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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
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
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
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
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
Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter
Mathematical Problems in Engineering 7
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
Before filteringAfter filtering
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
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
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
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
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
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
Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter
Mathematical Problems in Engineering 7
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
Before filteringAfter filtering
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
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
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
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
Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter
Mathematical Problems in Engineering 7
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
Before filteringAfter filtering
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
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
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
Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter
Mathematical Problems in Engineering 7
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
Before filteringAfter filtering
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
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
Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter
Mathematical Problems in Engineering 7
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
Before filteringAfter filtering
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
Figure 9 Comparison of decoupling accuracy of four groupsrsquo matched filter
Mathematical Problems in Engineering 7
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
Before filteringAfter filtering
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
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
8 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 9
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
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
Mathematical Problems in Engineering 9
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
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