disturbance observer-based integral backstepping control for a...

Research ArticleDisturbance Observer-Based Integral Backstepping Control for aTwo-Tank Liquid Level System Subject to External Disturbances

Xiangxiang Meng Haisheng Yu Herong Wu and Tao Xu

College of Automation Qingdao University Qingdao China

Correspondence should be addressed to Haisheng Yu yuhs163com

Received 30 August 2019 Revised 21 November 2019 Accepted 27 November 2019 Published 20 January 2020

Academic Editor Laurent Dewasme

Copyright copy 2020 Xiangxiang Meng et al -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A novel method of disturbance observer-based integral backstepping control is proposed for the two-tank liquid level system withexternal disturbances -e problem of external disturbances can be settled in this paper Firstly the mathematical model of thetwo-tank liquid level system is established based on fluid mechanics and principle of mass conservation Secondly an integralbackstepping control strategy is designed in order to ensure liquid level tracking performance by making the tracking errorsconverge to zero in finite time -irdly a disturbance observer is designed for the two-tank liquid level system with externaldisturbances Finally the validity of the proposed method is verified by simulation and experiment By doing so the simulationand experimental results prove that the scheme of disturbance observer-based integral backstepping control strategy can suppressexternal disturbances more effective than the disturbance observer-based sliding mode control method and has better dynamicand steady performance of the two-tank liquid level system

1 Introduction

In the last decade or so the two-tank liquid level controlsystem has been widely used in industrial production Fromthe point of view of control the liquid level system repre-sented complex multivariable nonlinear problems which isone of the important fields of nonlinear control theoryapplication -e liquid level system is affected by externaldisturbances and the parameters are of great uncertaintywhich makes the control task more complicated [1] So theprecise control of the two-tank liquid level system is verydifficult Sliding mode control is a robust control method foruncertain systems which maintains high robustness againstvarious uncertainties such as external disturbances andmeasurement errors For a class of nonlinear systems withmultiple input and multiple output a second-order slidingmode control algorithm [2] was proposed which takes theinput and output of nonlinear systems as the standard formof controller design In [3] feedback linearization andsliding mode algorithm are designed for a controller basedon feedback linearization for a four-tank system By

inserting a boundary layer around the sliding surfacebuffeting associated with sliding mode control can be re-duced which is far better than traditional proportionalintegral (PI) control A fractional-order proportional inte-gral-differential sliding surface sliding mode controller [4]for liquid level control in the two-tank system was proposedIn [5] for a class of uncertainty and unknown disturbance ofmultiple-input-multiple-output data-driven sliding modecontrol problem of nonlinear discrete systems Weng andGao used the nonparametric dynamic linearization tech-nique and the second-order discrete sliding mode observerand proposed a discrete sliding mode control law based onthe proportion integration differentiation (PID) slidingsurface in order to obtain the faster transient response andthe smaller steady-state tracking error Benamor and Mes-saoud [6] studied the robust adaptive sliding mode controllaw for uncertain discrete systems with unknown time-varying time-delay input Finally the verification results ofthe control law on the actual system showed that althoughthere was no overmodulation phenomenon there was stillchattering phenomenon Although the advanced control

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6801205 22 pageshttpsdoiorg10115520206801205

algorithm can obtain better control performance in thesimulation research its structure is often complex and thereare many undetermined parameters so it is difficult to bepopularized in practice limiting its application in practice-erefore it is necessary to select a control strategy withrelatively simple method and good control performance

For the two-tank liquid level system a model-basedbackstepping controller and an adaptive backsteppingcontroller were designed [7] In order to illustrate the ef-fectiveness of the adaptive inverse control design a detailedexperimental comparison is made with the proportionalintegral controller A nonlinear generalized predictivecontrol and a backstepping algorithm were applied andtested in [8 9] Khalili et al [10] have developed a robustbackstepping sliding mode controller for tracking control ofa 2-DOF piezoelectric micro-operating system to eliminatechattering phenomenon in the control process Combiningthe finite time current observer with the adaptive back-stepping control scheme a control mechanism with highcost performance and strong robustness was obtained -eresults show that the control scheme can successfully esti-mate the unknown current providing a possibility for therealization of the current-free sensor controller [11] Anactive queue management scheme to control networkcongestion in combination with Hinfin theory and integralbackstepping technique to ensure better tracking perfor-mance and asymptotically stable of all signal probabilities inthe closed-loop system is given in [12] In [13 14] theauthors have applied nonlinear backstepping to ship controlfor quite a long time And the strong coupling characteristicof underactuated system was resolved by a virtual controller[15]

In practice more and more attention has been paid tothe influence of external interference on nonlinear systemsand the suppression of interference-e distributed adaptivecommand-filtered backstepping scheme [16] based on theneural network was presented which can ensure that thetracking error of the container reaches the desired originneighborhood and all signals in the closed-loop system arebounded An adaptive output feedback control problem [17]was studied for a class of uncertain nonlinear systems withinput delay and disturbance In [18ndash21] the generalframework of the nonlinear system can be obtained by usingthe disturbance observer-based control technology and theapplication of this method in the industrial field was il-lustrated In [22] when the static feedback cannot guaranteethe closed-loop stability the disturbance observer wasallowed to feedback as a dynamic system Liu et al [23 24]studied the speed tracking control problem of the syn-chronous motor drive system under a matched and un-matched interference a terminal sliding mode controlmethod and a port control Hamiltonian control methodbased on nonlinear disturbance observer which was pro-posed to realize the speed and current tracking control of thepermanent magnet synchronous motor drive system Inaddition an adaptive sliding mode control strategy based oninterference observer [25] was proposed to solve theproblems of multiple actuator faults parameter uncertaintyand external disturbances in a quadrotor helicopter

Different mechanisms compensate actuator fault parameteruncertainty and external disturbances respectively adoptan adaptive scheme to adjust actuator fault and parameteruncertainty and design a disturbance observer to attenuateexternal disturbance Moreover the multivariable distur-bance observer was proposed to improve the antidisturbanceperformance of traditional advanced feedback control [26]In [27] a robust control method based on the finite timedisturbance observer was proposed to track the output of thethree-tank system in the presence of mismatched uncer-tainties -en a robust control strategy with time delaycompensation was designed for multiple-input-multiple-output processes [28] with matching uncertainty and pro-cess delay Based on the mixed fuzzy reasoning systemartificial hydrocarbon network was used in the

Tank 1

Tank 2

a1

a2

a4

a3

Pump 1 Pump 2Storage tank

Figure 1 -e schematic diagram of the two-tank process

Table 1 Adjustable parameters of the system mode

Parameters Value Unita1 02 cm2

a2 03 cm2

a3 03 cm2

a4 03 cm2

A1 196 cm2

A2 196 cm2

2 Mathematical Problems in Engineering

defuzzification stage which was called fuzzy molecularcontrol [29] Gouta et al [30] proposed an adaptive controland a generalized predictive control method of the coupledtwo-tank system which are to minimize the multilevel costfunction defined on the prediction layer Aiming at the

problem of four-tank control [31] a set of disturbanceuncertainty suppression control laws were compared andproved whose control requirements were usually expressedin the literature in the form of a set value sequence -euncertainty class was defined as the union of four subclasses

19999

199995

20

200005

20001

t (s)

Reference x1

Integral backstepping control x1Sliding mode control x1

0

0 100 200

20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x 1 (c

m)

Figure 2 -e liquid-level curve of tank 1

0

2

4

6

8

10

12

14

16

18

158

1585

159

1595

16

1605

161

1615

162

Time (s)

Reference x2


0 20 40 60 80 100 120 140 160 180 200

x 2 (c

m)

0 200100


Table 2 -e sliding mode controller adjustable parameters

Parameters Valuem1 1n1 01b1 005m2 1n2 01b2 006

Table 3 -e integral backstepping controller adjustableparameters

Parameters Valuek1 1c1 00001k2 1c2 00001

Table 4 -e disturbance observer parameters

Parameters Valueh1 10h2 minus 1h3 1h4 minus 001

Mathematical Problems in Engineering 3

unknown disturbance parameter uncertainty measure-ment error and neglected dynamics In addition to theabove literature the second-order sliding controller [32]was successfully used to adjust the liquid level of the two-tank coupling liquid level system and the computer sim-ulation results show that the controller can adjust the liquid

level with little difference in performance A real-timeimplementation method of fuzzy coordinated classicalproportional integral (PI) control scheme was proposedExperimental results show that the controller structuretrack parameter changed quickly and had good perfor-mance when load disturbance and the set value changed

Time (s)

0

5

10

15

20

25

x (c

m)

Sliding mode control x1Sliding mode control x2

0 20 40 60 80 100 120 140 160 180 200

Figure 4 -e liquid-level curve of tank 1 after adding disturbance

Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



u 1 (c

m3 )

Integral backstepping controller u1Sliding mode controller u1

100 105 110 1200

100

200

300

50

100

150

200

250

300

20 12060 16010040 140 2000 80 180Time (s)

115

Figure 6 -e input curves of controller u1

100 120 14020

30

40

50

60

u 2 (c

m3 )


0

50

100

150

200

250

300

20 40 60 80 100 120 140 160 180 2000Time (s)



[33] -e performance of an active disturbance rejectioncontrol method in coupling tank system control wasstudied [34] -e results show that compared with othercontrollers the active disturbance rejection control methodwas effective in improving time domain measurement andsuppressing interference but it lacked experimental veri-fication Smida et al [35] adopted the observation schemecombining the high-gain observer and sliding mode ob-server to improve the robustness of state estimation qualityand to reconstruct the disturbance waveform in a betterway

In this paper mathematical model for the two-tankliquid level system is established based on the principle ofhydromechanics and the principle of mass conservation-en an integral backstepping control method for the two-tank liquid level system and disturbance observer is de-veloped Furthermore research studies of the integralbackstepping control method and the system with dis-turbance have been carried out on the innovative experi-mental platform for the complex control system of four-tank NTC-I type in quantity Simulation and experimentresults prove this suggested control strategy and distur-bance attenuation strategy highly effective compared withthe disturbance observer-based sliding mode controlstrategy of the two-tank liquid level system Moreover the

controller has fewer adjusting parameters simpler struc-ture and easier implementation And the steady-state anddynamic performance of the proposed controller are bothfar better than some complex algorithms listed in thereferences such as [32] -e results show that the proposedmethod has high-blooded dynamic and steady-state per-formances In practical industrial application the factoryworkshop will change the given value irregularly accordingto production demand However it is not realistic toreadjust the controller parameters every time when thegiven value is changed -erefore the control methodproposed in this paper has a wide range of practical ap-plication prospects

2 System Description and Modeling

-e schematic drawing in Figure 1 represents the model of atwo-tank liquid level system -is system consists of twotanks two level sensors (the level sensor is located at the topof each tank) two orifices (the orifice at the bottom of eachtank) two pumps a storage tank and four manual valves-e two tanks have same cross sections but the cross sec-tions of the four manual valves are different In this ex-perimental device the pump 1 feeds tank 1 and the outflowof tank 1 turns into partial input of tank 2-e pump 2 feeds

x 1 (c

m)

Integral backstepping control x1

Sliding mode control x1Reference x1

100 120 140

20

205

21

215

22

20 40 60 80 100 120 140 160 180 2000Time (s)

0

5

10

15

20

25



100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)

Integral backstepping control x1Integral backstepping control x2

0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)

Reference x1Sliding mode control x1


100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004



tank 2 and the effluent from tank 2 is discharged into thestorage tank

It can be known from the law of conservation of massand the time rate of change of liquid in each tank is givenby

ddt

ρAihi(t)1113858 1113859 ρqin(t) minus ρqout(t) (1)

where ρ is density of liquid Ai is the cross section of tank i(the unit is cm2) ai is the cross section of the outlet manualvalve i hi(t) is the height of liquid inside tank i (the unit iscm) i isin 1 2 3 4 the qin(t) is the output flow rate ofelectric control valve and qout(t) is the output flow rate ofthe tank at the bottom (the unit is cm3s)

By the Bernoulli equation in hydrodynamics the liquidflow velocity qout(t) of flowing out of the tank at the bottom

0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2

Sliding mode control x2



20 40 60 80 100 120 140 160 180 2000Time (s)


Figure 14 Experimental platform


is related to the cross-sectional area ai of the manual controlvalve at the bottom of the tank and the liquid level height inthe tank So the outflow velocity of the valve outlet the tankcan be expressed as

qout(t) ai

2ghi(t)

1113969

(2)

where g is the gravitational accelerationAccording to the conservation of mass principle the

differential equation of tank 1 and tank 2 can be written asfollows

_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)

where q1 and q2 are the output flow of the pump 1 and pump2 respectively

Define the system state and input asxi hi

ui qi1113896 (4)

where i isin 1 2 -us the mathematical model of the two-tank liquid

level system can be expressed as

_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)

3 Controller Design of the Two-Tank LiquidLevel System

To facilitate the calculation and interpretation of the maincontents in this section the following constants are defined

A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)

Based on the situation above the system described byequation (5) can be rewritten as follows

_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)

31 e Integral Backstepping Controller Design and StabilityAnalysis -e accuracy of the control target will be quantifiedby the liquid level tracking errors e1 and e2 of tank 1 and tank2 respectively And tracking errors are as follows

e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)

-en first-order derivative of equation (8) can be writtenas

Figure 15 Wincc liquid-level monitoring interface




_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)

Define the first-order derivative of the errors as

_e1 minus k1e1 minus c1 1113946t

0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)

where k1 gt 0 k2 gt 0 c1 gt 0 and c2 gt 0Substitute equation (10) into equation (9) and end up

with

_x1 _x1d minus k1e1 minus c1 1113946t

0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)

By combining equations (7) and (11) the controller canbe computed as

u1 _x1d + A

x1

radicminus k1e1 minus c1 1113938

t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1

The liquid-level curve of tank 1The liquid-level curve of tank 2

Figure 16 -e liquid-level curves of sliding mode control




Hence the integral backstepping controller can be de-rived from equations (6) (8) and (12) -e result is

u1 a1

2gx1

1113968A11113872 1113873 + _x1d minus k1 x1 minus x1d( 1113857 minus c1 1113938

t

0 x1 minus x1d( 1113857dt

a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

Select the Lyapunov function V as

V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)

Taking the first-order time derivative for equation (15)we can obtain

_V _V1 + _V2 e1 _e1 + e2 _e2 (16)

-en we can apply equation (10) to equation (16) andthe result of the equation (16) can be calculated as

191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Figure 17 -e liquid-level curves of integral backstepping control




_V e1 minus k1e1 minus c1 1113946t

0e1dt1113888 1113889 + e2 minus k2e2 minus c2 1113946

t

0e2dt1113888 1113889

minus k1e21 minus c1 1113946

t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)

Obviously V is positive definite and _V is negativedefinite and both satisfy Lyapunov stability theorem sosystem (5) is asymptotically stable

Remark 1 Assume that there exists a scalar function V ofthe state x with continuous first-order derivatives suchthat

(i) V(x) is positive definite(ii) _V(x) is negative definite(iii) V(x)⟶infin as x⟶infin

-en the equilibrium at the xd is globally asymptoticallystable [36] xd is the expected value of the liquid level

32 Design and Stability Analysis of the Controller with theDisturbance Observer -is paper considers a class of

nonlinear systems with exogenous disturbances which canbe described as

_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)

where d1 and d2 are exogenous disturbances and they arebounded

d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)

where D1 and D2 are known positive constantsConstruct the disturbance observers as

1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)

where h1 h2 h3 and h4 are the adjustable parametersWith the estimated disturbance 1113957d1 and 1113957d2 the distur-

bance estimated errors are defined as

191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance

Figure 18 -e liquid-level curves of tank 1 with disturbance on sliding mode control


1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)

By combining equations (19) and (21) the first-orderderivative of the disturbance estimation errors can be cal-culate as

1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)

there λ1 and λ2 are both the nonlinear disturbance observergains

-en the observer error equation is given as

_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance

Figure 19 -e liquid-level curves of tank 1 with disturbance on integral backstepping control


H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)

It is easy to prove that the observer is asymptoticallystable with proper choice of h1 h2 h3 and h4

Remark 2 If the linearized system is strictly stable (ie if alleigenvalues of H are strictly in the left-half complex plane)then the equilibrium point is asymptotically stable (for theactual nonlinear system) [36]

Finally according to equations (13) (18) (20) and (21)the controller for the nonlinear systemwith disturbances canbe obtained as follows

u1 a1

2gx1


t

0 x1 minus x1d( 1113857dt minus 1113954d1

a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1

The input curve of sliding mode controller u1The input curve of sliding mode controller u2

Figure 20 -e input curves of the sliding mode controller


In this case tracking error equation (8) can be rewrittenas

e1prime x1 minus x1d

e2prime x2 minus x2 d

⎧⎨

⎩ (28)

Consider the rate of error change the time derivative of(28) can be expressed as

_e1prime _x1 minus _x1d


⎧⎨

⎩ (29)

Choose a Lyapunov function Vprime for nonlinear system(18) as

V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)

Vprime V1prime + V2prime 12

eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)

-en the first-order time derivative of the selectedLyapunov function (31) is obtained as

_Vprime _V1prime + _V2prime e1prime _e1prime + 1113957d1prime_1113957d1prime + e2prime _e2prime + 1113957d2prime

_1113957d2prime (32)

By combining equations (10) and (32) (32) can be re-written as

_Vprime e1prime minus k1e1prime minus c1 1113946t

0e1primedt1113888 1113889 + e2prime minus k2e2prime minus c2 1113946

t

0e2primedt1113888 1113889

minus λ11113957d21 minus λ21113957d

22 minus k1e

prime21 minus c1 1113946

t

0eprime21 dt minus k2e

prime22 minus c2

middot 1113946t

0eprime22 dt minus λ11113957d

21 minus λ21113957d

22

(33)

Obviously Vprime is positive definite In addition fromequation (33) one can know that _Vprime is negative definiteHence it satisfied Lyapunov stability theorem which meanssystem (18) is asymptotically stable

191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



4 Simulation Results and Analysis

In order to verify the effectiveness of the above controlstrategy we compared it with the disturbance observer-based sliding mode control method

-e disturbance observer-based sliding mode controllercan be described as

u1 A1

a3minus m1sgn s1( 1113857 + n1 + b1( 1113857 x1 minus x1d( 1113857 + n1b1 1113946 e1dt1113876 1113877 + _x1d +

a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)

where b1 b2 m1 m2 n1 and n2 are positive constants-e comparative study results of the two control strat-

egies are carried out in Simulink environment -en theadjustable parameters of the two-tank coupled liquid levelsystem are given in Table 1

-e desired equilibrium points are

x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



41 e Undisturbed Simulation Results

411 Sliding Mode Controller -e controlled liquid leveltrajectories of the two tanks are given in Figures 2 and 3respectively -e adjustable parameters are determined asshown in Table 2

As shown in Figures 2 and 3 the dynamic responsetime of the liquid-level curve of tank 1 is short its risetime is tr 10 s but overshoot occurs its overshoot isσ 00000025 and there has always been a steady-state error of 00002 cm Although the liquid-level curveof tank 2 did not overshoot its dynamic stage rose slowlywhen approaching the equilibrium point and the risetime is tr 100 s

412 Integral Backstepping Controller -e controlled liquidlevel trajectories of the two tanks are shown in Figures 2 and3 respectively -e adjustable parameters are determined asshown in Table 3

As shown in Figures 2 and 3 -e rise time of liquid-levelcurve of tank 1 is also tr 10 s Although overshoot occursand overshoot is σ 00000025 the steady-state error iszero-e liquid-level curve of tank 2 did not overshoot and itsdynamic response time is short and its rise time is tr 10 s

42 Simulation Results with Disturbance -e simulationverification in this paper is obtained by injecting the disturbancein the form of step signal into the system without changing thecontroller parameters of the two-tank liquid level system -eadjustable parameters are determined as shown in Table 4

421 Sliding Mode Controller -e liquid-level curves andcontrol inputs trajectories of two tanks are given in Figures 4and 5 and Figures 6 and 7 respectively -e disturbances areadded to tank 1 and tank 2 at the simulation time of 100seconds respectively Simulation results show that the in-jection of external disturbances into a single tank has noeffect on the liquid level of another tank It can be seen fromFigures 8 and 9 that the recovery time is relatively long ofliquid levels in tank 1 and tank 2 after injection of externaldisturbance and the adjustment time is ts 50 s

422 Integral Backstepping Controller -e liquid-levelcurves and control inputs trajectories of two tanks are given inFigures 10 and 11 and Figures 6 and 7 respectively At thesimulation time of 100 seconds the disturbances are added totank 1 and tank 2 respectively -en the simulation resultsshow that the injection of external disturbances into a singletank can also achieve almost no effect on the liquid level of

191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1

The input curve of integral backstepping controller u1The input curve of integral backstepping controller u2

Figure 23 -e input curves of the integral backstepping controller


another tank It can be seen from Figures 8 and 9 that therecovery time is faster of liquid levels in tank 1 and tank 2 afterinjection of external disturbance and the adjustment time ists 10 s

43 Comparison of Simulation Results For the determina-tion and optimization of the parameters the Lyapunovstability theorem is used to determine the range of pa-rameters and then the specific parameter values are de-termined by the empirical trial and error method such as inFigures 12 and 13

Simulation results show that the two-tank liquid levelsystem with sliding mode control algorithm has a slightliquid level overshoot which not only has a certainamount of steady-state error but also takes a long time torestore stability after injection of disturbances In the two-tank liquid level system using the integral backsteppingcontrol algorithm based on the disturbance observer theliquid level has a slight overshoot but its steady-stateerror is zero and it can recover to a stable value in a veryshort time after injecting the external disturbancesAccording to the above discussion the integral back-stepping control method based on the disturbance ob-server has good dynamic stability and disturbancesuppression characteristics

5 Experimental Results and Analysis

In order to verify the performances of the propose controlstrategy experiments are developed on the Innovative ex-perimental platform for the complex control system of thefour-tank NTC-I type As shown in Figures 14 and 15 theyare the experimental platform and Wincc liquid-levelmonitoring interface respectively -e experimental appa-ratus is a typical process control object in industrial process-four tank For the object a feedback control system based onprogrammable logic controller (PLC) and Matlabsimulinkare constructed -e PLC part uses the Siemens CPU s7-300module to extend the special analog input -e modulecollects the actual water level of the four-capacity water tankand expands the dedicated analog output module feed water-e pump provides the actual required analog voltageCommunication between Matlab and PLCHMl industrialcontrol is realized through OLE for process control (OPC)communication technology-e control system is integratedwith some advanced system identification techniques andadvanced control methods that can be applied to the con-trolled object of the four-tank easily -e upper computeruses a common desktop computer and the operating systemuses Microsoftrsquos Windows XP system Other developmentsoftware used in this device includes WinCC V62 moni-toring software of SIEMENS down-bit STEP7 V 54 of PLC

191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Figure 24 -e liquid-level changed curves of sliding mode control


controller configuration programming and configurationcommunication mode and OPC server configurationSIMATIC NET software -e adjustable parameters of thetwo-tank liquid level system are given in Table 1


x1d 20 cm

x2d 16 cm(36)

51 e Undisturbed Experimental Results

511 Sliding Mode Controller -e controller adjustableparameters are given in Table 5

-e actual controlled liquid-level trajectories of the twotanks are shown in Figure 16 -e dynamic response time ofthe experimental curves of tank 1 and tank 2 are relatively fastrising time is tr 35 s and tr 45 s respectively but there isalways obvious chattering phenomenon of the liquid leveland the amplitude is plusmn05 cm

512 Integral Backstepping Controller -e integral back-stepping controller adjustable parameters are given inTable 6

-e actual controlled liquid-level trajectories of the twotanks are given in Figure 17 Under the same condition ofvalve opening the dynamic response time of tank 1 and tank2 liquid-level curves is still very fast and the rising time is tr

30 s and tr 40 s respectively Moreover there is no chat-tering phenomenon in the liquid-level curve under thiscontrol method and the steady-state process is very good

52 Experimental Results with Disturbance In this paperdisturbances are added to the two-tank liquid level systemand the controller parameters are unchanged -e distur-bance observer parameters are shown in Table 7

521 Sliding Mode Controller -e actual controlled liquid-level curves and control input trajectories of the two tankswith disturbances are shown in Figures 18ndash20 A certainamount of water is injected into tank 1 at 20 07 20 -eliquid level of tank 1 has apparent fluctuation and theadjustment time is ts 30 s At 17 26 40 a certain amountof water is injected into tank 2 the liquid level of tank 2 hasapparent fluctuation and the adjustment time is ts 30 s

522 Integral Backstepping Controller -e actual con-trolled liquid-level curves and control input trajectories of

191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Figure 25 -e liquid-level changed curves of integral backstepping control


the two tanks with disturbances are given in Figures 21ndash23A certain amount of water is injected into tank 1 and tank 2at 11 12 30 and 11 00 30 respectively -e liquid levels oftank 1 and tank 2 have hardly any fluctuation and theadjustment time is ts 15 s and ts 15 s respectively

As shown in Figures 24 and 25 without changing thecontroller parameters of the two-tank liquid level system Itstill has good dynamic and steady-state performance whenthe set value is changed

53 Comparison of Experimental Results -e sliding modecontrol algorithm is tested and verified on the equipmentplatform It can be seen from Figures 16 18 and 19 that thetwo-tank liquid level system controlled by the sliding modealgorithm always has the phenomenon of chattering es-pecially when the liquid level is disturbed externally As canbe seen from Figures 17 21 and 22 the disturbance observer-based integral backstepping control method not only has nochattering phenomenon but also exhibits good steady-stateperformance and disturbance-suppression characteristics inthe case of disturbance

6 Conclusions

In view of the two-tank liquid level system a methodology ofdisturbance observer-based integral backstepping control isproposed in this paper According to the principle of fluidmechanics and mass conservation the mathematical model ofthe two-tank liquid level system is established As can be seenfrom the simulation results when the external disturbance isadded to a single tank the liquid-level value of another tank hasno influence Furthermore it can be known from the experi-ment results that the given value can be set discretionarily andreached quickly-en the results of simulation and experimentverified that the disturbance observer-based integral back-stepping control methodology has good dynamic and staticperformance compared with the disturbance observer-basedsliding mode control method Moreover the steady-state anddynamic performance of the proposed controller are both farbetter than some complex algorithms listed in the references Itsstructure is simpler and adjustable parameters are fewer andeasier to realize -erefore the control method proposed in thispaper has a wide range of practical application prospectsHowever the controller proposed in this paper does not con-sider the influence of time delay on the system and it is onlylimited to the two-tank water level systemWhether it is suitablefor the four-tank water level system needs to be further verified

Further in addition to considering the influence ofdisturbance on the system the influence of time delay on thesystem is also considered and corresponding solution isgiven which is combined with the method proposed in thispaper to achieve better control effect -en the compositecontrol method is extended and applied to the four-tankliquid level system

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e authors declare no conflicts of interest

Authorsrsquo Contributions

MXX developed the theoretical results wrote the paperand performed the experiments YHS modified themathematical formula and grammar WHR ensured thehardware and software support and XT analyzed the al-gorithms All authors have read and approved the finalpaper

Acknowledgments

-is research was funded by the National Natural ScienceFoundation of China Grant no 61573203

References

[1] H Yu J Yu H Wu and H Li ldquoEnergy-shaping and integralcontrol of the three-tank liquid level systemrdquo NonlinearDynamics vol 73 no 4 pp 2149ndash2156 2013

[2] M K Khan and S K Spurgeon ldquoRobust MIMO water levelcontrol in interconnected twin-tanks using second ordersliding mode controlrdquo Control Engineering Practice vol 14no 4 pp 375ndash386 2006

[3] P P Biswas R Srivastava S Ray and A N Samanta ldquoSlidingmode control of quadruple tank processrdquo Mechatronicsvol 19 no 4 pp 548ndash561 2009

[4] E Govind Kumar and J Arunshankar ldquoControl of nonlineartwo-tank hybrid system using sliding mode controller withfractional-order PI-D sliding surfacerdquo Computers amp ElectricalEngineering vol 71 pp 953ndash965 2018

[5] Y PWeng and XW Gao ldquoData-driven sliding mode controlof unknown MIMO nonlinear discrete-time systems withmoving PID sliding surfacerdquo Journal of the Franklin Institutevol 354 no 15 pp 6463ndash6502 2017

[6] A Benamor and H Messaoud ldquoRobust adaptive sliding modecontrol for uncertain systems with unknown time-varyingdelay inputrdquo ISA Transactions vol 79 pp 1ndash12 2018

[7] H Pan H Wong V Kapila and M S de Queiroz ldquoEx-perimental validation of a nonlinear backstepping liquid levelcontroller for a state coupled two tank systemrdquo ControlEngineering Practice vol 13 no 1 pp 27ndash40 2005

[8] H Gouta S Hadj Said and F Mrsquosahli ldquoModel-Based Pre-dictive and Backstepping controllers for a state coupled four-tank system with bounded control inputs a comparativestudyrdquo Journal of the Franklin Institute vol 352 no 11pp 4864ndash4889 2015

[9] J-H Huo T Meng R-T Song and Z-H Jin ldquoAdaptiveprediction backstepping attitude control for liquid-filledmicro-satellite with flexible appendagesrdquo Acta Astronauticavol 152 pp 557ndash566 2018

[10] A Khalili Z Mohamed and M Basri ldquoEnhanced back-stepping sliding mode controller for motion tracking of anonlinear 2-DOF piezo-actuated micromanipulation systemrdquoMicrosystem Technologies vol 25 no 3 pp 1ndash13 2019

[11] T K Nizami A Chakravarty and C Mahanta ldquoAnalysis andexperimental investigation into a finite time current observerbased adaptive backstepping control of buck convertersrdquoJournal of the Franklin Institute vol 355 pp 4771ndash5350 2018


[12] Y Liu X Liu Y Jing Z Zhang and X Chen ldquoCongestiontracking control for uncertain TCPAQM network based onintegral backsteppingrdquo ISA Transactions vol 89 pp 131ndash1382019

[13] J Peng and R Dubay ldquoAdaptive fuzzy backstepping controlfor a class of uncertain nonlinear strict-feedback systemsbased on dynamic surface control approachrdquo Expert SystemsWith Applications vol 120 pp 239ndash252 2019

[14] C Fu W Hong H Lu L Zhang X Guo and Y TianldquoAdaptive robust backstepping attitude control for a multi-rotor unmanned aerial vehicle with time-varying outputconstraintsrdquo Aerospace Science and Technology vol 78pp 593ndash603 2018

[15] X Shi Y Cheng C Yin S Dadras and X Huang ldquoDesign offractional-order backstepping sliding mode control forquadrotor UAVrdquo Asian Journal of Control vol 21 no 1pp 156ndash171 2019

[16] L Zhao J Yu H Yu and C Lin ldquoNeuroadaptive contain-ment control of nonlinear multiagent systems with inputsaturationsrdquo International Journal of Robust and NonlinearControl vol 29 no 9 pp 2742ndash2756 2019

[17] J Ma S Xu Y Li Y Chu and Z Zhang ldquoNeural networks-based adaptive output feedback control for a class of uncertainnonlinear systems with input delay and disturbancesrdquo Journalof the Franklin Institute vol 355 no 13 pp 5503ndash5519 2018

[18] A Vahidi-Moghaddam A Rajaei and M Ayati ldquoDistur-bance-observer-based fuzzy terminal sliding mode control forMIMO uncertain nonlinear systemsrdquo Applied MathematicalModelling vol 70 pp 109ndash127 2019

[19] X K Chen S Komada and T Fukuda ldquoDesign of a nonlineardisturbance observerrdquo IEEE Transactions on IndustrialElectronics vol 47 pp 429ndash436 2000

[20] W-H Chen ldquoDisturbance observer based control for non-linear systemsrdquo IEEEASME Transactions on Mechatronicsvol 9 no 4 pp 706ndash710 2004

[21] J Yang S Li and W-H Chen ldquoNonlinear disturbance ob-server-based control for multi-input multi-output nonlinearsystems subject to mismatching conditionrdquo InternationalJournal of Control vol 85 no 8 pp 1071ndash1082 2012

[22] W-H Chen J Yang L Guo and S Li ldquoDisturbance-ob-server-based control and related methods-an overviewrdquo IEEETransactions on Industrial Electronics vol 63 no 2pp 1083ndash1095 2016

[23] X Liu H Yu J Yu and L Zhao ldquoCombined speed andcurrent terminal sliding mode control with nonlinear dis-turbance observer for PMSM driverdquo IEEE Access vol 6pp 29594ndash29601 2018

[24] X Liu H Yu J Yu and Y Zhao ldquoA novel speed controlmethod based on port-controlled Hamiltonian and distur-bance observer for PMSM drivesrdquo IEEE Access vol 7pp 111115ndash111123 2019

[25] B Wang X Yu L Mu and Y Zhang ldquoDisturbance observer-based adaptive fault-tolerant control for a quadrotor heli-copter subject to parametric uncertainties and external dis-turbancesrdquo Mechanical Systems and Signal Processingvol 120 pp 727ndash743 2019

[26] P Zhou W Dai and T Y Chai ldquoMultivariable disturbanceobserver based advanced feedback control design and itsapplication to a grinding circuitrdquo IEEE Transactions onControl Systems Technology vol 4 pp 1471ndash1485 2014

[27] Z-J Yang and H Sugiura ldquoRobust nonlinear control of athree-tank system using finite-time disturbance observersrdquoControl Engineering Practice vol 84 pp 63ndash71 2019

[28] D H Shah and D M Patel ldquoDesign of sliding mode controlfor quadruple-tank MIMO process with time delay com-pensationrdquo Journal of Process Control vol 76 pp 46ndash61 2019

[29] H Ponce P Ponce H Bastida and A Molina ldquoA novelrobust liquid level controller for coupled-tanks systems usingartificial hydrocarbon networksrdquo Expert Systems with Ap-plications vol 42 no 22 pp 8858ndash8867 2015

[30] H Gouta S Hadj Saıd A Turki and F MrsquoSahli ldquoExperi-mental sensorless control for a coupled two-tank system usinghigh gain adaptive observer and nonlinear generalized pre-dictive strategyrdquo ISA Transactions vol 87 pp 187ndash199 2019

[31] C Huang E Canuto and C Novara ldquo-e four-tank controlproblem comparison of two disturbance rejection controlsolutionsrdquo Isa Transactions vol 71 pp 252ndash271 2017

[32] F A Khadra and J A Qudeiri ldquoSecond order sliding modecontrol of the coupled tanks systemrdquo Mathematical Problemsin Engineering vol 2015 Article ID 167852 9 pages 2015

[33] N Kanagaraj P Sivashanmugam and S ParamasivamldquoFuzzy coordinated PI controller application to the real-timepressure control processrdquo Advances in Fuzzy Systemsvol 2008 Article ID 691808 9 pages 2008

[34] F Abukhadra ldquoActive disturbance rejection control of acoupled-tank Systemrdquo Journal of Engineering vol 2018Article ID 7494085 6 pages 2018

[35] F Smida S Hadj Saıd and F Mrsquosahli ldquoRobust high-gainobservers based liquid levels and leakage flow rate estima-tionrdquo Journal of Control Science and Engineering vol 2018Article ID 8793284 8 pages 2018

[36] J J Slotine and W P Li Applied nonlinear control Prentice-Hall Englewood Cliffs NJ USA 1991


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algorithm can obtain better control performance in thesimulation research its structure is often complex and thereare many undetermined parameters so it is difficult to bepopularized in practice limiting its application in practice-erefore it is necessary to select a control strategy withrelatively simple method and good control performance

For the two-tank liquid level system a model-basedbackstepping controller and an adaptive backsteppingcontroller were designed [7] In order to illustrate the ef-fectiveness of the adaptive inverse control design a detailedexperimental comparison is made with the proportionalintegral controller A nonlinear generalized predictivecontrol and a backstepping algorithm were applied andtested in [8 9] Khalili et al [10] have developed a robustbackstepping sliding mode controller for tracking control ofa 2-DOF piezoelectric micro-operating system to eliminatechattering phenomenon in the control process Combiningthe finite time current observer with the adaptive back-stepping control scheme a control mechanism with highcost performance and strong robustness was obtained -eresults show that the control scheme can successfully esti-mate the unknown current providing a possibility for therealization of the current-free sensor controller [11] Anactive queue management scheme to control networkcongestion in combination with Hinfin theory and integralbackstepping technique to ensure better tracking perfor-mance and asymptotically stable of all signal probabilities inthe closed-loop system is given in [12] In [13 14] theauthors have applied nonlinear backstepping to ship controlfor quite a long time And the strong coupling characteristicof underactuated system was resolved by a virtual controller[15]

In practice more and more attention has been paid tothe influence of external interference on nonlinear systemsand the suppression of interference-e distributed adaptivecommand-filtered backstepping scheme [16] based on theneural network was presented which can ensure that thetracking error of the container reaches the desired originneighborhood and all signals in the closed-loop system arebounded An adaptive output feedback control problem [17]was studied for a class of uncertain nonlinear systems withinput delay and disturbance In [18ndash21] the generalframework of the nonlinear system can be obtained by usingthe disturbance observer-based control technology and theapplication of this method in the industrial field was il-lustrated In [22] when the static feedback cannot guaranteethe closed-loop stability the disturbance observer wasallowed to feedback as a dynamic system Liu et al [23 24]studied the speed tracking control problem of the syn-chronous motor drive system under a matched and un-matched interference a terminal sliding mode controlmethod and a port control Hamiltonian control methodbased on nonlinear disturbance observer which was pro-posed to realize the speed and current tracking control of thepermanent magnet synchronous motor drive system Inaddition an adaptive sliding mode control strategy based oninterference observer [25] was proposed to solve theproblems of multiple actuator faults parameter uncertaintyand external disturbances in a quadrotor helicopter

Different mechanisms compensate actuator fault parameteruncertainty and external disturbances respectively adoptan adaptive scheme to adjust actuator fault and parameteruncertainty and design a disturbance observer to attenuateexternal disturbance Moreover the multivariable distur-bance observer was proposed to improve the antidisturbanceperformance of traditional advanced feedback control [26]In [27] a robust control method based on the finite timedisturbance observer was proposed to track the output of thethree-tank system in the presence of mismatched uncer-tainties -en a robust control strategy with time delaycompensation was designed for multiple-input-multiple-output processes [28] with matching uncertainty and pro-cess delay Based on the mixed fuzzy reasoning systemartificial hydrocarbon network was used in the

Tank 1

Tank 2

a1

a2

a4

a3

Pump 1 Pump 2Storage tank

Figure 1 -e schematic diagram of the two-tank process

Table 1 Adjustable parameters of the system mode

Parameters Value Unita1 02 cm2

a2 03 cm2

a3 03 cm2

a4 03 cm2

A1 196 cm2

A2 196 cm2




19999

199995

20

200005

20001

t (s)

Reference x1


0

0 100 200

20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x 1 (c

m)


0

2

4

6

8

10

12

14

16

18

158

1585

159

1595

16

1605

161

1615

162

Time (s)

Reference x2


0 20 40 60 80 100 120 140 160 180 200

x 2 (c

m)

0 200100











Time (s)

0

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



u 1 (c

m3 )


100 105 110 1200

100

200

300

50

100

150

200

250

300

20 12060 16010040 140 2000 80 180Time (s)

115


100 120 14020

30

40

50

60

u 2 (c

m3 )


0

50

100

150

200

250

300

20 40 60 80 100 120 140 160 180 2000Time (s)








x 1 (c

m)



100 120 140

20

205

21

215

22

20 40 60 80 100 120 140 160 180 2000Time (s)

0

5

10

15

20

25



100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










19999

199995

20

200005

20001

t (s)

Reference x1


0

0 100 200

20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x 1 (c

m)


0

2

4

6

8

10

12

14

16

18

158

1585

159

1595

16

1605

161

1615

162

Time (s)

Reference x2


0 20 40 60 80 100 120 140 160 180 200

x 2 (c

m)

0 200100











Time (s)

0

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



u 1 (c

m3 )


100 105 110 1200

100

200

300

50

100

150

200

250

300

20 12060 16010040 140 2000 80 180Time (s)

115


100 120 14020

30

40

50

60

u 2 (c

m3 )


0

50

100

150

200

250

300

20 40 60 80 100 120 140 160 180 2000Time (s)








x 1 (c

m)



100 120 140

20

205

21

215

22

20 40 60 80 100 120 140 160 180 2000Time (s)

0

5

10

15

20

25



100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










Time (s)

0

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



u 1 (c

m3 )


100 105 110 1200

100

200

300

50

100

150

200

250

300

20 12060 16010040 140 2000 80 180Time (s)

115


100 120 14020

30

40

50

60

u 2 (c

m3 )


0

50

100

150

200

250

300

20 40 60 80 100 120 140 160 180 2000Time (s)








x 1 (c

m)



100 120 140

20

205

21

215

22

20 40 60 80 100 120 140 160 180 2000Time (s)

0

5

10

15

20

25



100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018








u 1 (c

m3 )


100 105 110 1200

100

200

300

50

100

150

200

250

300

20 12060 16010040 140 2000 80 180Time (s)

115


100 120 14020

30

40

50

60

u 2 (c

m3 )


0

50

100

150

200

250

300

20 40 60 80 100 120 140 160 180 2000Time (s)








x 1 (c

m)



100 120 140

20

205

21

215

22

20 40 60 80 100 120 140 160 180 2000Time (s)

0

5

10

15

20

25



100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













x 1 (c

m)



100 120 140

20

205

21

215

22

20 40 60 80 100 120 140 160 180 2000Time (s)

0

5

10

15

20

25



100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018








100 120 140158

16

162

164

166

168

17

x 2 (c

m)



0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100 120 140 160 180 2000Time (s)


Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)



Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018








Time (s)


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

x (c

m)


0 20 40 60 80 100 120 140 160 180 200Time (s)

0

5

10

15

20

25

x 1 (c

m)



100 120 140199

1995

20

2005

201

0 50 100195

20

205

n1 = 005

n1 = 006

n1 = 004





ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










ddt




0

2

4

6

8

10

12

14

16

18

0 20 40 60 80155

16

165

100 120 140

x 2 (c

m)

159

1595

16

1605

161

n1 = 005

n1 = 007

n1 = 006

Reference x2




20 40 60 80 100 120 140 160 180 2000Time (s)





qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









qout(t) ai

2ghi(t)

1113969

(2)



_h1(t) a3

A1q1 minus

a1

A1

2gh1(t)

1113969

_h2(t) a4

A2q2 +

a1

A2

2gh1(t)

1113969

minusa2

A2

2gh2(t)

1113969

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)



ui qi1113896 (4)



_x1 minusa1

A1

2gx1

1113968+

a3

A1u1

_x2 a1

A2

2gx1

1113968minus

a2

A2

2gx2

1113968+

a4

A2u2

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(5)



A a1

A1

2g

1113968

B a3

A1

C a1

A2

2g

1113968

D a2

A2

2g

1113968

E a4

A2

(6)


_x1 minus Ax1

radic+ Bu1

_x2 Cx1

radicminus D

x2

radic+ Eu2

⎧⎪⎨

⎪⎩(7)


e1 x1 minus x1d

e2 x2 minus x2d1113896 (8)






_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018








_e1 _x1 minus _x1d

_e2 _x2 minus _x2d1113896 (9)



0e1dt


0e2dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


with


0e1dt


0e2dt

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(11)


u1 _x1d + A

x1


t

0 e1dt

B

u2 _x2d + D

x2

radicminus C

x1


t

0 e2dt

E

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(12)

191111 165710165530 165850 170030 170210 170350 170530

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


V1 12

e21

V2 12

e22

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(14)

V V1 + V2 12

e21 +

12

e22 (15)


_V _V1 + _V2 e1 _e1 + e2 _e2 (16)


191111 102700102520 102840 103020 103200 103340 103520

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1








t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










t

0e2dt1113888 1113889


t

0e21dt minus k2e

22 minus c2 1113946

t

0e22dt

(17)







_x1 minus Ax1

radic+ Bu1 + d1

_x2 Cx1

radicminus D

x2

radic+ Eu2 + d2

⎧⎨

⎩ (18)


d11113868111386811138681113868

1113868111386811138681113868leD1

d21113868111386811138681113868

1113868111386811138681113868leD2

⎧⎨

⎩ (19)


1113954x_1 minus A1113954x1

1113968+ Bu1 + 1113954d1 + h1 x1 minus 1113954x1( 1113857

1113954d_

1 h2 x1 minus 1113954x1( 1113857

⎧⎪⎨

⎪⎩(20)

1113954x_ 2 Cx1

radicminus D

1113954x2

1113968+ Eu2 + 1113954d2 + h3 x2 minus 1113954x2( 1113857

1113954d_

2 h4 x2 minus 1113954x2( 1113857

⎧⎪⎨

⎪⎩(21)



191111 200338200158 200518 200658 200838 201018 201158

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018








1113957x1 x1 minus 1113954x1

1113957d1 d1 minus 1113954d1

⎧⎨

⎩

1113957x2 x2 minus 1113954x2

1113957d2 d2 minus 1113954d2

⎧⎨

⎩

(22)


1113957d_

1 _d1 minus 1113954d_

1 minus λ11113957d1

1113957d_

2 _d2 minus 1113954d_

2 minus λ21113957d2

⎧⎪⎨

⎪⎩(23)



_zi Hizi (24)

Here

z1

z2

⎡⎢⎣ ⎤⎥⎦ 1113957x1

1113957d1

⎡⎢⎣ ⎤⎥⎦z3

z4

⎡⎢⎣ ⎤⎥⎦ 1113957x2

1113957d2

⎡⎢⎣ ⎤⎥⎦

_z1

_z2

⎡⎢⎣ ⎤⎥⎦

_1113957x1

_1113957d1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

_z3

_z4

⎡⎢⎣ ⎤⎥⎦

_1113957x2

_1113957d2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

i 1 2

(25)

So one can get

_z1

_z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

minusA

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z1

z2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

_z3

_z4

⎡⎢⎢⎣ ⎤⎥⎥⎦ minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z3

z4

⎡⎢⎢⎣ ⎤⎥⎥⎦

191111 172306172126 172446 172626 172806 172946 173126

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance



H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018








H1 minus

A

2minus h1 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

H2 minus

D

2minus h2 0

0 1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ (26)




u1 a1

2gx1


t


a3A1( 1113857

u2 a2

2gx2

1113968A21113872 1113873 minus a1

2gx1


t


a4A2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(27)

191111 202826202646 203006 203146 203326 203506 203646

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2u1







⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











⎧⎨

⎩ (28)




⎧⎨

⎩ (29)


V1prime 12

eprime21 +

12

1113957d21

V2prime 12

eprime22 +

12

1113957d22

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(30)


eprime21 +

12

1113957d21 +

12

eprime22 +

12

1113957d22 (31)



_1113957d2prime (32)




t

0e2primedt1113888 1113889


22 minus k1e


t


prime22 minus c2

middot 1113946t


21 minus λ21113957d

22

(33)


191111 110853110713 111033 111213 111353 111533 111713

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance






u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











u1 A1


a12gx1

1113968

A1minus 1113954d11113896 1113897

u2 A2


a22gx2

1113968

A2minus

a12gx1

1113968

A2minus 1113954d21113896 1113897

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(34)




x1d 20 cm

x2d 16 cm(35)

191111 105659105519 105839 110019 110159 110339 110519

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1


Disturbance











191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018
















191111 212335212155 212515 212655 212835 213015 213155

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

136

150

0

8

16

24

32

40

48

56

64

72

80

88

96

104

112

120

128

144

u2 u1









191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













191111 175158175018 175338 175518 175658 175838 180018

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1






x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










x1d 20 cm

x2d 16 cm(36)










191111 164323164143 164503 164643 164823 165003 165143

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x2

x1







6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











6 Conclusions



Data Availability






Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









































Journal of












Journal of







Volume 2018






Volume 2018








disturbance observer-based integral backstepping control for a...

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