Influence of Seeker Disturbance Rejection and Radome Error ...downloads.hindawi.com/journals/mpe/2018/1890426.pdf · MathematicalProblemsinEngineering 0 2 4 6 8 10 Unstable Stable

Research ArticleInfluence of Seeker Disturbance Rejection and Radome Error onthe Lyapunov Stability of Guidance Systems

Kaiwei Chen 1 Qunli Xia 1 Xiao Du 2 and Yuemin Yao 3

1School of Aerospace Engineering Beijing Institute of Technology Beijing 100081 China2China Academy of Launch Vehicle Technology Beijing 100076 China3Beijing Institute of Space Long March Vehicle Beijing 100076 China

Correspondence should be addressed to Kaiwei Chen kwillchanoutlookcom

Received 18 March 2018 Accepted 15 May 2018 Published 20 June 2018

Academic Editor Calogero Orlando

Copyright copy 2018 Kaiwei Chen et al This 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

This study analyzes the effects of disturbance rejection and radome error on the stability of guidance systems First disturbancerejection rate transfer function (DRRTF) models are established Second the Lyapunov stability of a proportional navigationguidance system is proposed The passivity theorems are introduced for the analysis of the Lyapunov stability of the nonlineartime-varying system Finally the influence of the different DRRTF models on the stability of the guidance system is analyzed bymathematical simulations The simulation results indicate that the stable boundary of the guidance system varies and its toleranceto disturbance differs significantly for various disturbance torque types and radome slope

1 Introduction

The stabilized platform of a gimbaled seeker isolates thebody motion caused by missile maneuvering and vibrationduring flight However the isolation is incomplete due towirepulling inside the seeker and friction among componentsConsequently missile disturbance coupling enters the seekerthereby affecting the accuracy of the seeker output [1ndash3]and leading to the disturbance rejection effect The seekerdisturbance rejection effect then forms a parasitic loopwithinthe missile guidance loop that is the disturbance rejectionrate parasitic loop (DRRPL)

Radomes are attached to the nose of radar homingmissiles The refraction of the electromagnetic wave by theradome causes an error in the line-of-sight (LOS) angle thatis the radome refraction angle which also creates a parasiticloop with the guidance homing loop In certain situationsthe disturbance rejection and radome error act to reduce thestability of the guidance system and cause an increase in missdistance [4ndash7]

Several stability analysis approaches for guidance systemshave been suggested in the literature

(1) The first and most commonly used approach isbased on the ldquofrozen timerdquo assumption transforming the

guidance system into a time-invariant system and analyzingthe stability by the Routh criterion [8ndash11] However whenthe distance between the missile and the target is shortor the dynamics of the guidance system are relatively fastthe ldquofrozen timerdquo assumption is no longer valid Moreoverapplying the Routh criterion to analyze the stability generatesa large error

(2) The second approach assumes that the terminal guid-ance time is infinite and uses the stability analysis method oftime-varying systems on the basis of infinite time such as thePopov and circle criteria [4 12 13] These stability conditionsare defined over an infinite-horizon time domain and thusare not suitable for the stability analysis of a homing guidanceloop which is only defined over a finite time interval

(3) The third approach analyzes stability by using thefinite time stability theory of time-varying systems [14ndash17]This approach is practical in real engagement

This study evaluates the Lyapunov stability over a finitetime interval on the basis of the linearized model of theproportional navigation (PN) guidance system The distur-bance rejection rate is the transfer function which representsthe incorrect angular rate caused by an attitude disturbanceapplied to the missilersquos body [18] Therefore this study

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1890426 9 pageshttpsdoiorg10115520181890426

2 Mathematical Problems in Engineering

Target

O

T

Seeker Boresight

Missile Body Centerline

Inertial Reference

xtxs

qt

qs

r

xb

Figure 1 Basic geometry in pitch plane

Stabilized Loop

u

Disturbance Torque Loop

Correction NetworkCorrection Network

Tracking Loop

H(s)

Back EMF Loop

Rate Gyro Dynamics

qt

qs

+minus

+minus

+minusminus

k1

q

G1(s) G2(s) k2

KE

GD(s)

TD1

Ls + RKT

Tem

1

Js

1

s

qs

qs

r +minus

+minus

Figure 2 Control loop model of a gimbaled seeker

establishes the disturbance rejection rate transfer function(DRRTF) models under the joint action of the disturbancetorque and the radome error A PN guidance systemmodel isestablished on the basis of DRRTFThe passivity theorems foranalyzing the Lyapunov stability of nonlinear time-varyingsystems are convenient for engineering application becausethey donot need to build the state spacemodels and constructLyapunov functions [19ndash21] Therefore this study analyzesthe Lyapunov stability of the PN guidance system on the basisof the passivity theorems

This paper is structured as follows Section 2 establishesthe DRRTF and PN guidance systemmodels which considerthe different disturbance torque types and radome errorslope Section 3 defines the uniform asymptotical stabilityof the PN guidance system and introduces the Lyapunovstability represented by passivity theorems Section 4 ana-lyzes the stability of the PN guidance system on the basisof passivity theorems Section 5 investigates the influence ofspring torque damping torque and radome error slope on theLyapunov stability of the PN guidance system by simulationsSection 6 concludes this paper

2 Problem Formulation

21 Modeling of the DRRTF Figure 1 shows the basic geom-etry in pitch plane In Figure 1 119902119905 represents the LOS angle119902119904 is the seeker rotation angle 120599 denotes the missile attitudeangle120593119903 refers to the gimbal angle and 120576 stands for the seeker

LOS error angle From geometry the detector error angle canbe written as follows

120576 = 119902119905 minus 119902119904 (1)Figure 2 presents a typical control loop model of a

gimbaled seekerIn Figure 2 119871 is the inductance of armature winding

119877 denotes the total resistance 119870119879 represents the torqueconstant 119870119864 stands for the back electromotive force (EMF)119879119890119898(119905) refers to the electromagnetic torque 119879119863 signifies thetotal load torque of the motor and load 119869 indicates thetotal motor rotational inertia and 119902 symbolizes the LOS raterequired for guidance

Figure 2 shows that the model mainly includes the sta-bilized tracking disturbance torque and back EMF loopsThe disturbance torque loop is caused by wire pulling andfriction among rotation joints which happens in the seekerrotation process In Figure 2 various types of disturbancetorque are equivalent to the transfer function 119866119863(119904) Thedisturbance torque is the main factor leading to problems ofthe disturbance rejection rate of the gimbaled seeker includ-ing the spring and damping torques A real gimbaled seekerdisturbance torque is a nonlinear model A simplified modelis adopted to analyze the influence of disturbance torque onseeker disturbance rejection rate and output accuracy Theexpression is as follows [22]

119866119863 (119904) =119870119899119904+ 119870120596 (2)

Mathematical Problems in Engineering 3

Inertial Reference


Seeker Boresight

O

Apparent Target

Target

O

T

qs

qt

lowast qlowast

r

Δq

Figure 3 Basic geometry for radome analysis

where 119870119899 is the spring torque coefficient and 119870120596 denotes thedamping torque coefficient

The occurrence of the disturbance torque couples themissile and seeker motions thereby causing DRRPL andreducing the accuracy of seeker control The DRRTF 119877(119904) isdefined as follows

119877 (119904) =Δ 119902 (119904)120599 (119904)

(3)

where Δ 119902(119904) is the additional rotation angular rate of theplatform caused by the missile motion in relation to theinertial space From the definition the smaller the amplitudeof 119877(119904) the stronger the seekerrsquos isolation capability to thedisturbance of the missile

In accordance with Figure 2 assuming 119867(119904) = 1 andignoring the effect of back EMF the seeker DRRTF can beeduced as

119877 (119904) =Δ 119902 (119904)120599 (119904)

= 119904119866119863 (119904) (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 119866119863 (119904) (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(4)

With the damping torque 119866119863(119904) = 119870120596 and spring torque119866119863(119904) = 119870119899119904models substituted into (4) theDRRTFmodels

under the effect of the damping and spring torques are shownin

119877V (119904) =Δ 119902V (119904)120599 (119904)

= 119904119870120596 (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 119870120596 (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(5)

119877119904 (119904) =Δ 119902119904 (119904)120599 (119904)

= 119870119899 (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 119870119899 (119871119904 + 119877) + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(6)

22 Radome Parasitic Effects in a Stabilized Seeker Theexistence of radome affects the seeker measurement on thereal target position Figure 3 shows the basic missilendashtargetgeometry under the existence of the radome error

In Figure 3 Δ119902 denotes the radome refraction angle 120576lowastis the error angle between the seeker centerline and apparentLOS and 119902lowast represents the apparent LOS angle caused by theradome error

The refraction angle Δ119902 varies with the seeker gimbalangle 120593119903 For preliminary analysis the refraction angle isassumed to be linearly proportional to the gimbal angle

Δ119902 = 119877119889119900119898120593119903 (7)

where119877119889119900119898 is the radome slope 119877119889119900119898 is generally regarded asa constant despite being variable during flight [23 24]

In accordance with the angular relationships in Figure 3119902lowast can be expressed as follows

119902lowast = 119902119905 + (119902119904 minus 120599) 119877119889119900119898 (8)

Considering a stable seeker tracking 120576 is generally a smallparameterThus 119902119904 asymp 119902119905Then (8) can be rewritten as follows

119902lowast cong 119902119905 + (119902119905 minus 120599) 119877119889119900119898 = 119902119905 (1 + 119877) minus 120599119877119889119900119898 (9)

Generally speaking 119877119889119900119898 ≪ 1 Hence (9) can be simpli-fied into

119902lowast = 119902119905 minus 120599119877119889119900119898 (10)

Equation (10) shows thatmissile attitude is fed back to theseeker due to the existence of radome error thereby generat-ing a measurement error of 119902119905 Consequently a radome errorparasitic loop is formed in the guidance loop

The influence of the correction network on DRRTFcharacteristics fluctuates on a small scale within the commonoperating frequencies of the seeker (less than 5Hz) therebyfailing to transform its basic amplitude-phase characteristicsTherefore the analysismodel of the disturbance rejection ratecan be simplified to demonstrate intuitively its influence onthe guidance and control system Ignoring the impact of timedelay process the correction network effect of back EMFhigh-frequency dynamics and small quantity and assuming1198661(119904) = 1 1198662(119904) = 1 119877 asymp 1 and 119871 asymp 0 the equivalent gainof the stabilized loop is determined to be 1198702 = 11986621198962119870119879119869In addition the equivalent gain of the tracking loop is 1198701 =11986611198961 Figure 4 presents the educed seekermodel with radomeeffects


minus

+

Stabilized LoopTracking Loop

+

+ minus

minusRadome Error Parasitic LoopR

1

s

1

s

1

s

GD(s)

K1 K2

qcqs

qt

Rdom

qminus

++

+

qs

r

Figure 4 Simplified seeker model with radome effects

Flight ControlSystemNoise Filter

+ ++

AerodynamicsDRRTF

Parasitic Loop

Guidance Loop

sat 1

s2

1

Vc(TF minus t)qt qt

minus

1

Tg

5s + 1

NVc

ac1

(Tg

5s + 1)

3am

R(s)

Ts + 1

Vm

Figure 5 Guidance system model with parasitic effects

Table 1 First-order DRRTF

Disturbance type DRRTF Equivalent coefficient

Spring torque 119877119904 (119904) =119870119904 minus 119870119903119879119904119904 + 1

119870119904 =11987011989911987011198702

Damping torque 119877V (119904) =119904119870V minus 119870119903119879119904119904 + 1

119870V =11987012059611987011198702

Figure 4 shows that the DRRTF including the radomeslope can be educed as (11) with 119870119903 = 119877119889119900119898(1 minus 119877119889119900119898)

119877 (119904) =Δ 119902 (119904)120599 (119904)

= minus119870119903 + 119904119866119863 (119904) 11987011198702(111987011198702) 1199042 + ((119866119863 (119904) + 1198702) 11987011198702) 119904 + 1

(11)

11987011198702 ≫ 119870119899 is observed when the time delay processand correction network are ignored To simplify the analysison the basis of the seeker control system performance andignoring the high-frequency dynamics theDRRTF under theeffect of both radome slope and different types of disturbancetorque can be simplified as a first-order model as shown inTable 1 where 119879119904 is the equivalent time constant

23 Guidance System Model Figure 5 presents a block dia-gram of the guidance loop with the disturbance rejection

at +

minus

1

TF minus t

N

s [(( s5

+ 1)4 minus NVc

Vm

R (s) (Ts + 1)]am

Figure 6 Dimensionless guidance system model

and radome parasitic loop The noise filter is considereda first-order model the flight control system a third-ordermodel and the seeker dynamics a first-order model with theequivalent time constant of 119879119904 = 1198791198925

In Figure 5 119886119905 is the target acceleration 119886119898 denotes themissile acceleration 119881119888 represents the closing velocity 119881119898stands for themissile velocity119879119865minus119905 signifies the time to go119879119892refers to the guidance time constant 119879120572 indicates the missileturning rate time constant and 119873 symbolizes the effectivenavigation ratio of the proportional guidance

The dimensionless method is adopted to reduce theparameter Assuming 119904 = 119904119879119892 119879119865 = 119879119865119879119892 119905 = 119905119879119892and 119879120572 = 119879120572119879119892 the equivalent model of the dimensionlessguidance system can be obtained as shown in Figure 6

3 Lyapunov Stability of the Guidance System

31 Uniform Asymptotic Stability of the Guidance System Theguidance system shown in Figure 6 is a linear time-varying


system In this study passivity theorems are used to analyzethe Lyapunov stability of the equilibrium point of the systemWhen studying the Lyapunov stability we consider the caseof zero input Let the reference input 119886119905 in Figure 6 be zeroand the guidance system is considered as follows

= 119891 (119905 119909) (12)

If

119891 (119905 0) = 0 forall119905 ge 0 (13)

then the origin is the equilibrium point of (12) at 119905 = 0 Theequilibrium point 119909 = 0 of the system (12) is

(i) uniformly stable if each 120585 gt 0 has 120575 = 120575(120585) gt 0independent of 1199050 such that

1003817100381710038171003817119909 (1199050)1003817100381710038171003817 lt 120575 997904rArr 119909 (119905) lt 120585 forall119905 ge 1199050 ge 0 and (14)

(ii) uniformly asymptotically stable if it is uniformlystable and has a positive constant 119888 independent of 1199050such that for all 119909(1199050) lt 119888 119909(119905) rarr 0 as 119905 rarr infinuniformly in 1199050 that is for each 120578 gt 0 119879 = 119879(120578) gt 0is observed such that

119909 (119905) lt 120578 forall119905 ge 1199050 + 119879 (120578) forall1003817100381710038171003817119909 (1199050)

1003817100381710038171003817 lt 119888 (15)

Therefore if the guidance system is uniformly asymp-totically stable at some initial moment 1199050 then the systemhas Lyapunov stability at all 1199050 in the defined interval andthe convergence of the system will not deteriorate with anincrease in 1199050

32 Lyapunov Stability Represented by Passivity TheoremsPassivity theorems involve the concepts of passivity andpositive real The definitions of passivity and positive real areprovided in the following Proof of the relevant lemma andtheorem is presented in [19]

321 Passivity of a Time-Varying Memoryless FunctionWhen the output of the system is independent of the statevariables the system can be expressed as 119910 = ℎ(119905 119906) that isa memoryless function If ℎ is a scalar function that satisfiesthe inequality

1205721199062 le 119906ℎ (119905 119906) le 1205731199062 (16)

for all (119905 119906) where 120572 and 120573 are real numbers with 120573 ge 120572 thenℎ belongs to the sector [120572 120573] If ℎ(119905 119906) belongs to the sector[0infin] that is 119906ℎ(119905 119906) ge 0 then the system 119910 = ℎ(119905 119906) ispassive If ℎ(119905 119906) belongs to the sector (0infin) then119910 = ℎ(119905 119906)is strictly passive

322 Strictly Positive Real Transfer Functions A single-inputndashsingle-output linear system transfer function 119866(119904) isstrictly positive real function if and only if

(1) 119866(infin) gt 0(2) all poles of 119866(119904) are in Re[119904] lt 0 and(3) Re[119866(j120596)] gt 0 forall120596 gt 0

G(s)

h(t u)y u

-

Figure 7 Feedback connection

The second condition requires all poles of 119866(119904) to be in theopen left-half complex plane whereas the third conditionrequires the Nyquist plot of 119866(j120596) to lie in the closed right-half complex plane

Lemma 1 provides the relationship between positive realand passivity properties

Lemma 1 The linear time-invariant minimal realization = 119860119909 + 119861119906

119910 = 119862119909 + 119863119906(17)

with119866(119904) = 119862(119904119868minus119860)minus1119861+119863 is strictly passive if119866(119904) is strictlypositive real function

323 Passivity Theorems Consider the feedback connectionof Figure 7 where 119866(119904) is a time-invariant dynamical systemandℎ(119905 119906) is a time-varyingmemoryless functionTheorem 2presents the method of analyzing the Lyapunov stability offeedback systems this method uses the passivity propertiesof the feedback components

Theorem 2 Consider the feedback connection of a strictlypassive time-invariant dynamical system 119866(119904) with a passivetime-varying memoryless function ℎ(119905 119906) Then the origin ofthe closed-loop system is uniformly asymptotically stable

Furthermore if 119866(119904) is a strictly positive real transferfunction (from Lemma 1) then the dynamical system isstrictly passive According to Theorem 2 the origin of theclosed-loop system is uniformly asymptotically stable

324 Loop Transformations When 119866(119904) or ℎ(119905 119906) does notsatisfy the conditions of Theorem 2 the feedback systemcan be transformed into an equivalent feedback systemwhich satisfies the conditions through loop transformationsSuppose119866(119904) is a time-invariant dynamical system and ℎ(119905 119906)is a time-varying memoryless function that belongs to thesector [120572 120573] Then the feedback system can be reconfiguredinto a feedback system that is connected by 119866(119904) and ℎ(119905 119906)through loop transformations as shown in Figure 8 whereℎ(119905 119906) is a memoryless function that belongs to the sector[0infin] We can now apply Theorem 2 if 119866(119904) satisfies theconditions of the respective theorem In this case the systemsshown in Figures 7 8(a) and 8(b) are equivalent in terms ofstability

4 Stability Analysis of the Guidance System

The guidance system shown in Figure 6 is transformedinto a feedback system as shown in Figure 7 wherein the


ℎ(t u) G(s)-

(a) Equivalent system

ℎ(t u)

G(s)-

-

-

+

+ +

+

+

+

G(s)

ℎ(t u)

minus

( minus )minus1

(b) Loop transformations

Figure 8 Bock diagrams of equivalent feedback system

linear time-invariant system transfer function is as fol-lows

119866 (119904) = 119873119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)]

(18)

whereas the time-varying memoryless function is as fol-lows

ℎ (119905 119906) = 1119879119865 minus 119905

(19)

Let 119905119896 be any moment of the terminal guidance Wediscuss the stability of guidance systems in the boundedclosed domain 119905 isin [0 119905119896] Then ℎ(119905 119906) belongs to the sector[1119879119865 1(119879119865 minus 119905119896)] and is not passive With the use of trans-formations to reconfigure the system into the form shownin Figure 8(b) where ℎ(119905 119906) is a time-varying memorylessfunction that belongs to the sector [0infin]119866(119904) is transformedinto119866 (119904)

=119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873 (119879119865 minus 119905119896)

119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873119879119865(20)

In accordance with Theorem 2 if 119866(119904) is strictly positivereal then the feedback system satisfies the Lyapunov stabilitycondition that is the origin of the original PN guidancesystem is uniformly asymptotically stable over the 119905 isin [0 119905119896]interval The following are three conditions for 119866(119904) to bestrictly positive real

(i) Substituting theDRRTF inTable 1 into (14) shows that119866(infin) = 1 gt 0 for each DRRTF Therefore the firstcondition is satisfied

(ii) Expanding the characteristic equation of119866(119904) into thefollowing form

119886119899119904119899 + 119886119899minus1119904

119899minus1 + sdot sdot sdot + 11988621199042 + 1198861119904

1 + 1198860 = 0 (21)

Then for the second condition the Routh criterion can beused to obtain the constraint inequality that the eigenvaluesare all on the open left-half complex plane

(iii) The third condition holds when the Nyquist plot of119866(j120596) lies in the closed right-half complex plane

The previous analysis shows that the parameters thataffect the stability of the PN guidance system include 119873119881119888119881119898 119879120572 119879119865 119905119896 and 119877(119904)

5 Simulations and Analysis

This section analyzes the influence of different DRRTFmodels caused by disturbance torque and radome error onthe stability of the guidance system through simulationsThetypical PN guidance system parameters (119873 = 4119881119888119881119898 = 15119879120572 = 3 119879119865 = 10 and 119879119892 = 05) equivalent gains (1198701 = 101198702 = 10) and disturbance torque coefficients (119870119899 = 0001119870120596 = 0001) are selected In accordance with the Routhcriterion the second condition under the abovementionedparameters is satisfied To examine the third condition take119870119903 equal tominus003 and 003 respectively and theNyquist plotsof119866(j120596) corresponding to different DRRTFmodels and 119905119896 arepresented in Figure 9

Figure 9 illustrates that the critical stability time 119905119896119888for different DRRTF models is different under the sameconditions Regardless of the disturbance torque type thecritical stability time of the system when 119870119903 = 003 is greaterthan that when 119870119903 = minus003 Therefore when 119870119903 = minus003 theconstraint on the terminal guidance time when the guidancesystem satisfies the Lyapunov stability conditions is stricterthan that when 119870119903 = 003

119905119906119899119904 = 119879119865 minus 119905119896 indicates unstable flight time Figure 10shows the uniformly asymptotically stable boundary of thesystem determined by 119870119904 or 119870V and 119905119906119899119904 under different 119879119865when 119873 = 4 and 119879120572 = 3 The regions above and below thecurve are stable and unstable respectively


minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s

0 2 4 6 8 10minus2Real Axis

Kr = minus003

Kr = 003

tk = 3 Stabletk = 332 Critical statetk = 4 Unstable


(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003


tk = 8 Stabletk = 845 Critical state

tk = 9 Unstable


(b) Damping torque

Figure 9 Nyquist plots

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque

Figure 10 Stable boundary corresponding to different 119879119865

Figure 10 illustrates that different 119879119865 have little effect onthe unstable flight time of the guidance system The time ofinstability can be delayed by extending the terminal guidancetime Under the action of spring torque the unstable flighttime 119905119906119899119904 increases rapidly with the increase of 119870119904 When 119870119904increases to a certain value the guidance system cannot meetthe stability conditions during the entire terminal guidancetime Moreover when 119870119903 = minus003 the unstable flight time ishigher and the stability regions are smaller than those when119870119903 = 003 The change law of the stable boundary of theguidance system under the action of the damping momentis similar to that of the spring torque Only when 119870119903 = 003

the unstable flight time decreases slightly and then increasesrapidly with the increase of119870V

Figure 11 shows the uniformly asymptotically stableboundary of the system determined by 119870119903 and 119905119906119899119904 underdifferent disturbances when 119879119865 = 10

Figure 11 illustrates that when 119870119903 lt 0 the unstable flighttime is considerably larger than that when 119870119903 gt 0 andincreases sharply with the increase of |119870119903| When 119870119903 lt 0the tolerance of the guidance system to the radome erroris approximately 3 which is less than that when 119870119903 gt 0(higher than 8) When 119870119903 gt 0 the unstable flight timedecreases slowly as |119870119903| increases which is beneficial to


0

2

4

6

8

10

Unstable

Stable

Spring torqueDamping torque

t uns

000 005 010minus005

Kr

Figure 11 Stable boundary corresponding to different119870119903

0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque


the guidance system However when |119870119903| exceeds a certainlimit the guidance system immediately becomes unstableindicating that the tolerance of the guidance system to theradome error is also limited

Figure 12 presents the uniformly asymptotically stableboundary of the systemdetermined by119879120572 and119870119904 or119870V underdifferent119870119903 when 119879119865 = 10 and 119905119906119899119904 = 7

Figure 12 shows that the stability regions of the guidancesystem decrease with the increase of 119879120572 119879120572 is related to thealtitude of the missilersquos flight and 119879120572 is relatively large at highaltitude If 119879119892 is small then the guidance system will likelylose stability when the missile is flying at a high altitudeTherefore the decelerating response of the guidance systemin high airspace should be given considerable attention

Moreover under the action of spring torque the stabilityregions when 119870119903 lt 0 are smaller and the stability regionswhen 119870119903 gt 0 are larger than that when 119870119903 = 0 Under theaction of damping torque positive and negative 119870119903 lead to astability regions reduction When 119870119903 decreases to a certainvalue and 119879120572 is large the guidance system will not be stableTherefore the disturbance rejection rate index should bestrictly constrained in engineering practice

From the above analysis the stability regions for anegative slope are smaller than that for a positive slope andlager values of 119879120572 lead to a smaller range of stability Thesetrends are consistent with those obtained in [8] using theRouth criterion However there will be differences in thestability margin


6 Conclusions

The influence of the disturbance rejection rate and radomeerror slope on the Lyapunov stability of the PN guidancesystem is analyzed by passivity theorems Results of thesimulation show that different DRRTF models have differenteffects on the stability of the guidance system The stableboundary and tolerance of the PN guidance system to thespring torque damping torque and radome slope differsignificantly The unstable flight time caused by negativeradome slope is higher than that due to positive radomeslopeThe constraint of disturbance rejection rate indexwhenthe radome slope is negative and is stricter than that whenradome slope is positive These results provide a theoreticalbasis for the overall design and engineering application ofguidance systems and radar seekers

Data Availability

The data used to support the findings of this study (in txtformat) are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] P J Kennedy and R L Kennedy ldquoDirect versus indirect line ofsight (LOS) stabilizationrdquo IEEE Transactions on Control SystemsTechnology vol 11 no 1 pp 3ndash15 2003

[2] Y S Kwon H Y Hwang and Y S Choi ldquoStabilization loopdesign on direct drive gimbaled platform with low stiffness andheavy inertiardquo in Proceedings of the International Conferenceon Control Automation and Systems (ICCAS rsquo07) pp 320ndash325IEEE Seoul South Korea October 2007

[3] S Mondal S Sadhu and A Banerjee ldquoPlatformmotion distur-bances attenuation in a missile seeker subsystem using InternalModel Controlrdquo in Proceedings of the 2013 IEEE InternationalConference on Control Automation Robotics and EmbeddedSystems CARE 2013 ind December 2013

[4] I Klein and I Rusnak ldquoLoop-shaping approach to mitigateradome effects in homing missilesrdquo Journal of Guidance Con-trol and Dynamics vol 40 no 7 pp 1787ndash1793 2017

[5] S S Han S K Jang and S J Lee ldquoRadome compensationusing adaptive particle filterrdquo in Proceedings of the 17th IFACSymposium on Automatic Control in Aerospace ACArsquo 2007 pp43ndash48 fra June 2007

[6] C-L Lin and Y-H Hsiao ldquoAdaptive feedforward control fordisturbance torque rejection in seeker stabilizing looprdquo IEEETransactions on Control Systems Technology vol 9 no 1 pp108ndash121 2001

[7] S Mondal S Sadhu and S Talukdar ldquoPlatform motiondisturbances attenuation in a missile seeker subsystem usingdisturbance observer techniquesrdquo in Proceedings of the 10thInternational Conference on Intelligent Systems and ControlISCO 2016 ind January 2016

[8] FWNesline and P Zarchan ldquoRadome inducedmiss distance inaero-dynamically controlled homingmissilesrdquo in Proceedings ofthe Collection of Technical Papers - AIAA Guidance and ControlConference 1984 pp 1984-1845

[9] P Zarchan Tactical and Strategic Missile Guidance AIAAReston Va USA 2012

[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] Y Du Q Xia and Z Wang ldquoEffect of seeker disturbance rejec-tion performance on the control system stabilityrdquo in Proceedingsof the 3rd International Symposium on Systems and Control inAeronautics and Astronautics ISSCAA2010 pp 1032ndash1035 chnJune 2010

[12] H Eguchi H Kubo and T Yamashita ldquoRobust stability of guid-ance and control system for homing missilesrdquo in Proceedings ofthe 29th Aerospace Sciences Meeting Reno NV USA 1991

[13] O Goldan and S Gutman ldquoAdjoint stability and miss distancein proportional navigationrdquo Journal of Guidance Control andDynamics vol 35 no 4 pp 1089ndash1093 2012

[14] M Guelman ldquoThe stability of proportional navigation systemsrdquoin Proceedings of the Guidance Navigation and Control Confer-ence pp 586ndash590 usa August 1990

[15] P Gurfil M Jodorkovsky and M Guelman ldquoFinite timestability approach to proportional navigation systems analysisrdquoJournal of Guidance Control and Dynamics vol 21 no 6 pp853ndash861 1998

[16] D-Y Rew M-J Tahk and H Cho ldquoShort-time stability ofproportional navigation guidance looprdquo IEEE Transactions onAerospace and Electronic Systems vol 32 no 3 pp 1107ndash11151996

[17] T Tanaka and H Eguchi ldquoHyperstable range in homingmissilesrdquo inProceedings of the Guidance Navigation andControlConference 1990 pp 591ndash600 usa August 1990

[18] 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

[19] H K Khalil Nonlinear Systems Third Edition Prentice-HallInc Upper Saddle River NJ 2002

[20] C I Byrnes A Isidori and J C Willems ldquoPassivity feedbackequivalence and the global stabilization of minimum phasenonlinear systemsrdquo Institute of Electrical and Electronics Engi-neers Transactions onAutomatic Control vol 36 no 11 pp 1217ndash1240 1991

[21] A Teel et al ldquoInput-output stabilityrdquo inThe Control Handbookp 4250 CRC Press 1995

[22] X Du andQ Xia ldquoThe research of guidance performance of thephased array seeker with platform for air-to-air missilerdquo Optik- International Journal for Light and Electron Optics vol 127 no22 pp 10322ndash10334 2016

[23] FW Nesline andM L Nesline ldquoWing size vs radome compen-sation in aerodynamically controlled radar homing missilesrdquoJournal of Guidance Control and Dynamics vol 9 no 6 pp645ndash649 1986

[24] P Zarchan andH Gratt ldquoAdaptive radome compensation usingditherrdquo Journal of Guidance Control and Dynamics vol 22 no1 pp 51ndash57 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

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OptimizationJournal of



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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Target

O

T

Seeker Boresight


Inertial Reference

xtxs

qt

qs

r

xb

Figure 1 Basic geometry in pitch plane

Stabilized Loop

u

Disturbance Torque Loop

Correction NetworkCorrection Network

Tracking Loop

H(s)

Back EMF Loop

Rate Gyro Dynamics

qt

qs

+minus

+minus

+minusminus

k1

q

G1(s) G2(s) k2

KE

GD(s)

TD1

Ls + RKT

Tem

1

Js

1

s

qs

qs

r +minus

+minus

Figure 2 Control loop model of a gimbaled seeker

establishes the disturbance rejection rate transfer function(DRRTF) models under the joint action of the disturbancetorque and the radome error A PN guidance systemmodel isestablished on the basis of DRRTFThe passivity theorems foranalyzing the Lyapunov stability of nonlinear time-varyingsystems are convenient for engineering application becausethey donot need to build the state spacemodels and constructLyapunov functions [19ndash21] Therefore this study analyzesthe Lyapunov stability of the PN guidance system on the basisof the passivity theorems

This paper is structured as follows Section 2 establishesthe DRRTF and PN guidance systemmodels which considerthe different disturbance torque types and radome errorslope Section 3 defines the uniform asymptotical stabilityof the PN guidance system and introduces the Lyapunovstability represented by passivity theorems Section 4 ana-lyzes the stability of the PN guidance system on the basisof passivity theorems Section 5 investigates the influence ofspring torque damping torque and radome error slope on theLyapunov stability of the PN guidance system by simulationsSection 6 concludes this paper

2 Problem Formulation

21 Modeling of the DRRTF Figure 1 shows the basic geom-etry in pitch plane In Figure 1 119902119905 represents the LOS angle119902119904 is the seeker rotation angle 120599 denotes the missile attitudeangle120593119903 refers to the gimbal angle and 120576 stands for the seeker

LOS error angle From geometry the detector error angle canbe written as follows

120576 = 119902119905 minus 119902119904 (1)Figure 2 presents a typical control loop model of a

gimbaled seekerIn Figure 2 119871 is the inductance of armature winding

119877 denotes the total resistance 119870119879 represents the torqueconstant 119870119864 stands for the back electromotive force (EMF)119879119890119898(119905) refers to the electromagnetic torque 119879119863 signifies thetotal load torque of the motor and load 119869 indicates thetotal motor rotational inertia and 119902 symbolizes the LOS raterequired for guidance

Figure 2 shows that the model mainly includes the sta-bilized tracking disturbance torque and back EMF loopsThe disturbance torque loop is caused by wire pulling andfriction among rotation joints which happens in the seekerrotation process In Figure 2 various types of disturbancetorque are equivalent to the transfer function 119866119863(119904) Thedisturbance torque is the main factor leading to problems ofthe disturbance rejection rate of the gimbaled seeker includ-ing the spring and damping torques A real gimbaled seekerdisturbance torque is a nonlinear model A simplified modelis adopted to analyze the influence of disturbance torque onseeker disturbance rejection rate and output accuracy Theexpression is as follows [22]

119866119863 (119904) =119870119899119904+ 119870120596 (2)


Inertial Reference


Seeker Boresight

O

Apparent Target

Target

O

T

qs

qt

lowast qlowast

r

Δq




119877 (119904) =Δ 119902 (119904)120599 (119904)

(3)



119877 (119904) =Δ 119902 (119904)120599 (119904)

= 119904119866119863 (119904) (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 119866119863 (119904) (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(4)



119877V (119904) =Δ 119902V (119904)120599 (119904)

= 119904119870120596 (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 119870120596 (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(5)

119877119904 (119904) =Δ 119902119904 (119904)120599 (119904)

= 119870119899 (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 119870119899 (119871119904 + 119877) + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(6)




Δ119902 = 119877119889119900119898120593119903 (7)



119902lowast = 119902119905 + (119902119904 minus 120599) 119877119889119900119898 (8)




119902lowast = 119902119905 minus 120599119877119889119900119898 (10)




minus

+


+

+ minus


1

s

1

s

1

s

GD(s)

K1 K2

qcqs

qt

Rdom

qminus

++

+

qs

r



+ ++

AerodynamicsDRRTF

Parasitic Loop

Guidance Loop

sat 1

s2

1

Vc(TF minus t)qt qt

minus

1

Tg

5s + 1

NVc

ac1

(Tg

5s + 1)

3am

R(s)

Ts + 1

Vm





119870119904 =11987011989911987011198702


119870V =11987012059611987011198702


119877 (119904) =Δ 119902 (119904)120599 (119904)

= minus119870119903 + 119904119866119863 (119904) 11987011198702(111987011198702) 1199042 + ((119866119863 (119904) + 1198702) 11987011198702) 119904 + 1

(11)



at +

minus

1

TF minus t

N

s [(( s5

+ 1)4 minus NVc

Vm

R (s) (Ts + 1)]am









= 119891 (119905 119909) (12)

If

119891 (119905 0) = 0 forall119905 ge 0 (13)






1003817100381710038171003817 lt 119888 (15)




1205721199062 le 119906ℎ (119905 119906) le 1205731199062 (16)




G(s)

h(t u)y u

-





119910 = 119862119909 + 119863119906(17)









ℎ(t u) G(s)-


ℎ(t u)

G(s)-

-

-

+

+ +

+

+

+

G(s)

ℎ(t u)

minus

( minus )minus1




119866 (119904) = 119873119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)]

(18)


ℎ (119905 119906) = 1119879119865 minus 119905

(19)


=119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873 (119879119865 minus 119905119896)

119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873119879119865(20)




119886119899119904119899 + 119886119899minus1119904


1 + 1198860 = 0 (21)









minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s


Kr = minus003

Kr = 003



(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003



tk = 9 Unstable


(b) Damping torque


0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque







0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018









Inertial Reference


Seeker Boresight

O

Apparent Target

Target

O

T

qs

qt

lowast qlowast

r

Δq




119877 (119904) =Δ 119902 (119904)120599 (119904)

(3)



119877 (119904) =Δ 119902 (119904)120599 (119904)

= 119904119866119863 (119904) (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 119866119863 (119904) (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(4)



119877V (119904) =Δ 119902V (119904)120599 (119904)

= 119904119870120596 (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 119870120596 (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(5)

119877119904 (119904) =Δ 119902119904 (119904)120599 (119904)

= 119870119899 (119871119904 + 119877)119904 [119869119904 (119871119904 + 119877) + 1198662 (119904) 1198962119870119879] + 119870119899 (119871119904 + 119877) + 1198661 (119904) 1198662 (119904) 11989611198962119870119879

(6)




Δ119902 = 119877119889119900119898120593119903 (7)



119902lowast = 119902119905 + (119902119904 minus 120599) 119877119889119900119898 (8)




119902lowast = 119902119905 minus 120599119877119889119900119898 (10)




minus

+


+

+ minus


1

s

1

s

1

s

GD(s)

K1 K2

qcqs

qt

Rdom

qminus

++

+

qs

r



+ ++

AerodynamicsDRRTF

Parasitic Loop

Guidance Loop

sat 1

s2

1

Vc(TF minus t)qt qt

minus

1

Tg

5s + 1

NVc

ac1

(Tg

5s + 1)

3am

R(s)

Ts + 1

Vm





119870119904 =11987011989911987011198702


119870V =11987012059611987011198702


119877 (119904) =Δ 119902 (119904)120599 (119904)

= minus119870119903 + 119904119866119863 (119904) 11987011198702(111987011198702) 1199042 + ((119866119863 (119904) + 1198702) 11987011198702) 119904 + 1

(11)



at +

minus

1

TF minus t

N

s [(( s5

+ 1)4 minus NVc

Vm

R (s) (Ts + 1)]am









= 119891 (119905 119909) (12)

If

119891 (119905 0) = 0 forall119905 ge 0 (13)






1003817100381710038171003817 lt 119888 (15)




1205721199062 le 119906ℎ (119905 119906) le 1205731199062 (16)




G(s)

h(t u)y u

-





119910 = 119862119909 + 119863119906(17)









ℎ(t u) G(s)-


ℎ(t u)

G(s)-

-

-

+

+ +

+

+

+

G(s)

ℎ(t u)

minus

( minus )minus1




119866 (119904) = 119873119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)]

(18)


ℎ (119905 119906) = 1119879119865 minus 119905

(19)


=119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873 (119879119865 minus 119905119896)

119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873119879119865(20)




119886119899119904119899 + 119886119899minus1119904


1 + 1198860 = 0 (21)









minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s


Kr = minus003

Kr = 003



(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003



tk = 9 Unstable


(b) Damping torque


0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque







0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018









minus

+


+

+ minus


1

s

1

s

1

s

GD(s)

K1 K2

qcqs

qt

Rdom

qminus

++

+

qs

r



+ ++

AerodynamicsDRRTF

Parasitic Loop

Guidance Loop

sat 1

s2

1

Vc(TF minus t)qt qt

minus

1

Tg

5s + 1

NVc

ac1

(Tg

5s + 1)

3am

R(s)

Ts + 1

Vm





119870119904 =11987011989911987011198702


119870V =11987012059611987011198702


119877 (119904) =Δ 119902 (119904)120599 (119904)

= minus119870119903 + 119904119866119863 (119904) 11987011198702(111987011198702) 1199042 + ((119866119863 (119904) + 1198702) 11987011198702) 119904 + 1

(11)



at +

minus

1

TF minus t

N

s [(( s5

+ 1)4 minus NVc

Vm

R (s) (Ts + 1)]am









= 119891 (119905 119909) (12)

If

119891 (119905 0) = 0 forall119905 ge 0 (13)






1003817100381710038171003817 lt 119888 (15)




1205721199062 le 119906ℎ (119905 119906) le 1205731199062 (16)




G(s)

h(t u)y u

-





119910 = 119862119909 + 119863119906(17)









ℎ(t u) G(s)-


ℎ(t u)

G(s)-

-

-

+

+ +

+

+

+

G(s)

ℎ(t u)

minus

( minus )minus1




119866 (119904) = 119873119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)]

(18)


ℎ (119905 119906) = 1119879119865 minus 119905

(19)


=119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873 (119879119865 minus 119905119896)

119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873119879119865(20)




119886119899119904119899 + 119886119899minus1119904


1 + 1198860 = 0 (21)









minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s


Kr = minus003

Kr = 003



(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003



tk = 9 Unstable


(b) Damping torque


0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque







0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018










= 119891 (119905 119909) (12)

If

119891 (119905 0) = 0 forall119905 ge 0 (13)






1003817100381710038171003817 lt 119888 (15)




1205721199062 le 119906ℎ (119905 119906) le 1205731199062 (16)




G(s)

h(t u)y u

-





119910 = 119862119909 + 119863119906(17)









ℎ(t u) G(s)-


ℎ(t u)

G(s)-

-

-

+

+ +

+

+

+

G(s)

ℎ(t u)

minus

( minus )minus1




119866 (119904) = 119873119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)]

(18)


ℎ (119905 119906) = 1119879119865 minus 119905

(19)


=119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873 (119879119865 minus 119905119896)

119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873119879119865(20)




119886119899119904119899 + 119886119899minus1119904


1 + 1198860 = 0 (21)









minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s


Kr = minus003

Kr = 003



(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003



tk = 9 Unstable


(b) Damping torque


0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque







0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018









ℎ(t u) G(s)-


ℎ(t u)

G(s)-

-

-

+

+ +

+

+

+

G(s)

ℎ(t u)

minus

( minus )minus1




119866 (119904) = 119873119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)]

(18)


ℎ (119905 119906) = 1119879119865 minus 119905

(19)


=119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873 (119879119865 minus 119905119896)

119904 [(1199045 + 1)4 minus 119873 (119881119888119881119898) 119877 (119904) (119879120572119904 + 1)] + 119873119879119865(20)




119886119899119904119899 + 119886119899minus1119904


1 + 1198860 = 0 (21)









minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s


Kr = minus003

Kr = 003



(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003



tk = 9 Unstable


(b) Damping torque


0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque







0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018









minus6

minus4

minus2

0

2

4

6Im

agin

ary

Axi

s


Kr = minus003

Kr = 003



(a) Spring torque

minus6

minus4

minus2

0

2

4

6

Imag

inar

y A

xis

Kr = minus003

Kr = 003



tk = 9 Unstable


(b) Damping torque


0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

001 002 003 004 005 006 007000Ks

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

(a) Spring torque

0

3

6

9

12

15

Unstable

Unstable

Stable

Stable

t uns

Kr = minus003

Kr = 003

TF = 10

TF = 7

TF = 15

0002 0004 00060000Kv

(b) Damping torque







0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018









0

2

4

6

8

10

Unstable

Stable


t uns

000 005 010minus005

Kr


0 1 2 3 4 5 6

Stable

Unstable

T

Kr = minus003

Kr = 0

Kr = 003

000

002

004

006

008

010

012

Ks

(a) Spring torque

Stable

Unstable

0000

0005

0010

0015

0020K

v

0 1 2 3 4 5 6T

Kr = minus003

Kr = 0

Kr = 003

(b) Damping torque








6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018









6 Conclusions


Data Availability




References
































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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