INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 17, Issue: 6, Month: December, Year: 2022 Article Number: 4957, https://doi.org/10.15837/ijccc.2022.6.4957 CCC Publications Observer-based feedback control of interval-valued fuzzy singular system with time-varying delay and stochastic faults H. Jerbi, M. Kchaou, O. Alshammari, R. Abassi, D. Popescu Houssem Jerbi* College of Engineering University of Hail Po.Box 2440, Hail, Kingdom of Saudi Arabia. *Corresponding author: h.jerbi@uoh.edu.sa Mourad Kchaou College of Engineering University of Hail Po.Box 2440, Hail, Kingdom of Saudi Arabia. mouradkchaou@gmail.com Obaid Alshammari College of Engineering University of Hail Po.Box 2440, Hail, Kingdom of Saudi Arabia. os.alshammari@uoh.edu.sa Rabeh Abassi College of Engineering University of Hail Po.Box 2440, Hail, Kingdom of Saudi Arabia. r.abbassi@uoh.edu.sa Dumitru.Popescu Faculty of Automatics and Computers, University Politehnica of Bucharest, Romania dumitru.popescu@acse.pub.ro Abstract There are countless applications of non-linear systems that incorporate delay and algebraic equations. Despite current improvements in control theory, stochastic actuator defects still pose challenges when it comes to these systems. Furthermore, when it is not possible to measure the states of the system, and when uncertainties affect the system under investigation, the problem becomes even more complex. This paper is concerned with fault-tolerant observer-based controller synthesis for non-linear delayed singular systems with uncertainties and stochastic actuator failures. On the basis of interval valued models, a new Lyapunov-Krasovskii functional is built to develop a less conservative criterion to ensure that the closed-loop system is admissible in the mean-square sense. In addition, as these matrices are coupled with multiple variables, finding the parametric matrices of the observer and controller in terms of the obtained condition is more complex and challenging. The proposed method employs the matrix inequality decoupling technique to resolve this issue. Eventually, simulations are carried out to demonstrate the applicability of the proposed method. Keywords: Singular system, (IVF) model , time varying delay, probabilistic faults, reliable observer-based control. https://doi.org/10.15837/ijccc.2022.6.4957 2 1 Introduction This section includes the literature review, notations, and acronyms used in the document, as well as an outline of the publication and its goals. 1.1 Literature Review Singular systems which are described by a couple of algebraic and differential equations, are char- acterized by their different modes namely finite dynamic modes, infinite non-dynamic modes, and infinite dynamic modes, respectively. The infinite dynamic modes has the feature to destroy the stability and the performances of the system. Thus, the admissibility, that includes stability, reg- ularity, and non-impulsiveness/causality should be verified when dealing with this class of systems. As a consequence, the investigation of singular systems is both theoretically and practically impor- tant [1, 4, 39]. It is worth noting that time delays are common in many physical plants, and they can have a substantial negative impact on performance and even the stability of practical systems [7, 27, 32, 42, 43, 44]. Singular models and time-delay phenomena are general enough to enable some fundamental results from the theory of state-space systems to be extended to this class of systems (see for instance [3, 5, 10, 15, 17, 45]). On the other hand, the research on nonlinear systems [9] is an extremely hard issue due to their inherent complexity. Due to its rigorous mathematical structure, the T-S fuzzy model [14] has recently been applied to handle nonlinear complex systems, since this model has been known for its powerful approximation of smoothly nonlinear systems. In many cases, uncertainty arises such as partially unknown parameters, unpredictable disturbances in the system, varying interpretations of linguistic variables of the fuzzy models, etc. It is worth noting, however, that the membership functions of type-1 fuzzy sets are well-known, and the control problem can not be handled directly for nonlinear plants with parameter uncertainties. Research on (IVF) fuzzy systems has attracted attention due to the advantages of (IVF) sets over type 1 fuzzy sets in dealing with parameter uncertainties, and many control design results have been developed [24, 29]. Among oth- ers, in [33], the fuzzy decentralized output feedback control is investigated using the event-triggered approach for interval type-2 fuzzy systems against input saturation. The study in [18] discussed the issue of filter design for IVF systems with D-stability constraint. The authors in [13] investigated the discrete-time interval-valued fuzzy systems with actuator faults to study the reliable non-fragile con- trol problem with H∞ performance. Very recently, the IVF approach has been extended to non-linear singular systems with uncertainties. To mention a few, the admissibilization for IVF singular systems is studied in [6]. Based on the LMI approach, the dynamic output-feedback control design issue is investigated in [30] for singular interval-valued fuzzy systems. We emphasize that, the evolution of industrial engineering has led to numerous practical plants becoming quite complex with failures. Control communities become concerned about this issue as ac- tuator/sensor failures can adversely affect system performance. The goal of this issue is to introduce the concept of fault-tolerant control (FTC) and fault diagnosis as critical approaches for designing reliable controllers that are capable of maintaining the critical functionality of systems subject to problems and failures [47, 50]. Kavarian et al develop a method for designing fault-tolerant controllers for power systems subject to random changes and actuator failures in [12]. The FTC method for wind-diesel hybrid systems with time-varying bounded sensor faults has been proposed in [11]. In [37], the reliable observer-based control problem for discrete-time Takagi-Sugeno fuzzy systems with time-varying delay and stochastic actuator faults is formulated from the input-output approach. We also report on some results relating to FTC for singular systems. For systems with actuator and/or sensor faults, sliding mode control was used [25]. The reliable control problem for nonlinear singularly perturbed systems with random actuator failures is discussed in [38]. It is worth pointing out that, actuator faults sometimes manifest themselves in a stochastic manner since the faults may occur occa- sionally with a random phenomenon, and a deterministic model cannot adequately describe actuator faults. As such, it is more appropriate and significant to consider a fault factor that obeys a certain distribution of probabilities. Up till now, the FTC based on a stochastic scenario has been the subject of many published results. Among others, we cite, the authors in [21] proposed a resilient and reliable controller for Markovian jump systems subject to stochastic intermittent actuator faults and stochastic https://doi.org/10.15837/ijccc.2022.6.4957 3 controller gain fluctuations. For a class of networked control systems with random occurred actuator failures, an FTC control scheme was designed in [36]. The authors in [31] utilized an observer to develop a reliable consensus protocol for a multiagent system against stochastic actuator failures. In [35], a reliable fuzzy H∞ control law was synthesized and applied for permanent magnet synchronous motor suffered from random actuator faults We know that, to the best of our knowledge, the problem FTC for IVF systems employing a reliable controller with stochastic faults has not been completely addressed in the literature, which motivates us to carry out this study. In addition, it is also emphasized that, there are numerous complex plants with non-linearities where the state variables are not accessible due to a variety of factors, such as the lack of sensors that are able to detect specific states, or the increasing number of sensors making the system more complicated. Various approaches have been used in output feedback design, including static output feedback [26], dynamic output feedback [16, 30], observer adaptive control [22, 23], and fuzzy observer- based approaches [20, 28, 41]. Designing observer-based controllers by the IVF T-S fuzzy models is a crucial issue that needs to be extensively explored so that many result have been recently published [8, 48, 49]. To the extent of our knowledge, a few research efforts have been made on IVF singular systems with unmeasurable premise variables, time-varying delay and randomly occurring actuator failures. This establishes the second motivation for the present work. 1.2 Objective and Outline The discussion above has inspired us to study, in this study, the observer-based FTC problem for a class of uncertain non-linear singular systems against the random event of actuator failures. This article is noteworthy for the following features. (i) As an alternative to existing control schemes developed for type-1 fuzzy singular systems with delay and actuator failure [38, 40, 46], this study addresses a novel reliable controller design for interval-valued fuzzy systems which may exhibit actuator faults represented by stochastic variables with Bernoulli distribution (ii) Contrary to the work in [33], which considers a fuzzy decentralized observer-based event-triggered control for interval type-2 fuzzy systems assuming that the premise variables are measurable, this work supposes that the premise variables are unknown. (iii) Unlike the existing studies where the Finsler lemma is used in [2] to design the controller and observer gains, this work formulate a feasible control strategy for the considered control problem using the decoupling matrix procedure. After setting out the introduction and aims of the study, the paper is organized as follows: In Section 2, the model and assumptions as well as the problem characterization are provided. Our main findings are presented and discussed in Section 3. The focus of this section is on the development of a new delay-dependent stochastic admissibility criterion for the system under steady using a new Lyapunov-Krasovskii functional. In addition, the design of the controller and observer gains is carried out by employing the matrix inequality decoupling technique to resolve the bilinear matrix inequality problem. Section 4 presents numerical simulations as a mean of exhibiting the potential applications of the suggested control strategy and validating its significance. In Section 5, we present some conclusions regarding the obtained results and some suggestions for future research. 1.3 Notations Table 1 lists the notations and acronyms that should be used in this study. 2 Preliminaries and Problem Statement The aim of this section is to introduce some preliminaries that facilitate understanding of our proposal and state the problem we are investigating. https://doi.org/10.15837/ijccc.2022.6.4957 4 Table 1: List of notations and acronyms used in the paper. Symbol Acronym/Notation R set of the real numbers X ∈ Rn n-dimensional Euclidean space X ∈ Rn×m n×m real matrix X > 0 real symmetric positive definite matrix X ‖X‖ norm of the matrix X X> transpose of the matrix X sym(X) X + X> λ() eigenvalue of a matrix E mathematical expectation ∗ term that is induced by symmetry r number of if-then rules LMI linear matrix inequality BMI bi-linear matrix inequality IVF Interval-valued fuzzy T-S Takagi-Sugeno 2.1 IVF Model Consider a class of non-linear singular system which can be described by the following IVF Model: Ri : If θ1(x(t)) is M1i and If θ2(x(t)) is M 2 i · · ·If θs(x(t)) is M s i , Then  Eẋ(t) = Aix(t) + Adix(t−d(t)) + BiuF (t) y(t) = Cix(t) x(t) = φ(t),∀t ∈ [−d2 0], (1) where Mki is an IVF set of rule i corresponding to the premise variable θi(x(t)), k = 1, 2, · · · ,s ; k is the number of premise variables, and i ∈ S , {1, 2, . . . ,r} is the number of rules. x(t) ∈ Rn, y(t) ∈ Rny , and uF (t) ∈ Rm, define, respectively, the state, the output and input vectors. Matrices Ai, Adi, and Bi in model (1) are known with appropriate dimensions. d(t) stands for the time varying delay, and φ(t) defines the initial state for all t ∈ [−d2 0]. 2.2 Assumptions and Resulting Model A1 d(t) is a continuous function such that 0 < d1 ≤ d(t) ≤ d2, 0 ≤ ḋ(t) ≤ dr (2) where d1 represents the lower delay bound, d2 stands for upper delay bounds, and dr is the delay variation rate. A2 Singular matrix E satisfies rank(E) = q < n. A3 The actuator fault has the following form: uF (t) = Γu(t) = m∑ s=1 γsΣsu(t) (3) where Γ = diag{γ1,γ2, · · · ,γm} is a diagonal matrix with m uncorrelated random variables, and Σs = diag{0, · · · , 0︸ ︷︷ ︸ s−1 , 1, 0, · · · , 0︸ ︷︷ ︸ s−m }. https://doi.org/10.15837/ijccc.2022.6.4957 5 Assume that γs ∈{0, 1} is a Bernoulli distributed stochastic variable with a probabilistic density function ps(γs), where the expectation and variance of γs are respectively, defined as γ̄s and πs. Define Γ̄ = E{Γ} = diag{γ̄1, γ̄2, · · · , γ̄m}. It follows that for any matrix Z > 0 E { (Γ − Γ̄) } = ∑m s=1(γs − γ̄s)Σs = 0 E { (Γ − Γ̄)>Z(Γ − Γ̄) } = ∑m s=1 π 2 sΣ>s ZΣs (4) Based on the IVF approach, the following interval defines the firing strength of the ith rule: Mi = [∏s k=1 ωMki (θ(x(t))) ∏s k=1 ω̄Mki (θ(x(t))) ] = [ µ i (x(t)) µ̄i(x(t)) ] where µ i (x(t)) ≥ 0, and µ̄i(x(t)) ≥ 0 represent, respectively, the the lower and upper membership functions. Accordingly, ωMki (θ(x(t))) ≥ 0, and ω̄Mki (θ(x(t))) ≥ 0 stands, respectively, for the lower and upper grades of membership. Therewith, the non-linear singular system can be described as  Eẋ(t) = r∑ i=1 µi(x(t)) ( Aix(t) + Adix(t−d(t)) + BiuF (t) ) y(t) = r∑ i=1 µi(x(t))Cix(t), (5) µi(x(t)) denotes the grade of membership of the ith local system defined as µi(x(t)) = αi(x(t))µi(x(t)) + ᾱi(x(t))µ̄i(x(t)), r∑ i=1 µi(x(t)) = 1 where αi(x(t)) and ᾱi(x(t) are two weighting coefficient functions satisfying 0 ≤ αi(x(t)), ᾱi(x(t)) ≤ 1, αi(x(t)) + ᾱi(x(t)) = 1 (6) Note that, by introducing weighting coefficient functions, we can represent any time-variant or time- invariant unmeasured parameters of the general non-linear system. Moreover, these functions are not necessarily known but exist and satisfy (6). 2.3 Fuzzy Observer-based Controller Generally, the system’s states cannot be fully measured in practice. To estimate the state variables of system (5), an observer is required. We will proceed to address this issue by considering the following IVF observer for system (5): Ri : If θ1(x̂(t)) is M1i and If θ2(x̂(t)) is M 2 i · · ·If θs(x̂(t)) is M s i , Then{ E ˙̂x(t) = Alx̂(t) + BluF (t) + Ll(y(t) − ŷ(t)) ŷ(t) = Clx̂(t), where x̂(t) denotes the state of the observer and ŷ(t) is the observer output. Ll, (l ∈ S) is the observer gain to be evaluated. The whole fuzzy observer can be inferred as follows:  E ˙̂x(t) = r∑ l=1 µl(x̂(t)) ( Alx̂(t) + Bl uF (t) + Ll(y(t) − ŷ(t)) ) ŷ(t) = r∑ l=1 µl(x̂(t))(Clx̂(t)) (7) To increase the flexibility of fuzzy controller design, we propose a controller that uses exclusive mem- bership functions that would not be shared with (5). The state-feedback IVF controller has the following structure: Ri : If ϑ1(x̂(t)) is N 1i and If ϑ2(x(t)) is N 2 i · · ·If ϑs(x̂(t)) is N s i , Then u(t) = Kjx̂(t) https://doi.org/10.15837/ijccc.2022.6.4957 6 where ϑ(t) = [ϑ1(x̂(t)),ϑ2(x̂(t)), . . . ,ϑs(x̂(t))] is the premise variable vector, N kcj , kc = 1, 2, · · · ,s stands for the IVF sets, and Kj is the state feedback gain matrix of rule j. Following is the firing interval for the jth rule: N i = [ νj(x̂(t)) ν̄j(x̂(t)) ] , i ∈ S where νj(x̂(t)) = s∏ kc=1 ωN ki (ϑ(t)) ≥ 0, ν̄j(x̂(t)) = s∏ kc=1 ω̄N kc i (ϑ(t)) ≥ 0, (8) νj(x̂(t)) and ν̄j(x̂(t)) represent,respectively, the lower and upper membership functions. ωN kc i (ϑ(t)) ≥ 0 and ω̄N kc i (ϑ(t)) ≥ 0 are the lower and upper grades of membership of θ(t) in N j i, respectively. The global fuzzy model will be defined as follows: u(t) = r∑ j=1 νj(x̂(t))(Kjx̂(t)) (9) νj(x̂(t)) = β j (x̂(t))νj(x̂(t)) + β̄j(x̂(t))ν̄j(x̂(t)) r∑ l=1 (β l (x̂(t))νl(x̂(t)) + β̄l(x̂(t))ν̄l(x̂(t))) , νj(x̂(t)) ≥ 0, r∑ i=1 νj(x̂(t)) = 1 (10) With the error e(t) = x(t) − x̂(t), together with (5), (7), and (9) we obtain the following closed-loop system: Ẽ ˙̃x(t) = r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂l ( Ãijlx̃(t) + Ãdix̃(t−d(t)) + B̃ilΓ̃K̃jx̃(t) ) (11) where x̃(t) = [x>(t), e>(t)]>, Γ̃ = Γ − Γ̄, and Ãijl = [ Ai + BiΓ̄Kj −BiΓ̄Kj (Ai −Al) + (Bi −Bl)Γ̄Kj −Ll(Ci −Cl) Ai − (Bi −Bl)Γ̄Kj −LlCl ] , Ãdi = [ Adi 0 Adi 0 ] B̃il = [ Bi (Bi −Bl) ] , K̃j = [ Kj −Kj ] Ẽ = [ E 0 0 E ] Remark 1. Generally, the fuzzy observers and fuzzy controllers based on type-1 fuzzy share the same membership functions of the model. However, the membership functions used to describe the IVF systems are unknown except for their bounds. Thus, in this study, the membership functions of fuzzy observers are calculated based on the estimated state variable x̂(t) rather than x(t). 2.4 Problem Statement It is our primary objective in this paper to design an IVF observer-based controller that maintains the closed-loop system admissible in the event of random actuator failure in non-linear singular systems expressed by the IVF model as specified in (5). Let us begin by stating the following lemma that will be used in our main results. Lemma 2. [34] Let P be a symmetric matrix such that E>LPEL > 0. Then, matrix PE + U >XV > is non singular so that (PE + U>XV >)−1 = P̄E> + V X̄U where P̄ ∈ Rn×n is a symmetric matrix verifying E>RP̄ER = (E > LPEL) −1, X̄ ∈ R(n−q)×(n−q) is a non singular matrix satisfying X̄ = (V >V )−1V −1(UU>)−1, U be a full row rank matrix so that UE = 0 , and V be a full column rank matrix such that EV = 0. EL and ER are full column rank matrices with E = ELE>R. https://doi.org/10.15837/ijccc.2022.6.4957 7 3 Main Results 3.1 Admissibility Analysis As we will see below, we are able to obtain sufficient delay-dependent conditions for the closed-loop system stated in (11) to be stochastically admissible. Theorem 3. For given constants d1, d2 and 0 ≤ dr < 1, assume (A1) holds. System (11) is admissible, if there exist positive matrices Pk ∈ Rn×n, Q̃ ∈ R2n×2n, R̃1 ∈ R2n×2n, R̃2 ∈ R2n×2n, R̃3 > 0, Q̃a = [ Q̃11 Q̃12 Q̃>12 Q̃22 ] ∈ R4n×4n, S̃a = [ S̃11 S̃12 S̃>12 S̃22 ] ∈ R4n×4n, non-singular matrices Xk ∈ Rn×n, M̃k, Ñk, T̃k, and Ṽk, (k = 1, 2) such that the following inequality holds : Ψ̃ijl(Ẽ,Ãijl,Ãdi) =   ψ̃11ijl(Ẽ,Ãijl,Ãdi) √ d1 2 ψ̃ijlR̃1 √ d12 2 ψ̃ijlR̃2 √ d12ψ̃ijlR̃3 ψ̃15i Iπ ⊗ ψ̃16ijl ∗ −R̃1 0 0 0 0 ∗ ∗ −R̃2 0 0 0 ∗ ∗ ∗ −R̃3 0 0 ∗ ∗ ∗ ∗ −ψ̃55 0 ∗ ∗ ∗ ∗ ∗ −ψ̃66   < 0 (12) where d12 = d2 −d1, R̃ = d1 2 R̃1 + d12 2 R̃2 + d12R̃3, Π1 = P1E + U>X1V >, Π2 = P2E + U>X2V >, ψ̃11ijl(Ẽ,Ãijl,Ãdi) =   Ψ̃11i Ψ̃12i Ψ̃13i 0 S̃12 0 ∗ Ψ̃22i 0 Ψ̃24i 0 Ψ̃26i ∗ ∗ Ψ̃33i −Q̃12 Ψ̃35i 0 ∗ ∗ ∗ Ψ̃44i 0 0 ∗ ∗ ∗ ∗ Ψ̃55i −S̃12 ∗ ∗ ∗ ∗ ∗ Ψ̃66i   ψ̃ijl = col { Ã>ijl à > di 0 0 0 0 } ψ̃15 = [ ψ̃151 ψ̃152 ψ̃153 ψ̃154 ] ψ̃16ijl = [ π1(B̃ilΣ1K̃j)>, . . . ,πm(B̃ilΣmK̃j)> ] ψ̃55 = diag { R̃1, R̃2, R̃3, R̃3 } ψ̃66 = diag { R̃−1, · · · , R̃−1 } , ψ̃151 = √ d1 2 col { ẼM̃1 0 ẼM̃2 0 0 0 } ψ̃152 = √ d12 2 col { 0 0 ẼÑ1 0 ẼÑ2 0 } , ψ̃153 = √ d12 col { 0 ẼT̃1 0 0 0 ẼT̃2 } ψ̃154 = √ d12 col { 0 ẼṼ2 0 ẼṼ1 0 0 } Iπ = col { I 0 0 0 0 0 } Ψ̃11i = Q̃11 + S̃11 + Q̃1 + sym(M̃1Ẽ) + sym(Π̃>Ãijl) Ψ̃12i = (Π̃>Ãdi) Ψ̃13i = Q̃12 −M̃1Ẽ + (M̃2Ẽ)> Ψ̃15i = −Ñ1Ẽ + (Ñ2Ẽ)> + S̃12 Ψ̃22i = −(1 −hd)Q̃1 + sym(T̃1Ẽ) − sym(Ṽ2Ẽ) Ψ̃24i = −Ṽ2Ẽ + (Ṽ1Ẽ)> Ψ̃26i = −T̃1Ẽ + (T̃2Ẽ)> Ψ̃33i = Q̃22 − Q̃11 − sym(M̃2Ẽ) + sym(Ñ1Ẽ) Ψ̃35i = −Ñ1Ẽ + Ñ2Ẽ Ψ̃44i = −Q̃22 + sym(Ṽ1Ẽ) Ψ̃55i = S̃22 − S̃11 − sym(Ñ2Ẽ) Ψ̃66i = −S̃22 − sym(T̃2Ẽ) Π̃ = diag{Π1, Π2} Matrices U ∈ R(n−q)×(n) and V ∈ R(n)×(n−q) are of full rank such that UE = 0 and EV = 0. Proof. First, we will prove the stability of system (11). For this purpose, we choose the following https://doi.org/10.15837/ijccc.2022.6.4957 8 Lyapunov-Krasovsky functional: V (x̃(t)) = V1(x̃(t)) + V2(x̃(t)) + V3(x̃(t)) V1(x̃(t)) = x̃T (t)Ẽ>P̃Ẽx̃(t) V2(x̃(t)) = ∫ t t−d12 ζ̃>1 (s)Q̃aζ̃1(s)ds + ∫ t t−d22 ζ̃>2 (s)S̃aζ̃ > 2 (s)ds + ∫ t t−d(t) x̃>(s)Qx̃(s)ds V3(x̃(t)) = ∫ 0 −d12 ∫ t t+θ ˙̃x>(s)Ẽ>R̃1Ẽ ˙̃x(s)dsdθ + ∫ −d12 −d22 ∫ t t+θ ˙̃x>(s)Ẽ>R̃2Ẽ ˙̃x(s)dsdθ + ∫ −d1 −d2 ∫ t t+θ ˙̃x>(s)Ẽ>R̃3Ẽ ˙̃x(s)dsdθ (13) where ζ̃1(t) = [ x̃T (t) x̃>(t− d12 ) ]> , ζ̃2(t) = [ x̃T (t) x̃>(t− d22 ) ]> , Define the infinitesimal operator L of V (x̃(t)) as follows: L(V (x̃(t))) = lim ∆−→0+ 1 ∆ {E{V (x̃(t + ∆))|x̃(t)}−V (x̃(t))} (14) Evaluating the derivative of V (x̃(t)) along the solutions of system (11), and noting UE = 0, it results in E{LV1(x̃(t))} = E{2x̃T (t)Ẽ>P̃Ẽ ˙̃x(t)} = 2x̃T (t)Π̃>(Ãijlx̃(t) + Ãdix̃(t−d(t)) + B̃ilΓ̃K̃jx̃(t)) = 2x̃T (t)Π̃>(Ãijlx̃(t) + Ãdix̃(t−d(t))) E{LV2(x̃(t))}≤ ζ̃>1 (t)Q̃aζ̃1(t) − ζ̃ > 1 (t− d1 2 )Q̃aζ̃1(t− d1 2 ) + ζ̃>2 (t)S̃aζ̃2(t) − ζ̃ > 2 (t− d2 2 )S̃aζ̃2(t− d2 2 ) + x̃T (t)Q̃1x̃(t) − (1 −hd)x̃>(t−d(t))Q̃1x̃(t−d(t)) E{LV3(x̃(t))} = d1 2 (Ãijlx̃(t) + Ãdix̃(t−d(t)))>R̃1(Ãijlx̃(t) + Ãdix̃(t−d(t))) + d12 2 (Ãijlx̃(t) + Ãdix̃(t−d(t)))>R̃2(Ãijlx̃(t) + Ãdix̃(t−d(t))) + d12(Ãijlx̃(t) + Ãdix̃(t−d(t)))>R̃3(Ãijlx̃(t) + Ãdix̃(t−d(t))) + E { (B̃ilΓ̃K̃jx̃(t))>R̃(B̃ilΓ̃K̃jx̃(t)) } − ∫ t t−d12 ˙̃x>(s)Ẽ>R̃1Ẽ ˙̃x(s)ds− ∫ t−d12 t−d22 ˙̃x>(s)Ẽ>R̃2Ẽ ˙̃x(s)ds− ∫ t−d1 t−d2 ˙̃x>(s)Ẽ>R̃3Ẽ ˙̃x(s)ds (15) Let ζ̃3(t) = [ x̃>(t− d12 ) x̃ >(t− d22 ) ]> , ζ̃4(t) = [ x̃>(t−d(t)) x̃>(t−d2) ]> ζ̃5(t) = [ x̃>(t−d1) x̃>(t−d(t)) ]> , M̃ = [ M̃>1 M̃ > 2 ]> , Ñ = [ Ñ>1 Ñ > 2 ]> , T̃ = [ T̃>1 T̃ > 2 ]> , and Ṽ = [ Ṽ >1 Ṽ > 2 ]> . By defining the following expression: ∫ t t−d12 [ Ẽẋ(s) ξ1(t) ]>[ R̃1 M̃ ∗ M̃ẼR̃−11 Ẽ >M̃> ][ Ẽẋ(s) ξ1(t) ] ds ≥ 0, (16) ∫ t−d12 t−d22 [ Ẽẋ(s) ξ2(t) ]>[ R̃2 Ñ ∗ ÑẼR̃−12 Ẽ >Ñ> ][ Ẽẋ(s) ξ2(t) ] ds ≥ 0 (17) https://doi.org/10.15837/ijccc.2022.6.4957 9 we know that − ∫ t t−d12 ẋ(s)>Ẽ>R̃1Ẽẋ(s)ds ≤ 2ξ>1 (t)M̃ [ Ẽ − Ẽ ] ξ1(t) + d1 2 ξ>1 (t)M̃ẼR̃ −1 1 Ẽ >M̃>ξ1(t) (18) − ∫ t−d12 t−d22 ẋ(s)>Ẽ>R̃2Ẽẋ(s)ds ≤ 2ξ>3 (t)Ñ [ Ẽ − Ẽ ] ξ3(t) + d1 2 ξ>3 (t)ÑẼR̃ −1 2 Ẽ >Ñ>ξ3(t) (19) Moreover, we have − ∫ t−d1 t−d2 ˙̃x>(s)Ẽ>R̃3Ẽ ˙̃x(s)ds = − ∫ t−d(t) t−d2 ˙̃x>(s)Ẽ>R̃3Ẽ ˙̃x(s)ds− ∫ t−d1 t−d(t) ˙̃x>(s)Ẽ>R̃3Ẽ ˙̃x(s)ds (20) and − ∫ t−d(t) t−d2 ẋ(s)>Ẽ>R̃3Ẽẋ(s)ds ≤ 2ξ>4 (t)T̃ [ Ẽ − Ẽ ] ξ4(t) + (d2 −d1)ξ>4 (t)T̃ẼR̃ −1 3 Ẽ >T̃>ξ4(t) − ∫ t−d1 t−d(t) ẋ(s)>Ẽ>R̃3Ẽẋ(s)ds ≤ 2ξ>5 (t)Ṽ [ Ẽ − Ẽ ] ξ5(t) + (d2 −d1)ξ>5 (t)Ṽ ẼR̃ −1 3 Ẽ >Ṽ >ξ5(t) (21) Then, from (4), we get E { (B̃ilΓ̃K̃jx̃(t))>R̃(B̃ilΓ̃K̃jx̃(t)) } = x̃>(t) { m∑ s=1 π2s(B̃ilΣsK̃j) >R̃B̃ilΣsK̃j } x̃(t) (22) Let ξ(t) = [ x̃T (t) xT (t−d(t)) xT (t− d1 2 ) xT (t−d1) xT (t− d2 2 ) xT (t−d2) ]> . Combining (15)-(22), yields E{LV (x̃(t))}≤ r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂lξ T (t) ( ψ̃11ijl(Ẽ,Ãijl,Ãdi) + ψ̃ijlR̃ψ̃>ijl + m∑ s=1 π2sIπ ⊗ (B̃ilΣsK̃j) >R̃(I>π ⊗ (B̃ilΣsK̃j) ) ξ(t) (23) Hence, by performing the complement of Schur to (12), yields ψ̃11ijl(Ẽ,Ãijl,Ãdi) + ψ̃ijlR̃ψ̃>ijl + m∑ s=1 π2sIπ ⊗ (B̃ilΣsK̃j) >R̃(I>π ⊗ (B̃ilΣsK̃j) < 0 Thus, it is obvious that E{LV (x̃(t))} < 0, and system (11) is stochastically stable. Next, we prove the regularity and impulse-free properties of system (11). From (12), we know Ψ̃11ijl < 0 which implies that sym(Π̃>Ãijl) + sym(M̃1Ẽ) < 0 (24) For matrix Ẽ, there exist M and N, two non-singular matrices such that E = MẼN = [ I2q 0 0 0 ] , Âijl = MÃijlN = [ Âijl11 Âijl12 Âijl21 Âijl22 ] , Π̂ = M−T Π̃N = [ Π̂11 Π̂12 Π̂21 Π̂22 ] M̂1 = M−TM1M−1 = [ M̂111 M̂112 M̂121 M̂122 ] (25) Based on lemma 2, we know that Ẽ>Π̃ = Π̃>Ẽ which implies that Π̂12 = 0, using (25). Pre- and post-multiplying (24) by N> and N, respectively, the inequality shown below holds sym (Π̂>22Âijl22) < 0 which means that Âijl22 is non-singular and system (11) is regular and impulse free, according to the definitions stated in [4]. https://doi.org/10.15837/ijccc.2022.6.4957 10 3.2 Observer Design As a result of the bilinear terms involved in condition (12), it is difficult to parameterize the controller and observer gains. In the sequel, we provide the procedure to synthesize the gains K̄i, and L̄i. Theorem 4. For given constants d1, d2 and 0 ≤ dr < 1, assume ( A1) holds. System (11) is stochastically admissible, if there exist scalars α, β, λc, c = 1, 2, 3, 4 and matrices P̄k > 0, Jk > 0, M̄k, N̄k, T̄k V̄k, X̄k, k = 1, 2, Q̄ > 0, R̄1 > 0, R̄2 > 0, R̄3 > 0, Q̄a = [ Q̄11 Q̄12 Q̄>12 Q̄22 ] > 0, S̄a = [ S̄11 S̄12 S̄>12 S̄22 ] > 0, Λ, W̄l, Yj ∈ Rm×n, and Fl ∈ Rn×ny , the following LMIs hold under the condition µ̂l −σlν̂l ≥ 0 where σl is positive scalar for i,j, l ∈ S:  Ξijl − Λ < 0 σjΞijj −σjΛ + Λ < 0 σl(Ξijl − Λ) + σj(Ξilj − Λ) + 2Λ < 0, l > j (26) where Π̄1 = P̄1E> + U>X̄1V >, Π̄2 = P̄2E> + U>X̄2V >, Π̄ = diag{Π̄1, Π̄2}, Π̆1 = diag{Π̄1, Π̄1} Ξijl =   Ξ̄ijl αΥ1ij Υ2 βΥ3l Υ4l ∗ −α sym(Π̄1) + J1 0 0 0 ∗ ∗ −J1 0 0 ∗ ∗ ∗ −β sym(W̄l) + J2 0 ∗ ∗ ∗ ∗ −J2   (27) Ξ̄ijl =   Ξ̄11ijl √ d1 2 ψ̄ √ d12 2 ψ̄ √ d12ψ̄ ψ̄15 Iπ ⊗ ψ̄16ij ∗ ψ̄22 0 0 0 0 ∗ ∗ ψ̄33 0 0 0 ∗ ∗ ∗ ψ̄44 0 0 ∗ ∗ ∗ ∗ −ψ̄55 0 ∗ ∗ ∗ ∗ ∗ −ψ̄66   Υ1ij = col { B̄Γij 0 0 0 0 0 √ d1 2 B̄Γij √ d12 2 B̄Γij √ d12B̄Γij 0 B̄>Σij } Υ3l = col { F̄l 0 0 0 0 0 √ d1 2 F̄l √ d12 2 F̄l √ d12F̄l 0 0 } ψ̄ = col { Ā>ijl Ā > di 0 0 0 0 } ψ̄151 = √ d1 2 col { Ē>M̄1 0 Ē>M̄2 0 0 0 } ψ̄152 = √ d12 2 col { 0 0 Ē>N̄1 0 Ē>N̄2 0 } , ψ̄153 = √ d12 col { 0 Ē>T̄1 0 Ē>T̄2 0 0 } ψ̄154 = √ d12 col { 0 Ē>V̄1 Ē>V̄2 0 0 0 } B̄Σij = [ π1(B̄1Σij) >, . . . ,πm(B̄mΣij) > ] ψ̄16ij = [ π1(B̄1Σij × [1 − 1]) >, . . . ,πm(B̄mΣij × [1 − 1]) > ] Υ2 = col { Π̄>12 0 0 0 0 0 } Υ4l = col { C̄>il 0 0 0 0 0 } ψ̄22 = λ21R̃1 −λ1 sym(Π̄) ψ̄33 = λ22R̃2 −λ2 sym(Π̄) ψ̄44 = λ23R̃3 −λ3 sym(Π̄) ψ̄15 = [ ψ̄151 ψ̄152 ψ̄153 ψ̄154 ] ψ̄55 = diag { R̄1, R̄2, R̄3, R̄3 } ψ̄66 = λ24R̃ −λ4 sym(Π̆1), https://doi.org/10.15837/ijccc.2022.6.4957 11 Ξ̄11ijl =   Ξ̂11ijl Ξ̂12i Ξ̂13i 0 S̄12 0 ∗ Ξ̂22i 0 Ξ̂24i 0 Ξ̂26i ∗ ∗ Ξ̂33i −Q̄12 Ξ̂35i 0 ∗ ∗ ∗ Ξ̂44i 0 0 ∗ ∗ ∗ ∗ Ξ̂55i −S̄12 ∗ ∗ ∗ ∗ ∗ Ξ̂66i   Ξ̄11ijl = Q̄11 + S̄11 + Q̄1 + sym(M̄1Ē>) + sym(Āijl) Ξ̄12i = Ādi Ξ̄13i = Q̄12 −M̄1Ē> + (M̄2Ē>)> Ξ̄15i = −N̄1Ē> + (N̄2Ē>)> Ξ̄22i = −(1 −hd)Q̄1 + sym(T̄1Ē>) − sym(V̄2Ē>) Ξ̄24i = V̄2Ē> − (V̄1Ē>)> Ξ̄26i = −T̄1Ē> + (T̄2Ē>)> Ξ̄33i = Q̄22 − Q̄11 − sym(M̄2Ē>) + sym(N̄1Ē>) Ξ̄35i = −N̄1Ē> + N̄2Ē> Ξ̄44i = −Q̄22 + sym(V̄1Ē>) Ξ̄55i = S̄22 − S̄11 − sym(N̄2Ē>) Ξ̄66i = −S̄22 − sym(T̄2Ē>) Āijl = [ AiΠ̄1 + BiΓ̄Yj −BiΓ̄Yj (Ai −Al)Π̄1 + (Bi −Bl)Γ̄Yj −Fl(Ci −Cl) AiΠ̄2 − (Bi −Bl)Γ̄Yj −FlCl ] , Ādi = [ AdiΠ̄1 0 AdiΠ̄1 0 ] B̄Γij = [ BiΓ̄Yj (Bi −Bl)Γ̄Yj ] , B̄sΣij = [ BiΣsYj (Bi −Bl)ΣsYj ] , Π̄12 = [ 0 Π̄1 − Π̄2 ] , F̄l = [ 0 Fl ] C̄il = [ W̄l(Ci −Cl) − (Ci −Cl)Π̄1 W̄lCi −CiΠ̄1 ] Moreover, parameters Kj and Ll are computed by Kj = YjΠ̄−11 and Ll = FlW̄ −1 l , respectively. Proof. In order to obtain less conservative results, the slack matrix provided below is used. r∑ i=1 r∑ j=1 r∑ l=1 µiν̂j ( ν̂l − µ̂l ) Λ = 0 (28) where Λ is an arbitrary matrix with appropriate dimensions. Then, it can be known that r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂lΞijl = r∑ i=1 r∑ j=1 r∑ l=1 { µiν̂jµ̂lΞijl + µiν̂j ( ν̂l − µ̂l ) Λ } = r∑ i=1 r∑ j=1 r∑ l=1 { µiν̂j(µ̂l + σlν̂l −σlν̂l)Ξijl + µiν̂j ( ν̂l − µ̂l + σlν̂l −σlν̂l ) Λ } = r∑ i=1 r∑ j=1 r∑ l=1 µiν̂j { ν̂l ( σlΞijl −σlΛ + Λ ) + (µ̂l −σlν̂l) ( Ξijl − Λ )} = r∑ i=1 µi ( r∑ j=1 ν̂2j (σjΞijj −σjΛ + Λ) + r−1∑ j=1 r∑ l=j+1 ν̂jν̂l(σjΞijl −σjΛ + Λ + σlΞilj −σlΛ + Λ) + r∑ i=1 r∑ j=1 ν̂i(µ̂l −σlν̂l)(Ξijl − Λ) ) (29) In light of the conditions in (26), the expression shown below holds. r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂lΞijl < 0 (30) From Ξijl < 0, we know that −λ1 sym(Π̄) + λ21R̄1 < 0, and thus Π̄ is non-singular. https://doi.org/10.15837/ijccc.2022.6.4957 12 Moreover, for P̄1 > 0 , it is easy to verify that E>LP̄1EL > 0. So according to Lemma 2 we have Π̄−1k = (PkE + U >XkV >) = Πk, k = 1, 2 Π̄−1 = diag{Π1, Π2} Using the fact that for any λp > 0, p = 1, 2, 3 the following condition holds: 0 ≤ (λpΠ̄ − R̃−1p ) >R̃p(λpΠ̄ − R̃−1p ) = R̃ −1 p −λp sym(Π̄) + λ 2 pΠ̄ >R̃pΠ̄ Then, −R̃−1p ≤−λp sym(Π̄) + λ 2 pΠ̄ >R̃pΠ̄ Define Z =  I 0 0 0 00 I I 0 0 0 0 0 I I   > , and Γijl = [ I Υ1ijΠ̄−11 Υ3lW̄ −1 l ] . Multiplying Z> and its transposition in left and right sides to (30), respectively, to get r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂l  Ξ̄ijl αΥ1ij + Υ2 βΥ3l + Υ4il∗ −α sym(Π̄1) 0 ∗ ∗ −β sym(W̄l)   < 0 (31) Then, by congruence transformation by Γijl, we know that r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂l(Ξ̄ijl + sym (Υ1ijΠ̄−11 Υ > 2 + Υ3lW̄ −1 l Υ > 4il)) < 0 (32) Let Yj = KjΠ̄1 and F̄l = LlW̄l. From the terms Υ1ijΠ̄−11 Υ>2 , and Υ3lW̄ −1 l Υ > 4il one can calculate F̄lW̄ −1 l C̄il = [ 0 0 F̄l(Ci −Cl) −Ll(Ci −Cl)Π̄1 F̄lCi −LlCiΠ̄2 ] , B̄ΓijΠ̄−11 Π̄12 = [ 0 BiΓ̄Yj −BiΓ̄KjΠ2 0 (Bi −Bl)Γ̄Yj − (Bi −Bl)Γ̄KjΠ2 ] B̄sΣijΠ̄ −1 1 Π̄12 = [ 0 BiΣsYj −BiΣsKjΠ2 0 (Bi −Bl)ΣsYj − (Bi −Bl)ΣsKjΠ2 ] ,s = 1, 2, · · ·m Thus, it can be concluded that Āijl + B̄ΓijΠ̄−11 Π̄12 + F̄lW̄ −1 l C̄il = ÃijlΠ̄, B̄ s Σij × [1 − 1] + B̄ s ΣijΠ̄ −1 1 Π̄12 = B̃ilΣsK̃jΠ̄ (33) Using (33), and performing the congruence transformation to (32) by diag(Π, Π, Π, Π, Π, Π,I,I,I,I) and its transpose r∑ i=1 r∑ j=1 r∑ l=1 µiν̂jµ̂lΨ̃ijl(Ẽ,Ãijl,Ãdi) < 0 holds, using the following expressions: Q̃uv = Π̄−TQ̄uvΠ̄−1, S̃uv = Π̄−TS̄uvΠ̄−1, W̃k = Π̄−TW̄kΠ̄−1, M̃k = Π̄−TM̄kΠ̄−1, Ṽk = Π̄−TV̄kΠ̄−1, T̃k = Π̄−TT̄kΠ̄−1, R̃c = Π̄−1R̄cΠ̄−T , (u,v,k = 1, 2, c = 1, 2, 3), Hence, according to Theorem 3, closed-loop system (11) is stochastically admissible. Remark 5. Here, we address the issue of mismatched property between µi(x(t)) and νi(x̂(t)). By introducing a slack matrix Λ to express (28) which is combined with the condition µ̂l − σlν̂l ≥ 0 to get relaxed conditions as stated in (26). This technique incorporates membership function information into stability conditions, leading to less conservative conditions. Remark 6. Our approach for designing the controller and observer gains relies on decoupling the BMI terms by using slack matrices as defined in (33). Comparatively, this approach differs from that presented in [2], in which the Finsler lemma is investigated and auxiliary variables Ti are employed to validate CiTi = [1 0] and reach the desired gains. https://doi.org/10.15837/ijccc.2022.6.4957 13 4 Numerical examples In this section, we provide a concise description of the computational framework and present two examples to illustrate the effectiveness and advantages of the proposed control scheme. 4.1 Computational Framework and Algorithm Computing experiments were performed using the Matlab programming language and a computer with the following characteristics:(i) [OS] Windows 10 Enterprise for 64 bits; (ii) [RAM] 8 Gigabytes; and (iii) [Processor] Intel(R) Core(TM) i7-4790T CPU @ 2.70 GigaHertz. A detailed explanation of the design procedure is given in Algorithm 1 and the flowchart displayed in Figure 1. Using Yalmip software and the optimization toolbox mosek, the algorithm 1 was executed. Algorithm 1 Procedure design 1: Describe the non-linear system by the IVF system. 2: Choose the parameters of the actuator fault model described in (3). 3: Determine the gains Kj, Ll of the LMI established in Theorem 4. 4: Design the observer indicated in (7) 5: Apply the designed control law expressed in (9) to the model. Eẋ(t) = g(x(t) + ∆x(t), x(t − d(t)), uF (t))    Eẋ(t) = r∑ i=1 µi(x(t)) { Aix(t) + Adix(t − d(t)) + BiuF (t) } y(t) = r∑ i=1 µi(x(t))Cix(t) µ̄i(x(t)), µi(x(t))    E ˙̂x(t) = r∑ i=1 µi(x̂(t)) { Aix̂(t) + Li(y(t) − ŷ(t)) + BiuF (t) } ŷ(t) = r∑ i=1 µi(x̂(t))Cix(t) u(t) = r∑ j=1 νj(x̂(t))Kjx̂(t) ν̄i(x̂(t)), νi(x̂(t)) Theorem 3.2 u(t) = r∑ j=1 νj(x̂(t))Kjx̂(t)Γ Ki Li uF (t) Figure 1: Flowchart of the control procedure. The flowchart displayed in Figure 1 provides a clear description of the proposed design procedure. 4.2 Example 1 The efficiency and correctness of the suggested control scheme is illustrated with the example provided from [30] with the following matrices: E =  1 0 00 1 0 0 0 0   , A1 =   0 1 09.8 0 1 0 −1 −0.6   A2 =  0 1 00 0 1 0 −1 0.6   Ad1 = Ad2 =   0 0 00 0 0 −0.13 0 0   , B1 =  0 1 11 0 0 1 0 1   , B2 =  0 1 11 0 0 0 1 1   C21 = [ 1 1 0 1 0 1 ] , C22 = [ 1 1 0 1 0 0 ] https://doi.org/10.15837/ijccc.2022.6.4957 14 The lower and upper bounds of membership functions of the corresponding interval-valued TS fuzzy model are listed in Table 5. Lower membership functions Upper membership functions µ1(x2) = 0.95 − 0.95 1 + exp ( x2 + 4.5 8 ) µ2(x2) = 1 −µ1(x2) µ̄1(x2) = 0.95 − 0.925 1 + ( x2 + 3.5 8 ) µ̄2(x2) = 1 − µ̄2(x2) Table 2: Lower and upper membership functions of the plant. The weighting functions are chosen as αi = sin2(x2(t)) and ᾱi = 1 − αi for i = 1, 2. For the afore- mentioned interval type-2 fuzzy singular system, we are going to design an observer-based IVF con- troller that makes a closed-loop system stochastically admissible. Accordingly, we have defined the lower and upper bounds of membership functions for interval-valued fuzzy controllers in Table 6. To determine what constitutes membership functions, non-linear weight functions are selected as β j (x2(t)) = cos2(x2(t)) and β̄j(x1(t)) = 1 −βj(x2(t)). Lower membership functions Upper membership functions ν1(x2) = 1 − 1 1 + exp ( x2 + 5 2 ) ν2(x2) = 1 −ν1(x2) ν̄1(x2) = 1 − 1 1 + exp ( x2 + 4 2 ) ν̄2(x2) = 1 − ν̄1(x2) Table 3: Lower and upper membership functions of the controller. In this example, it is assumed that d(t) = 0.2 + 0.1 sin(t), which provides that d1 = 0.1, d2 = 0.3, dr = 0.1. Define EL = ER =  1 00 1 0 0   , U = V T = [0 0 1] Let σ1 = 0.1, σ2 = 0.1, α = 0.01, β = 0.002, λ1 = 0.1, λ2 = 5, λ3 = 12, and λ4 = 0.001. To demonstrate the efficacy of the proposed control strategy, two cases are considered using the previous values. case 1 The actuator faults are assumed to occur randomly with a Bernoulli-distribution satisfying with Γ̄ = diag{0.6, 0.55, 0.65}. case 2 For this case, the controller is designed for a system without actuator faults, i.e. we take Γ̄ = diag{1, 1, 1}. According to the algorithm above, the LMIs (26) in Theorem 4, are solved for both cases with parameters displayed in table 4. https://doi.org/10.15837/ijccc.2022.6.4957 15 Case parameters Case 1 K1 =  −31.437 −12.916 21.486−30.49 −12.922 20.022 24.519 12.621 −25.486   , K2 =  −31.594 −12.971 21.314−30.656 −12.977 19.833 24.7 12.686 −25.253   , L1 =   5.3154 2.08635.1943 0.2834 −0.92299 1.5574   , L2 =   4.1972 2.19515.1622 0.033923 −0.44853 0.36828   Case 2 K1 =  −11.896 −6.8293 −0.20661−11.698 −7.0008 −1.3811 11.704 7.7511 −2.5207   , K2 =  −11.932 −6.8332 −0.20605−11.727 −7.0052 −1.3798 11.741 7.755 −2.5222   , L1 =   4.5107 1.58446.7738 −3.0472 −1.3806 1.9677   , L2 =   3.5379 2.30548.0944 −3.6425 −0.94405 0.57315   Table 4: Controller and observer gains for both cases Three scenarios are considered in the numerical simulations: 1. The reliable observer-based controller obtained in the first case is applied to a healthy system; 2. The same observer-based controller designed in the first case is performed to a system suffering from actuator failure; 3. A system with an actuator failure is controlled by the unreliable observer-based controller de- veloped in the second case. For all cases, the simulations are undergone with initial condition φ(t) = [−0.73140, 0.5140, −0.5237]>, t ∈ [−0.3 0], and the results are depicted in Figures 2-4, from where the inputs as well as system and its re- lated observer state variables are displayed. We observe that, when the reliable control law is applied the system dynamics are stabilized despite uncertainties and stochastic actuator failures, however, when the unreliable control law is applied to the system with failures, we see that the performance of the system is degraded. Thus, The simulations validate that the proposed control scheme is effective in accommodating actuator faults in the system and shows its robustness in spite of uncertainty. https://doi.org/10.15837/ijccc.2022.6.4957 16 0 1 2 3 4 5 6 7 8 t -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 x1 x̂1 (a) The curves of x1(t), and x̂1(t) 0 1 2 3 4 5 6 7 8 t -0.4 -0.2 0 0.2 0.4 0.6 0.8 x2 x̂2 (b) The curves of x2(t), and x̂2(t) 0 1 2 3 4 5 6 7 8 t -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x3 x̂3 (c) Trajectories of x3(t), and x̂3(t) 0 0.5 1 1.5 2 2.5 3 -1 0 1 2 0 0.5 1 1.5 2 2.5 3 -0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 0 0.5 1 (d) Input trajectories Figure 2: Simulation plots for failure mode 0 1 2 3 4 5 6 7 8 t -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x1 x̂1 (a) The curves of x1(t), and x̂1(t) 0 1 2 3 4 5 6 7 8 t -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x2 x̂2 (b) The curves of x2(t), and x̂2(t) 0 1 2 3 4 5 6 7 8 t -4 -3 -2 -1 0 1 2 3 x3 x̂3 (c) Trajectories of x3(t), and x̂3(t) 0 0.5 1 1.5 2 2.5 3 -2 0 2 0 0.5 1 1.5 2 2.5 3 -0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 -0.5 0 0.5 1 (d) Input trajectories Figure 3: Simulation plots for normal mode https://doi.org/10.15837/ijccc.2022.6.4957 17 0 1 2 3 4 5 6 7 8 t -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 x1 x̂1 (a) The curves of x1(t), and x̂1(t) 0 1 2 3 4 5 6 7 8 t 0 0.1 0.2 0.3 0.4 0.5 0.6 x2 x̂2 (b) The curves of x2(t), and x̂2(t) 0 1 2 3 4 5 6 7 8 t -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 x3 x̂3 (c) Trajectories of x3(t), and x̂3(t) exam 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 4 -2 -1 0 1 (d) Input trajectories Figure 4: Simulation plots for non-reliable controller gains in case 2 4.3 Bio-economic System Using a bio-economic system adapted from [19], the proposed control scheme can be shown to be both efficient and correct. The bio-economic system is stated as  ż1(t) = −0.5z1(t) + 0.15z2(t) − 0.01z21 (t) −E(t)z1(t) + u1(t), ż2(t) = 0.5z1(t) − 0.1z2(t), 0 = E(t)(z1(t) − 50) + u2(t), (34) with z1(t) and z2(t) being the population density of, respectively, immature and mature species at time t. E(t) corresponds to the harvest effort on the immature population, u1(t) and u2(t) state, respectively, the capture of an immature population and the government regulation, by means of a tax or subsidy, of a biological resource. Based on model (34), the following model is obtained by translating the positive equilibriums to zero [19] :  ẋ1(t) = −1.25x1(t) + 0.15x2(t) − 50x3(t) − 0.01x21(t) −x1(t)x3(t) + u1(t), ẋ2(t) = 0.5x1(t) − 0.1x2(t), 0 = −0.75x1(t) + x1(t)x3(t) + u2(t), (35) Assume that x1(t) ∈ [−10 10], and 0 ≤ ∆x1(t) ≤ 2. Given the uncertainty associated with the parameter ∆x1(t), it is evident that the IT-2 T-S fuzzy system should be adopted to model non- linear system (35). The lower and upper bounds of membership functions of the corresponding IT-2 TS fuzzy model are listed in Table 5. Lower membership functions Upper membership functions µ1 = 12 −x1(t) 22 µ̄1 = 12 − (x1(t) + 2) 22 µ2 = 10 + x1(t) 22 µ̄2 = 10 + (x1(t) + 2) 22 Table 5: Lower and upper membership functions of the plant. https://doi.org/10.15837/ijccc.2022.6.4957 18 The weighting functions are chosen as αi = sin2(x2(t)) and ᾱi = 1 −αi for i = 1, 2. The interval-valued fuzzy system (1) is defined by Ai = cĀi, Adi = (1 − c)Āi and the following matrices: E =  1 0 00 1 0 0 0 0   , Ā1 =  −1.15 0.15 −400.5 −0.1 0 −0.75 0 −10   , Ā2 =  −1.37 0.15 −620.5 −0.1 0 −0.75 0 12   , B1 = B2 =  1 00 0 0 1   C21 = [ 1 0 0 0 1 1 ] , C22 = [ 1 0 0 0 1 −1 ] This example aims to design a fuzzy controller (9) that guarantees the admissibility of closed-loop systems. To accomplish this goal, table 6 lists the lower and upper bounds of membership functions to characterize the interval-valued fuzzy controller. Lower membership functions Upper membership functions ν1(x1) = 0.3e−(x 2 1/0.35) ν̄1(x1) = ν1(x1) ν2(x1) = 1 − 0.3e−(x 2 1/0.35) ν̄2(x1) = ν2(x1) Table 6: Lower and upper membership functions of the controller. To well determine the membership functions, the non-linear weight functions are selected as β j (x2(t)) = cos2(x2(t)) and β̄j(x2(t)) = 1 −βj(x2(t)). When dealing with the reliable observer-based feedback control problem for the bio-economic system under a stochastic fault, we can solve the LMIs (26) in Theorem 4, with c = 0.98, d1 = 0.2, d2 = 0.5, dr = 0.3, λ1 = 3, λ2 = 11, λ3 = 3, λ4 = 11, and Γ̄ = diag{0.5, 0.6}, to obtain a feasible solution that involves the following corresponding observer and controller gains: K1 = [ −8.9716 −0.058112 −224 0.24388 −0.0037578 −973.93 ] , K2 = [ −9.0421 −0.05894 −434.76 0.23901 −0.0035811 −982.23 ] , L1 =   1.9782 −0.315570.48519 0.84128 −0.42174 1.4254   , L2 =   3.1661 0.250490.44715 1.317 −0.85805 −2.3441   (36) For initial condition φ(t) = [10, 0.5, 12]>, t ∈ [−0.5 0], and d(t) = 0.2 + 0.3|sin(t)|, figure 5 shows the simulation results of the resulting closed-loop system, which was achieved by applying the developed fault-tolerant controller (36) to an uncertain system (34). This figure records the output responses of the system, and the control input uf (t). It can be concluded that, under actuator faults, uncertainties, and unmeasured states, the suggested controller would retain the closed-loop system dynamically stable. https://doi.org/10.15837/ijccc.2022.6.4957 19 0 5 10 15 20 25 30 -2 0 2 4 6 8 10 12 (a) The curves of y1(t), and ŷ1(t) 0 5 10 15 20 25 30 35 40 45 50 -15 -10 -5 0 5 10 15 (b) The curves of y2(t), and ŷ2(t) 0 2 4 6 8 10 12 14 -25 -20 -15 -10 -5 0 5 10 15 20 (c) Trajectories of uf1 (t), and u f 2 (t) Figure 5: Simulation plots for bio-economic system 5 Comparative Explanations For non-linear singular systems exhibiting uncertain states with time varying delay as well as stochastic actuator failures, this paper explores the fault-tolerant observer-based control issue. The advantages over existing methods are listed below. 1. In comparison with existing findings [14, 20, 28, 37, 41], the outcomes developed in this paper are more realistic and general since the fuzzy model incorporates the intrinsic uncertainty of the system. Moreover, fuzzy systems and fuzzy controllers have different premise membership functions. 2. Despite having studied the observer-based control problem for interval-valued fuzzy systems [8, 48, 51], the present investigation differed with the following features: • our approach is more general since the system under consideration is a singular one, in which E 6= I, • typically, stochastic actuator failure is considered to cope with the reliable control problem for this class of systems. 3. Compared to the fuzzy static and dynamic controllers proposed in [6, 30], the observer-based control strategy developed in this study assumes that the premise variables are unmeasurable as suggested in [51] 6 Conclusion An attempt is made in this study to deliver solutions to the main challenges that come up when dealing with non-linear singular systems, such as uncertainty, time-varying delay, and stochastic ac- tuator failures. Based on an IVF model that exploits both the lower and upper membership functions to adequately characterize the uncertainties, an IVF observer is designed to estimate the unmeasured states, and then an IVF controller is synthesized to stabilize the system under consideration. The analysis of the existing of such observer-based controller is carried out by involving an appropriate https://doi.org/10.15837/ijccc.2022.6.4957 20 Lyapunov-Krasovskii functional, and the key point of the conceived control scheme lies in the use of decoupling matrix technique to establish a set of feasible LMI-based constraints so that the closed-loop system is stochastically admissible. The proposed control scheme has been validated by two numerical simulations. As part of prospective research topics, we will extend the suggested developments to non-linear systems with event-triggered output feedback control problems. Acknowledgment This research has been funded by Research Deanship in University of Hail-Saudi Arabia through project number RG-21 119 Author contributions The authors contributed equally to this work. Conflict of interest The authors declare no conflict of interest. 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Non-fragile fault- tolerant control for nonlinear markovian jump systems with intermittent actuator fault. Nonlinear Analysis: Hybrid Systems, 32:337–350, May 2019. [51] Tao Zhao, Zhenbo Wei, Songyi Dian, and Jian Xiao. Observer-based H∞ controller design for interval -2 t–s fuzzy systems. Neurocomputing, 177:9–25, 2016. Copyright ©2022 by the authors. Licensee Agora University, Oradea, Romania. This is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International License. Journal’s webpage: http://univagora.ro/jour/index.php/ijccc/ This journal is a member of, and subscribes to the principles of, the Committee on Publication Ethics (COPE). https://publicationethics.org/members/international-journal-computers-communications-and-control Cite this paper as: Jerbi, H.; Kchaou, M.; Alshammari, O.; Abassi, R.; Popescu, D. (2022). Observer-based feedback control of interval-valued fuzzy singular system with time-varying delay and stochastic faults, Inter- national Journal of Computers Communications & Control, 17(6), 4957, 2022. https://doi.org/10.15837/ijccc.2022.6.4957 Introduction Literature Review Objective and Outline Notations Preliminaries and Problem Statement IVF Model Assumptions and Resulting Model Fuzzy Observer-based Controller Problem Statement Main Results Admissibility Analysis Observer Design Numerical examples Computational Framework and Algorithm Example 1 Bio-economic System Comparative Explanations Conclusion