Research ArticleHigh Accuracy Attitude Control System Design for Satellite withFlexible Appendages

Wenya Zhou,1,2 Haixu Wang,1 Zhengwei Ruan,1 Zhigang Wu,1,2 and Enmei Wang1

1 School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116023, China2 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, China

Correspondence should be addressed to Wenya Zhou; zwy@dlut.edu.cn

Received 27 August 2014; Revised 29 September 2014; Accepted 29 September 2014; Published 4 November 2014

Academic Editor: Hui Zhang

Copyright © 2014 Wenya Zhou et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In order to realize the high accuracy attitude control of satellite with flexible appendages, attitude control system consisting of thecontroller and structural filter was designed. When the low order vibration frequency of flexible appendages is approximating thebandwidth of attitude control system, the vibration signal will enter the control system throughmeasurement device to bring impacton the accuracy or even the stability. In order to reduce the impact of vibration of appendages on the attitude control system, thestructural filter is designed in terms of rejecting the vibration of flexible appendages. Considering the potential problem of in-orbitfrequency variation of the flexible appendages, the design method for the adaptive notch filter is proposed based on the in-orbitidentification technology. Finally, the simulation results are given to demonstrate the feasibility and effectiveness of the proposeddesign techniques.

1. Introduction

Since the satellite was designed with compact structure andsimple configuration at early stage, satisfied attitude controlaccuracy can be achieved while considering the satellite asthe rigid body during the design of attitude control system[1–3]. However, the satellite structure is becoming morecomplicated and the appendage is becoming bigger in sizeand more in quantity in order to satisfy the modern spacemissions. These flexible appendages are apt to vibrate dueto the attitude maneuver of satellite or external disturbance.When the vibration signal enters the attitude control systemof satellite through the measurement device, the attitudecontrol accuracy will be influenced or even the instability ofsystemmight be caused [4, 5].Many scholars have performeddeeper and further studies with respect to the design of highaccuracy attitude control system of satellites with flexibleappendages [6–8].

During the design of satellite structure, the vibrationmodes of its flexible appendages are usually obtained throughground test. While performing the design of attitude controlsystem for satellite, frequency isolation method, that is, the

bandwidth of attitude control system will be kept far awayfrom the first order mode vibration frequency of flexibleappendages, will be adopted in order to avoid the influenceof appendages vibration on the attitude control [9]. But whenthe flexible appendage is big, the first order mode vibrationfrequency will be close to the instinct frequency of the rigid;therefore, the above method will not be applicable any more.Aiming at such low frequency vibration rejection problem,structural filter is one of the effective methods to reject thevibration [10, 11]. In [12–14], Wie et al. presented the designmethod of structural filter. However, these methods aredeveloped for fixed frequency.When the satellite is operatingin orbit, the vibration frequency of flexible appendages mightchange due to the failure or damage. Such variation offrequency is hardly predicable and will directly threaten thecontrol accuracy of satellite attitude.Therefore it is powerlessto use the above structural filter to improve the controlaccuracy.

In this paper, with the pitch channel as an example, theattitude control system including the adaptive structural filteris developed for the satellite with flexible appendages. Inorder to reduce the impact of vibration of appendages on

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 695758, 6 pageshttp://dx.doi.org/10.1155/2014/695758

2 Mathematical Problems in Engineering

the attitude control system, the structural filter is designedto reject the vibration of flexible appendages. Consideringthe potential problem of in-orbit frequency variation of theflexible appendages, the design method for the adaptivenotch filter is proposed based on the in-orbit identificationtechnology. The in-orbit vibration frequency of appendagescan be obtained by above technology and structural filter canbe generated adaptively so that the purpose of improving thecontrol accuracy of satellite attitude can be realized.This is themain contribution of this paper. The simulation results showthat the proposed attitude control system can still achievegood attitude control accuracy when the vibration frequencyof the flexible appendages changed.

2. Attitude Control System Design

2.1. Pitch Channel DynamicsModel for Satellite. The equationof pitch channel dynamic model for satellite with flexibleappendages is given by

𝐼𝑦𝜃 − 2

4

∑

𝑖=1

𝐵𝑟𝑖 𝑞𝑖 (𝑡) = 𝑇𝑦,

𝑞𝑖 (𝑡) + 2𝜁𝜔𝑖 𝑞𝑖 (𝑡) + 𝜔2

𝑖𝑞𝑖 (𝑡) − 𝐵𝑟𝑖

𝜃 = 0,

(1)

where 𝜃 is the pitch-axis pointing error output, 𝐼𝑦 =

7750 kg⋅m2 is the satellite pitch inertia, 𝑇𝑦 is the pitch-axis reaction wheel control torque input, 𝑞𝑖 is the 𝑖th orderdisplacement in broad sense for antenna flexible vibration,𝜔𝑖 is the 𝑖th flexible mode frequency in rad/s, and 𝜁 isthe passive damping ratio assumed to be 0.01. 𝐵𝑟𝑖 is the𝑖th coupling coefficient; it describes the influence degree ofthe 𝑖th vibration mode and the center of the antenna rigid.Equation (1) can be expressed by Laplace transform:

𝐼𝑦𝑠2𝜃 (𝑠) − 2𝐵𝑟1𝑠

2𝑞1 (𝑠) − ⋅ ⋅ ⋅ − 2𝐵𝑟𝑛𝑠

2𝑞𝑛 (𝑠) = 𝑇𝑦 (𝑠) , (2)

𝑞𝑖 (𝑠) =𝐵𝑟𝑖𝑠2

𝑠2 + 2𝜁𝜔𝑖 + 𝜔2

𝑖

𝜃 (𝑠) , (𝑖 = 1, 2, . . . , 𝑛) , (3)

where 𝑠 is the Laplace transform variable.Substituting (3) into (2), simplify the equation and select

the fore 4 order modes; the transfer function of pitch channelis obtained as follows:

𝜃 (𝑠)

𝑇𝑦 (𝑠)=

1

𝐼𝑦𝑠2 − 2∑

4

𝑖=1(𝐵2

𝑟𝑖𝑠4/ (𝑠2 + 2𝜁𝜔𝑖𝑠 + 𝜔

2

𝑖))

. (4)

Data of ground test for the fore 4 order vibration mode offlexible appendages are summarized in Table 1.

2.2. Control Law Design. The system block diagram of pitchchannel control system for satellite is shown as in Figure 1.

Since the external disturbance 𝑑 is in cycle, there is nostatic error in the system output under normal situation. PDcontrol law is used:

𝑢 = 𝐾𝑃𝑒 + 𝐾𝐷 𝑒. (5)

Table 1: Modal data for the pitch axis.

Mode number 𝜔𝑖rad/s 𝐵

𝑟𝑖

1 0.569 61.092 4.438 −3.183 12.475 0.8064 23.281 −0.459

PD controller can be designed by the root locus method,where𝐾𝑃,𝐾𝐷 are the proportion and differential coefficients,respectively, with𝐾𝑃 = 68, 𝐾𝐷 = 1000.

Without the structure filter, the open loop magnitudecurve of pitch channel is shown in Figure 2 with solid line.

It can be seen that when the frequency is higher than1.77 rad/s, the system magnitude will go above the 0 dB lineagain. Obviously, the vibration signal, whose frequency ishigher than 1.77 rad/s, will be amplified once it enters thecontrol system. It will seriously affect the accuracy of theattitude control. Therefore, we need to design structure filterto reduce the effects on the pitch attitude control channelcaused by the vibration.

2.3. Structure Filter Design. Notch filter is one of the commonstructural filters. It has a good inhibition for fixed frequencyvibrations and there is no effect on the system bandwidth.Therefore, the notch filter is used in this study for rejectingthe vibration of appendages and its general expression is asfollows:

𝑠2/𝜔2

𝑧+ 2𝜁𝑧/𝜔𝑧 + 1

𝑠2/𝜔2𝑝+ 2𝜁𝑝/𝜔𝑝 + 1

, (6)

where𝜔𝑧,𝜔𝑝, 𝜁𝑧, 𝜁𝑝 are the undetermined parameters of filter.Notch filter for different purpose can be obtained throughadjusting these parameters such as band elimination filter,band-pass filter, low-pass filter, high-pass filter, and phaseleading/lagging filter. In order to reduce the influence fromthe fore two order modes on attitude control system, theexpressions of notch filter are selected based on the frequencycorresponding to the peak:

𝑆1 (𝑠) =

𝑠2/𝜔2

𝑧1

+ 2𝜁𝑧1

/𝜔𝑧1

+ 1

𝑠2/𝜔2𝑝1

+ 2𝜁𝑝1

/𝜔𝑝1

+ 1,

𝑆2 (𝑠) =

𝑠2/𝜔2

𝑧2

+ 2𝜁𝑧2

/𝜔𝑧2

+ 1

𝑠2/𝜔2𝑝2

+ 2𝜁𝑝2

/𝜔𝑝2

+ 1,

(7)

where 𝜁𝑧1

= 𝜁𝑧2

= 0.01, 𝜁𝑝1

= 𝜁𝑝2

= 0.99, 𝜔𝑧1

= 2.88, 𝜔𝑝1

=

(1 − 𝜎)𝜔𝑧1

, 𝜔𝑧2

= 4.71, and 𝜔𝑝2

= (1 − 𝜎)𝜔𝑧2

.The above filters are called minimum-phase notch filters.

𝜎 is the frequency regulating factor and equals 0.25. Themagnitude plot of open loop system with notch filters isshown in Figure 2with dashed line.Obviously, themagnitudewhich is higher than the bandwidth frequency is suppressedbelow the 0 dB line through introducing the notch filter.

Select 𝜃 = 0.5∘ as the input of system and there is a signal

in the following format in the measurement loop. Figure 3

Mathematical Problems in Engineering 3

Disturbancestorque

d

PD

Controller

e u

Baseline design: n

Filter

Measurement noise

Saturation

SF𝜃 𝜃

KP = 68

KD = 1000

1

Iys2

𝜔g = 50 rad/s𝜁g = 0.7

Gyro

++ + ++

++

𝜔2g

s2 + 2𝜁g𝜔gs + 𝜔2g

∑i

2B2ris

2

Iy(s2 + 2𝜁𝜔is + 𝜔2

i

−

Figure 1: Block diagram of the model pitch-axis control system.

Mag

nitu

de (d

B)

Bode diagram

Frequency (rad/s)10−2 10−1 100 101 102 103

−80

−60

−40

−20

0

20

40

60

80

0.136 rad/s

2.88 rad/s

4.71 rad/s1.77 rad/s

Figure 2: Bode magnitude plot of open loop system with/withoutthe notch filter.

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Ang

le (d

eg)

PD

60 65 70 750.45

0.50.55

−0.1

PD + SF

Figure 3: Response curves of pitch angle.

0 10 20 30 40 50 60 70 80 90 100Time (s)

55 60 65 70

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Torq

ue (N

·m) −0.4

−0.20

0.20.4

PDPD + SF

Figure 4: Control torque curves.

gives the system output response curves. The dashed linedenotes the system output with only PD control and the solidline with PD control plus notch filter. The correspondingtorque curves are shown in Figure 4.Themeanings of dashedline and solid line are the same as the above mentioned.

And the measurement noise is 𝑛 = 𝐴1 sin(𝜔𝑑𝑡), where𝐴1 = 0.1 and 𝜔𝑑 = 2.80 rad/s.

From Figures 3 and 4, the output responds and controltorque are improved evidently by introducing the notch filter.

3. Adaptive Notch Filter Design

The flexible appendages are susceptible to be damagedbecause the satellite will work in the environment of vacuum,weight loss, temperature variation, and intense radiation sothat the vibration mode might be changed. If this happens,the system response curve is given as in Figure 5 by assumingthat the first order vibration frequency 𝜔1 is changed from0.569 rad/s to 0.435 rad/s.

4 Mathematical Problems in Engineering

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Ang

le (d

eg)

−0.1

Figure 5: System response curve when 𝜔1= 0.435 rad/s.

It can be seen that when the first order vibrationfrequency of flexible appendages changes, the pitch angleresponse curve is divergent. This shows that the notch filteris effective for certain fixed frequency signal; however, whenthis frequency changes, the filter will be powerless. In orderto ensure that the high accuracy of attitude control still can beachieved under the above condition, it is required to estimatethe vibration frequency of in-orbit flexible appendages and toredesign the structural filter based on the estimated value.

3.1. LSM Identification. The least squares method (LSM) isoften used tomodel identification [15]. Assume the differenceequation as

𝐴(𝑧−1) 𝑦 (𝑘) = 𝐵 (𝑧

−1) 𝑢 (𝑘 − 𝑇𝑑) + 𝜀 (𝑘) , (8)

where 𝐴(𝑧−1), 𝐵(𝑧−1) are discrete unit operator polynomials

and 𝜀(𝑘) is external disturbance. Consider

𝐴(𝑧−1) = 1 + 𝑎1𝑧

−1+ 𝑎2𝑧−2

+ ⋅ ⋅ ⋅ + 𝑎𝑛𝑎

𝑧−𝑛𝑎 ,

𝐵 (𝑧−1) = 𝑏0 + 𝑏1𝑧

−1+ 𝑏2𝑧−2

+ ⋅ ⋅ ⋅ + 𝑏𝑛𝑏

𝑧−𝑛𝑏 ,

(9)

where 𝑛𝑎, 𝑛𝑏 are the orders of structure, respectively, and 𝑇𝑑

the order of pure delay link.The identification process is to determine the 𝑛𝑎 + 𝑛𝑏 +

𝑇𝑑 parameters including 𝑎1, 𝑎2, . . . , 𝑎𝑛𝑎

; 𝑏0, 𝑏1, . . . , 𝑏𝑛𝑏

throughdetected inputs and outputs.

Rewrite the identification model equation (8) into

𝑦 (𝑘) = −𝑎1𝑦 (𝑘 − 1) − 𝑎𝑛𝑎

𝑦 (𝑘 − 𝑛𝑎)

+ 𝑏0𝑢 (𝑘 − 𝑇𝑑) + 𝑏𝑛𝑏

𝑢 (𝑘 − 𝑇𝑑 − 𝑛𝑏) + 𝜀 (𝑘)

= 𝜑𝑇(𝑘) 𝜙 + 𝜀 (𝑘) ,

(10)

Structure Flexible

Signal

Systemidentify

modelContinuous

model

System poles

Centerfrequency

Discrete

modelfilter

sample

Figure 6: Realization process of adaptive notch filter.

where 𝜑(𝑘) is the observation vector composed of input andoutput signals and 𝜙 is the coefficient matrix composed of𝑎1, 𝑎2, . . . , 𝑎𝑛

𝑎

; 𝑏0, 𝑏1, . . . , 𝑏𝑛𝑏

. Consider

𝜑 (𝑘) = [−𝑦 (𝑘 − 1) , . . . , −𝑦 (𝑘 − 𝑛𝑛𝑎

) ,

𝑢 (𝑘 − 𝑇𝑑) , . . . , 𝑢 (𝑘 − 𝑇𝑑 − 𝑛𝑛𝑎

)]𝑇

,

𝜙 = [𝑎1, . . . , 𝑎𝑛𝑎

, 𝑏0, . . . , 𝑏𝑛𝑏

]𝑇

.

(11)

The coefficients of object 𝑎1, 𝑎2, . . . , 𝑎𝑛𝑎

; 𝑏0, 𝑏1, . . . , 𝑏𝑛𝑏

canbe obtained based on measured input and output signalsin 𝑛 groups; that is, 𝜙 = (Φ

𝑇Φ)−1Φ𝑇𝑌, where 𝑌 =

[𝑦(1) 𝑦(2) ⋅ ⋅ ⋅ 𝑦(𝑛)]𝑇, Φ = [𝜑

𝑇(1) 𝜑

𝑇(2) ⋅ ⋅ ⋅ 𝜑

𝑇(𝑛)]𝑇

.

3.2. Design of Adaptive Structural Filter. The vibration modelcan be obtained by the above LSM identification. The centerfrequency 𝜔𝑛 of adaptive filter can be calculated throughsolving poles of vibration model. Consider

𝜔𝑛 =1

𝑇𝑠

√ln 𝑧𝑟 ln 𝑧∗𝑟, (12)

where the sampling time 𝑇𝑠 = 0.1, 𝑧𝑟, 𝑧∗

𝑟is one pair

of conjugate poles with the maximum imaginary part. Thecenter frequency of adaptive filter is 𝜔𝑛 = 2.27 rad/s. It canbe known that the system magnitude is slightly increasingwith the vibration frequency of flexible appendages movingforward.Thismakes itmore difficult to design the notch filter.Theminimum-phase notch filter was designed by similar wayand the frequency regulating factor is taken as 𝜎 = 0.4. Theexpression of filter is given as follows:

𝑆1 (𝑠) =𝑠2/𝜔2

𝑛+ 2𝜁1/𝜔𝑛 + 1

𝑠2/(1 − 𝜎)2𝜔2𝑛+ 2𝜁2/ (1 − 𝜎) 𝜔𝑛 + 1

, (13)

where 𝜁1 = 0.01 and 𝜁2 = 0.99.The realization process of adaptive notch filter is shown

in Figure 6 [16].

Mathematical Problems in Engineering 5

65 70 75 80

0.450.5

0.55

0 10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Ang

le (d

eg)

−0.1

PDPD + ASF

+ SF

Figure 7: Response curves of pitch angle.

70 75 80 85

0 10 20 30 40 50 60 70 80 90 100Time (s)

PDPD + ASF

+ SF

−3

−2

−1

0

1

2

3

Torq

ue (N

·m)

−0.2−0.1

00.10.2

Figure 8: Control torque curves.

When the in-orbit frequency of flexible appendagesvaries, the output response of attitude control system withadaptive notch filter is shown in Figure 7. The control torquecurve is shown in Figure 8.

In Figure 7, solid line and dashed line represent thesystem output curves with notch filter and that with adaptivenotch filter, respectively. It is obvious that the system outputwith notch filter is divergent, whereas that with adaptivenotch filter is convergent and control accuracy can bemaintained. In Figure 8, the meanings of dashed line andsolid line are the same as the above mentioned. The controltorque can bemaintained stable with the adaptive notch filter.

The amplitude of torque curve is increasing while comparingwith that in Figure 4. This is caused by the increased gainamplitude of the system due to the changes of flexiblevibration frequency.

4. Conclusions

Notch filter is introduced into the control system designto improve the attitude control accuracy in this study inwhich the pitch angle attitude control system for satellitewith flexible appendages is taken as the object. It showsthat notch filter can well reject the appendages vibrationwith fixed frequency, but it will become powerless when thevibration frequency of flexible appendages varied. During thedesign of notch filter, the identification technology is adoptedto obtain the real-time estimate of vibration frequency offlexible appendages and the notch filter will be generatedadaptively based on the estimate value. This kind of filter canwell reject the potential uncertain flexible vibration so thatthe high accuracy attitude control can be ensured and systemreliability is improved.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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6 Mathematical Problems in Engineering

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