Research Article Adaptive Fixed-Time Stability Control and Parameters Identification for Chaotic Oscillation in Second Order Power System Caoyuan Ma , 1,2 Wenbei Wu , 1,2 Zhijie Li , 1,2 Yuzhou Cheng , 1,2 and Faxin Wang 1,2 1 Jiangsu Province Laboratory of Mining Electric and Automation, China University of Mining and Technology, 221116, China 2 School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221008, China Correspondence should be addressed to Yuzhou Cheng; [email protected] Received 11 July 2018; Revised 28 October 2018; Accepted 14 November 2018; Published 29 November 2018 Academic Editor: Dragan Poljak Copyright © 2018 Caoyuan Ma et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the novel adaptive fixed-time stability control for chaotic oscillation in second order power system is proposed. e settling time of fixed-time control can be adjusted to the desired value without knowing the initial condition, while the finite time control depends on that. en, we develop a parameter identification method of fixed-time depending on synchronous observer with adaptive law of parameters, which can guarantee these uncertain parameters to be identified effectively. Finally, some numerical results demonstrate the effectiveness and practicability of the scheme. 1. Introduction Over the past few decades, the mechanism of bifurcation and chaos in power system has been intensively studied. In 1980, Kopell [1] first studied how to turn a three-machine power system into a two-degree-of-freedom system and analyzed the bifurcation and chaos of the system by Meilenikov method, which marked the beginning of chaos research in power system. In 1990, American scientists Ott, Grebogi, and Yorke [2] used the OGY method to achieve the true sense of chaos control. In [3], Wang carried out chaotic control on a simplified second order model of power system and adopted the classical backstepping control algorithm to control the chaotic power system to a stable flat point. At present, many literatures have made theoretical analyses of chaos control, while the study of chaotic oscillation control strategy in power system has just started. Power system, as a typical nonlinear and nonautonomous system, contains many complex nonlinear electromechan- ical oscillations such as low frequency oscillation, subsyn- chronous oscillation, bifurcation, and chaos oscillation [4–6]. When periodic load disturbance reaches a certain amplitude, chaos oscillation phenomena in power system will happen, which is an aperiodic, irregularity, paroxysmal and sudden ill-conditioned electromechanical chaotic oscillation [7, 8]. In serious cases, chaos oscillation will lead to interconnection of power system and a serious threat to the safety of the power grid [9]. In recent years, with the development of the nonlinear system control theory, some useful control methods have been widely used in power system. In [10], a kind of adaptive- feedback control method was proposed to control the chaos in power system, which systematize and structure the design process of the system’s Lyapunov functions and controllers through the reverse design. But its structure is very complex, and the complexity of regression matrix would become stronger especially when the nonlinear damping existed for the system parameters uncertainty. In [11], the authors discussed the fuzzy control of chaotic power system with uncertainly. However, it needs to adjust the control rules and parameters constantly, which affect dynamic response and steady state precision. In [12], Zhao designed a fuzzy sliding mode variable structure controller, which is robust to disturbances, maintaining the advantages of fuzzy control and sliding mode control. In [13], by using the sliding mode control, a nonsingular terminal sliding-mode controller with Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 2162483, 9 pages https://doi.org/10.1155/2018/2162483
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Research ArticleAdaptive Fixed-Time Stability Control andParameters Identification for Chaotic Oscillation inSecond Order Power System
CaoyuanMa 12 Wenbei Wu 12 Zhijie Li 12 Yuzhou Cheng 12 and FaxinWang 12
1 Jiangsu Province Laboratory of Mining Electric and Automation China University of Mining and Technology 221116 China2School of Electrical and Power Engineering China University of Mining and Technology Xuzhou 221008 China
Correspondence should be addressed to Yuzhou Cheng 836754437qqcom
Received 11 July 2018 Revised 28 October 2018 Accepted 14 November 2018 Published 29 November 2018
Academic Editor Dragan Poljak
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
In this paper the novel adaptive fixed-time stability control for chaotic oscillation in second order power system is proposed Thesettling time of fixed-time control can be adjusted to the desired value without knowing the initial condition while the finite timecontrol depends on that Then we develop a parameter identification method of fixed-time depending on synchronous observerwith adaptive law of parameters which can guarantee these uncertain parameters to be identified effectively Finally some numericalresults demonstrate the effectiveness and practicability of the scheme
1 Introduction
Over the past few decades the mechanism of bifurcation andchaos in power system has been intensively studied In 1980Kopell [1] first studied how to turn a three-machine powersystem into a two-degree-of-freedom system and analyzedthe bifurcation and chaos of the system by Meilenikovmethod which marked the beginning of chaos research inpower system In 1990 American scientists Ott Grebogi andYorke [2] used the OGY method to achieve the true sense ofchaos control In [3] Wang carried out chaotic control on asimplified second order model of power system and adoptedthe classical backstepping control algorithm to control thechaotic power system to a stable flat point At present manyliteratures have made theoretical analyses of chaos controlwhile the study of chaotic oscillation control strategy inpower system has just started
Power system as a typical nonlinear and nonautonomoussystem contains many complex nonlinear electromechan-ical oscillations such as low frequency oscillation subsyn-chronous oscillation bifurcation and chaos oscillation [4ndash6]When periodic load disturbance reaches a certain amplitudechaos oscillation phenomena in power system will happen
which is an aperiodic irregularity paroxysmal and suddenill-conditioned electromechanical chaotic oscillation [7 8]In serious cases chaos oscillation will lead to interconnectionof power system and a serious threat to the safety of the powergrid [9]
In recent years with the development of the nonlinearsystem control theory some useful control methods havebeen widely used in power system In [10] a kind of adaptive-feedback control method was proposed to control the chaosin power system which systematize and structure the designprocess of the systemrsquos Lyapunov functions and controllersthrough the reverse design But its structure is very complexand the complexity of regression matrix would becomestronger especially when the nonlinear damping existedfor the system parameters uncertainty In [11] the authorsdiscussed the fuzzy control of chaotic power system withuncertainly However it needs to adjust the control rulesand parameters constantly which affect dynamic responseand steady state precision In [12] Zhao designed a fuzzysliding mode variable structure controller which is robustto disturbances maintaining the advantages of fuzzy controland sliding mode control In [13] by using the sliding modecontrol a nonsingular terminal sliding-mode controller with
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 2162483 9 pageshttpsdoiorg10115520182162483
2 Mathematical Problems in Engineering
nonlinear disturbance observer was proposed for chaoticoscillation in power system which can shorten the reachingtime and weaken system chattering However these methodscannot guarantee stability in a certain period of time andcannot guarantee convergence of speed error responses underuncertain parameters Recently the finite-time control andsynchronization of the chaotic power system have attractedinterests of many researchers [14ndash18] In [19] the authorsstudied the chaos control of power system based on thefinite-time stability theory In [20] an adaptive finite-timestability controller was presented for chaotic power systemwith uncertain parameters
Although the finite-time control is better than thosedescribed above in control effect the convergence rate ofthe finite-time control is extremely dependent on the initialconditions In the power system the initial conditions canhardly be given and estimated which may result in differentconvergence time and lead to the deterioration of the systemrsquosperformance To solve this problem Polyakov [21] firstproposed the definition of fixed-time stability The settlingtime of fixed-time control can be adjusted to the desired valuewithout knowing the initial condition Subsequently fixedtime control began to be used in some complex systems andshowed good performance [22ndash26]
According to the discussion of the above the paperpresents an adaptive fixed-time control strategy which canaccelerate the convergence time suppress chaos in the powersystem and avoid voltage collapse The main advantages ofthe proposed controller are that it can guarantee the systemstable in fixed time without depending on initial state andthe settling time can be calculated directly This paper firstdiscusses the use of adaptive fixed-time stability analysiswith uncertain parameters for chaotic oscillation in secondorder power system The structure of this paper is arrangedas follows In Section 2 we introduce the model of chaoticoscillation in second order power system some useful def-initions and lemmas In Section 3 the novel adaptive fixed-time stability control for chaotic oscillation in second orderpower system is proposed and some numerical examples areprovided to demonstrate the effectiveness and practicabilityof the results Section 4 proposes a parameter identificationmethod of fixed-time depending on synchronous observerwith adaptive law of parameters At last we make someconclusions
2 Model Description
Without the influence of the internal factors such as themoment of inertia of the equivalent system the dimension-less mathematical model of chaotic oscillation in secondorder power system is considered [27]
= 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867 + 119875120585119867 cos119891119905
minus 120575119867 cos 119911119905 sin 120590 (119905)(1)
minus3 minus2 minus1 0 1 2 3
minus2
minus1
0
1
2
Figure 1 Chaotic attractor in (119909 119910)-space
where 120590(119905) = 1205901 minus 1205902 is the relative angle between thesystem (1) equivalent generator and the system 2 equivalentgenerator q axis potential 119903119886119889 119867 is equivalent moment ofinertia 119896119892 sdot 1198982 120596(119905) is relative angular velocity 119903119886119889119904 119863is equivalent damping coefficient 119873 sdot 119898 sdot 119904119903119886119889 119875119898 is themechanical power of the equivalent generator 1 119882 119875120585 isdisturbance power amplitude119882 120575 = Δ119901119875max is electromag-netic power perturbation amplitude and119891 is the frequency ofthe disturbance power 119867119911 Δ119901 is increased electromagneticpower In this case 119875120585 and 120575 are the perturbation amplitudesof the two perturbed terms in the system
The bifurcation and chaos phenomena of the abovesystem (1) are fully investigated in Refs [28] For example ifgiven 119911 = 08119867119911 119875120585 = 2119882 119875max = 100119882 119867 = 100119896119892 sdot 1198982119863 = 40119873 sdot 119898 sdot 119904119903119886119889 119891 = 1119867119911 120575 ge 127 the chaotic behaviorcan be found in Figure 1
The chaotic oscillation of the power system poses a seri-ous threat to the safe operation of the power grid Thereforefinding an appropriate control method to solve this problemis significant In this paper we eliminate the chaos in second-order chaotic oscillation system by employing the adaptivefixed-time controller The following definition and lemma areintroduced in advance which are required for proving mainresults
Definition 1 (see [21]) Consider the following dynamic sys-tem
= 119891 (119909) (2)
where 119909 isin 119877119899 is the system state and 119891 is a smoothnonlinear function If for any initial condition there exists afixed settling time 1198790 which is not connected with the initialcondition such that
lim119905997888rarr1198790
119909 (119905) = 0 (3)
and 119909(119905) equiv 0 if 119905 ge 1198790 then the system (2) is said to befixed-time stable
Lemma 2 (see [29]) Suppose there exists a continuous func-tion 119881(119905) [0infin) 997888rarr [0infin) such that
Mathematical Problems in Engineering 3
(1) 119881 is positive definite(2) There exist real numbers 119888 gt 0 and 0 lt 120588 lt 1 such that
119881 (119905) le minus119888119881120588 (119905) 119905 ge 1199050 (4)
then one has
1198811minus120588 (119905) le 1198811minus120588 (1199050) minus 119888 (1 minus 120588) (119905 minus 1199050) 1199050 le 119905 le 119905lowast (5)
and
119881 (119905) = 0 119905 ge 119905lowast (6)
of which
119905lowast = 1199050 + 1198811minus120588 (1199050)119888 (1 minus 120588) (7)
Lemma 3 (see [21]) If there exists a continuous radicallyunbounded function 119881 119877119873 997888rarr 119877+⋃ |0| such that
(1) 119881(119909) = 0 lArrrArr 119909 = 0(2) any solution 119909(119905) satisfied the inequality119863lowast119881(119909(119905)) leminus[120572119881119901(119909(119905)) + 120573119881119902(119909(119905))]119896 for some 120572 120573 119901 119902 119896 gt0 119901119896 lt 1 and 119902119896 gt 1 where 119863 lowast 119881(119909(119905)) denotes the
upper right hand derivative of the function 119881(119909(119905))then the origin is globally fixed time stable and the followingestimate holds
119879 (1199090) le 1120572119896 (1 minus 119901119896) + 1120573119896 (119902119896 minus 1) forall1199090 isin 119877119899 (8)
Lemma 3 presents quite a conservative settling timeestimate A more accurate estimate is provided in the nextlemma Consider the case where constants p and q are of theform 119901 = 1 minus 12120574 and 119902 = 1 + 12120574 120574 gt 1Lemma 4 (see [30]) If 1199091 1199092 119909119873 ge 0 then
119873sum119894=1
119909120578119894 ge (119873sum119894=1
119909119894)120578
0 lt 120578 le 1119873sum119894=1
119909120589119894 ge 1198731minus120589(119873sum119894=1
119909119894)120589
120589 gt 1(9)
3 Main Results
31 Fixed-Time Chaotic Oscillation Control of Power System
Theorem 5 The fixed-time stability of chaotic oscillation insecond order power system can be achieved by adding thefollowing controller 11990611199061 = 119875max119867 sin 120590 minus 119875119898119867 minus 119875120585119867 cos119891119905 + 120575119867 cos 119911119905 sin 120590
minus 1198961 sign (120596) |120596|120572 minus 1198961120596120573(10)
where 0 lt 120572 lt 1 120573 gt 1 1198961 gt 0 1198961 is the tuning parameterfeedback gain of the terminal attractor and
1198961 = |120596|120572+1 + 120596120573+1 minus (1198961 minus 1198921)120572 minus (1198961 minus 1198921)120573 (11)
where 119892 is the arbitrary positive constant
The system is described as
= 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867 + 119875120585119867 cos119891119905
minus 120575119867 cos 119911119905 sin 120590 (119905) + 1199061(12)
Proof For analysis convenience we select the Lyapunovcandidate function
1198811 (119905) = 121205962 + 12 (1198961 minus 1198921)2 (13)
we can get the derivative of the system trajectory by usingthe design of the controller 1199061 and the corresponding tuningparameters
1 (119905) = 120596 + (1198961 minus 1198921) 1198961= minus1198631198671205962 minus 1198921 |120596|120572+1 minus 1198921120596120573+1 minus (1198961 minus 1198921)120572+1minus (1198961 minus 1198921)120573+1
le minus1198921 |120596|120572+1 minus 1198921120596120573+1 minus (1198961 minus 1198921)120572+1minus (1198961 minus 1198921)120573+1
= minus2(12)(120572+1)1198921 (121205962)(12)(120572+1)
minus 2(12)(120572+1) [12 (1198961 minus 1198921)2](12)(120572+1)
minus 2(12)(120573+1)1198921 (121205962)(12)(120573+1)
minus 2(12)(120573+1) [12 (1198961 minus 1198921)2](12)(120573+1)
le minus1198981 (121205962)(12)(120572+1) + [12 (1198961 minus 1198921)2]
(12)(120572+1)minus 1198991 (121205962)
(12)(120573+1) + [12 (1198961 minus 1198921)2](12)(120573+1)
(14)
where 1198981 = min2(12)(120572+1)1198921 2(12)(120572+1) 1198991 =min2(12)(120573+1)1198921 2(12)(120573+1)
Thus it follows from Lemma 4 that
1 (119905)le minus1198981 (121205962) + [12 (1198961 minus 1198921)2]
(12)(120572+1)
minus 2(1minus120573)21198991 (121205962) + [12 (1198961 minus 1198921)2](12)(120573+1)
= minus11989811198811(12)(120572+1) minus 2(1minus120573)211989911198811(12)(120573+1)
(15)
Thus it follows from Lemma 2 where 0 lt 120572 lt 1 120573 gt 1
4 Mathematical Problems in Engineering
50 100 150 200t (s)
minus3
minus2
minus1
0
1
2
3
(a)
50 100 150 200
minus2
minus1
0
1
2
t (s)
(b)
Figure 2The chaotic state trajectory of the system (1) without control (a) The timing diagram of 120590 (b)The timing diagram of 120596
0 2 4 6 8 10 12 14 16 18 20
Time
Syste
m v
aria
bles
minus02
minus015
minus01
minus005
0
005
01
015
02
025
03
(a)
0 2 4 6 8 10 12 14 16 18 20Tim
k1
e
05
045
04
035
03
025
015
01
005
0
Tuni
ng p
aram
eter
k1
(b)
Figure 3 (a)The evolution of the controlled state variables 120590 and 120596 (b) Tuning parameter 1198961
The system (12) will be stable in fixed time 1198791 where1198791 le 11198981 (1 minus (120572 + 1) 2)
+ 12(1minus120573)21198991 ((120573 + 1) 2 minus 1)= 21198981 (1 minus 120572) +
2(120573+1)21198991 (120573 minus 1)
(16)
when 119905 ge 1198791 120596 equiv 0 1198961 = 1198921Therefore based on the adaptivecontrol strategy the chaotic system will be stable for a fixedtime
32 Numerical Examples In this section numerical resultsare performed to demonstrate the effectiveness and practica-bility of the scheme We give 119911 = 08119867119911 119875120585 = 2119882 119875max =100119882119867 = 100119896119892sdot1198982119863 = 40119873sdot119898sdot119904119903119886119889119891 = 1119867119911120572 = 08
120573 = 11 1198961(0) = 0 and the initial value of the variables is tobe 120590 120596 = 12058715 0
Figures 2(a) and 2(b) are the response chaotic trajectoriesof the systemvariables120590 and120596when the second order chaoticoscillation system is not controlled From Figure 2 it is clearthat the state of the system variables 120596 and 120590 is aperiodic andis always in a state of instability
Figure 3(a) is the response curves of the system variables120590 and 120596 after the fixed-time controllers 1199061 are applied to thesecond order chaotic oscillation system Figure 3(b) showsthe evolution of tuning parameter of terminal attractors1198961 From Figures 2 and 3(a) it is clear that when thesystem is chaotic before 1 s without the controller once thecontroller is applied the variables 120590 and120596 become stable aftertransition process 42 s simultaneously The adaptive fixed-time controller synchronizes the chaotic system to a stablestate without chaos and the system eventually converges tothe originThus the simulation results show that the designedcontroller is feasible and achieves the desired effect
Mathematical Problems in Engineering 5
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
nonlinear optimalfinite-timefixed-time
Figure 4 Comparison of dynamic response of different controlmodes with 120572 = 08 and 120573 = 11
To ensure a fair comparison with other control methodswe set constant initial conditions and tuning parameter fthe terminal attractor 1198961 and investigated 120596 convergence forfixed-time control finite-time control and nonlinear optimalcontrol numerically As shown in Figure 4 convergence timefor the proposed fixed-time controller is significantly lessthan that for the other controllers The system state variable120596 achieves stable state at 42 s under the action of fixed-timecontrollers However the system state variable 120596 achievesstable state at 5 s and 53 s under the action of finite-timecontroller and nonlinear optimal control respectively Thusthe fixed-time method has the better capacity to handle anonlinear system in a short time
To explore the relationship between the convergence timeand the values of the parameters 120572 and 120573 experimentally weselect system state variable 120596 to demonstrate the convergencetime Figures 5(a) and 5(b) are respectively convergence for120596 that increases with increasing 120572 and 120573 parameter valuesunder the fixed-time controller In Figure 5(a) the parametervalues are 120573 = 18 and 120572 = 03 05 07 09 In Figure 5(b)the parameter values are 120572 = 08 and 120573 = 11 13 15 17In Figures 5(a) and 5(b) the system parameters and othercontroller parameters and tuning parameters of the terminalattractor are consistent with the previous sections Thesimulation results clearly show that changing the controllerparameters 120572 and 120573 can change the time of the system statevariable 120596 to reach the steady state And the smaller the 120572and 120573 values of the system are the faster the convergencetime will be Moreover the influence of 120572 on the convergencetime of the system state variable 120596 is more than the influenceof 120573 on it The simulation results are consistent with thetheoretical analysis of the maximum stable time 1198791 of thesystem in the previous section Thus the values of 120572 and 120573also affect the stability value of the system state That is tosay we can get the size of the system state variables to thenumerical value we need by controlling the size of 120572 and120573
4 Fixed-Time Synchronization andParameters Identification
In the long-termoperation of the grid changes in theworkingenvironment will affect the parameters of the grid modelTherefore it is very important to identify the parameters ofthe chaotic model of the power grid Without consideringthe effects of higher harmonic disturbances electromagneticinterference etc according to the previous analysis theuncertain response system is given as follows
(119905) = 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867
minus 120575119867 cos 119911119905 sin120590 (119905)= 119886 sin 120590 minus 119887120596 + 119888 minus 119889 cos 119911119905 sin 120590
(17)
The following describes the identification of unknownparameters in the model
Theorem 6 Construct a synchronization parameter observer[31]
120596 = 119886 sin 120590 minus 120596 + 119888 minus 119889 cos 119911119905 sin 120590 + 1199062 (18)
Take the controller
1199062 = minus11989622119890 minus 1198962 sign (119890) |119890|120572 minus 1198962 sign (119890) |119890|120573 (19)
and the following adaptive law of parameters
119886 = minusℎ1119890 sin 119909119887 = minusℎ2119890119910119888 = minusℎ3119890119889 = minusℎ4119890 cos 119911119905 sin 119909
(20)
where 119890 = minus 120596 is the synchronization error between theobserver state variable and the original system state variableℎ119894 (119894 = 1 2 3 4) is the arbitrary positive constant 1198962 = |119890|120572+1+|119890|120573+1minus(1198962minus1198922)120572minus(1198962minus1198922)120573 is tuning parameter feedback gainand 1198922 is the arbitrary positive constant The meaning of ldquo119910rdquo isthe system state variable 120596 the meaning of ldquo119909rdquo is the systemstate variable 120590 Then the state variables in the observer (18)can be synchronized with the state variables in the system (17)and the unknown parameters 119886 119887 119888 119889 can be identified
Proof The error system can be obtained from (18)ndash(20)
119890 = 1198901 sin 119909 + 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11990621198901 = minusℎ1119890 sin 1199091198902 = minusℎ21198901199101198903 = minusℎ31198901198904 = minusℎ4119890 cos 119911119905 sin 119909
(21)
6 Mathematical Problems in Engineering
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=03
=05
=07
=09
(a)
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=11
=13
=15
=17
(b)
Figure 5 The effect of the control parameter on the response speed of the variable 120596 (a) The variations of 120596 with 120573 = 18 and 120572 =03 05 07 09 (b) The variations of 120596 with 120572 = 08 and 120573 = 11 13 15 17
where 119890119895 = (119895 = 1 2 3 4) is the error between the estimatedand the actual value of the unknown parameter
We select the Lyapunov candidate function
1198812 (119905) = 121198902 +4sum119895=1
12ℎ119895 1198902119895 + 12 (1198962 minus 1198922)2 (22)
The derivative of the subsystem trajectory in (22) can beobtained
2 (119905) = 119890 119890 + 4sum119895=1
1ℎ119895 119890119895 119890119895 + (1198962 minus 1198922) 1198962 = 119890 (1198901 sin 119909+ 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11989622119890 minus 1198962 sign (119890) |119890|120572minus 1198962 sign (119890) |119890|120573) + 1ℎ1 1198901 (minusℎ1119890 sin 119909) +
1ℎ2sdot 1198902 (minusℎ2119890119910) + 1ℎ3 1198903 (minusℎ3119890) +
1ℎ4sdot 1198904 (minusℎ4119890 cos 119911119905 sin 119909) + (1198962 minus 1198922) [|119890|120572+1 + |119890|120573+1minus (1198962 minus 1198922)120572 minus (1198962 minus 1198922)120573] le minus1198922 |119890|120572+1minus 1198922 |119890|120573+1 minus (1198962 minus 1198922)120572+1 minus (1198962 minus 1198922)120573+1le minus1198982 (121198902)
(12)(120572+1)
+ [12 (1198962 minus 1198922)2](12)(120572+1) minus 1198992 (121198902)
(12)(120573+1)
+ [12 (1198962 minus 1198922)2](12)(120573+1)
(23)
where 1198982 = min2(12)(120572+1)1198922 2(12)(120572+1) 1198992 =min2(12)(120573+1)1198922 2(12)(120573+1) Thus it follows from Lemma 4that
2 (119905)le minus1198982 (121198902) + [12 (1198962 minus 1198922)2]
(12)(120572+1)
minus 2(1minus120573)21198992 (121198902) + [12 (1198962 minus 1198922)2](12)(120573+1)
= minus11989821198812(12)(120572+1) minus 2(1minus120573)211989921198812(12)(120573+1)
(24)
We can obtain stable time of the error system (21) byLemma 3
1198792 le 11198982 (1 minus (120572 + 1) 2)+ 12(1minus120573)21198992 ((120573 + 1) 2 minus 1)
= 21198982 (1 minus 120572) +2(120573+1)21198992 (120573 minus 1)
(25)
which means that 119890 equiv 0 1198962 = 1198922 119886 = 119886 = 119887 119888 = 119888 119889 = 119889when 119905 ge 1198792
Then we compare the parameter identification methodbased on the fixed-time synchronization observer with theparameter identification method which did not introduce thefixed-time control If given 120590 = 12058715 120596 = 0 ℎ1 = ℎ2 =ℎ3 = ℎ4 = 50 120572 = 08 120573 = 11 the simulation results of theidentification method are shown in Figure 6The blue dashedline indicates the identification curve of the synchronizationparameter observer (SPO) and the red solid line represents
Mathematical Problems in Engineering 7
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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2 Mathematical Problems in Engineering
nonlinear disturbance observer was proposed for chaoticoscillation in power system which can shorten the reachingtime and weaken system chattering However these methodscannot guarantee stability in a certain period of time andcannot guarantee convergence of speed error responses underuncertain parameters Recently the finite-time control andsynchronization of the chaotic power system have attractedinterests of many researchers [14ndash18] In [19] the authorsstudied the chaos control of power system based on thefinite-time stability theory In [20] an adaptive finite-timestability controller was presented for chaotic power systemwith uncertain parameters
Although the finite-time control is better than thosedescribed above in control effect the convergence rate ofthe finite-time control is extremely dependent on the initialconditions In the power system the initial conditions canhardly be given and estimated which may result in differentconvergence time and lead to the deterioration of the systemrsquosperformance To solve this problem Polyakov [21] firstproposed the definition of fixed-time stability The settlingtime of fixed-time control can be adjusted to the desired valuewithout knowing the initial condition Subsequently fixedtime control began to be used in some complex systems andshowed good performance [22ndash26]
According to the discussion of the above the paperpresents an adaptive fixed-time control strategy which canaccelerate the convergence time suppress chaos in the powersystem and avoid voltage collapse The main advantages ofthe proposed controller are that it can guarantee the systemstable in fixed time without depending on initial state andthe settling time can be calculated directly This paper firstdiscusses the use of adaptive fixed-time stability analysiswith uncertain parameters for chaotic oscillation in secondorder power system The structure of this paper is arrangedas follows In Section 2 we introduce the model of chaoticoscillation in second order power system some useful def-initions and lemmas In Section 3 the novel adaptive fixed-time stability control for chaotic oscillation in second orderpower system is proposed and some numerical examples areprovided to demonstrate the effectiveness and practicabilityof the results Section 4 proposes a parameter identificationmethod of fixed-time depending on synchronous observerwith adaptive law of parameters At last we make someconclusions
2 Model Description
Without the influence of the internal factors such as themoment of inertia of the equivalent system the dimension-less mathematical model of chaotic oscillation in secondorder power system is considered [27]
= 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867 + 119875120585119867 cos119891119905
minus 120575119867 cos 119911119905 sin 120590 (119905)(1)
minus3 minus2 minus1 0 1 2 3
minus2
minus1
0
1
2
Figure 1 Chaotic attractor in (119909 119910)-space
where 120590(119905) = 1205901 minus 1205902 is the relative angle between thesystem (1) equivalent generator and the system 2 equivalentgenerator q axis potential 119903119886119889 119867 is equivalent moment ofinertia 119896119892 sdot 1198982 120596(119905) is relative angular velocity 119903119886119889119904 119863is equivalent damping coefficient 119873 sdot 119898 sdot 119904119903119886119889 119875119898 is themechanical power of the equivalent generator 1 119882 119875120585 isdisturbance power amplitude119882 120575 = Δ119901119875max is electromag-netic power perturbation amplitude and119891 is the frequency ofthe disturbance power 119867119911 Δ119901 is increased electromagneticpower In this case 119875120585 and 120575 are the perturbation amplitudesof the two perturbed terms in the system
The bifurcation and chaos phenomena of the abovesystem (1) are fully investigated in Refs [28] For example ifgiven 119911 = 08119867119911 119875120585 = 2119882 119875max = 100119882 119867 = 100119896119892 sdot 1198982119863 = 40119873 sdot 119898 sdot 119904119903119886119889 119891 = 1119867119911 120575 ge 127 the chaotic behaviorcan be found in Figure 1
The chaotic oscillation of the power system poses a seri-ous threat to the safe operation of the power grid Thereforefinding an appropriate control method to solve this problemis significant In this paper we eliminate the chaos in second-order chaotic oscillation system by employing the adaptivefixed-time controller The following definition and lemma areintroduced in advance which are required for proving mainresults
Definition 1 (see [21]) Consider the following dynamic sys-tem
= 119891 (119909) (2)
where 119909 isin 119877119899 is the system state and 119891 is a smoothnonlinear function If for any initial condition there exists afixed settling time 1198790 which is not connected with the initialcondition such that
lim119905997888rarr1198790
119909 (119905) = 0 (3)
and 119909(119905) equiv 0 if 119905 ge 1198790 then the system (2) is said to befixed-time stable
Lemma 2 (see [29]) Suppose there exists a continuous func-tion 119881(119905) [0infin) 997888rarr [0infin) such that
Mathematical Problems in Engineering 3
(1) 119881 is positive definite(2) There exist real numbers 119888 gt 0 and 0 lt 120588 lt 1 such that
119881 (119905) le minus119888119881120588 (119905) 119905 ge 1199050 (4)
then one has
1198811minus120588 (119905) le 1198811minus120588 (1199050) minus 119888 (1 minus 120588) (119905 minus 1199050) 1199050 le 119905 le 119905lowast (5)
and
119881 (119905) = 0 119905 ge 119905lowast (6)
of which
119905lowast = 1199050 + 1198811minus120588 (1199050)119888 (1 minus 120588) (7)
Lemma 3 (see [21]) If there exists a continuous radicallyunbounded function 119881 119877119873 997888rarr 119877+⋃ |0| such that
(1) 119881(119909) = 0 lArrrArr 119909 = 0(2) any solution 119909(119905) satisfied the inequality119863lowast119881(119909(119905)) leminus[120572119881119901(119909(119905)) + 120573119881119902(119909(119905))]119896 for some 120572 120573 119901 119902 119896 gt0 119901119896 lt 1 and 119902119896 gt 1 where 119863 lowast 119881(119909(119905)) denotes the
upper right hand derivative of the function 119881(119909(119905))then the origin is globally fixed time stable and the followingestimate holds
119879 (1199090) le 1120572119896 (1 minus 119901119896) + 1120573119896 (119902119896 minus 1) forall1199090 isin 119877119899 (8)
Lemma 3 presents quite a conservative settling timeestimate A more accurate estimate is provided in the nextlemma Consider the case where constants p and q are of theform 119901 = 1 minus 12120574 and 119902 = 1 + 12120574 120574 gt 1Lemma 4 (see [30]) If 1199091 1199092 119909119873 ge 0 then
119873sum119894=1
119909120578119894 ge (119873sum119894=1
119909119894)120578
0 lt 120578 le 1119873sum119894=1
119909120589119894 ge 1198731minus120589(119873sum119894=1
119909119894)120589
120589 gt 1(9)
3 Main Results
31 Fixed-Time Chaotic Oscillation Control of Power System
Theorem 5 The fixed-time stability of chaotic oscillation insecond order power system can be achieved by adding thefollowing controller 11990611199061 = 119875max119867 sin 120590 minus 119875119898119867 minus 119875120585119867 cos119891119905 + 120575119867 cos 119911119905 sin 120590
minus 1198961 sign (120596) |120596|120572 minus 1198961120596120573(10)
where 0 lt 120572 lt 1 120573 gt 1 1198961 gt 0 1198961 is the tuning parameterfeedback gain of the terminal attractor and
1198961 = |120596|120572+1 + 120596120573+1 minus (1198961 minus 1198921)120572 minus (1198961 minus 1198921)120573 (11)
where 119892 is the arbitrary positive constant
The system is described as
= 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867 + 119875120585119867 cos119891119905
minus 120575119867 cos 119911119905 sin 120590 (119905) + 1199061(12)
Proof For analysis convenience we select the Lyapunovcandidate function
1198811 (119905) = 121205962 + 12 (1198961 minus 1198921)2 (13)
we can get the derivative of the system trajectory by usingthe design of the controller 1199061 and the corresponding tuningparameters
1 (119905) = 120596 + (1198961 minus 1198921) 1198961= minus1198631198671205962 minus 1198921 |120596|120572+1 minus 1198921120596120573+1 minus (1198961 minus 1198921)120572+1minus (1198961 minus 1198921)120573+1
le minus1198921 |120596|120572+1 minus 1198921120596120573+1 minus (1198961 minus 1198921)120572+1minus (1198961 minus 1198921)120573+1
= minus2(12)(120572+1)1198921 (121205962)(12)(120572+1)
minus 2(12)(120572+1) [12 (1198961 minus 1198921)2](12)(120572+1)
minus 2(12)(120573+1)1198921 (121205962)(12)(120573+1)
minus 2(12)(120573+1) [12 (1198961 minus 1198921)2](12)(120573+1)
le minus1198981 (121205962)(12)(120572+1) + [12 (1198961 minus 1198921)2]
(12)(120572+1)minus 1198991 (121205962)
(12)(120573+1) + [12 (1198961 minus 1198921)2](12)(120573+1)
(14)
where 1198981 = min2(12)(120572+1)1198921 2(12)(120572+1) 1198991 =min2(12)(120573+1)1198921 2(12)(120573+1)
Thus it follows from Lemma 4 that
1 (119905)le minus1198981 (121205962) + [12 (1198961 minus 1198921)2]
(12)(120572+1)
minus 2(1minus120573)21198991 (121205962) + [12 (1198961 minus 1198921)2](12)(120573+1)
= minus11989811198811(12)(120572+1) minus 2(1minus120573)211989911198811(12)(120573+1)
(15)
Thus it follows from Lemma 2 where 0 lt 120572 lt 1 120573 gt 1
4 Mathematical Problems in Engineering
50 100 150 200t (s)
minus3
minus2
minus1
0
1
2
3
(a)
50 100 150 200
minus2
minus1
0
1
2
t (s)
(b)
Figure 2The chaotic state trajectory of the system (1) without control (a) The timing diagram of 120590 (b)The timing diagram of 120596
0 2 4 6 8 10 12 14 16 18 20
Time
Syste
m v
aria
bles
minus02
minus015
minus01
minus005
0
005
01
015
02
025
03
(a)
0 2 4 6 8 10 12 14 16 18 20Tim
k1
e
05
045
04
035
03
025
015
01
005
0
Tuni
ng p
aram
eter
k1
(b)
Figure 3 (a)The evolution of the controlled state variables 120590 and 120596 (b) Tuning parameter 1198961
The system (12) will be stable in fixed time 1198791 where1198791 le 11198981 (1 minus (120572 + 1) 2)
+ 12(1minus120573)21198991 ((120573 + 1) 2 minus 1)= 21198981 (1 minus 120572) +
2(120573+1)21198991 (120573 minus 1)
(16)
when 119905 ge 1198791 120596 equiv 0 1198961 = 1198921Therefore based on the adaptivecontrol strategy the chaotic system will be stable for a fixedtime
32 Numerical Examples In this section numerical resultsare performed to demonstrate the effectiveness and practica-bility of the scheme We give 119911 = 08119867119911 119875120585 = 2119882 119875max =100119882119867 = 100119896119892sdot1198982119863 = 40119873sdot119898sdot119904119903119886119889119891 = 1119867119911120572 = 08
120573 = 11 1198961(0) = 0 and the initial value of the variables is tobe 120590 120596 = 12058715 0
Figures 2(a) and 2(b) are the response chaotic trajectoriesof the systemvariables120590 and120596when the second order chaoticoscillation system is not controlled From Figure 2 it is clearthat the state of the system variables 120596 and 120590 is aperiodic andis always in a state of instability
Figure 3(a) is the response curves of the system variables120590 and 120596 after the fixed-time controllers 1199061 are applied to thesecond order chaotic oscillation system Figure 3(b) showsthe evolution of tuning parameter of terminal attractors1198961 From Figures 2 and 3(a) it is clear that when thesystem is chaotic before 1 s without the controller once thecontroller is applied the variables 120590 and120596 become stable aftertransition process 42 s simultaneously The adaptive fixed-time controller synchronizes the chaotic system to a stablestate without chaos and the system eventually converges tothe originThus the simulation results show that the designedcontroller is feasible and achieves the desired effect
Mathematical Problems in Engineering 5
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
nonlinear optimalfinite-timefixed-time
Figure 4 Comparison of dynamic response of different controlmodes with 120572 = 08 and 120573 = 11
To ensure a fair comparison with other control methodswe set constant initial conditions and tuning parameter fthe terminal attractor 1198961 and investigated 120596 convergence forfixed-time control finite-time control and nonlinear optimalcontrol numerically As shown in Figure 4 convergence timefor the proposed fixed-time controller is significantly lessthan that for the other controllers The system state variable120596 achieves stable state at 42 s under the action of fixed-timecontrollers However the system state variable 120596 achievesstable state at 5 s and 53 s under the action of finite-timecontroller and nonlinear optimal control respectively Thusthe fixed-time method has the better capacity to handle anonlinear system in a short time
To explore the relationship between the convergence timeand the values of the parameters 120572 and 120573 experimentally weselect system state variable 120596 to demonstrate the convergencetime Figures 5(a) and 5(b) are respectively convergence for120596 that increases with increasing 120572 and 120573 parameter valuesunder the fixed-time controller In Figure 5(a) the parametervalues are 120573 = 18 and 120572 = 03 05 07 09 In Figure 5(b)the parameter values are 120572 = 08 and 120573 = 11 13 15 17In Figures 5(a) and 5(b) the system parameters and othercontroller parameters and tuning parameters of the terminalattractor are consistent with the previous sections Thesimulation results clearly show that changing the controllerparameters 120572 and 120573 can change the time of the system statevariable 120596 to reach the steady state And the smaller the 120572and 120573 values of the system are the faster the convergencetime will be Moreover the influence of 120572 on the convergencetime of the system state variable 120596 is more than the influenceof 120573 on it The simulation results are consistent with thetheoretical analysis of the maximum stable time 1198791 of thesystem in the previous section Thus the values of 120572 and 120573also affect the stability value of the system state That is tosay we can get the size of the system state variables to thenumerical value we need by controlling the size of 120572 and120573
4 Fixed-Time Synchronization andParameters Identification
In the long-termoperation of the grid changes in theworkingenvironment will affect the parameters of the grid modelTherefore it is very important to identify the parameters ofthe chaotic model of the power grid Without consideringthe effects of higher harmonic disturbances electromagneticinterference etc according to the previous analysis theuncertain response system is given as follows
(119905) = 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867
minus 120575119867 cos 119911119905 sin120590 (119905)= 119886 sin 120590 minus 119887120596 + 119888 minus 119889 cos 119911119905 sin 120590
(17)
The following describes the identification of unknownparameters in the model
Theorem 6 Construct a synchronization parameter observer[31]
120596 = 119886 sin 120590 minus 120596 + 119888 minus 119889 cos 119911119905 sin 120590 + 1199062 (18)
Take the controller
1199062 = minus11989622119890 minus 1198962 sign (119890) |119890|120572 minus 1198962 sign (119890) |119890|120573 (19)
and the following adaptive law of parameters
119886 = minusℎ1119890 sin 119909119887 = minusℎ2119890119910119888 = minusℎ3119890119889 = minusℎ4119890 cos 119911119905 sin 119909
(20)
where 119890 = minus 120596 is the synchronization error between theobserver state variable and the original system state variableℎ119894 (119894 = 1 2 3 4) is the arbitrary positive constant 1198962 = |119890|120572+1+|119890|120573+1minus(1198962minus1198922)120572minus(1198962minus1198922)120573 is tuning parameter feedback gainand 1198922 is the arbitrary positive constant The meaning of ldquo119910rdquo isthe system state variable 120596 the meaning of ldquo119909rdquo is the systemstate variable 120590 Then the state variables in the observer (18)can be synchronized with the state variables in the system (17)and the unknown parameters 119886 119887 119888 119889 can be identified
Proof The error system can be obtained from (18)ndash(20)
119890 = 1198901 sin 119909 + 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11990621198901 = minusℎ1119890 sin 1199091198902 = minusℎ21198901199101198903 = minusℎ31198901198904 = minusℎ4119890 cos 119911119905 sin 119909
(21)
6 Mathematical Problems in Engineering
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=03
=05
=07
=09
(a)
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=11
=13
=15
=17
(b)
Figure 5 The effect of the control parameter on the response speed of the variable 120596 (a) The variations of 120596 with 120573 = 18 and 120572 =03 05 07 09 (b) The variations of 120596 with 120572 = 08 and 120573 = 11 13 15 17
where 119890119895 = (119895 = 1 2 3 4) is the error between the estimatedand the actual value of the unknown parameter
We select the Lyapunov candidate function
1198812 (119905) = 121198902 +4sum119895=1
12ℎ119895 1198902119895 + 12 (1198962 minus 1198922)2 (22)
The derivative of the subsystem trajectory in (22) can beobtained
2 (119905) = 119890 119890 + 4sum119895=1
1ℎ119895 119890119895 119890119895 + (1198962 minus 1198922) 1198962 = 119890 (1198901 sin 119909+ 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11989622119890 minus 1198962 sign (119890) |119890|120572minus 1198962 sign (119890) |119890|120573) + 1ℎ1 1198901 (minusℎ1119890 sin 119909) +
1ℎ2sdot 1198902 (minusℎ2119890119910) + 1ℎ3 1198903 (minusℎ3119890) +
1ℎ4sdot 1198904 (minusℎ4119890 cos 119911119905 sin 119909) + (1198962 minus 1198922) [|119890|120572+1 + |119890|120573+1minus (1198962 minus 1198922)120572 minus (1198962 minus 1198922)120573] le minus1198922 |119890|120572+1minus 1198922 |119890|120573+1 minus (1198962 minus 1198922)120572+1 minus (1198962 minus 1198922)120573+1le minus1198982 (121198902)
(12)(120572+1)
+ [12 (1198962 minus 1198922)2](12)(120572+1) minus 1198992 (121198902)
(12)(120573+1)
+ [12 (1198962 minus 1198922)2](12)(120573+1)
(23)
where 1198982 = min2(12)(120572+1)1198922 2(12)(120572+1) 1198992 =min2(12)(120573+1)1198922 2(12)(120573+1) Thus it follows from Lemma 4that
2 (119905)le minus1198982 (121198902) + [12 (1198962 minus 1198922)2]
(12)(120572+1)
minus 2(1minus120573)21198992 (121198902) + [12 (1198962 minus 1198922)2](12)(120573+1)
= minus11989821198812(12)(120572+1) minus 2(1minus120573)211989921198812(12)(120573+1)
(24)
We can obtain stable time of the error system (21) byLemma 3
1198792 le 11198982 (1 minus (120572 + 1) 2)+ 12(1minus120573)21198992 ((120573 + 1) 2 minus 1)
= 21198982 (1 minus 120572) +2(120573+1)21198992 (120573 minus 1)
(25)
which means that 119890 equiv 0 1198962 = 1198922 119886 = 119886 = 119887 119888 = 119888 119889 = 119889when 119905 ge 1198792
Then we compare the parameter identification methodbased on the fixed-time synchronization observer with theparameter identification method which did not introduce thefixed-time control If given 120590 = 12058715 120596 = 0 ℎ1 = ℎ2 =ℎ3 = ℎ4 = 50 120572 = 08 120573 = 11 the simulation results of theidentification method are shown in Figure 6The blue dashedline indicates the identification curve of the synchronizationparameter observer (SPO) and the red solid line represents
Mathematical Problems in Engineering 7
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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Mathematical Problems in Engineering 3
(1) 119881 is positive definite(2) There exist real numbers 119888 gt 0 and 0 lt 120588 lt 1 such that
119881 (119905) le minus119888119881120588 (119905) 119905 ge 1199050 (4)
then one has
1198811minus120588 (119905) le 1198811minus120588 (1199050) minus 119888 (1 minus 120588) (119905 minus 1199050) 1199050 le 119905 le 119905lowast (5)
and
119881 (119905) = 0 119905 ge 119905lowast (6)
of which
119905lowast = 1199050 + 1198811minus120588 (1199050)119888 (1 minus 120588) (7)
Lemma 3 (see [21]) If there exists a continuous radicallyunbounded function 119881 119877119873 997888rarr 119877+⋃ |0| such that
(1) 119881(119909) = 0 lArrrArr 119909 = 0(2) any solution 119909(119905) satisfied the inequality119863lowast119881(119909(119905)) leminus[120572119881119901(119909(119905)) + 120573119881119902(119909(119905))]119896 for some 120572 120573 119901 119902 119896 gt0 119901119896 lt 1 and 119902119896 gt 1 where 119863 lowast 119881(119909(119905)) denotes the
upper right hand derivative of the function 119881(119909(119905))then the origin is globally fixed time stable and the followingestimate holds
119879 (1199090) le 1120572119896 (1 minus 119901119896) + 1120573119896 (119902119896 minus 1) forall1199090 isin 119877119899 (8)
Lemma 3 presents quite a conservative settling timeestimate A more accurate estimate is provided in the nextlemma Consider the case where constants p and q are of theform 119901 = 1 minus 12120574 and 119902 = 1 + 12120574 120574 gt 1Lemma 4 (see [30]) If 1199091 1199092 119909119873 ge 0 then
119873sum119894=1
119909120578119894 ge (119873sum119894=1
119909119894)120578
0 lt 120578 le 1119873sum119894=1
119909120589119894 ge 1198731minus120589(119873sum119894=1
119909119894)120589
120589 gt 1(9)
3 Main Results
31 Fixed-Time Chaotic Oscillation Control of Power System
Theorem 5 The fixed-time stability of chaotic oscillation insecond order power system can be achieved by adding thefollowing controller 11990611199061 = 119875max119867 sin 120590 minus 119875119898119867 minus 119875120585119867 cos119891119905 + 120575119867 cos 119911119905 sin 120590
minus 1198961 sign (120596) |120596|120572 minus 1198961120596120573(10)
where 0 lt 120572 lt 1 120573 gt 1 1198961 gt 0 1198961 is the tuning parameterfeedback gain of the terminal attractor and
1198961 = |120596|120572+1 + 120596120573+1 minus (1198961 minus 1198921)120572 minus (1198961 minus 1198921)120573 (11)
where 119892 is the arbitrary positive constant
The system is described as
= 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867 + 119875120585119867 cos119891119905
minus 120575119867 cos 119911119905 sin 120590 (119905) + 1199061(12)
Proof For analysis convenience we select the Lyapunovcandidate function
1198811 (119905) = 121205962 + 12 (1198961 minus 1198921)2 (13)
we can get the derivative of the system trajectory by usingthe design of the controller 1199061 and the corresponding tuningparameters
1 (119905) = 120596 + (1198961 minus 1198921) 1198961= minus1198631198671205962 minus 1198921 |120596|120572+1 minus 1198921120596120573+1 minus (1198961 minus 1198921)120572+1minus (1198961 minus 1198921)120573+1
le minus1198921 |120596|120572+1 minus 1198921120596120573+1 minus (1198961 minus 1198921)120572+1minus (1198961 minus 1198921)120573+1
= minus2(12)(120572+1)1198921 (121205962)(12)(120572+1)
minus 2(12)(120572+1) [12 (1198961 minus 1198921)2](12)(120572+1)
minus 2(12)(120573+1)1198921 (121205962)(12)(120573+1)
minus 2(12)(120573+1) [12 (1198961 minus 1198921)2](12)(120573+1)
le minus1198981 (121205962)(12)(120572+1) + [12 (1198961 minus 1198921)2]
(12)(120572+1)minus 1198991 (121205962)
(12)(120573+1) + [12 (1198961 minus 1198921)2](12)(120573+1)
(14)
where 1198981 = min2(12)(120572+1)1198921 2(12)(120572+1) 1198991 =min2(12)(120573+1)1198921 2(12)(120573+1)
Thus it follows from Lemma 4 that
1 (119905)le minus1198981 (121205962) + [12 (1198961 minus 1198921)2]
(12)(120572+1)
minus 2(1minus120573)21198991 (121205962) + [12 (1198961 minus 1198921)2](12)(120573+1)
= minus11989811198811(12)(120572+1) minus 2(1minus120573)211989911198811(12)(120573+1)
(15)
Thus it follows from Lemma 2 where 0 lt 120572 lt 1 120573 gt 1
4 Mathematical Problems in Engineering
50 100 150 200t (s)
minus3
minus2
minus1
0
1
2
3
(a)
50 100 150 200
minus2
minus1
0
1
2
t (s)
(b)
Figure 2The chaotic state trajectory of the system (1) without control (a) The timing diagram of 120590 (b)The timing diagram of 120596
0 2 4 6 8 10 12 14 16 18 20
Time
Syste
m v
aria
bles
minus02
minus015
minus01
minus005
0
005
01
015
02
025
03
(a)
0 2 4 6 8 10 12 14 16 18 20Tim
k1
e
05
045
04
035
03
025
015
01
005
0
Tuni
ng p
aram
eter
k1
(b)
Figure 3 (a)The evolution of the controlled state variables 120590 and 120596 (b) Tuning parameter 1198961
The system (12) will be stable in fixed time 1198791 where1198791 le 11198981 (1 minus (120572 + 1) 2)
+ 12(1minus120573)21198991 ((120573 + 1) 2 minus 1)= 21198981 (1 minus 120572) +
2(120573+1)21198991 (120573 minus 1)
(16)
when 119905 ge 1198791 120596 equiv 0 1198961 = 1198921Therefore based on the adaptivecontrol strategy the chaotic system will be stable for a fixedtime
32 Numerical Examples In this section numerical resultsare performed to demonstrate the effectiveness and practica-bility of the scheme We give 119911 = 08119867119911 119875120585 = 2119882 119875max =100119882119867 = 100119896119892sdot1198982119863 = 40119873sdot119898sdot119904119903119886119889119891 = 1119867119911120572 = 08
120573 = 11 1198961(0) = 0 and the initial value of the variables is tobe 120590 120596 = 12058715 0
Figures 2(a) and 2(b) are the response chaotic trajectoriesof the systemvariables120590 and120596when the second order chaoticoscillation system is not controlled From Figure 2 it is clearthat the state of the system variables 120596 and 120590 is aperiodic andis always in a state of instability
Figure 3(a) is the response curves of the system variables120590 and 120596 after the fixed-time controllers 1199061 are applied to thesecond order chaotic oscillation system Figure 3(b) showsthe evolution of tuning parameter of terminal attractors1198961 From Figures 2 and 3(a) it is clear that when thesystem is chaotic before 1 s without the controller once thecontroller is applied the variables 120590 and120596 become stable aftertransition process 42 s simultaneously The adaptive fixed-time controller synchronizes the chaotic system to a stablestate without chaos and the system eventually converges tothe originThus the simulation results show that the designedcontroller is feasible and achieves the desired effect
Mathematical Problems in Engineering 5
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
nonlinear optimalfinite-timefixed-time
Figure 4 Comparison of dynamic response of different controlmodes with 120572 = 08 and 120573 = 11
To ensure a fair comparison with other control methodswe set constant initial conditions and tuning parameter fthe terminal attractor 1198961 and investigated 120596 convergence forfixed-time control finite-time control and nonlinear optimalcontrol numerically As shown in Figure 4 convergence timefor the proposed fixed-time controller is significantly lessthan that for the other controllers The system state variable120596 achieves stable state at 42 s under the action of fixed-timecontrollers However the system state variable 120596 achievesstable state at 5 s and 53 s under the action of finite-timecontroller and nonlinear optimal control respectively Thusthe fixed-time method has the better capacity to handle anonlinear system in a short time
To explore the relationship between the convergence timeand the values of the parameters 120572 and 120573 experimentally weselect system state variable 120596 to demonstrate the convergencetime Figures 5(a) and 5(b) are respectively convergence for120596 that increases with increasing 120572 and 120573 parameter valuesunder the fixed-time controller In Figure 5(a) the parametervalues are 120573 = 18 and 120572 = 03 05 07 09 In Figure 5(b)the parameter values are 120572 = 08 and 120573 = 11 13 15 17In Figures 5(a) and 5(b) the system parameters and othercontroller parameters and tuning parameters of the terminalattractor are consistent with the previous sections Thesimulation results clearly show that changing the controllerparameters 120572 and 120573 can change the time of the system statevariable 120596 to reach the steady state And the smaller the 120572and 120573 values of the system are the faster the convergencetime will be Moreover the influence of 120572 on the convergencetime of the system state variable 120596 is more than the influenceof 120573 on it The simulation results are consistent with thetheoretical analysis of the maximum stable time 1198791 of thesystem in the previous section Thus the values of 120572 and 120573also affect the stability value of the system state That is tosay we can get the size of the system state variables to thenumerical value we need by controlling the size of 120572 and120573
4 Fixed-Time Synchronization andParameters Identification
In the long-termoperation of the grid changes in theworkingenvironment will affect the parameters of the grid modelTherefore it is very important to identify the parameters ofthe chaotic model of the power grid Without consideringthe effects of higher harmonic disturbances electromagneticinterference etc according to the previous analysis theuncertain response system is given as follows
(119905) = 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867
minus 120575119867 cos 119911119905 sin120590 (119905)= 119886 sin 120590 minus 119887120596 + 119888 minus 119889 cos 119911119905 sin 120590
(17)
The following describes the identification of unknownparameters in the model
Theorem 6 Construct a synchronization parameter observer[31]
120596 = 119886 sin 120590 minus 120596 + 119888 minus 119889 cos 119911119905 sin 120590 + 1199062 (18)
Take the controller
1199062 = minus11989622119890 minus 1198962 sign (119890) |119890|120572 minus 1198962 sign (119890) |119890|120573 (19)
and the following adaptive law of parameters
119886 = minusℎ1119890 sin 119909119887 = minusℎ2119890119910119888 = minusℎ3119890119889 = minusℎ4119890 cos 119911119905 sin 119909
(20)
where 119890 = minus 120596 is the synchronization error between theobserver state variable and the original system state variableℎ119894 (119894 = 1 2 3 4) is the arbitrary positive constant 1198962 = |119890|120572+1+|119890|120573+1minus(1198962minus1198922)120572minus(1198962minus1198922)120573 is tuning parameter feedback gainand 1198922 is the arbitrary positive constant The meaning of ldquo119910rdquo isthe system state variable 120596 the meaning of ldquo119909rdquo is the systemstate variable 120590 Then the state variables in the observer (18)can be synchronized with the state variables in the system (17)and the unknown parameters 119886 119887 119888 119889 can be identified
Proof The error system can be obtained from (18)ndash(20)
119890 = 1198901 sin 119909 + 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11990621198901 = minusℎ1119890 sin 1199091198902 = minusℎ21198901199101198903 = minusℎ31198901198904 = minusℎ4119890 cos 119911119905 sin 119909
(21)
6 Mathematical Problems in Engineering
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=03
=05
=07
=09
(a)
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=11
=13
=15
=17
(b)
Figure 5 The effect of the control parameter on the response speed of the variable 120596 (a) The variations of 120596 with 120573 = 18 and 120572 =03 05 07 09 (b) The variations of 120596 with 120572 = 08 and 120573 = 11 13 15 17
where 119890119895 = (119895 = 1 2 3 4) is the error between the estimatedand the actual value of the unknown parameter
We select the Lyapunov candidate function
1198812 (119905) = 121198902 +4sum119895=1
12ℎ119895 1198902119895 + 12 (1198962 minus 1198922)2 (22)
The derivative of the subsystem trajectory in (22) can beobtained
2 (119905) = 119890 119890 + 4sum119895=1
1ℎ119895 119890119895 119890119895 + (1198962 minus 1198922) 1198962 = 119890 (1198901 sin 119909+ 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11989622119890 minus 1198962 sign (119890) |119890|120572minus 1198962 sign (119890) |119890|120573) + 1ℎ1 1198901 (minusℎ1119890 sin 119909) +
1ℎ2sdot 1198902 (minusℎ2119890119910) + 1ℎ3 1198903 (minusℎ3119890) +
1ℎ4sdot 1198904 (minusℎ4119890 cos 119911119905 sin 119909) + (1198962 minus 1198922) [|119890|120572+1 + |119890|120573+1minus (1198962 minus 1198922)120572 minus (1198962 minus 1198922)120573] le minus1198922 |119890|120572+1minus 1198922 |119890|120573+1 minus (1198962 minus 1198922)120572+1 minus (1198962 minus 1198922)120573+1le minus1198982 (121198902)
(12)(120572+1)
+ [12 (1198962 minus 1198922)2](12)(120572+1) minus 1198992 (121198902)
(12)(120573+1)
+ [12 (1198962 minus 1198922)2](12)(120573+1)
(23)
where 1198982 = min2(12)(120572+1)1198922 2(12)(120572+1) 1198992 =min2(12)(120573+1)1198922 2(12)(120573+1) Thus it follows from Lemma 4that
2 (119905)le minus1198982 (121198902) + [12 (1198962 minus 1198922)2]
(12)(120572+1)
minus 2(1minus120573)21198992 (121198902) + [12 (1198962 minus 1198922)2](12)(120573+1)
= minus11989821198812(12)(120572+1) minus 2(1minus120573)211989921198812(12)(120573+1)
(24)
We can obtain stable time of the error system (21) byLemma 3
1198792 le 11198982 (1 minus (120572 + 1) 2)+ 12(1minus120573)21198992 ((120573 + 1) 2 minus 1)
= 21198982 (1 minus 120572) +2(120573+1)21198992 (120573 minus 1)
(25)
which means that 119890 equiv 0 1198962 = 1198922 119886 = 119886 = 119887 119888 = 119888 119889 = 119889when 119905 ge 1198792
Then we compare the parameter identification methodbased on the fixed-time synchronization observer with theparameter identification method which did not introduce thefixed-time control If given 120590 = 12058715 120596 = 0 ℎ1 = ℎ2 =ℎ3 = ℎ4 = 50 120572 = 08 120573 = 11 the simulation results of theidentification method are shown in Figure 6The blue dashedline indicates the identification curve of the synchronizationparameter observer (SPO) and the red solid line represents
Mathematical Problems in Engineering 7
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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Submit your manuscripts atwwwhindawicom
4 Mathematical Problems in Engineering
50 100 150 200t (s)
minus3
minus2
minus1
0
1
2
3
(a)
50 100 150 200
minus2
minus1
0
1
2
t (s)
(b)
Figure 2The chaotic state trajectory of the system (1) without control (a) The timing diagram of 120590 (b)The timing diagram of 120596
0 2 4 6 8 10 12 14 16 18 20
Time
Syste
m v
aria
bles
minus02
minus015
minus01
minus005
0
005
01
015
02
025
03
(a)
0 2 4 6 8 10 12 14 16 18 20Tim
k1
e
05
045
04
035
03
025
015
01
005
0
Tuni
ng p
aram
eter
k1
(b)
Figure 3 (a)The evolution of the controlled state variables 120590 and 120596 (b) Tuning parameter 1198961
The system (12) will be stable in fixed time 1198791 where1198791 le 11198981 (1 minus (120572 + 1) 2)
+ 12(1minus120573)21198991 ((120573 + 1) 2 minus 1)= 21198981 (1 minus 120572) +
2(120573+1)21198991 (120573 minus 1)
(16)
when 119905 ge 1198791 120596 equiv 0 1198961 = 1198921Therefore based on the adaptivecontrol strategy the chaotic system will be stable for a fixedtime
32 Numerical Examples In this section numerical resultsare performed to demonstrate the effectiveness and practica-bility of the scheme We give 119911 = 08119867119911 119875120585 = 2119882 119875max =100119882119867 = 100119896119892sdot1198982119863 = 40119873sdot119898sdot119904119903119886119889119891 = 1119867119911120572 = 08
120573 = 11 1198961(0) = 0 and the initial value of the variables is tobe 120590 120596 = 12058715 0
Figures 2(a) and 2(b) are the response chaotic trajectoriesof the systemvariables120590 and120596when the second order chaoticoscillation system is not controlled From Figure 2 it is clearthat the state of the system variables 120596 and 120590 is aperiodic andis always in a state of instability
Figure 3(a) is the response curves of the system variables120590 and 120596 after the fixed-time controllers 1199061 are applied to thesecond order chaotic oscillation system Figure 3(b) showsthe evolution of tuning parameter of terminal attractors1198961 From Figures 2 and 3(a) it is clear that when thesystem is chaotic before 1 s without the controller once thecontroller is applied the variables 120590 and120596 become stable aftertransition process 42 s simultaneously The adaptive fixed-time controller synchronizes the chaotic system to a stablestate without chaos and the system eventually converges tothe originThus the simulation results show that the designedcontroller is feasible and achieves the desired effect
Mathematical Problems in Engineering 5
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
nonlinear optimalfinite-timefixed-time
Figure 4 Comparison of dynamic response of different controlmodes with 120572 = 08 and 120573 = 11
To ensure a fair comparison with other control methodswe set constant initial conditions and tuning parameter fthe terminal attractor 1198961 and investigated 120596 convergence forfixed-time control finite-time control and nonlinear optimalcontrol numerically As shown in Figure 4 convergence timefor the proposed fixed-time controller is significantly lessthan that for the other controllers The system state variable120596 achieves stable state at 42 s under the action of fixed-timecontrollers However the system state variable 120596 achievesstable state at 5 s and 53 s under the action of finite-timecontroller and nonlinear optimal control respectively Thusthe fixed-time method has the better capacity to handle anonlinear system in a short time
To explore the relationship between the convergence timeand the values of the parameters 120572 and 120573 experimentally weselect system state variable 120596 to demonstrate the convergencetime Figures 5(a) and 5(b) are respectively convergence for120596 that increases with increasing 120572 and 120573 parameter valuesunder the fixed-time controller In Figure 5(a) the parametervalues are 120573 = 18 and 120572 = 03 05 07 09 In Figure 5(b)the parameter values are 120572 = 08 and 120573 = 11 13 15 17In Figures 5(a) and 5(b) the system parameters and othercontroller parameters and tuning parameters of the terminalattractor are consistent with the previous sections Thesimulation results clearly show that changing the controllerparameters 120572 and 120573 can change the time of the system statevariable 120596 to reach the steady state And the smaller the 120572and 120573 values of the system are the faster the convergencetime will be Moreover the influence of 120572 on the convergencetime of the system state variable 120596 is more than the influenceof 120573 on it The simulation results are consistent with thetheoretical analysis of the maximum stable time 1198791 of thesystem in the previous section Thus the values of 120572 and 120573also affect the stability value of the system state That is tosay we can get the size of the system state variables to thenumerical value we need by controlling the size of 120572 and120573
4 Fixed-Time Synchronization andParameters Identification
In the long-termoperation of the grid changes in theworkingenvironment will affect the parameters of the grid modelTherefore it is very important to identify the parameters ofthe chaotic model of the power grid Without consideringthe effects of higher harmonic disturbances electromagneticinterference etc according to the previous analysis theuncertain response system is given as follows
(119905) = 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867
minus 120575119867 cos 119911119905 sin120590 (119905)= 119886 sin 120590 minus 119887120596 + 119888 minus 119889 cos 119911119905 sin 120590
(17)
The following describes the identification of unknownparameters in the model
Theorem 6 Construct a synchronization parameter observer[31]
120596 = 119886 sin 120590 minus 120596 + 119888 minus 119889 cos 119911119905 sin 120590 + 1199062 (18)
Take the controller
1199062 = minus11989622119890 minus 1198962 sign (119890) |119890|120572 minus 1198962 sign (119890) |119890|120573 (19)
and the following adaptive law of parameters
119886 = minusℎ1119890 sin 119909119887 = minusℎ2119890119910119888 = minusℎ3119890119889 = minusℎ4119890 cos 119911119905 sin 119909
(20)
where 119890 = minus 120596 is the synchronization error between theobserver state variable and the original system state variableℎ119894 (119894 = 1 2 3 4) is the arbitrary positive constant 1198962 = |119890|120572+1+|119890|120573+1minus(1198962minus1198922)120572minus(1198962minus1198922)120573 is tuning parameter feedback gainand 1198922 is the arbitrary positive constant The meaning of ldquo119910rdquo isthe system state variable 120596 the meaning of ldquo119909rdquo is the systemstate variable 120590 Then the state variables in the observer (18)can be synchronized with the state variables in the system (17)and the unknown parameters 119886 119887 119888 119889 can be identified
Proof The error system can be obtained from (18)ndash(20)
119890 = 1198901 sin 119909 + 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11990621198901 = minusℎ1119890 sin 1199091198902 = minusℎ21198901199101198903 = minusℎ31198901198904 = minusℎ4119890 cos 119911119905 sin 119909
(21)
6 Mathematical Problems in Engineering
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=03
=05
=07
=09
(a)
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=11
=13
=15
=17
(b)
Figure 5 The effect of the control parameter on the response speed of the variable 120596 (a) The variations of 120596 with 120573 = 18 and 120572 =03 05 07 09 (b) The variations of 120596 with 120572 = 08 and 120573 = 11 13 15 17
where 119890119895 = (119895 = 1 2 3 4) is the error between the estimatedand the actual value of the unknown parameter
We select the Lyapunov candidate function
1198812 (119905) = 121198902 +4sum119895=1
12ℎ119895 1198902119895 + 12 (1198962 minus 1198922)2 (22)
The derivative of the subsystem trajectory in (22) can beobtained
2 (119905) = 119890 119890 + 4sum119895=1
1ℎ119895 119890119895 119890119895 + (1198962 minus 1198922) 1198962 = 119890 (1198901 sin 119909+ 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11989622119890 minus 1198962 sign (119890) |119890|120572minus 1198962 sign (119890) |119890|120573) + 1ℎ1 1198901 (minusℎ1119890 sin 119909) +
1ℎ2sdot 1198902 (minusℎ2119890119910) + 1ℎ3 1198903 (minusℎ3119890) +
1ℎ4sdot 1198904 (minusℎ4119890 cos 119911119905 sin 119909) + (1198962 minus 1198922) [|119890|120572+1 + |119890|120573+1minus (1198962 minus 1198922)120572 minus (1198962 minus 1198922)120573] le minus1198922 |119890|120572+1minus 1198922 |119890|120573+1 minus (1198962 minus 1198922)120572+1 minus (1198962 minus 1198922)120573+1le minus1198982 (121198902)
(12)(120572+1)
+ [12 (1198962 minus 1198922)2](12)(120572+1) minus 1198992 (121198902)
(12)(120573+1)
+ [12 (1198962 minus 1198922)2](12)(120573+1)
(23)
where 1198982 = min2(12)(120572+1)1198922 2(12)(120572+1) 1198992 =min2(12)(120573+1)1198922 2(12)(120573+1) Thus it follows from Lemma 4that
2 (119905)le minus1198982 (121198902) + [12 (1198962 minus 1198922)2]
(12)(120572+1)
minus 2(1minus120573)21198992 (121198902) + [12 (1198962 minus 1198922)2](12)(120573+1)
= minus11989821198812(12)(120572+1) minus 2(1minus120573)211989921198812(12)(120573+1)
(24)
We can obtain stable time of the error system (21) byLemma 3
1198792 le 11198982 (1 minus (120572 + 1) 2)+ 12(1minus120573)21198992 ((120573 + 1) 2 minus 1)
= 21198982 (1 minus 120572) +2(120573+1)21198992 (120573 minus 1)
(25)
which means that 119890 equiv 0 1198962 = 1198922 119886 = 119886 = 119887 119888 = 119888 119889 = 119889when 119905 ge 1198792
Then we compare the parameter identification methodbased on the fixed-time synchronization observer with theparameter identification method which did not introduce thefixed-time control If given 120590 = 12058715 120596 = 0 ℎ1 = ℎ2 =ℎ3 = ℎ4 = 50 120572 = 08 120573 = 11 the simulation results of theidentification method are shown in Figure 6The blue dashedline indicates the identification curve of the synchronizationparameter observer (SPO) and the red solid line represents
Mathematical Problems in Engineering 7
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
nonlinear optimalfinite-timefixed-time
Figure 4 Comparison of dynamic response of different controlmodes with 120572 = 08 and 120573 = 11
To ensure a fair comparison with other control methodswe set constant initial conditions and tuning parameter fthe terminal attractor 1198961 and investigated 120596 convergence forfixed-time control finite-time control and nonlinear optimalcontrol numerically As shown in Figure 4 convergence timefor the proposed fixed-time controller is significantly lessthan that for the other controllers The system state variable120596 achieves stable state at 42 s under the action of fixed-timecontrollers However the system state variable 120596 achievesstable state at 5 s and 53 s under the action of finite-timecontroller and nonlinear optimal control respectively Thusthe fixed-time method has the better capacity to handle anonlinear system in a short time
To explore the relationship between the convergence timeand the values of the parameters 120572 and 120573 experimentally weselect system state variable 120596 to demonstrate the convergencetime Figures 5(a) and 5(b) are respectively convergence for120596 that increases with increasing 120572 and 120573 parameter valuesunder the fixed-time controller In Figure 5(a) the parametervalues are 120573 = 18 and 120572 = 03 05 07 09 In Figure 5(b)the parameter values are 120572 = 08 and 120573 = 11 13 15 17In Figures 5(a) and 5(b) the system parameters and othercontroller parameters and tuning parameters of the terminalattractor are consistent with the previous sections Thesimulation results clearly show that changing the controllerparameters 120572 and 120573 can change the time of the system statevariable 120596 to reach the steady state And the smaller the 120572and 120573 values of the system are the faster the convergencetime will be Moreover the influence of 120572 on the convergencetime of the system state variable 120596 is more than the influenceof 120573 on it The simulation results are consistent with thetheoretical analysis of the maximum stable time 1198791 of thesystem in the previous section Thus the values of 120572 and 120573also affect the stability value of the system state That is tosay we can get the size of the system state variables to thenumerical value we need by controlling the size of 120572 and120573
4 Fixed-Time Synchronization andParameters Identification
In the long-termoperation of the grid changes in theworkingenvironment will affect the parameters of the grid modelTherefore it is very important to identify the parameters ofthe chaotic model of the power grid Without consideringthe effects of higher harmonic disturbances electromagneticinterference etc according to the previous analysis theuncertain response system is given as follows
(119905) = 120596 (119905) (119905) = minus119875max119867 sin 120590 (119905) minus 119863119867120596 (119905) + 119875119898119867
minus 120575119867 cos 119911119905 sin120590 (119905)= 119886 sin 120590 minus 119887120596 + 119888 minus 119889 cos 119911119905 sin 120590
(17)
The following describes the identification of unknownparameters in the model
Theorem 6 Construct a synchronization parameter observer[31]
120596 = 119886 sin 120590 minus 120596 + 119888 minus 119889 cos 119911119905 sin 120590 + 1199062 (18)
Take the controller
1199062 = minus11989622119890 minus 1198962 sign (119890) |119890|120572 minus 1198962 sign (119890) |119890|120573 (19)
and the following adaptive law of parameters
119886 = minusℎ1119890 sin 119909119887 = minusℎ2119890119910119888 = minusℎ3119890119889 = minusℎ4119890 cos 119911119905 sin 119909
(20)
where 119890 = minus 120596 is the synchronization error between theobserver state variable and the original system state variableℎ119894 (119894 = 1 2 3 4) is the arbitrary positive constant 1198962 = |119890|120572+1+|119890|120573+1minus(1198962minus1198922)120572minus(1198962minus1198922)120573 is tuning parameter feedback gainand 1198922 is the arbitrary positive constant The meaning of ldquo119910rdquo isthe system state variable 120596 the meaning of ldquo119909rdquo is the systemstate variable 120590 Then the state variables in the observer (18)can be synchronized with the state variables in the system (17)and the unknown parameters 119886 119887 119888 119889 can be identified
Proof The error system can be obtained from (18)ndash(20)
119890 = 1198901 sin 119909 + 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11990621198901 = minusℎ1119890 sin 1199091198902 = minusℎ21198901199101198903 = minusℎ31198901198904 = minusℎ4119890 cos 119911119905 sin 119909
(21)
6 Mathematical Problems in Engineering
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=03
=05
=07
=09
(a)
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=11
=13
=15
=17
(b)
Figure 5 The effect of the control parameter on the response speed of the variable 120596 (a) The variations of 120596 with 120573 = 18 and 120572 =03 05 07 09 (b) The variations of 120596 with 120572 = 08 and 120573 = 11 13 15 17
where 119890119895 = (119895 = 1 2 3 4) is the error between the estimatedand the actual value of the unknown parameter
We select the Lyapunov candidate function
1198812 (119905) = 121198902 +4sum119895=1
12ℎ119895 1198902119895 + 12 (1198962 minus 1198922)2 (22)
The derivative of the subsystem trajectory in (22) can beobtained
2 (119905) = 119890 119890 + 4sum119895=1
1ℎ119895 119890119895 119890119895 + (1198962 minus 1198922) 1198962 = 119890 (1198901 sin 119909+ 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11989622119890 minus 1198962 sign (119890) |119890|120572minus 1198962 sign (119890) |119890|120573) + 1ℎ1 1198901 (minusℎ1119890 sin 119909) +
1ℎ2sdot 1198902 (minusℎ2119890119910) + 1ℎ3 1198903 (minusℎ3119890) +
1ℎ4sdot 1198904 (minusℎ4119890 cos 119911119905 sin 119909) + (1198962 minus 1198922) [|119890|120572+1 + |119890|120573+1minus (1198962 minus 1198922)120572 minus (1198962 minus 1198922)120573] le minus1198922 |119890|120572+1minus 1198922 |119890|120573+1 minus (1198962 minus 1198922)120572+1 minus (1198962 minus 1198922)120573+1le minus1198982 (121198902)
(12)(120572+1)
+ [12 (1198962 minus 1198922)2](12)(120572+1) minus 1198992 (121198902)
(12)(120573+1)
+ [12 (1198962 minus 1198922)2](12)(120573+1)
(23)
where 1198982 = min2(12)(120572+1)1198922 2(12)(120572+1) 1198992 =min2(12)(120573+1)1198922 2(12)(120573+1) Thus it follows from Lemma 4that
2 (119905)le minus1198982 (121198902) + [12 (1198962 minus 1198922)2]
(12)(120572+1)
minus 2(1minus120573)21198992 (121198902) + [12 (1198962 minus 1198922)2](12)(120573+1)
= minus11989821198812(12)(120572+1) minus 2(1minus120573)211989921198812(12)(120573+1)
(24)
We can obtain stable time of the error system (21) byLemma 3
1198792 le 11198982 (1 minus (120572 + 1) 2)+ 12(1minus120573)21198992 ((120573 + 1) 2 minus 1)
= 21198982 (1 minus 120572) +2(120573+1)21198992 (120573 minus 1)
(25)
which means that 119890 equiv 0 1198962 = 1198922 119886 = 119886 = 119887 119888 = 119888 119889 = 119889when 119905 ge 1198792
Then we compare the parameter identification methodbased on the fixed-time synchronization observer with theparameter identification method which did not introduce thefixed-time control If given 120590 = 12058715 120596 = 0 ℎ1 = ℎ2 =ℎ3 = ℎ4 = 50 120572 = 08 120573 = 11 the simulation results of theidentification method are shown in Figure 6The blue dashedline indicates the identification curve of the synchronizationparameter observer (SPO) and the red solid line represents
Mathematical Problems in Engineering 7
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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
6 Mathematical Problems in Engineering
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=03
=05
=07
=09
(a)
0 2 4 6 8 10 12 14 16 18 20
Time
004
002
0
minus002
minus004
minus006
minus008
minus01
minus012
minus014
Syste
m v
aria
ble
=11
=13
=15
=17
(b)
Figure 5 The effect of the control parameter on the response speed of the variable 120596 (a) The variations of 120596 with 120573 = 18 and 120572 =03 05 07 09 (b) The variations of 120596 with 120572 = 08 and 120573 = 11 13 15 17
where 119890119895 = (119895 = 1 2 3 4) is the error between the estimatedand the actual value of the unknown parameter
We select the Lyapunov candidate function
1198812 (119905) = 121198902 +4sum119895=1
12ℎ119895 1198902119895 + 12 (1198962 minus 1198922)2 (22)
The derivative of the subsystem trajectory in (22) can beobtained
2 (119905) = 119890 119890 + 4sum119895=1
1ℎ119895 119890119895 119890119895 + (1198962 minus 1198922) 1198962 = 119890 (1198901 sin 119909+ 1198902119910 + 1198903 + 1198904 cos 119911119905 sin 119909 + 11989622119890 minus 1198962 sign (119890) |119890|120572minus 1198962 sign (119890) |119890|120573) + 1ℎ1 1198901 (minusℎ1119890 sin 119909) +
1ℎ2sdot 1198902 (minusℎ2119890119910) + 1ℎ3 1198903 (minusℎ3119890) +
1ℎ4sdot 1198904 (minusℎ4119890 cos 119911119905 sin 119909) + (1198962 minus 1198922) [|119890|120572+1 + |119890|120573+1minus (1198962 minus 1198922)120572 minus (1198962 minus 1198922)120573] le minus1198922 |119890|120572+1minus 1198922 |119890|120573+1 minus (1198962 minus 1198922)120572+1 minus (1198962 minus 1198922)120573+1le minus1198982 (121198902)
(12)(120572+1)
+ [12 (1198962 minus 1198922)2](12)(120572+1) minus 1198992 (121198902)
(12)(120573+1)
+ [12 (1198962 minus 1198922)2](12)(120573+1)
(23)
where 1198982 = min2(12)(120572+1)1198922 2(12)(120572+1) 1198992 =min2(12)(120573+1)1198922 2(12)(120573+1) Thus it follows from Lemma 4that
2 (119905)le minus1198982 (121198902) + [12 (1198962 minus 1198922)2]
(12)(120572+1)
minus 2(1minus120573)21198992 (121198902) + [12 (1198962 minus 1198922)2](12)(120573+1)
= minus11989821198812(12)(120572+1) minus 2(1minus120573)211989921198812(12)(120573+1)
(24)
We can obtain stable time of the error system (21) byLemma 3
1198792 le 11198982 (1 minus (120572 + 1) 2)+ 12(1minus120573)21198992 ((120573 + 1) 2 minus 1)
= 21198982 (1 minus 120572) +2(120573+1)21198992 (120573 minus 1)
(25)
which means that 119890 equiv 0 1198962 = 1198922 119886 = 119886 = 119887 119888 = 119888 119889 = 119889when 119905 ge 1198792
Then we compare the parameter identification methodbased on the fixed-time synchronization observer with theparameter identification method which did not introduce thefixed-time control If given 120590 = 12058715 120596 = 0 ℎ1 = ℎ2 =ℎ3 = ℎ4 = 50 120572 = 08 120573 = 11 the simulation results of theidentification method are shown in Figure 6The blue dashedline indicates the identification curve of the synchronizationparameter observer (SPO) and the red solid line represents
Mathematical Problems in Engineering 7
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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
6
4
2
0
minus2
minus40 50 100 150 200
ga
t
FSPOSPO
(a)ga
t
minus09996
minus09998
minus10000
minus10002
minus10004
minus10006
minus10008
300 350 400
FSPOSPO
(b)
6
4
2
0
minus2
minus4
gb
0 50 100 150 200
t
FSPOSPO
(c)
gb
t300 350 400
FSPOSPO
minus03992
minus03994
minus03996
minus03998
minus04000
minus04002
minus04004
(d)
6
4
2
0
minus2
minus4
gc
50 0 100 150 200
t
FSPOSPO
(e)
gc
t300 350 400
FSPOSPO
02010
02005
02000
01995
01990
(f)6
4
2
0
minus2
gd
0 50 100 150 200
t
FSPOSPO
(g)
gd
t300 350 400
FSPOSPO
minus12790
minus12795
minus12800
minus12805
(h)
Figure 6 Identifications of the uncertain parameters 119886 119887 119888 and 119889 (a) The identification of parameter a (b) The partial magnification ofFigure (a) (c) The identification of parameter b (d) The partial magnification of Figure (c) (e) The identification of parameter c (f) Thepartial magnification of Figure (e) (g)The identification of parameter d (h) The partial magnification of Figure (g)
the identification curve of the fixed-time synchronizationparameter observer (FSPO)
To measure the dynamic and static performance of theidentification system we define plusmn3 of the identified targetvalue to identify the stable area As shown in the experimentalsimulation we observe that both methods can accuratelyidentify the parameters to the target value of 119886 = minus1 119887 = minus04119888 = 02 119889 = minus128
There are obvious differences between the rapidity andstability of identification The parameters a b c and dare driven to the target value for a long time the over-shoot of identification curve is large with obvious chatteringphenomenon in the stable region by SPO However thetarget value of FSPO approach is faster and the overshoot
of identification curve is less with no obvious chatteringphenomenon basically The data of adjustment time andabsolute error in the stable area is shown in Table 1
In summary compared with the parameter identificationbased on synchronous observer the FSPO has the followingadvantages
The speed of identification is accelerated and the sta-bility of the system is improved without obvious overshootand chattering phenomenon Multiple Object synchronousidentification is realized and the dynamic and static per-formance of the identification system are good But theparameter identification time is still long so we are studyingmore superior methods to optimize parameter identifica-tion
8 Mathematical Problems in Engineering
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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
Table 1 Performance comparison of SPO and FSPO
Identification parameters Adjustment time ts Absolute ErrorSPO FSPO SP0 FSPO
a 12802 7250 006 lt001b 15741 8697 010 lt001c 19392 9488 010 lt001d 16644 8573 010 lt001
5 Conclusions
In this paper we have studied the adaptive fixed-time stabilityof chaotic oscillation in second order power system Anefficient adaptive controller is proposed which can stabilizethe chaotic oscillation in second order power system in fixedtime Then the uncertain parameters can be identified infixed time by synchronous observer with adaptive law ofparametersThe simulation results show the effectiveness andfeasibility of the proposed method In the high-order systemswith interference the convergence time of the proposedmethod may be slow but it is still superior to other methodsThe improved method is being studied to solve this problemby us In addition considering that noise perturbation isubiquitous we will investigate a methodology to solve noiseperturbation in our future work and further research thefixed-time stability of fourth-order power system
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare no conflicts of interest
Authorsrsquo Contributions
CaoyuanMaandWenbeiWu contributed equally to thisworkand should be considered co-first authors
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (71573256) and National Key Researchand Development Plan of China (2017YFC0804408)
References
[1] N Kopell and J Washburn ldquoChaotic motions in the two-degree-of-freedom swing equationsrdquo Institute of Electrical andElectronics Engineers Transactions on Circuits and Systems vol29 no 11 pp 738ndash746 1982
[2] E Ott C Grebogi and J A Yorke ldquoControlling chaosrdquo PhysicalReview Letters vol 64 no 11 pp 1196ndash1199 1990
[3] B H Wang Q Zhang C W Yang and W Yang Chaoticoscillation control of electric power system based on adaptivebackstepping Electric Power Automation Equipment
[4] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994
[5] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005
[6] FMin YWangG Peng EWang and J A Auth ldquoBifurcationschaos and adaptive backstepping sliding mode control of apower system with excitation limitationrdquo AIP Advances vol 6no 8 2016
[7] M M Zirkohi T Kumbasar and T-C Lin ldquoHybrid adaptivetype-2 fuzzy tracking control of chaotic oscillation damping ofpower systemsrdquo Asian Journal of Control vol 19 no 3 pp 1114ndash1125 2017
[8] C P Uzunoglu Y Babacan F Kacar and M Ugur ldquoModelingand Suppression of Chaotic Ferroresonance in a Power Systemby Using Memristor-based Systemrdquo Electric Power Componentsand Systems vol 44 no 6 pp 638ndash645 2016
[9] X Li and C A Canizares ldquoChaotic behavior observations ina power system modelrdquo in Proceedings of the IEEE BucharestPowerTech Innovative Ideas Toward the Electrical Grid of theFuture Bucharest Romania July 2009
[10] D Huang ldquoAdaptive-feedback control algorithmrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 73no 6 2006
[11] H Mokayed and A H Mohamed ldquoA robust thresholdingtechnique for generic structured document classifier using ordi-nal structure fuzzy logicrdquo International Journal of InnovativeComputing Information and Control vol 10 no 4 pp 1543ndash1554 2014
[12] H Zhao Y Ma S Liu and Y Yue ldquoFuzzy sliding mode variablestructure control of chaotic power system with uncertaintyrdquoJournal of Computational Information Systems vol 7 no 6 pp1959ndash1966 2011
[13] L Yuan K-Y Wei B-X Hu and N Wang ldquoNonsingularterminal sliding-mode controller with nonlinear disturbanceobserver for chaotic oscillation in power systemrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 3316ndash3320 July 2016
[14] Y Sun L Zhao M Liu W Weng and Q Meng ldquoFinite-timeleader-following consensus problem of multi-agent systemsrdquoBjog An International Journal of Obstetrics amp Gynaecology vol119 no 1 pp 6043ndash6046 2012
[15] Y Sun F LiuW Li andH Shi ldquoFinite-timeflocking of Cucker-Smale systemsrdquo in Proceedings of the 34th Chinese ControlConference CCC 2015 pp 7016ndash7020 China July 2015
[16] Y Dong and F Yang ldquoFinite-time stability and boundedness ofswitched nonlinear time-delay systems under state-dependentswitchingrdquo Complexity vol 21 no 2 pp 267ndash275 2015
Mathematical Problems in Engineering 9
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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
[17] T Jing and F Chen ldquoFinite-time lag synchronization of delayedneural networks via periodically intermittent controlrdquo Com-plexity vol 21 no S1 pp 211ndash219 2016
[18] K Mathiyalagan and K Balachandran ldquoFinite-time stability offractional-order stochastic singular systemswith time delay andwhite noiserdquo Complexity vol 21 no S2 pp 370ndash379 2016
[19] H Zhao Y-J Ma S-J Liu S-G Gao and D Zhong ldquoControl-ling chaos in power systembased on finite-time stability theoryrdquoChinese Physics B vol 20 no 12 Article ID 120501 2011
[20] N Cai W Li and Y Jing ldquoFinite-time generalized synchro-nization of chaotic systems with different orderrdquo NonlinearDynamics vol 64 no 4 pp 385ndash393 2011
[21] A Polyakov ldquoNonlinear feedback design for fixed-time sta-bilization of linear control systemsrdquo IEEE Transactions onAutomatic Control vol 57 no 8 pp 2106ndash2110 2012
[22] 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
[23] 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
[24] J Fu and JWang ldquoFixed-time coordinated tracking for second-order multi-agent systems with bounded input uncertaintiesrdquoSystems amp Control Letters vol 93 pp 1ndash12 2016
[25] S E Parsegov A E Polyakov and P S Shcherbakov ldquoFixed-time consensus algorithm for multi-agent systems with inte-grator dynamicsrdquo in Proceedings of the 4th IFAC Workshopon Distributed Estimation and Control in Networked SystemsNecSys 2013 pp 110ndash115 September 2013
[26] A Polyakov D Efimov and W Perruquetti ldquoFinite-time andfixed-time stabilization Implicit Lyapunov function approachrdquoAutomatica vol 51 pp 332ndash340 2015
[27] D Q Wei and X S Luo ldquoPassivity-based adaptive control ofchaotic oscillations in power systemrdquoChaos Solitonsamp Fractalsvol 31 no 3 pp 665ndash671 2007
[28] Q Zhang and B-H Wang ldquoControlling power system chaoticoscillation by time-delayed feedbackrdquoPower SystemTechnologyvol 28 no 7 pp 23ndash26 2004
[29] Y Sun X Wu L Bai Z Wei and G Sun ldquoFinite-time syn-chronization control and parameter identification of uncertainpermanentmagnet synchronousmotorrdquoNeurocomputing 2015
[30] H K Khalil Nonlinear systems Prentice-Hall Inc UpperSaddle River NJ 3rd edition 2002
[31] S M Wang C Y Yue and H G Luo ldquoIdentification ofparameters in Liursquos chaotic systems using unknown parameterobserversrdquo Journal of Huazhong University of Science and Tech-nology Natural Science Edition Huazhong Keji Daxue XuebaoZiran Kexue Ban vol 35 no 6 pp 47ndash49 2007
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
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