Proceedings of Advances in Control and Optimization of Dynamic Systems
ACODS-2012
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Three Axis Attitude Control by Fuzzy logic
based Controller using Magnetic Torquers
Kanu Priya Govila, Ramkiran Venkat ,P. Natarajan and Parameswaran K.

During the last two decades, several magnetic attitude
controllers have been developed and some of them have been
widely exploited in practical attitude control projects.
Magnetic attitude control approach was first suggested by [1]
and has been subsequently followed by many other works that
employ torque rods to provide control actuation, as in [2] and
[3]. A periodic controller was proposed in [4] and [5] that uses
magnetic torque rods to tackle the control problem by
extending the classical linear quadratic regulator technique for
periodic systems. A constant gain controller was developed
[8].
Abstract – A scheme which utilizes fuzzy logic for the
development of a three-axis attitude control system for
magnetically actuated satellites is suggested in this paper, which
calculates the magnetic dipole moment based on the attitude and
rate errors.
This controller which couples fuzzy logic with the LQR leads
to less computational effort as compared to conventional LQR
method. Since, the system is a time-varying system classical PD
/PID controller fails to provide good results. Performance of the
fuzzy logic based controller is compared with conventional LQR
controller.
Index Terms—Attitude control, Fuzzy
Moments, Optimal Control (Infinite horizon)
The area of magnetic attitude control using intelligent
methods is still to be explored. Fuzzy attitude controller for
magnetically actuated satellites are proposed [6] with two
limiting assumptions: bang-off-bang method of actuation, and
activating only one torque coil out of the three coils at an
instant. These assumptions simplify the computation of the
mechanical/control torque at each instant. An estimation of the
producible mechanical torque is used to calculate the magnetic
dipole moment. Based on a similar limiting assumption, a
fuzzy attitude controller[7] has been suggested for spinstabilized satellites with active magnetic actuation.
control,Magnetic
I. INTRODUCTION
T
he primary objective of this work is to develop control
laws for three-axis stabilization of a magnetically actuated
satellite. The interaction between the Earth’s magnetic field
and a magnetic field generated by a set of coils in the satellite
can be used for actuation to control satellite. Magnetic
torquing was found attractive for generating control torques on
small satellites since magnetic torquers are relatively
lightweight, require low power and are inexpensive. However,
the aforesaid principle is inherently nonlinear and difficult to
use because torques can only be generated perpendicular to the
direction of the geo-magnetic field vector. Thus, at most
instances the satellite can be controlled in only two axes which
prevents simultaneous control of all three axes using magnetic
torquers. But fortunately because of the satellite’s motion, the
third axis(which is non-controllable) becomes controllable
with time, therefore the required control torque can be
achieved over time.
Kanu Priya Govila scientist in ISRO Satellite Center
Bengaluru,INDIA (e-mail: kanug @isac.gov.in).
Ramkiran Venkat
scientist in ISRO Satellite Center
Bengaluru,INDIA (e-mail: [email protected])
P.Natarajan Division head of CDAD in ISRO Satellite Center
Bengaluru,INDIA (e-mail: [email protected])
K. Parameswaran - Group head of CSG in ISRO Satellite Center
Bengaluru,INDIA (e-mail: [email protected])
© ACODS-2012
Confined computation capacity limits design of the system.
So we modified the work carried out by [5] with a fuzzy
controller. As fuzzy logic controller is known to have
achieved the best overall performance under various
conditions while being less computationally demanding. A
control algorithm based on fuzzy control rules was designed in
order to allow for the choice of a magnetic torquer coil that
will achieve the best results, given a local geo-magnetic field
vector. The intention of this controller design is to define a set
of control rules and to implement them in such a way so as to
make the central values equidistant, resulting in a system that
covers the universe of discourse. Performance of the fuzzy
logic based controller is compared with conventional LQR
controller.
in
Section 2 illustrates the need of FBLC over LQR
controller. Next section gives an overview of spacecraft
attitude dynamics and kinematics. Fuzzy membership function
and controller design is elaborated in section 4. Results and
conclusion are described in Section 5 and 6 respectively.
in
in
in
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Proceedings of Advances in Control and Optimization of Dynamic Systems
ACODS-2012
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𝑥 (t) = A(t)𝑥(𝑡) + B(t)u(t),
Where here
𝑥 𝑡 = [𝜙 𝜃 𝜓 𝜙 𝜃 𝜓 ]′
x(t) is the state vector,
II. FUZZY BASED LQR CONTROLLER (FBLC) OVER LQR
CONTROLLER
Consider A(t) and B(t) be the state matrix and input matrix
respectively for a given plant. Let A(t) ,B(t) be periodic
matrices. Then the controller design can be made in the
following ways-
(5)
The linearized model considered for the system is given by
𝐴=
0
0
0
−4𝜔02 𝜎1
0
0
A. LQR Constant
In LQR-constant controller, time-averaged values of A(t)
and B(t) are calculated and those values are used in the LQR
gain calculation. But, at times it so happens the system goes
into the unstable region when the Eigen values of [A(t)B(t)*kavg] become positive, where ‘kavg’ is calculated from
Aavg and Bavg in ARE.
0
0
0
0
3𝜔02 𝜎2
0
0
0
0
0
0
𝜔02 𝜎3
0
0
0
0
𝐵(𝑡) =
B. LQR Periodic
In LQR-periodic controller, the instantaneous values of A(t)
and B(t) are used to calculate the gain. But the computational
load on the system increases, as ARE has to be solved at every
instant.
1
0
0
0
0
−𝜔𝑜 (1 + 𝜎3 )
−
0
0
0
𝐼11
𝑏 3(𝑡)
𝐼33
C. FBLC
FBLC controller optimizes the control algorithm between
LQR-constant and LQR-periodic according to the requirement
of the model.
𝐼𝑦 −𝐼𝑧
𝐼𝑥
,𝜎3 =
𝐼𝑥 −𝐼𝑦
𝐼𝑧
,𝜎3 =
𝐼𝑧
𝑏2 (𝑡)
(7)
𝐼22
𝑏1 (𝑡)
𝐼33
𝐼𝑥 −𝐼𝑦
−
𝐼11
𝑏1 (𝑡)
0
−
0
0
1
𝜔𝑜 (1 − 𝜎1 ) (6)
0
0
0
0
0
𝑏3 (𝑡)
𝐼22
𝑏2 (𝑡)
𝜎1 =
0
1
0
0
0
0
0
.
IV. FUZZY MEMBERSHIP FUNCTION AND CONTROLLER
DESIGN
The triangular fuzzy membership functions have been used
with the apex of the triangle corresponding to a central value.
In the system model, only the B matrix (i.e. magnetic field )is
time variant hence fuzzy logic has been used on values of B.
For all the three components of the magnetic field B, the
maximum and minimum values were assigned as the two
extreme central values and the remaining range space of B
was divided by two other equidistant central values.
III. ATTITUDE DYNAMICS AND KINEMATICS
The attitude dynamics of the small satellite in the inertial
frame is given by Eq. (1) as
𝑠𝑐𝑟𝑓
𝑇 = 𝐼. 𝜔 𝑖𝑟𝑓 + 𝜔
𝑠𝑐𝑟𝑓
𝑖𝑟𝑓
× 𝐼. 𝜔
𝑠𝑐𝑟𝑓
𝑖𝑟𝑓
(1)
where spacecraft angular velocity has been expressed in
Eq. (2) in the inertial frame
𝜔
𝑠𝑐𝑟𝑓
𝑖𝑟𝑓
=𝜔
𝑙𝑜𝑟𝑓
𝑖𝑟𝑓
For the three components of the magnetic field B the gains
were calculated for all possible combinations of the four
central values in each case using LQR method giving a total of
64 possible gain values. For any other intermediate value of Bcomponents within two central values as shown in Fig 1. the
gain was calculated in the following manner:
𝑠𝑐𝑟𝑓
+ 𝜔 𝑙𝑜𝑟𝑓
(2)
Equation (3) gives the attitude kinematics. Here 𝜙 𝜃 𝜓 are
the Euler angles
𝜔
𝑠𝑐𝑟𝑓
𝑖𝑟𝑓
=
𝜙 − 𝜓𝑠𝑖𝑛𝜃−𝜔0 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜓
𝜃 𝑐𝑜𝑠𝜙 + 𝜓𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝜃 − 𝜔0 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜓 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜓
−𝜃 𝑠𝑖𝑛𝜙 + 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝜃𝜓 − 𝜔0 (𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜓 − 𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝜓 )
Consider that at a given instant the B-components are
indicated as shown in the Fig 1. by the three back arrow
headed lines. Then, the gain is directly given by the relation:
(3)
The above Euler angles has been linearized and expressed
in Eq. (4)
𝜔
𝑠𝑐𝑟𝑓
𝑖𝑟𝑓
𝜙 − 𝜔0 𝜓
= 𝜃 − 𝜔0
𝜓 − 𝜔0 𝜙
𝐺𝐴𝐼𝑁 = 𝑥2 ∗ 𝑦2 ∗ 𝑧1 ∗ 𝑘(1,2,2) + 𝑥2 ∗ 𝑦2 ∗ 𝑧2 ∗ 𝑘(1,2,3)+ . . . . . . ..
On the right-hand side is the summation of the product of
all possible combinations of the LQR gains (k(1,2,3)) with the
three corresponding membership function values (eg:x2,y2,z2)
at the intercepts(of the membership function with the
instantaneous B-components).
(4)
This linearized angular velocity vector is substituted in the
attitude dynamics equation to obtain the linear state space
equation which has explained below.
In general , any control model can be defined by the statespace equation:
© ACODS-2012
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Proceedings of Advances in Control and Optimization of Dynamic Systems
ACODS-2012
Fig 1 . Membership function of magnetic field
V.
RESULTS
Fig 2. Angles and rates as observed in LQR constant controller
Different controllers (LQR periodic, LQR constant and
FBLC controller) has been tested with a set of initial
conditions with a specific inertia matrix with all three
magnetic torquers switching with 60% duty cycle and the
results are specified below.
For simulation
Inertia tensor = diagonal(2.01,1.324,2.025)
Disturbance torque =
Gravity Gradient-+1e-8*[sin(𝜔𝑜 𝑡),cos(𝜔𝑜 𝑡),-sin(𝜔𝑜 𝑡)]
Initial conditionsAngles =(7 ,8,-9)
Rates = (0.02/sec,-0.05/sec,0.01/sec)
Orbital Period = 5712 sec
Inclination = 92
Maximum dipole capacity =2 Amp.m2
Fig 3. Dipole moment as observed in LQR constant controller
© ACODS-2012
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Proceedings of Advances in Control and Optimization of Dynamic Systems
ACODS-2012
Fig 4. Angles and rates as observed in LQR periodic controller
Fig 6. Angles and rates as observed in FBLC
Fig 5. Dipole moment as observed in LQR periodic controller
Fig 7. Dipole moment as observed in FBLC
© ACODS-2012
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[2]
COMPARISON TABLE
Steady
state
error
(deg)
Steady
state
error
(deg/sec)
Settling
time (orbit)
LQR
constant
2.5
LQR
periodic
0.5
FBLC
4e-3
1e-3
2.5e-3
[3]
1.0
[4]
[5]
2.5
1.0
1.0
[6]
[7]
[8]
VI. CONCLUSION
1. Steady state error in LQR is less compared to LQR
constant controller, but in every cycle we have to compute a
solution for ARE , which increases the computational load on
the processor.
2. FBLC has comparable steady state error with LQR, but
with less computational effort on the processor.
3. In LQR constant settling time is more than LQR periodic
and FBLC so power consumption is more, as well as
sometimes it enters region of instability.
TERMINOLOGY
𝜔𝑜 - the orbital frequency of the satellite.
𝐼11 , 𝐼22 , 𝐼33 -Moments of Inertia of the satellite in the x, y, z
directions respectively.
𝑏1 (𝑡) , 𝑏2 (𝑡), 𝑏3 (𝑡) - the magnetic fields in the x,y,z body
directions respectively generated using IGRF model.
𝜙 -roll angle
𝜃 - pitch angle
𝜓 - yaw angle.
𝑠𝑐𝑟𝑓
𝜔 𝑖𝑟𝑓 - angular velocity of the spacecraft with respect to
inertial reference frame
𝑙𝑜𝑟𝑓
𝜔 𝑖𝑟𝑓 - angular velocity of the orbital reference frame with
respect to inertial reference frame
𝑠𝑐𝑟𝑓
𝜔 𝑙𝑜𝑟𝑓 - angular velocity of the spacecraft with respect to
orbital reference frame
ARE -Algebraic Riccati equation
REFERENCES
[1]
F. Martel, P. K. Pal, and M. L. Psiaki, “Active magnetic control system
for gravity gradient stabilized spacecraft,” in Proc. 2nd Annual
AIAA/USU Conference on Small Satellites, Utah State University, USA,
1988.
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5
C. Arduini and P. Baiocco, “Active magnetic damping attitude control
for gravity gradient stabilized spacecraft,” Journal of Guidance, Control,
and Dynamics, vol. 20, pp. 117–122, Jan. 1997.
R. Wisniewski and M. Blanke, “Fully magnetic attitude control for
spacecraft subject to gravity gradient,” Automatica, vol. 35, no. 7, pp.
1201–1214, 1999.
R. Wisniewski, “Linear time varying approach to satellite attitude
control using only electromagnetic actuation,” in Proc. AIAA Guidance,
Navigation and Control. Conf., vol. 23, AIAA, New Orleans, 1997, pp.
243–251.
M. L. Psiaki, “Magnetic torquer attitude control via asymptotic periodic
linear quadratic regulation,” Journal of Guidance, Control, and
Dynamics, vol. 24, pp. 386–393, Mar. 2001.
W. H. Steyn, “Fuzzy control for a non-linear MIMO plant subject to
control constraints,” IEEE Transactions on Systems, Man, and
Cybernetics, vol. 24, no. 10, pp 1565-1571, October 1994.
B. Petermann, Attitude control of small satellites using fuzzy logic,
Master Thesis, McGill University, Montreal, 1997.
R. Wisniewski ,"Satellite Attitude control using only electromagnetic
actuation", Ph.D Thesis,1996