CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 7, NO. …

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 7, NO. 2, MARCH 2021 307

Cascaded Sliding-mode Observer and ItsApplications in Output Feedback Control Part II:

Adaptive Output Feedback ControlTianhao Wen, Yang Liu , Member, IEEE, Q. H. Wu, Fellow, IEEE, Fellow, CSEE, and Luonan Qiu

Abstract—Part II proposes a cascaded sliding-mode observerbased output feedback controller for control of multi-inputmulti-output (MIMO) system. The controller, designed based onfeedback linearization control strategy, requires the informationof the states and perturbations of the system for realization ofdisturbance rejection. The observer studied in part I [1] is thenutilized to provide the accurate estimates of states and pertur-bations. As is proved, the proposed observer-based controllercan ensure Lyapunov stability of the closed-loop system. Also, itcan be used for output tracking control. Simulation studies arecarried out on a single-wind-energy-conversion-system-infinite-bus (SWNCSIB) system to test the performances of the proposedobserver-based output feedback controller.

Index Terms—Sliding-mode observer, stability, outputfeedback control.

NOMENCLATURE

z Derivative of variable/vector z with respect totime.

z, ˆz Observed value of the state variable/ statevector z.

z z − z.˜z z − ˆz.diag(b) Diagonal matrix with the ith diagonal element

equal to the ith element of the column/rowvector b.

blockdiag Block diagonal matrix with Vi the ith diagonal(V1, . . . , Vn) block.∂A The boundary of set A.A The closure of set A.λmax(P ) The maximum eigenvalue of the symmetric

matrix P .

Manuscript received August 20, 2020; revised October 30, 2020; acceptedDecember 4, 2020. Date of online publication December 21, 2020; date ofcurrent version December 29, 2020. This work was supported in part bythe State Key Program of National Natural Science Foundation of Chinaunder Grant No. U1866210, the National Natural Science Foundation of Chinaunder Grant No. 51807067 and Young Elite Scientists Sponsorship Programby CSEE under Grant No. CSEE-YESS-2018.

T. H. Wen, Y. Liu (corresponding author, email: [email protected];ORCID: https://orcid.org/0000-0002-1228-9038), Q. H. Wu, and L. N. Qiuare with the School of Electric Power Engineering, South China Universityof Technology, Guangzhou 510640, China.

DOI: 10.17775/CSEEJPES.2020.04170

λmin(P ) The minimum eigenvalue of the symmetricmatrix P .

AT The transpose of matrix A.LFh(x) The Lie derivative of h(x) with respect to

F (x).ε High-gain parameter of the observer (positive

and sufficiently small).O(ε) Infinitesimal of the same order of ε.‖A‖ The 2-norm of matrix A.

I. INTRODUCTION

AUTO disturbance rejection control strategy is useful forprevention of degradation of overall system performances

caused by unmodelled dynamics, external disturbances, orparameter uncertainties [2]. Feedback linearization control isone of its realization. But, the feedback linearization con-troller needs the information of all the system states andperturbations, which are practically difficult to be measured.As a result, observers are introduced to estimate those statesand perturbations, and the cascade of observers and feedbacklinearization controllers constitutes typical types of outputfeedback controllers.

Equipped with fast and adjustable convergence rate ofobservation error, the high-gain observer is capable of pro-viding accurate estimates of the states and perturbations forfeedback linearization controllers. Therefore, compared withthe system controlled by feedback linearization controllerwhich can get access to the real values of the states andperturbations, the system controlled by high-gain observerbased output feedback controller has the same equilibrium andnearly identical stability region, once the perturbations andtheir derivatives with respect to time, as well as the output ofthe controller, are bounded by values irrelevant of the high-gain parameter [3]. Also, the states of the system can beattracted to a neighbourhood of the equilibrium whose radiusis smaller when the high-gain parameter is smaller [3].

In recent years, the properties and potentials of the high-gain observer based output feedback controller have beenresearched. For example, Reference [4] introduced a high-gain observer based output feedback controller for nonlinearinterconnected systems. Reference [5] studied the stability ofinput-affine systems controlled by a sliding-mode observerbased output feedback controller. The dynamics of the sliding-mode observer studied in [5] are identical to the dynamics

2096-0042 © 2020 CSEE

308 CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 7, NO. 2, MARCH 2021

of the high-gain observer when its states are sliding on thesliding surface. In addition, Reference [6] put forward a high-gain observer-based excitation control system for control ofsynchronous generator. Furthermore, Reference [7] proposedan observer-based multiloop controller for double-fed induc-tion generator based wind farm (DFIG-WF). The observerinstalled in each control loop is the sliding-mode observerdescribed in [5]. Reference [8] presented an application ofhigh-gain observer-based output feedback controller to theimprovement of the low voltage ride-through performances ofDFIG-WF. Last but not least, Reference [9] proposed a high-gain observer-based Bang-Bang constant funnel controller(BBCFC). The controller is robust to external disturbances,and it utilizes maximum control energy to stabilize the system.

However, the high-gain observer may exert negative influ-ences on the feedback linearization controller, when the systemto be controlled has high relative degree. The influences aremainly originated from the peaking phenomenon [10] and theamplification of noise at the input of the observer [11]. Thedisturbance of the system can cause suddent change in theestimation error of the perturbation, which leads to the surgeof the states of the high-gain observer. The surge becomesmore significant when the dimension of the observer is larger.Besides, the noise can generate high-frequency oscillationsin the output of the controller. Simple solutions to overcomeproblems resulted from peaking phenomenon include addingsaturation blocks between observer outputs and controllerinputs [11]. But the saturation block cannot suppress thepeaking in the states of the high-gain observer to an acceptablelevel.

In recent years, other types of output feedback controllers,which do not rely on high-gain observers for estimation ofstates and perturbations, have been put forward. For instance,Reference [12] proposed a continuous finite-time output reg-ulator for systems subjected to mismatched disturbances. Thecontroller is designed via an inductive procedure, and thehigh-order sliding-mode observer described in [13] is adoptedfor the observation of disturbances. However, Reference [12]made the assumption that the states of the system to becontrolled are easy to be measured. In addition, Reference [14]described a quasi-continuous sliding-mode controller. It canensure finite-time stability of the closed-loop system. Thequasi-continuous sliding-mode controller requires high-orderderivatives of the outputs of the system, which are obtainedfrom the high-order sliding-mode observer. But severe chat-tering may occur in the output of the controller. Furthermore,Reference [15] proposed a nonsingular fast terminal sliding-mode controller for stabilizing uncertain time-varying and non-linear third-order systems. The disturbance observer designedthere, which is different from the high-order sliding-mode ob-server proposed in [13], can ensure the finite-time convergenceof the observation error. Meanwhile, the controller does notsuffer from chattering problem. But the states of the third-order system need to be measured.

In this paper (part II), a new cascaded sliding-mode observerbased output feedback controller is proposed. The proposedoutput feedback controller consists of a feedback linearizationcontroller for multi-input multi-output system and cascaded

sliding-mode observers proposed in part I [1]. As is analyzedand proved in section II, the proposed controller can not onlyensure Lyapunov stability of the closed loop system but alsobe applied to output tracking control. Compared with tradi-tional high-gain/sliding-mode observer based output feedbackcontroller, the proposed controller is more suitable for systemshaving higher relative degree. Because the proposed observerhas smaller gain coefficients, and the peaking phenomenonoccurred in the proposed observer is less severe, adverseeffects of the proposed observer on feedback linearizationcontroller are expected to be reduced.

This part is organized as follows. In section II, the basicoperation principle of the proposed cascaded sliding-modeobserver based output feedback controller is briefly introduced,and the stability of the overall system controlled by theproposed controller is proved. Section III presents simulationstudies on a single wind-energy-conversion-system infinite-bus (SWNCSIB) system for testing the performances of theproposed observer and output feedback controller. Section IVcontains the conclusions.

II. CASCADED SLIDING-MODE OBSERVER BASED OUTPUTFEEDBACK CONTROLLER FOR

MULTI-INPUT-MULTI-OUTPUT (MIMO) SYSTEM

For the following MIMO input-affine systemx = F (x) +G(x)u

y = H(x)(1)

where x ∈ RN , u, y ∈ Rp, G(x) = [g1(x), g2(x), . . .,gp(x)] and H(x) = [h1(x), h2(x), . . . , hp(x)]T. Assumethat firstly, F (0) = H(0) = 0, and F (·), G(·) and H(·)are sufficiently smooth in x. Secondly, there exists an opendomain Ω ⊆ RN containing the origin, such that the system(1) has a vector relative degree ρ1, ρ2, . . . , ρp for all x ∈ Ω.Thirdly, ρ1 + . . . + ρp = N (no zero dynamic). Via thecoordinate transformation ξ = T (x), where

ξ = [ξ1, ξ2, . . . , ξp]T

T (x) =[TT1 (x),TT

2 (x), . . . ,TTp (x)

]Tξj = Tj(x) =

[hj(x),LF hj(x), . . . ,Lρj−1F hj(x)

]T(2a)

the system (1) becomes

dξdt

= Aξ +B(Ξ(x) + Υ(x)u) (2)

where

Υ(x) =

Lg1L

ρ1−1F h1(x) . . . LgpL

ρ1−1F h1(x)

Lg1Lρ2−1F h2(x) . . . LgpL

ρ2−1F h2(x)

......

Lg1LρP−1F hp(x) . . . LgpL

ρP−1F hp(x)

Ξ(x) =

[Lρ1F h1(x) Lρ2F h2(x) . . . LρPF hp(x)

]TA = blockdiag(A1,A2, . . . ,Ap)

Ai(i = 1, . . . , p) is the antishift matrix of dimension ρiB = blockdiag(B1,B2, . . . ,Bp)

WEN et al.: CASCADED SLIDING-MODE OBSERVER AND ITS APPLICATIONS IN OUTPUT FEEDBACK CONTROL PART II: ADAPTIVE OUTPUT FEEDBACK CONTROL 309

Bi(i = 1, . . . , p) = ([0, . . . , 0, 1]1×ρi)T

Suppose that there exists a constant diagonal matrix Λ0

satisfying

max‖R‖61

∥∥(Υ(x)− Λ0)RΛ−10

∥∥ 6 ι < 1 (3)

for all x ∈ Ω. Then, consider the following state feedbacklinearization controller for system (1)

u = Λ−10 (−V ξ −Ψ) (4)

where

Ψ = Ξ(x) + (Υ(x)− Λ0)u

and

V = blockdiag(V1,V2, . . . , Vp)

is to be designed such that (A −BV ) is Hurwitz. As (3) issatisfied,

∥∥I + (Υ(x)− Λ0)Λ−10

∥∥ is invertible for all x ∈ Ω,so that Ψ is well-defined. If both Ψ and ξ can be measured,then controller (4) ensures that 0 is a locally exponentiallystable equilibrium for (1). However, in many practical cases,only the output y is available. As a result, observer is neededto estimate ξ and Ψ. According to (2) and (4), the system canbe regarded as a combination of p subsystems, each of whichhas the state equation

ξj = Ajξj +Bj(Ψj + λ0,juj) (j = 1, . . . , p) (5)

to estimate the value of its own states ξj and perturbationΨj . In this section only the case where every subsystem isequipped with the proposed observer is analyzed. Denote ξthe estimated value of ξ and Ψ the estimated value of Ψ. Withξ and Ψ in (4) replaced by ξ and Ψ, and the consideration ofsaturation on the output of the controller, (4) is changed to

uj = Ujgε?(−Vj ξj + Ψj

Ujλ0,j

)(j = 1, . . . , p) (6)

where Uj is the saturation level to be designed and gε? is anodd function defined by [16]

gε?(y) =

y (0 6 y 6 1)

y +y − 1

ε?− 0.5y2 + 1

ε?(1 < y 6 1 + ε?)

1 + 0.5ε? (y > 1 + ε?)

As shown in [16], |gε?(y)− sat(y, 1)|6 ε?/2 for all y ∈ R.Besides, even when controller (6) is used, condition (3) canensure a well-defined Ψ.

Regard Ψi as the fictitious state of the ith subsystem, andsuppose that all the states of the ith subsystem (fictitious stateincluded) of (2) are partitioned into ni groups. Denote rj,i(j =1, . . . , ni, i = 1, . . . , p) the number of state variables in thejth group of the ith subsystem, and ξk,ji(k = 1, . . . , rj,i)

the kth state variable of the jth group of the ith subsystem.Then, the dynamics of (1) controlled by (6), combined withthe dynamics of the proposed observer, are represented by thestate equations below,

˙ξj˙Ψj

˙ξj

= Fobs

(ξj , Ψj ,

ˆξj , uj , hj

(T−1(ξ)

))(j = 1, . . . , p)

ξ1ξ2...ξp

= A

ξ1ξ2...ξp

+

B

Ψ1

Ψ2

...Ψp

−U1gε?((V1ξ1 + Ψ1)/(U1λ0,1))

U2gε?((V2ξ2 + Ψ2)/(U2λ0,2))...

Upgε?((Vpξp + Ψp)/(Upλ0,p))

(7)

ˆξj =

[ˆξr1,j ,1j ,

ˆξr2,j ,2j , . . . ,

ˆξrnj−1,j ,(nj−1)j

]Tˆξ =

[ˆξT1 ,

ˆξT2 , . . . ,

ˆξTp

]Tξj =

[ξ1,1j , . . . , ξrnj,j

−1,(nj)j

]TΨj = ξrnj,j

,(nj)j

Theorem 2. Under the parameter design rules of the proposedcascaded sliding-mode observer, and the design of V makingA−BV Hurwitz, if for initial conditions ξ(0) ∈ Wc ⊆ T (Ω),[(ξ(0))T, (Ψ(0))T]T ∈ RN+p and ˆ

ξ(0) ∈ Rn1+...+np−p,where

Wc := ξ ∈ RN |W (ξ) < c

W (ξ) = ξTP0ξ and P0 is the symmetric positive definitematrix satisfying

(A−BV )TP0 + P0(A−BV ) = −I (8)

the following choice of Uj is adopted,

Uj > maxξ∈Wc

∣∣∣∣Vjξj + Jjλ0,j

∣∣∣∣J =

[J1 J2 . . . Jp

]T=(I + (Υ(T−1(ξ))− Λ0)Λ−10

)−1(Ξ(T−1(ξ))

− (Υ(T−1(ξ))− Λ0)Λ−10 V ξ) (9)

then,(a2) ∀ε > 0, there exists a local solution (in Filippov sense)

for (7), starting at [(ξ(0))T, (ξ(0))T, (Ψ(0))T, (ˆξ(0))T]T, and

maximally forward-extendible. Moreover, along the solutionand its forward-extensions, Ψ exists almost everywhere.

(b2) Assume that ni > 2 for all i = 1, . . . , p, then ∃ε∗0 > 0,such that for all 0 < ε < ε∗0, the local solution of (7) starting at

where Ψj , regarded as the perturbation of the jth subsystem,is the jth row of Ψ, λ0,j is the jth diagonal element of Λ0 anduj is the jth component of u. Due to the resemblance between(B1) in Appendix B and (5), each subsystem can be equippedwith a proposed cascaded sliding-mode observer, cascadedhigh-gain observer [10] or traditional high-gain observer [11]

where Fobs(·) stands for the right-hand side of the stateequation of the proposed observer (see (B2)), and


[(ξ(0))T, (ξ(0))T, (Ψ(0))T, (ˆξ(0))T]T will be attracted to the

intersection of the sliding surfaces⋂pi=1

⋂ni−1j=0 Sj,i, where

Sj,i :

hi(T

−1(ξ))− ξ1,1i = 0 (j = 0)

ξrj,i,ji −ˆξrj,i,ji = 0 (j = 1, . . . , ni − 1)

(c2) Under the assumptions stated in (b2), if ∀ε ∈ (0, ε∗0),‖Ψ‖ < γ1 almost everywhere along the portion of localsolution of (7) sliding on

⋂pi=1

⋂ni−1j=0 Sj,i, then ∃ε∗∗∗0 < ε∗0

and µ? > 0, such that ξ(t) is uniformly bounded, andlim

t→+∞‖ξ(t)‖ 6 µ?ε for all ε ∈ (0, ε∗∗∗0 ).

Note that ε is the high-gain parameter of the proposedobserver.

Proof. To analyze the existence of local solution, we trans-form (7) to differential inclusion via replacing all the sgn(·)functions in Fobs with S(·)

S(a)

= 1 (a > 0)

∈ [−1, 1] (a = 0)

= −1 (a < 0)

Obviously, the differential inclusion is a Filippov-type dif-ferential inclusion [17], because for all ε > 0, its right-handside is compact, convex and upper semi-continuous at set

C :

(t,[ξT, ξT, ΨT,

ˆξT]T)

∈ R× R2N+

p∑j=1

nj

∣∣∣∣∣ξ ∈ Wc

Since (0, [(ξ(0))T, (ξ(0))T, (Ψ(0))T, (

ˆξ(0))T]T) ∈ C, ac-

cording to Theorem 1(§7,CH2) in [17], for every ε > 0, thereexists a local solution

φε : [0, ωε)→ R2N+

p∑j=1

nj

for (7) with

φε(0) =[(ξ(0))T, (ξ(0))T, (Ψ(0))T, (

ˆξ(0))T

]Tand φε(ωε) ∈ C. In addition, according to Theorem 2 (§7,CH2)in [17], φε can be maximally extended forward up to theboundary of C. Moreover, as the solution of Filippov-typedifferential inclusion is absolutely continuous in t [17], andΨ is at least C1 in ξ, ξ and Ψ,

dΨ

dt=

[∂Ψ

∂ξ,∂Ψ

∂ξ,∂Ψ

∂Ψ

] ξ˙ξ˙Ψ

for almost all t ∈ [0, ωε).

Next we prove (b2). On one hand, because u is bounded,

supξ∈Wc

∥∥Aξ +B(Ξ(T−1(ξ)) + Υ(T−1(ξ))u)∥∥ = γ3 <∞

and γ3 is irrelevant to ε. On the other hand, according to (a1)of Theorem 1 in part I [1], ‖[ξT, ΨT,

ˆξT]T‖ will not become

infinite in finite time if ξ and u are bounded. As a result, forthe maximal forward-extension of φε, denoted as φεm,

φεm : [0, ωεm)→ R2N+

p∑j=1

nj

ωεm > ω

φεm([0, ωε)) = φ([0, ωε)) φεm(ωεm) ∈ ∂C

either ωεm = +∞ or ξ(ωεm) ∈ ∂Wc. And, for ξ(0),

∃Tξ(0) =1

γ3dist(ξ(0), ∂Wc) =

1

γ3inf

κ∈∂Wc

‖ξ(0)− κ‖ > 0

such that ξ(t) ∈ Wc for all t ∈ [0, Tξ(0)) and all ε > 0. Notethat Tξ(0) is irrelevant to ε. Since it is assumed that ni > 2for all i = 1, . . . , p, under the parameter design rules (a)∼(c)of the observer listed in part I [1], and the following choiceof Mj,i, the saturation level of ξrj,i,ji used in the observer,

Mj,i = β0 + supξ∈Wc

∣∣ξrj,i,ji∣∣ (j = 1, . . . , ni − 1, i = 1, . . . , p)

where β0 > 0 can be tuned, we can conclude from theproof of (b1) and (c1) of Theorem 1 in part I [1] that∃ε∗0 > 0, T ∗0 ∈ [0, Tξ(0)) and µk,j,i > 0(k = 1, . . . , rj,i, j =1, . . . , ni − 1, i = 1, . . . , p), s.t ∀ε ∈ (0, ε∗0) ⇒ φεm(t) ∈⋂pi=1

⋂ni−1j=0 Sj,i for all t ∈ (T ∗0 , ωε), and |ξk,ji(t)| < µk,j,iε

for all t ∈ (T ∗0 , ωε). Note that a smaller ε∗0 gives a smaller T ∗0 .Now we prove (c2). For every ε ∈ (0, ε∗0), the functions

sgneq(hi(T−1(ξ)) − ξ1,1i)(i = 1, . . . , p) and sgneq(ξrj,i,ji −

ˆξrj,i,ji) (j = 1, . . . , ni − 1) are continuous in t for allt ∈ (T ∗0 , ωε). This can be proved via Lemma 2 in part I [1].Therefore, Ψ exists and is continuous in t for all t ∈ (T ∗0 , ωε).And, ‖Ψ‖ 6 γ1 for all t ∈ (T ∗0 , ωε). Based on the proof of(c1) of Theorem 1 in part I [1], we can show that ∃ε∗∗0 < ε∗0,T ∗∗0 ∈ (T ∗0 , Tξ(0)), µk,ni,i > 0(k = 1, . . . , rni,i, i = 1, . . . , p)and µψi

> 0, s.t ∀ε ∈ (0, ε∗∗0 ) and ∀t ∈ (T ∗∗0 , ωε), we have|ξk,(ni)i(t)| < µk,ni,iε. Hence, for t ∈ (T ∗∗0 , ωε), the equationbelow is satisfied,

Λ−10 (V ξ + Ψ) = Λ−10 V ξ + Ξ(T−1(ξ))− (Υ(T−1(ξ))

− Λ0)UGε?(U−1Λ−10 (V ξ + Ψ))+O(ε)(10)

where U = diag([U1, . . . ,Up]), and

Gε?(U−1Λ−10 (V ξ + Ψ)) =

gε?((V1ξ1 + Ψ1)/(U1λ0,1))

gε?((V2ξ2 + Ψ2)/(U2λ0,2))...

gε?((Vpξp + Ψp)/(Upλ0,p))

(11)

Because of (3), it can be derived from (10) that [16]

Λ−10 (V ξ + Ψ) = Υ(T−1(ξ))−1(V ξ + Ξ(T−1(ξ))) +O(ε)

= Λ−10 (V ξ + J) +O(ε)

As a result,

Λ−10 (V ξ(t) + Ψ(t)) = Λ−10 (V ξ(t) + J(t)) +O(ε) (12)

for all t ∈ (T ∗∗0 , ωε). Based on (12), we can find that ∃ε?0 <ε∗∗0 , s.t ∀ε ∈ (0, ε?0) and ∀t ∈ (T ∗∗0 , ωε),

|(Viξi + Ψi)/λ0,i| < Ui

for all i = 1, . . . , p. So, for φε,

dW (ξ)

dt= −‖ξ‖2 + 2ξTP0B(V ξ + Ψ)


6− 1

λmax(P0)+k0√Wε (ε∈(0, ε?0), t ∈ (T ∗∗0 , ωε)) (13)

for some constant k0 > 0. If ε < min√c

λmax(P0)k0, ε?0, then

dWdt |t=t0 < 0 once φε hits ∂Wc at some instant t0. Therefore,φε cannot escape Wc, and it can be extended at t0. As aresult, for φεm, the maximal extension of φε, ωεm = +∞.According to (13), dW

dt < 0 if W > (λmax(P0)ε)2. Therefore,∀ε ∈ (0,min

√c

λmax(P0)k0, ε?0), φεm will finally be attracted to

set ξ ∈ RN | ‖ξ‖ 6 λmax(P0)k0√

λmin(P0)ε

With ε∗∗∗0 = min√c

λmax(P0)k0, ε?0 and µ? = λmax(P0)k0√

λmin(P0), the

proof of (c2) is completed.Remarks.1. As the dynamic of the observer is very fast for small

ε, the condition∥∥Ψ∥∥ < γ1 stated in (c2) is not easy to be

satisfied.2. Under all assumptions and conditions stated in Theorem

2, if furthermore, Ψ is locally Lipschitz continuous in ξ, ξand Ψ, then it can be proved that the proposed observer-basedcontroller makes the origin of (1) a locally exponentially stableequilibrium point. But the details of the proof will not bediscussed in this paper.

3. For output tracking control, the output of system (1)should be redefined as

ye = H(x)− yref yref = [y1,ref, . . . , yp,ref]T

where yi,ref(i = 1, . . . , p) is the reference value of yi = hi(x).Suppose that all the references are sufficiently smooth. Then,it is obvious that the vector relative degree of (1) remains un-changed, while the coordination transformation (2a) becomes

Tj =[hj(x)− yj,ref, . . . ,L

ρj−1F hj(x)− y(ρj−1)j,ref

]Twhere y(ρj−1)j,ref is the (ρj−1)th derivative of yj,ref with respectto time. Note that now the perturbation contains high-orderderivatives of the output reference signals.

III. SIMULATION STUDIES

Simulation studies are carried out on a single-wind-energy-conversion-system-infinite-bus (SWNCSIB) system with thedouble-fed induction generator based wind energy conversionsystem (DFIGWNCS) controlled by a four-loop observerbased output feedback controller. The structure of the SWNC-SIB system is shown in Fig. 1. The mathematical model ofthe DFIGWNCS in Appendix A is adopted for observer andcontroller design. With the output y = [ωr, iqr, Udc, Qg]

T andinput u = [udr, uqr, udg, uqg]

T, the DFIGWNCS has the vectorrelative degree 2, 1, 2, 1. Now consider that the measuredvalue of iqr and Qg are filtered by a second-order filterrespectively. The second-order filter has the transfer function

G(s) =1

s2 + 2ζωns+ ω2n

B120(Infinite Bus)

T1 T2T3

B4160

DFIG

GSC RSCShuntfilter

RL filter

B25-1 B25-2

Tline2

Tline1

Fig. 1. Structure of the SWNCSIB system.

whose minimum realization is

ϕ1 = ϕ2

ϕ2 = −ω2nϕ1 − 2ζωnϕ2 + uin

yfiltered = ϕ1

where uin and yfiltered are the input and output of the fil-ter respectively. With the new output ymod = [ωr, iqr,filtered,Udc, Qg,filtered]T, where iqr,filtered and Qg,filtered are filtered valueof iqr and Qg, and the input u, the DFIGWNCS now hasthe vector relative degree 2, 3, 2, 3. Under the coordinatetransformation (2a), the state equation of the DFIGWNCS,combined with the state equations of the second-order filters,can be rewritten into

ξ1,11 = ξ2,11ξ2,11 = Ψ1(t, ξ∗,u) + λ0,1udr

ξ1,21 = Ψ1ξ1,12 = ξ2,12ξ2,12 = ξ1,22ξ1,22 = Ψ2(t, ξ∗,u) + λ0,2uqr

ξ2,22 = Ψ2ξ1,13 = ξ2,13ξ2,13 = Ψ3(t, ξ∗,u) + λ0,3udg

ξ1,23 = Ψ3ξ1,14 = ξ2,14ξ2,14 = ξ1,24ξ1,24 = Ψ4(t, ξ∗,u) + λ0,4uqg

ξ2,24 = Ψ4

(14)

ξ∗ =[ξ1,11 , ξ2,11 , ξ1,12 , ξ2,12 , ξ1,22 , ξ1,13 ,

ξ2,13 , ξ1,14 , ξ2,14 , ξ1,24]T

ξ1,11 = ωr − ωr,ref ξ1,12 = iqr,filtered − iqr,ref

ξ1,13 = Udc − Udc,ref ξ1,14 = Qg,filtered −Qg,ref

when the shaft spring constant of the two-mass system ofthe DFIGWNCS is considered to be sufficiently large. Moredetails of the coordinate transformation are discussed in [7].

In the simulation studies, two types of output feedback con-trollers are considered. The first type is a cascaded observer-based output feedback controller. Its structure is shown inFig. 2. While the proposed cascaded sliding-mode observer isinstalled in rotor speed control loop and q-axis rotor currentcontrol loop, a cascaded high-gain observer is used in GSCreactive power control loop, and the observer in dc-link voltage


Secomd-order filter

Secomd-order filter

iqr

iqr,filtered

ωr

ωr,ref

iqr,ref

Udc

Udc,ref

Qg,filtered

Qg,ref

uqg

udg

uqr

udr uqr udg uqg

r1,1=2 r2,1=1

r1,2=2

r1,3=2

r1,4=2 r2,4=2

r2,3=1

r2,2=1 r3,2=1

udr

Qg

+

−

+

−

+

−

+

−

Sliding-modeobserver






High-gainobserver

High-gainobserver

High-gainobserver

Feedbacklinearizationcontroller (6)

ξ2,11

^

ξ2,12

^ξ1,22

^

ξ2,13

^

ξ2,14

^

vs,ref

Qs,ref

iqr,ref

Qs

s1

s1

vs

− − +

+ + −Kdroop

Fig. 2. Structure of the four-loop observer-based output feedback controller.

control loop is a sliding-mode observer connected with a high-gain observer. For the second type, the observer installed ineach control loop is a conventional sliding-mode observerproposed in [5]. Note that for both types of controllers, iqr,refis calculated from a ‘droop-control’ like algorithm shown inFig. 2, and

Qg,ref =

0 (0.5 < vs < 1.5)

−3(vs − 0.5) (0.4 < vs 6 0.5)

0.3 (vs 6 0.4)

−3(vs − 1/5) (1.5 6 vs < 1.6)

−0.3 (vs > 1.6)

Besides, all the sgn(·) functions in the observers are re-placed by fac(·, 0.001) in the simulation, where

fac(z, χ) =

1 (z > χ)

z/χ (|z| 6 χ)

−1 (z < −χ)

In the simulation studies, the wind speed condition is shownin Fig. 3. As MPPT is not considered, ωr,ref is simply fixedto 1.1 pu. A three-phase fault occurs near the terminal ofTline2 connected to B25-2 at t = 1.0 s and lasts for 500 ms.The faulted line is tripped at t = 1.3 s and switched onoperation again at t = 1.6 s. The fault resistance is 16 Ω.The simulation is undertaken via MATLAB/Simulink and theresults are demonstrated in Fig. 4 to Fig. 6. Simulation resultsof the variables whose subscript contains ‘CAO’ are obtainedwhen the DFIGWNCS is controlled by the cascaded observer-based output feedback controller. Meanwhile, the results of thevariables whose subscript contains ‘CVO’ are recorded whenthe DFIGWNCS is controlled by the conventional sliding-mode observer-based output feedback controller.

According to Fig. 4(a)–(f), the states of the proposed ob-server rapidly approach the sliding surfaces ξ2,11 −

ˆξ2,11 = 0,

ξ2,12 −ˆξ2,12 = 0 and ξ1,22 −

ˆξ1,22 = 0 after the fault occurs

in SWNCSIB system and is cleared. However, in numericalsimulation, they fail to slide ideally on the sliding surfaces,and chattering happens in the proposed observer, which givesrise to the extremely high frequency oscillations in Fig. 4(b),5(d) and 5(f). Since it is designed that k1,22 > k2,12 > k2,11 ,the oscillation in Fig. 4(f) is the most severe. According toFig. 4(p)–4(q), the proposed observer makes the estimatedvalue of the states ξ1,11 , ξ2,11 , ξ1,12 , ξ2,12 and ξ1,22 tracetheir real value. The peaks in those figures are resulted fromthe change of network topology of SWNCSIB system, whichbrings about sudden change of the fictitious state ξ1,21 andξ1,32 representing the perturbations of DFIGWNCS.

0 0.5 1 1.5 2 32.5Time (s)

11

11.2

11.4

11.6

11.8

12

Win

d s

pee

d (

m·s

−1)

Fig. 3. Wind speed condition.

Figure 5 gives the simulation results of the performancesof the cascaded observer-based output feedback controller. Itcan be seen from Fig. 5(a)–(d) that the difference betweenωr,CAO, Udc,CAO, iqr,filtered,CAO, Qg,filtered,CAO and their respective


1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

1 2 31.5 2.5t /s

−0.1

0

0.1

−0.6

0

0.6

×10−3

×104

×10−4×10−5

−50

0

50

−1

0

1

−2

0

2

4

−150

0

150

−2

0

2

4

6

ξ 1,1

1 a

nd i

ts e

stim

ate

(g) ξ1,11 and its estimate

−2

0

2

(h) Estimation error of ξ1,11

Est

mat

e of ξ 2

,11

Est

mat

e of ξ 2

,12

Est

mat

e of ξ 1

,22

(a) Estmate of ξ2,11

(c) Estmate of ξ2,12

(e) Estmate of ξ1,22

ξ 2,1

1−ξ 2

,11

^^ξ2,11

^

ξ2,11

^

ξ2,12

^

ξ2,12

^

ξ1,22

^

ξ1,22

^

ξ1,11

^

ξ1,11

ξ 2,1

2−ξ 2

,12

^^

ξ 1,2

2−ξ 1

,22

^^

ξ 1,1

1−ξ 1

,11

^

(b) ξ2,11−ξ2,11

^ ^

(d) ξ2,12−ξ2,12

^ ^

(f) ξ1,22−ξ1,22

^ ^

1 2 3t /s

−0.2

0

0.2

(i) ξ1,12 and its estimate

−5

0

5

×10−5

(j) Estimation error of ξ1,12

−0.1

0

0.1

−0.06

0

0.06

ξ 1,1

2 a

nd i

ts e

stim

ate

ξ 2,1

1 a

nd i

ts e

stim

ate

ξ 1,1

2−ξ 1

,12

^ξ 2

,11−ξ 2

,11

^

1.5 2.5 1 2 3t /s

1.5 2.5

(l) Estimation error of ξ2,11

1 2 3t /s

1.5 2.51 2 3t /s

(k) ξ2,11 and its estimate

1.5 2.5

ξ1,12

^

ξ1,12

ξ2,11

^

ξ2,11

reference quickly becomes small after the fault occurs andis cleared. But the difference is not exactly zero, which ispartly resulted from that the observation error of the observer

−30

0

30

60

−4

−2

0

2

4

6

−2

0

2

4

−4

−2

0

2

4

ξ 2,1

2 a

nd

its

est

imate

ξ 1,2

2 a

nd

its

est

imate

ξ 2,1

2−ξ 2

,12

^ξ 1

,22−ξ 1

,22

^

(n) Estimation error of ξ2,12

1 2 3t /s

1.5 2.5

(p) Estimation error of ξ1,22

1 2 3t /s

1.5 2.5

1 2 3t /s

(m) ξ2,12 and its estimate

1.5 2.5

1 2 3t /s

(o) ξ1,22 and its estimate

1.5 2.5

×104×104

ξ2,12

^

ξ2,12

ξ1,22

^

ξ1,22

Fig. 4. Simulation results of the cascaded sliding-mode observers.

(a) Rotor speed (b) DC-link voltage

1 1.5 2 32.5t (s)

1 1.5 2 32.5t (s)

1 1.5 2 32.5t (s)

1 1.5 2 32.5t (s)

1 1.5 2 32.5t (s)

1 1.5 2 32.5t (s)

−0.7−0.6−0.5−0.4−0.3−0.2−0.1

0

(c) q-axis rotor current

−0.05

0

0.05

0.1

(d) Reactive power output of GSC

0

1

2

3

4

5

P (

MW

)

(e) Active power output of DFIGWNCS

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Vs,CAO

Vs,CVO

PCAO

PCVO

(f) Voltage of the bus B4160

i qr (p

.u.)

Vs

(p.u

.)Q

g (

p.u

.)

ωr (p

.u.)

Udc

(V)

1.0996

1.0998

1.1

1.1002

1.1004

1.1006

3400

3500

3600

3700

3800

3900

4000

iqr,filtered,CAO

iqr,filtered,CVO

iqr,ref

Qg,filtered,CAO

Qg,filtered,CVO

Qg,ref

ωr,CAO

ωr,CVO

ωr,ref

Udc,CAO

Udc,CVO

Udc,ref

Fig. 5. Simulation results of the SWNCSIB system.

cannot be totally eliminated. In a short time after the faultis cleared, the output of the dc-link voltage control loop issaturated, which is part of the reason why Udc rise from 3600 V


(a) Rotor speed

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

t (s)1 1.5 2 32.5

35003600370038003900400041004200

(b) DC-link voltage

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

(c) q-axis rotor current

−0.06

−0.04

−0.02

0

0.02

0.04

(d) Reactive power output of GSC

−0.8

−0.6

−0.4

−0.2

(e) q-axis rotor current

−0.2

−0.1

0

0.1

(f) Reactive power output of GSC

−1

0

1

2

3

4

5

(g) Active power output of DFIGWNCS

0.6

0.8

1

1.2

(h) Voltage of the bus B4160

1.1006

1.1004

1.1002

1.1

1.0998

1.0996

ωr (p

.u.)

i qr (p

.u.)

i qr (p

.u.)

P (

MW

)

Udc

(V)

Qg (

p.u

.)Q

g (

p.u

.)V

s (p

.u.)

Vs,CVO

Vs,CAO

PCVO

PCAO

ωr,CVO

ωr,ref

ωr,CAOUdc,CVO

Udc,ref

Udc,CAO

Qg,ref

Qg,filtered,CVO

Qg,ref

Qg,filtered,CAO

iqr,ref

iqr,filtered,CVO

iqr,ref

iqr,filtered,

Fig. 6. Simulation results of the SWNCSIB system.

to nearly 4100 V at 1.3 s. Simulation result of the activepower output of the DFIGWNCS contains both low frequencyoscillations and high frequency oscillations. The low frequencyoscillations are the consequence of the interaction betweengenerator shaft and turbine shaft. Note that the speed of windturbine ωt and the twisting angle θ are internal dynamics ofthe DFIGWNCS and cannot be observed by the proposedobserver or traditional high-gain observer. The high-frequencyoscillations are originated from power electronic devices.

Performances of the conventional sliding-mode observer-based output feedback controller are also presented in Fig. 5.The controller cannot ensure stability of the DFIGWNCSafter the fault occurs. The difference between ωr,CVO, Udc,CVO,iqr,filtered,CVO, Qg,filtered,CVO and their respective reference di-verges at t = 1.0s.

In order to make further comparisons between the twocontrollers, the above-mentioned fault resistance is increasedto 20Ω, and the simulation results are demonstrated in Fig. 6.As shown in Fig. 6, both controllers can stabilize the DFIG-

WNCS after the fault occurs and is cleared. Compared tothe conventional observer-based output feedback controller,the cascaded observer-based output feedback controller hasbetter tracking performance, and brings about less severe low-frequency oscillations in ωr, Udc, iqr,filtered and Qg,filtered. How-ever, the reduction of the peaking in Udc at t = 1.3 is slightwhen the cascaded observer-based output feedback controlleris used. Besides, although the difference between ωr,CVO andωr,ref is larger, the conventional sliding-mode observer-basedoutput feedback controller makes the low frequency oscillationof the total output active power smaller. This might partlybecause forcing ωr to trace its reference will amplify thetorsional oscillation in the drive-train. In all, the advantagesof the cascaded observer-based output feedback controller arenot so obvious in control of the DFIGWNCS as the relativedegree of the DFIGWNCS is not large enough to make theperformances of the conventional sliding-mode observer-basedoutput feedback controller totally deteriorated.

IV. CONCLUSION

Part II of this series of papers has proposed a new cascadedsliding-mode observer based output feedback controller. Sim-ulation studies on a SWNCSIB system have demonstrated thatthe proposed observer can provide accurate estimates of statesand perturbations for feedback linearization controller and hasfast convergence rate of observation error. As a result, thecontroller can ensure the stability of the closed-loop systemand make the outputs of the system track their references.

Apart from the control of double-fed induction generator,the proposed controller can be used for control of permanentmagnetic synchronous generator (PMSG) and traditional syn-chronous generator (SG). For PMSG, a multiloop observer-based controller similar to that for the DFIGWNCS can bedesigned. For SG, a cascaded sliding-mode observer-basedexcitation control system can be worked out to regulate therotor speed or rotor angle. Since the relative degree of SG withrotor angle as output is high, the advantage of the proposedobserver in regard to the suppression of peaking phenomenonwill be more significant.

The proposed observer-based controller can also be appliedto the control of multimachine power system, since multima-chine power system can be treated as a MIMO system in theform of (1). But this issue was not discussed in detail in thispart due to the limitation of space.

But for a system with internal dynamics, the internal dy-namics cannot be observed and regulated by the proposedobserver based controller. To avoid unstable internal dynamic,the structure of the controller needs to be redesigned. We makethis problem as our future research.

APPENDIX A

MATHEMATICAL MODEL OF DFIGWNCS

The model, originated from [18], is given by

y = [wr, iqr, Udc, Qg]T


ddt

idriqridgiqgωrωtθUdc

=

ωb

%Lr0 0 0

0ωb

%Lr0 0

0 0 −ωb

Lg0

0 0 0 −ωb

Lg

0 0 0 00 0 0 0

udruqrudguqg

+

ωb

%Lr

(−Rridr + sωs

(%Lriqr +

L2m

Lsims

))ωb

%Lr(−Rriqr − sωs%Lridr)

ωb

Lg(−Rgidg + ωsLgiqg + vs)

ωb

Lg(−Rgiqg − ωsLgidg)

1

2Hg

(Kshθ +Dsh(ωt − ωr)−

Lmvsidr

Ls

)1

2Ht(Tm −Kshθ −Dsh(ωt − ωr))

ωb(ωt − ωr)

1

CUdc(vsidg − Pr)

where idr is the d-axis rotor current, iqr is the q-axis rotorcurrent, idg is the d-axis current of GSC, iqg is the q-axiscurrent of GSC, ωr is the rotor speed, ωt is the turbine speed,ωs is the synchronous speed, ωb = 120π, s is the slip, θis the twisting angle, Udc is the dc-link voltage, C is thecapacitance of the dc-link capacitor, Ls is the stator inductance,Lr is the rotor inductance, Lm is the mutual inductance, Lg isthe total inductance of GSC, Rr is the rotor resistance, Rgis the total resistance of GSC, Hg is the generator inertiaconstant, Ht is the turbine inertia constant, Ksh is the shaftspring constant, Dsh is the shaft mutual damping coefficient,vs is the stator voltage, Tm is the mechanic torque, ims =

(Lsiqs + Lmiqr) /Lm, % = 1 − L2m

LrLs, Pr is the active power

consumed by rotor winding, Qs is the stator reactive poweroutput, and Qg is the reactive power output of GSC.

APPENDIX B

STRUCTURE AND STATE-SPACE MODEL OF THE CASCADEDSLIDING-MODE OBSERVER PROPOSED IN PART I [1]

For a single-input-single-output (SISO) system which hasor can be transformed into the following form,

x = Ax+B(b(x)u+ d(t,x))

y = Cx;(B1)

where x ∈ RM is the state vector, xi(i = 1, 2, . . . ,M)are the states, A is the antishift matrix, B = [0, . . . , 0, 1]T,C = [1, 0, . . . , 0], u is the control input, y is the output,b(x) is a nonlinear function and d(t,x) represents externaldisturbances, model uncertainties and nonlinearities, we define

the fictitious state xM+1 = Ψ(t,x, u) = (b(x)−b0)u+d(t,x)and partition all the states(xM+1 included) randomly into ngroups. The fictitious state stands for the perturbation. Then,the cascaded sliding-mode observer, whose state equation isshown below, is proposed in part I [1] to estimate the statesand perturbation of the SISO system.

Block 1:

˙x1,1 = x2,1 + α1,1(y − x1,1) + k1,1sgn(y − x1,1)˙x2,1 = x3,1 + α2,1(y − x1,1) + k2,1sgn(y − x1,1)

. . .˙xr1,1 = αr1,1(y − x1,1) + kr1,1sgn(y − x1,1)

Block 2:

˙xr1,1 = x1,2 + αr1,1(xr1,1 − ˆxr1,1)

+ kr1,1sgn(xr1,1 − ˆxr1,1)˙x1,2 = x2,2 + α1,2(xr1,1 − ˆxr1,1)

+ k1,2sgn(xr1,1 − ˆxr1,1)˙x2,2 = x3,2 + α2,2(xr1,1 − ˆxr1,1)

+ k2,2sgn(xr1,1 − ˆxr1,1)

. . .˙xr2,2 = αr2,2(xr1,1 − ˆxr1,1)

+ kr2,2sgn(xr1,1 − ˆxr1,1)

(B2)

. . . . . .

Block n:

˙xrn−1,n−1 = x1,n

+ αrn−1,n−1(xrn−1,n−1 − ˆxrn−1,n−1)

+ krn−1,n−1sgn(xrn−1,n−1 − ˆxrn−1,n−1)˙x1,n = x2,n

+ α1,n(xrn−1,n−1 − ˆxrn−1,n−1)

+ k1,nsgn(xrn−1,n−1 − ˆxrn−1,n−1)˙x2,n = x3,n

+ α2,n(xrn−1,n−1 − ˆxrn−1,n−1)

+ k2,nsgn(xrn−1,n−1 − ˆxrn−1,n−1)

. . .˙xrn−1,n = xrn,n + b0u

+ αrn−1,n(xrn−1,n−1 − ˆxrn−1,n−1)

+ krn−1,nsgn(xrn−1,n−1 − ˆxrn−1,n−1)˙xrn,n = αrn,n(xrn−1,n−1 − ˆxrn−1,n−1)

+ krn,nsgn(xrn−1,n−1 − ˆxrn−1,n−1)

The variables αj,i, kj,i, αri,i and kri,i(j = 1, . . . , ri, i = 1,. . . , n) in (B2) are parameters to be designed based on rulesdemonstrated in part I [1]. The design rules will introduce ahigh-gain parameter to the proposed observer. xj,i is the jthstate variable of the ith group, and xj,i is its observed value.

Block 1y

−M1 −M2 −Mn−1

M1 M2 Mn−1

Block 2 Block n

u

x1,1~xr1,1^ ^

xr1,1^ xr2,2

^ xrn−1,n−1^

xr1,1,x1,2~xr2,2^ ^ ^ xrn−1,n−1,x1,n~xrn,n

^ ^ ^

Fig. B1. Block diagram of the observer.


ri is the number of states in the ith group. Although both xri,iand ˆxri,i are the observed value of xri,i, only xri,i is providedfor the feedback controller studied in this part.

APPENDIX C

PARAMETERS OF THE SWNCSIB SYSTEM

TABLE CIDFIGWNCS

Name ValueBase freqency/Hz 60Rated power/MVA 4.16Rated Voltage/kV 3.6Stator resistance/p.u. 0.0079Stator inductance/p.u. 4.4794Rotor resistance/p.u. 0.025Rotor inductance/p.u. 4.8Mutual inductance/p.u. 4.4DC-link capacitance/F 0.016DC-link nominal voltage/V 3600Resistance of the RL filter/Ω 0.005Inductance of the RL filter/H 0.001Generator inertia constant/s 0.91Turbine inertia constant/s 4.32Shaft spring constant/p.u. 3.16Shaft mutual damping

1.5Coefficient/p.u.

TABLE CIITRANSFORMERS AND LINES

Name ValueT1Ratio 120/25Rated Power/MVA 3.6Leakage resistance/p.u. 0.004Leakage inductance/p.u. 0.16T2Ratio 25/4.16Rated Power/MVA 3.6Leakage resistance/p.u. 0.004Leakage inductance/p.u. 0.16T3Ratio 4.16/1.8Rated Power/MVA 3.6Leakage resistance/p.u. 0.004Leakage inductance/p.u. 0.08Tline1 & Tline2r/Ω·km−1 0.01273l/mH·km−1 0.9337Length/km 20

REFERENCES

[1] T. H. Wen, Y. Liu, Q. H. Wu, and L. N. Qiu, “Cascaded sliding-modeobserver and its applications in output feedback control part I: observerdesign and stability analysis,” in CSEE Journal of Power and EnergySystems, doi: 10.17775/CSEEJPES.2020.04160.

[2] J. Yang, T. Li, C. J. Liu, S. H. Li, and W. H. Chen, “Nonlinearityestimator-based control of a class of uncertain nonlinear systems,” IEEETransactions on Automatic Control, vol. 65, no. 5, pp. 2230–2236, May2020.

[3] A. N. Atassi and H. K. Khalil, “A separation principle for the stabiliza-tion of a class of nonlinear systems,” IEEE Transactions on AutomaticControl, vol. 44, no. 9, pp. 1672–1687, Sep. 1999.

[4] L. Jiang and Q. H. Wu, “Decentralized nonlinear adaptive output-feedback controller based on high gain state and perturbation observer,”in Proceedings of the International Conference on Control Applications,2002, pp. 611–616.

[5] L. Jiang and Q. H. Wu, “Nonlinear adaptive control via sliding-modestate and perturbation observer,” IEE Proceedings-Control Theory andApplications, vol. 149, no. 4, pp. 269–277, Jul. 2002.

[6] L. Jiang, Q. H. Wu, and J. Y. Wen, “Decentralized nonlinear adaptivecontrol for multimachine power systems via high-gain perturbationobserver,” IEEE Transactions on Circuits and Systems I: Regular Papers,vol. 51, no. 10, pp. 2052–2059, Oct. 2004.

[7] Y. Liu, Q. H. Wu, X. X. Zhou, and L. Jiang, “Perturbation observer basedmultiloop control for the DFIG-WT in multimachine power system,”IEEE Transactions on Power Systems, vol. 29, no. 6, pp. 2905–2915,Nov. 2014.

[8] X. Lin, K. S. Xiahou, Y. Liu, and Q. H. Wu, “Design and hardware-in-the-loop experiment of multiloop adaptive control for DFIG-WT,” IEEETransactions on Industrial Electronics, vol. 65, no. 9, pp. 7049–7059,Sep. 2018.

[9] Y. Liu, K. S. Xiahou, Q. H. Wu, H. Pang, and G. F. Tang, “Robustbang-bang control of disturbed nonlinear systems based on nonlinearobservers,” CSEE Journal of Power and Energy Systems, vol. 6, no. 1,pp. 193–202, Mar. 2020.

[10] H. K. Khalil, “Cascade high-gain observers in output feedback control,”Automatica, vol. 80, pp. 110–118, Jun. 2017.

[11] H. K. Khalil, Nonlinear Systems, Upper Saddle River: Prentice Hall,2001.

[12] S. H. Li, H. B. Sun, J. Yang, and X. H. Yu, “Continuous finite-timeoutput regulation for disturbed systems under mismatching condition,”IEEE Transactions on Automatic Control, vol. 60, no. 1, pp. 277–282,Jan. 2015.

[13] A. Levant, “Higher-order sliding modes, differentiation and output-feedback control,” International Journal of Control, vol. 76, no. 9–10,pp. 924–941, Jun. 2003.

[14] A. Levant and B. Shustin, “Quasi-continuous MIMO sliding-modecontrol,” IEEE Transactions on Automatic Control, vol. 63, no. 9, pp.3068–3074, Sep. 2018.

[15] S. Mobayen and F. Tchier, “Nonsingular fast terminal sliding-modestabilizer for a class of uncertain nonlinear systems based on disturbanceobserver,” Scientia Iranica, vol. 24, no. 3, pp. 1410–1418, May/Jun.2017.

[16] D. Chowdhury and H. K. Khalil, “Practical synchronization in networksof nonlinear heterogeneous agents with application to power systems,”IEEE Transactions on Automatic Control, vol. 66, no. 1, pp. 184–198,Jan. 2021.

[17] A. F. Filippov, Differential Equations with Discontinuous RighthandSides, Norwell: Kluwer Academic Publishers, 1988.

[18] Y. Liu, Q. H. Wu, and X. X. Zhou, “Co-ordinated multiloop switchingcontrol of DFIG for resilience enhancement of wind power penetratedpower systems,” IEEE Transactions on Sustainable Energy, vol. 7, no.3, pp. 1089–1099, Jul. 2016.

[19] M. A. Mahmud, H. R. Pota, M. Aldeen, and M. J. Hossain, “Partial feed-back linearizing excitation controller for multimachine power systems toimprove transient stability,” IEEE Transactions on Power Systems, vol.29, no. 2, pp. 561–571, Mar. 2014.

[20] J. Chen, L. Jiang, W. Yao, and Q. H. Wu, “Perturbation estimationbased nonlinear adaptive control of a full-rated converter wind turbinefor fault ride-through capability enhancement,” IEEE Transactions onPower Systems, vol. 29, no. 6, pp. 2733–2743, Nov. 2014.

[21] Y. Liu, Z. H. Lin, K. S. Xiahou, M. S. Li, and Q. H. Wu, “On thestate-dependent switched energy functions of DFIG-based wind powergeneration systems,” CSEE Journal of Power and Energy Systems, vol.6, no. 2, pp. 318–328, Jun. 2020.

[22] D. Astolfi and G. Casadei, “Stabilization of nonlinear systems in pres-ence of filtered output via extended high-gain observers,” Automatica,vol. 110, pp. 108594, Dec. 2019.

[23] J. Hernandez and J. P. Barbot, “Sliding observer-based feedback controlfor flexible joints manipulator,” Automatica, vol. 32, no. 9, pp. 1243–1254, Sep. 1996.

[24] S. Mobayen and J. Ma, “Robust finite-time composite nonlinear feed-back control for synchronization of uncertain chaotic systems withnonlinearity and time-delay,” Chaos, Solitons & Fractals, vol. 114, pp.46–54, Sep. 2018.

[25] R. Sanchis and H. Nijmeijer, “Sliding controller-sliding observer designfor non-linear systems,” European Journal of Control, vol. 4, no. 3, pp.208–234, Dec. 1998.

[26] V. I. Utkin, Sliding Modes in Control and Optimization, Berlin: SpringerVerlag, 1992.

[27] J. Chen, E. Prempain, and Q. H. Wu, “Observer-based nonlinear controlof a torque motor with perturbation estimation,” International Journalof Automation and Computing, vol. 3, no. 1, pp. 84–90, Mar. 2006.


[28] S. Mobayen, “Design of LMI-based sliding mode controller with anexponential policy for a class of underactuated systems,” Complexity,vol. 21, no. 5, pp. 117–124, May/Jun. 2016.

[29] S. Mobayen, M. J. Yazdanpanah, and V. J. Majd, “A finite-time trackerfor nonholonomic systems using recursive singularity-free FTSM,” inProceedings of the 2011 American Control Conference, 2011, pp. 1720–1725.

Tianhao Wen received a B.E. degree in ElectricalEngineering from Huazhong University of Scienceand Technology(HUST), Wuhan, China in 2018.He is currently pursuing the Ph.D degree in SouthChina University of Technology. His research inter-ests include nonlinear observers and power systemtransient stability analysis.

Yang Liu (M’18) received a B.E. degree and a Ph.Ddegree in Electrical Engineering from South ChinaUniversity of Technology (SCUT), Guangzhou,China, in 2012 and 2017, respectively. He is cur-rently a Lecturer in the School of Electric PowerEngineering, SCUT. His research interests includethe areas of power system stability analysis andcontrol, control of wind power generation systems,and nonlinear control theory. He has authored or co-authored more than 30 peer-reviewed SCI journalpapers.

Q. H. Wu (M’91–SM’97–F’11) obtained a Ph.D.degree in Electrical Engineering from The Queen’sUniversity of Belfast (QUB), Belfast, U.K. in 1987.He worked as a Research Fellow and subsequentlya Senior Research Fellow in QUB from 1987 to1991. He joined the Department of MathematicalSciences, Loughborough University, Loughborough,U.K. in 1991, as a Lecturer, subsequently he wasappointed Senior Lecturer. In September, 1995, hejoined The University of Liverpool, Liverpool, U.K.to take up his appointment to the Chair of Electrical

Engineering in the Department of Electrical Engineering and Electronics.Currently, he is now with the School of Electric Power Engineering, SouthChina University of Technology, Guangzhou, China, as a DistinguishedProfessor and the Director of Energy Research Institute of the University.Professor Wu has authored and coauthored more than 540 technical pub-lications, including 350 journal papers, 20 book chapters and 3 researchmonographs published by Springer. He is a Fellow of IEEE, Fellow of IET,and Chartered Engineer. His research interests include nonlinear adaptivecontrol, mathematical morphology, evolutionary computation, power qualityand power system control and operation.

Luonan Qiu received a B.E. degree in Energy Powerand Mechanical Engineering from North China Elec-tric Power University, Beijing, China, in 2017. Sheis currently pursuing the Ph.D degree in SouthChina University of Technology. Her main researchinterests include power system stability analysis andcontrol.

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 7, NO. …

Documents

Transcript of CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 7, NO. …