An Efficient Method to Control the Amplitude of The Limit Cycle in Satellite Attitude Control System

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Amirkabir University of Technology (Tehran Polytechnic) Vol. 48, No. 1, Spring 2016, pp An Efficient Method to Control the Amplitude of The Limit Cycle in Satellite Attitude Control System M. Sabet
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Amirkabir University of Technology (Tehran Polytechnic) Vol. 48, No. 1, Spring 2016, pp An Efficient Method to Control the Amplitude of The Limit Cycle in Satellite Attitude Control System M. Sabet Rasekh 1, S. K. Y. Nikravesh 2* and N. Ghahramani 3 1- PhD Educated, Electrical and Electronic Engineering Faculty, Amirkabir University of Technology, Tehran, Iran 2- Professor, Electrical and Electronic Engineering Faculty, Amirkabir University of Technology, Tehran, Iran 3-Associate Professor, Control Group Malek-Ashtar University of Technology, Tehran, Iran Received 26 July 2015, Accepted 3 January, 2016 ABSTRACT In this paper, an efficient method is presented to control the attitude of a satellite with ON-OFF actuator. The main objective of this method is to control the amplitude of the limit cycle which commonly appears in the steady state of such systems, while simultaneously by consideration of real actuator constraints, it reduces the fuel consumption of system. The proposed method is a combination of a command modifier (which is based on the set point and the required accuracy of pointing, by means of optimization, it calculates the desired limit cycle), a phase plane controller (PPC) and a compensator (to compensate for the real constraints of the actuator). The effectiveness and outperformance of the proposed method is approved in comparison with the previous methods of attitude control problem through the closed loop simulation and the stability and robustness of the closed loop system is fully analyzed and illustrated by simulations. KEYWORDS Limit Cycle, On/Off Actuator, Phase Plane Controller, Attitude Control, Robustness. Corresponding Author, Vol. 48, No. 1, Spring M. Sabet Rasekh, S. K. Y. Nikravesh and N. Ghahramani 1. INTRODUCTION Limit cycle is one of the common characteristics of nonlinear dynamical systems which generally arises in the systems with: uncertain transport delay, ON-OFF actuators, and friction [1-3]. The use of ON-OFF actuators is very common in a large amount of systems such as satellite attitude control systems. In situations in which the amplitude of external noise and disturbance are large, or high control effort is needed, ON-OFF actuators are commonly used. Many of ON-OFF actuators used in attitude control systems are of reaction thruster types. In these actuators, exiting of the gas particles from nuzzles causes a reaction force on system [4]. Attitude control systems, which have ON-OFF actuators, generally converge to a stable limit cycle in their steady state. In the literature, two main reasons are presented for causing this limit cycle. The first reason relates to physical characteristics of ON-OFF actuators. These actuators often have a minimum on-time which means that after the actuator turns ON, there will be no possibility to turn it off in time (thruster valves must stay open over a finite time interval). Therefore, the energy delivered to the system unlike to the case of using proportional actuators- has a minimum positive value. This cancels the possibility of reaching to the equilibrium state and staying there [2, 5]. The second reason, which causes the limit cycle, is due to the command system. Generally, commanding to the ON-OFF actuator, needs some intermediate systems (such as modulator or bangbang switches) to change the continuous control command generated by the controller to the ON-OFF command. These intermediate systems generally have a minimum duty cycle or minimum on-time, like the minimum on-time of the actuator, causes a limit cycle. [6] Obviously, in any satellite attitude control system, a limit cycle is not desirable, because the limit cycle amplitude has an immediate effect on the satellite pointing error1.it means that by increasing the amplitude of the limit cycle, the pointing error will increase. Furthermore, the ON-OFF actuators need propellants; hence, in a limit cycle condition, satellite fuel wastes increases rapidly. Thus, for a certain application, more fuel is needed. The limit cycle frequency and number of pulses in each cycle directly affects loss of fuel. In other words, with increasing frequency of the limit cycle, fuel consumption 1 Pointing error in each control channel is defined as, where is the command angle and is the angular position of satellite in that channel. 12 will increase. The Limit cycle also appears in other attitude control systems such as missiles [7]. To reach a stable equilibrium state, the limit cycle, if possible, should be suppressed. In some researches, some proportional actuators (such as momentum exchange devise as reaction wheels), in addition to thrusters, are used in order to suppress the limit cycle [8]. Obviously, additional proportional actuator makes it possible to eliminate the limit cycle; nevertheless, simultaneously, it requires to add drivers, power supply system and controller to the system; this, in turn, leads to an increase in the system weight by occupying a considerable part of the satellite volume, which is not allowed in some applications. Without proportional actuators, suppressing limit cycle is impossible. Therefore, to avoid the negative role of the limit cycle, its amplitude as well as its frequency should be reduced [2, 9-10]. Of course, it is hard to decrease simultaneously the limit cycle amplitude and its frequency, because they are usually in conflict, i.e. if the limit cycle amplitude reduces, its frequency will increase and vice versa. Hence, a tradeoff is needed between these two criteria [11]. There are two major methods to analyze and control the limit cycle of ON-OFF systems: 1) The describing function method, which is an approximate method based on semi-linearization. In this method, the nonlinear section (i.e. ON-OFF subsystem) is approximated by ratio of Furrier transformation of its output to its sinusoidal input with variable frequency [12-13]. 2) Phase plane method, in which, by analyzing the limit cycle in phase plane, proper command for attitude control system is generated [14-16]. The attitude control systems have different types of Limit cycles. The more desirable one is the 2-pulse limit cycle (a limit cycle that has a positive pulse, a negative pulse and two coast regions in each cycle). The 2-pulse limit cycle has the best condition from fuel consumption and robustness point of views [17-18]. In the best knowledge of the authors, despite many researches carried out on this area, there is not yet an effective method to conduct the system to a limit cycle with desired amplitude and frequency characteristics. In this paper an effective method for this problem is presented based on the optimization and phase plane method. This method guarantees the pointing accuracy of the satellite in addition to optimizing the fuel consumption Vol. 48, No. 1, Spring 2016 An Efficient Method to Control the Amplitude of The Limit Cycle in Satellite Attitude Control System and its robustness against satellite and actuator model variations. The rest of the paper is organized as follows. Section 2 fully presents the statement of the problem. In section 3, the system model components are derived. In section 4 the controller design procedure is fully described, followed by section 5, which presents the effectiveness of the proposed method through simulation. Finally, section 6 concludes the paper. 2. PROBLEM STATEMENT AND ASSUMPTIONS A satellite attitude control system with ON-OFF actuator, as mentioned before, in steady state converges to a limit cycle and a 2-pulse limit cycle is the most robust type and has the best fuel consumption condition in contrast or with other types of limit cycles. Amplitude of this limit cycle should satisfy pointing requirement of the system. On the other hand, the limit cycle frequency in order to have less fuel consumption, should decrease as big as possible. Other assumptions of this problem are as follows. The satellite is rigid and non-spinning. The thrust force delivered by actuators are the only moment signal acting on the body. The sensors have sufficient resolution and their noises are negligible. The actual ON-OFF actuators have some imperfections such as delay, minimum on-time and having dynamics in both turning ON and OFF phases. Therefore, design of the proper controller should be done considering these constraints. In order to design an attitude control system, we are required to have in advance both the angular position and angular velocity of the system in all three control channels. In this paper, it is assumed that these relevant variables are measured through a precise inertial navigation system which will be in turn to the control system. The main objective here is to design this controller to fulfill 1) an on-orbit stabilization with a fine attitude regulation in which the pointing error should not exceed a predefined value; 2) the closed loop system should converge to a 2-pulse limit cycle with a minimum fuel consumption subject to the aforementioned assumptions. In the next section, the mathematical model of the attitude control system and ON-OFF actuators are represented. 3. MODELING A. Satellite Attitude Control System Model A satellite attitude control system model is composed of the following six coupled first order nonlinear differential equations [8]: = + = + = + = + tan ( sin + cos ) = cos sin = ( sin + cos ) In which [ ] and [ ] are Euler angels and system angular velocities in roll, pitch and yaw channels, respectively. [ ], [ ] and, [ ] are system inertial momentums, actuators output thrust levels and control commands in each channel, respectively. Obviously, if approaches90, a singularity will happen in the system. In such conditions, the above model known as Euler angle model is not suitable to use and the quaternion based model is more convenient [8]. Since the Euler angles used in this paper are far from the singularity point, the Euler angles model is used to design the controllers and in addition, the linearized model of the system can be used in this case. In the linearized case, the attitude control equation in each channel is described as: [19] (1) = (0) = (0) = (2) where and are the angular position and angular velocity at the actuator switching moment, is the actuator output thrust level, and is a constant coefficient depends on the inertial momentum. B. The ON-OFF Actuator Model In the baseline problem considered in this paper, the actuator switchings are subject to some restrictions as stated below. The real cold-gas actuator has a delay time, a rise time,, a minimum on-time,, and a fall time, in each pulse. The actuator behavior in rise time and fall time can be approximated by a first order or second order polynomial function, based on the accuracy required. During the delay time and the minimum-on-time the actuator's output is considered constant. From now on, the sum of these times is named as minimum pulse width, (3) Vol. 48, No. 1, Spring M. Sabet Rasekh, S. K. Y. Nikravesh and N. Ghahramani To obtain in real cases we should run some experimental tests for determining the proper values of rise time, delay time, fall time and minimum-on-time in an average sense. 4. THE ATTITUDE CONTROLLER DESIGN According to the modeling of the plant and the actuator in the previous section, the controller design procedure is presented in this section. First, the cost function, which is the base of the controller optimality, will be described, and then, by solving the optimal control problem, a suboptimal limit cycle is derived. Then the controller is presented and finally a compensator to compensate the actuator constraints is represented. A. Defining the Cost Function In the method presented in this paper, for the system to have a limit cycle with desired characteristics, first the modification of command is done. It means that rather than an equilibrium state, the limit cycle trajectory with desired characteristics will exert to the system as a modified command. Then, a controller will be designed to force the system to track this optimal limit cycle. Indeed, the regulation problem will convert to a tracking problem. Albeit this change makes the problem more difficult, it makes it possible to reach the control goals. Furthermore, it is obvious that the robustness of the method against noise and model changes will improve. As mentioned earlier a 2-pulse limit cycle has a positive pulse (L1), a negative pulse (L3) and two OFF regions or coast region (L2, L4), as shown in Fig. 1. As this figure depicts, in the coast region, the angular velocity of the system is constant and in the conditions that the actuator is on, the trajectory of the system is indeed a parabolic path. Fig.1. A 2-pulse limit cycle in the phase plane To have an optimal 2-pulse limit cycle, a cost function is defined as: = + (4) = (5) = ( + ) (6) in which and are the fuel consumption in regions L1 and L3 and and are the coast times of L2 and L4 segments, respectively. To optimize this cost function, one can divide it into two separate sections. Thus, optimization can be done through two separate optimization problems in ON and OFF regions as stated below. min & ( ) & min & ( ) (7) The optimization should be done according to both the required pointing accuracy and the plant constraints. B. The Suboptimal Limit Cycle To have a 2-pulse limit cycle as shown in Fig. 1, inequalities (8) and (9) should hold. 0 (8) where and are the angular velocity in L4 and L2 respectively. Inequality (8) is obtained according to the characteristics of the 2-pulse limit cycle which has a positive and a negative pulse. Therefore, the angular velocity in one of the coast regions is positive and in the other one is negative. Inequality (9) is obtained based on the actuator minimum ON-time. As mentioned earlier, the minimum ON-time of the actuator makes changes in angular velocity of the plant in each pulse get greater than a minimum value equals to with =. In addition to these constraints, the maximum value of the attitude error is one of the other constraints that should be considered. Obviously, if the system converges to a limit cycle with zero mean and an amplitude equals to the maximum permissible attitude error, the attitude error constraint should be met. Thus, if a controller is able to satisfy these three constraints simultaneously, it is guaranteed that the system converges to a desirable twopulse-limit cycle. Now, with regard to these constraints, the optimization problem is solved as stated below. According to equation (2), the system s trajectory in positive, negative and OFF regions can be described respectively as: = + (10) (9) 14 Vol. 48, No. 1, Spring 2016 An Efficient Method to Control the Amplitude of The Limit Cycle in Satellite Attitude Control System = + (11) = + (12) Thus, if the limit cycle's amplitude becomes equal to A, the trajectories L1 and L3 can be described as (13) and (14), respectively. L1: = (13) L3: = + (14) Hence, and are obtained as: = + 2 (15) = + 2 and becomes = ( )( ) (16) (17) Now, by differentiating equation (17) (for briefness, the straightforward procedure of differentiating and the mathematical operations are omitted), the optimal value of this function will be obtained when either θ orθ is zero; this means that if the angular velocity of the system becomes zero and the actuator is OFF, theoretically, the system should keep its state fixed. It is obvious that this situation will not be possible in real world. Therefore, this problem has no optimal solution though it is possible to obtain a suboptimal solution. On the other hand, in order to minimize, the width of each pulse ( and ) no matter positive or negative must be made equal to the minimum pulse width and under the constraint mentioned in (9) ( = ). Therefore, the desired limit cycle can be explained as 1) Trajectory described by = ; (positive pulse) 2) Trajectory described by = ; (coast region) 3) Trajectory described by = + ; (negative pulse) 4) Trajectory described by = ; (coast region) in which or should be as close as possible to axis and also the condition (9) should be satisfied. The value A is the desired limit cycle amplitude which is equal to the maximum allowable pointing error. Now, by having the desired trajectory in hand, a controller is needed to bring the system trajectory to this desired limit cycle. C. The Controller Design The goal here is to design a controller which leads the system to the desired limit cycle. A good choice for this purpose would be a phase plane controller (PPC). A PPC generates the ON-OFF command based on the comparison between the state variables of the system (the angular position and the angular velocity) and some switching curves. The PPC controller has some advantages such as: This controller directly generates the desired ON- OFF commands. Thus, there is no need to an intermediate subsystem (such as modulator) to change the control command to an ON-OFF command. The PPC controller can be simply implemented on flight computers. Since the suboptimal limit cycle was obtained based on the phase plane trajectories, the PPC design is much simpler than the other control methods. The PPC is able to provide a good feeling of dynamics and physical performance of the system for the user. To design the PPC controller, the following steps should be taken: Two switching curves according to eq. (13) and (14) are taken since the optimized limit cycle is determined with these equations. The controller should somehow operate that the system from any initial condition go toward the switching curves. Therefore: If the initial state of the system is between the two switching curves, considering that the error of the attitude of the system is less than the allowable limit, the actuator remains off till the state reaches one of the switching curves. If the initial condition is out of the area between the switching curves, since the error is exceeded from the allowable error limit, the state should bring to one of the switching curves, using bang-off-bang command. Having reached the switching curves, in a suitable timing, the required commands must be given to have the desired limit cycle. For this reason, two offboundaries are considered in =± and another one is on the =0 axis. Whenever the system's trajectory reaches one of the switching curves and the actuator is ON or ON, an OFF command should be sent to the actuator. Furthermore if the system's trajectory reaches one of the switching Vol. 48, No. 1, Spring M. Sabet Rasekh, S. K. Y. Nikravesh and N. Ghahramani curves and the actuator is OFF, an ON command (L1) or ON command (L2) should be executed. To have a desired limit cycle, the PPC controller should consist of two switching curves described by (13) and (14) and two OFF boundaries given by =±. When the system trajectory meets this off boundary, the actuator should turn off. For more explanations, this controller is depicted graphically in Fig. 2. This controller works well using an ideal actuator, and in the presence of uncertainties or real constraints in actuators, a compensator might be used along with the PPC controller in order to improve its performance. approximated by a linear function, then (18) and (19) would hold. = (18) = + + (19) At the time instance =0, we will have the initial values =, = and =0, and at the time = we have: ( ) = + (20) ( ) = + + (21) For ( ) to be on the trajectory (13), the following equation should be satisfied: ( ) = + + = + (22) Therefore, when the actuator s output level reaches its nominal value, the states will lie on (13) whenever satisfies (23). 16 Fig. 2. Switching curves of optimal controller (: = : = + ) D. Design of a Compensator Based Constrained Controller As mentioned before, the On-OFF actuators have some imperfections and constraints due to their physical nature [20-21]. Since in the steady state the actuator generates pulses with a minimum pulse width, the actuator constraints have more effects
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