Design of Static Output Feedback Controller for Fractional...

Research ArticleDesign of Static Output Feedback Controller for Fractional-OrderT-S Fuzzy System

Zhile Xia

Department of Mathematics TaiZhou University ZheJiang TaiZhou 318000 China

Correspondence should be addressed to Zhile Xia zhilexia163com

Received 12 April 2020 Revised 18 May 2020 Accepted 20 May 2020 Published 15 June 2020

Guest Editor Amr Elsonbaty

Copyright copy 2020 Zhile Xia )is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)is paper studies the design of fuzzy static output feedback controllers for two kinds of fractional-order T-S fuzzy systems )efractional order α satisfies 0lt αlt 1 and 1le αlt 2 Based on the fractional order theory matrix decomposition technique andprojection theorem four new sufficient conditions for the asymptotic stability of the system and the corresponding controllerdesign methods are given All the results can be expressed by linear matrix inequalities and the relationship between fuzzysubsystems is also considered )ese have great advantages in solving the results and reducing the conservatism Finally asimulation example is given to show the effectiveness of the proposed method

1 Introduction

Compared with the integer-order system the fractional-ordersystem has many advantages for example the integer order is aspecial case of the fractional-order system the fractional-ordersystem has global properties and the system state depends notonly on the current time but also on the past time it can ac-curately describe the memory and genetic properties in physicsand engineering )erefore in recent decades fractional-ordercontrol systems have also beenwidely concerned and importantresults have been obtained For example Chen et al [1ndash3] madea comprehensive introduction and in-depth study of fractional-order systems Specifically the paper [1] considered the robuststability and stabilization of fractional-order linear systems withpolyhedral uncertainties )e results obtained are not onlyapplicable to the fractional order α satisfying 0lt αlt 1 but also tothe fractional order α satisfying 1le αlt 2 For the fractional-order neural network system with time-varying delay the lit-erature [2] solved a challenging problem the delay-dependentstability and stabilization of the fractional-order delay system)e literature [3] studied the stability and synchronization of thedelayed neural network and a series of very important resultswere given including delay-independent stability criterionmeasurable algebra criterion and synchronization criterionGallegos and Duarte-Mermoud [4] investigated the stability of

fractional-order and integral-order coupled systems in whichthe concepts of dissipativity and passivity were extended and thetheorems of small gain and passivity for correlated systems wereobtained Liu et al [5] used the fractional filtering method tostudy the backstepping controller design of the actuator faultfuzzy neural network system with completely unknown pa-rameters andmodes Combinedwith the fractional adaptive lawthe tracking error and compensation tracking error converged toa small enough area Based on Lyapunov functions and thecomparison principle Jia et al [6] designed delayed statefeedback control and coupling state feedback control for frac-tional-order memristor-based neural networks with time delayTo sum up the control problem of the fractional-order system isa hot issue at present attracting a large number of researchers toparticipate in it

In addition the fuzzy theory opens up a new way to dealwith the fuzzy uncertainty in nature with strict mathematicalmethods and proposes the method of the membershipfunction to describe the degree of the element set [7] Whenthe membership degree is 1 it means that the elementbelongs to the set completely when the membership degreeis 0 it means that the element does not belong to the set atall and when the membership degree belongs to the interval(0 1) it means that the element belongs to the set but doesnot completely belong to the set Obviously this is a

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7898109 9 pageshttpsdoiorg10115520207898109

generalization of the classical set theory Fuzzy control is basedon the fuzzy theory By using expert language fuzzy control canachieve good and effective control for some complex nonlinearcontrol objects which are difficult to establish themathematicalmodel accurately [8 9] So far two main fuzzy models havebeen proposed )e first model is based on the input-outputmodel proposed in document [9] )e disadvantage is thatthere is a lot of useful information which is not used )esecond model is the T-S fuzzy model proposed by Japanesescholars Takagi and Sugeno in 1985 [10 11] In [12ndash14] it isproved that this kind of fuzzy system has the ability of universalapproximation Some recent literature studies are listed to showthat the T-S fuzzy system has beenwidely studied For examplebased on the T-S fuzzy model in [15] the design of the fuzzyfilter for the discrete-time nonlinear networked system isstudied and a new method is proposed to ensure that thefiltering error system is dissipative by using themethod of eventtriggering and quantization For continuous-time nonlineartime-delay systems by using repetitive control Rathinasamyet al [16] proposed an improved output feedback controllerdesign method )is method ensures that the system can trackthe given reference signal within the allowable error range Liet al [17] studied the finite-time Hinfin control problem and thesufficient conditions based on LMI were given )e researchresults of the output feedback can be found in the literaturestudies [18ndash28] )ere are also a large number of referencesrelated to the T-S fuzzymodel whichwe cannot be enumeratedhere

Compared with many research results of the integer-order T-S fuzzy system the research results of the fractional-order T-S fuzzy system are less For the fractional-order T-Sfuzzy system Zheng et al [29] studied the stabilization of thechaotic system by using the adaptive method When theorder of the fractional order considered satisfies 1le αlt 2and 0lt αlt 1 respectively the stability conditions based onLMI were proposed in [30 31] Huang et al [32] studied thestabilization of the state feedback that can be applied to boththe following situations 1le αlt 2 and 0lt αlt 1 However inthe practical engineering field the state of the system is oftendifficult to obtain directly At this time the state observer oroutput feedback control methods need to be consideredBased on the T-S fuzzy model Duan et al [33] studied thedesign problem of the nonfragile observer by using singularvalue decomposition of thematrix and put forward sufficientconditions of linear matrix inequalities which guaranteethat the corresponding closed-loop system is stochasticasymptotically stable when the fuzzy antecedent variablesare unmeasurable Duan and Li [34] considered the designproblem of dynamic output feedback controllers and pro-posed sufficient conditions for the system to be asymptot-ically stable Karthick et al [35] designed a dynamic outputfeedback controller by means of interference suppressionand quantization and Lin et al and Ji et al [36 37] studiedthe design of the fuzzy static output feedback controller )edesign of the static fuzzy output feedback controller willincrease the fuzzy relation in the stabilization condition by acertain multiple because the product term of coefficientmatrices BiFjCk appears while the design state feedbackonly contains the product term of coefficient matrices

BiFj(i j k 1 2 r) where r is an integer indicating thenumber of fuzzy rules For the traditional T-S fuzzy systemChaibi et al [38] used the matrix decomposition techniqueto effectively separate the coefficient term BiFjCk and ef-fectively reduced the fuzzy relation of one layer and pro-posed the design method based on the linear matrixinequality In this paper we try to extend the research resultsof Chaibi et al [38] to the fractional-order T-S fuzzy system

In conclusion this paper studies the fuzzy static outputfeedback control of the fractional-order T-S fuzzy system)e order includes two cases 0lt αlt 1 and 1le αlt 2According to the theory of fractional differential correlationand the projection theorem several new stabilization designmethods are given All the results are expressed by the linearmatrix inequality )e main contributions of this paper areas follows (1) the control method of the fuzzy static outputfeedback is extended from the integer-order fuzzy system tothe fractional-order fuzzy system so it is more practical (2)the condition of system stabilization and controller designmethods are all expressed by LMIs )erefore LMI toolboxof Matlab can be used to program and calculate (3) outputmatrix C1 C2 Cr of the system can still be completelydifferent (4) for some similar Lyapunov matrices onlysymmetry or antisymmetry is required no further diagonalis required and stability conditions are relaxed and (5) thestabilization conditions are also considered Finally a nu-merical simulation example is given to show the effective-ness of the proposed method

Notations )roughout this paper Rn denotes thenminusdimensional Euclidean space Rntimesm is the set of all n times m

real matrices I and O represent the identity matrix and thezero matrix with appropriate dimensions Pgt 0(Plt 0)

represents the positive definite (negative definite) matrix theright superscript T denotes the transposition of the matrixthe symbol sym S means S + ST the diagonal block matrixis denoted by diag A1 A2 An1113864 1113865 and the symbol lowast showsthe symmetric part of the block matrix

2 Basic Definition and Lemmas

Several definitions of fractional-order differential are givenin monograph [39] among which Caputo and Rie-mannndashLiouville fractional-order differential are widely usedIn this paper we adopt the definition of Caputo fractional-order differential

Definition 1 (see Podlubny [39]) )e Caputo-type frac-tional derivative of order α gt 0 for a function f(t) is definedas follows

Dαf(t)

1Γ(m minus α)

1113946t

0

f(m)(τ)

(t minus τ)α+1minusmdτ (1)

where Γ(middot) is the gamma function defined asΓ(q) 1113938

infin0 tqminus1eminustdt and the integer m satisfies the condition

m minus 1lt αlemNext we give the following lemmas that will be used in

this paper

2 Mathematical Problems in Engineering

Lemma 1 (see Zhang et al [40 41]) Let A isin Rntimesn be a realmatrix )e fractional-order system Dαx(t) Ax(t) with0lt αlt 1 is asymptotically stable if and only if there exist realsymmetric positive definite matrices P11 and P21 and skew-symmetric matrices P12 and P22 such that

P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

1113944

2

i11113944

2

j1sym θij otimes APij1113872 11138731113872 1113873lt 0

(2)

where θ11 a b

b a1113890 1113891 θ12

b a

minusa b1113890 1113891 θ21

a b

minusb a1113890 1113891 and

θ22 minusb a

minusa minus b1113890 1113891 a sin θ1 b cos θ1 θ1 πα2

Lemma 2 (see Chilali et al [42]) Let A isin Rntimesn be a realmatrix )e fractional-order system Dαx(t) Ax(t) with1le αlt 2 is asymptotically stable if and only if there exists asymmetric positive definite matrix P isin Rntimesn such that

c AP + PAT( 1113857 d AP minus PAT( 1113857

lowast c AP + PAT( 1113857⎡⎣ ⎤⎦lt 0 (3)

where c sin θ2 and d cos θ2 θ2 π minus πα2

Lemma 3 (see Chaibi et al [38]) For matrices T Q U andW with proper dimensions scalar ξ if inequality

T ξQ + WTU

lowast minussym(ξU)⎡⎣ ⎤⎦lt 0 (4)

holds then

T + sym(QW)lt 0 (5)

Lemma 4 (see Gahinet and Apkarian [43]) Given a sym-metric matrixZ0 isin Rmtimesm and twomatricesX Y of columnmthere exists a matrix Z such that the LMI

Z0 + sym XTZY1113872 1113873lt 0 (6)

holds if and only if the following two projection inequalitiesare satisfied

XTperpZ0Xperp lt 0

YTperpZ0Yperp lt 0

(7)

where Xperp and Yperp are arbitrary matrices whose columns formbases of the null bases of X and Y respectively

3 Problem Description and Formation

Consider the following fractional-order T-S fuzzy systemsPlant rule i if θ1(t) is μi1(t) and and θp(t) is μip(t)

then

Dαx(t) Aix(t) + Biu(t)

y(t) Cix(t) i 1 2 r1113896 (8)

where the order α satisfies 0lt αlt 1 or 1le αlt 2 r is thenumber of if-then rules θj(t) and μij(t)(j 1 2 p) arethe premise variables and the fuzzy sets respectivelyx(t) isin Rn u(t) isin Rl and y(t) isin Rm are the state themeasurable output and the controller respectively andAi Bi and Ci are matrices with appropriate dimensions

Using fuzzy reasoning technology the final output offuzzy system (8) is

Dαx(t) 1113944r

i1hi(θ(t)) Aix(t) + Biu(t)( 1113857

y(t) 1113944r

i1hi(θ(t))Cix(t)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

where θ(t) [θ1(t) θ2(t) θp(t)] and hi(θ(t))

ωi(θ(t))1113936ri1 ωi(θ(t)) ωi(θ(t)) Πp

j1μij(θj(t))μij(θj(t)) is the membership degree of θj(t) in fuzzy setμij(t)

According to the definition of the membership functionwe can easily get the following properties

ωi(θ(t)) ge 0

0le hi(θ(t))le 1 i 1 2 r 1113944r

i1hi(θ(t)) 1

(10)

For system (8) we will design the following fuzzy staticoutput feedback controller

u(t) 1113944r

i1hi(θ(t))Fiy(t) (11)

where Fi(i 1 2 r) are the matrices with proper di-mensions to be designed

By bringing controller (11) into system (8) we can getthe closed-loop system

Dαx(t) 1113944

r

i11113944

r

j11113944

r

k1hi(θ(t))hj(θ(t))hk(θ(t)) Ai + BiFjCk1113872 1113873x(t)

(12)

)e main purpose of this paper is to design static fuzzycontroller (11) for system (8) so that the correspondingclosed-loop system (12) is asymptotically stable

4 Design of the Fuzzy Static OutputFeedback Controller

In this section the design methods of the fuzzy static outputfeedback controller for fractional-order fuzzy system (8) aregiven and the corresponding closed-loop system is guar-anteed to be asymptotically stable Now we can prove thefollowing results

Theorem 1 Fractional-order closed-loop system (12) with0lt αlt 1 is asymptotically stable if there is scalar ξ gt 0 properdimensional matrices U G1i G2i G3i G4i Ki(i 1 2 r)

Mathematical Problems in Engineering 3

symmetric matrices P11 P21 and skew-symmetric matricesP21 P22 such that the following LMIs are satisfied

P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

(13)

Ωii lt 0 i 1 2 r (14)

Ωij +Ωji lt 0 i 1 2 r minus 1 j i + 1 r (15)

In addition the controller gain matrices can be designedas

Fi KiUminus1

i 1 2 r (16)

where

Ωij

minussym G1i( 1113857 Γ1 + G1iATj minus GT

2i 0 Γ2

lowast sym G2iATj1113872 1113873 minusΓT2 0 ξQij + WT

ij

lowast lowast minussym G3i( 1113857 Γ1 + G3iATj minus GT

4i

lowast lowast lowast sym G4iATj1113872 1113873

lowast lowast lowast lowast minussym(ξU)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Γ1 aP11 + bP12 + aP21 minus bP22

Γ2 minusbP11 + aP12 + bP21 + aP22

a sinπα2

1113874 1113875

b cosπα2

1113874 1113875

Qij diag G1iCTj G2iC

Tj G3iC

Tj G4iC

Tj1113966 1113967

U diag U U U U

Wij diag 0 KTj B

Ti 0 K

Tj B

Ti1113960 1113961 0 K

Tj B

Ti 0 K

Tj B

Ti1113960 11139611113966 1113967

(17)

Proof Let

Ω

minussym G1i( 1113857 Γ1 + G1iAT

i minus GT

2i 0 Γ2

lowast sym G2iAT

i1113874 1113875 minusΓT2 0 ξQi + WT

iuU

lowast lowast minussym G3i( 1113857 Γ1 + G3iAT

i minus GT

4i

lowast lowast lowast sym G4iAT

i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

where Qi diag G1iCT

i G2iCT

i G3iCT

i G4iCT

i1113882 1113883 Wiu

diag [0 Uminus T1113864 KT

i BT

i 0 UminusTKT

i BT

i ] [0 Uminus TKT

i BT

i 0 Uminus TKT

i

BT

i ] and [Ai Bi Ci Ki G1i G2i G3i G4i] 1113936ri1 hi(θ(t))[Ai

Bi Ci Ki G1i G2i G3i G4i]


According to inequalities (14) and (15) and the prop-erties of the membership function we can get

Ω 1113944

r

i11113944

r

i1hi(θ(t))hj(θ(t))Ωij

1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r

ji+1hi(θ(t))hj(θ(t)) Ωij +Ωji1113872 1113873lt 0

(19)

Replacing KiUminus1 with Fi in Ω we obtain


i minus GT

2i 0 Γ2lowast sym G2iA

T

i1113874 1113875 minusΓT2 0 ξQi + WT

ifU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lt 0 (20)

where Wif diag [0 FT

i BT

i 0 FT

i BT

i ] [0 FT

i BT

i 0 FT

i BT

i ]1113882 1113883 According to Lemma 3 if inequality (20) holds then


i minus GT


T

i1113874 1113875 minusΓT2 0


i minus GT

4i


i1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym QiWif1113872 1113873lt 0 (21)

Let

1113957A 1113944r

i11113944

r

j11113944

r

k1hi(θ(t))hj(θ(t))hk(θ(t)) Ai + BiFjCk1113872 1113873

(22)

)en 1113957A Ai + BiFiCi)erefore inequality (21) can be rewritten as

0 Γ1 0 Γ2lowast 0 minus ΓT2 0

lowast lowast 0 Γ1lowast lowast lowast 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusI 1113957AT 0 0

0 0 minusI 1113957AT

⎡⎢⎣ ⎤⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠lt 0

(23)

Taking Ψ 1113957A I 0 00 0 1113957A I

1113890 1113891and using Lemma 4 we have

Ψ

0Γ1 0 Γ2lowast 0 minusΓT2 0

lowastlowast 0 Γ1lowastlowast lowast 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ΨT +sym Ψ

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusI1113957AT 0 0

0 0 minusI1113957AT

⎡⎢⎣ ⎤⎥⎦ΨT

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lt0

which is equivalent to

1113957AΓ1 + ΓT1 1113957AT 1113957AΓ2 minus ΓT2 1113957A

T

lowast 1113957AΓ1 + ΓT1 1113957AT

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦lt 0 (24)

)e above inequality can be rewritten as

sym1113957AΓ1 1113957AΓ2

minus1113957AΓ2 1113957AΓ1⎡⎣ ⎤⎦

⎧⎨

⎩

⎫⎬

⎭ lt 0 (25)

According to the definition of the Kronecker productone has from (25)

syma minusb

b a1113890 1113891otimes 1113957AP111113872 1113873 +

b a

minusa b1113890 1113891otimes 1113957AP121113872 11138731113896

+a b

minusb a1113890 1113891otimes 1113957AP211113872 1113873 +

minusb a

minusa minusb1113890 1113891otimes 1113957AP221113872 11138731113897lt 0

(26)

According to Lemma 1 if inequalities (13) and (26) holdclosed-loop system (12) with 0lt αlt 1 is asymptoticallystable )is completes the proof

Remark 1 Compared with [19ndash26] this paper studies thedesign of the fuzzy static output feedback controller for frac-tional-order fuzzy systems In general the fuzzy static outputfeedback controller will generate the term BiFjCk

(i j k 1 2 r) which increases the fuzzy r-times


relationship At the same time the controller gain matrix islocated between the two matrices which makes the design ofthe control more difficult In this paper Lemma 3 is used toingeniously separate this item and eliminate the above-mentioned difficulties Furthermore the relationship betweenfuzzy systems is considered

Considering the relationship between fuzzy subsystemswe can prove the following conclusions

Theorem 2 Fractional-order closed-loop system (12) with0lt αlt 1 is asymptotically stable if there is scalar ξ gt 0 properdimensional matrices U G1i G2i G3i G4i Ki Zij(i 1 2

r) symmetric matrices P11 P21 Zii and skew-symmetric ma-trices P21 P22 such that (13) and the following LMIs hold

Ωii ltZii i 1 2 r (27)

Ωij +Ωji ltZij + ZTij i 1 2 r minus 1 j i + 1 r

Z

Z11 Z12 middot middot middot Z1r

ZT12 Z22 middot middot middot Z2r

⋮ ⋮ ⋱ ⋮ZT1r ZT

2r middot middot middot Zrr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0

(28)

If the above matrix inequalities hold the controller gainmatrices can be designed as

Fi KiUminus 1

i 1 2 r (29)

Proof According to inequalities (27) and (28) we can easilyget

1113944

r

i11113944

r

j1hi(θ(t))hj(θ(t))Ωij

1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r

ji+1hi(θ(t))hj(θ(t)) Ωij +Ωji1113872 1113873

lt 1113944r

i1h2i (θ(t))Zii + 1113944

rminus1

i11113944

r

ji+1hi(θ(t))hj(θ(t)) Zij + Z

Tij1113872 1113873

h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859

middot Z h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859T

(30)

If Zlt 0 holds there is 1113936ri1 1113936

rj1 hi(θ(t))hj(θ(t))Ωij lt 0

ie inequality (19) holds According to the proof process of)eorem 1 closed-loop system (12) with 0 asymptoticallystable )is completes the proof

For fractional-order system (12) with order greater thanzero and less than one two new controller design methodsare proposed Using the similar method when the order isgreater than or equal to 1 but less than 2 we can prove thefollowing results

Theorem 3 Fractional-order closed-loop system (12) with1le αlt 2 is asymptotically stable if there is scalar ξ gt 0 properdimensional matrices U G1i G2i G3i G4i Ki(i 1 2 r)symmetric positive definite matrices Pi(i 1 2 r) andthe following LMIs are satisfied

Ξii lt 0 i 1 2 r (31)

Ξij + Ξji lt 0 i 1 2 r minus 1 j i + 1 r

Fi KiUminus 1

i 1 2 r(32)

where

Ωij

minussym G1i( 1113857 cPi + G1iATj minus GT

2i 0 dPi

lowast sym G2iATj1113872 1113873 minusdPi 0 ξQij + WT

ij

lowast lowast minussym G3i( 1113857 cPi + G3iATj minus GT

4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

c sin(π minus πα2) d cos(π minus πα2) and the definitions ofother symbols are the same as those in )eorem 1

Proof Let Pi 1113936ri1 hi(θ(t))Pi Because every Pi is a sym-

metric positive definite matrix and the membership function isgreater than 0 Pi is also a symmetric positive definite matrix

In the process of proving )eorem 2 let Γ1 cPi andΓ2 dPi starting from inequalities (31) and (32) we can getthe following results from (24)

c 1113957AijkPi + Pi1113957A

T

ijk1113874 1113875 d 1113957AijkPi minus Pi1113957A

T

ijk1113874 1113875

lowast c 1113957AijkPi + Pi1113957A

T

ijk1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (34)

According to Lemma 2 )eorem 3 ensures that closed-loop system (12) with 1le αlt 2 is asymptotically stable )iscompletes the proof

Similar to )eorem 1 we can prove the followingresult


Theorem 4 Fractional-order closed-loop system (12) is as-ymptotically stable if there is scalar ξ gt 0 proper dimensionalmatrices U G1i G2i G3i G4i Ki Zij(i 1 2 r) andsymmetric positive definite matrices Pi(i 1 2 r) suchthat the following linear matrix inequalities hold

Ξii ltZii i 1 2 r (35)

Ξij +Ωji ltZij + ZTij i 1 2 r minus 1 j i + 1 r

(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)

)e corresponding controller can be selected as

Fi KiUminus 1

i 1 2 r (38)

Proof )e proof process of)eorem 4 is exactly the same asthat of )eorem 2 so it is omitted here

5 Numerical Example

In order to illustrate the effectiveness of the proposedmethods a numerical simulation example is given in thispart Since the simulation methods are similar we onlyverify )eorem 1

Example 1 For system (9) the corresponding simulationparameters are selected as follows

A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)

Membership functions are taken as h1(θ(t)) 05(1minus

sin(x1(t))) h2(θ(t)) 1 minus h1(θ(t))Take fractional order α 075 and initial condition of the

system x0 [1 minus3 18]T When the controller u(t) 0 the

state trajectory of the system is shown in Figure 1 Obviouslythe system is unstable

According to)eorem 1 the following feasible solutionscan be obtained by using the linear matrix inequality toolboxand making ξ 1

P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41

minus193450 minus670576 minus21058


18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)

Gain matrices of the controller can be obtained bycalculation

F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)


Using fuzzy controller (11) the state trajectories ofclosed-loop system (12) can be obtained as shown in Fig-ure 2 As can be seen from Figure 2 the controller designedis effective

6 Conclusion

For the fractional-order T-S fuzzy system the controllerdesign methods with order in two different intervals arestudied Four theorems are given to ensure that the closed-loop system is asymptotically stable )e result is expressedby the linear matrix inequality which fully considers thefeasibility and conservatism In order to consider the fea-sibility the condition of each theorem is the strict linearmatrix inequality )is can be solved directly by the linearmatrix inequality toolbox of Matlab In order to reduce theconservatism we try to make the matrix correspond to the

fuzzy rule )eorems 2 and 4 further consider the rela-tionship between fuzzy subsystems

Data Availability

)e simulation results of this paper can be obtained byMATLAB software

Conflicts of Interest

)e author declares that there are no conflicts of interest

References

[1] L Chen R Wu Y He and L Yin ldquoRobust stability andstabilization of fractional-order linear systems with polytopicuncertaintiesrdquo Applied Mathematics and Computationvol 257 pp 274ndash284 2015

[2] L Chen T Huang J A Tenreiro Machado A M LopesY Chai and R Wu ldquoDelay-dependent criterion for as-ymptotic stability of a class of fractional-order memristiveneural networks with time-varying delaysrdquo Neural Networksvol 118 pp 289ndash299 2019

[3] L Chen J Cao R Wu J A Tenreiro Machado A M Lopesand H Yang ldquoStability and synchronization of fractional-order memristive neural networks with multiple delaysrdquoNeural Networks vol 94 pp 76ndash85 2017

[4] J A Gallegos and M A Duarte-Mermoud ldquoA dissipativeapproach to the stability of multi-order fractional systemsrdquoIMA Journal of Mathematical Control and Informationvol 37 no 1 pp 143ndash158 2018

[5] H Liu Y P Pan J D Cao H X Wang and Y ZhouldquoAdaptive neural network backstepping control of fractional-order nonlinear systems with actuator faultsrdquo IEEE Trans-actions on Neural Networks and Learning Systems pp 1ndash12 Inpress 2020

[6] J Jia X Huang Y Li J Cao and A Alsaedi ldquoGlobal sta-bilization of fractional-order memristor-based neural net-works with time delayrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 31 no 3 pp 997ndash10092020

[7] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[8] X J Zeng and M G Singh ldquoApproximation theory of fuzzysystems SISO caserdquo IEEE Transactions on Systems Man andcybernetics vol 24 no 2 pp 332ndash342 1994

[9] L X Wang ldquoFuzzy systems are universal approximationsrdquo inProceedings of the IEEE International Conference on FuzzySystems pp 1163ndash1170 San Diego CA USA March 1992

[10] T Takagi andM Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactionson Systems Man and Cybernetics vol SMC-15 no 1pp 116ndash132 1985

[11] M Sugeno and T Yasukawa ldquoA fuzzy-logic-based approachto qualitative modelingrdquo IEEE Transactions on Fuzzy Systemsvol 1 no 1 pp 7ndash25 1993

[12] L-X Wang and J M Mendel ldquoFuzzy basis functions uni-versal approximation and orthogonal least-squares learningrdquoIEEE Transactions on Neural Networks vol 3 no 5pp 807ndash814 1992

[13] L Wang Adaptive Fuzzy System and Control Design andStability Analysis National Defense Industry Press BeijingChina 1995

5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

Figure 1 )e state trajectories of the system when the controlleru(t) 0

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)

Figure 2 State trajectories of the system after adding thecontroller


[14] S G Cao N W Rees and G Feng ldquoAnalysis and design offuzzy control systems using dynamic fuzzy global modelsrdquoFuzzy Sets and Systems vol 75 no 1 pp 47ndash62 1995

[15] Z Chen B Zhang Y Zhang and Z Zhang ldquoDissipative fuzzyfiltering for nonlinear networked systems with limitedcommunication linksrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 50 no 3 pp 962ndash971 2020

[16] S Rathinasamy S Palanisamy and K BoomipalaganldquoModified repetitive control design for nonlinear systems withtime delay based on T-S fuzzy modelrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 50 no 2pp 646ndash655 2020

[17] Y Li L Liu and G Feng ldquoFinite-time Hinfin controller syn-thesis of T-S fuzzy systemsrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 50 no 5 pp 1956ndash19632020

[18] Y-C Chang S-S Chen S-F Su and T-T Lee ldquoStatic outputfeedback stabilization for nonlinear interval time-delay sys-tems via fuzzy control approachrdquo Fuzzy Sets and Systemsvol 148 no 3 pp 395ndash410 2004

[19] D Huang and S K Nguang ldquoStatic output feedback con-troller design for fuzzy systems an ILMI approachrdquo Infor-mation Sciences vol 177 no 14 pp 3005ndash3015 2007

[20] H N Wu ldquoAn ILMI approach to robust static outputfeedback fuzzy control for uncertain discrete-time nonlinearsystemsrdquo Automatica vol 44 no 9 pp 2333ndash2339 2008

[21] S-W Kau H-J Lee C-M Yang C-H Lee L Hong andC-H Fang ldquoRobust fuzzy static output feedback control ofT-S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Setsand Systems vol 158 no 2 pp 135ndash146 2007

[22] J Dong and G-H Yang ldquoStatic output feedback control of aclass of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009

[23] H-Y Chung and S-M Wu ldquoHybrid approaches for regionalTakagi-Sugeno static output feedback fuzzy controller de-signrdquo Expert Systems with Applications vol 36 no 2pp 1720ndash1730 2009

[24] J C Lo andM L Lin ldquoRobustHinfin nonlinear control via fuzzystatic output feedbackrdquo IEEE Transactions on Circuits andSystems I Fundamental )eory and Applications vol 50no 11 pp 1494ndash1502 2003

[25] D Huang and S K Sing Kiong Nguang ldquoRobust Hinfin staticoutput feedback control of fuzzy systems an ILMI approachrdquoIEEE Transactions on Systems Man and Cybernetics Part B(Cybernetics) vol 36 no 1 pp 216ndash222 2006

[26] H N Wu and H Y Zhang ldquoReliable mixed L2Hinfin fuzzystatic output feedback control for nonlinear systems withsensor faultsrdquo Automatica vol 41 no 11 pp 1925ndash19322005

[27] H J Lee and DW Kim ldquoFuzzy static output feedbackmay bepossible in LMI frameworkrdquo IEEE Transactions on FuzzySystems vol 17 no 5 pp 1229-1230 2009

[28] C H Fang Y S Liu S W Kau et al ldquoA new LMI-basedapproach to relaxed quadratic stabilization of T-S fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 3 pp 386ndash397 2006

[29] Y Zheng Y Nian and D Wang ldquoControlling fractionalorder chaotic systems based on Takagi-Sugeno fuzzy modeland adaptive adjustment mechanismrdquo Physics Letters Avol 375 no 2 pp 125ndash129 2010

[30] J Li and Y Li ldquoRobust stability and stabilization of fractionalorder systems based on uncertain Takagi-Sugeno fuzzy modelwith the fractional order 1le αlt 2rdquo Journal of Computing andNonlinear Dynamics vol 8 no 4 pp 1ndash7 2013

[31] Y Li and J Li ldquoStability analysis of fractional order systemsbased on T-S fuzzy model with the fractional order α0 lt α lt1rdquo Nonlinear Dynamics vol 78 no 4 pp 2909ndash2919 2014

[32] X Huang Z Wang Y Li and J Lu ldquoDesign of fuzzy statefeedback controller for robust stabilization of uncertainfractional-order chaotic systemsrdquo Journal of the FranklinInstitute vol 351 no 12 pp 5480ndash5493 2014

[33] R Duan J Li and J Chen ldquoMode-dependent non-fragileobserver-based controller design for fractional-order T-Sfuzzy systems with Markovian jump via non-PDC schemerdquoNonlinear Analysis Hybrid Systems vol 34 pp 74ndash91 2019

[34] R Duan and J Li ldquoObserver-based non-PDC controllerdesign for T-S fuzzy systems with the fractional-order 0 lt α lt0rdquo IET Control )eory amp Applications vol 12 no 5pp 661ndash668 2018

[35] S A Karthick R Sakthivel Y K Ma S Mohanapriya andA Leelamani ldquoDisturbance rejection of fractional-order T-Sfuzzy neural networks based on quantized dynamic outputfeedback controllerrdquo Applied Mathematics and Computationvol 361 pp 846ndash857 2019

[36] C Lin B Chen and Q-G Wang ldquoStatic output feedbackstabilization for fractional-order systems in T-S fuzzymodelsrdquo Neurocomputing vol 218 pp 354ndash358 2016

[37] Y Ji L Su and J Qiu ldquoDesign of fuzzy output feedbackstabilization for uncertain fractional-order systemsrdquo Neuro-computing vol 173 pp 1683ndash1693 2016

[38] R Chaibi E Tissir A Hmamed et al ldquoStatic output feedbackcontroller for continuous-time fuzzy systemsrdquo InternationalJournal of Innovative Computing Information and Controlvol 15 no 4 pp 1469ndash1484 2019

[39] L Podlubny Fractional Differential Equations AcademiePress New York NY USA 1999

[40] X F Zhang and Y Q Chen ldquoD-stability based LMI criteria ofstability and stabilization for fractional order systemsrdquo inProceedings of the ASME 2015 ASMEIEEE InternationalConference on Mechatronic and Embedded Systems andApplications Boston MA USA August 2015

[41] B Li and X Zhang ldquoObserver-based robust control of (0 lt αlt1) fractional-order linear uncertain control systemsrdquo IETControl )eory amp Applications vol 10 no 14 pp 1724ndash17312016

[42] M Chilali P Gahinet and P Apkarian ldquoRobust poleplacement in LMI regionsrdquo IEEE Transactions on AutomaticControl vol 44 no 12 pp 2257ndash2270 1999

[43] P Gahinet and P Apkarian ldquoA linear matrix inequalityapproach toHinfin controlrdquo International Journal of Robust andNonlinear Control vol 4 no 4 pp 421ndash448 1994


generalization of the classical set theory Fuzzy control is basedon the fuzzy theory By using expert language fuzzy control canachieve good and effective control for some complex nonlinearcontrol objects which are difficult to establish themathematicalmodel accurately [8 9] So far two main fuzzy models havebeen proposed )e first model is based on the input-outputmodel proposed in document [9] )e disadvantage is thatthere is a lot of useful information which is not used )esecond model is the T-S fuzzy model proposed by Japanesescholars Takagi and Sugeno in 1985 [10 11] In [12ndash14] it isproved that this kind of fuzzy system has the ability of universalapproximation Some recent literature studies are listed to showthat the T-S fuzzy system has beenwidely studied For examplebased on the T-S fuzzy model in [15] the design of the fuzzyfilter for the discrete-time nonlinear networked system isstudied and a new method is proposed to ensure that thefiltering error system is dissipative by using themethod of eventtriggering and quantization For continuous-time nonlineartime-delay systems by using repetitive control Rathinasamyet al [16] proposed an improved output feedback controllerdesign method )is method ensures that the system can trackthe given reference signal within the allowable error range Liet al [17] studied the finite-time Hinfin control problem and thesufficient conditions based on LMI were given )e researchresults of the output feedback can be found in the literaturestudies [18ndash28] )ere are also a large number of referencesrelated to the T-S fuzzymodel whichwe cannot be enumeratedhere

Compared with many research results of the integer-order T-S fuzzy system the research results of the fractional-order T-S fuzzy system are less For the fractional-order T-Sfuzzy system Zheng et al [29] studied the stabilization of thechaotic system by using the adaptive method When theorder of the fractional order considered satisfies 1le αlt 2and 0lt αlt 1 respectively the stability conditions based onLMI were proposed in [30 31] Huang et al [32] studied thestabilization of the state feedback that can be applied to boththe following situations 1le αlt 2 and 0lt αlt 1 However inthe practical engineering field the state of the system is oftendifficult to obtain directly At this time the state observer oroutput feedback control methods need to be consideredBased on the T-S fuzzy model Duan et al [33] studied thedesign problem of the nonfragile observer by using singularvalue decomposition of thematrix and put forward sufficientconditions of linear matrix inequalities which guaranteethat the corresponding closed-loop system is stochasticasymptotically stable when the fuzzy antecedent variablesare unmeasurable Duan and Li [34] considered the designproblem of dynamic output feedback controllers and pro-posed sufficient conditions for the system to be asymptot-ically stable Karthick et al [35] designed a dynamic outputfeedback controller by means of interference suppressionand quantization and Lin et al and Ji et al [36 37] studiedthe design of the fuzzy static output feedback controller )edesign of the static fuzzy output feedback controller willincrease the fuzzy relation in the stabilization condition by acertain multiple because the product term of coefficientmatrices BiFjCk appears while the design state feedbackonly contains the product term of coefficient matrices

BiFj(i j k 1 2 r) where r is an integer indicating thenumber of fuzzy rules For the traditional T-S fuzzy systemChaibi et al [38] used the matrix decomposition techniqueto effectively separate the coefficient term BiFjCk and ef-fectively reduced the fuzzy relation of one layer and pro-posed the design method based on the linear matrixinequality In this paper we try to extend the research resultsof Chaibi et al [38] to the fractional-order T-S fuzzy system

In conclusion this paper studies the fuzzy static outputfeedback control of the fractional-order T-S fuzzy system)e order includes two cases 0lt αlt 1 and 1le αlt 2According to the theory of fractional differential correlationand the projection theorem several new stabilization designmethods are given All the results are expressed by the linearmatrix inequality )e main contributions of this paper areas follows (1) the control method of the fuzzy static outputfeedback is extended from the integer-order fuzzy system tothe fractional-order fuzzy system so it is more practical (2)the condition of system stabilization and controller designmethods are all expressed by LMIs )erefore LMI toolboxof Matlab can be used to program and calculate (3) outputmatrix C1 C2 Cr of the system can still be completelydifferent (4) for some similar Lyapunov matrices onlysymmetry or antisymmetry is required no further diagonalis required and stability conditions are relaxed and (5) thestabilization conditions are also considered Finally a nu-merical simulation example is given to show the effective-ness of the proposed method

Notations )roughout this paper Rn denotes thenminusdimensional Euclidean space Rntimesm is the set of all n times m

real matrices I and O represent the identity matrix and thezero matrix with appropriate dimensions Pgt 0(Plt 0)

represents the positive definite (negative definite) matrix theright superscript T denotes the transposition of the matrixthe symbol sym S means S + ST the diagonal block matrixis denoted by diag A1 A2 An1113864 1113865 and the symbol lowast showsthe symmetric part of the block matrix

2 Basic Definition and Lemmas

Several definitions of fractional-order differential are givenin monograph [39] among which Caputo and Rie-mannndashLiouville fractional-order differential are widely usedIn this paper we adopt the definition of Caputo fractional-order differential

Definition 1 (see Podlubny [39]) )e Caputo-type frac-tional derivative of order α gt 0 for a function f(t) is definedas follows

Dαf(t)

1Γ(m minus α)

1113946t

0

f(m)(τ)

(t minus τ)α+1minusmdτ (1)

where Γ(middot) is the gamma function defined asΓ(q) 1113938

infin0 tqminus1eminustdt and the integer m satisfies the condition

m minus 1lt αlemNext we give the following lemmas that will be used in

this paper



P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

1113944

2

i11113944

2


(2)

where θ11 a b

b a1113890 1113891 θ12

b a

minusa b1113890 1113891 θ21

a b

minusb a1113890 1113891 and

θ22 minusb a




lowast c AP + PAT( 1113857⎡⎣ ⎤⎦lt 0 (3)



T ξQ + WTU


holds then

T + sym(QW)lt 0 (5)


Z0 + sym XTZY1113872 1113873lt 0 (6)


XTperpZ0Xperp lt 0

YTperpZ0Yperp lt 0

(7)




then


y(t) Cix(t) i 1 2 r1113896 (8)



Dαx(t) 1113944r

i1hi(θ(t)) Aix(t) + Biu(t)( 1113857

y(t) 1113944r

i1hi(θ(t))Cix(t)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)





ωi(θ(t)) ge 0

0le hi(θ(t))le 1 i 1 2 r 1113944r

i1hi(θ(t)) 1

(10)


u(t) 1113944r




Dαx(t) 1113944

r

i11113944

r

j11113944

r


(12)







P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

(13)

Ωii lt 0 i 1 2 r (14)



Fi KiUminus1

i 1 2 r (16)

where

Ωij


2i 0 Γ2


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



a sinπα2

1113874 1113875

b cosπα2

1113874 1113875


Tj G3iC

Tj G4iC

Tj1113966 1113967

U diag U U U U

Wij diag 0 KTj B

Ti 0 K

Tj B

Ti1113960 1113961 0 K

Tj B

Ti 0 K

Tj B

Ti1113960 11139611113966 1113967

(17)

Proof Let

Ω


i minus GT

2i 0 Γ2

lowast sym G2iAT

i1113874 1113875 minusΓT2 0 ξQi + WT

iuU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

where Qi diag G1iCT

i G2iCT

i G3iCT

i G4iCT

i1113882 1113883 Wiu


i BT

i 0 UminusTKT

i BT

i ] [0 Uminus TKT

i BT

i 0 Uminus TKT

i

BT





Ω 1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


(19)



i minus GT


T

i1113874 1113875 minusΓT2 0 ξQi + WT

ifU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lt 0 (20)


i BT

i 0 FT

i BT

i ] [0 FT

i BT

i 0 FT

i BT



i minus GT


T

i1113874 1113875 minusΓT2 0


i minus GT

4i


i1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym QiWif1113872 1113873lt 0 (21)

Let

1113957A 1113944r

i11113944

r

j11113944

r


(22)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎣ ⎤⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠lt 0

(23)

Taking Ψ 1113957A I 0 00 0 1113957A I


Ψ



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ΨT +sym Ψ

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusI1113957AT 0 0

0 0 minusI1113957AT

⎡⎢⎣ ⎤⎥⎦ΨT

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lt0



T

lowast 1113957AΓ1 + ΓT1 1113957AT

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦lt 0 (24)


sym1113957AΓ1 1113957AΓ2

minus1113957AΓ2 1113957AΓ1⎡⎣ ⎤⎦

⎧⎨

⎩

⎫⎬

⎭ lt 0 (25)


syma minusb

b a1113890 1113891otimes 1113957AP111113872 1113873 +

b a


+a b


minusb a


(26)











Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0

(28)


Fi KiUminus 1

i 1 2 r (29)


1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


lt 1113944r

i1h2i (θ(t))Zii + 1113944

rminus1

i11113944

r


Tij1113872 1113873

h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859


(30)






Ξii lt 0 i 1 2 r (31)


Fi KiUminus 1

i 1 2 r(32)

where

Ωij


2i 0 dPi


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)





c 1113957AijkPi + Pi1113957A

T


T

ijk1113874 1113875


T

ijk1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (34)







(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)


Fi KiUminus 1

i 1 2 r (38)


5 Numerical Example



A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)






P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41



18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)


F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)



6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)



































P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

1113944

2

i11113944

2


(2)

where θ11 a b

b a1113890 1113891 θ12

b a

minusa b1113890 1113891 θ21

a b

minusb a1113890 1113891 and

θ22 minusb a




lowast c AP + PAT( 1113857⎡⎣ ⎤⎦lt 0 (3)



T ξQ + WTU


holds then

T + sym(QW)lt 0 (5)


Z0 + sym XTZY1113872 1113873lt 0 (6)


XTperpZ0Xperp lt 0

YTperpZ0Yperp lt 0

(7)




then


y(t) Cix(t) i 1 2 r1113896 (8)



Dαx(t) 1113944r

i1hi(θ(t)) Aix(t) + Biu(t)( 1113857

y(t) 1113944r

i1hi(θ(t))Cix(t)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)





ωi(θ(t)) ge 0

0le hi(θ(t))le 1 i 1 2 r 1113944r

i1hi(θ(t)) 1

(10)


u(t) 1113944r




Dαx(t) 1113944

r

i11113944

r

j11113944

r


(12)







P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

(13)

Ωii lt 0 i 1 2 r (14)



Fi KiUminus1

i 1 2 r (16)

where

Ωij


2i 0 Γ2


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



a sinπα2

1113874 1113875

b cosπα2

1113874 1113875


Tj G3iC

Tj G4iC

Tj1113966 1113967

U diag U U U U

Wij diag 0 KTj B

Ti 0 K

Tj B

Ti1113960 1113961 0 K

Tj B

Ti 0 K

Tj B

Ti1113960 11139611113966 1113967

(17)

Proof Let

Ω


i minus GT

2i 0 Γ2

lowast sym G2iAT

i1113874 1113875 minusΓT2 0 ξQi + WT

iuU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

where Qi diag G1iCT

i G2iCT

i G3iCT

i G4iCT

i1113882 1113883 Wiu


i BT

i 0 UminusTKT

i BT

i ] [0 Uminus TKT

i BT

i 0 Uminus TKT

i

BT





Ω 1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


(19)



i minus GT


T

i1113874 1113875 minusΓT2 0 ξQi + WT

ifU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lt 0 (20)


i BT

i 0 FT

i BT

i ] [0 FT

i BT

i 0 FT

i BT



i minus GT


T

i1113874 1113875 minusΓT2 0


i minus GT

4i


i1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym QiWif1113872 1113873lt 0 (21)

Let

1113957A 1113944r

i11113944

r

j11113944

r


(22)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎣ ⎤⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠lt 0

(23)

Taking Ψ 1113957A I 0 00 0 1113957A I


Ψ



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ΨT +sym Ψ

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusI1113957AT 0 0

0 0 minusI1113957AT

⎡⎢⎣ ⎤⎥⎦ΨT

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lt0



T

lowast 1113957AΓ1 + ΓT1 1113957AT

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦lt 0 (24)


sym1113957AΓ1 1113957AΓ2

minus1113957AΓ2 1113957AΓ1⎡⎣ ⎤⎦

⎧⎨

⎩

⎫⎬

⎭ lt 0 (25)


syma minusb

b a1113890 1113891otimes 1113957AP111113872 1113873 +

b a


+a b


minusb a


(26)











Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0

(28)


Fi KiUminus 1

i 1 2 r (29)


1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


lt 1113944r

i1h2i (θ(t))Zii + 1113944

rminus1

i11113944

r


Tij1113872 1113873

h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859


(30)






Ξii lt 0 i 1 2 r (31)


Fi KiUminus 1

i 1 2 r(32)

where

Ωij


2i 0 dPi


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)





c 1113957AijkPi + Pi1113957A

T


T

ijk1113874 1113875


T

ijk1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (34)







(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)


Fi KiUminus 1

i 1 2 r (38)


5 Numerical Example



A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)






P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41



18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)


F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)



6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)



































P11 P12

minusP12 P111113890 1113891gt 0

P21 P22

minusP22 P211113890 1113891gt 0

(13)

Ωii lt 0 i 1 2 r (14)



Fi KiUminus1

i 1 2 r (16)

where

Ωij


2i 0 Γ2


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



a sinπα2

1113874 1113875

b cosπα2

1113874 1113875


Tj G3iC

Tj G4iC

Tj1113966 1113967

U diag U U U U

Wij diag 0 KTj B

Ti 0 K

Tj B

Ti1113960 1113961 0 K

Tj B

Ti 0 K

Tj B

Ti1113960 11139611113966 1113967

(17)

Proof Let

Ω


i minus GT

2i 0 Γ2

lowast sym G2iAT

i1113874 1113875 minusΓT2 0 ξQi + WT

iuU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

where Qi diag G1iCT

i G2iCT

i G3iCT

i G4iCT

i1113882 1113883 Wiu


i BT

i 0 UminusTKT

i BT

i ] [0 Uminus TKT

i BT

i 0 Uminus TKT

i

BT





Ω 1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


(19)



i minus GT


T

i1113874 1113875 minusΓT2 0 ξQi + WT

ifU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lt 0 (20)


i BT

i 0 FT

i BT

i ] [0 FT

i BT

i 0 FT

i BT



i minus GT


T

i1113874 1113875 minusΓT2 0


i minus GT

4i


i1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym QiWif1113872 1113873lt 0 (21)

Let

1113957A 1113944r

i11113944

r

j11113944

r


(22)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎣ ⎤⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠lt 0

(23)

Taking Ψ 1113957A I 0 00 0 1113957A I


Ψ



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ΨT +sym Ψ

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusI1113957AT 0 0

0 0 minusI1113957AT

⎡⎢⎣ ⎤⎥⎦ΨT

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lt0



T

lowast 1113957AΓ1 + ΓT1 1113957AT

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦lt 0 (24)


sym1113957AΓ1 1113957AΓ2

minus1113957AΓ2 1113957AΓ1⎡⎣ ⎤⎦

⎧⎨

⎩

⎫⎬

⎭ lt 0 (25)


syma minusb

b a1113890 1113891otimes 1113957AP111113872 1113873 +

b a


+a b


minusb a


(26)











Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0

(28)


Fi KiUminus 1

i 1 2 r (29)


1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


lt 1113944r

i1h2i (θ(t))Zii + 1113944

rminus1

i11113944

r


Tij1113872 1113873

h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859


(30)






Ξii lt 0 i 1 2 r (31)


Fi KiUminus 1

i 1 2 r(32)

where

Ωij


2i 0 dPi


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)





c 1113957AijkPi + Pi1113957A

T


T

ijk1113874 1113875


T

ijk1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (34)







(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)


Fi KiUminus 1

i 1 2 r (38)


5 Numerical Example



A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)






P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41



18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)


F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)



6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)



































Ω 1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


(19)



i minus GT


T

i1113874 1113875 minusΓT2 0 ξQi + WT

ifU


i minus GT

4i


i1113874 1113875


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

lt 0 (20)


i BT

i 0 FT

i BT

i ] [0 FT

i BT

i 0 FT

i BT



i minus GT


T

i1113874 1113875 minusΓT2 0


i minus GT

4i


i1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym QiWif1113872 1113873lt 0 (21)

Let

1113957A 1113944r

i11113944

r

j11113944

r


(22)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+ sym

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎣ ⎤⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠lt 0

(23)

Taking Ψ 1113957A I 0 00 0 1113957A I


Ψ



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ΨT +sym Ψ

G1i 0

G2i 0

0 G3i

0 G4i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusI1113957AT 0 0

0 0 minusI1113957AT

⎡⎢⎣ ⎤⎥⎦ΨT

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lt0



T

lowast 1113957AΓ1 + ΓT1 1113957AT

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦lt 0 (24)


sym1113957AΓ1 1113957AΓ2

minus1113957AΓ2 1113957AΓ1⎡⎣ ⎤⎦

⎧⎨

⎩

⎫⎬

⎭ lt 0 (25)


syma minusb

b a1113890 1113891otimes 1113957AP111113872 1113873 +

b a


+a b


minusb a


(26)











Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0

(28)


Fi KiUminus 1

i 1 2 r (29)


1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


lt 1113944r

i1h2i (θ(t))Zii + 1113944

rminus1

i11113944

r


Tij1113872 1113873

h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859


(30)






Ξii lt 0 i 1 2 r (31)


Fi KiUminus 1

i 1 2 r(32)

where

Ωij


2i 0 dPi


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)





c 1113957AijkPi + Pi1113957A

T


T

ijk1113874 1113875


T

ijk1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (34)







(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)


Fi KiUminus 1

i 1 2 r (38)


5 Numerical Example



A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)






P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41



18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)


F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)



6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)








































Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0

(28)


Fi KiUminus 1

i 1 2 r (29)


1113944

r

i11113944

r


1113944r

i1h2i (θ(t))Ωii + 1113944

rminus1

i11113944

r


lt 1113944r

i1h2i (θ(t))Zii + 1113944

rminus1

i11113944

r


Tij1113872 1113873

h1(θ(t)) h2(θ(t)) hr(θ(t))1113858 1113859


(30)






Ξii lt 0 i 1 2 r (31)


Fi KiUminus 1

i 1 2 r(32)

where

Ωij


2i 0 dPi


ij


4i



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)





c 1113957AijkPi + Pi1113957A

T


T

ijk1113874 1113875


T

ijk1113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (34)







(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)


Fi KiUminus 1

i 1 2 r (38)


5 Numerical Example



A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)






P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41



18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)


F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)



6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)





































(36)

Z



⋮ ⋮ ⋱ ⋮ZT1r ZT


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦lt 0 (37)


Fi KiUminus 1

i 1 2 r (38)


5 Numerical Example



A1

minus1 1 2

1 0 1

minus2 minus1 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A2

minus6 minus3 4

1 0 0

minus2 minus2 2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B1

08

minus3

38

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B2

2

minus1

2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C1 minus2 1 01113858 1113859

C2 4 minus3 21113858 1113859

(39)






P11 P21

479949 minus49576 101684

minus49576 96889 79151

101684 79151 253468

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P12 P22

0 minus10326 54116

10326 0 minus21499

minus54116 21499 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G11

173758 minus22678 minus119404

minus08326 276204 minus86416

150427 230184 174726

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G12

63362 minus35964 minus150566

06207 328234 145413

92638 339112 341965

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G21

96221 minus168697 minus252363

minus204052 44510 minus251853

216206 424479 276159

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G22

111564 minus56354 minus52083

minus83296 70806 minus48982

102057 255957 243865

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G31

251420 85343 minus886732

132014 349178 minus1759623

796915 1841028 387372

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G32

237128 1928158 2549889

minus1637837 311085 3640394

minus2757049 minus3479581 366636

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G41



18301 minus290457 minus381671

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

G42



67847 98925 minus07920

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K1 61395

K2 minus215572

U 1011738

(40)


F1 Kminus11 U

minus1 00607

F2 Kminus12 U

minus1 minus02131

(41)



6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)



































6 Conclusion



Data Availability




References














5 10 15 20 25 30 35 400t (s)

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es


ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

Stat

es

5 10 15 20 25 30 35 400t (s)


































Design of Static Output Feedback Controller for Fractional...

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