Limits of Achievable Performance of Controlled ... - CiteSeerX

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

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Limits of Achievable Performance of Controlled Combustion Processes Andrzej Banaszuk, Prashant G. Mehta, Member, IEEE, Clas A. Jacobson, Senior Member, IEEE, and Alexander I. Khibnik

Sensitivity function. Performance bandwidth. Control bandwidth. Lower frequency for control bandwidth. Higher frequency for control bandwidth.

Abstract—This paper presents a fundamental limitations-based analysis to quantify limits on obtainable performance for active control of combustion (thermoacoustic) instability. Experimental data from combustor rigs and physics-based models are used to motivate the relevance of both the linear and nonlinear thermoacoustic models. For linear models, Bode integral-based analysis is used to explain peak-splitting observed in experiments. It is shown that large delay in the feedback loop and limited actuator bandwidth are the primary factors that limits the effectiveness of the active control. Explicit bounds on obtainable performance in the presence of delay, unstable dynamics, and limited controller bandwidth are obtained. A multi-input describing function framework is proposed to extend this analysis to the study of nonlinear models that also incorporate the effects of noise. The fundamental limitations are interpreted for a modified sensitivity function defined with respect to noise balance. The framework is applied to the analysis of linear thermoacoustic models with nonlinear ON-OFF actuators and Gaussian noise. The results of the analysis are well-supported by experiments and model simulations. In particular, we reproduce in model simulations and explain analytically the peak-splitting phenomenon observed in experiments.

Relative degree of open-loop transfer function. Required attenuation level for sensitivity function over performance bandwidth. Power-spectral density function of the input disturbance. Power-spectral density function of the combustor pressure. Frequency. Time. Standard deviation of noise. Real part of unstable pole. Time delay.

Index Terms—Active control, combustion instability, describing function, fundamental limitation, nonlinear analysis.

I. INTRODUCTION

NOMENCLATURE

f

On level for ON–OFF actuator. Transfer function representing combustor, fuel line, and valve dynamics. Controller transfer function. Linear transfer function in feedback loop. Static nonlinearity in feedback loop. Random-input describing function gain. Amplitude of the sinusoidal component of pressure. dc Bias of the pressure. Combustor pressure. Gaussian component of the pressure. Input Gaussian noise.

Manuscript received September 6, 2005. Manuscript received in final form May 9, 2006. Recommended by Associate Editor D. Rivera. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Grants F49620-01-C-0021 and FA9550-04-C-0042. A. Banaszuk, C. A. Jacobson, and A. I. Khibnik are with United Technologies Research Center, East Hartford, CT 06108 USA (e-mail: banasza@utrc. utc.com; [email protected]; [email protected]). P. G. Mehta is with the Department of Mechanical and Industrial Engineering, University of Illinois, Urbana Champaign, IL 61801 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2006.879980

E

MPHASIS on reducing the level of pollutants created by gas turbine combustors has led to the development of premixed combustor designs, especially for industrial applications. Premixing large amounts of air with the fuel prior to its injection into the combustor, greatly reduces peak temperatures within the combustor and leads to lower NOx emissions. However, premixed combustors are susceptible to the so-called thermoacoustic combustion instabilities. These instabilities occur due to a destabilizing feedback coupling between acoustics and combustion (unsteady heat release). It causes large pressure oscillation in the combustor that detrimentally affects the combustor durability and raises environmental noise pollution [23]. Active combustion instability control (ACIC) with fuel modulation has proven to be an effective approach for reducing pressure oscillations in combustors. Promising experimental results have been reported by researchers at United Technologies Research Center (UTRC) [10], [11], Seimens kWU [25], [40], ABB/Alstom [33], Honeywell Inc. [2], Westinghouse/Georgia Institute of Technology [36], and the U.S. Department of Energy [35]. However, the achieved reduction of pressure oscillation varies between these experiments from 6 to 20 dB. In many cases, the attenuation of the oscillation at primary frequency is accompanied by excitation of the oscillation in some other frequency band [18], [30], [38]. This phenomenon is commonly referred to as secondary peaking or peak splitting.

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A satisfactory explanation of the different attenuation levels and peak-splitting phenomena has not been presented in the literature. Much of the theoretical attention in the area of ACIC has focused on control design [6], [7], [9], [17], [21], [30]—that is inherently dependent on the dynamics considered in the model or present in the experiment—and not so much on factors that actually limit the achievable performance. One of the reasons for this is that the thermoacoustic oscillations frequently arise as a limit cycle that requires nonlinear models of combustion dynamics. This limits the mathematical tools available for both control design as well as the analysis of resulting dynamics. In this paper, we investigate the factors that determine achievable reduction of the level of pressure oscillation in combustors using fuel control. Our studies have been motivated by experience with ACIC in the experiments conducted at UTRC [10], [24]. These experiments range from studies done is subscale single nozzle combustors to full scale engine tests carried out at different operating conditions: high (fuel rich) versus low (fuel lean) equivalence ratio condition, varying flow velocities, low versus high turbulence noise environments, proportional versus ON–OFF actuators for controlling combustion instabilities, etc. This wide spectrum of operating conditions together with an even wider range for published literature in the area of ACIC led us to examine the fundamental limitations of ACIC. The contribution of this work is as follows. To the best of our knowledge, we are the first to address the issue of fundamental limitations in linear settings as applied to ACIC. The initial results were reported by us in [4] and [5]. We are also the first to systematically quantify the role and importance of nonlinearity in combustion instability models and its effect on dynamics and control of the instability. In particular, we outline a describing function-based framework to obtain the approximate solutions of nonlinear combustion models in the presence of a strong noise. We formally employ the framework to understand the fundamental limitations associated with ACIC using ON-OFF actuators. The results of analysis are well supported by the results of experiments and model simulations. In particular, we reproduce in models and analytically explain the peak-splitting phenomenon observed in combustion experiments both in linear and nonlinear settings. The outline of this paper is as follows. In Section II, we present a physics-based model of the UTRC experimental setup together with a framework for model identification and its validation with experiment data. In Section III, we discuss fundamental limitations for linear ACIC and apply it to the particular model of the UTRC experiment. In Section IV, we demonstrate the shortcomings of standard linear analysis to satisfactory explain the ACIC experimental observations. We show the important effects of noise and nonlinearity and this motivates the development of the following sections. In Section V, we propose a framework for nonlinear analysis of ACIC systems that incorporates the effects of noise. In Section VI, we apply the framework to explain the fundamental limitations associated with some of the experimental results. Finally, we present conclusions in Section VII. II. COMBUSTION DYNAMICS—THE OPEN-LOOP SYSTEM Combustion dynamics arise due to a feedback coupling between the acoustic modes of the combustor cavity and the unsteady heat released due to combustion of fuel-air mixture.

TABLE I DESCRIPTION OF PARAMETERS IN THE THERMOACOUSTIC MODEL (1)

The resulting feedback interconnection is typically referred to as a thermoacoustic loop. In the simplest setting considered here, the acoustics is modeled by the bulk Hemholtz mode of the combustor cavity. The precise physical mechanisms underlying the unsteady heat release are complex- and reduced-order models for the same are not well-understood. Here, the unsteady heat release is modeled as a fluctuation in the equivalence ratio (normalized fuel/air ratio) expressed as a nonlinear function of acoustic velocity input. Only the simplest two effects are considered to model the functional relationship. One is the bulk fluid convection effect that is modeled by a time delay and the other is the effect due to time-delay and nonlinearities in the burning rate, where the latter that is modeled by a static saturation nonlinearity. The resulting thermoacoustic model equations arise as

(1) where is the combustor chamber (modeled as a capacitance) is the upstream nozzle mass velocity, is pressure, is the fuel mass flow the downstream exit mass velocity, input, and is used to model the stochastic turbulent flow velocity in the nozzle—assumed to be a broadband white noise. in The heat release function (1) captures in a reduced-order fashion, the nonlinear effects due to combustion. The parameter represents the cumulative time delay—primary delay due to convection plus delay because of chemistry and duel-air mixing. For additional details on the model and explicit characteristics of the forcing term, see [34]. The parameters in the model (1) are described in Table I. The model of (1) is the simplest possible thermoacoustic model. Both the distributed acoustic modes as well as distributed (and unsteady) fluid dynamics and flame area variations, have been neglected. We consider this model here because: 1) the model was experimentally found to be relevant to the combustion instabilities being studied [1] and 2) it is useful for carrying out control-oriented analysis and studying fundamental limitations without adding unnecessary complexity. In [12], [14], [15], [22], [32], and [38] that carry out controloriented nonlinear analysis with thermoacoustic models, pressure oscillation in combustors is attributed to a Hopf bifurcation

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Fig. 1. UTRC combustion experimental setup: single-nozzle and three-nozzle combustors.

leading to a limit-cycle. However, the presence of a broadband complicates this picture. On turbulent velocity disturbance one hand, it allows for a purely linear (and stable) explanation of the observed pressure oscillation—lightly damped Hemholtz oscillator driven by large noise. On the other hand, noise in the linearly unstable Hopf bifurcation regime can lead to complex dynamics requiring analysis tools [29] that are not immediately amenable to a control-oriented analysis. Combustor experiments in academic settings [22], [38] are typically done in combustors with a low-noise environment and small acoustic damping. For the corresponding models, a Hopf bifurcationtype nonlinear analysis is applicable. An industrial combustor generally has a larger turbulent noise (due to higher Reynolds number) and a larger acoustic damping (due to the presence of liners) and exhibits oscillations in a noise driven regime. Obviously, the latter case is more interesting to us and provides the motivation for this paper; see also [8], [28], and [31] for additional details on the effect of noise on pressure oscillations in combustors. A. UTRC Combustion Instability Experimental Setup An industrial engine is equipped with an annular combustor comprising of several premixing fuel nozzles arranged along the circumference. The ACIC experiments used sector embodiments of the annular combustor. Fig. 1 depicts a four-megawatt single-nozzle combustor and a three-nozzle sector combustor. In either setup, experiments were carried out at realistic operating conditions and between 10%–17% of the net fuel was modulated for control using linear proportional or nonlinear ON-OFF fuel valves. Additional details on the experiments appear in [10] and [24]. B. Identification and Validation of a Linear Model From Experiment Control-oriented thermoacoustic models are identified by fitting the experimentally obtained frequency response from

Fig. 2. Feedback control of thermoacoustic plant in the presence of noise: a pressure measurement is used to obtain the fuel valve control input.

fuel valve input to the pressure sensor output. Frequency response experiments were carried out in a range of operating conditions with both the single-nozzle and sector (three-nozzle) combustors. For the single-nozzle combustor operating at the high equivalence ratio condition, a linear model consisting of a lightly damped second-order system together with a (large) delay, was found to be adequate for control. In the following, we describe the identification and validation of the linearity hypothesis with this model. At the high equivalence ratio condition, the pressure oscillations observed in the single nozzle combustor are relatively small and the proportional actuator used for control operates in its linear range. Therefore, it was hypothesized that a linear plant and controller model may be used to analyze the behavior of the controlled system. Fig. 2 depicts the structure of the feedback control system. Fig. 3 compares the experimentally obtained frequency response to it’s model fit. The identified model arises as a second-order lightly damped oscillator with a delay of ms chosen to match the phase roll-off in the 300–400 Hz frequency range. As the identified model is linear and stable, a

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Fig. 4. Square root of the PSD of pressure from experiment and from model simulation.

III. LINEAR CONTROL ANALYSIS

Fig. 3. Results of the linear model identification of the frequency response obtained in the single nozzle rig experiment.

model of external noise is needed to account for the pressure oscillations observed in the experiments. We use the model structure for noise in (1) together with the identified model to estimate a noise model. In particular, a white noise model is built at the plant input (see Fig. 2) by inverting the identified model suitably to match the experimentally obtained power spectral density (PSD) of the uncontrolled pressure. Fig. 4 shows that the identified noise model allows us to match the experimentally obtained pressure PSD with the results of the model simulations using SIMULINK. In the combustion experiment, wide-band turbulent air velocity fluctuation in the nozzle is one of the sources for the presence of noise. The identified plant model includes the fuel valve actuator dynamics together with the thermoacoustic model dynamics of (1). The frequency response of the actuator is effectively flat over a wide frequency band about the resonant thermoacoustic frequency . As a result, additional states are not needed and a second order model with delay consistent with (1) is sufficient. Finally, feedback control is used to validate the implicit linearity hypothesis and the noise model. An observer-based phase-shifting controller is used both in the experiment and in the model simulations. Fig. 5 compares the experimentally obtained pressure PSD with the PSD obtained from simulations with various phase-shifting controllers. The fact that the two PSDs are identical implies that out plant and noise models are valid and suitable for control design and prediction at the high equivalence ratio condition in single-nozzle combustor. At the low equivalence ratio condition, the effect of nonlinearity does become significant and these results will be discussed as part of Sections IV onwards.

In this section, the fundamental limitations associated with the feedback control of combustion instabilities are discussed. The theory is applied to obtain bounds on achievable performance in the high equivalence ratio experiments where linear plant and control models are adequate. In particular, the analysis helps explain the peak-splitting phenomenon observed in our and other ACIC experiments. In frequency domain, the closedloop transfer function from the noise model to the pressure measurement is given by (2) where (3) denotes the sensitivity function. The control objective is to: 1) stabilize the closed loop and 2) shape the sensitivity function with the objective of reducing the noise driven pressure oscillation. In particular, the controller attenuates the noise and amplifies the noise at frequencies where otherwise. Fig. 6 depicts the experimentally obtained Nyquist diagram for the controlled single-nozzle combustor. The attenuation and excitation frequency bands are also shown. The effect of the phase-shifting controller is to rotate the diagram so that the attenuation is maximized at the resonant frequency . The presence of a large delay in the loop makes it difficult to achieve broadband attenuation of pressure oscillations—the sidelobes in the diagram are the regions of secondary peaks. This observation in our closed-loop combustion experiments [10], [24] together with a wide range of performance results in the ACIC literature [2], [25], [33], [35], [36], and [40], motivated us to study the fundamental limitations of ACIC. Our objective is to better understand—in a controller independent fashion—the effect of delay, limited actuator bandwidth and

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Fig. 5. Results of validation using feedback control: effect of a phase-shifting controller on pressure PSD in experiment and simulation.

1) Any stable minimum-phase dynamics are invertible in the controller. In fact, an infinite bandwidth controller always inverts these dynamics. These dynamics do not affect conservation equations—which determine the performance limits. 2) The unmodeled high-frequency thermoacoustic dynamics are open-loop stable. In closed-loop, they are gain stabilized by rolling off the control gain at higher frequencies. Thus, the effect of these higher frequency dynamics is not important once control bandwidth restrictions are imposed. Fundamental limitations in obtainable performance (and robustness) are determined by certain conservation laws that govern the balance of negative and positive areas under the sensitivity (and complementary sensitivity) frequency response. These laws are used to obtain controller-independent bounds on performance and robustness with any linear time-invariant (LTI) controller. For the sensitivity function, obtainable performance bounds can be derived from the celebrated Bode integral formula (4) Fig. 6. Nyquist diagram for the phase-shifting controller with optimal phase shift.

authority, and plant dynamics (unstable poles) on the achievable performance and study the resulting tradeoffs. We remark that ACIC employing more sophisticated control design approaches in models has also appeared recently in [3], [9], [17], and [21] to get the best possible performance in the face of the previous factors. A. Framework for Studying Limits on Achievable Performance An identified second-order model with a time-delay (see Section II-B) is used as a baseline thermoacoustic model. The actuator and the high-frequency thermoacoustic dynamics are both neglected because of the following.

is the real part of the resonant unstable pole-pair; where right-hand side is zero for open-loop stable plant. The integral over a cerformula shows that noise attenuation tain frequency band is always accompanied by noise amplifiover some other frequency band. In the cation presence of unstable poles, a larger penalty is paid in terms of sensitivity amplification. Fig. 7 provides a graphical representation of the area formula: sensitivity reduction (negative area in the integral) is always accompanied by sensitivity amplification (positive area). The performance objective for ACIC is to shape the sensitivity function so that it is small at and near the resonant frequency , i.e., (5)

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j ( )j <  for ! 2 1! ; j ( )j  ((!)=(! ))

Fig. 8. Closed-loop performance specification: S j! for ! > ! , and L |! for ! < ! .

jL(|!)j  ((! )=((!))

width case does not imply any peaking [16], [39]. For the finite control bandwidth case, the area formula together with the performance constraints in Fig. 8, can be manipulated (see [19] and [39]) to obtain Fig. 7. Typical sensitivity function showing positive and negative areas.

for , where is the performance bandwidth centered at . Meeting this performance objective creates negative area in the integral which leads to noise amplification at some other frequencies. If the control bandwidth were infinite, the positive area may be distributed over a wide frequency band so amplification at any given frequency may be designed to be arbitrarily small. However, if the control bandwidth is finite (so the loop rolls off beyond certain low and high frequencies), the positive area would have to be accommodated in a smaller frequency band (where loop gain is high) and this would necessarily result in peaking of the sensitivity function. To model the effect due to finite control bandwidth, we to satisfy the require the loop inequality (6) for high frequencies . Here, it is assumed that and (relative degree of at least two). We impose a similar constraint on the loop gain (7) for low frequencies

and define the frequency band (8)

as the control bandwidth. Fig. 8 illustrates the finite control bandwidth performance specification. The loop gain constraints for in (6) and (7) imply that and . Analysis of performance limitations interpreted as peaking in the sensitivity function magnitude is provided next. In the absence of any right-half plane zeros, the infinite control band-

(9)

where (10) gives the peaking as maximum harmonic amplification relative to the uncontrolled response. This formula shows that the following factors are responsible for the peaking in the sensitivity function: ; 1) desired performance-bandwidth product 2) limitation on the control bandwidth relative to the required performance bandwidth as represented by the amplifying ; term 3) real part of the unstable pole as represented by ; the larger the growth rate, the larger the peaking; 4) combustion response time delay ; one can verify that the lower bound on the sensitivity peaking [right-hand side of (9)] is an increasing function of the delay. Figs. 9–11 show lower bounds on sensitivity peaking obtained by using the inequality in (9). Fig. 9 shows that the peaking increases as ratio of control to performance bandwidth decreases. Figs. 10 and 11 show that the peaking is accentuated by the increase in the delay and the growth rate of instability, respectively. In Section VI, we will use the above together with suitable analogue from nonlinear analysis, to comment on the peak-splitting phenomenon observed in the ACIC experiments.

IV. LIMITATIONS OF STANDARD ANALYSIS In the industrial ACIC settings at UTRC, the linearity hypothesis and subsequent control-oriented analysis of the preceding

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Fig. 9. kS k -effect of varying .

Fig. 10. kS k -effect of varying delay 

section applies only to a limited set of operating conditions. For most operating conditions of practical interest, the linearity hypothesis is not applicable due to the following factors. 1) Nonlinearity in Thermoacoustic Loop: For the experimentally observed amplitudes, the acoustics in a thermoacoustic model are typically well-described by linear models [see for e.g., (1)]. However, most physical mechanisms responsible for the heat release dynamics in a thermoacoustic loop are nonlinear. Nonlinear dynamics such as Hopf bifurcation leading to limit cycles are widely seen and reported in thermoacoustic literature [12], [14],

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Fig. 11. kS k -effect of varying damping ratio of resonant poles.

[15], [22], [32], [38]. The simplest nonlinear models of the unsteady heat release include a static saturating function that captures saturation in burning rate as a function of equivalence ratio. In addition, nonlinearities can arise due to quadratic jet dump losses at the [26] and [27]. For a few operating conditions such as single-nozzle high-equivalence ratio condition, the linear models are adequate because the loop is linearly stable and nonlinearity is sufficiently benign for the amplitudes of noise present. 2) Nonlinearity in Actuation System: In industrial settings, the high power requirements of fuel modulation due to control, means that the actuator essentially operates in its saturated nonlinear range. Next, ON–OFF actuator is a popular and cheap fuel actuator that is widely used for ACIC. As the name suggests, it has only two ranges of operation: ON or OFF. In order to keep the ideas simple, we show experimental results and carry-out analysis for the case of linear thermoacoustic plant model controlled with a linear controller and an ON-OFF actuator. In the final section, we formally extend these ideas to the more general situations of interest. A. Peak Splitting With ON-OFF Actuators in Sector Combustor Experiments ACIC experiments in sector combustor [24] used ON–OFF actuators to meet the large actuator authority requirement. The resulting closed-loop feedback system was, thus, nonlinear. Experimental results obtained with a linear controller showed peak splitting for a range of operating conditions. Fig. 12 depicts the PSD of the pressure oscillations with one, two, and three fuel nozzles operating. In order to understand the nonlinear effects because of ON–OFF actuators, the operating conditions for the uncontrolled case are specifically chosen to verify the linearity hypothesis.

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Fig. 12. PSD of pressure signal with ON-OFF control of one, two, or three liquid fuel nozzles showing the peak-splitting phenomenon observed in experiment and simulation.

A linear thermoacoustic and noise model are identified from experiments using the procedure described in Section II. The thermoacoustic model now includes a larger time delay of ms and a second-order linear system with resonant Hz. Before presenting our approach to frequency the analysis of peak-splitting in nonlinear settings, we discuss the important role of noise which limits the ability of a standard Hopf-bifurcation type analysis in predicting the experimental results. B. Limitations of a Standard Nonlinear Analysis In this section, we exercise the standard sinusoidal input describing function method for predicting the amplitude and the frequency of the limit cycle in the presence of ON–OFF actuators. We show that such an analysis, which fails to account for the noise, cannot explain the peak splitting observed in experiments. In the absence of noise, a first-order harmonic balance of the closed-loop gives (11) where (12) is the describing function gain of the ON-OFF valve modeled as a static relay nonlinearity with . Fig. 13 shows a graphical technique for determining the existence of (limit cycling) solutions of (11). The intersections of the Nyquist diagram for the transfer function with the plot of indicates the possibility of two limit cycles with frequencies of 183 and 238 Hz. Using (12), the corresponding limit cycle amplitudes are found to be about 1 psi for the three actuated nozzles and a third of that value for one nozzle. Simulations with the model without noise, are in agreement with the analysis and two attractors with dominant frequencies at 183 and 238 Hz are seen to co-exist. Fig. 14 shows

the time traces, phase-portraits, and PSD for the two attractors. The phase portraits are constructed by embedding the attractors using the delay coordinate technique [37]. Three time shifts of the pressure measurement delayed by 0, 1.5, and 3.5 ms, respectively, are used. Next, a set of simulations are carried out now with the noise model included. Fig. 15 shows a comparison of the resulting pressure PSD with and without noise for one to three actuated fuel nozzles. The two PSDs with and without noise are rather different. In particular, the shapes of PSD, the frequencies of the peaks, and the values of pressure PSD at the peaks strongly depend on the presence of noise and on the number of actuated nozzles. Qualitatively, there are two ways to interpret the effect of noise on the observed PSD. 1) Dynamical Systems Interpretation: Without noise, the attractors are relatively small in size. The presence of a large noise causes the state of the dynamical system to visit regions in phasespace, where the effect of saturated control is small and the behavior is determined by open-loop dynamics. The power spectrum represents an average statistic of the motion of the system state. In the case of large noise relative to the saturated control, one observes more of the open-loop spectral characteristics in the closed-loop spectrum. 2) Control-Theoretic Interpretation: The effect of the noise can be interpreted as dither that leads to the effective (average) linearization of the nonlinearity [13], [41]. The dither reduces the effective gain of the valve with respect to the periodic signal. The ON–OFF valve effectively works as a proportional low gain valve recovering more of the open-loop characteristics in the closed-loop system. The inability of the standard describing function technique to analytically predict the frequencies and level of pressure PSD peaks in the presence of noise motivated the framework presented in the following section. The results can be used to interpret the fundamental limitations in the regime of nonlinear noise driven systems such as the ones encountered in ACIC. The framework is presented with the example studied in this section linear thermoacoustic model controlled with nonlinear ON–OFF actuators.

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Fig. 13. Graphical determination of the amplitude and frequencies of the possible limit cycles for a controlled thermoacoustic model with a standard describing function method.

V. RANDOM-INPUT-BASED DESCRIBING FUNCTION ANALYSIS WITH ON–OFF ACTUATORS We begin by introducing the random-input-based describing function framework [20]. The framework considers a loop balance with respect to the noise in addition to the balance with respect to sinusoidal signals. It allows for an approximate analysis of a linear system in feedback with static nonlinearity in the presence of noise. The framework is applicable to the study of linear acoustics in feedback with a (static) nonlinear heat release, or a linear thermoacoustics controlled by (static) nonlinear actuator, or some combination of the two. It is assumed that all signals in the feedback loop under consideration, can be decomposed as a combination of the three signals—constant bias, harmonic components of a limit cycle, and Gaussian noise. Consider a decomposition of the output pressure signal as (13) is dc bias, is sinusoidal signal (harmonic where component of a limit cycle) of amplitude and frequency , is a Gaussian random process with zero mean and variand ance . Using the random-input describing function method, the is approxoutput of a scalar nonlinear element imated as (14) where the individual gains

(15)

(16)

(17) are obtained through averaging ([20, p. 371]). With no noise and zero bias , the last formula reduces to the standard sinusoidal input-describing function gain. The describing function approximation is used to interpret the pressure time-series data in (13) for both open- and closed-loop thermoacoustic system. The time-series is either an output of , or it is a pure self-exa stable noise-driven system , or it is some combination of the cited oscillation . Denote the PSD of pressure as two are defined by (13), the input noise is assumed to be a zero-mean Gaussian process with variance and . The noise may additionally contain sinusoidal PSD signals with amplitude , and dc signal with bias . For the feedback loop depicted in Fig. 16 with linear transfer function and feedback nonlinearity , the balance equations for the two distinct noise-driven cases of interest to us are: 1) stable noise driven system

(18)

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Fig. 14. Time traces, phase-portraits, and PSD for the co-existing attractors of the closed-loop system with three actuated nozzles (no noise case).

(19) (20)

A. Dynamics of Sector Combustor Model With ON-OFF Actuators The amplitude of the limit cycle is obtained by solving (23) (26)

(21) 2) self-excited oscillations with noise

(22) (23)

where is the describing function gain with respect to the fundamental harmonic of the limit cycle. The following theorem summarizes certain useful results and bounds for the describing function corresponding to the relay nonlinearity. linear and Theorem V.1: Consider the loop in Fig. 16 with , a relay nonlinearity. In stable in feedback with , we have the limit as

(24)

(27)

(25)

(28)

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Fig. 15. Model simulation: effects of noise and actuator authority on the pressure power spectra under closed-loop control with ON–OFF valve.

Note that the previous bounds are independent of the dynamics . The stability of together with the saturating naof ture of the feedback nonlinearity , is used to ensure that is a bounded signal for bounded noise . The above theorem implies the following for dynamics of the feedback loop drawn in Fig. 16. , the feedback loop admits a gain 1) In the limit as margin of 2 with respect to the Gaussian noise at the fundamental frequency of limit cycle. To see this, use (26) to obtain

Fig. 16. Schematic of a feedback loop with nonlinearity.

Next, there is an amplitude-independent bound

(33) (29) for the describing function gain corresponding to the limit cycle. Proof of Theorem V.1: Equations (27) and (28) are easily obtained by direct computation using (16) and (17), respectively. In order to show the amplitude-independent bound (29), we express (17) as

at the fundamental frequency (28)

of the limit cycle. Using (34)

i.e., the loop gain margin with respect to the noise is . such that for all , the loop does 2) There exists a not support a limit cycle. This is because the bound in (29) implies that (26) can not admit a solution for (35)

(30) where the first part is zero because (31) independent of follows by:

and

. Equation (29) then

(32)

, the feedback loop behaves as the 3) In the limit as . open loop , the appearance of the limit cycle stabilizes the For loop with respect to the noise. This leads to a bounded inputoutput response with respect to the Gaussian noise input. In particular, the PSD of the pressure output is obtained by solving is stable (24) which is well-posed because the loop and in particular from the above considerations. For , and the large noise stabilizes as the loop with respect to the limit cycle. This causes the limit cycle to disappear and the feedback system behaves as a stable noise-driven system. Next, we provide numerical results for the analysis of the feedback loop in Fig. 16 at intermediate values of . In the aband the closed-loop system exhibit a limit sence of noise, resulting from the solution of (33). For cycle with amplitude

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Fig. 17. Amplitude of the limit cycle in the presence of the Gaussian noise.

values of , the amplitude tions satisfies the analytic relation

Fig. 18. Describing function gains in the presence of Gaussian noise.

of the self-excited oscilla-

(36) It then follows that and the presence of Gaussian input noise has the effect of suppressing the self-excited oscillations. Fig. 17 shows the numerically computed solution of the integral equation (36). At a numerically determined critical , the limit cycle disappears . For value of , the balance considers only the Gaussian signals in the feedback loop. The corresponding random input describing function gain is given by a closed-form expression (37) For the values of , the gain is numerically computed using (15)–(17). Fig. 18 shows the gains and as a function of . , the limit cycle We summarize the conclusions. For is present and the describing function gain with respect to the limit cycle is given by (38) monotonically increases between 0 and . We have cally decreases for

and monotoni-

(39) The final inequality ensures that the loop with respect to the Gaussian noise is stable for all

. The

Fig. 19. Comparison of the pressure PSD from the random input describing function calculation and from the simulation for (a) single nozzle and (b) threenozzles case.

second equality in (39) implies that the largest loop gain occurs at the critical value . Here, the loop is neutrally stable. VI. FUNDAMENTAL LIMITATIONS IN SECTOR COMBUSTOR ACIC EXPERIMENTS In the models of the ACIC experiments with ON-OFF actuators, Gaussian balance yields an approximation of the feedback loop with respect to the noise. The analysis above shows that the loop yields a well-posed closed-loop system for all . Note that such is the case independent of the dynamics of the open loop , amplitude of the

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Fig. 20. Model simulation with noise showing the effects of delay and performance bandwidth on pressure PSD under closed-loop control.

Fig. 21. Open- and closed-loop pressure PSD for the single-nozzle and sector combustors.

limit cycle , and values of . The resulting sensitivity function is stable and one can formally write down an area formula which gives peak splitting for the approximation. Under the assumption that the approximation yields a good representation of the nonlinear model, this explains the peak splitting seen in the PSD of the ACIC experiments with ON-OFF actuators. Fig. 19 shows that the comparison between the approximation and the nonlinear model is indeed good for our case. The PSD of the pressure time-series obtained from simulation of the nonlinear model matches closely to the PSD obtained with the frequency domain balance [using (24)]. The above considerations give a formal framework for extending the fundamental limitations analysis for control of thermoacoustic loops. One considers the modified sensitivity function with respect to the noise balance. Peak splitting is a consequence of the area formula as applied to the modified sensitivity function. For the case of ON-OFF nonlinearity with stable, we showed that the modified sensitivity function is stable and well-posed independent of the dynamics of and the noise (variance). We expect this to be true for a larger class of nonlinearities. Analysis provided in this paper indicates that the peaking phenomenon observed in ACIC experiments is to a large extent

inevitable for combustion systems with large delay controlled with actuators of limited bandwidth. This is reflected in the fact that the sensitivity with the linear actuator case or the modified sensitivity function with the nonlinear ON-OFF actuator achieves values exceeding 1. For the high-equivalence ratio-operating condition, the damping of the thermoacoustic plant is relatively large which leads to broadband pressure oscillations. Requiring a high-performance bandwidth for its suppression invariably leads to peak splitting. In addition to the performance bandwidth effect, delay limits the performance in ACIC. Fig. 20 shows that in the two cases with no delay and smaller performance bandwidth requirement have little performance limitations in terms of the peaking effect. The results are obtained using simulations of nonlinear ACIC models of the type described above with a large noise variance . We also use the analysis to explain the difference between the experimental results obtained in single-nozzle and sector combustors (see Fig. 21). Compared to the sector combustor, the single nozzle combustor shows higher open-loop oscillations but in a narrower band of frequencies. Such is the case because of lower damping of the thermoacoustics in the single nozzle combustor. Next, the plant delay identified from the fre-

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quency responses is higher in the sector combustor than in the single-nozzle combustor. As a result, limitations and peaking in the sector combustor—with its broadband performance objective for a thermoacoustic plant with large delay—is more severe than in the single nozzle combustor. VII. CONCLUSION In the first part of this paper, we carried out linear analysis of combustion instability control in UTRC experiments. Linear analysis is determined to be applicable for the high-equivalence ratio conditions. For these experiments, the theory of fundamental limitations is used to conclude that large delay in the feedback loop and limited actuator bandwidth are the primary factors that limit the effectiveness of the active control. Explicit bounds on fundamental limitations imposed upon performance due to the presence of delay, unstable dynamics, and limited controller bandwidth are obtained. In the second part of this paper, it has been shown that random-input describing function analysis is an appropriate tool to study the effect of the input noise in a nonlinear model. The describing function analysis predicts that for sufficiently large levels of noise, the loop effectively behaves linearly with the feedback gain dependent upon the variance of the Gaussian noise. As a result, the fundamental limitations studied in the first part of this paper can be extended to explain the peak-splitting phenomenon even in the nonlinear case. In particular, the peak-splitting phenomenon observed in our experiment and reported by other researchers is explained with the aid of this analysis. The theory of fundamental limitations, thus, is relevant and has practical implications to ACIC. In industrial combustors with their noisy environment, peaking in the sensitivity function is unavoidable due to large delays and limited actuator bandwidth. In certain cases—such as the high-equivalence ratio condition in the single-nozzle combustor—where the open-loop pressure oscillations are relatively small/narrow band and control objectives are modest, the required (absolute) level of controlled pressure oscillations for safe operation can be met with ACIC. In many cases of industrial importance, however, performance is fundamentally limited. Here, either better passive design changes are needed or actuation mechanism with smaller delay/higher bandwidth needs to be developed. The latter can possibly be achieved by using injection of fuel closer to the flame area or by increasing mixing rates. ACKNOWLEDGMENT The authors would like to thank DARPA and the contract monitor Dr. W. Scheuren for sponsoring a significant portion of this work. Pratt and Whitney/Turbo Power and Marine graciously provided test hardware for the sector rig. They also acknowledge the help and support from UTRC dynamics and combustion control team, with special thanks to J. Cohen, R. Hibshman, W. Proscia, and T. Anderson. REFERENCES [1] T. J. Anderson, W. Sowa, and S. Morford, “Dynamic flame structure in a low nox premixed combustor,” in Proc. ASME Gas Turbine Aerospace Congr., 1998.

[2] B. Anson, I. Critchley, J. Schumacher, and M. Scott, “Active control of combustion dynamics for lean premixed gas fired systems,” American Society of Mechanical Engineers, Amsterdam, The Netherlands, ASME Paper GT-2002-30068, 2002. [3] A. Banaszuk, K. Ariyur, M. Krstic, and C. Jacobson, “An adaptive algorithmfor control of combustion instability,” Automatica, vol. 40, no. 11, pp. 1965–1972, Nov. 2004. [4] A. Banaszuk, C. A. Jacobson, A. I. Khibnik, and P. G. Mehta, “Linear and nonlinear analysis of controlled combustion processes—Part I: Linear analysis,” in Proc. Conf. Contr. Appl., 1999, pp. 199–205. [5] ——, “Linear and nonlinear analysis of controlled combustion processes—Part II: Nonlinear analysis,” in Proc. Conf. Contr.Appl., 1999, pp. 206–212. [6] G. Bloxsidge, A. Dowling, N. Hooper, and P. Langhorne, “Active control of acoustically driven combustion instability,” J. Theoretical Appl. Mech., vol. 6, pp. 161–175, 1987. [7] ——, “Active control of reheat buzz,” AIAA J., vol. 26, no. 7, pp. 783–790, Jul. 1988. [8] V. Burnley, “Nonlinear Combustion Instabilities and Stochastic Sources,” Ph.D. dissertation, California Inst. Technol., Pasadena, CA, 1996. [9] Y. C. Chu, K. Glover, and A. P. Dowling, “Control of combustion oscillations via h-infinity loopshaping, -analysis and integral quadratic constraints,” Automatica, vol. 39, no. 2, pp. 219–231, Feb. 2003. [10] J. M. Cohen, N. M. Rey, C. Jacobson, and T. Anderson, “Active control of combustion instability in a liquid-fueled low-nox combustor,” ASME J. Eng. Gas Turbines Power, vol. 121, no. 2, pp. 281–284, Apr. 1999. [11] J. M. Cohen, J. H. Stufflebeam, and W. Proscia, “The effect of fuel/air mixing on actuation authority in an active combustion instability control system,” ASME J. Eng. Gas Turbines Power, vol. 23, no. 3, pp. 537–542, Jul. 2001. [12] F. Culick, W. Lin, C. Jahnke, and J. Sterling, “Modeling for active control of combustion and thermally driven oscillations,” in Proc. Amer. Contr. Conf., 1991, pp. 2939–2948. [13] C. Desoer and S. Shahruz, “Stability of dithered non-linear systems with backlash and hysteresis,” Int. J. Contr., vol. 43, no. 4, pp. 1045–1060, 1986. [14] A. Dowling, “Nonlinear acoustically coupled combustion oscillations,” in Proc. 2nd AIAA/CEAS Aeroacoustics Conf., May 1996. [15] ——, “Nonlinear self-excited oscillations of a ducted flame,” J. Fluid Mech., vol. 346, pp. 271–290, Sep. 1997. [16] J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1992. [17] S. Evesque and A. P. Dowling, “LMS algorithm for adaptive control of combustion oscillations,” Combust. Sci. Technol., vol. 164, pp. 65–93, 2001. [18] M. Fleifil, J. P. Hathout, A. M. Annaswamy, and A. F. Ghoniem, “The origin of secondary peaks with active control of thermoacoustic instability,” Combust. Sci. Tech., vol. 133, pp. 227–265, 1998. [19] J. Freudenberg and D. Iooze, “A sensitivity tradeoff for plants with time delay,” IEEE Trans. Autom. Contr., vol. AC-32, no. 2, pp. 99–104, Feb. 1987. [20] A. Gelb and W. V. Velde, Multiple-Input Describing Functions and Nonlinear System Design. New York: McGraw-Hill, 1968. [21] J. P. Hathout, A. M. Annaswamy, and A. F. Ghoniem, “Modeling and control of combustion instability using fuel injection,” presented at the NATO/RTO Active Contr. Symp., Braunschweig, Germany, 2000. [22] J. Hathout, A. Annaswamy, M. Fleifil, and A. Ghoniem, “A modelbased active control design for thermoacoustic instability,” Combustion Sci. Technol., vol. 132, pp. 99–105, 1998. [23] J. Hermann, S. Gleis, and D. Vortmeyer, “Active instability control (AIC) of spray combustors by modulation of the liquid fuel flow rate,” Combustion Sci. Technol., vol. 118, pp. 1–25, 1996. [24] J. Hibshman, J. Cohen, A. B. T. Anderson, and H. Alholm, “Active control of combustion instability in a liquid-fueled sector combustor,” presented at the ASME Turbo Expo., Indianapolis, IN, 1999. [25] S. Hoffmann, G. Weber, H. Judith, J. Hermann, and A. Orthmann, “Application of active combustion instability control to siemens heavy duty gas turbines,” presented at the Symp. AVT Panel Gas Turbine Engine Combustion, Emission Alternative Fuels, Lisbon, Portugal, 1998. [26] U. Ingard, “Acoustic nonlinearity of an orifice,” J. Acoustical Soc. Amer., vol. 42, no. 1, 1967. [27] U. Ingard and S. Labate, “Acoustic circulation effects and the nonlinear impedance of orifices,” J. Acoustical Soc. Amer., vol. 22, no. 3, Mar. 1950.

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[28] C. A. Jacobson, A. I. Khibnik, A. Banaszuk, J. Cohen, and W. Proscia, “Active control of combustion instabilities in gas turbine engines for low emissions—Part I: Physics-based and experimentally identified models of combustion instability,” in Proc. Appl. Vehicle Technol. Panel Symp. Active Contr. Technol., 2000. [29] L. Arnold, N. S. Namachchivaya, and K. R. Schenk, “Toward an understanding of stochastic bifurcation: Case study,” Int. J. Bifurcation Chaos, vol. 6, no. 11, pp. 1947–1979, 1996. [30] P. Langhorne, A. Dowling, and N. Hooper, “Practical active control system of combustion ocsillations,” J. Propulsion Power, vol. 6, no. 3, pp. 324–333, 1988. [31] T. C. Lieuwen, “Experimental investigation of limit cycle oscillations in an unstable gas turbine combustor,” J. Propulsion Power, vol. 18, no. 1, pp. 61–67, Jun. 1998. [32] R. Murray, C. Jacobson, R. Casas, A. Khibnik, C. J. Jr., R. Bitmead, A. Peracchio, and W. Proscia, “System identification for limit cycling systems: A case study for combustion instabilities,” in Proc. Amer. Contr. Conf., 1998, pp. 2004–2008. [33] C. Paschereit, E. Gutmar, and W. Weisenstein, “Control of combustion driven oscillations by equivalence ratio modulations,” American Society of Mechanical Engineers, Indianapolis, IN, ASME Paper 99-GT118, 1999. [34] A. A. Peracchio and W. Proscia, “Nonlinear heat-release/acoustic model for thermoacoustic instability in lean premixed combustors,” ASME J. Eng. Gas Turbines Power, vol. 121, no. 3, pp. 415–421, Jul. 1999. [35] G. A. Richards and M. C. Janus, “Characterization of oscillations during premix gas turbine combustion,” ASME J. Eng. Gas Turbines Power, vol. 120, pp. 294–302, 1998. [36] S. S. Sattinger, Y. Neumeier, A. Nabi, B. T. Zinn, D. J. Amos, and D. D. Darling, “Sub-scale demonstration of the active feedback control of gas-turbine combustion instabilities,” ASME J. Eng. Gas Turbines Power, vol. 122, pp. 262–68, 2000. [37] T. Sauer, J. Yorke, and M. Casdagli, “Embedology,” J. Stat. Phys., vol. 65, no. 3-4, pp. 579–616, 1991. [38] W. Saunders, M. Vaudrey, B. Eisenhauer, U. Vandsburger, and C. Fannin, “Perspectives on linear compensator designs for active combustion control,” presented at the 37th AIAA Aerospace Sci. Meeting, Reno, Nevada, 1999. [39] M. Seron, J. Braslavsky, and G. Goodwin, Fundamental Limitations in Filtering and Control. New York: Springer-Verlag, 1997. [40] J. Seume, N. Vortmeyer, W. Krause, J. Hermann, C. Hantschk, and P. Zangl, “Application of active combustion instability control to a heavy duty gas turbine,” ASME J. Eng. Gas Turbines Power, vol. 120, pp. 721–726, 1998. [41] G. Zames, “Dither in nonlinear systems,” IEEE Trans. Autom. Contr., vol. AC-21, no. 5, pp. 660–667, Oct. 1976. Andrzej Banaszuk (M’99–SM’05) received the Ph.D. degree in electrical engineering from Warsaw University of Technology, Warsaw, Poland, and the Ph.D. degree in mathematics from the Georgia Institute of Technology, Atlanta, GA, in 1989 and 1995, respectively. He is currently a Control and Embedded Systems Group Leader at United Technologies Research Center (UTRC). Since joining UTRC in 1997, he has conducted research in control-oriented modeling, nonlinear model validation, parameter identification, and linear, nonlinear, and adaptive control design for unsteady flow and combustion phenomena affecting operations of aeroengines. Specifically, he worked on modeling, identification, and control of turbomachinery flutter, rotating stall, combustion instability, flow separation, and mixing. Before coming to UTRC, he held academic positions at Warsaw University of Technology, Warsaw, Poland, Georgia Institute of Technology, Atlanta, GA, University of Colorado at Boulder, Boulder, and University of California at Davis, Davis, where he performed research in linear and nonlinear control theory. His recent interest is

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design and control of large networks of dynamical systems with applications to surveillance networks, people estimation, and emergency egress control. He is an author of 37 journal papers, over 50 conference papers, and holds 3 patents. From 1999 to 2002, he was an Associate Editor of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY and in 2004, he was appointed to serve on the Board of Governors of the IEEE Control Systems Society.

Prashant G. Mehta (S’00–M’04) received the Ph.D. degree in applied mathematics from Cornell University, Ithaca, NY, in 2004. He is currently an Assistant Professor at the Department of Mechanical and Industrial Engineering, University of Illinois (UIUC), Urbana-Champaign. Prior to joining UIUC, he was a Research Engineer at the United Technologies Research Center (UTRC), East Hartford, CT. His research interests include dynamical systems and control, including its applications to stochastic dynamics and control of network problems, fundamental limitations in control of nonequilibrium dynamic behavior, and multiscale and symmetry-aided analysis of interconnected systems. Dr. Mehta is a recipient of the Outstanding Achievement Award from the UTRC for his contributions in developing dynamical systems methods to obtain practical solutions to problems in aero-engines.

Clas A. Jacobson (SM’80) received the Ph.D. degree in electrical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1986. He is currently Director of the Systems Department at the United Technologies Research Center (UTRC), East Hartford, CT, where he is responsible for Research and Development involving systems engineering. He was an Associate Professor at Northeastern University, Boston, MA, from 1986 to 1995. He has published over 50 papers and holds 2 patents.

Alexander I. Khibnik received the M.S. degree in mathematics from Moscow State University, Moscow, Russia, and the P.D. degree in mathematics and physics from the Institute of Biological Physics, Russian Academy of Sciences (RAS), Pushchino, Russia, in 1973 and 1985, respectively. He is currently a Staff Engineer in Controls and Diagnostics Systems at Pratt and Whitney Aircraft, Hartford, CT, where he is responsible for development and demonstration of real-time structural health monitoring of rotor disks and blades in aircraft engines. His current interests include prognostics and health management (PHM) of engine and aircraft structural components including sensing of dynamic loads and responses, advanced signal processing, and data mining, reasoning methods, development of structural health indices, and their integration with conditionbased maintenance. Before 2006, he was with the United Technologies Research Center (UTRC), East Hartford, CT, where he was a Staff Scientist conducting research in nonlinear dynamics and controls, diagnostics and prognostics, robust design, fleet, and supply chain optimization. Before 1997, he held academic positions with the Institute of Mathematical Problems in Biology of RAS and Cornell University, Ithaca, NY, where he conducted research in nonlinear dynamical systems and chaos theory and their applications to physics, chemistry, and biology. He has authored three books (two of them as an editor), more than 30 technical papers, and several software packages. He holds one patent and one pending patent.