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

Linear Active Disturbance Rejection Control for Lever-Type Electric Erection System Based on... Hindawi Journal of Control Science and Engineering Volume 2019, Article ID 3742694, 7 pages https://doi.org/10.1155/2019/3742694 Research Article Linear Active Disturbance Rejection Control for Lever-Type Electric Erection System Based on Approximate Model Hailong Niu , Qinhe Gao , Shengjin Tang , and Wenliang Guan Xi’anHighTechnologyInstitute,Xi’an710025,China Correspondence should be addressed to Qinhe Gao; qhgao201@126.com Received 26 December 2018; Accepted 8 March 2019; Published 27 March 2019 Academic Editor: Paolo Mercorelli Copyright © 2019 Hailong Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 tfi ting, the approximate model is built, which is consisting of theservo drivesystemwithvelocityclosed-loopandthelever-typeerectionmechanism.Theproportionalcontrollawwithvelocity feedforward is designed to improve the trajectory tracking performance. eTh 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. 1. Introduction the control target for the electric erection system is making the erection loads track the planned trajectory accurately. In some weaponry and engineering machinery, the erection Many control methods, such as adaptive backstepping sliding system is the important part. During the erection process, modecontrol[5],ADRC[6],andsoon,havebeenapplied theforce betweenloadsandthe actuator isvaryingand to mechatronic servo systems, while most of them are based meanwhile there are friction, parameter variation, and exter- on current control for the motor and not suitable for the nal disturbances. The traditional hydraulic erection system commercial servo driver. The commercial servo driver has usually uses the multistage hydraulic cylinder as actuator to inner current controller and usually can be congfi ured at drive the erection loads, which has shock at changing stage position, velocity, and torque control mode for users. and will aeff ct the rapidity and smoothness of the erection The concept of active disturbance rejection control process. Fuzzyslidingmodecontrol [1]andadaptivesliding (ADRC) was firstly proposed by Han [7], which is inde- mode control [2] have been applied to control the erec- pendence of the precise mathematical model. ADRC can tion system. However these control algorithms take valve- estimate and compensate for the generalized disturbances controlled cylinder system as the control object with the caused by the model error and external disturbances by the displacement of the valve core as input and the displacement extended state observer (ESO). Gao [8] developed a concise of the cylinder rod as output. eTh erection angle is converted linear ESO (LESO) and linear ADRC (LADRC), which have from the displacement of the cylinder rod based on the fewer parameters and are easy to adjust. The convergence and kinematical analysis of the erection mechanism, which always stability of LADRC have been proved theoretically [9–13] and ignores the flexibility of the erection mechanism. its effectiveness has been verified in many applications [14– The electric cylinder is starting to be used in the erection 18], which illustrate that LADRC has theoretical integrity and system, with the development of the electric cylinder and practicability. the servo motor [3]. Also, combining the single-stage electric This paper designs LADRC based on the approximate cylinder and lever-type erection mechanism can shorten the models of the servo drive system with velocity closed-loop stroke andavoid shocks,which isconducivetorapid and and the lever-type erection mechanism, which are built smooth erection [4]. As a typical mechatronic servo system, by means of system identification and curve tfi ting. The 2 Journal of Control Science and Engineering Motion Terminal Encoder of erection angle controller board IPC Servo driver Electric cylinder + Erection mechanism + Loads Figure 1: Composition of the electric erection system. significant control and disturbance-rejection performance of LADRC are verified on the experimental platform, compared with PI controller. In the rest of this paper, Section 2 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. eTh conclusions are given in Section 5. 2. Composition and Modelling of Electric Erection System −10 0.8 0.9 1 1.1 1.2 1.3 1.4 2.1. Composition of Electric Erection System. Figure 1 [4] time (sec) shows the electric erection system, which is mainly composed Desired of the controller, the servo driver, the electric cylinder, the Actual erection mechanism, loads, and encoders. The servo driver configured at velocity control mode Figure 2: Step response curve of servo motor. receives the analog voltage command of -10V∼+10V. eTh encoder outputs pulse signals, whose resolution is 0.05 .The model error can be estimated and compensated for by means motion controller can acquire the signal of angular encoder of the control algorithm in next section. and calculate the control signal of servo driver according The proportional model can be expressed as to the control program. The control program is written in MATLAB/Simulink, which can be compiled to codes and 푊 (푠 ) 퐺 = =퐾 (1) 1 Ω loaded to the motion controller, and the sample time is 1 푈 (푠 ) millisecond. where W(s)and U(s) are Laplace transform of motor speed and control input, respectively, and 퐾 is velocity gain. 2.2. Modelling of Electric Erection System. There are inner The first-order inertia model can be expressed as velocity controller and current controller in the servo driver 푊 (푠 ) 퐾 configured at velocity control mode, whose parameters are 퐺 = = (2) not disclosed. er Th e are also nonlinear factors, such as 푈 (푠 ) 푇 푠+1 friction and delay. er Th efore, it is difficult to build the where 푇 is time constant. accurate model of the velocity loop including servo driver and The rigid body model of electric cylinder can be expressed servo motor. as According to the step response curve of the servo motor 휃 퐿 in Figure 2, proportional model [19] and first-order inertia (3) 푆= model [16] can describe the velocity loop approximately. The 2휋푖 speed (r/min) 0.22 0.15 0.3 0.79 0.8 Journal of Control Science and Engineering 3 0.75 A x 0.15 0.07 Figure 3: Structure and size of lever-type erection mechanism. where S is the extension length of the electric cylinder, 휃 is in engineering implementation. The structure of LADRC is the rotation angle of the servo motor, i is the reduction ratio shown in Figure 4, where u is the control law, b is the control of the reducer, and L is the lead of the ball screw. gain, and 푧 ...푧 arethe output oftheLESO. 1 𝑛+1 As isshowninFigure3,the lever-type erection mech- eTh controltargetistotracktheplanned trajectory with anism is consisting of the triangular arm O BC and the the tracking error in the range of ±0.2 . connecting rod AB,and O C represents the electric cylinder. Based on the rigid body model of the electric cylinder and 3.1. Construction of Linear Extended State Observer. Based on the structure and size of the lever-type erection mechanism, the rfi st-order system model (7), the second-order extended the analytical expression can be obtained, which is compli- state-space representation can be obtained as cated and inconvenient for building the system model and designing the control algorithm. eTh refore, an approximate 푥 ̇ =푥 +푏 푢 1 2 1 polynomial expression obtained by curve tfi ting is used to express the relationship [20]. The expression is [4] 푥 ̇ =ℎ (7) 휃 =푓 휃 푦=푥 𝑒 𝑚 −9 3 −7 2 (4) = 1.391 × 10 휃 − 5.863 × 10 휃 + 0.001763휃 휃 𝑚 𝑚 𝑚 푏 =퐾 1 Ω − 0.005364 where b is the control gain and h is derivative of the where the units of 휃 and 휃 are both radian (rad). 1 𝑚 𝑒 generalized disturbances. Based on the proportional model (1) and the approximate The second-order LESO can be constructed as polynomial expression (6), the first-order system model can be expressed as 푧 ̇ =푧 +2휔 푥 −푧 +푏 푢 1 2 1 1 1 1 휃  (8) 휃 =퐾 푢 (5) 𝑒 Ω 푧 =휔 푥 −푧 2 1 1 1 where u is the control command of the servo driver. where z and z estimate the erection angle and generalized 1 2 Similarly, based on the first-order inertia model (2) and disturbances, respectively, and 휔 is the observer gain. (6), the second-order system model can be expressed as Similarly, based on the second-order system model, the third-order extended state-space representation can be 2 2 푑 푓휃 /휃푑 휃 1 퐾 𝑚 𝑚 2 Ω 𝑚 ̈ ̇ ̇ obtained as 휃 = 휃 − 휃 + 푢 (6) 𝑒 𝑒 2 𝑒 푇 푇 푑푓 휃 /휃푑  Ω Ω 𝑚 𝑚 𝑚 푥 ̇ =푥 1 2 3. Linear Active Disturbance Rejection 푥 ̇ =푥 +푏 푢 2 3 2 Controller Design 푥 ̇ =ℎ (9) The core technology of ADRC is estimating and compen- 푦=푥 sating for the generalized disturbances including the model error and external disturbances, based on the ESO and error Ω 𝑚 feedback control. The LADRC, using the LESO, is simpli- 푏 = fication of ADRC, with fewer parameters and being easier Ω 𝑚 0.1261 푑휃 푑푓 푑휃 푑푓 푑휃 푑푓 푑휃 푑푓 4 Journal of Control Science and Engineering Table 1: Main parameters of experimental platform. Parameters/Units Symbols Values −1 −1 Velocity gain/(rad⋅s ⋅V ) 퐾 20.944 Time constant/s 푇 0.0061 Reduction ratio 푖 2 Lead of ball screw/m 퐿 0.005 Disturbance compensation Target position u Trajectory Control 0 1/b Plant planning law n+1 z ···z 1 n LESO Figure 4: Structure of LADRC. Hence, the third-order LESO can be constructed as 1,2..., ∀푡≥ 푇 >0 and 휔> 0. Furthermore, 휎 = O(1/휔 ) 1 𝑖 for some positive integer k. 푧 =푧 +3휔 푥 −푧 1 2 2 1 1 Thistheorem hasbeenprovedbyZheng [9].Moreover,if 푧 ̇ =푧 +3휔 푥 −푧 +푏 푢 (10) 2 3 2 1 1 2 the generalized disturbances show a slow dynamics compared 푧 ̇ =휔 푥 −푧 with that of the observed system, which means h=0, the 3 1 1 observer estimation error can converge to zero [21]. where z , z ,and z estimate the erection angle, angular 1 2 3 Sincethereferencesignals arealwaysbounded, h is velocity, and generalized disturbances, respectively, and 휔 is bounded with the condition that the generalized disturbances the observer gain. is dieff rentiable [22]. Consequently, when there is unknown The estimated generalized disturbances can be compen- model error, the estimation error of LESO can converge to a sated for by the error feedback control as constant in the ni fi te time and its upper bound monotonously decreases with the observer gain. 푢 −푧 0 𝑑 푢= (11) 4. Experiments and Results where u is the control law which is designed on the basis 4.1. Description of Experimental Platform. To verify the ee ff c- of the controlled system, 푧 is the estimated generalized disturbances, and b is the control gain. tiveness and control performance of the proposed controller, experiments are performed on the experimental platform 3.2. Design of Control Law. The erection system is required to showninFigure4. tracktheplannedtrajectoryaccuratelytorealize smooth and Table 1 shows the main parameters of the experimental fast erection. eTh planned trajectory includes the reference platform. 퐾 , i,and L are confirmed by the parameters of erection angle, angular velocity, and angular acceleration. eTh servo driver and electric cylinder, respectively, while 푇 is resolution of the erection angle encoder is not very high, so identiefi dbythestep response curveinFigure2. thederivativeofanglesignalwillbringbignoisetothecontrol In the process of experiment, the erection system should system. Hence, the control law is designed, which takes the track the planned trajectory to realize the erection of 60 observer output z as feedback and the reference angular within 10s, which is expressed as velocity as feedforward, and expressed as 휋 휋 휋 휃=− cos $ 푡%+ 푢 =푘 휃 − 푧 + 휃 (12) 6 10 6 0 𝑝 1 휋 휋 where 푘 is the proportional coefficient, 휃 is the reference 휃= sin $ 푡% (14) 60 10 erection angle, and 휃 is the reference angular velocity. 휋 휋 휃= cos $ 푡% 3.3. Convergence Analysis of Linear Extended State Observer. 600 10 For the convergence of LESO, define the estimation error as To verify the eeff ctiveness and performance of LADRC, second-order LADRC and third-order LADRC are tested on 푥  =푥 −푧 , 푖 = 1,2... (13) 𝑖 𝑖 𝑖 the experimental system, respectively. Meanwhile, PID con- Theorem 1. Assuming h is bounded, there exist a constant troller is introduced as a comparison to test the performance 휎 >0andafinite T >0suchthat |푥  (푡)| ≤ 휎 , 푖= of LADRC. Due to the noise of low-resolution encoder, the 𝑖 1 𝑖 𝑖 Journal of Control Science and Engineering 5 Table 2: Tuned control parameters of three controllers. Controller Control parameters Symbols Values Observer gain 휔 20 Second-order LADRC Proportional coefficient 푘 12 𝑝1 Observer gain 휔 45 Third-order LADRC Proportional coefficient 푘 750 𝑝2 Proportional coefficient 푘 200 PI Integral coefficient 푘 1200 Filter coefficient 휏 0.0125 70 0.2 0.15 0.1 Zoom in 0.05 54.5 30 54.3 −0.05 54.1 7.98 8 8.02 −0.1 −0.15 0 −0.2 02468 10 02468 10 time (sec) time (sec) Third-order LADRC Second-order LADRC Planned trajectory Second-order LADRC PI Third-order LADRC PI Figure 5: Tracking curves of erection angle. Figure 6: Tracking error curves of erection angle. derivative element may lead to system instability. er Th efore, PI controller with first-order low pass filter is designed, which than PI controller and the tracking accuracy of second-order canbeexpressed as LADRC is the best of three controllers. According to the velocity variation of planned trajectory, the tracking error of 휏 푧 ̇ +푧 =푦 𝑓 𝑓 PI controller increases with the erection velocity, since there (15) is phase lag of the filter and no derivative element in the 푢 =푘 '휃 − 푧 *+푘 - '휃 − 푧 * controller which influences the dynamic response capability 𝑃 𝑓 𝐼 𝑓 of the controller. However, the tracking error of third-order where 휏 is the filter coefficient, z is the output of rfi st-order f LADRChas theoppositetrend,since LESOis more sensitive lowpassfilter, k is the proportional coefficient, and k is the to the noise of low-resolution encoder at low speed. P I integral coefficient. In Table 3, minimum, maximum, and terminal error of After parameters tuning in simulation and on the exper- three controllers during the erection are compared. imental platform, the tuned control parameters of three According to Table 3, second-order LADRC can keep controllers can be obtained as shown in Table 2. The increase the tracking error within the range of encoder resolution of these parameters would make the system instability. and the terminal error is almost zero. Although third-order LADRC has larger error and terminal error of -0.05 ,itis 4.2. Results of Trajectory Tracking. Figures5and6show still within the range of ±0.2 andmeets therequirement the trajectory tracking curves and tracking error curves, of tracking error. In theory, increasing the observer gain respectively. eTh tracking error is defined as the value of and proportional coefficient can reduce the tracking error; plannedtrajectoryminus theactualvalueat thesametime. however this will lead to the erection system chattering with The stair-step of actual curves and burrs of tracking error high frequency. curves are caused by the low resolution of the erection angle These results show that, for the electric erection system, encoder. LADRC can control the tracking error within the required From Figures 5 and 6, it can be seen visually that LADRC range and second-order LADRC has the best tracking accu- can track the planned trajectory with smaller tracking error racy. erection angle (deg) tracking error (deg) 6 Journal of Control Science and Engineering 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 0.2 0.08 0.15 0.06 0.1 0.04 0.05 0.02 −0.05 −0.02 −0.1 −0.04 −0.15 −0.06 −0.2 02468 10 02468 10 time (sec) time (sec) Second-order LADRC Without disturbance Third-order LADRC With disturbance PI Figure 8: Two tracking error curves of second-order LADRC. Figure 7: Tracking error curves with disturbance on control output. 0.15 4.3. Results of Disturbance Rejection. In order to test the 0.1 disturbance-rejection performance of three controllers, a disturbance of control output is manually set on the erection 0.05 system. Starting from the time of 5s, the control output is reducedby10%,whichisequaltoreducingthevelocitygainof servodriverby10%.Thiscantesttheperformanceofrejecting the change of intrinsic parameters. −0.05 Figure 7 shows the tracking error curves with disturbance on control output. Also the tracking error curves with −0.1 disturbance are contrasted to their respective curves without disturbanceasshownin Figures8,9,and 10,respectively. −0.15 It is apparent that the eeff ct on second-order LADRC, 02468 10 caused by the reduction of control output, is minimum time (sec) of three controllers. After the disturbance occurs, LADRC canreducethe trackingerrortothe leveloferror without Without disturbance disturbance aer ft a process of adjustment. Especially, second- With disturbance order LADRC has smaller amplitude and shorter duration Figure 9: Two tracking error curves of third-order LADRC. 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 aeft r an adjustment process. erection mechanism. LADRC can successfully realize trajec- These results indicate that second-order LADRC has tory tracking control, based on the approximate models of strong robustness to the disturbance caused by the change of velocity loop and erection mechanism. Significant trajectory intrinsic parameters. tracking and disturbance-rejection performance are achieved in the erection experiments. Based on the experimental results, second-order LADRC has better tracking accuracy 5. Conclusion andstrongerrobustnesstothedisturbancecausedbythe eTh lever-type electric erection system is a complicated posi- change of intrinsic parameters than PI controller and third- tion servo system consisting of servo system and lever-type order LADRC. tracking error (deg) tracking error (deg) tracking error (deg) Journal of Control Science and Engineering 7 0.15 [6] Y.Zuo,J.Zhang,C.Liu,and T. 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Zeng et al., “Research on frequency- Without disturbance band characteristics and parameters configuration of linear With disturbance active disturbance rejection control for second-order systems,” Control eTh ory and Applications ,vol.30, no.12, pp.1630–1640, Figure 10: Two tracking error curves of PI controller. [11] W. C. Xue and Y. Huang, “On frequency-domain analysis of ADRC for uncertain system,” in Proceedings of the American Data Availability Control Conference, pp. 6652–6657, Washington, DC, USA, The data used to support the findings of this study are avail- [12] Y. Huang and W. Xue, “Active disturbance rejection control: able from the author [Hailong Niu, hailong9281@163.com] methodology and theoretical analysis,” ISA Transactions,vol. upon request. 53,no.4, pp.963–976,2014. [13] Q. Zheng and Z. Gao, “Active disturbance rejection control: Conflicts of Interest between the formulation in time and the understanding in frequency,” Control Theory and Technology ,vol.14,no.3,pp. eTh authors declare that there are no conflicts of interest 250–259, 2016. regarding the publication of this paper. [14] J. Tao, Q. Sun, Z. Chen, and Y. He, “LADRC-based trajectory tracking control for a parafoil system,” Journal of Harbin Acknowledgments Engineering University,vol.39,no.3,pp.510–516,2018. [15] Z. Ma, X. Zhu, and Z. Zhou, “A lateral-directional control This work was supported by the National Natural Science method combining rudder and propeller for full-wing solar- Foundation of China under Grant no. 617034 and Natural powered UAV,” Acta Aeronautica et Astronautica Sinica,vol.39, Science Foundation of Shaanxi Province under Grant no. no.3,Article ID321633,2018. 2017JQ6015. 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Linear Active Disturbance Rejection Control for Lever-Type Electric Erection System Based on Approximate Model

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Hindawi Publishing Corporation
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Copyright © 2019 Hailong Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract

Hindawi Journal of Control Science and Engineering Volume 2019, Article ID 3742694, 7 pages https://doi.org/10.1155/2019/3742694 Research Article Linear Active Disturbance Rejection Control for Lever-Type Electric Erection System Based on Approximate Model Hailong Niu , Qinhe Gao , Shengjin Tang , and Wenliang Guan Xi’anHighTechnologyInstitute,Xi’an710025,China Correspondence should be addressed to Qinhe Gao; qhgao201@126.com Received 26 December 2018; Accepted 8 March 2019; Published 27 March 2019 Academic Editor: Paolo Mercorelli Copyright © 2019 Hailong Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 tfi ting, the approximate model is built, which is consisting of theservo drivesystemwithvelocityclosed-loopandthelever-typeerectionmechanism.Theproportionalcontrollawwithvelocity feedforward is designed to improve the trajectory tracking performance. eTh 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. 1. Introduction the control target for the electric erection system is making the erection loads track the planned trajectory accurately. In some weaponry and engineering machinery, the erection Many control methods, such as adaptive backstepping sliding system is the important part. During the erection process, modecontrol[5],ADRC[6],andsoon,havebeenapplied theforce betweenloadsandthe actuator isvaryingand to mechatronic servo systems, while most of them are based meanwhile there are friction, parameter variation, and exter- on current control for the motor and not suitable for the nal disturbances. The traditional hydraulic erection system commercial servo driver. The commercial servo driver has usually uses the multistage hydraulic cylinder as actuator to inner current controller and usually can be congfi ured at drive the erection loads, which has shock at changing stage position, velocity, and torque control mode for users. and will aeff ct the rapidity and smoothness of the erection The concept of active disturbance rejection control process. Fuzzyslidingmodecontrol [1]andadaptivesliding (ADRC) was firstly proposed by Han [7], which is inde- mode control [2] have been applied to control the erec- pendence of the precise mathematical model. ADRC can tion system. However these control algorithms take valve- estimate and compensate for the generalized disturbances controlled cylinder system as the control object with the caused by the model error and external disturbances by the displacement of the valve core as input and the displacement extended state observer (ESO). Gao [8] developed a concise of the cylinder rod as output. eTh erection angle is converted linear ESO (LESO) and linear ADRC (LADRC), which have from the displacement of the cylinder rod based on the fewer parameters and are easy to adjust. The convergence and kinematical analysis of the erection mechanism, which always stability of LADRC have been proved theoretically [9–13] and ignores the flexibility of the erection mechanism. its effectiveness has been verified in many applications [14– The electric cylinder is starting to be used in the erection 18], which illustrate that LADRC has theoretical integrity and system, with the development of the electric cylinder and practicability. the servo motor [3]. Also, combining the single-stage electric This paper designs LADRC based on the approximate cylinder and lever-type erection mechanism can shorten the models of the servo drive system with velocity closed-loop stroke andavoid shocks,which isconducivetorapid and and the lever-type erection mechanism, which are built smooth erection [4]. As a typical mechatronic servo system, by means of system identification and curve tfi ting. The 2 Journal of Control Science and Engineering Motion Terminal Encoder of erection angle controller board IPC Servo driver Electric cylinder + Erection mechanism + Loads Figure 1: Composition of the electric erection system. significant control and disturbance-rejection performance of LADRC are verified on the experimental platform, compared with PI controller. In the rest of this paper, Section 2 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. eTh conclusions are given in Section 5. 2. Composition and Modelling of Electric Erection System −10 0.8 0.9 1 1.1 1.2 1.3 1.4 2.1. Composition of Electric Erection System. Figure 1 [4] time (sec) shows the electric erection system, which is mainly composed Desired of the controller, the servo driver, the electric cylinder, the Actual erection mechanism, loads, and encoders. The servo driver configured at velocity control mode Figure 2: Step response curve of servo motor. receives the analog voltage command of -10V∼+10V. eTh encoder outputs pulse signals, whose resolution is 0.05 .The model error can be estimated and compensated for by means motion controller can acquire the signal of angular encoder of the control algorithm in next section. and calculate the control signal of servo driver according The proportional model can be expressed as to the control program. The control program is written in MATLAB/Simulink, which can be compiled to codes and 푊 (푠 ) 퐺 = =퐾 (1) 1 Ω loaded to the motion controller, and the sample time is 1 푈 (푠 ) millisecond. where W(s)and U(s) are Laplace transform of motor speed and control input, respectively, and 퐾 is velocity gain. 2.2. Modelling of Electric Erection System. There are inner The first-order inertia model can be expressed as velocity controller and current controller in the servo driver 푊 (푠 ) 퐾 configured at velocity control mode, whose parameters are 퐺 = = (2) not disclosed. er Th e are also nonlinear factors, such as 푈 (푠 ) 푇 푠+1 friction and delay. er Th efore, it is difficult to build the where 푇 is time constant. accurate model of the velocity loop including servo driver and The rigid body model of electric cylinder can be expressed servo motor. as According to the step response curve of the servo motor 휃 퐿 in Figure 2, proportional model [19] and first-order inertia (3) 푆= model [16] can describe the velocity loop approximately. The 2휋푖 speed (r/min) 0.22 0.15 0.3 0.79 0.8 Journal of Control Science and Engineering 3 0.75 A x 0.15 0.07 Figure 3: Structure and size of lever-type erection mechanism. where S is the extension length of the electric cylinder, 휃 is in engineering implementation. The structure of LADRC is the rotation angle of the servo motor, i is the reduction ratio shown in Figure 4, where u is the control law, b is the control of the reducer, and L is the lead of the ball screw. gain, and 푧 ...푧 arethe output oftheLESO. 1 𝑛+1 As isshowninFigure3,the lever-type erection mech- eTh controltargetistotracktheplanned trajectory with anism is consisting of the triangular arm O BC and the the tracking error in the range of ±0.2 . connecting rod AB,and O C represents the electric cylinder. Based on the rigid body model of the electric cylinder and 3.1. Construction of Linear Extended State Observer. Based on the structure and size of the lever-type erection mechanism, the rfi st-order system model (7), the second-order extended the analytical expression can be obtained, which is compli- state-space representation can be obtained as cated and inconvenient for building the system model and designing the control algorithm. eTh refore, an approximate 푥 ̇ =푥 +푏 푢 1 2 1 polynomial expression obtained by curve tfi ting is used to express the relationship [20]. The expression is [4] 푥 ̇ =ℎ (7) 휃 =푓 휃 푦=푥 𝑒 𝑚 −9 3 −7 2 (4) = 1.391 × 10 휃 − 5.863 × 10 휃 + 0.001763휃 휃 𝑚 𝑚 𝑚 푏 =퐾 1 Ω − 0.005364 where b is the control gain and h is derivative of the where the units of 휃 and 휃 are both radian (rad). 1 𝑚 𝑒 generalized disturbances. Based on the proportional model (1) and the approximate The second-order LESO can be constructed as polynomial expression (6), the first-order system model can be expressed as 푧 ̇ =푧 +2휔 푥 −푧 +푏 푢 1 2 1 1 1 1 휃  (8) 휃 =퐾 푢 (5) 𝑒 Ω 푧 =휔 푥 −푧 2 1 1 1 where u is the control command of the servo driver. where z and z estimate the erection angle and generalized 1 2 Similarly, based on the first-order inertia model (2) and disturbances, respectively, and 휔 is the observer gain. (6), the second-order system model can be expressed as Similarly, based on the second-order system model, the third-order extended state-space representation can be 2 2 푑 푓휃 /휃푑 휃 1 퐾 𝑚 𝑚 2 Ω 𝑚 ̈ ̇ ̇ obtained as 휃 = 휃 − 휃 + 푢 (6) 𝑒 𝑒 2 𝑒 푇 푇 푑푓 휃 /휃푑  Ω Ω 𝑚 𝑚 𝑚 푥 ̇ =푥 1 2 3. Linear Active Disturbance Rejection 푥 ̇ =푥 +푏 푢 2 3 2 Controller Design 푥 ̇ =ℎ (9) The core technology of ADRC is estimating and compen- 푦=푥 sating for the generalized disturbances including the model error and external disturbances, based on the ESO and error Ω 𝑚 feedback control. The LADRC, using the LESO, is simpli- 푏 = fication of ADRC, with fewer parameters and being easier Ω 𝑚 0.1261 푑휃 푑푓 푑휃 푑푓 푑휃 푑푓 푑휃 푑푓 4 Journal of Control Science and Engineering Table 1: Main parameters of experimental platform. Parameters/Units Symbols Values −1 −1 Velocity gain/(rad⋅s ⋅V ) 퐾 20.944 Time constant/s 푇 0.0061 Reduction ratio 푖 2 Lead of ball screw/m 퐿 0.005 Disturbance compensation Target position u Trajectory Control 0 1/b Plant planning law n+1 z ···z 1 n LESO Figure 4: Structure of LADRC. Hence, the third-order LESO can be constructed as 1,2..., ∀푡≥ 푇 >0 and 휔> 0. Furthermore, 휎 = O(1/휔 ) 1 𝑖 for some positive integer k. 푧 =푧 +3휔 푥 −푧 1 2 2 1 1 Thistheorem hasbeenprovedbyZheng [9].Moreover,if 푧 ̇ =푧 +3휔 푥 −푧 +푏 푢 (10) 2 3 2 1 1 2 the generalized disturbances show a slow dynamics compared 푧 ̇ =휔 푥 −푧 with that of the observed system, which means h=0, the 3 1 1 observer estimation error can converge to zero [21]. where z , z ,and z estimate the erection angle, angular 1 2 3 Sincethereferencesignals arealwaysbounded, h is velocity, and generalized disturbances, respectively, and 휔 is bounded with the condition that the generalized disturbances the observer gain. is dieff rentiable [22]. Consequently, when there is unknown The estimated generalized disturbances can be compen- model error, the estimation error of LESO can converge to a sated for by the error feedback control as constant in the ni fi te time and its upper bound monotonously decreases with the observer gain. 푢 −푧 0 𝑑 푢= (11) 4. Experiments and Results where u is the control law which is designed on the basis 4.1. Description of Experimental Platform. To verify the ee ff c- of the controlled system, 푧 is the estimated generalized disturbances, and b is the control gain. tiveness and control performance of the proposed controller, experiments are performed on the experimental platform 3.2. Design of Control Law. The erection system is required to showninFigure4. tracktheplannedtrajectoryaccuratelytorealize smooth and Table 1 shows the main parameters of the experimental fast erection. eTh planned trajectory includes the reference platform. 퐾 , i,and L are confirmed by the parameters of erection angle, angular velocity, and angular acceleration. eTh servo driver and electric cylinder, respectively, while 푇 is resolution of the erection angle encoder is not very high, so identiefi dbythestep response curveinFigure2. thederivativeofanglesignalwillbringbignoisetothecontrol In the process of experiment, the erection system should system. Hence, the control law is designed, which takes the track the planned trajectory to realize the erection of 60 observer output z as feedback and the reference angular within 10s, which is expressed as velocity as feedforward, and expressed as 휋 휋 휋 휃=− cos $ 푡%+ 푢 =푘 휃 − 푧 + 휃 (12) 6 10 6 0 𝑝 1 휋 휋 where 푘 is the proportional coefficient, 휃 is the reference 휃= sin $ 푡% (14) 60 10 erection angle, and 휃 is the reference angular velocity. 휋 휋 휃= cos $ 푡% 3.3. Convergence Analysis of Linear Extended State Observer. 600 10 For the convergence of LESO, define the estimation error as To verify the eeff ctiveness and performance of LADRC, second-order LADRC and third-order LADRC are tested on 푥  =푥 −푧 , 푖 = 1,2... (13) 𝑖 𝑖 𝑖 the experimental system, respectively. Meanwhile, PID con- Theorem 1. Assuming h is bounded, there exist a constant troller is introduced as a comparison to test the performance 휎 >0andafinite T >0suchthat |푥  (푡)| ≤ 휎 , 푖= of LADRC. Due to the noise of low-resolution encoder, the 𝑖 1 𝑖 𝑖 Journal of Control Science and Engineering 5 Table 2: Tuned control parameters of three controllers. Controller Control parameters Symbols Values Observer gain 휔 20 Second-order LADRC Proportional coefficient 푘 12 𝑝1 Observer gain 휔 45 Third-order LADRC Proportional coefficient 푘 750 𝑝2 Proportional coefficient 푘 200 PI Integral coefficient 푘 1200 Filter coefficient 휏 0.0125 70 0.2 0.15 0.1 Zoom in 0.05 54.5 30 54.3 −0.05 54.1 7.98 8 8.02 −0.1 −0.15 0 −0.2 02468 10 02468 10 time (sec) time (sec) Third-order LADRC Second-order LADRC Planned trajectory Second-order LADRC PI Third-order LADRC PI Figure 5: Tracking curves of erection angle. Figure 6: Tracking error curves of erection angle. derivative element may lead to system instability. er Th efore, PI controller with first-order low pass filter is designed, which than PI controller and the tracking accuracy of second-order canbeexpressed as LADRC is the best of three controllers. According to the velocity variation of planned trajectory, the tracking error of 휏 푧 ̇ +푧 =푦 𝑓 𝑓 PI controller increases with the erection velocity, since there (15) is phase lag of the filter and no derivative element in the 푢 =푘 '휃 − 푧 *+푘 - '휃 − 푧 * controller which influences the dynamic response capability 𝑃 𝑓 𝐼 𝑓 of the controller. However, the tracking error of third-order where 휏 is the filter coefficient, z is the output of rfi st-order f LADRChas theoppositetrend,since LESOis more sensitive lowpassfilter, k is the proportional coefficient, and k is the to the noise of low-resolution encoder at low speed. P I integral coefficient. In Table 3, minimum, maximum, and terminal error of After parameters tuning in simulation and on the exper- three controllers during the erection are compared. imental platform, the tuned control parameters of three According to Table 3, second-order LADRC can keep controllers can be obtained as shown in Table 2. The increase the tracking error within the range of encoder resolution of these parameters would make the system instability. and the terminal error is almost zero. Although third-order LADRC has larger error and terminal error of -0.05 ,itis 4.2. Results of Trajectory Tracking. Figures5and6show still within the range of ±0.2 andmeets therequirement the trajectory tracking curves and tracking error curves, of tracking error. In theory, increasing the observer gain respectively. eTh tracking error is defined as the value of and proportional coefficient can reduce the tracking error; plannedtrajectoryminus theactualvalueat thesametime. however this will lead to the erection system chattering with The stair-step of actual curves and burrs of tracking error high frequency. curves are caused by the low resolution of the erection angle These results show that, for the electric erection system, encoder. LADRC can control the tracking error within the required From Figures 5 and 6, it can be seen visually that LADRC range and second-order LADRC has the best tracking accu- can track the planned trajectory with smaller tracking error racy. erection angle (deg) tracking error (deg) 6 Journal of Control Science and Engineering 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 0.2 0.08 0.15 0.06 0.1 0.04 0.05 0.02 −0.05 −0.02 −0.1 −0.04 −0.15 −0.06 −0.2 02468 10 02468 10 time (sec) time (sec) Second-order LADRC Without disturbance Third-order LADRC With disturbance PI Figure 8: Two tracking error curves of second-order LADRC. Figure 7: Tracking error curves with disturbance on control output. 0.15 4.3. Results of Disturbance Rejection. In order to test the 0.1 disturbance-rejection performance of three controllers, a disturbance of control output is manually set on the erection 0.05 system. Starting from the time of 5s, the control output is reducedby10%,whichisequaltoreducingthevelocitygainof servodriverby10%.Thiscantesttheperformanceofrejecting the change of intrinsic parameters. −0.05 Figure 7 shows the tracking error curves with disturbance on control output. Also the tracking error curves with −0.1 disturbance are contrasted to their respective curves without disturbanceasshownin Figures8,9,and 10,respectively. −0.15 It is apparent that the eeff ct on second-order LADRC, 02468 10 caused by the reduction of control output, is minimum time (sec) of three controllers. After the disturbance occurs, LADRC canreducethe trackingerrortothe leveloferror without Without disturbance disturbance aer ft a process of adjustment. Especially, second- With disturbance order LADRC has smaller amplitude and shorter duration Figure 9: Two tracking error curves of third-order LADRC. 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 aeft r an adjustment process. erection mechanism. LADRC can successfully realize trajec- These results indicate that second-order LADRC has tory tracking control, based on the approximate models of strong robustness to the disturbance caused by the change of velocity loop and erection mechanism. Significant trajectory intrinsic parameters. tracking and disturbance-rejection performance are achieved in the erection experiments. Based on the experimental results, second-order LADRC has better tracking accuracy 5. Conclusion andstrongerrobustnesstothedisturbancecausedbythe eTh lever-type electric erection system is a complicated posi- change of intrinsic parameters than PI controller and third- tion servo system consisting of servo system and lever-type order LADRC. tracking error (deg) tracking error (deg) tracking error (deg) Journal of Control Science and Engineering 7 0.15 [6] Y.Zuo,J.Zhang,C.Liu,and T. 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