CHAPTER 5 FUZZY LOGIC FOR ATTITUDE CONTROL

Transcription

1 104 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).

2 105 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 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

3 106 control laws, one for each magneto-torquer (M x, M y, and M z 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 Inputs A total of six inputs were used, they are as given below: X 1 = X 2 = X 3 = X 4 = X 5 = X 6 = ω x, angular rate about x axis ω y, angular rate about y axis k (ω z - ω spin ), rate error N x, estimated control torque about x axis N y, estimated control torque about y axis N z, estimated control torque about z axis 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.: X 1 m P (X 1 ) and X 4 m N (X 4 ) (5.1)

4 107 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 X 1, X 2, X 3 Figure 5.3 Membership function for variables X 4, X 5 Figure 5.4 Membership function for variable X 6

5 108 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 M z controller. A set of intuitive rules is used to describe the M z controller. In the initial spin up mode the polarity of the constant current that is passed to M x and M y coils are solely determined from the sign and magnitude of the transverse earth s magnetic field component Rules Rules for spin axis controller and transverse axes controller are given in the Tables 5.1 and 5.2 respectively. Table 5.1 M z fuzzy variable control rule Rule X 1 X 2 X 4 X 5 U R 1 P - P R 2 P - N R 3 N - N R 4 N - N R 5 - P - P - 1 R 6 - P - N + 1 R 7 - N - P + 1 R 8 - N - N - 1

6 109 Table 5.2 M x, M y fuzzy variable control rule Rule X 1 X 2 X 3 X 4 X 5 X 6 U R 1 P - - P - Z -1 R 2 P - - N - Z +1 R 3 N - - P - Z +1 R 4 N - - N - Z - 1 R 5 - P - - P Z - 1 R 6 - P - - N Z + 1 R 7 - N - - P Z + 1 R 8 - N - - N Z - 1 R P - - P +1 R P - - N - 1 R N - - P -1 R 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: R 1 : µ 1 = m P (X 1 ).m P (X 2 ).m Z (X 3 ) (5.2) where m i are the membership functions. The truth-value obtained is then used to scale the output: R 1 : y 1 = 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:

7 110 N N i i i y (y ) sgn y min 1, y (5.4) i 1 i 1 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 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 M y = (sign of B x ) Current polarity M x = (sign of B y ) 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.

8 111 B y SISO FLC M x B x SISO FLC M y 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 B y and B x respectively. For increasing the rate, fuzzy logic controller gives the same sign as its input magnetic field for M y controller, while the polarity given by M x 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 B x is negative then M y is negative If B x is positive then M y is positive. Also for M x controller: If B y is negative then M x is positive If B y is positive then M x is negative Similarly for spinning down the rules are slightly modified. In this M x 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

9 112 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 B y, B x 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 X1 B y X2 B x Output Y 1 Torque value along y axis Y 2 Torque value along z axis Thirteen membership functions (2 trapezoidal membership, 11 triangular membership functions) are used for inputs B y and B x. 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

10 113 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: N i i y sgn y min 1, y (5.5) i 1 In short the simulation steps to be performed are the following: i) Obtain the magnetometer measurement of the geomagnetic field vector B. ii) iii) Compute the attitude and attitude error (from desired attitude) Maps the entire fuzzy input variable into their fuzzy set. iv) Evaluate the fuzzy rule to get M x 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.

11 114 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 Satellite Configuration The proposed micro-satellite has only diagonal elements in inertial matrix and there are no other elements in off diagonal matrix I (5.6)

12 115 The moment of inertia along x, y and z-axes are given below: I x = kg-m 2 I y = kg-m 2 I z = kg-m 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. ω x = 6 deg/sec ω y = 6 deg/sec ω z = 6 deg/sec 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 Am 2 in spin axis and 4.5 Am 2 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.

13 116 z (deg/sec) x (deg/sec) y (deg/sec) Figure 5.7 Detumbling with initial spin up fuzzy controller response Spin Rate Controller Response The significance of spin rate controller is to maintain angular rate about the spin axis within a specified band (8 0.5rpm), 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.

14 117 z (deg/sec) x (deg/sec) y (deg/sec) Figure 5.8 Spin rate controller response -spin up x (deg/sec) y (deg/sec) z (deg/sec) Figure 5.9 Spin rate controller response -spin down

15 Spin Axis Orientation Controller Response The fuzzy controller response for spin axis orientation controller in orbit normal position is given in Figure 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. Declination (deg) Samples Figure 5.10 Spin axis orientation controller response orbit normal 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

16 119 of the order of 10 km/sec. Other disturbances are due to magnetic flux, solar winds and solar radiation pressure torque (approximately 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 ( Nm), gravity gradient torque ( N-m) and solar pressure ( Nm) 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. z (deg/sec) x (deg/sec) y (deg/sec) Figure 5.11 Detumbling with initial spin up fuzzy controller response along with external disturbance

17 120 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 The SAOC controller response for declination to reduce from 140 deg to 0 deg is given in Figure z (deg/sec) x (deg/sec) y (deg/sec) Figure 5.12 Spin rate controller response-spin up along with external disturbance

18 121 z (deg/sec) x (deg/sec) y (deg/sec) Figure 5.13 Spin rate controller response-spin down along with external disturbance Declination (deg) Samples Figure 5.14 Spin axis orientation control orbit normal along with external disturbance

19 122 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.

20 123 Settling time (sec) Magnetic torquer capacity (Am 2 ) Figure 5.15 Effect of torquer capacity on settling time detumbling with initial spin up Settling time (sec) Figure 5.16 Effect of torquer capacity on settling time spin rate control (spin up) Magnetic torquer capacity (Am 2 )

21 124 Settling time (sec) Magnetic torquer capacity (Am 2 ) Figure 5.17 Effect of torquer capacity on settling time spin rate control (spin down) Settling time (sec) Magnetic torquer capacity (Am 2 ) Figure 5.18 Effect of torquer capacity on settling time spin axis orientation control (orbit normal)

22 125 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.