Closed-loop Vectoring Control of a Turbulent Jet

AIAA 2002-3073

1st Flow Control Conference 24-26 June 2002, St. Louis, Missouri

AIAA-2002-3073

Closed-loop Vectoring Control of a Turbulent Jet Using Periodic Excitation Rapoport', D., Fono', I., Cohen', K. and Seifert', A. Dep. of Fluid Mech. and Heat Transfer

Downloaded by UNIVERSITY OF CINCINNATI on December 7, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2002-3073

Faculty of Engineering, Tel-Aviv University, ISRAEL Abstract Closed-loop active flow control strategies were studied experimentally using periodic excitation in order to vector a turbulent jet. Jet deflection was achieved by attaching a short, wide-angle diffuser at the jet exit and introducing periodic excitation from a segmented slot around the circumference of the round turbulent jet. Conventional and modem control methods were applied in order to quickly and smoothly transition between different jet vectoring angles. It was found that the frequency response of the zero-mass-flux Piezo-electric actuator could be made flat up to about 0.5kI-I~. However, the jet can optimally respond at 30-5OHn This is still an order of magnitude faster than any conventional thrust vectoring system. System identification procedures were applied in order to approximate the system's transfer function and characterize its open- and closed-loop dynamics. Based on the transfer function, a linear controller was designed that enabled fast and smooth transitions between stationary deflection angles and maintained desired jet vectoring angles under varying system conditions. The linear controller was tested over the entire range of available deflection angles and its performance is evaluated and discussed.

slot width, lmm periodic momentum at slot exit,

P4*& distance from jet exit to diffiser exit, measured along diffiser wall pressure, [pa] root mean square of pressure fluctuations, [pa] mean of actuator's cavity pressure, Pal low-pass fittered cavity pressure, [pa]

- N

Y

AN

jet cross-section area, xr2 active slot area, xh(2r+h)/4

z

c,

periodic excitation momentum

2,

Axhl

[ml

6

coefficient, J V ( ~ A ~ , ~ U , ~ Cp,cav

jet radius, 19.5mm Reynolds number based on jet diameter, U,Dl v time, [SI jet mean velocity, [ d s ] root mean square of the velocity fluctuations, [ d s ] kinematic viscosity axial direction, x=O at jet exit and diffuser inlet, [mm] imaginary source of deflected jet, x,,= -D/2, [mm] horizontal direction, y=O at jet centerline, [mm] vertical direction, z=O at jet centerline, jet center of linear momentum, [mm] jet deflection angle,

cavity pressure coefficient,

P

P~",",,, /uj

D

jet diameter, 39mm

F" f

reduced frequency, jZ IU. excitation frequency, [Hz]

Q

density diffuser half-angle, a =30, [deg]

' Graduate student Consultant IDF officer Corresponding author: Seifertliiene,tau,ac.il,Associate fellow AIAA

1

American Institute of Aeronautics and Astronautics

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Control related parameters denominator of TF 4s) numerator of TF

f H(s) H(jw) K


KP Ki K,,

frequency, [Hzl TF in the Laplace domain complex frequency response gain proportional gain integral gain derivative gain

Tz

complex variable of the Laplace transform, s = 0 + j w radial fiequency, 2 ~ f[ d, s e c ] time constant of IMC controller time-constant of TF zero, [sec]

rn 4“

time-constant of TF pole, [sec] damping ratio of TF pole

Subscripts cav diff e slot

actuator’s cavity diffuser wall conditions at jet exit condition at slot exit

S

co

2

Terminology AM Baseline HW RMS

TF

Amplitude Modulation No control, e C p O . 0 Hot-wire root mean square of fluctuating value transfer function 1. Introduction

Conventional thrust vectoring systems are capable of significant jet deflection angles, enabling controlled flight at high angles of attack resulting in super maneuverability. However, the complexity and weight penalty associated with this added capability is huge, thrust penalty usually occurs as well and the slow response of the turning vanes or movable nozzle makes the search for a simpler and faster responding alternative worthwhile (Refs. 1-6).Active control of turbulent jets is not new. For several decades, researchers introduced controlled excitation into the shear layers of laminar or turbulent jets. This was performed mainly in order to gain better understanding of the underlying physics and the importance of large coherent structures to their mixing, and their noise generation (Refs. 7,8). Several studies were conducted to explore the role of coherent structures in the control of flow separation (Refs. 9,101.

AIAA-2002-3073 Several years ago, it W ~ Ssuggested (Ref. 1 1 ) to combine the two approaches and attach a short, wide-angle diffiser to the exit of a turbulent jet. Then, excite the separated shear layer in order to promote reattachment of the jet to the diffuser walls. Controlling the entire circumference of the jet enhances the spreading rate of the entire jet, reduces the noise and increases the thrust efficiency. Introducing the periodic excitation to a fraction of the jet circumference, moves the entire jet towards the controlled side (or sector) of the diffuser wall, therefore the jet is deflected up to the same angle as the diffiser wall, instead of being just slowed down by it. It was further shown that the deflection angle is proportional to the magnitude of the excitation as defined by the oscillatory momentum coefficient (Ref. 12). It was clearly demonstrated that the presence of a wide diffuser at the jet exit enhances the capability of the periodic excitation to vector the jet. Additional modes of excitation were discovered and the optimal values of the most significant parameters were identified (Ref. 12). Initial transient response tests of the jet deflection as a result of applying standard control inputs (step and impulse) demonstrated that the jet responds quickly (in the order of 10-20msec, 50-100Hz) and more importantly that the response is not sensitive to the jet speed (and Reynolds number). The response has several common features with second order, under -damped systems (Ref. 12). Similar responses were previously seen in external aerodynamic applications of active separation control using periodic excitation (Refs. 13-14). The present investigation makes use the existing jet facility, its piezoelectric actuators and its established modes of excitation in order to perform the closed-loop active flow control experiment, with the aim of smooth and rapid transition from one state (Le., jet deflection angle) to another. The experiment first focused on the identification of the system’s static and dynamic responses to generic control inputs, including testing the linearity of the input/output relationship over a significant portion of the parameter space. It continued with a search for simple yet comprehensive methods for real-time detection of the system’s state. Initially, a single hot-wire probe was used to measure the shear layer velocity variations as an indication of the jet deflection angle (system output). Additional measuring techniques that were subsequently used are steady and unsteady diffuser surface and actuator’s cavity pressures. The primary control input was the amplitude of the cavity pressure fluctuations, having a flat frequency response up to about OSkHz. Innovative control approaches were used to supplement traditional methods of closed-loop control in order to smooth the transients associated with step changes in the control input. MATLAB and SIMU LINK simulations assisted in the design and optimization of the controller

2 American Institute of Aeronautics and Astronautics


AIAA-2002-3073 and a state-of-the-art dSPACE DSP control hardware and real-time software were used to perform the experiments. The study included the following stages: 1. Static characterization experiments of the actuators and jet response to standard control input. (Sec. 3.1) 2. Development of a dynamic model for the response of the turbulent jet to the control input (Sec. 3.2) 3. Design and analysis of the closed-loop control system (Sec. 3.2.3) 4. Closed-loop control of the turbulent jet (Sec. 3.2.3) 5 . Sensitivity and robustness tests (Sec. 3.2.3). 2. ExDeriment SetuD

2.1 Jet

2.3 Actuator A zero-niass-flux Piezo-electric actuator, which resonated at 725Hz, was used to generate the periodic excitation. The velocity fluctuations, u\lo,, exiting the slot were measured using a hot-wire positioned at the slot exit, in the absence of main jet flow. An unsteady pressure transducer was mounted in the actuator's cavity to measure the static and dynamic cavity pressures. The p',,signal was used as an indication of the actuator's performance, while the low-pass filtered cavity pressure signal, Pcav,lpf, (using 5* order Butterworth filter with

200Hz cutoff) was used to close the control loop as discussed in Sec. 3.1.2. 2.4 Instrumentation and Data Acauisition The jet flow field measurements were acquired using a single hot-wires. Hot-wire calibrations were performed at the core of the jet exit and data repeatability was often checked. The data acquisition sequence included measuring the jet baseline flow before and after acquiring controlled jet data. A check of the two baseline data sets also sewed as a check of the hot-wire calibration. Hot-wire, static and dynamic pressure data were acquired using a 12-bit A/D converter. The sampling rate of the dynamic data was 8192Hz, and the data were low-pass filtered prior to sampling. The frequency response of the hot-wire and unsteady pressure transducer were better than 20kHz. The hot-wire was mounted on a three-axis traverse system. All of the experimental instruments were interfaced with a desktop computer.

The jet facility consists of a DC fan connected to a 76.2mm diameter settling chamber, leading to a 3.151 contraction that is followed by a straight aluminum pipe with an inside diameter of 38mm and an outside diameter of 40mm. The thickness of the pipe wall, at the jet exit, was machined to bring it gradually down to OSmm, in order to reduce the gap between the jet flow and the excitation slot. The jet exit diameter is therefore 39mm (Figs. 2.la and 2.lb). A segmented slot, Imm wide, surrounded the jet. Each segment of the slot was connected to a cavity allowing independent excitation of every 45' of the jet circumference. The excitation was introduced through the slot surrounding the upper 90' of the jet circumference (Figs. 2.la and 2.lb). Trip grit (#36) was placed inside the aluminum pipe, 200 mrn from the jet exit, to ensure that the flow at the jet exit was turbulent over the velocity range of interest ( I O d s 14ds). Jet exit velocity was determined by the static pressure in the jet-settling chamber that was found to be 3% below the dynamic pressure at the jet exit.

2.5 ExDerimental Uncertainty The velocities measured with the hot-wire are accurate to within 2%. The hot-wire position (x, y, or z) is accurate to within f0.04mm, the C,is accurate to within 15%, and the jet deflection angle estimates, S are accurate to within fl'.

2.2 Diffuser

2.6 Baseline Jet and Exoerimental Conditions

A short, wide-angle difiser was attached to the exit of the jet with a half-angle of 4=30'and length of Lz0.58D. Previous publications (Ref. 12) include the effects of various diffuser spreading angles and length, but presently the shortest yet effective diffuser that was tested to date, was used. An unsteady pressure transducer was installed in the diffuser wall, 2mm from the jet exit in line with the active segment of the actuator, to measure the dynamic response of the jet vectoring. The transducer was mounted inside a miniature cavity that was connected through a 0.5rnrn diameter orifice to the flow exposed side of the diffuser.

The baseline uncontrolled jet flow was measured and found to be well-behaved, circular turbulent jet. The boundary layers upstream of the jet exit were tripped to assure turbulent flow at the jet exit and minimize Reynolds number effects. All data presented in this manuscript were acquired at a jet exit velocity between 1Ods and 1 4 d s . 3. Discussion of Results 3.1 Static ResDonse of the Plant 3.1.1 Actuator Static Response Static testing of the actuator output and actuator-flow interaction were performed in order to characterize its 3



response and establish its gain, offset and linearity. These tests revealed that the resonance frequency of the actuator is around 725Hz, and this frequency was used throughout the present investigation. The actuator was characterized using unsteady pressure sensor installed inside the actuator’s chamber (Fig. 2.lb) and by a hot-wire placed at the slot communicating the actuator and the jet flow. Special care was taken to align the hotwire in the actuator’s slot such that the blowing and suction related peak velocities were identical, and the resulting velocity signal matched an ideally rectified sine wave. The static and the dynamic actuator performances were then measured. These established a linear relationship between the excitation voltage provided to the actuator and the resulting level of pressure fluctuations in the actuator’s cavity, ptcav(Fig. 2.2). A linear relationship between the excitation voltage and the resulting pressure fluctuations is advantageous from the control aspect. The measured level of pressure fluctuations was used as the primary control input instead of the excitation voltage, because it compensates for variations in the actuator’s performance, and it still maintains system closed-loop performance, regardless of the actuator’s status (as long as sufficient control authority is maintained). The RMS velocity, z&,~,,at the slot exit as measured by a hot-wire in the absence of jet flow, is shown in Fig. 2.3. The maximum u’ output of the actuator, utmax,using streamwise excitation, is about 1 8 d s . The effect of the jet exit velocity or the presence of a diffuser changes the u:~,~,, by less than 5%. These hot-wire measurements established a linear relationship between the cavity pressure fluctuations level and the slot velocity fluctuations. Although it is established that the u\,,>, (or Cp) is the leading flow control input parameter, the current experiment used plcnv as the primary input parameter due to its linear relationship with uLo, and the ability to monitor it when the hot-wire was removed. 3.1.2 Jet Static Response The static jet flow response to the control input was measured when a quarter of the jet circumference was excited periodically with the wide-angle diffuser being present (Fig. 2.1). The result of the excitation, with a mean jet exit velocity of 1 2 d s and a range of excitation levels (indicated by the cavity pressure fluctuations in the legend of Fig. 3.1), was a gradual spreading and vectoring of the jet towards the excited shear layer (i.e., negative z direction), at angles up to IO” for excitation momentum coefficient of up to 7% (Fig. 3.2). These results are in good quantitative agreement with the data of Ref. 12 while the jet deflection angle was estimated from a single vertical velocity profile measured on y=O

AIAA-2002-3073 and the imaginary source of the jet was always taken at xrS=-0.5D.

While the data presentation at Fig. 3.2 is physically meaningful and can also account for variations in the jet exit velocity (as will be done in the next section), it is not convenient to be used for closed-loop control, where a linear relationship is always an advantage. It was found out that the jet deflection angle is almost linearly dependant on the level of the cavity pressure fluctuations (Fig. 3.3), which are linearly related also to the slot R M S velocity. The latter, and therefore also P‘~~,, , are in turn proportional to This finding

&.

indicates that the cavity pressure fluctuations could be used as a convenient control input. The data presented in Fig. 3.3 was acquired 15 months apart and provides an indication of the data repeatability. Furthermore, between the acquisition of the first and second data sets, one out of the multiple Piezo actuators driving the cavity was damaged. However, since the control input parameter was the ptcov, it resulted in only increased excitation voltage, with no effect on the deflection angle. The next step in the static jet characterization was a search for sensor type and optimal placement for “closing the loop”. It is clear that intrusive velocity sensor such as hot-wire cannot be used beyond preliminary lab experiments and therefore, it was used only as an indicator of the jet state rather than a sensor for closing the loop. The diffuser skin friction measured by hot-films is another possibility that was rejected due to the assumption that the jet does not fully reattach to the diffuser and therefore significant changes in skin friction are not expected. The next logical option to consider was the diffiser pressure. It is a local quantity but it is directly related to the forces acting on the diffuser and jet. Therefore, an unsteady pressure sensor was also placed on the diffuser wall (in addition to the one installed in the actuator’s cavity), 2mm From the active excitation slot (as indicated on Fig. 2.la). Static tests revealed that the mean diffuser surface pressures are reduced as the jet is vectored (Fig. 3.4). However, a similar and even stronger response was measured in the actuator’s cavity. This response makes perfect sense if one thinks of the jet deflection mechanism in terms of (the somewhat oversimplified similarity to) ideal comer flow. The reduced pressure is generated because of the curvature of the jet streamlines as the jet turns closer to the diffuser wall (or vise versa) but the suction peak has to be the strongest at the comer, i.e., where the actuator slot is located. The finding that the low-pass filtered pressure signal measured in the actuator’s cavity is stronger, while the sensor placed at the diffuser wall also exhibits a high level of fluctuations at the actuator’s resonance frequency (725Hz)makes the use of the diffuser sensor redundant. In addition, the

4



sensor placed at the actuator’s cavity is more protected from external hazards. Therefore, it was decided to use the lowpass filtered cavity pressure, PCa,,,w, as the primaty indicator of thejet deflection angie (Fig. 3.5). The hot-wire measured velocity signal, just opposite the active slot and at a distance of one jet diameter downstream of the jet exit, was used as a local indicator for the success of the control strategy. Clearly, the cavity pressure is a better and more convenient choice for positioning a sensor for closed-loop control purposes, as it is non-intrusive and eliminates the need for at least two separate sensors; one to monitor the actuator and the other to sense the flow state. The actuator’s cavity pressure sensor reading is both statically and dynamically similar to the diffuser pressure sensor reading, only the signal is larger since the sensor communicates with the external flow at the comer, where the flow turns so the pressure suction peaks there.

3.1.3 The Effect of Jet Mean Exit Velocity Certain jet flow control applications might require constant jet deflection angle under varying operating conditions. Examples range from thermal protection, industrial processing, chemical reactions and pollutant dispersion in the atmosphere. To be able to control in close-loop and maintain a constant jet deflection angle one needs, initially, to statically explore the effect of the jet exit velocity on the resulting jet deflection angle. Certainly, an additional sensor should be added to the system to monitor the jet exit velocity by, as an example, measuring the static (or more accurately the low-pass filtered) pressure in the jet plenum. Experiments were conducted to determine the static jet deflection angles by varying the cavity pressure fluctuation levels at three jet exit mean velocities. The data presented in Fig. 3.6a indicates that, as expected, the jet deflection angle decreased as the jet exit velocity was increased for a constant level ofp‘,, . However, when the jet deflection data is re-plotted against pTcOv/Uj, that is proportional to the square root of the oscillatory momentum coefficient,

G,the

data

coalesces to a single curve (Fig. 3.6b). Although physically meaningful, C , is not a convenient control parameter. Still, it provides a guideline as to the required control methodology; measure U j, alongside with

p‘crtv, and use the ratio p‘co,,/Uj as the control input. As an indicator for the jet deflection angle we suggest to use the cavity pressure coefficient Cp,cav= p c u v , l l , e u , , /.~Fig. ~ 3 . 6 ~clearly shows that, when plotting Cp,cavvs. the required jet deflection angle for the three different jet exit velocities, the data collapses

AIAA-2002-3073 into a single curve. Nevertheless, it is not a linear relationship. The curve in Fig. 3 . 6 ~was used to establish the correlation to derive the required Cp,cav for a desired jet deflection angle. Multiplying the result by ,-~j provides the set-point for the closed-loop system. Fig. 3.6d presents three velocity profiles, measured by a hot-wire at x/D=1 on y‘0, acquired when the controller was required to maintain a closed-loop deflection angle of 7O, regardless of the jet mean exit velocity. The excellent agreement between the three velocity profiles clearly demonstrates the success of the approach currently selected. The details of the closedloop control methodology are discussed in Sec. 3.2.3, however for completeness it was decided to include the closed-loop deflection tests in this section.

3.2 Dvnamic Resaonse of the Plant 3.2.1 Preliminary Actuator and Flow Dynamic Response A dynamic characterization of the actuator response is the initial step discussed in this section (Figs. 3.7a-c). These tests revealed that the jet response, as measured by the RMS of the cavity pressure fluctuations, plcaV, resembles a first order system with a time constant of about 5msec (thin line in Fig. 3.7b). The dynamic actuator response was modeled using MATLAB, and Fuzzy logic was used to construct a modified step, with the aim of generating an “ideal” step response (i.e., shortening the rise time of the actuators cavity pressure fluctuations). Following an empirical tuning of the theoretical “ideal” step, the desired response was obtained (thick lines in Figs. 3.7a and 3.7b, corresponding to the “ideal” step input signal and the resulting cavity pressure fluctuation, respectively). It was found that the current Piezo actuator could reach steady-state operation within 2-3 msec (Fig. 3.7b). Next, the flow response, as measured by a hot-wire placed in the flow at x/D=l and in the shear layer opposite the active segment of the actuator, was characterized (Fig. 3 . 7 ~ ) in order to evaluate its dynamics (the effect of the “ideal” step actuator input signal on the flow response). It was found that the flow response resembles a second order system with an initial inverse response and an overshoot. The former is a typical response of a system with a transmission zero at the right-hand side of the complex plane. When using the “ideal” instead of the “regular” actuator input, the severity of the inverse response increased (compare thick with thin lines in Fig. 3.7c), as did the magnitude of the overshoot. These undesired effects were accompanied by a small increase in the rise time. The conclusion drawn from this test was that one would need to use a more gradually evolving signal (Le., rate limited) than the “idealyyor even “regular” actuator step input signals in order to smooth the undesired transients.

5



Figs. 3.8a-c present the effect of a 20msec ramp input signal to the actuator (compare thick to thin signals in Fig. 3.8a) on the cavity pressure fluctuations response (as shown in Fig. 3.8b) and on the resulting flow response (Fig. 3.8c, measured by a hot-wire as in the data of Fig. 3.7~).The milder flow response, without the inverse and overshoot, indicate that if one would like to obtain rapid but still smooth transitions between different stationary jet deflection angles, a rate limiter or low-pass filter would need to be included in the closedloop controller. Even with the rate limiter, the flow response indicates that a 30-50Hz frequency response could be achieved in real, full size systems (for a proportional increase in size and speed), which is an order of magnitude faster than conventional thrust vectoring and control surface frequency response. The rate limited step changes were also found to improve the linearity of the system. The flow response was measured for several magnitudes of step and it was found that the normalized response of the modified step is linear while the flow response to the regular step was non-linear, indicative again that a rate limited controller would perform reasonably well even though the system response is highly non-linear. 3.2.2 System Identification of Jet Vectoring Using Frequency Sweep (“chirp”) Linearization The static response of the jet deflection, as measured by the actuator’s mean pressure, Pcav,mean, to the

magnitude of the pressure fluctuations, pfcav,is nonlinear (Fig. 3.4). In order to find a meaningful linear model of the plant, the first step is to establish an operating point and a range around which the response curve is approximately linear. The chosen operating point was pIcm =360Pa and the operating range from plCav of 250Pa to 465Pa at Uj=12mls. All subsequent experiments related to the identification of the plant were performed at this range of excitation levels. Modeling The purpose of dynamic modeling is to establish a mathematical description of the physical system (plant). Experimental modeling typically assumes an a-priori form of the model and then uses available measurements to estimate the coefficient values that cause the assumed form to best fit the data. The assumed plant transfer function was H ( s ) = B ( s ) l A ( s ) , that is a rational function of polynomials in s (the Laplace variable). The estimation of the polynomial coefficients was performed in the complex frequency domain. The experimental setup of Figs. 3.9a and 3.9b was used. The input to the plant was an Amplitude Modulated (AM) sine wave at 725Hz, the resonance of the Piezo actuators. The

AIAA-2002-3073 amplitude of the 725Hz signal (without modulation) was statically adjusted to provide an excitation of pICav=360Pa to the actuators, the chosen operating point, resulting in a steady state jet deflection of 6 ~ 7 ” (Fig. 3.3). The AM signal was a swept sine (“chirp”) in the range of 1Hz to 9OHz with a linear increase in frequency over a sweep-time of 90 seconds (Fig. 3.10a showing the initial 2.5 seconds and Fig. 3.11a showing the last 0.05 seconds of the AM signal). This frequency range was selected because the jet response at higher frequencies was significantly attenuated (Figs. 3.10d and 3.1 Id present the hot-wire measured velocity response). The amplitude of the “chirp” signal was adjusted to provide 30% modulation to cover a partial, linear operating range of the plant as mentioned above. The resulting excitation signal to the actuators is shown in Figs. 3.10b and Fig. 3.11b. Note that the 725Hz excitation cannot be seen in the plot resolution of Fig. 3.10b. Data Acauisition The input AM signal, together with the plant responses (cavity pressure and hot-wire measured velocity at x/D=l) were amplified, low-pass filtered (200Hz for cavity pressure and 3 . 8 W for the hot-wire) and acquired at a sampling rate of 8192Hz with a 12 bits A/D board. The resulting data were ensemble-averaged over 10 realizations. The low number of realizations was sufficient to remove random noise from the cavity pressure signal (Figs. 3 . 1 0 ~and 3.11~)but certainly not enough for the noisy turbulent jet velocity signal (Figs. 3.10d and 3.11d). However, the hot-wire signal was only used as an independent indicator for the jet deflection and not as a sensor for closed-loop operation. The hotwire was positioned at x/D=l downstream of the jet exit at a point, z, in the shear layer such that U = Umw / 2 while pfCmwas held steady at 360Pa. Fig. 3.10d shows the first 2.5 seconds of the HW signal with the AM signal operating, indicating that the jet closely follows the input signal, while Fig. 3.1 Id presents the final 0.05 seconds of the velocity data, showing the lack of flow response to the AM signal at 90Hz. Figs. 3 . 1 0 ~and 3 . 1 1 ~present the cavity pressure signal, showing the lack of symmetry (with a mean less than zero, indicating negative cavity pressure and therefore jet deflection in conjunction with the static deflection tests shown in Fig. 3.5), which is a result of the varying jet deflection angle. Also note the inverse relationship between the amplitude of the cavity pressure signal and the low-pass filtered cavity pressure, PCm,&,, i.e., the higher the amplitude of the cavity pressure fluctuations, the more negative the low-pass filtered cavity pressure becomes. This is due to larger jet deflection angle that caused increased acceleration


AIAA-2002-3073

around the jet-diffuser comer, lowering the local static pressure. The cavity low-pass filtered pressure is also plotted on Figs. 3 . 1 0 ~and 3.11c, with the ordinate on the right hand side of those charts. Note that this ordinate is smaller than the left hand side ordinate by a factor of ten. Remembering that the Pcm,bfis indicative of the jet

.s2 + 2.2.3.IO-3

(2.3.

.0.935s+ I

This can be written in the form: TFpiant(s)= K .

deflection angle (Fig. 3 4 , one can note that for low modulating frequency the jet deflection angle follows closely the amplitude of the excitation signal. For high modulating frequency (Fig. 3.11~) the amplitude of Pca,,,lpJ is significantly reduced and a phase shill of Downloaded by UNIVERSITY OF CINCINNATI on December 7, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2002-3073

1.395.I0-~s -I

= 0.773x

r,s-l

r,2s2 +2r,,cs+1

where: K = 0.773 5;

almost 180' develops between the input excitation and the resulting jet deflection angle. Approximately 80' of the phase lag is the result of the low-pass filter used. This result is physically reasonable, as the jet with a time constant of about IOmsec, develops an increasingly larger phase lag with increasing modulating frequency. In addition, the magnitude of the jet deflection angle fluctuations is attenuated as the AM frequency increases. When the modulating frequency approaches IOOHz,the jet will no longer follow the changes in the A M signal.

= 13 9 . IO--'sec I

rll = -= 2.3

sec

w,,= 434rad I sec

69.2 Hz

Jl=

WIt

C = 0.935 Figs. 3.12a and 3.12b present the Bode plots of the plant as indicated by the actuator's cavity pressure. It demonstrates the good correspondence between the measured and modeled data. .

Data Processinq The ensemble averaged input stimulus and plant response signals were transformed to the frequency domain by FFT. From the transformed data only the relevant frequency components from 1Ht to 90Hz were considered for modeling, corresponding to the frequency range of the input stimulus (AM signal). The FFT of the response signal divided by the FFT of the input AM signal is the measured complex planf frequency response, H ( j e ) . The measured Bode plot of the plant (bfugnirude and Phase of H ( j u ) ) is presented in Figs. 3.12a and 3.12b. The frequency response is flat with the -3dB point (amplitude ratio of 0.707) at about 60Hz. The phase lag between the AM signal and the cavity low-pass filtered signal is initially 180'' as shown in Figs 3.1Oa-c. It increases by additional 100' at 60Hz, but part of this increase is due to the Butterworth filter. Transfer Function The plant transfer function was found from the complex frequency response, N ( j w ) , by assuming a transfer function of the form H ( s )= B ( s ) / A ( s ) and numerically searching for the coefficients of B(s) and A(s) that best fit the data. The order of the numerator and denominator of the transfer function were arrived at by perforniing several iterations of the above procedure and choosing the orders that provided the best correspondence with both the magnitude and the phase of the plant frequency response. The resulting transfer function of the plant is:

Trans~ortDelav The form of the transfer function H ( s )= B(s)/A(s) assumes zero transport delay. If the measured signal has significant delay, like in the case of the velocity signal measured by a hot-wire positioned 39mm downstream of the jet exit (x/D=l), the frequency response data should be corrected before it can be modeled. The effect of a pure time delay on the frequency response is a linear increase of phase with increasing frequency, according to e - J u x t , where Td is the delay time. To eliminate the effect of the delay, the phase of the measured frequency response should be decreased linearly with frequency according to

Hcorrecte~( jo)= H ( j o ) . Once the transfer function coefficients are found from the corrected data. the delay time is set as an attribute of the transfer function object. The model frequency response, including the delay time, was then computed. Consequently, the computed frequency response can be compared directly to the original (uncorrected) hot-wire data (Figs. 3.13a and 3.13b). In the case of the hot-wire velocity data (with a mean exit velocity of 12mls). a delay time of 6.5msec provided the best fit. This is consistent with a typical convection speed for the large coherent structures of 0.6Uj (Ref. 12). Following are the transfer functions of the plant, as determined by the hot-wire: TF- HW'(s) =

HWsignnl(s)

AMsignal(s)

s'

-4.273.105 s-2.836. IO6 +8.57.1OJs+ 6.027. IO5

+ 421s'

I American Institute of Aeronautics and Astronautics

e-0.00(i5s


Figs. 3.13a and 3.13b show the computed Bode plots corresponding to the above transfer function superimposed on the measured HW data.

AlAA-2002-3073 again on the real plant. It should be mentioned that three independent gains need to be tuned and this process could be time consuming and is difficult to optimize.

PI Controller with Rate Feedback An additional problem of the three-term PID controller is that the derivative term amplifies noise. Furthermore, it can lead to saturation of the actuator’s input when fast rising error signals occur. A PI controller avoids these problems, but its step-response is characterized by a larger overshoot. To overcome this, a minor feedback loop was added with a signal proportional to the rate-ofchange of the jet angle, indicated by Pcav,,,f . The rate feedback provided suflicient damping to reduce the overshoot to an acceptable level. Fig. 3.17 shows the Simlink model of the PI controller with rate feedback, and Fig. 3.18 presents the simulated step-responses of the PI controller with and without rate feedback. The simulation clearly shows that the rate feedback is effective in reducing the [arge overshoot typical of PI controllers. Fig. 3.19 shows the simulated and measured closedloop step response for the PI controller with rate feedback The response is not perfectly matched to the model. It does not exhibit the inverse response nor the overshoot predicted by the model and the lag time is shorter than the modeled response.

3.2.3 Implementation of the Closed-loop Control Rapid ProtoNDing The closed-loop control of the turbulent jet was realized using rapid prototyping methodology. In the framework of this methodology, the controller model is first simulated, tuned for optimal closed-loop step response and then implemented on a dedicated DSP hardware and software system. The DSP board presently used has realtime computing capability and integrated high-speed ADC and DAC YO channels. These YO channels communicate with the plant’s sensors and actuators. The DSP board software also provides the graphical user interface to perform the experiments and to acquire realtime data. Fig. 3.15 shows the process flow chart for the rapid prototyping stage of the study. Controller Design and ODtimization The controller was designed for the specific plant model described in Sec. 2. It was designed to meet several closed-loop performance criteria, e.g., fast rise-time, minimal inverse response, minimal overshoot and zero steady-state error for a step change in the input (Le., the required jet deflection angle). Furthermore, the controller should perform well under a wide range of operating conditions, including random noise, changing jet exit velocity, partial loss of actuator output and more, that is, it should be robust To achieve these goals three controller configurations were investigated: 1. PID Proportional, Integral, Derivative; 2. PI with Rate Feedback; 3. IMC Internal Model Control. The following text contains description of the controllers considered for use and those eventually used in the course of the present study.

IMC - Internal Model Controller A fundamental problem controllers have to overcome is the inherent uncertainty in plant models. If the plant model was perfect (i.e., matches the plant under all operating conditions), closed-loop control was not needed, and the transfer function of the controller was simply the inverse of the plant model, resulting in an ideal step response. However, real systems are impossible to model perfectly and therefore, when attempting to control the real plant in closed-loop, feedback is needed to take into account the model inaccuracies (Ref. 16). In the IMC configuration, the controller’s output simultaneously enters the plant and its model. Fig 3.20 shows the closed-loop configuration with IMC controller. To account for model uncertainty, as mentioned above, the difference between the responses of the plant and the model is used as thefeedback signal. The input to the controller is the difference between this feedback signal and the required set-point. The transfer function of the controller is the inverse of the model, with an additional real pole at -2, where R is the effective IMC controller time-constant, as will become evident later. The multiplicity of this pole, n, is the difference between the order of the model’s denominator and numerator. The IMC controller should be based on a reasonably accurate plant model. If that is not the case, the

-

-

PID Controller The PID (Proportional, Integral, Derivative) controller’s input is the closed-loop error, Le., the difference between the ser-point and the plant’s controlled variable (Ref. 15). The output is the sum of a proportional, an integral and a derivative term, each with a corresponding gain (Fig. 3.16). The integral term guaranties zero steady-state error. The transfer function of the PID controller, G,(s), in the Laplace domain is given by: G,(s)= K , +-+ Ki Kd * S S

The closed-loop step response is optimized (tuned) for a given plant by proper selection of K,, Ki and Kd . This should be done initially in a simulation tool, and 8



advantage of the IMC controller is significantly diminished. In the framework of the present investigation, the plant’s model included a zero at the right-hand side of the complex plane; therefore, the model cannot be inverted LIS is because a right-hand side pole always results in an unstable system. To solve the latter problem, the model was partitioned into its minimum phase (i.e., without the right-hand side zero) and its nonminimum phase components as follows: -- 203.8s - 1.461.IO5 TFpjom = pcu~,ip/(s) AM.vigmu/(s) S’ +813.3s+l.89.1O5 -

is the non-minimum phase part of the model that is also called the all pass part ( JVW). P ~ ( s ) i sthe minimum phase part of the model and therefore it is invertible. Finally, the IMC controller transfer function was defined by: pA(s)

IP~(s)/=

=-.

1

Is + I

-

- 0.005 s 2 - 4 s - 927 .58 s

+ 716 .85

- 0.005 s 2 - 4 s - 927 .58 1 s ’ + (716 .85A + 1 ) + 716 .85

The advantage of the IMC controller is its simplicity: only one parameter, A , needs to be “tuned” to achieve optimal step response. Fig. 3.21 compares the optimal step responses of the IMC controller and the PI controller with rate feedback. It clearly demonstrates that similar results were achieved by both configurations. Real-time Inielementation and Testinq Once the simulation results met the performance criteria, the Sirnulink model of the controller was compiled and downloaded into the DSP board. The excitation signal to the plant’s Piezo actuators was provided by one of the ICbit, 200kHz (max. rate) DAC channels of the DSP board. A 16-bit, 250kHz (max. rate) ADC channel of the same board acquired in real-time the output signal of the unsteady actuator’s cavity pressure sensor. The lowpass filtered cavity pressure signal, Pcm,rp/, was used as the primary indication of the jet deflection angle, Le., the plant output. Fig. 3.22 shows the diagram of the complete closed-loop system. Monitoring of the real-time closed-loop performance of the physical plant was then performed. The same work environment was used for on-line tuning of the

AIAA-2002-3073 controller parameters in order to achieve optimal performance. Concurrently the DSP board acquired the relevant time-history data that was saved on disk for post-processing and documentation. Selection of the IMC Controller Time-Constant The main advantage of the IMC controller over the PI controller with rate feedback is, that in order to achieve optimal closed-loop performance of the former, only one parameter, the time-constant h, should be “tuned”. Figs. 3.23a-c present the closed-loop step response of the cavity pressure, the HW velocity and the error between the pressure set-point and the controlled variable, Pcm,lp/, for three different values of A. Note the linearity of the responses, as the low-to-high and high-to-low jet deflection transitions are almost identical for every A. An additional important feature of the IMC controller is that even though there is a certain deadtime, only minimal inverse response was observed, This is partly due to the rate-limiting effect of the controller’s time-constant, and partly to the low-pass filter in the loop. The data presented in Fig. 3.23 is also an indication that, when operating within the linear range of the static flow response curve (Fig. 3.3, the designed IMC controller performs well. Based on these results and limiting the overshoot to lo%, 71=0.004 was selected. Fig 3 . 2 3 ~shows that the system settles to within 5% of its final value in about 20msec and that there is no steady state error. If a faster response deemed essential, a smaller h could have been used, with the penalty of a larger overshoot and a stronger inverse response. Closed-looD Plant Reseonse The closed-loop frequency response of the plant was measured by exciting the system with a swept sine (“chirp”) signal with frequencies between 1Hz and 9OHz. The measured and modeled Bode plots of the closed-loop system with IMC controller are shown in Figs. 3.24a and 3.24b. The bandwidth of the closed-loop system, as indicated by the -3dB point, is approximately 50Hz, not much different from the open-loop response. The amplitude ratio indicates a slightly underdamped system, commensurate with the 10% overshoot mentioned before. The phase lag at 50Hz is about 140”. The measured and modeled step responses of the closedloop system are shown in Fig. 3.25. The excellent agreement speaks for itself. The main benefit of the current implementation of the IMC controller and the selected h. are in reducing the inverse response and guarantying zero steady-state error, and not necessarily a wider bandwidth. The latter task could be achieved by a selection of a smaller h. that perhaps should be accompanied by a higher cutoff


frequency low-pass filter for the cavity pressure signal used as a sensor for the jet deflection angle.


Ooeration Outside the Design Envelone A linear controller hnctions less than optimally outside the linear range of the system. The IMC controller was designed to operate in the jet deflection range of 5" to 9", where the static sensitivity of the Pcm,b, to the jet

AIAA-2002-3073 in the "forced push" vs. "natural restore" responses, is clearly illustrated by considering the Pcav,,pf and HW velocity responses presented in Fig. 3.29. A detailed analysis of the time-constants of the highto-low transitions was conducted. It utilized the fact that the natural logarithm of the exponential decay of the jet deflection back to its natural state is linear. Fig. 3.30 presents the calculated time-constants for all the step changes measured, either returning to zero or to a different, but smaller jet deflection angle. It was found that the timeconstants decrease as the mean deflection angle increases (Le., the mid point of the step range), in agreement with the higher static sensitivity of Pcm,rp/ to

deflection angle is close to linear (Fig. 3.5). Nevertheless, operation at a wider range of jet deflection angles, outside the linear range, was also tested, in order to quantify the degradation in the controller's performance and to determine if a more sophisticated control methodology should be attempted. The results the jet deflection angle, follow. The time-constants multiplied by three, represent the Figs. 3.26a and 3.26b present the response of Pcav,lp/ time it takes a first order system to reach 95% of its final and HW measured flow velocity to a 2" step increase in state. This happens between 10 to 45msec after the the jet deflection angle. Figs. 3.27a and 3.27b present command for the highest and lowest mean jet deflection data for the same step size but for decreasing jet angles, respectively, corresponding to bandwidths of deflection angles. While operating inside the design SoHz and 1OHz. The nature of this type of response can space (i.e., 5" to go), the jet response is quick and be altered by using a "push-pull" controller utilizing smooth, as desired. It settles to the new state within two actuators, each situated on an opposite side of the jet 25-30msec. The HW velocity signal is characterized by centerline, instead of the "push-release" mode that was a larger overshoot than the Pc,,,l,/ signal. This result is used throughout this investigation. Figs. 3.3 la-d present features of the transient PCm,rp/ not general, however, as the HW position was not adjusted for this data set as it should have been (Le., be at the same mean velocity in the mid range of the deflections), and the normalization created some artifacts. As the mean deflection of the jet decreases, the response becomes more sluggish, in accordance with the lower sensitivity at this operating range. This trend could easily be seen from the pressure, as well as from the H W velocity signals, for increasing and decreasing jet deflection angles. Fig. 3.28a presents normalized Pcm,+fresponse to step changes from zero to gradually increasing jet deflection angles. The 0" (not deflected) jet angle is a problematic set-point, as the sensitivity at this point is very low. Indeed, when transitioning between 0" and 2", the response is sluggish and accompanied by a steadystate error. As the deflection range increases, the response is faster, and for the low-to-high transitions it is associated with an increasing overshoot. Note that the high-to-low transitions (steps terminating at zero deflection - the uncontrolled, natural state) are inherently different from the low-to-high transitions (Fig. 3.28b). Physically, this finding can be interpreted as follows. The jet has to be forced to change its natural, uncontrolled state. Currently, this is achieved by introducing the periodic excitation at one quarter of the jet circumference only (Figs. 2.la and 2.lb). Therefore, the only restoring force is the tendency of the jet to return to its natural, uncontrolled state. This difference

response to low-to-high transitions in the deflection angle command. Fig. 3.31a shows the time lag between the command and the time the Pca,,lp/ signal increases beyond 10% of the set-point. The data presented in Fig. 3.3 la shows that this lag is typically Smsec, close to the design point. It increases to about lOmsec for large step changes and low mean deflection angles. For 0" to 2" deflection it increases to 20msec. Fig. 3.31b presents the time interval it takes the response to increase from 10% to 90% of the set-point. The data shows that when the system operates close to the design point, it transitions between these two levels in about 5-7msec. This interval increases in a similar manner to the lag time for small mean deflections (2" and smaller). Fig. 3.3 IC shows the time it takes the plant to cross the set-point (measured from the command). The data indicates that for the majority of the tested cases (with the exception of those with mean deflections equal to or smaller than 2"), the system crosses the set-point within 12-18msec from the command. Whether these intervals are sufficiently short or not remains for the applicator to determine. The response overshoot (Fig. 3.31d) is lower than 15% of the set-point, except the large 0" to 8" step, reaching 20%. Overall, it could be stated that the degradation in the controller performance was not as severe as one could expect from a linear controller operating well outside its designed linear range. If these performances are sufficient or not, is still an open question. If there is an


2.

application that needs better off -design closed-loop performance, then robust control methods should be considered. Robustness was not an issue at the onset and a possible tradeoff may take place between controller complexity and robustness without having to sacrifice the performance at design conditions. Possible robust strategies include linear approaches (Ref. 17) and nonlinear approaches such as Artificial Neural Networks and Fuzzy Logic.

3. 4.

5.


4. Conclusions

Closed-loop control of jet vectoring due to the introduction of periodic excitation over a quarter of the circumference, at the junction between the exit from a circular jet and the entrance to a short, wide-angle diffuser was demonstrated experimentally. Detailed static and dynamic identification tests revealed that a single pressure sensor located in the actuator’s cavity can be used as the control authority indicator (“input”), while its low-pass filtered signal may serve as a jet deflection indicator (“output”). This is because the static pressure at the jet-diffiser junction is reduced as the flow turning angle increases, and vise versa It was demonstrated that the Piezo actuator currently used to vector the jet can reach steady state within a few milliseconds from the command, but this rate is too fast compared to the jet response time-constant and may therefore cause undesirable transient response. Thereafter, the system was modeled, including its typical inverse response. Closed-loop controllers were designed, simulated and implemented on the plant. It was found that Internal Model Controller (IMC) provided adequate, yet simple, tool for controlling smoothly and rapidly the jet deflection angle. Closedloop bandwidth was 50Hz as indicated by the -3dB frequency response point and the performance of the IMC controller over the entire available range of jet deflection angles was evaluated and shown to perform reasonably well, application dependent. Acknowledement The experiments were performed at the Steve and Mary Meadow aerodynamics laboratory. Partial finding by the Israel Science Foundation, the Fleischman family fund and the Meadow and lazar^^ knds are gratefilly acknowledged.

7.

8.

9.

10.

11.

12.

1999. 13. Amitay M., Smith D.R., Kibens, V.. Parekh, D.E. and Glezer, A., 2001. “Aerodynamic flow control over an

14. 15.

16.

References 1.

6.

17.

Davis. M.R., “Variable Control of Jet Decay”, AlAA Jozimal. Vol. 20, No. 5. 1982.

AIAA-2002-3073 Strykowski, P. J. and Witcoxin, P. J.. “Mixing enhancement due to global oscillations in jets with annular countefflow“. AIAA Journal, Vol. 31. No. 3. March 1993. Strykowski, P.J. and Krothapalli, A., “The Countercumt Mixing Layer: Strategies for ShearLayer Control”. AIAA paper 93-3260. 1993. Strykowski. P.J., Krothapalli, A.. and Forliti. D. J.. “Countefflow Thrust Vectoring of Supersonic Jets“. AIAA paper 96-01 15, 1996. Raman, G. and Cornelius, D.. ”Jet Mixing Control using Excitation from Miniature Oscillating Jet“, AIAA Journal, Vol. 33. No. 2, 1995. Smith, B. L., and Glezer, A., “Vectoring and SmallScale Motions Effected in Free Shear Flows Using Synthetic Jet Actuators”. AIAA paper 97-0213. 1997. Also: Smith, B.L. and Glezer, A., “The Formation and Evolution of Synthetic Jets”, Phys. of Fluids, Vol. IO, No. 9, September 1998, pp. 2281 -2297. Crow. S. C. and Champagne. F. H.. “Orderly Structure in jet turbulence”, Journal of Fluid Mechanics Vol. 48, part3, 1971, pp. 547-591. Zaman, K.B.M.Q. and Hussain. A.K.M.F. “Turbulence suppression in free shear flows by controlled excitation”, Journal of Fluid Mechanics, Vol. 103, 1981, pp. 133-159. Seifert, A., Bachar, T., Koss. D., Shepshelovich, M. and Wygnanski, I., “Oscillatory Blowing. a Tool to Delay Boundary Layer Separation”. AIR4 Journal. VOI.31, NO. 1 I , 1993, pp. 2052-2060. Seifert, A., Darabi, A. and Wygnanski, I.. ”On the delay of airfoil stall by periodic excitation“, Journal of Aircraft. Vol. 33, No. 4, 1996. pp. 69 1-699. Seifert, A. and Wygnanski. I., “Apparatus and Method for Controlling the Motion of a Solid Body or Fluid Stream”, Israeli Patent 1 16668. Submitted 1.1.1996. Granted 6.12.1998 (US Patent granted 2001). European patent pending. Pack, L.G. and Seifert. A., “Periodic Excitation for Jet Vectoring and Enhanced Spreading”, (AIAA paper 99 0672). accepted for publication in J. Aircrqfi. July

unconventional airfoil using synthetic jet actuators”. AIAA J. 39. pp. 361 -370. Darabi, A,. PhD thesis, TeI-Aviv University 2001. Franklin, G.F., Powel. J.D. and Emami-Naeini. A.. “Feedback Control of Dynamic Systems”, AddisonWesley, New York, 1994. Morari, M. and Zafiriou. E., “Robust Process Control“. Englewood Cliffs. N.J. Prentice Hall. 1989. Barbu, V. and Sritharan, S.S., “Hm-control theory of fluid dynamics”, Proc. R SOC.London Vol. 454. 1998. pp. 3009-3033.

11



3073-2002-AIAA

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Fig. 2.lc A view of the jet exit, diffuser excitation slot and instrumentation.

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Fig. 2.2 - Static actuator calibration showing the cavity

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Cavity P', Pa Fig. 3.4 Static response of actuator s cavity and d i f h e r mean pressures to actuator s cavity pressure fluctuations level.

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Fig. 3.1 Baseline and controlled velocity vertical profiles, @ x/D=l .

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Fig. 3.2 Jet deflection angle vs. excitation oscillatory momentum coefficient.

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Fig. 3.3 Jet deflection angle vs. actuator s cavity pressure fluctuation level also showing data repeatability 13 American Institute of Aeronautics and Astronautics

3073-2002-AIAA

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Time, ms

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Fig. 3.11 - Last 50 milliseconds of AM c h q signal (a), excitation signal to Piezo actuators (b), the resulting actuator s cavity pressure (Pcav) and its low-pass filtered part (Pcav lpf) (c) and HW measured velocity (d). 17


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.-6

3

06

z

-

r 04 P 5

m

2

ao -2.0 4.0

02

0 -0 2


I

4.0

12

-12.0

- 04

0.05

0.00

0.10

0.15

0.20

-14.0 I00

IO

1

Time, see

(a> 1.4

.:...\. .( -HW, L=O 003

h

1.2

i

1.0

-HW,

\

0.8

1 :

.. .. . ..

0.6 0.4

c

0.2 0.0

.i.

....

. .. . . . .. ... .

... . .

.

.. ..

.

.....

......

....... .. . ~~

.. .. .. .... ...... .

....

....

....... - . . .

, a + .

0 2

.. . ..

L=0004

HW, L=O 005

.

*

.c

.

.o 4 !

0 05

000

0 10

0 15

Fig. 3.24 Closed-loop measured plant Bode and its best fitted model. (a) amplitude ratio. (b) phase lag.

(b) 2

7

1

.I’

I

-Error, L=O003

i

04

-

Error, L=OOO3

14 12

-Error, L=O 003

\\ %-

00

i / I I

10

I

08

‘ iI

7

06 04

-1 2

..

i

02

0 05

-1;

-measured

00

4

0 00

100

Frequency, Hz

-

0 20

Time, sec

1

10

1

I

0 10

0 15

Time, sec

0 20

0.2 0 00

0 01

(c>

0 02

0 03

0 04

mme, sec

-

-

Fig. 3.25 Modeled and measured closed-loop step

Fig. 3.23 Selection criteria for the IMC controller time constant. i,. (a) Pcav.lpf. (b) HW velocilj. (c) Pcav.lpf error.

response.

21

American Institute of Aeronautics and A S ~ ~ O M U ~ ~ C S

3071-2002-AI AA '51

g

r

g

10

I

-*P

1.0

U

U

E

1

2

E

w

i

-n

-

c P

w

0.5

0.5

E 0

0

z

z

0.0

00


1

-05

-05

f

011

012

013

014

015

016

001

017

(a) 1

1

5

7

003

002

004

005

006

004

005

006

lime, SIX

lime, sec

I I

r;

I

!

-05 011

-05 012

013

014

015

016

'

017

OM

001

003

Time, tec

Time, sec

(b)

-

Fig. 3.26 - Closed-loop small jet deflection step change

Fig. 3.27 Closed-loop small jet deflection step change DOWN response, (a) Inverted Pcav,lpf, (b) HW velocity

UP response. (a) Inverted PcavJpf. (b) HW velocity @ s/D= 1

..

_-

--

-InputStep

-,r

P

1.o

0.5

1.0

3 n

-Input

J

I

0.o

-0.5

L

i-oto8deg

i

0.50

(a)

-0 to 4 deg -0to 6 deg

- 8to 0 deg

3.--

Step

- Oto2deg

0.5

(D

E

I1

5i

0.0

-0.5 I

0.55

0.60

0.65

0.70

0.75

- 6to 0 deg

0.80

Normalized Time [4.5*sec]

Fig. 3.28a - Closed-loopjet deflection step change, deflection increasing from zero UP.note that Pcav,lpf inverted.

J

0.00

(b)

, 0.05

0.10

0.15

0.20

0.25

0.30

Normalized Time [4.5*sec]

Fig. 3.28b - Closed-loop jet deflection step change. deflection decreasing to zero DOWN. note that Pcav,lpf inverted. 22

American Tnstitute of Aeronautics and Astronautics

3073-2002- AM 1.z

1.1

=-

1.o

0.9

5 0.8

! 0.7 n u

-E Q

0

2

0.6

0.5

0.4 0.3

0.2 Downloaded by UNIVERSITY OF CINCINNATI on December 7, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2002-3073

0.1

0.0 -0.1

c

4 0.1

0.0

0.2

0.4

0.3

0.5

0.8

0.7

0.8

4.3 1.0

0.9

Normalized Time lrpl Q 1/43 rec]

Fig. 3.29 A Comparison of HW and Pcav,lpf step response.

____________________- - - - -

o’Mi

-

RIse Ume of Low to High Jet deflection (from Stop to 1st

0.07 ._ . - - - - - - - -

0.06 _.

,

- - ssc!o.!~?g!

------------ ---- - - - --

---------------

---_---_---_-_--_--

0.05-.

t

a- 0.04 _. . E is 0,03-. 0.02.0.01

.-

.. -

0.00.

-------------------------------- ------ --- --- ---- -- - - -- - - - -- - - - - - - - - - -.

---- - -._ -

7

-

1

.

-

-

1

,

---------

-

7

-

7

r

3

a .

.

.

Closed-loop Vectoring Control of a Turbulent Jet

Closed-loop Vectoring Control of a Turbulent Jet

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