Optimized Fuzzy-Quaternion Attitude control of ...

AIAA 2016-2552 SpaceOps Conferences 16-20 May 2016, Daejeon, Korea 14th International Conference on Space Operations

Optimized Fuzzy-Quaternion Attitude control of Satellite in Large maneuver Hossein Ghadiri1 and Mohammad Sadeghi2 Space Research Institute, Tehran, Iran, Zip Code Alireza Abbaspour3 Florida International University, Miami, Florida, 33172, USA and

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

Reza Esmaelzadeh 4 Technology Development University Holding, Tehran, Iran In this paper a fuzzy-quaternion controller is designed for attitude control of a satellite, then the fuzzy memberships are tuned in an intelligent way by using particle swarm optimization (PSO) algorithm. Due to the satellite nonlinear behavior, classic methodologies cannot control satellite. The simulation result show that the designed controller can accurately control the satellite attitude in severe maneuvers. To evaluate the controller robustness in presence of uncertainties, 20 percent uncertainties were considered in inertias of momentum through the simulations. The simulation results show that the optimized fuzzy logic controller (OFLC) can control the satellite in large maneuvers in desirable time. In addition, the simulation results demonstrated that the proposed design is robust against uncertainties and have quite better performance than quaternion proportional-derivative (PD) controller in satellite motion control.

Nomenclature AE α ANFIS CoA e FLC Kdi Kpi MFs PD PSO Ti θ

= = = = = = = = = = = = =

Direction cosine error matrix Angle between primary Euler vector and its latter (angle error) Adaptive network based fuzzy inference system Center of area Euler axis Fuzzy logic controller Derivative control gain Proportional control gain Membership functions Proportional- Derivative control Particle swarm optimization Torque Principal rotation angle

I. Introduction

T

HE Main goal of satellite attitude control is providing the capability of large and fast maneuvers with enough accuracy and sufficient stability. The nonlinear system dynamic, huge moments, and actuators saturation are the major difficulties of controlling such systems. Since the linear model is only valid in small maneuvers, nonlinear dynamic equation must be considered for large maneuvers. PD controllers are the most common controllers which are used in satellite control. But according to the problems mentioned above, using this controller for large maneuvers may severely degrade the system performance. Long settling time and large overshoot are the main problems of attitude control by using PD controllers. Also the PD PhD student, Space Research Institute, Tehran, Iran ([email protected]). PhD student, Space Research Institute, Tehran, Iran (mohammed.sadeghi.mut @gmail.com). 3 PhD student, Electrical & Computer Eng., Florida International University, Miami, FL, USA ([email protected]). 4 Director of Educational Department, Technology Development University Holding, Tehran, Iran ([email protected]). 1 1

2

American Institute of Aeronautics and Astronautics

Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

controller cannot control the unpredicted nonlinear properties of dynamic model, which may degrade the control performance critically. To find a more robust controller to tolerate these uncertainties, various method such as fuzzycompensator, fuzzy self-gain-tuner, fuzzy-PD controller and adaptive neural networks were introduced [1-6]. In these papers, the proportional and derivative gains of the fuzzy-PD controller were modified by the two-input fuzzy controller. Guangzheng et al., introduced a fuzzy gain tuner based on that adjusts the two parameters of PD controller on-line [7]. Complementary fuzzy logic method was designed to compensate the command tracking error [2, 8]. Despite the fact that gains tuning helps to improve the PD controller’s performance, these controllers are still vulnerable in large maneuvers. To overcome this problem, quaternion PD control is introduced. Quaternion PD control has some advantages such as fast command tracking and less computation time, but it also has some deficiencies such as weak performance in presence of uncertainty and actuator delays in large maneuvers. In this paper, fuzzy logic is used to make the quaternion PD control more robust against uncertainties and actuator delays. Unlike the simplicity in obtaining the fuzzy rules, obtaining optimal fuzzy Membership Functions (MFs) is quite difficult and time consuming. To tackle this problem, various techniques have been investigated for MFs tuning. An adaptive network based fuzzy inference system (ANFIS) was introduced in [9], where the adaptive neural network was used to learn the mapping between the inputs and outputs, and a Sugeno-type of fuzzy system was generated based on the neural network. In another approach, a quantum neural network was used to learn the space data of a Tagaki-Sugeno FLC [10]. Using optimization algorithms for tuning the membership functions have also received great attention [11-15]. Pratihar et. al introduced genetic algorithm for off-line tuning of the fuzzy membership functions of a motion planning controller for a mobile robot in the presence of moving obstacles [11]. An automated design of a fuzzy controller using genetic algorithms was introduced to control the wall-following behavior in a mobile robot [12]. A teaching–learning-based optimization (TLBO) is used to tune the membership functions algorithm to control a power network system and improve the THD and voltage sag indices of a sensitive load in the network [13]. Emotional learning algorithm is used to tune and optimize the fuzzy membership controller in order to control the load frequency for micro grids [14]. In the current work, the particle swarm optimization (PSO) technique is used to optimize the MFs’ parameters. PSO algorithm received a lot of attention in various application such as in image processing [16], and in system modeling [17]. The PSO algorithm is a population based stochastic optimization technique developed by Eberhart and Kennedy [18, 19]. Due to the PSO fast convergence, global optimality, and simplicity in its application, this algorithm is used to tune the fuzzy MFs in this work. The paper is organized as follows: in section 2, a brief overview of mathematical model of satellite is provided, whereas the attitude control, and combination of fuzzy and PSO explained in section 3, while the numerical simulation has been done to evaluate the proposed control design in section 4. Finally the research conclusion is available on section 5.

II. Satellite Mathematical Model Satellite equation of motion is defined by two sets of equations. One of them is kinematic equations that include the angular velocities and satellite attitude; the other one is a set of kinetics equations including the rate of angular velocities with respect of external and internal moments. The kinematics describes the body’s orientation in space which is obtained through integration of the angular velocity. The kinematic equations are expressed by separate integrations of the attitude quaternion vector and its scalar parts [20]. 3 2 1   q1   0  q1    q  0 1 2   q2   2  1  3 . (1)  q3  2  2 1 0 3   q3        q4   1 2 3 0   q4  where the attitude quaternion formula can be derived from the Euler axis, e, and principal rotation angle, θ, as follows:   q1  e1sin   , 2

  q2  e2 sin   , 2 2

American Institute of Aeronautics and Astronautics

(2)

  q3  e3 sin   , 2   q4  cos   . 2 while the Kinetic Equations can be expressed as follow:



 



Tx  hx   y hz  z hy   y hwz  z hwy , Ty  hy  z hx  x hz   z hwx  x hwz  ,



 

(3)



Tz  hz  x hy   y hx  x hwy   y hwx .

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

where T = [Tx, Ty, Tz] is the matrix of environmental and control moments. In the current work four reactions wheel are used as satellite control input.

III. Control System The attitude control guarantees the desired performance of satellite. In big maneuvers which are used for the stereo imaging, accurate rate control is really essential. On the other hand, there are some limitation in the satellite energy sources, because its energy is obtained from the sun beams and according to solar cell efficiencies, there is a restriction for energy consumption. Thus, the desired control system should provide the correct rate of commands while using minimum energy. The overall view of proposed control design is depicted on Fig. 1, while the fuzzy quaternion diagram is shown on Fig.2. In this design, the fuzzy controller input are the quaternion error and satellite angular velocities.

Fig. 1. Overall view of control structure

A. Design of Fuzzy-Quaternion Controller This fuzzy PD controller design is based on the desired parameter control error and the derivative the desired parameter control error. Therefore, the fuzzy rules are defined considering this assumption and the experiences obtained from the dynamic responses. In order to obtain the best results, Minimum Mamdani (And method) is used to design the fuzzy controller. This inference engine received the most of attention among researcher due to its low computational load which makes it suitable for real time applications [3]. As it can be seen on Fig.2, three fuzzy controllers are designed based on the mentioned method. In order to design a fuzzy controller, firstly, the shape and number of MFs for should be defined. It is already found that complex forms of fuzzy MFs are not suitable for fuzzy tuning algorithm; moreover, complicated fuzzy membership functions are not suitable for fast computation (real time application)[21]. For these reasons, in this work we selected triangular shapes for the MFs. Seven MFs is selected for the proposed fuzzy design and the selected MFs are shown on Fig.3; where, the linguistic values are abbreviations of {Large Negative, Medium Negative…Large Positive}, respectively.

3

American Institute of Aeronautics and Astronautics

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

Fig. 2. Fuzzy-quaternion controller

Fig.3.

Membership functions

In the second step of fuzzy controller design, the fuzzy rules should be derived. As we previously mentioned, the proposed fuzzy rules are obtained based on the designer experience and PD controller assumption. In the fuzzy PD controller two parameters should be controlled, one is the error of the desired parameter which is denoted by e and its variation ratio denoted by e . In the current work, AND method is used to obtain rules. Seven MFs is selected for this design, therefore, 49 rules should be defined that can be seen on Table.1. Table 1. Fuzzy Laws e

NB

NM

NS

ZE

PS

PM

PB

NB NM

ZE NS

PS ZE

PS PS

PM PS

PM PM

PB PM

PB PB

NS

NS

NS

ZE

PS

PS

PM

PM

ZE PS PM PB

NM NM NB NB

NS NM NM NB

NS NS NM NM

ZE NS NS NM

PS ZE NS NS

PS PS ZE NS

PM PS PS ZE

𝒆̇

The third step in this design is to tune the MFs of the fuzzy-PD quaternion controller. Tuning these MFs is usually done by trial and error which is a time consuming and frustrating process; thus, we introduced PSO algorithm to tune MFs based on our defined objectives. B. PSO Algorithm Particle Swarm Optimization (PSO) algorithm is a stochastic optimization algorithm based on a swarm of particles searching over the space. In this optimization algorithm, each of particles position and velocity are used to solve the 4

American Institute of Aeronautics and Astronautics

optimization problem. The velocities of these particles are modified based on a set of rules that govern the swarm dynamic. These rules can be obtained based on the experience of each particle and its neighbors in the network structure in the swarm. Adding the velocity to the current position modifies the position of each particle. While the particles are searching in the defined environment, fitness values are applied to particles the based on their closeness to the desired objective. This process is repeated to obtain the best global position and the particles will eventually concentrate around the best solution. The global optimality of the PSO searching has direct relation with the number of particles; therefore, increasing the particle number will increase the chance of finding the global optimal point. The particle velocity and behavior, size of searching space and the inertia weight are the other factors that can influence on finding the global optimal point. The velocity related to each particle can be calculated with the following equations [22]. (4) vi (k  1)  w.vi ( k )  c1 .r1 (k )( xg  xi (k ))  c2 .r2 (k )( xip  xi (k ))

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

xi (k  1)  xi ( k )  vi ( k  1)

(5)

where vi(k) is the particle i velocity at time step k , w is particle inertia weight, xg is the particle best global position at time step k, xp is the best experience particle has had up to time step k, xi(k) is the current location of particle i , and c1 , c2 are constants usually equal each to other, r1 , r2 are random numbers within [0, 1] those represent random fiction [23]. In order to reduce the searching time, the searching space is limited to be within a certain range of vimin≤vi≤vimax. .

A cost function is needed to define the priorities in the optimization process J  xQxT  uRuT dt.

(6)

where x is the tracking error , u is the torques of reaction wheels and Q and R are the coefficient matrices that are choose by designer to obtain desired performance.

IV. Simulation Results In this study, a new control design for satellite attitude control is introduced. The proposed control strategy uses PSO algorithm to optimize the designed fuzzy-PD controller for the system. This section provides the simulation results and analyzing factors to demonstrate the advantages of the proposed controller. Several aspects can be considered in analyzing of control performance. In proposed control system design, the system attitude is the most important factor in the controller. Second factor for investigation of control laws is  angle error.  is attitude maneuver about Euler axis of rotation. This parameter can be defined as[20]: 1 cos    trace  AE   1 . 2





(7)

where AE is the direction cosine error matrix, detailed information about the this matrix can be found in [20]. According to (7), the desired condition is reached when the primary vector and its latter coincided on each other. EULERINT is regarded as a criterion for the evaluation of attitude control and it is defined as below: (8) EULERINT  a dt. The third factor is the investigation of system robustness against uncertainties. 20 percent of inertia moment considered as uncertainty in the system. The fourth factor which is used to analyze the control performance is control effort. Four reaction wheels are used to control the satellite dynamic; therefore, control effort can be expressed as sum of integral squares of each reaction wheel. 4

Control Effort   Ti 2 dt . i 1

(9)

After introducing the considered factors for the control performance analysis, the proposed optimal fuzzy logic controller (OFLC) controller is compared with quaternion PD controller (PDC) through the simulation in MATLAB SIMULINK software is done. The proportional and derivative control gains are obtained from pole-placement method as follows: (10) kdi  2n , k pi  n 2 , i  1, 2,..., n 5

American Institute of Aeronautics and Astronautics

Where  is the damping ratio, n is the undamped natural frequency. The proportional and derivative control gains

obtained by assuming   1 and n  1

I  diag 1000,500,700 ,



K p  diag 1000,500,700 ,



Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

Kd  diag  2000,1000,1400 .  The results of simulations are depicted on Fig. 4-8. The results of attitude control performance are shown on Fig.46. As it is already mentioned, minimum energy consuming is another important factor for analyzing control performance, so the result of control effort comparison is shown on Fig. 7. EULERINT diagram of these two controllers are compared in Fig. 8. In the current simulation the system performance is tested in presence of 20 percent uncertainties in inertia moment. The effect of uncertainties in control systems are shown by dashed lines in following figures. Figs. 4-6 show that OFLC controller has better performance than quaternion PD method. It has less tracking error; moreover, it is more robust in presence of uncertainties. According to Fig.7, OFLC has less control effort that leads to reduce energy consumption. Less energy consumption is an important advantage in satellite system that has a limited access to energy sources. As it can be seen the OFLC controller controls the satellite attitude in less than 15 seconds without any chattering, while the quaternion PD controller is incapable of accurate control during large maneuvers. Fig. 8 shows that EULERINT of OFLC controller is less than EULERINT of quaternion PD controller. This means that by using OFLC controller, satellite experiences less angular distance.

Maneuver

Maneuver 1 Maneuver 2 Maneuver 3 Maneuver 4

Table 2. Desired maneuver satellite Explanation

[Roll , Pitch, Yaw] Degree From [0 0 0] [60 60 0] [0 0 0] [60 -60 0]

Nadir to point 1 in 50 s point 1 to Nadir in 50 s Nadir to point 2 in 50 s point 2 to Nadir in 50 s

Fig. 4. Attitude along X axis during the maneuver.

Fig. 5. Attitude along Y axis during the maneuver. 6

American Institute of Aeronautics and Astronautics

To [60 60 0] [0 0 0] [60 -60 0] [0 0 0]

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

Fig.6

Attitude along Z axis during the maneuver.

Fig.7. Control Effort of Fuzzy-Quaternion Controller

Fig.8. Displacement angle during the maneuver.

V. Conclusion At the present study, an optimal fuzzy-quaternion controller is designed for a satellite which uses four reaction wheels setup. The fuzzy rules and membership function are obtained based on designer experience, and then the membership functions are optimized by PSO algorithm. To analyze the control performance of the introduced method, some analyze factor are defined. Simulation results show that OFLC controller can control the satellite in large maneuvers in acceptable time, and its performance such as tracking accuracy and robustness to uncertainties is far better than conventional controller like quaternion or classic control. Furthermore, the simulation results show that the proposed design has less control effort which leads to less energy consumption.

References [1] [2] [3] [4] [5] [6]

B.-S. Chen, C.-S. Wu, and Y.-W. Jan, "Adaptive fuzzy mixed H 2/H∞ attitude control of spacecraft," Aerospace and Electronic Systems, IEEE Transactions on, vol. 36, pp. 1343-1359, 2000. P. Guan, X. Liu, F. Lara-Rosano, and J. Chen, "Adaptive fuzzy attitude control of satellite based on linearization," in American Control Conference, 2004. Proceedings of the 2004, 2004, pp. 1091-1096. A. Abaspour, M. Sadeghi, and H. Sadati, "Using fuzzy logic in dynamic inversion flight controller with considering uncertainties," in 13th Iranian Conference on Fuzzy Systems (IFSC), 2013. J.-m. Zheng, S.-d. Zhao, and S.-g. Wei, "Application of self-tuning fuzzy PID controller for a SRM direct drive volume control hydraulic press," Control Engineering Practice, vol. 17, pp. 1398-1404, 2009. A. Abaspour, S. H. Sadati, and M. Sadeghi, "Nonlinear optimized adaptive trajectory control of helicopter," Control Theory and Technology, vol. 13, pp. 297-310, 2015. M. R. Soltanpour, M. H. Khooban, and M. R. Khalghani, "An optimal and intelligent control strategy for a class of nonlinear systems: adaptive fuzzy sliding mode," Journal of Vibration and Control, vol. 22, pp. 159175, 2016.

7

American Institute of Aeronautics and Astronautics

[7]

[8] [9] [10] [11] [12] [13]

Downloaded by 85.198.8.149 on May 25, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-2552

[14]

[15] [16] [17]

[18] [19] [20] [21] [22] [23]

P. Guangzheng, W. Xuesong, and X. Yang, "Study on fuzzy PD control of planar two-link flexible manipulator," in TENCON'02. Proceedings. 2002 IEEE Region 10 Conference on Computers, Communications, Control and Power Engineering, 2002, pp. 1542-1545. C.-H. Cheng, S.-L. Shu, and P.-J. Cheng, "Attitude control of a satellite using fuzzy controllers," Expert Systems with Applications, vol. 36, pp. 6613-6620, 2009. J.-S. R. Jang, "ANFIS: adaptive-network-based fuzzy inference system," Systems, Man and Cybernetics, IEEE Transactions on, vol. 23, pp. 665-685, 1993. R. Fullér, "Neural fuzzy systems," 1995. D. K. Pratihar, K. Deb, and A. Ghosh, "A genetic-fuzzy approach for mobile robot navigation among moving obstacles," International Journal of Approximate Reasoning, vol. 20, pp. 145-172, 1999. M. Mucientes, D. L. Moreno, A. Bugarín, and S. Barro, "Design of a fuzzy controller in mobile robotics using genetic algorithms," Applied Soft Computing, vol. 7, pp. 540-546, 2007. M. R. Khalghani and M. H. Khooban, "A novel self-tuning control method based on regulated bi-objective emotional learning controller's structure with TLBO algorithm to control DVR compensator," Applied Soft Computing, vol. 24, pp. 912-922, 2014. M. R. Khalghani, M. H. Khooban, E. Mahboubi-Moghaddam, N. Vafamand, and M. Goodarzi, "A self-tuning load frequency control strategy for microgrids: Human brain emotional learning," International Journal of Electrical Power & Energy Systems, vol. 75, pp. 311-319, 2016. A. Abaspour, N. T. Parsa, and M. Sadeghi, "A New Feedback Linearization-NSGA-II based Control Design for PEM Fuel Cell," International Journal of Computer Applications, vol. 97, 2014. M. G. Omran, A. P. Engelbrecht, and A. Salman, "Particle swarm optimization for pattern recognition and image processing," in Swarm intelligence in data mining, ed: Springer, 2006, pp. 125-151. N. Kwok, Q. Ha, T. Nguyen, J. Li, and B. Samali, "A novel hysteretic model for magnetorheological fluid dampers and parameter identification using particle swarm optimization," Sensors and Actuators A: Physical, vol. 132, pp. 441-451, 2006. J. Kennedy, J. F. Kennedy, R. C. Eberhart, and Y. Shi, Swarm intelligence: Morgan Kaufmann, 2001. M. Clerc and J. Kennedy, "The particle swarm-explosion, stability, and convergence in a multidimensional complex space," Evolutionary Computation, IEEE Transactions on, vol. 6, pp. 58-73, 2002. M. J. Sidi, Spacecraft dynamics and control: a practical engineering approach vol. 7: Cambridge university press, 1997. C.-L. Lin, H.-Z. Hung, Y.-Y. Chen, and B.-S. Chen, "Development of an integrated fuzzy-logic-based missile guidance law against high speed target," Fuzzy Systems, IEEE Transactions on, vol. 12, pp. 157-169, 2004. R. C. Eberhart and Y. Shi, "Comparing inertia weights and constriction factors in particle swarm optimization," in Evolutionary Computation, 2000. Proceedings of the 2000 Congress on, 2000, pp. 84-88. C.-J. Lin and S.-J. Hong, "The design of neuro-fuzzy networks using particle swarm optimization and recursive singular value decomposition," Neurocomputing, vol. 71, pp. 297-310, 2007.

8

American Institute of Aeronautics and Astronautics