Abstract
This paper presents a high-precision control scheme based on active disturbance rejection control (ADRC) to improve the stabilization accuracy of an inertially stabilized platform (ISP) for aerial remote sensing applications. The ADRC controller is designed to suppress the effects of the disturbance on the stabilization accuracy that consists of a tracking differentiator, a nonlinear state error feedback, and an extended state observer. By the ADRC controller, the effects of both the internal uncertain dynamics and the external multisource disturbances on the system output are compensated as a total disturbance in real time. The disturbance rejection ability of the ADRC is analyzed by simulations. To verify the method, the experiments are conducted. The results show that compared with the conventional PID controller, the ADRC has excellent capability in disturbance rejection, by which the effect of main friction disturbance on the control system can be weakened seriously and the stabilization accuracy of the ISP is improved significantly.
1. Introduction
It is of crucial importance for an aerial remote sensing system to obtain the high-resolution images under a nonideal multisource disturbance environment. The inertially stabilized platform (ISP) is a key component for an aerial remote sensing system, which is used to hold and control the line of sight (LOS) of the imaging sensors keeping steady relative to the tracking object in the inertial space [1, 2]. If the high-resolution images of the imaging sensor were obtained, the ISP should have the capability to effectively isolate the attitude changes of aircraft in three axes; besides, it is required to suppress the multisource disturbances whatever inside or outside of the aircraft body so that the high control precision can be gotten. Therefore, the most critical performance metric for the ISP is disturbance rejection.
Generally, the ISP is placed between the aircraft and imaging sensor served as a physical interface. It is a principal issue for the control system of the ISP to minimize the effects of disturbances introduced on the ISP [1]. In [2], a proportional integral control scheme is proposed to realize zero steady-state error of angular output when facing disturbances from carrier. In [3], a hybrid adaptive fuzzy proportional-integral-derivative (PID) controller is developed for a servo system with nonlinear property and uncertainties. In [4], a new current loop inside rate loop is particularly designed to suppress the influences of voltage fluctuation and motor back electromotive force on control precision. In [5], the model uncertainties and unmeasurable disturbances existing objectively are solved by introducing internal model control for a two-axis ISP system. In [6], an approach to compensate disturbing torque based on acceleration feedforward for the three-axis ISP is proposed. In [7], a parameter identification method for LuGre model based on multilevel coordinate search algorithm was presented. In [8], a compound control scheme based on both of the model reference adaptive control (MRAC) and PID control methods is proposed.
The active disturbance rejection control (ADRC) was proposed to deal with the nonlinear systems, where both uncertain dynamics and disturbances exist. The ADRC is composed of three components, that is, a tracking differentiator (TD), a nonlinear state error feedback (NLSEF), and an extended state observer (ESO). The key in the ADRC is to online estimate the total uncertainty, which lumps the internal uncertain dynamics and the external disturbances by the ESO. The ADRC can solve quickly and effectively the problems of nonlinear, strong coupling, and so on [9, 10]. The ADRC not only can improve the dynamic performance of the system but also can achieve good adaptability and robustness [11, 12]. A large number of simulation experiments show that the ADRC algorithm is simple to detect and compensate the disturbance of the control object automatically [13, 14]. This design methodology requires very little information about the plant dynamics, is very easy to tune, and has a very good disturbance rejection capability. In [15], a feedback linearization based the ESO control law is proposed for the trajectory tracking control of a flexible joint robotic system. In [16], the effectiveness of the ESO in the estimation of the states and uncertainties is demonstrated. In [17], an ADRC for the antenna pointing control of a large flexible satellite system is designed, by which the high pointing accuracy and rotation speed are achieved.
In this paper, to improve the ability of disturbance rejection of an aerial three-axis ISP, a disturbance rejection scheme based on the ADRC is proposed. To verify the method, the simulations and experiments are carried out, respectively. The rest of this paper is arranged as follows. Section 2 first introduces the background of this work. Section 3 explains the ADRC principle and algorithm that includes three parts: the TD, the NLSEF, and the ESO. Section 4 demonstrates the simulation analysis, including ADRC controller as well as the PID controller. The ADRC experiment is in Section 5, including the static base experiment and the moving base experiment. In the end, a brief conclusion is given in Section 6.
2. Background
2.1. Aerial Remote Sensing System
Figure 1 shows a schematic diagram to illustrate the important effect of the ISP on improving the image quality in an aerial remote sensing system. In general, an aerial remote sensing system consists of an aircraft vehicle, an imaging sensor, a three-axis ISP, and a position and orientation system (POS). During the work, the ISP is used as a physical interface to separate the imaging sensor and the aircraft. Depending on the ISP, the serious influences of various disturbances on the imaging sensors are effectively suppressed, by which the LOS can be kept stable all along and the high-resolution images are gotten eventually.
2.2. The Role of Three-Axis ISP System
Figure 2 shows the schematic diagram of an aerial remote sensing system. As shown in this figure, during application, the ISP is installed on the floor of the airplane. Meanwhile, the imaging sensor is installed on the inner azimuth gimbal of the ISP, and the POS is installed on the top of the imaging sensor. When the aircraft rotates or jitters, the high-precision attitude reference information is measured by the POS in real time, which is then provided to the ISP control system to make the LOS of imaging sensor to achieve the high stabilization precision.
2.3. Working Principle of Three-Axis ISP System
Figure 3 shows the schematic diagram of the three-axis ISP’s principle. As shown in the figure, the ISP consists of three gimbals, which are azimuth gimbal (A-gimbal), pitch gimbal (P-gimbal), and roll gimbal (R-gimbal) from the outside to the inside. Three rate gyros installed on the three different gimbals are used to measure inertial angular rate of P-gimbal, R-gimbal, and A-gimbal, which are used to improve the stabilization precision through compensating the difference between the rate command input and the angular rate of the gimbal. On the other aspects, three photoelectric encoders installed on different gimbals are used to measure the relative angular position changes between two adjacent gimbals. Three servo motors installed on different gimbals are used to drive R-gimbal, P-gimbal, and A-gimbal rotating inversely to compensate the attitude changes to three gimbals in real time.
3. Design of the ADRC Controller
3.1. Tracking Differentiator (TD)
For a second-order system [18], we get if the solution of (1) satisfies
So, for any boundary integrable variable, , and any constants, and , the solution of the system can meet the following relationship:
Therefore, assuming that a function can be found to make (1) asymptotically stable at the origin, a nonlinear tracking differentiator of in the form of (3) can be constructed, in which the same function is used [19].
Kasim and Riyadh [20] have made some investigations on traditional structures and essential properties of the nonlinear TD. A kind of second-order nonlinear TD based on second-order bang-bang switch system has been proposed: where is the desired trajectory and is its derivative. Note that the parameter is an application dependent and it is set accordingly to speed up or slow down the transient profile. Then, is denoted as the TD of .
In order to avoid chattering near the origin, changing the sign function to linear saturation function sat, then the modified TD is represented by [21–23]
The TD gets the differentiated signal by computing the integration of the relational viable. So it can efficiently weaken the effect of noise, while the classical differential component enlarges it and the differentiated signal obtained by this way is far from the real one. In some sense, the differential component of the TD is feasible physically, while differential components of the classical methods are not feasible physically [21].
3.2. Nonlinear State Error Feedback (NLSEF)
For the first-order system [22],
is the sum of the perturbations of the system, and linear feedback is used for the control object:
Then, the closed-loop system can be expressed:
The closed-loop equation of the finishing system is obtained [23]:
If the state quantity converges, then is needed, that is to say, it needs to be satisfied:
It is easy to get
As long as , there must be . The state parameter converges, and the system disturbance can also be well suppressed. But with the convergence of the state quantity , it will gradually approach the range of , which leads to the suppression of the disturbance of the system and the steady-state error of the system.
The ADRC reduces the steady-state error of the system by adopting appropriate nonlinear functions. According to the different error states, the nonlinear function can make the feedback gain appropriate increase and decrease, so as to reduce the steady-state error of the system, and effectively avoid the disadvantages caused by adopting the fixed feedback gain.
Nonlinear feedback is used to control objects:
Then, the controlled object is converted into a closed loop:
Then get
When , the state is convergent, which requires
Then get
According to the analysis of the upper form, when , the state converges, and with the further convergence, state can converge to the range of , so the steady-state error of the system will not be greater than .
It is easier to know the linear feedback obtained the maximum steady-state error compared with that when and , there are
For systems subject to uncertainties, the steady-state error obtained by using nonlinear feedback is much smaller than that by using linear feedback. In addition, in nonlinear feedback, if the value is changed slightly, the can rapidly approach zero.
The TD generates the arranged transition process and its derivatives. The outputs of ESO estimate the states of the control system. By comparing the difference between the outputs of TD and those of ESO, the NLSEF control law is used to drive the state trajectory to the desired reference signal. Its mathematic expression is given as [24]
Thus, the nonlinear combination of system state error feedback is formed: where , and are adjustable parameters. where is defined as a modified exponential gain function [25].
3.3. Extended State Observer (ESO)
The performance of the ESO is very important to the performance of the controller [18, 20]. The state observer is used to determine the internal state of the system by observing the external state of the system in different situations.
If the function is not known and the above description is not valid, the way to expand the state observer is to expand into a new state variable :
In this way, nonlinear objects can be regarded as new linear objects [25]:
Then, the state observer is established:
In this way, the provided parameters , , and are selected properly, the state variables of the system can be estimated in real time by , , and , and the real-time action amount of the acceleration of the system is measured. Equation (25) can not only achieve good control effect but also avoid high-frequency tremor. where is the length of the linear section.
If contains time and unknown disturbances , it is available [17]:
The real-time action of acceleration acting on the system can be obtained:
Thus, (28) can become the extended state observer of the system. The ESO is a dynamic process that does not take advantage of system parameters and specific models but uses the information between inputs and outputs of the system to estimate the system state variables directly.
3.4. The ADRC-Based Three-Loop Compound Controller
Figure 4 shows the block diagram of the ADRC-based three-loop compound controller. As shown in this figure, the blocks of G-cur, G-spe, and G-pos indicate the controllers from the inner loop to the outer loop, which are the current loop, rate loop, and position loop, respectively; the PWM block indicates the pulse width modulation, which is used for the current amplification to drive the torque motor; the symbol , , , and are the inductance of a torque motor, resistance, motor torque coefficient, and transition ratio from the torque motor to the gimbal, respectively. Besides, , and indicate the moment of inertia of the motor and the moment of the gimbal inertia along the rotation axis, respectively.
As shown in this figure, in order to improve the control accuracy of the three-axis ISP system, the ADRC controller is designed. One of the inputs of the ADRC is the tracking instruction of angular position of the ISP control system, , and another input of the ADRC, , is the real angular position of the ISP measured by the POS. The control amount calculated by the ADRC is acted on the ISP position loop.
In Figure 4, the generalized inputs, and , are first obtained by the TD. The generalized outputs of the ISP control system, and , are obtained by the ESO. is the angle value measured by POS in the position loop of the three-axis ISP system, is the derivative of the angle, and is the compensatory amount of the ESO to the disturbance in the ISP control system. The compared errors between the generalized inputs and the generalized outputs are inputted to the NLSEF to generate the control amount , which are compared with to generate the control amount . The control amount after compensation is acted on the position loop of the ISP. Thus, the disturbance rejection ability and the control accuracy of the ISP control system are improved.
4. Simulation Analysis
4.1. Angular Position Response to Step Input
In order to verify the validity of ADRC algorithm, MATLAB/Simulink simulation is firstly carried out. The main parameters of the NLSEF control law are . Normally, the value of is from 0.5 to 0.75, and the value of is from 1.25 to 1.5. The value of is the same as the integral step length as 0.01 generally. The parameters tuned by the experiential method are as follows: and .
In the simulation, the pitch gimbal system is chosen as the control object. When the given input angle instruction is , , the disturbance resistance performance of the PID and the ESO/PD controller is simulated by adding the disturbance model; the results are shown in Figure 5.
(a) Global curve
(b) Local magnification curve
In Figure 5(a), the PID and the ESO/PD composite controllers can track the sinusoidal signals in different position precision. In the local magnification diagram of Figure 5(b), we can see that the PID has approximately 0.01° errors in tracking the sinusoidal instruction, but the ESO/PD compound controller tracking curve almost coincided with the sinusoidal instruction. Therefore, the ESO/PD controller has strong disturbance resistance ability, which can effectively suppress the disturbance effect, and its tracking error is almost zero.
Under the influence of disturbance, the comparison of angular position error between the PID and the ESO/PD controller is shown in Figure 6. From Figure 6, we can see that the angular error of PID fluctuates slowly from −0.02° to 0.025°. When the disturbance amplitude is large, the tracking error curve of the PID will increase correspondingly, while the angular error of the ESO/PD composite controller is stable at 0°. Therefore, the ESO/PD controller has better anti-interference ability than the PID controller.
Figure 7 shows the comparison of position feedback to step angle between conventional PID and ADRC controller. When the track reference is a step signal with magnitude 1° which lasts for 10 s, the ADRC controller has a good dynamic performance compared to the conventional PID controller. As shown in this figure, compared to the conventional PID controller, the ADRC controller has nearly no any overshoot and can more easily reach stabilization state with much shorter adjusting time.
4.2. Angular Position Response to Sinusoidal Input
Figure 8 shows the comparison of position feedback to sinusoidal signal between conventional PID and ADRC controller. When the track reference is sinusoidal signal with amplitude 1° and frequency 0.1 Hz, the ADRC controller can achieve better disturbance rejection performance compared to the conventional PID controller. As shown in this figure, compared to the conventional PID controller, the tracking curve of the ADRC controller coincides almost with the sine command.
5. Experimental Verification
In order to verify the validity and practicability of the ADRC, the experiments are carried out. Figure 9 shows the picture of moving base experiment system. In the experiments, the three-axis ISP is placed on a moving cart. The experimental system includes a three-axis ISP, a data acquisition and processing system (upper computer), and a 28 V DC power supply. The weight of the ISP is 40 kg and its maximum support load is 80 kg. The maximum leveling rotation angle range of the ISP is ±5°; the maximum heading rotation angle range is ±25°.
Set the tracking reference angle of the roll gimbal system as 0°; sampling points is 5000, and the sampling frequency is 50 Hz. As shown in Figure 10, in leveling experiments, the results obtained by three different methods are as follows: for the PID controller, the RMS (root mean square) error is 0.2673°; for the ESO/PD controller, the RMS error is 0.1315°, which is decreased up to 50.8% compared to the PID controller; for the ADRC controller, the RMS error is 0.0826°, which are decreased up to 69.1% and 37.2% compared to the PID controller and the ESO/PD controller, respectively. The results show that the ADRC control method has better control precision and interference suppression capability compared to the PID and the ESO/PD.
(a) Global curve
(b) Local enlarged drawing
6. Conclusion
This paper investigates the high-precision control method for a three-axis inertially stabilized platform (ISP). Based on the analysis, a disturbance rejection control scheme based on the active disturbance rejection control (ADRC) is proposed to realize high-precision position servo control, which is robust to the multisource disturbances. Then, the numerical simulations and the experimental validations are conducted to analyze and verify the scheme, respectively. The results show that the ADRC control method can get a better control accuracy and have strong disturbance rejection ability than the conventional PID methods. Under the moving base experimental condition, the stabilization precision of the ISP obtained by the ADRC controller is improved up to 69.1% compared to the conventional PID control method.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This project is supported in part by the National Natural Science Foundation of China (Grant nos. 51775017, 51375036), by the Beijing Natural Science Foundation (Grant no. 3182021), and by the Open Research Fund of the State Key Laboratory for Manufacturing Systems Engineering (sklms2018005).