Fuzzy Observer Design for State and Fault Estimation for Takagi-Sugeno Implicit Models

Manal Ouzaz; Abdellatif El Assoudi; El Hassane El Yaagoubi

doi:10.5391/IJFIS.2023.23.1.1

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 1-10

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.1

Fuzzy Observer Design for State and Fault Estimation for Takagi-Sugeno Implicit Models

Manal Ouzaz, Abdellatif El Assoudi , and El Hassane El Yaagoubi

Laboratory of High Energy Physics and Condensed Matter, Faculty of Sciences Ain Chock, Hassan II University of Casablanca, Casablanca, Morocco

Correspondence to :
Manal Ouzaz (manal.ouzaz@gmail.com)

Received: September 3, 2021; Revised: January 24, 2023; Accepted: March 7, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

In this study, we develop a fuzzy observer for a class of discrete-time nonlinear implicit models that are described by the Takagi–Sugeno structure and affected by actuator and sensor faults with unmeasurable premise variables satisfying the Lipschitz constraints. This study is based on separating dynamic and static equations in discrete-time Takagi–Sugeno implicit models. The design of a fuzzy observer is proposed to estimate unknown states, actuators, and sensor faults simultaneously. It is designed by considering the fault variables constituted by the actuator and sensor faults as auxiliary state variables. The observer gain is calculated by studying the exponential convergence of the state estimation error using the Lyapunov theory and the stability condition given as a linear matrix inequality. Simulation results demonstrated the effectiveness and validity of the proposed method.

Keywords: Discrete-time systems, Fault detection, Linear matrix inequalities, Observers, Takagi-Sugeno model

1. Introduction

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Over the past decades, the need for high reliability and safety in industrial applications has led to a surge in the use of model-based fault diagnosis techniques in automated processes. Therefore, the field of unknown states and fault observer design for nonlinear systems has attracted considerable attention from researchers owing to its important role in fault detection and diagnosis (FDD) and the design of fault-tolerant control (FTC) strategies [1–3]. Several studies have investigated the fault-detection filter design problem for a class of nonlinear Markov jump systems [4, 5].

There are numerous studies related to explicit and implicit nonlinear systems in both continuous-time and discrete-time cases. Regarding the continuous-time case, we may cite [6–9] for explicit models and [10–13] for implicit models, which are also referred to as singular or differential-algebraic systems. Likewise, in the discrete-time case, several studies exist on explicit and implicit structures, for example, [14–16]. Extensive literature exists on the topic of fuzzy unknown input observer and its applications to FDD and FTC.

An effective manner to solve the various fuzzy observers raised previously is to write the convergence conditions in the linear matrix inequality (LMI) form [17]. Recall that the fuzzy Takagi–Sugeno (TS) approach [18–20], which is recognized as a powerful tool for describing the global behaviors of nonlinear systems, has received considerable attention over the last few decades.

Nonlinear systems can be represented as the average weighted sum of linear systems using TS fuzzy systems [21]. TS models are widely used for the analysis and controller synthesis of nonlinear systems through the direct Lyapunov method [14]. For instance, in [22], an adaptive event-triggered controller design algorithm for TS fuzzy systems under multiple cyber-attacks was studied using the Lyapunov stability theory. The primary benefit of this formalism is the utilization of linear control methodologies in the study of complex nonlinear systems once the TS fuzzy models are obtained (e.g., [1, 19, 20]).

The implicit model, also called the singular model or descriptor model, is a general dynamic model that has been used to depict electrical systems, such as biological systems, mechanical systems, and chemical processes, for example, [23–26]. Moreover, in [10, 11] the authors extended the ordinary TS fuzzy model [18] to define an implicit TS fuzzy model.

Simultaneous state and fault estimation is increasingly appealing, as it allows us to obtain both state and fault information in a single design. In [27], a fuzzy observer design to simultaneously estimate the state and fault variables for continuous-time Takagi–Sugeno implicit models (TSIMs) with actuator and sensor faults was developed. In addition, a state and fault observer based on Euler discretization for TSIMs was proposed in [28].

This study aims to develop an observer design for simultaneous state and fault estimation for a class of discrete-time nonlinear implicit models (DNIMs) with actuator and sensor faults, as described by the TS structure.

This study proposes a novel methodology for designing a fuzzy observer for DNIMs described by discrete-time Takagi–Sugeno implicit models (DTSIMs) with unmeasurable premise variables satisfying Lipschitz constraints. The proposed approach presents a novel contribution to simultaneously estimating state, actuator, and sensor faults. The approach begins by separating the dynamic and static relations in the DTSIM. Subsequently, we consider both sensor and actuator faults as auxiliary state variables to construct an augmented system. Subsequently, an observer is developed for the given augmented system. The representation of an implicit system by separating it into differential and algebraic equations demonstrates the physical properties of the process. A differential equation constitutes the dynamics of the system, whereas an algebraic equation translates the interconnections and static constraints. This technique allows us to use different mathematical tools in an explicit form to synthesize the observer for implicit systems. The exponential stability of the state estimation error is studied using the Lyapunov theory, and the stability condition is given in terms of only one LMI.

The rest of this paper is organized as follows: Section 2 introduces the class of DNIMs described by the TS structure affected by actuator and sensor faults. Section 3 presents the main results of the fuzzy observer design for the considered DTSIMs that estimate the states, and actuator and sensor faults simultaneously. Section 4 illustrates the effectiveness of the proposed method via a single-link flexible joint robot application.

The following notation have been adopted throughout this paper:

• Matrix P > 0 (or P < 0) indicates that P is symmetric and positive definite (or negative definite).
• P^T denotes the transposition of P. The symbol I (or 0) represents an identity matrix (or zero matrices) with appropriate dimensions.

Let • Rⁿ and Rⁿ^×^m denote the spaces of n-dimensional real vectors and n × m real matrices, respectively.

• (A*BC)=(ABTBC).

2. System Description

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

In this study, the following class of DNIMs with actuator and sensor faults was adopted:

(1){Mzk+1=A(zk)zk+B(zk)uk+Fa(zk)fka,yk=Czk+Duk+Dafka+Fsfks,

where zkT=[Zk1TZk2T]∈Rn denotes the state vector with Zk1∈Rn1 is the vector of differential variables, Zk2∈Rn2 is the vector of algebraic variables with n₁ + n₂ = n, u_k ∈ R^m is the input vector which is known (measured), fka∈Rna and fks∈Rns are respectively the actuator fault and the sensor fault, y_k ∈ R^p is the measurement vector.

A(z_k) ∈ Rⁿ^×ⁿ, B(z_k) ∈ Rⁿ^×^m, and F_a(z_k) ∈ Rⁿ^×ⁿ^{_^a} are nonlinear matrix functions. C ∈ R^p^×ⁿ, D ∈ R^p^×^m, D_a ∈ R^p^×ⁿ^_a, F_s ∈ R^p^×ⁿ^_s are the known real constant matrices.

M ∈ Rⁿ^×ⁿ with rank(M) = n₁ is a real known constant matrix assumed to be of the form

(2)M=(I000).

If this is not the case (M is not of the form (2)), without loss of generality, we can determine two invertible matrices, M₁ and M₂, with appropriate dimensions such that the following relation is satisfied (see [23] or [29] for more details):

(3)M1MM2=(I000).

The DNIM described above in (1) can be represented using the sector nonlinearity approach [19, 20] by the following DTSIM with unmeasurable premise variables satisfying the Lipschitz constraints:

(4){Mzk+1=∑i=1qφi(zk)(Aizk+Biuk+Faifka),yk=Czk+Duk+Dafka+Fsfks,

where

A_i ∈ Rⁿ^×ⁿ, B_i ∈ Rⁿ^×^m, C ∈ R^p^×ⁿ, D ∈ R^p^×^m, D_a ∈ R^p^×ⁿ^_a, F_ai ∈ Rⁿ^×ⁿ^_a, F_s ∈ R^p^×ⁿ^_s are known constant matrices with

(5){Ai(Ai11Ai12Ai21Ai22); Bi=(Bi1Bi2),[-0.15pc]Fai=(Fai1Fai2); C=(C10),

where Ai22 is assumed to be invertible. q is the number of submodels and φ_i(z_k) are the weighting functions that verify the convex sum properties and are expressed as

(6){∑i=1qφi(zk)=10≤φi(zk)≤1i=1,…, q.

They ensure the transition between the contributions of each submodeland are expressed as follows:

(7){Mzk+1=Aizk+Biuk+Faifka,yk=Czk+Duk+Dafka+Fsfks.

The following assumptions are made before presenting the primary result:

Assumption 1

Suppose that

• (M, A_i) is regular, i.e., det(zM – A⁼) ≠ 0 ∀z ∈ C.
• All sub-models (7) are impulse observable and detectable.

As previously mentioned, we proceed to the separation between differential and algebraic equations in each sub-model (7), as the dynamic characteristics of the implicit system (regularity, causality, stabilizability, and detectability) remain constant [26].

This separation into differential and algebraic equations enables us to leverage certain analytical and design tools developed for systems described by differential equations. This significantly facilitates the study of the observer synthesis of complex nonlinear implicit systems.

From (5), the sub-models (7) can be rewritten as follows:

(8){Zk+11=Ai11Zk1+Ai12Zk2+Bi1uk+Fai1fka,0=Ai21Zk1+Ai22Zk2+Bi2uk+Fai2fka,yk=C1Zk1+Duk+Dafka+Fsfks.

From (8) and using the fact that the matrices Ai22 are invertible, the algebraic equations can be solved directly for algebraic variables to obtain

(9)Zk2=JiZk1+Kiuk+Laifka,

where

(10){Ji=-(Ai22)-1Ai21,Ki=-(Ai22)-1Bi2,Lai=-(Ai22)-1Fai2.

Thus, substituting the resulting expression of Zk2 (9) in (8) we obtain the following model:

(11){Zk+11=MiZk1+Niuk+Paifka,Zk2=JiZk1+Kiuk+Laifka,yk=C1Zk1+Duk+Dafka+Fsfks,

where

(12){Mi=Ai11+Ai12Ji,Ni=Bi1+Ai12Ki,Pai=Fai1+Ai12Lai.

Let

(13)fk=((fka)T(fks)T).

Thus, Model (11) can be written in the following form:

(14){Zk+11=MiZk1+Niuk+Pifk,Zk2=JiZk1+Kiuk+Lifk,yk=C1Zk1+Duk+Tfk,

where

(15)Pi=(Pai0); Li=(Lai0); T=(DaFs).

The weighting functions φ_i(z_k), i = 1, …, q can be rewritten as

(16)φi(zk)=φi(Zk1,Zk2=JiZk1+Kiuk+Lifk)=φi(λk),

with

(17)λkT=[Zk1TfkTukT]T.

Thus, by aggregating the resulting sub-models (14), the global fuzzy model is obtained

(18){Zk+11=∑i=1qφi(λk) (MiZk1+Niuk+Pifk),Zk2=∑i=1qφi(λk) (JiZk1+Kiuk+Lifk),yk=C1Zk1+Duk+Tfk.

Assumption 2

Suppose that f_k is a constant unknown fault signal per time interval, that is,

(19)fk+1=fk, k∈[T1 T2],∀T1, T2∈R+.

To make the main contribution, we rewrite the system (18) under the equivalent augmented state representation given by

(20){ξk+11=∑i=1qφi(μk) (Φiξk1+Ψiuk),ξk2=∑i=1qφi(μk) (Ωiξk1+Kiuk),yk=Rξk1+Duk,

where

(21){ξk1T=[Zk1TfkT],ξk2=Zk2,μkT=[ξk1TukT],Φi=(MiPi0I),Ψi=(Ni0),Ωi=(JiLi),R=(C1T).

To design an observer for the system (1), we introduce the following matrices:

(22){Φ0=1q∑i=1qΦi,Ψ0=1q∑i=1qψi,Φ¯i=Φi-Φ0,Ψ¯i=Ψi-Ψ0.

From (22), (20) is equivalent to the following form:

(23){ξk+11=Φ0ξk1+Ψ0uk+∑i=1qφi(μk) (Φ¯iξk1+Ψ¯iuk),ξk2=∑i=1qφi(μk) (Ωiξk1+Kiuk),yk=Rξk1+Duk.

3. Main Results

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Based on the transformation of DNIM (1) into its equivalent form (23), the proposed fuzzy observer for (1) is expressed as follows:

(24){ξ^k+11=Φ0ξ^k1+Ψ0uk-H(y^k-yk)+∑i=1qφi(μ^k) (Φ¯iξ^k1+Ψ¯iuk),ξ^k2=∑i=1qφi(μ^k) (Ωiξ^k1+Kiuk),y^k=Rξ^k1+Duk,

where (ξ^k1,ξ^k2), ŷ_k and μ̂_k are the estimates of (ξ^k1,ξ^k2), y_k and μ_k respectively. Matrix H is determined such that (ξ^k1,ξ^k2) converges toward (ξk1,ξk2) exponentially.

To obtain the condition for the exponential convergence of the observer (24), we define the state estimation error as follows:

(25)ɛk=(ɛk1ɛk2)=(ξ^k1-ξk1ξ^k2-ξk2).

From (23) and (24), the dynamics of the estimation error can be described as follows:

(26){ɛk+11=Γ0ɛk1+Δ,ɛk2=∑i=1qφi(μ^k)Ωiɛk1+∑i=1q(φi(μ^k)-φi(μk)) (Ωiξk1+Kiuk),

where

(27)Γ0=Φ0-HR,

and

(28)Δ=∑i=1q(Φ¯iΔk1+Δk2uk),

with

(29){Δk1=φi(μ^k)ξ^k1-φi(μk)ξk1,Δk2=Ψ¯i(φi(μ^k)-φi(μk)).

Therefore, to demonstrate the convergence of ɛ_k to zero, it is sufficient to prove that ɛk1 converges to zero.

Assumption 3

Assume that the following conditions hold:

(30){∣Δk1∣<δi∣ɛk1∣,∣Δk2∣<βi∣ɛk1∣,∣uk∣<γ,

where δ_i and β_i are positive scalar Lipschitz constants and γ > 0.

Using Assumption 3, the term Δ can be bound as follows:

(31)∣Δ∣<θ∣ɛk1∣,

where

(32)θ=∑i=1q(σ(Φ¯i)δi+βiγ),

where σ(Φ̄_i) denotes the maximum singular value of matrix Φ̄_i.

The main results of this study are presented as the following theorem:

Theorem 1

Under Assumption 3, the system (26) is globally exponentially stable if given ρ > 0 there exist the matrices P >0, Q > 0, and W verifying the following LMI:

(33)(Σ**WR-P*PΦ0-WR0-Q)<0,

where

(34)Σ=Φ0TPΦ0-Φ0TWR-RTWTΦ0+θ2P+θ2Q-ρ2P.

The gain that stabilizes the estimation error is given by

(35)H=P-1W.

Proof of Theorem 1

Consider the following standard Lyapunov function:

(36)Vk=ɛk1TPɛk1, P>0.

The variation in V_k along the trajectory of (26) is given as follows:

(37)ΔVk=Vk+1-Vk.

From (26), we have

(38)ΔVk=ɛk1T(Γ0TPΓ0-P)ɛk1+ΔTPΔ+ΔTPΓ0ɛk1+ɛk1TΓ0TPΔ.

Lemma 1

For any matrices X and Y with appropriate dimensions, the following property holds for a definite positive matrix J:

(39)XTY+YTX≤XTJ-1X+YTJY.

For Q > 0, by applying Lemma 1, (38) becomes

(40)ΔVk<ɛk1T(Γ0TPΓ0-P)ɛk1+ΔTPΔ+ɛk1TΓ0TPQ-1PΓ0ɛk1+ΔTQΔ.

Considering (31), (40) becomes

(41)ΔVk<ɛk1T(Γ0TPΓ0-P+Γ0TPQ-1PΓ0)+θ2P+θ2Q)ɛk1.

Estimation error convergence exponentially lacks if the condition in [29], cited in [20], is satisfied:

(42)ΔVk<ɛk1T(Γ0TPΓ0-P+Γ0TPQ-1PΓ0+θ2P+θ2Q)ɛk1<(ρ2-1)Vk, ρ<1,

which leads to the following condition:

(43)Γ0TPΓ0+Γ0TPQ-1PΓ0+θ2P+θ2Q-ρ2P<0,ρ<1.

By substituting Γ₀ from (27) into (43), we can establish the LMI condition (33) of Theorem 1 using the Schur complement and the following change of variables:

(44)W=PH.

Thus, from the Lypunov stability theory, if the LMI condition (33) is satisfied, then the system (26) is globally exponentially stable. This concludes the proof of Theorem 1.

4. Application to a Single-Link Flexible Joint Robot

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
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References
Biographies

To illustrate the performance of the proposed fuzzy observer design, we considered a single-link flexible joint robot. Considering the model given in [16] and assuming that it is affected by simultaneous actuator and sensor faults, we obtain a DNIM with an unmeasurable premise variable in the following form:

(45){Mzk+1=A(zk)zk+Buk+Faifka,yk=Czk+Fsfks,

where z_k = (z₁_k, z₂_k, z₃_k, z₄_k, z₅_k, z₆_k)^T represents the state vector, z₁_k and z₂_k are the angles of rotation of the motor and link, respectively; z₃_k and z₄_k are the angular velocities, z₅_k and z₆_k are their angular accelerations. u_k ∈ R and y_k ∈ R² are the control variables and the measured output vector, respectively. fka is the actuator fault, and fks is the sensor fault.

A(zk)=(10Te000010Te000010Te000010Te-k1TeJmk1TeJm-βTeJm0-Te0k1TeJLη000-Te),B=(0000k2TeJm0), Fa=B, Fs=(010),C=(100000001000000100),M=(100000010000001000000100000000000000),

where T_e > 0 is the sampling time and

(46)η=-k1TeJL-mgbTeJLsin(z2k)z2k.

To write model (45) as a TS model with an unmeasurable premise variable, we consider the nonlinearities of the term η ∈ [η_min, η_max] of matrix A(z_k). We can then transform this nonlinear term into the following form:

(47)η=α1ηmax+α2ηmin,

where

(48){α1=η-ηminηmax-ηmin,α2=ηmax-ηηmax-ηmin.

Therefore, the obtained global TS fuzzy model is expressed as follows:

(49){Mzk+1=∑i=12φi(zk) (Aizk+Buk+Faifka),yk=Czk+Fsfks,

with

A1=(10Te000010Te000010Te000010Te-k1TeJmk1TeJm-βTeJm0-Te0k1TeJLηmax000-Te),A2=(10Te000010Te000010Te000010Te-k1TeJmk1TeJm-βTeJm0-Te0k1TeJLηmin000-Te).

The membership functions are as follows:

(50){φ1(zk)=α1,φ2(zk)=α2.

In this case,

zk1=(z1k, z2k, z3k, z4k)T; zk2=(z5k, z6k)T,

and A122=A222=(-Te00-Te) are invertible.

We rewrite the model (49) in the equivalent form (20) to apply the fuzzy observer proposed in (24) for the single-link flexible joint robot.

The values and definitions of the physical parameters are given in [12] and we assume T_e = 0.01s.

The expressions for the actuator and sensor faults are shown in Figure 4. Considering Theorem 1 with ρ = 0.98, the observer gain obtained is as follows:

H=(0.6618-0.02050.00660.11100.2610-0.62512.87350.4394-0.28790.0943-0.12490.94171.88990.35310.6777-3.15040.48850.0763).

Simulation results with initial conditions

ξk1=[0.00 0.3142 0.00 0.00 0.02 0.00]T,ξk2=[15.7159 -16.6906]T,ξ^k1=[0.00 0.3342 0.00 0.00 0.00 0.00]T,ξ^k2=[16.2564 -17.7286]T

are shown in Figures 1–4 where the input is u_k = sin(k). The dashed lines denote the state variables and the actuator and sensor faults estimated by the proposed fuzzy observer (24) with gain H.

As shown in Figure 4, the actuator and sensor faults are applied during the interval [0 40 s], and they are also applied simultaneously at intervals [10 s 15 s] and [25 s 30 s]; Nevertheless, as shown in Figures 1–3, the proposed fuzzy observer effectively estimates the states of the flexible robot in the entire interval [0 40s]. From Figure 4, it is evident that the fault estimations applied to the flexible robot were accurate.

Noise can lead to compromised performance or even instability. A second simulation was performed with centered measurement noise to verify the effectiveness of this approach.

The simulation results in Figures 5–8 demonstrate the effectiveness of the proposed fuzzy observer, even with noise.

5. Conclusion

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

This study presents a novel methodology to design a state and fault fuzzy observer for a class of DNIMs described by a TS structure with unmeasurable premise variables that satisfy Lipschitz constraints. This approach permits the simultaneous estimation of the fault and system state variables. This method is based on separating dynamic and static relations in DTSIMs. The convergence is studied using the Lyapunov theory, and the stability condition is given in terms of only one LMI. The efficacy of the proposed fuzzy observer is demonstrated by a simulation of a single-link flexible joint robot, serving as an illustrative application.

Conflict of Interest

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
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Biographies

Fig. 1.

State variables z₁_k and z₂_k with their estimates.

Fig. 2.

State variables z₃_k and z₄_k with their estimates.

Fig. 3.

State variables z₅_k and z₆_k with their estimates.

Fig. 4.

Actuator fault fka and sensor fault fks and their estimates.

Fig. 5.

State variables z₁_k and z₂_k with noise and their estimates.

Fig. 6.

State variables z₃_k and z₄_k with noise and their estimates.

Fig. 7.

State variables z₅_k and z₆_k with noise and their estimates.

Fig. 8.

Actuator fault fka and sensor fault fks with noise and their estimates.

References

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Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

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Biographies

Go to

Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Manal Ouzaz received an Electrical Engineering degree from the National High School of Electricity and Mechanics (ENSEM) in Morocco in 2010. Her research interests include observer design, fault detection and fuzzy control.

E-mail: manal.ouzaz@gmail.com

Abdellatif El Assoudi is a professor in the Department of Electrical Engineering at the National High School of Electricity and Mechanics (ENSEM) at Hassan II University of Casablanca (Morocco).

E-mail: a.elassoudi@ensem.ac.ma

El Hassane El Yaagoubi is a professor in the Department of Electrical Engineering at the National High School of Electricity and Mechanics (ENSEM) at Hassan II University of Casablanca (Morocco).

E-mail: h.elyaagoubi@ensem.ac.ma

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 1-10

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.1

Fuzzy Observer Design for State and Fault Estimation for Takagi-Sugeno Implicit Models

Manal Ouzaz, Abdellatif El Assoudi , and El Hassane El Yaagoubi

Laboratory of High Energy Physics and Condensed Matter, Faculty of Sciences Ain Chock, Hassan II University of Casablanca, Casablanca, Morocco

Correspondence to:Manal Ouzaz (manal.ouzaz@gmail.com)

Received: September 3, 2021; Revised: January 24, 2023; Accepted: March 7, 2023

Abstract

Keywords: Discrete-time systems, Fault detection, Linear matrix inequalities, Observers, Takagi-Sugeno model

1. Introduction

The following notation have been adopted throughout this paper:

• Matrix P > 0 (or P < 0) indicates that P is symmetric and positive definite (or negative definite).
• P^T denotes the transposition of P. The symbol I (or 0) represents an identity matrix (or zero matrices) with appropriate dimensions.

Let • Rⁿ and Rⁿ^×^m denote the spaces of n-dimensional real vectors and n × m real matrices, respectively.

• $(\begin{matrix} A & * \\ B & C \end{matrix}) = (\begin{matrix} A & B^{T} \\ B & C \end{matrix})$ .

2. System Description

In this study, the following class of DNIMs with actuator and sensor faults was adopted:

(1)

{\begin{array}{l} M z_{k + 1} = A (z_{k}) z_{k} + B (z_{k}) u_{k} + F_{a} (z_{k}) f_{k}^{a}, \\ y_{k} = C z_{k} + D u_{k} + D_{a} f_{k}^{a} + F_{s} f_{k}^{s}, \end{array}

where $z_{k}^{T} = [\begin{array}{l} Z_{k}^{1^{T}} & Z_{k}^{2^{T}} \end{array}] \in R^{n}$ denotes the state vector with $Z_{k}^{1} \in R^{n_{1}}$ is the vector of differential variables, $Z_{k}^{2} \in R^{n_{2}}$ is the vector of algebraic variables with n₁ + n₂ = n, u_k ∈ R^m is the input vector which is known (measured), $f_{k}^{a} \in R^{n_{a}}$ and $f_{k}^{s} \in R^{n_{s}}$ are respectively the actuator fault and the sensor fault, y_k ∈ R^p is the measurement vector.

M ∈ Rⁿ^×ⁿ with rank(M) = n₁ is a real known constant matrix assumed to be of the form

(2)

M = (\begin{matrix} I & 0 \\ 0 & 0 \end{matrix}) .

(3)

M_{1} M M_{2} = (\begin{matrix} I & 0 \\ 0 & 0 \end{matrix}) .

The DNIM described above in (1) can be represented using the sector nonlinearity approach [19, 20] by the following DTSIM with unmeasurable premise variables satisfying the Lipschitz constraints:

(4)

{\begin{array}{l} M z_{k + 1} = \sum_{i = 1}^{q} φ_{i} (z_{k}) (A_{i} z_{k} + B_{i} u_{k} + F_{a i} f_{k}^{a}), \\ y_{k} = C z_{k} + D u_{k} + D_{a} f_{k}^{a} + F_{s} f_{k}^{s}, \end{array}

where

A_i ∈ Rⁿ^×ⁿ, B_i ∈ Rⁿ^×^m, C ∈ R^p^×ⁿ, D ∈ R^p^×^m, D_a ∈ R^p^×ⁿ^_a, F_ai ∈ Rⁿ^×ⁿ^_a, F_s ∈ R^p^×ⁿ^_s are known constant matrices with

(5)

{\begin{array}{l} A_{i} (\begin{matrix} A_{i}^{11} & A_{i}^{12} \\ A_{i}^{21} & A_{i}^{22} \end{matrix}); & B_{i} = (\begin{matrix} B_{i}^{1} \\ B_{i}^{2} \end{matrix}), \\ [- 0.15 p c] F_{a i} = (\begin{matrix} F_{a i}^{1} \\ F_{a i}^{2} \end{matrix}); & C = (\begin{matrix} C^{1} & 0 \end{matrix}), \end{array}

where $A_{i}^{22}$ is assumed to be invertible. q is the number of submodels and φ_i(z_k) are the weighting functions that verify the convex sum properties and are expressed as

(6)

{\begin{array}{l} \sum_{i = 1}^{q} φ_{i} (z_{k}) = 1 \\ 0 \leq φ_{i} (z_{k}) \leq 1 \\ i = 1, \dots, q . \end{array}

They ensure the transition between the contributions of each submodeland are expressed as follows:

(7)

{\begin{array}{l} M z_{k + 1} = A_{i} z_{k} + B_{i} u_{k} + F_{a i} f_{k}^{a}, \\ y_{k} = C z_{k} + D u_{k} + D_{a} f_{k}^{a} + F_{s} f_{k}^{s} . \end{array}

The following assumptions are made before presenting the primary result:

Assumption 1

Suppose that

• (M, A_i) is regular, i.e., det(zM – A⁼) ≠ 0 ∀z ∈ C.
• All sub-models (7) are impulse observable and detectable.

From (5), the sub-models (7) can be rewritten as follows:

(8)

{\begin{array}{l} Z_{k + 1}^{1} = A_{i}^{11} Z_{k}^{1} + A_{i}^{12} Z_{k}^{2} + B_{i}^{1} u_{k} + F_{a i}^{1} f_{k}^{a}, \\ 0 = A_{i}^{21} Z_{k}^{1} + A_{i}^{22} Z_{k}^{2} + B_{i}^{2} u_{k} + F_{a i}^{2} f_{k}^{a}, \\ y_{k} = C^{1} Z_{k}^{1} + D u_{k} + D_{a} f_{k}^{a} + F_{s} f_{k}^{s} . \end{array}

From (8) and using the fact that the matrices $A_{i}^{22}$ are invertible, the algebraic equations can be solved directly for algebraic variables to obtain

(9)

Z_{k}^{2} = J_{i} Z_{k}^{1} + K_{i} u_{k} + L_{a i} f_{k}^{a},

where

(10)

{\begin{array}{l} J_{i} = - {(A_{i}^{22})}^{- 1} A_{i}^{21}, \\ K_{i} = - {(A_{i}^{22})}^{- 1} B_{i}^{2}, \\ L_{a i} = - {(A_{i}^{22})}^{- 1} F_{a i}^{2} . \end{array}

Thus, substituting the resulting expression of $Z_{k}^{2}$ (9) in (8) we obtain the following model:

(11)

{\begin{array}{l} Z_{k + 1}^{1} = M_{i} Z_{k}^{1} + N_{i} u_{k} + P_{a i} f_{k}^{a}, \\ Z_{k}^{2} = J_{i} Z_{k}^{1} + K_{i} u_{k} + L_{a i} f_{k}^{a}, \\ y_{k} = C^{1} Z_{k}^{1} + D u_{k} + D_{a} f_{k}^{a} + F_{s} f_{k}^{s}, \end{array}

where

(12)

{\begin{array}{l} M_{i} = A_{i}^{11} + A_{i}^{12} J_{i}, \\ N_{i} = B_{i}^{1} + A_{i}^{12} K_{i}, \\ P_{a i} = F_{a i}^{1} + A_{i}^{12} L_{a i} . \end{array}

Let

(13)

f_{k} = (\begin{matrix} {(f_{k}^{a})}^{T} \\ {(f_{k}^{s})}^{T} \end{matrix}) .

Thus, Model (11) can be written in the following form:

(14)

{\begin{array}{l} Z_{k + 1}^{1} = M_{i} Z_{k}^{1} + N_{i} u_{k} + P_{i} f_{k}, \\ Z_{k}^{2} = J_{i} Z_{k}^{1} + K_{i} u_{k} + L_{i} f_{k}, \\ y_{k} = C^{1} Z_{k}^{1} + D u_{k} + T f_{k}, \end{array}

where

(15)

P_{i} = (\begin{matrix} P_{a i} & 0 \end{matrix}); L_{i} = (\begin{matrix} L_{a i} & 0 \end{matrix}); T = (\begin{matrix} D_{a} & F_{s} \end{matrix}) .

The weighting functions φ_i(z_k), i = 1, …, q can be rewritten as

(16)

\begin{array}{l} φ_{i} (z_{k}) = φ_{i} (Z_{k}^{1}, Z_{k}^{2} = J_{i} Z_{k}^{1} + K_{i} u_{k} + L_{i} f_{k}) \\ = φ_{i} (λ_{k}), \end{array}

with

(17)

λ_{k}^{T} = {[\begin{matrix} Z_{k}^{1^{T}} & f_{k}^{T} & u_{k}^{T} \end{matrix}]}^{T} .

Thus, by aggregating the resulting sub-models (14), the global fuzzy model is obtained

(18)

{\begin{array}{l} Z_{k + 1}^{1} = \sum_{i = 1}^{q} φ_{i} (λ_{k}) (M_{i} Z_{k}^{1} + N_{i} u_{k} + P_{i} f_{k}), \\ Z_{k}^{2} = \sum_{i = 1}^{q} φ_{i} (λ_{k}) (J_{i} Z_{k}^{1} + K_{i} u_{k} + L_{i} f_{k}), \\ y_{k} = C^{1} Z_{k}^{1} + D u_{k} + T f_{k} . \end{array}

Assumption 2

Suppose that f_k is a constant unknown fault signal per time interval, that is,

(19)

f_{k + 1} = f_{k}, k \in [T_{1} T_{2}], \forall T_{1}, T_{2} \in R^{+} .

To make the main contribution, we rewrite the system (18) under the equivalent augmented state representation given by

(20)

{\begin{array}{l} ξ_{k + 1}^{1} = \sum_{i = 1}^{q} φ_{i} (μ_{k}) (Φ_{i} ξ_{k}^{1} + Ψ_{i} u_{k}), \\ ξ_{k}^{2} = \sum_{i = 1}^{q} φ_{i} (μ_{k}) (Ω_{i} ξ_{k}^{1} + K_{i} u_{k}), \\ y_{k} = R ξ_{k}^{1} + D u_{k}, \end{array}

where

(21)

{\begin{array}{l} {ξ_{k}^{1}}^{T} = [\begin{matrix} {Z_{k}^{1}}^{T} & {f_{k}}^{T} \end{matrix}], ξ_{k}^{2} = Z_{k}^{2}, \\ μ_{k}^{T} = [\begin{matrix} {ξ_{k}^{1}}^{T} & u_{k}^{T} \end{matrix}], \\ Φ_{i} = (\begin{matrix} M_{i} & P_{i} \\ 0 & I \end{matrix}), \\ Ψ_{i} = (\begin{matrix} N_{i} \\ 0 \end{matrix}), \\ Ω_{i} = (\begin{matrix} J_{i} & L_{i} \end{matrix}), \\ R = (\begin{matrix} C^{1} & T \end{matrix}) . \end{array}

To design an observer for the system (1), we introduce the following matrices:

(22)

{\begin{array}{l} Φ_{0} = \frac{1}{q} \sum_{i = 1}^{q} Φ_{i}, \\ Ψ_{0} = \frac{1}{q} \sum_{i = 1}^{q} ψ_{i}, \\ {\bar{Φ}}_{i} = Φ_{i} - Φ_{0}, \\ {\bar{Ψ}}_{i} = Ψ_{i} - Ψ_{0} . \end{array}

From (22), (20) is equivalent to the following form:

(23)

{\begin{array}{l} ξ_{k + 1}^{1} = Φ_{0} ξ_{k}^{1} + Ψ_{0} u_{k} + \sum_{i = 1}^{q} φ_{i} (μ_{k}) ({\bar{Φ}}_{i} ξ_{k}^{1} + {\bar{Ψ}}_{i} u_{k}), \\ ξ_{k}^{2} = \sum_{i = 1}^{q} φ_{i} (μ_{k}) (Ω_{i} ξ_{k}^{1} + K_{i} u_{k}), \\ y_{k} = R ξ_{k}^{1} + D u_{k} . \end{array}

3. Main Results

Based on the transformation of DNIM (1) into its equivalent form (23), the proposed fuzzy observer for (1) is expressed as follows:

(24)

{\begin{array}{l} {\hat{ξ}}_{k + 1}^{1} = Φ_{0} {\hat{ξ}}_{k}^{1} + Ψ_{0} u_{k} - H ({\hat{y}}_{k} - y_{k}) + \sum_{i = 1}^{q} φ_{i} ({\hat{μ}}_{k}) ({\bar{Φ}}_{i} {\hat{ξ}}_{k}^{1} + {\bar{Ψ}}_{i} u_{k}), \\ {\hat{ξ}}_{k}^{2} = \sum_{i = 1}^{q} φ_{i} ({\hat{μ}}_{k}) (Ω_{i} {\hat{ξ}}_{k}^{1} + K_{i} u_{k}), \\ {\hat{y}}_{k} = R {\hat{ξ}}_{k}^{1} + D u_{k}, \end{array}

where ( ${\hat{ξ}}_{k}^{1}, {\hat{ξ}}_{k}^{2}$ ), ŷ_k and μ̂_k are the estimates of ( ${\hat{ξ}}_{k}^{1}, {\hat{ξ}}_{k}^{2}$ ), y_k and μ_k respectively. Matrix H is determined such that ( ${\hat{ξ}}_{k}^{1}, {\hat{ξ}}_{k}^{2}$ ) converges toward ( $ξ_{k}^{1}, ξ_{k}^{2}$ ) exponentially.

To obtain the condition for the exponential convergence of the observer (24), we define the state estimation error as follows:

(25)

ɛ_{k} = (\begin{matrix} ɛ_{k}^{1} \\ ɛ_{k}^{2} \end{matrix}) = (\begin{matrix} {\hat{ξ}}_{k}^{1} - ξ_{k}^{1} \\ {\hat{ξ}}_{k}^{2} - ξ_{k}^{2} \end{matrix}) .

From (23) and (24), the dynamics of the estimation error can be described as follows:

(26)

{\begin{array}{l} ɛ_{k + 1}^{1} = Γ_{0} ɛ_{k}^{1} + Δ, \\ ɛ_{k}^{2} = \sum_{i = 1}^{q} φ_{i} ({\hat{μ}}_{k}) Ω_{i} ɛ_{k}^{1} + \sum_{i = 1}^{q} (φ_{i} ({\hat{μ}}_{k}) - φ_{i} (μ_{k})) (Ω_{i} ξ_{k}^{1} + K_{i} u_{k}), \end{array}

where

(27)

Γ_{0} = Φ_{0} - H R,

and

(28)

Δ = \sum_{i = 1}^{q} ({\bar{Φ}}_{i} Δ_{k}^{1} + Δ_{k}^{2} u_{k}),

with

(29)

{\begin{array}{l} Δ_{k}^{1} = φ_{i} ({\hat{μ}}_{k}) {\hat{ξ}}_{k}^{1} - φ_{i} (μ_{k}) ξ_{k}^{1}, \\ Δ_{k}^{2} = {\bar{Ψ}}_{i} (φ_{i} ({\hat{μ}}_{k}) - φ_{i} (μ_{k})) . \end{array}

Therefore, to demonstrate the convergence of ɛ_k to zero, it is sufficient to prove that $ɛ_{k}^{1}$ converges to zero.

Assumption 3

Assume that the following conditions hold:

(30)

{\begin{array}{l} ∣ Δ_{k}^{1} ∣ < δ_{i} ∣ ɛ_{k}^{1} ∣, \\ ∣ Δ_{k}^{2} ∣ < β_{i} ∣ ɛ_{k}^{1} ∣, \\ ∣ u_{k} ∣ < γ, \end{array}

where δ_i and β_i are positive scalar Lipschitz constants and γ > 0.

Using Assumption 3, the term Δ can be bound as follows:

(31)

∣ Δ ∣ < θ ∣ ɛ_{k}^{1} ∣,

where

(32)

θ = \sum_{i = 1}^{q} (σ ({\bar{Φ}}_{i}) δ_{i} + β_{i} γ),

where σ(Φ̄_i) denotes the maximum singular value of matrix Φ̄_i.

The main results of this study are presented as the following theorem:

Theorem 1

Under Assumption 3, the system (26) is globally exponentially stable if given ρ > 0 there exist the matrices P >0, Q > 0, and W verifying the following LMI:

(33)

(\begin{matrix} Σ & * & * \\ W R & - P & * \\ P Φ_{0} - W R & 0 & - Q \end{matrix}) < 0,

where

(34)

Σ = Φ_{0}^{T} P Φ_{0} - Φ_{0}^{T} W R - R^{T} W^{T} Φ_{0} + θ^{2} P + θ^{2} Q - ρ^{2} P .

The gain that stabilizes the estimation error is given by

(35)

H = P^{- 1} W .

Proof of Theorem 1

Consider the following standard Lyapunov function:

(36)

V_{k} = {ɛ_{k}^{1}}^{T} P ɛ_{k}^{1}, P > 0.

The variation in V_k along the trajectory of (26) is given as follows:

(37)

Δ V_{k} = V_{k + 1} - V_{k} .

From (26), we have

(38)

Δ V_{k} = {ɛ_{k}^{1}}^{T} (Γ_{0}^{T} P Γ_{0} - P) ɛ_{k}^{1} + Δ^{T} P Δ + Δ^{T} P Γ_{0} ɛ_{k}^{1} + {ɛ_{k}^{1}}^{T} Γ_{0}^{T} P Δ .

Lemma 1

For any matrices X and Y with appropriate dimensions, the following property holds for a definite positive matrix J:

(39)

X^{T} Y + Y^{T} X \leq X^{T} J^{- 1} X + Y^{T} J Y .

For Q > 0, by applying Lemma 1, (38) becomes

(40)

Δ V_{k} < {ɛ_{k}^{1}}^{T} (Γ_{0}^{T} P Γ_{0} - P) ɛ_{k}^{1} + Δ^{T} P Δ + {ɛ_{k}^{1}}^{T} Γ_{0}^{T} P Q^{- 1} P Γ_{0} ɛ_{k}^{1} + Δ^{T} Q Δ .

Considering (31), (40) becomes

(41)

Δ V_{k} < {ɛ_{k}^{1}}^{T} (Γ_{0}^{T} P Γ_{0} - P + Γ_{0}^{T} P Q^{- 1} P Γ_{0}) + θ^{2} P + θ^{2} Q) ɛ_{k}^{1} .

Estimation error convergence exponentially lacks if the condition in [29], cited in [20], is satisfied:

(42)

\begin{array}{l} Δ V_{k} < {ɛ_{k}^{1}}^{T} (Γ_{0}^{T} P Γ_{0} - P + Γ_{0}^{T} P Q^{- 1} P Γ_{0} + θ^{2} P + θ^{2} Q) ɛ_{k}^{1} \\ < (ρ^{2} - 1) V_{k}, ρ < 1, \end{array}

which leads to the following condition:

(43)

\begin{array}{l} Γ_{0}^{T} P Γ_{0} + Γ_{0}^{T} P Q^{- 1} P Γ_{0} + θ^{2} P + θ^{2} Q - ρ^{2} P < 0, \\ ρ < 1. \end{array}

By substituting Γ₀ from (27) into (43), we can establish the LMI condition (33) of Theorem 1 using the Schur complement and the following change of variables:

(44)

W = P H .

Thus, from the Lypunov stability theory, if the LMI condition (33) is satisfied, then the system (26) is globally exponentially stable. This concludes the proof of Theorem 1.

4. Application to a Single-Link Flexible Joint Robot

(45)

{\begin{array}{l} M z_{k + 1} = A (z_{k}) z_{k} + B u_{k} + F_{a i} f_{k}^{a}, \\ y_{k} = C z_{k} + F_{s} f_{k}^{s}, \end{array}

where z_k = (z₁_k, z₂_k, z₃_k, z₄_k, z₅_k, z₆_k)^T represents the state vector, z₁_k and z₂_k are the angles of rotation of the motor and link, respectively; z₃_k and z₄_k are the angular velocities, z₅_k and z₆_k are their angular accelerations. u_k ∈ R and y_k ∈ R² are the control variables and the measured output vector, respectively. $f_{k}^{a}$ is the actuator fault, and $f_{k}^{s}$ is the sensor fault.

\begin{array}{l} A (z_{k}) = (\begin{matrix} 1 & 0 & T_{e} & 0 & 0 & 0 \\ 0 & 1 & 0 & T_{e} & 0 & 0 \\ 0 & 0 & 1 & 0 & T_{e} & 0 \\ 0 & 0 & 0 & 1 & 0 & T_{e} \\ \frac{- k_{1} T_{e}}{J_{m}} & \frac{k_{1} T_{e}}{J_{m}} & \frac{- β T_{e}}{J_{m}} & 0 & - T_{e} & 0 \\ \frac{k_{1} T_{e}}{J_{L}} & η & 0 & 0 & 0 & - T_{e} \end{matrix}), \\ B = (\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ \frac{k_{2} T_{e}}{J_{m}} \\ 0 \end{matrix}), F_{a} = B, F_{s} = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}), \\ C = (\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}), \\ M = (\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}), \end{array}

where T_e > 0 is the sampling time and

(46)

η = \frac{- k_{1} T_{e}}{J_{L}} - \frac{m g b T_{e}}{J_{L}} \frac{s i n (z_{2 k})}{z_{2 k}} .

(47)

η = α_{1} η_{m a x} + α_{2} η_{m i n},

where

(48)

{\begin{matrix} α_{1} = \frac{η - η_{m i n}}{η_{m a x} - η_{m i n}}, \\ α_{2} = \frac{η_{m a x} - η}{η_{m a x} - η_{m i n}} . \end{matrix}

Therefore, the obtained global TS fuzzy model is expressed as follows:

(49)

{\begin{array}{l} M z_{k + 1} = \sum_{i = 1}^{2} φ_{i} (z_{k}) (A_{i} z_{k} + B u_{k} + F_{a i} f_{k}^{a}), \\ y_{k} = C z_{k} + F_{s} f_{k}^{s}, \end{array}

with

\begin{array}{l} A_{1} = (\begin{matrix} 1 & 0 & T_{e} & 0 & 0 & 0 \\ 0 & 1 & 0 & T_{e} & 0 & 0 \\ 0 & 0 & 1 & 0 & T_{e} & 0 \\ 0 & 0 & 0 & 1 & 0 & T_{e} \\ \frac{- k_{1} T_{e}}{J_{m}} & \frac{k_{1} T_{e}}{J_{m}} & \frac{- β T_{e}}{J_{m}} & 0 & - T_{e} & 0 \\ \frac{k_{1} T_{e}}{J_{L}} & η_{m a x} & 0 & 0 & 0 & - T_{e} \end{matrix}), \\ A_{2} = (\begin{matrix} 1 & 0 & T_{e} & 0 & 0 & 0 \\ 0 & 1 & 0 & T_{e} & 0 & 0 \\ 0 & 0 & 1 & 0 & T_{e} & 0 \\ 0 & 0 & 0 & 1 & 0 & T_{e} \\ \frac{- k_{1} T_{e}}{J_{m}} & \frac{k_{1} T_{e}}{J_{m}} & \frac{- β T_{e}}{J_{m}} & 0 & - T_{e} & 0 \\ \frac{k_{1} T_{e}}{J_{L}} & η_{m i n} & 0 & 0 & 0 & - T_{e} \end{matrix}) . \end{array}

The membership functions are as follows:

(50)

{\begin{matrix} φ_{1} (z_{k}) = α_{1}, \\ φ_{2} (z_{k}) = α_{2} . \end{matrix}

In this case,

z_{k}^{1} = {(z_{1 k}, z_{2 k}, z_{3 k}, z_{4 k})}^{T}; z_{k}^{2} = {(z_{5 k}, z_{6 k})}^{T},

and $A_{1}^{22} = A_{2}^{22} = (\begin{matrix} - T_{e} & 0 \\ 0 & - T_{e} \end{matrix})$ are invertible.

We rewrite the model (49) in the equivalent form (20) to apply the fuzzy observer proposed in (24) for the single-link flexible joint robot.

The values and definitions of the physical parameters are given in [12] and we assume T_e = 0.01s.

The expressions for the actuator and sensor faults are shown in Figure 4. Considering Theorem 1 with ρ = 0.98, the observer gain obtained is as follows:

H = (\begin{matrix} 0.6618 & - 0.0205 & 0.0066 \\ 0.1110 & 0.2610 & - 0.6251 \\ 2.8735 & 0.4394 & - 0.2879 \\ 0.0943 & - 0.1249 & 0.9417 \\ 1.8899 & 0.3531 & 0.6777 \\ - 3.1504 & 0.4885 & 0.0763 \end{matrix}) .

Simulation results with initial conditions

\begin{array}{l} ξ_{k}^{1} = {[0.00 0.3142 0.00 0.00 0.02 0.00]}^{T}, \\ ξ_{k}^{2} = {[15.7159 - 16.6906]}^{T}, \\ {\hat{ξ}}_{k}^{1} = {[0.00 0.3342 0.00 0.00 0.00 0.00]}^{T}, \\ {\hat{ξ}}_{k}^{2} = {[16.2564 - 17.7286]}^{T} \end{array}

are shown in Figures 1–4 where the input is u_k = sin(k). The dashed lines denote the state variables and the actuator and sensor faults estimated by the proposed fuzzy observer (24) with gain H.

Noise can lead to compromised performance or even instability. A second simulation was performed with centered measurement noise to verify the effectiveness of this approach.

The simulation results in Figures 5–8 demonstrate the effectiveness of the proposed fuzzy observer, even with noise.

5. Conclusion

Abstract
1. Introduction
2. System Description
3. Main Results
4. Application to a Single-Link Flexible Joint Robot
5. Conclusion

Fig 1.

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