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CHAPTER 5

FUZZY LOGIC FOR ATTITUDE CONTROL

5.1 INTRODUCTION

Fuzzy control is one of the most active areas of research in the

application of fuzzy set theory, especially in complex control tasks, which do

not lend themselves to control by conventional methods, because of lack of

quantitative data regarding input-output relations (Driankov and Hellendoorn

1995). It involves the use of expert’s knowledge in order to accomplish the

control task.

The present interest in fuzzy logic control is largely due to the

successful application of it to a variety of consumer products and industrial

systems. Basically, fuzzy logic is a generalization of binary logic. Fuzzy sets,

membership functions and rules are three primary elements of a fuzzy logic

system. In the IF – THEN rules (which are logical statements), linguistic

terms are used to incorporate vagueness and ambiguity. This logical inference

using fuzzy set is known as fuzzy logic.

Fuzzy logic is much closer in spirit to human thinking and natural

language than the traditional logical systems. It provides an effective means

of capturing the approximate, inexact nature of the real world. It is the

generalization of the classical logic in which there is a smooth transition from

TRUE to FALSE (Bart 1992).

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Development of fuzzy logic for the attitude controller of micro

satellites is dealt in this chapter.

5.2 FLC SYSTEM FOR SATELLITE ATTITUDE CONTROL

Fuzzy control is one of the expanding application fields of fuzzy set

theory. It is an expanding field in space technology (Chiang and Jang 1994).

Fuzzy controllers differ from classical math-model controller. Fuzzy

controllers do not require a mathematical model of how control outputs

functionally depend on control inputs and therefore especially suited for

situations where the system is too complex to model. Fuzzy controllers also

differ in the type of uncertainty they represent.

5.3 IMPLEMENTATION

In this section fuzzy controller is designed for detumbling with

initial spin up, spin rate control and spin axis orientation control with the help

of Fuzzy Toolbox, Matlab.

5.3.1 Detumbling with Initial Spin Up Controller

The input variables for the fuzzy controllers are the measured state

variables of the satellite and the estimated control torques. This choice of

input variables will make it possible to regulate the state variables while

considering the control torque constraints. The torques can be estimated using

a cross product between moment and magnetic field.

The intention of this controller design is to define a set of control

rules and to implement them in such a way as to make boundaries between

them less strict, resulting in a system that cover a large universe of discourse

with a relatively small rule base. A block diagram of the proposed fuzzy

controller is shown in Figure 5.1. The controller consists of three fuzzy

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control laws, one for each magneto-torquer (Mx, My, and Mz coils). Each

control law embodies a fuzzy rule base to decide on the control desirability

and output level when using the corresponding torquer (Steyn 1996).

Figure 5.1 Block diagram of the fuzzy controller

5.3.1.1 Inputs

A total of six inputs were used, they are as given below:

X1 = ωx, angular rate about x axis

X2 = ωy, angular rate about y axis

X3 = k (ωz - ωspin), rate error

X4 = Nx, estimated control torque about x axis

X5 = Ny, estimated control torque about y axis

X6 = Nz, estimated control torque about z axis

5.3.1.2 Membership functions

These variables are then mapped into fuzzy sets (ex. for positive for

negative). The fuzzy set values are obtained from membership functions e.g.:

X1 mP(X1) and X 4 mN(X4) (5.1)

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These values then act accordingly to choose the torquer polarity.

The membership functions for each input variable are shown in Figures 5.2,

5.3 and 5.4 respectively. The reasons for choosing the functions in this

specific format are a need to limit the number of fuzzy sets and to obtain a

linear mapping in the normal operating region of the system.

Figure 5.2 Membership function for variables X1, X2, X3

Figure 5.3 Membership function for variables X4, X5

Figure 5.4 Membership function for variable X6

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The amount of overlap between the different fuzzy sets is optimised

through simulation. The saturation point of each input variable is set using an

engineering knowledge of the system and optimised using simulation trails.

In the detumbling mode both the transverse rates and estimated

control torques are given as a input to Mz controller. A set of intuitive rules is

used to describe the Mz controller. In the initial spin up mode the polarity of

the constant current that is passed to Mx and My coils are solely determined

from the sign and magnitude of the transverse earth’s magnetic field

component.

5.3.1.3 Rules

Rules for spin axis controller and transverse axes controller are

given in the Tables 5.1 and 5.2 respectively.

Table 5.1 Mz fuzzy variable control rule

Rule X1 X2 X4 X5 U

R1 P - P - - 1

R2 P - N - + 1

R3 N - N - + 1

R4 N - N - - 1

R5 - P - P - 1

R6 - P - N + 1

R7 - N - P + 1

R8 - N - N - 1

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Table 5.2 Mx, My fuzzy variable control rule

Rule X1 X2 X3 X4 X5 X6 U R1 P - - P - Z -1 R2 P - - N - Z +1

R3 N - - P - Z +1 R4 N - - N - Z - 1

R5 - P - - P Z - 1

R6 - P - - N Z + 1 R7 - N - - P Z + 1

R8 - N - - N Z - 1

R9 - - P - - P +1 R10 - - P - - N - 1

R11 - - N - - P -1

R12 - - N - - N +1

Rule evaluation is performed using correlation-product encoding, i.e

the conjunctive (AND) combination of the antecedent fuzzy sets. For example

rule 1:

R1: µ1 = mP(X1).mP(X2).mZ(X3) (5.2)

where mi are the membership functions.

The truth-value obtained is then used to scale the output:

R1 : y1 =1 .U (5.3)

When the result of all the rules is known, the final value is obtained by

disjunctively (OR) combining the rule values:

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N

1i

iN

1i

ii y,1minysgn)y(y (5.4)

The disjunction method can be described as a kind of signed

Lukasiewicz OR logic. It is chosen to maximally negatively correlate the rule

outputs. For example, opposing rule outputs (different in sign) cancel each

other to deliver a small rule base output.

Above said detumbling with initial spin up controller is developed

using FIS Editor, Fuzzy logic Toolbox, Matlab.

5.3.2 Spin Rate Controller

The spin rate control is done to keep the satellite spinning at

constant rate, even when there is a disturbance. During normal mission of the

spacecraft, spin rate along maximum moment of inertia axis i.e. spin axis is

required to be maintained within the specified band.

The transverse axes torquers are actuated till the specified spin rate

is reached. Current polarities of the torquers in transverse axes are:

Current polarity My = (sign of Bx)

Current polarity Mx = (sign of By)

where +, - sign depends on whether it is spin up or spin down mode.

Fuzzy spin rate controller basically consists of two Single Input

Single Output (SISO) controllers. Spin rate control involves two modes, spin

up and spin down.

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By Mx

Bx My

Figure 5.5 Block diagram of fuzzy spin rate controller

The input variables for this fuzzy controller are transverse

component of earth’s magnetic field By and Bx respectively. For increasing

the rate, fuzzy logic controller gives the same sign as its input magnetic field

for My controller, while the polarity given by Mx controller is directly

opposite to its input magnetic field sign. The implementation process is

explained in the Figure 5.5.

For spin up, the rules are:

If Bx is negative then My is negative

If Bx is positive then My is positive.

Also for Mx controller:

If By is negative then Mx is positive

If By is positive then Mx is negative

Similarly for spinning down the rules are slightly modified. In this

Mx controller follows the same sign as it input magnetic field, while other

follows opposite to that of input. Rule evaluation is performed using

correlation product encoding i.e conjunctive (AND) combination of the

antecedent fuzzy sets. The membership function for input variable is shown in

SISO FLC

SISO FLC

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Figure 5.6. The overlapping is made minimum to sharply define a boundary

for the polarity that could be applied to the torquers.

Figure 5.6 Membership function for variables By, Bx

5.3.3 Spin Axis Orientation Controller

The attitude control for this mode fuzzy is implemented by direct

mapping of torquer values to the given magnetic field.

For this research work, the magnetic field along transverse axes is

taken as input and required torque is produced along the transverse axes.

Input Output

X1 – By Y1 – Torque value along y axis

X2 – Bx Y2 – Torque value along z axis

Thirteen membership functions (2 trapezoidal membership,

11 triangular membership functions) are used for inputs By and Bx. In the

output 13 triangular membership functions are used to represent the linguistic

terms. A scale mapping is performed using triangular membership function,

which transfers the range of input variable into corresponding universe of

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discourse. An element of each set is mapped on to the domain corresponding

to the linguistic variable.

Knowledge to perform deductive reasoning is called inference. The

truth-value for each premise of every rule is computed. Inference mechanism

helps to draw conclusion from the rule base, which can produce an output

from the collection of If-Then rules. In the present work the inference

mechanism is designed using Mamdani (Min – Max) method.

The resulting fuzzy set must be converted to a number that can be

sent to the process as a control signal. This operation is called

“defuzzification”. In this work, centroid method of defuzzification is used.

All the rule outputs are then combined disjunctively (OR) to obtain

the crisp rule base output:

iN

1i

i y,1minysgny (5.5)

In short the simulation steps to be performed are the following:

i) Obtain the magnetometer measurement of the geomagnetic

field vector B.

ii) Compute the attitude and attitude error (from desired attitude)

iii) Maps the entire fuzzy input variable into their fuzzy set.

iv) Evaluate the fuzzy rule to get Mx for generate the required

torque.

v) Compute attitude and attitude error for the influence made on

satellite dynamics due to generated torque.

vi) Repeat steps 2, 3 and 4 till attitude error is zero.

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The optimality and performance (control response time) of the

controller are therefore solely dependent on the choice of membership

function.

5.4 SIMULATION RESULTS

Attitude control is an area of concern for a variety of applications

ranging from autonomous vehicles to spacecraft. Accurate attitude control of

spacecraft and other vehicles is crucial to the success of their mission.

However, due to the high degree of non-linearity, it is very difficult to ensure

performance or stability without a significant amount of effort and luck. This

work provides a new methodology for developing controllers in a faster and

more cost effective manner. Because of its flexibility or generic nature the

fuzzy attitude controller, which utilizes fuzzy logic and angular estimate

feedback, can be easily ported to other systems.

The performance of the proposed fuzzy controller is evaluated

through simulation. Also disturbance torque is added to the simulation, to

show the robustness of the fuzzy controller.

5.4.1 Satellite Configuration

The proposed micro-satellite has only diagonal elements in inertial

matrix and there are no other elements in off diagonal matrix.

442010.1000338694.1000255427.1

I (5.6)

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The moment of inertia along x, y and z-axes are given below:

Ix = 1.255427 kg-m2

Iy = 1.338694 kg-m2

Iz = 1.442010 kg-m2

5.4.2 Initial Rates

Since the beginning of each simulation, an initial angular rate of

6 degree per second for each axis is maintained and these rates are decided

according to the specification given by the launch vehicle team.

ωx = 6 deg/sec

ωy = 6 deg/sec

ωz = 6 deg/sec

5.4.3 Detumbling With Initial Spin Up Response

The response of the controller is shown in the Figure 5.7. This

shows that the transverse rates are found to be decaying within

1.5 orbits. An orbit takes a time period of 6500 seconds. Also the desired spin

rate of 8±0.5 rpm is achieved in five orbits with a torquer capacity of 9 Am2

in spin axis and 4.5 Am2 in transverse axes respectively. The value of gain in

rate error equation is found to be 14.9 through numerous simulation trials. A

manual tuning is employed to find such gain. A compromise between the

shape, size of membership function and the rate error gain is made to obtain a

better performance, even in presence of erroneous magnetometer

measurements.

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Figure 5.7 Detumbling with initial spin up fuzzy controller response

5.4.4 Spin Rate Controller Response

The significance of spin rate controller is to maintain angular rate

about the spin axis within a specified band (80.5rpm), even in the presence

of external disturbances. Since spin stabilized satellites necessitates a constant

angular rate about spin axis for its own stabilization. The performance of spin

rate controller in both, spin up mode and spin down mode are given in

Figures 5.8 and 5.9 respectively.

z (

deg/

sec)

x (de

g/se

c)

y (

deg/

sec)

Time (in orbits)

Time (in orbits)

Time (in orbits)

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Figure 5.8 Spin rate controller response -spin up

Figure 5.9 Spin rate controller response -spin down

z (

deg/

sec)

x (de

g/se

c)

y (

deg/

sec)

x (de

g/se

c)

z (

deg/

sec)

y (de

g/se

c)

Time (in orbits)

Time (in orbits)

Time (in orbits)

Time (in orbits)

Time (in orbits)

Time (in orbits)

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5.4.5 Spin Axis Orientation Controller Response

The fuzzy controller response for spin axis orientation controller in

orbit normal position is given in Figure 5.10. The input is taken in the form of

samples. Two samples are taken per second so the settling time is obtained by

dividing the sample by 2.

Figure 5.10 Spin axis orientation controller response – orbit normal

5.4.6 Effects of Disturbances

The universe is not an ideal place and things like uncertainties or

disturbances are constantly present. Disturbances in space come in a variety

of forms. Currently there are 6,650 known objects (debris) weighing

approximately 3,000,000 kilograms orbiting the earth. The debris are

composed of paint flecks, refuse, waste, solid rocket effluent, etc. There are

also millions of smaller (fewer than 10 cm) untraceable objects with velocities

Dec

linat

ion

(deg

)

Samples

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of the order of 10 km/sec. Other disturbances are due to magnetic flux, solar

winds and solar radiation pressure torque (approximately 1 10-5 Nm).

In order to test the robustness and flexibility of the proposed

scheme, the controller has been simulated offline with various external

environmental disturbances. Through out the simulation, initial conditions

were set to the same value as mentioned earlier. Magnitude of various

disturbances: aerodynamic drag (3.23810-12Nm), gravity gradient torque

(3.01610-7 N-m) and solar pressure (2.78310-7Nm) are included in the

simulation, which still shows similar results. This explains flexibility and

robustness of the controller.

The response of the controller to a disturbance is obtained as shown

in the Figure 5.11 in the detumbling with initial spin up mode.

Figure 5.11 Detumbling with initial spin up fuzzy controller response

along with external disturbance

z (

deg/

sec)

x (de

g/se

c)

y (

deg/

sec)

Time (in orbits)

Time (in orbits)

Time (in orbits)

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The response of the controller to a disturbance is obtained for spin

rate controller in the spin up and spin down modes as shown in the

Figures 5.12 and 5.13. The SAOC controller response for declination to

reduce from 140 deg to 0 deg is given in Figure 5.14.

Figure 5.12 Spin rate controller response-spin up along with external

disturbance

z (

deg/

sec)

x (de

g/se

c)

y (

deg/

sec)

Time (in orbits)

Time (in orbits)

Time (in orbits)

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Figure 5.13 Spin rate controller response-spin down along with external

disturbance

Figure 5.14 Spin axis orientation control – orbit normal along with

external disturbance

z (

deg/

sec)

x (de

g/se

c)

y (

deg/

sec)

Time (in orbits)

Time (in orbits)

Time (in orbits)

Dec

linat

ion

(deg

)

Samples

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The response of the controller to an external disturbance up to many

a times of the worst case disturbance torque has been simulated. The response

of the controller is almost the same and the controller is insensitive to external

disturbances of the order of 20 times the worst-case external disturbance

torque. This sensitivity analysis proves the robustness of the fuzzy controller.

5.5 CONCLUSION

In this chapter the satellite attitude controller was designed using

fuzzy controller for both the detumbling with initial spin up and spin rate

control modes. Here the application of fuzzy logic to the proposed satellite

(ANUSAT) is explained. The design is implemented in Matlab using fuzzy

toolbox and the responses are plotted for various attitude modes. The settling

time taken for detumbling with initial spin-up is around 37,202 s and that of

SAOC is 28,525 s. The robustness of the controller has been proved by

simulating the response of the satellite’s attitude in the presence of external

disturbances. The disturbances for which the sensitivity of the controller is

tested are for many times of the order of worst cases external torques. The

responses prove that the controller is less sensitive to twenty times of external

disturbances torques. The results are given in the form of responses and were

plotted using Matlab. The fuzzy controller was also simulated for three

torquer capacities to analyze its effect on the satellite attitude. Figures 5.15

through 5.18 shows the effect of torquer capacities on the settling time for

various modes of the attitude control system.

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Figure 5.15 Effect of torquer capacity on settling time – detumbling with

initial spin up

Figure 5.16 Effect of torquer capacity on settling time – spin rate control

(spin up)

Sett

ling

time

(sec

)

Magnetic torquer capacity (Am2)

Settl

ing

time

(sec

)

Magnetic torquer capacity (Am2)

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Figure 5.17 Effect of torquer capacity on settling time – spin rate control

(spin down)

Figure 5.18 Effect of torquer capacity on settling time – spin axis

orientation control (orbit normal)

Settl

ing

time

(sec

)

Magnetic torquer capacity (Am2)

Settl

ing

time

(sec

)

Magnetic torquer capacity (Am2)

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The inference from the simulation results are that the time taken for

settling increases as the torquer capacity is decreased and decreases if the

torquer capacity is increased. This shows that the use of fuzzy control, an

intelligent control, does not have any impact on the power system of the

satellite and is just a control technique effective in improving the response of

the system. The time taken for settling can be improved by optimizing the

parameters of the fuzzy controller and can be an effective control technique

than the conventional controllers.