Design of observer-based non-fragile load frequency control for power systems with electric vehicles

ISA Transactions 91 (2019) 21–31

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Design of observer-based non-fragile load frequency control for power systems with electric vehicles ∗

D. Aravindh a , R. Sakthivel b , , B. Kaviarasan d , S. Marshal Anthoni d , Faris Alzahrani c a

Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore 641407, India Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India c Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Department of Mathematics, Anna University Regional Campus, Coimbatore 641046, India b

highlights • A load frequency controller with multiple time delays is proposed for power systems. • The proposed method does not require any information on states of the power system. • Delay-dependent stability criterion is developed for the resulting closed-loop system.

article

info

Article history: Received 24 January 2018 Received in revised form 23 December 2018 Accepted 24 January 2019 Available online 7 February 2019 Keywords: Load frequency control Finite-time stability Power systems Electric vehicles Non-fragile control

a b s t r a c t This paper establishes an observer-based finite-time non-fragile load frequency control design using electric vehicles for power systems with modeling uncertainties and external disturbances. A state space representation of the addressed power systems together with dynamic interactions of electric vehicles is formulated. A full-order observer-based non-fragile controller is designed to ensure finitetime boundedness and satisfactory finite-time H∞ performance of the considered system. By constructing an augmented Lyapunov–Krasovskii functional and employing Wirtinger-based integral inequality, the required conditions are obtained in terms of linear matrix inequalities. The desired non-fragile load frequency control law is presented via the observer-based feedback approach. Simulations are given to show the effectiveness of the proposed control scheme. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction The active power and frequency control is often referred to as load frequency control (LFC). In power systems, voltage and frequency deviations often exist due to oscillations and mismatch of generations, unexpected events and uncertainties. These factors could damage equipment, degrade the load performance and cause transmission lines to be over loaded, which ultimately leads to unstable conditions for the power systems. The primary objective of LFC method is to maintain zero steady state errors for frequency deviation and good tracking of load demands in power systems. Based on these facts, LFC design has received considerable research focus to obtain satisfactory performance of control systems [1–3]. In recent years, many LFC schemes have been proposed for various kinds of power systems, such as electric vehicles (EVs), smart grid systems, multi-area thermal systems and wind farms [4–7]. In particular, power systems with EVs have attracted considerable ∗ Corresponding author. E-mail address: [email protected] (R. Sakthivel). https://doi.org/10.1016/j.isatra.2019.01.031 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

research interests due to their environmentally friendly characteristics, such as lower green house emission and noise pollution. Using bidirectional power electronic devices and vehicle-to-grid technology, an aggregation of EVs plays the role of generating power source to assist conventional power units to respond rapidly to attain the requirement of LFC. Therefore, EVs participate in LFC to assist power units to promptly suppress the variations of system disturbances. Due to the nonlinearity of various components in power systems, a linear model obtained by linearization around an operating point like LFC is usually adopted for the controller design. In most of the existing LFC design approaches for the EV models, it is assumed that all the states of the models could be measured by using sensors. However, because of the high cost and maintenance of sensors, it is not easy to consider this assumption for the actual EV models. To overcome this difficulty, observerbased approach is introduced and consequently, several important works are proposed to estimate the unmeasurable states of many real-time systems via observer-based control technique in recent years [8–12]. An observer-based non-fragile control scheme for stochastic time delay systems is proposed in [13], where a new set

22

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

of linear matrix inequality (LMI)-based conditions is developed for designing observer-based non-fragile controller without assuming any constraints on system matrices. Miao and Li [14] derived a set of sufficient conditions in terms of LMIs for designing an observerbased controller for stochastic time delay systems with the aid of Lyapunov stability theory. A robust LFC design problem for multi-area interconnected power systems has been investigated in [15] and [16] according to the constrained population extremal optimization and the adaptive population extremal optimization, respectively. Similarly, few interesting works based on the extremal optimization method have been reported for some practical systems, for instance see [17,18] and references cited therein. It should be pointed out that finite-time stability and asymptotic stability are totally different concepts. In addition, the design of finite-time control for real process has received much attention due to its practical importance. For the past few years, the concepts of finite-time stability and stabilization have received much attention for various classes of dynamical systems, such as Markov jump systems [19], stochastic systems [20], nonlinear systems [21] and power systems [22]. By using the average dwell-time approach, the problem of finite-time boundedness for switched neutral systems with unknown time-varying disturbance is studied in [23], where a set of sufficient conditions that guarantees finite-time boundedness with H∞ disturbance attenuation level of the closed-loop system is derived. Shi et al. [24] developed a mode-dependent finitetime H∞ controller that can guarantee the finite-time boundedness with a prescribed H∞ performance of the switched systems. Following these seminal works, it is more interesting to study the concepts of finite-time stability and stabilization for power systems to effectively deal with uncertainties. Specifically, a system is finite-time stable if its state is not larger than the prescribed bounds in the fixed time interval for the given bound on the initial condition. Hence, it is possible to make all the signals in the power systems being experience better performance over a prescribed finite time period even the frequency oscillates widely from its scheduled value. Thus, the investigation of finite-time stability and stabilization of power systems is of great importance from the practical point of view. On the other hand, in real-time process, it is not possible to obtain the controllers exactly because of existence of some unavoidable uncertainties in their coefficients due to aging of components, parameter’s re-adjustment process and network environment circumstances. Precisely, small perturbations in the control coefficients may destabilize the systems [25–27]. Therefore, it is important to consider gain fluctuations during controller design. Recently, non-fragile control approaches are proposed to deal with the controller gain fluctuation issues, which can ensure the stability of the considered systems [28–32]. In many practical control systems, time delay often exists, which is an important source of instability and poor system performance. Further, the exact delay value is not known in advance, which can only be estimated via a controller design process. Thus, research on practical control systems with time delay has gained a remarkable attention from research communities. Following this concept, many results about LFC design for power systems with time delays have been reported. On the other hand, robustness of control systems subject to disturbances and uncertainties has always been a central issue in the feedback control design. The main advantage of the H∞ control method is to design robust controllers with respect to disturbances and to obtain good system performances with respect to unstructured uncertainties. Therefore, the consideration of robust H∞ performance in the study of non-fragile stabilization of power systems has potential benefit from both theoretical and practical perspectives. However, to the best of our knowledge, the issue of observerbased non-fragile control design with gain fluctuation for an EV

model with external disturbances and multiple time delays in the controller design over a finite domain has not yet been fully considered. Motivated by the above discussions, the aim of this paper is to investigate the finite-time stabilization issue for an EV model with external disturbances and multiple time delays in the controller design by using an observer-based non-fragile H∞ control technique. Therefore, the problem under consideration can reflect more realistic dynamical behaviors. Moreover, the significant contributions of this study can be summarized in the following aspects: (i) Inspired by the work in [6], the LFC problem of an isolated power system with multiple time delays in the control input is considered for which an observer-based non-fragile control strategy that can ensure the system stability fluctuated by the load demands within a desired finite-time period is proposed. (ii) A state-space mathematical model of the considered power system integrated with EVs is formulated, which represents the dynamic interactions of EVs with multiple time delays. For this model, a full-order state observer is constructed since the states of power system are assumed to be unavailable for the stability analysis. (iii) Based on the Lyapunov–Krasovskii stability theory and the LMI approach, a delay-dependent criterion for finite-time boundedness of the formulated system is established by using the Wirtinger-based integral inequality. (iv) Compared with the traditional LFC schemes, the proposed scheme can offer the advantages of low complexity, ease of implementation, systems operating over a finite time interval and robust performance in the presence of multiple time delays, disturbance and controller gain fluctuations. The result of this study reveals that the proposed LFC method guarantees that all the signals in the considered power systems experience better performance as the frequency oscillates widely from its scheduled value. Finally, numerical simulations are given to illustrate the effectiveness of the proposed control design. 2. Problem formulation In this study, we consider an isolated power system model with two input delays as presented in [6], whose transfer function model is given in Fig. 1. In particular, we consider one input delay in the dynamics of EVs and the other in the dynamics of the plant involving reheated thermal turbine. In this system model, the control signal is divided into two parts, namely Pcg and Pce , from the power set-point Pc by the participation factors αg and αe . The divided control signals could regulate the outputs of the reheated thermal turbine and EVs to sustain the overall system frequency at the prescribed value. Within these setups, we first consider a time delay between the transmission of the control signal Pcg and the reheated thermal turbine. Then, the power command sent from the control center to the generator can be given as ϵg (t) = Pcg (t − d) = αg Pc (t − d), where d is the network-induced time delay and the dynamics of the reheated thermal generation can be expressed as 1 1 f (t) + αg Pc (t − d), Rg Tg Tg 1 P˙ r (t) = − Pr (t) + Xg (t), Tt Tt 1 Kr T t − Kr P˙ g (t) = − Pg (t) + Xg (t) + Pr (t), Tr Tr Tt Tt Tr X˙ g (t) = −

1

Tg 1

Xg (t) −

(1)

where the system parameters are listed in Table 1. In a similar way, we consider the second time delay in the EVs communication channel. Here, the power command received at the

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

23

Fig. 1. Transfer function model of a power system integrating EVs with two input time delays [6].

EVs can be defined as ϵ (t) = αe Pc (t − h), where h is the time delay between the control center and the aggregator of EVs. It is widely seen in the literature [5,6] that the dynamics of the considered EVs are represented by a first-order differential equation with a time constant Te and a gain Ke , whose output power deviation is given by P˙ e (t) = −

1 Te

Pe (t) +

Ke Te

αe Pc (t − h).

(2)

On the other hand, in the study of LFC problem of power systems, the area control error is generally defined as ACE(t) = bf (t) that is employed to guarantee zero steady state error of the frequency deviation. In continuation, the state variable, control input and external disturbance of the considered ]power system [ T are chosen as x(t) = f (t) Xg (t) Pr (t) Pg (t) Pe (t)∫ ζ (t) , u(t) = Pc (t) and w (t) = Pl (t), respectively, where ζ (t) = ACE(t)dt and w (t) ∫∞ satisfies the condition 0 w T (t)w (t)dt < δ with δ > 0. According to the above formulation, the state space representation of the considered power system model can be expressed as follows: x˙ (t) = Ax(t) + Bh u(t − h) + Bd u(t − d) + Bw w (t), y(t) = Cx(t),

(3)

[ ]T where y(t) is the output vector defined as y(t) = f (t) ζ (t) , ⎡ D ⎤ 1 1 ⎡ ⎤ 0 0 0 −M M M 0 1 1 ⎢− Rg Tg − Tg 0 0 0 0⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ 0 ⎥ 1 1 ⎢ 0 − 0 0 0⎥ ⎢ ⎥ T T ⎥ ⎢ t t , B = A = ⎢ ⎢ 0 ⎥, h Tt −Kr Kr 1 ⎥ ⎢ ⎥ − 0 0 ⎥ ⎢ 0 Tt Tr Tt Tr Tr ⎣ αe Ke ⎦ ⎣ 0 Te 0 0 0 − T1e 0⎦ b

0

0

0

⎡ 1⎤ −M αg ⎢ Tg ⎥ ⎢ 0 ⎥ [ ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎢ 0 ⎥ Bd = ⎢ 0 ⎥, Bw = ⎢ and C = ⎥ 0 ⎢0⎥ ⎢ 0 ⎥ ⎣0⎦ ⎣ 0 ⎦ ⎡

0

0

0

0

Table 1 Nomenclature. M, D f , Rg αg , αe Tg , Tt K r , Tr K e , Te ACE, b Pl Pe , Pg Pr Xg Pcg , Pce Pc

Inertia constant and load damping coefficient Frequency deviation and governor droop characteristic Reheated thermal turbine and EVs participation factors Speed governor and thermal turbine time constants Reheated gain and time constant EVs gain and time constant Area control error and frequency bias constant local power demand EVs and turbine output power deviations Intermediate thermal power output deviation Governor valve position deviation Reheated thermal turbine and EVs control inputs Local control input

the controller gain matrices. Supposing that uncertainties or gain fluctuations exist in the controller gain matrices, the conventional control scheme could not give the desired system performance. It is, therefore, necessary to design a controller that is insensitive to the aforementioned issues. Hence, we consider the controllers of the system (3) in the following form:

{

u(t − h) = − (K1 + ∆K1 (t)) xˆ (t − h),

(5)

u(t − d) = − (K2 + ∆K2 (t)) xˆ (t − d),

where matrices satisfy[ ∆K1 (t) and ∆]K2 (t) are some [ perturbed ] ing ∆K1 (t) ∆K2 (t) = MF (t) N1 N2 in which M and Ni (i = 1, 2) are constant matrices and F (t) is a time-varying matrix satisfying F T (t)F (t) ≤ I. Now, we define the estimation error as e(t) = x(t) − xˆ (t). Then, by using relations from (3)–(5), the following augmented system is formulated: x˙¯ (t) = A¯ x¯ (t) + B¯ h1 x¯ (t − h) + B¯ h2 (t)x¯ (t − h) + B¯ d1 x¯ (t − d)

⎤

+B¯ d2 (t)x¯ (t − d) + B¯ w w(t), 0 0

0 0

0 0

0 0

]

0 . 1

where

[

B¯ h2 (t) = B¯ d1 =

[

where xˆ (t) ∈ R6 is the estimation of x(t); yˆ (t) ∈ R2 is the observer output; L is the observer gain matrix; K1 and K2 are

[

0

]

0 , 0

0

[

A¯ =

A 0

]

LC , A − LC

[ −Bh MF (t)N1

−Bd K2

B¯ w = (4)

]

xˆ (t) , e(t)

x¯ (t) =

0 0 In this study, we assume that the state variables of (3) are not available for measurement. Therefore, in order to estimate the unavailable states, we consider the following state observer dynamics for the system (3):

⎧ x˙ˆ (t) = Axˆ (t) + Bh u(t − h) + Bd u(t − d) + L[y(t) − yˆ (t)], ⎪ ⎪ ⎪ ⎪ ⎨yˆ (t) = C xˆ (t), ⎪ ⎪u(t − h) = −K1 xˆ (t − h), ⎪ ⎪ ⎩ u(t − d) = −K2 xˆ (t − d),

(6)

B¯ h1 =

[ −Bh K1 0

]

0 , 0

]

0 , 0

B¯ d2 (t) =

[

−Bd MF (t)N2 0

]

0 , 0

]

0 . Bw

Furthermore, we define z(t) = ζ (t) = E¯ x¯ (t) as an observation [ ] vector, where E¯ = E 0 . Lemma 2.1 ([10]). For any matrix Π ∈ Rq×n with q < n and full row rank (rank(Π ) = q), there exists a singular value decomposition

24

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

√

⎡ [Σ ]10×10 ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

√

h(Γ1T + Γ2T (t)) −P6−1

∗ ∗ ∗

w1 c1 + δ (1 − e

−α T

d(Γ1T + Γ2T (t)) 0 −P7−1

ϵ1 P1 (Bˆ h + Bˆ d )Σ1

Σ2T

0 0

0 0 0

−ϵ1 I ∗

∗ ∗

) < λ1 c2 e

−α T

⎤ ⎥ ⎥ ⎥ < 0, ⎦

(9)

−ϵ1 I

,

(10)

Box I.

of Π as Π = U [S 0]V T , where S ∈ Rn×n is a diagonal matrix with non-negative diagonal elements in decreasing order, and U ∈ Rq×q and V ∈ Rn×n are unitary matrices. Lemma 2.2 ([10]). For a given matrix Π ∈ Rq×n with q < n and full row rank, that is, rank(Π ) = q, assume that X ∈ Rn×n is a symmetric matrix, then there exists a matrix Xˆ ∈ Rq×q [satisfying Π ] X = Xˆ Π X11 0 if and only if X can be described as X = V V T , where 0 X22 X11 ∈ Rq×q , X22 ∈ R(n−q)×(n−q) , and V ∈ Rn×n is the unitary matrix. Definition 2.3 ([23]). Given time constant T > 0, positive scalars δ , c1 and c2 with c1 < c2 , and positive definite matrix R¯ > 0, system (6) is said to be finite-time bounded with respect to (c1 , c2 , T , R¯ , δ ) if there exist feedback controllers in the form of (5) such that the following condition holds:

{

sup

− max{h,d}≤s≤0

(7)

Definition 2.4 ([6]). For a given time constant T > 0, the considered system (6) is said to have finite-time H∞ performance with respect to (0, c2 , T , γ , R¯ , δ ), where c2 > 0, γ > 0, δ > 0 and R¯ is a positive definite matrix, if the system (6) is finite-time bounded and the following inequality holds: T

∫

z T (t)z(t)dt ≤ γ 2

T

∫

wT (t)w(t)dt .

(8)

0

0

• the corresponding closed-loop system is finite-time bounded. • under zero initial ∫ T condition, the inequality (8) holds for any w(t) satisfying 0 w T (t)w (t)dt < δ . Lemma 2.6 ([31]). For any constant matrix Θ > 0, the following inequality holds for all continuously differentiable function x : [a, b] → Rn : b

xT (s)Θ x(s)ds ≥

(b−a) a

where Φ =

]T

b

[∫

x(s)ds a

∫

b

x(s)ds − a

Σ1,1 = P1 A¯ + A¯ T P1 + P2 + P3 + hP4 + dP5

−

4 h

P6 −

4 d

P7 − α P1 ,

4

Σ1,3 = P1 B¯ d1 +

d

P7 ,

Σ1,2 = P1 B¯ h1 +

Σ1,8 =

6 h2

P6 ,

4 h

P6 ,

Σ1,9 =

6 h2

P7 ,

4

P6 , h 4 6 6 Σ2,8 = − 2 P6 , Σ3,3 = −P3 − P7 , Σ3,9 = − 2 P7 , h d d 4 6 Σ4,4 = − P4 , Σ4,5 = 2 P4 , h h 12 4 6 Σ5,5 = − 3 P4 , Σ6,6 = − P5 , Σ6,7 = 2 P5 , h d d 12 12 12 Σ7,7 = − 3 P5 , Σ8,8 = − 3 P6 , Σ9,9 = − 3 P7 , d h d ¯ Σ10,10 = −α I , Γ1 = [A¯ B¯ h1 B¯ d1 0 · · · 0    Bw ],

Σ2,2 = −P2 −

times ], 0 

6

Definition 2.5 ([23]). For a given time constant T , system (6) is said to be robustly finite-time bounded with an H∞ disturbance attenuation level γ if there exist controllers in (5) such that

∫

Theorem 3.1. For given positive constants h, d, c1 , c2 , T , δ and symmetric matrix R¯ > 0, the considered power system (3) is finitetime bounded with respect to (c1 , c2 , T , R¯ , δ ) under the observerbased non-fragile controller (5) if there exist positive constants ϵ1 , α , λi (i = 1, 2, 3, . . . , 7) and symmetric matrices Pi > 0 (i = 1, 2, 3, . . . , 7) such that the conditions (see Eqs. (9) and (10)) given in Box I are satisfied, where

Σ1,10 = P1 B¯ w ,

}

x¯ T (s)R¯ x¯ (s), x˙¯ T (s)R¯ x˙¯ (s) ≤ c1 ⇒ x¯ T (t)R¯ x¯ (t) < c2 ,

∀ t ∈ [0, T ].

under the observer-based non-fragile controller (5) are presented in the following theorem.

2 (b − a)

Θ

b

[∫

]

x(s)ds +3Φ T ΘΦ , a

b

∫

s

∫

x(u)duds. a

Γ2 (t) = [0 B¯ h2 (t) B¯ d2 (t)

In this section, we will focus on design of an observer-based non-finite-time H∞ controller for power system (3) with multiple time delays. First, we will present a detailed procedure for obtaining observer-based non-fragile finite-time controller for system (3). In particular, finite-time boundedness conditions of system (3)

· ··

7

¯T Σ1 = [M

0 

· ··

9

[ ¯ = M Bˆ h =

[

M 0

−Bh 0

0 ] ,

0 , 0

[ N¯ i =

Bˆ d =

0 

· ··

7

]

]

times

Σ2 = [0 N¯ 1 N¯ 2

T

times

0 , 0

Ni 0

[ −Bd 0

0 0

] ]

times

0 ],

(i = 1, 2),

0 , 0

λ1 = λmin (P¯ 1 ),

λ2 = λmax (P¯ 1 ), λ3 = λmax (P¯ 2 ), λ4 = λmax (P¯ 3 ), λ5 = λmax (P¯ 4 ), λ6 = λmax (P¯ 5 ), λ7 = λmax (P¯ 6 ), λ8 = λmax (P¯ 7 ) and w1 = λ2 + hλ3 + dλ4 +

a

3. Main results

0 

h2 2

λ5 +

d2 2

λ6 +

h2 2

λ7 +

d2 2

λ8 .

Proof. In order to prove the required result, it is sufficient to prove that the augmented system (6) is finite-time bounded. Let us consider the following Lyapunov–Krasovskii functional candidate for the augmented system (6): V (x¯ (t), t) =

3 ∑ i=1

Vi (x¯ (t), t),

(11)

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

where

−

V1 (x¯ (t), t) =¯x (t)P1 x¯ (t), T

V2 (x¯ (t), t) = V3 (x¯ (t), t) =

∫

t

∫

t −h t

x¯ (s)P3 x¯ (s)ds, T

x¯ (s)P2 x¯ (s)ds + t

∫

t

∫

T

T

x¯ (u)P5 x¯ (u)duds

t

∫

t −h t

∫

t

x˙¯ T (u)P6 x˙¯ (u)duds,

]

Pa 0 (a = 1, 2, . . . , 7) are symmetric positive 0 Pa definite matrices. Calculating the time derivatives of Vi (x¯ (t), t) (i = 1, 2, 3) along the trajectories of system (6), we can obtain V˙ 1 (x¯ (t), t) = x¯ T (t)P1 x˙¯ (t) + x˙¯ T (t)P1 x¯ (t),

(12)

V˙ 2 (x¯ (t), t) = x¯ T (t) [P2 + P3 ] x¯ (t) − x¯ T (t − h)P2 x¯ (t − h)

− x¯ T (t − d)P3 x¯ (t − d), V˙ 3 (x¯ (t), t) = x¯ T (t) [hP4 + dP5 ] x¯ (t) + x˙¯ T (t) [hP6 + dP7 ] x˙¯ (t) ∫ t ∫ t x¯ T (s)P5 x¯ (s)ds − x¯ T (s)P4 x¯ (s)ds − t −d t −h ∫ t ∫ t T ˙ ˙ − x¯ (s)P6 x¯ (s)ds − x˙¯ T (s)P7 x˙¯ (s)ds. t −h

(13)

T

x¯ (s)P4 x¯ (s)ds ≤ − t −h

1 h

T

g1 (t) P4 g1 (t) −

[

3

[ g1 (t) −

h

2 h

T

x¯ (s)P5 x¯ (s)ds ≤ −

− t −d

1 d

T

g3 (t) P5 g3 (t) −

[

2

t

]T

3 d

[ g3 (t) −

2 d

t

x˙¯ T (s)P6 x˙¯ (s)ds ≤ −

− t −h

−

1 h 3

x˙¯ (s)ds

h

t

x˙¯ (s)ds − t −h

[∫

[∫

t

x˙¯ (s)ds

2 h

]

x˙¯ (s)ds −

2 h

g5 (t) ,

−

d

t −d

−

3

]T

[∫

d

t

]

[∫

x˙¯ (s)ds −

× P7 t −d

≤−

1 d

(17)

T

2 d

− 21

(21)

(i = 1, 2, . . . , 7). Then,

1

(22)

where λ1 = λmin (P¯ 1 ). Moreover, from (11), we can have V (x¯ (0), 0) = x¯ (0)P1 x¯ (0) + T

∫

]

∫

t

x¯ T (0)P2 x¯ (0)ds

t

x¯ T (0)P3 x¯ (0)ds +

+ t −d t

∫

]

t −d

]

g6 (t) ,

[x¯ (t) − x¯ (t − d)] P7 [x¯ (t) − x¯ (t − d)]

t

∫

t

∫

t −h

x¯ T (0)P4 x¯ (0)duds s

t

∫

x¯ T (0)P5 x¯ (0)duds +

+

d

t

]

e−α s w T (s)w (s)ds ,

t −h

x˙¯ (s)ds

x˙¯ (s)ds − g6 (t)

T

∫

≥λmin (P¯ 1 )x¯ T (t)R¯ x¯ (t) ≥λ1 x¯ T (t)R¯ x¯ (t),

t −d T

2

t −d

By integrating the above inequality from 0 to T , we can get

1

t

P7

t −d

[∫

[V (x¯ (t), t)e−αt ]

V (x¯ (t), t) ≥ x¯ T (t)R¯ 2 P¯ 1 R¯ 2 x¯ (t)

h

x˙¯ (s)ds

d dt

(20)

On the other hand, choose P¯ i = R¯ Pi R¯ it is easy to compute from (11) that

]

2

t

times

− 21

× P6 x¯ (t) − x¯ (t − h) − g5 (t) ,

[∫

times

0

1

1 x˙¯ T (s)P7 x˙¯ (s)ds ≤ −

6

0 · · 0],  ·

[ ] V (x¯ (t), t) < eα T V (x¯ (0), 0) + δ (1 − e−α T ) .

g5 (t)

[

t

B¯ w ],

0 · · 0  ·

times

[

[x¯ (t) − x¯ (t − h)]T P6 [x¯ (t) − x¯ (t − h)] h [ ]T 3 2 − x¯ (t) − x¯ (t − h) − g5 (t) h h

∫

E1 = [1 0 · · 0], E2 = [0 1 0 · · 0],  ·  · 9 times 8 times ¯ ], Γ = [ A B¯ h1 B¯ d1 E3 = [0 0 1 0 · · · 0 1   

V (x¯ (t), t)e−α T < V (x¯ (0), 0) + α

]

t −h T

t −h

≤−

Γ (t) = [Σ ]10×10 + E1T P1 B¯ h2 (t)E2 + E1T P1 B¯ d2 (t)E3 + E2T B¯ Th2 (t)P1 E1 + E3T B¯ Td2 (t)P1 E1 + (Γ1T + Γ2T (t)) [hP6 + dP7 ] (Γ1 + Γ2 (t)),

< e−αt wT (t)α I w(t). (16)

P6

t

× P6

g4T (t) g5T (t) g6T (t) w T (t) ,

]

t −h

[∫

g3T (t)

V˙ (x¯ (t), t) − α V (x¯ (t), t) − w T (t)α I w (t) < 0 (or)

g4 (t)

d

∫

g2T (t)

]T

g2 (t)

× P5 g3 (t) − g4 (t) ,

[∫

g1T (t)

x¯ T (t − d)

]

(15)

]T

h

t

where ψ T (t) = x¯ T (t) x¯ T (t − h)

(19)

and the elements of [Σ ]10×10 are given in the statement of the theorem. According to Lemma 2.7 in [31], there exists a positive scalar ϵ1 such that E1T P1 B¯ h2 (t)E2 + E1T P1 B¯ d2 (t)E3 + E2T B¯ Th2 (t)P1 E1 + ¯M ¯ T Bˆ Th P1T E1 + ϵ1−1 (E2T N1T N1 E2 + E3T N2T E3T B¯ Td2 (t)P1 E1 ≤ ϵ1 E1T P1 Bˆ h M N2 E3 ). With the aid of Schur complement, Γ (t) can be equivalently written as the matrix in the left-hand side of (9). Hence, it follows from (9) that Γ (t) < 0. Then, we can have

]

2

[

7

(14)

× P4 g1 (t) − g2 (t) ,

∫

t −h

∫t

7

t −d

t

∫t ∫s x¯ (s)ds, g (t) = t −h t −h x¯ (u)duds, g3 (t) = ∫t ∫s ∫t ∫s 2 x¯ (s)ds, g4 (t) = t −d t −d x¯ (u)duds, g5 (t) = t −h t −h x˙¯ (u)duds, t −d ∫t ∫s g6 (t) = t −d t −d x˙¯ (u)duds. ∫t

where g1 (t) =

Γ2 (t) = [0 B¯ h2 (t) B¯ d2 (t)

By applying Lemma 2.6 to each integral terms of (14), we can get

−

(18)

V˙ (x¯ (t), t) − α V (x¯ (t), t) − w T (t)α I w (t) ≤ ψ T (t)Γ (t)ψ (t),

s

where Pa =

∫

g6 (t)

Then, by combining the relations from (12)–(18), we can obtain

s

x˙¯ T (u)P7 x˙¯ (u)duds,

+

[

s

t −d

s

∫ +

t −d

d

d

t

∫

x¯ (u)P4 x¯ (u)duds +

t

2

t −d

t −h

∫

x¯ (t) − x¯ (t − d) −

d

]T

[ ] 2 × P7 x¯ (t) − x¯ (t − d) − g6 (t) ,

t

∫

T

3

25

[

s

× P6 x˙¯ (0)duds +

∫

t t −h

∫

t t −d

t

∫

x˙¯ T (0) s

t

∫

x˙¯ T (0)P7 x˙¯ (0)duds s

[ ≤ λmax (P¯ 1 ) + hλmax (P¯ 2 ) + dλmax (P¯ 3 )

26

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

√ √ T √ √ T ˆ1 Σ ˆ 2T ˆ1 ˆ ]10×10 ϵ2 ( h + d)(Bˆ h + Bˆ d )Σ ϵ1 (Bˆ h + Bˆ d )Σ hΓˆ 1 dΓˆ 1 [Σ ⎢ ˆ ∗ P − 2 X 0 0 0 0 6 ⎢ ⎢ ∗ ∗ Pˆ 7 − 2X 0 0 0 ⎢ ⎢ ∗ ∗ ∗ −ϵ1 I 0 0 Γˆ = ⎢ ⎢ ∗ ∗ ∗ ∗ −ϵ1 I 0 ⎢ ∗ ∗ ∗ ∗ ∗ −ϵ2 I ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ β1 R¯ −1 < X < R¯ −1 , 0 < Pˆ i < βi R¯ −1 (i = 2, 3, . . . , 7), ] [ h2 d2 h2 d2 1 + hβ2 + dβ3 + β4 + β5 + β6 + β7 c1 + δ (1 − e−αT ) < c2 e−αT , β1 2 2 2 2

⎡

ˆ 2T Σ

Γˆ 3T

⎤

0 0 0 0 0

0 0 0 0 0 0 −I

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎦

−ϵ2 I ∗

(25)

(26) (27)

Box II.

h2

d2

h2

λmax (P¯ 4 ) + λmax (P¯ 5 ) + λmax (P¯ 6 ) 2 2 2 ] { } d2 + λmax (P¯ 7 ) sup x¯ T (s)R¯ x¯ (s), x˙¯ T (s)R¯ x˙¯ (s) +

2

d2

h2

h2

d2

≤ λ2 + hλ3 + dλ4 + λ5 + λ6 + λ7 + λ8 2 2 2 2 } { T T x¯ (s)R¯ x¯ (s), x˙¯ (s)R¯ x˙¯ (s) × sup

]

− max{h,d}≤s≤0

≤ w1 c1 ,

(23) 2

2

where w1 = λ2 + hλ3 + dλ4 + h2 λ5 + d2 λ6 + on the relations (21)–(23), we can obtain eα T w1 c1 + δ (1 − e−α T )

[

x¯ (t)R¯ x¯ (t) ≤

λ1

h2 2

λ7 +

d2 2

λ8 . Based (24)

Thus, if the condition (10) holds, it is obvious that x¯ (t)R¯ x¯ (t) < c2 , ∀t ∈ [0, T ]. Therefore, according to Definition 2.3, the power system (3) is finite-time bounded under the observer-based non( ) fragile controller (5) with respect to c1 , c2 , T , R¯ , δ . This completes the proof. T

Now, by utilizing the results obtained in Theorem 3.1, we derive the sufficient conditions ensuring finite-time boundedness of system (3) with an acceptable H∞ performance index. As a foundation, Theorem 3.1 provides an LMI-based sufficient criterion for the finite-time boundedness of the augmented system (6). Further, by taking the H∞ performance into account, in a similar way to Theorem 3.1, we present the following theorem. Before proceeding further, for brevity, we denote the H∞ performance index as J(t) = z T (t)z(t) − γ 2 w T (t)w (t). Theorem 3.2. Consider the multiple input delayed power system (3). For prescribed scalars h > 0, d > 0 and symmetric matrix R¯ > 0, the considered system (3) is finite-time bounded with a satisfactory H∞ performance index via the observer-based non-fragile controller (5) if there exist positive constants α , ϵ1 , ϵ2 , γ , βi (i = 1, 2, 3, . . . , 7), positive definite matrices X , Pˆ i (i = 2, 3, . . . , 7) and any appropriate dimensioned matrices W1 , W2 , W3 such that the conditions (see Eqs. (25)–(27) in Box II) hold: where

ˆ 1,1 = Θ1 + Θ1T + Pˆ 2 + Pˆ 3 + hPˆ 4 + dPˆ 5 − Σ ˆ 1,2 = Θ2 + Σ ˆ 1,8 = Σ

6 h2

Pˆ 6 ,

4 h

Pˆ 6 ,

ˆ 1,9 = Σ

ˆ 2,2 = −Pˆ 2 − Σ

4 h

Pˆ 6 ,

ˆ 1 , 3 = Θ3 + Σ 6 h2

Pˆ 7 ,

4 d

4 h

Pˆ 6 −

4 d

Pˆ 7 ,

6 h2

Pˆ 7 ,

Pˆ 6 ,

ˆ 3,3 = −Pˆ 3 − Σ

4

Pˆ 7 ,

12 h3

12 h3

ˆ 4,4 = − Pˆ 4 , Σ

ˆ 4,5 = Σ

h

4

Pˆ 4 ,

Pˆ 5 ,

ˆ 6,6 = − Pˆ 5 , Σ ˆ 8,8 Σ

ˆ 10,10 = −γ 2 I , Σ

h 12 = − 3 Pˆ 6 , h

6 h2

Γˆ 1 = [Θ1 Θ2 Θ3

6

0 · · 0]T ,  ·

ˆ 1 = [M ¯T Σ

9

[

AX Θ1 = 0

Θ2 =

[ −Bh W1

Bˆ d =

0

[ −Bd 0

ˆ 2 = [0 N¯ 1 X Σ

6

Pˆ 5 , h2 12 = − 3 Pˆ 7 , d ¯ 0 · · ·   0 Bw ],

ˆ 6,7 = Σ ˆ 9,9 Σ

Pˆ 4 ,

times

N¯ 2 X

times

0 · · 0],  · 7

times

]

W2 C , AX − W2 C

]

0 , 0

]

0 , 0

Θ3 =

[ −Bd W3

Γˆ 3T = [E¯ X

0

]

0 , 0

Bˆ h =

[ −Bh 0

4 d

Pˆ 7 ,

]

0 , 0

0 · · 0]T .  · 9

times

Moreover, the desired controller and observer gain matrices are ob−1 −1 T tained by K1 = W1 X −1 , K2 = W3 X −1 and L = W2 USX11 S U , respectively. Proof. According to Theorem 3.1, the augmented system (6) is finite-time bounded. Now, we will prove the H∞ performance of augmented system (6) under zero initial condition. Based on the proof of Theorem 3.1, it can be easily computed that V˙ (x¯ (t), t) − α V (x¯ (t), t) + J(t) < ψ T (t)[Γ¯ ]ψ (t), where Γ¯ = Γ (t) + Γ3T Γ3 with Γ3T = [E¯

0 

(28)

· ··

9

times

T 0 ] and the

elements of Γ (t) are defined in Theorem 3.1 but Γ (t)10×10 = 2 −γ [ I. By using ] Schur complement, Γ¯ can be rewritten as Γ¯ = Γ (t) Γ3T . Then, following the similar lines in the proof of The∗ −I orem 3.1, Γ (t) can be written as the matrix in (9) with Σ10×10 = −γ 2 I. It should be noted that the matrix Γ¯ consists of nonlinear terms by means of two unknown matrices. To make it linear, let −1 X = P1 and pre- and post-multiply Γ¯ by diag{X , . . . , X , I , I , I },

  

ˆ 1,10 = B¯ w , Σ

ˆ 2 ,8 = − Σ

ˆ 7,7 = − Σ

] .

6 d2

ˆ 5,5 = − Σ

− max{h,d}≤s≤0

[

T

ˆ 3,9 = − Σ

12 times where X = diag{X , X }. After that, the term CX can be rewritten as −1 −1 T Xˆ C with the aid of Lemmas 2.1 and 2.2, where Xˆ = USX11 S U . Now, by introducing some new variables Pˆ i = X Pi X (i = 2, 3, . . . , 7), W1 = K1 X , W2 = LXˆ , W3 = K2 X and using the facts −X P6−1 X < Pˆ 6 − 2X and −X P7−1 X < Pˆ 7 − 2X , Γ¯ can be equivalently written as Γˆ in (25). Besides, it is noted from

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

27

Fig. 2. Closed-loop system performance according to Theorem 3.2. 1

1

1

1

Theorem 3.1 that λ1 < R¯ − 2 P1 R¯ − 2 < λ2 and 0 < R¯ − 2 Pi R¯ − 2 < λi+1 (i = 2, 3, . . . , 7). According to the relationships X = P1−1 and −1 ¯ −1 Pˆ i = X Pi X (i = 2, 3, . . . , 7), it can be obtained that λ2 R < −1 ¯ −1 − 2 − 1 ¯ X < λ1 R and 0 < Pˆ i < λ1 λi+1 R , (i = 2, 3, . . . , 7). Here, 1 by setting, λ1 = 1, λ− = β1 and λi+1 ≤ βi (i = 2, 3, . . . , 7), 2 the constraint (26) can easily be obtained. Moreover, based on the above settings, the condition (10) can be equivalently viewed as (27) which is the desired condition. Thus, if the inequalities (25)–(28) hold, it can be concluded that the augmented system (6) is finite-time bounded with a desired H∞ performance level according to Definition 2.5, which completes the proof. In what follows, we present the finite-time boundedness criterion of system (3) with a desired H∞ performance index in the absence of gain fluctuations, that is, ∆K1 (t) = ∆K2 (t) = 0. Corollary 3.3. For given positive scalars h, d and symmetric matrix R¯ > 0, the considered power system (3) under the proposed controller (5) without gain fluctuations is finite-time bounded with a satisfactory H∞ disturbance attenuation level subject to (c1 , c2 , T , R¯ , δ ) if there exist positive constants α , γ , β , positive definite matrices X , Pˆ i (i = 2, 3, . . . , 7) and any appropriate dimensioned matrices W1 , W2 , W3 such that the inequality (27) and the following condition hold:

⎡ ˆ ]10×10 [Σ ⎢ ∗ ⎢ ⎣ ∗ ∗

√

hΓˆ 1T Pˆ 6 − 2X

∗ ∗

√

dΓˆ 1T 0 Pˆ 7 − 2X

∗

Γˆ 3T

⎤

0 ⎥ ⎥ < 0, 0 ⎦ −I

(29)

where the above parameters are mentioned in Theorem 3.2. Furthermore, the non-fragile control and the observer gain matrices are −1 −1 T calculated by K1 = W1 X −1 , K2 = W3 X −1 and L = W2 USX11 S U , respectively. Proof. The proof of this corollary is similar to that of Theorem 3.2, so it is not presented here. Remark 3.4. In [6] and [8], the authors investigated the stability problem of power systems with EVs using LFC design, where the

effect of uncertainties were not considered. In this study, we include the uncertain factors in the power system model and employ an LFC design that consists of two different time delays. Therefore, the proposed system is more general than those in [6] and [8], and the considered LFC design is more applicable than those in [2– 4]. Further, a small amount of works about power systems has been examined the stability issue along with H∞ performance via non-fragile control strategy. However, those works are invalid if the system states are not measurable. To tackle this situation, an observer-based non-fragile LFC design is proposed in this paper for power systems. Hence, the analysis technique and the proposed system model in this paper deserve much attention to meet the demands more effectively. Remark 3.5. As we know that the delay-dependent stability criterion is more effective than the delay-independent one, wherein conservatism plays a vital role. In the task of reducing conservatism of the proposed stability criterion, Wirtinger-based integral inequality has been widely accepted as an effective method and employed in many time delay systems compared to some existing methods, such as Jensen’s inequality, reciprocally convex approach and free-weighting matrix approach. Thus, we adopt the Wirtinger-based integral inequality in this paper to obtain a tighter bound for the derivative of considered Lyapunov functional. Though the advantage of the proposed method is not displayed explicitly, it seems that the obtained results are less conservative than that of employing the aforementioned existing methods. However, the conservatism of proposed results could be further reduced by applying advanced integral inequalities and novel control algorithms. Remark 3.6. In order to validate the system, an observer-based robust non-fragile control is designed to operate the output power of EVs such that the LFC requirement is satisfied. Specifically, the purpose of employing such controller is to regulate the original thermal power plant when it could not receive any control signal

28

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

Fig. 6. Response of the system output according to Theorem 3.2. Fig. 3. Measured and observer output responses of system (3) according to Theorem 3.2.

be mentioned that the proposed model of power system is flexible to be deployed in hydro and thermal turbines.

Fig. 4. Time history of x¯ T (t)R¯ x¯ (t) according to Theorem 3.2.

from the control center. The main objective of EVs is to reheat the thermal turbine units to provide stability on fluctuations of load demands. They are coordinated for high value services, such as ancillary services, which can potentially lower the running cost, despite a higher initial cost. However, the reheated thermal turbine is designed for EV coordination, which is used to define the EV requirements for power system ancillary services. The reheated thermal turbine EV coordination infrastructure shown in Fig. 1 is composed of centralized coordination layer and local functions at the grid interface. To combine the needs of the power system and those of the EV users, the EV coordinator generates a power schedule for the single vehicles. Based on the same scheme, all EVs are expected to respond according to the received schedules. Besides, EVs have small-sized batteries in comparison with traditional electrical grid units, but they are able to modulate their charging rate very quickly. The coordination of EVs load of public or private charging stations is an energy storage solution to facilitate photovoltaic integration in low voltage distribution grids. It should

Remark 3.7. In some practical situations, the states of a power system are all not measurable when formulating its mathematical model. So the state feedback control approach might not possible to assure the stability of power system in this case. To handle this issue, the observer-based control techniques are often applied [10– 13], where the unmeasurable system states are estimated from the relative outputs of the system. However, the parameters in the observer-based control might have some variations owing to additive unknown noise and environmental influence. For this reason, non-fragile control, which is more insensitive to some errors or variations in the gain of feedback control [25–28], is proposed for the considered power system to smoothly achieve the desired performance. Also in the LFC problem of power systems, the presence of time delays might often cause the interchange powers and frequencies oscillate widely from the scheduled values. Based on the above discussions, in this study, an observer-based nonfragile LFC design for power systems with multiple time delays and external disturbance is developed to attain the required system performance within a prescribed finite time period. 4. Numerical example In this section, we show that the obtained theoretical results can be used to solve the LFC problem of an EV via computerized simulations. For this purpose, we conduct simulations by applying the proposed observer-based non-fragile controller to the considered multiple time delayed EV model. For convenience, the parameter values of the EV model (3) are borrowed from [6] and given as follows: Tt = 0.8, Kr = 5, Tr = 10, Tg = 0.08, Rg = 2.4, M = 0.1667, b = 0.425, D = 0.0083, αg = 0.9, αe = 0.1, Te = 1 and Ke = 1. In practice, due to the change of load demand of the EV, gain fluctuation behavior frequently exists. Thus, the parameters representing gain fluctuations are taken as

Fig. 5. Control responses of system (3) according to Theorem 3.2.

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

29

Fig. 7. Closed-loop system performance according to Corollary 3.3.

M = 0.01[1 2 0 5 − 2 1], N1 = 0.01, N2 = 0.01 and F (t) = 0.5 cos(t). The rest of parameters involved in the simulation are selected as T = 2, α = 1, c1 = 0.5, δ = 4 and R¯ = I. Then, by solving the LMI-based conditions in Theorem 3.2, a set of feasible solutions can be obtained and subsequently, the controller and observer gain matrices can be computed as [ ] K1 = 6.3214 2.7215 −12.7071 40.1196 31.3395 2.2821 , [ ] K2 = 0.0235 0.0355 −0.1540 0.4829 0.3725 0.0272 and [ ]T 0.7879 1.0432 −0.0138 0.0744 0.0711 −0.1091 L= . −0.0888 −0.0153 −0.0072 0.0046 0.0022 0.2937

Moreover, with the above said designing parameter values, the optimal finite-time bound value of c2 and the optimized disturbance attenuation level are calculated as c2 = 1.8862 and γ = 2.1325, respectively. For the simulation purposes, initial conditions for exact states and observer states are chosen as x(0) = [0.1 0.1 0.5 0.1 0.4 0.6]T and xˆ (0) = [0.4 0.2 0.3 0.2 0.0 0.4]T , respectively. Fig. 2(a), 2(b) and 2(c), respectively show the actual state trajectories, estimated state trajectories and corresponding error trajectories of the power system (3) under the observerbased non-fragile finite-time controller (5). The result in Fig. 2 depicts that the error system states converge to zero in a finite-time interval with the obtained observers. The corresponding measured and observation output responses are shown in Fig. 3. The time history of x¯ T (t)R¯ x¯ (t) is shown in Fig. 4 from which it can be seen that the finite-time boundedness of system (6) is guaranteed. Furthermore, Figs. 5 and 6 provide the control and output responses, respectively. In addition, let us consider the uncertain free case, that is ∆K1 (t) = 0 and ∆K2 (t) = 0. Now, by solving the LMIs obtained in Corollary 3.3 using MATLAB LMI control tool box, we can obtain feasible solutions for h = 2, d = 2 and γ = 3, and the optimum finite-time bound value is found to be c2 = 1.4674. The corresponding gain matrices are estimated as [ ] K1 = 5.6263 2.7676 −12.6591 40.0736 31.3012 2.2782 ,

Fig. 8. Measured and observer output responses of system (3) according to Corollary 3.3.

Fig. 9. Time history of x¯ T (t)R¯ x¯ (t) according to Corollary 3.3.

K2 = 0.0658

[

1.2900 −0.0894

[ L=

0.0357 0.0252 −0.0144

−0.1524 −0.0133 −0.0075

0.4793 0.0748 0.0046

0.3697 0.0722 0.0021

0.0270

]

and

]T −0.1103 . 0.2967

Based on the above gain matrices, the actual state trajectories, estimated state trajectories and estimation error trajectories of the

30

D. Aravindh, R. Sakthivel, B. Kaviarasan et al. / ISA Transactions 91 (2019) 21–31

Fig. 10. Control responses of system (3) according to Corollary 3.3.

References

Fig. 11. Response of the system output according to Corollary 3.3.

power system are shown in Fig. 7(a), 7(b) and 7(c) respectively. Fig. 8 displays the responses of measured output and observer output vectors. Fig. 9 represents the time history of x¯ T (t)R¯ x¯ (t) according to Corollary 3.3. The corresponding control and output responses are shown in Figs. 10 and 11, respectively. From these simulation results, it can be concluded that the proposed observer-based non-fragile finite-time controller (5) effectively stabilizes the considered power system with multiple input delays in a finite-time interval whether gain fluctuation occurs in the controller design or not. This illustrates the applicability and significance of the newly proposed control design. 5. Conclusion In this paper, a novel delay-dependent H∞ finite-time boundedness condition has been derived for dealing with LFC issue of an EV model by using an observer-based non-fragile control scheme. In particular, the proposed control design has taken not only multiple time delays into account but also modeling uncertainties. More precisely, a new of set of sufficient conditions for obtaining the proposed observer-based H∞ finite-time controller has been established in terms of LMIs with the use of Lyapunov stability theory. Finally, the effectiveness of the proposed control design under all possible deregulated scenarios, such as multiple time delays, modeling uncertainties and external disturbances has been demonstrated by using a numerical example. Moreover, it is important and significant to apply the proposed method for a more complicated model with nonlinearity factors, which has a wider application scope in practical engineering. This could be our future direction of research.

Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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