A Model Reference Adaptive Control / PID Compound Scheme on Disturbance Rejection for an Aerial Inertially Stabilized Platform

This paper describes a method to suppress the effect of nonlinear and time-varying mass unbalance torque disturbance on the dynamic performances of an aerial inertially stabilized platform (ISP). To improve the tracking accuracy and robustness of the ISP, a compound control scheme based on both of model reference adaptive control (MRAC) and PID control methods is proposed. The dynamic model is first developed which reveals the unbalance torque disturbance with the characteristic of being nonlinear and time-varying. Then, the MRAC/PID compound controller is designed, in which the PID parameters are adaptively adjusted based on the output errors between the reference model and the actual system. In this way, the position errors derived from the prominent unbalance torque disturbance are corrected in real time so that the tracking accuracy is improved. To verify the method, the simulations and experiments are, respectively, carried out.The results show that the compound scheme has good ability in mass unbalance disturbance rejection, by which the system obtains higher stability accuracy compared with the PID method.


Introduction
For an aerial remote sensing system, due to the serious effects of internal and external disturbances, the movement of the aircraft is not ideal that makes the sensor's line of sight (LOS) jitter, eventually resulting in the degradation of images quality [1][2][3][4].In order to obtain high-resolution images and satisfy the requirements of high photo overlapping ratio, the sensor's LOS must be strictly controlled.Therefore, inertially stabilized platform (ISP) is a key component for the highprecision aerial remote sense imaging system, which is used to hold and control the LOS of the imaging sensor to keep steady relative to the inertial space or the tracked target [5,6].The ISP with high control precision is indispensable for isolating disturbances derived from diverse sources [7,8], particularly for the case of swings of three angular attitudes of aircraft.It is a principal issue for the control system of ISP of how to minimize the effects of disturbances introduced on the ISP [2].
The most critical performance metric for an ISP is torque disturbance rejection.It is difficult for the conventional PID control method in low speed servo motion to suppress these complex nonlinear disturbances.It is hard for the traditional feedforward control method to further improve the ISP's dynamic performance [9].Therefore, there is continuous interest for researchers to develop the control methods with higher accuracy and stability by various disturbances rejection.The development of computer technology and advanced intelligent control theory provide a new way for the control of complex dynamic uncertain systems and the disturbance rejection.They have been gradually used in the control of ISP, such as neural network [10], genetic algorithm [11,12], fuzzy control [13,14], robust control [15], state compensation control [16], and autodisturbance rejection control [17].Predictive control and fuzzy control are the effective methods to optimize the control of uncertain systems.The combination of the two methods can enable the system to have the quicker dynamic response and smaller overshot [11][12][13][14].For a system characterized by nonlinear and timevarying behavior, the issue of stability performance becomes very prominent.In [1], the decoupling compensation controller obtains a good result in which the angular velocity coupling, torque coupling, and moment of inertia coupling are considered.In [18], a three-closed loop PID compound control scheme is applied to a two-axis ISP to obtain the desired results.In [19], a feedforward compensation scheme is proposed to achieve vibration rejection of ISP.In [20], an active disturbance rejection control strategy is put forward to improve robustness of electrode regulator system.
Disturbances arise from diverse sources; for example, the angular motion and linear vibration of aircraft platform generate the disturbance torques due to mass unbalance and gimbal geometry [4,5].Generally, the influence of unbalance torque is prominent over other disturbances.Since the centroid is not exactly coincident with the center of the rotating axis, the mass unbalance torque will occur when ISP operates, which will severely degrade the system control accuracy.To compensate the mass unbalance torque, the static equilibrium test is required before the operation.However, due to many nonideal cases such as different imaging sensors being installed interchangeably, it is hard to completely eliminate the unbalance torque by static equilibrium mass correction.Moreover, since the mass unbalance torque is a nonlinear and time-varying disturbance, it is difficult for the conventional control to solve it [21].Therefore, it is necessary to compensate the mass unbalance torque by using the intelligent control methods which have strong adaptive disturbance rejection ability.Previously, some methods have been proposed to compensate the unbalance torques, such as the adaptive control based on disturbance observer [22], the neural network control [10], the feedforward control [23], independent mechanisms [24], fuzzy inference mechanism [25], iterative feedback tuning of fuzzy control [26], adaptive neural network control [27], and so on.
Model reference adaptive control (MRAC) can restrain the influences of external disturbance by effectively revising the model parameter errors.MRAC does not need the online identification of the mathematical model of mass unbalance torque, by which the time of adaptive control is greatly shortened.Therefore, MRAC is appropriate for the occasion of parameters change [21].In MRAC, an adaptive reference model needs to be designed which can achieve the desired performance index with the same order of the plant.In [28], a MRAC system based on the certainty equivalent (CE) principle for the first-order delay system is proposed.MRAC/PID compound control scheme is a combination of adaptive control and traditional PID control, which can make the PID parameters of the nonlinear time-varying uncertain systems adjusted in real time, so as to improve the system robustness and control accuracy.MRAC/PID controller owns the great robust ability for the nonlinear, hysteresis, and variable parameters systems.Compared with the conventional PID control method, the MRAC/PID controller can tune the PID parameters automatically and make the system stable in the whole working range [29].
In this paper, to improve the tracking accuracy and robustness of an aerial inertially stabilized platform, a MRAC/PID compound control scheme is proposed to weaken the influence of prominent unbalance torque disturbance.The dynamic model is first developed which reveals the unbalance torque disturbance with the characteristic of being nonlinear and time-varying.Then, a MRAC/PID compound controller is designed and simulation analysis is conducted.To verify the method, the experiments are carried out.

Background
2.1.Aerial Remote Sensing System.Figure 1 shows the schematic diagram of an aerial remote sensing system.Generally, an aerial remote sensing system consists of four main components, a three-axis ISP, an imaging sensor, a position and orientation system (POS), and the aviation platform.When applied, the three-axis ISP is mounted on the aviation platform, and the imaging sensor and POS are mounted on inner azimuth gimbal of the ISP.When the aviation platform rotates or jitters, the control system of three-axis ISP gets the high-precision attitude reference information measured by POS and then routinely controls the LOS of imaging sensor to achieve accurate pointing and stabilizing relative to ground level and flight track.The POS, which is mainly composed of three main components, that is, inertial measurement unit (IMU), GPS receiving antenna, and data processing system, is used to provide an accurate reference of position and attitude  in inertial space for control system of ISP and imaging sensor through measuring the angular movement of imaging sensor.

Working Principal of ISP.
Figure 2 shows the schematic diagram of the three-axis ISP principle.We can see that the ISP consists of three gimbals, which are azimuth gimbal (-gimbal), pitch gimbal (-gimbal), and roll gimbals (gimbal).Among them, the -gimbal is assembled on the -gimbal and can rotate around   axis.Likewise, the gimbal is assembled on the -gimbal and can rotate around   axis.The -gimbal is assembled on the base of aviation platform and can rotate around   axis.From Figure 2, we can see the relationships between three gimbals:   ,   , and   , respectively, stand for rate gyro that measures inertial angular rate of -gimbals, -gimbals, and -gimbals.  ,   , and   , respectively, stand for photoelectric encoder which measures relative angular between gimbals.  ,   , and   , respectively, stand for gimbals servo motor which drives gimbals, -gimbal, and -gimbal to keep these three gimbals steady in inertial space. and , respectively, represent accelerometers installed on the -gimbal and -gimbal used to measure the gimbals' rotary angular acceleration.  and   represent encoders installed on two leveling gimbals to detect the gimbals' rotary angular position.

Three-Closed Loop Compound Control Scheme.
Conventional stabilization techniques employ rate gyros, rate integrating gyros, or rate sensors to sense rate disturbances about the LOS. Figure 3 shows the block diagram of traditional three-loop control system for ISP.In Figure 3, the blocks of -pos, -spe, and -cur separately represent the controllers in the position loop, speed loop, and current loop; the PWM block represents the power amplification used for the current amplification to drive the torque motor;  represents the inductance of a torque motor and  represents the resistance;   represents the torque coefficient of the motor and  is the transition ratio from the torque motor to the gimbals;   represents the moment of inertia of the motor and   represents the moment of inertia of the gimbals along the rotation axis.

Dynamic Modeling of the Mass Unbalance Torque
Due to the influence of mechanical structure design and machining accuracy, three-axe ISP's center of mass is not coincident with the center of rotating shaft completely.So, when there is acceleration that acted on the ISP gimbals, the mass unbalance torque disturbance occurs [23], which will severely degrade the system control accuracy.Mass unbalance torque under the static base is caused by both of the gravity and gimbal acceleration.
The mass imbalance produces LOS jitter when the payload center of gravity is not centered on an axis of rotation for the gimbals.Linear vibration, acting through the lever arm of the center of gravity offset, thus produces torque disturbances.When an ISP is working at a flying aircraft, due to both effects of gravity acceleration and the motion acceleration of ISP's gimbals and imaging sensors, the mass imbalance torque occurs.Figure 5 shows the schematic diagram of the functional mechanism of mass imbalance torque under moving base and static base, respectively.
In Figure 5(a),   and   , respectively, stand for the eccentric lever arms of mass center of ISP relative to a horizontal rotation axis ( or ) and the vertical axis-.  and   , respectively, stand for the horizontal and vertical interference accelerated speed during movement acting on the gimbal and  stands for acceleration of gravity.If we take the counterclockwise direction as a positive direction, imbalance torque can be expressed as where  im is the unbalance torque of the moving base and  is the total mass of the frame and the load, where  motor is the extra unbalance torque produced by motor. is the transmission ratio.The unbalance torque caused by gravity can be expressed as Since the leveling angles of ISP are changed in a small range of about  = ±5 ∘ , the cosine and sine function values of  are approximately equal to 1 and 0, respectively, so the unbalance torque caused by gravity can be further expressed as

MRAC/PID Compound Controller Design
4.1.Adaptive Model Reference Control Principal.Figure 6 shows the structure of an adaptive model reference control (MRAC) system.It is composed of four components, that is, control process, controller, reference model, and adjustment mechanism block [30].
The adaptive model reference control (MRAC) technique is used for devising a controller based on the information   ,   , (), and ().The adaptive mechanism automatically adjusts controller parameters so that the behavior of the closed loop control plant output   closely follows that of   of the reference model.Parameters and structure of reference model are specified on the base of requirements of control performance.The adjustment mechanism of MRAC system is constructed by adaptive control rule, which performs the algorithms as follows.
Tracking error is defined as where  is error of input and output,   is the output of the reference model, and   is the output of the actual system.
Thus,  will eventually be close to a constant or 0. If  is bounded, then   is bounded.So the control system is convergent under this controller.

MRAC/PID Compound Controller
. PID control is one of the most commonly used control methods in engineering.It has the advantages of simple algorithm and high reliability, but it needs more accurate system model [31].When the input signal is applied to a variable structure control system, the system should be stable while altering controller coefficients according to control error signal.According to Figure 3, the bandwidth of the current loop is much larger than the bandwidth of the stable loop, so the current loop is regarded as a proportional component with coefficient of 1.The transfer functions of the -pos and -spe are too small and can be equivalent to 1.   ((   + 1)/) is the transfer function of -cur, and /( PWM  + 1) is the transfer function of the PWM [32].In the simulation, the values of these parameters are too small to be ignored for 0, such as , ,   ,  PWM , and   .So the ISP control system is established as the two-order system model.
Based on [29,30], the controller is designed as follows: where the time rate of change of  stands for proportional to negative gradient of , where  stands for the controller parameter vector.The components of / stand for the sensitivity derivatives of the error with respect to .The parameter  stands for known as the adaptation gain.
Considering an aerial ISP system described by secondorder model /( 2 + 1 + 2 ), the closed loop transfer function is () From ( 9) and required performance of system, we obtain a reference model as follows: where where / = , / = 1,  = /, and Then   /,   /, and   / can be derived by If the adaptive control law is given, the system can guarantee the stability of the system under the action of the model reference adaptive PID controller and can achieve the control goal.Thus, a new PID control based on the reference model is formed.The basic structure of the MRAC/PID compound control system is shown in Figure 7.In Figure 7, the blocks of -pos, -spe, and -cur separately represent the controllers; the PWM block represents the power amplification used for the current amplification to drive the torque motor.

Tracking Performance.
According to the adaptive model reference control (MRAC) method, through the parameters adjustment simulation environment, the system can get the ideal output.
The PID parameter values of the reference model are selected by the parameter tuning or the test calculation method, as shown in Figures 8 and 9.
So the result of the PID parameter value is selected and taken out by the reference of the automatic adjustment:   = 10,   = 0, and   = 4. Based on the fixed parameters in the three-loop control system of ISP, the reference model is Further, by (14), the control strategy of the adaptive PID controller is adjusted by the MRAC/PID compound controller, and the stability and accuracy of the system are improved continuously.
First, the responses for the step input are analyzed.Figure 10 shows the system response curve and its partial   enlarged detail for the traditional PID controller and the MRAC/PID compound controller.As seen in the figures, compared with PID control, the stability time and accuracy of the MRAC/PID compound control are much shorter and higher.From Figures 10 and 11, we see that the MRAC/PID compound control scheme can obviously improve the accuracy of the control system.Under the same unbalance torque disturbance conditions, the position output peak-valley errors of MRAC/PID compound scheme and traditional PID control method are +0.0075∘ to −0.006 ∘ and +0.035 ∘ to −0.05 ∘ , respectively, meaning that the position accuracy is improved up to 84.1% after MRAC/PID scheme applied.The simulation results illustrate the compound scheme has high disturbance rejection ability compared with the PID controller.

Experimental Verification
In order to validate the proposed MRAC/PID compound scheme, the experiments are then performed, which are conducted on a three-axis ISP.
According to the specific functional requirements of the system, the system hardware circuit is designed.Figure 12 Journal of Sensors  shows the hardware control system circuit connection diagram of three-axis ISP.The main functional devices are DC torque motor, POS, gyroscope, encoder, and so forth.
The main program flowchart is shown in Figure 13.After the system is reset and the external device is initialized, the external device interrupt is opened.The six interrupt management procedures include ADC sequencer ADCSEQ interrupt, interrupt timer CPU-Timer 0, SPI receiving interrupt, SCI A sending/receiving interrupt, SCI B sending/receiving interrupt, and SCI C receiving interrupt using the software  Figure 14 shows the picture of the experimental system.The main parameters of the ISP are as follows: maximum load and self-weight are, respectively, 80 kg and 40 kg, the maximum leveling rotation angle range is ±5 ∘ , and the maximum heading rotation angle range is ±25 ∘ .In the experiments, the artificial load is 20 kg and the power supply voltage is 28 V.As The experiments are designed aiming at rejecting unbalanced torque disturbance.From (4), we know that the unbalance torque is nonlinear and time-varying with characteristic of cosine function.Therefore, the cosine interference component is artificially added into the current loop of the gimbal system to represent the unbalance torque disturbance.Thus, we can evaluate the disturbance rejection ability of the compound scheme.
Figure 15 shows the experiment comparison between the PID and adaptive MRAC/PID controllers when the interference current  = 0.5 cos(0.4)+ cos(0.9).We can see that, for the MRAC/PID and PID controllers, the output error ranges of the attitude angle are about ±1 ∘ and ±2 ∘ , respectively.Obviously, compared with the PID controller, the position accuracy is improved up to nearly 50% by the MRAC/PID compound controller.

Conclusion
In this paper, to suppress the nonlinear and time-varying mass unbalance torque disturbance of the aerial-axis inertially stabilized platform (ISP), a compound control scheme based on both of model reference adaptive control (MRAC) and PID control methods is proposed.In this way, the tracking accuracy and stability of the ISP are improved significantly.To verify the method, the simulations and experiments are, respectively, carried out.The results show that MRAC/PID compound scheme is good at the disturbance rejection.Compared with the PID, the tracking accuracy of the MRAC/PID compound controller is improved by about 50%.

Figure 1 :
Figure 1: Schematic diagram of an aerial remote sensing system.

Figure 2 :
Figure 2: Schematic diagram of the three-axis ISP's working principle.

Figure 3 :
Figure 3: A block diagram of traditional three-loop control system for ISP.

Figure 4 :
Figure 4: The process model of the ISP control scheme block diagram.

Figure 5 :Figure 6 :
Figure 5: The geometrically schematic diagram to show the functional mechanism of mass imbalance torque: (a) under moving base condition and (b) under static base condition.

Figure 8 :
Figure 8: Automatic adjustment of PID parameters for reference model design.

Figure 9 :
Figure 9: Automatic adjustment of PID parameters model.
Figure 11  shows the parameters variation curves of the MRAC/PID compound controller.

Figure 10 :Figure 11 :
Figure 10: System response curves and partial enlarged detail for the (a) traditional PID controller and (b) MRAC/PID compound controller.

Figure 14 :
Figure 14: The three-axis ISP experimental system.

Figure 15 :
Figure 15: Experiment comparison between the PID and adaptive MRAC/PID controllers under unbalance disturbance.
shows the process model of the ISP control scheme block diagram.  ,   , and   are the PID controllers for position loop of azimuth gimbal, pitch gimbal, and roll gimbal, respectively.  ,   , and   are the PID controllers for rate loop of azimuth gimbal, pitch gimbal, and  3 =   .The values of PID controller parameters are determined,   ,   , and   , in