Direct Fuzzy Logic Controller for Nano-Satellite

Journal of Control Engineering and Technology JCET

JCET Vol. 4 Iss. 3 July 2014 PP. 210-219 www.vkingpub.com © American V-King Scientific Publish 210

Direct Fuzzy Logic Controller for Nano-Satellite 1Mohammed Chessab Mahdi, 2Abdal-Razak Shehab, 3Mohammed. J. F Al Bermani

1Foundation of Technical Education Technical Institute of Kufa-Iraq 2University of Kufa - College of Engineering-Iraq 3University of Kufa - College of Engineering-Iraq

[email protected]; [email protected]; [email protected]

Abstract-In this paper, a direct fuzzy controller is applied to

the attitude stabilization control of a CubeSat. The Takagi-

Sugeno (T-S) model fuzzy controller is evaluated for attitude

control of magnetic actuated satellites based on the attitude

error including error in the angles and their rates. The

detailed design procedure of the fuzzy control system is

presented. Simulation results show that precise attitude

control is accomplished and the time of satellite maneuver is

shortened in spite of the uncertainty in the system.

Keywords- Fuzzy Logic Controller; PID Controller; Attitude

Control System; Nanosatellite; KufaSat.

I. INTRODUCTION

CubeSats are a class of research spacecraft called Nano

Satellites. The first CubeSats were launched in June 2003 on

a Russian Eurockot, and approximately 75 CubeSats have

been placed into orbit as of August 2012.

The Iraqi student satellite project Kufasat was started in

2012. The launch of the satellite is planned in the near

future. The main tasks for Kufasat will be to perform dust

density measurement and remote sensing. The project is

sponsored by the University of Kufa and it will be the first

Iraqi satellite to fly in space. Kufasat is a Nano-satellite

based on the CubeSat concept. In accordance with CubeSat

specifications, its mass is restricted to 1.3 kg, and its size is

restricted to a cube measuring 10×10×10 cm3. It also

contains 1.5 m long gravity gradient boom, which will be

used for passive attitude stabilization. The satellite attitude

control problem includes attitude stabilization and attitude

maneuver. Attitude stabilization is the process of

maintaining a desired attitude, and the attitude maneuver is

the re-orientation process of changing one attitude to

another [1]. In general, attitude stabilization systems are

classified as active or passive. Active attitude stabilization

requires power while passive do not require any power. The

simplicity and low cost of active magnetic control makes it

an attractive option for small satellites in Low Earth Orbit

(LEO).

A gravity gradient stabilized satellite has limited

stability and pointing capabilities (± 5º) so, magnetic coils

are added to improve both the three-axis stabilization and

pointing properties. Magnetic coils around the satellite's

XYZ axes can be fed with bidirectional constant current

electrical power to generate a magnetic dipole moment ,

which will interact with the geomagnetic field vector to

generate a satellite torque by taking the cross product [2] ,

which is used to control the attitude of the satellite.

Gravity gradient stabilization has been used in attitude

control since the early 1960s [3], but accurate three-axis

control cannot been achieved using gravity gradient

stabilization alone. Gravity gradient stabilization combined

with magnetic torquing has gained increased attention as an

attractive attitude control system (ACS) for small

inexpensive satellites and is also proposed in this satellite

[4].

Because both the direction and the strength of the

geomagnetic field vary as the satellite orbits Earth, the

magnetic control is both non-linear and time dependent.

Attitude control in two-axes only can be achieved when

using three magnetic torquers because the magnetic torques

are constrained on a plane perpendicular to the local

magnetic field. In this paper, a comparison between two

attitude control laws: Proportional-Integral-Derivative

controller (PID) and Fuzzy logic controller (FLC) are

presented. Two well-known fuzzy logic control methods are

Mamdani‟s Fuzzy Inference Systems (FIS) and Takagi-

Sugeno‟s T-S Method. Mamdani type fuzzy inference gives

an output that is a fuzzy set while Sugeno-type inference

gives an output that is either constant or a linear (weighted)

mathematical expression.

A novel approach is presented using direct fuzzy logic

control with the Takagi-Sugeno (T-S) model to control the

magnetic coil current directly.

II. DYNAMIC MODEL

The mathematical model of a satellite‟s attitude is

described by kinetic and kinematic equations of motion

[5]which are now presented. The kinetic equations of

motion for a satellite in LEO is

(1)

where

is the angular velocity of a body-fixed reference

frame to an inertial reference frame.

is the principal axis moment of inertia matrix with

respect to a body-fixed frame .

, where diag is a 3x3 diagonal matrix .

is the total external torque acting on a satellite

expressed in body-fixed reference frame components found

in equation (2) .



(2)

where is the total gravity gradient torque, is

the total magnetic torque , and is the total

disturbance torques , all expressed as components in the

body-fixed frame, Fb.

Equation (1) can be broken down by components into

roll, pitch, and yaw dynamic equations respectively, as

shown in Figure 1 and restated as equations 3a, 3b, and 3c.

Fig. 1 Roll (φ), Pitch (θ) and Yaw (ψ) Angles in body frame

(3a)

(3b)

(3c)

where ωx, ωy, and ωz are angular velocity component of

expressed in the body-fixed frame, Ix, Iy, and Iz, are

the principal moments of inertia expressed in the body-fixed

reference frame, and Tx, Ty, and Tz are the components of

the total external torque, given in Eq.(2), expressed in the

body- fixed frame. Equations (3a, 3b, 3c) are known as

Euler‟s equations of motion for a rigid body [6]. If the Euler

angles ϕ, θ, and ψ are small in magnitude, the relationship

between body angular velocity and Euler angular rates may

be approximated [7] as

(4)

where ωo is the orbital angular velocity and

are the time rates of change of ϕ, θ, and ψ .

A. Gravity Gradient Torque:

The gravity gradient torque, using the aforementioned

small Euler angle approximation and taking principal axes

as reference axes, is given [3] by

(5)

where TGx , TGy, and TGz are the gravity torque (from

Eq. 2) components about the roll, pitch, and yaw axes,

respectively.

B. Magnetic Field Torque:

The magnetic coil produces a magnetic dipole when

currents flow through its windings, which is proportional to

the ampere-turns and the area enclosed by the coil. The

torque generated by the magnetic coils can be modeled

as

(6)

where is the generated magnetic moment vector inside

the body (and written as components in the Body-fixed

Frame, Fb) and is the local geomagnetic

field vector . Eq (7) shows the components of when

written in Fb.

(7)

where Nk is number of windings in the magnetic coil, ik is

the coil current, Ak is the span area of the coil, and k = x, y,

z (i.e. the x, y, z values for these quantities). The magnetic

torque can be represented as

(8)

where Tmx , Tmy, and Tmz are the magnetic torque

components about the roll, pitch, and yaw axes, respectively,

mx, my, and mz are the corresponding components of the

magnetic moments, and Bφ , Bθ , and Bψ is the Earth‟s

magnetic field expressed in body-fixed frame, Fb. After

combining Equations (3), (4), (5), and (8) the final

linearized attitude dynamic model of the satellite, including

the gravity gradient and magnetic coil torques written in

body frame components, becomes

(Roll) (9a)

(Pitch) (9b)

(Yaw) (9c)



, and are the second time derivatives of ϕ, θ, and ψ.

Disturbance torques , magnitudes such as solar

radiation pressure torque and residual magnetic dipole

torque are negligible and therefore have been neglected.

If the system angular position and velocity states are

given as: ; (Roll), ;

(Pitch) ; (Yaw) ,

then the linear system can be expressed as a state space

model

(10)

Where

(11)

(12)

is the control signal defined by

(13)

III. PID CONTROLLER DESIGN

The Proportional-Integral-Derivative controller (PID

controller) is the most widely used feedback control

approach [8].It is also one of the simplest control algorithms.

In the absence of knowledge of the underlying process,

employing one or many PID controllers is often the best

choice to achieve a system designer‟s control objectives. A

typical PID control system structure is shown in Figure 2,

where Kp is the proportional gain, Kd is the derivative gain,

and Ki is the integral gain, where each gain is assumed to be

a positive scalar.

By appropriately adjusting these gains, the desired

output can be achieved while maintaining system stability.

It can be seen that in a PID controller, the error signal e(t) is

used to generate the proportional, integral, and derivative

with the resulting signals weighted and summed to form the

process defined as a „plant model‟ feedback error control

signal u(t) applied.

Fig. 2 PID Controlled System

A mathematical description of the PID controller per [9]

can be found as:

(14)

where u(t) is the plant model input signal and is

defined as the error signal, , where

is the input reference signal. In Equation (14), the

proportional action is related to the error present and is used

to reduce the system‟s rise time. The integral action based

on past error, is used to reduce the system‟s steady state

error, that is, make the system final value closely match its

desired value. Finally, the derivative action, related to

future behavior of error and it is used to improve system

stability. It is also used to drive system overshoot

performance error and improve transient response.

Parameters values used to dynamically simulate Kufasat

performance are listed in Table (1), which explain that the

moment of inertia for one axis, Iz, is significantly smaller

than the other two due to the deployed configuration.

TABLE 1 KUFASAT MOI, ORBITAL ANGULAR VELOCITY AND MASS OF THE GG

BOOM PARAMETERS

Ix

(kg m2)

Iy

(kg m2)

Iz

(kg m2) ωo (rad/sec)

Tip

Mass(g)

0.1043 0.1020 0.0031 1.083*10-3 40

The PID controller is tuned by selecting parameters KP,

Ki, and Kd that give an acceptable closed-loop response. A



desirable response is often characterized by the measures of

settling time, overshoot, and steady state error to mention a

few. Many PID tuning methods have been proposed over

the years, ranging from the simple, but most famous

Ziegler-Nichols tuning method, to the more modern simple

internal model control (SIMC) tuning rules by Skogestad

[10]. In this work, all PID parameters are obtained using

Ziegler-Nichols tuning method. These parameter values are

listed in Table (2) and the post-tuning performance

parameters are listed in Table (3).

TABLE 2 PID CONTROLLER PARAMETERS

Controller type Proportional

gain (Kp)

Derivative

gain (Kd)

Integral

gain (Ki)

PID controller for Roll 0.00216 0.04 0.000004

PD controller for Pitch 0.00064 0.052 ------

PID controller for Yaw 0.0006 0.003 0.000003

TABLE 3 POST-TUNING PERFORMANCE PARAMETERS

Angle

Controller

Delay Time

(Td) sec

Rise Time

(Tr) sec

Peak Time

(Tp) sec

Settling

Time

(TS) sec

Peak

Overshoot

PO%

Steady State

Error %

Roll PID 3 5 10 50 10 0.0

Pitch PD 2.5 4.5 8 10 2.6 0.3

Yaw PID 2 2.5 5 15 21 0.1

The high level attitude control PID Controller (s) SIMULINK block diagram is shown in Figure 3 with further details

reflected in Figure 4.

Fig. 3 Simulink block diagram of satellite model with PID controller

Fig. 4 Complete Simulink diagram of satellite model with PID controller



IV. FUZZY LOGIC CONTROLLER DESIGN

Fuzzy logic controller (FLC) system have been

successfully applied to a wide variety of problems [11]. For

nonlinear problems, many existing experiments have

demonstrated that FLC systems have good performance even

with additive noise [12].

The primary advantage of the FLC is the ability to easily

incorporate heuristic rule-based knowledge from experts in

the control strategy [12] . As a result, fuzzy control is usually

applied to a complex system whose dynamic model is not

well defined or not available at all.

FLC offers the following important benefits, compared to

conventional control techniques:

Developing a FLC is cheaper than developing PID

which need to a sensor for the feedback information.

FLCs are more robust than PID controllers because

they can cover a much wider range of operating

conditions than PID controllers.

FLCs are customizable, because it is easier to

understand and modify their rules, which are

expressed in linguistic terms [12], [13].

Two well-known fuzzy control methods are Mamdani‟s

Fuzzy Inference Systems (FIS) and Takagi-Sugeno‟s T-S

Method [14]. Mamdani‟s method is widely accepted for

capturing expert knowledge. It allows us to describe this

expertise in more intuitive, more like human manner.

However, the Mamdani method‟s fuzzy inference concept

entails substantial computational effort.

The T-S method, originally proposed by Takagi and

Sugeno [15], is computationally efficient and works well with

optimization and adaptive techniques. This makes it very

attractive to solving control problems, especially those using

dynamic nonlinear systems. The most fundamental difference

between the Mamdani and the T-S methods is the way the

crisp output is generated from the fuzzy inputs. While the

Mamdani method uses the fuzzy output "defuzzification", the

T-S method uses weighted average to compute the crisp

output [16]. So the T-S method is typically used when output

membership functions are either linear, constant, or both.

Three T-S Multi Input Single Output (MISO) FLCs, one

for each coil, are used to control each coil‟s current switching

(on, off) and polarity. Each FLC has nine rules and three

linguistic variables. Each FLC‟s three linguistic variables are

for its two inputs, the error (E) and change-in-error (CE), and

it‟s one output, (U) which represents the control action. Each

variable (E, CE, and U) is then represented by a membership

function, as show in Figure 5. Each FLC‟s output is used to

control the roll, pitch and yaw angles, respectively, through

the associated coil current. These variables per FLC are

mapped into three fuzzy set Positive (PO), Negative (NE) and

Zero (ZE) rules. The output variable indicates the desired

magneto-torquer polarity, (HI) for positive polarity and (LO)

for negative polarity. These rules are given in Table (4)

which allows describing the dynamics of the controller.

TABLE 4 RULE BASE FOR THE CONTROLLER OF ROLL, PITCH AND YAW ANGLES

(a) Membership Function for Ephi (Error)

(b) Membership Function for CEphi (Change of Error)

(c) output U

(d) output surface

Fig. 5 Membership functions used in FLC for (a) error E, (b) change of error

CE, (c) output U, and (d) output surface



The attitude control/FLC set up SIMULINK block diagram is shown in Figure 6.

Fig. 6 Satellite model and Fuzzy Logic Controller SIMULINK Diagram

V. SIMULATION

In this section, several simulations of the proposed direct fuzzy control are presented. The FLC Scaling factors were

selected based on trial and error. The values of these scaling factors are listed in Table (5).

TABLE 5 SCALING FACTORS VALUES

Scaling factor GE

Phi

GCE

phi

GU

phi

GE

theta

GCE

theta

GU

theta

GE

psi

GCE

psi

GU

psi

Value 200 2 5 205 30 5 50 1.5 5

where GE is the error scaling factor, GCE is the change-in-error scaling factor, and GU is the output scaling factor. The coil

parameters values used for the Kufasat simulations are listed in Table (6):

TABLE 6 COIL PARAMETERS

Parameter Symbol Value Unit

Length a 85 mm

Width b 75 mm

Mass M 20 g

Wire diameter D 0.1016 mm

Wire resistivity 1.68×10-7 Ω/m

No of turns N 305 turn

Power at full load P 100 mW

Voltage at full load V 4.5 Volt

Coil resistance R 211 Ohm

Coil current I 21.5 mA

Min Temperature Tmin -60 Co

Max Temperature Tmax 80 Co



A. Stabilization Test:

In this test, (0 to 1) rad step input was applied on two cases, first using the PID controller and the second using the FLC

controller. Figure 7 show the system response when using PID and FLC controllers, when the initial conditions of the roll,

pitch, and yaw angles are equal to (0, 0, 0) degrees.

Fig. 7 Attitude response using three PID and three FLC controllers with a 1 rad step input



B. Attitude Control Maneuver (ACM) Test

The Direct Fuzzy Controller (DFC) is designed for any initial Euler orientation and any desired reference attitude. In this

section, the fuzzy controller is tested to achieve different orientations. Figures 8and 9 illustrate Kufasat attitude response to a

small and a large ACM with PID and FLC controllers.

Fig. 8 Response to a small ACM from [0° 5° 0°] to [5° 0° 10°] with three PID and three FLC controllers



Fig. 9 Response to a large ACM from [-20° 50° -10°] to [+20° 0° 60°] with PID controller

The FLC has better performance in terms of percent

overshoot and rise time and minimal steady state error. Even

though the PID controller produces the response with lower

delay and rise times compared with the fuzzy logic

controller, it has a long settling time due high peak

overshoot. In addition to that FLC is controllable and more

stable than PID controller when the system is under effect of

Attitude Maneuver (AM). It is clear from results that the

settling time of FLC is less than the settling time of PID,

that mean the fuzzy controller consume shorter execution

time than the PID.

VI. CONCLUSION

In this paper, a simple direct fuzzy logic controller for

Kufasat attitude control is developed. Then its performance

compared with conventional PID controller. Several

fundamental observations were made based on simulation

and analysis of both controller types.

First, the FLC has good performance, in terms of

minimum overshoot, short rise time and minimal steady

state error. The three FLCs were built using simple direct

fuzzy controller logic with a reduced number of fuzzy rules.



This simple structure can reduce the calculation time of the

control action, hence improving the reliability of system.

Second, the FLC has better performance in terms of

percent overshoot and rise time. It is observed that FLC is

controllable and more stable than PID controller when the

system is under effect of Attitude Maneuver (AM). Third,

because roll and yaw are strongly coupled, roll takes a

longer time to achieve acceptable appointing accuracy. Fourth, even though the PID controller produces the

response with lower delay and rise times compared with the

fuzzy logic controller, it has a long settling time due high

peak overshoot. Fifth, CubeSats operate on a strict power

budget due to tight power requirements and limitations.

These systems have relatively limited (small solar array area

available and, limited battery mass and volume allocation).

Kufasat uses two lithium ion cells in parallel as a 2W peak

power and 1.5W average secondary battery. Due to these

power limitations, only one magneto-torquer coil can be

switched on at a time. A control algorithm must be

modified in future to allow for the choice of the coil that

will achieve the best results, given the local geomagnetic

field vector.

REFERENCES

[1] Scrivener, S. L. and Thomson, R C,” Survey of Time-

Optimal Attitude Maneuvers” Journal of Guidance Control

and Dynamics, vol. 17, pp225-233,1994.

http://dx.doi.org/10.2514/3.21187

[2] W. H. Steyn,” Fuzzy Control for a Non-Linear MIMO Plant

Subject to Control Constraints” IEEE Transaction on

Systems, Man, and Cybernetics, Vol.24, No.10, October

1994. http://dx.doi.org/10.1109/21.310540

[3] Hughes, P. C, “Spacecraft Attitude Dynamics”. John Wiley,

NY, 1986.

[4] Skullestad, A and Olsen, K.” Control of GravityGradient

Stabilized Satellite Using Fuzzy Logic,Modeling,

Identification and Control”, VOL.22, NO. 141-152, 2001.

[5] M. Paluszek, P. Bhatta, P. Griesemer, J. Mueller and S.

Thomas, ”Spacecraft Attitude and Orbit Control”,Princeton

Satellite Systems, Inc, 2009.

[6] Marcel J. Sidi “Spacecraft Dynamics and Control APractical

Engineering Approach“, Cambridge University Press, 1997.

[7] Bryson, A. E.”Control of Spacecraft and Aircraft”, New

Jersey, USA: Princeton University press.

[8] Boiko, Igor.” Non-parametric Tuning of PIDControllers”,

Springer, 2013. http://dx.doi.org/10.1007/978-1-4471-4465-6

[9] Katsuhiko Ogata,”Modern Control Engineering”, Pearson

Education International, 2002.

[10] Skogestad, S and Grimholt, C.” The SIMC Method for

Smooth PID Controller Tuning, in PID Control in the Third

Millennium: Lessons Learned and NewApproaches” edited

by Ramon Vilanova,Antonio Visioli, Springer, pp. 147-175,

2012. http://dx.doi.org/10.1007/978-1-4471-2425-2_5

[11] Elmer P. Dadios. ” Fuzzy Logic - Controls,

Concepts,Theories and Applications”, InTech, 2012.

[12] K. M. Passino & Stephen Yurkovich ,”Fuzzy Control”,

Addison Wesley Longman, Inc.1998.

[13] Huaguang Zhang Derong Liu “Fuzzy Modeling andFuzzy

Control “,ISBN-10 0-8176-4491-1 2006.

[14] Yuanyuan Chai, Limin Jia, “Mamdani Model based Adaptive

Neural Fuzzy Inference System and itsApplication”,

International Journal of Computational Intelligence, pp. 22-

29, 2009.

[15] Takagi, T. & Sugeno, M.”Fuzzy Identification of Systems

and Its Applications to Modeling and Control”, Proceedings

of the IEEE Transactions on Systems, Man and Cybernetics,

Vol. 15, 1985.

[16] Haman .A, Geogranas.N.D,”Comparison of Mamdani and

Sugeno Fuzzy Inference Systems for Evaluating the Quality

of Experience of Hapto- Audio-Visual

Applications”,IEEEInternational Workshop on Haptic

Audio Visual Environments and their Applications, 2008.

BIOGRAPHY

Mohammed Chessab Mahdi had his B.Sc.

degree in control and system engineering

from University of Technology –Baghdad at

1984 and had his M.Sc. degree in space

technology from University of Kufa at 2013.

He is full time lecture in Technical Institute of

Kufa -Foundation of Technical Education–

Iraq and member of KufaSat team - space

research unit-Faculty of Engineering

University of Kufa. He has good skills in the design and modeling

of attitude determination and control systems using Matlab

program. He has been published more than 6 researches.

Dr. Abd AL-Razak Shehab body of receive

his B.Sc. from Baghdad University at 1987 ,

M.Sc. and Ph.D. from Saint Petersburg

polytechnic government university (Russia

federal) at 2000 and 2004

respectively .Currently he is full time lecture in

electrical engineering department (Head of

department since 2007) – Faculty of

Engineering –Kufa University–Iraqi ministry of

high education and scientific research. He has good skills in the

design and modeling of control systems and switched reluctance

motor (SRM). He has been published more than 9 researches.

Dr. Mohammed. J. F Al Bermani body of

receive his B.Sc. from Baghdad University at

1971, M.Sc. from Aston University,

Birmingham, UK at 1983 and Ph.D. from

Baghdad University at 1997. Currently he is

full time lecture and head of KufaSat team -

space research unit-Faculty of Engineering

University of Kufa. Iraqi ministry of high

education and scientific research. He has good

skills in Attitude determination and six degrees of freedom

dynamics of spacecraft. He has been published more than 25

researches

http://dx.doi.org/10.2514/3.21187

http://dx.doi.org/10.1109/21.310540

http://dx.doi.org/10.1007/978-1-4471-4465-6

http://dx.doi.org/10.1007/978-1-4471-2425-2_5

Direct Fuzzy Logic Controller for Nano-Satellite

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