. Culick for giving him an opportunity to spend asabbatical year at the California Inst. of Technology, Pasadena, CA, where the work on the synchronization-based observer was initiated.

References

1Culick, F.E.C., “Combustion instabilities in propulsion systems,” in Culick, F.E.C. et al. (Eds.) Unsteady Combustion,Kluwer Academic Publishers, The Netherlands, pp. 173-241.

2Culick, F.E.C., “Some recent results for nonlinear acoustics in combustion chambers,” AIAA Journal, Vol. 32, No. 1,1994, pp. 146-169.

3Dowling, A.P., “Nonlinear Acoustically-Coupled Combustion Oscillations,” AIAA Paper 96-1749, 2nd AIAA/CEASAeroacoustics Conference, State College, PA, May 6-8, 1996.

4Annaswamy, A.M., Fleifil, M., Rumsey, J.W., Prasanth, R., Hathout, J.-P., and Ghoniem, A.F., “Thermoacoustic insta-bility: Model-based optimal control designs and experimental validation,” IEEE Transactions on Control Systems Technology,Vol. 8, No. 6, 2000, pp. 905-918.

5Annaswamy, A.M., and Ghoniem, A.F., “Active control in combustion systems,” IEEE Control Systems Magazine,Vol. 15, No. 6, 1995, pp. 49-63.

6Culick, F.E.C, “Nonlinear behavior of acoustic waves in combustion chambers,” Acta Astronautica, Vol. 3, 1976, pp. 714-757.

7Fung, Y.-T., and Yang, V., “Active control of nonlinear pressure oscillations in combustion chambers,” Journal ofPropulsion and Power, Vol. 8, 1992, pp. 1282-1289.

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8Fung, Y.-T., Yang, V., and Sinha, A., “Active control of combustion instabilities with distributed actuators,” CombustionScience Technology, Vol. 78, 1991, pp. 217-245.

9Krstic, M., Krupadanam, A., and Jacobson, C., “Self-tuning control of a nonlinear model of combustion instabilities,”IEEE Transactions on Control Systems Technology, Vol. 7, No. 4, 1999, pp. 424-436.

10Mettenleiter, M., Haile, E., and Candel, S., “Adaptive control of aeroacoustic instabilities,” Journal of Sound andVibration, Vol. 230, No. 4, 2000, pp. 761-789.

11Evesque, S., Park, S., Riley, A.J., Annaswamy, A.M., and Dowling, A.P., “Adaptive combustion instability control withsaturation: theory and validation,” Journal of Propulsion and Power, Vol. 20, No. 6, 2004, pp. 1086-1095.

12Dowling, A.P., and Morgans, A.S., “Feedback control of combustion oscillations,” Annual Reviews in Fluid Mechanics,Vol. 37, 2005, pp. 151-182.

13Singh, S.N., Yim, N., and Wells, W.R., “Direct adaptive and neural control of wing-rock motion of slender delta wings,”Journal of Guidance, Control, and Dynamics, Vol. 18, No. 1, 1995, pp. 25-30.

14Monahemi, M.M., and Krstic, M., “Control of wing rock motion using adaptive feedback linearization,” Journal ofGuidance, Control, and Dynamics, Vol. 19, No. 4, 1996, pp. 905-912.

15Jain, H., Kaul, V., and Ananthkrishnan, N., “Parameter estimation of unstable, limit cycling systems using adaptivefeedback linearization: Example of delta wing roll dynamics,” Journal of Sound and Vibration, Vol. 287, Nos. 4-5, 2005,pp. 939-960.

16Ananthkrishnan, N., Deo, S., and Culick, F.E.C., “Reduced-order modeling and dynamics of nonlinear acoustic waves ina combustion chamber,” Combustion Science and Technology, Vol. 177, No. 2, 2005, pp. 221-248.

17Neumeier, Y., and Zinn, B.T., “Active control of combustion instabilities using real time identification of unstablecombustor modes,” IEEE Conference on Control Applications, 1995, pp. 691-698.

18Monopoli, R.V., “Adaptive control for systems with hard saturation,” Proceedings of the IEEE Conference on Decisionand Control, Houston, TX, 1975, pp. 841-843.

19Karason, S.P., and Annaswamy, A.M., “Adaptive control in the presence of input constraints,” IEEE Transactions onAutomatic Control, Vol. 39, No. 11, 1994, pp. 2325-2330.

20Johnson, E.N., and Calise, A.J., “Limited authority adaptive flight control for reusable launch vehicles,” AIAA Journalof Guidance, Control and Dynamics, Vol. 26, No. 6, 2003, pp. 906-913.

21Tandale, M.D., and Valasek, J., “Adaptive dynamic inversion control with actuator saturation constraints applied totracking spacecraft maneuvers,” Proceedings of the 2004 American Control Conference, Boston, MA, 2004.

22Lavretsky, E., and Hovakimyan, N., “Positive µ-modification for stable adaptation in the presence of input constraints,”Proceedings of the 2004 American Control Conference, Boston, MA, 2004, pp. 2545-2550.

23Tandale, M.D., and Valasek, J., “Adaptive dynamic inversion control of a linear scalar plant with constrained controlinputs,” Proceedings of the 2005 American Control Conference, Portland, OR, 2005.

24Yang, B.-J., Calise, A.J., and Craig, J.I., “Adaptive output feedback control with input saturation,” Georgia Institute ofTechnology, School of Aerospace Engineering, Atlanta, GA 30332.

25Jahnke, C.C., and Culick, F.E.C., “Application of dynamical systems theory to nonlinear combustion instabilities,”Journal of Propulsion and Power, Vol. 10, 1994, pp. 508-517.

26Ananthkrishnan, N., “A nonlinear estimator for acoustic wave dynamics in a combustion chamber using synchronizationtheory,” Guggenheim Jet Propulsion Center, Documents on Active Control of Combustion Instabilities, Document Number CI01-03, California Inst. of Technology, Pasadena, CA, Mar. 2001.

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−0.1 −0.05 0 0.05 0.1 0.15

−0.1

−0.05

0

0.05

0.1

0.15

ETA__1

ET

A__

1__D

OT

(a)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.06

−0.04

−0.02

0

0.02

0.04

0.06

ETA__2

ET

A__

2__D

OT

(b)

−0.015 −0.01 −0.005 0 0.005 0.01 0.015−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

ETA__3

ET

A__

3__D

OT

(c)

−8 −6 −4 −2 0 2 4 6 8

x 10−3

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

ETA__4

ET

A__

4__D

OT

(d)

Figure 1. Open-loop simulation showing limit cycle oscillations in (a) first mode, (b) second mode, (c) thirdmode, and (d) fourth mode.

Figure 2. Block diagram of adaptive feedback linearization scheme.

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−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

ETA__1

ET

A__

1__D

OT

(a)

0 100 200 300 400 500 600 700 8000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Nondimensional time

2−N

orm

of s

tate

err

or

(b)

0 500 1000 1500−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Nondimensional time

Con

trol

inpu

t com

man

ded

(c)

0 500 1000 1500−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Nondimensional time

Con

trol

inpu

t ava

ilabl

e

(d)

0 100 200 300 400 500 600 700 800−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Nondimensional time

Par

amet

er e

rror

1

(e)

0 100 200 300 400 500 600 700 800−3

−2

−1

0

1

2

3

Nondimensional time

Par

amet

er e

rror

2

(f)

Figure 3. Simulation results for a saturation limit of ±0.05, with no control hedging (a) first mode plant states,η1 and η1, (b) 2-norm of the state error vector e, (c) control commanded, uc, (d) actual control available, u,(e) controller parameter error ε1, and (f) controller parameter error ε2.

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−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

ETA__1

ET

A__

1__D

OT

(a)

0 500 1000 15000

0.02

0.04

0.06

0.08

0.1

0.12

Nondimensional time

2−N

orm

of s

tate

err

or

(b)

0 500 1000 1500

−0.1

−0.05

0

0.05

0.1

0.15

Nondimensional time

Con

trol

inpu

t com

man

ded

(c)

0 500 1000 1500

−0.1

−0.05

0

0.05

0.1

0.15

Nondimensional time

Con

trol

inpu

t ava

ilabl

e

(d)

0 500 1000 1500−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Nondimensional time

Par

amet

er e

rror

1

(e)

0 500 1000 1500−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Nondimensional time

Par

amet

er e

rror

2

(f)

Figure 4. Simulation results for a saturation limit of ±0.005, with no control hedging (a) first mode plant states,η1 and η1, (b) 2-norm of the state error vector e, (c) control commanded, uc, (d) actual control available, u,(e) controller parameter error ε1, and (f) controller parameter error ε2.

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−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

ETA__1

ET

A__

1__D

OT

(a)

0 100 200 300 400 500 600 700 8000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Nondimensional time

2−N

orm

of s

tate

err

or

(b)

0 500 1000 1500

−0.1

−0.05

0

0.05

0.1

0.15

Nondimensional time

Con

trol

inpu

t com

man

ded

(c)

0 500 1000 1500

−0.1

−0.05

0

0.05

0.1

0.15

Nondimensional time

Con

trol

inpu

t ava

ilabl

e

(d)

0 100 200 300 400 500 600 700 800

−0.4

−0.2

0

0.2

0.4

0.6

Nondimensional time

Par

amet

er e

rror

1

(e)

0 100 200 300 400 500 600 700 800

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Nondimensional time

Par

amet

er e

rror

2

(f)

Figure 5. Simulation results for a saturation limit of ±0.005 with control hedging (a) first mode plant states,η1 and η1, (b) 2-norm of the state error vector ep, (c) control commanded, uc, (d) actual control available, u,(e) controller parameter error ε1, and (f) controller parameter error ε2.

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0 500 1000 1500

−0.1

−0.05

0

0.05

0.1

0.15

Nondimensional time

Hed

ge s

igna

l

Figure 6. Hedge signal, δh, for a saturation limit of ±0.005.

−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

η1

ζ 1

(a)

−6 −4 −2 0 2 4 6

x 10−4

−6

−4

−2

0

2

4

6x 10

−4

η2

ζ 2

(b)

Figure 7. Projection of plant-observer dynamics on (a) η1 − ζ1 plane, and (b) η2 − ζ2 plane.

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−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

ETA__1__DOT

Est

imat

e of

ET

A__

1__D

OT

Figure 8. Projection of closed-loop trajectories on the actual and estimated η1 plane.

0 100 200 300 400 500 600 700 800−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Nondimensional time

Par

amet

er e

rror

1

(a)

0 100 200 300 400 500 600 700 800−1.5

−1

−0.5

0

0.5

1

1.5

Nondimensional time

Par

amet

er e

rror

2

(b)

Figure 9. Simulation results with band-limited differentiator showing (a) controller parameter error, ε1, and(b) controller parameter error, ε2.

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0 100 200 300 400 500 600 700 800

−3

−2

−1

0

1

2

3

x 10−4

Nondimensional time

Sum

of n

onlin

ear

term

s

Figure 10. Sum of nonlinear terms.

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