research article the effects of matched filter on stable...

Research ArticleThe Effects of Matched Filter on Stable Performance ofSemistrapdown Inertially Stabilized Platform

Feng Liu and Hua Wang

School of Astronautics Beihang University (BUAA) Beijing 100191 China

Correspondence should be addressed to Feng Liu lfjssx126com

Received 4 March 2016 Revised 2 May 2016 Accepted 26 June 2016

Academic Editor Romain Aubry

Copyright copy 2016 F Liu and H WangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To enhance the optimization performance of matched filter and further improve line of sight (LOS) stability of platform in inertialspace the proposed matched filter algorithm is conducted by adjusting matched filter coefficients of first-order low pass filterutilizing the regional search method based on invariance principle The coefficients of the fraction molecule and denominator ofproposed regional search algorithm are altered instead of denominator coefficients only beingmodified Simulations are performedto verify the validity of inside factors performedwith stabilization controlmodel and quartz rate sensor (QRS)mathematical modelThe stable angular error is sharply alleviated so the decoupling accuracy of airborne semistrapdown inertially stabilized platformis largely promoted The optimization matched filter can effectively increase stability of LOS in inertial space

1 Introduction

Photoelectric sensor is a device of great importance fordetecting and tracking targets in semistrapdown stabilizedplatform [1] LOS of the photoelectric sensor is affected whenflying body attitude changes because the photoelectric sensoris directly connected with flying body in [2] which leads tounstable optical axis andnonideal tracking effectThenoise ofsensor is attenuated using matched filter in general The rategyro sensor can measure azimuth and pitch angular velocityWhile the angular velocity is directly fed back to actuator andmakes LOS reversely deflect so as to achieve stabilization theattitude information is fed back to the closed loop accordingto space coordinate transformation when the information ismeasured by inertia devices of flying bodies which makeframes reduce vibration by flying bodies disturbance

Scholars put forward some opinions on increasing stabil-ity In 1993 strapdown platform model was investigated anddecoupling results of QRS and FOG sensor were obtainedin [3] In recent years a controller was established basedon offline initialization to get the optimal controller andmodeling errorswere solved by optimization filters in [4]Thestability of parasitic loop induced by disturbance rejection

effect (DRE) of a semistrapdown homing seeker (SSHS) wasemployed in [5] The sensorsrsquo dynamic errors of strapdowndetector and rate gyro based on guidance system wereaddressed in [6] The matching of rate gyro and dynamicswere researched utilizing constraining nonlinear minimiza-tion optimization method in [7] A newly continuously dif-ferentiable friction model and filtered regression estimationparameter were introduced the stability of the proposedmethods was proved in [8] The matched filter was expressedin order to suppress and compensate the imperfect influenceof nonlinear friction force factors in [9ndash11] for instancethe static friction force of the frame and motor dead zonephenomenon

Previousmatched filter researches are just mostly focusedon the change of the denominator coefficient while themolecular coefficient of the first-order low-pass filter is afixed parameter However to further improve the optimiza-tion performance the proposed regional search algorithmdynamically limits the search area and reduces the searchcomplexity of the algorithm in time and space so theoperation efficiency of the algorithm is greatly improved

The overall paper is organized as follows Section 1addresses the research purpose Section 2 presents the control

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8389350 9 pageshttpdxdoiorg10115520168389350

2 Mathematical Problems in Engineering

minus

minus

120585

120596

1205960

1205962

Td

Gc(s)

Gd(s)

Gg(s)

GJ(s)

Figure 1 Control model of semistrapdown platform

model of semistrapdown stabilization Section 3 developsmatched filter optimization algorithms Section 4 proves theefficiency of the proposed matched filter optimization algo-rithm and Section 5 summarizes theoretical and practicalengineering significance of the study

2 Semistrapdown Stabilization Control Model

21 Stabilization Principle The two-axis and two-frameminiature semistrapdown stabilization platform has itsadvantages which makes it become a great tool for the inte-gration of investigation and combat In [12] themathematicalmodel of semistrapdown stabilization control is shown inFigure 1

In this system 120596 is angular velocity instruction 1205962is

angular velocity of stabilized platform under inertial axisflying bodies attitude disturbance angular velocity is 120596

0 120578

denotes measurement noise 119879119889is disturbance torque 119866

119869is

control object transfer function 119866119888is transfer function of

speed loop controller in [13] 119866119889represents transfer function

of measurement rate link and 119866119892represents gyro transfer

function The control model of semistrapdown platform isshown in Figure 1 The angular velocity of semistrapdownstabilized platform control model under inertial axis isextracted as

1205962

=

119866119869sdot 119866119888sdot 120596 + 119866

119869sdot 119866119888sdot (119866119889minus 119866119892) sdot 1205960+ 119866119869sdot 119879119889minus 119866119869sdot 119866119888sdot 120578

1 + 119866119869sdot 119866119888sdot 119866119889

(1)

where 1205960= 119860 sin(2120587119891119905) 119891 is body disturbance frequency

and 119860 denotes the maximum amplitude of the angularvelocity of the flying bodies

When 120596 119879119889 and 120585 are ignored we have

1205962=

119866119869sdot 119866119888(119866119889minus 119866119892)

1 + 119866119869sdot 119866119888sdot 119866119889

1205960 (2)

The disturbance of body to the LOS is eliminated when119866119892= 119866119889from (2) which is ideally equivalent to the complete

decoupling of semistrapdown stabilization

22 Simplified Control Model The closed-loop structure ofstable rate is simplified as shown in Figure 2

minus +120596 1205960Gd

Gg(s)

Gc(s)GJ(s)

Figure 2 The closed-loop structure of stable rate

Table 1 The related data of the QRS model

Dynamic component Expression

Compensation 700(11990450 + 1)

119904(1199041000 + 1)

Inertia 005Differentiator 119904

(1199047560 + 1)

Gyro 1

1199042194551198905 + 11990431546 + 1

Converter 1199047854 + 1

(11990418850) sdot (119904235781198908 + 11990413398 + 1)

3 The Optimization of MatchedFilter Algorithm

In order to make 119866119892= 119866119889 matched filter algorithm is pro-

posed in engineering as shown in (2) where119866119889is conducted

under the matched filter according to invariance principleThe transfer function of matched filter is assumed to be

119866 =

119886 sdot 119904 + 1

119887 sdot 119904 + 1

(119886 lt 119887) (3)

The related data of the QRS model of matched filteralgorithm is based on [3] as reflected in Table 1

31 Algorithm 1 1198660represents the transfer function before

matched filter (matched filter is not used) which can beexpressed as

1198660=

1205960

1205961

=

(119866119889sdot 119866119877119863minus 119866119892) sdot 119866119888sdot 119866119869

1 + 119866119889sdot 119866119877119863sdot 119866119888sdot 119866119869

(4)

The transfer function after matched filter is

1198661=

1205960

1205961

=

(119866119889sdot 119866119877119863sdot 119866119898minus 119866119892) sdot 119866119888sdot 119866119869

1 + 119866119889sdot 119866119877119863sdot 119866119898sdot 119866119888sdot 119866119869

(5)

Theobjective function can be established by the nonlinearconstrained optimization algorithm of mechanical optimiza-tion design scheme [14]

119891 (119909) = min (10038161003816100381610038161198661

1003816100381610038161003816minus10038161003816100381610038161198660

1003816100381610038161003816) (6)

where the constraint condition is 119886 ge 119887 ge 0 At last 119886 =0 119887 = 003 is obtained by simulation

32 Algorithm 2 The flowchart of Algorithm 2 is indicatedin Figure 3

(i) The First Step In order to compare with Algorithm 1 theinterval [0 001] is regarded as research subject The interval

Mathematical Problems in Engineering 3

a b isin 0001 0002 0003 001

a = x b gt x x isin 0001 0002 0003 001

[a b] isin [0001 0004] [0 0003]

a isin 0 00001 00002 0001 b isin 0003 000031 0004

[a b] = [0 0003]

a = 0 b isin 0003 0000301 000309

[a b] = [0 000304]

Figure 3 Flowchart of Algorithm 2

a b isin 0001 0002 0003 0004 0005 0006 0007 0008 0009 001

[a b] isin [0001 0004] [0 0003]

a = 0

b gt 0

a = 0001

b gt 0001

a = 0002

b gt 0002

a = 0003

b gt 0003

a = 0004

b gt 0004

a = 0005

b gt 0005

a = 0006

b gt 0006

a = 0007

b gt 0007

a = 0008

b gt 0008

a = 0009

b gt 0009

b = 0003 b = 0004 b = 0005 b = 0006 b = 0007 b = 0008 b = 0009 b = 001b = 001b = 001

Figure 4 Algorithm structure diagram of the first step

[0 001] is divided into 10 equal parts there are eleven num-bers from 0 0001 to 001 Let 119886 119887 isin 0 0001 0002 00110 kinds of situations are displayed in line two of Figure 4when 119887 is greater than 119886 The optimal matching of everysituation is acquired by simulationThen the best decouplingcharacteristics are regarded as a new group then 10 groupsare reflected in line 3 of Figure 4 Followed by analogy thebetter of two groups is shown in Figure 4

The better simulation effect of [0001 0004] and[0 0003] is obtained from Figure 4 then the better result issearched from [0 0001] and [0003 0004]

(ii) The Second Step The interval [0 0001] is divided into10 equal parts there are eleven numbers from 0 00001to 0001 Let 119886 isin 0 0001 0002 001 Similarly theinterval [0003 0004] is divided into 10 equal parts where


a = 0

b1 isin b

a = 00001

b2 isin b

a = 00002

b3 isin b

a = 00003

b4 isin b

a = 00004

b5 isin b

a = 00005

b6 isin b

a = 00006

b7 isin b

a = 00007

b8 isin b

a = 00008

b9 isin b

a = 00009

b10 isin b

a = 0001

b11 isin b

b1 = 0003 b2 = 00031 b3 = 00032 b4 = 00033 b5 = 00034 b6 = 00035 b7 = 00036 b8 = 00037 b9 = 00038 b10 = 00039 b11 = 0004

[a b] = [0 0003]

a isin 0 00001 00002 00003 00004 00005 00006 00007 00008 00009 0001

b isin 0003 00031 00032 00033 00034 00035 00036 00037 00038 00039 0004

Figure 5 Algorithm structure diagram of the second step

b1 = 0003 b2 = 000301 b3 = 000302 b4 = 000303 b5 = 000304 b6 = 000305 b7 = 000306 b8 = 000307 b9 = 000308 b10 = 000309

[0 000304] [0 000303] [0 000305] [0 0003][a b] =

[a b] = [0 000304]

a = 0

b isin 0003 000301 000302 000303 000304 000305 000306 000307 000308 000309

Figure 6 Algorithm structure diagram of the third step

there are eleven numbers from 0003 00031 to 0004 Let119887 isin 0003 00031 0004 thus eleven kinds of situationsare illustrated in line two of Figure 5 when 119887

119894isin 119887 119894 =

1 2 11The optimalmatching of every situation is gainedby simulation Then the best decoupling characteristics areregarded as a new group and 11 groups are reflected in line 3of Figure 5 Followed by analogy the best group is shown inFigure 5

The last result of Figure 5 is completely consistentwith theresult of Algorithm 1 by simulink However we hope to finda better result by search method(iii) The Third Step Let 119887 isin 0003 000301 000302 000309 and 119886 = 0 Ten kinds of situations are shown inline 2 of Figure 6 Then four groups of the better decouplingcharacteristics are selected they are reflected in line 3 ofFigure 6 Followed by analogy the best group is shown in


Friction

QRScompensation

TransferFcn

+

+

+

+

+

+

minus

+

minus

ConstantGimbalinertia

Differentiator Matchedfilter RD converter Integrator

Flyingbodiesmotion

Quartz ratesensor

Sensornoise input

ratePlatform

Figure 7 Optimization model of inertial stabilization platform

Time

[0003]

[0001

0004

]

[0002

0005

]

[0003

0006

]

[0004

0007

]

[0005

0008

]

[0006

0009

]

[0007

001]

[0008

001]

[0009

001]

08070605040302010

(a)

08070605040302010

Time (s)

[00003

]

[0000100031

]

[0000200032

]

[0000300033

]

[0000400034

]

[0000500035

]

[0000600036

]

[0000700037

]

[0000800038

]

[0000900039

]

[0001

0004

]

(b)Time

0350345034033503303250320315031030503

[00003

]

[0000301

]

[0000302

]

[0000303

]

[0000304

]

[0000305

]

[0000306

]

[0000307

]

[0000308

]

[0000309

]

(c)

Figure 8 Step response time of three steps

Figure 6 The last result of Figure 6 is a perfect result bysimulink

4 Validation Test and Simulation Analysis

In order to better explain the validity of the algorithm takingthe closed-loop stability control system into considerationthe simulation model in [15] is shown in Figure 7

41 Simulation Experiment Validations

411 The Step Simulation Experiments Considering thespeed of reaching the steady state of the system the stepresponse is presented in Figure 8

The time of reaching the steady state is very principal forengineering application It is clear that the [0001 0004] and[0 0003] are excellent among ten group coefficients of the


Mag

nitu

de (d

B)

Phas

e (de

g)

Frequency (rads) Frequency (rads)

0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

Bode diagram Bode diagram

(a)

Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108


[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

(b)

Mag

nitu

de (d

B)

Phas

e (de

g)


360

180

0

minus180

minus360

100 101 102 103 105104 106


[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

0

minus50

minus100

minus150

minus200

minus250

minus300100 101 102 103 105104 106

[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

(c)

Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter


Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1

Before filteringAfter filtering

Plat

form

rate

(deg

s)

Captive carry (times)

60

40

20

0

minus20

minus40

minus600 02 04 06 08 1


Free flight number 1 (times) Free flight number 2 (times)

Free flight number 3 (times)

Figure 10 The platform rate before matched filter and after matched filter

first step in Figure 8(a) the result of simulation experimentsis consistent with the result of Algorithm 1 and the timeof [0001 0004] and [0 0003] when arriving at the steadystate is shorter than others The step response is indicated inFigure 8(b) the reaching speed of the steady state about thecoefficient [0 0003] is faster than others The coefficient of[0 000304] is the best among ten coefficients and it can beconfirmed based on Figure 8(c)

412 Bode Diagram Simulation Experiments The matchingeffect of the matched filter plays an indispensable role in

engineering It is helpful even if there is a little improvementas is shown in Figure 9

The coefficients [0001 0004] and [0 0003] ofFigure 8(a) are very prominent The coefficient [0 0003] ofthe second step is indicated in Figure 8(b) where it is shownthat the optimization result is in accordance with the resultof Algorithm 1 The coefficients [0 000304] [0 000303][0 000305] and [0 0003] are better from Bode diagramof the closed-loop control simulation But the coefficient[0 000304] is the best Figure 8(c) can be illustrated bythis truth Meanwhile the coefficient [0 000304] has a


Table 2 The input rate of flying bodies

Environment Rate sdot 120596119898

Degs HzFree flight number 1 130 5Free flight number 2 260 25Free flight number 3 430 5Captive carry 315 50

06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]

Figure 11 Stable angle error under different matched filter

relatively higher decoupling accuracy and the noise also canbe decreased so 119866

119898= 1(000304119904 + 1) is the best matched

filter

42 Stable Error beforeMatched Filter and afterMatched Filter

421 The Rate Comparison of Platform before and afterMatched Filter The related data of flying bodies angularvelocity motion is given based on [3] which is used as theverification test when the input signal is the unit step signalas shown in Table 2

As shown in Figure 10 the rate of stable platform beforematched filter and after matched filter is as follows

Simulation results from Figure 10 show that the angularrate of the platform is declined by 90 so the optimizationresults are very good

422 The Comparison of Stable Angle Error under DifferentMatched Filter The stable angle errors of the four groupsrsquomatched filter of Figure 8 are compared utilizing searchmethod and their differences are revealed in Figure 11

Stable angle error is obviously distinct using dissimilarmatched filter from Figure 11 the solid line stands for the bestmatched filter its error of the stable angle is smaller

After matchingBefore matching

4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)

Figure 12 Stable angular error

43 Effect on the Stable Error of Sensor Measurement NoiseThe angular velocity of the semistrapdown stabilized plat-form is obtained by rate gyroscope and related calculationfrom Figure 1 and (1) The measurement noise of the sensorhas influence on stabilization of semistrapdown stabilizedplatform The amplification factor of the measurement ratecan be enlarged but not without limitation The matchedfilter algorithm is proposed in order to reduce the noise inengineering

In Figure 12 the measurement noise of the sensor has agreat effect on the stability of the angle error and the stabilityof the angle error is diminished by nearly 70 after matchedfilter

5 Conclusions

(1) Measurement rate is matched by first-order low passfilter based on invariance principle Simulations showthat the angular rate of the platform is lessened by90aftermatched filter Not only canwe get the resultof Algorithm 1 but also we can obtain The optimalmatching which can promote decoupling accuracy asfar as possible

(2) The measurement noise of sensor has huge influenceon the stable error The stability of the angle error isdecreased by nearly 70 after matched filter

(3) The stability of LOS can be strengthened based onthe above simulation results So it provides theoreticalfoundations for designing and optimization of themicrostable platform which has a strong guidingsignificance in engineering


Competing Interests

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

References

[1] S Yin H Jia Y Zhang and H Gao ldquoSemi-strapdown stabiliza-tion of optical imaging seekerrdquo Infrared and Laser Engineeringvol 40 no 1 pp 129ndash148 2011

[2] X Zhou Z Zhang and D Fan ldquoImproved angular velocityestimation using MEMS sensors with applications in miniatureinertially stabilized platformsrdquo Chinese Journal of Aeronauticsvol 24 no 5 pp 648ndash656 2011

[3] R T Rudin ldquoStrapdown stabilization for imaging seekersrdquoin Proceedings of the 2nd Annual AIAA SDIO InterceptorTechnology Conference pp 1ndash10 June 1993

[4] Z-R Tsai ldquoNeural-fuzzy digital strategy of continuous-timenonlinear systems using adaptive prediction and random-local-optimization designrdquo Mathematical Problems in Engineeringvol 2013 Article ID 836414 12 pages 2013

[5] S Jianmei C Gaohua C Xianxiang and K Lixia ldquoStabilityregion analysis of the parasitic loop of the semi-strapdownhoming seekerrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems and Control Engineeringvol 226 no 4 pp 550ndash562 2012

[6] S-A Jang C-K Ryoo K Choi and M-J Tahk ldquoGuidancealgorithms for tactical missiles with strapdown seekerrdquo inProceedings of the SICEAnnual Conference pp 2616ndash2619 IEEETokyo Japan August 2008

[7] Y Yifang Research on Guidance and Control Technology forStrapdown Guided Munition Beijing Institute of TechnologyBeijing China 2015

[8] Z-J Fu W-D Xie and X-B Ning ldquoAdaptive nonlinear tire-road friction force estimation for vehicular systems based on anovel differentiable friction modelrdquo Mathematical Problems inEngineering vol 2015 Article ID 201062 7 pages 2015

[9] Z Zhiyong A Study on Key Measurement and Control Problemsof Electro-Optical Stabilization Servo Mechanism National Uni-versity of Defense Technology Changsha China 2006

[10] J Song G Cai L Kong and J Fan ldquoPrecision analysis of thesemi-strapdown homing guided systemrdquo Journal of AerospaceEngineering vol 27 no 1 pp 151ndash167 2014

[11] J Xu J Wang T Song and K-R Hu ldquoA disturbance observer-based inhibition method for disturbance rejection rate ofseekerrdquo Acta Armamentarii vol 35 no 11 pp 1790ndash1798 2014

[12] A Lawrence Modern Inertial Technology Navigation Guid-ance and Control Springer Science and Business Media 2012

[13] R Yin RWang X Y Zhou X Y Peng andKWang ldquoDynamicmodeling and nonlinear decoupling control of inertial sta-bilized platform for aerial remote sensing systemrdquo AdvancedMaterials Research vol 898 pp 807ndash813 2014

[14] S S Rao and S S Rao Engineering Optimization Theory andPractice John Wiley and Sons 2009

[15] Moore and Holly MATLAB for Engineers Prentice Hall Press2014

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Stochastic AnalysisInternational Journal of


minus

minus

120585

120596

1205960

1205962

Td

Gc(s)

Gd(s)

Gg(s)

GJ(s)

Figure 1 Control model of semistrapdown platform

model of semistrapdown stabilization Section 3 developsmatched filter optimization algorithms Section 4 proves theefficiency of the proposed matched filter optimization algo-rithm and Section 5 summarizes theoretical and practicalengineering significance of the study

2 Semistrapdown Stabilization Control Model

21 Stabilization Principle The two-axis and two-frameminiature semistrapdown stabilization platform has itsadvantages which makes it become a great tool for the inte-gration of investigation and combat In [12] themathematicalmodel of semistrapdown stabilization control is shown inFigure 1

In this system 120596 is angular velocity instruction 1205962is

angular velocity of stabilized platform under inertial axisflying bodies attitude disturbance angular velocity is 120596

0 120578

denotes measurement noise 119879119889is disturbance torque 119866

119869is

control object transfer function 119866119888is transfer function of

speed loop controller in [13] 119866119889represents transfer function

of measurement rate link and 119866119892represents gyro transfer

function The control model of semistrapdown platform isshown in Figure 1 The angular velocity of semistrapdownstabilized platform control model under inertial axis isextracted as

1205962

=

119866119869sdot 119866119888sdot 120596 + 119866

119869sdot 119866119888sdot (119866119889minus 119866119892) sdot 1205960+ 119866119869sdot 119879119889minus 119866119869sdot 119866119888sdot 120578

1 + 119866119869sdot 119866119888sdot 119866119889

(1)

where 1205960= 119860 sin(2120587119891119905) 119891 is body disturbance frequency

and 119860 denotes the maximum amplitude of the angularvelocity of the flying bodies

When 120596 119879119889 and 120585 are ignored we have

1205962=

119866119869sdot 119866119888(119866119889minus 119866119892)

1 + 119866119869sdot 119866119888sdot 119866119889

1205960 (2)

The disturbance of body to the LOS is eliminated when119866119892= 119866119889from (2) which is ideally equivalent to the complete

decoupling of semistrapdown stabilization

22 Simplified Control Model The closed-loop structure ofstable rate is simplified as shown in Figure 2

minus +120596 1205960Gd

Gg(s)

Gc(s)GJ(s)

Figure 2 The closed-loop structure of stable rate

Table 1 The related data of the QRS model

Dynamic component Expression

Compensation 700(11990450 + 1)

119904(1199041000 + 1)

Inertia 005Differentiator 119904

(1199047560 + 1)

Gyro 1

1199042194551198905 + 11990431546 + 1

Converter 1199047854 + 1

(11990418850) sdot (119904235781198908 + 11990413398 + 1)

3 The Optimization of MatchedFilter Algorithm

In order to make 119866119892= 119866119889 matched filter algorithm is pro-

posed in engineering as shown in (2) where119866119889is conducted

under the matched filter according to invariance principleThe transfer function of matched filter is assumed to be

119866 =

119886 sdot 119904 + 1

119887 sdot 119904 + 1

(119886 lt 119887) (3)

The related data of the QRS model of matched filteralgorithm is based on [3] as reflected in Table 1

31 Algorithm 1 1198660represents the transfer function before

matched filter (matched filter is not used) which can beexpressed as

1198660=

1205960

1205961

=

(119866119889sdot 119866119877119863minus 119866119892) sdot 119866119888sdot 119866119869

1 + 119866119889sdot 119866119877119863sdot 119866119888sdot 119866119869

(4)

The transfer function after matched filter is

1198661=

1205960

1205961

=

(119866119889sdot 119866119877119863sdot 119866119898minus 119866119892) sdot 119866119888sdot 119866119869

1 + 119866119889sdot 119866119877119863sdot 119866119898sdot 119866119888sdot 119866119869

(5)

Theobjective function can be established by the nonlinearconstrained optimization algorithm of mechanical optimiza-tion design scheme [14]

119891 (119909) = min (10038161003816100381610038161198661

1003816100381610038161003816minus10038161003816100381610038161198660

1003816100381610038161003816) (6)

where the constraint condition is 119886 ge 119887 ge 0 At last 119886 =0 119887 = 003 is obtained by simulation

32 Algorithm 2 The flowchart of Algorithm 2 is indicatedin Figure 3

(i) The First Step In order to compare with Algorithm 1 theinterval [0 001] is regarded as research subject The interval


a b isin 0001 0002 0003 001

a = x b gt x x isin 0001 0002 0003 001

[a b] isin [0001 0004] [0 0003]

a isin 0 00001 00002 0001 b isin 0003 000031 0004

[a b] = [0 0003]

a = 0 b isin 0003 0000301 000309

[a b] = [0 000304]


a b isin 0001 0002 0003 0004 0005 0006 0007 0008 0009 001

[a b] isin [0001 0004] [0 0003]

a = 0

b gt 0

a = 0001

b gt 0001

a = 0002

b gt 0002

a = 0003

b gt 0003

a = 0004

b gt 0004

a = 0005

b gt 0005

a = 0006

b gt 0006

a = 0007

b gt 0007

a = 0008

b gt 0008

a = 0009

b gt 0009

b = 0003 b = 0004 b = 0005 b = 0006 b = 0007 b = 0008 b = 0009 b = 001b = 001b = 001






a = 0

b1 isin b

a = 00001

b2 isin b

a = 00002

b3 isin b

a = 00003

b4 isin b

a = 00004

b5 isin b

a = 00005

b6 isin b

a = 00006

b7 isin b

a = 00007

b8 isin b

a = 00008

b9 isin b

a = 00009

b10 isin b

a = 0001

b11 isin b

b1 = 0003 b2 = 00031 b3 = 00032 b4 = 00033 b5 = 00034 b6 = 00035 b7 = 00036 b8 = 00037 b9 = 00038 b10 = 00039 b11 = 0004

[a b] = [0 0003]

a isin 0 00001 00002 00003 00004 00005 00006 00007 00008 00009 0001

b isin 0003 00031 00032 00033 00034 00035 00036 00037 00038 00039 0004


b1 = 0003 b2 = 000301 b3 = 000302 b4 = 000303 b5 = 000304 b6 = 000305 b7 = 000306 b8 = 000307 b9 = 000308 b10 = 000309

[0 000304] [0 000303] [0 000305] [0 0003][a b] =

[a b] = [0 000304]

a = 0

b isin 0003 000301 000302 000303 000304 000305 000306 000307 000308 000309



119894isin 119887 119894 =




Friction

QRScompensation

TransferFcn

+

+

+

+

+

+

minus

+

minus



Flyingbodiesmotion

Quartz ratesensor

Sensornoise input

ratePlatform


Time

[0003]

[0001

0004

]

[0002

0005

]

[0003

0006

]

[0004

0007

]

[0005

0008

]

[0006

0009

]

[0007

001]

[0008

001]

[0009

001]

08070605040302010

(a)

08070605040302010

Time (s)

[00003

]

[0000100031

]

[0000200032

]

[0000300033

]

[0000400034

]

[0000500035

]

[0000600036

]

[0000700037

]

[0000800038

]

[0000900039

]

[0001

0004

]

(b)Time

0350345034033503303250320315031030503

[00003

]

[0000301

]

[0000302

]

[0000303

]

[0000304

]

[0000305

]

[0000306

]

[0000307

]

[0000308

]

[0000309

]

(c)









Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]


(a)

Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108


[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

(b)

Mag

nitu

de (d

B)

Phas

e (de

g)


360

180

0

minus180

minus360

100 101 102 103 105104 106


[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

0

minus50

minus100

minus150

minus200

minus250

minus300100 101 102 103 105104 106

[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

(c)



Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1


Plat

form

rate

(deg

s)


60

40

20

0

minus20

minus40

minus600 02 04 06 08 1













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










a b isin 0001 0002 0003 001

a = x b gt x x isin 0001 0002 0003 001

[a b] isin [0001 0004] [0 0003]

a isin 0 00001 00002 0001 b isin 0003 000031 0004

[a b] = [0 0003]

a = 0 b isin 0003 0000301 000309

[a b] = [0 000304]


a b isin 0001 0002 0003 0004 0005 0006 0007 0008 0009 001

[a b] isin [0001 0004] [0 0003]

a = 0

b gt 0

a = 0001

b gt 0001

a = 0002

b gt 0002

a = 0003

b gt 0003

a = 0004

b gt 0004

a = 0005

b gt 0005

a = 0006

b gt 0006

a = 0007

b gt 0007

a = 0008

b gt 0008

a = 0009

b gt 0009

b = 0003 b = 0004 b = 0005 b = 0006 b = 0007 b = 0008 b = 0009 b = 001b = 001b = 001






a = 0

b1 isin b

a = 00001

b2 isin b

a = 00002

b3 isin b

a = 00003

b4 isin b

a = 00004

b5 isin b

a = 00005

b6 isin b

a = 00006

b7 isin b

a = 00007

b8 isin b

a = 00008

b9 isin b

a = 00009

b10 isin b

a = 0001

b11 isin b

b1 = 0003 b2 = 00031 b3 = 00032 b4 = 00033 b5 = 00034 b6 = 00035 b7 = 00036 b8 = 00037 b9 = 00038 b10 = 00039 b11 = 0004

[a b] = [0 0003]

a isin 0 00001 00002 00003 00004 00005 00006 00007 00008 00009 0001

b isin 0003 00031 00032 00033 00034 00035 00036 00037 00038 00039 0004


b1 = 0003 b2 = 000301 b3 = 000302 b4 = 000303 b5 = 000304 b6 = 000305 b7 = 000306 b8 = 000307 b9 = 000308 b10 = 000309

[0 000304] [0 000303] [0 000305] [0 0003][a b] =

[a b] = [0 000304]

a = 0

b isin 0003 000301 000302 000303 000304 000305 000306 000307 000308 000309



119894isin 119887 119894 =




Friction

QRScompensation

TransferFcn

+

+

+

+

+

+

minus

+

minus



Flyingbodiesmotion

Quartz ratesensor

Sensornoise input

ratePlatform


Time

[0003]

[0001

0004

]

[0002

0005

]

[0003

0006

]

[0004

0007

]

[0005

0008

]

[0006

0009

]

[0007

001]

[0008

001]

[0009

001]

08070605040302010

(a)

08070605040302010

Time (s)

[00003

]

[0000100031

]

[0000200032

]

[0000300033

]

[0000400034

]

[0000500035

]

[0000600036

]

[0000700037

]

[0000800038

]

[0000900039

]

[0001

0004

]

(b)Time

0350345034033503303250320315031030503

[00003

]

[0000301

]

[0000302

]

[0000303

]

[0000304

]

[0000305

]

[0000306

]

[0000307

]

[0000308

]

[0000309

]

(c)









Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]


(a)

Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108


[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

(b)

Mag

nitu

de (d

B)

Phas

e (de

g)


360

180

0

minus180

minus360

100 101 102 103 105104 106


[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

0

minus50

minus100

minus150

minus200

minus250

minus300100 101 102 103 105104 106

[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

(c)



Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1


Plat

form

rate

(deg

s)


60

40

20

0

minus20

minus40

minus600 02 04 06 08 1













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










a = 0

b1 isin b

a = 00001

b2 isin b

a = 00002

b3 isin b

a = 00003

b4 isin b

a = 00004

b5 isin b

a = 00005

b6 isin b

a = 00006

b7 isin b

a = 00007

b8 isin b

a = 00008

b9 isin b

a = 00009

b10 isin b

a = 0001

b11 isin b

b1 = 0003 b2 = 00031 b3 = 00032 b4 = 00033 b5 = 00034 b6 = 00035 b7 = 00036 b8 = 00037 b9 = 00038 b10 = 00039 b11 = 0004

[a b] = [0 0003]

a isin 0 00001 00002 00003 00004 00005 00006 00007 00008 00009 0001

b isin 0003 00031 00032 00033 00034 00035 00036 00037 00038 00039 0004


b1 = 0003 b2 = 000301 b3 = 000302 b4 = 000303 b5 = 000304 b6 = 000305 b7 = 000306 b8 = 000307 b9 = 000308 b10 = 000309

[0 000304] [0 000303] [0 000305] [0 0003][a b] =

[a b] = [0 000304]

a = 0

b isin 0003 000301 000302 000303 000304 000305 000306 000307 000308 000309



119894isin 119887 119894 =




Friction

QRScompensation

TransferFcn

+

+

+

+

+

+

minus

+

minus



Flyingbodiesmotion

Quartz ratesensor

Sensornoise input

ratePlatform


Time

[0003]

[0001

0004

]

[0002

0005

]

[0003

0006

]

[0004

0007

]

[0005

0008

]

[0006

0009

]

[0007

001]

[0008

001]

[0009

001]

08070605040302010

(a)

08070605040302010

Time (s)

[00003

]

[0000100031

]

[0000200032

]

[0000300033

]

[0000400034

]

[0000500035

]

[0000600036

]

[0000700037

]

[0000800038

]

[0000900039

]

[0001

0004

]

(b)Time

0350345034033503303250320315031030503

[00003

]

[0000301

]

[0000302

]

[0000303

]

[0000304

]

[0000305

]

[0000306

]

[0000307

]

[0000308

]

[0000309

]

(c)









Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]


(a)

Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108


[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

(b)

Mag

nitu

de (d

B)

Phas

e (de

g)


360

180

0

minus180

minus360

100 101 102 103 105104 106


[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

0

minus50

minus100

minus150

minus200

minus250

minus300100 101 102 103 105104 106

[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

(c)



Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1


Plat

form

rate

(deg

s)


60

40

20

0

minus20

minus40

minus600 02 04 06 08 1













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Friction

QRScompensation

TransferFcn

+

+

+

+

+

+

minus

+

minus



Flyingbodiesmotion

Quartz ratesensor

Sensornoise input

ratePlatform


Time

[0003]

[0001

0004

]

[0002

0005

]

[0003

0006

]

[0004

0007

]

[0005

0008

]

[0006

0009

]

[0007

001]

[0008

001]

[0009

001]

08070605040302010

(a)

08070605040302010

Time (s)

[00003

]

[0000100031

]

[0000200032

]

[0000300033

]

[0000400034

]

[0000500035

]

[0000600036

]

[0000700037

]

[0000800038

]

[0000900039

]

[0001

0004

]

(b)Time

0350345034033503303250320315031030503

[00003

]

[0000301

]

[0000302

]

[0000303

]

[0000304

]

[0000305

]

[0000306

]

[0000307

]

[0000308

]

[0000309

]

(c)









Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]


(a)

Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108


[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

(b)

Mag

nitu

de (d

B)

Phas

e (de

g)


360

180

0

minus180

minus360

100 101 102 103 105104 106


[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

0

minus50

minus100

minus150

minus200

minus250

minus300100 101 102 103 105104 106

[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

(c)



Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1


Plat

form

rate

(deg

s)


60

40

20

0

minus20

minus40

minus600 02 04 06 08 1













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]

[0 0003][0001 0004][0002 0005][0003 0006][0004 0007]

[0005 0008][0006 0009][0007 001][0008 001][0009 001]


(a)

Mag

nitu

de (d

B)

Phas

e (de

g)


0

minus100

minus200

minus300

minus400

minus500

minus600

180

90

0

minus90

minus180

minus270

minus360100 102 104 106 108100 102 104 106 108


[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

[0 0003][00001 00031][00002 00032][00003 00033][00004 00034][00005 00035]

[00006 00036][00007 00037][00008 00038][00009 00039][0001 0004]

(b)

Mag

nitu

de (d

B)

Phas

e (de

g)


360

180

0

minus180

minus360

100 101 102 103 105104 106


[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

0

minus50

minus100

minus150

minus200

minus250

minus300100 101 102 103 105104 106

[0 0003][0 000301][0 000302][0 000303][0 000304]

[0 000305][0 000306][0 000307][0 000308][0 000309]

(c)



Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1


Plat

form

rate

(deg

s)


60

40

20

0

minus20

minus40

minus600 02 04 06 08 1













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus150 02 04 06 08 1

Plat

form

rate

(deg

s)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus100 02 04 06 08 1

Plat

form

rate

(deg

s)

20

15

10

5

0

minus5

minus10

minus15

25

minus200 02 04 06 08 1


Plat

form

rate

(deg

s)


60

40

20

0

minus20

minus40

minus600 02 04 06 08 1













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













06

05

04

03

02

01

0

minus01

minus02

minus030 01 02 03 04 05 06 07 08 09 1

Time (s)

Stab

le an

gle e

rror

(∘)

[0 000304][0 000303]

[0 000305][0 000300]




filter








4

3

2

1

0

minus1

minus2

minus30 1 2 3 4 5 6 7

Time (s)

Stab

le an

gula

r err

or(∘)




5 Conclusions





Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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