Modified ADRC for Inertial Stabilized Platform With Corrected Disturbance Compensation and Improved Speed Observer

In this article, a modified scheme of nonlinear active disturbance rejection control (ADRC) is proposed by correcting the disturbance compensation and improving the speed observer. The stability, disturbance rejection and tracking performance of the modified ADRC scheme are analyzed and compared with that of the typical ADRC scheme utilizing the describing function method and numerical method, which shows that the dynamic stiffness and tracking performance are significantly improved while the stability almost remains invariant. Both the simulations and experiments are conducted in inertial stabilized platform (ISP) control system. It turns out that the simulation and experimental results further support the analysis.


I. INTRODUCTION
Most industrial control systems often encounter the problems of uncertainty and external disturbances, which seriously restrict the system from obtaining excellent control accuracy. To address this issue, various methods have been proposed to estimate and compensate the disturbance and uncertainty in the past few decades, such as the disturbance observer-based control (DOBC) [1], active disturbance rejection control (ADRC) [2], internal model control (IMC) [3], robust control [4], etc.
Among these existing attempts, ADRC has a remarkable feature that neither the direct measurement of external disturbance nor the prior knowledge of the disturbance model is required. All the uncertain factors that affect the plant are attributed to generalized disturbance, which is estimated by the input and output of the plant, and eventually compensated [5], [6]. Therefore, ADRC is one of the best choices to deal with the control problems in nonlinear systems. For the The associate editor coordinating the review of this manuscript and approving it for publication was Xiaosong Hu . last decade, the superiority of ADRC has been demonstrated in fields of motion control of microelectromechanical gyroscopes [7], trajectory tracking control of flexible-joint robotic system [8], superconducting radio frequency cavities control [9], piezoelectric multimode vibrationcontrol [10], dc-dc power converter control [11], motion control of noncircular turning [12], the control for permanent magnet synchronous motor and asynchronous motor servo system [13], [14].
In recent years, theoretical studies on ADRC have drawn many attentions. Some of the achievements are rather prominent, such as the stability analysis with boundary disturbance and modelling uncertainties [15]- [17], the frequency-domain analysis [18], [19], the capability analysis of the extended state observer for nonlinear systems [20], [21], to name a few. Meanwhile, by combining ADRC with specific algorithms, such as fractional order control algorithm, intelligent algorithm, sliding mode control algorithm, the differential flatness theory, the performances of ADRC have been improved to a large extent [22]- [25]. Besides, a linear ADRC/non-linear ADRC switching control scheme is proposed and analyzed in [26]. VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ However, given the various kinds of uncertainties and disturbances encountered in practical ISP control systems, the challenges of ADRC are still existed in the insufficient disturbance rejection ability and tracking performance. Some efforts have been made to deal with these issues. In [27], both the disturbance rejection performance and tracking performance are found to be severely affected by the nonlinear parameters in a second-order single-input single-output plant. In [28], significant disturbance rejection ability of proposed ADRC within a three-axis ISP is obtained by adopting a genetic algorithm-based parameters tuning method. In [29], to improve the ability of rejecting multiple disturbances, a least mean square (LMS) based ADRC system is designed, enabling significant improvement in performance. These methods are essentially dedicated to optimizing the disturbance rejection and tracking performances by elaborately tuning the parameters of control system, without considering the intrinsic disturbance observation error and velocity feedback characteristics of ADRC.
In this article, inherent structure of nonlinear ADRC is modified to improve the disturbance rejection ability and tracking performance of two-axis ISP control system. The modification of the nonlinear ADRC is based on the correction of disturbance compensation and the improvement of speed observer. To verify the effectiveness of the proposed method, the disturbance rejection and tracking performance of the modified ADRC scheme are investigated through the simulations and experiments, compared with that of the typical nonlinear ADRC.
In the remainder of this article, the theoretical derivation of the modified ADRC scheme, i.e. the disturbance compensation correction and speed observer improvement are presented in Section II. In Section III, the performance analysis of the modified ADRC scheme is conducted by describing function method and numerical method. By simulations and experiments, Section IV investigates the modified ADRC controller for a practical ISP of photoelectric tracking system. Finally, the conclusions are presented in Section V.

II. MODIFIED ADRC DESIGN
In this article, the performance of nonlinear ADRC for a speed loop control system of ISP is investigated and then improved. The two-axis ISP examined in this article consists of a pitch gimbal and an azimuth gimbal. Pitch gimbal is the controlled plant because the imaging load is mounted on the pitch gimbal, whose performance is more likely to be affected by multi-source disturbances. The block diagram of a typical three-loop control system for pitch gimbal [30] is shown in Fig. 1. G-pos, G-spe and G-cur are the controllers in the position loop, speed loop and current loop, respectively. PWM is a power amplification to drive the torque motor, whose inductance and resistance are L and R, respectively. Besides, K T is the torque coefficient of the motor, and J is the load moment of inertia of the gimbal along the rotation axis. Since the feedback element of the speed loop is rate gyro, the speed loop can be regard as stability loop. In addition, since the current loop is equivalent to a proportional link, the system model can also be equivalent to a firstorder link, on account of its small time constant. Therefore, first order nonlinear ADRC is adopted as the research object. In fact, first order nonlinear ADRC is modified to improve the disturbance rejection ability and the tracking performance of the stability loop control system. The design procedure for the proposed controller is as follows: Firstly, the outer loop controller is designed based on the nominal integral model and the expected control performance; Secondly, appropriate ESO parameters and gyro estimation model are selected to enhance the observer's ability to estimate external disturbances and model uncertainties, so that high-accuracy control can be obtained. The premise of selecting ESO parameters and gyro estimation model is to ensure the dynamic performance of the system under the conditions of uncertain model and observer noise.  Fig. 2 is the block diagram of the nonlinear ADRC for a stability loop control system of ISP, based on the Laplace transform and describing function method. T d (s), G s (s), G c (s) and R(s) are the transfer functions of the disturbance, gyro, stability loop controller and stability loop input, respectively. U (s) is the input voltage of the current loop, K I is the equivalent proportionality coefficient of the current loop, and can be obtained by testing the step response of the current loop. ω(s) is the angular rate of the plant relative to the inertial space. Besides, Y (s) and U 0 (s) are the output of gyro and the stability loop controller, respectively. Z 1 (s) and Z 2 (s) are the output state of extended state observer (ESO), whose parameters consist of b 0 , β 1 , β 2 , α, and δ. Note that N (E) is the describing function of the fal nonlinearity, which is proposed by Han [5], and defined as fal(e, α, δ) = e δ 1−α = ke, |e| ≤ δ |e| α sgn(e), |e| > δ (1) N (E) is a real function of the input amplitude [18], and can be expressed as

A. DISTURBANCE COMPENSATION CORRECTION
where α = 0.5 and τ = arcsin(δ/E). The dotted frame in Fig. 2 is a simplified model for the current loop, motor and controlled plant. Since E(s)= Y(s)-Z 1 (s), the following can be obtained from Fig. 2: Assuming that R(s) = 0, the transfer function from T d (s) to Z 1 (s) can be denoted as (4), shown at the bottom of the page.
What can be found from (4) is that the system output is not zero, so the disturbance cannot be eliminated from the system theoretically.
In Fig. 2, the disturbance observed by ESO is expressed as Z 2 (s), which is used to compensate the influence of disturbance. The compensated controlled plant can be considered as an integral series type restraining the influence of internal and external disturbances. In an ISP control system, the controlled object is equivalent to a first order integral element. Therefore, after the compensation of ESO, Z 1 (s) can be expressed as According to Fig. 2, Z 1 (s) can be denotes as Combining (5) with (6), U 0 (s) can be denoted as U 0 (s) of the original system shown in Fig. 2 is From (7) and (8), the disturbance compensations are Z 2 (s)+β 1 E(s) and Z 2 (s), respectively. Therefore, the compensation quantity can be corrected by adding β 1 E(s). Assuming that the amount of compensation is ε(s), then we can get   3 shows the block diagram of the nonlinear ADRC after disturbance compensation correction. The following can be obtained from Fig. 3: Similarly, by assuming R(s) = 0, Z 1 (s)s can be described as Since G c (s) = -s, the transfer function from T d (s) to Z 1 (s) can be denoted as Therefore, the disturbance compensation accuracy has been significantly improved after disturbance compensation.

B. SPEED OBSERVER IMPROVEMENT
In the stability loop of ISP, the speed of the plant relative to the inertial space is measured by the rate gyro. In practice, the speed of the rate gyro will be lagged for its intrinsic characteristics. To reduce the effects of gyro lag and therefore get the real speed of the controlled plant, the estimated gyro transfer function G sEst (s) is introduced in the observation of angular rate, and the observation feedback Z 1 (s) is introduced as the speed feedback signal. After the improvement of speed observer, the block diagram of the system shown in Fig. 3 can be converted to the nonlinear ADRC depicted in Fig. 4, where the following formula can be obtained (13), as shown at the bottom of the page.
Assuming that T d (s) = 0, the transfer function from Y (s) to Z 1 (s) can be denoted as Considering Y (s) = G s (s)ω(s), we can get Given that b 0 = K I K T /J and G s (s) = G sEst (s), Z 1 (s) can be described as Apparently, the influence of gyro has been eliminated. For the sake of simplicity, in the subsequent parts of this article, typical nonlinear ADRC is described as ADRC-C, the nonlinear ADRC after disturbance compensation correction and speed observer improvement is described as ADRC-A.

III. FREQUENCY ANALYSIS OF MODIFIED ADRC
In this section, the stability, disturbance rejection ability and tracking performance of ISP control system are analyzed in frequency domain by using the describing function method and numerical method. The photoelectric tracking system composed of the two-axis ISP control system, CCD and slip ring is shown in Fig. 5. The biaxial rate gyroscope can detect the angular rates of the pitch and azimuth gimbals relative to the inertial space. The two-axis control circuit and the power amplifier circuit are integrated on a single circuit board.
The tuning of the parameters of ADRC is crucial to enhance the performance of nonlinear ADRC system. Generally, the constrains between the parameters of ADRC are as follows: 1) h is the sampling control step; 2) β 1 and β 2 are the extended state observer parameters when the observer is in a steady state, 2β 1 < β 2 < 10β 1 ; 3) δ should be greater than the peak noise of the gyro; 4) α should be in the range of (0,1]; 5) b 0 is relevant to the controlled plant, and b 0 = K I K T /J . Abiding by these restrictions, through theoretical calculation and off-line identification, some parameters of the plant can be expressed as the sampling period of the stability loop is 1 ms. According to the practical simulations and system debugging, the other parameters of the nonlinear ADRC investigated in this article can be set as h = 0.001, α = 0.5, δ = 0.1, β 1 = 125 rad/s, β 2 = 750 rad/s, the stability loop controller G c (s) is a PD controller, with the parameters of K p = 500, and K d = 10. Note that the constraints such as phase angle and amplitude margin can be acquired by the frequency correction method, based on the nominal integral model and expected control performance, then the optimal PD coefficients can be eventually obtained.

A. DISTURBANCE REJECTION ABILITY ANALYSIS
For a closed-loop system, disturbance rejection ability is usually evaluated by dynamic stiffness, which can be obtained by adopting numerical solution. According to (3), by assuming G p (s) = G s (s)/Js, the following can be derived (see details in Based on (17), block diagram of the nonlinear ADRC which originally depicted in Fig. 2 can be converted and shown in Fig. 6, where the transfer functions G r1 (s), G r2 (s), G f 1 (s), G f 2 (s), G w1 (s) and G w2 (s) are given as Similarly, by assuming G p (s) = G s (s)/Js, according to appendix C, the following can be obtained b 0 (s+G c (s)) Based on (19), block diagram of ADRC-A which originally depicted in Fig. 4 can also be converted and shown in Fig. 6, where the transfer functions can be given as For each transfer function G * (s), |G * | and θ * are defined as its magnitude and phase angel, respectively. To simplify the calculation, by assuming r(t) = 0, when the output of the plant in nonlinear system y(t) = y 0 sin(ωt), for e(t), n(t), w(t), f(t), u(t), v(t) and q(t) shown in Fig. 6, we can get After going through N (E), only the amplitude of e(t) is changed, hence Then After deduction, we can get   After going through G p (s), q(t) is converted to y(t), then Furthermore, the disturbance T d (t) can be obtained and given as After deduction and simplification, the amplitude and phase of T d (t) can be described as Then the amplitude of the amplitude-frequency response of system compliance is According to the numerical solutions, the compliance magnitude frequency response of ADRC-C and ADRC-A can be obtained and shown in Fig. 7. When the frequency is 2 Hz (12.56rad/s), the compliance magnitude of ADRC-C and ADRC-A is −41.22 dB and −50.19 dB, respectively. Since the compliance is the reciprocal of the stiffness, the dynamic stiffness of ADRC-C and ADRC-A are 115 Nm/(rad/s) and 323 Nm/(rad/s), respectively. VOLUME 8, 2020

B. TRACKING PERFORMANCE ANALYSIS
In the analysis of the tracking performance, the influence of external disturbance is ignored. The tracking performance is reflected by the relative tracking accuracy, which can be obtained by dividing angular rate tracking error by the reference angular rate. Numerical method is adopted to pursue the tracking error frequency response of the nonlinear ADRC system, since the output of the nonlinear block N (E) is related to its input E. To facilitate the analysis of tracking performance, summing points are moved and the linear parts are combined based on Fig. 6, the adjusted block diagram for the analysis of tracking performance can be shown as Fig. 8. G r1 (s) in Fig. 8 is identical to that in Fig.6, while G m (s) and G n (s) are converted to To obtain the value of the describing function, supposing e(t) = Esinωt and r(t) = r 0 sin(ωt+θ r ), the following can be obtained from Fig. 8 e 1 (t) = r(t)−y 1 (t) where where |G r1 (jω)|, |G m (jω)| and |G n (jω)| are the magnitudes of G r1 (s), G m (s) and G n (s), and θ r1 , θ m and θ n are the phase angles of G r1 (s), G m (s) and G n (s), respectively. Note that all the six parameters vary with the frequency ω.
Combining (33) with (34), after deduction and simplification, the phase of the input r(t) can be described as Hence the amplitude of input r 0 can be given as (37), shown at the bottom of the next page.
Therefore, the amplitude response of the tracking error in nonlinear ADRC system can be expressed as When (18) and (20) are substituted into the formula from (32) to (38), the amplitude-frequency responses of the relative angular rate tracking error of ADRC-C and ADRC-A are shown in Fig. 9. When a sinusoidal wave with the frequency of 2 Hz and the amplitude of 5 • /s is input to the system, the relative angular rate tracking errors of ADRC-C and ADRC-A are −17.2 dB and −24.15 dB, respectively. Therefore, after modification, tracking performance of ADRC has improved by 55%.

C. STABILITY ANALLYSIS
By assuming R(s) = 0, nonlinear control system ARDC-C and ADRC-A shown in Fig. 10 can be regarded as a basic negative feedback structure consisting of a linear partial L(s) and a nonlinear dynamic partial N (E). Specifically, the linear dynamic partial of ARDC-C and ADRC-A are defined as L 1 (s) and L 2 (s), respectively.
From (18) and (32), as well as Fig. 8 and Fig. 10, the following can be obtained Jb 0 (k d +1) K I K T G s (s) ·s 2 + Jb 0 (k p +β 1 ) Likewise, from (20) and (32), as well as Fig. 8 and Fig. 10, the following can be obtained the following can be obtained Jb 0 (k d +1) K I K T G s (s) ·s 2 + Jb 0 k p K I K T G s (s) +β 1 (k d +1) s+k p β 1 (40)  According to Fig. 10, the stability of the closed-loop system is determined by N (E) and L(s). When the curve of −1/N (E) intersects the Nyquist curve of L(jω), the system will be unstable [18]. By ignoring the influence of gyro transmission, and assuming G s (s) = 1, linear dynamic partial of the basic negative feedback L 1 (s) is identical to that of L 2 (s). In addition, N (E) of ADRC-C is identical to that of ADRC-A. Therefore, the stability is consistent.
When G s (s) = 1, the stability of the closed-loop system is analyzed on the basis of Nyquist criterion, as shown in Fig. 11. Since the curve L 1 (jω) coincides with the curve L 2 (jω), and the two curves has no intersection with the curve of −1/N (E), the closed-loop system is stable. Note that S follows −(1/k) = −δ α−1 . When δ = 0.1 and α = 0.5, the coordinate of S is (−0.316, 0).
In addition, when G s (s) = 1/(0.000016s 2 +0.0056s+1), N (E) keeps invariant while L 1 (s) is different with L 2 (s), so the stability of ADRC-A is different with that of ADRC-C. When G s (s) = 1/(0.000016s 2 +0.0056s+1), the stability of the closed-loop system of ADRC-C and ADRC-A are also analyzed based on Nyquist criterion, as shown in Fig. 12. Obviously, the curve of −1/N (E) has no intersection with the Nyquist curve L 1 (jω) and L 2 (jω), which reveals that the system is stable. Besides, the points L 1 and L 2 are the intersections of L 1 (jω) and L 2 (jω) with the real axis, where L 1 (jω) and L 2 (jω) are the Nyquist curve of ADRC-C, and ADRC-A, respectively. The starting point S of −1/N (E) moves along the real axis as δ varies. When S coincides with  L 1 or L 2 , the system will become unstable. Once the system stability margin is expressed by the distance between S and L 1 /L 2 , the stability margin will be slightly decreased, since the distance changes from 0.2473 to 0.2455.

A. SIMULATION INVESTIGATION
The simulations are based on the inertial stabilization platform of photoelectric tracking system, as shown in Fig. 13. The simulations of disturbance rejection ability, tracking performance and robustness are conducted by MAT-LAB/Simulink. As mentioned above, the parameters are set as α = 0.5, h = 0.001, δ = 0.1, β 1 = 125 rad/s, β 2 = 750 rad/s, K I = 5, K T = 1.4 Nm/A, J = 0.5 kg.m 2 , b 0 = K I K T /J = 14, respectively. Besides, the stabilized

1) DISTURBANCE REJECTION ABILITY
To analyze the disturbance rejection ability of ADRC system, the system input is set as 1 • /s. After the system works steadily at a constant speed, a step disturbance signal with the amplitude of 0.1 Nm is added at the time of 0.5 s. Fig. 14 shows the angular rate response curve under the square wave for ADRC-C and ADRC-A. What can be obtained from Fig. 14 is that, the peak rate error of ADRC-C and ADRC-A are 0.0734 • /s and 0.0438 • /s, respectively. Therefore, after modification, the setting time decreases from 0.047 s to 0.041 s.
Similarly, to analyze the disturbance rejection ability of the ADRC system, the system input is reset as 0 • /s. Then a sinusoidal disturbance signal with the amplitude of 0.1 Nm and the frequency of 2 Hz is added. Fig. 15 shows the   A are 0.143 • /s and 0.051 • /s, respectively. In other words, by adopting ADRC-A, the mean square error of the position error decreases from 0.14 mrad to 0.05 mrad. In addition, error curve of the observer output under the sinusoidal disturbance signal is depicted in Fig. 17. According to Fig. 17, by adopting ADRC-A, peak error of the observer output of ADRC has decreased by 87.9%.
Obviously, compared with ADRC-C, ADRC-A has a remarkable improvement in disturbance rejection ability for the typical step and sinusoidal disturbances. This is mainly due to the significant improvement in disturbance observation (as shown in Fig. 17), which allows the system to respond more quickly and accurately to disturbances.

2) TRACKING PERFORMANCE
To analyze the tracking performance of ADRC system shown in Fig. 13, a square wave is adopted as the system input, with the amplitude of 5 • /s, frequency of 1 Hz (duty ratio: 50%). The torque disturbance is 0 Nm. Fig. 18 shows the angular rate response curve under the square wave for ADRC-C and ADRC-A. For ADRC-C and ADRC-A, the overshoot amounts are 58% and 1.94%, and the corresponding adjustment time are 0.099 s and 0.016 s, respectively. In addition, output torque of motor under square wave input is also depicted in Fig. 19. Obviously, by adopting ADRC-A, the output is more stable, implying that the noise and the mechanical wear can be effectively reduced, during the speed adjustment process of the pitch gimbal.  Similarly, a sinusoidal wave is adopted as the system input, with the amplitude of 5 • /s, frequency of 2 Hz. The torque disturbance is 0 Nm. Fig. 20 shows the angular rate error curve under the sinusoidal wave for ADRC-C and ADRC-A. Obviously, the peak angular rate errors of ADRC-C and ADRC-A are 0.69 • /s and 0.31 • /s, respectively.
By contrast, significant improvement in overshoot, adjustment time and tracking accuracy of ADRC-A can be found, which can be attributed to improved control accuracy and smaller fluctuation of the controlled quantity as shown in Fig. 19.

3) ROBUSTNESS
The robustness of the ADRC system is analyzed from the following aspects: VOLUME 8, 2020

a: GYRO ESTIMATION MODEL G sEst (s)
The empirical transfer function of the accurate gyro estimation model can be expressed as G sEst (s) = 1/(0.000016s 2 +0.0056s+1). However, the deviation of the gyro estimation model often occurs in practice. To examine the influence of the deviation to the performance of the ADRC system, two different estimation errors are added, hence the transfer functions of the gyro estimation model are set as G sEst1 (s) = 1/(0.00001s 2 +0.0044s+1) and G sEst2 (s) = 1/(0.000028s 2 +0.0074s+1), respectively. At first, a square signal with the amplitude of 5 • /s and frequency of 1 Hz is adopted as the system input. The disturbance torque is 0 Nm. The angular rate response with different transfer function of gyro estimation model is shown in Fig. 21. According to Fig. 21, the overshoot changes from 12.8% to 11.9%, and the adjustment time changes from 0.049 s to 0.070 s, correspondingly. Then, a sinusoidal signal with the amplitude of 5 • /s and frequency of 2 Hz is adopted as the system input. The disturbance torque is 0 Nm. Therefore, angular rate error curve with different transfer function of gyro estimation model is shown in Fig. 22. According to Fig. 22, the peak angular rate errors of the system changes from 0.32 • /s to 0.396 • /s.

b: EXTERNAL DISTURBANCE
When the system input is set as 0 • /s, sinusoidal wave disturbances are added, with the frequency of 2 Hz, and the amplitude of 0.13 Nm and 0.16 Nm, respectively. The angular position error curve under sinusoidal disturbances with different amplitude is shown in Fig. 23. According to Fig. 23, the mean square errors of the angular position error are 0.066mrad and 0.082 mrad, respectively. In the simulations, both the estimation model G sEst (s) and the external disturbances have been changed, while the ADRC system remains robust.

B. EXPERIMENTAL INVESTIGATION
Experiments are carried out in a two-axis ISP of photoelectric tracking system located in an unmanned ground vehicle (UGV), by taking the stability loop of pitch gimbal. Fig. 24 is the picture of experimental equipment. The ISP is equipped with a visible light camera and a gyro. A direct current torque motor is used as the actuator for the inertially stabilized platform. The hardware design of the control board adopts DSP+FPGA architecture. The control quantity is calculated in DSP28335, and then output to FPGA in digital coding form. Then the FPGA calculates the code value and carries out pulse width modulation using triangle wave. DSP code is written in C language, and FPGA code is written in VHDL language. Sampling period of the stable loop control software running in DSP is 1ms, and the actual system operation time is 0.43 ms, both of which meet the requirements of real-time capability. Finally, motor is controlled by the pulse width modulation wave with certain duty ratio. Stewart platform is used to simulate the disturbance of the photoelectric tracking system during the movement of unmanned vehicle, under the actual load condition. When a square wave with the amplitude of 5 • /s and a period of 1 s (duty ratio: 50%) is input to the ADRC system, the angular rate errors of ADRC-C and ADRC-A are shown in Fig. 25. For ADRC-C and ADRC-A, the peak angular position errors after integral are 1.14 • and 0.87 • , respectively. After modification, the angular position tracking accuracy improved by 23.7%. When a sinusoidal wave with the amplitude of 5 • /s and a period of 0.5 s is input to the ADRC system, the angular rate errors of ADRC-C and ADRC-A are shown in Fig. 26. The peak angular position error of ADRC-C and ADRC-A after integral are 0.60 • and 0.43 • , implying that the angular position tracking accuracy improved by 28.3%. Besides, according to Fig. 26.b, the mean square error of system tracking error (MSESTE) is 0.49 • /s. In another modified scheme of ADRC [31], when a signal of 1 • and 2 Hz is input, MSESTE is 0.146 • /s, which is equivalent to 0.73 • /s when a signal of 5 • and 2 Hz is input (through linear fitness). By contrast, MSESTE is smaller by adopting our method.
When it comes to the stability accuracy, the vibration of the UGV is simulated by Stewart platform. When a sinusoidal wave is input to the system, with the amplitude of 2 • and a period of 1s, the angular position error relative to the inertial space of ADRC-C and ADRC-A are shown in Fig. 27. Note that the parameters of the sinusoidal wave are identical to that of the main components of the disturbance a UGV encounters when moving on a typical road. The stability accuracy represents the mean square error of the angular position error relative to the inertial space. Stability accuracy of ADRC-C and ADRC-A are 0.061 mrad and 0.038 mrad, respectively. Therefore, by adopting ADRC-A, the stability accuracy improved by 37.7%. In general, in contrast to ADRC-C, the tracking performance and the disturbance compensation ability of the ISP is significantly improved by adopting ADRC-A. This is mainly due to the improvement of disturbance estimation accuracy, compensation accuracy, as well as the state estimation accuracy of angular rate. As a result, high quality tracking images can be obtained by the photoelectric tracking system, through better isolation of the disturbances caused by the movement of the UGV. What's more, stabilized tracking of the targets can be accomplished, because of improved tracking performance.

V. CONCLUSION
Nonlinear ADRC is modified to improve the disturbance rejection ability and tracking performance, based on the disturbance compensation correction and the speed observer improvement. In contrast to the typical ADRC scheme, the dynamic stiffness and tracking performance of the modified ADRC scheme are significantly improved while the stability almost remains invariant. Specifically, by adopting the modified ADRC scheme, the angular position tracking accuracy and the stability accuracy have improved by 23.7% and 37.7% respectively, for the practical ISP control system. In the future, the improvement of high-order ADRC should be further studied.

ADRC
Active disturbance rejection control ADRC-C From (3), after derivation and simplification, the following can be obtained From (3), the following can be obtained Combining (42) with (43), we can get (13), the following can be obtained Combining (14) with (46), we can get He is currently an Engineer with the North Automatic Control Technology Institute. His research interests include design of servo control systems, automatic control algorithm, and motor drive. He has published some articles in the relevant areas. She is currently an Associate Research Fellow with the North Automatic Control Technology Institute, Taiyuan, China. Her research interests include automatic control technologies, cooperative control, and route planning.