Application of Two Chaos Methods to Higher Harmonic Control Data

Application of Two Chaos Methods to Higher Harmonic Control Data Marti M. Sarigul-Klijn Ramcsh Kolar Comrnaiider Resenr-ch Professor U.S. Nnvy Naval Aviation Depot, Alanieda, CA E. Roberts Wood Profe..~sor Dept. of Aeronautics and Astronautics U. S. Naval Postgraduate School Monterey, CA This paper presents two new data analysis methods which will reduce H i ~ h e Harmonic r Control (HHC) night test reouirements. HHC is an active svstem which suuuresses helieouter vibrations. These methods.. orieinallv .. .. . used in the \tudy of C ~ D D S ,are the I'oinciarc srctiun iand the Van d ~ 1'r01 plane. Using flight test data iturn a \lrl)nnnrll D4,sgla%OII-f,,\ IIII(' lest 3irrr;afl. 1hr.s~rnrthn~lPsh,,wlhc libllnsing: - I ) the linlitv 111IIIIC rihrution reduction; 2) determination of HHC controller type; and 3) that the HHC controller transfer matrix is predictable and repeatahlc when defined in the "Rotor Time Domain.'' (This domain is a concept introduced in this uauer. These benefits do not require an HHC equiu~edhelieader, althouda they do require an instrumentation Introduction ost helicopters vibrate. In the search to reduce these vibraas Hi*er Harmonic 'OnMtiOns an active 'yStem trul (HHC) has been tcsted. There have been numerous theoretical analvses. seven wind tunnel investiaations. and the four flight tests of HHC to date. Common t o both the wind tunnel and flight test programs has been the extensive amount of testing required to make HHC work. This paper presents two data analysis methods which will reduce HHC testrequirements. These methods, the Poincare section and the Van der Pol plane, are adapted from methods used to study chaotic systems and are describcd in the first section of this paper. ~h~ second section briefly reviews HHC. In the third the chaos methods are applied to flight test data compared with ~~~~i~~frequency domain methods. The Methods of Chaos 1, I From the phase plane method developed in 1904 by ~~~~i Poincare to the pseudo phase space method proposed in 1980 by Floris Takens (Ref. 1) the study of chaotic dynamic systems has inspired new mcthods of analysis. The concept of chaos and geometric methods have revolutionized ,he investigation of problems in nonlinear dynamics. T o understand the two methods used here, some review of the early methods is necessary. T o facilitate that review, flight test data is used. The data is from an accelerometer, which senses vertical direction accelerations under the right pilot's seat of a McDonnell Douglas OH-6A helicopter. This OH-6A was used in the first successful flight test of HHC. The conditions of flight are level flight at 60 kts with the HHC system turned off. ~h~ signal was digitized at I230 Hertz for 5 secs, giving a total o f 6 1 5 0 data points. Figure I presents a portion of the time history of this signal. The Dimensional Plane The first step in understanding either the Poincare section or the Van der Pol plane is to examine the two-dimensional ( 2 ~ ) phase plane. In the classical 2D phase plane. amplitude is plotted on the horizontal axis and the velocity is plotted on the vertical axis. The plot forms a trajectory of a moving point, which represents the history of the system. However, in a typical flight test, only strain amplitudes or accelerations are recorded. Depending on the signal, to get amplitude and velocity requires either interration o r differentiation. Both of these Drocesses have the ekect of filtering the signal. Differentiation'will amplify the high frequency noise and attenuate the low frequency signal. Integration amplifies the low frequency components while attenuating the high frequency components. T o alleviate problem, (Ref. ') and (Ref. advanced a method for constructing phase plane diagrams using fake observables generated from a single experimental measurea single measured me"t, fx (ti), x ff2J , . . .I APRIL 1993 CHAOS METHODS APPLICATION T O HHC 69 TIME (SEC) Fig. 1 Portion of the time history of OH-6A right pilot seat vertical direction accelerometer data. a new array of numbers is constructed as follows: (X (ti). X (ti - E ) , . . .) 01 (X (ti), x (ti + E)) where: x ( t ) is the measured variable E is the embedding time r is the sampling time for each time ti. This method of forming fake observahles and plotting is referred to as pseudo phase plane method and the two-dimensional space in which it is drawn is called the pseudo phase space. T o illustrate this, consider the following example: Fig. 2 Two-Dimensional pseudo phase plane with Fig. I data plottcd along horizontal axis and fake observable plotted along vertical axis. amplitude, velocity, acceleration, etc. may be plotted in the pseudo phase plane to study the behavior of a given nonlinear dynamic system. Figure 2 presents the accelerometer data in the 2 0 pseudo phase plane using an embedding time of 1 0 samples. The data forms a tangle of trajectories that appears inadequate to yield any relevant information. Figure 3 presents the Fourier power spectrum for the accelerometer data. Notice that the 32.36 Hertz (4lrev) component is much stronger than the other components and hence is the predominant vibration. An embedding time of 10 samples is used throl~ghoutthis paper. This is based on a data sampling rate of 1230 Hertz, and a quarter of the predominant vibration period (11128 seconds). ... Series 1: 1.43,1.40,1.35,1.32,1.28,1.31, Series 2: 1.40,1.35,1.32, 1.28, 1.31, . . . Let Series 1 he the digitized values of a signal. Then Series 2 is the fake observable formed by displacing the time series by one sample. This displacement is the embedding time. To form a trajectory, plot successive columns of pairs with the points in Series I as the x-coordinate and the corresponding points in Series 2 as the y-coordinate. In this case, the first pair is (1.43, 1.40) and the second pair is (1.40, 1.35). An important parameter, then, is the embedding time. It depends to a large extent on the system dynamics, and several embedding times should be compared before a final choice is made. A good choice is about one-quarter of the period of the most predominant frequency of the observable. For example, for a simple sinusoidal, an embedding time equal to a quarter of its period yields a function proportional to its derivative (note that in a classical phase plane, displacement is plotted against its derivative). The nrincioal . advantage of Takens' method is that a single observable is adequate to construct the trajectories that can capture the svstem dvnamics. Also the method eliminates the need or of the observed signal. Furtherfor d i f f e r e n t i a t i ~ ~integration more, Takens shows that any representative quantity such as . - lorlot 0 ' ' 30 60 90 FREQUENCY (HZ) Fig. 3 Fourier power spectrum for OH-6A right pilot seat vertical direction accelerometer data. M.M. SARlGUL-KLIJN 70 F(T JOURNAL O F THE AMERICAN HELICOPTER SOCIETY portion o i a trajectory from a single frequency oscillation. The observable, the fake observable, and time are plotted on the three axes. The trajectory wraps around the time axis with each wrap equal to the period of the oscillation. In Fig. 4(b), the time axis is bent to form a torus. Speeding up the rate of plotting, i.e., adjusting the time taken for one cycle around the torus, results in R g . 4(c). Onc trip now takes exactly one period. This rate of plotting results in the trajectories retracing themselves. The trajectories revisit the exact same space, referrcd to here as an "attractor." This def~nitionis not to be confused with the formal mathematical definition of an attractor. Notice the plott~ngconvention with the observable and fake observable in the cross \ccllo~l;~l planc 111 lhc torus 'IIICI lime ax15 ilrollntl thc lurus. 'I ihc Iahci~nx111 thc azilll~~thal p11~11ion ; I ~ O I I Iihr: tc~ru\is I I I I I I ~ I I U the labeling of helicopter iotor blade azimuthal position. Figure 5 depicts the attractor of the pilot seat acceleration data in toroidal phase space. A new parameter is the rate at which the trajectory is plotted around the torus. Traditionally, the plotting rate is constant. A new concept, introduced hcrc, is to synchronize the plotting rate with that of the forcing function, in this case thc helicopler rotor. The resulting trajectories are in the "Rotor Time Domain." Plotting the trajectories at a constant rate results in the "Clock Time Domain." The 32.36 Hertz used in Fig. 5(a) is the average 4Irev obtained by Fourier analysis performed over several samples taken from the time history. - E) TlME F(T1 Fig. 4 Construction of toroidal phase space with a simple sinusoidal plotted along the F(T) axis and fake observahlc plotled along F(T-E) axis. The trajfetory wraps around an apparent cylinder. Part (a) plots only s portion of the data. In (h), toroidal phase space is formed by connfeting ends of the cylinder. In (c), speeding up the rate of platling gives an attractor. Thc trajectories retrace themselves, over and over. Toraidal Phase Space Now consider toroidal phase spacc (Refs. 3-5). As the first step of construction, consider Fig. 4(a), which shows a small Poincare Seclion A plane is shown in Fig. 5(b). This 2D surface intersecting trajectories orthogonally is an example of a Poincare Section (PS). A PS corresponds to strobing the data at a specific azimuth angle and plotting the points where the trajectory pierces this section. The PS takes a slice through the torus, revealing the internal structure at this location. For a given torus, an infinite number of azimuth angles are available. The azimuth angle at which to take the PS is a parameter that needs to be selected. Figure 6 presents the PS of Fig. 5 at 270 deg azimuth. This signal produces intersections that are bounded to a small area when plottcd in the Rotor Time Domain. The intersections fill the entire plane when plotted in the Clock Time Domain. Figure Fig. 5 Taroidal phase space with data from Fig. 1. In (a) thc rate of plotting is fixed at 32.36 Herlr, known as plotting in clack time domain. In (b) the rate of platting is at Urev, as the rotor's rotational rate varies the plotting rstc varies. Plotting rate is four times around the torus for each ralor revolution. This is rcferred to as plotting in rotor time domain. CHAOS METHODS APPLICATION T O HHC 71 (b) Fig. 6 Poinrare sections of Fig. 5 with sections at 270" azimuth. Numbers indicate when the trajectories intersect with the Poincarc plane. Zeros indicate intersection in the first half second, ones the next half second, and nines indicate the last half secnnd. 4(c) shows that a pure sinusoidal oscillation yields only one point on the PS. Although not shown, a signal composed of noise fills the entire PS plane regardless of which time domain it is plotted in. Pol (VDP) plane captures the internal structure in a single plane. Figure 7 presents the VDP construction (Ref. 5). This process is cquivalcnt to untwisting the trajectories on the 2D pseudo phase plane at a prescribed rate. Mathematically, the VDP plane is computed by the iollowing transformation: Van der Pal Plane Several PS taken at different azimuth angles of the torus reveal the internal structure of an attractor in toroidal phase space. Rather than constructing an infinite number of PS, the Van der U = X cos (wf)- Y sir1 (wf) V = X sin (wt) + Y cos (wf) Fig. 7 (:onslructic,n ,,1'thc Van der 1'01 plane. Te,ruicl:al phusc <pare allrerl~,ris firrl ~,l~,llcd 2 s in (a). In (I,), I'oinrnrc wcli#miare liahun ul IXII', 2711",and U" iwi~nulh.In (L), I'oinurv s e t ' l i ~ nUIC~ n,lulcd: IXIYurim~nlh*ertit,n is nut nl1;aInl. 2711 :rximulh vurlia,n is rulalrd YO, the l l ' i l ~ i n ~ ~ a ~ l h section is rntstcd 180'. In (d), sections at all azimuth angles are combined to form Van dcr Pol plane. 72 M.M. SARIGUL-KLIJN where X and Y are the coordinates of a trajectory in the 2D pseudo phase plane, U and V are the coordinates of a trajectory in the VDP plane and w is the rate of untwisting. The rate o r untwisting, w, is similar in nature to the plotting rate around the torus. The untwisting rate may cither be at a constant rate or at a rate synchronized with that of the rotor. The rate of untwisting in Fig. 7 is in the Rotor Time Domain. Unlike the PS, the VDP plane method is effective only for signals with a single predominant frequency. In the present case, Fig. 3 shows that the principal excitation (32.36 Hertz) is at a single frequency. Our investigations show that the VDP plane is an effective analysis method for most helicopter signals. HHC This section reviews the McDonnell Douglas OH-6A test program, the HHC control law, and HHC controller types. In forward flight the primary excitation to the fuselage occurs at the blade passage frequency, which is at 4Irev (about 32.36 Hertz) for the four bladed OH-6A rotor. Excitation exists also at higher harmonics or 4Irev. that is at Slrev, l2/rcv, etc. Figure 3 shows that the major source (about 60 percent of the total energy) of OH-6A pilot seat vibration is essentially at a single frequency (32.36 Hertz). An active system can suppress vibrations at a single frequency. The McDonnell DnugIadNASNArmy HHC Test Pmgram HHC was first successfully flight tested in a joint McDonnell DouclaslNASAIArmv test oroaram during 1982 to 1984. One C was to minimize the of the main objectives of thd H ~ program 4Irev vibrations under the pilot seat. HHC superimposes a 4lrev oscillation through a set of three electro-hydraulic actuators attached to the helicopter stationary swashplate. Tilting the swashplate results in feathering of the rotor bladcs. In open loop testing, the phase and amplitude of swashplate excitation is set manually. In closed loop operation, an onboard digital computer controls the actuators. The computer determines the required swashplate tilting to reduce the vibration simultaneously in the longitudinal, lateral, and vertical directions as scnsed under the pilot's seat by three acceleronietel-s. In either case, the swash- JOURNAL OF THE AMERICAN HELICOPTER SOCIETY plate excitation generates new incremental rotor blade airloads to eliminate the 4Irev vibration. Figure 8 shows a sketch or the HHC systcm installed in the OH-6A. Figure 9 presents a typical test result for lateral only 4Irev excitation of thc swashplate. In this open loop test, the swashplate was oscillated laterally at a fixed amplitude of 39.33 deg with the helicopter a1 60 kts assigned forward indicated airspeed. The figure depicts the 4Irev vcrtical and lateral direction accelerations in g's measured by accclero~netersmounted under the pilot's seat. The controller input phase was varied in increments of 30 deg over a total range of 0 to 360 deg. The input controller phase refers to the phase of the swashplate tilting in relation to the main rotor azimuth position. Notice that 360 deg of input phase to the controller corresponds only to 90 deg of rotation of the rotor since the input HHC excitation is at 4lrev. Wood el, nl., (Ref. 6) and Straub and Byrns (Ref. 7) summarize othcr results from this flight test. Airspccds tested included, in addition to hover, 4 0 to I00 kts in 10 kt increments. For all airspccds, maximum vibration occurs at about 90 dcg controller phase and minimum vibration at about 300 deg manual controller phase for lateral only HHC. Vibration reduction also occurred for longitudinal only and collective only 41rev excitation or the swashplate. These flight test results were similar to the lateral only excitation. Optimal vihration reduction occurred with simultaneous oscillation of the swashplate in the lateral, longitudinal, and collective directions. Finally, they (Refs. 6 and 7) conclude that the HHC system did not increase vibrations at other frequencies, adversely affect blade loads or adversely afrect helicopter performance. HHC Control Lam This section briefly describes the control law used in HHC. To describc a vibration at a single frequency requires two quantities, amplitude, and phase. However the amplitudes or the sine and cosine parts of a vibration are simpler to use in numerical computations. Hence, six quantities-a sine and cosine part each directions-are cnough for the vertical, lateral, and lo~~gitudinal to describe helicopter vibrations at a single location at a single frequency. Accordingly, the rollowing control law applies lor HHC: 0.40 AIRSPEED - 60 KIAS LATERAL COllTROL INPUT iO.3J0 I / \VERTICAL - nnc ow I a)FEEOlllCK ACCELtROMtlERS PILOT SEAll m R l E M E T R I OAlA PACUOE @HIGH FREOUEYCI HHC ACTUATORS 131 @MICROCOMPUTER I - LAIEML nnc or\\/ M l Y U A L COYTROLLER PHASE ~ i g 8. Prinlnry elements of the HHC system in the OH-6A (Ref. 6). - DEGREES Fig. 9 Variation of pilot seat acceleration with HHC cnntroller input phase. Urev vertical and lateral dircetion sceelcromctcrs data from the right pilot seat. Obtained by Fourier analysis using a HP 5423 ~pectralanalyzer. The baseline vertirirl direction test point is the same as in Fig. 1 (Ref. 6). CHAOS METHODS APPLICATION T O HHC APRIL 1993 z = zo + Tq where, for the four bladed OH-6A z is a 6 x I vector of measured 4Ircv (32 Hertz) vibrations (g'sj and zo is a 6 x 1 vector of baseline 4lrev (32 Hertz) vibrations (g's). The I and I~. , vectors consist of the sine and cosine comoonents of lateral, longitudinal, and vertical direction vibrations, for a total of six elements. These vectors reoresent vibrations at a specific location on the helicopter (underneath thc pilot's seat) at a single frequency (4lrev). T is a 6 x 6 control resoonse matrix that relates the swasholate k movement to the vibration response of the helicopter. ~ h 36 elements of the T matrix relate swasholate excitation to the rcsulting helicopter vibration. Each element in thc T matrix is in units of the z vector divided by units of the 0 vector. In the case of the OH-6A, the units for the T matrix were g'slin. 0 is a 6 x I vector of swashplate Wrev (32 Hertz) movement. It consists of the sine and cosine components of lateral, longitudinal, and collective swashplate excitation (in). This control law assumes a linear transfer relationship between command 4Irev swasholate motion and 4Irev fuselaae vibrations. The equalion states' that the helicopter 4lrev rcsponse (vector z) consists of a baseline response (vector zoj plus a response related to the 4Irev swashplate inputs (vector 0) by a The transfer matrix T and the baseline vitransfer matrix brations z. depend on night conditions such as forward speed. A current issue is whether the transfer matrix, T, changcs predictably with flight conditions and whether it is repeatable at the same flight condition. T is repeatable if the values for all 36 elements of the matrix rcmain the same each time they are measured under similar night conditions. It is non-repeatable if the 36 elements of the matrix change significantly each timc an estimate is made for the matrix, cven though test flight conditions, such as airspeed, do not change. If T i s repeatable, it may be predictable. T is predictable if small changes in test conditions result in oredictable changes in the values of its elements. For example, T is predictable, the values of Tmeasured at 70 kts are between the values of T measurcd at 6 0 and 80 kts. (n. ii HHC Controller p p e s ~h~ characteristics of the T matrix determine the controller type needed for a HHC system. If the Tmatrix is repeatable and does not change with flight conditions, then a fixed gain control system is adequate. A fixed gain control system uses only one set - - of -~- values ~- for ~~- the elemcnts of the Tmatrix for all flight conditions. This system represents an open loop system since there is no ~~~- direct feedhack -~~~~~~~ of a measured resoonse. Identification of the T matrix is done off line since the characteristics of T and zo are assumed invariant. This controller type is the simplest and cheapest. If the T matrix is repeatable but changes predictably with measured flight conditions, such as airspeed, then a scheduled gain control system is adequate. A scheduled gain control systcm uses oredetermined libraries of T matrices. with selection I his \y\tcn~i \ cl;r\ch;fing~nga ~ l hmu;irurcd f l ~ g hrc'ndilio~ls. ~ 1 I c I I i l c ~ i l r ~ t Ii n I I 1 1 r h e c done off line. The rcquircment for sensors to measure the input variables such as airspeed, sideslip, etc. make this system more complcx than the fixed gain system. A slightly more complex system would be a fuzzy logic controller. Like the scheduled gain controller, it too would rcquire that thc T matrix be repeatable and change predictably with measured flight conditions. The fuzzy logic controller also uses a predetermined library of T matrices. Unlike a scheduled gain controller that switches abruptly from one matrix to another as ~~~~ ~~~~~~ ~~~~ ~~~~~~~ ~ ~ ~ 73 flight conditions change, the fuzzy logic controller allows a gradual transition from one Tmatrix to another. Its operation is much smoother than a scheduled gain controller. If the T matrix is either non-repeatable or non-predictable, then an adaptive gain control system is required. This systcm is closed loop and uses on line identification of the T matrix. Online identification continuously updates the characteristics of thc matrix with time. The update time is normally in the order of once every rotor revolution. Methods of updating the matrices include Kalman estimators and Least Mean Square adaptive inverse control. Finally, to limit the rate at which the control law changcs, many adaptive controllers add caution terms. The resulting control systcm is a complicated system and sometimes is unstable where it increases vibrations instead of decreasing them. Finally, even the best of the adaptive systems cannot follow rapid flight maneuvers such as 180 deg pedal turn from forward to backward flight at a slow airspeed. During such a maneuver, an adaptive gain HHC may increase vibrations for a short time instead of reducing them, due to its relatively slow update time. Results This section presents the two chaos methods applied to HHC flight test data. Three main results are achieved and are described later in this section. Finally, a comparison of these methods is made with frequency domain methods. These results are based on a qualitative relationship that exists between the z vectors in the HHC control law and trajectories drawn in toroidal phase space. Consider a trajectory of a vibration drawn in toroidal ~ h a s esoace with the ~ l o t t i-n rrate at the frequency of the z vector. The single attractor represents two of the six elements of the z vectors. In the PS or VDP lane, the horizontal location of the attractor, with appropriatc scaling, is the cosine part of the vibration. The vertical location of the I \ the une p;irl o i llle :~ttrnaor.:i#aln u ~ t h.tppr~pr~;llc \c;~l~ng. v~hrali<~n. kurlhcr. ~ n ofc lhc p r ~ n i ~ ;p~~d l\ ~ a t i l ; of ~ ~]he c s I'S ;ind VDP plane presentations is'that they displa; amplitude and phase simultaneously. The distancc from the origin givcs the ampl~tudewhile the clock position about the origin indicates the response phase. Thus three PS or VDP planes, one each for lateral, vertical, and longitudinal direction vibration respectively, describe thc z vectors fully. With HHC off, a PS or VDP plane of fl~ghttest data represents the z, vector, or the baseline response. With Ihe new HHC On, a PS Or VDP plane Iepresents the vector, 'yStem Finally, in the HHC control law, the nature of the T matrix Can be deduced from the nature of z and z, vectors and their response to the swashplate excitation, 0. This is because the HHC control law assumes a linear static transfer rclationshi~ (matrix T ) between thc commanded swashplate movement (vcdlor 0) and the fuselage vibrations (vectors z). We now consider Fig:10. It presents VDPs of Fig. 9 (vertical direction accelero~neterdata) for various manual HHC controller phases. Accelerations in the lateral and longitudinal directions are not presented here to save space. The PS method yields similar results but is not oresented here for the same reason. The cili,ct of !Ill(: I, 1,) IIIOVC lhc :,llr~~ctur. A, the I I I ~ I I I U ; Ic<.~~trc~llcr ~ phaw c h i l n s ~ \lhc , atlr.~ctur, h ~ i t s11s ~,osltit,nin the planc. /\I*$,. ihe attractor is bounded to a small arda of the phasespace when plotted in the rotor time domain. Although predictability of the trajectories within the attractor is not possible, the overall attractor location is fixed for each controller phase. Minimum 4lrev vibration occurs at 300 deg manual controller phase. This attractor is closest to the origin. The mean distance of the center of the attractor from the or~ginof the plot is an indication of the magnitude of vibration. A smaller distance means smaller vibration levels. Maximum 41rcv vibration occurs 74 JOURNAL OF THE AMERICAN HELICOPTER SOCIETY M.M. SARIGUL-KLIJN ( BASELINE I O Fig. 10 Van der Pol Plancs of vertical direction accelerometer data taken during level night with f 0.33' of lateral 4lrev swashplate excitation at 60 kts assigned airspeed. The manual controller phase is indicated on each figure. Rate of untwisting, o,is 4lrev. at 90 deg manual controller phase. The attractor center is at the greatest distance from the origin, indicating greater vibrations. The effect of HHC is to move the attractor lrom the baseline position. The attractor moves predictably with changes in manual controller phase in both the PS and the VDP plane. It may be noticed that the attractors at some manual controller phases arc slightly larger than the others. Figure I1 presents the time histories o l the vertical direction accelerometer signal during baseline testing and HHC on testing at 300 deg manual controller phase. This is the same data as in Fig. 10. The PS and VDP methods are very sensitive to small changes in vibrations at other than the plotting rate, in this case Wrev. Notice for the 300 deg controller lest point, the mean amplitude of acceleration varies up and down in the time history. Integrations of this accelerometer data showed that the helicopter changed its base altilude by 67 ft in five sec. Also, the mean amplitude was 1.087 g's as compared to 0.9997 g's during baseline testing. This low frequency variation in mean acceleration amplitude will enlarge the attractor in either the PS or in the VDP plane. The cause of thc low frequency variation may be due lo factors such as air turbulence, pilot longitudinal control input or instrument error. ! Rotor Time Domain versus Clock Time Domain As discussed earlier, helicopter vibrations are non-repeatable when defined in the clock time domain. As presented in the toroidal phase space section, Fig. 5 illustrates the two methods of plotting about the toms, the clock and rotor time domains, using the pilot seat vertical direction acceleration signal. In the specific context of time domain, it is expedient to revisit Fig. 6 , which is the PS o l F i g . 5. The method of presentation shown in Fig. 6 is believed to he new. The number 0 marks the first half second duration of trajectories which intersect the PS. The number I marks the next half second duration o i intersections, the number 2 the next hall second duration, and so on to the number 9. Plotting in the clock time domain, part (a), causes the trajectories to first intersect the PS near the positivc x axis. They then move clockwise past the negative y axis, the negative x axis and finally end near the positive y axis. The phase, indicated by the clock position ahout the origin, does not remain constant. Plotting in the rotor time domain, part (b), causes all of the trajectories to intersect the PS near the positive x axis and the phase remains relatively constant. The pilot seat lateral and longitudinal direction accelerations, although not presented here, look similar to Figs. 5 and 6. L 0 1 2 3 4 5 TIME (SEC) TIME (SEC) Fig. 11 Time histories of baselinc and 300° manual controller phase acceleration data. (a) Bascline or HHC off, and (h) HHC on with manual eonlraller phase set at 300". APRIL 1991 CHAOS METHODS APPLICATION TO HHC It is important to emphasize that the variation in rotor rotational rate accounts for the difference in the clock versus the rotor time domain presentations. It is taken up in Fig. 12, which oresents the main rotor rum for the flight - condition considered here. About a one percent rpm change occurred in one sec. Although not seen in this figure, the rotational rate of the main rotor varied by up to two percent. This amount of variation is typical for helicopter rotors. Figures 5 and 6 highlight the importance of defining the HHC control law in the rotor time domain. In contrast, defining the z vectors in the clock time domain results in a vector which is non-repeatable even at one flight condition and one swashplate c~cit.ition,0. 'I'hv 7' tn~;i~r~x is. Inen, non-repc.i~shlu.llcncc, lhr clock llmc <inn~.iin require.; thc usc of an ;i~l.iplivcgniu rulllroller to chase the constantly changing phase. The rotor time domain allows use of fixed, scheduled or fuzzy logic controllers since phase remains relatively constant. From a practical standpoint, Figs. 5 and 6 show the importance of constant rotor rpm. The HHC actuators should move in synchronization with the main rotor. The HHC computer must have accurate information on main rotor azimuth position through a reliable pipper system. ~ i m /oft HHC Performance The PS and VDP methods clearly show the limit of HHC improvement. The PS and VDP methods provide a visual indication of the vibratory energy in a signal. In both the PS and VDP methods, the area of a circle ccnrewd at the origin of the phase plane and encompassing the entire attractor is a measure of the vibratory energy. The greater the area of the circle, the greater the energy. This measure of energy is the same as the area under a Fourier Power Spectral Density (PSD) plot in the frequency domain (Parseval Theorem, Ref. 8). Also, as in the PSD, units of energy are the measured variable, hut squared. Consider Fig. 13, which is the VDP representation of vertical direction vibration under the right pilot's seat for a flight condition of 100 kts. This closed loop test point was one of the best reductions achieved with HHC and required computer controlled cyclic and collective excitation of the swashplate. In this figure, rate of untwisting, w, is 41rev. With HHC on, the centcr of the attractor is near the origin of the phase plane. Notice that the placement of the attractor is the limit of HHC vibration re- 2 U 1 2 3 4 5 TIME (SEC) Fig. 12 Main rotor rpm for the OH-6A. Same tcsl point as for ICigs. 5 and 6. 75 duction. Once the center of the atlractor is at the origin, then HHC can reduce vibrations no further. Longitudinal and lateral direction vibration data showed similar results. The PS method also shows that the placement of the attractor is the limit of HHC vibration reduction. Thus. to determine best HHC performance requires only one sampleof baseline (HHC off) vibration data at &ch test &ndition. Simply generate a PS or VDP with this data and move the resulting attractor to the origin. This approach does not require a HHC system installed in the helicopter. Determination of Controller 5 p e from Flight Test The PS and VDP methods allow rapid determination of HHC controller type from a few minutes duration of flight test data. The only measured flight condition, which was changed in the OH-6A flight test was airspeed. Figure 14 presents the PS representation:? of pilot seat Certical and lateral acceleration data for different airspeeds. The rate of plotting around t l ~ cturoidal phase space is Qrev and the s e c t i h s are at I80 deg azimuth. The flight conditions are level flight with HHC off. The VDP method produces similar results and is not presented here to save space. Observe that as airspeed changes from 60 kts (left column) to 100 kts (right column), the amplitude and phase of the response changes smoothly and predictably. Based on these observations, it may he concluded that a scheduled gain controller with only two or three transfer matrices T can accommodate the changes in the response shown in Fig. 14. A fuzzy logic controller would allow gradual transition as airspeed changes. Note that these results apply since thc rotor time domain is used. The analysis reported in this paper may also be performed in real time and requires only a few minutes of maneuvering flight data. The controller type is determined by observing the movement of the atlractor of a measured variable, such as vertical, lateral, and longitudinal direction vibration. This result requires an instrumentation system but does not need a HHC system installed in thc aircraft. Comparison to Fouricr Methods The VDP and PS methods are compared here with Fouricr frequency domain methods. In all the methods, though the same information exists, the presentation is different. Each gives a different insight. As shown in Fig. 3, one of the major advantages of Fourier methods is that they give accurate amplitude (in this case, amplitude squared) versus frequency information. However, Fourier analysis methods have important restrictions on their practical application to helicopter data. One restriction is that frequency resolution is the inverse of data acquisition time. For example, a Fourier transform of five secs of data give a minimum difference between adjacent frequencies of 115 Hertz. Taking longer periods of data increases the frequency resolution, thus making the difference between adjacent frequencies smaller. In any case to get adequate frequency resolution in helicopter flight test, the data acquisition time must encompass several rotor revolutions. Since the rotor speed varies slightly with time, the phase information is an averagc, which has no physical meaning. In a sense, the Fourier tl-ansform works in the clock time domain. The only solution is to calculate phase relative to the rotor azimuthal position and to limit the data record length to one rotor revolution in time. Unfortunately, frequency resolution suffers. For example, the 41rev (32 Hertz) amplitude and phase information from one rotor revolution of data corresponds to an average of all the amplitude and phase information between 28 Hcrtz and 36 Hertz. Thus these restrictions prevent accurate determination of phase using Fouricr methods. Unlike Fourier analysis, a PS or VDP plane shows in one figurc both amplitude and phase of thc single vibration in which 76 M.M. SARlGULKLIJN JOURNAL O F THE AMERICAN HELICOPTER SOCIETY (b) Fig. 13 Limit of HHC performances by comparison of Van der Pol planes. Using computer controlled cyclic and collective excitation of the swashplate, this pilot seat vertical direction acceleration data taken at 100 kts assigned airspeed was one of the best reductions achieved with HHC. Lcft figure is baseline response and right figure is closed loop response. we are most interested. They also use short data records as compared to Fourier analysis. Furthermore, they can monitor the performance of a HHC system in real time, since they need only one observable and a time delayed fake observable. Conclusion The geometric methods of chaos were applied, for the first time, to study flight test data. The data used here were from the McDonnell Douglas OH-6A HHC test helicopter. New engineering applications of the chaos methods were demonstrated with the HHC flight test data. A technique based on a relationship between the chaos methods (the Poincare section and Van der Pol plane) and vibration amolitude and ohase is oresented. Three main results are reported: 1) the limits of HHC vibration reduction; 2) determination of HHC controller type from a few minutes of flight test data (for the OH-6A, a scheduled gain or fuzzy logic controller is adequate for steady level flight); and 3) that the HHC control law transfer matrix, T, is repeatable and predictable in the Rotor Time Domain hut is non-reoeatable in the Clock Time Domain. As a result, these techniques will reduce HHC flight test requirements. Further, although instrumentation is required, a HHC system is not required. These approaches also have poten- tial applications to other vibration control and flight testing problems. Not included in this paper are the application of other chaos methods to HHC data. Among these methods are higher dimension phase spaces, Lyapunov exponents, and Fractal Correlation Dimension. For a description of them, their application to 13 different helicopter signals, and a detailed review of previous HHC work, see Sarigul-Klijn (Ref. 9). Acknowledgments The authors would like to thank the Naval Air Systems Command and the NPS Foundation for funding a portion of this project. Additional funding was provided by McDonnell Douglas Helicopter Company (MDHC). From MDHC, we thank the assistance of Dr. Friedrich Straub, who provided the HHC data, Mr. Roger Gould, who converted the data to VAX format, Mr. Don Duel, who provided insight into classical time and frequency domain analysis, and Mr. Scott Kehl, who provided information on MDHC instrumentation. We acknowledge the assistance of Professor Nesrin Sarigul-Klijn of the University of California, Davis, who provided valuable suggestions. The authors also appreciate the constructive criticism of the reviewers. APRIL 1993 CHAOS METHODS APPLICATION TO H H C 77 Fig. 14 Determination of HHC controller types by Poincare sections of pilot seat accelerations. T a p row is vertical acceleration and bottom raw is lateral acceleration. The left column is 60 kts, the middle is SO kts, the right is 100 kts assigned airspeed. References ' ~ a k e n s ,F., "Detecting Strange Attractors in Turbulence." in k c n i x Notes in Mnrltenmticr, Springer Verlag, Voi. 898, 1981, pp. 366-381. 'packard, N. H., Crutchfieid, J. P., Farmer, J. D., and Shaw, R. S.. "Geometry from a Time Series," Physical Review Letters, Voi. 45, (9). Se 1980. 'Abraham, R. H. and Shaw. C. D.. Dyna~nics:the Geomerv ofBebnvior, Aerial Press, Santa Cruz, CA, 1983, Chapters 4 and 5. 4 ~ a o nF. , C., Clzaotic Vibmtions. John Wiley and Sons, New York, NY. 1987, Chapter 2. 5~hompson,J. M. T. and Stewart, H.B.. 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