The Applications of Smart Structures for Vibration Suppression in Spacecraft

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AIAA 95-3848

The Applications of Smart Structures for

Vibration Suppression in Spacecraft

William B. Harrington, Jr. and Brij N. Agrawal

Naval Postgraduate School

Monterey, CA

AIAA 1995 Space Pro g ram s and

Techn ologies Con ferenc e

September 26-28, 1995/HuntsviIle, A L

For permiss lon to copy or republ ish, contact the Amer ican Ins t i tute of Aeronaut i cs an d As t ronaut i cs

370 L'Enfant Prom enade, S.W., Washington, D.C.20024



T H E A P P L I C A T I O N S O F S M A R T S T R U C T U R E S FOR V I B R A T I O N S U P P R E S S IO N

IN S PACECRAFT

William B. Hmington, Jr.'

Brij N. Agrawal"Naval Postgraduate School

Monterey, CA, USA

A b s t r a c t

This paper presents the smart structures technology prog ram at the N aval P ostgraduate School.

The research program consists of vibration dam ping of space sttuctu res using piezoc eram ic sensors and

actuators, vibration isolation of space structures using H- wave absorbing control, and the development

of paramet er optimization techniques to minimize antenna pointing errors. Seve ral new con trol laws are

exper imen tally implemented using an air-bearing flexible spacecraft simulator (FSS ) repres enting a

space craft with smart structures. The major empha sis has been to im prove performance of flexible

space craft antenna using the latest smart structure technology. Both analytical and exp erime ntal research

is performed i n the Spac ecraft Controls Laboratory.

I. Introduction

Seve ral future civilian and military spacecraft will require acti ve dam pin g for large flexible

structures, vibration isolation of optical payloads, and sh ape control of reflector and optical bench es to

meet perform ance requirements. For these type of applications, smart structures is a promising

technology. In general, smart structures are the system elem ents that sense the dynamic state and chan ge

the system 's structural properties, such as its natural frequencies and its damping , to meet given

performance objectives.

The re are several types of embedded senso rs and actuators which can be used for vibration

control. Th e embed ded sens ors are piezoelectric deformation sensors, strain gages , and fibe r optic

senso rs. ' The em bedd ed actuators are piezoceramic w afers, electrostrictive ceram ic wafers, piezoceramicpolymer film and shape memory metal wires. For the research program at the Naval Postgradu ate

Scho ol, piezoceram ic sensors and actuators are currently used, additionally a Vision Server camera

system is used as an external optical sensor for the FSS. Piezoceramic sensors have a high strain

sensitivity , a low noise baseline, low to moderate temperature sensitivity, and an ease of implem entati on.

Piezo ceram ic actuators have high stiffness, sufficient stress to control vibration, good linearity,

temperature insensitivity, are easy to implement, and minimize pow er consum ption.

structure of Flexible Spac e Simulator (FSS) shown in Figure 1. This discussion provides an overview of

the research program on sm art structures at the Naval Postgraduate School.

11. Research Program 1990 - 1995

Initial motivation for research in smart structures was to provide active dam ping to the flexible

The Flexible Spacecraft Simulator (FSS) simu lates attitude motion abo ut the pitch axis of a

spacecraft It consists of a single degree-of-freedom rigid central body, repres enting the spacecraft

central body, and a multiple degree-of-freedom flexible appen dage, representing an antenna reflector

with a flexibl e support structur e. Piezo ceram ic sensors and actuators are used to provide active damping

to the flexible support structure. The entire system is floated on air pads ove r a finely ground granite

* Lieutenant Commander, United States Navy, Student Member AIAA..Professor, Department of Aeronautics and Astronautics, Associate Fellow AIAA

a



table to simulate a microgravity environment. The central body has two thrusters and a momentum

wheel as its actuators.

I

.'

igure 1. NPS Flexible Spacecraft Simulator FSS)

L

Piezoceramic sensors and actuators are located on the flexible appendage as shown in Figure 2.

The piezoceramic wafers are bonded to the surface of the flexible arm and the voltage developed from

the sensor s is fed to the actuators in such a way that motion of the flexible rmis damped. Figure 3

illustrates the orientation of a piezoceramic wafer on an arm and the alignment of its axis that describes

the electro-mechanical relationships.'

The piezoceramic wafers in a sensory m ode produce a charge between their electrodes that isdirectly proportion al to the lateral strains. It is given by L

e = AEd,, (E \ + E * 1 (1)

where A is the lateral area of the piezo wafers, E is Young's modulus of the wafe r, d,l is the lateral

charge coefficient, and E, and E~ are the strain values in the lateral direc i:ms resp ectiv ely. Th ecapacitance for a piezoceramic wafer as shown in Figure 3 is given by

D Ac=

where D is the dielectric constant of the piezoceramic and t i s th e th ickness of t h e w a f e r

Figure 2. Sensor A ctuator Placement f o r FSS Beam

2



I

Figure 3. Piezoceramic Mounting on FSS Beam

The voltage V produced by a sensor under strain is given by

When using piezoceramic wafers as actuators, the attachment geometry is similar to the sensor geometry

shown in Figure 3. Th e contro l voltage , e,, is applied to the wafers and the lateral strain that is

devel oped can act to control the bending of the beam. The electric field that is developed by the wafer isgiven by

vca= 4)

Care must be taken not to induce a strong electric field that is opposed to the piezo’s poling direction as

that can dama ge the material by depolarizing it. Typical field limits by mos t materia ls are between 500

and 1000volts/mm.

One of the first methods employed to dam p the flexible appen dage is positive position feedback

(PPF). In order to imple men t this control law on the FSS, it is modeled a s a second order sy stem:

is the modal coordinate of the structure

w is the structural resonant frequency

< is the structural dam ping ratio

7j is the modal coordinate of the compensator

e , is the actuator applied voltage

e , is the senso r sensed voltage

<, is the actuator damp ing ratio

0 s the actuator frequency

k is the actua tor gain

For this method of control initially an analog compensator was used. Figure 4 show s the damping with

and without a PPF controller.

3



t , , , , . ,. ,. = . . I .*I

F i g u r e 4. Posit ive Position Feedb ack Response

By setting the actuato r frequency equal to the structure fundamental frequ ency, a 90 phase shift is

attained in the region w here W / O n = 1 . on, s the fundamental structural frequency. This equation

is added into the system state-space representation as an additional two states. Figure 5 illustrates that

the effect of this type of compen sation is an active damping region in the area of W / W = 1 withactive stiffness for frequencies ab ove this region and active flexibility in the frequencies below this

region.

I

F ig u re 5. PPF G a in a n d P h a se P lot s

Thus it is important to set the actuator frequency at the fundamental structural frequency to avoid an

increase in flexible mode strength for lower frequencies. Thi s concept was verified in a study conducted

to comp are the control authority between single and multiple senso dactu ator systems.

first three m odes of a cantilevered beam were to be controlled using two different scheme s. First, using

three collocated sensor dactua tors pairs connected to three comp ensators which were tuned to the beam's.

first three modal frequ encies, and second , using a single collocated sensodac tuator pair connected to the

three comp ensato rs. Figures 6a and 6b illustrate the block diagrams of these two control schemes.

An analysis was performed on a multi-modal vibration suppression s y ~ t e m . ~pecifically, the

F ig u re 6a. Mult i -SensodActua tor Arrangement

4



I N W OUlPlJT

SENSOR

44

Compmator 3

ACTUATOR

Figure 6b. Single Sensor/Actuator Arrangement

Figure 7 shows the response of the beam under th e influence of three actuators, at the base, the

middle, and a t the tip of the beam. The effectiveness of this type of system is seen in how quickly all

three modes are suppressed. Figure 8 indicates that, using the second s ystem setup , a sufficient amount

of control is obtained w ith the single actuator located at the root of the beam. Th e effect of actuator

placement is evident when Figure 8 is compared with Figures 9 and 10which show the beam 's response

with actuators located in the middle and at the tip of the beam, feeding their respective compensators

with the sam e gains. How ever, their performance can be improved by increasing the gains. Individualmodal responses ca n be tailored by varying the gains such that the damping of one mod e is maximized in

relation to the other modes.

II8

111 . .

Figure 7. PPF Three Actuators Mode

II.

-a .

,111, ,<<

Figure 9. Middle Mounted Actuator

.I .

... 2

8... X

11.1. .*

Figure 8. PPF R oot Mounted Actuator

111 WtM - ,,"1 Y. . 11..11, cz<> ..I .01

I

IS

-a .

,, .. ...

Figure 10. Tip Mounted A ctuator

5



The next step was to integrate digital control into the spacecraf t s imula tion ~ y s t e m . ~he FSS

as connected to a Digital Equi pmen t Corporation VAX 3100 computer system and the Integrated

Syste ms, Inc. AC-1 00 real time control processor. Alon g with Matrixx, the V A X computer system was

used to compa re digital performance of derivative control, integral control and pos itive position fee dback

control.

The procedure was to explore the performance of different control approach es using a digitalcontroller and compa ring the results to those obtained previously using "modified" PPF control. Open

and closed loop mn s were performed. Most of the mns attempted to excite the arm's fundamental mode

without appreciably exciti ng the higher m odes , however, many mns did have multiple mode excitation.

was dam ping ratio, <. The dam ping ratio was calculated by the log decrem ent meth od:

Th e primary performance indicator used for comparison between PPF and Phase Lead models

where Ai is the initial amplitude, Af is the final amplitude, and n is the number of cycles between the two

amplitudes measured. The observed damping ratios were small enough to assume that the damped

frequency was equal to the natural frequency.

a 167% increase i n damp ing fo r the multimode response. The best PPF multimode response was

obtained by da mpi ng only the first mod e, attcinpts to control the second m ode tended to deg rade overall

damping performance. Figures 11 and 1 2 are the beam responses for the PPF controller.

The PPF contro ller was able to obtain a 187% increase in damping fo r the fundamental mod e and

Phas e lead controllers with sensor-actuator phase angles appro achin g 90 were first attempted.

However, choosing controller gains low enough so as not to cause instability in the ann's response

resulted in unacceptably low dam ping ratios. The instability is believed to be caused by the time delays

in the contro ller circuitry and the associated phase shifts through the digital controller. At a phase angle

of 88.88" a 42% increase in dam pin g was achieved, and with the same phase angle, the multimode

response resulted in a 32% increase in damping. The sensor-actuator phase relationship was thenreduced to 60 . This allowed the use of higher controller gains while preserving the contr oller's ability

to dam p higher modes. Figures 13 and 14 show favorable performance. Th e funda men tal mode response

was a 72% increase in damping with a 100% increase i n damping for the multimode response.

I 1 I

Figure 11. PPF Fundamental Mode

1I 1 x . I L n - "

D I

Figure 12. PPF M ultimode

6



- --- I

Control Mode

Modified PPF

Phase Lead (@=90")

Phase Lead (@=60")

Integral

Figure 13. Phase Lead Fundam ental Mode Figure 14.Phase Lead Multimode

The integral controller was originally studied with unsatisfacto ry results. Co ntin uou s activation

of the contro ller was allowin g integration of small system biases, resulting in degraded co ntrolle r

performance. A modified approach was taken whereby the actuator was not activated until immediatelybefore it was required. Figures 15 and 16 show t he favorable results. A 139% increase in dam ping was

the fundam ental mode response with a 100% increase in damp ing for the multimode response.

- I1 . .

> *

i .1i

iI

1

i

. . l . l l

.,.I ., -.-I .I I ,4 n r I m

,.- I.

Figure 15. Integral Fundam ental Mode Figure 16. Integral Multimode

A summary of the comparison results, measured as damp ing ratios, (, s shown in Table 1

Open Loop Closed Loop Closed Loop

(lst/2nd Modes) (First Mode) (Higher Modes)

0.0142/0.0183 0.0408 0.0489

0.0142/0.0183 0.0202 0.0241

0.0142/0.0 183 0.0243 0.0367

0.0142/0.0183 0.034 0.0367

7



A study was initiated to solve for optimal control gains for multiple-input multiple-output (M IM O)

systems using PPF.' Stability criterion was established for t h e two, second or der, system and

compensator equations as:

a - n , n 2 C r G C > 0 7)

where Q is the modal natural frequency matrix, a, and a? are consta nts representing ac tuator and sensor

sensitivity respectively, C is the sensor st ate vector, and G is the feedback gain matrix. Notice thematrix should be positive defin ite and the stability criterion in Eq. (7) does not depend on the structural

properties of the compensator.

suggested to minimize the follo wing cost function by the feedback gains being subje ct to the constraint

equation, Eq. (7).

Motivated by general optimization of a Linear Quadratic Reg ulator (LQR) cost function it is

p [ p , p 2 ",p , ] is a vector of design parameters consist ing of f e e d b a c k gains

x = [ q , * , 5 , < l r s a state vector for the sys tem in first order state space form

This approach was implem ented on the FSS w i t h the in ten t of minimizing th e flexible arm tip deflection

and rotation. An optimization algorithm u sing Homotopic No nlinea r progr amm ing was successfully

applied to designing an optimized PPF compensator for control of the flexible spacecraft model using

adaptive structures. The results verify that the proposed method c an be used to improve the performance

over the conventional PPF approach.

cantilevered beam6 A major advantage of this method is that it does not involve truncation into a finite

dimensional mathematical model. A closed loop scattering matrix was derived which gives the

relationship between incom ing waves, outgo ing waves, sensor, and actuator. The control law was

determined by minimizing the H.. norm of this matrix. The H- wave abs orbin g controller contributedsignificant damping to the structure, especially at its fundamental m ode of 1 Hz.

This approach describes the structural response of the system in terms of an elastic distu rbanc e

which travels through the structure. ' In this approach compe nsato rs wer e designed to reduc e the effects

of incom ing waves on outgoing waves. The advantages of the wave abso rbing control are: relative ease

of implementation, broad-band control, and no requirement for a finite eleme nt or modal model.

intricate and uses a transformation for the state vector from be am cros s sectional properties to traveling

waves. In this process, the control function derived is essentially & based.

ecently, the effects of H_ optimized w ave absorbing control were studied on the FSS with a

The sequence that takes place in deriving the scattering matrix in terms of the wave amplitudes is

The control law is effectively a "half differenti ator". It is similar to velocity fe edbac k with a 45 instead

of 90 phase lead. In order to implem ent this control law, the function & had to be estimate d by a

rational transfer function. Th e functi on used was:

8



Free Response

H_ Control

Derivative C ontrol

(s+104) (s+10-2) (s+100) (s+102)

Sf 0 - ~ ) ( ~ +io-^)(^+ I O I ) ( ~ +o 3 )= 10)

The s-func tions were then transformed to the z-domain (discretized) prior to form ing the closed loop

system. Th e relationship between the continuous domain (s) and the discreet domain (z) is:

T

z = ewhere T is the samplin g period. The transfer functions were transformed to the discreet dom ain using aTustin transformation for the structure and a matched pole-zero technique for the controller. Th e system

was then simulated on the FSS with the following results:

Frequency Damping Ratio

0.95 0.0039

0.98 0.037

0.96 0.039

Table 2

HI. Current Work

The implementation of a MIMO optimal controller using Linear Quadratic Gaussian (L QG ) control

theory is being analyzed on the FSS. A two actuator, three sensor arrangement is currently being

evaluated as a viable control method for flexible spacecraft structures. The experim ental setup involves a

“L” shaped flexible arm attached to the main body of the FSS. Tw o nearly collo cated sensor/actuator

pairs are mounted on the two beams of the ann at each beam’s respective root as show n in Figure 17. A

Vision Server camera system is mounted overhead th e granite table and sen ses position and rotation

information on the tip of the flexible arm. The sensor information is then fed back into the LQG

controller whi ch, along with the sensor information fro m the piezo wafers, minimizes the rotation of the

tip.

5 en so r

a c t u a t o r : /

i

Figure 17. FSS Experimental Setup for LQG Nis io n S erver Tes t in g

9



The Vision Se rver camera system consists of a single infrared CC D camera situated above the

test area with Real Ti me Innovations. Inc. Pointgrabber software interface for the real-time oper atingsystem, VxWorks.

Anot her area of interest is antenna shape control. At this time feasibility stud ies and literature

research is being condu cted to evalu ate whether piezoceramics can be used for structural shap e contro l,

namely in the antenna regime. No rmall y, antennas use multiple feedhorn s to alter their bea m pattern,

using reflecto r shape control by using smart structures a single feedhorn can be used for a variety of

beam patterns. This research is in the developmental stage and looks promi sing in the near future.

IV. Future Work

Future work will involve more emphasis on vibration isolation and antenna shap e contro l. Also,

application speci fic expe rimen tal work will be the main thrust of the Spacec raft Con trols Laboratory

Team. New interest in the use of smar t structures for vibration reduction of rotorcraft holds so me

promise with the use of piezocera mic and magnetostrictive actuators. Furth er analy sis into the state-of-

the-art actuation devices is to be studied and tested for possible implementation into sm art structures

V. Conclusions

Th e main thrust to date in the Spacecraft C ontrols Laboratory design te am has been to analyze,

test and evaluate various control sch emes for smart structure technology. Positive Position Feedba ck has

been used extensively in the sm art structures area and is a good single mode damp ing contro l law. Strain

Rate Feedback SRF) has also been tested and is a good way to get generalized d am ping for all modes .

Ciassicai PID conirol has been used and compared against PPF and SRF and for certain types of

applications a nd is a suitable control schem e. Wa ve absorbing H-co ntrol has been analyzed and tested

and showe d good results as a low authority controller. MIMO contro l using LQG is expected to produce

satisfactory results and should fare to be a viable control scheme as well. Th e NF’S Spacecraft Controls

Laboratory d esign team is proactiv e in its research with state-of-the-art materials and con cepts and plans

on expan ding its capabilities in the near future.

VI. References

I ) Bronowicki, Allen and Betros, Robert, “Design and Implementation of Active Struc tures” , Active StructuresWorkshop , TRW Space and Electronics Group , Redondo Beach , CA.

(2) Agrawal, B.N., Bang, H., and Jones, E., IAF-9 2-0319, “Application of Piezoelectric Actuators and Sensors i n

the Vibration Control of Flexible Spacecraf t Structures”,43rd Congress of the International AstronauticalFederation, Washington, DC, 1992.

(3) Newman, Scott M., “Active Damping Control of a Flexible Space Structure using Piezoelectric Sensors and

Actuators”, Master’s Thesis, Naval Postgraduate School, December 1992.

4) Feuerstein, Mark G., “A Comparison of Different Control Methods for Vibration Suppression of FlexibleStructures Using Piezoelectric Actuators”. Mas ter’s Thesis, Naval Postgraduate School, June 1994.

(5) Agrawal, B.N. and Bang, H., IAF-94-1.4.202. “Adaptive Structures for Large Precision A ntennas”, 45thCongress of the International Astronautical Federation, Jerusalem, Israel, 1994 .

(6) Strong, Ronald E., “Control of Flexible Spacecraft Structures using H-Infinity Wave Absorbing Control”,Master’s Thesis, December 1994.

(7) von Flotow, A H ., “Traveling Wave Control for Large Spacecraft Structures , Journal of guidance” , Vol. 9, No.4, July-August 1986,pp. 462-468.

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