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Faculty of Engineering and Information Sciences

2018

Delay-dependent Fault-Tolerant Shape Control for Stochastic Distribution Delay-dependent Fault-Tolerant Shape Control for Stochastic Distribution

Systems Systems

Yu Cheng Beihang University

Long Chen Nanjing University of Information Science and Technology

Tao Li Nanjing University of Information Science and Technology

Haiping Du University of Wollongong, hdu@uow.edu.au

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Recommended Citation Recommended Citation Cheng, Yu; Chen, Long; Li, Tao; and Du, Haiping, "Delay-dependent Fault-Tolerant Shape Control for Stochastic Distribution Systems" (2018). Faculty of Engineering and Information Sciences - Papers: Part B. 1156. https://ro.uow.edu.au/eispapers1/1156

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Delay-dependent Fault-Tolerant Shape Control for Stochastic Distribution Delay-dependent Fault-Tolerant Shape Control for Stochastic Distribution Systems Systems

Abstract Abstract A new adaptive delay-dependent fault-tolerant shape control strategy is proposed for the stochastic distribution system with time-delay, which can effectively make the output probability density function (PDF) track a given target distribution. In this framework, firstly, a system model based on PDF with the actuator failure is constructed. And then, an adaptive delay-dependent fault-tolerant controller, which include a normal controller and an adaptive compensation controller is designed. The normal controller can track the given output distribution with an optimized performance index when there is no actuator fault, and the adaptive compensation controller can reduce the negative effects caused by the actuator fault. Numerical simulations are used to show the effectiveness of the proposed method.

Disciplines Disciplines Engineering | Science and Technology Studies

Publication Details Publication Details Y. Cheng, L. Chen, T. Li & H. Du, "Delay-dependent Fault-Tolerant Shape Control for Stochastic Distribution Systems," IEEE Access, vol. 6, pp. 12727-12735, 2018.

This journal article is available at Research Online: https://ro.uow.edu.au/eispapers1/1156

Received January 8, 2018, accepted January 30, 2018, date of publication February 8, 2018, date of current version March 19, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2803779

Delay-Dependent Fault-Tolerant Shape Controlfor Stochastic Distribution SystemsYU CHENG 1, LONG CHEN2, TAO LI2, AND HAIPING DU3, (Senior Member, IEEE)1School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China2Nanjing University of Information Science and Technology, School of Information and Control, Nanjing 210044, China3School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, NSW 2522, Australia

Corresponding author: Yu Cheng (chengyi1976@126.com)

This work was supported in part by the National Natural Science Foundation of China under Grant 61573189, in part by the NaturalScience Foundation for Distinguished Young Scholars of Jiangsu Province under Grant BK20140045, in part by the Six Talent peaksProject in Jiangsu Province under Grant 2015-DZXX-013, and in part by the 333 Project of Jiangsu Province, the InnovationTeam Plan of Jiangsu Province.

ABSTRACT A new adaptive delay-dependent fault-tolerant shape control strategy is proposed for thestochastic distribution system with time-delay, which can effectively make the output probability densityfunction (PDF) track a given target distribution. In this framework, first, a system model based on PDFwith the actuator failure is constructed. And then, adaptive delay-dependent fault-tolerant controllers, whichinclude a normal controller and an adaptive compensation controller is designed. The normal controller cantrack the given output distribution with an optimized performance index when there is no actuator fault, andthe adaptive compensation controller can reduce the negative effects caused by the actuator fault. Numericalsimulations are used to show the effectiveness of the proposed method.

INDEX TERMS Stochastic distribution system, fault-tolerant shape control, adaptive control.

I. INTRODUCTIONIt is known that the control design of stochastic system hasextensive applications in practice [1]. For Gaussian stochasticsystems, the control objectives are described by mean andvariance, which may fully characterize the Gaussian stochas-tic process [2], [3]. However, the nonlinear and uncertain areuniversal in most of the practical systems, which make thesystem states to obey the non-Gaussian stochastic distribu-tion [4]. In this case, the mean and variance are not sufficientto completely describe the statistic properties of stochasticvariables. To deal with this problem, the output PDF shapecontrol strategy is proposed and applied to both Gaussianand non-Gaussian dynamic systems in [5]. Compared withthe traditional control methods based on mean and variance,the PDF shape control can utilize more complete characteris-tic information of stochastic systems.

In the PDF shape control scheme, the dynamic relationshipbetween control input and output PDF can be described viaB-spline approximation methods. In this framework, thisshape control problem can be transformed into a trackingproblem with finite number of weights in the B-spline model.Several types of B-spline models have been derived by differ-ent B-spline approximations, which included linear B-splineapproximation [6], square-root B-spline approximation [7],

rational B-spline approximation [8], and square root rationalB-spline (SR-RBS) approximation [9]. It is noted that thedynamic weights in SR-RBS model are independent witheach other [10], thus it has a great advantage compared withthe other three approximation methods. Therefore, the shapecontrol based on SR-RBS model attracts more and moreattention [11], [12].

In recent years, due to the requirements of high relia-bility of complex systems, fault-tolerant control has drawnextensive attention [13]–[17], [28]. There are a few workabout fault-tolerant control for stochastic distribution sys-tem (SDS). For example in [18] and [19], non-GaussianSDS is approximated by rational square root B-spline model.Based the obtained system, the adaptive observer is designedto estimate the fault. Then, the fault-tolerant controller isdesigned via the estimated information. In [9] and [11],the linear singular non-Gaussian SDS is considered. Thenbased on the diagnostic information, some optimal fault-tolerant control algorithms are designed. However, it is worthpointing out that few researches on fault-tolerant shape con-trol for SR-RBS model in the existing literature.

In addition, time-delay is very common in actual indus-trial production process. The existence of time-delay affectsthe system stability [20], [21]. Therefore, it is necessary to

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Y. Cheng et al.: Delay-Dependent Fault-Tolerant Shape Control for SDSs

consider the effect of time-delay in fault tolerant control [22].In [22], a new delay-dependent fault diagnosis algorithm isproposed for continuous SDS with time-delays. In order toimprove the fault detection sensitivity in [23], an optimal faultdetection observer is designed by fully considering the infor-mation of time-delays. However, it is noted that time-delaysare not considered in fault-tolerant shape control (FTSC)in [12].

Therefore, inspired by the above discussions, in this paper,a new delay-dependent FTSC scheme is studied for SDS withtime-delay, where the partial loss of actuator effectivenessis considered. In this framework, the SR-RBS expansiontechnique is used to approximate the output PDF, thus theoriginal stochastic system is transformed into a dynamicweight system with time-delay. Based on this, a new delay-dependent FTSC strategy is proposed, which included twoaspects, namely a normal shape control law and an adaptivecompensation shape control law. The main contributions ofthis paper include: (i) The effect of time-delay is consideredin the designed fault tolerant shape control based on squareroot rational B-spline approximation. (ii) Delay-dependentguaranteed cost performance index is used to effectivelymake the output PDF track a given target distribution.(iii) A delay-dependent adaptive estimation algorithm isderived to compensate the fault online. Finally, numericalexamples are presented to demonstrate the validity of theproposed design.Notation: In order to facilitate the description, the sym-

metric position of the symmetric matrix is represented bythe ‘‘*’’. And the superscripts ‘‘T ’’, ‘‘−1’’, ‘‘†’’ stand formatrix transposition, inverse, and pseudo inverse respectively,and define ‘‘sym(AB) = AB+ BTAT ’’.

II. PROBLEM DESCRIPTIONFor a dynamic stochastic system, u(t) is the control input,and the output η(t) ∈ [a, b] is assumed to be an uniformlybounded stochastic process. Then, under the action of u(t),the probability P of the output η(t) lying in [a, δ](a ≤ δ ≤ b)is defined as follows.

P(a ≤ η(t) ≤ δ|u(t)) =

δ∫a

ϕ(y, u(t))dy (1)

where ϕ(y, u(t)) is the PDF of the output η(t) under thecontrol input u(t) at time t , and y is the variable of thePDF function.

The output PDF ϕ(y, u(t)) is assumed to be measurable,continuous, and bounded. The following SR-RBS model isusually used to approximate ϕ(y, u(t)).

√ϕ(y, u(t)) =

n∑i=1

vi(u(t))Ci(y)√n∑

i=1,j=1vi(u(t))vj(u(t))

b∫aCi(y)Cj(y)dy

(2)

where C(y) = [C1(y),C2(y), . . . ,Cn(y)], Ci(y) ≥ 0, denotesthe independent basis function vector given in advance,

v(t) = [v1(u(t)), v2(u(t)), . . . , vn(u(t))]T , v(t) 6= 0, is thedynamic weight vector and vi(u(t))(i = 1, 2, ..n) are inde-pendent from each other.

A. SYSTEM MODELIn this paper, the dynamic relationship between u(t) and v(t)can be described as follows [12]:{

x(t) = Ax(t)+ Adx(t − d(t))+ Gg(x(t))+ Bu(t)v(t) = Ex(t)

(3)

where x(t) is the state vector, d(t) is the time-varying delayand satisfies 0 ≤ d(t) ≤ d , where d is a constant time-delay and 0 ≤ d(t) ≤ µ ≤ 1. A, Ad , G, B, E are systemmatrices with appropriate dimension. The initial value of stateis expressed as x(t) = φ(t)(−d ≤ t ≤ 0).Remark 1: Time delays are also considered in [9] and [25],

where time delays came from the sampling intervals anditerative learning observer respectively. A derivative of time-varying delay is not considered in [25] and a constant timedelay is studied in [9]. However, in this paper, the slowvarying time delay is considered, which ismore a general casethan those considered in [9] and [25].

For any x1(t) and x2(t), the nonlinear function g(x(t)) meetsthe following conditions:{

g(0) = 0||g(x1(t))− g(x2(t))||2 ≤ ||U (x1(t)− x2(t))||2

(4)

where U is a given constant matrix.Considering the loss of actuator control effectiveness,

the dynamic system can be written as:x(t) = Ax(t)+ Adx(t − d(t))+ Gg(x(t))+ Bωu(t)v(t) = Ex(t)√ϕ(y, u(t)) =

C(y)v(t)√vT (t)Lv(t)

(5)

Here, ω can be described as follows [25]:

ω = diag{ω1, ω2, . . . , ωm}, ωi ∈ [ωi, ωi],

0 ≤ ωi ≤ ωi ≤ 1

where m is the number of inputs, ωi is an unknown con-stant, ωi and ωi are the lower and upper bounds of ωi, and0 ≤ ωi ≤ ωi ≤ 1 represents the loss of control effectiveness.When ωi = ωi = 1 or 0 < ωi < 1, it indicates that theith actuator is fault-free or the ith actuator is partially disabledrespectively. When ωi = 0, it indicates that the ith actuator istotally disabled.

To derive the delay-dependent FTSC strategy for thesystem (5), a desired PDF can be described as√

ϕ∗(y, u(t)) =C(y)vg(t)√vTg (t)Lvg(t)

, y ∈ [a, b] (6)

where vg(t) is the desired weight vector related to C(y), L =b∫aCT (y)C(y)dy. Thus, the target of the shape control becomes

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finding the control u(t) so that ϕu(y, t) can track ϕ∗(y, t).If vε(t) = v(t)− vg(t)→ 0, then

√ϕu(y, t)−

√ϕ∗(y, t)→ 0.

Therefore, the control problem of FTSC for output PDF canbe formulated as tracking vg(t) by v(t).

B. CONTROL OBJECTIVESFor the system (5), the design of controller in this paper istwofold.

(1) Under normal situation, the system is asymptoticallystable, and the dynamic weight v(t) can track the desiredweight vg(t) with zero steady-state error.

limt→∞

v(t)− vg(t) = 0 (7)

And for givenmatrices Z1 > 0, Z2 > 0, Z3 > 0, the controllercan achieve the best performance by minimizing the upperbound of the cost function as follows

Jt =∫∞

0[ξT (t)Z1ξ (t)+ xT (t)Z2x(t)+ uT (t)Z3u(t)]dt (8)

where ξ (t) =∫ t0 (v(τ )− vg(τ ))dτ

(2) In the case of actuator failure, the closed-loop system isstable and the dynamic weight v(t) can also track the desiredweight vg(t) with zero steady-state error.In order to obtain delay-dependent results, the following

Lemma is needed.Lemma 1 [24]: For any matrix R > 0, x ∈ [α1, α2],

the following inequality holds:

−

∫ α2

α1

xT (s)Rx(s)ds ≤1

α2 − α1$ Tχ$ (9)

where

χ =

−4R −2R 6R∗ −4R 6R∗ ∗ −12R

$ T= [xT (α2) xT (α1) xT ], x =

1α2 − α1

∫ α2

α1

xT (s)ds

Remark 2: In [11] and [27], the premise of fault-tolerantcontrol is the observer design, which is not needed in ourproposed method. Meanwhile, it is worth pointing out thattime delay is not considered in [11] and [27]. For time-varyingdelay, a new technique based on Lemma 1 will be used in thesubsequent delay-dependent fault-tolerant control design.

III. MAIN RESULTSA. OUTPUT PDF TRACKING FOR FAULT-FREE CASELet x(t) = [ξT (t) xT (t)]T , system (5) can be written as

˙x(t) = Ax(t)+ Ad x(t−d(t))+ Gg(x(t))+ Bωu(t)+ Hvg(t)

(10)

where

g(x(t)) = g(x(t)), A =[0 E0 A

], Ad =

[0 00 Ad

]G =

[0G

], B =

[0B

], H =

[−I0

]

In the normal case, ω = I in system (10), the followingcontroller is designed

uN (t) = KN x(t)− BT (BBT )†Gg(x(t)) (11)

where KN is the controller gain. Thus, the closed-system canbe written as

˙x(t) = (A+ BKN )x(t)+ Ad x(t − d(t))+ Hvg(t) (12)

For the closed-system (12), the following result can beobtained via the method in [26].Theorem 1: Given constants γ > 0, 0 ≤ µ ≤ 1,

the matrices Z = diag{Z1,Z2} > 0 and Z3 > 0, if thereexist a constant κ > 0, matrices P > 0, Q > 0, so that thefollowing inequality holds

� =

�11 PAd PH∗ −(1− µ)Q 0∗ ∗ −γ I

< 0 (13)

where

�11 = sym(P(A+ BKN ))+ Q+ Z + KTN Z3KN + κI

then, system (12) is asymptotically stable, v(t) can convergeto the weight vg(t) and the performance index (8) has an upperbound

Jt ≤ xT (0)Px(0)+∫ 0

−dxT (s)Qx(s)ds+ γ

∫∞

0vTg (t)vg(t)dt

(14)

Proof: For the closed-loop system (12), the followingLyapunov candidate is chosen:

V (x(t)) = xT (t)Px(t)+∫ t

t−d(t)xT (s)Qx(s)ds

where P > 0 and Q > 0. By used the similar method in [12],we can obtain

V (x(t))− γ vTg (t)vg(t) ≤ θT (t)�θ (t) (15)

where θ (t) = [xT xT (t − d(t)) vTg (t)]T ,

� =

sym(P(A+ BKN ))+ Q PAd PH∗ −(1− µ)Q 0∗ ∗ −γ I

For any θ (t) 6= 0, if (13) holds, we can get

V (x(t)) < −κ||x(t)||2 + γ ||vg(t)||2 (16)

Thus, V (x(t)) < 0 only if ||x(t)|| >√γκ||vg(t)|| holds, which

means

||x(t)|| ≤ max{||x(0)||,

√γ

κ||vg(t)||} (17)

Next, we will prove that the equilibrium point of (12) forgiven vg(t) is unique via the method in [26].Assuming that x1(t), x2(t) are two trajectories of

system (12), and letting ε(t) = x1(t) − x2(t) and initialcondition ε(t) = 0, we can obtain

ε(t) = (A+ BKN )ε(t)+ Adε(t − d(t)) (18)

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The following Lyapunov candidate is chosen:

V (ε(t)) = εT (t)Pε(t)+∫ t

t−d(t)εT (s)Qε(s)ds

Similar to (15), we can get

V (ε(t)) = θT1 (t)�1θ1(t)

where θ1(t) = [εT (t) ε(t − d(t))]T and

�1 =

[sym(P(A+ BKN ))+ Q PAd

∗ −(1− µ)Q

]From (13), it can be seen that V (ε(t)) < −κ||ε(t)||2,so ε(t) = 0 is the unique asymptotically stable equi-librium point of the system (18), which means that theclosed-loop system (12) also has a unique stable equilibrium.So limt→∞ ξ (t) = ξε , and limt→∞ ξ (t) = 0, where ξε is theequilibrium of ξ (t). Consequently, limt→∞[v(t)− vg(t)] = 0holds, which means that v(t) converges to the desired weightvg(t).

Finally, we will prove that the tracking performance isguaranteed

Jt =∫∞

0[ξT (t)Z1ξ (t)+ xT (t)Z2x(t)+ uTZ3u(t)]dt

≤ −

∫∞

0V (x(t))dt + γ

∫∞

0vTg (t)vg(t)dt

≤ − limt→∞

[V (x(t))− V (x(0))]+ γ∫∞

0vTg (t)vg(t)dt

≤ xT (0)Px(0)+∫ 0

−dxT (s)Qx(s)ds

+ γ

∫∞

0vTg (t)vg(t)dt

This completes the proof.It is clear that inequality (13) in Theorem 1 is not linear

matrix inequality (LMI), the similar method can be foundin [26]. In order to obtain a feasible LMI condition, an optimaltracking controller is designed as follows.Theorem 2: For system (12), given constants γ > 0,

0 ≤ µ ≤ 1, the matrices Z = diag{Z1,Z2} > 0 and Z3 > 0,if there exist nonsingular matrix V ,31 > 0, 32 > 0, P > 0,

Q > 0, constant κ > 0 and any matrix W , so that thefollowing optimization problem has feasible solutions:

minκ,P,Q,V ,W

{31 + trace(32)}

Subject to the following LMIs (19), as shown at the bottomof this page, where

∫ 0−d x(s)x

T (s)ds = NNT . Then, KN =WV−1 is an optimal control gain, which ensures the mini-mization of the guaranteed cost (14) for system (12).Proof: Pre-multiplying (13) by ψ−1, and post-multiplying

by ψ−T = (ψ−1)T , the following inequality holds: �1 Ad H∗ −(1− µ)Q 0∗ ∗ −γ I

< 0 (22)

where

ψ =

P 0 0∗ I 0∗ ∗ I

�1 = sym((A+ BKN )V )+ V T (Q+ Z + KT

N Z3KN + κI )V

V = P−T , KN = WV−1

Applying the Schur complement to (20), as shown at thebottom of this page, and (21), as shown at the bottom of thispage, we can get

xT (0)Px(0) < 31, NT QN < 32

and ∫ 0

−dxT (s)Qx(s)ds =

∫ 0

−dtr(xT (s)Qx(s))ds

= tr(Q1/2∫ 0

−dxT (s)x(s)dsQ1/2)

= tr(Q1/2NNT Q1/2)

= tr(NT QN )

< tr(32)

Thus

Jt ≤ 31 + tr(32)+ γ∫∞

0vTg (t)vg(t)dt (23)

sym(AV + BW ) Ad H V T Q V T W T V T

∗ −(1− µ)Q 0 0 0 0 0∗ ∗ −γ I 0 0 0 0∗ ∗ ∗ −Q 0 0 0∗ ∗ ∗ ∗ −Z−1 0 0∗ ∗ ∗ ∗ ∗ −Z−13 0∗ ∗ ∗ ∗ ∗ ∗ −κ−1I

< 0 (19)

[−31 xT (0)PPx(0) −P

]< 0 (20)[

−32 NT QQN −Q

]< 0 (21)

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So, the minimization of {31+trace(32)} implies the mini-mization of the guaranteed cost index in (23). This completesthe proof.

B. FAULT-TOLERANT SHAPE CONTROL FORACTUATOR FAULT CASEIn order to reduce the negative effect caused by the actuatorfault, an adaptive fault compensation control law uad (t) isadded to the normal control law uN (t). In the normal case,uad (t) = 0, while uad (t) 6= 0 is in the faulty case.

The target model is incorporated in the proposed FTSC asfollows:{˙x(t) = Ax(t)+ Ad x(t − d(t))+ Gg(x(t))+ Bωr(t)v(t) = Ex(t)

(24)

where ω = diag{ω1, ω2, . . . , ωm} denotes the estimatedvalue of control effectiveness factor, v(t) is the estimatedvalue of v(t), r(t) is compensation term. Let x(t) =[ξT (t) xT (t)]T , x(t) denotes the state of the augmented systemwhen fault occurs.

So we can get the following augmented system:

˙x(t) = Ax(t)+ Ad x(t−d(t))+ Gg(x(t))+ Bωr(t)+ Hvg(t)

(25)

where ξ (t) =∫ t0 (v(τ ) − vg(τ ))dτ , and A, Ad , G, B, H are

same with (10).Let the state error vector of the augmented system

e(t) = x(t) − x(t), the control input is then written asu(t) = r(t) − Fe(t). Let B = [b1, b2, . . . , bm], r(t) =[r1(t), r2(t), . . . , rm(t)]T and ge(t) = g(x(t)) − g(x(t)), so,the state error of the augmented system can be written as:

e(t) = (A+ BωF)e(t)+ Ade(t − d(t))+ Gge(t)

+

m∑i=1

biωiui(t) (26)

where ωi = ωi − ωi.For the state error system (26), we can obtain the following

result.Theorem 3: Given constants 0 ≤ µ ≤ 1, β1, β2, β3, β4, β5

and time delay upper bound d , if there exist matrices P > 0,Q > 0, R > 0, M1, N1, such that the followinginequality holds (27), as shown at the bottom of this page,

where

811 = Q+ UTU −4dR+ sym(M1A+ N1)

812 = M1Ad + β1(M1A+ N1)T

822 = −(1− µ)Q+ sym(β1M1Ad )814 = P−M1 + β3(M1A+ N1)T

824 = −β1M1 + β3(M1Ad )T

844 = dR− sym(β3M1)

815 =6dR+ β4(M1A+ N1)T

816 = M1G+ β5(M1A+ N1)T

826 = β1M1G+ (β5M1Ad )T

836 = β2M1G846 = β3M1G− β5MT

1856 = β4M1G866 = sym(β5M1G)− I

then the system (26) is stable with F = (M1Bω)†N1, v(t)converges to vg(t). And ωi is determined according to thefollowing adaptive estimation algorithm.

˙ωi=Proj[ωi,ωi]{−liζT (t)Mbiui(t)}

=

0,

{if ωi = ωi, −liζ

T (t)Mbiui(t) ≤ 0or ωi = ωi, −liζ T (t)Mbiui(t) ≥ 0

−liζ T (t)× Mbiui(t), otherwise

(28)

where

ζ T (t) = [eT (t) eT (t − d(t)) eT (t − d) eT (t) eT (t) gTe (t)]

M = [M1 β1M1 β2M1 β3M1 β4M1 β5M1]T

Proj{·} denotes the projection operator, and li > 0 is theadaptive learning gain.Proof: The following Lyapunov candidate is chosen:

V (e(t)) = eT (t)Pe(t)+∫ t

t−d(t)eT (s)Qe(s)ds

+

∫ 0

−d

∫ t

t+βeT (s)Re(s)ds+

m∑i=1

ω2i

li

+

∫ t

0[‖ Ue(s) ‖2 − ‖ ge(s) ‖2]ds

8 =

811 812 −2dR+ β2(M1A+ N1)T 814 815 816

∗ 822 β2ATdMT1 824 β4ATdM

T1 826

∗ ∗ −4dR −β2M1

6dR 836

∗ ∗ ∗ 844 −β4MT1 846

∗ ∗ ∗ ∗ −12dR 856

∗ ∗ ∗ ∗ ∗ 866

< 0 (27)

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By differentiating V (e(t)) along the trajectory of the sys-tem (26), we can obtain

V (e(t)) = 2eT (t)Pe(t)+ eT (t)Qe(t)

− (1− d(t))eT (t − d(t))Qe(t − d(t))

+ eT (t)dRe(t)−∫ t

t−deT (s)Re(s)ds

+ eT (t)UTUe(t)− gTe (t)ge(t)+ 2m∑i=1

ωi ˙ωi

li

(29)

By using Lemma 1, the following inequality holds:

−

∫ t

t−deT (s)Re(s)ds ≤

1d$ Tχ$ (30)

where

χ =

−4R −2R 6R∗ −4R 6R∗ ∗ −12R

$ T= [eT (t) eT (t − d) eT (t)], e(t) =

1d

∫ t

t−deT (s)ds

For any matrix M , the following equation holds

2ζ T (t)M [(A+ BωF)e(t)+ Ade(t − d(t))

+ Gge(t)+m∑i=1

biωiui(t)− e(t)] = 0 (31)

thus

V (e(t)) = 2eT (t)Pe(t)+ eT (t)Qe(t)

− (1− d(t))eT (t − d(t))Qe(t − d(t))

+ eT (t)dRe(t)−∫ t

t−deT (s)Re(s)ds

+ eT (t)UTUe(t)− gTe (t)ge(t)+ 2m∑i=1

ωi ˙ωi

li

+ 2ζ T (t)M [(A+ BωF)e(t)+ Gge(t)

+ Ade(t − d(t))+m∑i=1

biωiui(t)− e(t)]

≤ ζ T (t)8ζ (t)+ 2m∑i=1

ωiζT (t)Mbiui(t)

+ 2m∑i=1

ωi ˙ωi

li(32)

where

8 =

811 812 813 814 815 816∗ 822 ATdM

T3 824 ATdM

T5 826

∗ ∗ −4dR −M3

6dR 836

∗ ∗ ∗ 844 −MT5 846

∗ ∗ ∗ ∗ −12dR 856

∗ ∗ ∗ ∗ ∗ 866

811 = Q+ UTU −4dR+ sym(M1(A+ BωF))

812 = M1Ad + [M2(A+ BωF)]T

822 = −(1− µ)Q+ sym(M2Ad )

813 = −2dR+ [M3(A+ BωF)]T

814 = P−M1 + [M4(A+ BωF)]T

824 = −M2 + (M4Ad )T

844 = dR− sym(M4)

815 =6dR+ [M5(A+ BωF)]T

816 = M1G+ [M6(A+ BωF)]T

826 = M2G+ (M6Ad )T

836 = M3G

846 = M4G−MT6

856 = M5G

866 = sym(M6G)− I

It is noted that ωi are constants, so ˙ωi = ˙ωi holds. If (28) ischosen as the adaptive algorithm to estimate ωi, there will betwo cases: 1) when ω = ωi or ω = ωi, obviously ˙ωi = 0,and 2

∑mi=1 ωiζ

T (t)Mbiui(t) ≤ 0; 2) otherwise, the followinginequality is true:

2m∑i=1

ωi ˙ωi

li≤ −2

m∑i=1

ωiζT (t)Mbiui(t) (33)

Noted that M2 = β1M1,M3 = β2M1,M4 = β3M1,M5 =

β4M1,M6 = β5M1, and M1BωF = N1. Then we can get

V (e(t)) ≤ ζ T (t)8ζ (t)

Therefore, V (e(t)) ≤ 0 if (27) holds , which means theerror system (26) is stable. So V (e(t)) ∈ L∞, it implies thate(t) ∈ L∞. From (26), it can be seen that e(t) ∈ L∞, and∫

∞

0||e(t)||2 dt ≤ V (0)− V (∞) ≤ ∞

Hence, e(t) ∈ L∞ ∩ L2, using the Barbalat Lemma [17],we can get limt→∞ e(t) = 0, it means that the dynamicweight v(t) converges to the desired weight vg(t). This com-pletes the proof.

The signal r(t) can be computed as

r(t) = ω−1[KN x(t)− BT (BBT )†Gg(x(t))]

So

u(t) = r(t)− Fe(t)

= ω−1[KN x(t)− BT (BBT )†Gg(x(t))]− Fe(t)

= KN x(t)− BT (BBT )†Gg(x(t))

+ ω−1(I − ω)KN x(t)

+ (KN − F)e(t)− BT (BBT )†Gge(t)

+ ω−1(I − ω)BT (BBT )†Gg(x(t))

= uN (t)+ uad (t) (34)

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FIGURE 1. Control effectiveness estimation with actuator faults (d=1s).

where

uN (t) = KN x(t)− BT (BBT )†Gg(x(t))

uad (t) = ω−1(I − ω)KN x(t)+ (KN − F)e(t)

− BT (BBT )†Gge(t)

+ ω−1(I − ω)BT (BBT )†Gg(x(t))

It can be seen from the above that the FTSC strategyincludes a normal shape control law and an adaptive com-pensation shape control law. In the normal case ω = I ,uad (t) = 0, when faults occur, ω 6= I , uad (t) 6= 0, and uad (t)is activated for adaptive control compensation.

IV. SIMULATIONIn order to verify the effectiveness of the proposed method,in this section, the simulation is carried out under theMATLAB environment. Consider a stochastic system givenin [12], the PDF is formulated with the dynamic weightv(t) = [v1(t), v2(t), v3(t)] and correspondingly, the B-splinefunctions Ci(y), i = 1, 2, 3 are

C1(y) = 0.5(y− 5)2 I1 + (−y2 + 13y− 41.5)I2+ 0.5(y− 8)2 I3

C2(y) = 0.5(y− 6)2 I2 + (−y2 + 15y− 55.5)I3+ 0.5(y− 9)2 I4

C3(y) = 0.5(y− 7)2 I3 + (−y2 + 17y− 71.5)I4+ 0.5(y− 10)2 I5

where Ii, (i = 1, 2, 3, 4, 5) are the unit pulse functionsdefined as

Ii =

{1, y ∈ [i+ 4, i+ 5]0, otherwise

And other parameters are set as follows:

A =

0 1 00 0 1−1 −2 −4

, B =

0.5 0.3 00 0.9 0.1−0.5 −0.2 0.1

,G =

0.1 0 00 0.1 00 0 0.1

, E =

0.9 0 00 0.9 00 0 0.9

,Ad = diag{0.2, 0.2, 0.2},

g(x) =1

1+ e−2x− 0.5

FIGURE 2. Control effectiveness estimation with actuator faults (d=0.1s).

FIGURE 3. Tracking performance on weights with actuator faults (d=1s).

FIGURE 4. Tracking performance on weights with actuator faults(d=0.1s).

The desired PDF φ∗(y, t) is chosen based on (6), with vg =[5, 3, 6]T . The initial condition in the dynamic weight systemis given by (5) with the above matrices and x(0) = [1, 1, 1]T .To validate the time-delay influence, we consider two caseswith d = 1s and d = 0.1s.

A. ACTUATOR FAULT CASEWe assume that the control effectiveness factors are reducedfrom 100% to 40%, 60%, and 80%, i.e., ω1 = 0.4, ω2 = 0.6,and ω3 = 0.8, respectively. Based on Theorem 3, whend = 1s and d = 0.1s, these control effectiveness factorsare estimated, which are shown in Figure 1 and Figure 2,respectively. Figure 3 and Figure 4 illustrate the trackingperformance on the weights, and the output PDF distribu-tions are plotted in Figure 5 and Figure 6. From the aboveresults, we can see that the control effectiveness factors can be

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Y. Cheng et al.: Delay-Dependent Fault-Tolerant Shape Control for SDSs

FIGURE 5. Tracking performance on PDF shape with actuator faults(d=1s).

FIGURE 6. Tracking performance on PDF shape with actuator faults(d=0.1s).

FIGURE 7. Control effectiveness estimation for delay-dependent anddelay-independent case.

estimated as ω1 = 0.4, ω2 = 0.6, and ω3 = 0.8. And fromFigure 5 and Figure 6, we can see that the proposed FTSCcan make the output PDF track a given PDF. Compared withthe case d = 1s, the tracking performance is better whend = 0.1s.

B. COMPARISON BETWEEN DELAY-DEPENDENT CASEAND DELAY-INDEPENDENT CASEIn order to further show delay-dependent result has less con-servatism than delay-independent result, we further show thecomparison results in Figures 7-9 about the control effec-tiveness factors estimation, the tracking of weights and thetracking performance on output PDF shape, respectively.

FIGURE 8. Tracking performance on weights for delay-dependent anddelay-independent case.

FIGURE 9. Tracking performance on PDF shape for delay-dependent anddelay-independent case.

From these simulations, we can see that delay-dependenttracking will have more effectiveness than delay-independentcase. On the other hand, the tracking performance will beworse when time-delay is increased.

V. CONCLUSIONIn this paper, the FTSC for stochastic distribution systemwith time-delay is explored. Based on the adaptive controltechnique, an adaptive FTSC strategy is proposed to treat thepartial loss of actuator effectiveness. The simulation resultsshow that the proposed strategy can effectively make theoutput PDF track the given target distribution. Future workwill be performed by adopting consensus control [29], [30]to treat stochastic distribution system.

REFERENCES[1] G. Gregorcic and G. Lightbody, ‘‘Local model network identification

with Gaussian processes,’’ IEEE Trans. Neural Netw., vol. 18, no. 5,pp. 1404–1423, Sep. 2007.

[2] H. Li, L. Bai, L. Wang, Q. Zhou, and H. Wang, ‘‘Adaptive neural controlof uncertain nonstrict-feedback stochastic nonlinear systems with outputconstraint and unknown dead zone,’’ IEEE Trans. Syst., Man, Cybern.,Syst., vol. 47, no. 8, pp. 2048–2059, Aug. 2017.

[3] Y. G. Niu, D. W. C. Ho, and X. Wang, ‘‘Robust H∞ control for nonlinearstochastic systems: A sliding-mode approach,’’ IEEE Trans. Autom. Con-trol, vol. 53, no. 7, pp. 1695–1701, Aug. 2008.

[4] L. Guo and H.Wang, ‘‘Minimum entropy filtering for multivariate stochas-tic systems with non-Gaussian noises,’’ IEEE Trans. Autom. Control,vol. 51, no. 4, pp. 695–700, Apr. 2006.

[5] H. Wang, Bounded Dynamic Stochastic Systems: Modelling and Control.London, U.K.: Springer-Verlag, 2000.

12734 VOLUME 6, 2018

Y. Cheng et al.: Delay-Dependent Fault-Tolerant Shape Control for SDSs

[6] L. Guo and H. Wang, ‘‘Fault detection and diagnosis for general stochasticsystems using B-spline expansions and nonlinear filters,’’ IEEE Trans.Circuits Syst. I, Reg. Papers, vol. 52, no. 8, pp. 1644–1652, Aug. 2005.

[7] Y. Yi, L. Guo, and H. Wang, ‘‘Constrained PI tracking control for outputprobability distributions based on two-step neural networks,’’ IEEE Trans.Circuits Syst. I, Reg. Papers, vol. 56, no. 7, pp. 1416–1426, Jul. 2009.

[8] H. Wang and H. Yue, ‘‘A rational spline model approximation and controlof output probability density functions for dynamic stochastic systems,’’Trans. Inst. Meas. Control, vol. 25, no. 2, pp. 93–105, 2003.

[9] L. Yao, J. Qin, H. Wang, and B. Jiang, ‘‘Design of new fault diagnosisand fault tolerant control scheme for non-Gaussian singular stochasticdistribution systems,’’ Automatica, vol. 48, no. 9, pp. 2305–2313, 2012.

[10] J.-L. Zhou, H. Yue, and H. Wang, ‘‘Shaping of output probability densityfunctions based on the rational square-root B-spline model,’’ Acta Autom.Sinica, vol. 31, no. 3, pp. 344–351, 2005.

[11] L. Yao, J. Qin, A. Wang, and H. Wang, ‘‘Fault diagnosis and fault-tolerantcontrol for non-Gaussian non-linear stochastic systems using a rationalsquare-root approximationmodel,’’ IET Control Theory Appl., vol. 7, no. 1,pp. 116–124, 2013.

[12] T. Li, G. Li, and Q. Zhao, ‘‘Adaptive fault-tolerant stochastic shape controlwith application to particle distribution control,’’ IEEE Trans. Syst., Man,Cybern., Syst., vol. 45, no. 12, pp. 1592–1604, Dec. 2015.

[13] X.-Z. Jin, Z. Zhao, and Y.-G. He, ‘‘Insensitive leader-following consen-sus for a class of uncertain multi-agent systems against actuator faults,’’Neurocomputing, vol. 272, pp. 189–196, Jan. 2018.

[14] L. Yao and L. Feng, ‘‘Fault diagnosis and fault tolerant control fornon-Gaussian time-delayed singular stochastic distribution systems,’’Int. J. Control, Autom. Syst., vol. 14, no. 2, pp. 435–442, 2016.

[15] J. Qiu, Y. Wei, and L. Wu, ‘‘A novel approach to reliable control ofpiecewise affine systems with actuator faults,’’ IEEE Trans. Circuits Syst.II, Exp. Briefs, vol. 64, no. 8, pp. 957–961, Aug. 2017.

[16] T. Li and W. X. Zheng, ‘‘Networked-based generalised H∞ fault detectionfiltering for sensor faults,’’ Int. J. Syst. Sci., vol. 46, no. 5, pp. 831–840,2015.

[17] X.-Z. Jin, Y.-G. He, andY.-G. He, ‘‘Finite-time robust fault-tolerant controlagainst actuator faults and saturations,’’ IET Control Theory Appl., vol. 11,no. 4, pp. 550–556, 2017.

[18] L.-N. Yao, A. P. Wang, and H. Wang, ‘‘Fault detection, diagnosis andtolerant control for non-Gaussian stochastic distribution systems usinga rational square-root approximation model,’’ Int. J. Model., Identificat.Control, vol. 3, no. 2, pp. 162–172, 2008.

[19] L. Yao, V. Cocquempot, and H. Wang, ‘‘Fault diagnosis and fault tolerantcontrol for non-Gaussian singular stochastic distribution systems,’’ inProc.8th IEEE Int. Conf. Control Autom. (ICCA), Xiamen, China, Jun. 2010,pp. 1949–1954.

[20] F. Lei, X. Xu, T. Li, and G. Song, ‘‘Attitude tracking control for Mars entryvehicle via T-S model with time-varying input delay,’’ Nonlinear Dyn.,vol. 85, no. 3, pp. 1749–1764, 2016.

[21] Q. K. Shen, B. Jiang, and P. Shi, ‘‘Active fault-tolerant control againstactuator fault and performance analysis of the effect of time delay due tofault diagnosis,’’ Int. J. Control, Autom. Syst., vol. 15, no. 2, pp. 537–546,2017.

[22] T. Li, Y. Yi, L. Guo, and H. Wang, ‘‘Delay-dependent fault detectionand diagnosis using B-spline neural networks and nonlinear filters fortime-delay stochastic systems,’’ Neural Comput. Appl., vol. 17, no. 4,pp. 405–411, 2008.

[23] L. Guo, Y. M. Zhang, H. Wang, and J. C. Fang, ‘‘Observer-based optimalfault detection and diagnosis using conditional probability distributions,’’IEEE Trans. Signal Process., vol. 54, no. 10, pp. 3712–3719, Oct. 2006.

[24] A. Seuret and F. Gouaisbaut, ‘‘Wirtinger-based integral inequality: Appli-cation to time-delay systems,’’ Automatica, vol. 49, no. 9, pp. 2860–2866,2013.

[25] R. Sakthivel, H. R. Karimi, M. Joby, and S. Santra, ‘‘Resilient sampled-data control for Markovian jump systems with an adaptive fault-tolerantmechanismm,’’ IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 64, no. 11,pp. 1312–1316, Nov. 2017, 10.1109/TCSII.2017.2669102.2017.

[26] L. Chen, T. Li, Z. Dai, and L. Zhong, ‘‘Probability density functionshape control for stochastic systems with time-delay,’’ in Proc. 36th Chin.Control Conf., Dalian, China, Jul. 2017, pp. 1729–1733.

[27] S. K. Kommuri, M. Defoort, H. R. Karimi, and K. C. Veluvolu,‘‘A robust observer-based sensor fault-tolerant control for PMSM in elec-tric vehicles,’’ IEEE Trans. Ind. Electron., vol. 63, no. 12, pp. 7671–7682,2016.

[28] X. Jin, J. Qin, Y. Shi, and W. X. Zheng, ‘‘Auxiliary fault tolerantcontrol with actuator amplitude saturation and limited rate,’’IEEE Trans. Syst., Man, Cybern., Syst., to be published, doi:10.1109/TSMC.2017.2752961.2017.

[29] J. Qin, W. Fu, H. Gao, and W. X. Zheng, ‘‘Distributed k-means algorithmand fuzzy C-means algorithm for sensor networks based on multiagentconsensus theory,’’ IEEE Trans. Cybern., vol. 47, no. 3, pp. 772–783,Mar. 2017.

[30] J. Qin, W. Fu, W. X. Zheng, and H. Gao, ‘‘On the bipartite con-sensus for generic linear multiagent systems with input saturation,’’IEEE Trans. Cybern., vol. 47, no. 8, pp. 1948–1958, Aug. 2017, doi:10.1109/TCYB.2016.2612482.2017.

YU CHENG received the B.Sc. degree from theDepartment of Automatic Control, Kunming Uni-versity of Science and Technology, Kunming,China, in 1999, and the M.Sc. degree in con-trol theory and control engineering from TsinghuaUniversity, Beijing, in 2002. He is currently pur-suing the Ph.D. degree with Beihang University.His research interests include fault detection anddiagnosis.

LONG CHEN received the B.Sc. degree from theNanjing University of Information Science andTechnology in 2013, where he is currently pursu-ing the M.S. degree. His research interests includefault detection and diagnosis and optimal control.

TAO LI received the Ph.D. degree from South-east University, Nanjing, China. He is currentlya Professor with the Nanjing University of Infor-mation Science and Technology, Nanjing. Hehas published over 50 papers in journals withover 800 citations. His current research interestsinclude fault detection and fault-tolerant controlfor time-delay systems. He was a recipient of theOutstanding Young Scholar of Jiangsu Province,China. He has completed the visiting scholar fel-

lowships with the University of Alberta, Edmonton, AB, Canada, the Uni-versity of Western Sydney, South Penrith, NSW, Australia, The Universityof Hong Kong, and The City University of Hong Kong.

HAIPING DU (M’09–SM’17) received the Ph.D.degree in mechanical design and theory fromShanghai Jiao Tong University, Shanghai, China,in 2002. He was a Post-Doctoral Research Asso-ciate with the University of Hong Kong from2002 to 2003 and with the Imperial College Lon-don from 2004 to 2005, and was a Research Fellowwith the University of Technology, Sydney, from2005 to 2009. He is currently a Professor with theSchool of Electrical, Computer and Telecommu-

nications Engineering, University of Wollongong, Wollongong, NSW, Aus-tralia. His research interests include vibration control, vehicle dynamics andcontrol systems, robust control theory and engineering applications, electricvehicles, robotics and automation, and smart materials and structures. Hewas a recipient of the Australian Endeavour Research Fellowship 2012. Heis a Subject Editor of the Journal of Franklin Institute, an Associate Editor ofthe IEEE Control Systems Society Conference, an Editorial Board Memberfor some international journals, such as the Journal of Sound and Vibration,IMechE Journal of Systems and Control Engineering, and the Journal of LowFrequency Noise, Vibration and Active Control, and a Guest Editor of IETControl Theory and Application,Mechatronics, and Advances inMechanicalEngineering.

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