# Linear Active Disturbance Rejection Control for Lever-Type Electric Erection System Based on Approximate Model

Authors: Hailong Niu; Qinhe Gao; Shengjin Tang; Wenliang Guan
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

Linear active disturbance rejection control (LADRC) algorithm is proposed to realize accurate trajectory tracking for the lever-type electric erection system. By means of system identification and curve fitting, the approximate model is built, which is consisting of the servo drive system with velocity closed-loop and the lever-type erection mechanism. The proportional control law with velocity feedforward is designed to improve the trajectory tracking performance. The experimental results verify that, based on approximate model, LADRC has better tracking accuracy and stronger robustness to the disturbance caused by the change of intrinsic parameters compared with PI controller.


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## Body

## 1. Introduction

In some weaponry and engineering machinery, the erection system is the important part. During the erection process, the force between loads and the actuator is varying and meanwhile there are friction, parameter variation, and external disturbances. The traditional hydraulic erection system usually uses the multistage hydraulic cylinder as actuator to drive the erection loads, which has shock at changing stage and will affect the rapidity and smoothness of the erection process. Fuzzy sliding mode control ‎[1] and adaptive sliding mode control ‎[2] have been applied to control the erection system. However these control algorithms take valve-controlled cylinder system as the control object with the displacement of the valve core as input and the displacement of the cylinder rod as output. The erection angle is converted from the displacement of the cylinder rod based on the kinematical analysis of the erection mechanism, which always ignores the flexibility of the erection mechanism.The electric cylinder is starting to be used in the erection system, with the development of the electric cylinder and the servo motor ‎[3]. Also, combining the single-stage electric cylinder and lever-type erection mechanism can shorten the stroke and avoid shocks, which is conducive to rapid and smooth erection ‎[4]. As a typical mechatronic servo system, the control target for the electric erection system is making the erection loads track the planned trajectory accurately. Many control methods, such as adaptive backstepping sliding mode control ‎[5], ADRC ‎[6], and so on, have been applied to mechatronic servo systems, while most of them are based on current control for the motor and not suitable for the commercial servo driver. The commercial servo driver has inner current controller and usually can be configured at position, velocity, and torque control mode for users.The concept of active disturbance rejection control (ADRC) was firstly proposed by Han ‎[7], which is independence of the precise mathematical model. ADRC can estimate and compensate for the generalized disturbances caused by the model error and external disturbances by the extended state observer (ESO). Gao ‎[8] developed a concise linear ESO (LESO) and linear ADRC (LADRC), which have fewer parameters and are easy to adjust. The convergence and stability of LADRC have been proved theoretically ‎[9–13] and its effectiveness has been verified in many applications ‎[14–18], which illustrate that LADRC has theoretical integrity and practicability.This paper designs LADRC based on the approximate models of the servo drive system with velocity closed-loop and the lever-type erection mechanism, which are built by means of system identification and curve fitting. The significant control and disturbance-rejection performance of LADRC are verified on the experimental platform, compared with PI controller.In the rest of this paper, Section2 introduces the composition of the electric erection system and builds the approximate model. Section 3 designs LADRC based on the approximate model. Section 4 provides the experimental results of trajectory tracking and disturbance rejection. The conclusions are given in Section 5.


## 2. Composition and Modelling of Electric Erection System

### 2.1. Composition of Electric Erection System

Figure1 ‎[4] shows the electric erection system, which is mainly composed of the controller, the servo driver, the electric cylinder, the erection mechanism, loads, and encoders.Figure 1
Composition of the electric erection system.The servo driver configured at velocity control mode receives the analog voltage command of -10V~+10V. The encoder outputs pulse signals, whose resolution is 0.05°. The motion controller can acquire the signal of angular encoder and calculate the control signal of servo driver according to the control program. The control program is written in MATLAB/Simulink, which can be compiled to codes and loaded to the motion controller, and the sample time is 1 millisecond.


### 2.2. Modelling of Electric Erection System

There are inner velocity controller and current controller in the servo driver configured at velocity control mode, whose parameters are not disclosed. There are also nonlinear factors, such as friction and delay. Therefore, it is difficult to build the accurate model of the velocity loop including servo driver and servo motor.According to the step response curve of the servo motor in Figure2, proportional model ‎[19] and first-order inertia model ‎[16] can describe the velocity loop approximately. The model error can be estimated and compensated for by means of the control algorithm in next section.Figure 2
Step response curve of servo motor.The proportional model can be expressed as(1)G1=WsUs=KΩwhereW(s) andU(s) are Laplace transform of motor speed and control input, respectively, and KΩ is velocity gain.The first-order inertia model can be expressed as(2)G2=WsUs=KΩTΩs+1where TΩ is time constant.The rigid body model of electric cylinder can be expressed as(3)S=θmL2πiwhereS is the extension length of the electric cylinder, θm is the rotation angle of the servo motor,i is the reduction ratio of the reducer, andL is the lead of the ball screw.As is shown in Figure3, the lever-type erection mechanism is consisting of the triangular armO1BC and the connecting rodAB, andO2C represents the electric cylinder.Figure 3
Structure and size of lever-type erection mechanism.Based on the rigid body model of the electric cylinder and the structure and size of the lever-type erection mechanism, the analytical expression can be obtained, which is complicated and inconvenient for building the system model and designing the control algorithm. Therefore, an approximate polynomial expression obtained by curve fitting is used to express the relationship ‎[20]. The expression is ‎[4](4)θ^e=fθm=1.391×10-9θm3-5.863×10-7θm2+0.001763θm-0.005364where the units of θm and θe are both radian (rad).Based on the proportional model (1) and the approximate polynomial expression (6), the first-order system model can be expressed as(5)θ˙e=KΩdfθmdθmuwhereu is the control command of the servo driver.Similarly, based on the first-order inertia model (2) and (6), the second-order system model can be expressed as(6)θ¨e=d2fθm/dθm2dfθm/dθm2θ˙e2-1TΩθ˙e+KΩTΩdfθmdθmu


## 2.1. Composition of Electric Erection System

Figure1 ‎[4] shows the electric erection system, which is mainly composed of the controller, the servo driver, the electric cylinder, the erection mechanism, loads, and encoders.Figure 1
Composition of the electric erection system.The servo driver configured at velocity control mode receives the analog voltage command of -10V~+10V. The encoder outputs pulse signals, whose resolution is 0.05°. The motion controller can acquire the signal of angular encoder and calculate the control signal of servo driver according to the control program. The control program is written in MATLAB/Simulink, which can be compiled to codes and loaded to the motion controller, and the sample time is 1 millisecond.


## 2.2. Modelling of Electric Erection System

There are inner velocity controller and current controller in the servo driver configured at velocity control mode, whose parameters are not disclosed. There are also nonlinear factors, such as friction and delay. Therefore, it is difficult to build the accurate model of the velocity loop including servo driver and servo motor.According to the step response curve of the servo motor in Figure2, proportional model ‎[19] and first-order inertia model ‎[16] can describe the velocity loop approximately. The model error can be estimated and compensated for by means of the control algorithm in next section.Figure 2
Step response curve of servo motor.The proportional model can be expressed as(1)G1=WsUs=KΩwhereW(s) andU(s) are Laplace transform of motor speed and control input, respectively, and KΩ is velocity gain.The first-order inertia model can be expressed as(2)G2=WsUs=KΩTΩs+1where TΩ is time constant.The rigid body model of electric cylinder can be expressed as(3)S=θmL2πiwhereS is the extension length of the electric cylinder, θm is the rotation angle of the servo motor,i is the reduction ratio of the reducer, andL is the lead of the ball screw.As is shown in Figure3, the lever-type erection mechanism is consisting of the triangular armO1BC and the connecting rodAB, andO2C represents the electric cylinder.Figure 3
Structure and size of lever-type erection mechanism.Based on the rigid body model of the electric cylinder and the structure and size of the lever-type erection mechanism, the analytical expression can be obtained, which is complicated and inconvenient for building the system model and designing the control algorithm. Therefore, an approximate polynomial expression obtained by curve fitting is used to express the relationship ‎[20]. The expression is ‎[4](4)θ^e=fθm=1.391×10-9θm3-5.863×10-7θm2+0.001763θm-0.005364where the units of θm and θe are both radian (rad).Based on the proportional model (1) and the approximate polynomial expression (6), the first-order system model can be expressed as(5)θ˙e=KΩdfθmdθmuwhereu is the control command of the servo driver.Similarly, based on the first-order inertia model (2) and (6), the second-order system model can be expressed as(6)θ¨e=d2fθm/dθm2dfθm/dθm2θ˙e2-1TΩθ˙e+KΩTΩdfθmdθmu


## 3. Linear Active Disturbance Rejection Controller Design

The core technology of ADRC is estimating and compensating for the generalized disturbances including the model error and external disturbances, based on the ESO and error feedback control. The LADRC, using the LESO, is simplification of ADRC, with fewer parameters and being easier in engineering implementation. The structure of LADRC is shown in Figure4, whereu0 is the control law,b is the control gain, and z1…zn+1 are the output of the LESO.Figure 4
Structure of LADRC.The control target is to track the planned trajectory with the tracking error in the range of ±0.2°.


### 3.1. Construction of Linear Extended State Observer

Based on the first-order system model (7), the second-order extended state-space representation can be obtained as(7)x˙1=x2+b1ux˙2=hy=x1b1=KΩdfθmdθmwhereb1 is the control gain andh is derivative of the generalized disturbances.The second-order LESO can be constructed as(8)z˙1=z2+2ω1x1-z1+b1uz˙2=ω12x1-z1wherez1 andz2 estimate the erection angle and generalized disturbances, respectively, andω1 is the observer gain.Similarly, based on the second-order system model, the third-order extended state-space representation can be obtained as(9)x˙1=x2x˙2=x3+b2ux˙3=hy=x1b2=KΩTΩdfθmdθmHence, the third-order LESO can be constructed as(10)z˙1=z2+3ω2x1-z1z˙2=z3+3ω22x1-z1+b2uz˙3=ω23x1-z1wherez1,z2, andz3 estimate the erection angle, angular velocity, and generalized disturbances, respectively, andω2 is the observer gain.The estimated generalized disturbances can be compensated for by the error feedback control as(11)u=u0-zdbwhereu0 is the control law which is designed on the basis of the controlled system, zd is the estimated generalized disturbances, andb is the control gain.


### 3.2. Design of Control Law

The erection system is required to track the planned trajectory accurately to realize smooth and fast erection. The planned trajectory includes the reference erection angle, angular velocity, and angular acceleration. The resolution of the erection angle encoder is not very high, so the derivative of angle signal will bring big noise to the control system. Hence, the control law is designed, which takes the observer outputz1 as feedback and the reference angular velocity as feedforward, and expressed as(12)u0=kpθ-z1+θ˙where kp is the proportional coefficient,θ is the reference erection angle, and θ˙ is the reference angular velocity.


### 3.3. Convergence Analysis of Linear Extended State Observer

For the convergence of LESO, define the estimation error as(13)x~i=xi-zi,i=1,2…Theorem 1.
Assumingh is bounded, there exist a constant σi >0 and a finiteT1 >0 such that x~it≤σi, i=1,2…,∀t≥T1>0 andω >0. Furthermore, σi=O(1/ωk) for some positive integerk.This theorem has been proved by Zheng ‎[9]. Moreover, if the generalized disturbances show a slow dynamics compared with that of the observed system, which meansh=0, the observer estimation error can converge to zero ‎[21].Since the reference signals are always bounded,h is bounded with the condition that the generalized disturbances is differentiable ‎[22]. Consequently, when there is unknown model error, the estimation error of LESO can converge to a constant in the finite time and its upper bound monotonously decreases with the observer gain.


## 3.1. Construction of Linear Extended State Observer

Based on the first-order system model (7), the second-order extended state-space representation can be obtained as(7)x˙1=x2+b1ux˙2=hy=x1b1=KΩdfθmdθmwhereb1 is the control gain andh is derivative of the generalized disturbances.The second-order LESO can be constructed as(8)z˙1=z2+2ω1x1-z1+b1uz˙2=ω12x1-z1wherez1 andz2 estimate the erection angle and generalized disturbances, respectively, andω1 is the observer gain.Similarly, based on the second-order system model, the third-order extended state-space representation can be obtained as(9)x˙1=x2x˙2=x3+b2ux˙3=hy=x1b2=KΩTΩdfθmdθmHence, the third-order LESO can be constructed as(10)z˙1=z2+3ω2x1-z1z˙2=z3+3ω22x1-z1+b2uz˙3=ω23x1-z1wherez1,z2, andz3 estimate the erection angle, angular velocity, and generalized disturbances, respectively, andω2 is the observer gain.The estimated generalized disturbances can be compensated for by the error feedback control as(11)u=u0-zdbwhereu0 is the control law which is designed on the basis of the controlled system, zd is the estimated generalized disturbances, andb is the control gain.


## 3.2. Design of Control Law

The erection system is required to track the planned trajectory accurately to realize smooth and fast erection. The planned trajectory includes the reference erection angle, angular velocity, and angular acceleration. The resolution of the erection angle encoder is not very high, so the derivative of angle signal will bring big noise to the control system. Hence, the control law is designed, which takes the observer outputz1 as feedback and the reference angular velocity as feedforward, and expressed as(12)u0=kpθ-z1+θ˙where kp is the proportional coefficient,θ is the reference erection angle, and θ˙ is the reference angular velocity.


## 3.3. Convergence Analysis of Linear Extended State Observer

For the convergence of LESO, define the estimation error as(13)x~i=xi-zi,i=1,2…Theorem 1.
Assumingh is bounded, there exist a constant σi >0 and a finiteT1 >0 such that x~it≤σi, i=1,2…,∀t≥T1>0 andω >0. Furthermore, σi=O(1/ωk) for some positive integerk.This theorem has been proved by Zheng ‎[9]. Moreover, if the generalized disturbances show a slow dynamics compared with that of the observed system, which meansh=0, the observer estimation error can converge to zero ‎[21].Since the reference signals are always bounded,h is bounded with the condition that the generalized disturbances is differentiable ‎[22]. Consequently, when there is unknown model error, the estimation error of LESO can converge to a constant in the finite time and its upper bound monotonously decreases with the observer gain.


## 4. Experiments and Results

### 4.1. Description of Experimental Platform

To verify the effectiveness and control performance of the proposed controller, experiments are performed on the experimental platform shown in Figure4.Table1 shows the main parameters of the experimental platform. KΩ,i, andL are confirmed by the parameters of servo driver and electric cylinder, respectively, while TΩ is identified by the step response curve in Figure 2.Table 1
Main parameters of experimental platform.
Parameters/Units Symbols Values Velocity gain/(rad·s−1·V−1) K Ω 20.944 Time constant/s T Ω 0.0061 Reduction ratio i 2 Lead of ball screw/m L 0.005In the process of experiment, the erection system should track the planned trajectory to realize the erection of 60° within 10s, which is expressed as(14)θ=-π6cos⁡π10t+π6θ˙=π260sin⁡π10tθ¨=π3600cos⁡π10tTo verify the effectiveness and performance of LADRC, second-order LADRC and third-order LADRC are tested on the experimental system, respectively. Meanwhile, PID controller is introduced as a comparison to test the performance of LADRC. Due to the noise of low-resolution encoder, the derivative element may lead to system instability. Therefore, PI controller with first-order low pass filter is designed, which can be expressed as(15)τz˙f+zf=yu′=kPθ-zf+kI∫θ-zfwhereτ is the filter coefficient,zf is the output of first-order low pass filter,kP is the proportional coefficient, andkI is the integral coefficient.After parameters tuning in simulation and on the experimental platform, the tuned control parameters of three controllers can be obtained as shown in Table2. The increase of these parameters would make the system instability.Table 2
Tuned control parameters of three controllers.
Controller Control parameters Symbols Values Second-order LADRC Observer gain ω 1 20 Proportional coefficient k p 1 12  Third-order LADRC Observer gain ω 2 45 Proportional coefficient k p 2 750  PI Proportional coefficient k P 200 Integral coefficient k I 1200 Filter coefficient τ 0.0125


### 4.2. Results of Trajectory Tracking

Figures5 and 6 show the trajectory tracking curves and tracking error curves, respectively. The tracking error is defined as the value of planned trajectory minus the actual value at the same time. The stair-step of actual curves and burrs of tracking error curves are caused by the low resolution of the erection angle encoder.Figure 5
Tracking curves of erection angle.Figure 6
Tracking error curves of erection angle.From Figures5 and 6, it can be seen visually that LADRC can track the planned trajectory with smaller tracking error than PI controller and the tracking accuracy of second-order LADRC is the best of three controllers. According to the velocity variation of planned trajectory, the tracking error of PI controller increases with the erection velocity, since there is phase lag of the filter and no derivative element in the controller which influences the dynamic response capability of the controller. However, the tracking error of third-order LADRC has the opposite trend, since LESO is more sensitive to the noise of low-resolution encoder at low speed.In Table3, minimum, maximum, and terminal error of three controllers during the erection are compared.Table 3
Minimum, maximum, and terminal error of three controllers.
Controller Minimum error Maximum error Terminal error Second-order LADRC -0.04° 0.04° ≈0° Third-order LADRC -0.09° 0.12° -0.05° PI -0.18° 0.11° -0.05°According to Table3, second-order LADRC can keep the tracking error within the range of encoder resolution and the terminal error is almost zero. Although third-order LADRC has larger error and terminal error of -0.05°, it is still within the range of ±0.2° and meets the requirement of tracking error. In theory, increasing the observer gain and proportional coefficient can reduce the tracking error; however this will lead to the erection system chattering with high frequency.These results show that, for the electric erection system, LADRC can control the tracking error within the required range and second-order LADRC has the best tracking accuracy.


### 4.3. Results of Disturbance Rejection

In order to test the disturbance-rejection performance of three controllers, a disturbance of control output is manually set on the erection system. Starting from the time of 5s, the control output is reduced by 10%, which is equal to reducing the velocity gain of servo driver by 10%. This can test the performance of rejecting the change of intrinsic parameters.Figure7 shows the tracking error curves with disturbance on control output. Also the tracking error curves with disturbance are contrasted to their respective curves without disturbance as shown in Figures 8, 9, and 10, respectively.Figure 7
Tracking error curves with disturbance on control output.Figure 8
Two tracking error curves of second-order LADRC.Figure 9
Two tracking error curves of third-order LADRC.Figure 10
Two tracking error curves of PI controller.It is apparent that the effect on second-order LADRC, caused by the reduction of control output, is minimum of three controllers. After the disturbance occurs, LADRC can reduce the tracking error to the level of error without disturbance after a process of adjustment. Especially, second-order LADRC has smaller amplitude and shorter duration of the adjustment process than third-order LADRC. With the disturbance, the tracking error of PI controller is slightly larger than that without the disturbance after an adjustment process.These results indicate that second-order LADRC has strong robustness to the disturbance caused by the change of intrinsic parameters.


## 4.1. Description of Experimental Platform

To verify the effectiveness and control performance of the proposed controller, experiments are performed on the experimental platform shown in Figure4.Table1 shows the main parameters of the experimental platform. KΩ,i, andL are confirmed by the parameters of servo driver and electric cylinder, respectively, while TΩ is identified by the step response curve in Figure 2.Table 1
Main parameters of experimental platform.
Parameters/Units Symbols Values Velocity gain/(rad·s−1·V−1) K Ω 20.944 Time constant/s T Ω 0.0061 Reduction ratio i 2 Lead of ball screw/m L 0.005In the process of experiment, the erection system should track the planned trajectory to realize the erection of 60° within 10s, which is expressed as(14)θ=-π6cos⁡π10t+π6θ˙=π260sin⁡π10tθ¨=π3600cos⁡π10tTo verify the effectiveness and performance of LADRC, second-order LADRC and third-order LADRC are tested on the experimental system, respectively. Meanwhile, PID controller is introduced as a comparison to test the performance of LADRC. Due to the noise of low-resolution encoder, the derivative element may lead to system instability. Therefore, PI controller with first-order low pass filter is designed, which can be expressed as(15)τz˙f+zf=yu′=kPθ-zf+kI∫θ-zfwhereτ is the filter coefficient,zf is the output of first-order low pass filter,kP is the proportional coefficient, andkI is the integral coefficient.After parameters tuning in simulation and on the experimental platform, the tuned control parameters of three controllers can be obtained as shown in Table2. The increase of these parameters would make the system instability.Table 2
Tuned control parameters of three controllers.
Controller Control parameters Symbols Values Second-order LADRC Observer gain ω 1 20 Proportional coefficient k p 1 12  Third-order LADRC Observer gain ω 2 45 Proportional coefficient k p 2 750  PI Proportional coefficient k P 200 Integral coefficient k I 1200 Filter coefficient τ 0.0125


## 4.2. Results of Trajectory Tracking

Figures5 and 6 show the trajectory tracking curves and tracking error curves, respectively. The tracking error is defined as the value of planned trajectory minus the actual value at the same time. The stair-step of actual curves and burrs of tracking error curves are caused by the low resolution of the erection angle encoder.Figure 5
Tracking curves of erection angle.Figure 6
Tracking error curves of erection angle.From Figures5 and 6, it can be seen visually that LADRC can track the planned trajectory with smaller tracking error than PI controller and the tracking accuracy of second-order LADRC is the best of three controllers. According to the velocity variation of planned trajectory, the tracking error of PI controller increases with the erection velocity, since there is phase lag of the filter and no derivative element in the controller which influences the dynamic response capability of the controller. However, the tracking error of third-order LADRC has the opposite trend, since LESO is more sensitive to the noise of low-resolution encoder at low speed.In Table3, minimum, maximum, and terminal error of three controllers during the erection are compared.Table 3
Minimum, maximum, and terminal error of three controllers.
Controller Minimum error Maximum error Terminal error Second-order LADRC -0.04° 0.04° ≈0° Third-order LADRC -0.09° 0.12° -0.05° PI -0.18° 0.11° -0.05°According to Table3, second-order LADRC can keep the tracking error within the range of encoder resolution and the terminal error is almost zero. Although third-order LADRC has larger error and terminal error of -0.05°, it is still within the range of ±0.2° and meets the requirement of tracking error. In theory, increasing the observer gain and proportional coefficient can reduce the tracking error; however this will lead to the erection system chattering with high frequency.These results show that, for the electric erection system, LADRC can control the tracking error within the required range and second-order LADRC has the best tracking accuracy.


## 4.3. Results of Disturbance Rejection

In order to test the disturbance-rejection performance of three controllers, a disturbance of control output is manually set on the erection system. Starting from the time of 5s, the control output is reduced by 10%, which is equal to reducing the velocity gain of servo driver by 10%. This can test the performance of rejecting the change of intrinsic parameters.Figure7 shows the tracking error curves with disturbance on control output. Also the tracking error curves with disturbance are contrasted to their respective curves without disturbance as shown in Figures 8, 9, and 10, respectively.Figure 7
Tracking error curves with disturbance on control output.Figure 8
Two tracking error curves of second-order LADRC.Figure 9
Two tracking error curves of third-order LADRC.Figure 10
Two tracking error curves of PI controller.It is apparent that the effect on second-order LADRC, caused by the reduction of control output, is minimum of three controllers. After the disturbance occurs, LADRC can reduce the tracking error to the level of error without disturbance after a process of adjustment. Especially, second-order LADRC has smaller amplitude and shorter duration of the adjustment process than third-order LADRC. With the disturbance, the tracking error of PI controller is slightly larger than that without the disturbance after an adjustment process.These results indicate that second-order LADRC has strong robustness to the disturbance caused by the change of intrinsic parameters.


## 5. Conclusion

The lever-type electric erection system is a complicated position servo system consisting of servo system and lever-type erection mechanism. LADRC can successfully realize trajectory tracking control, based on the approximate models of velocity loop and erection mechanism. Significant trajectory tracking and disturbance-rejection performance are achieved in the erection experiments. Based on the experimental results, second-order LADRC has better tracking accuracy and stronger robustness to the disturbance caused by the change of intrinsic parameters than PI controller and third-order LADRC.



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