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Dynamic stability of rolling missile with proportional navigation & PI autopilot considering parasitic radome loop
Accepted Manuscript Dynamic stability of rolling missile with proportional navigation & PI autopilot considering parasitic radome loop
Duo Zheng, Defu Lin, Xinghua Xu, Song Tian
PII: DOI: Reference:
S1270-9638(17)30549-7 http://dx.doi.org/10.1016/j.ast.2017.03.036 AESCTE 3975
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
11 January 2016 13 November 2016 29 March 2017
Please cite this article in press as: D. Zheng et al., Dynamic stability of rolling missile with proportional navigation & PI autopilot considering parasitic radome loop, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.036
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Dynamic Stability of Rolling Missile with Proportional Navigation & PI Autopilot Considering Parasitic Radome Loop Duo Zheng1, Defu Lin1, Xinghua Xu2, Song Tian1 1ǃSchool of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China 2ǃBeijing Institute of Automatic Control Equipment, Beijing, 100074, People’s Republic of China
Abstract The dynamic stability of coning motion for a radar homing rolling missile is studied in this paper, and the parasitic radome loop is considered and introduced into the guidance system of the radar homing rolling missile. A mathematical model with complex summation considering the parasitic radome loop is proposed, and the sufficient and necessary conditions for the dynamic stability of the coning motion for the rolling missile are analytically derived by using the stability criterion of third-order characteristic equation with complex coefficients. Numerical simulations with different parameters are conducted to demonstrate the effectiveness of the proposed stability condition. The stability condition proposed in this paper can be used in the guidance and control system design of a radar homing rolling missile while considering the parasitic radome loop. Keywords Rolling missile, PI autopilot, dynamic stability, parasitic radome loop, proportional navigation
1. Introduction For all radar homing missiles, the radome is designed and equipped to protect the seeker antenna both mechanically and thermally. However, when the radar signal passes through the radome, the radar wave will be bent or refracted by the radome, thereby generating a false LOS rate. The introduction of the false LOS rate coupled with angular body motion into the guidance system adds a parasitic feedback loop to the missile’s guidance system, which in turn affects the miss distance (MD) and even the dynamic stability of the radar homing missile. As a result, the parasitic radome loop must be considered in the guidance system design of radar homing missiles. A detailed analysis of the radome error slope (RES) was conducted by Nesline and Zarchan
[1]
; in their
papers, the missile guidance system was replaced by the linear fifth-order binomial, and the radome error
slope was introduced to the linear fifth-order binomial. The stability condition of the system was proposed based on the Routh-Hurwitz stability criterion, and the miss distance result from the radome error slope was analysed using the adjoint method. The miss distances of a radar homing missile induced by the radome error slope were investigated under various parameter conditions, and a sensitivity analysis of system to the radome error slope was performed by assuming its step and sinusoidal change during the flight time
[2]
.A
fundamental trade-off between the radome compensation and wing size was studied for a radar-guided aerodynamically controlled missile; it was shown that radome compensation can be used to reduce the wing size, thereby reducing a missile’s drag and weight
[3]
. The radome error slope was considered one of the
seeker component imperfections, and its effect on the guidance system was studied for high-altitude radarguided air defence missiles
[4]
. Brown, et al. indicated that both the positive and negative boresight error
shifts induced by radome refraction are important in the design because these shifts will increase the miss distance and may actually lead to system instability
[5]
. Both the gimbal seeker and electronically scanned
arrays contribute to the radome problem. Peterson D, et al. studied the radome boresight error in electronically scanned arrays and found that radome-induced boresight error and boresight error slope are usually the dominant errors. The radome error not only limits the beam pointing accuracy but also affects the beam-shape. A negative slope leads to an under damped guidance loop, possibly creating instabilities during the performance of manoeuvres. A positive slope leads to an overly damped system and an underresponsive guidance loop [6]. A perturbation analysis of the parasitic radome loop for a BTT missile was also proposed. Because BTT missiles exhibit inertial and aerodynamic cross-coupling, the equations of the missile and guidance system were linearized around the operating point to conveniently determine the poles and stability margins. The effects of the radome error slope on the stability were studied under different coupling conditions
[7]
. According to the above-described studies, the missile homing performance will be
seriously degraded by the radome error slope. To increase both the homing performance and target intercept accuracy, some researchers have studied in-flight radome error compensation to overcome the restrictions imposed by the radome[6, 8-15]. The rolling missile cross-coupling between the pitch and yaw channels may lead to a further dynamic instability in the form of a divergent coning motion. There have been many studies on the dynamic stability of a spinning missile[16-21]. The sufficient and necessary conditions of dynamic stability for canard-control spinning missiles with a rate loop, an attitude autopilot and PI acceleration autopilots were analytically
derived and further verified by numerical simulations
[16-18]
. The coning motion instability of a spinning
missile induced by the hinge moment was considered, and the sufficient and necessary condition of the conning motion stability was derived and validated through nonlinear six degrees-of-freedom simulation [19]. The dynamic stability of spinning projectiles induced by backlash of actuator was studied in the reference [20]
. The dynamic stability for a rolling flight vehicle using a continuous actuator was proposed by
Koohmaskan et al. [21], who considered the dynamics of the actuator in the system model, analytically derived the sufficient and necessary condition based on the Routh-Hurwitz stability criterion, and examined the derived stability condition in a simulation. In another study [22], a novel optimal control algorithm based on stability analyses was developed to suppress the coning motion. The coning motion stability of spinning missiles with strapdown seekers was studied by Lin D et al.; in their study, the rolling missile is equipped with a strapdown seeker. The parasitic loop induced by the scaling factor error is considered, and the sufficient and necessary stability condition is proposed using a complex summation method[23]. Although there are many references on the radome error slope, none of these references has considered its effect on a spinning missile. The dynamic stability of rolling missile equipped with radar seeker may be worse because of the radome error slope and the cross coupling of the pitch and yaw. This paper focuses on the dynamic stability of a rolling missile using a radar seeker with PI acceleration autopilot, considering the parasitic radome loop. The mathematical models of the radome error slope, the airframe, and the system equation in a form of complex summation are obtained. Based on the stability criterion with complex coefficients, the sufficient and necessary condition of the dynamic stability for a rolling missile considering the radome error slope is derived analytically. A proof of the third-order characteristic equation with complex coefficient is presented in the Appendix. The numerical simulation is further used to examine the proposed stability conditions. The rest of this paper is organized as follows. In the problem formulation section, simplified mathematical models of the radome and airframe are presented with reasonable assumptions. In the section describing the stability condition of the rolling missile considering the parasitic radome loop, the sufficient and necessary condition of dynamic stability is analytically derived by using the stability criterion with complex coefficients. In the simulation results section, the results of simulation studies used for verifying the stability condition proposed are presented. The conclusions are presented in the conclusions section.
2. Problem formulation 2.1.Radome problem formulation The physics of radome refraction are shown in Fig. 1, where ߟ
denotes the refraction angle. It can be
observed that the radar wave is refracted by the radome. The target appears to be displaced from its true position because of the radome refraction angle. The radome refraction angle varies with the seeker gimbal angle. The radome error slope is defined as the rate of change of the radome refraction angle with gimbal angle and is given as
Rs
dK dT B
(1)
where ߠ is the gimbal angle of the seeker.
x Apparent Target angle ction
K Refra
Radar Antenna
x
Target
Radome
Fig. 1 Radome refraction physics
Supposing the ܴ௦ is a constant, the refraction angle is linearly proportional to the gimbal angle according to the radome slope. It can be expressed as
K
RsT B
(2)
The missile-target geometry is shown in Fig. 2, where ݍand ݍᇱ is the real LOS angle and the false LOS angle caused by radome refraction, respectively, ߝ is the error angle, ߶ெ is the seeker angle relative to the reference line, and ߴ is the pitch angle,
Apparent target Target
K H
-
IM
q
Seeker
TB
qc Body axis
Reference Line
Fig. 2 Planar missile-target geometry
According to Fig. 2, the real LOS angle can be expressed as
q
qc K IM H K
(3)
Substituting Eq. (2) into Eq. (3) yields
q IM H RsT B
(4)
The gimbal angle can be expressed as
T B IM -
(5)
Substituting Eq. (5) into Eq. (4) yields
q IM H Rs IM - H 1 Rs IM Rs-
(6)
Because radome slope is small (that is, ܴ௦ )ͳ ا, Eq. (6) can be simplified as
q H IM Rs-
(7)
The false LOS angle due to radome refraction can be expressed as
qc IM H
(8)
Substituting Eq. (8) into Eq. (7), the false LOS angle can be solved as
qc q Rs-
(9)
Because the radome slope is assumed to be constant, the false LOS rate with radome refraction can be expressed as
qc q Rs-
(10)
where ݍሶ is the LOS rate, ݍሶ ᇱ is the false LOS rate, and ߴሶ is the pitch rate. Expanding Eq. (10) to the yaw channel, the PN guidance system of the roll missile with PI acceleration considering the effect of radome is shown in Fig. 3.
Rs ka
s
c
kZ qy
qz
N cVc
a yc
N cVc a zc
kI s k kp I s
kp
V c1
e1 Command Generation
e2
V c2
Command Resolution & Actuators Dynamics
-
V1
ay
Airframe Motion
V1
\
az
kZ s
c
ka
Rs
Fig. 3 PN system of a rolling missile with PI acceleration considering the parasitic radome loop
2.2.Airframe Equations of Lateral Motion Ignoring rolling motion, the rolling missile dynamics equations
[18]
can be simplified based on the
following assumptions: (a) The velocity remains constant over a short interval in the flight; (b) The rolling rate remains constant over a short interval in the flight; (c) The gravity effect is ignored; (d) The coefficients of aerodynamic moments are linearized; and (e) The variables ߰ǡ ߰ ǡ ߙǡ ߚǡ ߶ǡ ܽ݊݀ሶ ߰ are small, i.e., ܿ ߰ݏൎ ͳ, ܿ߰ݏ ൎ ͳ, ܿ ߙݏൎ ͳ, ܿ ߚݏൎ ͳ, ߙ݊݅ݏൎ ߙ, ߚ݊݅ݏൎ ߚ, and ߶݊݅ݏൎ ߶. Under these assumptions, the equations of rolling missile can be formulated as 1 1 °mVmT cos\ V QSCL sin I cos E sin D QSCM sin I sin E T cos E sin D ° ° 1 1 °mVm\ V QSCL sin I sin E QSCM sin I cos E sin D T sin E ° ® 1 1 ° A\ CJsin E cos D QSLM M sin D QSLM Z L / Vm Z y QSLM V V 2 - cos\ C A - 2 sin i \ cos\ QSLM S ° sin I sin I ° 1 1 ° A - cos\ -\ sin D QSLM M sin E cos D QSLM Z L / Vm Z z QSLM V V 1 \ sin i \ CJ\ C A -\ - sin si \ QSLM QS S °¯ sin I sin I
(11) where ܵ is the reference area, ܮis the reference length, ܳ is the dynamic pressure, ܸ is the missile velocity, ܣis the lateral moment of inertia, ܥis the longitudinal moment of inertia, ܥ , ܥ , and ܥெ are the drag, lift, and Magnus force coefficient, respectively, ܯெ , ܯௌ , and ܯఠ are the coefficient of the Magnus moment, the static moment, and the damping moment, respectively, ܯఙ is the coefficient of the
control moment, ߴ, ߰ and ߛ are the pitch, yaw and roll angle, respectively, ߠ and ߰ are the flight-path angle and the heading angle, respectively, ߙ and ߚ are the angle of attack and sideslip in non-rolling coordinates, respectively, ߙ and ߚ are the actuator responses in the non-rolling coordinate system, ߶ is the total angle of attack, and ܶ is the thrust force. Eq. (11) can be further simplified to
§J d · Q QS QS T cLc D cMc ¨ D °T ¸E mV mV V mV m m m © m ¹ ° ° §J d · Q QS QS T °\ V cLc E cMc ¨ E ¸D ° mVm mVm mVm © Vm ¹ ° ® §J d · § L · C QSL QSL QSL QSL Q QS Q Q ° M ScD M Mc ¨ M Z ¨ ¸- MVV1 J ¸E °- A JI A A V A V A m ¹ m ¹ © © ° ° §J d · § L · Q L Q QSL Q C QSL QSL QSL MVV 2 M Sc E M Mc ¨ M Z ¨ ¸\ °\ J¸D A A A A A ° © Vm ¹ © Vm ¹ ¯
(12)
ᇱ are the derivatives of the lift and the Magnus force coefficients, respectively, and ܯௌᇱ where ܥᇱ and ܥெ ᇱ and ܯெ are the derivatives of the static and Magnus moments, respectively.
Defining a1
b21
C J , b22 A
QS T , a2 CLc mVm mVm
§J d · QS CMc ¨ ¸ , b11 mVm © Vm ¹
§ L· QSL M Z ¨ ¸ , and cV A © Vm ¹
QSL M Sc , b12 A
§J d · QSL M Mc ¨ ¸, A © Vm ¹
QSL M V , Eq. (12) can be rewritten as A
T a1D a2 E ° °\ V a1E a2D ® °- b11D b12 E b21\ b22- cV V 1 °\ b E b D b - b \ c V 11 12 21 22 V 2 ¯
(13)
It can be obtained that ߴ ൌ ߠ ߙ and ߰ ൌ ߰ ߚ when ߴ, ߰, ߙ, and ߚ are small values; therefore, the response of the airframe can be further expressed as
°- T D a1D a2 E D ® °¯\ \ V E a1E a2D E
(14)
From Eqs. (13) and (14), the motion of the airframe can be described by using the non-rolling angle of attack ߙ and the non-rolling sideslip angle ߚ. Following the same formulations ߴ ൌ ߠ ߙ and ߰ ൌ ߰ ߚ, Eq. (13) can be rewritten as
D - T b11D b12 E b21\ b22- cV V 1 a1D a2 E ° ® ° ¯ E \ \ V b11E b12D b21- b22\ cV V 2 a1E a2D
(15)
Substituting Eq. (14) into Eq. (15) yields
b11 b21a2 b22 a1 D b12 b21a1 b22a2 E b22 a1 D b21 a2 E cV V1 b11 b21a2 b22 a2 E b12 b21a1 b22a2 D b22 a1 E b21 a2 D cV V 2
°D ® °¯ E
(16)
2.3.Command Generation 1ǃ Guidance Command Generation From Fig. 3, the PN guidance command with the parasitic radome loop is obtained as
ª a yc º «a » ¬ zc ¼
ª N cVc q y N cVc Rs - º » « ¬ N cVc qz N cVc Rs \ ¼
(17)
where ܰ ᇱ is the effective navigation ratio, ܸ is the closing velocity, ܸ is the missile velocity, and ܴ௦ is the radome error slope. 2ǃ Flight Command Generation According to Fig. 3, the actuator signals in the non-rolling coordinate system, denoted by ݁ଵ and ݁ଶ in the pitch and yaw channels, respectively, can be expressed as
ª e1 º «e » ¬ 2¼
ª§ kI «¨ k p s «© «§ kI «¨ k p s ¬©
kI · º · § ¸ a yc ¨ k p ¸ ka a y c- kZ- » s ¹ ¹ © » » kI · · § ¸ azc ¨ k p ¸ ka az c\ kZ\ » s ¹ ¹ © ¼
(18)
where ݇ is the gain of the proportional item, ݇ூ is the gain of the integral regulator, ݇ఠ is the gain of the damping loop, ܿ is the distance between the accelerometer installation location and the centre of mass. For a canard-control missile, note that the coordinate system mentioned above is such that a positive angle of a servo produces a positive-going angle of attack and positive acceleration in the pitch, along with a positive-going sideslip angle and negative acceleration in the yaw. Therefore, the input commands to the actuators can be described as the following formations for a canard-control missile:
ªV c1 º «V » ¬ c2 ¼
ª1 0 º ª e1 º «0 1» «e » ¬ ¼¬ 2¼
ª§ kI · kI · º § «¨ k p s ¸ a yc ¨ k p s ¸ ka a y c- kZ- » ¹ © ¹ «© » « § » kI · k § I · « ¨ k p ¸ azc ¨ k p ¸ ka az c\ kZ\ » s s ¹ © ¹ ¬ © ¼
Considering ܽ௬ ൌ ܸ ߠሶ and ܽ௭ ൌ െܸ ߰ሶ and substituting Eq. (13) into ܽ௬ and ܽ௭ , one has
(19)
Vm a1D a2 E
°a y ® ° ¯ az
(20)
Vm a1E a2D
3ǃ Command Resolution and Actuator Response The rolling of a missile induces harmonic responses in the rudder angles under constant commands. Because the servo-system can be modelled by a second-order system with natural frequency ͳȀܶ௦ , damping ratio ߤ௦ , command transmission delay ߬, and constant roll rate (i.e., ɀሶ ൌ Ͳ), the relationship between the input and output of the servo-system in the non-rolling body system for the rolling missile can be expressed as
ªV 1 º «V » ¬ 2¼
ª cos J d ks kr « ¬ sin J d
sin J d º ªV c1 º cos J d »¼ «¬V c 2 »¼
(21)
where ݇௦ is the gain of the servosystem and ݇ is the dynamic gain of the servo-system under rolling and can be expressed as
kr
1
1 T J 2P T J 2 s
2 2
2
s s
(22)where ܶ௦ is the time constant of the servo-system, ߛ is defined as the steady-state deviation angle caused by the delay of the servo-system dynamics and rotation, and
Jc
§ arccos ¨¨ ¨ ©
1 Ts2J 2
1 T J 2P T J 2 s
2 2
s s
2
· ¸ ¸ ¸ ¹
(23)
Therefore, the total delay angle can be calculated as ɀௗ ൌ ߛ ߬ ή ߛሶ , where ߬ is the time delay of the control system.
3. Dynamic Stability Condition of Rolling Missiles Considering Parasitic Radome Loop It is known from the mathematical model for each component of the acceleration autopilot that both the actuator responses and the airframe responses induce cross-coupling between pitch and yaw and that the attitudes are coupled into the guidance system because of the parasitic radome loop. Therefore, a complex summation model is required to conveniently perform the stability analysis. Eq. (21) can be expressed as
V
ks kr V c1 sin J d V c 2 cos J d i k s kr V c1 cos J d V c 2 sin J d
(24)
ks kr cos J d i sin J d V c 2 i V c1 where ɐ ൌ ߪଶ ݅ ή ߪଵ is defined as the complex deflection angle. Substituting Eq. (17) into Eq. (19), the following formations can be obtained:
ª§ kI · kI · º § » «¨ k p s ¸ N cVc q y N cVc Rs - ¨ k p s ¸ ka a y c- kZ¹ © ¹ «© » » « § kI · kI · § « ¨ k p ¸ N cVc qz N cVc Rs \ ¨ k p ¸ ka az c\ kZ\ » s ¹ s ¹ © ¬ © ¼
ªV c1 º «V » ¬ c2 ¼
(25)
Without loss of generality, the LOS rate inputs ݍሶ ௬ and ݍሶ ௭ are assumed to be zero in the stability analysis; thus, Eq.(25) can be simplified as
kI ª º « k p N cVc Rs - k I N cVc Rs - k p ka a y k p ka cs- ka s a y k I ka c- kZ- » « » (26) « k N cV R \ k N cV R \ k k a k k cs\ k I k a k k c\ k \ » I c s p a z p a a z I a Z c s «¬ p »¼ s
ªV c1 º «V » ¬ c2 ¼
Substituting Eq. (20) into Eq. (26) yields
ªV c1 º «V » ¬ c2 ¼
ª k p N cVc Rs - k I N cVc Rs - k p kaVmT ka k IVmT k p ka c- k I ka c kZ - º « » ¬« k p N cVc Rs \ k I N cVc Rs \ k p kaVm\ V ka k IVm\ V k p ka c\ k I ka c kZ \ ¼» (27)
Substituting Eqs. (13) and (14) into Eq. (27) yields
ªV c1 º ª k p N cVc Rs a1D a2 E D k I N cVc Rs - k p kaVm a1D a2 E ka k IVmT «V » = « ¬ c 2 ¼ «¬ k p N cVc Rs a1E a2D E k I N cVc Rs \ k p kaVm a1E a2D ka k IVm\ V
k k c b E b D b a D a E D b a E a D E c V k p ka c b11D b12 E b21 a1E a2D E b22 a1D a2 E D cV V 1 p a
11
12
21
1
2
22
1
2
1
V
(28)
2
k I ka c kZ a1D a2 E D º » k I ka c kZ a1E a2D E ¼»
Substituting Eq. (28) into Eq. (24) with the definitions ߜ ൌ ߚ ݅ ή ߙ, ߯ ൌ ߰ ݅ ή ߴ, and ߮ ൌ ߰ ݅ ή ߠ, one obtains
k p kaVm a1 i a2 G ka k IVmM k p ka c b22 i b21 G ½ ° ° °° ck p ka ª¬ b11 b21a2 b22a1 i b12 b21a1 b22a2 º¼ G °° k s kr cos J d i sin J d ® ¾ ° ck p ka cV V kZ ck I ka a1 i a2 G kZ ck I ka G ° ° ° ¯°k p N cVc Rs a1 i a2 G k p N cVc Rs G k I N cVc Rs F ¿°
V
(29) Because ߯ ൌ ߮ ߜ, one has
k p kaVm a1 i a2 G ka k IVmM k p ka c b22 i b21 G ½ ° ° ck k b b a b a i b b a b a G ª º ° ° p a ¬ 11 21 2 22 1 12 21 1 22 2 ¼ ° ° ks kr cos J d i sin J d ® ¾ °ck p ka cV V kZ ck I ka a1 i a2 G kZ ck I ka G ° ° ° °k p N cVc Rs a1 i a2 G k p N cVc Rs G k I N cVc Rs M G ¿ ° ¯
V
(30) From Eq. (16), the complex summation equation of the airframe motion can also be obtained:
ia2 G cV V ª¬ b11 b21a2 b22a1 b12 b21a1 b22a2 º¼ G b22 ib21 G a1 ia
G
(31)
By defining ܭൌ ܿఙ ݇௦ ݇ and ܭᇱ ൌ ܭሺܿߛݏௗ ݅ ή ߛ݊݅ݏௗ ሻ, and substituting Eq. (29) into Eq. (31), one has
ªVm ka N cVc Rs º¼ kI K c
G p1 iq iq1 G p2 iq i 2 G ¬
1 ck p ka K c
M
0
(32)
where
p1 iq1
p2 iq2
a1 +ia2 b22 ib21
cka k p K c b22 ib21 ª¬ kZ ck I ka k p NVc Rs º¼ K c 1 ck p ka K c 1 ck p ka K c
¬ª b11 b21a2 b22 a1 i b12 b21a1 b22 a2 ¼º
Vm k p ka K c 1 ck p ka K c
ck p ka K c
ª b11 b21a2 b22 a1 i b12 b21a1 b22 a2 ¼º 1 ck p ka K c ¬
k
Z
ck I ka k p N cVc Rs K c 1 ck p ka K c
a1 ia2
By denoting the sate variables ݔൌ ሾ߮ can be expressed as
ߜ
(33)
a1 ia2 (34)
k I K cN cVc Rs 1 ck p ka K c
ߜሶ ሿ, the state equation of the PN guidance for a rolling missile
x
ªM º «G » « » ¬«G »¼
ª º « » 0 0 a1 ia2 « » ª\ º « » «G » 0 0 1 « »« » « ª¬Vm ka N cVc Rs º¼ k I K c » «¬ G »¼ D 2 i E 2 D1 iE1 » « 1 ck p ka K c ¬« ¼»
(35)
The characteristic equation can be further obtained as (see Appendix 1)
s3 p1 iq1 s 2 p2 iq2 s a1 ia2
(Vm ka N cVc Rs )kI K c 1 ck p ka K c
0
(36)
Defining
p3 iq3
ª¬Vm ka N cVc Rs º¼ k I K c a1 ia2 1 ck p ka K c
(37)
The characteristic equation can be rewritten as
s3 p1 iq1 s 2 p2 iq2 s p3 iq3 0
(38)
Theorem 1: The third-order characteristic equation with complex coefficients has the following form:
D s
s3 p1 iq1 s 2 p2 iq2 s p3 iq3 0
(39)
where ߙ , ߚ אԹ, ݅ ൌ ͳǡʹǡ͵, and the necessary and sufficient conditions for stability are
° p1 ! 0 ° 2 2 ® p1 p2 p1 p3 p1q1q2 q2 ! 0 ° 2 ° p12 p2 p1 p3 p1q1q2 q22 p1 p2 p3 p32 p1q2 q3 p12 q3 p1 p3q1 p3q2 ! 0 ¯
(40)
Proof. See Appendix 2. According to theorem 1, the sufficient and necessary condition for all the roots of Eq. (38) in the half-left of the complex plane is that the three inequalities (40) are positive. Therefore, the sufficient and necessary condition of dynamic stability for a rolling missile can be expressed as
° p1 ! 0 ° 2 2 ® p1 p2 p1 p3 p1q1q2 q2 ! 0 ° 2 ° p12 p2 p1 p3 p1q1q2 q22 p1 p2 p3 p32 p1q2 q3 p12 q3 p1 p3q1 p3q2 ! 0 ¯
(41)
4. Simulation Results In this section, the sufficient and necessary conditions of dynamic stability for a rolling missile are
examined by using numerical simulation. The parameters of the rolling missile are listed in Table 1. Table 1 Parameters of a rolling missile Parameters
Value
Parameters
Value
ܿᇱ
10.03
ᇱ ܿெ
-1.103
-0.6589
ᇱ ݉ெ
-5.357
݉௦ᇱ ܥΤܣ
3.148×10
݉ఠ
1.777
ܸ ሺ݉Ȁݏሻ
1200
݉ఙ
0.0546
݇௦
10
ܶ௦ ሺݏሻ
0.016
ߤ௦
0.5
߬ሺݏሻ
0.015
-3
According to Table 1 and setting the rolling rate as Ͷ ή ʹɎ Τݏ, the coefficients can be calculated as ܽଵ ൌ ͳǤͳʹͶͲ , ܽଶ ൌ ͲǤͲͲͲͺ , ܾଵଵ ൌ െͳͶͲǤͳ͵ , ܾଵଶ ൌ Ǥͳͺ͵ , ܾଶଵ ൌ ͲǤͲͻͳ , ܾଶଶ ൌ െʹǤͶͲʹͶ , and ܿఙ ൌ ͳͳǤͷʹͲ. (1) Ignoring the effect of the radome When the parasitic radome loop is ignored, the stability of the rolling missile depends on the PI autopilot. Supposing ൌ Ͳ and ൌ ͳ, the parameters of the PI autopilot are ூ ൌ Ͳ, ൌ ͲǤͲͷ, and ఠ ൌ ͲǤ; these parameters are chosen according to the conventional assumption that the frequency bandwidth of the rudder is 3.5 times that of the autopilot and the stability limit for a rolling missile. 3
80
ay
60
az
2
40
1
20
D (deg)
Lateral Accleration(m/s2)
100
0
0 -20
-1
-40 -60
-2
-80 -100 0.0
0.5
1.0
1.5
2.0
2.5
t(s)
(a) Curves of ܽ௬ and ܽ௭
3.0
-3
-3
-2
-1
0
1
2
3
E (deg)
(b) The relationship between Ƚ and Ⱦ
Fig. 4 Stable conning motion while ignoring the parasitic radome loop
Fig. 4 shows that the initial disturbance decays and the coning motion approaches the zero point; i.e., the rolling missile is proved to be stable. (2) Considering the effect of the radome Using the same parameters of PI autopilot without considering the effect of the radome, according to the dynamic stability condition of the rolling missile (41), the calculated upper bound is ܰ ᇱ ܸ ሺെܴ௦ ሻ ൏ ͳǤͷͶͳͻ,
when the rolling rate is Ͷ ή ʹɎ Τ ݏ. Numerical simulations with ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͻ , ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͷͶͳͻ, and ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͲ are conducted to verify the stability condition obtained using our proposed method in the different initial values. 100
2000
60
az
1000
40
500
20
D (deg)
Lateral Accleration(m/s2)
80
ay
1500
0
0
-20
-500
-40
-1000
-60
-1500 -2000 0.0
-80
0.5
1.0
1.5
2.0
2.5
3.0
-100 -100
-80
-60
-40
-20
0
20
40
60
80
100
E (deg)
t(s)
(a) Curves of ܽ௬ and ܽ௭
(b) The relationship between Ƚ and Ⱦ
Fig. 5 Unstable coning motion with ܰԢ ܸܿ ሺെܴ௦ ሻ ൌ ͳǤͻ 3
80
ay
60
az
2
40
1
20
D(deg)
Lateral Acceleration(m/s2)
100
0
0 -20
-1
-40 -60
-2
-80 -100 0.0
0.5
1.0
1.5
2.0
t(s)
(a) Curves of ܽ௬ and ܽ௭
2.5
3.0
-3
-3
-2
-1
0
1
2
E(deg)
(b) The relationship between Ƚ and Ⱦ
Fig. 6 Critical state of coning motion with ܰԢ ܸܿ ሺെܴ௦ ሻ ൌ ͳǤͷͶͳͻ
3
3
60
ay
2
az
1
20
D (deg)
Lateral Acceleration(m/s2)
40
0
0
-20
-1
-40
-2
-60 0.0
0.5
1.0
1.5
2.0
2.5
3.0
-3
-3
-2
-1
(a) Curves of ܽ௬ and ܽ௭
0
1
2
3
E(deg)
t(s)
(b) The relationship between Ƚ and Ⱦ
Fig. 7 Stable coning motion with ܰԢ ܸܿ ሺെܴ௦ ሻ ൌ ͳǤͲ
For ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͻ, which is out of bounds, Fig. 5 shows that the lateral acceleration and nutation angle are divergent under the initial disturbance. At ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͷͶͳͻ, which is the exact critical gain, Fig. 6 shows the lateral acceleration and nutation angle with initial disturbance are neutrally stable. At ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͲ, which satisfies the stability condition, Fig. 7 shows that the initial disturbance decays and the coning motion approach the zero point. Evidently, the sufficient and necessary conditions of the rolling missile are consistent with the simulation results, which can be used as a reference for guiding the system design of a radar homing rolling missile, considering the effect of the radome on the guidance system. Note that when using the same PI autopilot parameters, the rolling missile is stable when ignoring the effect of the radome; however, when considering the effect of the radome, the stability of the rolling missile is related to the value of ܰ ᇱ ܸ ሺെܴ௦ ሻ. Thus, even if the PI autopilot is designed to be stable, the parasitic radome loop may lead to the instability of the rolling missile. This proves that the parasitic radome loop should be considered in the design of the guidance and control system for the rolling missile.
5. Conclusions In this paper, by using the stability criterion of the third-order characteristic equation with complex coefficients, the sufficient and necessary condition of the dynamic stability of a rolling missile with PN and PI acceleration considering the parasitic radome loop is analytically derived, and the stability condition further verified in a numerical simulation based on the nonlinear model. In the design of a rolling missile equipped with a radar seeker, the effect of the radome on the guidance system must be considered. The applicability of the design parameters according to the stability conditions obtained in this paper must be
checked. The stability conditions given in this paper can be applied to the engineering design of a guidance and control system for a radar homing rolling missile considering the parasitic radome loop. Future work will consider more topologies of the autopilot in this framework.
Funding This work was supported by the Natural Science Foundation of China (No.61172182), the open research project of the Beijing key laboratory of high dynamic navigation technology (Gran No. HDN2014103) and the fundamental research project of Beijing Institute of Technology (Grant No.20130142017).
Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this article.
Appendix A. Derivation of the characteristic equation of the system for a rolling missile Defining
A
ª º « » 0 0 a1 ia2 « » « » 0 0 1 « » « ª¬ vka N cVc Rs º¼ k I K c » p2 iq2 p1 iq1 » « 1 ck p ka K c «¬ »¼
(A1)
According to the calculation formula of the characteristic equation, the characteristic equation can be solved as follows:
sI A
s 0
a1 ia2 s
0 1
ª¬ vk I ka N cVc Rs k I º¼ K c 1 ck p ka K c
p2 iq2
s p1 iq1
ª vk I ka N cVc Rs k I º¼ K c s 2 ª¬ s p1 iq1 º¼ a1 ia2 ¬ ¬ª p2 iq2 s ¼º 1 ck p ka K c s 3 p1 iq1 s 2 p2 iq2 s a1 ia2 where ܫis the ͵ ൈ ͵ unit matrix.
( vka N cVc Rs )k I K c 1 ck p ka K c
0
(A2)
Appendix B. Stability condition of the third-order differential equation with complex coefficients Assume the third-order characteristic equation with complex coefficients is
D s O0 s 3 O1s 2 O2 s O3 where Ok
(B1)
Ok1 Ok 2 Dk iEk , k 1,2,3. .
According to [24], if there exist
Q s O11s 2 O22 s O31
(B2)
The J-fraction can be constructed as Eq. (B3)
Q s D s
where ݉ଵ ൌ
ఎభభ ఒభభ
ˈ݉ଶ ൌ
ఎయభ ఎమభ
ˈଷ ൌ
ఎరభ
r1 s m1
ˈݎଵ ൌ
ఒబ ఒభభ
(B3)
r2 s m2
ˈݎଶ ൌ
ఒభభ ఎమభ
r3 s m3
ˈݎଷ ൌ
ఎమభ ఎరభ
. If ݎ ሺ݊ ൌ ͳǡʹǡ͵ሻ exists and is positive,
then the system determined by characteristic equation (B1) is stable. The coefficients can be determined using the following method, with the coefficient matrix given as
O0 O11 K11 K21 K31 K41 where ߟଵଵ ൌ ఎమభ ఒమమ ିఒభభ ఎమమ ఎమభ
ఒభభ ఒభ ିఒబ ఒమమ ఒభభ
, ߟଵଶ ൌ
, ߟଷଶ ൌ ߣଷଵ , ߟସଵ ൌ
ఒభభ ఒమ ିఒబ ఒయభ ఒభభ
, ߟଵଷ ൌ ߣଷ , ߟଶଵ ൌ
ఎమభ ఎయమ ିఎయభ ఎమమ ఎమభ
O1 O2 O3 O22 O31 K12 K13 K22 K32
(B4)
ఒభభ ఎభమ ିఒమమ ఎభభ ఒభభ
, ߟଶଶ ൌ
ఒభభ ఎభయ ିఒయభ ఎభభ ఒభభ
, ߟଷଵ ൌ
.
For the third-order characteristic equation with complex coefficients (B1), let ߣ ൌ ͳ, ߣଵ ൌ ଵ ݅ݍଵ , ߣଶ ൌ ଶ ݅ݍଶ , ߣଷ ൌ ଷ ݅ݍଷ , ߣଵଵ ൌ ߙଵ , ߣଶଶ ൌ ݅ߚଶ , and ɉଷଵ ൌ ߙଷ ; thus, the parameters in this continued fraction can be obtained and expressed as
r1
p1
r2
p q
(B5) 2 1 2
p1 p3 p1q1q2 q22 / r12
(B6)
r3
ª p 2 p p p p q q q 2 p p p q 2 p q q p 2 q p p q p q 2 º / r 6r 3 1 3 1 1 2 2 1 2 3 3 1 2 3 1 3 1 3 1 3 2 ¬« 1 2 ¼» 1 1
(B7)
The sufficient and necessary stability condition can be further simplified as
° p1 ! 0 ° 2 2 ® p1 p2 p1 p3 p1q1q2 q2 ! 0 ° 2 ° p12 p2 p1 p3 p1q1q2 q22 p1 p2 p3 p32 p1q2 q3 p12 q3 p1 p3q1 p3q2 ! 0 ¯
(B8)
Reference [1] Nesline F W, Zarchan P. Radome Induced Miss Distance in Aerodynamically Controlled Homing Missile[J]. AIAA. 1984, 84(1): 845. [2] Miwa S, Kouya S. Radome effect on the miss distance of a radar homing missile[J]. Electronics and Communications in Japan(Part I Communications). 1998, 81(7): 14-22. [3] Nesline F W, Nesline M L. Wing size vs radome compensation in aerodynamically controlled radar homing missiles[J]. Journal of Guidance, Control, and Dynamics. 1986, 9(6): 645-649. [4] Nesline F W, Zarchan P. Missile guidance design tradeoffs for high-altitude air defense[J]. Journal of Guidance Control & Dynamics. 1983, 6(3): 207-212. [5] Brown R, Santi G. Missile system for low altitude air defence[C]. 1984. [6] Peterson D, Otto J, Douglas K. Radome boresight error and compensation techniques for electronically scanned arrays[R]. DTIC Document, 1993. [7] Stallard D. Analysis of the parastitic radome loop in a bank-to-turn homing missile[J]. AIAA Pap. 1985: 176-187. [8] Yost D J, Weckesser L B, Mallalieu R C. Technology Survey of Radomes for Anti-Air Homing Missiles[J]. Johns Hopkins University Applied Physics Laboratory, Laurel, MD. 1980. [9] Lin C F. Bank-to-turn guidance performance analysis with in-flight radome error compensation[C]. 1984. [10] Yueh W R, Lin C F. Guidance performance analysis with in-flight radome error calibration[J]. Journal of Guidance, Control, and Dynamics. 1985, 8(5): 666-669. [11] Lin J, Chau Y. Radome slope compensation using multiple-model Kalman filters[J]. Journal of Guidance, Control, and Dynamics. 1995, 18(3): 637-640. [12] Willman W. Radome compensation using adaptive dither[C]. 1998.
[13] Zarchan P, Gratt H. Adaptive Radome Compensation Using Dither[J]. Journal of Guidance, Control, and Dynamics. 1999, 22(1): 51-57. [14] Tiwari P K, Vora P, Bhattacharjee R N. In flight radome error compensation through simulated test data[Z]. 2005. [15] Song T L, Lee D G, Shin S J. Active homing performance enhancement with multiple model radome slope estimation[J]. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. 2005, 219(3): 217-224. [16] Yan X, Yang S, Zhang C. Coning Motion of Spinning Missiles Induced by the Rate Loop[J]. Journal of Guidance, Control, and Dynamics. 2010, 33(5): 1490-1499. [17] Yan X, Yang S, Xiong F. Stability Limits of Spinning Missiles with Attitude Autopilot[J]. Journal of Guidance, Control, and Dynamics. 2011, 34(1): 278-283. [18] Li K, Yang S, Zhao L. Stability of Spinning Missiles with an Acceleration Autopilot[J]. Journal of Guidance, Control, and Dynamics. 2012, 35(3): 774-786. [19] Zhou W, Yang S, Dong J. Coning motion instability of spinning missiles induced by hinge moment[J]. Aerospace Science and Technology. 2013, 30(1): 239-245. [20] Zhou W, Yang S, Zhao L. Limit Circular Motion of Spinning Projectiles Induced by Backlash of Actuators[C]. 2014. [21] Koohmaskan Y, Arvan M R, Vali A R, et al. Dynamic stability conditions for a rolling flight vehicle applying continuous actuator[J]. Aerospace Science and Technology. 2015, 42: 451-458. [22] Mao X, Yang S. Optimal Control of Coning Motion of Spinning Missiles[J]. Journal of Aerospace Engineering. 2015, 28(2). [23] He S, Lin D, Wang J. Coning motion stability of spinning missiles with strapdown seekers[J]. The Aeronautical Journal. 2016, 120(1232): 1566-1577. [24] Frank E. On the zeros of polynomials with complex coefficients[J]. Bulletin of the American Mathematical Society. 1946, 52(2): 144-157.
Duo Zheng, Defu Lin, Xinghua Xu, Song Tian
PII: DOI: Reference:
S1270-9638(17)30549-7 http://dx.doi.org/10.1016/j.ast.2017.03.036 AESCTE 3975
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
11 January 2016 13 November 2016 29 March 2017
Please cite this article in press as: D. Zheng et al., Dynamic stability of rolling missile with proportional navigation & PI autopilot considering parasitic radome loop, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.036
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Dynamic Stability of Rolling Missile with Proportional Navigation & PI Autopilot Considering Parasitic Radome Loop Duo Zheng1, Defu Lin1, Xinghua Xu2, Song Tian1 1ǃSchool of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China 2ǃBeijing Institute of Automatic Control Equipment, Beijing, 100074, People’s Republic of China
Abstract The dynamic stability of coning motion for a radar homing rolling missile is studied in this paper, and the parasitic radome loop is considered and introduced into the guidance system of the radar homing rolling missile. A mathematical model with complex summation considering the parasitic radome loop is proposed, and the sufficient and necessary conditions for the dynamic stability of the coning motion for the rolling missile are analytically derived by using the stability criterion of third-order characteristic equation with complex coefficients. Numerical simulations with different parameters are conducted to demonstrate the effectiveness of the proposed stability condition. The stability condition proposed in this paper can be used in the guidance and control system design of a radar homing rolling missile while considering the parasitic radome loop. Keywords Rolling missile, PI autopilot, dynamic stability, parasitic radome loop, proportional navigation
1. Introduction For all radar homing missiles, the radome is designed and equipped to protect the seeker antenna both mechanically and thermally. However, when the radar signal passes through the radome, the radar wave will be bent or refracted by the radome, thereby generating a false LOS rate. The introduction of the false LOS rate coupled with angular body motion into the guidance system adds a parasitic feedback loop to the missile’s guidance system, which in turn affects the miss distance (MD) and even the dynamic stability of the radar homing missile. As a result, the parasitic radome loop must be considered in the guidance system design of radar homing missiles. A detailed analysis of the radome error slope (RES) was conducted by Nesline and Zarchan
[1]
; in their
papers, the missile guidance system was replaced by the linear fifth-order binomial, and the radome error
slope was introduced to the linear fifth-order binomial. The stability condition of the system was proposed based on the Routh-Hurwitz stability criterion, and the miss distance result from the radome error slope was analysed using the adjoint method. The miss distances of a radar homing missile induced by the radome error slope were investigated under various parameter conditions, and a sensitivity analysis of system to the radome error slope was performed by assuming its step and sinusoidal change during the flight time
[2]
.A
fundamental trade-off between the radome compensation and wing size was studied for a radar-guided aerodynamically controlled missile; it was shown that radome compensation can be used to reduce the wing size, thereby reducing a missile’s drag and weight
[3]
. The radome error slope was considered one of the
seeker component imperfections, and its effect on the guidance system was studied for high-altitude radarguided air defence missiles
[4]
. Brown, et al. indicated that both the positive and negative boresight error
shifts induced by radome refraction are important in the design because these shifts will increase the miss distance and may actually lead to system instability
[5]
. Both the gimbal seeker and electronically scanned
arrays contribute to the radome problem. Peterson D, et al. studied the radome boresight error in electronically scanned arrays and found that radome-induced boresight error and boresight error slope are usually the dominant errors. The radome error not only limits the beam pointing accuracy but also affects the beam-shape. A negative slope leads to an under damped guidance loop, possibly creating instabilities during the performance of manoeuvres. A positive slope leads to an overly damped system and an underresponsive guidance loop [6]. A perturbation analysis of the parasitic radome loop for a BTT missile was also proposed. Because BTT missiles exhibit inertial and aerodynamic cross-coupling, the equations of the missile and guidance system were linearized around the operating point to conveniently determine the poles and stability margins. The effects of the radome error slope on the stability were studied under different coupling conditions
[7]
. According to the above-described studies, the missile homing performance will be
seriously degraded by the radome error slope. To increase both the homing performance and target intercept accuracy, some researchers have studied in-flight radome error compensation to overcome the restrictions imposed by the radome[6, 8-15]. The rolling missile cross-coupling between the pitch and yaw channels may lead to a further dynamic instability in the form of a divergent coning motion. There have been many studies on the dynamic stability of a spinning missile[16-21]. The sufficient and necessary conditions of dynamic stability for canard-control spinning missiles with a rate loop, an attitude autopilot and PI acceleration autopilots were analytically
derived and further verified by numerical simulations
[16-18]
. The coning motion instability of a spinning
missile induced by the hinge moment was considered, and the sufficient and necessary condition of the conning motion stability was derived and validated through nonlinear six degrees-of-freedom simulation [19]. The dynamic stability of spinning projectiles induced by backlash of actuator was studied in the reference [20]
. The dynamic stability for a rolling flight vehicle using a continuous actuator was proposed by
Koohmaskan et al. [21], who considered the dynamics of the actuator in the system model, analytically derived the sufficient and necessary condition based on the Routh-Hurwitz stability criterion, and examined the derived stability condition in a simulation. In another study [22], a novel optimal control algorithm based on stability analyses was developed to suppress the coning motion. The coning motion stability of spinning missiles with strapdown seekers was studied by Lin D et al.; in their study, the rolling missile is equipped with a strapdown seeker. The parasitic loop induced by the scaling factor error is considered, and the sufficient and necessary stability condition is proposed using a complex summation method[23]. Although there are many references on the radome error slope, none of these references has considered its effect on a spinning missile. The dynamic stability of rolling missile equipped with radar seeker may be worse because of the radome error slope and the cross coupling of the pitch and yaw. This paper focuses on the dynamic stability of a rolling missile using a radar seeker with PI acceleration autopilot, considering the parasitic radome loop. The mathematical models of the radome error slope, the airframe, and the system equation in a form of complex summation are obtained. Based on the stability criterion with complex coefficients, the sufficient and necessary condition of the dynamic stability for a rolling missile considering the radome error slope is derived analytically. A proof of the third-order characteristic equation with complex coefficient is presented in the Appendix. The numerical simulation is further used to examine the proposed stability conditions. The rest of this paper is organized as follows. In the problem formulation section, simplified mathematical models of the radome and airframe are presented with reasonable assumptions. In the section describing the stability condition of the rolling missile considering the parasitic radome loop, the sufficient and necessary condition of dynamic stability is analytically derived by using the stability criterion with complex coefficients. In the simulation results section, the results of simulation studies used for verifying the stability condition proposed are presented. The conclusions are presented in the conclusions section.
2. Problem formulation 2.1.Radome problem formulation The physics of radome refraction are shown in Fig. 1, where ߟ
denotes the refraction angle. It can be
observed that the radar wave is refracted by the radome. The target appears to be displaced from its true position because of the radome refraction angle. The radome refraction angle varies with the seeker gimbal angle. The radome error slope is defined as the rate of change of the radome refraction angle with gimbal angle and is given as
Rs
dK dT B
(1)
where ߠ is the gimbal angle of the seeker.
x Apparent Target angle ction
K Refra
Radar Antenna
x
Target
Radome
Fig. 1 Radome refraction physics
Supposing the ܴ௦ is a constant, the refraction angle is linearly proportional to the gimbal angle according to the radome slope. It can be expressed as
K
RsT B
(2)
The missile-target geometry is shown in Fig. 2, where ݍand ݍᇱ is the real LOS angle and the false LOS angle caused by radome refraction, respectively, ߝ is the error angle, ߶ெ is the seeker angle relative to the reference line, and ߴ is the pitch angle,
Apparent target Target
K H
-
IM
q
Seeker
TB
qc Body axis
Reference Line
Fig. 2 Planar missile-target geometry
According to Fig. 2, the real LOS angle can be expressed as
q
qc K IM H K
(3)
Substituting Eq. (2) into Eq. (3) yields
q IM H RsT B
(4)
The gimbal angle can be expressed as
T B IM -
(5)
Substituting Eq. (5) into Eq. (4) yields
q IM H Rs IM - H 1 Rs IM Rs-
(6)
Because radome slope is small (that is, ܴ௦ )ͳ ا, Eq. (6) can be simplified as
q H IM Rs-
(7)
The false LOS angle due to radome refraction can be expressed as
qc IM H
(8)
Substituting Eq. (8) into Eq. (7), the false LOS angle can be solved as
qc q Rs-
(9)
Because the radome slope is assumed to be constant, the false LOS rate with radome refraction can be expressed as
qc q Rs-
(10)
where ݍሶ is the LOS rate, ݍሶ ᇱ is the false LOS rate, and ߴሶ is the pitch rate. Expanding Eq. (10) to the yaw channel, the PN guidance system of the roll missile with PI acceleration considering the effect of radome is shown in Fig. 3.
Rs ka
s
c
kZ qy
qz
N cVc
a yc
N cVc a zc
kI s k kp I s
kp
V c1
e1 Command Generation
e2
V c2
Command Resolution & Actuators Dynamics
-
V1
ay
Airframe Motion
V1
\
az
kZ s
c
ka
Rs
Fig. 3 PN system of a rolling missile with PI acceleration considering the parasitic radome loop
2.2.Airframe Equations of Lateral Motion Ignoring rolling motion, the rolling missile dynamics equations
[18]
can be simplified based on the
following assumptions: (a) The velocity remains constant over a short interval in the flight; (b) The rolling rate remains constant over a short interval in the flight; (c) The gravity effect is ignored; (d) The coefficients of aerodynamic moments are linearized; and (e) The variables ߰ǡ ߰ ǡ ߙǡ ߚǡ ߶ǡ ܽ݊݀ሶ ߰ are small, i.e., ܿ ߰ݏൎ ͳ, ܿ߰ݏ ൎ ͳ, ܿ ߙݏൎ ͳ, ܿ ߚݏൎ ͳ, ߙ݊݅ݏൎ ߙ, ߚ݊݅ݏൎ ߚ, and ߶݊݅ݏൎ ߶. Under these assumptions, the equations of rolling missile can be formulated as 1 1 °mVmT cos\ V QSCL sin I cos E sin D QSCM sin I sin E T cos E sin D ° ° 1 1 °mVm\ V QSCL sin I sin E QSCM sin I cos E sin D T sin E ° ® 1 1 ° A\ CJsin E cos D QSLM M sin D QSLM Z L / Vm Z y QSLM V V 2 - cos\ C A - 2 sin i \ cos\ QSLM S ° sin I sin I ° 1 1 ° A - cos\ -\ sin D QSLM M sin E cos D QSLM Z L / Vm Z z QSLM V V 1 \ sin i \ CJ\ C A -\ - sin si \ QSLM QS S °¯ sin I sin I
(11) where ܵ is the reference area, ܮis the reference length, ܳ is the dynamic pressure, ܸ is the missile velocity, ܣis the lateral moment of inertia, ܥis the longitudinal moment of inertia, ܥ , ܥ , and ܥெ are the drag, lift, and Magnus force coefficient, respectively, ܯெ , ܯௌ , and ܯఠ are the coefficient of the Magnus moment, the static moment, and the damping moment, respectively, ܯఙ is the coefficient of the
control moment, ߴ, ߰ and ߛ are the pitch, yaw and roll angle, respectively, ߠ and ߰ are the flight-path angle and the heading angle, respectively, ߙ and ߚ are the angle of attack and sideslip in non-rolling coordinates, respectively, ߙ and ߚ are the actuator responses in the non-rolling coordinate system, ߶ is the total angle of attack, and ܶ is the thrust force. Eq. (11) can be further simplified to
§J d · Q QS QS T cLc D cMc ¨ D °T ¸E mV mV V mV m m m © m ¹ ° ° §J d · Q QS QS T °\ V cLc E cMc ¨ E ¸D ° mVm mVm mVm © Vm ¹ ° ® §J d · § L · C QSL QSL QSL QSL Q QS Q Q ° M ScD M Mc ¨ M Z ¨ ¸- MVV1 J ¸E °- A JI A A V A V A m ¹ m ¹ © © ° ° §J d · § L · Q L Q QSL Q C QSL QSL QSL MVV 2 M Sc E M Mc ¨ M Z ¨ ¸\ °\ J¸D A A A A A ° © Vm ¹ © Vm ¹ ¯
(12)
ᇱ are the derivatives of the lift and the Magnus force coefficients, respectively, and ܯௌᇱ where ܥᇱ and ܥெ ᇱ and ܯெ are the derivatives of the static and Magnus moments, respectively.
Defining a1
b21
C J , b22 A
QS T , a2 CLc mVm mVm
§J d · QS CMc ¨ ¸ , b11 mVm © Vm ¹
§ L· QSL M Z ¨ ¸ , and cV A © Vm ¹
QSL M Sc , b12 A
§J d · QSL M Mc ¨ ¸, A © Vm ¹
QSL M V , Eq. (12) can be rewritten as A
T a1D a2 E ° °\ V a1E a2D ® °- b11D b12 E b21\ b22- cV V 1 °\ b E b D b - b \ c V 11 12 21 22 V 2 ¯
(13)
It can be obtained that ߴ ൌ ߠ ߙ and ߰ ൌ ߰ ߚ when ߴ, ߰, ߙ, and ߚ are small values; therefore, the response of the airframe can be further expressed as
°- T D a1D a2 E D ® °¯\ \ V E a1E a2D E
(14)
From Eqs. (13) and (14), the motion of the airframe can be described by using the non-rolling angle of attack ߙ and the non-rolling sideslip angle ߚ. Following the same formulations ߴ ൌ ߠ ߙ and ߰ ൌ ߰ ߚ, Eq. (13) can be rewritten as
D - T b11D b12 E b21\ b22- cV V 1 a1D a2 E ° ® ° ¯ E \ \ V b11E b12D b21- b22\ cV V 2 a1E a2D
(15)
Substituting Eq. (14) into Eq. (15) yields
b11 b21a2 b22 a1 D b12 b21a1 b22a2 E b22 a1 D b21 a2 E cV V1 b11 b21a2 b22 a2 E b12 b21a1 b22a2 D b22 a1 E b21 a2 D cV V 2
°D ® °¯ E
(16)
2.3.Command Generation 1ǃ Guidance Command Generation From Fig. 3, the PN guidance command with the parasitic radome loop is obtained as
ª a yc º «a » ¬ zc ¼
ª N cVc q y N cVc Rs - º » « ¬ N cVc qz N cVc Rs \ ¼
(17)
where ܰ ᇱ is the effective navigation ratio, ܸ is the closing velocity, ܸ is the missile velocity, and ܴ௦ is the radome error slope. 2ǃ Flight Command Generation According to Fig. 3, the actuator signals in the non-rolling coordinate system, denoted by ݁ଵ and ݁ଶ in the pitch and yaw channels, respectively, can be expressed as
ª e1 º «e » ¬ 2¼
ª§ kI «¨ k p s «© «§ kI «¨ k p s ¬©
kI · º · § ¸ a yc ¨ k p ¸ ka a y c- kZ- » s ¹ ¹ © » » kI · · § ¸ azc ¨ k p ¸ ka az c\ kZ\ » s ¹ ¹ © ¼
(18)
where ݇ is the gain of the proportional item, ݇ூ is the gain of the integral regulator, ݇ఠ is the gain of the damping loop, ܿ is the distance between the accelerometer installation location and the centre of mass. For a canard-control missile, note that the coordinate system mentioned above is such that a positive angle of a servo produces a positive-going angle of attack and positive acceleration in the pitch, along with a positive-going sideslip angle and negative acceleration in the yaw. Therefore, the input commands to the actuators can be described as the following formations for a canard-control missile:
ªV c1 º «V » ¬ c2 ¼
ª1 0 º ª e1 º «0 1» «e » ¬ ¼¬ 2¼
ª§ kI · kI · º § «¨ k p s ¸ a yc ¨ k p s ¸ ka a y c- kZ- » ¹ © ¹ «© » « § » kI · k § I · « ¨ k p ¸ azc ¨ k p ¸ ka az c\ kZ\ » s s ¹ © ¹ ¬ © ¼
Considering ܽ௬ ൌ ܸ ߠሶ and ܽ௭ ൌ െܸ ߰ሶ and substituting Eq. (13) into ܽ௬ and ܽ௭ , one has
(19)
Vm a1D a2 E
°a y ® ° ¯ az
(20)
Vm a1E a2D
3ǃ Command Resolution and Actuator Response The rolling of a missile induces harmonic responses in the rudder angles under constant commands. Because the servo-system can be modelled by a second-order system with natural frequency ͳȀܶ௦ , damping ratio ߤ௦ , command transmission delay ߬, and constant roll rate (i.e., ɀሶ ൌ Ͳ), the relationship between the input and output of the servo-system in the non-rolling body system for the rolling missile can be expressed as
ªV 1 º «V » ¬ 2¼
ª cos J d ks kr « ¬ sin J d
sin J d º ªV c1 º cos J d »¼ «¬V c 2 »¼
(21)
where ݇௦ is the gain of the servosystem and ݇ is the dynamic gain of the servo-system under rolling and can be expressed as
kr
1
1 T J 2P T J 2 s
2 2
2
s s
(22)where ܶ௦ is the time constant of the servo-system, ߛ is defined as the steady-state deviation angle caused by the delay of the servo-system dynamics and rotation, and
Jc
§ arccos ¨¨ ¨ ©
1 Ts2J 2
1 T J 2P T J 2 s
2 2
s s
2
· ¸ ¸ ¸ ¹
(23)
Therefore, the total delay angle can be calculated as ɀௗ ൌ ߛ ߬ ή ߛሶ , where ߬ is the time delay of the control system.
3. Dynamic Stability Condition of Rolling Missiles Considering Parasitic Radome Loop It is known from the mathematical model for each component of the acceleration autopilot that both the actuator responses and the airframe responses induce cross-coupling between pitch and yaw and that the attitudes are coupled into the guidance system because of the parasitic radome loop. Therefore, a complex summation model is required to conveniently perform the stability analysis. Eq. (21) can be expressed as
V
ks kr V c1 sin J d V c 2 cos J d i k s kr V c1 cos J d V c 2 sin J d
(24)
ks kr cos J d i sin J d V c 2 i V c1 where ɐ ൌ ߪଶ ݅ ή ߪଵ is defined as the complex deflection angle. Substituting Eq. (17) into Eq. (19), the following formations can be obtained:
ª§ kI · kI · º § » «¨ k p s ¸ N cVc q y N cVc Rs - ¨ k p s ¸ ka a y c- kZ¹ © ¹ «© » » « § kI · kI · § « ¨ k p ¸ N cVc qz N cVc Rs \ ¨ k p ¸ ka az c\ kZ\ » s ¹ s ¹ © ¬ © ¼
ªV c1 º «V » ¬ c2 ¼
(25)
Without loss of generality, the LOS rate inputs ݍሶ ௬ and ݍሶ ௭ are assumed to be zero in the stability analysis; thus, Eq.(25) can be simplified as
kI ª º « k p N cVc Rs - k I N cVc Rs - k p ka a y k p ka cs- ka s a y k I ka c- kZ- » « » (26) « k N cV R \ k N cV R \ k k a k k cs\ k I k a k k c\ k \ » I c s p a z p a a z I a Z c s «¬ p »¼ s
ªV c1 º «V » ¬ c2 ¼
Substituting Eq. (20) into Eq. (26) yields
ªV c1 º «V » ¬ c2 ¼
ª k p N cVc Rs - k I N cVc Rs - k p kaVmT ka k IVmT k p ka c- k I ka c kZ - º « » ¬« k p N cVc Rs \ k I N cVc Rs \ k p kaVm\ V ka k IVm\ V k p ka c\ k I ka c kZ \ ¼» (27)
Substituting Eqs. (13) and (14) into Eq. (27) yields
ªV c1 º ª k p N cVc Rs a1D a2 E D k I N cVc Rs - k p kaVm a1D a2 E ka k IVmT «V » = « ¬ c 2 ¼ «¬ k p N cVc Rs a1E a2D E k I N cVc Rs \ k p kaVm a1E a2D ka k IVm\ V
k k c b E b D b a D a E D b a E a D E c V k p ka c b11D b12 E b21 a1E a2D E b22 a1D a2 E D cV V 1 p a
11
12
21
1
2
22
1
2
1
V
(28)
2
k I ka c kZ a1D a2 E D º » k I ka c kZ a1E a2D E ¼»
Substituting Eq. (28) into Eq. (24) with the definitions ߜ ൌ ߚ ݅ ή ߙ, ߯ ൌ ߰ ݅ ή ߴ, and ߮ ൌ ߰ ݅ ή ߠ, one obtains
k p kaVm a1 i a2 G ka k IVmM k p ka c b22 i b21 G ½ ° ° °° ck p ka ª¬ b11 b21a2 b22a1 i b12 b21a1 b22a2 º¼ G °° k s kr cos J d i sin J d ® ¾ ° ck p ka cV V kZ ck I ka a1 i a2 G kZ ck I ka G ° ° ° ¯°k p N cVc Rs a1 i a2 G k p N cVc Rs G k I N cVc Rs F ¿°
V
(29) Because ߯ ൌ ߮ ߜ, one has
k p kaVm a1 i a2 G ka k IVmM k p ka c b22 i b21 G ½ ° ° ck k b b a b a i b b a b a G ª º ° ° p a ¬ 11 21 2 22 1 12 21 1 22 2 ¼ ° ° ks kr cos J d i sin J d ® ¾ °ck p ka cV V kZ ck I ka a1 i a2 G kZ ck I ka G ° ° ° °k p N cVc Rs a1 i a2 G k p N cVc Rs G k I N cVc Rs M G ¿ ° ¯
V
(30) From Eq. (16), the complex summation equation of the airframe motion can also be obtained:
ia2 G cV V ª¬ b11 b21a2 b22a1 b12 b21a1 b22a2 º¼ G b22 ib21 G a1 ia
G
(31)
By defining ܭൌ ܿఙ ݇௦ ݇ and ܭᇱ ൌ ܭሺܿߛݏௗ ݅ ή ߛ݊݅ݏௗ ሻ, and substituting Eq. (29) into Eq. (31), one has
ªVm ka N cVc Rs º¼ kI K c
G p1 iq iq1 G p2 iq i 2 G ¬
1 ck p ka K c
M
0
(32)
where
p1 iq1
p2 iq2
a1 +ia2 b22 ib21
cka k p K c b22 ib21 ª¬ kZ ck I ka k p NVc Rs º¼ K c 1 ck p ka K c 1 ck p ka K c
¬ª b11 b21a2 b22 a1 i b12 b21a1 b22 a2 ¼º
Vm k p ka K c 1 ck p ka K c
ck p ka K c
ª b11 b21a2 b22 a1 i b12 b21a1 b22 a2 ¼º 1 ck p ka K c ¬
k
Z
ck I ka k p N cVc Rs K c 1 ck p ka K c
a1 ia2
By denoting the sate variables ݔൌ ሾ߮ can be expressed as
ߜ
(33)
a1 ia2 (34)
k I K cN cVc Rs 1 ck p ka K c
ߜሶ ሿ, the state equation of the PN guidance for a rolling missile
x
ªM º «G » « » ¬«G »¼
ª º « » 0 0 a1 ia2 « » ª\ º « » «G » 0 0 1 « »« » « ª¬Vm ka N cVc Rs º¼ k I K c » «¬ G »¼ D 2 i E 2 D1 iE1 » « 1 ck p ka K c ¬« ¼»
(35)
The characteristic equation can be further obtained as (see Appendix 1)
s3 p1 iq1 s 2 p2 iq2 s a1 ia2
(Vm ka N cVc Rs )kI K c 1 ck p ka K c
0
(36)
Defining
p3 iq3
ª¬Vm ka N cVc Rs º¼ k I K c a1 ia2 1 ck p ka K c
(37)
The characteristic equation can be rewritten as
s3 p1 iq1 s 2 p2 iq2 s p3 iq3 0
(38)
Theorem 1: The third-order characteristic equation with complex coefficients has the following form:
D s
s3 p1 iq1 s 2 p2 iq2 s p3 iq3 0
(39)
where ߙ , ߚ אԹ, ݅ ൌ ͳǡʹǡ͵, and the necessary and sufficient conditions for stability are
° p1 ! 0 ° 2 2 ® p1 p2 p1 p3 p1q1q2 q2 ! 0 ° 2 ° p12 p2 p1 p3 p1q1q2 q22 p1 p2 p3 p32 p1q2 q3 p12 q3 p1 p3q1 p3q2 ! 0 ¯
(40)
Proof. See Appendix 2. According to theorem 1, the sufficient and necessary condition for all the roots of Eq. (38) in the half-left of the complex plane is that the three inequalities (40) are positive. Therefore, the sufficient and necessary condition of dynamic stability for a rolling missile can be expressed as
° p1 ! 0 ° 2 2 ® p1 p2 p1 p3 p1q1q2 q2 ! 0 ° 2 ° p12 p2 p1 p3 p1q1q2 q22 p1 p2 p3 p32 p1q2 q3 p12 q3 p1 p3q1 p3q2 ! 0 ¯
(41)
4. Simulation Results In this section, the sufficient and necessary conditions of dynamic stability for a rolling missile are
examined by using numerical simulation. The parameters of the rolling missile are listed in Table 1. Table 1 Parameters of a rolling missile Parameters
Value
Parameters
Value
ܿᇱ
10.03
ᇱ ܿெ
-1.103
-0.6589
ᇱ ݉ெ
-5.357
݉௦ᇱ ܥΤܣ
3.148×10
݉ఠ
1.777
ܸ ሺ݉Ȁݏሻ
1200
݉ఙ
0.0546
݇௦
10
ܶ௦ ሺݏሻ
0.016
ߤ௦
0.5
߬ሺݏሻ
0.015
-3
According to Table 1 and setting the rolling rate as Ͷ ή ʹɎ Τݏ, the coefficients can be calculated as ܽଵ ൌ ͳǤͳʹͶͲ , ܽଶ ൌ ͲǤͲͲͲͺ , ܾଵଵ ൌ െͳͶͲǤͳ͵ , ܾଵଶ ൌ Ǥͳͺ͵ , ܾଶଵ ൌ ͲǤͲͻͳ , ܾଶଶ ൌ െʹǤͶͲʹͶ , and ܿఙ ൌ ͳͳǤͷʹͲ. (1) Ignoring the effect of the radome When the parasitic radome loop is ignored, the stability of the rolling missile depends on the PI autopilot. Supposing ൌ Ͳ and ൌ ͳ, the parameters of the PI autopilot are ூ ൌ Ͳ, ൌ ͲǤͲͷ, and ఠ ൌ ͲǤ; these parameters are chosen according to the conventional assumption that the frequency bandwidth of the rudder is 3.5 times that of the autopilot and the stability limit for a rolling missile. 3
80
ay
60
az
2
40
1
20
D (deg)
Lateral Accleration(m/s2)
100
0
0 -20
-1
-40 -60
-2
-80 -100 0.0
0.5
1.0
1.5
2.0
2.5
t(s)
(a) Curves of ܽ௬ and ܽ௭
3.0
-3
-3
-2
-1
0
1
2
3
E (deg)
(b) The relationship between Ƚ and Ⱦ
Fig. 4 Stable conning motion while ignoring the parasitic radome loop
Fig. 4 shows that the initial disturbance decays and the coning motion approaches the zero point; i.e., the rolling missile is proved to be stable. (2) Considering the effect of the radome Using the same parameters of PI autopilot without considering the effect of the radome, according to the dynamic stability condition of the rolling missile (41), the calculated upper bound is ܰ ᇱ ܸ ሺെܴ௦ ሻ ൏ ͳǤͷͶͳͻ,
when the rolling rate is Ͷ ή ʹɎ Τ ݏ. Numerical simulations with ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͻ , ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͷͶͳͻ, and ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͲ are conducted to verify the stability condition obtained using our proposed method in the different initial values. 100
2000
60
az
1000
40
500
20
D (deg)
Lateral Accleration(m/s2)
80
ay
1500
0
0
-20
-500
-40
-1000
-60
-1500 -2000 0.0
-80
0.5
1.0
1.5
2.0
2.5
3.0
-100 -100
-80
-60
-40
-20
0
20
40
60
80
100
E (deg)
t(s)
(a) Curves of ܽ௬ and ܽ௭
(b) The relationship between Ƚ and Ⱦ
Fig. 5 Unstable coning motion with ܰԢ ܸܿ ሺെܴ௦ ሻ ൌ ͳǤͻ 3
80
ay
60
az
2
40
1
20
D(deg)
Lateral Acceleration(m/s2)
100
0
0 -20
-1
-40 -60
-2
-80 -100 0.0
0.5
1.0
1.5
2.0
t(s)
(a) Curves of ܽ௬ and ܽ௭
2.5
3.0
-3
-3
-2
-1
0
1
2
E(deg)
(b) The relationship between Ƚ and Ⱦ
Fig. 6 Critical state of coning motion with ܰԢ ܸܿ ሺെܴ௦ ሻ ൌ ͳǤͷͶͳͻ
3
3
60
ay
2
az
1
20
D (deg)
Lateral Acceleration(m/s2)
40
0
0
-20
-1
-40
-2
-60 0.0
0.5
1.0
1.5
2.0
2.5
3.0
-3
-3
-2
-1
(a) Curves of ܽ௬ and ܽ௭
0
1
2
3
E(deg)
t(s)
(b) The relationship between Ƚ and Ⱦ
Fig. 7 Stable coning motion with ܰԢ ܸܿ ሺെܴ௦ ሻ ൌ ͳǤͲ
For ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͻ, which is out of bounds, Fig. 5 shows that the lateral acceleration and nutation angle are divergent under the initial disturbance. At ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͷͶͳͻ, which is the exact critical gain, Fig. 6 shows the lateral acceleration and nutation angle with initial disturbance are neutrally stable. At ܰ ᇱ ܸ ሺെܴ௦ ሻ ൌ ͳǤͲ, which satisfies the stability condition, Fig. 7 shows that the initial disturbance decays and the coning motion approach the zero point. Evidently, the sufficient and necessary conditions of the rolling missile are consistent with the simulation results, which can be used as a reference for guiding the system design of a radar homing rolling missile, considering the effect of the radome on the guidance system. Note that when using the same PI autopilot parameters, the rolling missile is stable when ignoring the effect of the radome; however, when considering the effect of the radome, the stability of the rolling missile is related to the value of ܰ ᇱ ܸ ሺെܴ௦ ሻ. Thus, even if the PI autopilot is designed to be stable, the parasitic radome loop may lead to the instability of the rolling missile. This proves that the parasitic radome loop should be considered in the design of the guidance and control system for the rolling missile.
5. Conclusions In this paper, by using the stability criterion of the third-order characteristic equation with complex coefficients, the sufficient and necessary condition of the dynamic stability of a rolling missile with PN and PI acceleration considering the parasitic radome loop is analytically derived, and the stability condition further verified in a numerical simulation based on the nonlinear model. In the design of a rolling missile equipped with a radar seeker, the effect of the radome on the guidance system must be considered. The applicability of the design parameters according to the stability conditions obtained in this paper must be
checked. The stability conditions given in this paper can be applied to the engineering design of a guidance and control system for a radar homing rolling missile considering the parasitic radome loop. Future work will consider more topologies of the autopilot in this framework.
Funding This work was supported by the Natural Science Foundation of China (No.61172182), the open research project of the Beijing key laboratory of high dynamic navigation technology (Gran No. HDN2014103) and the fundamental research project of Beijing Institute of Technology (Grant No.20130142017).
Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this article.
Appendix A. Derivation of the characteristic equation of the system for a rolling missile Defining
A
ª º « » 0 0 a1 ia2 « » « » 0 0 1 « » « ª¬ vka N cVc Rs º¼ k I K c » p2 iq2 p1 iq1 » « 1 ck p ka K c «¬ »¼
(A1)
According to the calculation formula of the characteristic equation, the characteristic equation can be solved as follows:
sI A
s 0
a1 ia2 s
0 1
ª¬ vk I ka N cVc Rs k I º¼ K c 1 ck p ka K c
p2 iq2
s p1 iq1
ª vk I ka N cVc Rs k I º¼ K c s 2 ª¬ s p1 iq1 º¼ a1 ia2 ¬ ¬ª p2 iq2 s ¼º 1 ck p ka K c s 3 p1 iq1 s 2 p2 iq2 s a1 ia2 where ܫis the ͵ ൈ ͵ unit matrix.
( vka N cVc Rs )k I K c 1 ck p ka K c
0
(A2)
Appendix B. Stability condition of the third-order differential equation with complex coefficients Assume the third-order characteristic equation with complex coefficients is
D s O0 s 3 O1s 2 O2 s O3 where Ok
(B1)
Ok1 Ok 2 Dk iEk , k 1,2,3. .
According to [24], if there exist
Q s O11s 2 O22 s O31
(B2)
The J-fraction can be constructed as Eq. (B3)
Q s D s
where ݉ଵ ൌ
ఎభభ ఒభభ
ˈ݉ଶ ൌ
ఎయభ ఎమభ
ˈଷ ൌ
ఎరభ
r1 s m1
ˈݎଵ ൌ
ఒబ ఒభభ
(B3)
r2 s m2
ˈݎଶ ൌ
ఒభభ ఎమభ
r3 s m3
ˈݎଷ ൌ
ఎమభ ఎరభ
. If ݎ ሺ݊ ൌ ͳǡʹǡ͵ሻ exists and is positive,
then the system determined by characteristic equation (B1) is stable. The coefficients can be determined using the following method, with the coefficient matrix given as
O0 O11 K11 K21 K31 K41 where ߟଵଵ ൌ ఎమభ ఒమమ ିఒభభ ఎమమ ఎమభ
ఒభభ ఒభ ିఒబ ఒమమ ఒభభ
, ߟଵଶ ൌ
, ߟଷଶ ൌ ߣଷଵ , ߟସଵ ൌ
ఒభభ ఒమ ିఒబ ఒయభ ఒభభ
, ߟଵଷ ൌ ߣଷ , ߟଶଵ ൌ
ఎమభ ఎయమ ିఎయభ ఎమమ ఎమభ
O1 O2 O3 O22 O31 K12 K13 K22 K32
(B4)
ఒభభ ఎభమ ିఒమమ ఎభభ ఒభభ
, ߟଶଶ ൌ
ఒభభ ఎభయ ିఒయభ ఎభభ ఒభభ
, ߟଷଵ ൌ
.
For the third-order characteristic equation with complex coefficients (B1), let ߣ ൌ ͳ, ߣଵ ൌ ଵ ݅ݍଵ , ߣଶ ൌ ଶ ݅ݍଶ , ߣଷ ൌ ଷ ݅ݍଷ , ߣଵଵ ൌ ߙଵ , ߣଶଶ ൌ ݅ߚଶ , and ɉଷଵ ൌ ߙଷ ; thus, the parameters in this continued fraction can be obtained and expressed as
r1
p1
r2
p q
(B5) 2 1 2
p1 p3 p1q1q2 q22 / r12
(B6)
r3
ª p 2 p p p p q q q 2 p p p q 2 p q q p 2 q p p q p q 2 º / r 6r 3 1 3 1 1 2 2 1 2 3 3 1 2 3 1 3 1 3 1 3 2 ¬« 1 2 ¼» 1 1
(B7)
The sufficient and necessary stability condition can be further simplified as
° p1 ! 0 ° 2 2 ® p1 p2 p1 p3 p1q1q2 q2 ! 0 ° 2 ° p12 p2 p1 p3 p1q1q2 q22 p1 p2 p3 p32 p1q2 q3 p12 q3 p1 p3q1 p3q2 ! 0 ¯
(B8)
Reference [1] Nesline F W, Zarchan P. Radome Induced Miss Distance in Aerodynamically Controlled Homing Missile[J]. AIAA. 1984, 84(1): 845. [2] Miwa S, Kouya S. Radome effect on the miss distance of a radar homing missile[J]. Electronics and Communications in Japan(Part I Communications). 1998, 81(7): 14-22. [3] Nesline F W, Nesline M L. Wing size vs radome compensation in aerodynamically controlled radar homing missiles[J]. Journal of Guidance, Control, and Dynamics. 1986, 9(6): 645-649. [4] Nesline F W, Zarchan P. Missile guidance design tradeoffs for high-altitude air defense[J]. Journal of Guidance Control & Dynamics. 1983, 6(3): 207-212. [5] Brown R, Santi G. Missile system for low altitude air defence[C]. 1984. [6] Peterson D, Otto J, Douglas K. Radome boresight error and compensation techniques for electronically scanned arrays[R]. DTIC Document, 1993. [7] Stallard D. Analysis of the parastitic radome loop in a bank-to-turn homing missile[J]. AIAA Pap. 1985: 176-187. [8] Yost D J, Weckesser L B, Mallalieu R C. Technology Survey of Radomes for Anti-Air Homing Missiles[J]. Johns Hopkins University Applied Physics Laboratory, Laurel, MD. 1980. [9] Lin C F. Bank-to-turn guidance performance analysis with in-flight radome error compensation[C]. 1984. [10] Yueh W R, Lin C F. Guidance performance analysis with in-flight radome error calibration[J]. Journal of Guidance, Control, and Dynamics. 1985, 8(5): 666-669. [11] Lin J, Chau Y. Radome slope compensation using multiple-model Kalman filters[J]. Journal of Guidance, Control, and Dynamics. 1995, 18(3): 637-640. [12] Willman W. Radome compensation using adaptive dither[C]. 1998.
[13] Zarchan P, Gratt H. Adaptive Radome Compensation Using Dither[J]. Journal of Guidance, Control, and Dynamics. 1999, 22(1): 51-57. [14] Tiwari P K, Vora P, Bhattacharjee R N. In flight radome error compensation through simulated test data[Z]. 2005. [15] Song T L, Lee D G, Shin S J. Active homing performance enhancement with multiple model radome slope estimation[J]. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. 2005, 219(3): 217-224. [16] Yan X, Yang S, Zhang C. Coning Motion of Spinning Missiles Induced by the Rate Loop[J]. Journal of Guidance, Control, and Dynamics. 2010, 33(5): 1490-1499. [17] Yan X, Yang S, Xiong F. Stability Limits of Spinning Missiles with Attitude Autopilot[J]. Journal of Guidance, Control, and Dynamics. 2011, 34(1): 278-283. [18] Li K, Yang S, Zhao L. Stability of Spinning Missiles with an Acceleration Autopilot[J]. Journal of Guidance, Control, and Dynamics. 2012, 35(3): 774-786. [19] Zhou W, Yang S, Dong J. Coning motion instability of spinning missiles induced by hinge moment[J]. Aerospace Science and Technology. 2013, 30(1): 239-245. [20] Zhou W, Yang S, Zhao L. Limit Circular Motion of Spinning Projectiles Induced by Backlash of Actuators[C]. 2014. [21] Koohmaskan Y, Arvan M R, Vali A R, et al. Dynamic stability conditions for a rolling flight vehicle applying continuous actuator[J]. Aerospace Science and Technology. 2015, 42: 451-458. [22] Mao X, Yang S. Optimal Control of Coning Motion of Spinning Missiles[J]. Journal of Aerospace Engineering. 2015, 28(2). [23] He S, Lin D, Wang J. Coning motion stability of spinning missiles with strapdown seekers[J]. The Aeronautical Journal. 2016, 120(1232): 1566-1577. [24] Frank E. On the zeros of polynomials with complex coefficients[J]. Bulletin of the American Mathematical Society. 1946, 52(2): 144-157.