Research ArticleFixed-Time Stability of the Hydraulic Turbine Governing System

CaoyuanMa1 Chuangzhen Liu 1 Xuezi Zhang1 Yongzheng Sun 2

Wenbei Wu 1 and Jin Xie1

1School of Electrical and Power Engineering China University of Mining and Technology Xuzhou 221116 China2School of Mathematics China University of Mining and Technology Xuzhou 221008 China

Correspondence should be addressed to Yongzheng Sun yzsuncumteducn

Received 11 August 2017 Revised 16 December 2017 Accepted 21 December 2017 Published 30 January 2018

Academic Editor Ton D Do

Copyright copy 2018 Caoyuan Ma 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

This paper studies the problemof fixed-time stability of hydraulic turbine governing systemwith the elasticwater hammer nonlinearmodel To control and improve the quality of hydraulic turbine governing system a new fixed-time control strategy is proposedwhich can stabilize the water turbine governing system within a fixed time Compared with the finite-time control strategy wherethe convergence rate depends on the initial state the settling time of the fixed-time control scheme can be adjusted to the requiredvalue regardless of the initial conditions Finally we numerically show that the fixed-time control ismore effective than and superiorto the finite-time control

1 Introduction

Hydropower as a low-cost zero-polluting and renewableenergy source has been deeply developed since the twentiethcentury [1] With the reserves of coal natural gas andother nonrenewable energy sources decreasing graduallyand the serious environmental problems caused by powergeneration hydropower is becoming an increasingly largeproportion of the electricity structure According to thecurrent projections provided by international hydropowerindustries referring to the next thirty years a significantgrowth in the sector is expected [2] As a typical nonlinearcomplex system and an important part of hydraulic powergeneration system hydraulic turbine governing system is ahydraulic mechanical electrical integrated control system[3] The normal operation of the water turbine governingsystem is essential to the whole hydraulic power systemand it even affects the safe and stable operation of therelated power grid thus affecting the power quality and thepower consumption experience of the users In view of thehigh proportion of the hydropower system in the electricitystructure the research of the hydraulic turbine governingsystem is of great importance

At present the control method commonly used inwater turbine control system mainly includes the following

nonlinear control [4ndash7] sliding mode control [8ndash10] PIDcontrol [11ndash13] fuzzy control [14] fault tolerant control [15]predictive control [16 17] and finite-time control [18] Thesecontrol methods have important theoretical and practicalsignificance for the control of hydraulic turbine governingsystem but they also have their own defects For examplefeedback control has the time delay problem The nonlinearcontrol is targeted and each nonlinear control strategy is onlysuitable for solving some special nonlinear system controlproblems PID control is difficult to balance the stabilitytime and overshoot When the initial state of the systemdeviates from the equilibrium point it is difficult for thecontrol system to restore the system to the equilibrium pointfuzzy control is difficult to adapt to the requirements oflarge-scale adjustment and it needs to constantly adjust thecontrol rules and parameters The effect of fault tolerantcontrol is greatly influenced by the delay of fault detectionand separation and the long time delay will cause seriousstability problem Predictive controlrsquos accuracy is not veryhigh and the optimization process needs to be performedonline repeatedly in finite-time control the stability of thesystem is affected by the initial state of the system All theabove control strategies can ensure the exponential stabilityof the system while the adjustment time affected by theinitial state of the system is not always short enough The

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1352725 10 pageshttpsdoiorg10115520181352725

2 Mathematical Problems in Engineering

fixed-time stability control not only ensures the exponentialstability and the shorter adjustment time but also has strongerrobustness and disturbance rejection ability than the abovecontrol strategies

The definition of the fixed-time stability was firstly pro-posed by Polyakov in [19] and this definition was evolvedfrom the definition of finite-time stability Finite-time controltheory has been widely used in Cucker-Smale systems [20]complex dynamic network systems [21] PMSM [22] delayneural networks systems [23] chaotic systems [24] and soforth Compared with the finite-time control the fixed-timecontrol has the characteristic that the maximum adjustmenttime is not affected by the initial conditions In view of manyadvantages of fixed-time control the control method hasbeen widely used in multiagent systems [25] aircraft systems[26] robot systems [27] neural network systems [28ndash31] andchaotic systems [32 33]

In [16] a six-dimensional nonlinear mathematical modelfor the elastic water hammer of hydraulic turbine governingsystem is presented Based on the six-dimensional nonlinearmathematical model this paper analyzes the system runningstate without controllers Comparing the operation statusof the system under the fixed-time control strategy and thefinite-time control strategy we find that the control strategyused in this paper can directly calculate the systemrsquos settlingtimeThe settling time is independent of the initial state of thesystem In conclusion whether the initial state of the systemchanges or not we can use the fixed-time control strategy tomake the system achieve stable state quickly

2 Fixed-Time Stability of Hydraulic TurbineGoverning System

21 System Modeling and Preliminaries For the convenienceof analysis we give some necessary definitions and lemmasin advance

Definition 1 (see [19]) Consider the following nonlineardynamic system

= 119891 (119909) (1)

where 119909 isin 119877119899 is the system state 119891 is a smooth nonlinearfunction If for any initial condition there exists a fixedsettling time1198790 which is not connectedwith initial conditionsuch that

lim119905rarr1198790

119909 (119905) = 0 (2)

and 119909(119905) equiv 0 if 119905 ge 1198790 then this nonlinear dynamic systemis said to be fixed-time stable

Lemma 2 (see [34]) Suppose there exists a continuous func-tion such that 119881(119905) [0infin) rarr [0infin) such that

(1) 119881 is positive definite(2) there exist real numbers 119888 gt 0 and 0 lt 120588 lt 1 such that

(119905) le minus119888119881120588 (119905) 119905 ge 1199050 (3)

and then one has

1198811minus120588 (119905) le 1198811minus120588 (1199050) minus 119888 (1 minus 120588) (119905 minus 1199050) 1199050 le 119905 le 119905lowast119881 (119905) = 0 119905 ge 119905lowast (4)

of which

119905lowast = 1199050 + 1198811minus120588 (1199050)119888 (1 minus 120588) (5)

Lemma 3 (see [19]) If there exists a continuous radicallyunbounded function 119881 119877119899 rarr 119877+ cup |0| such that

(1) 119881(119909) = 0 hArr 119909 = 0(2) any solution 119909(119905) satisfied the inequality 119863lowast119881(119909(119905)) leminus120572119881119901(119909(119905)) minus 120573119881119902(119909(119905)) for some 120572 120573 and 119901 = 1 minus12120574 119902 = 1 + 12120574 and 120574 gt 1 where 119863lowast119881(119909(119905))

denotes the upper right-hand derivative of the function119881(119909(119905))Then the origin is globally fixed-time stable and the

following estimate holds

119879 (1199090) le 119879max fl120587120574radic120572120573 forall1199090 isin 119877119873 (6)

Lemma 4 (see [35]) If 1199091 1199092 119909119873 ge 0 then119873sum119894=1

119909120578119894 ge (119873sum119894=1

119909119894)120578

0 lt 120578 le 1119873sum119894=1

119909120579119894 ge 1198731minus120579(119873sum119894=1

119909119894)120579

120579 gt 1(7)

22 Main Results Here we use the nonlinear model of thewater turbine strike system proposed in [18]

1 = 11990922 = 11990933 = minus11988601199091 minus 11988611199092 minus 11988621199093 + 119910120575 = 1205960120596 = 1119879119886119887 [119898119905 minus 119863120596 minus 119875119890]119910 = minus 119910119879119910

(8)

where

119875119890 = 11986410158401199021198811199041199091015840119889Σ

sin 120575 + 119881211990421199091015840119889Σ minus 119909119902Σ1199091015840119889Σ119909119902Σ sin 2120575

119898119905 = 1198873119910 + (1198870 minus 11988601198873) 1199091 + (1198871 minus 11988611198873) 1199092+ (1198872 minus 11988621198873) 1199093

Mathematical Problems in Engineering 3

1199091015840119889Σ = 1199091015840119889 + 119909119879 + 12119909119871119909119902Σ = 119909119902 + 119909119879 + 121199091198711198870 = 24119890119910119890119902ℎℎ1205961198793119903 1198871 = minus24119890119890119910119890119902ℎ1198792119903 1198872 = 3119890119910119890119902ℎℎ120596119879119903 1198873 = minus119890119890119910119890119902ℎ 1198860 = 24119890119902ℎℎ1205961198793119903 1198861 = 241198792119903 1198862 = 3119890119902ℎℎ120596119879119903

(9)

11990911199092 and1199093 are state variables 120575 is the generator rotor angle120596 is the relative value of generator speed 119910 is the incrementaldeviation of the guide vane opening119863 is generator dampingcoefficient 119890 is intermediate variable 119890119902ℎ is the first-orderpartial derivative value of flow rate with respect towater head119890119910 is the first-order partial derivative value of torque withrespect to wicket gate 1198641015840119902 is the transient internal voltageof the armature ℎ120596 is characteristic coefficient of waterdiversion system 119898119905 is torque relative value of hydraulicturbine 119879119910 is relay reaction time constant 1199091015840119889 is the directaxis transient reactance 119909119902 is the quartered axis reactance119909119879 is the short circuit reactance of the transformer 119909119871 is thereactance of the electric transmission line and 119881119904 is the busvoltage at infinity

From system (8) we can see that 119875(0 0 0 119898 0 0) is apoint of equilibrium of the system where 119898 is a constant Inorder to make the system fast and stable to the equilibriumpoint 119875 the fifth and sixth subsystems of model (8) are addedwith the controllers 119906120596 and 119906119910 and the controlled system isformed as follows

1 = 11990922 = 11990933 = minus11988601199091 minus 11988611199092 minus 11988621199093 + 119910120575 = 1205960120596 = 1119879119886119887 [119898119905 minus 119863120596 minus 119875119890] + 119906120596119910 = 1119879119910 (minus119910 + 119906119910)

(10)

Theorem 5 The hydraulic turbine governing system (10) canbecome stable in a fixed time under the following controllers

119906119910 = minus1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120572 minus 1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120573119906120596 = minus119898119905 minus 119875119890 minus 1198873 sdot 119910119879119886119887 minus 1198962sign (120596) |120596|120572

minus 1198962sign (120596) |120596|120573 (11)

where the parameters 120572 and 120573 satisfy 0 lt 120572 lt 1 and 120573 gt 1 andthe parameters 1198961 and 1198962 are tuning parameters of the terminalattractor

Proof Here we use the two-step method of two steps toprove that the system is stable for the fixed time for thesixth subsystem of system (10) we put 119906119910 into the controlledsubsystem and we can have the following relationship

119889119910119889119905 = 1119879119910 (minus119910 minus 1198961sign (119910)10038161003816100381610038161199101003816100381610038161003816120572 minus 1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120573) (12)

The Lyapunov function is constructed as follows

1198811 (119905) = 11991022 (13)

The derivative along the trajectory of the sixth subsystem in(10) can be obtained

1198891198811 (119905)119889119905 = 119910119889119910119889119905 = 119910119879119910 [minus119910 minus 1198961sign (119910)10038161003816100381610038161199101003816100381610038161003816120572

minus 1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120573] = minus 1119879119910 (1199102 + 1198961 10038161003816100381610038161199101003816100381610038161003816120572+1

+ 1198961 10038161003816100381610038161199101003816100381610038161003816120573+1) le minus 1198961119879119910 [(1199102)(120572+1)2 + (1199102)(120573+1)2]

= minus 1198961119879119910 [(11991022 )(120572+1)2 sdot (12)

minus(120572+1)2 + (11991022 )(120573+1)2

sdot (12)minus(120573+1)2] = minus 1198961119879119910 sdot 2

(120572+1)2119881(120572+1)21 (119905) minus 1198961119879119910sdot 2(120573+1)2119881(120573+1)21 (119905)

(14)

where

119898 = 1198961119879119910 sdot 2(120572+1)2

119899 = 1198961119879119910 sdot 2(120573+1)2

119901 = 120572 + 12 119902 = 120573 + 12

(15)

4 Mathematical Problems in Engineering

According to Lemma 3 we know that the sixth subsystem inmodel (10) is stable in fixed time

1199051 = 1205871198791199101198961 (120573 minus 120572) sdot 2(120572+120573minus2)4 (16)

which means that the system state variable 119910 satisfies thefollowing relation 119910 = 0 when 119905 ge 1199051 And

1 = 11990922 = 11990933 = minus11988601199091 minus 11988611199092 minus 11988621199093120575 = 1205960120596 = 1119879119886119887 [(1198870 minus 11988601198873) 1199091 + (1198871 minus 11988611198873) 1199092+ (1198872 minus 11988621198873) 1199093 minus 119863120596 minus 119875119890] + 119906120596

(17)

To this end we select the following Lyapunov function

1198812 (119905) = 12059622 (18)

Thus

1198891198812 (119905)119889119905 = 120596119889120596119889119905 = 120596 [minus119863120596119879119886119887 minus 1198962sign (120596) |120596|120572

minus 1198962sign (120596) |120596|120573] = minus1198631205962119879119886119887 minus 1198962 |120596|120572+1

minus 1198962 |120596|120573+1 le minus1198962 |120596|120572+1 minus 1198962 |120596|120573+1= minus1198962 [2(120572+1)2 sdot (12059622 )

(120572+1)2 + 2(120573+1)2

sdot (12059622 )(120573+1)2] = minus1198962 sdot 2(120572+1)2119881(120572+1)22 (119905) minus 1198962

sdot 2(120573+1)2119881(120573+1)22 (119905)

(19)

where

119898 = 1198962 sdot 2(120572+1)2119899 = 1198962 sdot 2(120573+1)2119901 = 120572 + 12 119902 = 120573 + 12

(20)

According to Lemma 3 we can show that the fourth and fifthsubsystems in model (10) are stable in fixed time

1199052 = 1205871198962 (120573 minus 120572) sdot 2(120572+120573minus2)4 (21)

whichmeans that when 119905 ge 1199052 then 120596 = 0 and 120575 = 0 In otherwords when 119905 ge 1199051 + 1199052 the value of 120575 tends to be stable

To sum up when 119905 ge 1199053 where 1199053 = 1199051 + 1199052 + Δ119905 and Δ119905 isthe time from 119910 = 0 to 119906120596 acting onmodel (10) the hydraulicturbine governing system (10) is stable under the controllers119906119910 and 119906120596 That is the system is stable in a fixed time and thetheorem is proved

3 Numerical Simulations

In this section numerical results are provided to verify thetheoretical results The system parameters and controllerparameters in this paper are 1205960 = 314 119879119886119887 = 80 119863 = 051198641015840119902 = 135 1199091015840119889Σ = 115 119909119902Σ = 1474 119879119910 = 01 119881119904 = 10119890119902ℎ = 05 119890119910 = 10 119879119903 = 10 ℎ120596 = 20 119903 = 0 1198860 = 24 1198861 = 241198862 = 3 1198870 = 24 1198871 = 336 1198872 = 3 and 1198873 = minus14 respectivelyThe simulation sampling time is 00001 s and the initial statesare (1199091 1199092 1199093 120575 120596 119910) = (01 01 01 01 01 01)

Figures 1(a)ndash1(c) are the response curves of the systemvariables 120575 120596 and 119910 when the hydraulic turbine governingsystem is not controlled From Figure 1 it is clear that thesteady state of the system state variable 119910 is about 06 s beforebeing controlledThe state of the system variables 120575 and120596 areaperiodic and are always in a state of instability

Figures 2(a)ndash2(b) are the response curves of the fixed-time controllers 119906120596 and 119906119910 respectively In this simulation119906119910 acts on the hydraulic turbine governing system in 03 sand 119906120596 acts on the hydraulic turbine governing system in075 s Figures 3(a)ndash3(c) are the response curves of the systemvariables 120575 120596 and 119910 after the fixed-time controllers 119906119910 and119906120596 are applied to the hydraulic turbine governing systemrespectively From Figure 3 it is clear that when the systemis coupled with 119906119910 in 03 s and 119906120596 in 075 s the system statevariable 119910 reaches a stable state at 0326 s and the system statevariables 120575 and 120596 achieve stable state at 2 s simultaneouslyThe simulation results show that the system can achieve astable state in a short time by fixed-time controllers and thecontrol effect is achieved

In order to make a fair comparison between fixed-time control and finite-time control the parameters initialconditions and the tuning parameters of the terminal attrac-tor are the same in this paper Figures 4(a)ndash4(c) are thecomparison of response curves of system state variables 120575 120596and 119910 with fixed-time controllers and finite-time controllersrespectively From Figure 4 it is obvious that the settlingtime of system state variable 119910 under the action of fixed-timecontrollers is mostly equal to the settling time of system statevariable 119910 under the action of the finite-time controllers Thesystem state variables 120575 and 120596 achieve stable state at 2 s underthe action of fixed-time controllers simultaneously And thesystem state variables 120575 and 120596 achieve stable state at 207 sunder the action of finite-time controllers simultaneouslyThe fixed-time controllers stabilize the nonlinear systemfaster than finite-time controllers do Thus the fixed-timemethod has the better capacity to handle a nonlinear systemin a short time

To explore the relationship between the settling time andthe values of the parameters 120572 and 120573 experimentally we

Mathematical Problems in Engineering 5

minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

1 2 3 4 5 6 7 8 9 100Time (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

05

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 1 The response of the system without controller

minus075

minus050

minus025

000

025

050

075

100

125

150

175

u

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus005

minus004

minus003

minus002

minus001

000

001

002

uy

01 02 03 04 05 06 07 08 09 1000Time (s)

(b)

Figure 2 The response of the fixed-time controllers 119906120596 and 119906119910

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus015

minus010

minus005

000005010015020025030035040

045

1 2 3 4 5 6 7 8 9 100Time (s)

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 3 The response of the system with fixed-time controllers

select system state variables 120575 120596 and 119910 to demonstrate thesettling time Figures 5(a)ndash5(c) and Figures 6(a)ndash6(c) arerespectively the effects of different control parameters 120572 and120573 on the state variables 120575 120596 and 119910 under the fixed-timecontrollers In Figure 5 the parameter values are 120573 = 15and120572 = 03 04 05 06 07 In Figure 6 the parameter values

0

5

10

15

20

25

30

35

40

45

50

55

60

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

minus01

00

01

02

03

04

minus0015minus0010minus00050000

191817

(b)

Finite timeFixed time

000

002

004

006

008

010

y

05 10 15 20 25 30 35 40 45 5000Time (s)

(c)

Figure 4 Comparison of converging speed of fixed-time and finite-time controllers

Mathematical Problems in Engineering 7

0

5

10

15

20

25

30

35

40

45

50

55

60

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

= 03

= 04

= 05

= 06

= 07

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 5 Response of 120572 of the system controlled by fixed-time controllers

are 120572 = 05 and 120573 = 11 13 15 17 19 In Figures 5 and 6the system parameters and other controller parameters andtuning parameters of the terminal attractor are consistentwith the previous sections The simulation results clearlyshow that changing the controller parameters 120572 and 120573 canchange the time of the system state variables 120575 and 120596 to reachthe steady state But the time of the system state variable 119910 toreach the steady state is almost the same And the smaller the120572 and 120573 value of the system are the faster the settling timewill be Moreover the influence of 120573 on the settling time ofthe system state variables 120575 and 120596 is less than the influenceof 120572 on the settling time of the system state variables 120575 and120596 The simulation results are consistent with the theoreticalanalysis of the maximum stable time 1199053 of the system in theprevious section Moreover the values of 120572 and 120573 also affectthe stability value of the system state variable 120575 That is to

say we can get the size of the system state variable 120575 to thenumerical value we need by controlling the size of 120572 and 120573

In order to explore the effect of the initial state ofhydropower system with fixed-time controllers we com-pared the response of three different initial conditionsof the hydropower system Figures 7(a)ndash7(c) show theresponse of the system state variables 120575 120596 and 119910 atdifferent initial conditions with the fixed-time controllers119906120596 and 119906119910 respectively From Figures 7(a)ndash7(c) it isclear that when the initial states of system are 1198781 =(008 008 008 008 008 008) 1198782 = (01 01 01 01 0101) and 1198783 = (012 012 012 012 012 012) and thesystem is coupled with 119906119910 at 03 s and 119906120596 at 075 s the systemstate variables 120575 are stable at 192 s 2 s and 222 s respectivelyThe system states variables 120596 are also stable at 192 s 2 s and222 s respectively The system state variables 119910 are all stable

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

Applied MathematicsJournal of

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Mathematical Problems in Engineering 3

1199091015840119889Σ = 1199091015840119889 + 119909119879 + 12119909119871119909119902Σ = 119909119902 + 119909119879 + 121199091198711198870 = 24119890119910119890119902ℎℎ1205961198793119903 1198871 = minus24119890119890119910119890119902ℎ1198792119903 1198872 = 3119890119910119890119902ℎℎ120596119879119903 1198873 = minus119890119890119910119890119902ℎ 1198860 = 24119890119902ℎℎ1205961198793119903 1198861 = 241198792119903 1198862 = 3119890119902ℎℎ120596119879119903

(9)

11990911199092 and1199093 are state variables 120575 is the generator rotor angle120596 is the relative value of generator speed 119910 is the incrementaldeviation of the guide vane opening119863 is generator dampingcoefficient 119890 is intermediate variable 119890119902ℎ is the first-orderpartial derivative value of flow rate with respect towater head119890119910 is the first-order partial derivative value of torque withrespect to wicket gate 1198641015840119902 is the transient internal voltageof the armature ℎ120596 is characteristic coefficient of waterdiversion system 119898119905 is torque relative value of hydraulicturbine 119879119910 is relay reaction time constant 1199091015840119889 is the directaxis transient reactance 119909119902 is the quartered axis reactance119909119879 is the short circuit reactance of the transformer 119909119871 is thereactance of the electric transmission line and 119881119904 is the busvoltage at infinity

From system (8) we can see that 119875(0 0 0 119898 0 0) is apoint of equilibrium of the system where 119898 is a constant Inorder to make the system fast and stable to the equilibriumpoint 119875 the fifth and sixth subsystems of model (8) are addedwith the controllers 119906120596 and 119906119910 and the controlled system isformed as follows

1 = 11990922 = 11990933 = minus11988601199091 minus 11988611199092 minus 11988621199093 + 119910120575 = 1205960120596 = 1119879119886119887 [119898119905 minus 119863120596 minus 119875119890] + 119906120596119910 = 1119879119910 (minus119910 + 119906119910)

(10)

Theorem 5 The hydraulic turbine governing system (10) canbecome stable in a fixed time under the following controllers

119906119910 = minus1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120572 minus 1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120573119906120596 = minus119898119905 minus 119875119890 minus 1198873 sdot 119910119879119886119887 minus 1198962sign (120596) |120596|120572

minus 1198962sign (120596) |120596|120573 (11)

where the parameters 120572 and 120573 satisfy 0 lt 120572 lt 1 and 120573 gt 1 andthe parameters 1198961 and 1198962 are tuning parameters of the terminalattractor

Proof Here we use the two-step method of two steps toprove that the system is stable for the fixed time for thesixth subsystem of system (10) we put 119906119910 into the controlledsubsystem and we can have the following relationship

119889119910119889119905 = 1119879119910 (minus119910 minus 1198961sign (119910)10038161003816100381610038161199101003816100381610038161003816120572 minus 1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120573) (12)

The Lyapunov function is constructed as follows

1198811 (119905) = 11991022 (13)

The derivative along the trajectory of the sixth subsystem in(10) can be obtained

1198891198811 (119905)119889119905 = 119910119889119910119889119905 = 119910119879119910 [minus119910 minus 1198961sign (119910)10038161003816100381610038161199101003816100381610038161003816120572

minus 1198961sign (119910) 10038161003816100381610038161199101003816100381610038161003816120573] = minus 1119879119910 (1199102 + 1198961 10038161003816100381610038161199101003816100381610038161003816120572+1

+ 1198961 10038161003816100381610038161199101003816100381610038161003816120573+1) le minus 1198961119879119910 [(1199102)(120572+1)2 + (1199102)(120573+1)2]

= minus 1198961119879119910 [(11991022 )(120572+1)2 sdot (12)

minus(120572+1)2 + (11991022 )(120573+1)2

sdot (12)minus(120573+1)2] = minus 1198961119879119910 sdot 2

(120572+1)2119881(120572+1)21 (119905) minus 1198961119879119910sdot 2(120573+1)2119881(120573+1)21 (119905)

(14)

where

119898 = 1198961119879119910 sdot 2(120572+1)2

119899 = 1198961119879119910 sdot 2(120573+1)2

119901 = 120572 + 12 119902 = 120573 + 12

(15)

4 Mathematical Problems in Engineering

According to Lemma 3 we know that the sixth subsystem inmodel (10) is stable in fixed time

1199051 = 1205871198791199101198961 (120573 minus 120572) sdot 2(120572+120573minus2)4 (16)

which means that the system state variable 119910 satisfies thefollowing relation 119910 = 0 when 119905 ge 1199051 And

1 = 11990922 = 11990933 = minus11988601199091 minus 11988611199092 minus 11988621199093120575 = 1205960120596 = 1119879119886119887 [(1198870 minus 11988601198873) 1199091 + (1198871 minus 11988611198873) 1199092+ (1198872 minus 11988621198873) 1199093 minus 119863120596 minus 119875119890] + 119906120596

(17)

To this end we select the following Lyapunov function

1198812 (119905) = 12059622 (18)

Thus

1198891198812 (119905)119889119905 = 120596119889120596119889119905 = 120596 [minus119863120596119879119886119887 minus 1198962sign (120596) |120596|120572

minus 1198962sign (120596) |120596|120573] = minus1198631205962119879119886119887 minus 1198962 |120596|120572+1

minus 1198962 |120596|120573+1 le minus1198962 |120596|120572+1 minus 1198962 |120596|120573+1= minus1198962 [2(120572+1)2 sdot (12059622 )

(120572+1)2 + 2(120573+1)2

sdot (12059622 )(120573+1)2] = minus1198962 sdot 2(120572+1)2119881(120572+1)22 (119905) minus 1198962

sdot 2(120573+1)2119881(120573+1)22 (119905)

(19)

where

119898 = 1198962 sdot 2(120572+1)2119899 = 1198962 sdot 2(120573+1)2119901 = 120572 + 12 119902 = 120573 + 12

(20)

According to Lemma 3 we can show that the fourth and fifthsubsystems in model (10) are stable in fixed time

1199052 = 1205871198962 (120573 minus 120572) sdot 2(120572+120573minus2)4 (21)

whichmeans that when 119905 ge 1199052 then 120596 = 0 and 120575 = 0 In otherwords when 119905 ge 1199051 + 1199052 the value of 120575 tends to be stable

To sum up when 119905 ge 1199053 where 1199053 = 1199051 + 1199052 + Δ119905 and Δ119905 isthe time from 119910 = 0 to 119906120596 acting onmodel (10) the hydraulicturbine governing system (10) is stable under the controllers119906119910 and 119906120596 That is the system is stable in a fixed time and thetheorem is proved

3 Numerical Simulations

In this section numerical results are provided to verify thetheoretical results The system parameters and controllerparameters in this paper are 1205960 = 314 119879119886119887 = 80 119863 = 051198641015840119902 = 135 1199091015840119889Σ = 115 119909119902Σ = 1474 119879119910 = 01 119881119904 = 10119890119902ℎ = 05 119890119910 = 10 119879119903 = 10 ℎ120596 = 20 119903 = 0 1198860 = 24 1198861 = 241198862 = 3 1198870 = 24 1198871 = 336 1198872 = 3 and 1198873 = minus14 respectivelyThe simulation sampling time is 00001 s and the initial statesare (1199091 1199092 1199093 120575 120596 119910) = (01 01 01 01 01 01)

Figures 1(a)ndash1(c) are the response curves of the systemvariables 120575 120596 and 119910 when the hydraulic turbine governingsystem is not controlled From Figure 1 it is clear that thesteady state of the system state variable 119910 is about 06 s beforebeing controlledThe state of the system variables 120575 and120596 areaperiodic and are always in a state of instability

Figures 2(a)ndash2(b) are the response curves of the fixed-time controllers 119906120596 and 119906119910 respectively In this simulation119906119910 acts on the hydraulic turbine governing system in 03 sand 119906120596 acts on the hydraulic turbine governing system in075 s Figures 3(a)ndash3(c) are the response curves of the systemvariables 120575 120596 and 119910 after the fixed-time controllers 119906119910 and119906120596 are applied to the hydraulic turbine governing systemrespectively From Figure 3 it is clear that when the systemis coupled with 119906119910 in 03 s and 119906120596 in 075 s the system statevariable 119910 reaches a stable state at 0326 s and the system statevariables 120575 and 120596 achieve stable state at 2 s simultaneouslyThe simulation results show that the system can achieve astable state in a short time by fixed-time controllers and thecontrol effect is achieved

In order to make a fair comparison between fixed-time control and finite-time control the parameters initialconditions and the tuning parameters of the terminal attrac-tor are the same in this paper Figures 4(a)ndash4(c) are thecomparison of response curves of system state variables 120575 120596and 119910 with fixed-time controllers and finite-time controllersrespectively From Figure 4 it is obvious that the settlingtime of system state variable 119910 under the action of fixed-timecontrollers is mostly equal to the settling time of system statevariable 119910 under the action of the finite-time controllers Thesystem state variables 120575 and 120596 achieve stable state at 2 s underthe action of fixed-time controllers simultaneously And thesystem state variables 120575 and 120596 achieve stable state at 207 sunder the action of finite-time controllers simultaneouslyThe fixed-time controllers stabilize the nonlinear systemfaster than finite-time controllers do Thus the fixed-timemethod has the better capacity to handle a nonlinear systemin a short time

To explore the relationship between the settling time andthe values of the parameters 120572 and 120573 experimentally we

Mathematical Problems in Engineering 5

minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

1 2 3 4 5 6 7 8 9 100Time (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

05

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 1 The response of the system without controller

minus075

minus050

minus025

000

025

050

075

100

125

150

175

u

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus005

minus004

minus003

minus002

minus001

000

001

002

uy

01 02 03 04 05 06 07 08 09 1000Time (s)

(b)

Figure 2 The response of the fixed-time controllers 119906120596 and 119906119910

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus015

minus010

minus005

000005010015020025030035040

045

1 2 3 4 5 6 7 8 9 100Time (s)

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 3 The response of the system with fixed-time controllers

select system state variables 120575 120596 and 119910 to demonstrate thesettling time Figures 5(a)ndash5(c) and Figures 6(a)ndash6(c) arerespectively the effects of different control parameters 120572 and120573 on the state variables 120575 120596 and 119910 under the fixed-timecontrollers In Figure 5 the parameter values are 120573 = 15and120572 = 03 04 05 06 07 In Figure 6 the parameter values

0

5

10

15

20

25

30

35

40

45

50

55

60

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

minus01

00

01

02

03

04

minus0015minus0010minus00050000

191817

(b)

Finite timeFixed time

000

002

004

006

008

010

y

05 10 15 20 25 30 35 40 45 5000Time (s)

(c)

Figure 4 Comparison of converging speed of fixed-time and finite-time controllers

Mathematical Problems in Engineering 7

0

5

10

15

20

25

30

35

40

45

50

55

60

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

= 03

= 04

= 05

= 06

= 07

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 5 Response of 120572 of the system controlled by fixed-time controllers

are 120572 = 05 and 120573 = 11 13 15 17 19 In Figures 5 and 6the system parameters and other controller parameters andtuning parameters of the terminal attractor are consistentwith the previous sections The simulation results clearlyshow that changing the controller parameters 120572 and 120573 canchange the time of the system state variables 120575 and 120596 to reachthe steady state But the time of the system state variable 119910 toreach the steady state is almost the same And the smaller the120572 and 120573 value of the system are the faster the settling timewill be Moreover the influence of 120573 on the settling time ofthe system state variables 120575 and 120596 is less than the influenceof 120572 on the settling time of the system state variables 120575 and120596 The simulation results are consistent with the theoreticalanalysis of the maximum stable time 1199053 of the system in theprevious section Moreover the values of 120572 and 120573 also affectthe stability value of the system state variable 120575 That is to

say we can get the size of the system state variable 120575 to thenumerical value we need by controlling the size of 120572 and 120573

In order to explore the effect of the initial state ofhydropower system with fixed-time controllers we com-pared the response of three different initial conditionsof the hydropower system Figures 7(a)ndash7(c) show theresponse of the system state variables 120575 120596 and 119910 atdifferent initial conditions with the fixed-time controllers119906120596 and 119906119910 respectively From Figures 7(a)ndash7(c) it isclear that when the initial states of system are 1198781 =(008 008 008 008 008 008) 1198782 = (01 01 01 01 0101) and 1198783 = (012 012 012 012 012 012) and thesystem is coupled with 119906119910 at 03 s and 119906120596 at 075 s the systemstate variables 120575 are stable at 192 s 2 s and 222 s respectivelyThe system states variables 120596 are also stable at 192 s 2 s and222 s respectively The system state variables 119910 are all stable

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

According to Lemma 3 we know that the sixth subsystem inmodel (10) is stable in fixed time

1199051 = 1205871198791199101198961 (120573 minus 120572) sdot 2(120572+120573minus2)4 (16)

which means that the system state variable 119910 satisfies thefollowing relation 119910 = 0 when 119905 ge 1199051 And

1 = 11990922 = 11990933 = minus11988601199091 minus 11988611199092 minus 11988621199093120575 = 1205960120596 = 1119879119886119887 [(1198870 minus 11988601198873) 1199091 + (1198871 minus 11988611198873) 1199092+ (1198872 minus 11988621198873) 1199093 minus 119863120596 minus 119875119890] + 119906120596

(17)

To this end we select the following Lyapunov function

1198812 (119905) = 12059622 (18)

Thus

1198891198812 (119905)119889119905 = 120596119889120596119889119905 = 120596 [minus119863120596119879119886119887 minus 1198962sign (120596) |120596|120572

minus 1198962sign (120596) |120596|120573] = minus1198631205962119879119886119887 minus 1198962 |120596|120572+1

minus 1198962 |120596|120573+1 le minus1198962 |120596|120572+1 minus 1198962 |120596|120573+1= minus1198962 [2(120572+1)2 sdot (12059622 )

(120572+1)2 + 2(120573+1)2

sdot (12059622 )(120573+1)2] = minus1198962 sdot 2(120572+1)2119881(120572+1)22 (119905) minus 1198962

sdot 2(120573+1)2119881(120573+1)22 (119905)

(19)

where

119898 = 1198962 sdot 2(120572+1)2119899 = 1198962 sdot 2(120573+1)2119901 = 120572 + 12 119902 = 120573 + 12

(20)

According to Lemma 3 we can show that the fourth and fifthsubsystems in model (10) are stable in fixed time

1199052 = 1205871198962 (120573 minus 120572) sdot 2(120572+120573minus2)4 (21)

whichmeans that when 119905 ge 1199052 then 120596 = 0 and 120575 = 0 In otherwords when 119905 ge 1199051 + 1199052 the value of 120575 tends to be stable

To sum up when 119905 ge 1199053 where 1199053 = 1199051 + 1199052 + Δ119905 and Δ119905 isthe time from 119910 = 0 to 119906120596 acting onmodel (10) the hydraulicturbine governing system (10) is stable under the controllers119906119910 and 119906120596 That is the system is stable in a fixed time and thetheorem is proved

3 Numerical Simulations

In this section numerical results are provided to verify thetheoretical results The system parameters and controllerparameters in this paper are 1205960 = 314 119879119886119887 = 80 119863 = 051198641015840119902 = 135 1199091015840119889Σ = 115 119909119902Σ = 1474 119879119910 = 01 119881119904 = 10119890119902ℎ = 05 119890119910 = 10 119879119903 = 10 ℎ120596 = 20 119903 = 0 1198860 = 24 1198861 = 241198862 = 3 1198870 = 24 1198871 = 336 1198872 = 3 and 1198873 = minus14 respectivelyThe simulation sampling time is 00001 s and the initial statesare (1199091 1199092 1199093 120575 120596 119910) = (01 01 01 01 01 01)

Figures 1(a)ndash1(c) are the response curves of the systemvariables 120575 120596 and 119910 when the hydraulic turbine governingsystem is not controlled From Figure 1 it is clear that thesteady state of the system state variable 119910 is about 06 s beforebeing controlledThe state of the system variables 120575 and120596 areaperiodic and are always in a state of instability

Figures 2(a)ndash2(b) are the response curves of the fixed-time controllers 119906120596 and 119906119910 respectively In this simulation119906119910 acts on the hydraulic turbine governing system in 03 sand 119906120596 acts on the hydraulic turbine governing system in075 s Figures 3(a)ndash3(c) are the response curves of the systemvariables 120575 120596 and 119910 after the fixed-time controllers 119906119910 and119906120596 are applied to the hydraulic turbine governing systemrespectively From Figure 3 it is clear that when the systemis coupled with 119906119910 in 03 s and 119906120596 in 075 s the system statevariable 119910 reaches a stable state at 0326 s and the system statevariables 120575 and 120596 achieve stable state at 2 s simultaneouslyThe simulation results show that the system can achieve astable state in a short time by fixed-time controllers and thecontrol effect is achieved

In order to make a fair comparison between fixed-time control and finite-time control the parameters initialconditions and the tuning parameters of the terminal attrac-tor are the same in this paper Figures 4(a)ndash4(c) are thecomparison of response curves of system state variables 120575 120596and 119910 with fixed-time controllers and finite-time controllersrespectively From Figure 4 it is obvious that the settlingtime of system state variable 119910 under the action of fixed-timecontrollers is mostly equal to the settling time of system statevariable 119910 under the action of the finite-time controllers Thesystem state variables 120575 and 120596 achieve stable state at 2 s underthe action of fixed-time controllers simultaneously And thesystem state variables 120575 and 120596 achieve stable state at 207 sunder the action of finite-time controllers simultaneouslyThe fixed-time controllers stabilize the nonlinear systemfaster than finite-time controllers do Thus the fixed-timemethod has the better capacity to handle a nonlinear systemin a short time

To explore the relationship between the settling time andthe values of the parameters 120572 and 120573 experimentally we

Mathematical Problems in Engineering 5

minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

1 2 3 4 5 6 7 8 9 100Time (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

05

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 1 The response of the system without controller

minus075

minus050

minus025

000

025

050

075

100

125

150

175

u

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus005

minus004

minus003

minus002

minus001

000

001

002

uy

01 02 03 04 05 06 07 08 09 1000Time (s)

(b)

Figure 2 The response of the fixed-time controllers 119906120596 and 119906119910

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus015

minus010

minus005

000005010015020025030035040

045

1 2 3 4 5 6 7 8 9 100Time (s)

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 3 The response of the system with fixed-time controllers

select system state variables 120575 120596 and 119910 to demonstrate thesettling time Figures 5(a)ndash5(c) and Figures 6(a)ndash6(c) arerespectively the effects of different control parameters 120572 and120573 on the state variables 120575 120596 and 119910 under the fixed-timecontrollers In Figure 5 the parameter values are 120573 = 15and120572 = 03 04 05 06 07 In Figure 6 the parameter values

0

5

10

15

20

25

30

35

40

45

50

55

60

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

minus01

00

01

02

03

04

minus0015minus0010minus00050000

191817

(b)

Finite timeFixed time

000

002

004

006

008

010

y

05 10 15 20 25 30 35 40 45 5000Time (s)

(c)

Figure 4 Comparison of converging speed of fixed-time and finite-time controllers

Mathematical Problems in Engineering 7

0

5

10

15

20

25

30

35

40

45

50

55

60

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

= 03

= 04

= 05

= 06

= 07

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 5 Response of 120572 of the system controlled by fixed-time controllers

are 120572 = 05 and 120573 = 11 13 15 17 19 In Figures 5 and 6the system parameters and other controller parameters andtuning parameters of the terminal attractor are consistentwith the previous sections The simulation results clearlyshow that changing the controller parameters 120572 and 120573 canchange the time of the system state variables 120575 and 120596 to reachthe steady state But the time of the system state variable 119910 toreach the steady state is almost the same And the smaller the120572 and 120573 value of the system are the faster the settling timewill be Moreover the influence of 120573 on the settling time ofthe system state variables 120575 and 120596 is less than the influenceof 120572 on the settling time of the system state variables 120575 and120596 The simulation results are consistent with the theoreticalanalysis of the maximum stable time 1199053 of the system in theprevious section Moreover the values of 120572 and 120573 also affectthe stability value of the system state variable 120575 That is to

say we can get the size of the system state variable 120575 to thenumerical value we need by controlling the size of 120572 and 120573

In order to explore the effect of the initial state ofhydropower system with fixed-time controllers we com-pared the response of three different initial conditionsof the hydropower system Figures 7(a)ndash7(c) show theresponse of the system state variables 120575 120596 and 119910 atdifferent initial conditions with the fixed-time controllers119906120596 and 119906119910 respectively From Figures 7(a)ndash7(c) it isclear that when the initial states of system are 1198781 =(008 008 008 008 008 008) 1198782 = (01 01 01 01 0101) and 1198783 = (012 012 012 012 012 012) and thesystem is coupled with 119906119910 at 03 s and 119906120596 at 075 s the systemstate variables 120575 are stable at 192 s 2 s and 222 s respectivelyThe system states variables 120596 are also stable at 192 s 2 s and222 s respectively The system state variables 119910 are all stable

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

1 2 3 4 5 6 7 8 9 100Time (s)

minus04

minus03

minus02

minus01

00

01

02

03

04

05

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 1 The response of the system without controller

minus075

minus050

minus025

000

025

050

075

100

125

150

175

u

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus005

minus004

minus003

minus002

minus001

000

001

002

uy

01 02 03 04 05 06 07 08 09 1000Time (s)

(b)

Figure 2 The response of the fixed-time controllers 119906120596 and 119906119910

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus015

minus010

minus005

000005010015020025030035040

045

1 2 3 4 5 6 7 8 9 100Time (s)

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 3 The response of the system with fixed-time controllers

select system state variables 120575 120596 and 119910 to demonstrate thesettling time Figures 5(a)ndash5(c) and Figures 6(a)ndash6(c) arerespectively the effects of different control parameters 120572 and120573 on the state variables 120575 120596 and 119910 under the fixed-timecontrollers In Figure 5 the parameter values are 120573 = 15and120572 = 03 04 05 06 07 In Figure 6 the parameter values

0

5

10

15

20

25

30

35

40

45

50

55

60

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

minus01

00

01

02

03

04

minus0015minus0010minus00050000

191817

(b)

Finite timeFixed time

000

002

004

006

008

010

y

05 10 15 20 25 30 35 40 45 5000Time (s)

(c)

Figure 4 Comparison of converging speed of fixed-time and finite-time controllers

Mathematical Problems in Engineering 7

0

5

10

15

20

25

30

35

40

45

50

55

60

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

= 03

= 04

= 05

= 06

= 07

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 5 Response of 120572 of the system controlled by fixed-time controllers

are 120572 = 05 and 120573 = 11 13 15 17 19 In Figures 5 and 6the system parameters and other controller parameters andtuning parameters of the terminal attractor are consistentwith the previous sections The simulation results clearlyshow that changing the controller parameters 120572 and 120573 canchange the time of the system state variables 120575 and 120596 to reachthe steady state But the time of the system state variable 119910 toreach the steady state is almost the same And the smaller the120572 and 120573 value of the system are the faster the settling timewill be Moreover the influence of 120573 on the settling time ofthe system state variables 120575 and 120596 is less than the influenceof 120572 on the settling time of the system state variables 120575 and120596 The simulation results are consistent with the theoreticalanalysis of the maximum stable time 1199053 of the system in theprevious section Moreover the values of 120572 and 120573 also affectthe stability value of the system state variable 120575 That is to

say we can get the size of the system state variable 120575 to thenumerical value we need by controlling the size of 120572 and 120573

In order to explore the effect of the initial state ofhydropower system with fixed-time controllers we com-pared the response of three different initial conditionsof the hydropower system Figures 7(a)ndash7(c) show theresponse of the system state variables 120575 120596 and 119910 atdifferent initial conditions with the fixed-time controllers119906120596 and 119906119910 respectively From Figures 7(a)ndash7(c) it isclear that when the initial states of system are 1198781 =(008 008 008 008 008 008) 1198782 = (01 01 01 01 0101) and 1198783 = (012 012 012 012 012 012) and thesystem is coupled with 119906119910 at 03 s and 119906120596 at 075 s the systemstate variables 120575 are stable at 192 s 2 s and 222 s respectivelyThe system states variables 120596 are also stable at 192 s 2 s and222 s respectively The system state variables 119910 are all stable

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

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Decision SciencesAdvances in

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AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Mathematical Problems in Engineering

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 100Time (s)

(a)

minus015

minus010

minus005

000005010015020025030035040

045

1 2 3 4 5 6 7 8 9 100Time (s)

(b)

000

002

004

006

008

010

y

1 2 3 4 5 6 7 8 9 100Time (s)

(c)

Figure 3 The response of the system with fixed-time controllers

select system state variables 120575 120596 and 119910 to demonstrate thesettling time Figures 5(a)ndash5(c) and Figures 6(a)ndash6(c) arerespectively the effects of different control parameters 120572 and120573 on the state variables 120575 120596 and 119910 under the fixed-timecontrollers In Figure 5 the parameter values are 120573 = 15and120572 = 03 04 05 06 07 In Figure 6 the parameter values

0

5

10

15

20

25

30

35

40

45

50

55

60

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

Finite timeFixed time

05 10 15 20 25 30 35 40 45 5000Time (s)

minus01

00

01

02

03

04

minus0015minus0010minus00050000

191817

(b)

Finite timeFixed time

000

002

004

006

008

010

y

05 10 15 20 25 30 35 40 45 5000Time (s)

(c)

Figure 4 Comparison of converging speed of fixed-time and finite-time controllers

Mathematical Problems in Engineering 7

0

5

10

15

20

25

30

35

40

45

50

55

60

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

= 03

= 04

= 05

= 06

= 07

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 5 Response of 120572 of the system controlled by fixed-time controllers

are 120572 = 05 and 120573 = 11 13 15 17 19 In Figures 5 and 6the system parameters and other controller parameters andtuning parameters of the terminal attractor are consistentwith the previous sections The simulation results clearlyshow that changing the controller parameters 120572 and 120573 canchange the time of the system state variables 120575 and 120596 to reachthe steady state But the time of the system state variable 119910 toreach the steady state is almost the same And the smaller the120572 and 120573 value of the system are the faster the settling timewill be Moreover the influence of 120573 on the settling time ofthe system state variables 120575 and 120596 is less than the influenceof 120572 on the settling time of the system state variables 120575 and120596 The simulation results are consistent with the theoreticalanalysis of the maximum stable time 1199053 of the system in theprevious section Moreover the values of 120572 and 120573 also affectthe stability value of the system state variable 120575 That is to

say we can get the size of the system state variable 120575 to thenumerical value we need by controlling the size of 120572 and 120573

In order to explore the effect of the initial state ofhydropower system with fixed-time controllers we com-pared the response of three different initial conditionsof the hydropower system Figures 7(a)ndash7(c) show theresponse of the system state variables 120575 120596 and 119910 atdifferent initial conditions with the fixed-time controllers119906120596 and 119906119910 respectively From Figures 7(a)ndash7(c) it isclear that when the initial states of system are 1198781 =(008 008 008 008 008 008) 1198782 = (01 01 01 01 0101) and 1198783 = (012 012 012 012 012 012) and thesystem is coupled with 119906119910 at 03 s and 119906120596 at 075 s the systemstate variables 120575 are stable at 192 s 2 s and 222 s respectivelyThe system states variables 120596 are also stable at 192 s 2 s and222 s respectively The system state variables 119910 are all stable

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 7

0

5

10

15

20

25

30

35

40

45

50

55

60

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

= 03

= 04

= 05

= 06

= 07

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

= 03

= 04

= 05

= 06

= 07

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 5 Response of 120572 of the system controlled by fixed-time controllers

are 120572 = 05 and 120573 = 11 13 15 17 19 In Figures 5 and 6the system parameters and other controller parameters andtuning parameters of the terminal attractor are consistentwith the previous sections The simulation results clearlyshow that changing the controller parameters 120572 and 120573 canchange the time of the system state variables 120575 and 120596 to reachthe steady state But the time of the system state variable 119910 toreach the steady state is almost the same And the smaller the120572 and 120573 value of the system are the faster the settling timewill be Moreover the influence of 120573 on the settling time ofthe system state variables 120575 and 120596 is less than the influenceof 120572 on the settling time of the system state variables 120575 and120596 The simulation results are consistent with the theoreticalanalysis of the maximum stable time 1199053 of the system in theprevious section Moreover the values of 120572 and 120573 also affectthe stability value of the system state variable 120575 That is to

say we can get the size of the system state variable 120575 to thenumerical value we need by controlling the size of 120572 and 120573

In order to explore the effect of the initial state ofhydropower system with fixed-time controllers we com-pared the response of three different initial conditionsof the hydropower system Figures 7(a)ndash7(c) show theresponse of the system state variables 120575 120596 and 119910 atdifferent initial conditions with the fixed-time controllers119906120596 and 119906119910 respectively From Figures 7(a)ndash7(c) it isclear that when the initial states of system are 1198781 =(008 008 008 008 008 008) 1198782 = (01 01 01 01 0101) and 1198783 = (012 012 012 012 012 012) and thesystem is coupled with 119906119910 at 03 s and 119906120596 at 075 s the systemstate variables 120575 are stable at 192 s 2 s and 222 s respectivelyThe system states variables 120596 are also stable at 192 s 2 s and222 s respectively The system state variables 119910 are all stable

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

05 10 15 20 25 30 35 40 45 5000Time (s)

175 225 250200

3637383940

= 11

= 13

= 15

= 17

= 19

(a)

minus02

minus01

00

01

02

03

04

05

05 10 15 20 25 30 35 40 45 5000Time (s)

= 11

= 13

= 15

= 17

= 19

170175180185190165

minus0015

minus0010

minus0005

0000

(b)

= 11

= 13

= 15

= 17

= 19

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

y

(c)

Figure 6 Response of 120573 of the system controlled by fixed-time controllers

at 0326 s These simulation results show that when the initialstate of the system changes the settling times of the systemstate variables have changed but they are not more thantheoretical deduction time 1199053 That is to say the simulationresults are consistent with the theoretical derivation

4 Conclusions

In this paper to ensure the safe and stable operation ofhydraulic turbine governing system a new control methodbased on the fixed-time theory is proposed Comparedwith the finite-time control method the hydraulic turbinegoverning system under the fixed-time controllers has moreadvantages better robustness fast response ability and thesetting time to reach the stable state being regardless ofthe initial state Finally the effectiveness and superiority of

the proposed control method are verified by the simulationresults Note that time delay may influence the dynamicbehavior of the system the fixed-time control of hydraulicturbine governing system with time delay is our futuredirection

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Authorsrsquo Contributions

Caoyuan Ma Yongzheng Sun and Chuangzhen Liu con-ceived and designed the experiments Chuangzhen Liu XueziZhang Wenbei Wu and Jin Xie performed the experiments

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(a)

minus020

minus015

minus010

minus005

000

005

010

015

020

025

030

035

040

045

050

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

(b)

S1 = 008

S2 = 010

S3 = 012

05 10 15 20 25 30 35 40 45 5000Time (s)

000

002

004

006

008

010

012

y

0302

000

001

002

(c)

Figure 7 The response of the system with the different 119878

and analyzed the data Chuangzhen Liu and Xuezi Zhangwrote the paper

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2017XKZD11)

References

[1] R E Grumbine and J Xu ldquoMekong hydropower developmentrdquoScience vol 332 no 6026 pp 178-179 2011

[2] B Xu D Chen H Zhang F Wang X Zhang and Y WuldquoHamiltonian model and dynamic analyses for a hydro-turbinegoverning systemwith fractional item and time-lagrdquoCommuni-cations in Nonlinear Science and Numerical Simulation vol 47pp 35ndash47 2017

[3] D J 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 Mesnage M Alamir N Perrissin-Fabert and Q AlloinldquoNonlinear model-based control for minimum-time start ofhydraulic turbinesrdquo European Journal of Control vol 34 pp 24ndash30 2017

[5] W C Guo J D Yang M J Wang and X Lai ldquoNonlinear mod-eling and stability analysis of hydro-turbine governing systemwith sloping ceiling tailrace tunnel under load disturbancerdquoEnergy Conversion andManagement vol 106 pp 127ndash138 2015

[6] D Chen C Ding Y Do X Ma H Zhao and Y Wang ldquoNon-linear dynamic analysis for a Francis hydro-turbine governingsystem and its controlrdquo Journal ofThe Franklin Institute vol 351no 9 pp 4596ndash4618 2014

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

[7] H H Li D Y Chen H Zhang F Wang and D Ba ldquoNonlinearmodeling and dynamic analysis of a hydro-turbine governingsystem in the process of sudden load increase transientrdquoMechanical Systems and Signal Processing vol 80 pp 414ndash4282016

[8] C Xu and D Qian ldquoGovernor design for a hydropower plantwith an upstream surge tank by GA-based Fuzzy reduced-ordersliding moderdquo Energies vol 8 no 12 pp 13442ndash13457 2015

[9] X Yuan Z Chen Y Yuan and Y Huang ldquoDesign of fuzzysliding mode controller for hydraulic turbine regulating systemvia input state feedback linearization methodrdquo Energy vol 93pp 173ndash187 2015

[10] X Yuan Z Chen Y Yuan Y Huang X Li and W Li ldquoSlidingmode controller of hydraulic generator regulating system basedon the inputoutput feedback linearizationmethodrdquoMathemat-ics and Computers in Simulation vol 119 pp 18ndash34 2016

[11] C Li N Zhang X Lai J Zhou and Y Xu ldquoDesign of afractional-order PID controller for a pumped storage unit usinga gravitational search algorithm based on the Cauchy andGaussian mutationrdquo Information Sciences vol 396 pp 162ndash1812017

[12] Y Xu J Zhou X Xue W Fu W Zhu and C Li ldquoAn adaptivelyfast fuzzy fractional order PID control for pumped storagehydro unit using improved gravitational search algorithmrdquoEnergy Conversion and Management vol 111 pp 67ndash78 2016

[13] C Li Y Mao J Zhou N Zhang and X An ldquoDesign of a fuzzy-PID controller for a nonlinear hydraulic turbine governingsystem by using a novel gravitational search algorithm based onCauchymutation andmass weightingrdquoApplied Soft Computingvol 52 pp 290ndash305 2017

[14] S Simani S Alvisi and M Venturini ldquoData-driven design ofa fault tolerant fuzzy controller for a simulated hydroelectricsystemrdquo IFAC-PapersOnLine vol 28 no 21 pp 1090ndash10952015

[15] S Simani S Alvisi and M Venturini ldquoFault tolerant control ofa simulated hydroelectric systemrdquo Control Engineering Practicevol 51 pp 13ndash25 2016

[16] R Zhang D Chen and X Ma ldquoNonlinear predictive control ofa hydropower system modelrdquo Entropy vol 17 no 9 pp 6129ndash6149 2015

[17] Z Xiao S Meng N Lu and O P Malik ldquoOne-Step-AheadPredictive Control for Hydroturbine Governorrdquo MathematicalProblems in Engineering vol 2015 Article ID 382954 2015

[18] B Wang L Yin S Wang S Miao T Du and C ZuoldquoFinite time control for fractional order nonlinear hydroturbinegoverning system via frequency distributedmodelrdquoAdvances inMathematical Physics vol 2016 Article ID 7345325 2016

[19] A Polyakov ldquoNonlinear feedback design for fixed-time stabi-lization of linear control systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Automatic Control vol 57no 8 pp 2106ndash2110 2012

[20] Y Sun and W Lin ldquoA positive role of multiplicative noise onthe emergence of flocking in a stochastic Cucker-Smale systemrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 25no 8 Article ID 083118 2015

[21] Y Z Sun S Y Leng Y C Lai C Grebogi andW Lin ldquoClosed-loop control of complex networks a trade-off between timeand energyrdquo Physical Review Letters vol 119 no 19 Article ID198301 2017

[22] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015

[23] J Huang C Li T Huang and X He ldquoFinite-time lag synchro-nization of delayed neural networksrdquoNeurocomputing vol 139pp 145ndash149 2014

[24] J Wu Z-c Ma Y-z Sun and F Liu ldquoFinite-time synchroniza-tion of chaotic systems with noise perturbationrdquo Kybernetikavol 51 no 1 pp 137ndash149 2015

[25] H F Hong W W Yu G H Wen and X H Yu ldquoDistributedrobust fixed-time consensus for nonlinear and disturbed mul-tiagent systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1464ndash1473 2017

[26] J Gao and Y Cai ldquoFixed-time control for spacecraft attitudetracking based on quaternionrdquo Acta Astronautica vol 115article no 5463 pp 303ndash313 2015

[27] Y N Yang C C Hua J P Li and X P Guan ldquoFixed-time coordination control for bilateral telerobotics system withasymmetric time-varying delaysrdquo Journal of Intelligent RoboticSystems vol 86 no 3-4 pp 447ndash466 2017

[28] X Ding J Cao A Alsaedi F E Alsaadi and T Hayat ldquoRobustfixed-time synchronization for uncertain complex-valued neu-ral networks with discontinuous activation functionsrdquo NeuralNetworks vol 90 pp 42ndash55 2017

[29] C Hu J Yu Z Chen H Jiang and T Huang ldquoFixed-timestability of dynamical systems and fixed-time synchronizationof coupled discontinuous neural networksrdquo Neural Networksvol 89 pp 74ndash83 2017

[30] J D Cao and R X Li ldquoFixed-time synchronization of delayedmemristor-based recurrent neural networksrdquo Science ChinaInformation Sciences vol 60 no 3 Article ID 032201 2017

[31] Y Wan J Cao G Wen and W Yu ldquoRobust fixed-timesynchronization of delayed Cohen-Grossberg neural networksrdquoNeural Networks vol 73 pp 86ndash94 2016

[32] JNi L Liu C Liu XHu and S Li ldquoFast fixed-timenonsingularterminal sliding mode control and its application to chaossuppression in power systemrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 64 no 2 pp 151ndash155 2017

[33] J Ni L Liu C Liu X Hu and T Shen ldquoFixed-time dynamicsurface high-order sliding mode control for chaotic oscillationin power systemrdquo Nonlinear Dynamics vol 86 no 1 pp 401ndash420 2016

[34] S P Bhat and D S Bernstein ldquoFinite-time stability of con-tinuous autonomous systemsrdquo SIAM Journal on Control andOptimization vol 38 no 3 pp 751ndash766 2000

[35] H K Khalil and JWGrizzle Nonlinear systems third editionPrentice Hall Upper Saddle River 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

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

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom