Direct Fuzzy Logic Controller for Nano-Satellite https://www.researchgate.net/.../Mohammed.../Direct-Fuzzy-Logic-Controller-for-Nan...

Journal of Control Engineering and Technology

JCET

Direct Fuzzy Logic Controller for Nano-Satellite 1

Mohammed Chessab Mahdi, 2Abdal-Razak Shehab, 3Mohammed. J. F Al Bermani 1

Foundation of Technical Education Technical Institute of Kufa-Iraq 2 University of Kufa - College of Engineering-Iraq 3 University 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 TakagiSugeno (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.

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].

Keywords- Fuzzy Logic Controller; PID Controller; Attitude Control System; Nanosatellite; KufaSat.

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 TakagiSugeno‟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.

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 novel approach is presented using direct fuzzy logic control with the Takagi-Sugeno (T-S) model to control the magnetic coil current directly. II.

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.

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

DYNAMIC MODEL

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) .

to

generate a satellite torque by taking the cross product [2] , which is used to control the attitude of the satellite.

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


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A. Gravity Gradient Torque:

(2) where

is the total 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

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.

(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 as

generated by the magnetic coils can be modeled (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.

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

(3a) (3b)

(7)

(3c)

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

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

(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

(4) where

ωo

is the orbital angular velocity and are the time rates of change of ϕ , θ, and ψ .

(Roll)

(9a)

(Pitch)

(9b)

(Yaw)

(9c)



,

and

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then the linear system can be expressed as a state space model

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.

(10) Where

If the system angular position and velocity states are given as: ; (Roll), ; (Pitch) ; (Yaw) ,

(11)

(12) is the control signal defined by (13) III.

A mathematical description of the PID controller per [9] can be found as:

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.

(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.

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.

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) 0.1043

Iy (kg m2) 0.1020

Iz (kg m2) 0.0031

ωo (rad/sec) 1.083*10-3

Tip Mass(g) 40

The PID controller is tuned by selecting parameters KP, Ki, and Kd that give an acceptable closed-loop response. A

Fig. 2 PID Controlled System



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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

PID controller for Roll

Proportional gain (Kp) 0.00216

Derivative gain (Kd) 0.04

PD controller for Pitch

0.00064

0.052

------

PID controller for Yaw

0.0006

0.003

0.000003

Controller type

Integral gain (Ki) 0.000004

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.

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

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.

(a)

(b)

Membership Function for Ephi (Error)

Membership Function for CEphi (Change of Error)

(c)

(d)

output U

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



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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

1.68×10-7

Ω/m

Wire resistivity 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



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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



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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



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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.



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This simple structure can reduce the calculation time of the control action, hence improving the reliability of system.

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.

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 REFERENCES

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BIOGRAPHY

Scrivener, S. L. and Thomson, R C,” Survey of TimeOptimal Attitude Maneuvers” Journal of Guidance Control and Dynamics, vol. 17, pp225-233,1994. http://dx.doi.org/10.2514/3.21187 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 Hughes, P. C, “Spacecraft Attitude Dynamics”. John Wiley, NY, 1986. Skullestad, A and Olsen, K.” Control of GravityGradient Stabilized Satellite Using Fuzzy Logic,Modeling, Identification and Control”, VOL.22, NO. 141-152, 2001. M. Paluszek, P. Bhatta, P. Griesemer, J. Mueller and S. Thomas, ”Spacecraft Attitude and Orbit Control”,Princeton Satellite Systems, Inc, 2009. Marcel J. Sidi “Spacecraft Dynamics and Control APractical Engineering Approach“, Cambridge University Press, 1997. Bryson, A. E.”Control of Spacecraft and Aircraft”, New Jersey, USA: Princeton University press. Boiko, Igor.” Non-parametric Tuning of PIDControllers”, Springer, 2013. http://dx.doi.org/10.1007/978-1-4471-4465-6 Katsuhiko Ogata,”Modern Control Engineering”, Pearson Education International, 2002. 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 Elmer P. Dadios. ” Fuzzy Logic - Controls, Concepts,Theories and Applications”, InTech, 2012. K. M. Passino & Stephen Yurkovich ,”Fuzzy Control”, Addison Wesley Longman, Inc.1998. Huaguang Zhang Derong Liu “Fuzzy Modeling andFuzzy Control “,ISBN-10 0-8176-4491-1 2006. Yuanyuan Chai, Limin Jia, “Mamdani Model based Adaptive Neural Fuzzy Inference System and itsApplication”, International Journal of Computational Intelligence, pp. 2229, 2009. 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. Haman .A, Geogranas.N.D,”Comparison of Mamdani and Sugeno Fuzzy Inference Systems for Evaluating the Quality of Experience of HaptoAudio-Visual Applications”,IEEEInternational Workshop on Haptic Audio Visual Environments and their Applications, 2008.

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
