Control of Nonlinear Harmonic Coupling in Pulsed Jet Injection

Cory Hendrickson and Robert M’Closkey, Member, IEEE,

Abstract— This paper addresses control of nonlinear har-monic coupling in a pulsed jet injection experiment using arepetitive based control strategy for periodic reference tracking.A data driven approach identifies the harmonic interactionaround a particular operating condition and generates a modelfor controller analysis and design. The single-input, single-output nonlinear plant is modeled as a nonlinear map of theinteraction between N harmonics of the periodic reference. Themap is compensated using a multi-input multi-output controllerwhich asymptotically tracks the periodic reference up to theNth harmonic.

I. INTRODUCTION

Periodic signals in the form of disturbances or referenceinputs commonly appear in a wide variety of engineeringsystems such as robotics, industrial machinery, and computerdisk drives. Repetitive control is an effective method forasymptotic disturbance rejection or reference tracking in suchsystems. All forms of repetitive control are united, directlyor indirectly, by Francis and Wonham’s Internal ModelPrinciple (IMP) which requires a model of the disturbanceor reference to be included in the feedback loop for perfectrejection or tracking [1]. If a periodic reference has an infinteFourier series, an infinite number of models at the harmonicsof the reference are required for perfect tracking. The mostcommon method of repetitive control includes a time delay inthe feedback loop which places an infinite number of poleson the imaginary axis at the harmonics of the disturbanceor reference. Reserchers using time delay repetitive controlhave focused primarily on disturbance rejection in areassuch as industrial machinery [2], AC power supplies [3] andcomputer disk drives [4], [5].

In most applications, either the reference has a finiteFourier series or the plant has a finite bandwidth, therefore, afinite number of internal models are required for asymptotictracking. In this case, modulated-demodulated control canbe used as an alternative to time delay repetitive control[6], [7]. Modulated-demodulated control, sometimes referredto as adaptive feedforward control or adaptive feedforwardcancellation [8], [9], shifts the spectrum of “high” fre-quency oscillations down to baseband which includes DC,operates at baseband, then shifts the baseband spectrumback to high frequency. Modulated-demodulated control canbe advantageous in implementation because low-bandwidthcompensators are used to control high frequency oscillations.

This work is sponsored by the National Science Foundation under grantno. CBET-0755104

C. Hendrickson and R.T. M’Closkey are with the Department of Mechan-ical and Aerospace Engineering, University of California, Los Angeles, LosAngeles, CA, 90095 USA rtm@seas.ucla.edu

A limited amount of research has focused on repetitivecontrol for nonlinear plants. Some studies have used slidingmodes to eliminate nonlinearity [10],[11] and others considerapproximate I/O linearized nonlinear systems [12]. Most,however, consider only well defined and modeled nonlinearsystems.

We consider a single-input single-output periodic referencetracking problem in a pulsed jet injection experimentalstudy. At high amplitudes of forcing, we must contend withsignificant nonlinear coupling between harmonics of theperiodic reference. The proposed control strategy transformsthe plant into a static nonlinear map of the harmonic couplingidentified at a given operating condition. A multi-input multi-output (MIMO) controller based on modulated-demodulatedcontrol is used to stabilize a feedback loop around themap to asymptotically track a periodic pulse-like referencein the measured jet velocity. Actively controlling a jet toform sharply defined pulses has the potential to significantlyincrease mixing in a variety of flowfields. A challenge inthe present application is an incomplete theoretical under-standing of the flowfield dynamics. Indeed, the point ofthese experiments is to help guide numerical and theoreticalanalysis of the flow.

II. PULSED JET EXPERIMENT

The control strategy developed in this paper provides aframework to study the sensitivity of pulsed jet mixednessto the shape of the pulse’s temporal velocity profile [13].An effective control system, which tracks a precise periodicreference, facilitates an evaluation of how pulse width andpulse transition effect jet mixedness. The jet response can beoptimized based on reference pulse width and the sharpnessof velocity transition.

A. Actuation System

The experimental apparatus is shown in Figure 1. Regu-lated compressed air flows into a plenum, or mixing chamber,then through a smoothly contracted nozzle into quiescentsurroundings. A pressure regulator maintains a constant meanvelocity of 8ms−1. A lightweight piston actuator sits at thebottom of the plenum beneath a flexible rubber seal. Thepiston is rigidly connected to a Ling LVS-100 voice coilmodal shaker which drives the piston axially in line the thejet. Acoustic forcing perturbs the jet temporal velocity profileto form square wave like pulses. The apparatus is equippedwith a microphone located at the top of the plenum anda hotwire anemometer placed in the center of the jet atthe nozzle exit. The control system is implemented usingMatlab’s XPC Target application with a 25kHz sampling

rate. Two, 8-pole low-pass Chebyshev filters with 10kHzcorner frequencies filter the microphone and hotwire signalsfor anti-aliasing.

Voice CoilAmplifier

Signal ConditioningCompressed

Air

Voice Coil

PressureRegulator

Piston

Plenum

PistonMotion

Nozzle

Air Flow

Hotwire

Filtering/DSPPiston

Support Structure

Microphone

Fig. 1. Pulsed jet injection experimental setup using a piston to activelycontrol the temporal velocity waveform of a jet at the nozzle exit.

Prior pulsed jet injection control studies used a singlehotwire measurement to successfully track a periodic refer-ence using modulated-demodulated control [14] [15]. Theseexperiments, however, were performed with peak-to-peakvelocity perturbations less than 1ms−1, conditions for whicha linear plant model was adequate for controller design.At larger forcing amplitudes, though, nonlinear interactionbetween neighboring harmonics of the fundamental forcingfrequency can produce instability when using the priormodulated-demodulated control method which ignores thecoupling. For example, Figure 2 shows how forcing condi-tions beyond 1ms−1 peak-to-peak alter the systems’ velocityfrequency response. The frequency response is measuredusing a band-limited white noise identification input with anamplitude set to perturb the jet velocity by urms = 0.5ms−1.The identification signal is summed with a 100Hz sinusoid(the fundamental pulse repetition rate in these experimentsis 100Hz so all periodic waveforms will have a large Fourierseries component at 100Hz). The sinusoid’s amplitude isvaried to traverse a range of possible forcing amplitudes andit is evident that the system’s frequency response deviatessignificantly from the nominal, or unperturbed, frequencyresponse when the sinusoid amplitude exceeds 1ms−1. Thetrend shows reduction in gain and more phase lag withgreater peak-to-peak perturbations. Alternatively, Figure 2bshows that the pressure frequency response is essentiallyunchanged in the presence of the 100Hz sinusoid at anyamplitude.

The coherence of the frequency response data, shown inFigure 3, provides further insight into the system behavior[16]. The velocity coherence with a 1.5ms−1 100Hz addedtone decreases from the coherence with the nominal exci-tation, particularly above 1kHz, which indicates a greaterportion of the output is driven by nonlinear dynamics. Thepressure coherence with a 1.5ms−1 100Hz added tone,however, matches the nominal excitation below 2kHz. Asmall deviation between the two forcing conditions occurs

beyond 2kHz where the microphone frequency response issmall which indicates the change is due to a low signal tonoise ratio, not nonlinear dynamics. This evidence suggestswe can separate the plant into a nonlinear velocity systemand a linear pressure system with measurement available atboth points.

500 1000 1500 2000 2500 3000

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102

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b)

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102

Magnitude (m/

500 1000 1500 2000 2500 3000 -200

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0

100

200

Phase (deg)

a)

Fig. 2. Actuation frequency response obtained with urms = 0.5ms−1

white noise identification input and an additional 100hz sinusoid perturba-tion at various amplitudes. 100Hz perturbation amplitudes: Blue - 0ms−1,Red - 1.0ms−1, Green - 1.5ms−1. a) Velocity, b) Pressure

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Frequency (Hz)

Cohe

renc

e

Velocity 0ms 1

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Pressure 0ms 1

Pressure 1.5ms 1

Fig. 3. Coherence of the empirical frequency response data in Figure 2.A decrease in the coherence with a 1.5ms−1 100Hz added tone comparedto the nominal excitation provides evidence for nonlinear coupling in thevelocity system.

The pressure and velocity frequency responses roll off inmagnitude after a plenum mode near 1.8kHz so very little in-fluence over the jet velocity is possible above approximately2.2kHz. As such, we specify the periodic reference to bea square wave truncated at 2.2kHz to avoid saturation ofthe actuator amplifier. It is important to retain all harmonicswithin the actuation bandwidth in order to form periodicreferences that have sharp velocity transitions and limitedringing. Sharp velocity transitions and little ringing arethought to create strong vortex rings which increase jetmixing.

P (s)mic

P (s)vel

C (s)mic

demodry

P (s)mic

P (s)vel

u

yvel

ymic

u

Open-Loop ID of nonlinear coupling

yvel

ymic

1 1

1

2N

2N

P (s)mic

P (s)vel

u

C (s)mic

demody

C (s)vel

rvel,d-

Asymptotic tracking

1

2N

2N

ymic 1

K

vel,d

vel,d

vel,d

a) b) c)

Fig. 4. a) Open-loop plant. b) Block diagram for identification of harmonic coupling. The inner control loop establishes an operating condition whichdrives yvel,d close to the desired waveform. c) Block diagram for asymptotic tracking of the Fourier series coefficients of the desired periodic velocitywaveform. The components are listed in a vector denoted rvel,d. The dashed system is modeled as a constant linear map which quantifies the harmoniccoupling between the controlled frequency bands. Scalar and vector signals are denoted by < 1 > and < 2N > respectively.

III. CONTROL STRATEGY

Maintainting loop gain magnitude larger than one overthe entire usable actuation bandwidth is not possible dueto large phase delay (see Figure 2). One component ofthe phase delay is the result of transport lag caused bythe physical distance between the piston actuator and themicrophone/hotwire. Since periodic reference tracking re-quires control only at the fundamental forcing frequencyand its harmonics, the control strategy presented here breaksthe wideband control problem into narrow bands centeredat the fundamental and each harmonic within the 2.2kHzactuation bandwidth. Although the disturbance spectrum willonly be attenuated in a narrow band about each harmonic, theprimary reason for using feedback to shape the hotwire signalis the uncertainty associated with the plant dynamics: it issimply not possible to identify a plant model of sufficientlyhigh fidelity that its “inverse” provides the correct open-loop forcing conditions. Thus, feedback is used to force thehotwire measurement to asymptotically track the periodicreference within the actuator bandwidth.

The main challenge in modeling the system is quantifyingthe nonlinear coupling that occurs between frequency “chan-nels” in the hotwire signal. The physical mechanism causingthe nonlinear coupling is not well understood, but becausethe measurements are repeatable, we can formulate modelsfrom the data that can be used for controller analysis anddesign.

A. Control System Architecture

Figure 4 outlines the identification and control strategywe employ to contend with the nonlinear coupling in theplant dynamics at high forcing amplitudes. An “inner” loop isclosed using the microphone measurement, denoted ymic inFigure 4, using the modulated-demodulated control methoddescribed in [14] because this measurement is largely inde-pendent of the the forcing conditions. The system dynamicsfrom the perspective of ymic are linear and so possess atransfer function denoted Pmic. The time constant associatedwith each frequency channel in the Pmic loop is typicallydesigned to be about 0.1 seconds. The system dynamicsfrom the perspective of the hotwire measurement, denotedyvel, however, are nonlinear to an extent that cannot be

ignored in the controller design. In order to identify thenonlinearity, though, an approximate operating conditionsufficiently close to the desired operating point must beestablished. This is accomplished by specifying the referencecommand in the Pmic loop to approximately drive yvelto the desired periodic waveform. The reference command,denoted as rmic,d in Figure 4b, is selected at each of the Nfrequencies of control as Pmic(jωn)P̄−1

vel (jωn)rvel,n wherervel,n is the Fourier series coefficient at frequency ωn of thedesired hotwire waveform, and where P̄vel represents theexperimental velocity transfer function chosen from Figure2 according to the RMS forcing condition.

B. Inner Loop

The inner loop controller, labeled Cmic(s), usesmodulated-demodulated control to track the reference rmic,d.As outlined in Figure 5a, Cmic(s) consists of N independentcontrol loops, each designed track the Fourier series coeffi-cients of the inner loop reference at a single harmonic. TheFourier series coefficients are arranged in the vector rmic,d

rmic,d =

ri1rq1...rqN

, (1)

where constants ri1 and rq1, for example, are the real andimaginary parts of the inner loop reference Fourier seriescoefficient at the fundamental frequency. The control effortof each loop is summed and scaled by gmic to form theactuator input.

Figure 5b shows the details of a modulated-demodulatedcontrol loop for control at a single frequency, denotedωn. The measurement ymic is split into an in-phase andquadrature branch, demodulated down to a baseband whichincludes DC, compensated in the baseband, then modulatedback to the measurement band. References [14] and [15]provide a detailed analysis of the dynamics of this systemfrom two perspectives, the measurement perspective and thebaseband perspective. From the measurement perspective, themodulated-demodulated controller can be represented as a

2cos( t)

2sin( t)

cos( t)

sin( t)

u--

-

n

n

n

n

1s

1s

R

R

I

I

n

LPF

LPF

rni

rnq

ymic

P (s)

C (s)1

C (s)2

C (s)N

r1i

r1q

r2i

r2q

rNi

rNq

ymicgp

uN

u1

u2

{ {{ {

{ {

Details of C (s) n

C (s)mic

mic

yni

ynq

uni

unq

n

n

n

n

a)

b)

Fig. 5. a) Block diagram of the inner loop modulated-demodulated controller. Each control loop within Cmic(s) tracks the Fourier series coefficientof the microphone reference at a single harmonic. b) Block diagram of modulated-demodulated control at a single frequency. The demodulated hotwiresignals yi

n and yqn track the real and imaginary part of the ωn Fourier series coefficient, ri

n and rqn, respectively.

linear time-invariant (LTI) system with transfer function

Cn(s) = gmic

[LPF (s− jωn)(Rn − jIn)

s− jωn

+LPF (s+ jωn)(Rn + jIn)

s+ jωn] (2)

Poles are placed at ±jωn as required by the Internal ModelPrinciple for perfect reference tracking of a periodic signalat ωn [1]. From the baseband perspective, the system can beviewed as a two-input, two-output compensated plant withinputs uin and uqn and outputs yin and yqn, and a two-input,two-output baseband compensator. The reference for a singleharmonic is simply defined by the constants rin and rqn.

C. Baseband System

Similar to baseband analysis of modulated-demodulatedcontrol, the system from rmic,d to yvel,d in Figure 4b can beviewed from the baseband perspective as a MIMO plant. TheMIMO system has 2N inputs (the in-phase and quadraturereference signals for the demodulated microphone measure-ment) and 2N outputs (the in-phase and quadrature signalsof the demodulated hotwire measurement). The 2N outputsof the yvel,d vector are formed by demodulation and low-pass filtering, as in Figure 5b, at each frequency of control.For the present study we are interested in the low-frequencyproperties of this arrangement and its model identification isfacilitated by the stable operating point maintained by theinner loop. Interestingly, we will show that a linear map issufficient to capture the low frequency relationship betweenthe “channels.” Note that this map represents the couplingbetween frequency bands, something akin to intermodulationdistortion, which is a nonlinear phenomenon.

1) Identification: Identification of the MIMO map isachieved by applying N , independent, low bandwidth ran-dom signals to the in-phase reference inputs of the inner loop.The low bandwidth identification inputs are added to thereference used to establish a suitable operating point. The 2Nsignals associated with in-phase and quadrature componentsof the demodulated hotwire signal are recorded in responseto these inputs. The inner loop introduces a constant time

delay, denoted td across all channels, which can be identifiedusing cross-correlation of the input and output signals. Forexample, Figure 6 shows the input-output cross-correlation at100Hz and 2200Hz. The 100Hz cross-correlation of ri1 withyi1 and yq1 is shown in blue and red, respectively, and the2200Hz cross-correlation of ri22 with yi22 and yq22 is shownin cyan and magenta, respectively. Both frequencies havetime delays of approximately td ≈ 0.99s. The time delayis approximately equal because we specify the convergencerate of the inner loop to be equivalent at all frequencies.

15 10 5 0 5 10 15

1

0

1

2

3

4X: 0.9924Y: 4.5

Time (s)

Cro

ssC

orre

latio

n (V

2 s)

Fig. 6. Cross-correlation of ri1 with yi

1 (blue), ri1 with yq

1 (red), ri22 with

yi22 (cyan), and ri

22 with yq1 (magenta). y is shifted by td ≈ 0.99s in

relation to r.

At any given operating point we assume the input-outputdata can be captured by

y(t) = Kr(t− td),

where K is a constant 2N×2N real matrix that characterizesthe nonlinear coupling amongst harmonics in the hotwiresignal. In-phase inputs alone are sufficient to identify thecoupling at a given operating condition because the responseto a quadrature reference input can be shown to be anti-symmetric with respect to the response to an in-phase input.For example, the in-phase and quadrature signals at the nth

frequency in the hotwire signal are related to perturbationsat the pth frequency of the microphone reference by[

yinyqn

]=[αin,p −αqn,pαqn,p αin,p

] [riprqp

](3)

The constant, real coefficient αn,p with subscripts n and pcharacterizes the coupling between inputs at frequency p tothe outputs at frequency n. The matrix K is composed of theidentified α’s assembled in 2× 2 antisymmetric blocks as in(3). The coupling between input p and output n is describedby the [2p − 1 : 2p, 2n − 1 : 2n] block of K. Identificationof the α’s is carried out using a simple least squares criteriaon the data sequences,

yin(t) = αi1,nri1(t− td) + · · ·+ αiN,nr

iN (t− td)

yqn(t) = αq1,nri1(t− td) + · · ·+ αqN,nr

iN (t− td),

(4)

for n = 1, 2, . . . , N after the delay has been removed fromthe output data by a suitable shift, i.e.

αls = arg min ‖Y − αR‖2 (5)

where

Y =

yi1(t)yq1(t)

...yqN (t)

and R =

ri1(t− td)rq1(t− td)

...rqN (t− td)

(6)

and

α =

αi1,1 αi1,2 . . . αi1,Nαq1,1 αq1,2 . . . αq1,N

......

. . ....

αqN,1 αqN,2 . . . αqN,N

. (7)

2) Identification Error: The identification error is quanti-fied in terms of the least squares error using a separate setof identification data from which αls was derived,

E = Y′− αlsR

′, (8)

where the alternate data set is denoted with the superscriptprime. Figure 7 shows an example identification time seriesand the resulting error for a single output at a 20% dutycycle, ∆V = 2.5s−1 operating condition. The duty cycle,denoted β, is the ratio of the temporal pulse width τ to thewaveform period T , β = τ

T , and ∆V is the peak-to-peakvelocity perturbation. This identification, and all subsequentexperimental results, use an N = 22 controller with afundamental frequency of 100Hz. The set of measured data,yi

′

1 , is shown in solid blue along with the approximatedoutput using αls, shown in dashed red, and the resultingerror, shown in cyan.

The 2-norm of the RMS identification error, ‖ERMS‖2

indicates the accuracy of the identification when comparedto the 2-norm of the RMS output, ‖Y

RMS‖2. The β = 0.2,

∆V = 2.5s−1 identification presented above, for example,has values ‖E

RMS‖2 = 0.0028 and ‖Y

RMS‖2 = 0.0120.

The prediction error is small compared to the measuredperturbation of K. This error could potentially be the result

of higher order nonlinear dynamics, however, the linearmodel of (4) captures the majority of the nonlinear couplingand provides a simple framework for compensation.

0 50 100 1506

4

2

0

2

4

6x 10 3

y 1i

Time (s)

Fig. 7. Identification time series for the yi1 channel of the β = 0.2,

∆V = 2.5s−1 operating condition. The prediction using (4), shown indashed red, is compared to an independent set of measured data, shown insolid blue. The error is shown in cyan.

3) Harmonic Coupling: There is a graphical way to viewthe coupling between frequency channels by computing themaximum singular value of each 2×2 subblock (3) of K andreplacing the subblock with a color that denotes the valueof the norm. Figure 8 illustrates two such maps, one at theunforced operating condition and the other at a β = 0.2,∆V = 2.5s−1 operating condition. At the unforced operatingcondition, K is essentially block diagonal which indicateslittle-to-no harmonic coupling. This is representative of a lin-ear response in which an input perturbation at one frequencyonly effects that same frequency in the output. In contrast,however, at the β = 0.2, ∆V = 2.5s−1 operating condition,the identified K shows a strong level of harmonic couplingamong frequencies above the n = 12 index.

D. Outer Loop

For demonstration purposes, we choose a model inversemethod to design the 2N×2N outer loop controller, Cvel(s),shown in Figure 4c. The controller is simply

Cvel(s) = gvelK−1 1sI (9)

where gvel is a positive gain used to control the outer loopconvergence rate and integral control is used to provideasymptotic tracking of rvel,d. In practice, gvel ≤ gmic/10to ensure the inner loop dynamics have a negligible effecton closed-loop stability. As defined here, Cvel(s) tracks thervel,d reference at a single operating condition mapped by K,however, a controller formed by convex optimization usinga set of maps can potentially control a range of forcingconditions instead of a single point. This method is left forfuture work.

In general, K must be well conditioned to use (9). If Kis ill-conditioned, an inversion of K using a truncated setof singular values can be used to avoid amplifying possiblemodeling errors. The singular values associated with small

y

2 4 6 8 10 12 14 16 18 20 22

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2

a)a)

b)b)

Fig. 8. Static nonlinearity map of K at a) the unforced operating conditionand b) at a 20% duty cycle, ∆V = 2.5ms−1 operating condition.Harmonic coupling significantly increases above ∆V = 1ms−1.

plant gain, such as those from the n = 4, 5, . . . , 10 diagonalin Figure 8, need not be truncated. The unforced operatingcondition map, for example, has a condition number of5.5 due to variation of the plant gain over the 100Hz to2.2kHz band. The β = 0.2, ∆V = 2.5ms−1 map hasa condition number of 9.3, which is close enough to theunforced operating condition to safely use a full matrixinversion to form Cvel(s).

IV. REFERENCE TRACKING RESULTS

The results for asymptotic tracking of a ∆V = 2.5ms−1

tapered square wave are presented here with duty cycles β =0.2 and β = 0.4. A tapered, or trapezoidal, square wavereference is used in place of typical square wave to reduceoscillation in the reference by removing discontinuities. Atrade off is made between the level of ringing and the rateof transition, which is slightly slower with a tapered squarewave. The inner loop and outer loop time constants are setto τmic = 0.1s and τvel = 1s, respectively.

Figure 9 shows the progression from the operating condi-tion established for plant identification to the asymptoticallytracked measured waveform. The operating condition in redand the asymptotically tracked measurement in blue arecompared to the reference in black. The initial operatingconditions in red are sufficiently close to the velocity ref-erences to identify accurate harmonic coupling maps forboth duty cycles. After identification, the outer loop slowlycompensates the outputs to form the measured waveforms in

0.006 0.008 0.01 0.012 0.014 0.016

7

8

9

10

11

Velocity (m/s)

Time (s)

0.006 0.008 0.01 0.012 0.014 0.016

7

8

9

10

11

Velocity (m/s)

a)

b)

Fig. 9. Square wave time series comparing the operating condition (red) andthe asymptotically tracked output (blue) to the reference (black) at ∆V =2.5ms−1. a) β = 0.2, b) β = 0.4.

blue which, as can be seen in Figure 9, closely follow theperiodic references.

A closer look at the spectra of yvel,d and rvel,d in Figure10 reveals the Fourier series coefficients of yvel,d matchup with the Fourier series of rvel,d at all frequencies ofcontrol. There are small periodic deviations, however, whichare the result of harmonics excited beyond 2.2kHz. As theseharmonics lie beyond the bandwidth of the actuation system,they are uncontrollable. Small non-periodic errors are alsopresent which come from noise that falls between the narrowbands of control. Nevertheless, these errors have little impacton the shape of the velocity waveform.

V. CONCLUSION

In this paper, we presented a strategy to identify andcontrol nonlinear coupling for periodic reference trackingin a pulsed jet injection experimental study. A data drivenapproach is needed, whereby a model for the nonlinearcoupling is identified around a particular forcing condition.In the baseband coordinates, the plant can be modeled as alinear map which captures the coupling between frequencychannels. A MIMO controller was synthesized based on theidentified map to asymptotically track a periodic square wavereference at its first 22 harmonics.

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100

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0

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Frequency (Hz)

Magnitude (m/s/V)

Phase (deg)

a)

b)

Fig. 10. Spectra of yv (blue x’s) and rv (black circles) at each of theharmonics of the 100Hz fundamental frequency. a) β = 0.2, b) β = 0.4.

ACKNOWLEDGMENT

The authors gratefully acknowledge the contribution ofProf. Ann Karagozian.

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