Márcia L C Peixoto
email: marcialcpeixoto@gmail.com

Pedro H S Coutinho
email: coutinho.p92@gmail.com

Anh-Tu Nguyen
email: tnguyen@uphf.fr

Thierry-Marie Guerra
email: guerra@uphf.fr

Reinaldo M Palhares
email: rpalhares@ufmg.br

Fault Estimation for Nonlinear Parameter-Varying Time-Delayed Systems

Keywords: Fault estimation, nonlinear parameter-varying systems, time-varying delays, Lyapunov-Krasovskii functional, linear matrix inequalities

A fault estimation method for a class of nonlinear parameter-varying systems subject to time-varying delay and unmeasured nonlinearities is presented. The unmeasured time-varying parameters are effectively handled using a sector-based condition approach. A gain-scheduling intermediate estimator is proposed to simultaneously estimate the system state and the unknown faults. Design conditions are derived based on Lyapunov-Krasovskii functional and integral inequality techniques. These conditions, expressed as linear matrix inequalities, ensure that the estimation error dynamics are input-to-state stable with respect to the time-derivative of the faults. Moreover, it is demonstrated that for the case of piecewise constant faults, the estimation error dynamics are exponentially stable. As a corollary result, conditions are also presented to design gain-scheduling intermediate estimators for nonlinear parameter-varying systems without time-varying delays. Three physically motivated examples are provided to demonstrate the effectiveness and practical interests of the proposed nonlinear estimation method.

Introduction

Modern control systems require more sophisticated design requirements concerning safety, reliability, and maintainability. Possible occurrences of sensor and actuator faults can lead to closed-loop performance degradation or even instability [START_REF] Du | Actuator fault estimation and accommodation for switched systems with time delay: Discrete-time case[END_REF][START_REF] Lan | A new strategy for integration of fault estimation within fault-tolerant control[END_REF]. To this end, fault-tolerant control (FTC) techniques have been proposed to ensure desirable closed-loop requirements despite the presence of faults [START_REF] Bessa | TS fuzzy reconfiguration blocks for fault tolerant control of nonlinear systems[END_REF][START_REF] Bessa | Passivation blocks for fault tolerant control of nonlinear systems[END_REF][START_REF] Zhu | Distributed finite-time fault estimation and fault-tolerant control for cyber-physical systems with matched uncertainties[END_REF]. However, a key point for the correct operation of several FTC strategies is a well-designed fault detection and isolation (FDI) scheme, which is able to provide precise and correct information about the faults. Although FDI schemes are designed to provide residual signals to indicate the fault occurrence and the information of its type and location, the exact information about the magnitude and the shape of the fault cannot be obtained, which has motivated the development of fault estimation techniques to provide more precise information about the fault [START_REF] Zhu | Fault Estimation for a Class of Nonlinear Systems Based on Intermediate Estimator[END_REF].

Within this context, observer-based fault estimation techniques have been widely studied, including sliding mode observers [START_REF] Edwards | Sliding mode observers for fault detection and isolation[END_REF][START_REF] Liu | Fault estimation sliding-mode observer with digital communication constraints[END_REF], adaptive observers [START_REF] Jiang | Fault accommodation for nonlinear dynamic systems[END_REF][START_REF] Yan | Adaptive fault estimation for cyber-physical systems with intermittent dos attacks[END_REF], and unknown input observers [START_REF] García | Robust fault estimation based on interval Takagi-Sugeno unknown input observer[END_REF][START_REF] Du | Robust fault estimation observer design for switched systems with unknown input[END_REF]. However, these techniques generally require the so-called matching condition to be satisfied, which may be restrictive, especially for nonlinear systems. To overcome this issue, Zhu et al. [START_REF] Zhu | Fault Estimation for a Class of Nonlinear Systems Based on Intermediate Estimator[END_REF] have proposed the use of intermediate estimators to estimate both the states and the faults of nonlinear systems with Lipschitz nonlinearities. Other results concerning intermediate estimators have been derived for nonlinear systems with Lipschitz nonlinearities subject to stochastic faults [START_REF] Wang | Fault estimation for nonlinear systems by an intermediate estimator with stochastic failure[END_REF], linear systems with multiple faults and disturbances [START_REF] Zhu | Fault-tolerant control for linear systems with multiple faults and disturbances based on augmented intermediate estimator[END_REF], cooperative fault-tolerant tracking control of multi-agent systems [START_REF] Zhu | Cooperative fault tolerant tracking control for multiagent systems: An intermediate estimatorbased approach[END_REF], and attack reconstruction for cyber-physical systems [START_REF] Zhu | Performance guaranteed attack reconstruction for cyber-physical systems: A new intermediate estimator-based framework[END_REF]. However, the issue of intermediate estimatorbased fault estimation has not been addressed for time-delayed (nonlinear) systems in the open literature. Fault estimation results have been developed for time-delayed systems. Sliding mode observers have been proposed for state and parameter estimation of nonlinear time-delayed systems in [START_REF] Yan | State and parameter estimation for nonlinear delay systems using sliding mode techniques[END_REF], and for the estimation of sensor and actuator faults of time-delay Markov jump systems in [START_REF] Yang | Actuator and sensor fault estimation for time-delay markov jump systems with application to wheeled mobile manipulators[END_REF]. Moreover, the fault diagnosis problem has been addressed for descriptor linear parameter varying (D-LPV) systems with time delay based on an adaptive polytopic observer [START_REF] Hamdi | Observerbased fault diagnosis for time-delay LPV descriptor systems[END_REF]. However, for observer design, these results require well-known matching conditions. Without requiring specific matching conditions, a proportional multiple integral observer has been proposed to es-timate both sensor and actuator faults of time-delayed Takagi-Sugeno (TS) fuzzy systems in [START_REF] You | Robust fault estimation for Takagi-Sugeno fuzzy systems with state time-varying delay[END_REF]. Using the concept of intermediate estimators, the fault accommodation problem has been addressed by [START_REF] Zhu | Fault accommodation for linear systems with time-varying delay[END_REF] for linear systems with time-varying delays, and by [START_REF] Wang | Fault estimation for continuous-time non-linear switched systems with time-varying delay based on intermediate estimator[END_REF] for switched systems with time-varying delays. More recently, the actuator and sensor fault estimation problem has been also addressed for linear neutral-type systems in [START_REF] Wei | Simultaneous actuator and sensor fault estimation for neutral-type systems via intermediate observer[END_REF]. Most of the existing fault estimation results are related to linear systems and only a few results are available for nonlinear time-delay systems, e.g., D-LPV systems or TS fuzzy systems. Accordingly, the first motivation of this work is to address the fault estimation problem for a class of nonlinear time-delayed systems using observers without requiring matching conditions. This problem is addressed here regarding a polytopic embedding of nonlinear systems. It is known that nonlinear dynamical systems can be equivalently represented by polytopic quasi-LPV systems or TS fuzzy models [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: a Linear Matrix Inequality Approach[END_REF]. However, although these representations are useful to derive constructive and numerically implementable conditions for designing observers [START_REF] Coutinho | A sufficient condition to design unknown input observers for nonlinear systems with arbitrary relative degree[END_REF], there is an important issue that should be accounted into the observer design, i.e., the necessity to deal with unmeasured scheduling functions or premise variables in TS fuzzy systems. The results available for fault estimation and diagnosis for LPV systems [START_REF] Hamdi | Observerbased fault diagnosis for time-delay LPV descriptor systems[END_REF] and TS fuzzy models [START_REF] You | Robust fault estimation for Takagi-Sugeno fuzzy systems with state time-varying delay[END_REF] assume that the scheduling functions are measured or dependent on the output variables, which makes the design easier but limited to specific classes of nonlinear dynamical systems. To avoid this restriction, the second motivation of this work is to provide constructive observer design conditions considering a polytopic representation of nonlinear time-delay systems with unmeasured nonlinearities.

For properly handling the unmeasured premise variables, the nonlinear system is rewritten as a TS fuzzy model with nonlinear consequent parts (N-TS) [START_REF] Coutinho | A multiple-parameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF][START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF][START_REF] Nguyen | Avoiding Unmeasured Premise Variables in Designing Unknown Input Observers for Takagi-Sugeno Fuzzy Systems[END_REF]. Overall, the main advantage of the N-TS fuzzy models over the standard TS fuzzy models is the possibility to obtain a model with a fewer number of fuzzy rules, which reduces the representation's complexity [START_REF] Coutinho | A multiple-parameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF]. In the specific context of observer design, the N-TS fuzzy model provides a suitable alternative for handling the unmeasured premise variables by subsuming the unmeasured terms in a local nonlinearity. Although this representation has shown to be effective for designing observers [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF][START_REF] Nguyen | Avoiding Unmeasured Premise Variables in Designing Unknown Input Observers for Takagi-Sugeno Fuzzy Systems[END_REF] and unknown input observers [START_REF] Peixoto | Unknown input observers for time-varying delay Takagi-Sugeno fuzzy systems with unmeasured nonlinear consequents[END_REF] for N-TS fuzzy models, there are no results available for timedelay N-TS fuzzy models and N-LPV systems concerning fault estimation problems, which is the scope of this work.

This work addresses the fault estimation problem for a class of nonlinear time-delay systems considering intermediate observers. The main contributions can be summarized as follows.

• A new class of gain-scheduling intermediate observers is proposed to simultaneously estimate the state and fault, without requiring matching conditions. Notice that results on intermediate observers are only available for linear time-delay systems [START_REF] Zhu | Fault accommodation for linear systems with time-varying delay[END_REF][START_REF] Wei | Simultaneous actuator and sensor fault estimation for neutral-type systems via intermediate observer[END_REF].

• Time-delay nonlinear systems are represented by a specific nonlinear parameter-varying (N-LPV) form, which allows circumventing the assumption of measured scheduling functions or premise variables in [START_REF] Hamdi | Observerbased fault diagnosis for time-delay LPV descriptor systems[END_REF][START_REF] You | Robust fault estimation for Takagi-Sugeno fuzzy systems with state time-varying delay[END_REF].

• Constructive and numerically implementable conditions are derived in the form of linear matrix inequalities (LMIs) for gain-scheduling intermediate observer design such that the error dynamics is input-to-state stable (ISS) with respect to the fault time-derivative. A corollary result for N-LPV systems without time-varying delays is also presented.

This paper is organized as follows. The class of nonlinear time-delay parameter varying systems and the problem definition are described in Section 2. Section 3 presents the main results on gain-scheduling intermediate observer design. In Section 4, three numerical examples with physically motivated systems are presented to illustrate the interests of the proposed approach. Finally, Section 5 concludes the paper.

Notation. N is the set of natural numbers and N ≤m is the set of natural numbers less than or equal to m; R n denotes the n-dimensional Euclidean space, and R m×n is the set of all m×n real matrices. S n + is the set of symmetric and positive-definite matrices of order n. The notation X ≻ 0 (X ≺ 0) means that X is a positive (negative) definite matrix, and He{X} = X + X ⊤ . The identity matrix of order n is denoted by I n and the null matrix of order n × m by 0 n×m . If the dimensions of both identity and null matrices are straightforwardly deduced, they are omitted. Let x be a vector, its ith entry is denoted by x i . L ∞ is the space of all essentially bounded functions whose norm is defined as ∥f ∥ ∞ = ess sup x∈[a,b] |f (x)|; the symbol • in a matrix means that position can assume any value. diag(A, B) denotes a block diagonal matrix whose elements are A and B. The time dependency of the variables is dropped when convenient.

Problem Formulation 110

This section first describes the class of nonlinear time-delay systems to be considered. Then, a nonlinear intermediate observer is constructed for its simultaneous estimation of system states and faults.

System Description

Consider a nonlinear parameter-varying system with time-varying delays

ẋ(t) = F (α(t), x(t)) + A d (α(t))x(t -τ (t)) + h(y(t), u(t)) + Ef (t) y(t) = Cx(t) x(t) = φ(t), t ∈ [-τ , 0], (1)
where

x ∈ R nx is the state, u ∈ R nu is the input, y ∈ R ny is the output, f ∈ R n f is the fault signal, α = [α 1 , α 2 , . . . , α N ] ⊤ ∈ R N is
A(α) A d (α) G(α) = N i=1 α i A i A di G i ,
where N is the number of vertices of the polytopic domain, and the unit simplex Λ is given by

Λ = α ∈ R N : N i=1 α i = 1, α i ≥ 0, i ∈ N ≤N . (2)
The following assumptions are considered for system [START_REF] Du | Actuator fault estimation and accommodation for switched systems with time delay: Discrete-time case[END_REF].

115 Assumption 1. The time-varying delay τ (t) is known and satisfies τ ≤ τ (t) ≤ τ , ∀t ∈ R, where the lower bound τ and the upper bound τ are given.

Assumption 2. The fault signal f (t) satisfies ḟ (t) ∈ L ∞ , for ∀t ∈ R, i.e., ∥ ḟ (t)∥ ∞ ≤ δ, for some positive scalar δ.

Remark 1. The boundedness condition for time-varying delays in Assumption 1 is commonly considered in the literature on time-delayed systems [START_REF] Fridman | Introduction to Time-Delay Systems, Systems & Control: Foundations & Applications[END_REF], as delays in engineering applications always remain within their lower and upper bounds. Concerning Assumption 2, descriptor sliding mode observers such as those described in [START_REF] Liu | Fault estimation sliding-mode observer with digital communication constraints[END_REF][START_REF] Lee | Robust H ∞ sliding mode descriptor observer for fault and output disturbance estimation of uncertain systems[END_REF] typically require prior knowledge of fault bounds and their derivatives. The adaptive observer presented in [START_REF] Jiang | Fault accommodation for nonlinear dynamic systems[END_REF] also requires that both the faults and their first derivatives remain bounded. However, the parameter δ in Assumption 2 can be unknown. This flexibility enables us to handle a broad range of fault signals, including unbounded faults such as ramp-type faults. Consequently, Assumption 2 is less restrictive when compared to the fault constraints imposed in other related works [START_REF] Liu | Fault estimation sliding-mode observer with digital communication constraints[END_REF][START_REF] Jiang | Fault accommodation for nonlinear dynamic systems[END_REF][START_REF] Lee | Robust H ∞ sliding mode descriptor observer for fault and output disturbance estimation of uncertain systems[END_REF].

Assumption 3. There exist positive constants k ij , for i ∈ N ≤n ϕ , j ∈ N ≤nx , such that 0 ≤ ∂ϕ i (x) ∂x j ≤ k ij , i ∈ N ≤n ϕ , j ∈ N ≤nx , ∀x ∈ D, (3)
where D ⊆ R nx is a compact set containing the origin x = 0.

Remark 2. Condition (3) is not restrictive for practical uses. Indeed, if we consider system (1) with F (α, x) = A(α)x + G(α) φ(x), where φ(x) is any differentiable function with k ij ≤ ∂ φi (x)/∂x j ≤ k ij , for ∀x ∈ D, and some scalars k ij and k ij . Applying a simple loop transformation ϕ(x) = φ(x) -Kx, where K = [k ij ] is the element-wise lower bound of the Jacobian matrix ∂ φ(x)/∂x, we have

F (α, x) = (A(α) + G(α)K)x + G(α)ϕ(x). Then, condition (3)
is satisfied with k ij = k ij -k ij , and we can apply the same proposed observer design procedure in this case. This procedure is illustrated in the example shown in Section 4.2.

Problem Definition

Let us define the following intermediate variable:

ζ(t) = f (t) -ωE ⊤ x(t), (4)
where ω is a predefined positive scalar. From (1) and ( 4), the dynamics of the intermediate variable is defined as

ζ(t) = -ωE ⊤ Eζ(t) -ωE ⊤ A(α) + ωEE ⊤ x(t) + h(y(t), u(t)) -ωE ⊤ [A d (α)x(t -τ (t)) + G(α)ϕ(x(t))] + ḟ (t). (5)
To estimate the state x(t), the intermediate variable ζ(t), and the fault f (t), we propose the following gain-scheduled intermediate observer:

ẋ(t) = A(α)x(t) + A d (α)x(t -τ (t)) + X(α) -1 L(α)(y(t) -ŷ(t)) + h(y(t), u(t)) + E f (t) + G(α)ϕ x(t) + H(α)(y(t) -C x(t)) (6) ζ(t) = -ωE ⊤ E ζ(t) -ωE ⊤ A(α) + ωEE ⊤ x(t) + A d (α)x(t -τ (t)) + h(y(t), u(t)) + G(α)ϕ x(t) + H(α)(y(t)-C x(t)) (7) ŷ(t) = C x(t) (8) f (t) = ζ(t) + ωE ⊤ x(t) (9)
where x(t), ζ(t), ŷ(t) and f (t) are the estimates of x(t), ζ(t), y(t) and f (t), respectively. The parameter-dependent matrices to be designed X(α), L(α) and H(α) are of the form

X(α) L(α) H(α) = N i=1 α i X i L i H i . (10)
We define the estimation errors as

ε x (t) = x(t) -x(t), (11)
ε ζ (t) = ζ(t) -ζ(t), (12)
ε f (t) = f (t) -f (t). ( 13
)
Then, the error dynamics can be represented by

εx (t) = A(α) -X(α) -1 L(α)C ε x (t) + A d (α)ε x (t -τ (t)) + G(α)g(α, x, ε x ) + Eε f (t) εζ (t) = -ωE ⊤ Eε ζ (t) -ωE ⊤ A(α) + ωEE ⊤ ε x (t) + ḟ (t) -ωE ⊤ A d (α)ε x (t -τ (t)) -ωE ⊤ G(α)g(α, x, ε x ), (14)
where

g(α, x, ε x ) = ϕ(x) -ϕ x + (I + H(α)C)ε x . ( 15
) Let ε(t) = [ε ⊤ x (t) ε ⊤ ζ (t)
] ⊤ be the augmented error vector. Using the definition of the intermediate variable in (4), the error system ( 14) can be reformulated as follows

ε(t) = A (α)ε(t) + A d (α)ε(t -τ (t)) + G (α)g(α, x, ε x ) + B ḟ (t), (16)
where

A (α) = A(α) -X(α) -1 L(α)C+ωEE ⊤ E -ωE ⊤ A(α) + ωEE ⊤ -ωE ⊤ E A d (α) = A d (α) 0 -ωE ⊤ A d (α) 0 , G (α) = G(α) -ωE ⊤ G(α) , B = 0 I .
Remark 3. The proposed gain-scheduled intermediate observer introduced in (6)-( 9) allows to deal with the simultaneous state and fault estimation without resorting to matching conditions, different from the sliding mode observers [START_REF] Edwards | Sliding mode observers for fault detection and isolation[END_REF][START_REF] Liu | Fault estimation sliding-mode observer with digital communication constraints[END_REF][START_REF] Lee | Robust H ∞ sliding mode descriptor observer for fault and output disturbance estimation of uncertain systems[END_REF], adaptive observers [START_REF] Jiang | Fault accommodation for nonlinear dynamic systems[END_REF][START_REF] Yan | Adaptive fault estimation for cyber-physical systems with intermittent dos attacks[END_REF], and unknown input observers [START_REF] García | Robust fault estimation based on interval Takagi-Sugeno unknown input observer[END_REF][START_REF] Du | Robust fault estimation observer design for switched systems with unknown input[END_REF]. Besides that, the proposed intermediate observer does not require any strict rank assumption over the system (1), constituting an effective solution to deal with the estimation problem for nonlinear time-delay systems.

The estimation problem for the N-LPV system (1) is stated as follows.

Problem 1. Consider the nonlinear parameter-varying time-delay system (1)

under Assumptions 1-3. Design the parameter-dependent matrices X(α), L(α), and H(α) of the nonlinear intermediate observer ( 6)-( 9) such that the error dynamics ( 16) is input-to-state stable (ISS) with respect to ḟ (t).

Further details regarding ISS stability can be found in [32, Chapter 4.9].

Nonlinear Intermediate Observer Design for Fault Estimation

This section presents the design of intermediate observers for N-LPV systems subject to faults. For observer design, the following lemma is useful to deal with the nonlinear function ϕ(x) by exploiting the sector condition and the boundedness of its Jacobian matrix as stated in Assumption 3.

Lemma 1 (Adapted from [START_REF] Wang | Observer design for parameter varying differentiable nonlinear systems, with application to slip angle estimation[END_REF]).

Let K = [K ⊤ 1 K ⊤ 2 • • • K ⊤ n ϕ ] ⊤ , with K i = [k i1 k i2 • • • k in ], i ∈ N ≤n ϕ , with k ij defined in Assumption 3. If the function ϕ(x) : R nx → R n ϕ
satisfies Assumption 3, then the following sector condition holds for ε x in (11) and g(α, x, ε x ) in (15):

g(α, x, ε x ) ⊤ [g(α, x, ε x ) -K(I + H(α)C)ε x ] ≤ 0. ( 17
)
Note that condition (17) can be represented in the following quadratic form

ε x g(α, x, ε x ) ⊤ H(α) ⊤ 0 0 I 0 -1 2 K ⊤ -1 2 K I H(α) 0 0 I ε x g(α, x, ε x ) ≤ 0
where

H(α) = I -H(α)C.
Remark 4. Regarding Assumption 3, since the function ϕ(x) possesses a bounded gradient with respect to x, the matrix norm of K, defined in Lemma 1, can be related to the Lipschitz constant. However, employing the Lipschitz property often results in overestimated bounds, introducing conservatism into the design procedure. In this paper, we have utilized the sector-bounded prop-165 erty, as stated in Lemma 1, to derive less conservative design conditions.

Time-Delay Nonlinear Parameter-Varying Systems

The following theorem provides LMI-based conditions to design an intermediate observer for system (1).

Theorem 1. Given positive scalars ϵ, β and ω, if there exist matrices

P ∈ S 3nx + , Q 1 , Q 2 , R 1 , R 2 ∈ S nx + , Γ ∈ S n f + , matrices Y 1 , Y 2 ∈ R 2nx×2nx , X i ∈ R nx×nx , H i ∈ R nx×ny , L i ∈ R nx×ny , F 1i , F 2i ∈ R n f ×n f ,
and a positive scalar γ, such that following inequalities hold:

Θ ii (τ ) ≺ 0, i = j Θ ij (τ ) + Θ ji (τ ) ≺ 0, i < j (18)
for all i, j ∈ N ≤N and τ ∈ {τ , τ }, with

Θ ij (τ ) = Ψ(τ ) -H j + He{Υ ij + Π ij } -e -2β τ G ⊤ Φ(τ )G ⋆ e -β τ Y ⊤ 1 0 G -R 2 , (19)
Θ ij (τ ) = Ψ(τ ) -H j + He{Υ ij + Π ij } -e -2β τ G ⊤ Φ(τ )G ⋆ 0 e -β τ Y ⊤ 2 G -R 2 . ( 20
)
The matrices Ψ(τ ) and Φ(τ ), with τ ∈ {τ , τ }, are defined as

Ψ(τ ) = G ⊤ 1 (τ )P G 0 + G ⊤ 0 P G 1 (τ ) + 2βG ⊤ 1 (τ )P G 1 (τ ) -e -2βτ G ⊤ 2 R 1 G 2 + G ⊤ 0 (τ 2 R 1 + τ 2 e -2β τ R 2 )G 0 + G ⊤ 5 ΓG 5 + v⊤ 2 2βΓv 2 + Q + F , Φ(τ ) = R 2 0 0 R 2 + τ -τ (t) τ R 2 Y 2 Y ⊤ 2 0 + τ (t)-τ τ 0 Y 1 Y ⊤ 1 R 2 , (21)
where

τ = τ -τ , R 1 = diag(R 1 , 3R 1 ), R 2 = diag(R 2 , 3R 2 ), R 1 = diag(R 1 , 0 2nx ), R 2 = diag(R 2 , 0 2nx ), Q = diag 0 nx , Q 1 , e -2βτ (Q 2 -Q 1 ), 0 nx , -e -2βτ Q 2 , 0 3nx+3n f +n ϕ , F = diag(0 8nx , 0 2n f , -γI n f , 0 n ϕ ), Υ ij = (ϵv 1 + v 2 + v 4 ) ⊤ (A ij v 2 + X j A di v 4 + X j Ev 2 + X j G i ṽ -X j v 1 ) , A ij = X j A i -L j C + ωX j EE ⊤ , Π ij = (F 1j v1 + F 2j v2 ) ⊤ (E i -v1 + v3 ) , E i = -ωE ⊤ Ev 2 + A i +ωEE ⊤ v 2 +A di v 4 +G i ṽ , G 1 (τ ) =   v 2 τ v 6 (τ (t) -τ )v 7 + (τ -τ (t))v 8   , (22)
G 0 =   v 1 v 2 -v 3 v 3 -v 5   , G 2 = v 2 -v 3 v 2 + v 3 -2v 6 , G 3 = v 3 -v 4 v 3 + v 4 -2v 7 , G 4 = v 4 -v 5 v 4 + v 5 -2v 8 , G 5 = v1 v2 , Γ = 0 Γ Γ 0 , G = G 3 G 4 , v κ = 0 nx×(κ-1)nx I nx 0 nx×(8-κ)nx 0 nx×3n f 0 nx×n ϕ , vl = 0 n f ×8nx 0 n f ×(1-l)n f I n f 0 n f ×(3-l)n f 0 n f ×n ϕ , ṽ = 0 n ϕ ×(8nx+3n f ) I n ϕ , H j = v 2 ṽ ⊤ I -H j C 0 0 I ⊤ 0 -K ⊤ 2 -K 2 I I -H j C 0 0 I v 2 ṽ ,
with κ ∈ N ≤8 and l ∈ N ≤3 . Then, the intermediate observer ( 6)-( 9), with the parameter-dependent gains given as in [START_REF] Yan | Adaptive fault estimation for cyber-physical systems with intermittent dos attacks[END_REF], ensures that the error system ( 16) is ISS with respect to ḟ (t). Moreover, the ellipsoid

R ∞ = ε x ∈ R nx , ε ζ ∈ R n f : ε ⊤ x P ε x + ε ⊤ ζ Γε ζ ≤ γδ 2 2β , (23)
is attractive with respect to system [START_REF] Zhu | Performance guaranteed attack reconstruction for cyber-physical systems: A new intermediate estimator-based framework[END_REF], where P = U P U ⊤ and U = [I 0 0].

Proof. For stability analysis, consider the proposed Lyapunov-Krasovskii functional (LKF) candidate given by

V (ε xt , εxt , ε ζt , t) = V 1 (ε xt , t) + V 2 (ε xt , t) + V 3 ( εxt , t) + V 4 (ε ζ , t) (24)
with

V 1 (ε xt , t) =    ε x (t) t t-τ ε x (s)ds t-τ t-τ ε x (s)ds    ⊤ P    ε x (t) t t-τ ε x (s)ds t-τ t-τ ε x (s)ds    V 2 (ε xt , t) = t t-τ e -2β(t-s) ε ⊤ x (s)Q 1 ε x (s)ds + t-τ t-τ e -2β(t-s) ε ⊤ x (s)Q 2 ε x (s)ds V 3 ( εxt , t) = τ 0 -τ t t+θ e -2β(t-s) ε⊤ x (s)R 1 εx (s)dsdθ + τ -τ -τ t t+θ e -2β(t-s) ε⊤ x (s)R 2 εx (s)dsdθ V 4 (ε ζ , t) = ε ⊤ ζ (t)Γε ζ (t)
and τ = τ -τ . Since the matrices P , Q 1 , Q 2 , R 1 and R 2 are positive definite, the LKF candidate defined in ( 24) is also positive definite. Consider the augmented vector

ξ(t)= ξ(t) ν ⊤ 1 (t) ν ⊤ 2 (t) ν ⊤ 3 (t) ε⊤ ζ (t) ε ⊤ ζ (t) ḟ ⊤ (t) g ⊤ (α, x, ε x ) ⊤ (25)
with

ξ(t) = ε⊤ x (t) ε ⊤ x (t) ε ⊤ x (t -τ ) ε ⊤ x (t -τ (t)) ε ⊤ x (t -τ ) ν 1 (t) = 1 τ t t-τ ε x (s)ds ν 2 (t) = 1 τ (t) -τ t-τ t-τ (t) ε x (s)ds ν 3 (t) = 1 τ -τ (t) t-τ (t) t-τ ε x (s)ds. ( 26
)
The time-derivative of V 1 (ε xt , t) along the solution of system ( 14) is given by

V1 (ε xt , t) = ξ ⊤ (t) G ⊤ 1 (τ )P G 0 + G ⊤ 0 P G 1 (τ ) ξ(t). ( 27
)
The expression of V2 (ε xt , t) can be computed as

V2 (ε xt , t) = ε ⊤ x (t)Q 1 ε x (t) + ε ⊤ x (t -τ ) e -2βτ (Q 2 -Q 1 ) ε x (t -τ ) -ε ⊤ x (t -τ ) e -2βτ Q 2 ε x (t -τ ) -2β t t-τ e -2β(t-s) ε ⊤ x (s)Q 1 ε x (s)ds -2β t-τ t-τ e -2β(t-s) ε ⊤ x (s)Q 2 ε x (s)ds = ξ ⊤ (t) Qξ(t) -2βV 2 (ε xt , t). ( 28
)
The computation of V3 ( εxt , t) results in

V3 ( εxt , t) = ε⊤ x (t)(τ 2 R 1 + τ 2 e -2β τ R 2 ) εx (t) -τ t t-τ e -2β(t-s) ε⊤ x (s)R 1 εx (s)ds -τ t-τ t-τ e -2β(t-s) ε⊤ x (s)R 2 εx (s)ds -2βτ 0 -τ t t+θ e -2β(t-s) ε⊤ x (s)R 1 εx (s)dsdθ -2β τ -τ -τ t t+θ e -2β(t-s) ε⊤ x (s)R 2 εx (s)dsdθ. ( 29
)
From the definition of the matrix G 0 , expression (29) can be rewritten as

V3 ( εxt , t) = ξ ⊤ (t)G ⊤ 0 (τ 2 R 1 + τ 2 e -2β τ R 2 )G 0 ξ(t) -τ t t-τ e -2β(t-s) ε⊤ x (s)R 1 εx (s)ds -τ t-τ t-τ e -2β(t-s) ε⊤ x (s)R 2 εx (s)ds -2βτ 0 -τ t t+θ e -2β(t-s) ε⊤ x (s)R 1 εx (s)dsdθ -2β τ -τ -τ t t+θ e -2β(t-s) ε⊤ x (s)R 2 εx (s)dsdθ. ( 30
)
According to the Wirtinger-based integral inequality (see Lemma 2 in Appendix A), it follows from (30) that

V3 ( εxt , t) ≤ ξ ⊤ (t)G ⊤ 0 (τ 2 R 1 + τ 2 e -2β τ R 2 )G 0 ξ(t) -e -2βτ ξ ⊤ (t)G ⊤ 2 R 1 G 2 ξ(t) -e -2β τ ξ ⊤ (t) G 3 G 4 ⊤ τ τ (t)-τ R 2 0 0 τ τ -τ (t) R 2 G 3 G 4 ξ(t) -2βτ 0 -τ t t+θ e -2β(t-s) ε⊤ x (s)R 1 εx (s)dsdθ -2β τ -τ -τ t t+θ e -2β(t-s) ε⊤ x (s)R 2 εx (s)dsdθ. ( 31
)
Moreover, we define ϱ = τ (t)-τ τ . Then, by employing the delay-dependent reciprocally convex lemma (see Lemma 3 in Appendix A) with

X 1 = R 2 - Y 1 R -1 2 Y ⊤ 1 and X 2 = R 2 -Y ⊤ 2 R -1 2 Y 2 , one has that V3 ( εxt , t) ≤ ξ ⊤ (t)G ⊤ 0 (τ 2 R 1 + τ 2 e -2β τ R 2 )G 0 ξ(t) -e -2βτ ξ ⊤ (t)G ⊤ 2 R 1 G 2 ξ(t) -2βτ 0 -τ t t+θ e -2β(t-s) ε⊤ x (s)R 1 εx (s)dsdθ -2β τ -τ -τ t t+θ e -2β(t-s) ε⊤ x (s)R 2 εx (s)dsdθ -e -2β τ ξ ⊤ (t)G ⊤ (Φ(τ ) -Φ 0 (τ ))G ξ(t), (32)
where

Φ 0 (τ ) = τ -τ (t) τ Y 1 R -1 2 Y ⊤ 1 0 0 τ (t)-τ τ Y ⊤ 2 R -1 2 Y 2 .
Finally, the computation of V4 (ε ζt , t) results in

V 4 (ε ζt , t) = ξ ⊤ (t)G ⊤ 5 ΓG 5 ξ(t). ( 33
)
Substituting the expressions of V1 (ε xt , t) in [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF], V2 (ε xt , t) in (28), V4 (ε ζt , t) in [START_REF] Wang | Observer design for parameter varying differentiable nonlinear systems, with application to slip angle estimation[END_REF] and the upper bound of V3 ( εxt , t) in [START_REF] Khalil | Nonlinear Systems[END_REF], into the expression of V (ε xt , εxt , ε ζt , t), it follows that

V (ε xt , εxt , ε ζt , t) + 2βV (ε xt , εxt , ε ζt , t) ≤ ξ ⊤ (t)Ξ(τ )ξ(t), (34)
with

Ξ(τ ) = G ⊤ 1 (τ )P G 0 + G ⊤ 0 P G 1 (τ ) + 2βG ⊤ 1 (τ )P G 1 (τ ) + G ⊤ 0 (τ 2 R 1 + τ 2 e -2β τ R 2 )G 0 -e -2βτ G ⊤ 2 R 1 G 2 + G ⊤ 5 ΓG 5 + v⊤ 2 2βΓv 2 + Q -e -2β τ G ⊤ (Φ(τ ) -Φ 0 (τ ))G .
Moreover, it follows from the sector condition ( 17) that

ξ ⊤ (t)H (α)ξ(t) ≤ 0, (35)
with

H (α) = v 2 ṽ ⊤ H(α) 0 0 I ⊤ 0 -1 2 K ⊤ -1 2 K I H(α) 0 0 I v 2 ṽ .
Therefore, we have from ( 34) and ( 35) that

V (ε xt , εxt , ε ζt , t) + 2βV (ε xt , εxt , ε ζt , t) ≤ ξ ⊤ (t) (Ξ(τ ) -H (α)) ξ(t). (36)
Note that the following zero-equalities hold for any matrices

X(α) ∈ R nx×nx , F 1 (α) ∈ R n f ×n f and F 2 (α) ∈ R n f ×n f : Υ(α)ξ(t) = 0 and Π(α)ξ(t) = 0 (37) with Υ(α) = X ⊤ (α)Υ 1 (α), Π(α) = [F 1 (α)v 1 + F 2 (α)v 2 ] ⊤ [E (α) -v1 + v3 ] , X (α) = ϵX ⊤ (α)v 1 + X ⊤ (α)v 2 + X ⊤ (α)v 4 , Υ 1 (α) = A(α)v 2 + A d (α)v 4 + Ev 2 + G(α)ṽ -v 1 , A(α) = A(α) -X(α) -1 L(α)C + ωEE ⊤ , E (α) = -ωE ⊤ Ev 2 + A(α) + ωEE ⊤ v 2 + A d (α)v 4 + G(h)ṽ . (38)
Hence, from ( 36) and [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], it follows that

V (t) + 2βV (t) ≤ ξ ⊤ (t) (Ξ(τ ) -H (α) + He{Υ(α) + Π(α)}) ξ(t). (39)
To ensure that system ( 16) is ISS with respect to ḟ (t), it suffices that

V (t) ≤ -2βV (t) + γ ḟ ⊤ (t) ḟ (t). ( 40
)
It follows from ( 39) and (40) that

V (t) + 2βV (t) ≤ ξ ⊤ (t) (Ξ(τ ) -H (α) + He{Υ(α) + Π(α)} + F ) ξ(t), (41)
with F = diag(0 8nx , 0 2n f , -γI n f , 0 n ϕ ). Based on the convexity with respect to τ and the Schur complement lemma, it follows that

Θ(α, τ ) = Ψ(τ ) -H (α) + He{Υ(α) + Π(α)} -e -2β τ G ⊤ Φ(τ )G ⋆ e -β τ Y ⊤ 1 0 G -R 2 ≺ 0, (42)
with τ ∈ {τ , τ } and the matrices Ψ(τ ) and Φ(τ ) are defined as in [START_REF] Zhu | Fault accommodation for linear systems with time-varying delay[END_REF]. The parameter-dependent matrices X(α), L(α), H(α), F 1 (α), and F 2 (α) are parameterized as

X(α) L(α) H(α) F 1 (α) F 2 (α) = N i=1 α i X i L i H i F 1i F 2i .
Hence, the expressions Θ(α, τ ) in ( 42) can be rewritten as

Θ(α, τ ) = N i=1 N j=1 α i α j Θ ij (τ ) = N i=1 α 2 i Θ ii (τ ) + N i=1 N j>i α i α j Θ ij (τ ). ( 43
)
It is clear from [START_REF] Seuret | Stability of Linear Systems With Time-Varying Delays Using Bessel-Legendre Inequalities[END_REF] and (43) that condition (41) holds if conditions in [START_REF] Yang | Actuator and sensor fault estimation for time-delay markov jump systems with application to wheeled mobile manipulators[END_REF] are verified. Once the condition [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] is satisfied, by the comparison lemma [START_REF] Khalil | Nonlinear Systems[END_REF], it follows that

V (t) ≤ e -2β(t-t 0 ) V (t 0 )+ γ 2β 1-e -2β(t-t 0 ) ∥ ḟ (t)∥ 2 ∞ . (44)
As t → ∞, we have

V (t) ≤ γ 2β ∥ ḟ (t)∥ 2 ∞ ≤ γδ 2 2β . Since ε ⊤ x (t) P ε x (t) + ε ⊤ ζ (t)Γε ζ (t) ≤ V (t)
, then the ellipsoid ( 23) is attractive with respect to system [START_REF] Zhu | Performance guaranteed attack reconstruction for cyber-physical systems: A new intermediate estimator-based framework[END_REF]. Please notice that the first block of Eq. [START_REF] Hamdi | Observerbased fault diagnosis for time-delay LPV descriptor systems[END_REF] and Eq. ( 20) can be expressed as

τ 2 R 1 + τ 2 e -2β τ R 2 -ϵX(α) -ϵX(α) ⊤ ≺ 0.
Since R 1 and R 2 are positive definite matrices, one can conclude that X(α)+ X(α) ⊤ ≻ 0, ensuring the existence of the inverse of matrix X(α). This 175 concludes the proof.

Remark 5. The volume of the attractive ellipsoid (23) depends on three parameters, i.e., the induced gain γ, the decay rate β, and the assumed bound δ of ḟ (t). For any fault signals, to reduce the ultimate bound of the estimation error, the volume of the attractive ellipsoid can be minimized by minimizing the induced gain γ and maximizing the decay rate β. In particular, notice that for time-invariant fault signals, i.e., ḟ (t) = 0, the attractive ellipsoid tends to the origin, thus ensuring the exponential convergence of the estimation errors to zero. Additionally, the values of β, ϵ, and ω can be determined through a linear grid line-search process. The line-search procedure systematically varies and evaluates different values of β, ϵ, and ω to find the combination that yields the best performance for the induced gain γ.

Delay-Free Nonlinear Parameter-Varying Systems

The conditions presented in Theorem 1 can be adapted to deal with delayfree nonlinear parameter-varying systems. To this end, the following error dynamics can be derived considering system (1) with A d (α) ≡ 0:

ε(t) = A (α)ε(t) + G (α)g(α, x(t), ε x (t)) + B ḟ (t), (45)
where the matrices A (α), G (α) and B are as in [START_REF] Zhu | Performance guaranteed attack reconstruction for cyber-physical systems: A new intermediate estimator-based framework[END_REF].

Corollary 1. Given positive scalars ϵ, β and ω, if there exist matrices

P ∈ S nx + , Γ ∈ S n f + , matrices X i ∈ R nx×nx , L i ∈ R nx×ny , F 1i , F 2i ∈ R n f ×n f ,
and a positive scalar γ, such that following inequalities hold

Θ ii ≺ 0, i = j Θ ij + Θ ji ≺ 0, i < j (46)
for all i, j ∈ N ≤N , with

Θ ij = Q + X j M ij + M ⊤ ij X ⊤ j -ϖ ⊤ H j ϖ Q =         0 nx×nx P 0 nx×n f 0 nx×n f 0 nx×n f 0 nx×n ϕ P βP 0 nx×n f 0 nx×n f 0 nx×n f 0 nx×n ϕ 0 n f ×nx 0 n f ×nx 0 n f ×n f Γ 0 n f ×n f 0 n f ×n ϕ 0 n f ×nx 0 n f ×nx Γ βΓ 0 n f ×n f n f ×n ϕ 0 n f ×nx 0 n f ×nx 0 n f ×n f 0 n f ×n f -γI n f 0 n f ×n ϕ 0 n ϕ ×nx 0 n ϕ ×nx 0 n ϕ ×n f 0 n ϕ ×n f 0 n ϕ ×n f 0 n ϕ ×n ϕ         X j =         ϵI nx 0 nx×n f I nx 0 nx×n f 0 n f ×nx F 1j 0 n f ×nx F 2j 0 n f ×nx 0 n f ×n f 0 n ϕ ×nx 0 n ϕ ×n f         M ij = -X j A i,j 0 nx×n f X j E 0 nx×n f X j G i 0 n f ×nx -ωE ⊤ (A i + ωEE ⊤ ) -I n f -ωE ⊤ E I n f -ωE ⊤ G i ϖ = 0 nx×nx I nx 0 nx×n f 0 nx×n f 0 nx×n f 0 nx×n ϕ 0 n ϕ ×nx 0 n ϕ ×nx 0 n ϕ ×n f 0 n ϕ ×n f 0 n ϕ ×n f I n ϕ H j = I nx -H j C 0 nx×n ϕ 0 n ϕ ×nx I n ϕ ⊤ 0 nx×nx -1 2 K ⊤ -1 2 K I n ϕ I nx -H j C 0 nx×n ϕ 0 n ϕ ×nx I n ϕ and A ij = X j A i -L j C + ωX j EE ⊤ .
Then, the intermediate observer ( 6)-( 9), with A d (α) ≡ 0, ensures the delay-free system (45) is ISS with respect to ḟ (t). Moreover, the ellipsoid

R ∞ = ε x ∈ R nx , ε ζ ∈ R n f : ε ⊤ x P ε x + ε ⊤ ζ Γε ζ ≤ γδ 2 /β , (47)
is attractive with respect to system (45).

190

Proof. For stability analysis, we consider the Lyapunov function

V (t) = ε ⊤ x (t)P ε x (t) + ε ⊤ ζ (t)Γε ζ (t). ( 48
)
To ensure that the system is ISS with respect to ḟ (t), it suffices that

V (t) ≤ -βV (t) + γ ḟ ⊤ (t) ḟ (t). ( 49
)
Note that

V (t) + βV (t) -γ ḟ ⊤ (t) ḟ (t) = ξ ⊤ (t)Qξ(t), (50)
with ξ = ε⊤ x (t) ε ⊤ x (t) ε⊤ ζ (t) ε ⊤ ζ (t) ḟ ⊤ (t) g ⊤ (α, x, ε x ) . It follows from the sector condition defined in (17) that ξ ⊤ (t)ϖ ⊤ H (α)ϖξ(t) ≤ 0, (51)
with

H (α) = I nx -H(α)C 0 nx×n ϕ 0 n ϕ ×nx I n ϕ ⊤ 0 nx×nx -1 2 K ⊤ -1 2 K I n ϕ I nx -H(α)C 0 nx×n ϕ 0 n ϕ ×nx I n ϕ .
Therefore, we have from ( 50) and (51) that

V (t) + βV (t) -γ ḟ ⊤ (t) ḟ (t) ≤ ξ ⊤ (t)Qξ(t) -ξ ⊤ (t)ϖ ⊤ H (α)ϖξ(t). (52)
Note also that the following zero-equality holds for any matrices

X(α) ∈ R nx×nx , F 1 (α) ∈ R n f ×n f and F 2 (α) ∈ R n f ×n f : X (α)M (α)ξ(t) = 0, (53)
with

X (α) =         ϵI nx 0 nx×n f I nx 0 nx×n f 0 n f ×nx F 1 (α) 0 n f ×nx F 2 (α) 0 n f ×nx 0 n f ×n f 0 n ϕ ×nx 0 n ϕ ×n f         , M (α) = -X(α) A(α) 0 nx×n f X(α)E 0 nx×n f X(α)G(α) 0 n f ×nx E(α) -I n f -ωE ⊤ E I n f -ωE ⊤ G(α) , A(α) = X(α)A(α) -L(α)C + ωX(α)EE ⊤ , E(α) = -ωE ⊤ (A(α) + ωEE ⊤ ).
It follows from ( 52) and (53) that

V (t) + βV (t) -γ ḟ ⊤ (t) ḟ (t) ≤ ξ ⊤ (t)Θ(α)ξ(t) < 0, ( 54
) with Θ(α) = Q + X (α)M (α) + M ⊤ (α)X ⊤ (α) -ϖ ⊤ H (α)ϖ. ( 55
)
The parameter-dependent matrices X(α), L(α), H(α) , F 1 (α), and F 2 (α) are parameterized as

X(α) L(α) H(α) F 1 (α) F 2 (α) = N i=1 α i X i L i H i F 1i F 2i .
Hence, the expressions Θ(α) in ( 55) can be rewritten as

Θ(α) = N i=1 N j=1 α i α j Θ ij = N i=1 α 2 i Θ ii + N i=1 N j>i α i α j Θ ij . ( 56
)
It is clear from ( 55) and ( 56) that conditions in (46) imply (54). By the comparison lemma, this latter, in turn, yields

V (t) ≤ e -β(t-t 0 ) V (t 0 ) + γ β 1-e -β(t-t 0 ) ∥ ḟ (t)∥ 2 ∞ . (57)
As t → ∞, we have

V (t) ≤ γ β ∥ ḟ (t)∥ 2 ∞ ≤ γδ 2 β . Since ε ⊤ x (t)P ε x (t) + ε ⊤ ζ (t)Γε ζ (t) ≤ V (t)
, then the ellipsoid (47) is attractive with respect to system (45). This concludes the proof.

Numerical Examples

This section presents three numerical examples to illustrate the practical interests of the proposed design conditions. For the first example, Corol-195 lary 1 is employed to estimate the delay-free dynamics of a satellite launch vehicle affected by an actuator fault. For the second and third examples, the conditions in Theorem 1 are applied to simultaneously estimate the fault and the states of an electronic circuit system and an air-fuel ratio control system of a spark-ignition (SI) engine, respectively. All the design conditions 200 are implemented in Matlab using the Yalmip parser with the semidefinite programming solver MOSEK.

Example 1: Pitch Dynamics of Satellite Launch Vehicle

Consider the model of a satellite launch vehicle along the pitch axis [START_REF] Zhu | Fault Estimation for a Class of Nonlinear Systems Based on Intermediate Estimator[END_REF][START_REF] Das | Robust and fault tolerant controller for attitude control of a satellite launch vehicle[END_REF]

ẋ1 (t) = x 2 (t) + 0.1 sin(x 1 (t)) ẋ2 (t) = 2.94u(t) + 2.94f (t) y(t) = x 1 (t) (58)
where x 1 is the pitch angle, x 2 is the pitch rate, and u is the input command along the pitch axis. The fault f (t) can be interpreted as an actuator deflection, modeling uncertainties or disturbances [START_REF] Zhu | Fault Estimation for a Class of Nonlinear Systems Based on Intermediate Estimator[END_REF]. The nonlinear system (58) can be represented in the form (1) with one vertex, and

A = 0 1 0 0 , A d = 0 0 0 0 , G = 0 0 , E = 0 2.94 , C = 1 0 , h(y, u) = 0.1 sin(x 1 ) 2.94u . ( 59
)
Initially, the aim is to evaluate the influence of the parameter ω and the decay rate β on the induced gain γ obtained by solving Corollary 1.

Considering ϵ = 1 given, Corollary 1 is solved using different values of ω and β and the results are depicted in Figure 1. Clearly, one can notice that there exists a trade-off between the values of β, ω, and γ. Specifically, in this case, higher values of the decay rate β and also ω generally lead to a larger induced gain γ.

The purpose now is to compare the proposed approach with other techniques present in the literature. As discussed in [START_REF] Zhu | Fault Estimation for a Class of Nonlinear Systems Based on Intermediate Estimator[END_REF], the observer matching condition is not satisfied, i.e., rank(CE) ̸ = rank(E). Consequently, the adaptive observer proposed in [START_REF] Jiang | Fault accommodation for nonlinear dynamic systems[END_REF] cannot be applied for fault estimation purposes. Hence, Corollary 1 is compared with the intermediate observer design in [START_REF] Zhu | Fault Estimation for a Class of Nonlinear Systems Based on Intermediate Estimator[END_REF]. Solving the LMI conditions in Corollary 1 with ϵ = 1, ω = 0.7 and β = 1, we obtain X = 0.1278 -0.0017 -0.0113 0.0005 , L = 5.0148 -0.1135 , and γ = 1.0674. In the simulation, three types of fault signals are considered, i.e., constant, ramp, and sinusoidal, which are described in Table 1.

For illustrations, we have considered the initial condition as x(0) = [0 0] ⊤ , x(0) = [0 0] ⊤ , and the input control as u(t) = -0.5963y(t). approach and by the method proposed in [6, Theorem 1]. Figure 3 shows the states and their estimates. The method in [6, Theorem 1] was solved considering ε = 1, ω = 0.3 and l g = 0.1. We can see that the proposed approach presents a better estimation accuracy than the method in [6, Theorem 1].

Example 2: Electronic Circuit System

Consider the electronic circuit system depicted in Figure 4, which can be described for the following dynamical equations [START_REF] Yang | Fault estimation and fault tolerant control for discrete-time nonlinear systems with perturbation by a mixed design scheme[END_REF]: where x j , j ∈ N ≤3 , are the voltages on the capacitors with

ẋ1 (t) = - 1 R 1 C 1 x 1 (t -τ (t)) + 1 10R 2 C 1 x 1 (t)x 2 (t) - 1 R 3 C 1 (u(t) + f (t)) ẋ2 (t) = - 1 R 5 C 2 x 2 (t) + 1 10R 4 C 2 x 1 (t) ẋ3 (t) = - 1 R 7 C 3 x 1 (t) - 1 R 6 C 3 x 3 (t) + 1 10R 8 C 3 x 2 2 (t) y(t) = x 1 (t), (60)
x 1 ∈ [-2, 2], x 2 ∈ [-2, 2]
, u is the voltage source, f is the fault signal occurring in the input voltage source. The model parameters are

R 1 = 0.6kΩ, R 2 = 66.667Ω, R 3 = 0.4kΩ, R 4 = 16.667Ω, R 5 = 0.2kΩ, R 6 = 0.5Ω, R 7 = R 8 = 0.1kΩ, C 1 = C 3 = 10mF and C 2 = 1mF.

225

Taking φ(x) = x 2 2 as the unmeasured nonlinearity, it is evident that its Jacobian is bounded as

k j ≤ ∂ φ(x) ∂x j ≤ k j , ∀j ∈ N ≤3 , with k 1 = 0, k 2 = -4, k 3 = 0, k 1 = 0, k 2 = 4
and k 3 = 0. Thus, Assumption 3 cannot be directly applied to the nonlinearity φ(x) = x 2 2 . However, in accordance with Remark 2, a loop transformation can be employed by defining ϕ(x) = φ(x) -Kx, with K = [0 -4 0], leading to ϕ(x) = x 2 + 4x 2 . Consequently, the boundedness condition of the Jacobian in (3) is satisfied with k j = k j -k j , for j ∈ N ≤3 , and K = 0 8 0 . Therefore, after employing this loop transformation, the system (60) is written as We define the scheduling parameter as ρ x = x 1 . Then, the electronic circuit system (61) can be rewritten as in [START_REF] Du | Actuator fault estimation and accommodation for switched systems with time delay: Discrete-time case[END_REF] with

ẋ1 (t) = - 1 R 1 C 1 x 1 (t -τ (t)) + 1 10R 2 C 1 x 1 (t)x 2 (t) - 1 R 3 C 1 (u(t) + f (t)) ẋ2 (t) = - 1 R 5 C 2 x 2 (t) + 1 10R 4 C 2 x 1 (t) (61) ẋ3 (t) = - 1 R 7 C 3 x 1 (t) - 1 R 6 C 3 x 3 (t) - 4 10R 8 C 3 x 2 (t) + 1 10R 8 C 3 (x 2 2 (t) + 4x 2 (t)) y(t) = x 1 (t) - + R 5 R 4 C 2 + - x 2 × - 1 10 × - 1 10 - + R 2 R 1 τ (t) C 1 + - x 1 R 3 - + u - + R 6 R 7 C 3 + - x 3 R 8 × - 1
A(α) =    0 1 10R 2 C 1 ρ 0 1 10R 4 C 2 -1 R 5 C 2 0 -1 R 7 C 3 -4 10R 8 C 1 -1 R 6 C 3    , h(y, u) =   -1 R 3 C 1 0 0   , A d =   -1 R 1 C 1 0 0 0 0 0 0 0 0   , E =   -1 R 3 C 1 0 0   , G =   0 0 1 10R 8 C 1   , C = 1 0 0 .
Applying the sector nonlinearity approach [24, Chapter 2], system (61) can be represented in the form (1) with 2 vertices, whose state-space matrices are given by

A 1 =   0 -0.3 0 6 -5 0 -1 -0.4 -0.2   , A 2 =   0 0.3 0 6 -5 0 -1 -0.4 -0.2   , A d =   -0.1667 0 0 0 0 0 0 0 0   , E =   -0.025 0 0   , G =   0 0 1   .
The corresponding time-varying parameters are given by

α 1 (ρ x ) = 2 -x 1 4 , α 2 (ρ x ) = 1 -α 1 (ρ x ).
The nonlinearity x 2 2 cannot be measured, which raises technical difficulties for the fault estimation design. It should be noted that the approaches presented in [START_REF] Hamdi | Observerbased fault diagnosis for time-delay LPV descriptor systems[END_REF][START_REF] You | Robust fault estimation for Takagi-Sugeno fuzzy systems with state time-varying delay[END_REF] cannot be applied in this case study because they are not designed to handle unmeasured nonlinearity. For this example, we design a gain-scheduled intermediate observer for system (60) such that the error dynamics is ISS with respect to ḟ (t) for a maximum bound of the time-delay. To this end, applying Theorem 1 with β = 0.01, ω = 140, ϵ = 10, and τ = 0, the maximum delay reached is τ = 0.52s. For this case, the following observer matrices are obtained:

X 1 =   0.
  , H 1 =   • -0.0001 •   , H 2 =   • 0.0001 •   ,
and γ = 0.004. For comparison purposes, the fault signal given by

f (t) =          0, 0 ≤ t <
) 62
Notice that according to Assumption 2, by taking the time-derivative of the fault signal, one can compute ∥ ḟ ∥ ∞ ≤ δ with δ = 0.007. Figures 5a-5c depict the system states and their estimates for an initial condition x(0) = [0.5 0.6 0.75] ⊤ and x(0) = [0 0 0] ⊤ . Figure 5d shows the fault signal and its estimation. Figure 5e presents the input control trajectory defined as u(t) = 2.5 sin(0.5πt) cos(t). Finally, Figure 5f depicts the time-varying delay given by τ (t) = 0.26 + 0.26 cos(tπ/4). We can see that the states of the electronic circuit (60) and the fault signal are accurately estimated despite the effect of the time-varying delay τ (t). This example is concerned with the estimation problem of an air-fuel ratio (AFR) control system, which is composed of three parts: the cylinder mass air flow dynamics, the fuel mass flow dynamics, and the AFR sensor dynamics. Figure 6 shows the schematic diagram of a classical cylinder gasoline engine. The cylinder air flow dynamics can be described by the following speed density equation:

ṁcyl (t) = (s 1 p col (t) + s 2 ) V d 120RT col N e (t), (63)
where N e is the engine speed [rpm], p col is the intake manifold pressure [Pa],

V d is the volume displacement [m 3 ], T col is the manifold temperature [ • K], R is the constant for perfect gas [kJ/kg • K]. This relation gives an estimate of the cylinder air mass flow through cylinders, where the two constant parameters are given by s 1 = 0.0281 and s 2 = -1.6208. The AFR in the cylinder λ cyl can be defined as

λ cyl (t) = m cyl (t) λ s m f (t) , (64)
where λ s is the air-fuel stoichiometric ratio. Thus, the time derivative of λ cyl (t) is given by

λcyl (t) = 1 λ s ṁcyl (t) m f (t) - ṁf (t) m f (t) λ cyl (t), (65)
where ṁf (t) is the mass fuel flow into the cylinders [kg/s]. Using the mean value model for a four-cylinder engine, we have [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] ṁf (t) = N

) 67
For the AFR control system, a time-varying delay, depending on the engine speed N e , due to the transport of the gas flow from the cylinder to the lambda sensor is unavoidable [START_REF] Lauber | Air-fuel ratio control in a gasoline engine[END_REF]. Hence, to measure the AFR, the following lambda sensor dynamics is considered [START_REF] Lauber | Air-fuel ratio control in a gasoline engine[END_REF][START_REF] Kim | Developing a fault tolerant powertrain control system by integrating design of control and diagnostics[END_REF]:

λmes (t) = - 1 τ λ λ mes (t) + 1 τ λ λ cyl (t -τ (N e )) ( 68
) with τ (N e ) = 86.6 N e (t) . ( 69
)
The dynamics of the engine speed N e can be described as [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] Ṅe

(t) = 1 I e (T ind (t) -T br (t) -T f (t)) , (70)
where T br is the brake torque, which is assumed to be regulated by the speed controller T br (t) = 0.1(N e (t) -r) + 0.01 t 0

(N e (s) -r)ds,

where r is the reference signal, T f represents a lumped fault of the braking system and other torque frictions, I e is the rotational moment of the engine inertia. The effective indicated torque T ind is modeled as [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF] T

ind (t) = H u η i ṁcyl (t) λ s N e (t) 1 λ cyl(t) , (72)
where H u is the fuel energy constant, and η i is the thermal efficiency multiplier. The AFR dynamics can be then represented from subsystems (67), ( 68) and (70) as

ẋ1 (t) = - x 1 (t)x 3 (t) 30 + ṁcyl (t)x 3 (t) 30λ s 1 ṁf (t) ẋ2 (t) = - 1 τ λ x 2 (t) + 1 τ λ x 1 (t -τ (t)) ẋ3 (t) = H u η i (s 1 p col (t) + s 2 )V d 120λ s RT col I e 1 x 1 (t) - T br (t) I e - T f (t) I e (73)
where x 1 (t) = λ cyl (t), x 2 (t) = λ mes (t) and x 3 (t) = N e (t). Table 2 provides the engine parameters. The nonlinear system described in (73) can be reformulated in a nonlinear parameter-varying time-delay system as

  ẋ1 (t) ẋ2 (t) ẋ3 (t)   =   -x 3 30 0 0 0 -1 τ λ 0 0 0 0     x 1 (t) x 2 (t) x 3 (t)   +   0 0 0 1 τ λ 0 0 0 0 0     x 1 (t -τ (t)) x 2 (t -τ (t)) x 3 (t -τ (t))   +   ṁcyl x 3 30λs 0 0 0 0 -1 Ie   1 ṁf T br +   0 0 Huη i (s 1 p col (t)+s 2 )V d 120λsRT col Ie   1 x 1 (t) +   0 0 -1 Ie   T f (t) y(t) = 0 1 0 0 0 1   x 1 (t) x 2 (t) x 3 (t)   . (74)
For a standard operating condition of a naturally aspirated SI engine, i.e., without a turbocharger, we have the physical bounds of the engine state variables as x 1 ∈ [0.5, 1.5], x 3 ∈ [1000, 4000] and p col (t) ∈ [3 × 10 4 , 2 × 10 5 ].Hence, the lower and upper bounds of the time-varying delay can be derived from (69) as τ = 0.02 and τ = 0.1. For system (74), ρ 1 (t) = x 3 and ρ 2 (t) = p col (t) 240 are selected as scheduling functions. Considering the unmeasured nonlinearity φ(x) = 1 x 1 , Assumption 3 cannot be directly applied. However, by employing the loop-transformation as described in Remark 2, such that ϕ(x) = φ(x) -Kx, where K = -0.44 0 0 , we obtain ϕ(x) = 1

x 1 + 4x 1 that satisfies the sector condition (3) with K = 3.56 0 0 . Applying the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], the engine model (74) can be represented as an N-LPV system in (1) with four vertices (77)

The initial conditions are x(0) = [0.55 0.6 2400] ⊤ and x(0) = [0.5 0.7 2200] ⊤ . The time-varying delay is computed as in (69). Figure 7 depicts the intake manifold pressure p col (t), the input signal u(t), the reference signal r(t), and the corresponding break torque T br (t) given by (71). Figures 8a-8c show the system states and their corresponding estimates. Lemma 3 (Delay-dependent reciprocally convex inequality [START_REF] Seuret | Stability of Linear Systems With Time-Varying Delays Using Bessel-Legendre Inequalities[END_REF]). Let R 1 , R 2 ∈ R n×n be symmetric positive definite matrices. If there exist symmetric matrices X 1 , X 2 ∈ R n×n and matrices Y 1 , Y 2 ∈ R n×n such that

R 1 0 0 R 2 -ϱ X 1 Y 1 Y ⊤ 1 0 -(1 -ϱ) 0 Y 2 Y ⊤ 2 X 2 ≥ 0
holds for ϱ ∈ B. Then, the following inequality

1 ϱ R 1 0 0 1 1-ϱ R 2 ≥ R 1 0 0 R 2 + (1 -ϱ) X 1 Y 2 Y ⊤ 2 0 + ϱ 0 Y 1 Y ⊤ 1 X 2 (A.1)
holds for all ϱ ∈ (0, 1) ⊂ R.

Figure 1 :

 1 Figure 1: Example 1. Induced gain γ obtained by solving Corollary 1 for different values of ω and β.

Figure 2 :

 2 Figure 2: Example 1. (a) Fault f (t) (straight gray line) and its estimation f (t) obtained by Corollary 1 (blue dashed line) and fZhu (t) obtained by [6] (dashed-dotted red line); (b) Fault estimation error e f (t) obtained by Corollary 1 (blue dashed line) and e Zhu (t) obtained by [6] (dashed-dotted red line).

Figure 3 :

 3 Figure 3: Example 1. System states (straight black line) and their estimates. Comparison between the proposed method and [6, Theorem 1].

10 fFigure 4 :

 104 Figure 4: Diagram of the electronic circuit system.

4. 3 .

 3 Example 3: Air-Fuel Ratio Control System for an SI Engine 235

  Real and estimated states.

  Real and estimated states.

  Time-varying delay τ (t).

Figure 5 :

 5 Figure 5: Example 2. Estimation performance obtained for the electronic circuit system (60) under the effect of the fault f (t) and the time-varying delay τ (t).

  in Theorem 1 with ϵ = 0.5, β = 0.1 and ω = 0.35, a gain-scheduled intermediate observer is obtained, whose vertices are de-and γ = 5.4862 × 10 -6 . For simulations, the fault signal is given by T f (t) = 0.5πt), 0 ≤ t < 5, -5, 5 ≤ t < 10, 15 sin(πt), 10 ≤ t ≤ 20.

  250

Figure 8d presents the

  Figure8dpresents the simulation result for the fault estimation obtained with the proposed approach. This result illustrates the practical interest of the proposed fault estimation design for N-LPV systems with time-varying delays and unmeasured nonlinearities.

Figure 7 :

 7 Figure 7: Example 3. Trajectories of the manifold pressure (p col ), the load torque (T br ), the input (u) and the input reference (r).

255

  This paper has addressed the problem of simultaneous estimation of states and faults affecting nonlinear parameter-varying systems with time-varying delays. Design conditions have been proposed to design a gain-scheduling intermediate estimator to ensure that the error dynamics are ISS with respect to the time derivative of the fault. As a sub-product, a condition has been derived to design a gain-scheduling intermediate estimator for a nonlinear parameter-varying system without time-varying delays. Three numerical examples with physically motivated systems subject to faults have been considered to illustrate the application of the proposal. The results indicate the Real and estimated states.

  Real and estimated states.

  Real and estimated states.

  Fault and its estimation.

Figure 8 : 2 where Θ 1

 821 Figure 8: Example 3. Estimation performance obtained for the AFR engine model (73) under the effect of the fault T f .

Table 1 :

 1 Description of the fault signal f (t).

	t	[0s, 5s) [5s, 15s) [15s, 25s) [25s, 40s) [40s, 47s) [47s, 70s]
	f (t)	0	0.2t	0	2	0	2 sin(0.2t)

Table 2 :

 2 Parameter values of the AFR engine model.

	Description	Parameters Values	Units
	volume displacement	V d	0.6 × 10 -3 m 3
	manifold temperature	T col	293	• K
	constant for perfect gas	R	0.287	kJ/kg • K
	air-fuel stoichiometric ratio	λ s	14.67	-
	fuel energy constant	H u	18 × 10 5	Nm/Kg
	thermal efficiency multiplier η i	0.7	-
	rotational moment of Inertia I e	0.1307	kgm 2
	time constant	τ λ	0.1	s

Acknowledgements

This work was supported by the Brazilian agency CNPq (Grant numbers: 270 305578/2022-7, 150602/2022-7, 164692/2020-7).

Appendix A. Instrumental Lemmas

The following lemmas are used to develop the proposed results.