Stabilization of uncertain switched discrete-time systems against actuator faults and input saturation

Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Stabilization of uncertain switched discrete-time systems against actuator faults and input saturation✩ ∗

∗∗

R. Sakthivel a,b , , L. Susana Ramya c , Yong-Ki Ma d , , Muslim Malik e , A. Leelamani c a

Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea Department of Mathematics, Anna University - Regional Campus, Coimbatore 641 046, India d Department of Applied Mathematics, Kongju National University, Chungcheongnam-do 32588, South Korea e School of Basic Sciences, Indian Institute of Technology Mandi Kamand (H.P.), 175 005, India b c

article

info

Article history: Received 28 November 2018 Received in revised form 17 September 2019 Accepted 11 October 2019 Available online xxxx Keywords: Switched discrete-time systems Fault-tolerant control Actuator saturation Actuator faults

a b s t r a c t This paper studies the robust fault-tolerant stabilization problem for a class of uncertain switched discrete-time systems subject to time delays, actuator faults and input saturation by using delta operator approach. By constructing an appropriate Lyapunov– Krasovskii functionals in delta domain, a new set of sufficient conditions is provided for the stability of the delta operator time delay switched systems. More precisely, gain matrices of the fault-tolerant controller can be computed by solving the linear matrix inequalities and the proposed controller can robustly stabilize the uncertain delta operator time delay switched system for all admissible parameter perturbations. Further, in order to obtain a maximal estimate of the domain of attraction, an LMI based optimization algorithm is presented. It is noted that a better control performance can be obtained for small sampling periods by using delta operator approach than using shift operator. Finally, numerical examples with simulation results are presented to illustrate the effectiveness and potential of the developed control design technique. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Switched systems are class of hybrid systems, which can be used to describe a large number of practical systems with switching characteristics such as chemical processes, mechanical systems, computer controlled systems, power electronics, network control system and so on [1–6]. Therefore, for the past few decades the study of stability and stabilization of switched control systems have become an important research topic [7,8]. By applying the average dwelltime approach with Lyapunov–Krasovskii functional, the result in [5] considered the problem of exponential stability of uncertain discrete-time nonlinear switched systems with parameter uncertainties and randomly occurring delays via Takagi–Sugeno fuzzy model. By using the singular value decomposition approach, the authors in [9,10] obtained some new delay-dependent sufficient conditions for the finite-time stability of singular nonlinear switched systems subject to delay ✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.nahs.2019.100827. ∗ Corresponding author at: Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India. ∗∗ Corresponding author. E-mail addresses: [email protected] (R. Sakthivel), [email protected] (Y.-K. Ma). https://doi.org/10.1016/j.nahs.2019.100827 1751-570X/© 2019 Elsevier Ltd. All rights reserved.

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R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

and disturbances. On the other hand, when faults such as actuator or component failures occur in the control system, the overall system performance may alter which ranges from performance degradation to instability. Therefore, fault-tolerant control problems have received much attention and a number of interesting works have been obtained [11–17]. Recently, with the help of reduced-order observer method and switched Lyapunov function technique, the results in [18] proposed the output feedback fault-tolerant controller design for switched discrete-time systems with actuator fault. On the other hand, in many practical control systems, actuator saturations are ubiquitous and inevitable, and its existence may result in poor performance of a control system and even cause the instability [19–23]. Therefore, considerable attention has been devoted to the stabilization of control systems subject to saturating controllers [24–26]. By using the multiple Lyapunov functions approach, the results in [25] was established for stability analysis for a class of discrete-time switched systems with saturating actuators. Fu and Ma [23,27] obtained a set of sufficient conditions for the passive controller design for singular time-delay system with actuator saturation and nonlinear disturbance which guarantees that the singular time-delay system is regular, impulse free and asymptotically stable. Moreover, delta operator is utilized to replace the traditional shift operator because of its high-speed sampling. Because of its applications in many practical control systems, considerable attention have been paid on the study of delta operator systems [28–31]. In [32], by choosing an appropriate Lyapunov–Krasovskii functional in δ -domain, a new set of sufficient condition is obtained for asymptotic stability with a prescribed H∞ index for Takagi–Sugeno fuzzy systems with time-varying delays via delta operator approach. Yang et al. [33] proposed fault-detection filter for uncertain T–S fuzzy models based on the delta operator approach. Precisely, there exist works on fault-tolerant controller design with input saturation for discrete-time switched systems with time delays [34,35]. Moreover, fault-tolerant control design in the presence of time varying actuator faults and saturation for continuous-time systems have been reported, as in ,fan. But, it is of utmost importance to study the faulttolerant controller design problem for uncertain discrete-time switched systems subject to time delays, actuator faults and input saturation by using delta operator approach. Further in order to reduce the conservatism, a more general Lyapunov function for delta operator systems is used in the estimation of domain of attraction. Motivated by the discussions above, in this paper we focus on solving the fault-tolerant controller design problem for uncertain switched discrete-time systems subject to time delays, actuator faults and input saturation. The main contributions of this paper lie in the following facts:

• The switched time delay systems via delta operator approach is investigated in which the controller design is subject to actuator saturations and faults.

• By choosing an appropriate Lyapunov–Krasovskii functional in delta domain, a robust control scheme is obtained for the discrete-time switched systems.

• More precisely, we have designed the fault-tolerant controller such that the closed-loop poles are robustly asymptotically stable within a specified ellipsoid.

• A set of sufficient conditions for the existence of robust controller have been proposed in the form of LMIs, which depends on sampling period and state delay.

• The sampling period is an explicit parameter so that it is possible to analyze the effect of the robust controller with different sampling periods. 2. Problem formulation and preliminaries In this paper, we consider the following delta operator uncertain switched discrete-time delay system in the form

δ xk = Aˆ σ (k) xk + Aˆ dσ (k) xk−n + Bˆ σ (k) ufsk ,

(1) fs

where xk ∈ Rp is the state vector at the kth instant; n represents the state delay and uk ∈ Rm denotes the control input with actuator fault and saturation at the kth instant. Further, σ (k) : Z + → S = {1, 2, . . . , N } is a switching rule with N being the number of subsystems. For notational simplicity, in the sequel, for σ (k) = i ∈ S, we denote Aˆ σ (k) , Aˆ dσ (k) and Bˆ σ (k) as Aˆ i , Aˆ di and Bˆ i , respectively. Without loss of generality, uncertain real-valued matrices Aˆ i ∈ Rp×p , Aˆ di ∈ Rp×p and Bˆ i ∈ Rp×m are assumed to have the following form:

[

Aˆ i

]

Aˆ di

[

Bˆ i = Ai

Adi

]

[

Bi + Hi Dik E1i

E2i

E3i ,

]

(2)

where Ai , Adi and Bi are known real constant matrices; Hi , E1i , E2i and E3i denote the uncertain structure with Dik being the time varying matrix such that DTik Dik ≤ I. Now, the delta operator is defined as

{ δ xk =

,

dx dt xk+1 −xk T

T =0

,

T ̸ = 0,

where T ≥ 0 is the sampling period of the system (1). Further, the indicator function ξ (k) for i ∈ {1, 2, . . . , N } is defined as ξ (k) = (ξ1 (k), ξ2 (k), . . . , ξN (k))T , where

ξi (k) =

1, 0,

{

when the ith subsystem is active otherwise.

(3)

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

Substituting (3) into (1), we obtain δ xk = of saturation is described as follows:

∑N

i=1

3

ξi (k)(Aˆ i xk + Aˆ di xk−n + Bˆ i ufsk ). Then, the actuator fault model in the presence

fs

uk = Fi sat(uk ),

(4)

where sat(uk ) denotes the saturated feedback control input and Fi (i ∈ S) is continuous fault matrix of the form Fi = diag {fi1 , fi2 , . . . , fim } with 0 ≤ fdij ≤ fij ≤ fuij ≤ 1, (j ∈ {1, 2, . . . , m}) for some given constants fdij and fuij . For simplicity, define Fui = diag {fui1 , fui2 , . . . , fuim } and Fdi = diag {fdi1 , fdi2 , . . . , fdim }. Furthermore, define F0i = 12 (Fui +Fdi ), F1i = 1 (F − Fdi ). Now, the fault matrix Fi can be rewritten as Fi = F0i + F1i ηi , where ηi = diag {η˜ i1 , η˜ i2 , . . . , η˜ im } ∈ Rm×m , 2 ui −1 ≤ η˜ ij ≤ 1, j = 1, 2, . . . , m. Besides, the saturation term sat(·) is represented by a standard vector-valued function sat(uk ) = [sat(u1k ), sat(u2k ), . . . , sat(umk )]T , where sat(ujk ) = sgn(ujk ) min{1, |ujk |}, j ∈ {1, 2, . . . , m}. This paper proposes a systematic method to design a fault-tolerant saturated controller for the switched delta-operator system (1) using a state feedback control uk = Ki xk . If the designed control gain Ki is such that the matrix Aˆ i + Bˆ i Ki satisfies the stability condition, then the local stability of the closed-loop system (1) can be formally assured as follows: for every ϵ > 0, there exist δ > 0 such that, for every initial condition x0 contained in the closed ball Bδ = {xk Rp ∥ xk |≤ δ}, the corresponding solution x(k, x0 ) belongs to the closed ball Bϵ = {xk Rp ∥ xk |≤ ϵ}. Therefore, by choosing ϵ such that, for every xk ∈ Bϵ when the input is not saturated, we can obtain stability and determine the stable region. However, the closed ball Bδ might be quite small. To reduce the conservatism, we consider the stable region containing the saturated region. For this purpose, to ensure stability, we need to estimate the domain of attraction of the origin, which is defined as L(Li ) = {x0 ∈ Rn : limk→∞ φ (x0 , k) = 0}, for initial condition xk = x0 and the system state trajectory φ (x0 , k). In the case of the switched systems, the domain of attraction can be estimated{based on the ellipsoid approach. } For a symmetric positive definite matrix P > 0, let us define an ellipsoid as Ω (P) = xk ∈ Rn |xTk Pxk ≤ 1, P T = P > 0 . For a quadratic Lyapunov function V (k, xk ) = xTk Pxk , if the time derivative V˙ (k, xk ) is negative definite in Ω (P), then Ω (P) ⊆ L(Li ). In this case, Ω (P) is said to be contractively invariant. Thus, for the considered switched system, we can estimate L(Li ) by finding Ω (P). According to Lemma 1 in [36], for given matrices Ki and Li ∈ Rp×p satisfies −1 ≤ sat(Ki xk ) ≤ 1, then the sat(Ki xk ) can be represented as follows: m

sat(Ki xk ) =

2 ∑

m

β s ( ∆ s Ki + ∆ − s Li )xk , where

s=1

2 ∑

βs = 1 and βs > 0,

(5)

s=1

where ∆s ∈ Rm×m (for s = {1, 2, . . . , 2m }) is a diagonal matrix whose elements are either 0 or 1 and ∆− s = I − ∆s . By taking into consideration, loss of effectiveness of the actuators and saturated control input (5), the actuator fault model (4) can be rewritten as m

[ fs uk

= F0i

2 ∑

m

βs ∆s Ki + F1i ηi

s=1

2 ∑

m

βs ∆s Ki + F0i

s=1

2 ∑

m

βs ∆s Li + F1i ηi −

s=1

2 ∑

] βs ∆s Li xk . −

(6)

s=1

Now, by combining (1)–(6), the closed-loop system takes the form

δ xk = A˜ i xk + Aˆ di xk−n , ∑2m ∑2m ∑2m ∑2m − ˆ ˆ where A˜ i = Aˆ i + Bˆ i F0i s=1 βs ∆s Ki + Bˆ i F0i s=1 βs ∆− s Li + Bi F1i ηi s=1 βs ∆s Li . s=1 βs ∆s Ki + Bi F1i ηi

(7)

3. Main results In this section, based on Lyapunov stability theory and delta operator approach, a new design algorithm for faulttolerant controller will be developed for the asymptotic stability of the system (1) which contains the actuator fault and the input saturation in its design. m2

m

Let Xi > 0, Zi

m

m2

> 0, Zij1 > 0, Vi1 > 0, Sij1 > 0, Ri 1 > 0, Ri 2 > 0, Qi i, j ∈ {1, 2, . . . , N }; m1 = 1, 2, . . . , m2 ; m2 = 1, 2, . . . , (n − 1) be symmetric matrices such that Theorem 3.1. m1 +1

Rj

m +1 Qj 2

m1

< Ri <

m Qi 2

> 0 and Qin > 0 for (8)

, ∀i, j ∈ {1, 2, . . . , N }.

(9)

For T ̸ = 0 and for some positive scalars ϵi and αi , system (1) is asymptotically stabilized via the controller (4) within the contractively invariant set Ω (Pi ) if the following LMIs hold:

⎡ Λ1,1 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

CiT + ϵi Hi HiT −2Xi + nT 2 Zi1 + ϵi Hi HiT

∗ ∗ ∗

m

Adi Xi + 1n Zi 2 Adi Xi m −Vin − 1n Zi 2

∗ ∗

√

√

T CiT + ϵi T Hi HiT √ T ϵ√ i T Hi Hi T T Xi Adi −Xi + ϵi THi HiT

∗

⎤

T E4i 0 ⎥ ⎥ (E2i Xi )T ⎥ ⎥ < 0, 0 ⎦ −ϵi I

(10)

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R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

⎡ ˆ 1 ,1 Λ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗ [ −1 ∗

1 m2 Z n i

CiT −2Xi + nT 2 Zij1 + αi Hi HiT

Adi Xi m2

−Vin − 1n Zi ∗ ∗

∗ ∗ ∗ Lit −Xi

]

⎤

TCiT + Xi αi THi HiT T (Adi Xi )T −TXj + αi T 2 Hi HiT

T E4i 0 ⎥ ⎥ (E2i Xi )T ⎥ ⎥ < 0, 0 ⎦ −αi I

∗

< 0, (t = 1, 2, . . . , m),

(11)

(12)

∑m ˆ 1,1 = − 1 Xi − 1 Zim2 + Sij1 with Ci = Ai Xi + Bi F0i 2s=1 βs ∆s Wi + + Vi1 + ϵi Hi HiT and Λ T n ∑m ∑m ∑m ∑ ∑2m β ∆ W + B F η 2 β ∆− U and E4i = E1i Xi + E3i F0i 2s=1 βs ∆s Wi + E3i F0i 2s=1 βs ∆− U+ η Bi F0i s=1 βs ∆− s Ui + ∑s2m s i − i 1i i s=1 s s i ∑2m s i E3i F1i ηi s=1 βs ∆s Wi + E3i F1i ηi s=1 βs ∆s Ui . Also, the robust fault-tolerant controller gain matrices can be computed by Ki = Wi Xi−1 and Li = Ui Xi−1 . 1 m2 Z n i 2m Bi F1i i s=1

where Λ1,1 = Ci + CiT −

Proof. In order to prove the asymptotic stability of the system (1), it is enough to show that the closed-loop system (7) is asymptotically stable. Now, we choose the Lyapunov–Krasovskii functional (LKF) candidate as V (k, xk ) =

3 ∑

Vp (k, xk ),

(13)

p=1

where

( V1 (k, xk ) =xTk P(ξ (k))xk = xTk

N ∑

) ξi (k)Pi xk ,

i=1 n ∑

V2 (k, xk ) =T

xTk−m2 Q m2 (

ξ (k))xk−m2 = T

n ∑

( xTk−m2

m2 =1

m2 =1

N ∑

) m2 i (k)Qi

ξ

xk−m2

i=1

and V3 (k, xk ) = T

m2 n ∑ ∑

ξ (k))ek−m1 = T

eTk−m1 Rm1 (

m2 n ∑ ∑

( eTk−m1

m2 =1 m1 =1

m2 =1 m1 =1

N ∑

) m1 i (k)Ri

ξ

ek−m1 ,

i=1

with ek = xk+1 − xk . Subsequently, in the delta domain, the Lyapunov–Krasovskii function takes in the following form

δ V1 (k, xk ) = =

=

=

V1 (k + 1, xk+1 ) − V1 (k, xk ) T 1{ T 1

xTk+1 P(ξ (k + 1))xk+1 − xTk P(ξ (k))xk

⎧ ⎨

T ⎩ 1{ T

⎛ (T δ xk + xk )T ⎝

N ∑

}

⎞

(

ξj (k + 1)Pj ⎠ (T δ xk + xk ) − xTk

j=1

N ∑

ξi (k)Pi xk

⎭

i=1

(TxTk A˜ Ti + TxTk−n Aˆ Tdi + xTk )(TPj A˜ i xk + TPj Aˆ di xk−n + Pj xk ) − xTk Pi xk

= xTk [T A˜ Ti Pj A˜ i + A˜ Ti Pj + Pj A˜ i +

) ⎫ ⎬

1 T

}

(Pj − Pi )]xk + xTk [(T A˜ Ti + I)Pj Aˆ di ]xk−n

{ }T + xTk [(T A˜ Ti + I)Pj Aˆ di ]xk−n + xTk−n (T Aˆ Tdi Pj Aˆ di )xk−n . δ V2 (k, xk ) =

V2 (k + 1, xk+1 ) − V2 (k, xk ) T

⎫ ⎧ n n ⎨ ∑ ⎬ ∑ 1 = xTk+1−m2 Q m2 (ξ (k + 1))xk+1−m2 − T xTk−m2 Q m2 (ξ (k))xk−m2 T ⎭ T ⎩ m2 =1

m2 =1

(14)

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

=

n ∑

m2

xTk+1−m2 Qj

n ∑

xk+1−m2 −

m2 =1

m2

xTk−m2 Qi

5

xk−m2

m2 =1

= xTk Qj1 xk − xTk−n Qin xk−n +

n−1 ∑

m2 +1

xTk−m2 (Qj

m

− Qi 2 )xk−m2 .

(15)

m2 =1

It follows from (9) and the above equation that

δ V2 (k, xk ) < xTk Qj1 xk − xTk−n Qin xk−n .

(16)

Further, we have

δ V3 (k, xk ) =

V3 (k + 1, xk+1 ) − V3 (k, xk )

T ⎫ ⎧ m2 m2 n n ⎬ ⎨ ∑ ∑ ∑ ∑ 1 = T eTk+1−m1 Rm1 (ξ (k + 1))ek+1−m1 − T eTk−m1 Rm1 (ξ (k))ek−m1 ⎭ T ⎩ m2 =1 m1 =1

= neTk R1j ek +

m2 =1 m1 =1

m2 −1

n

∑ ∑

m1 +1

eTk−m1 Rj

ek−m1

m

n ∑

m2 =2 m1 =1

−

n m2 −1 ∑ ∑

eTk−m1 Ri 1 ek−m1 −

m

eTk−m2 Ri 2 ek−m2 .

(17)

m2 =1

m2 =2 m1 =1

By applying Jensen inequality in [37] to (17) and using (8), we can obtain

δ V3 (k, xk ) < neTk R1j ek −

n 1 ∑

n

m2

eTk−m2 Ri

m2 =1

n ∑

ek−m2

m2 =1

1

m

< n(xk − xk+1 )T R1j (xk − xk+1 ) − (xk−n − xk )T Ri 2 (xk−n − xk ) n

1

m

< nT 2 (δ xk )T R1j δ xk − (xk−n − xk )T Ri 2 (xk−n − xk ).

(18)

n

In addition, we can observe that

− 2δ xTk Pi {δ xk − A˜ i xk − Aˆ di xk−n } = 0

(19)

Combining (14), (16), (18) and (19) we can obtain that δ V (k, xk ) = Ψ T Υij Ψ , where Ψ = xTk

[

⎡ Υ11 Υij = ⎣ ∗ ∗

˜

1 m2 R n i

δ xTk

xTk−n

]T

and

˜ + I)Pj Aˆ di + 2 ⎦, where Υ11 = A˜ Ti Pj +Pj A˜ i +T A˜ Ti Pj A˜ i + 1 (Pj −Pi )− 1 Rm +Qj1 . According −2Pi + nT 2 R1j Pi Aˆ di T n i m ∗ T Aˆ Tdi Pj Aˆ di − Qin − 1n Ri 2 to the definition of asymptotic stability, system (7) will be asymptotically stable if Υij < 0. To obtain the required result under this circumstance, we need to discuss stability results under two cases, namely, for i = j and for i ̸ = j. Case(i): If i = j, by applying Schur complement to Υij , we can obtain ⎡ √ T⎤ m m A˜ Ti Pi + Pi A˜ i − 1n Ri 2 + Qi1 A˜ Ti Pi Pi Aˆ di + 1n Ri 2 T A˜ i ⎢ ⎥ 2 1 ⎢ ˆ ∗ −2Pi + nT Ri Pi Adi 0 ⎥ Υii = ⎢ √ T⎥ ⎢ ⎥. m ∗ ∗ −Qin − 1n Ri 2 T Aˆ di ⎦ ⎣ ATi Pi

(T ATi

∗

⎤

∗

−Pi−1

∗

m2

Let us denote Xi = Pi−1 ; Wi = Ki Pi−1 ; Ui = Li Pi−1 ; Vi1 = Pi−1 Qi1 Pi−1 ; Vin = Pi−1 Qin Pi−1 ; Zi1 = Pi−1 R1i Pi−1 ; Zi By pre and post multiplying Υii by diag {Xi , Xi , Xi , I } we can obtain that m2

Cˆ i + Cˆ iT − 1n Zi

⎡ ⎢ Υˆ ii = ⎢ ⎣

∗ ∗ ∗

+ Vi1

Cˆ iT

−2Xi + nT 2 Zi1 ∗ ∗

m2

Aˆ di Xi + 1n Zi Aˆ di Xi

−

Vin

− ∗

1 m2 Z n i

√

T Cˆ iT

√ 0

m

= Pi−1 Ri 2 Pi−1 .

⎤

⎥ ⎥, T ⎦ ˆ T Xi Adi −Xi

(20)

6

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

∑2m

where Cˆ i = Aˆ i Xi + Bˆ i F0i (20), we have

s=1 m2

⎡

Ci + CiT − 1n Zi

βs ∆s Wi + Bˆ i F0i + Vi1

Hi ⎢ Hi ⎥ [ ⎣ 0 ⎦ Dik E4i √ T Hi

s=1

ˆ βs ∆− s Ui + Bi F1i ηi

s=1

−2Xi + nT 2 Zi1 ∗ ∗

βs ∆s Wi + Bˆ i F1i ηi

√

m

0

E2i Xi

[

0 + E4i

0

]T

E2i Xi

0

s=1

βs ∆− s Ui . By using (2) in

⎤

⎤T

Hi T ⎢ Hi ⎥ Dik ⎣ ⎦ . √0 T Hi

⎡

]

∑2m

T CiT ⎥ √ 0 T ⎥+ T Xi Adi ⎦ −Xi

∗

⎤

⎡

∑2m

Adi Xi + 1n Zi 2 Adi Xi m −Vin − 1n Zi 2

CiT

∗ ∗ ∗

⎢ Υˆ ii = ⎢ ⎣

∑2m

(21)

By applying Lemma 2.4 in [38] and Schur complement to (21), it can be verified that (10) holds. Case(ii) : If i ̸ = j, by applying Schur complement to Υij , we can get m

− T1 Pi − 1n Ri 2 + Qj1 ⎢ ∗ Υij = ⎢ ⎣ ∗ ∗

⎡

A˜ Ti Pi

−2Pi + nT 2 R1j ∗ ∗

−

1 m2 R n i

T A˜ Ti + I

Pi Aˆ di

0

Qin

− ∗

1 m2 R n i

⎤

⎥ ⎥. T ˆ T Adi ⎦ −T Pj −1

m

m

Let us denote Xi = Pi−1 ; Wi = Ki Pi−1 ; Ui = Li Pi−1 ; Zi 2 = Pi−1 Ri 2 Pi−1 ; Zij1 = Pi−1 R1j Pi−1 ; Vin = Pi−1 Qin Pi−1 ; Sij1 = Pi−1 Qj1 Pi−1 . By pre and post multiplying Υij by diag {Xi , Xi , Xi , I } we can obtain m2

− T1 Xi − 1n Zi ⎢ ∗ Υˆ ij = ⎢ ⎣ ∗ ∗

⎡

+ Sij1

CiT

−2Xi + nT 2 Zij1 ∗ ∗

−

1 m2 Z n i

T Cˆ iT + Xi

Aˆ di Xi

0

Vin

− ∗

1 m2 Z n i

⎤

⎥ ⎥. ⎦ T (Aˆ di Xi ) −TXj

(22)

T

By using (2) in (22) and following the similar proof as the previous case, we can see that (11) holds. By the definition of linear region and Schur complement, if (12) holds then the ellipsoidal set Ω (Pi ) is an invariant set of system (7) satisfying Ω (Pi ) ⊂ L(Li ). Thus, it is concluded that the system (1) is asymptotically stabilized via the controller (4) within the contractively invariant set Ω (Pi ). Hence the proof is complete. When the sampling period T → 0, the system (7) takes the following form x˙ (t) = A˜ i x(t) + Adi x(t − h),

(23)

where h ≥ 0 is the constant time delay. It should be noted that (23) represents a continuous-time system for which the conditions for stability criteria is provided in the following Corollary 3.2. m

m

m

m

Corollary 3.2. Let Xi > 0, Zi 2 > 0, Zij1 > 0, Vi1 > 0, Sij1 > 0, Ri 1 > 0, Ri 2 > 0, Qi 2 > 0 and Qin > 0 for i, j ∈ {1, 2, . . . , N }; m1 = 1, 2, . . . , m2 ; m2 = 1, 2, . . . , (n − 1); be symmetric matrices such that (8) and (9) holds. System (23) is asymptotically stable within the contractively invariant set Ω (Pi ) if the following LMIs hold: m2

Ci + CiT − 1n Zi

⎡ ⎢ ⎢ ⎢ ⎣

+ Vi1 + ϵi Hi HiT

∗ ∗ ∗ ∗

m

CiT + ϵi Hi HiT −2Xi + ϵi Hi HiT

Adi Xi + 1n Zi 2 Adi Xi m −Vin − 1n Zi 2

∗ ∗ ∗

⎡ 1 m2 1 m2 CiT Z − n Zi + Sij1 n i T ⎢ ∗ −2Xi + αi Hi Hi Adi Xi ⎢ m ⎢ ∗ ∗ −Vin − 1n Zi 2 ⎣ ∗ ∗ ∗ ∗ ∗ ∗ [ ] −1 Lit < 0, (t = 1, 2, . . . , m), ∗ −Xi

∗ ∗

Xi 0 0 0

∗

0 0 0 − Xi

∗

T E4i 0 ⎥ ⎥ (E2i Xi )T ⎥ < 0, ⎦ 0 −ϵi I

⎤

T E4i 0 ⎥ ⎥ (E2i Xi )T ⎥ < 0. ⎦ 0 −αi I

⎤

(24)

. Further, it should be mentioned that when the sampling period is unity, that is when T = 1, system (7) becomes a discrete-time system of the form xk+1 = Aσ (k) xk + Adσ (k) xk−n ,

(25)

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

7

where Ai = A˜ i + I. The following Corollary 3.3 gives the sufficient conditions for the stability of the discrete-time system (25), which can be deduced from Theorem 3.1. m

m

m

m

Corollary 3.3. Let Xi > 0, Zi 2 > 0, Zij1 > 0, Vi1 > 0, Sij1 > 0, Ri 1 > 0, Ri 2 > 0, Qi 2 > 0 and Qin > 0 for i, j ∈ {1, 2, . . . , N }; m1 = 1, 2, . . . , m2 ; m2 = 1, 2, . . . , (n − 1); be symmetric matrices such that (8) and (9) holds. Then, the system (25) is asymptotically stable within the contractively invariant set Ω (Pi ) if the following LMIs hold:

⎡

V11

CTi + ϵi Hi HiT −2Xi + nZi1 + ϵi Hi HiT

⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗ ⎡ − Xi − ⎢ ⎢ ⎢ ⎢ ⎣ [

−1 ∗

∗ ∗ ∗ 1 m2 Z n i

∗ ∗ ∗ ∗ ]

Lit

−Xi

∗ ∗

+ Sij1

CTi −2Xi + nZij1 + αi Hi HiT

∗ ∗ ∗

CTi + ϵi Hi HiT ϵi Hi HiT Xi ATdi −Xi + ϵi Hi HiT

T E4i 0 ⎥ ⎥ (E2i Xi )T ⎥ < 0, ⎦ 0 −ϵi I

⎤

∗ 1 m2 Z n i

Adi Xi m2

−Vin − 1n Zi ∗ ∗

CiT + Xi αi Hi HiT (Adi Xi )T Xj + αi Hi HiT

∗

⎤

T E4i 0 ⎥ ⎥ (E2i Xi )T ⎥ ⎥ < 0, 0 ⎦ −αi I

< 0, (t = 1, 2, . . . , m), m2

where V11 = Ci + CTi − 1n Zi Bi F1i ηi

m

Adi Xi + 1n Zi 2 Adi Xi m −Vin − 1n Zi 2

∑2m

+ Vi1 +ϵi Hi HiT with Ci = Ai Xi + Bi F0i

(26)

∑2m

s=1

βs ∆s Wi + Bi F0i

∑2m

s=1

βs ∆− s Ui + Bi F1i ηi

∑2m

s=1

βs ∆s Wi +

− s=1 βs ∆s Ui .

Remark 1. By implementing the Lyapunov function technique, a robust controller design for discrete-time systems subject to actuator faults and saturation has been presented in [35]. Note that, if we take T = 1 in this present study, then the delta operator system (1) can be reduced to the discrete-time system with actuator faults and saturation as in [35]. And letting T → 0, system (1) is changed to the continuous-time system with actuator faults and saturation. Therefore, this present study includes some existing studies as special cases. Remark 2. In [39], the authors derived a set of conditions for obtaining a fault-tolerant control which ensures the quadratically D-stability of uncertain delta operator continuous-time switched systems. In the proposed work, the delta operator discrete-time switched control system is considered, where the control input is subject to saturation. It is noted that ignoring input saturation can lead to performance degradation and even instability of a closed-loop system. Further, a set invariance condition is established for the delta operator system with actuator saturation in terms of LMIs. Moreover, it should be mentioned that fault-tolerant controller with saturation has not been fully investigated yet. Motivated by these considerations, in this work, we investigate the robust fault-tolerant stabilization problem for uncertain switched discrete-time systems subject to time delays, actuator faults and input saturation by using delta operator approach. Remark 3. It should be noted that in the present study, the input saturation is assumed to be bounded by [−1, 1]. But in practice, the saturation limit may not be unity and in some cases, it is even non-symmetric. On the other hand, the direct mathematical analysis of stabilization with non-unit input saturation or non-symmetric input saturation is significantly complicated. As discussed in [40], by applying variable transformation technique and the virtue of Lemma 3.2 in [40], it is possible to link the non-symmetrical saturation constraints with normalized symmetric constraints. Thus, the proposed results under symmetrical saturation can be extended, in term of LMIs, to the system with nonsymmetric saturation constraints. Therefore, reliable control design for uncertain switched discrete-time systems with non-symmetric saturation is one of our future studies. 4. Numerical simulations In order to demonstrate the effectiveness of the obtained theoretical results and to highlight the superiority of the developed control design technique, two numerical examples with its simulation results are presented in this section. Example 1. Consider the switched delta operator system (7) consisting of two subsystems as described in [39] Subsystem 1:

[ ] [ ] [ −1.4 0.3 −1.8 −0.5 0.1 0.2 −2 0.3 ; B1 = 0.2 −1.1 ; H1 = −0.2 A1 = 0.9 0.1 0.3 −2 −0.7 0.2 0.1 [ ] [ ] −0.9 −0.1 0.5 0.3 −1.3 0.6 E11 = and E31 = . 0.6 −0.3 0.2 −0.4 0.2 −0.8

] −0.1 0.3 ; −0.2

8

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

Fig. 1. Switching law.

Subsystem 2:

[ ] [ ] [ −1.8 0.6 −1.2 −0.3 0.1 0.7 −1.5 0.7 ; B2 = 1.2 −0.8 ; H2 = 0.4 A2 = 1.2 0.6 2.3 −2.4 −0.4 0.6 −0.2 [ ] [ ] 1.3 −0.6 0.8 −0.5 −0.6 1.2 E12 = and E32 = . 0.1 0.4 −0.7 0.6 0.1 −0.4

0.3 −0.1 ; 0.5

]

of obtaining fault-tolerant parameters are taken Fu1 ]= [For the purpose ] [ ] [ controller ] against[ actuator,] the following [ ] [ 0.95 0 1 0 0.5 0 0.8 0 −0.1 0 0.3 0 ; Fu2 = ; Fd1 = ; Fd2 = ; η1 = and η2 = . 0 0.99 0 0.98 0 0.8 0 0.7 0 1 0 0.6 Further[to investigate the we assume n = 2 and the time delay system state matrices as ] effect of [ time delay in the ] system [ ] [ ] 0.1 0 0 0 0.1 0 0.05 −0.09 0.1 0.06 −0.1 0. 1 0 0.2 0 , Ad2 = 0 0 0.1 , E21 = Ad1 = and E22 = . We 0.02 −0.1 −.03 0.03 −0.09 −0.3 0 0 0 .2 0.1 0 0 choose an admissible switching law σ (k) which is shown in Fig. 1. Our goal is to design the fault-tolerant state feedback controller such that system (1) is asymptotically stable. By solving the stabilization problem as in Theorem 3.1 recursively by using MATLAB, a set of feasible solutions is obtained when the sampling period T = 0.25. From the obtained solutions, the state feedback control gain matrices can be computed as follows:

[ −0.9015 1.2368 [ −0.0436 L1 = −0.0511

K1 =

0.1391 −2.6952

−0.0401 0.2513

] [ ] −5.1903 −6.8205 −3.7961 3.1548 , K2 = , −0.5249 −6.7557 −9.1606 7.7480 ] [ ] 0.2341 0.6781 0.1850 0.4532 and L2 = . −0.0229 −0.1633 0.4899 0.0341

For the simulation purpose we arbitrarily choose the initial condition as [0.5 − 0.5 0.5]. The performance of the system (1) under the effect of actuator fault in the saturated controlled input is shown in Fig. 2. It is concluded from Fig. 2 that the system eventually gets stabilized even though it consumes more time under the presence of actuator faults in the saturated controlled input. Hence, we can conclude that the proposed controller (6) can effectively stabilize the system (1) even in the presence of time delay, parameter uncertainties, actuator faults and input saturation. Consider system (1) under the absence of actuator faults in the saturated control input. Then system (1) can be rewritten as

δ yk = Aˆ σ (k) yk + Aˆ dσ (k) yk−n + Bˆ σ (k) sat(uk ), where yk ∈ Rp is the state vector of the ith subsystem at the kth instant; n is the state delay of the system, uk ∈ Rm is the saturated control input of the ith subsystem at the kth instant and the other terms are similar to those in system (1). Let Θ (Pi ) = {yk ∈ Rp /yTk Pi yk ≤ 1} denote the ellipsoidal invariant set of the i th subsystem without the actuator faults in the saturated control input. Alternatively let Ω (Pi ) = {xk ∈ Rp /xTk Pi xk ≤ 1} denote the ellipsoidal invariant set of the ith subsystem with the presence of actuator faults in the saturated control input. Fig. 3 represents the ellipsoidal invariant sets of the ith subsystem with the presence and absence of actuator faults in the saturated control input. It can be seen from Fig. 3 that Ω (Pi ) ⊂ Θ (Pi ). The state responses of the system (1) with (0.5, −0.5, −0.5), (0.5, 0.5, −0.5), (−0.5, 0.5, −0.5) and (−0.5, −0.5, −0.5) as initial conditions are shown in Fig. 4. It can be observed from Fig. 4 that for various initial conditions inside Ω (Pi ), the system is stable and the state trajectories lie within the contractively invariant ellipsoid Ω (Pi ).

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

9

Fig. 2. State and control response of system (1) in the presence and absence of actuator faults and input saturation.

Fig. 3. Invariant ellipsoids in the presence and absence of actuator faults.

In order to show the significance of the delta operator system, Fig. 5 is presented. The state response of the systems with sampling periods 0.001, 0.75 and 1 is shown in Fig. 5. This figure depicts that the continuous-time system (23) and the discrete system (25) are stable. According to the definition of delta operator, δ x → x˙ (t) as T → 0, which is obvious from Fig. 5. Therefore we can conclude that the convergence process of δ x → x˙ (t) is a monotonous process. It should be mentioned that a delta operator system approaches the original continuous-time system when the sampling period is considerably small and in particular, a delta operator system unifies a discrete-time system and a continuous-time system. It is noted that system (1) maintains better performance even under the various initial conditions inside the region of convergence. The simulation results reveal that the system (1) under study in this paper is well stabilized by using the fault-tolerant controller with the presence of the actuator faults in the saturated control input. Example 2. To demonstrate the applicability of the developed results in this paper, we provide a real world example along with its simulation results. In particular, we consider the truck–trailer model as in [41] which consists of two subsystems with time delay n = 2. The system parameters are given as follows: Subsystem 1: 0.5104 −0.5104 0.5206

[ A1 =

0 0 −4

0 0 ; Ad1 = 0

]

[

0.2187 −0.2187 0.2231

0 0 0

0 0 ; B1 = 0

]

[ ] [ ] −1.4322 0.2 0.0036 ; H1 = −0.2 ; −0.0037 0.1

10

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

Fig. 4. State response of system (1) for various initial conditions inside Ω (P).

Fig. 5. State responses of system (1) for different sampling periods.

E11 = −0.9

−0.1

[

0.5 ; E21 = 0.05

]

−0.09

[

0.1 and E31 = 0.3

]

−1.3

[

0.6 .

]

Subsystem 2: 0.5104 −0.5104 0.8290

[ A2 =

E12 = 1.3

[

−0.6

0 0 −6.3694

0 0 ; Ad2 = 0

]

0.8 ; E22 = 0.06

]

[

[

0.2187 −0.2187 0.3553

−0.1

0 0 0

]

0.1 and E32

]

] [ ] −1.4322 0.7 0.0036 ; H2 = 0.4 ; −0.0059 −0.2 [ ] = −0.5 −0.6 1.2 .

0 0 ; B2 = 0

[

The purpose of this paper is to develop fault tolerant controller against actuator saturation for the considered model. In order to obtain the required result, it is assumed that Fu1 = 0.99; Fu2 = 1; Fd1 = 0.5; Fd2 = 0.8; η1 = 0.1 and η2 = 0.3. By solving the stabilization problem in Theorem 3.1 recursively by using MATLAB, a set of feasible solutions is obtained for a sampling period of T = 0.05. From the obtained solutions, the state feedback control gain matrices can be computed as K1 = 2.4819

−0.2205

0.0496 , K2 = 2.2064

L1 = 0.1554

−0.0232

0.0028 and L2 = 0.1866

[

[

]

]

[

[

−0.2137 0.0076

0.0289 ,

]

0.0471 .

]

For simulation purposes, we arbitrarily choose the initial condition as [0.5 − 0.5 10]. The state responses of the truck–trailer system with the presence and absence of actuator fault and input saturation is shown in Fig. 6. It can be concluded from Fig. 6 that the system is stabilized even in the presence of actuator faults in the saturated controlled input. Fig. 7 represents the ellipsoidal invariant sets of the ith subsystem of the truck–trailer model in the presence and absence of actuator faults in the saturated control input. It can be seen from Fig. 7 that Ω (Pi ) ⊂ Θ (Pi ). The results reveal that the considered truck–trailer model is well stabilized with the use of the fault-tolerant control law in the presence of the actuator faults and saturations. Remark 4. It should be mentioned that the delta operator system can be reduced to the traditional shift operator system when T = 1. The stability of the system (1) when T = 1 is shown in Fig. 5. Hence, the delta operator system is more general case than the traditional shift operator system. Moreover, the delta operator is a limiting case of the differential operator. Therefore, delta operator provides better performance with high-speed sampling while the shift operator may lead to numerical instability under the same conditions. 5. Conclusion In this paper, we have studied the fault-tolerant control problem of uncertain switched discrete-time systems with time delays against actuator failures and input saturation within the delta operator framework. By using Lyapunov functional

R. Sakthivel, L. Susana Ramya, Y.-K. Ma et al. / Nonlinear Analysis: Hybrid Systems 35 (2020) 100827

11

Fig. 6. State response of truck–trailer system in the presence and absence of actuator faults and input saturation.

Fig. 7. Invariant ellipsoids in the presence and absence of actuator faults.

theory, new design conditions for the fault-tolerant control have been established in LMI framework, which guarantee the asymptotic stability of the uncertain delta operator switched system. The main advantage of the paper lies in the fact that even in the presence of control input constraints like actuator faults and actuator saturation, the system can be well stabilized. At the end, two numerical examples with graphical results are provided to demonstrate the effectiveness of the proposed control design. Our future research topic deals with the study of finite-time dissipative control design for discrete-time switched fuzzy singular stochastic time-delay systems with LPV based uncertainties and nonlinear actuator faults. Acknowledgments The research of Yong-Ki Ma was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, South Korea (No. 2018R1D1A1B07049623). The work of L. Susana Ramya was supported by the Department of Science & Technology, Government of India through the Women Scientists Scheme-A under grant No. SR/WOS-A/PM-101/2016. References [1] Y. Liu, Y. Niu, Y. Zou, H.R. Karimi, Adaptive sliding mode reliable control for switched systems with actuator degradation, IET Control Theory Appl. 9 (2015) 1197–1204. [2] M. Wag, J. Qiu, M. Chadli, M. Wang, A switched system approach to exponential stabilization of sampled-data t-s fuzzy systems with packet dropouts, IEEE Trans. Cybern. 46 (2016) 3145–3156. [3] Y. Wu, T. Liu, Y. Wu, Y. Zhang, H∞ Output tracking control for uncertain networked control systems via a switched system approach, Internat. J. Robust Nonlinear Control 26 (2016) 995–1009. [4] M.J. Park, O.M. Kwon, S.G. Choi, Stability analysis of discrete-time switched systems with time-varying delays via a new summation inequality, Nonlinear Anal. Hybrid Syst. 23 (2017) 76–90. [5] P. Balasubramaniam, L.J. Banu, Robust stability criterion for discrete-time nonlinear switched systems with randomly occurring delays via TS fuzzy approach, Complexity 20 (2015) 49–61. [6] E. Noghreian, H.R. Koofigar, Adaptive output feedback tracking control for a class of uncertain switched nonlinear systems under arbitrary switching, Internat. J. Systems Sci. 49 (2018) 486–495. [7] H. Li, Z. Chen, Y. Sun, H.R. Karimi, Stabilization for a class of nonlinear networked control systems via polynomial fuzzy model approach, Complexity 21 (2015) 74–81. [8] Y. Dong, F. Yang, Finite-time stability and boundedness of switched nonlinear time-delay systems under state-dependent switching, Complexity 21 (2015) 267–275. [9] N.T. Thanh, P. Niamsup, H. Zou, V.N. Phat, Finite-time stability of singular nonlinear switched time-delay systems: A singular value decomposition approach, J. Franklin Inst. B 354 (2017) 3502–3518.

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