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Research ArticleImproved Linear Active Disturbance Rejection Control forLever-Type Electric Erection System with Varying Loads andLow-Resolution Encoder

Hailong Niu Qinhe Gao Zhihao Liu Shengjin Tang andWenliang Guan

Xirsquoan High Technology Institute Xirsquoan 710025 China

Correspondence should be addressed to Qinhe Gao qhgao201126com

Received 2 November 2018 Accepted 12 December 2018 Published 17 January 2019

Academic Editor Rafael Morales

Copyright copy 2019 Hailong Niu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The lever-type electric erection system is a novel kind of erection system and the experimental platform in this paper operateswith varying loads and low-resolution encoder For high accuracy trajectory tracking linear active disturbance rejection control(LADRC) is introduced An approximate model consisting of the servo system configured at velocity control mode and the lever-type erection mechanism is built by means of system identification and curve fitting Reduced-order LADRC based on the furthersimplified model is proposed to improve tracking accuracy and robustness As comparisons traditional LADRC and PID withhigh-gain tracking differentiator (HGTD) are designed Simulation and experimental results indicate that reduced-order LADRCcan realize higher trajectory tracking accuracywith low-resolution encoder and has better robustness to variation in erection loadscompared with traditional LADRC and PID with HGTD

1 Introduction

The erection system is an important component of weaponryand engineering machinery such as the rocket launcher andthe dump truck Typically the erection system needs totrack the planned trajectory However the trajectory trackingaccuracy could be affected by variation in erection loadsresolution of sensors and other disturbances Uncertaintiesalways exist in physical servo system and affect the trackingaccuracy Many nonlinear control methods are developed toachieve high accuracy trajectory tracking such as the adap-tive integral robust controller [1] the adaptive backsteppingcontroller with modified LuGre model [2] and the adaptiverepetitive controller [3]

Traditional erection systems are usually hydraulicallydriven by multistage hydraulic cylinder which has longstrokes and shock during the process of changing stageTime-varying integral adaptive sliding mode control [4] andflow-pressure compound control [5] have been adopted tocontrol the hydraulic erection systemWith the improvementof servo motor and drive mechanism the electric cylindercan be used as the actuator in the electric erection system

[6] The erection system in this paper combines the single-stage electric cylinder with the lever-type erection mech-anism which can shorten the strokes and avoid shockConsidering the modeling uncertainties varying loads andlow-resolution encoder of the experimental platform in thispaper traditional linear controllers are difficult to satisfythe performance demands in terms of tracking accuracyand robustness Therefore linear active disturbance rejectioncontrol (LADRC) is introduced

The concept of active disturbance rejection control(ADRC) was firstly proposed by Han [7] which inheritsfrom classic PID and has better performance in rejecting thedisturbances actively [8] The core of ADRC is estimatingthe generalized disturbances consisting of internal modeluncertainties and external disturbances and compensatingfor them such that a relatively low precision model isnecessary to design the control loop [9 10] The normalADRC consists of tracking differentiator (TD) extended stateobserver (ESO) nonlinear state error feedback (NLSEF) anddisturbance rejection (DR) [11] Gao [12] developed LADRCwhich is the simplification of ADRC by using linear extendedstate observer (LESO) and linear state error feedback (LSEF)

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9867467 13 pageshttpsdoiorg10115520199867467

2 Mathematical Problems in Engineering

Electric cylinder

Servo driver

Erection mechanism

Loads

Encoder of erection

angle

Encoder of motor

MotioncontrollerReference

Figure 1 Composition of the electric erection system

LADRC has fewer parameters and is easily tuned in practiceThe convergence and stability of LADRC have been provedin time domain [13 14] and frequency domain [15] [16]The validity has been verified in many applications [17ndash21] Consequently LADRC has theoretical completeness andpracticability

First this paper builds an approximate model of theelectric erection system which is composed of the servo sys-tem configured at velocity control mode and the lever-type erection mechanism Second reduced-order LADRC isproposed based on further simplified model As compar-isons traditional LADRC and PID with high-gain track-ing differentiator (HGTD) are designed for this systemFinally simulations and experiments are carried out to vali-date the effectiveness and robustness of the proposed control-lers

The rest of this paper is organized as follows Section 2describes the composition of the electric erection system andthe experimental platform Section 3 builds the approximatemodel of the erection system Section 4 designs reduced-order LADRC traditional LADRC and PID with HGTDfor the erection system Section 5 conducts the simulationand experimental verification The conclusions are given inSection 6

2 System Introduction

The electric erection system is a typical mechatronic servosystem which mainly consists of a controller a servo driveran electric cylinder an erection mechanism loads andencoders as shown in Figure 1

The motion controller acquires the signals of encodersand calculates the control signal according to the controlalgorithm The servo driver is configured at velocity controlmode and controls the speed of servomotor in response to theanalog voltage command of -10ndash+10 V The electric cylindercanmake rotationmotion of the servomotor to linearmotionof the pushrod via the reducer and ball screw and thenthrust the erection mechanism and loads The erection angle

and the motor rotation angle are detected by encoders andtransmitted to the controller

Figure 2 shows the experimental platform of the elec-tric erection system The motion controller in IPC isGTS800 produced by Googoltech and the servo driver isMCDKT3520CA1 produced by Panasonic The encoder oferection angle produced by Tamagawa outputs pulse sig-nals with the resolution of 005∘ The erection loads areadjustable from zero mass block (30 kg) to six mass blocks(180 kg)

3 Modeling of Electric Erection System

The electric erection system is a position servo system withthe speed command of the servo driver as input and theerection angle as output The control block diagram of theelectric erection system is shown in Figure 3

The servo driver and the servo motor can be consideredas a whole and called the servo system Therefore theapproximate model of the electric erection consists of twopartsmdashthe servo system and the erection mechanism

31 Servo System Model Figure 4 shows the control blockdiagram of the servo system

The commercial servo driver which is configured atvelocity control mode has inner velocity controller and cur-rent controller Some parameters of the controllers and servomotor are not public Also there are some uncertainties suchas friction parameter variation and delay So it is difficult tobuild a precise model of the servo system

Typically an approximate second-order model is devotedto describe the velocity loop of the servo system [19] and thetransfer function can be described as

119866 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (119879119904 + 1) (1)

where Θm(s) and U(s) are Laplace transform of the motorrotation angle 120579m and the speed command u respectively Kis the transfer coefficient and T is the time constant

Mathematical Problems in Engineering 3

Electric cylinder + Erection mechanism

+ Loads

IPCServo driver

Motion controller

Terminal board

Encoder of erection angle

Figure 2 Experimental platform of the electric erection system

Servodriver

Motioncontroller

Reference Servomotor

Erectionmechanism

amp Loads

Encoder

Erection angle

Speed command

Motor rotation angle

Figure 3 Control block diagram of the electric erection system

Current control

Velocity control

Current

Power drive

Encoder Motor rotation angle

Speed command

Load torque

Motor(electrical

part)

Motor(mechanical

part)

Velocity detection

Motor speed

Figure 4 Control block diagram of the servo system

In order to identify the model parameters the stepresponse method is applied to the servo system on theexperimental platform According to the input gain of speedcommand the unit of speed command can be converted fromV to rmin Step response tests of 100 200 300 and 400 rminare respectively carried out on the experimental platformThe output of the servo system is the motor rotation anglewith the unit of rad The experimental curves are shown inFigure 5

The model parameters can be identified by means of thesystem identification toolbox inMATLABThe identificationmodel of the servo system can be described as

119866 (119904) = 20944119904 (00158119904 + 1) (2)

32 Erection Mechanism Model The model of the erectionmechanism takes the motor rotation angle 120579m as the input

4 Mathematical Problems in Engineering

0 2 4 6 8

Time (s)

0

50

100Sp

eed

com

man

d (r

min

)

0 2 4 6 8Time (s)

0

100

200

0 2 4 6 8Time (s)

0

100

200

300

Spee

d co

mm

and

(rm

in)

0 2 4 6 8Time (s)

0

200

400

0

25

50

0

50

100

Mot

or ro

tatio

n an

gle (

rad)

0

50

100

150

0

100

200

Mot

or ro

tatio

n an

gle (

rad)

Figure 5 Step response curves of the servo system

O

y

e

O1

O2

A

x

B

C

Crsquo

Brsquo

Arsquo

Figure 6 Structure of the lever-type mechanism

and the erection angle 120579e as the output consisting of thetransmission mechanism in the electric cylinder and thelever-type mechanism

The rigid model of transmission mechanism can beexpressed as

119878 = 1205791198981198712120587119894 (3)

where S is the extended length of the pushrod L is the lead ofthe ball screw and i is the reduction ratio of the reducer

The lever-type mechanism is composed of the electriccylinder O2C the triangular arm O1BC and the connectingrod AB as shown in Figure 6 To analyze the kinematics ofthe lever-type mechanism a coordinate system is establishedin the vertical plane through the center of gravity where theoriginO is the rotation center of the erection loads the X axisis horizontal and the Y axis is vertical

Mathematical Problems in Engineering 5

Table 1 Values of structure size and initial coordinate

ParametersUnits Symbols Values

Length of O1B(O1Brsquo)m100381610038161003816100381610038161003816997888997888997888rarr1198741119861100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119874111986110158401003816100381610038161003816100381610038161003816) 08

Length of O1C(O1Crsquo) m100381610038161003816100381610038161003816997888997888997888rarr1198741119862100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119874111986210158401003816100381610038161003816100381610038161003816) 079

Length of BC(BrsquoCrsquo) m100381610038161003816100381610038161003816997888997888rarr119861119862100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119861101584011986210158401003816100381610038161003816100381610038161003816) 01261

Length of AB(ArsquoBrsquo)m100381610038161003816100381610038161003816997888997888rarr119860119861100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119860101584011986110158401003816100381610038161003816100381610038161003816) 03

Coordinate of O m (119909119900 119910119900) (00)Coordinate of O1 m (1199091199001 1199101199001) (015-015)Coordinate of O2 m (1199091199002 1199101199002) (022-037)Coordinate of A m (119909119860 119910119860) (0750)Coordinate of B m (119909119861 119910119861) (09463-02269)Coordinate of C m (119909119862 119910119862) (09146-03489)Initial length of electric cylinderm

100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816 06949

When the extended length of the electric cylinder is SpointsA B and Cmove to points Arsquo Brsquo and Crsquo the triangulararmO1BC turns the angle of 120573 and the erection angle is 120579e Inthe initial position S is zero and 120579e is zero The structure sizeand the coordinate of points A B and C are shown in Table 1

Ignoring the deformation of themechanism the kinemat-ics analysis can be expressed as

10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198742119862101584010038161003816100381610038161003816100381610038161003816 =

100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816 + 119878

cosang11987421198741119862 =100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816

2 + 100381610038161003816100381610038161003816997888997888997888rarr11987411198621003816100381610038161003816100381610038162 minus 100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816

2

2 100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816 sdot 100381610038161003816100381610038161003816997888997888997888rarr1198741119862100381610038161003816100381610038161003816

cosang119874211987411198621015840 =100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816

2 + 10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198741119862101584010038161003816100381610038161003816100381610038161003816

2 minus 10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198742119862101584010038161003816100381610038161003816100381610038161003816

2

2 100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816 sdot10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198741119862101584010038161003816100381610038161003816100381610038161003816

120573 = ang119874211987411198621015840 minus ang11987421198741119862[1199091198611015840 minus 11990911987411199101198611015840 minus 1199101198741] = [

cos120573 minus sin 120573sin 120573 cos 120573 ][

119909119861 minus 1199091198741119910119861 minus 1199101198741]

[11990911986010158401199101198601015840] = [cos 120579119890 minus sin 120579119890sin 120579119890 cos 120579119890 ][

119909119860119910119860]1003816100381610038161003816100381610038161003816100381699788899788899788811986010158401198611015840100381610038161003816100381610038161003816100381610038162 = (1199091198611015840 minus 1199091198601015840)2 + (1199101198611015840 minus 1199101198601015840)2

(4)

Solving inverse trigonometric and quadratic equation theexpression of 120579e can be obtained as

120579119890 = arccos(119886119888 minus 119887radic1198862 + 1198872 minus 11988821198862 + 1198872 )119886 = 1199091198611015840 = (119909119861 minus 1199091198741) cos120573 minus (119910119861 minus 1199101198741) sin 120573 + 1199091198741

119887 = 1199101198611015840 = (119909119861 minus 1199091198741) sin 120573 minus (119910119861 minus 1199101198741) cos 120573 + 1199101198741

119888 = 1199092119860 + 11990921198611015840 + 11991021198611015840 minus 10038161003816100381610038161003816100381610038161003816997888997888997888rarr11986010158401198611015840100381610038161003816100381610038161003816100381610038162

2119909119860(5)

The relationship between 120579e and 120579m can be obtained based on(3) and (5)However the analytical expression is complicatedIt is not convenient for building the systemmodel and design-ing the control algorithm So an approximate polynomialexpression [22] is used to express the relationship which isobtained by means of curve fitting

The maximum stroke of the electric cylinder is 03 m thereduction ratio i is two and the lead of ball screw L is 0005m Changing S from 0 to 03 120579m and 120579e can be calculatedaccording to (3) and (5) respectively Then curve fittingis performed in MATLAB and the approximate polynomialexpression is

120579119890 = 119891 (120579119898)= 1391 times 10minus91205793119898 minus 5863 times 10minus71205792119898 + 0001763120579119898minus 0005364

(6)

where the units of 120579m and 120579e are both rad the confidencebounds of the coefficients are 95 the RMSE is 00049 andthe SSE is 02419

The calculated curve and fitted curve are shown inFigure 7

33 System Model Based on (1) and (6) the approximatemodel of the electric erection system can be expressed in theform of differential equation as

11988921205791198901198891199052 = (1198892119891 (120579119898) 1198891205792119898119889119891 (120579119898) 119889120579119898 minus 1) 119889120579119890119889119905 + 119870119879

119889119891 (120579119898)119889120579119898 119906 (7)

For the convenience of designing control algorithm definethe state variable 1199091 = 120579119890 and 1199092 = 120579119890 and transform

6 Mathematical Problems in Engineering

Calculated curveFitting curve

200 400 600 8000m (rad)

minus020

02040608

112141618

2 e

(rad

)

Figure 7 Curves of the relationship between 120579e and 120579m

the differential equation to the state-space form The systemmodel is expressed as

1199091 = 11990921199092 = 1198861199092 + 119887119906119910 = 1199091

(8)

where a is themodel parameter and b is the control gainTheirexpressions are

119886 = 1198892119891 (120579119898) 1198891205792119898119889119891 (120579119898) 119889120579119898 minus 1119887 = 119870119879

119889119891 (120579119898)119889120579119898(9)

4 Design of Control Algorithm

The control target is to let the electric erection systemtrack the planned trajectory within the error range of plusmn02∘However there are several factors affecting the trackingaccuracy as follows(1) There are model uncertainties between the actualsystem and the approximate model in previous section(2)There is variation in erection loads(3)The low-resolution encoder of erection angle outputssignals with quantization noise(4) There are other external disturbances during theprocess of erection

In order to overcome above problems and realize thecontrol target LADRC is introduced to control the electricerection system which has low requirements for the mod-eling precision and can estimate and compensate for thegeneralized disturbances

41 Control System Structure Thenormal LADRC consists ofTD LESO LSEF andDR For the electric erection system the

suitable erection trajectory should be planned to improve thestability and rapidity of the erection processThe informationof velocity and acceleration is easy to be obtained so thereis no need to use TD LESO is used to estimate the statevariables and the generalized disturbances which is the coreof LADRC In order to improve tracking accuracy LSEF ismodified by combining state error feedback with velocity andacceleration feedforward The structure of the control systemis shown in Figure 8

In this paper half period cosine function is used to planthe erection trajectory which is expressed as

120579119903 = 1205792 minus 1205792 cos( 120587119905119891 119905)120579119903 = 1205871205792119905119891 sin(

120587119905119891 119905)120579119903 = 120587212057921199052

119891

cos( 120587119905119891 119905)(10)

where 120579119903 120579119903 and 120579119903 are respectively the reference positionvelocity and acceleration at the moment of t 120579 is the targetposition and tf is the duration of the erection process

42 LADRC Design

421 Reduced-Order LADRC For the servo system model(1) ignoring the dynamic characteristics of the velocity loop[23] the model can be further simplified as

1198661015840 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (11)

Besides the extended state-space form of the simplifiedmodel can be expressed as

1199091 = 1199092 + 1198871199031199061199092 = ℎ119910 = 1199091

(12)

119887119903 = 119870119889119891 (120579119898)119889120579119898 (13)

where x2 is the generalized disturbances including modeluncertainties and external disturbances h is the differentia-tion of the generalized disturbances and br is the control gain

The model uncertainties caused by the simplified model(12) can be estimated and compensated for by LADRC

Structure the second-order LESO as

119910 = 119911119889 + 2120596119900 (1199091 minus 119911119910) + 119887119903119906119889 = 1205962119900 (1199091 minus 119911119910)

(14)

where zy and zd estimate x1 and x2 respectively and 120596o is theobserver gain

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

Although br varies with 120579m based on (13) the encoder ofmotor can accurately measure 120579m in real time and br can becalculated in real time

The modified LSEF combines position error feedbackwith velocity and acceleration feedforward to improvedtracking accuracy and stability which is expressed as

1199060 = 119896119901 (120579119903 minus 119911119910) + 119896V 120579119903 + 119896119886 120579119903 (16)

where kp kv and ka are the coefficient of position errorfeedback velocity feedforward and acceleration feedforwardrespectively

422 Traditional LADRC Based on the second-order systemmodel (8) the third-order extended state-space representa-tion can be expressed as

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

where x3 is the generalized disturbances and hrsquo is thedifferentiation of the generalized disturbances

The third-order LESO can be structured as1 = 1199112 + 31205961015840119900 (1199091 minus 1199111)2 = 1199113 + 312059610158401199002 (1199091 minus 1199111) + 11988711990610158403 = 12059610158401199003 (1199091 minus 1199111)

(18)

where z1 z2 and z3 estimate x1 x2 and x3 respectively and120596rsquoo is the observer gainThe expressions of DR and modified LSEF are designed

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

1199060 = 1198961015840119901 (120579119903 minus 1199111) + 1198961015840V 120579119903 + 1198961015840119886 120579119903 (20)

where krsquop krsquov and krsquoa are the coefficient of position errorfeedback velocity feedforward and acceleration feedforwardrespectively

For both second-order and third-order LESO if thedifferentiation of the generalized disturbances is boundedthe estimation error of LESO is bounded and its upper bounddecreases with the observer gain [13] However LESO willbe more sensitive to noise with higher observer gain whichshould be rationally selected according to control target andcharacteristics of the actual system

43 PID Controller with HGTD Design In contrast toLADRC PID controller is applied to the electric erection sys-tem In consideration of the quantization noise of the low-resolution encoder HGTD [24] is introduced to filter thestair-step signals of the erection angle encoder and estimateits differential signals The structure of PID controller withHGTD is shown in Figure 9

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

where y1 and y2 are respectively the filtered signals and thedifferential signals of the erection angle and 120596h is the gain ofHGTD

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Parameters Symbols Values UnitsTransfer coefficient K 20944 radsdotsminus1 sdotVminus1Time constant T 00158 sReduction ratio i 2 ndashLead of ball screw L 0005 mTarget position 120579 1205873 (60) rad (deg)Duration of erection process tf 10 sSolver type ndash Fixed-step ndashSample time ndash 0001 s

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Observer gain 120596o 10Coefficient of position error feedback kp 10Coefficient of velocity feedforward kv 1

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Observer gain 120596o 20Coefficient of position error feedback kp 200Coefficient of velocity feedforward kv 0

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

1199061 = 119896119875 (120579119903 minus 1199101) + 119896119868int (120579119903 minus 1199101) + 119896119863 ( 120579119903 minus 1199102) (22)

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

In order to verify the performance of the proposed con-trollers simulations are conducted in MATLABSimulinkand experiments are carried out on the experimental platformof the electric erection system as shown in Figure 2 Thecontrol program of the experiments is written and compiledto codes in MATLABSimulink which can be loaded to theGTS800 motion controller The parameters of model andSimulink configuration are shown in Table 2

51 Simulation Results and Analysis Based on (1) and (6)the simulation model can be built in Simulink More-over the stair-step signals of the low-resolution encodercaused by quantization noise can be simulated by theblockmdashldquoQuantizerrdquomdashusing the round-to-nearest methodaccording to the resolution of encoder The tuned controlparameters of LADRC and PID are shown in Table 3

The trajectory tracking curves of reduced-order LADRCand traditional LADRC and PID with HGTD are shown inFigure 10The tracking error is defined as the value of planned

trajectory minus the simulation value without quantizationnoise at the same time

From Figure 10 it can be seen that each of reduced-order LADRC traditional LADRC and PID with HGTDcan track the planned trajectory within the required errorrange of plusmn02∘ and the terminal errors of them are almostzero Besides reduced-order LADRChas the highest trackingaccuracy among three controllers

Figure 11 shows the erection angle estimation error curvesof second-order LESO third-order LESO and HGTD Theestimation error is defined as the simulation value withoutquantization noise minus the estimation value at the sametime

In Figure 11 the erection angle estimation error ofsecond-order LESO can be controlled around 0∘ which is theminimum among three controllers Both third-order LESOand HGTD can control the estimation error within plusmn005∘which means that the maximum error is not exceeding theresolution of the encoder Also according to the plannedtrajectory in (10) the velocity is relatively low at the beginningand end For three controllers at low speed the estimationerror is larger than that at high speed Therefore both LESOandHGTD aremore sensitive to the quantization noise at lowspeed

Figure 12 shows the angular velocity estimation errorcurves of third-order LESO and HGTD The simulationangular velocity is obtained by calculating the differential ofthe simulation erection angle without quantization noiseThe

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

Figure 10 Trajectory tracking curves of reduced-order LADRC traditional LADRC and PID with HGTD (a) Trajectory tracking curves (b)Tracking error curves

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Figure 13 Trajectory tracking curves with 180 kg loads (a) Trajectory tracking curves (b) Tracking error curves

Table 4 Minimum maximum and terminal values of tracking error

Controller Minimum error Maximum error Terminal errorReduced-order LADRC -004∘ 006∘ asymp0∘Traditional LADRC -007∘ 007∘ asymp0∘PID with HGTD -005∘ 009∘ asymp0∘

estimation error is defined as the simulation value minus theestimation value at the same time

In Figure 12 it is obvious that the angular velocityestimation error of third-order LESO is larger than that ofHGTD Also at low speed third-order LESO and HGTDboth have larger angular velocity estimation error than that athigh speed which is similar to the erection angle estimationerror

Simulation results indicate that reduced-order LADRCtraditional LADRC and PID with HGTD all can realize thecontrol target for the electric erection system and reduced-order LADRC has the highest tracking accuracy Moreoverboth LESO and HGTD can filter the stair-step signals of low-resolution encoder and third-order LESO and HGTD alsocan estimate its differential signals which is beneficial toimprove the tracking accuracy

52 Experimental Results and Analysis The simulationresults have proven that LADRC is effective and reduced-order LADRChas the highest tracking accuracy Carrying outthree controllers on the experimental platform with the loadsof 30 kg 80 kg 130 kg and 180 kg respectively can verifytheir control performance in further Besides comparing theexperimental results with the simulation results can verify thevalidity of the approximate model using the same controlparameters shown in Table 2

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

From Figure 13 it can be seen visually that three con-trollers all can keep the trajectory tracking error within therequired error range of plusmn02∘ Table 4 shows the Minimummaximum and terminal error of three controllers in theprocess of erection

According to Figure 13 and Table 4 all tracking errorcurves of three controllers have different levels of fluctuationat the early stage of the erection process and then convergeto zero which is influenced by the low-speed performanceof the servo system and the estimation performance of LESOand HGTD at low speed Among three controllers reduced-order LADRC has minimum fluctuation range

Figure 14 shows trajectory tracking error curves ofreduced-order LADRC traditional LADRC and PID withHGTD under the conditions of 30 kg 80 kg 130 kg and 180kg loads respectively

As presented in Figure 14(a) for reduced-order LADRCthe error range and fluctuation trend are not affected by thevarying loads and the error curves are nearly the same withdifferent loads In Figures 14(b) and 14(c) the error curveshave slight differences on the fluctuation trend under thecondition of 180 kg loads In general all of three controllersare robust to variation in erection loads and reduced-orderLADRC has the best robustness

Figure 15 compares the experimental trajectory tack-ing curves and tracking error curves of three controllers

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

Figure 14 Trajectory tracking error curves with different loads (a) Reduced-order LADRC (b) Traditional LADRC (c) PID with HGTD

with the simulation curves under the condition of 180 kgloads

It can be seen from Figure 15 that for three controllersthere is no significant difference between the experimentalcurves and the simulation curves This result indicates thatthe approximate model of the electric erection system inSection 3 is valid to verify the control algorithm and tunecontrol parameters

Experimental results indicate that reduced-order LADRCcan control the trajectory tracking error within the requirederror range with low-resolution encoder and has robustnessto variation in erection loads Compared with traditionalLADRC and PID with HGTD reduced-order LADRC hashigher tracking accuracy and better robustness In additionthe validity of the approximate systemmodel proposed in thispaper is verified by comparison of experimental curves withsimulation curves

6 Conclusions

The lever-type electric erection system in this paper ismainly composed of the servo system configured at velocitycontrol mode and the lever-type erection mechanism Theapproximate model is built by means of system identificationand curve fitting Considering the modeling uncertaintiesvarying loads and the low-resolution encoder reduced-orderLADRC is proposed based on the further simplified systemmodelThe following conclusions can be drawn based on thesimulation and experimental results(1) For the lever-type electric erection system reduced-order LADRC has higher tracking accuracy with low-resolution encoder and better robustness to variation inerection loads compared with traditional LADRC and PIDwith HGTD(2)The approximate model is proved to be valid to verifythe control algorithm and tune control parameters

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Figure 15 Comparison of experimental curves with simulation curves (a) Reduced-order LADRC (b) Traditional LADRC (c) PID withHGTD

Future research will consider the trajectory planning ofthis system and the optimization of the dynamic performanceof this controller

Data Availability

The data used to support the findings of this studyare available from the corresponding author upon re-quest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

[1] J Yao Z Jiao D Ma and L Yan ldquoHigh-accuracy tracking con-trol of hydraulic rotary actuators with modeling uncertaintiesrdquoIEEEASME Transactions on Mechatronics vol 19 no 2 pp633ndash641 2014

[2] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[3] J Yao Z Jiao and D Ma ldquoA Practical Nonlinear AdaptiveControl of Hydraulic Servomechanisms with Periodic-LikeDisturbancesrdquo IEEEASME Transactions on Mechatronics vol20 no 6 pp 2752ndash2760 2015

[4] Z Xie J Xie W Z Du et al ldquoTime-varying integral adaptivesliding mode control for the large erecting systemrdquoMathemat-ical Problems in Engineering vol 2014 Article ID 950768 11pages 2014

[5] J T Feng Q H Gao Y J Shao and W X Qian ldquoFlowand pressure compound control strategy for missile hydraulicerection systemrdquoActaArmamentarii vol 39 no 2 pp 209ndash2162018

[6] Y G Liu X H Gao and X W Yang ldquoResearch of controlstrategy in the large electric cylinder position servo systemrdquoMathematical Problems in Engineering vol 2015 Article ID167628 6 pages 2015

[7] J Han ldquoAuto-disturbances-rejection controller and its applica-tionsrdquo Control and Decision vol 13 no 1 pp 19ndash23 1998

[8] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[9] B Kou F Xing C Zhang L Zhang Y Zhou and T WangldquoImproved ADRC for aMaglev PlanarMotor with a ConcentricWinding Structurerdquo Applied Sciences vol 6 no 12 Article ID419 2016

[10] G Herbst ldquoA simulative study on active disturbance rejectioncontrol (ADRC) as a control tool for practitionersrdquo Electronicsvol 2 no 3 pp 246ndash279 2013

[11] C F Fu Analysis and Design of Linear Active DisturbanceRejection Control [PhD thesis] North China Electric PowerUniversity Beijing China 2018

[12] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence pp 4989ndash4996 Denver Colo USA June 2003

[13] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknowndynamicsrdquo inProceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[14] Q Zheng and Z Gao ldquoActive disturbance rejection controlbetween the formulation in time and the understanding infrequencyrdquo Control Theory and Technology vol 14 no 3 pp250ndash259 2016

[15] D Yuan X J Ma Q H Zeng and X B Qiu ldquoResearch onfrequency-band characteristics and parameters configurationof linear active disturbance rejection control for second-ordersystemsrdquo Control Theory Appl vol 30 no 12 pp 1630ndash16402013 (Chinese)

[16] Y Huang and W Xue ldquoActive disturbance rejection controlmethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[17] J Yao and W Deng ldquoActive Disturbance Rejection AdaptiveControl of Hydraulic Servo Systemsrdquo IEEE Transactions onIndustrial Electronics vol 64 no 10 pp 8023ndash8032 2017

[18] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[19] Y Zheng D W Ma J Y Yao and J Hu ldquoActive disturbancerejection control for position servo system of rocket launcherrdquoActa Armamentarii vol 35 no 5 pp 597ndash603 2014

[20] D Qiu M Sun Z Wang Y Wang and Z Chen ldquoPracticalwind-disturbance rejection for large deep space observatoryantennardquo IEEE Transactions on Control Systems Technology vol22 no 5 pp 1983ndash1990 2014

[21] DWu T Zhao and K Chen ldquoResearch and industrial applica-tions of active disturbance rejection control to fast tool servosrdquoControl Theory Appl vol 30 no 12 pp 1534ndash1542 2013

[22] Y Liu Q Gao H Niu and X Cheng ldquoModeling and trajectoryplanning of leveraged balance on lifting mechanismrdquo Journal ofVibration and Shock vol 36 no 16 pp 212ndash217 2017

[23] Z W Xu J P Jiang and Z F Luo ldquoPermanent magnetsynchronous motor position servo system controlled by fuzzymodel algorithmic controlrdquo Transaction of China Electrotechni-cal Society vol 18 no 4 pp 99ndash102 2003

[24] H Feng and S Li ldquoA tracking differentiator based on Taylorexpansionrdquo Applied Mathematics Letters vol 26 no 7 pp 735ndash740 2013

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2 Mathematical Problems in Engineering

Electric cylinder

Servo driver

Erection mechanism

Loads

Encoder of erection

angle

Encoder of motor

MotioncontrollerReference

Figure 1 Composition of the electric erection system

LADRC has fewer parameters and is easily tuned in practiceThe convergence and stability of LADRC have been provedin time domain [13 14] and frequency domain [15] [16]The validity has been verified in many applications [17ndash21] Consequently LADRC has theoretical completeness andpracticability

First this paper builds an approximate model of theelectric erection system which is composed of the servo sys-tem configured at velocity control mode and the lever-type erection mechanism Second reduced-order LADRC isproposed based on further simplified model As compar-isons traditional LADRC and PID with high-gain track-ing differentiator (HGTD) are designed for this systemFinally simulations and experiments are carried out to vali-date the effectiveness and robustness of the proposed control-lers

The rest of this paper is organized as follows Section 2describes the composition of the electric erection system andthe experimental platform Section 3 builds the approximatemodel of the erection system Section 4 designs reduced-order LADRC traditional LADRC and PID with HGTDfor the erection system Section 5 conducts the simulationand experimental verification The conclusions are given inSection 6

2 System Introduction

The electric erection system is a typical mechatronic servosystem which mainly consists of a controller a servo driveran electric cylinder an erection mechanism loads andencoders as shown in Figure 1

The motion controller acquires the signals of encodersand calculates the control signal according to the controlalgorithm The servo driver is configured at velocity controlmode and controls the speed of servomotor in response to theanalog voltage command of -10ndash+10 V The electric cylindercanmake rotationmotion of the servomotor to linearmotionof the pushrod via the reducer and ball screw and thenthrust the erection mechanism and loads The erection angle

and the motor rotation angle are detected by encoders andtransmitted to the controller

Figure 2 shows the experimental platform of the elec-tric erection system The motion controller in IPC isGTS800 produced by Googoltech and the servo driver isMCDKT3520CA1 produced by Panasonic The encoder oferection angle produced by Tamagawa outputs pulse sig-nals with the resolution of 005∘ The erection loads areadjustable from zero mass block (30 kg) to six mass blocks(180 kg)

3 Modeling of Electric Erection System

The electric erection system is a position servo system withthe speed command of the servo driver as input and theerection angle as output The control block diagram of theelectric erection system is shown in Figure 3

The servo driver and the servo motor can be consideredas a whole and called the servo system Therefore theapproximate model of the electric erection consists of twopartsmdashthe servo system and the erection mechanism

31 Servo System Model Figure 4 shows the control blockdiagram of the servo system

The commercial servo driver which is configured atvelocity control mode has inner velocity controller and cur-rent controller Some parameters of the controllers and servomotor are not public Also there are some uncertainties suchas friction parameter variation and delay So it is difficult tobuild a precise model of the servo system

Typically an approximate second-order model is devotedto describe the velocity loop of the servo system [19] and thetransfer function can be described as

119866 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (119879119904 + 1) (1)

where Θm(s) and U(s) are Laplace transform of the motorrotation angle 120579m and the speed command u respectively Kis the transfer coefficient and T is the time constant

Mathematical Problems in Engineering 3

Electric cylinder + Erection mechanism

+ Loads

IPCServo driver

Motion controller

Terminal board

Encoder of erection angle

Figure 2 Experimental platform of the electric erection system

Servodriver

Motioncontroller

Reference Servomotor

Erectionmechanism

amp Loads

Encoder

Erection angle

Speed command

Motor rotation angle

Figure 3 Control block diagram of the electric erection system

Current control

Velocity control

Current

Power drive

Encoder Motor rotation angle

Speed command

Load torque

Motor(electrical

part)

Motor(mechanical

part)

Velocity detection

Motor speed

Figure 4 Control block diagram of the servo system

In order to identify the model parameters the stepresponse method is applied to the servo system on theexperimental platform According to the input gain of speedcommand the unit of speed command can be converted fromV to rmin Step response tests of 100 200 300 and 400 rminare respectively carried out on the experimental platformThe output of the servo system is the motor rotation anglewith the unit of rad The experimental curves are shown inFigure 5

The model parameters can be identified by means of thesystem identification toolbox inMATLABThe identificationmodel of the servo system can be described as

119866 (119904) = 20944119904 (00158119904 + 1) (2)

32 Erection Mechanism Model The model of the erectionmechanism takes the motor rotation angle 120579m as the input

4 Mathematical Problems in Engineering

0 2 4 6 8

Time (s)

0

50

100Sp

eed

com

man

d (r

min

)

0 2 4 6 8Time (s)

0

100

200

0 2 4 6 8Time (s)

0

100

200

300

Spee

d co

mm

and

(rm

in)

0 2 4 6 8Time (s)

0

200

400

0

25

50

0

50

100

Mot

or ro

tatio

n an

gle (

rad)

0

50

100

150

0

100

200

Mot

or ro

tatio

n an

gle (

rad)

Figure 5 Step response curves of the servo system

O

y

e

O1

O2

A

x

B

C

Crsquo

Brsquo

Arsquo

Figure 6 Structure of the lever-type mechanism

and the erection angle 120579e as the output consisting of thetransmission mechanism in the electric cylinder and thelever-type mechanism

The rigid model of transmission mechanism can beexpressed as

119878 = 1205791198981198712120587119894 (3)

where S is the extended length of the pushrod L is the lead ofthe ball screw and i is the reduction ratio of the reducer

The lever-type mechanism is composed of the electriccylinder O2C the triangular arm O1BC and the connectingrod AB as shown in Figure 6 To analyze the kinematics ofthe lever-type mechanism a coordinate system is establishedin the vertical plane through the center of gravity where theoriginO is the rotation center of the erection loads the X axisis horizontal and the Y axis is vertical

Mathematical Problems in Engineering 5

Table 1 Values of structure size and initial coordinate

ParametersUnits Symbols Values

Length of O1B(O1Brsquo)m100381610038161003816100381610038161003816997888997888997888rarr1198741119861100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119874111986110158401003816100381610038161003816100381610038161003816) 08

Length of O1C(O1Crsquo) m100381610038161003816100381610038161003816997888997888997888rarr1198741119862100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119874111986210158401003816100381610038161003816100381610038161003816) 079

Length of BC(BrsquoCrsquo) m100381610038161003816100381610038161003816997888997888rarr119861119862100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119861101584011986210158401003816100381610038161003816100381610038161003816) 01261

Length of AB(ArsquoBrsquo)m100381610038161003816100381610038161003816997888997888rarr119860119861100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119860101584011986110158401003816100381610038161003816100381610038161003816) 03

Coordinate of O m (119909119900 119910119900) (00)Coordinate of O1 m (1199091199001 1199101199001) (015-015)Coordinate of O2 m (1199091199002 1199101199002) (022-037)Coordinate of A m (119909119860 119910119860) (0750)Coordinate of B m (119909119861 119910119861) (09463-02269)Coordinate of C m (119909119862 119910119862) (09146-03489)Initial length of electric cylinderm

100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816 06949

When the extended length of the electric cylinder is SpointsA B and Cmove to points Arsquo Brsquo and Crsquo the triangulararmO1BC turns the angle of 120573 and the erection angle is 120579e Inthe initial position S is zero and 120579e is zero The structure sizeand the coordinate of points A B and C are shown in Table 1

Ignoring the deformation of themechanism the kinemat-ics analysis can be expressed as

10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198742119862101584010038161003816100381610038161003816100381610038161003816 =

100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816 + 119878

cosang11987421198741119862 =100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816

2 + 100381610038161003816100381610038161003816997888997888997888rarr11987411198621003816100381610038161003816100381610038162 minus 100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816

2

2 100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816 sdot 100381610038161003816100381610038161003816997888997888997888rarr1198741119862100381610038161003816100381610038161003816

cosang119874211987411198621015840 =100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816

2 + 10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198741119862101584010038161003816100381610038161003816100381610038161003816

2 minus 10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198742119862101584010038161003816100381610038161003816100381610038161003816

2

2 100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816 sdot10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198741119862101584010038161003816100381610038161003816100381610038161003816

120573 = ang119874211987411198621015840 minus ang11987421198741119862[1199091198611015840 minus 11990911987411199101198611015840 minus 1199101198741] = [

cos120573 minus sin 120573sin 120573 cos 120573 ][

119909119861 minus 1199091198741119910119861 minus 1199101198741]

[11990911986010158401199101198601015840] = [cos 120579119890 minus sin 120579119890sin 120579119890 cos 120579119890 ][

119909119860119910119860]1003816100381610038161003816100381610038161003816100381699788899788899788811986010158401198611015840100381610038161003816100381610038161003816100381610038162 = (1199091198611015840 minus 1199091198601015840)2 + (1199101198611015840 minus 1199101198601015840)2

(4)

Solving inverse trigonometric and quadratic equation theexpression of 120579e can be obtained as

120579119890 = arccos(119886119888 minus 119887radic1198862 + 1198872 minus 11988821198862 + 1198872 )119886 = 1199091198611015840 = (119909119861 minus 1199091198741) cos120573 minus (119910119861 minus 1199101198741) sin 120573 + 1199091198741

119887 = 1199101198611015840 = (119909119861 minus 1199091198741) sin 120573 minus (119910119861 minus 1199101198741) cos 120573 + 1199101198741

119888 = 1199092119860 + 11990921198611015840 + 11991021198611015840 minus 10038161003816100381610038161003816100381610038161003816997888997888997888rarr11986010158401198611015840100381610038161003816100381610038161003816100381610038162

2119909119860(5)

The relationship between 120579e and 120579m can be obtained based on(3) and (5)However the analytical expression is complicatedIt is not convenient for building the systemmodel and design-ing the control algorithm So an approximate polynomialexpression [22] is used to express the relationship which isobtained by means of curve fitting

The maximum stroke of the electric cylinder is 03 m thereduction ratio i is two and the lead of ball screw L is 0005m Changing S from 0 to 03 120579m and 120579e can be calculatedaccording to (3) and (5) respectively Then curve fittingis performed in MATLAB and the approximate polynomialexpression is

120579119890 = 119891 (120579119898)= 1391 times 10minus91205793119898 minus 5863 times 10minus71205792119898 + 0001763120579119898minus 0005364

(6)

where the units of 120579m and 120579e are both rad the confidencebounds of the coefficients are 95 the RMSE is 00049 andthe SSE is 02419

The calculated curve and fitted curve are shown inFigure 7

33 System Model Based on (1) and (6) the approximatemodel of the electric erection system can be expressed in theform of differential equation as

11988921205791198901198891199052 = (1198892119891 (120579119898) 1198891205792119898119889119891 (120579119898) 119889120579119898 minus 1) 119889120579119890119889119905 + 119870119879

119889119891 (120579119898)119889120579119898 119906 (7)

For the convenience of designing control algorithm definethe state variable 1199091 = 120579119890 and 1199092 = 120579119890 and transform

6 Mathematical Problems in Engineering

Calculated curveFitting curve

200 400 600 8000m (rad)

minus020

02040608

112141618

2 e

(rad

)

Figure 7 Curves of the relationship between 120579e and 120579m

the differential equation to the state-space form The systemmodel is expressed as

1199091 = 11990921199092 = 1198861199092 + 119887119906119910 = 1199091

(8)

where a is themodel parameter and b is the control gainTheirexpressions are

119886 = 1198892119891 (120579119898) 1198891205792119898119889119891 (120579119898) 119889120579119898 minus 1119887 = 119870119879

119889119891 (120579119898)119889120579119898(9)

4 Design of Control Algorithm

The control target is to let the electric erection systemtrack the planned trajectory within the error range of plusmn02∘However there are several factors affecting the trackingaccuracy as follows(1) There are model uncertainties between the actualsystem and the approximate model in previous section(2)There is variation in erection loads(3)The low-resolution encoder of erection angle outputssignals with quantization noise(4) There are other external disturbances during theprocess of erection

In order to overcome above problems and realize thecontrol target LADRC is introduced to control the electricerection system which has low requirements for the mod-eling precision and can estimate and compensate for thegeneralized disturbances

41 Control System Structure Thenormal LADRC consists ofTD LESO LSEF andDR For the electric erection system the

suitable erection trajectory should be planned to improve thestability and rapidity of the erection processThe informationof velocity and acceleration is easy to be obtained so thereis no need to use TD LESO is used to estimate the statevariables and the generalized disturbances which is the coreof LADRC In order to improve tracking accuracy LSEF ismodified by combining state error feedback with velocity andacceleration feedforward The structure of the control systemis shown in Figure 8

In this paper half period cosine function is used to planthe erection trajectory which is expressed as

120579119903 = 1205792 minus 1205792 cos( 120587119905119891 119905)120579119903 = 1205871205792119905119891 sin(

120587119905119891 119905)120579119903 = 120587212057921199052

119891

cos( 120587119905119891 119905)(10)

where 120579119903 120579119903 and 120579119903 are respectively the reference positionvelocity and acceleration at the moment of t 120579 is the targetposition and tf is the duration of the erection process

42 LADRC Design

421 Reduced-Order LADRC For the servo system model(1) ignoring the dynamic characteristics of the velocity loop[23] the model can be further simplified as

1198661015840 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (11)

Besides the extended state-space form of the simplifiedmodel can be expressed as

1199091 = 1199092 + 1198871199031199061199092 = ℎ119910 = 1199091

(12)

119887119903 = 119870119889119891 (120579119898)119889120579119898 (13)

where x2 is the generalized disturbances including modeluncertainties and external disturbances h is the differentia-tion of the generalized disturbances and br is the control gain

The model uncertainties caused by the simplified model(12) can be estimated and compensated for by LADRC

Structure the second-order LESO as

119910 = 119911119889 + 2120596119900 (1199091 minus 119911119910) + 119887119903119906119889 = 1205962119900 (1199091 minus 119911119910)

(14)

where zy and zd estimate x1 and x2 respectively and 120596o is theobserver gain

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

Although br varies with 120579m based on (13) the encoder ofmotor can accurately measure 120579m in real time and br can becalculated in real time

The modified LSEF combines position error feedbackwith velocity and acceleration feedforward to improvedtracking accuracy and stability which is expressed as

1199060 = 119896119901 (120579119903 minus 119911119910) + 119896V 120579119903 + 119896119886 120579119903 (16)

where kp kv and ka are the coefficient of position errorfeedback velocity feedforward and acceleration feedforwardrespectively

422 Traditional LADRC Based on the second-order systemmodel (8) the third-order extended state-space representa-tion can be expressed as

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

where x3 is the generalized disturbances and hrsquo is thedifferentiation of the generalized disturbances

The third-order LESO can be structured as1 = 1199112 + 31205961015840119900 (1199091 minus 1199111)2 = 1199113 + 312059610158401199002 (1199091 minus 1199111) + 11988711990610158403 = 12059610158401199003 (1199091 minus 1199111)

(18)

where z1 z2 and z3 estimate x1 x2 and x3 respectively and120596rsquoo is the observer gainThe expressions of DR and modified LSEF are designed

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

1199060 = 1198961015840119901 (120579119903 minus 1199111) + 1198961015840V 120579119903 + 1198961015840119886 120579119903 (20)

where krsquop krsquov and krsquoa are the coefficient of position errorfeedback velocity feedforward and acceleration feedforwardrespectively

For both second-order and third-order LESO if thedifferentiation of the generalized disturbances is boundedthe estimation error of LESO is bounded and its upper bounddecreases with the observer gain [13] However LESO willbe more sensitive to noise with higher observer gain whichshould be rationally selected according to control target andcharacteristics of the actual system

43 PID Controller with HGTD Design In contrast toLADRC PID controller is applied to the electric erection sys-tem In consideration of the quantization noise of the low-resolution encoder HGTD [24] is introduced to filter thestair-step signals of the erection angle encoder and estimateits differential signals The structure of PID controller withHGTD is shown in Figure 9

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

where y1 and y2 are respectively the filtered signals and thedifferential signals of the erection angle and 120596h is the gain ofHGTD

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Parameters Symbols Values UnitsTransfer coefficient K 20944 radsdotsminus1 sdotVminus1Time constant T 00158 sReduction ratio i 2 ndashLead of ball screw L 0005 mTarget position 120579 1205873 (60) rad (deg)Duration of erection process tf 10 sSolver type ndash Fixed-step ndashSample time ndash 0001 s

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Observer gain 120596o 10Coefficient of position error feedback kp 10Coefficient of velocity feedforward kv 1

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Observer gain 120596o 20Coefficient of position error feedback kp 200Coefficient of velocity feedforward kv 0

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

1199061 = 119896119875 (120579119903 minus 1199101) + 119896119868int (120579119903 minus 1199101) + 119896119863 ( 120579119903 minus 1199102) (22)

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

In order to verify the performance of the proposed con-trollers simulations are conducted in MATLABSimulinkand experiments are carried out on the experimental platformof the electric erection system as shown in Figure 2 Thecontrol program of the experiments is written and compiledto codes in MATLABSimulink which can be loaded to theGTS800 motion controller The parameters of model andSimulink configuration are shown in Table 2

51 Simulation Results and Analysis Based on (1) and (6)the simulation model can be built in Simulink More-over the stair-step signals of the low-resolution encodercaused by quantization noise can be simulated by theblockmdashldquoQuantizerrdquomdashusing the round-to-nearest methodaccording to the resolution of encoder The tuned controlparameters of LADRC and PID are shown in Table 3

The trajectory tracking curves of reduced-order LADRCand traditional LADRC and PID with HGTD are shown inFigure 10The tracking error is defined as the value of planned

trajectory minus the simulation value without quantizationnoise at the same time

From Figure 10 it can be seen that each of reduced-order LADRC traditional LADRC and PID with HGTDcan track the planned trajectory within the required errorrange of plusmn02∘ and the terminal errors of them are almostzero Besides reduced-order LADRChas the highest trackingaccuracy among three controllers

Figure 11 shows the erection angle estimation error curvesof second-order LESO third-order LESO and HGTD Theestimation error is defined as the simulation value withoutquantization noise minus the estimation value at the sametime

In Figure 11 the erection angle estimation error ofsecond-order LESO can be controlled around 0∘ which is theminimum among three controllers Both third-order LESOand HGTD can control the estimation error within plusmn005∘which means that the maximum error is not exceeding theresolution of the encoder Also according to the plannedtrajectory in (10) the velocity is relatively low at the beginningand end For three controllers at low speed the estimationerror is larger than that at high speed Therefore both LESOandHGTD aremore sensitive to the quantization noise at lowspeed

Figure 12 shows the angular velocity estimation errorcurves of third-order LESO and HGTD The simulationangular velocity is obtained by calculating the differential ofthe simulation erection angle without quantization noiseThe

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

Figure 10 Trajectory tracking curves of reduced-order LADRC traditional LADRC and PID with HGTD (a) Trajectory tracking curves (b)Tracking error curves

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Figure 13 Trajectory tracking curves with 180 kg loads (a) Trajectory tracking curves (b) Tracking error curves

Table 4 Minimum maximum and terminal values of tracking error

Controller Minimum error Maximum error Terminal errorReduced-order LADRC -004∘ 006∘ asymp0∘Traditional LADRC -007∘ 007∘ asymp0∘PID with HGTD -005∘ 009∘ asymp0∘

estimation error is defined as the simulation value minus theestimation value at the same time

In Figure 12 it is obvious that the angular velocityestimation error of third-order LESO is larger than that ofHGTD Also at low speed third-order LESO and HGTDboth have larger angular velocity estimation error than that athigh speed which is similar to the erection angle estimationerror

Simulation results indicate that reduced-order LADRCtraditional LADRC and PID with HGTD all can realize thecontrol target for the electric erection system and reduced-order LADRC has the highest tracking accuracy Moreoverboth LESO and HGTD can filter the stair-step signals of low-resolution encoder and third-order LESO and HGTD alsocan estimate its differential signals which is beneficial toimprove the tracking accuracy

52 Experimental Results and Analysis The simulationresults have proven that LADRC is effective and reduced-order LADRChas the highest tracking accuracy Carrying outthree controllers on the experimental platform with the loadsof 30 kg 80 kg 130 kg and 180 kg respectively can verifytheir control performance in further Besides comparing theexperimental results with the simulation results can verify thevalidity of the approximate model using the same controlparameters shown in Table 2

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

From Figure 13 it can be seen visually that three con-trollers all can keep the trajectory tracking error within therequired error range of plusmn02∘ Table 4 shows the Minimummaximum and terminal error of three controllers in theprocess of erection

According to Figure 13 and Table 4 all tracking errorcurves of three controllers have different levels of fluctuationat the early stage of the erection process and then convergeto zero which is influenced by the low-speed performanceof the servo system and the estimation performance of LESOand HGTD at low speed Among three controllers reduced-order LADRC has minimum fluctuation range

Figure 14 shows trajectory tracking error curves ofreduced-order LADRC traditional LADRC and PID withHGTD under the conditions of 30 kg 80 kg 130 kg and 180kg loads respectively

As presented in Figure 14(a) for reduced-order LADRCthe error range and fluctuation trend are not affected by thevarying loads and the error curves are nearly the same withdifferent loads In Figures 14(b) and 14(c) the error curveshave slight differences on the fluctuation trend under thecondition of 180 kg loads In general all of three controllersare robust to variation in erection loads and reduced-orderLADRC has the best robustness

Figure 15 compares the experimental trajectory tack-ing curves and tracking error curves of three controllers

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

Figure 14 Trajectory tracking error curves with different loads (a) Reduced-order LADRC (b) Traditional LADRC (c) PID with HGTD

with the simulation curves under the condition of 180 kgloads

It can be seen from Figure 15 that for three controllersthere is no significant difference between the experimentalcurves and the simulation curves This result indicates thatthe approximate model of the electric erection system inSection 3 is valid to verify the control algorithm and tunecontrol parameters

Experimental results indicate that reduced-order LADRCcan control the trajectory tracking error within the requirederror range with low-resolution encoder and has robustnessto variation in erection loads Compared with traditionalLADRC and PID with HGTD reduced-order LADRC hashigher tracking accuracy and better robustness In additionthe validity of the approximate systemmodel proposed in thispaper is verified by comparison of experimental curves withsimulation curves

6 Conclusions

The lever-type electric erection system in this paper ismainly composed of the servo system configured at velocitycontrol mode and the lever-type erection mechanism Theapproximate model is built by means of system identificationand curve fitting Considering the modeling uncertaintiesvarying loads and the low-resolution encoder reduced-orderLADRC is proposed based on the further simplified systemmodelThe following conclusions can be drawn based on thesimulation and experimental results(1) For the lever-type electric erection system reduced-order LADRC has higher tracking accuracy with low-resolution encoder and better robustness to variation inerection loads compared with traditional LADRC and PIDwith HGTD(2)The approximate model is proved to be valid to verifythe control algorithm and tune control parameters

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Figure 15 Comparison of experimental curves with simulation curves (a) Reduced-order LADRC (b) Traditional LADRC (c) PID withHGTD

Future research will consider the trajectory planning ofthis system and the optimization of the dynamic performanceof this controller

Data Availability

The data used to support the findings of this studyare available from the corresponding author upon re-quest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

[1] J Yao Z Jiao D Ma and L Yan ldquoHigh-accuracy tracking con-trol of hydraulic rotary actuators with modeling uncertaintiesrdquoIEEEASME Transactions on Mechatronics vol 19 no 2 pp633ndash641 2014

[2] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[3] J Yao Z Jiao and D Ma ldquoA Practical Nonlinear AdaptiveControl of Hydraulic Servomechanisms with Periodic-LikeDisturbancesrdquo IEEEASME Transactions on Mechatronics vol20 no 6 pp 2752ndash2760 2015

[4] Z Xie J Xie W Z Du et al ldquoTime-varying integral adaptivesliding mode control for the large erecting systemrdquoMathemat-ical Problems in Engineering vol 2014 Article ID 950768 11pages 2014

[5] J T Feng Q H Gao Y J Shao and W X Qian ldquoFlowand pressure compound control strategy for missile hydraulicerection systemrdquoActaArmamentarii vol 39 no 2 pp 209ndash2162018

[6] Y G Liu X H Gao and X W Yang ldquoResearch of controlstrategy in the large electric cylinder position servo systemrdquoMathematical Problems in Engineering vol 2015 Article ID167628 6 pages 2015

[7] J Han ldquoAuto-disturbances-rejection controller and its applica-tionsrdquo Control and Decision vol 13 no 1 pp 19ndash23 1998

[8] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[9] B Kou F Xing C Zhang L Zhang Y Zhou and T WangldquoImproved ADRC for aMaglev PlanarMotor with a ConcentricWinding Structurerdquo Applied Sciences vol 6 no 12 Article ID419 2016

[10] G Herbst ldquoA simulative study on active disturbance rejectioncontrol (ADRC) as a control tool for practitionersrdquo Electronicsvol 2 no 3 pp 246ndash279 2013

[11] C F Fu Analysis and Design of Linear Active DisturbanceRejection Control [PhD thesis] North China Electric PowerUniversity Beijing China 2018

[12] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence pp 4989ndash4996 Denver Colo USA June 2003

[13] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknowndynamicsrdquo inProceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[14] Q Zheng and Z Gao ldquoActive disturbance rejection controlbetween the formulation in time and the understanding infrequencyrdquo Control Theory and Technology vol 14 no 3 pp250ndash259 2016

[15] D Yuan X J Ma Q H Zeng and X B Qiu ldquoResearch onfrequency-band characteristics and parameters configurationof linear active disturbance rejection control for second-ordersystemsrdquo Control Theory Appl vol 30 no 12 pp 1630ndash16402013 (Chinese)

[16] Y Huang and W Xue ldquoActive disturbance rejection controlmethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[17] J Yao and W Deng ldquoActive Disturbance Rejection AdaptiveControl of Hydraulic Servo Systemsrdquo IEEE Transactions onIndustrial Electronics vol 64 no 10 pp 8023ndash8032 2017

[18] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[19] Y Zheng D W Ma J Y Yao and J Hu ldquoActive disturbancerejection control for position servo system of rocket launcherrdquoActa Armamentarii vol 35 no 5 pp 597ndash603 2014

[20] D Qiu M Sun Z Wang Y Wang and Z Chen ldquoPracticalwind-disturbance rejection for large deep space observatoryantennardquo IEEE Transactions on Control Systems Technology vol22 no 5 pp 1983ndash1990 2014

[21] DWu T Zhao and K Chen ldquoResearch and industrial applica-tions of active disturbance rejection control to fast tool servosrdquoControl Theory Appl vol 30 no 12 pp 1534ndash1542 2013

[22] Y Liu Q Gao H Niu and X Cheng ldquoModeling and trajectoryplanning of leveraged balance on lifting mechanismrdquo Journal ofVibration and Shock vol 36 no 16 pp 212ndash217 2017

[23] Z W Xu J P Jiang and Z F Luo ldquoPermanent magnetsynchronous motor position servo system controlled by fuzzymodel algorithmic controlrdquo Transaction of China Electrotechni-cal Society vol 18 no 4 pp 99ndash102 2003

[24] H Feng and S Li ldquoA tracking differentiator based on Taylorexpansionrdquo Applied Mathematics Letters vol 26 no 7 pp 735ndash740 2013

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Mathematical Problems in Engineering 3

Electric cylinder + Erection mechanism

+ Loads

IPCServo driver

Motion controller

Terminal board

Encoder of erection angle

Figure 2 Experimental platform of the electric erection system

Servodriver

Motioncontroller

Reference Servomotor

Erectionmechanism

amp Loads

Encoder

Erection angle

Speed command

Motor rotation angle

Figure 3 Control block diagram of the electric erection system

Current control

Velocity control

Current

Power drive

Encoder Motor rotation angle

Speed command

Load torque

Motor(electrical

part)

Motor(mechanical

part)

Velocity detection

Motor speed

Figure 4 Control block diagram of the servo system

In order to identify the model parameters the stepresponse method is applied to the servo system on theexperimental platform According to the input gain of speedcommand the unit of speed command can be converted fromV to rmin Step response tests of 100 200 300 and 400 rminare respectively carried out on the experimental platformThe output of the servo system is the motor rotation anglewith the unit of rad The experimental curves are shown inFigure 5

The model parameters can be identified by means of thesystem identification toolbox inMATLABThe identificationmodel of the servo system can be described as

119866 (119904) = 20944119904 (00158119904 + 1) (2)

32 Erection Mechanism Model The model of the erectionmechanism takes the motor rotation angle 120579m as the input

4 Mathematical Problems in Engineering

0 2 4 6 8

Time (s)

0

50

100Sp

eed

com

man

d (r

min

)

0 2 4 6 8Time (s)

0

100

200

0 2 4 6 8Time (s)

0

100

200

300

Spee

d co

mm

and

(rm

in)

0 2 4 6 8Time (s)

0

200

400

0

25

50

0

50

100

Mot

or ro

tatio

n an

gle (

rad)

0

50

100

150

0

100

200

Mot

or ro

tatio

n an

gle (

rad)

Figure 5 Step response curves of the servo system

O

y

e

O1

O2

A

x

B

C

Crsquo

Brsquo

Arsquo

Figure 6 Structure of the lever-type mechanism

and the erection angle 120579e as the output consisting of thetransmission mechanism in the electric cylinder and thelever-type mechanism

The rigid model of transmission mechanism can beexpressed as

119878 = 1205791198981198712120587119894 (3)

where S is the extended length of the pushrod L is the lead ofthe ball screw and i is the reduction ratio of the reducer

The lever-type mechanism is composed of the electriccylinder O2C the triangular arm O1BC and the connectingrod AB as shown in Figure 6 To analyze the kinematics ofthe lever-type mechanism a coordinate system is establishedin the vertical plane through the center of gravity where theoriginO is the rotation center of the erection loads the X axisis horizontal and the Y axis is vertical

Mathematical Problems in Engineering 5

Table 1 Values of structure size and initial coordinate

ParametersUnits Symbols Values

Length of O1B(O1Brsquo)m100381610038161003816100381610038161003816997888997888997888rarr1198741119861100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119874111986110158401003816100381610038161003816100381610038161003816) 08

Length of O1C(O1Crsquo) m100381610038161003816100381610038161003816997888997888997888rarr1198741119862100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119874111986210158401003816100381610038161003816100381610038161003816) 079

Length of BC(BrsquoCrsquo) m100381610038161003816100381610038161003816997888997888rarr119861119862100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119861101584011986210158401003816100381610038161003816100381610038161003816) 01261

Length of AB(ArsquoBrsquo)m100381610038161003816100381610038161003816997888997888rarr119860119861100381610038161003816100381610038161003816 (

1003816100381610038161003816100381610038161003816997888997888997888rarr119860101584011986110158401003816100381610038161003816100381610038161003816) 03

Coordinate of O m (119909119900 119910119900) (00)Coordinate of O1 m (1199091199001 1199101199001) (015-015)Coordinate of O2 m (1199091199002 1199101199002) (022-037)Coordinate of A m (119909119860 119910119860) (0750)Coordinate of B m (119909119861 119910119861) (09463-02269)Coordinate of C m (119909119862 119910119862) (09146-03489)Initial length of electric cylinderm

100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816 06949

When the extended length of the electric cylinder is SpointsA B and Cmove to points Arsquo Brsquo and Crsquo the triangulararmO1BC turns the angle of 120573 and the erection angle is 120579e Inthe initial position S is zero and 120579e is zero The structure sizeand the coordinate of points A B and C are shown in Table 1

Ignoring the deformation of themechanism the kinemat-ics analysis can be expressed as

10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198742119862101584010038161003816100381610038161003816100381610038161003816 =

100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816 + 119878

cosang11987421198741119862 =100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816

2 + 100381610038161003816100381610038161003816997888997888997888rarr11987411198621003816100381610038161003816100381610038162 minus 100381610038161003816100381610038161003816997888997888997888rarr1198742119862100381610038161003816100381610038161003816

2

2 100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816 sdot 100381610038161003816100381610038161003816997888997888997888rarr1198741119862100381610038161003816100381610038161003816

cosang119874211987411198621015840 =100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816

2 + 10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198741119862101584010038161003816100381610038161003816100381610038161003816

2 minus 10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198742119862101584010038161003816100381610038161003816100381610038161003816

2

2 100381610038161003816100381610038161003816997888997888997888997888rarr11987411198742100381610038161003816100381610038161003816 sdot10038161003816100381610038161003816100381610038161003816997888997888997888997888rarr1198741119862101584010038161003816100381610038161003816100381610038161003816

120573 = ang119874211987411198621015840 minus ang11987421198741119862[1199091198611015840 minus 11990911987411199101198611015840 minus 1199101198741] = [

cos120573 minus sin 120573sin 120573 cos 120573 ][

119909119861 minus 1199091198741119910119861 minus 1199101198741]

[11990911986010158401199101198601015840] = [cos 120579119890 minus sin 120579119890sin 120579119890 cos 120579119890 ][

119909119860119910119860]1003816100381610038161003816100381610038161003816100381699788899788899788811986010158401198611015840100381610038161003816100381610038161003816100381610038162 = (1199091198611015840 minus 1199091198601015840)2 + (1199101198611015840 minus 1199101198601015840)2

(4)

Solving inverse trigonometric and quadratic equation theexpression of 120579e can be obtained as

120579119890 = arccos(119886119888 minus 119887radic1198862 + 1198872 minus 11988821198862 + 1198872 )119886 = 1199091198611015840 = (119909119861 minus 1199091198741) cos120573 minus (119910119861 minus 1199101198741) sin 120573 + 1199091198741

119887 = 1199101198611015840 = (119909119861 minus 1199091198741) sin 120573 minus (119910119861 minus 1199101198741) cos 120573 + 1199101198741

119888 = 1199092119860 + 11990921198611015840 + 11991021198611015840 minus 10038161003816100381610038161003816100381610038161003816997888997888997888rarr11986010158401198611015840100381610038161003816100381610038161003816100381610038162

2119909119860(5)

The relationship between 120579e and 120579m can be obtained based on(3) and (5)However the analytical expression is complicatedIt is not convenient for building the systemmodel and design-ing the control algorithm So an approximate polynomialexpression [22] is used to express the relationship which isobtained by means of curve fitting

The maximum stroke of the electric cylinder is 03 m thereduction ratio i is two and the lead of ball screw L is 0005m Changing S from 0 to 03 120579m and 120579e can be calculatedaccording to (3) and (5) respectively Then curve fittingis performed in MATLAB and the approximate polynomialexpression is

120579119890 = 119891 (120579119898)= 1391 times 10minus91205793119898 minus 5863 times 10minus71205792119898 + 0001763120579119898minus 0005364

(6)

where the units of 120579m and 120579e are both rad the confidencebounds of the coefficients are 95 the RMSE is 00049 andthe SSE is 02419

The calculated curve and fitted curve are shown inFigure 7

33 System Model Based on (1) and (6) the approximatemodel of the electric erection system can be expressed in theform of differential equation as

11988921205791198901198891199052 = (1198892119891 (120579119898) 1198891205792119898119889119891 (120579119898) 119889120579119898 minus 1) 119889120579119890119889119905 + 119870119879

119889119891 (120579119898)119889120579119898 119906 (7)

For the convenience of designing control algorithm definethe state variable 1199091 = 120579119890 and 1199092 = 120579119890 and transform

6 Mathematical Problems in Engineering

Calculated curveFitting curve

200 400 600 8000m (rad)

minus020

02040608

112141618

2 e

(rad

)

Figure 7 Curves of the relationship between 120579e and 120579m

the differential equation to the state-space form The systemmodel is expressed as

1199091 = 11990921199092 = 1198861199092 + 119887119906119910 = 1199091

(8)

where a is themodel parameter and b is the control gainTheirexpressions are

119886 = 1198892119891 (120579119898) 1198891205792119898119889119891 (120579119898) 119889120579119898 minus 1119887 = 119870119879

119889119891 (120579119898)119889120579119898(9)

4 Design of Control Algorithm

The control target is to let the electric erection systemtrack the planned trajectory within the error range of plusmn02∘However there are several factors affecting the trackingaccuracy as follows(1) There are model uncertainties between the actualsystem and the approximate model in previous section(2)There is variation in erection loads(3)The low-resolution encoder of erection angle outputssignals with quantization noise(4) There are other external disturbances during theprocess of erection

In order to overcome above problems and realize thecontrol target LADRC is introduced to control the electricerection system which has low requirements for the mod-eling precision and can estimate and compensate for thegeneralized disturbances

41 Control System Structure Thenormal LADRC consists ofTD LESO LSEF andDR For the electric erection system the

suitable erection trajectory should be planned to improve thestability and rapidity of the erection processThe informationof velocity and acceleration is easy to be obtained so thereis no need to use TD LESO is used to estimate the statevariables and the generalized disturbances which is the coreof LADRC In order to improve tracking accuracy LSEF ismodified by combining state error feedback with velocity andacceleration feedforward The structure of the control systemis shown in Figure 8

In this paper half period cosine function is used to planthe erection trajectory which is expressed as

120579119903 = 1205792 minus 1205792 cos( 120587119905119891 119905)120579119903 = 1205871205792119905119891 sin(

120587119905119891 119905)120579119903 = 120587212057921199052

119891

cos( 120587119905119891 119905)(10)

where 120579119903 120579119903 and 120579119903 are respectively the reference positionvelocity and acceleration at the moment of t 120579 is the targetposition and tf is the duration of the erection process

42 LADRC Design

421 Reduced-Order LADRC For the servo system model(1) ignoring the dynamic characteristics of the velocity loop[23] the model can be further simplified as

1198661015840 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (11)

Besides the extended state-space form of the simplifiedmodel can be expressed as

1199091 = 1199092 + 1198871199031199061199092 = ℎ119910 = 1199091

(12)

119887119903 = 119870119889119891 (120579119898)119889120579119898 (13)

where x2 is the generalized disturbances including modeluncertainties and external disturbances h is the differentia-tion of the generalized disturbances and br is the control gain

The model uncertainties caused by the simplified model(12) can be estimated and compensated for by LADRC

Structure the second-order LESO as

119910 = 119911119889 + 2120596119900 (1199091 minus 119911119910) + 119887119903119906119889 = 1205962119900 (1199091 minus 119911119910)

(14)

where zy and zd estimate x1 and x2 respectively and 120596o is theobserver gain

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

Although br varies with 120579m based on (13) the encoder ofmotor can accurately measure 120579m in real time and br can becalculated in real time

The modified LSEF combines position error feedbackwith velocity and acceleration feedforward to improvedtracking accuracy and stability which is expressed as

1199060 = 119896119901 (120579119903 minus 119911119910) + 119896V 120579119903 + 119896119886 120579119903 (16)

where kp kv and ka are the coefficient of position errorfeedback velocity feedforward and acceleration feedforwardrespectively

422 Traditional LADRC Based on the second-order systemmodel (8) the third-order extended state-space representa-tion can be expressed as

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

where x3 is the generalized disturbances and hrsquo is thedifferentiation of the generalized disturbances

The third-order LESO can be structured as1 = 1199112 + 31205961015840119900 (1199091 minus 1199111)2 = 1199113 + 312059610158401199002 (1199091 minus 1199111) + 11988711990610158403 = 12059610158401199003 (1199091 minus 1199111)

(18)

where z1 z2 and z3 estimate x1 x2 and x3 respectively and120596rsquoo is the observer gainThe expressions of DR and modified LSEF are designed

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

1199060 = 1198961015840119901 (120579119903 minus 1199111) + 1198961015840V 120579119903 + 1198961015840119886 120579119903 (20)

where krsquop krsquov and krsquoa are the coefficient of position errorfeedback velocity feedforward and acceleration feedforwardrespectively

For both second-order and third-order LESO if thedifferentiation of the generalized disturbances is boundedthe estimation error of LESO is bounded and its upper bounddecreases with the observer gain [13] However LESO willbe more sensitive to noise with higher observer gain whichshould be rationally selected according to control target andcharacteristics of the actual system

43 PID Controller with HGTD Design In contrast toLADRC PID controller is applied to the electric erection sys-tem In consideration of the quantization noise of the low-resolution encoder HGTD [24] is introduced to filter thestair-step signals of the erection angle encoder and estimateits differential signals The structure of PID controller withHGTD is shown in Figure 9

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

where y1 and y2 are respectively the filtered signals and thedifferential signals of the erection angle and 120596h is the gain ofHGTD

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Parameters Symbols Values UnitsTransfer coefficient K 20944 radsdotsminus1 sdotVminus1Time constant T 00158 sReduction ratio i 2 ndashLead of ball screw L 0005 mTarget position 120579 1205873 (60) rad (deg)Duration of erection process tf 10 sSolver type ndash Fixed-step ndashSample time ndash 0001 s

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Observer gain 120596o 10Coefficient of position error feedback kp 10Coefficient of velocity feedforward kv 1

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Observer gain 120596o 20Coefficient of position error feedback kp 200Coefficient of velocity feedforward kv 0

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

1199061 = 119896119875 (120579119903 minus 1199101) + 119896119868int (120579119903 minus 1199101) + 119896119863 ( 120579119903 minus 1199102) (22)

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

In order to verify the performance of the proposed con-trollers simulations are conducted in MATLABSimulinkand experiments are carried out on the experimental platformof the electric erection system as shown in Figure 2 Thecontrol program of the experiments is written and compiledto codes in MATLABSimulink which can be loaded to theGTS800 motion controller The parameters of model andSimulink configuration are shown in Table 2

51 Simulation Results and Analysis Based on (1) and (6)the simulation model can be built in Simulink More-over the stair-step signals of the low-resolution encodercaused by quantization noise can be simulated by theblockmdashldquoQuantizerrdquomdashusing the round-to-nearest methodaccording to the resolution of encoder The tuned controlparameters of LADRC and PID are shown in Table 3

The trajectory tracking curves of reduced-order LADRCand traditional LADRC and PID with HGTD are shown inFigure 10The tracking error is defined as the value of planned

trajectory minus the simulation value without quantizationnoise at the same time

From Figure 10 it can be seen that each of reduced-order LADRC traditional LADRC and PID with HGTDcan track the planned trajectory within the required errorrange of plusmn02∘ and the terminal errors of them are almostzero Besides reduced-order LADRChas the highest trackingaccuracy among three controllers

Figure 11 shows the erection angle estimation error curvesof second-order LESO third-order LESO and HGTD Theestimation error is defined as the simulation value withoutquantization noise minus the estimation value at the sametime

In Figure 11 the erection angle estimation error ofsecond-order LESO can be controlled around 0∘ which is theminimum among three controllers Both third-order LESOand HGTD can control the estimation error within plusmn005∘which means that the maximum error is not exceeding theresolution of the encoder Also according to the plannedtrajectory in (10) the velocity is relatively low at the beginningand end For three controllers at low speed the estimationerror is larger than that at high speed Therefore both LESOandHGTD aremore sensitive to the quantization noise at lowspeed

Figure 12 shows the angular velocity estimation errorcurves of third-order LESO and HGTD The simulationangular velocity is obtained by calculating the differential ofthe simulation erection angle without quantization noiseThe

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

Figure 10 Trajectory tracking curves of reduced-order LADRC traditional LADRC and PID with HGTD (a) Trajectory tracking curves (b)Tracking error curves

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Figure 13 Trajectory tracking curves with 180 kg loads (a) Trajectory tracking curves (b) Tracking error curves

Table 4 Minimum maximum and terminal values of tracking error

Controller Minimum error Maximum error Terminal errorReduced-order LADRC -004∘ 006∘ asymp0∘Traditional LADRC -007∘ 007∘ asymp0∘PID with HGTD -005∘ 009∘ asymp0∘

estimation error is defined as the simulation value minus theestimation value at the same time

In Figure 12 it is obvious that the angular velocityestimation error of third-order LESO is larger than that ofHGTD Also at low speed third-order LESO and HGTDboth have larger angular velocity estimation error than that athigh speed which is similar to the erection angle estimationerror

Simulation results indicate that reduced-order LADRCtraditional LADRC and PID with HGTD all can realize thecontrol target for the electric erection system and reduced-order LADRC has the highest tracking accuracy Moreoverboth LESO and HGTD can filter the stair-step signals of low-resolution encoder and third-order LESO and HGTD alsocan estimate its differential signals which is beneficial toimprove the tracking accuracy

52 Experimental Results and Analysis The simulationresults have proven that LADRC is effective and reduced-order LADRChas the highest tracking accuracy Carrying outthree controllers on the experimental platform with the loadsof 30 kg 80 kg 130 kg and 180 kg respectively can verifytheir control performance in further Besides comparing theexperimental results with the simulation results can verify thevalidity of the approximate model using the same controlparameters shown in Table 2

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

From Figure 13 it can be seen visually that three con-trollers all can keep the trajectory tracking error within therequired error range of plusmn02∘ Table 4 shows the Minimummaximum and terminal error of three controllers in theprocess of erection

According to Figure 13 and Table 4 all tracking errorcurves of three controllers have different levels of fluctuationat the early stage of the erection process and then convergeto zero which is influenced by the low-speed performanceof the servo system and the estimation performance of LESOand HGTD at low speed Among three controllers reduced-order LADRC has minimum fluctuation range

Figure 14 shows trajectory tracking error curves ofreduced-order LADRC traditional LADRC and PID withHGTD under the conditions of 30 kg 80 kg 130 kg and 180kg loads respectively

As presented in Figure 14(a) for reduced-order LADRCthe error range and fluctuation trend are not affected by thevarying loads and the error curves are nearly the same withdifferent loads In Figures 14(b) and 14(c) the error curveshave slight differences on the fluctuation trend under thecondition of 180 kg loads In general all of three controllersare robust to variation in erection loads and reduced-orderLADRC has the best robustness

Figure 15 compares the experimental trajectory tack-ing curves and tracking error curves of three controllers

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

Figure 14 Trajectory tracking error curves with different loads (a) Reduced-order LADRC (b) Traditional LADRC (c) PID with HGTD

with the simulation curves under the condition of 180 kgloads

It can be seen from Figure 15 that for three controllersthere is no significant difference between the experimentalcurves and the simulation curves This result indicates thatthe approximate model of the electric erection system inSection 3 is valid to verify the control algorithm and tunecontrol parameters

Experimental results indicate that reduced-order LADRCcan control the trajectory tracking error within the requirederror range with low-resolution encoder and has robustnessto variation in erection loads Compared with traditionalLADRC and PID with HGTD reduced-order LADRC hashigher tracking accuracy and better robustness In additionthe validity of the approximate systemmodel proposed in thispaper is verified by comparison of experimental curves withsimulation curves

6 Conclusions

The lever-type electric erection system in this paper ismainly composed of the servo system configured at velocitycontrol mode and the lever-type erection mechanism Theapproximate model is built by means of system identificationand curve fitting Considering the modeling uncertaintiesvarying loads and the low-resolution encoder reduced-orderLADRC is proposed based on the further simplified systemmodelThe following conclusions can be drawn based on thesimulation and experimental results(1) For the lever-type electric erection system reduced-order LADRC has higher tracking accuracy with low-resolution encoder and better robustness to variation inerection loads compared with traditional LADRC and PIDwith HGTD(2)The approximate model is proved to be valid to verifythe control algorithm and tune control parameters

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Figure 15 Comparison of experimental curves with simulation curves (a) Reduced-order LADRC (b) Traditional LADRC (c) PID withHGTD

Future research will consider the trajectory planning ofthis system and the optimization of the dynamic performanceof this controller

Data Availability

The data used to support the findings of this studyare available from the corresponding author upon re-quest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

[1] J Yao Z Jiao D Ma and L Yan ldquoHigh-accuracy tracking con-trol of hydraulic rotary actuators with modeling uncertaintiesrdquoIEEEASME Transactions on Mechatronics vol 19 no 2 pp633ndash641 2014

[2] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[3] J Yao Z Jiao and D Ma ldquoA Practical Nonlinear AdaptiveControl of Hydraulic Servomechanisms with Periodic-LikeDisturbancesrdquo IEEEASME Transactions on Mechatronics vol20 no 6 pp 2752ndash2760 2015

[4] Z Xie J Xie W Z Du et al ldquoTime-varying integral adaptivesliding mode control for the large erecting systemrdquoMathemat-ical Problems in Engineering vol 2014 Article ID 950768 11pages 2014

[5] J T Feng Q H Gao Y J Shao and W X Qian ldquoFlowand pressure compound control strategy for missile hydraulicerection systemrdquoActaArmamentarii vol 39 no 2 pp 209ndash2162018

[6] Y G Liu X H Gao and X W Yang ldquoResearch of controlstrategy in the large electric cylinder position servo systemrdquoMathematical Problems in Engineering vol 2015 Article ID167628 6 pages 2015

[7] J Han ldquoAuto-disturbances-rejection controller and its applica-tionsrdquo Control and Decision vol 13 no 1 pp 19ndash23 1998

[8] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[9] B Kou F Xing C Zhang L Zhang Y Zhou and T WangldquoImproved ADRC for aMaglev PlanarMotor with a ConcentricWinding Structurerdquo Applied Sciences vol 6 no 12 Article ID419 2016

[10] G Herbst ldquoA simulative study on active disturbance rejectioncontrol (ADRC) as a control tool for practitionersrdquo Electronicsvol 2 no 3 pp 246ndash279 2013

[11] C F Fu Analysis and Design of Linear Active DisturbanceRejection Control [PhD thesis] North China Electric PowerUniversity Beijing China 2018

[12] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence pp 4989ndash4996 Denver Colo USA June 2003

[13] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknowndynamicsrdquo inProceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

[14] Q Zheng and Z Gao ldquoActive disturbance rejection controlbetween the formulation in time and the understanding infrequencyrdquo Control Theory and Technology vol 14 no 3 pp250ndash259 2016

[15] D Yuan X J Ma Q H Zeng and X B Qiu ldquoResearch onfrequency-band characteristics and parameters configurationof linear active disturbance rejection control for second-ordersystemsrdquo Control Theory Appl vol 30 no 12 pp 1630ndash16402013 (Chinese)

[16] Y Huang and W Xue ldquoActive disturbance rejection controlmethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[17] J Yao and W Deng ldquoActive Disturbance Rejection AdaptiveControl of Hydraulic Servo Systemsrdquo IEEE Transactions onIndustrial Electronics vol 64 no 10 pp 8023ndash8032 2017

[18] J Yao Z Jiao and D Ma ldquoAdaptive robust control of dc motorswith extended state observerrdquo IEEE Transactions on IndustrialElectronics vol 61 no 7 pp 3630ndash3637 2014

[19] Y Zheng D W Ma J Y Yao and J Hu ldquoActive disturbancerejection control for position servo system of rocket launcherrdquoActa Armamentarii vol 35 no 5 pp 597ndash603 2014

[20] D Qiu M Sun Z Wang Y Wang and Z Chen ldquoPracticalwind-disturbance rejection for large deep space observatoryantennardquo IEEE Transactions on Control Systems Technology vol22 no 5 pp 1983ndash1990 2014

[21] DWu T Zhao and K Chen ldquoResearch and industrial applica-tions of active disturbance rejection control to fast tool servosrdquoControl Theory Appl vol 30 no 12 pp 1534ndash1542 2013

[22] Y Liu Q Gao H Niu and X Cheng ldquoModeling and trajectoryplanning of leveraged balance on lifting mechanismrdquo Journal ofVibration and Shock vol 36 no 16 pp 212ndash217 2017

[23] Z W Xu J P Jiang and Z F Luo ldquoPermanent magnetsynchronous motor position servo system controlled by fuzzymodel algorithmic controlrdquo Transaction of China Electrotechni-cal Society vol 18 no 4 pp 99ndash102 2003

[24] H Feng and S Li ldquoA tracking differentiator based on Taylorexpansionrdquo Applied Mathematics Letters vol 26 no 7 pp 735ndash740 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

0 2 4 6 8

Time (s)

0

50

100Sp

eed

com

man

d (r

min

)

0 2 4 6 8Time (s)

0

100

200

0 2 4 6 8Time (s)

0

100

200

300

Spee

d co

mm

and

(rm

in)

0 2 4 6 8Time (s)

0

200

400

0

25

50

0

50

100

Mot

or ro

tatio

n an

gle (

rad)

0

50

100

150

0

100

200

Mot

or ro

tatio

n an

gle (

rad)

Figure 5 Step response curves of the servo system

O

y

e

O1

O2

A

x

B

C

Crsquo

Brsquo

Arsquo

Figure 6 Structure of the lever-type mechanism

The rigid model of transmission mechanism can beexpressed as

119878 = 1205791198981198712120587119894 (3)

Mathematical Problems in Engineering 5

Table 1 Values of structure size and initial coordinate

ParametersUnits Symbols Values

Ignoring the deformation of themechanism the kinemat-ics analysis can be expressed as

2

2

cos120573 minus sin 120573sin 120573 cos 120573 ][

119909119861 minus 1199091198741119910119861 minus 1199101198741]

(4)

Solving inverse trigonometric and quadratic equation theexpression of 120579e can be obtained as

2119909119860(5)

(6)

The calculated curve and fitted curve are shown inFigure 7

119889119891 (120579119898)119889120579119898 119906 (7)

6 Mathematical Problems in Engineering

Calculated curveFitting curve

200 400 600 8000m (rad)

minus020

02040608

112141618

2 e

(rad

)

Figure 7 Curves of the relationship between 120579e and 120579m

the differential equation to the state-space form The systemmodel is expressed as

1199091 = 11990921199092 = 1198861199092 + 119887119906119910 = 1199091

(8)

where a is themodel parameter and b is the control gainTheirexpressions are

119889119891 (120579119898)119889120579119898(9)

4 Design of Control Algorithm

120587119905119891 119905)120579119903 = 120587212057921199052

119891

cos( 120587119905119891 119905)(10)

42 LADRC Design

1198661015840 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (11)

Besides the extended state-space form of the simplifiedmodel can be expressed as

1199091 = 1199092 + 1198871199031199061199092 = ℎ119910 = 1199091

(12)

119887119903 = 119870119889119891 (120579119898)119889120579119898 (13)

Structure the second-order LESO as

(14)

where zy and zd estimate x1 and x2 respectively and 120596o is theobserver gain

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

(18)

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

trajectory minus the simulation value without quantizationnoise at the same time

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

Table 1 Values of structure size and initial coordinate

ParametersUnits Symbols Values

Ignoring the deformation of themechanism the kinemat-ics analysis can be expressed as

2

2

cos120573 minus sin 120573sin 120573 cos 120573 ][

119909119861 minus 1199091198741119910119861 minus 1199101198741]

(4)

Solving inverse trigonometric and quadratic equation theexpression of 120579e can be obtained as

2119909119860(5)

(6)

The calculated curve and fitted curve are shown inFigure 7

119889119891 (120579119898)119889120579119898 119906 (7)

6 Mathematical Problems in Engineering

Calculated curveFitting curve

200 400 600 8000m (rad)

minus020

02040608

112141618

2 e

(rad

)

Figure 7 Curves of the relationship between 120579e and 120579m

the differential equation to the state-space form The systemmodel is expressed as

1199091 = 11990921199092 = 1198861199092 + 119887119906119910 = 1199091

(8)

where a is themodel parameter and b is the control gainTheirexpressions are

119889119891 (120579119898)119889120579119898(9)

4 Design of Control Algorithm

120587119905119891 119905)120579119903 = 120587212057921199052

119891

cos( 120587119905119891 119905)(10)

42 LADRC Design

1198661015840 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (11)

Besides the extended state-space form of the simplifiedmodel can be expressed as

1199091 = 1199092 + 1198871199031199061199092 = ℎ119910 = 1199091

(12)

119887119903 = 119870119889119891 (120579119898)119889120579119898 (13)

Structure the second-order LESO as

(14)

where zy and zd estimate x1 and x2 respectively and 120596o is theobserver gain

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

(18)

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

trajectory minus the simulation value without quantizationnoise at the same time

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

Calculated curveFitting curve

200 400 600 8000m (rad)

minus020

02040608

112141618

2 e

(rad

)

Figure 7 Curves of the relationship between 120579e and 120579m

the differential equation to the state-space form The systemmodel is expressed as

1199091 = 11990921199092 = 1198861199092 + 119887119906119910 = 1199091

(8)

where a is themodel parameter and b is the control gainTheirexpressions are

119889119891 (120579119898)119889120579119898(9)

4 Design of Control Algorithm

120587119905119891 119905)120579119903 = 120587212057921199052

119891

cos( 120587119905119891 119905)(10)

42 LADRC Design

1198661015840 (119904) = Θ119898 (119904)119880 (119904) = 119870119904 (11)

Besides the extended state-space form of the simplifiedmodel can be expressed as

1199091 = 1199092 + 1198871199031199061199092 = ℎ119910 = 1199091

(12)

119887119903 = 119870119889119891 (120579119898)119889120579119898 (13)

Structure the second-order LESO as

(14)

where zy and zd estimate x1 and x2 respectively and 120596o is theobserver gain

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

(18)

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

trajectory minus the simulation value without quantizationnoise at the same time

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

Trajectoryplanning

Targetposition

DRModified

LSEF

LESO

Plant1bu

b

-y

r

rr

u0

zn+1z1 middot middot middot zn

Figure 8 Structure of the control system

Trajectoryplanning

Targetposition

PID

HGTD

Plantu1

y1

y2

y

r

r

Figure 9 Structure of PID controller with HGTD

With the estimation of the generalized disturbancesLADRC can compensate for them in real time as

119906 = 1199060 minus 119911119889119887119903 (15)

1 = 11990922 = 1199093 + 11988711990610158403 = ℎ1015840119910 = 1199091

(17)

(18)

in the same way as (15) and (16) which are shown as

1199061015840 = 11990610158400 minus 1199113119887 (19)

The expression of HGTD is

1199101 = 1199102 + 2120596ℎ (119910 minus 1199101)1199102 = 1205962ℎ (119910 minus 1199101) (21)

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

trajectory minus the simulation value without quantizationnoise at the same time

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

Table 2 Parameters of model and Simulink configuration

Table 3 Tuned control parameters of LADRC and PID

Controller Control parameters Symbols Values

Reduced-order LADRC

Coefficient of acceleration feedforward ka 0

Traditional LADRC

Coefficient of acceleration feedforward krsquoa 12

PID with HGTD

Gain of HGTD 120596ℎ 10Proportional coefficient kP 150

Integral coefficient kI 1500Derivative coefficient kD 15

The control law is expressed as

where kP kI and kD are the proportional integral andderivative coefficient respectively

5 Simulation and Experimental Verification

trajectory minus the simulation value without quantizationnoise at the same time

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

Reduced-order LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

86 100 2 4Time (s)

(a)

0 2 4 6 8 10Time (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Trac

king

erro

r (de

g)

Reduced-order LADRCTraditional LADRCPID with HGTD

(b)

0 2 4 6 8 10minus005

0

005Secondminusorder LESO

0 2 4 6 8 10minus005

0

005

Erec

tion

angl

e esti

mat

ion

erro

r (de

g)

Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus005

0

005HGTD

Figure 11 Erection angle estimation error curves of second-order LESO third-order LESO and HGTD

0 2 4 6 8 10minus2

0

2Thirdminusorder LESO

0 2 4 6 8 10Time (s)

minus2

0

2

Ang

ular

velo

city

estim

atio

n er

ror (

deg

s)

HGTD

Figure 12 Angular velocity estimation error curves of third-order LESO and HGTD

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

0 2 4 6 8 10Time (s)

0

10

20

30

40

50

60

70Er

ectio

n an

gle (

deg)

Reducedminusorder LADRCTraditional LADRC

PID with HGTDPlanned trajectory

145 150 15531

33

35

845 850 855565

567

569

Zoomin

Zoomin

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

PID with HGTDReducedminusorder LADRCTraditional LADRC

(b)

Table 4 Minimum maximum and terminal values of tracking error

estimation error is defined as the simulation value minus theestimation value at the same time

Figure 13 shows the experimental trajectory trackingcurves of reduced-order LADRC traditional LADRC

and PID with HGTD under the condition of 180 kgloads

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01Tr

acki

ng er

ror (

deg)

30 kg80 kg

130 kg180 kg

(a)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(b)

0 2 4 6 8 10Time (s)

minus01minus008minus006minus004minus002

0002004006008

01

Trac

king

erro

r (de

g)

30 kg80 kg

130 kg180 kg

(c)

with the simulation curves under the condition of 180 kgloads

6 Conclusions

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(a)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10

Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)Experiment

(b)

0 2 4 6 8 100

20

40

60

Erec

tion

angl

e (de

g)

ExperimentSimulationPlanned trajectory

0 2 4 6 8 10Time (s)

Time (s)

minus01

0

01

Trac

king

erro

r (de

g)

ExperimentSimulation

(c)

Data Availability

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China [Grant no 61703410]

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

References

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

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