- Email: [email protected]
Robust Control of the Missile Attitude Based on Quaternion Feedback
ROBUST CONTROL OF THE MISSILE ATTITUDE BASED ON QUATERNION FEEDBACK
Chanho Song, Sangjae Kim, Seung-Hwan Kim, and H. S. Nam
Agency for Defense Development 3-1-3, P.D. Box 35-3, Yusung, Daejon, Korea, 305-600 (E-mail mail : ks;[email protected]@unitel.co.kr) (I'el : +82-42-821-4411, Fax: +82-42-821-2224)
Abstract: In this paper, a robust control scheme based on the quaternion feedback for attitude control of the missiles employing thrust vector control is proposed. The control law consists of three parts: the nominal feedback part and an additional term compensating for the aerodynamic moment, and the third part for ensuring robustness to the plant uncertainties. For the proposed control scheme, stability analysis is given, and the performance is shown via computer simulation. Keywords: quaternion feedback, attitude control, uncertainty, stability analysis
1. INTRODUCTION
In general, most attitude control schemes of tactical missiles are based on the Euler angle feedback concept. However, modern satellites or spacecrafts have a trend forward using quaternion feedback instead of Euler angles. Moreover, it is reported that quaternion control enables the attitude change along the shortest path by the eigenaxis rotation (Weiss, 1993; Wie and Barba, 1985; Wie and Weiss, 1989). It can be achieved by matching the control torque vector to the eigenaxis of rotation which is not possible with Euler angle control because Euler angles are based on the concept of sequential rotation. Some remarkable results are found in Wie et al.'s work (Wie and Weiss,1989). However, similar research is hardly found in the area of attitude control for the tactical missiles operated in the low atmosphere. It seems due to a view that the quaternion feedback will not retain its advantage where the aerodynamic effects are not negligible. But Song et al. proposed a control scheme which might be prospective even in this case (Song, et al., 2000). In this paper, the results are extended to the case
where the plant uncertainties exist. Several control design methods for the uncertain dynamical systems are introduced in (Barmish, et al., 1983; Corless and Leitmann, 1981 ; Khalil, (1996». Most of them require that the uncertain system satisfies the so-called "matching condition". First, it will be shown that the plant uncertainty can be mode led so that it satisfies this condition. Secondly, a robust attitude control law based on the previous work for the uncertain system is proposed. It is constructed by invoking a robust stabilization method is given in (Khalil, (1996». This control law consists of three parts: the nominal feedback part and an additional term compensating for the aerodynamic moment which were discussed in (Song, et al., 2000), and the third part for ensuring the robustness to the plant uncertainties. In (Song, et al. , 2000), aerodynamic moment was assumed perfectly known so that the effect of aerodynamics to the missile motion could be perfectly cancelled. But this is not possible in practice, and there exist a variety of uncertainties in aerodynamics, inertia , torque generated by thrust,
SESSION 9
493
and so forth. These uncertainties will be taken into consideration in constructing the control law. For the proposed control scheme, stability analysis is given. FinalIy, it is shown via the computer simulation that the performance of the overalI control scheme is satisfactory.
missile with cruciform configuration, Ma in Eqn. (2) can be described as QSD 2 QSDC f(n,a)sin 4y + -z-;-C fP (n)lV 1 QSD 2 QSDC m(n,a) + -z-;-C mq (n)lV2
(8)
QSD 2 - QSDC n (n,p) + -z-;-C n, (n)lV3
2. MISSILE DYNAMICS WITH UNCERTAINTY Missile motion is described by the six degree of freedom equations which consist of the translational and rotational motion equations as folIows:
where D is the reference length, Ct. C"" C,. Cl", Cmq and Cn , are nondimensional moment coefficients, and y is the bank angle defmed by
v w
/3
(9)
r=-=-
a
mv+m(aJxv)-g=Fa(v)+FI(u) (1) JciJ={J)xJ{J)+Ma(v,{J)+M1(u)
wf is the linear (J) = [(J)\ ,{J)2, (J)3 f is the angular
where v = [u, v,
(2)
FinalIy, M, ,the control moment, is written as
velocity vector, (10)
velocity vector,
u = [0,,0 p,oyf is the control input vector, m is the mass of a missile, J is a inertia matrix, g is gravity, Fa and Ma are aerodynamic force and moment vectors, and F, and M, are control force and control moment vectors, respectively. Now, to simplify Eqs. (1) and (2), the folIowing assumptions are used: A1.Velocity and altitude of the missile are constant. A2.Gravity is neglected. A3 . v and w are much smalIer than u. A4. Missile body has the symmetrical cruciform. A5. F,(u) and M,(u) is linear to u and invertible. Under the AI-A4, the translational motion equation can be described as
a = (J)2 + Zo(Q,n,a) -
ZIO p
jJ = -(J)3 + Yo(Q,n,/3) + ZIOy
(3) (4)
where Tc is the control thrust force, l~ and Iy are the moment arms defined by the distance from c.g. to the position of the control thrust, and 0" 0", and 0;. are control inputs. Now, let's denote the uncertain elements by A.J,!1M a' Mp I:lJ, I:lg and MJ 2 which correspond to J,Ma,Bt,f,g and B2 , respectively. Furthermore, let u = [o"op,oyf, the the motion Eqs. (1) and (2) can be generalized in the folIowing form: (J +M)io = D.(J +M)lV+ Ma +l:lMa +(B\ + MJ\)u (11)
z=J(z)+!j'(z) + g(aJ)+~aJ)+(~ +~)u where
(12)
n is a skew symmetric matrix defmed by
where a is angle of attack, /3 is sideslip angle, Q is the dynamic pressure, n is Mach number, Zo(Q, n, a) and Yo (Q,n,/3) are functions given by Zo(Q,n,a)
= QS Cz(n,a)
mu QS Yo(Q,n,/3) = - Cy(n,/3)
mu
(5) (6)
where C.(n, a) and Cy(n,/3) are nondimensional aerodynamic coefficients, S is the reference area and Z" the control thrust coefficient is given by
Tc ZI=mu
SESSION 9
~ ],z=[a
/3f.
M/
3. UNCERTAINTY MODEL (7)
with the magnitude of the control thrust Tc' For the
494
and
In this section, it is shown that the plant uncertainties satisfies so-called "matching condition". In other
words, several uncertain terms can be lumped together. First, it follows from the matrix inverse lemma that
(13) with M
defined by
4. KINEMATICS Euler's rotational theorem states that the attitude of a rigid-body can change from one orientation to another by rotating the body about an axis called the Euler axis or eigenaxis. The defmition of quaternion comes from such eigenaxis rotation. The vector part of the quaternion indicates the direction of the eigenaxis and the scalar part indicates the rotation angle about the eigenaxis; Le., a quaternion is defined by
Eqs. (11) and (13) give
dJ
= f(w,t)+Mi +(G+i1G)u
where
M! =(r l -M)/lMa -MO./w+(r l -M)o.Mw, l i1G=r i1B l -M(Bl +L1Bl)'
few,!)
=rl (O./w),
G
=rl Bl .
Since G is a invertible square matrix, i1G can be written as
q;
= e; Sin(~ ),i = 1,2,3
q4
= cos(~)
j
(15)
. 1
MI and
(16)
It means that uncertain terms satisfy the matching condition. Now, the following lumped equation is obtained:
dJ = few, t) + G(u + 0)
(17)
where 0 is the lumped uncertain element.
2
=
2
1 T X 2
(23)
--(U
[qle, Q2e, Q3e, Q4e] between the current attitude and the commanded one is given by q1e
(18)
u='I'+v v
is an
additional term for retaining robustness to the plant uncertainties. Then,
dJ = f(w,t)+G'I'+G(o +v)
1
Let's denote the commanded attitude by a quatemion [qlc' q2c' Q3c' Q4c ] Then error quaternion
q2e q3e
Let the control input u be defmed:
is the nominal one and
1
X=-OX+-q4{U .
'I'
2
where rp is the magnitude of the eigenaxis rotation angle and a vector (el ,e2 ,e3) is the direction cosine of the eigenaxis with respect to the reference frame . The orientation equations in terms of quaternion are described by
q4
where
(22)
(19)
=
r 1 Q4.
Note that the differential equation of the error quaternion has the same form as Eqn. (23).
5. NOMINAL FEEDBACK CONTROLLER From Eqs. (19) and (23),
where 0 is given by
(20) (25) and uncertain terms 6. h, i1 g satisfy
(21)
where X,T =[QI.,Q2.,Q), ] is a vector part of the error quaternion given by Eqn. (23). Let's define x as
SESSION 9
49~
(31)
(26)
That is obtained by choosing the controller v satisfying the following inequality.
where
(32) Now, it is assumed that Eqn. (20) satisfies 118(x,t,lfI+v)ll::;p(x,t)+~g
f(rn,l)
Ilvll,
O::;~g <1.(33)
It means that uncertainty 8 is limited by the known function p and the controller v. Then, it follows from Eqs. (32) and (33) that
Let the nominal controller be in the form:
Put v to be
(i)1 = rn2 == rn3 =0, qle = q2e = q3e = 0, q4e = 1 . Now~ consider the following Lyapunov function candIdate V
f.J
1]
For the nominal system without 8, the equilibrium state is :
v=----1- ~ g 11 f.J 11 where
fiT
1] ~
(35)
p . Then (34) becomes
(v + c5) ::; -
77 --11 fill + P 1 It 1 + -6.g77-11 It 11 1-6. g
1-6. g
= - _77_11 fill + P 11 1- 6. g
fi
11 ::; - p 11 f.J 11 + p 11 f.J 11= 0
Some manipulations give
(28) Because V is a positive defmite, de crescent and radially unbounded function and V ::; 0 , the equilibriwn state is globally uniformly stable.
which satisfies the condition (32). But this control law is not defmed at 11 f.J 11= O. Practically, such a discontinuous controller is characterized by the phenomenon of chattering. A way of to avoid division by zero is to modify the control law as follows:
6. ROBUST CONTROLLER AND STABILITY ANALYSIS
v=
{ " ii;Ti" ' -1-6. g 77
(29)
,..
T
av ax .
=-G
fi
Vi
- l-~gi ~ 11 fill ,
=
1 1-6.gi where
496
SESSION 9
if 7711 fill< &
To control each of roll, pitch, and yaw channels, Eqn.(36) needs to be implemented per channel as follows:
&
means
if
Pi
11fill~ & (37)
if
__P_i_!:!"!",,
The .s~bility condition requires that the following condItIon should be satisfied.
(36)
This control law u=lfI+v stabilizes the system with uncertainty given by (26).
Pi
(30)
&
g
where 11
fill~
2
-~~'
In this section, an additional term v is designed through the following process(Khalil, (1996)). In case that 8 is not zero, Lyapunov derivative Vu of the uncertain system given by Eqn. (26) becomes:
if 7711
Pillfill<&
'
pitch,
yaw,
roll
channel,
respectively. From Eqs. (20) and (33), the following inequality is satisfied.
:f\~___________- j
1.0
I Ahi +A gi(-kpiqle - kdi(lJi) + Ag/Vi I ~ IAhi 1+ I Agi(-kpi qle -kdi (lJi )I+IAgiVi I ~Pi +IAgi Vi I
to.· !
I
0.'
t.
-RoIMtiO ••• • PIch M(jo .... v..,M(jo
I
;!!
0.2
So, P i is defined as follows:
:
O. O-f----.-~---,r_-_,_--,---..__-_l
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Tlme ,"oci
Figure 2. Roll,Pitch, and Yaw angle without uncertainty where A hi and A gi is given in Eqn. (21). 1.'
7. SIMULATION RESULTS
1..
To confirm the performance of the proposed quatemion feedback controller, simulations were carried out. For this study, we took a fictitious tactical missile model and designed the quatemion feedback controller as explained in Section 6. Actuator dynamics was neglected.
1.2
u
Figure I shows the block diagram of the nominal quatemion feedback control system. The desired attitude commands are assumed to be [60· ,60·,60· ] . Figure 2 shows the normalized step response with the nominal control 'I' for the system without uncertainty. Figure 3-5 show the responses of the 2 cases where the uncertainty exists: with 'I' only and with the robust controller ( 'I' + v). The magnitude of the uncertainty was assumed to be 50% of the nominal. The nominal control 'I' shows good performance in case that uncertainty does not exist. But in case that the uncertainties exist, as expected the robust controller gives the better performance than the control law with 'I' only.
'.'
0.' 0.'
0.2 0.0 -f'----r-~-,----r---,--__r----l 1.0 0.0 2.0 2.5 ' .0 0.' 1.'
T1mo)HC)
Figure 3. Roll angle with uncertainty
1.2
.:: i/ · ··· ·· {
~ 01 0.' 0.2
0.0 +----.-~---,r_--,-~---r--.__-~ 0.0
0.'
1.0
1.'
2.0
2..
3.0
Timo(!ecJ
Figure 4. Pitch angle with uncertainty
.'
1.8
,," ',
1.1
.,
i
1.4
'
'
~
1::11 1
I
>
Figure I. The nominal quatemion feedback controller
:
0.8
i
0.8
:
0."
f
..
\. "
. -- . a.-rion -R_~
0.2 0.0 .jL..---.-~-.----,-~---r--.__---l 0.0
0.5
1.0
1.'
2.0
2..
3.0
T1rre(secI
Figure 5. Yaw angle with uncertainty
SESSION 9
497
8. CONCLUSION In this paper, a robust quatemion feedback control scheme was proposed which can be used efficiently in the attitude control of the tactical missiles employing thrust vector control. It was shown that the plant uncertainty can be modeled so that the matching condition is satisfied. Furthermore, it was shown that even in presence of uncertainty the proposed control law makes the closed-loop system globally uniformly stable. Finally, it was shown via computer simulations that the proposed control scheme gives good performance.
REFERENCES Barmish, B.R., M. Corless and G. Leitmann. (1983). A new class of stabilizing controllers for uncertain dynamical systems. SIAM J. Control and
498
SESSION 9
Optimization, 21,246-255 . Corless, M. and G. Leitmann. (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control, 26, 1139-1144. Khalil,H.K. (1996). Nonlinear Systems, Prentice Hall. Siouris, G.M. (1993). Aerospace Avionics Systems:A Modern Synthesis, Academic Press. Song, C., H.S. Nam and S.H. Kim. (2000). Missile attitude control based on quatemion feedback, ASCC2000(Asian Control Conference}, 2000.7. Weiss, H: (1993). Quatemion-based rate/attitude tracking system with application to gimbal attitude control. J. Guidance, 16, 609-616. Wie, B. and P.M Barba. (1985). Quatemion feedback for spacecraft large angle maneuvers. J. Guidance, 8,360-365. Wie, B, H. Weiss and A. Arapostathis. (1989). Quatemion feedback regulator for spacecraft eigenaxis rotations . J. Guidance, 12, 375-380.
Chanho Song, Sangjae Kim, Seung-Hwan Kim, and H. S. Nam
Agency for Defense Development 3-1-3, P.D. Box 35-3, Yusung, Daejon, Korea, 305-600 (E-mail mail : ks;[email protected]@unitel.co.kr) (I'el : +82-42-821-4411, Fax: +82-42-821-2224)
Abstract: In this paper, a robust control scheme based on the quaternion feedback for attitude control of the missiles employing thrust vector control is proposed. The control law consists of three parts: the nominal feedback part and an additional term compensating for the aerodynamic moment, and the third part for ensuring robustness to the plant uncertainties. For the proposed control scheme, stability analysis is given, and the performance is shown via computer simulation. Keywords: quaternion feedback, attitude control, uncertainty, stability analysis
1. INTRODUCTION
In general, most attitude control schemes of tactical missiles are based on the Euler angle feedback concept. However, modern satellites or spacecrafts have a trend forward using quaternion feedback instead of Euler angles. Moreover, it is reported that quaternion control enables the attitude change along the shortest path by the eigenaxis rotation (Weiss, 1993; Wie and Barba, 1985; Wie and Weiss, 1989). It can be achieved by matching the control torque vector to the eigenaxis of rotation which is not possible with Euler angle control because Euler angles are based on the concept of sequential rotation. Some remarkable results are found in Wie et al.'s work (Wie and Weiss,1989). However, similar research is hardly found in the area of attitude control for the tactical missiles operated in the low atmosphere. It seems due to a view that the quaternion feedback will not retain its advantage where the aerodynamic effects are not negligible. But Song et al. proposed a control scheme which might be prospective even in this case (Song, et al., 2000). In this paper, the results are extended to the case
where the plant uncertainties exist. Several control design methods for the uncertain dynamical systems are introduced in (Barmish, et al., 1983; Corless and Leitmann, 1981 ; Khalil, (1996». Most of them require that the uncertain system satisfies the so-called "matching condition". First, it will be shown that the plant uncertainty can be mode led so that it satisfies this condition. Secondly, a robust attitude control law based on the previous work for the uncertain system is proposed. It is constructed by invoking a robust stabilization method is given in (Khalil, (1996». This control law consists of three parts: the nominal feedback part and an additional term compensating for the aerodynamic moment which were discussed in (Song, et al., 2000), and the third part for ensuring the robustness to the plant uncertainties. In (Song, et al. , 2000), aerodynamic moment was assumed perfectly known so that the effect of aerodynamics to the missile motion could be perfectly cancelled. But this is not possible in practice, and there exist a variety of uncertainties in aerodynamics, inertia , torque generated by thrust,
SESSION 9
493
and so forth. These uncertainties will be taken into consideration in constructing the control law. For the proposed control scheme, stability analysis is given. FinalIy, it is shown via the computer simulation that the performance of the overalI control scheme is satisfactory.
missile with cruciform configuration, Ma in Eqn. (2) can be described as QSD 2 QSDC f(n,a)sin 4y + -z-;-C fP (n)lV 1 QSD 2 QSDC m(n,a) + -z-;-C mq (n)lV2
(8)
QSD 2 - QSDC n (n,p) + -z-;-C n, (n)lV3
2. MISSILE DYNAMICS WITH UNCERTAINTY Missile motion is described by the six degree of freedom equations which consist of the translational and rotational motion equations as folIows:
where D is the reference length, Ct. C"" C,. Cl", Cmq and Cn , are nondimensional moment coefficients, and y is the bank angle defmed by
v w
/3
(9)
r=-=-
a
mv+m(aJxv)-g=Fa(v)+FI(u) (1) JciJ={J)xJ{J)+Ma(v,{J)+M1(u)
wf is the linear (J) = [(J)\ ,{J)2, (J)3 f is the angular
where v = [u, v,
(2)
FinalIy, M, ,the control moment, is written as
velocity vector, (10)
velocity vector,
u = [0,,0 p,oyf is the control input vector, m is the mass of a missile, J is a inertia matrix, g is gravity, Fa and Ma are aerodynamic force and moment vectors, and F, and M, are control force and control moment vectors, respectively. Now, to simplify Eqs. (1) and (2), the folIowing assumptions are used: A1.Velocity and altitude of the missile are constant. A2.Gravity is neglected. A3 . v and w are much smalIer than u. A4. Missile body has the symmetrical cruciform. A5. F,(u) and M,(u) is linear to u and invertible. Under the AI-A4, the translational motion equation can be described as
a = (J)2 + Zo(Q,n,a) -
ZIO p
jJ = -(J)3 + Yo(Q,n,/3) + ZIOy
(3) (4)
where Tc is the control thrust force, l~ and Iy are the moment arms defined by the distance from c.g. to the position of the control thrust, and 0" 0", and 0;. are control inputs. Now, let's denote the uncertain elements by A.J,!1M a' Mp I:lJ, I:lg and MJ 2 which correspond to J,Ma,Bt,f,g and B2 , respectively. Furthermore, let u = [o"op,oyf, the the motion Eqs. (1) and (2) can be generalized in the folIowing form: (J +M)io = D.(J +M)lV+ Ma +l:lMa +(B\ + MJ\)u (11)
z=J(z)+!j'(z) + g(aJ)+~aJ)+(~ +~)u where
(12)
n is a skew symmetric matrix defmed by
where a is angle of attack, /3 is sideslip angle, Q is the dynamic pressure, n is Mach number, Zo(Q, n, a) and Yo (Q,n,/3) are functions given by Zo(Q,n,a)
= QS Cz(n,a)
mu QS Yo(Q,n,/3) = - Cy(n,/3)
mu
(5) (6)
where C.(n, a) and Cy(n,/3) are nondimensional aerodynamic coefficients, S is the reference area and Z" the control thrust coefficient is given by
Tc ZI=mu
SESSION 9
~ ],z=[a
/3f.
M/
3. UNCERTAINTY MODEL (7)
with the magnitude of the control thrust Tc' For the
494
and
In this section, it is shown that the plant uncertainties satisfies so-called "matching condition". In other
words, several uncertain terms can be lumped together. First, it follows from the matrix inverse lemma that
(13) with M
defined by
4. KINEMATICS Euler's rotational theorem states that the attitude of a rigid-body can change from one orientation to another by rotating the body about an axis called the Euler axis or eigenaxis. The defmition of quaternion comes from such eigenaxis rotation. The vector part of the quaternion indicates the direction of the eigenaxis and the scalar part indicates the rotation angle about the eigenaxis; Le., a quaternion is defined by
Eqs. (11) and (13) give
dJ
= f(w,t)+Mi +(G+i1G)u
where
M! =(r l -M)/lMa -MO./w+(r l -M)o.Mw, l i1G=r i1B l -M(Bl +L1Bl)'
few,!)
=rl (O./w),
G
=rl Bl .
Since G is a invertible square matrix, i1G can be written as
q;
= e; Sin(~ ),i = 1,2,3
q4
= cos(~)
j
(15)
. 1
MI and
(16)
It means that uncertain terms satisfy the matching condition. Now, the following lumped equation is obtained:
dJ = few, t) + G(u + 0)
(17)
where 0 is the lumped uncertain element.
2
=
2
1 T X 2
(23)
--(U
[qle, Q2e, Q3e, Q4e] between the current attitude and the commanded one is given by q1e
(18)
u='I'+v v
is an
additional term for retaining robustness to the plant uncertainties. Then,
dJ = f(w,t)+G'I'+G(o +v)
1
Let's denote the commanded attitude by a quatemion [qlc' q2c' Q3c' Q4c ] Then error quaternion
q2e q3e
Let the control input u be defmed:
is the nominal one and
1
X=-OX+-q4{U .
'I'
2
where rp is the magnitude of the eigenaxis rotation angle and a vector (el ,e2 ,e3) is the direction cosine of the eigenaxis with respect to the reference frame . The orientation equations in terms of quaternion are described by
q4
where
(22)
(19)
=
r 1 Q4.
Note that the differential equation of the error quaternion has the same form as Eqn. (23).
5. NOMINAL FEEDBACK CONTROLLER From Eqs. (19) and (23),
where 0 is given by
(20) (25) and uncertain terms 6. h, i1 g satisfy
(21)
where X,T =[QI.,Q2.,Q), ] is a vector part of the error quaternion given by Eqn. (23). Let's define x as
SESSION 9
49~
(31)
(26)
That is obtained by choosing the controller v satisfying the following inequality.
where
(32) Now, it is assumed that Eqn. (20) satisfies 118(x,t,lfI+v)ll::;p(x,t)+~g
f(rn,l)
Ilvll,
O::;~g <1.(33)
It means that uncertainty 8 is limited by the known function p and the controller v. Then, it follows from Eqs. (32) and (33) that
Let the nominal controller be in the form:
Put v to be
(i)1 = rn2 == rn3 =0, qle = q2e = q3e = 0, q4e = 1 . Now~ consider the following Lyapunov function candIdate V
f.J
1]
For the nominal system without 8, the equilibrium state is :
v=----1- ~ g 11 f.J 11 where
fiT
1] ~
(35)
p . Then (34) becomes
(v + c5) ::; -
77 --11 fill + P 1 It 1 + -6.g77-11 It 11 1-6. g
1-6. g
= - _77_11 fill + P 11 1- 6. g
fi
11 ::; - p 11 f.J 11 + p 11 f.J 11= 0
Some manipulations give
(28) Because V is a positive defmite, de crescent and radially unbounded function and V ::; 0 , the equilibriwn state is globally uniformly stable.
which satisfies the condition (32). But this control law is not defmed at 11 f.J 11= O. Practically, such a discontinuous controller is characterized by the phenomenon of chattering. A way of to avoid division by zero is to modify the control law as follows:
6. ROBUST CONTROLLER AND STABILITY ANALYSIS
v=
{ " ii;Ti" ' -1-6. g 77
(29)
,..
T
av ax .
=-G
fi
Vi
- l-~gi ~ 11 fill ,
=
1 1-6.gi where
496
SESSION 9
if 7711 fill< &
To control each of roll, pitch, and yaw channels, Eqn.(36) needs to be implemented per channel as follows:
&
means
if
Pi
11fill~ & (37)
if
__P_i_!:!"!",,
The .s~bility condition requires that the following condItIon should be satisfied.
(36)
This control law u=lfI+v stabilizes the system with uncertainty given by (26).
Pi
(30)
&
g
where 11
fill~
2
-~~'
In this section, an additional term v is designed through the following process(Khalil, (1996)). In case that 8 is not zero, Lyapunov derivative Vu of the uncertain system given by Eqn. (26) becomes:
if 7711
Pillfill<&
'
pitch,
yaw,
roll
channel,
respectively. From Eqs. (20) and (33), the following inequality is satisfied.
:f\~___________- j
1.0
I Ahi +A gi(-kpiqle - kdi(lJi) + Ag/Vi I ~ IAhi 1+ I Agi(-kpi qle -kdi (lJi )I+IAgiVi I ~Pi +IAgi Vi I
to.· !
I
0.'
t.
-RoIMtiO ••• • PIch M(jo .... v..,M(jo
I
;!!
0.2
So, P i is defined as follows:
:
O. O-f----.-~---,r_-_,_--,---..__-_l
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Tlme ,"oci
Figure 2. Roll,Pitch, and Yaw angle without uncertainty where A hi and A gi is given in Eqn. (21). 1.'
7. SIMULATION RESULTS
1..
To confirm the performance of the proposed quatemion feedback controller, simulations were carried out. For this study, we took a fictitious tactical missile model and designed the quatemion feedback controller as explained in Section 6. Actuator dynamics was neglected.
1.2
u
Figure I shows the block diagram of the nominal quatemion feedback control system. The desired attitude commands are assumed to be [60· ,60·,60· ] . Figure 2 shows the normalized step response with the nominal control 'I' for the system without uncertainty. Figure 3-5 show the responses of the 2 cases where the uncertainty exists: with 'I' only and with the robust controller ( 'I' + v). The magnitude of the uncertainty was assumed to be 50% of the nominal. The nominal control 'I' shows good performance in case that uncertainty does not exist. But in case that the uncertainties exist, as expected the robust controller gives the better performance than the control law with 'I' only.
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Figure 3. Roll angle with uncertainty
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SESSION 9
497
8. CONCLUSION In this paper, a robust quatemion feedback control scheme was proposed which can be used efficiently in the attitude control of the tactical missiles employing thrust vector control. It was shown that the plant uncertainty can be modeled so that the matching condition is satisfied. Furthermore, it was shown that even in presence of uncertainty the proposed control law makes the closed-loop system globally uniformly stable. Finally, it was shown via computer simulations that the proposed control scheme gives good performance.
REFERENCES Barmish, B.R., M. Corless and G. Leitmann. (1983). A new class of stabilizing controllers for uncertain dynamical systems. SIAM J. Control and
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Optimization, 21,246-255 . Corless, M. and G. Leitmann. (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control, 26, 1139-1144. Khalil,H.K. (1996). Nonlinear Systems, Prentice Hall. Siouris, G.M. (1993). Aerospace Avionics Systems:A Modern Synthesis, Academic Press. Song, C., H.S. Nam and S.H. Kim. (2000). Missile attitude control based on quatemion feedback, ASCC2000(Asian Control Conference}, 2000.7. Weiss, H: (1993). Quatemion-based rate/attitude tracking system with application to gimbal attitude control. J. Guidance, 16, 609-616. Wie, B. and P.M Barba. (1985). Quatemion feedback for spacecraft large angle maneuvers. J. Guidance, 8,360-365. Wie, B, H. Weiss and A. Arapostathis. (1989). Quatemion feedback regulator for spacecraft eigenaxis rotations . J. Guidance, 12, 375-380.