An Experiment Control

ABSTRACT: In this paper, an experiment called the “jitter beam” simulates the inter- action of a pointing control system and a flexible structure. Noncolocation of a sensor and an actuator makes control difficult. A linear-quadratic-Gaussian design overcomes the noncolocation problem by use of a dy- namic model that includes structural bend- ing. The capability to deal with noncoloca- tion enables the jitter-beam control to coordinate, with a single torquer, the mo- tions of several points on the structure to achieve the pointing goal. The significant achievement of the experiment is the prac- tical demonstration of a control bandwidth two times higher than a critical bending fre- quency, a factor of 10 beyond what a rigid- body design can achieve.

Introduction

Interaction between control forces and structural vibrations can degrade perfor- mance for lightweight space structures. Over the last 10 years, there has been extensive progress in developing effective controls for large space structures (LSS). Early efforts in the Defense Advanced Research Projects AgencylActive Control of Space Structures programs involved damping of transient ex- citations, whereas more recent work has been devoted to suppression of vibrations from steady disturbances [I]-[3]. For additional references to early experimental work in control of flexible structures, see Ozgiiner et al. [4].

In recent years, the problem of LSS con- trol of flexible structures has broadened to include other situations where the control bandwidth and the modal frequency spec- trum overlap. New examples of controll structural interaction arise as demands for control performance outstrip capabilities for stiffer structures. In particular, interaction between the control system and structural dynamics is a major issue in active optical systems. Here, the object is to control mir-

Presented at the 1988 American Control Confer- ence, Atlanta, Georgia, June 15-17, 1988. Eric K. Parsons is with the Guidance and Control De- partment, Lockheed Palo Alto Research Labora- tory, 3251 Hanover Street, Palo Alto, CA 94304.

April 1989

De m onst ra t i n g Poi n t i ng a Flexible Structure Eric K. Parsons

rors rather than the structure itself. Never- theless, the control design must account for the destabilizing effect of structural dynam- ics.

Controversy over the merits of different control approaches depends on the relative importance of three factors: performance, robustness, and model accuracy. For given model accuracy, a control system can show robustness with no performance, or perfor- mance with no robustness. Colocated rate- feedback systems, such as low-authority control (LAC) [ 5 ] , or positivity approach [3] are examples emphasizing robust control systems. On the other hand, linear-quad- ratic-Gaussian (LQG) controllers, such as high-authority control (HAC), emphasize performance. Adaptive control can improve the model accuracy with time to increase performance and robustness capabilities. But, eventually, adaptive schemes will also reach some limit.

Lockheed has combined two approaches to achieve the desired compromise between robustness and performance-the so-called HAC/LAC approach [6]. Early experiments with vibration damping [7], [SI demonstrate the validity of such an approach.

This paper presents new results for the more difficult problem of active optics that interact with the structure. In particular, an experiment called the “jitter beam” simu- lates a pointing system on a two-dimensional structure. The objectives of the experiment are to demonstrate high-bandwidth beam steering in the presence of structural inter- actions, to evaluate factors that limit perfor- mance, and to define technology areas in need of improvement for future space appli- cation. Analytical results give insight into the closed-loop motion of the structure.

The jitter-beam experiment resembles small, stiff structures, such as active mirrors. Nevertheless, the control methodology ap- plies in principle to large structures and com- plex systems, such as segmented mirrors. The sparse modal frequency spectrum of stiff structures simplifies identification, but be- cause bending frequencies are high, sensor noise, actuator saturation, and computational speed constrain performance. In contrast, the low frequencies of large structures make hardware constraints a small concern, but the

0272-170818910400-0079 $01 .OO @ 1989 IEEE

dense frequency distribution complicates system identification.

One important achievement is a control bandwidth higher than critical bending fre- quencies. In contrast, classical designs his- torically have required the lowest vibration mode to be five times greater than the control system bandwidth.

Previous experiments in pointing flexible beams looked at control of vibration during slewing, for both colocated and noncolo- cated systems [9], [lo]. Slewing is a large, low-frequency disturbance that excites low- frequency bending modes. The present ex- periment addresses wideband control of steady disturbances, so the control interacts with more bending modes but has a smaller linear range. Other experiments with flexible beams investigate nonminimum-phase filter- ing [11] and wave propagation control [12] [I31 as alternatives to LQG control. Finally, several government projects will use active optics on flexible structures, but little infor- mation about these applications is published in readily available sources.

The remainder of the paper contains the following sections: experiment description, analytical model, experimental model, con- trol design, experimental results, and sug- gestions for further research.

Experiment Description The jitter-beam experiment simulates an

active pointing system that interacts with a two-dimensional structure. Figure 1 is a schematic of the experiment. The structure is a vertical cantilevered aluminum beam, 36 cm long and 7 cm wide. Mounted on the tip is an active mirror, which consists of a mir- ror mounted on the bottom of a pivoted proof-mass actuator. The actuator is de- signed to exert forces for controlling flexible structures, but that is not its function in this experiment. Instead, the proof-mass actuator allows us to simulate a pointing mirror that interacts strongly with its supporting struc- ture. Specifically, the acceleration of the proof mass causes a reaction force on the structure that excites structural motion.

A laser beam travels from a source to a fixed mirror located halfway up the alumi- num beam and reflects from there to the ac-

79

POINTING CONTROL SYSTEM

HARD'. .

- 1 I PREDICTED

MEASUREMENT

ARE

PIVOTED PROOF MASS

JITTER MIRROR

1 LINE-OF-SIGHT MEASUREMENT)

MIRROR

10

Fig. 1 . Jitter-beam experiment.

COMPUTATION (PROCESSOR) r

SMALL i &ERBEAM 8 STIFF ...... ~.. .......... ,... ~~~ ...., ~~~ .......

i 1 1 1 I 1 I , 1 1 1 ; : I 1 / 1 1 : I , , 1 1 1 1 1

tive mirror and finally to a detector fixed in the laboratory. The spot position on the de- tector simulates the line of sight of an optical system. Electromagnets torque the active mirror to control the spot position. A sensor measures the angular rate of the mirror rel- ative to the structure. In addition, a voice- coil-driven mirror between the source and the experiment disturbs the line of sight to test the disturbance rejection capability of the control system.

The object of the experiment is to control spot position with the active mirror at as high a bandwidth as possible. If the structure were rigid, the active mirror alone would accom- plish this. However, when the Structure moves, the displacement and rotation of the fixed mirror also affect spot position. Ac- cordingly, the control system must coordi- nate the motion of both the active and the fixed mirrors, using a single actuator not lo- cated near the fixed mirror.

To accomplish this, an array processor computes in real time a control law based on a bending model. The measurements do not feed back directly, but correct the comput- er's estimation of the bending response. This control is HAC. Analog feedback of the ac- tive mirror rate signal directly to the mirror torquer adds damping. This control is LAC.

Figure 2 compares the jitter-beam experi- ment with other structures and optical sys- tems. The jitter beam is a small, stiff struc- ture, like an active mirror. Stiff structures have a less dense distribution of bending fre- quencies, so they are relatively easy to iden- tify. On the other hand, large structures such

80

_- - - ,

as the Ten-Meter Telescope [14] and the Space Station have many low bending fre- quencies that cluster together. Accordingly, the jitter-beam experiment is a simple first test of identification and control that gives insight into more complicated, three-dimen- sional systems. One such system in Fig. 2 is the Advanced StructureslControls Inter- action Experiment (ASCIE), which is being built in the Lockheed Palo Alto Research Laboratory to develop controls for large seg- mented mirrors.

For stiff structures, hardware capability limits the c6ntrol performance. In particular, the maximum force of the actuators restricts the linear control range at high frequency. Sensor noise must be less than this range for control to be useful. Also, the real-time dig- ital computation needed for control is not yet feasible for bending frequencies higher than 1 kHz.

Analytical Model This section describes a dynamic model

that is useful for visualizing the structural motion. In particular, the model predicts the frequencies and shapes of the jitter-beam bending modes. Normally, finite-element analysis determines the natural modes for complex structures. But the modes for a can- tilever beam are well known [15] so that one can easily write the complete dynamic equa- tions for the jitter beam as the rotational dy- namic equation of the pivoted proof mass coupled with modal bending equations of the beam.

Figure 3 defines the geometry and coor- dinate system. The equations for force and toque exerted by the control actuator on the beam follow, where I is the moment of in- ertia of the actuator arm about the pivot, 8 the inertial angle of the actuator arm, m the actuator mass, b the moment arm of the ac- tuator about the pivot, g the acceleration of gravity, and x, the displacement of the tip of the beam.

f = -mbe - (1)

T = -16 - mbx, - mgbe ( 2 )

1000 c

p z y c m & E FLEXIBLE

COMPLEX IDENTIFICATION ADAPTIVE CONTROL

OF MODES

\\ MIRRO?i@ LOW NOISE (SENSOR) HIGH TORQUE (ACTUATOR)

Fig. 2 . Range of structures and control requirements.

IEEE Control Systems Mogorine

_ ~ - -~

Y

Fig. 3. Experiment geometry.

The torque T, which is the reaction on the beam from the torque applied to the actuator arm, is given by the following, wheref, is an electromagnetic force, r the moment arm off, relative to the pivot, K and D the spring and damping constants for actuator motion relative to the beam, and Or the angle of the beam at its tip.

- T = flC - K ( 0 - e,) - D(e - e,) (3)

Equation (2) is the rotational equation of the active mirror. The ith modal dynamic equation for the beam is shown, where q1 is the ith modal displacement, {, and U, the damping ratio and natural frequency, and 9,1 the angular and displacement mode shapes at the beam tip, and n the number of canti- lever modes retained in the model.

4, + 2f,U,ilr + d q l

= @bT + 9J, i = 1, n (4)

The mode shape is normalized as shown, where p is the beam’s mass density per unit length.

@p@ dy = 1

In Eqs. (1)-(3), the tip displacement and angle are equal to n, and 8,.

n

xr = * n q i (5)

4 = @bqr (6)

r = l

n

I = 1

Finally, the line-of-sight measurement, y , is the following, where x, is the fixed mirror displacement, 8, the fixed mirror rotation, and LI and L, the optical path lengths shown in Fig. 3.

y = 2x, - 2 ( ~ , + Lz)em + Lze (7)

where

n

xm = amqi 1 = 1

n

e, = C @kiql , = I

The preceding dynamic equations, to- gether with rough measurements of the phys- ical parameters, give the frequency response of the line of sight to actuator torque, shown in Fig. 4.

This response illustrates the problem of in- teraction between structure and control. At low frequency, only the active mirror moves, controlling spot position, and the cantilever beam appears rigid. However, higher fre- quencies excite vibration of the beam and, hence, the fixed mirror, which also affects the line of sight. Accordingly, to remove pointing disturbances at high frequency, the mirror torquer must coordinate motion of the active m i m r and a point on the structure far away. This noncolocation creates adverse phase lag in the frequency response of the line of sight.

One can understand the phase lag by in- terpreting response in Fig. 4 as that of a se- ries of spring masses, one for each bending mode. Below a resonance, the mode behaves as a spring, with a constant frequency re-

FREQUENCY ( H z )

1 10 1 0 0 1000

FREQUENCY (HI)

Fig. 4. Frequency response of the line of sight to active mirror torque (analytical model).

sponse and zero phase lag. Above the reso- nance, the mode behaves as a mass, with response lagging 180 deg and decreasing with frequency. Soon, the spring behavior of the next higher mode dominates and phase returns to 0 deg. However, for some bending shapes, the noncolocated fixed mirror moves the spot position in the direction opposite to the applied toque. Therefore, instead of re- turning to zero, the response will remain 180 deg out of phase. Then the further 180-deg loss at the next resonance makes high-gain feedback at that frequency unstable.

The shape of the structure undergoing forced motion gives further insight into the relationship of noncolocation and frequency response. The shape can be derived from the analytical mode shapes of a cantilever beam [ 151 and the complex frequency responses of 8 and q1 in the dynamic equations [Eqs. (1) - (6) ] . For accuracy, the bending equation [Eq. (4)] includes eight bending modes of the beam. In Fig. 5, the sketches Z1 to 24 and P1

to P5 show the shape of the structural vibra- tion caused by applying a sinusoidal torque to the active mirror in the direction shown in sketch 23. In each frame, e.g., P1, each line drawing is the shape for one forcing fre- quency. The family of line drawings in each frame then shows the change in shape and phase of the motion over a frequency range, indicated by projection lines to the frequency scale of the middle plot. The arrow for each family indicates the direction of increasing frequency. A “bump” on each line drawing shows the position of the fixed mirror on the cantilever beam.

The motions at the resonances P1 to P5 are pure mode shapes, and the motions at the zeros, Z1 to 24 , are combinations of the two resonant modes that bracket each zero in frequency. For example, the shapes for

are combinations of the resonant re- sponses for P4 and P5.

The shapes of the forced motion reveal how the motion of various points in the op- tical path contributes to the line of sight, and ultimately explain the frequency response in Fig. 4. In particular, the position and rota- tion of the fixed mirror, x, and e,, and the rotation of the active mirror, 0, determine the line of sight, and the sketches show that these quantities change in magnitude and di- rection with forcing frequency. The middle plot of Fig. 5 shows the contributions of x,, e,, and 0 to the line of sight quantitatively versus frequency. The solid line shows the net line-of-sight response. We normalize this to 1 to better illustrate the relative sizes of the contributing mirror motions at different forcing frequencies. The sign of the line-of-

April 1989

7 - __ ~

81

Fig. 5. Structural shape for forced motion and resulting mirror contributions to the line of sight.

sight response reverses at each pole and zero frequency, and the individual mirror re- sponses grow large at zero frequencies (a consequence of normalizing the line-of-sight response).

In the first five modes of the analytical model, the effect of noncolocation is benign; the fixed mirror moves the line of sight in the direction of the applied torque so that the phase lag never exceeds 180 deg in Fig. 4.

Experimental Model The LQG design method used to design

the high-authority controller requires an ac- curate state-space model of the dynamic sys-

tem. However, the analytical model devel- oped in the last section can only approximate the real experiment. To emphasize the point, Fig. 6 shows the measured frequency re- sponse of line of sight to actuator torque. The modal frequencies do not match exactly. Worse still, a zero between the 8-Hz and 55- Hz modes in the analytical model does not appear at all in the measurement.

Inspection of the 55-Hz response in Fig. 5 (P3) shows that the active and fixed mir- rors move the line of sight in opposite di- rections. The fixed mirror predominates in the model, producing a line-of-sight re- sponse in phase with the control torque. Ap- parently, the active mirror moves more in

the experiment than predicted, so that the line-of-sight response is out of phase. In this mode, the beam tip moves a lot, and its translation couples into active mirror rota- tion. The amount of rotation is very sensitive to local inertia properties of the beam and actuator. For instance, some unmodeled lump mass exists at the beam tip in the ac- tuator mount. In fact, slight detuning of the model will cause the active mirror to stop moving at all, as the inertial torque from tip translation exactly balances the applied con- trol torque. These problems with the simple jitter beam point out the difficulty of mod- eling complex structures.

Accordingly, to obtain a better model for

82

- - .

I € € € Control Systems Mogorine

- -

-ioo t

-720 1 I I

1 i o 100 10 FREQUENCY (Hz)

Fig. 6. sight for model and experiment.

Frequency response of the line of

the experiment, we adjust the parameters of the analytical model until its response matches the system response to an excita- tion, Such a process is known as system identification. A flat input spectrum will cause the low-frequency modes to dominate the response. Accordingly, shaping the spec- trum to excite a similar response at all fre- quencies enables the identification to match the system transfer function over a wider fre- quency range. The resulting input is a “chirp” of variable frequency and ampli- tude. A computer generates this complex signal.

Previous work [16] compared identifica- tion of the lowest two jitter-beam modes by maximum likelihood estimation (MLE), re- cursive least squares, and the eigenvector re- alization algorithm. Only MLE worked well with chirp inputs. In the present study, we use MLE to identify three more modes at high frequency.

In MLE, a Kalman filter reduces the ef- fects of sensor noise and system distur- bances. The resulting identification mini- mizes a log-likelihood function. To reduce computation, our identification neglects sen- sor noise and eliminates the filter. Then MLE becomes the same as least-squares estima- tion.

Because we lack confidence in the present analytical model, we choose to identify the system in modal form, which is sufficiently general to avoid constraining the solution ar- tificially. In addition, a matrix inversion that is necessary to compute a transition matrix from the analytical equations [Eqs. (1)-(7)] would add to the computational burden of MLE.

Limitations of computer hardware force us to identify the model sequentially rather than globally. First, the jitter-beam transfer func- tions range over 80 dB in gain-too wide to capture in a single measurement with the 15- bit conversion hardware. Also, a storage limit of 20,000 points, together with the sample time, restricts the time interval of the linear sweep. As a result, the excitation can span only a decade and a half of the fre- quency range with good resolution.

Fortunately, the maximum likelihood al- gorithm allows us to vary any parameters we choose while holding the others fixed. Re- ducing the number of parameters increases the speed and accuracy of the numerical op- timization. And, with the model in modal form, one can easily select appropriate pa- rameters to vary according to the excitation frequency range. However, because the pa- rameters depend on each other, tedious it- eration is necessary to find a consistent, global model. The burden is small for the jitter beam with its few modes. However, the experiment does not address the problem of accurate, efficient identification of large structures with many modes.

Specifically, we identify modal gain, damping ratio, and frequency for each of five modes-one mode at a time-exciting the re- sponse with a chirp spanning the bending frequency. Two additional parameters model the bias and scale factor of the measurement. The parameter estimates converge after re- peating the entire procedure once.

The frequency response of the identified model agrees well with the measurement in Fig. 6. Still, identifying 25 parameters and matching the response at 20,000 points re- quires considerable computation-20 min of computer time in this case. And, for large structures with many similar bending fre- quencies, computation will increase signifi- cantly, because the model has many inter- dependent parameters. Accordingly, efficient identification of large, complex space struc- tures, particularly on-line, remains a formi- dable problem worthy of research.

Control Design The control design strategy is to combine

high- and low-authority control. LAC is col- ocated rate feedback. For the jitter beam, feedback of the active mirror’s relative an- gular rate to mirror torque constitutes the low-authority controller.

The function of the low-authority control- ler is to add damping. This extra damping reduces the settling time after a disturbance. In the experiment, the low-frequency modes that are disturbed most strongly are also

damped most strongly by LAC. Moreover, damping stabilizes unmodeled modes close to the HAC bandwidth. Ideally, colocated sensors and actuators make the low-authority controller stable for any negative rate-feed- back gain. However, in practice, sensor and actuator bandwidths limit the amount of achievable damping, because modes above the bandwidth may become unstable if the rate gain is too high.

Figure 7(a) shows the effect of the rate- feedback gain on the line-of-sight frequency response. The feedback decreases the reso- nant peaks and will prevent a lower-band- width control system from destabilizing bending modes. However, it does little to compensate for the large phase lag arising from the noncolocation of the torquer and the fixed mirror. The crossover frequency must remain below the bending frequency at 50 Hz for stability. Although the low au- thority raises the maximum crossover fre- quency slightly, the bending frequency still restricts it, just as in classical rigid-body controllers.

To improve the performance of the posi- tion control is the function of HAC. To de- sign a high-authority controller, we apply standard techniques for discrete LQG syn- thesis to the identified dynamic model from the last section.

Figure 7(b) contrasts the open-loop fre- quency response of HAC designs and the position loop with LAC only. HAC alters the phase dramatically, stabilizing the 55-Hz and 128-Hz modes with phase lead and the 280-Hz mode with phase lag.

Transfer functions from the pointing com- mand to pointing angle and pointing error appear in Fig. 8. The pointing response bandwidth overlaps the first bending fre- quencies, so that the controller interacts strongly with these modes. The pointing er- ror transfer function shows a disturbance re- jection bandwidth of 80 Hz, which is above the first difficult mode at 55 Hz.

We use the analytical model described ear- lier to visualize the closed-loop motion. Spe- cifically, the response of any point on the structure is a sum of products of the closed- loop responses q, from Eq. (4) and the open- loop mode shapes +( y ) . The open-loop poles have low damping, so that the open-loop q, are either in phase or 180 deg out of phase with the forcing function at any frequency. As a result, points on the structure move in unison, and the open-loop forced vibration is a standing wave.

The closed-loop forced vibration is differ- ent. Feedback adds damping, which results in each closed-loop q, having a different phase between 0 and 180 deg at each forcing

83

LAC Open-Loop Transfer Function 40

RATE GAIN

2 0

0 - m 2 - 2 0

z 5 - 4 0

~ 60

-80

20

0

- * -20

z -40 U

4 -60

-80

- l o @

4 . 8 8 5 3 3 132 282

t t V

' ' ' ~ 1 ~ ~ ~ 1 ' ' " 1 1 1 1 1 ' ' '"'A

1 10 100

FREOUENCY (Hz)

HAC Open-Loop Transfer Function OPEN-LOO)' CROSSOVER FREOUENCY ( H r )

100

-40 -

100 : : : : : + : : : : :m.kt : : : : : *

-300 -

-400 - -500 -

-600 1000 100 10

I I ' ' ' ' ' " I

FREQUENCY ( H r )

Fig. 7. Bandwidth capabilities of low- and high-authority control.

POINTING-ERROR TRANSFER FUNCTION 20

10

0 - - 1 0

2 -20

z - 3 0 - 5 - 4 0

~ 50

- 60 - 70

- PREDICTION ......... MEASUREMENT

0 . 0 0 1 0 . 0 1 0 . 1 1 10 100 1000 FREQUENCY ( H z )

(b)

Fig. 8. Transfer firnctions of pointing response and pointing error for 80-Hz disturbance-rejection bandwidth.

frequency. Because the mode shapes @( y ) weight the contributions of qi to total motion differently at each y coordinate on the beam, the motion of different points is out of phase. As a result, the structure "wriggles," and the closed-loop forced vibration is a travel- ing wave. Mathematically, this more com- plex vibration comprises two simpler stand- ing waves-one in phase, and one 90 deg out of phase, with the forcing function.

In similar fashion to Fig. 5, Fig. 9 shows the shape of the in-phase vibration for var- ious frequencies of a sinusoidal pointing command. Again, the center plot shows quantitatively the contributions of the mir- rors to the closed-loop response of the line of sight. The solid curve showing the total response is flat, although the shape the con- trolled structure assumes varies dramatically over the frequency range, resembling some- what the open-loop shape in Fig. 5.

The amplitudes of mirror motions vary dramatically also, as the command fre- quency passes through the first two lightly damped closed-loop poles. These poles are close to the open-loop zeros seen in the mea- sured transfer function in Fig. 6. It appears that the high-authority design controls inter- action with the structure by changing the shape of the structure without adding a lot of damping. This feature results from the optimal design, which minimizes control ef- fort. To control the line of sight, a single

torquer coordinates the motion of two non- colocated points on the structure.

As discussed earlier, the total closed-loop motion consists of the in-phase component in Fig. 9 and a component 90 deg out of phase (not shown). The 90-deg motion is most significant near the closed-loop reso- nances. Within the control bandwidth, this motion has no effect on the line of sight, because otherwise the pointing response would not be in phase with the command. Thus, the 90-deg motion must resemble the open-loop motion at the transfer-function ze- ros, shown by ZI and 22 in Fig. 5. The motion of the real structure, consisting of the in-phase and 90-deg motions simulta- neously, would look complex.

Experimental Results The experiment tests the design in Fig. 9.

An array processor computes the LQG con- trol at a 2-kHz sampling rate. To measure the disturbance rejection bandwidth, we dis- turb the laser beam direction with the voice- coil-driven mirror in Fig. 1 and measure re- sidual spot motion on the detector. As an illustration, Fig. IO displays 10-Hz and 20- Hz disturbances with the controller on and off. Figure 9 shows the closed-loop struc- tural shape at these frequencies.

Similar measurements over a frequency range give the measured error frequency re- sponse shown by the dotted line in Fig. 8(b). The measured bandwidth is less than pre- dicted-only 50 Hz instead of 80 Hz. Satu- ration of actuator torque causes the discrep- ancy. The current ratings of the amplifier and the windings of the active mirror elec- tromagnets limit maximum torque. Distur- bances demand increasing control torque with frequency, decreasing the linear range of the controller. Eventually, at some fre- quency, the linear range falls below the level of the sensor noise itself. Then sensor noise saturates the control torque, and the band- width cannot increase.

Thus, the fundamental obstacle for high- frequency control of stiff structures is not bending, but hardware capability (i.e., ac- tuator power, sensor noise, and processor speed). For small structures such as active mirrors, this capability is even more critical, because the lowest bending frequencies are kiloHertz, not hundreds of Hertz as in the experiment.

It would perhaps be useful to lengthen the cantilever beam to bring more bending modes within the bandwidth of the control. Expres- sions for the cantilever-beam frequencies from [15] show that the number of reso- nances below a certain frequency increase

84 I€€€ ControJ Systems Mogorinr

- -

F 1 . 5 U

U7

t; 1.0

W

2 0.5 -I

0

z 0

3

t - 0

F - 0 . 5

rn - 1 . 0

I- z 0 - 1 . 5 U

- 2 . 0 ~ /JJz) FREQUENCY \\

Fig. 9. to line of sight.

Structural shape for closed-loop forced motion and resulting mirror contributions

I 10-Hz JITTER I 0.5

0.25 U)

’ -0.5

5 0

r\ OPEN LOOP -0.25 CLOSED LOOP

1 2 3

0.25 U)

5 0 9 -0.25

-0.5

I I I 0 0.1 0.2 0.3

TIME ( 5 )

Fig. 10. Measured jitter attenuation demonstrating 50-Hz bandwidth.

linearly with beam length. Moreover, a ver- tical suspension would allow this length to increase indefinitely, since gravity would stretch, rather than buckle, the beam. In fact, several experiments have studied vibration control of hanging beams [7], [13].

However, at some point, it is useless to study beams fuxther because their distribu- tion of bending frequencies is not typical of LSS. Instead, complex three-dimensional trusses exhibit large numbers or clusters of similar bending frequencies because of the dynamic interaction of repeated structural elements [14].

Conclusion The jitter-beam experiment demonstrates,

for the first time, pointing control on a flex- ible structure over a wide range of bending frequencies. The control bandwidth is higher than a critical bending frequency and a factor of 10 beyond what rigid-body design can achieve.

In addition, the experiment demonstrates the validity of the high/low-authority control approach for controlling interaction of active optics with the structure. Although the ex- periment is quite unlike a large system, the control methodology is applicable to large StNCtureS. However, the many similar bend- ing frequencies of large structures make them more difficult to identify, because the dy- namic model has many parameters that de- pend strongly on each other. As a result, identification will require more computation, and it may be less accurate.

The analytical results show the importance of knowing the natural mode shapes for un- derstanding the system dynamics and visu- alizing the controlled structural motion. Un- like the natural motion of structures, which is very intuitive and familiar from experi- ence, the motion of controlled structures is complex and perplexing. Examples are few and are seldom presented in enough detail to visualize the closed-loop motion clearly. We must gain experience to understand the con- trolled Structure as well as we understand the natural StlUCture today.

References [l] “ACOSS (Active Control of Space Struc-

tures) Six,” RADC-TR-80-3777, Interim Report, Charles Stark Draper Lab., Cam- bridge, MA, Jan. 1981.

121 J-N. Aubrun et al., “ACOSS (Active Con- trol of Space Structures) Five,” RADC-TR- 82-21, Final Technical Report, Mar. 1982. Lockheed Missiles and Space Company, Inc., Lockheed Palo Alto Research Labo- ratory, Palo Alto, Calif.

April 1989 85

[3] R. J . Benhabib, R. P. Iwens, and R. L. Jackson, “Stability of Large Space Struc- ture Control Systems Using Positivity Con- cepts,” J . Guid. Contr., vol. 4, pp. 487- 494, Sept./Oct. 1981. U. Ozguner, S. Yurkovich, J . Martin, and P. Kotnik, “Laboratory Facilities for Flex- ible Structure Control Experiments,’’ I€€€ Conrr. Sysr. Mag., vol. 8, pp. 27-33, Aug. 1988.

[5] J-N. Aubrun, “Theory of the Control of Structures by Low-Authority Controllers,” J. Guid. Conrr., vol. 3, pp. 444451, Sept./ Oct. 1980. Lockheed Missiles and Space Company, Inc., “Vibration Control of Space Struc- tures, VCOSS A: High and Low Authority Hardware Implementation.” AFWAL-TR- 83-3074, July 1983. D. B. Schaechter and D. B. Eldred, “Ex- perimental Demonstration of the Control of Flexible Structures,” J . Guid., Conrr., Dy- nam., vol. 7, pp. 527-534, Sept./Oct. 1984.

[8] J-N. Aubrun, M. G. Lyons, and M. J . Rat- ner, “Structural Control for a Circular Plate,” J . Guid., Conrr., Dynum., vol. 7, pp. 535-545, Sept./Oct. 1984.

[4]

161

[7]

[9] E. Barbieri, S. Yurkovich, and U . Ozguner, ”Control of Multiple-MirroriFlexible- Structures in Slew Maneuvers,” Proc. A I M Guid. Contr. Conf., Monterey, CA, p. 379, Aug. 1987. J-N. Juang, L. G. Horta, and H. H. Rob- ertshaw, “A Slewing Control Experiment for Flexible Structures,” J . Guid. Contr., vol. 9, pp. 599-607, Sept./Oct. 1986. B. Wie, “Active Vibration Control Synthe- sis for the Control of Flexible Structures Mast Flight System,” J. Guid. Conrr., vol. 11 , pp. 271-277, MayiJune 1988. A. H. von Flotow, “Traveling Wave Con- trol for Large Spacecraft Structures,” J. Guid. Contr., vol. 9, pp. 462-468, July/ Aug. 1986. A. H. von Flotow and B. Schafer, “Wave- Absorbing Controllers for a Flexible Beam,” J . Guid. Conrr., vol. 9, pp. 673- 680, Nov./Dec. 1986. J-N. Aubrun, K. Lorell, T. Mast, and J . Nelson, “Dynamic Analysis of the Actively Controlled Segmented Mirror of the W. M. Keck Ten Meter Telescope,” I€EE Conrr. Syst. Mug., vol. 7, pp. 3-10, Dec. 1987. T. R. Kane, P. W. Likins, and D. A. Lev-

(101

111

121

[13]

[I41

1151

inson, Spacecrufr Dynamics, New York: McGraw-Hill, pp. 318-320, 1983.

[16] B. Sridhar, J-N. Aubrun, and K. R. Lorell, “Identification Experiment for Control of Flexible Structures,” IEEE Conrr. Swt . Mag., vol. 5, pp. 29-35. May 1985.

Eric K. Parsons received the B.A.Sc. degree in en- gineering science from the University of Toronto in 1973 and the Ph.D. de- gree in aeronautics and astronautics from Stan- ford University in 1982. From 1979 to 1980, he was a consultant to NASA Ames Research Center. developing a pointing . -

system for the Shuttle In- frared Telescope Facility. Dr. Parsons is currently a Research Scientist at the Lockheed Palo Alto Research Laboratory. His research interests are precision pointing systems, identification and con- trol of flexible structures. and insensitive, re- duced-order digital control design.

Nominations for George S. Axelby Outstanding Paper Award

The award for the outstanding paper in the IEEE Transactions on Automatic Control is named the George S . Axelby Outstanding Paper Award. Professor Tamer Basar of the University of Illinois is Chairman of the Con- trol Systems Society subcommittee that ad- ministers the award. We are now soliciting nominations for the award from papers published in the IEEE Transactions on

Automatic Control from January 1987 through December 1988-the previous two calendar years. If you wish to nominate a paper for consideration for the award, send a letter stating the title and author(s) of the paper along with a brief one-paragraph ex- planation of why this paper should receive the award. The deadline for nominations is May 15, 1989. The nomination should be sent to:

Prof. Tamer Basar Axelby Award Chairman Coordinated Science Laboratory University of Illinois 1101 W. Springfield Ave. Urbana, IL 61801 Phone: (217) 333-3607 E-mail: tbasar@uicsl.csl.uiuc.edu

1989 SMC Conference The 1989 IEEE International Conference

on Systems, Man, and Cybernetics (SMC), with Joseph G. Wohl as the Conference Chair, will be held November 14-17, 1989, at the Hyatt Regency Hotel in Cambridge, Massachusetts. Located near the Massa- chusetts Institute of Technology, on the banks of the Charles River, the Hyatt is one of the most impressive hotels in the Boston- Cambridge area and provides an excellent setting for informal discussions and technical exchange. The conference program will include both contributed and invited papers, as well as tutorial and state-of-the-art presen- tations and workshops in those areas of interest to the IEEE SMC Society. These

include, but are not limited to: behavioral decision making, information and decision systems, command and control systems, knowledge-based systems, manufacturing systems, robotics, pattern recognition and im- age processing, neural systems, manual con- trol, human-computer interaction, systems methodology, design and simulation.

The conference theme, “Decision making in Large-scale Systems,” has been selected to reflect the increasing interest and importance of this topic in our evolving, ever-more- complex, computer-oriented society. Presen- tations dealing with theory, analysis, design, and simulation techniques for application to decision making in large-scale systems are

especially encouraged. Plenary and technical sessions will emphasize this year’s theme.

The Organizing Committee intends to arrange for workshops to be held in conjunc- tion with the conference. The date set aside for such workshops is on Tuesday, November 14. Potential organizers should contact the SMC Conference Coordinator: Daniel Serfaty, Alphatech, Inc., 111 Middlesex Tpk., Burl- ington, MA 01803; phone: (617) 273-3388.

For further information about the con- ference, contact: David L. Kleinman, Pro- gram Chairman, Electrical and Systems Engineering Dept., Box U-157, University of Connecticut, Storrs, CT 06268; phone: (203) 486-3066.

86 IEEE Control Systems Mogilrrfie