research article robust finite-time terminal sliding mode...

Research ArticleRobust Finite-Time Terminal Sliding Mode Control fora Francis Hydroturbine Governing System

Fengjiao Wu Junling Ding and Zhengzhong Wang

College of Water Resources and Architectural Engineering Northwest AampF University Yangling 712100 China

Correspondence should be addressed to Zhengzhong Wang wangzz0910163com

Received 4 May 2016 Accepted 27 July 2016

Academic Editor Enrique Onieva

Copyright copy 2016 Fengjiao Wu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The robust finite-time control for a Francis hydroturbine governing system is investigated in this paper Firstly the mathematicalmodel of a Francis hydroturbine governing system is presented and the nonlinear vibration characteristics are analyzed Then onthe basis of finite-time control theory and terminal sliding mode scheme a new robust finite-time terminal sliding mode controlmethod is proposed for nonlinear vibration control of the hydroturbine governing system Furthermore the designed controller hasgood robustness which could resist external random disturbances Numerical simulations are employed to verify the effectivenessand superiority of the designed finite-time sliding mode control scheme The approach proposed in this paper is simple and alsoprovides a reference for relevant hydropower systems

1 Introduction

In the last two decades the worldrsquos total energy demandhas dramatically increased Renewable energy has got moreand more attention [1] Many countries take the hydropowerdevelopment in the first place With the increase inhydropower stations the hydropower system security andstability face more challenges [2ndash5] As we know hydro-turbine governing system (HGS) plays an important role inmaintaining the safety stability and economical operationfor hydropower plant However HGS is a nonlinear time-varying and nonminimum phase systems [6 7] The internaluncertainty of the dynamics and variability of external envi-ronment disturbance increase the difficulty of HGS stabilityanalysis and control

Scholars have made many important contributions onthe stability analysis of HGS [8ndash11] Based on the abovetheoretical results stability control of HGS has become a hottopic recently Many control methods have been proposedsuch as the classical PID control [12] sliding mode control[13] intelligent control [14] and identification control [15ndash17] However all of the mentioned control methods arebased on the stability theory of Lyapunov stability theoremand asymptotic stability theory The dynamic quality of

the transition process is little considered From the viewof improving control quality and time optimization finite-time control technique could greatly improve the maximumdeviation and the transition time of the system and has betterrobustness and anti-interference capability [18 19]

Sliding mode control is an essentially nonlinear controlstrategy with a fast response good dynamic characteristicsand insensitivity to external changes and many other attrac-tive advantages [20 21] In the conventional sliding modecontrol process usually a linear sliding surface is selectedWhen the system reaches the sliding mode the trackingerror converges to zero and the asymptotic convergence ratecould be regulated by selecting sliding surface parametersHowever in any case the tracking error will not converge tozero within a finite time [22 23]

That is both finite-time control in improving the transi-tion process and sliding mode control in inhibiting externaldisturbances have potential advantages Researchers havetried to combine these two techniques and proposed finite-time terminal sliding mode (TSM) method But the conven-tional TSM is often selected as 119904 = 119909

2+ 120573119909119902119901

1 The form

of this sliding mode often causes the singularity problemaround the equilibrium To address this problem [24] pro-poses an improved version of TSM which is expressed as

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 2518734 8 pageshttpdxdoiorg10115520162518734

2 Journal of Control Science and Engineering

119904 = + 120573|119909|120574sign (119909) Then the authors also propose a fast

TSM 119904 = + 120572119909 + 120573|119909|120574sign (119909) which enables faster and

higher-precision tracking performance than common TSMUntil now some finite-time sliding mode control techniquesfor nonlinear systems have been proposed [25ndash27] Howeverto the best of our knowledge there is very little literaturecombining finite-time stability theory with sliding modecontrol for HGS Could finite-time control of nonlinear HGSbe implemented via sliding mode If the hypothesis is truewhat are the specific mathematical derivation and controllerforms There are no relevant results yet It is still an openproblem Research in this area should be meaningful andchallenging

In light of the above analysis there are several advantageswhich make our study attractive Firstly the mathematicalmodel of a Francis HGS is introduced Then based onfinite-time stability theory and sliding mode scheme anovel finite-time terminal sliding mode control method isdesigned for the stability control of HGS Furthermore thecontrol method could resist random disturbances whichshows the good robustness Simulation results indicate thedesigned finite-time sliding mode control scheme works wellcompared with the existing method

The remaining contents of our paper are organized asfollows In Section 2 the nonlinear model of a Francis HGSis introduced The design of finite-time controller for HGS ispresented in Section 3 Numerical simulations are drawn inSection 4 Section 5 concludes this paper

2 Nonlinear Modeling ofHydroturbine Governing System

Here a Francis turbine which is widely used in China isselected as the research object The hydroturbine governingsystem consists of four parts including the hydroturbinemodel generator model water diversion system model andhydraulic servo system model

21 Nonlinear Hydroturbine Model The dynamic character-istics of the hydroturbine could be expressed as

119872119905= 119872119905(119867119873 119886)

119876119905= 119876119905(119867119873 119886)

(1)

where 119872119905 119876119905 119867 119873 and 119886 represent the turbinersquos active

torque flow water head rotational speed and guide vaneopening respectively

Mark the relative deviations of the dynamic perfor-mance parameters 119872

119905 119876119905 119867119873 119886 as 119898

119905 119902119905 ℎ 119899 119910 respec-

tively The dynamic expression of the turbine in the stableoperating point using Taylor series expansion withmore thantwo times higher order items ignored

Δ119898119905=

120597119898119905

120597ℎ

Δℎ +

120597119898119905

120597119899

Δ119899 +

120597119898119905

120597119910

Δ119910

Δ119902119905=

120597119902119905

120597ℎ

Δℎ +

120597119902119905

120597119899

Δ119899 +

120597119902119905

120597119910

Δ119910

(2)

where 120597119898119905120597ℎ is the transfer coefficient of turbine torque on

the water head 120597119898119905120597119899 is the transfer coefficient of turbine

torque on the speed 120597119898119905120597119899 is the transfer coefficient of

turbine torque on the main servomotor stroke 120597119898119905120597119899 is

the transfer coefficient of turbine torque on the water head120597119902119905120597ℎ is the transfer coefficient of turbine flow on the

head 120597119902119905120597119899 is the transfer coefficient of turbine flow on the

speed 120597119902119905120597119910 is the transfer coefficient of turbine flow on the

main servomotor strokeIn order to facilitate the analysis and calculation let 119890

ℎ

119890119899 119890119910 119890119902ℎ

119890119902119899

119890119902119910

express 120597119898119905120597ℎ 120597119898

119905120597119899 120597119898

119905120597119910 120597119902

119905120597ℎ

120597119902119905120597119899 120597119902

119905120597119910 respectively Equation (2) can be rewritten as

follows

Δ119898119905= 119890ℎΔℎ + 119890

119899Δ119899 + 119890

119910Δ119910

Δ119902119905= 119890119902ℎ

Δℎ + 119890119902119899

Δ119899 + 119890119902119910

Δ119910

(3)

The Laplace transform of (3) can be described as

119898119905(119904) = 119890

ℎℎ (119904) + 119890

119899119899 (119904) + 119890

119910119910 (119904)

119902119905(119904) = 119890

119902ℎℎ (119904) + 119890

119902119899119899 (119904) + 119890

119902119910119910 (119904)

(4)

In practice 119890119899and 119890

119902119899are no longer considered sepa-

rately so (4) could be simplified as

119898119905(119904) = 119890

ℎℎ (119904) + 119890

119910119910 (119904)

119902119905(119904) = 119890

119902ℎℎ (119904) + 119890

119902119910119910 (119904)

(5)

22 Water Diversion System Model For a simple waterdiversion system when the effect of water and the elasticityof the pipe wall on the water hammer are small it can beconsidered as rigid water hammer It is assumed that thewater is incompressible liquid and pressure pipeline is rigidthen the relationship between water head and flow of waterdiversion system can be expressed as

ℎ = minus119879119908

119879119908

=

119876119903

119892119867119903

sum

119871119894

119865119894

(6)

where 119879119908is the water inertia time constant 119867

119903represents

the rated head 119876119903means the rated flow 119871

119894is the length of

each section of the water diversion pipeline 119865119894is the cross-

sectional area of each water diversion pipeline segmentThe transfer function of (6) is described as

119866ℎ(119904) =

ℎ (119904)

119902 (119904)

= minus119879119908119904 (7)

Considering the turbine module and the water diversionsystem module together the transfer function of the turbineand the water diversion system can be got as

119866119905(119904) =

119890119910minus (119890119902119910

119890ℎminus 119890119902ℎ

119890119910) 119879119908119904

1 + 119890119902ℎ

119879119908119904

(8)

Journal of Control Science and Engineering 3

23 Generator Model The second-order nonlinear model ofthe generator can be described as follows

120575 = 120596

0120596

=

1

119879119886119887

[119898119905minus 119898119890minus 119863120596]

(9)

where 120575 represents the generatorrsquos rotor angle 120596 is the rela-tive deviation of the rotational speed of the generator 119863 isthe damping coefficient of the generator When analyzingthe generator dynamic features if the impact of generatorspeed vibration on the torque is included in the genera-tor damping coefficient so the electromagnetic torque andelectromagnetic power are equal that is 119898

120576= 119875119890 and the

electromagnetic power is

119875119890=

1198641015840

119902119881119904

1199091015840

119889Σ

sin 120575 +

1198812

119904

2

1199091015840

119889Σminus 119909119902Σ

1199091015840

119889Σ119909119902Σ

sin 2120575 (10)

where 1198641015840

119902is transient electromotive force of 119902 axis and 119881

119904is

infinite system bus voltage of the power system

1199091015840

119889Σ= 1199091015840

119889+ 119909119879+

1

2

119909119871

119909119902Σ

= 119909119902+ 119909119879+

1

2

119909119871

(11)

where 1199091015840

119889is transient reactance of 119889 axis 119909

119902is synchronous

reactance of 119902 axis 119909119879is transformer short circuit reactance

and 119909119871is transmission line reactance respectively

24 Hydraulic Servo System Model When the turbine gen-erator works at rated conditions with disturbances thedifferential equation for the main relay can be expressed as

119889119910

119889119905

= (119906 minus 119910)

1

119879119910

(12)

where 119906 represents the output of speed governor In thispaper we assume 119906 = 0 so (12) can be rewritten as

119889119910

119889119905

= minus

1

119879119910

119910 (13)

Based on (8) and (13) the output torque of the turbinecould be given as

119905=

1

119890119902ℎ

119879119908

[minus119898119905+ 119890119910119910 +

119890119890119910119879119908

119879119910

119910] (14)

Combining (9)ndash(14) the nonlinear dynamic model of thehydroturbine governing system can be described as

120575 = 120596

0120596

=

1

119879119886119887

[119898119905minus 119863120596 minus

1198641015840

119902119881119904

1199091015840

119889Σ

sin 120575

minus

1198812

119904

2

1199091015840

119889Σminus 119909119902Σ

1199091015840

119889Σ119909119902Σ

sin 2120575]

119905=

1

119890119902ℎ

119879119908

[minus119898119905+ 119890119910119910 +

119890119890119910119879119908

119879119910

119910]

= minus

1

119879119910

119910

(15)

where 120575 120596119898119905 119910 are dimensionless variables and 120596

0 119879119886119887

119863 1198641015840

119902 1199091015840

119889Σ 119909119902Σ

119879119908 119879119910 119881119904 119890119902ℎ

119890119910 119890 are dimensionless param-

eters The parameters are selected as 1205960

= 300 119879119886119887

= 19119863 = 20 1198641015840

119902= 135 1199091015840

119889Σ= 125 119909

119902Σ= 1474 119879

119908= 08

119879119910

= 01 119881119904

= 10 119890119902ℎ

= 05 119890119910

= 10 and 119890 = 07In order to facilitate the analysis 119909 119910 119911 119908 are used toreplace 120575 120596119898

119905 119910 respectively In actual operation the

hydroturbine governing system is often affected by theuncertain load changes In practice the exact values of thesystem uncertainties are difficult to know However in mostpractical examples the upper bound of the nonlinear systemsuncertainties can be estimated and the states of the nonlinearsystems are globally bounded [28] In this paper the limi-tation value is set as 1 So the random load disturbance isconsidered 119889

1(119905) = 08 rand (1) 119889

2(119905) = 01 rand (1) 119889

3(119905) =

05 rand (1) 1198894(119905) = 09 rand (1) Substituting the parameters

into system (15) after some calculation the mathematicalmodel of hydroturbine governing system under the randomload disturbance could be described as

= 300119910 + 08 rand (1)

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1)

= minus25119911 + 66119908 + 05 rand (1)

= minus10119908 + 09 rand (1)

(16)

The time domain of system (16) is illustrated in Figure 1 Itis clear that the hydroturbine governing system is in unstableoperation and nonlinear vibration So it is necessary todesign corresponding controller to ensure the safe and stableoperation of the hydroturbine governing system

3 Designing of Finite-TimeTerminal Sliding Mode Controller

To get the main results the following lemma of finite-timestability is given firstly

Lemma 1 (see [29]) If there is a continuous positive definitefunction 119881(119905) satisfying the following differential inequality

(119905) le minus119888119881120578(119905) forall119905 ge 119905

0 119881 (119905

0) ge 0 (17)

where 119888 gt 0 0 lt 120578 lt 1 are two constants then for anygiven 119905

0 119881(119905) satisfies the following inequality

1198811minus120578

(119905) le 1198811minus120578

(119905) minus 119888 (1 minus 120578) (119905 minus 1199050) 1199050le 119905 le 119905

1

119881 (119905) equiv 0 forall119905 ge 1199051

(18)


0 2 4 6 8 100

100

200

300

400

500

600

700

800

t (s)

x

(a) 119909 minus 119905

0 2 4 6 8 10t (s)

0

01

02

03

04

05

y

(b) 119910 minus 119905

0 2 4 6 8 10t (s)

035

04

045

05

055

z

(c) 119911 minus 119905

0 2 4 6 8 10t (s)

0

001

002

003

004

005

006

w

(d) 119908 minus 119905

Figure 1 Time domain of system (16)

with 1199051given by

1199051= 1199050+

1198811minus120578

(1199050)

119888 (1 minus 120578)

(19)

Then the system could be stabilized in a finite-time 1199051

The design of sliding mode can generally be divided intotwo steps Firstly a sliding surface is constructed which isasymptotically stable and has good dynamic quality Sec-ondly a sliding mode control law is designed such thatthe arrival condition is satisfied thus the sliding mode isperformed on the switching surface

To control the unstable hydroturbine governing systemthe control inputs are added to system (16) one has

= 300119910 + 08 rand (1) + 1199061

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1) + 1199062

= minus25119911 + 66119908 + 05 rand (1) + 1199063

= minus10119908 + 09 rand (1) + 1199064

(20)

For the convenience of mathematical analysis using [1199091

1199092 1199093 1199094] instead of [119909 119910 119911 119908] the unified form of system

(20) could be presented as = 119891 (119909) + 119889 (119905) + 119906 (119905) (21)

where119909 = [119909

1 1199092 1199093 1199094]119879

119891 (119909)

=

[

[

[

[

[

[

[

[

300119910

minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

minus25119911 + 66119908

minus10119908

]

]

]

]

]

]

]

]

119889 (119905) =

[

[

[

[

[

[

08 rand (1)

01 rand (1)

05 rand (1)

09 rand (1)

]

]

]

]

]

]

119906 (119905) = [1199061 1199062 1199063 1199064]119879

(22)


The control target is to make the states 119909 = [1199091 1199092

1199093 1199094]119879 track the setting value 119909

119889= [1199091198891

1199091198892

1199091198893

1199091198894

] Thetracking errors are defined as follows

119890 = 119909 minus 119909119889 (23)

The error dynamics can be described as follows

119890 = minus 119909119889= 119891 (119909) + 119889 (119905) + 119906 minus 119909

119889 (24)

Then the terminal sliding mode is defined as

119904 = 120572119890 + int

119905

0

120573 |119890|119903 sat (119890) 119889120591 (25)

where 120572 120573 119903 are given positive real constants with 0 lt 119903 lt

1 And

sat (119890) =

sign (119890) |119890| gt 119896

119890

119896

|119890| le 119896

(26)

where 119896 is a positive constant the value of 119896 is generallysmall In general the saturation function can effectivelysuppress the chattering phenomenon

When the system state reaches the sliding surface thefollowing equality is satisfied

119904 = 119904 = 0 (27)

Based on (25) and (27) one has

119904 = 120572 119890 + 120573 |119890|119903 sat (119890) = 0 (28)

Once an appropriate sliding surface is established thenext step of the method is to construct an input signal 119906(119905)which can make the state trajectories reach to the slidingsurface 119904(119905) = 0 and stay on it forever The sliding modecontrol law is presented as follows

119906 (119905) = minus119891 (119909) minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904))

minus (

120573

120572

) |119890|119903 sat (119890) + 119909

119889

(29)

where 119896 119871 119906 are given positive constants with 0 lt 119906 lt

1 120585 is the bounded value of random perturbation with 120585 =

[08 05 01 09]119879

Theorem 2 If the terminal sliding surface is selected in theform of (28) and the control law is designed as (29) then thestate trajectories of the hydroturbine governing system (20) willconverge to the sliding surface in a finite time 119879 = (1120572119896(1 minus

119906)) ln ((119896|119904(0)|1minus119906

+ 119871)119871)

Proof Selecting the Lyapunov function 119881(119905) = |119904| and takingits time derivative one can obtain

(119905) = sign (119904) 119904 (30)

Substituting (28) into (30) there is

= sign (119904) (120572 119890 + 120573 |119890|119903 sat (119890))

= sign (119904) (120572 (119891 (119909) + 119889 (119905) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

+1003816100381610038161003816sign (119904) sdot 120572119889 (119905)

1003816100381610038161003816

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

(31)

Based on (26) and (29)

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

= sign (119904) (120572(119891 (119909) minus 119891 (119909)

minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904)) minus (

120573

120572

) |119890|119903 sat (119890)

+ 119909119889minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585 = sign (119904)

sdot (120572 (minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904)))) + 120572120585 = minus120572119896 |119904|

minus 120572120585 minus 120572119871 |119904|119906+ 120572120585 = minus120572119896 |119904| minus 120572119871 |119904|

119906le 0

(32)

According to Lyapunov stability theory and Lemma 1the state trajectories of the unstable hydroturbine governingsystem (20) will converge to 119904(119905) = 0 in a finite timeasymptotically Then the reaching time 119879 can be got asfollows

Based on inequality (32) one has

119889119881

119889119905

=

119889 |119904|

119889119905

le minus120572119896 |119904| minus 120572119871 |119904|119906 (33)

It is obvious that

119889119905 le

minus119889 |119904|

120572 (119896 |119904| + 119871 |119904|119906)

= minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(34)

Taking integral for both sides of (34) from 0 to 119905119903 one can

obtain

int

119905119903

0

119889119905 le int

119904(119905119903)

119904(0)

minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(35)

Then

119905119903le minus

1

120572119896 (1 minus 119906)

ln (120572119896 |119904|1minus119906

+ 120572119871)

10038161003816100381610038161003816100381610038161003816

119904(119905119903)

119904(0)

(36)

Setting 119904(119905119903) = 0 one gets

119905119903le

1

120572119896 (1 minus 119906)

ln(119896 |119904 (0)|

1minus119906+ 119871)

119871

(37)

So the states trajectories of system (20) will converge tothe sliding surface 119904(119905) = 0 in a finite time 119879 = (1120572119896(1 minus

119906)) ln ((119896|119904(0)|1minus119906

+ 119871)119871) This completes the proof


0 1 2 3 4 5minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

x

FTSMCESMC

t (s)

(a) 119909 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

y

0 1 2 3 4 5

FTSMCESMC

t (s)

(b) 119910 minus 119905

minus15

minus1

minus05

0

05

1

z

0 1 2 3 4 5

FTSMCESMC

t (s)

(c) 119911 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

w

0 1 2 3 4 5

FTSMCESMC

t (s)

(d) 119908 minus 119905

Figure 2 Responses of controlled hydroturbine governing system (20)

4 Numerical Simulations

The parameters of the sliding surface (28) are selected as 120572 =

4 120573 = 6 119903 = 04 The parameters of the sliding mode controllaw (29) are selected as 119896 = 1 119871 = 8 119906 = 04

According to (25) the sliding surfaces of the hydroturbinegoverning system are described as follows

119904119894= 4119890119894+ int

119905

0

61003816100381610038161003816119890119894

1003816100381610038161003816

04

119889120591 (119894 = 1 2 3 4) (38)

Subsequently the sliding mode control laws are given asfollows

1199061(119905) = minus119891

1(119909) minus (119904

1+ (08 + 8

10038161003816100381610038161199041

1003816100381610038161003816

04

) sign (1199041))

minus (

6

4

)10038161003816100381610038161198901

1003816100381610038161003816

04 sat (1198901) + 1199091198891

1199062(119905) = minus119891

2(119909) minus (119904

2+ (01 + 8

10038161003816100381610038161199042

1003816100381610038161003816

04

) sign (1199042))

minus (

6

4

)10038161003816100381610038161198902

1003816100381610038161003816

04 sat (1198902) + 1199091198892

1199063(119905) = minus119891

3(119909) minus (119904

3+ (05 + 8

10038161003816100381610038161199043

1003816100381610038161003816

04

) sign (1199043))

minus (

6

4

)10038161003816100381610038161198903

1003816100381610038161003816

04 sat (1198903) + 1199091198893

1199064(119905) = minus119891

4(119909) minus (119904

4+ (09 + 8

10038161003816100381610038161199044

1003816100381610038161003816

04

) sign (1199044))

minus (

6

4

)10038161003816100381610038161198904

1003816100381610038161003816

04 sat (1198904) + 1199091198894

(39)

Initial value of the hydroturbine governing system isselected as 119909 = [0 0 1205876 0] In this study the fixedpoint 119909

119889= [0 0 0 0] is set In order to illustrate the

superiority of the designed finite-time terminal sliding modecontrol (FTSMC) in this paper the existing sliding modecontrol (ESMC) method in [30] is performed also The statetrajectories of the controlled hydroturbine governing systemare shown in Figure 2

It is clear that the proposed method can stabilize thehydroturbine governing system in a finite time Comparedwith the technique presented in [30] the transition time is


shorter and the overshoot is less which show the effectivenessand superiority of the designed control scheme Comparedwith the conventional terminal sliding mode control algo-rithms it is obvious that the new method promoted in thispaper can resist external random disturbances which will beof great significance and very suitable for the stability controlof hydroturbine governing system

5 Conclusions

In this paper a finite-time control scheme was studied forthe nonlinear vibration control of a Francis hydroturbinegoverning system By using the modularization modelingmethod combined with the two-order generator model themathematical model of a Francis hydroturbine governingsystem was established firstly Then based on the finite-time theory and the terminal sliding mode control methoda novel robust finite-time terminal sliding mode controlschemewas proposed for the stability control of HGS Finallynumerical simulations verified the robustness and superiorityof the proposed method In the future this approach will beextended to other hydroturbine governing systems such ashydropower systems with fractional order or time delay

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Thisworkwas supported by the scientific research foundationof the National Natural Science Foundation (51509210 and51479173) the Science and Technology Project of ShaanxiProvincial Water Resources Department (Grant no 2015slkj-11) the 111 Project from the Ministry of Education of China(B12007) and Yangling Demonstration Zone TechnologyProject (2014NY-32)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178ndash179 2011

[2] B Lehner G Czisch and S Vassolo ldquoThe impact of globalchange on the hydropower potential of Europe a model-basedanalysisrdquo Energy Policy vol 33 no 7 pp 839ndash855 2005

[3] D Ling and Y Tao ldquoAn analysis of the Hopf bifurcationin a hydroturbine governing system with saturationrdquo IEEETransactions on Energy Conversion vol 21 no 2 pp 512ndash5152006

[4] H H Chernet K Alfredsen and G H Midttoslashmme ldquoSafety ofhydropower dams in a changing climaterdquo Journal of HydrologicEngineering vol 19 no 3 pp 569ndash582 2014

[5] C L Li J Z Zhou S Ouyang X L Ding and L ChenldquoImproved decomposition-coordination and discrete differ-ential dynamic programming for optimization of large-scalehydropower systemrdquo Energy Conversion and Management vol84 pp 363ndash373 2014

[6] X D Lai Y Zhu G L Liao X Zhang T Wang and W BZhang ldquoLateral vibration response analysis on shaft system ofhydro turbine generator unitrdquo Advances in Vibration Engineer-ing vol 12 no 6 pp 511ndash524 2013

[7] YXu ZH Li andXD Lai ldquoDynamicmodel for hydro-turbinegenerator units based on a databasemethod for guide bearingsrdquoShock and Vibration vol 20 no 3 pp 411ndash421 2013

[8] X Yu J Zhang and L Zhou ldquoHydraulic transients in the longdiversion-type hydropower station with a complex differentialsurge tankrdquo The Scientific World Journal vol 2014 Article ID241868 11 pages 2014

[9] W-Q Sun and D-M Yan ldquoIdentification of the nonlinearvibration characteristics in hydropower house using transferentropyrdquo Nonlinear Dynamics vol 75 no 4 pp 673ndash691 2014

[10] F-J Wang W Zhao M Yang and J-Y Gao ldquoAnalysis onunsteady fluid-structure interaction for a large scale hydraulicturbine II Structure dynamic stress and fatigue reliabilityrdquoJournal of Hydraulic Engineering vol 43 no 1 pp 15ndash21 2012

[11] Y Zeng L X Zhang Y K Guo J Qian and C L Zhang ldquoThegeneralized Hamiltonian model for the shafting transient anal-ysis of the hydro turbine generating setsrdquo Nonlinear Dynamicsvol 76 no 4 pp 1921ndash1933 2014

[12] W Tan ldquoUnified tuning of PID load frequency controller forpower systems via IMCrdquo IEEE Transactions on Power Systemsvol 25 no 1 pp 341ndash350 2010

[13] A Zargari R Hooshmand andMAtaei ldquoA new control systemdesign for a small hydro-power plant based on particle swarmoptimization-fuzzy sliding mode controller with Kalman esti-matorrdquoTransactions of the Institute ofMeasurement andControlvol 34 no 4 pp 388ndash400 2012

[14] Y K Bhateshvar HDMathur H Siguerdidjane and S BhanotldquoFrequency stabilization for multi-area thermal-hydro powersystem using genetic algorithm-optimized fuzzy logic con-troller in deregulated environmentrdquo Electric Power Componentsand Systems vol 43 no 2 pp 146ndash156 2015

[15] Z H Xiao Z P An S Q Wang and S Q Zeng ldquoResearch onthe NNARX model identification of hydroelectric unit basedon improved L-M algorithmrdquoAdvancedMaterials Research vol871 pp 304ndash309 2014

[16] C S Li J Z Zhou J Xiao and H Xiao ldquoHydraulic turbinegoverning system identification using T-S fuzzy model opti-mized by chaotic gravitational search algorithmrdquo EngineeringApplications of Artificial Intelligence vol 26 no 9 pp 2073ndash2082 2013

[17] Z H Chen X H Yuan H Tian and B Ji ldquoImprovedgravitational search algorithm for parameter identification ofwater turbine regulation systemrdquo Energy Conversion and Man-agement vol 78 pp 306ndash315 2014

[18] J Song and S P He ldquoObserver-based finite-time passive controlfor a class of uncertain time-delayed Lipschitz nonlinear sys-temsrdquo Transactions of the Institute of Measurement and Controlvol 36 no 6 pp 797ndash804 2014

[19] M J Cai and Z R Xiang ldquoAdaptive fuzzy finite-time controlfor a class of switched nonlinear systems with unknown controlcoefficientsrdquo Neurocomputing vol 162 pp 105ndash115 2015

[20] M P Aghababa ldquoRobust stabilization and synchronization of aclass of fractional-order chaotic systems via a novel fractionalsliding mode controllerrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 6 pp 2670ndash2681 2012

[21] A Marcos-Pastor E Vidal-Idiarte A Cid-Pastor and LMartınez-Salamero ldquoLoss-free resistor-based power factor cor-rection using a semi-bridgeless boost rectifier in sliding-mode


controlrdquo IEEE Transactions on Power Electronics vol 30 no 10pp 5842ndash5853 2015

[22] D Y Chen R F Zhang J C Sprott H T Chen and X YMa ldquoSynchronization between integer-order chaotic systemsand a class of fractional-order chaotic systems via sliding modecontrolrdquo Chaos vol 22 no 2 Article ID 023130 2012

[23] B J Zhang and H G Guo ldquoUniversal function projective lagsynchronization of chaotic systems with uncertainty by usingactive sliding mode and fuzzy sliding mode controlrdquo NonlinearDynamics vol 81 no 1-2 pp 867ndash879 2015

[24] S H Yu X H Yu B Shirinzadeh and Z H Man ldquoContinuousfinite-time control for robotic manipulators with terminalsliding moderdquo Automatica vol 41 no 11 pp 1957ndash1964 2005

[25] M Ou H Du and S Li ldquoFinite-time formation control ofmultiple nonholonomic mobile robotsrdquo International Journal ofRobust and Nonlinear Control vol 24 no 1 pp 140ndash165 2014

[26] S Khoo L H Xie S K Zhao and Z H Man ldquoMulti-surfacesliding control for fast finite-time leader-follower consensuswith high order SISO uncertain nonlinear agentsrdquo InternationalJournal of Robust and Nonlinear Control vol 24 no 16 pp2388ndash2404 2014

[27] L Li Q L Zhang J Li and G L Wang ldquoRobust finite-time119867infin

control for uncertain singular stochastic Markovian jumpsystems via proportional differential control lawrdquo IET ControlTheory amp Applications vol 8 no 16 pp 1625ndash1638 2014

[28] P F Curran and L O Chua ldquoAbsolute stability theory and thesynchronization problemrdquo International Journal of Bifurcationand Chaos in Applied Sciences and Engineering vol 7 no 6 pp1375ndash1382 1997

[29] J L Yin S Khoo Z HMan and X H Yu ldquoFinite-time stabilityand instability of stochastic nonlinear systemsrdquoAutomatica vol47 no 12 pp 2671ndash2677 2011

[30] D-Y Chen W-L Zhao X-Y Ma and R-F Zhang ldquoNo-chattering slidingmode control chaos in Hindmarsh-Rose neu-rons with uncertain parametersrdquo Computers and Mathematicswith Applications vol 61 no 10 pp 3161ndash3171 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



119904 = + 120573|119909|120574sign (119909) Then the authors also propose a fast

TSM 119904 = + 120572119909 + 120573|119909|120574sign (119909) which enables faster and

higher-precision tracking performance than common TSMUntil now some finite-time sliding mode control techniquesfor nonlinear systems have been proposed [25ndash27] Howeverto the best of our knowledge there is very little literaturecombining finite-time stability theory with sliding modecontrol for HGS Could finite-time control of nonlinear HGSbe implemented via sliding mode If the hypothesis is truewhat are the specific mathematical derivation and controllerforms There are no relevant results yet It is still an openproblem Research in this area should be meaningful andchallenging

In light of the above analysis there are several advantageswhich make our study attractive Firstly the mathematicalmodel of a Francis HGS is introduced Then based onfinite-time stability theory and sliding mode scheme anovel finite-time terminal sliding mode control method isdesigned for the stability control of HGS Furthermore thecontrol method could resist random disturbances whichshows the good robustness Simulation results indicate thedesigned finite-time sliding mode control scheme works wellcompared with the existing method

The remaining contents of our paper are organized asfollows In Section 2 the nonlinear model of a Francis HGSis introduced The design of finite-time controller for HGS ispresented in Section 3 Numerical simulations are drawn inSection 4 Section 5 concludes this paper

2 Nonlinear Modeling ofHydroturbine Governing System

Here a Francis turbine which is widely used in China isselected as the research object The hydroturbine governingsystem consists of four parts including the hydroturbinemodel generator model water diversion system model andhydraulic servo system model

21 Nonlinear Hydroturbine Model The dynamic character-istics of the hydroturbine could be expressed as

119872119905= 119872119905(119867119873 119886)

119876119905= 119876119905(119867119873 119886)

(1)

where 119872119905 119876119905 119867 119873 and 119886 represent the turbinersquos active

torque flow water head rotational speed and guide vaneopening respectively

Mark the relative deviations of the dynamic perfor-mance parameters 119872

119905 119876119905 119867119873 119886 as 119898

119905 119902119905 ℎ 119899 119910 respec-

tively The dynamic expression of the turbine in the stableoperating point using Taylor series expansion withmore thantwo times higher order items ignored

Δ119898119905=

120597119898119905

120597ℎ

Δℎ +

120597119898119905

120597119899

Δ119899 +

120597119898119905

120597119910

Δ119910

Δ119902119905=

120597119902119905

120597ℎ

Δℎ +

120597119902119905

120597119899

Δ119899 +

120597119902119905

120597119910

Δ119910

(2)

where 120597119898119905120597ℎ is the transfer coefficient of turbine torque on

the water head 120597119898119905120597119899 is the transfer coefficient of turbine

torque on the speed 120597119898119905120597119899 is the transfer coefficient of

turbine torque on the main servomotor stroke 120597119898119905120597119899 is

the transfer coefficient of turbine torque on the water head120597119902119905120597ℎ is the transfer coefficient of turbine flow on the

head 120597119902119905120597119899 is the transfer coefficient of turbine flow on the

speed 120597119902119905120597119910 is the transfer coefficient of turbine flow on the

main servomotor strokeIn order to facilitate the analysis and calculation let 119890

ℎ

119890119899 119890119910 119890119902ℎ

119890119902119899

119890119902119910

express 120597119898119905120597ℎ 120597119898

119905120597119899 120597119898

119905120597119910 120597119902

119905120597ℎ

120597119902119905120597119899 120597119902

119905120597119910 respectively Equation (2) can be rewritten as

follows

Δ119898119905= 119890ℎΔℎ + 119890

119899Δ119899 + 119890

119910Δ119910

Δ119902119905= 119890119902ℎ

Δℎ + 119890119902119899

Δ119899 + 119890119902119910

Δ119910

(3)

The Laplace transform of (3) can be described as

119898119905(119904) = 119890

ℎℎ (119904) + 119890

119899119899 (119904) + 119890

119910119910 (119904)

119902119905(119904) = 119890

119902ℎℎ (119904) + 119890

119902119899119899 (119904) + 119890

119902119910119910 (119904)

(4)

In practice 119890119899and 119890

119902119899are no longer considered sepa-

rately so (4) could be simplified as

119898119905(119904) = 119890

ℎℎ (119904) + 119890

119910119910 (119904)

119902119905(119904) = 119890

119902ℎℎ (119904) + 119890

119902119910119910 (119904)

(5)

22 Water Diversion System Model For a simple waterdiversion system when the effect of water and the elasticityof the pipe wall on the water hammer are small it can beconsidered as rigid water hammer It is assumed that thewater is incompressible liquid and pressure pipeline is rigidthen the relationship between water head and flow of waterdiversion system can be expressed as

ℎ = minus119879119908

119879119908

=

119876119903

119892119867119903

sum

119871119894

119865119894

(6)

where 119879119908is the water inertia time constant 119867

119903represents

the rated head 119876119903means the rated flow 119871

119894is the length of

each section of the water diversion pipeline 119865119894is the cross-

sectional area of each water diversion pipeline segmentThe transfer function of (6) is described as

119866ℎ(119904) =

ℎ (119904)

119902 (119904)

= minus119879119908119904 (7)

Considering the turbine module and the water diversionsystem module together the transfer function of the turbineand the water diversion system can be got as

119866119905(119904) =

119890119910minus (119890119902119910

119890ℎminus 119890119902ℎ

119890119910) 119879119908119904

1 + 119890119902ℎ

119879119908119904

(8)



120575 = 120596

0120596

=

1

119879119886119887

[119898119905minus 119898119890minus 119863120596]

(9)


120576= 119875119890 and the


119875119890=

1198641015840

119902119881119904

1199091015840

119889Σ

sin 120575 +

1198812

119904

2

1199091015840

119889Σminus 119909119902Σ

1199091015840

119889Σ119909119902Σ

sin 2120575 (10)

where 1198641015840


119904is


1199091015840

119889Σ= 1199091015840

119889+ 119909119879+

1

2

119909119871

119909119902Σ

= 119909119902+ 119909119879+

1

2

119909119871

(11)

where 1199091015840






119889119910

119889119905

= (119906 minus 119910)

1

119879119910

(12)


119889119910

119889119905

= minus

1

119879119910

119910 (13)


119905=

1

119890119902ℎ

119879119908

[minus119898119905+ 119890119910119910 +

119890119890119910119879119908

119879119910

119910] (14)


120575 = 120596

0120596

=

1

119879119886119887

[119898119905minus 119863120596 minus

1198641015840

119902119881119904

1199091015840

119889Σ

sin 120575

minus

1198812

119904

2

1199091015840

119889Σminus 119909119902Σ

1199091015840

119889Σ119909119902Σ

sin 2120575]

119905=

1

119890119902ℎ

119879119908

[minus119898119905+ 119890119910119910 +

119890119890119910119879119908

119879119910

119910]

= minus

1

119879119910

119910

(15)


0 119879119886119887

119863 1198641015840

119902 1199091015840

119889Σ 119909119902Σ

119879119908 119879119910 119881119904 119890119902ℎ



= 300 119879119886119887

= 19119863 = 20 1198641015840

119902= 135 1199091015840

119889Σ= 125 119909

119902Σ= 1474 119879

119908= 08

119879119910

= 01 119881119904

= 10 119890119902ℎ

= 05 119890119910




1(119905) = 08 rand (1) 119889

2(119905) = 01 rand (1) 119889

3(119905) =



= 300119910 + 08 rand (1)

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1)

= minus25119911 + 66119908 + 05 rand (1)

= minus10119908 + 09 rand (1)

(16)






0 119881 (119905

0) ge 0 (17)



1198811minus120578

(119905) le 1198811minus120578


1

119881 (119905) equiv 0 forall119905 ge 1199051

(18)


0 2 4 6 8 100

100

200

300

400

500

600

700

800

t (s)

x

(a) 119909 minus 119905

0 2 4 6 8 10t (s)

0

01

02

03

04

05

y

(b) 119910 minus 119905

0 2 4 6 8 10t (s)

035

04

045

05

055

z

(c) 119911 minus 119905

0 2 4 6 8 10t (s)

0

001

002

003

004

005

006

w

(d) 119908 minus 119905



1199051= 1199050+

1198811minus120578

(1199050)

119888 (1 minus 120578)

(19)




= 300119910 + 08 rand (1) + 1199061

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1) + 1199062

= minus25119911 + 66119908 + 05 rand (1) + 1199063

= minus10119908 + 09 rand (1) + 1199064

(20)




where119909 = [119909

1 1199092 1199093 1199094]119879

119891 (119909)

=

[

[

[

[

[

[

[

[

300119910

minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

minus25119911 + 66119908

minus10119908

]

]

]

]

]

]

]

]

119889 (119905) =

[

[

[

[

[

[

08 rand (1)

01 rand (1)

05 rand (1)

09 rand (1)

]

]

]

]

]

]

119906 (119905) = [1199061 1199062 1199063 1199064]119879

(22)




119889= [1199091198891

1199091198892

1199091198893

1199091198894


119890 = 119909 minus 119909119889 (23)


119890 = minus 119909119889= 119891 (119909) + 119889 (119905) + 119906 minus 119909

119889 (24)


119904 = 120572119890 + int

119905

0

120573 |119890|119903 sat (119890) 119889120591 (25)


1 And

sat (119890) =

sign (119890) |119890| gt 119896

119890

119896

|119890| le 119896

(26)



119904 = 119904 = 0 (27)


119904 = 120572 119890 + 120573 |119890|119903 sat (119890) = 0 (28)


119906 (119905) = minus119891 (119909) minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904))

minus (

120573

120572

) |119890|119903 sat (119890) + 119909

119889

(29)



[08 05 01 09]119879


119906)) ln ((119896|119904(0)|1minus119906

+ 119871)119871)


(119905) = sign (119904) 119904 (30)


= sign (119904) (120572 119890 + 120573 |119890|119903 sat (119890))

= sign (119904) (120572 (119891 (119909) + 119889 (119905) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

+1003816100381610038161003816sign (119904) sdot 120572119889 (119905)

1003816100381610038161003816

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

(31)


le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

= sign (119904) (120572(119891 (119909) minus 119891 (119909)

minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904)) minus (

120573

120572

) |119890|119903 sat (119890)

+ 119909119889minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585 = sign (119904)



119906le 0

(32)



119889119881

119889119905

=

119889 |119904|

119889119905

le minus120572119896 |119904| minus 120572119871 |119904|119906 (33)

It is obvious that

119889119905 le

minus119889 |119904|

120572 (119896 |119904| + 119871 |119904|119906)

= minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(34)


obtain

int

119905119903

0

119889119905 le int

119904(119905119903)

119904(0)

minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(35)

Then

119905119903le minus

1

120572119896 (1 minus 119906)

ln (120572119896 |119904|1minus119906

+ 120572119871)

10038161003816100381610038161003816100381610038161003816

119904(119905119903)

119904(0)

(36)

Setting 119904(119905119903) = 0 one gets

119905119903le

1

120572119896 (1 minus 119906)

ln(119896 |119904 (0)|

1minus119906+ 119871)

119871

(37)


119906)) ln ((119896|119904(0)|1minus119906



0 1 2 3 4 5minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

x

FTSMCESMC

t (s)

(a) 119909 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

y

0 1 2 3 4 5

FTSMCESMC

t (s)

(b) 119910 minus 119905

minus15

minus1

minus05

0

05

1

z

0 1 2 3 4 5

FTSMCESMC

t (s)

(c) 119911 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

w

0 1 2 3 4 5

FTSMCESMC

t (s)

(d) 119908 minus 119905






119904119894= 4119890119894+ int

119905

0

61003816100381610038161003816119890119894

1003816100381610038161003816

04

119889120591 (119894 = 1 2 3 4) (38)


1199061(119905) = minus119891

1(119909) minus (119904

1+ (08 + 8

10038161003816100381610038161199041

1003816100381610038161003816

04

) sign (1199041))

minus (

6

4

)10038161003816100381610038161198901

1003816100381610038161003816

04 sat (1198901) + 1199091198891

1199062(119905) = minus119891

2(119909) minus (119904

2+ (01 + 8

10038161003816100381610038161199042

1003816100381610038161003816

04

) sign (1199042))

minus (

6

4

)10038161003816100381610038161198902

1003816100381610038161003816

04 sat (1198902) + 1199091198892

1199063(119905) = minus119891

3(119909) minus (119904

3+ (05 + 8

10038161003816100381610038161199043

1003816100381610038161003816

04

) sign (1199043))

minus (

6

4

)10038161003816100381610038161198903

1003816100381610038161003816

04 sat (1198903) + 1199091198893

1199064(119905) = minus119891

4(119909) minus (119904

4+ (09 + 8

10038161003816100381610038161199044

1003816100381610038161003816

04

) sign (1199044))

minus (

6

4

)10038161003816100381610038161198904

1003816100381610038161003816

04 sat (1198904) + 1199091198894

(39)







5 Conclusions


Competing Interests


Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120575 = 120596

0120596

=

1

119879119886119887

[119898119905minus 119898119890minus 119863120596]

(9)


120576= 119875119890 and the


119875119890=

1198641015840

119902119881119904

1199091015840

119889Σ

sin 120575 +

1198812

119904

2

1199091015840

119889Σminus 119909119902Σ

1199091015840

119889Σ119909119902Σ

sin 2120575 (10)

where 1198641015840


119904is


1199091015840

119889Σ= 1199091015840

119889+ 119909119879+

1

2

119909119871

119909119902Σ

= 119909119902+ 119909119879+

1

2

119909119871

(11)

where 1199091015840






119889119910

119889119905

= (119906 minus 119910)

1

119879119910

(12)


119889119910

119889119905

= minus

1

119879119910

119910 (13)


119905=

1

119890119902ℎ

119879119908

[minus119898119905+ 119890119910119910 +

119890119890119910119879119908

119879119910

119910] (14)


120575 = 120596

0120596

=

1

119879119886119887

[119898119905minus 119863120596 minus

1198641015840

119902119881119904

1199091015840

119889Σ

sin 120575

minus

1198812

119904

2

1199091015840

119889Σminus 119909119902Σ

1199091015840

119889Σ119909119902Σ

sin 2120575]

119905=

1

119890119902ℎ

119879119908

[minus119898119905+ 119890119910119910 +

119890119890119910119879119908

119879119910

119910]

= minus

1

119879119910

119910

(15)


0 119879119886119887

119863 1198641015840

119902 1199091015840

119889Σ 119909119902Σ

119879119908 119879119910 119881119904 119890119902ℎ



= 300 119879119886119887

= 19119863 = 20 1198641015840

119902= 135 1199091015840

119889Σ= 125 119909

119902Σ= 1474 119879

119908= 08

119879119910

= 01 119881119904

= 10 119890119902ℎ

= 05 119890119910




1(119905) = 08 rand (1) 119889

2(119905) = 01 rand (1) 119889

3(119905) =



= 300119910 + 08 rand (1)

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1)

= minus25119911 + 66119908 + 05 rand (1)

= minus10119908 + 09 rand (1)

(16)






0 119881 (119905

0) ge 0 (17)



1198811minus120578

(119905) le 1198811minus120578


1

119881 (119905) equiv 0 forall119905 ge 1199051

(18)


0 2 4 6 8 100

100

200

300

400

500

600

700

800

t (s)

x

(a) 119909 minus 119905

0 2 4 6 8 10t (s)

0

01

02

03

04

05

y

(b) 119910 minus 119905

0 2 4 6 8 10t (s)

035

04

045

05

055

z

(c) 119911 minus 119905

0 2 4 6 8 10t (s)

0

001

002

003

004

005

006

w

(d) 119908 minus 119905



1199051= 1199050+

1198811minus120578

(1199050)

119888 (1 minus 120578)

(19)




= 300119910 + 08 rand (1) + 1199061

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1) + 1199062

= minus25119911 + 66119908 + 05 rand (1) + 1199063

= minus10119908 + 09 rand (1) + 1199064

(20)




where119909 = [119909

1 1199092 1199093 1199094]119879

119891 (119909)

=

[

[

[

[

[

[

[

[

300119910

minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

minus25119911 + 66119908

minus10119908

]

]

]

]

]

]

]

]

119889 (119905) =

[

[

[

[

[

[

08 rand (1)

01 rand (1)

05 rand (1)

09 rand (1)

]

]

]

]

]

]

119906 (119905) = [1199061 1199062 1199063 1199064]119879

(22)




119889= [1199091198891

1199091198892

1199091198893

1199091198894


119890 = 119909 minus 119909119889 (23)


119890 = minus 119909119889= 119891 (119909) + 119889 (119905) + 119906 minus 119909

119889 (24)


119904 = 120572119890 + int

119905

0

120573 |119890|119903 sat (119890) 119889120591 (25)


1 And

sat (119890) =

sign (119890) |119890| gt 119896

119890

119896

|119890| le 119896

(26)



119904 = 119904 = 0 (27)


119904 = 120572 119890 + 120573 |119890|119903 sat (119890) = 0 (28)


119906 (119905) = minus119891 (119909) minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904))

minus (

120573

120572

) |119890|119903 sat (119890) + 119909

119889

(29)



[08 05 01 09]119879


119906)) ln ((119896|119904(0)|1minus119906

+ 119871)119871)


(119905) = sign (119904) 119904 (30)


= sign (119904) (120572 119890 + 120573 |119890|119903 sat (119890))

= sign (119904) (120572 (119891 (119909) + 119889 (119905) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

+1003816100381610038161003816sign (119904) sdot 120572119889 (119905)

1003816100381610038161003816

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

(31)


le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

= sign (119904) (120572(119891 (119909) minus 119891 (119909)

minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904)) minus (

120573

120572

) |119890|119903 sat (119890)

+ 119909119889minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585 = sign (119904)



119906le 0

(32)



119889119881

119889119905

=

119889 |119904|

119889119905

le minus120572119896 |119904| minus 120572119871 |119904|119906 (33)

It is obvious that

119889119905 le

minus119889 |119904|

120572 (119896 |119904| + 119871 |119904|119906)

= minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(34)


obtain

int

119905119903

0

119889119905 le int

119904(119905119903)

119904(0)

minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(35)

Then

119905119903le minus

1

120572119896 (1 minus 119906)

ln (120572119896 |119904|1minus119906

+ 120572119871)

10038161003816100381610038161003816100381610038161003816

119904(119905119903)

119904(0)

(36)

Setting 119904(119905119903) = 0 one gets

119905119903le

1

120572119896 (1 minus 119906)

ln(119896 |119904 (0)|

1minus119906+ 119871)

119871

(37)


119906)) ln ((119896|119904(0)|1minus119906



0 1 2 3 4 5minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

x

FTSMCESMC

t (s)

(a) 119909 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

y

0 1 2 3 4 5

FTSMCESMC

t (s)

(b) 119910 minus 119905

minus15

minus1

minus05

0

05

1

z

0 1 2 3 4 5

FTSMCESMC

t (s)

(c) 119911 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

w

0 1 2 3 4 5

FTSMCESMC

t (s)

(d) 119908 minus 119905






119904119894= 4119890119894+ int

119905

0

61003816100381610038161003816119890119894

1003816100381610038161003816

04

119889120591 (119894 = 1 2 3 4) (38)


1199061(119905) = minus119891

1(119909) minus (119904

1+ (08 + 8

10038161003816100381610038161199041

1003816100381610038161003816

04

) sign (1199041))

minus (

6

4

)10038161003816100381610038161198901

1003816100381610038161003816

04 sat (1198901) + 1199091198891

1199062(119905) = minus119891

2(119909) minus (119904

2+ (01 + 8

10038161003816100381610038161199042

1003816100381610038161003816

04

) sign (1199042))

minus (

6

4

)10038161003816100381610038161198902

1003816100381610038161003816

04 sat (1198902) + 1199091198892

1199063(119905) = minus119891

3(119909) minus (119904

3+ (05 + 8

10038161003816100381610038161199043

1003816100381610038161003816

04

) sign (1199043))

minus (

6

4

)10038161003816100381610038161198903

1003816100381610038161003816

04 sat (1198903) + 1199091198893

1199064(119905) = minus119891

4(119909) minus (119904

4+ (09 + 8

10038161003816100381610038161199044

1003816100381610038161003816

04

) sign (1199044))

minus (

6

4

)10038161003816100381610038161198904

1003816100381610038161003816

04 sat (1198904) + 1199091198894

(39)







5 Conclusions


Competing Interests


Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 100

100

200

300

400

500

600

700

800

t (s)

x

(a) 119909 minus 119905

0 2 4 6 8 10t (s)

0

01

02

03

04

05

y

(b) 119910 minus 119905

0 2 4 6 8 10t (s)

035

04

045

05

055

z

(c) 119911 minus 119905

0 2 4 6 8 10t (s)

0

001

002

003

004

005

006

w

(d) 119908 minus 119905



1199051= 1199050+

1198811minus120578

(1199050)

119888 (1 minus 120578)

(19)




= 300119910 + 08 rand (1) + 1199061

= minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

+ 01 rand (1) + 1199062

= minus25119911 + 66119908 + 05 rand (1) + 1199063

= minus10119908 + 09 rand (1) + 1199064

(20)




where119909 = [119909

1 1199092 1199093 1199094]119879

119891 (119909)

=

[

[

[

[

[

[

[

[

300119910

minus

2

19

119910 +

1

19

119911 minus

1

19

(108 sin119909 + 0061 sin 2119909)

minus25119911 + 66119908

minus10119908

]

]

]

]

]

]

]

]

119889 (119905) =

[

[

[

[

[

[

08 rand (1)

01 rand (1)

05 rand (1)

09 rand (1)

]

]

]

]

]

]

119906 (119905) = [1199061 1199062 1199063 1199064]119879

(22)




119889= [1199091198891

1199091198892

1199091198893

1199091198894


119890 = 119909 minus 119909119889 (23)


119890 = minus 119909119889= 119891 (119909) + 119889 (119905) + 119906 minus 119909

119889 (24)


119904 = 120572119890 + int

119905

0

120573 |119890|119903 sat (119890) 119889120591 (25)


1 And

sat (119890) =

sign (119890) |119890| gt 119896

119890

119896

|119890| le 119896

(26)



119904 = 119904 = 0 (27)


119904 = 120572 119890 + 120573 |119890|119903 sat (119890) = 0 (28)


119906 (119905) = minus119891 (119909) minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904))

minus (

120573

120572

) |119890|119903 sat (119890) + 119909

119889

(29)



[08 05 01 09]119879


119906)) ln ((119896|119904(0)|1minus119906

+ 119871)119871)


(119905) = sign (119904) 119904 (30)


= sign (119904) (120572 119890 + 120573 |119890|119903 sat (119890))

= sign (119904) (120572 (119891 (119909) + 119889 (119905) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

+1003816100381610038161003816sign (119904) sdot 120572119889 (119905)

1003816100381610038161003816

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

(31)


le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

= sign (119904) (120572(119891 (119909) minus 119891 (119909)

minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904)) minus (

120573

120572

) |119890|119903 sat (119890)

+ 119909119889minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585 = sign (119904)



119906le 0

(32)



119889119881

119889119905

=

119889 |119904|

119889119905

le minus120572119896 |119904| minus 120572119871 |119904|119906 (33)

It is obvious that

119889119905 le

minus119889 |119904|

120572 (119896 |119904| + 119871 |119904|119906)

= minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(34)


obtain

int

119905119903

0

119889119905 le int

119904(119905119903)

119904(0)

minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(35)

Then

119905119903le minus

1

120572119896 (1 minus 119906)

ln (120572119896 |119904|1minus119906

+ 120572119871)

10038161003816100381610038161003816100381610038161003816

119904(119905119903)

119904(0)

(36)

Setting 119904(119905119903) = 0 one gets

119905119903le

1

120572119896 (1 minus 119906)

ln(119896 |119904 (0)|

1minus119906+ 119871)

119871

(37)


119906)) ln ((119896|119904(0)|1minus119906



0 1 2 3 4 5minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

x

FTSMCESMC

t (s)

(a) 119909 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

y

0 1 2 3 4 5

FTSMCESMC

t (s)

(b) 119910 minus 119905

minus15

minus1

minus05

0

05

1

z

0 1 2 3 4 5

FTSMCESMC

t (s)

(c) 119911 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

w

0 1 2 3 4 5

FTSMCESMC

t (s)

(d) 119908 minus 119905






119904119894= 4119890119894+ int

119905

0

61003816100381610038161003816119890119894

1003816100381610038161003816

04

119889120591 (119894 = 1 2 3 4) (38)


1199061(119905) = minus119891

1(119909) minus (119904

1+ (08 + 8

10038161003816100381610038161199041

1003816100381610038161003816

04

) sign (1199041))

minus (

6

4

)10038161003816100381610038161198901

1003816100381610038161003816

04 sat (1198901) + 1199091198891

1199062(119905) = minus119891

2(119909) minus (119904

2+ (01 + 8

10038161003816100381610038161199042

1003816100381610038161003816

04

) sign (1199042))

minus (

6

4

)10038161003816100381610038161198902

1003816100381610038161003816

04 sat (1198902) + 1199091198892

1199063(119905) = minus119891

3(119909) minus (119904

3+ (05 + 8

10038161003816100381610038161199043

1003816100381610038161003816

04

) sign (1199043))

minus (

6

4

)10038161003816100381610038161198903

1003816100381610038161003816

04 sat (1198903) + 1199091198893

1199064(119905) = minus119891

4(119909) minus (119904

4+ (09 + 8

10038161003816100381610038161199044

1003816100381610038161003816

04

) sign (1199044))

minus (

6

4

)10038161003816100381610038161198904

1003816100381610038161003816

04 sat (1198904) + 1199091198894

(39)







5 Conclusions


Competing Interests


Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119889= [1199091198891

1199091198892

1199091198893

1199091198894


119890 = 119909 minus 119909119889 (23)


119890 = minus 119909119889= 119891 (119909) + 119889 (119905) + 119906 minus 119909

119889 (24)


119904 = 120572119890 + int

119905

0

120573 |119890|119903 sat (119890) 119889120591 (25)


1 And

sat (119890) =

sign (119890) |119890| gt 119896

119890

119896

|119890| le 119896

(26)



119904 = 119904 = 0 (27)


119904 = 120572 119890 + 120573 |119890|119903 sat (119890) = 0 (28)


119906 (119905) = minus119891 (119909) minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904))

minus (

120573

120572

) |119890|119903 sat (119890) + 119909

119889

(29)



[08 05 01 09]119879


119906)) ln ((119896|119904(0)|1minus119906

+ 119871)119871)


(119905) = sign (119904) 119904 (30)


= sign (119904) (120572 119890 + 120573 |119890|119903 sat (119890))

= sign (119904) (120572 (119891 (119909) + 119889 (119905) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890))

+1003816100381610038161003816sign (119904) sdot 120572119889 (119905)

1003816100381610038161003816

le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

(31)


le sign (119904) (120572 (119891 (119909) + 119906 minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585

= sign (119904) (120572(119891 (119909) minus 119891 (119909)

minus (119896119904 + (120585 + 119871 |119904|119906) sign (119904)) minus (

120573

120572

) |119890|119903 sat (119890)

+ 119909119889minus 119909119889) + 120573 |119890|

119903 sat (119890)) + 120572120585 = sign (119904)



119906le 0

(32)



119889119881

119889119905

=

119889 |119904|

119889119905

le minus120572119896 |119904| minus 120572119871 |119904|119906 (33)

It is obvious that

119889119905 le

minus119889 |119904|

120572 (119896 |119904| + 119871 |119904|119906)

= minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(34)


obtain

int

119905119903

0

119889119905 le int

119904(119905119903)

119904(0)

minus

1

1 minus 119906

times

119889 |119904|1minus119906

120572 (119896 |119904|1minus119906

+ 119871)

(35)

Then

119905119903le minus

1

120572119896 (1 minus 119906)

ln (120572119896 |119904|1minus119906

+ 120572119871)

10038161003816100381610038161003816100381610038161003816

119904(119905119903)

119904(0)

(36)

Setting 119904(119905119903) = 0 one gets

119905119903le

1

120572119896 (1 minus 119906)

ln(119896 |119904 (0)|

1minus119906+ 119871)

119871

(37)


119906)) ln ((119896|119904(0)|1minus119906



0 1 2 3 4 5minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

x

FTSMCESMC

t (s)

(a) 119909 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

y

0 1 2 3 4 5

FTSMCESMC

t (s)

(b) 119910 minus 119905

minus15

minus1

minus05

0

05

1

z

0 1 2 3 4 5

FTSMCESMC

t (s)

(c) 119911 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

w

0 1 2 3 4 5

FTSMCESMC

t (s)

(d) 119908 minus 119905






119904119894= 4119890119894+ int

119905

0

61003816100381610038161003816119890119894

1003816100381610038161003816

04

119889120591 (119894 = 1 2 3 4) (38)


1199061(119905) = minus119891

1(119909) minus (119904

1+ (08 + 8

10038161003816100381610038161199041

1003816100381610038161003816

04

) sign (1199041))

minus (

6

4

)10038161003816100381610038161198901

1003816100381610038161003816

04 sat (1198901) + 1199091198891

1199062(119905) = minus119891

2(119909) minus (119904

2+ (01 + 8

10038161003816100381610038161199042

1003816100381610038161003816

04

) sign (1199042))

minus (

6

4

)10038161003816100381610038161198902

1003816100381610038161003816

04 sat (1198902) + 1199091198892

1199063(119905) = minus119891

3(119909) minus (119904

3+ (05 + 8

10038161003816100381610038161199043

1003816100381610038161003816

04

) sign (1199043))

minus (

6

4

)10038161003816100381610038161198903

1003816100381610038161003816

04 sat (1198903) + 1199091198893

1199064(119905) = minus119891

4(119909) minus (119904

4+ (09 + 8

10038161003816100381610038161199044

1003816100381610038161003816

04

) sign (1199044))

minus (

6

4

)10038161003816100381610038161198904

1003816100381610038161003816

04 sat (1198904) + 1199091198894

(39)







5 Conclusions


Competing Interests


Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

x

FTSMCESMC

t (s)

(a) 119909 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

y

0 1 2 3 4 5

FTSMCESMC

t (s)

(b) 119910 minus 119905

minus15

minus1

minus05

0

05

1

z

0 1 2 3 4 5

FTSMCESMC

t (s)

(c) 119911 minus 119905

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

01

w

0 1 2 3 4 5

FTSMCESMC

t (s)

(d) 119908 minus 119905






119904119894= 4119890119894+ int

119905

0

61003816100381610038161003816119890119894

1003816100381610038161003816

04

119889120591 (119894 = 1 2 3 4) (38)


1199061(119905) = minus119891

1(119909) minus (119904

1+ (08 + 8

10038161003816100381610038161199041

1003816100381610038161003816

04

) sign (1199041))

minus (

6

4

)10038161003816100381610038161198901

1003816100381610038161003816

04 sat (1198901) + 1199091198891

1199062(119905) = minus119891

2(119909) minus (119904

2+ (01 + 8

10038161003816100381610038161199042

1003816100381610038161003816

04

) sign (1199042))

minus (

6

4

)10038161003816100381610038161198902

1003816100381610038161003816

04 sat (1198902) + 1199091198892

1199063(119905) = minus119891

3(119909) minus (119904

3+ (05 + 8

10038161003816100381610038161199043

1003816100381610038161003816

04

) sign (1199043))

minus (

6

4

)10038161003816100381610038161198903

1003816100381610038161003816

04 sat (1198903) + 1199091198893

1199064(119905) = minus119891

4(119909) minus (119904

4+ (09 + 8

10038161003816100381610038161199044

1003816100381610038161003816

04

) sign (1199044))

minus (

6

4

)10038161003816100381610038161198904

1003816100381610038161003816

04 sat (1198904) + 1199091198894

(39)







5 Conclusions


Competing Interests


Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











5 Conclusions


Competing Interests


Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article robust finite-time terminal sliding mode...

Documents