Actuator and sensor fault estimation for T–S fuzzy fractional-order systems based on adaptive and sliding mode observers

Abstract

This study delves into the issue of estimating faults in actuators and sensors for a specific category of uncertain fractional-order nonlinear systems (FONSs). This category is characterized by Takagi-Sugeno (T-S) fuzzy model, which operates in the presence of local nonlinear functions and external disturbances. The fractional-order T-S fuzzy system is divided into two subsystems using linear coordinate transformations: The first subsystem includes the effects of disturbances but not sensor faults, while the second subsystem incorporates sensor faults but no disturbances. Subsequently, a T-S fuzzy fractional sliding mode observer (FSMO) is constructed based on the first subsystem to entirely neutralize the impact completely of external disturbances. For the second subsystem, a T-S fuzzy fractional adaptive observer (FAO) is formulates to estimate both actuator and sensor faults, leveraging the estimated output subsystem. The stability analysis of state estimation errors ventures into the domain of Lyapunov stability theory for FONSs, thereby enriching our comprehension of their dynamics and predictive abilities. The necessary conditions for stability are subsequently deduced and articulated in the form of linear matrix inequalities (LMIs). To corroborate the findings, an exemplifying scenario is demonstrated at the end.

AMS Subject Classifications: 34K20.

Keywords

FONS sliding mode observers LMIs T-S fuzzy systems

Introduction

As timeless as the classical calculus, the fractional-order calculus provides a novel viewpoint by broadening the notion of integro-differentiation to non-integer orders via the employment of fractional-order differintegral operators.^1,2 Its contemporary applications encompass a vast array of research domains, primarily due to the availability of potent tools^2,3 that are absent in the classical calculus.

Specifically, fractional calculus (FC) equips scholars with the ability to effectively model numerous intricate real-world issues such as COVID-19,⁴ HIV,⁵ AH1N1 influenza,⁶ and the Ebola virus.⁷ Furthermore, it has discovered fascinating recent applications in fields like diabetes⁸ and image processing.⁹

Fractional calculus instruments are exceedingly apt for the accurate modelling of many actual physical systems.^1,2 While fractional operators present appealing advantages for control system design, the analysis of the stability of closed-loop systems poses a significant obstacle. This issue stems from the dual character of differ integrals, acting as both a primary benefit and restriction, as elaborated in Muñoz Vázquez et al.^10,11 In studies like Matignon¹² and Petrás,¹³ scholars have introduced stability analyses for certain types of fractional linear systems (FOSs). However, wider concerns such as external stability or the contemplation of general FOSs remain uninvestigated in these studies.

The T-S fuzzy modeling approach is recognized as one of the most potent techniques in the literature for addressing intricate nonlinearity in FOSs.¹⁴ Consequently, a plethora of results concerning stability analysis and control theory for nonlinear FOSs based on the T-S fuzzy models have been established in the literature Lin et al.¹⁵, Mirzajani et al.¹⁶, and Liu et al.¹⁷ In Zheng et al.,¹⁸ a novel technique for controlling fractional order chaotic systems was introduced using the generalized T-S fuzzy model and adaptive adjustment mechanism. In Lin et al.,¹⁵ a stabilization issue was examined. In Ning and Hua,¹⁹ observer-based $H_{\infty}$ control and stabilization problem for T-S fuzzy FOSs with unknown input and time-delay were scrutinized. In Djeddi et al.,²⁰ a fractional order unknown input observer (FOUIO) was devised.

The sliding mode technique, renowned for its robustness, simplistic structure, and extensive applicability, has surfaced as a formidable instrument for managing matched uncertainties and nonlinearities. It has achieved considerable triumph in formulating observers for uncertain nonlinear systems. These observers, with their capacity to procure detailed fault information such as size and shape, have been employed to address a variety of concerns, including detection, estimation of sensor and actuator faults.

In Dadras and Momeni,²¹ the sliding mode technique was utilized to analyze the state estimation problem for a class of uncertain FONSs. In Pisano et al.,²² a second-order sliding mode observer was implemented to resolve fault estimation (FE) and detection problem in FOs. In Lee et al.,²³ the authors constructed a $H_{\infty}$ filter and sliding mode unknown input observer for linear time-invariant fractional-order dynamic systems, taking into account the appropriate initial memory effect. A robust fault identification scheme predicated on a fractional-order integral sliding mode observer was presented for turbofan engine sensors with uncertainties.²⁴ In the domain explored by Li and Zhang,²⁵ the emphasis was on formulating sliding mode observers (SMOs) for FOSs. They presented a condition, articulated through LMIs, that assures the suitability of the Sliding Mode (SM) dynamics. In a related context, Zhang et al.²⁶ ventured into the domain of $H_{\infty}$ approach.

However, the intricacy of computational tasks related to the determination of observer or controller gain matrices intensifies with the augmentation of fuzzy rules. To tackle this obstacle, researchers have integrated local nonlinear models that satisfy certain criteria^27–29 into the description of the fuzzy system. This strategy results in a decrease in the number of rules while preserving model precision. Importantly, the matter of fault estimation (FE) for T-S fuzzy FOSs incorporating local nonlinear functions along with considerations remains largely uncharted in the existing literature. This knowledge gap is what drives the present study.

Our approach builds upon and extends existing research in the field. For instance, Kchaou et al.³⁰ address synchronization in fractional-order systems, Vigneshwar et al.³¹ focus on $H_{\infty}$ fault detection in T-S systems, and Kchaou and Jerbi³² develop sliding mode control for systems with actuator failures. Unlike these studies, our method specifically targets fractional-order T-S fuzzy systems and uniquely integrates adaptive algorithms with sliding mode observers to enhance fault estimation accuracy and robustness, thereby providing a comprehensive solution for simultaneous state and fault estimation.

The Motivations behind our work are:

(1) Complex system dynamics: Modern control systems are increasingly complex, often involving nonlinearities and interacting components that can lead to unpredictable behaviors.

(2) Need for robust fault estimation: Ensuring the reliability and safety of these systems necessitates the accurate estimation of faults, particularly when both actuators and sensors are susceptible to failures.

(3) Improving estimation accuracy: Enhancing the accuracy of state and fault estimation can lead to significant improvements in system performance and reliability.

The main contributions of our work are:

(1) Novel FE approach: This work introduces a novel FE approach by amalgamating the adaptive algorithm theory and the sliding mode technique.

(2) Innovative FSMO and FAO construction: We pioneer the construction of fractional sliding mode observer and fractional adaptive observer for Takagi-Sugeno fuzzy fractional-order systems that feature local nonlinear models, external disturbances, and faults in both actuators and sensors.

(3) Simultaneous estimation capability: Contrary to conclusions from previous studies, the proposed observers are capable of simultaneously estimating system states, actuator faults, and sensor faults by utilizing the estimation data derived from the system’s output.

(4) Enhanced stability conditions: To ensure the stability of state estimation errors, we develop new sufficient conditions expressed through linear matrix inequalities (LMIs).

Preliminaries and problem statement

Preliminaries

Let’s begin by revisiting fundamental definitions and lemmas associated with fractional order derivatives.

Definition 1. (Podlubny² and Kilbas et al.³) The Riemann–Liouville (RL) integral of order $α > 0$ of a continuous function $f (t)$ is formally defined as:

{{}_{t_{0}}I}_{t}^{α} f (t) = \frac{1}{Γ (α)} \int_{t_{0}}^{t} (t - τ)^{α - 1} f (τ) d τ

(1)

with $t \geq t_{0}$ , $n - 1 \leq α < n$ , $n \in N^{+}$ . The Euler’s Gamma function $Γ (α)$ is defined as

Γ (α) = \int_{0}^{\infty} x^{α - 1} e^{- x} dx

(2)

Definition 2. (Podlubny² and Kilbas et al.³) The following fractional-order derivative of order $α > 0$ of a continuous function $f (t)$ , known as the Caputo derivative, is presented as follows:

{}_{t_{0}}^{C}{D_{t}^{α}} f (t) = \frac{1}{Γ (n - α)} \int_{t_{0}}^{t} (t - τ)^{n - α - 1} f^{(n)} (τ) d τ

(3)

where the superscript $C$ indicates the Caputo derivative, $t \geq t_{0}$ , $n - 1 \leq α < n$ , $n \in N^{+}$ .

Definition 3. (Podlubny²) The Mittag–Leffler (ML) function, characterized by two positive parameters $a$ and $b$ , is defined in the following manner:

E_{a, b} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (ka + b)}

(4)

where $z \in C$ . When $b = 1$ , one has $E_{a} (z) = E_{a, 1} (z)$ , furthermore, $E_{1, 1} (z) = e^{z}$ .

Lemma 1. (see Podlubny²) If $0 < a < 2$ , $b \in R$ , $\frac{a π}{2} < μ < min {π, a π}$ , and $λ > 0$ , then the estimate of Mittag–Leffler function is defined by

| E_{a, b} (z) | \leq \frac{λ}{1 + | z |}

(5)

where $μ \leq | \arg (z) | \leq π$ , $λ \geq 0$ .

Lemma 2. (Manuel et al.³³) Let $x (t) \in R^{n}$ be a vector comprising differentiable functions. For any moment $t \in [t_{0}, \infty)$ and $α \in (0, 1]$ , we have:

{}_{0}^{C}{D_{t}^{α}} (x^{T} (t) Px (t)) \leq 2 x^{T} (t) P ({}_{0}^{C}{D_{t}^{α}} x (t))

(6)

where $P \in R^{n \times n}$ is a positive definite matrix.

Lemma 3. (Gong³⁴) One supposes $V (t) : [0, \infty) \to R$ a continuous function such that its $α$ -order derivative satisfies the condition:

{}_{0}^{C}{D_{t}^{α}} V (t) \leq - k_{1} V (t) + k_{2}

(7)

where $0 < α < 1$ , $k_{1} > 0$ , and $k_{2} \geq 0$ . So

V (t) \leq V (0) E_{α} (- k_{1} t^{α}) + \frac{k_{2} d}{k_{1}}

(8)

where $d = max {1, λ}$ and $λ$ is defined as in Lemma 1.

Lemma 4. (Wang et al.³⁵) If an integrable function $f (t)$ has at least one point $t_{1} \in (0, t)$ where $f (t_{1}) \neq 0$ , then there exists a positive constant $T_{0}$ such that ${}_{0}^{C}{D_{t}^{- α}} | f (t) | \geq T_{0}$ for all $t \geq 0$ .

Problem statement

Let us consider the fractional-order nonlinear system with order $0 < α < 1$

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} x_{k} (t) = f_{1} (x_{k} (t)) + f_{2} (x_{k} (t)) ϕ_{k} (t, x_{k} (t)) \\ + f_{3} (x_{k} (t)) u_{k} (t) + f_{4} (x_{k} (t)) d_{k} (t) \end{matrix}

(9)

y_{k} (t) = h (x_{k} (t))

(10)

where $x_{k} (t) \in R^{n_{x}}$ , $u_{k} (t) \in R^{n_{u}}$ and $y_{k} (t) \in R^{n_{y}}$ are the state vector, control input vector and measurable output vector, respectively. $d_{k} (t) \in R^{n_{d}}$ represent the unknown external disturbances. $f_{i} (x_{k} (t))$ , $h (x_{k} (t))$ and $ϕ_{k} (t, x_{k} (t))$ are nonlinear functions $(i = 1, 2, 3, 4)$ .

Using the technique of sector nonlinearity,³⁶ we have:

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} x_{k} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{ki} x_{k} (t) + N_{ki} ϕ_{k} (t, x_{k} (t)) \\ + B_{ki} u_{k} (t) + E_{ki} d_{k} (t)] \end{matrix}

(11)

y_{k} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) C_{ki} x_{k} (t)

(12)

where $ν (t) = [\begin{matrix} ν_{1} (t) & ν_{2} (t) & . . . & ν_{g} (t) \end{matrix}]$ denotes the premise vector which is measurable

ψ_{i} (ν (t)) = \frac{σ_{i} (ν (t))}{\sum_{i = 1}^{r} σ_{i} (ν (t))}, σ_{i} (ν (t)) = \prod_{j = 1}^{g} M_{i j} (ν (t))

$ψ_{i} (ν (t))$ fulfills the condition:

ψ_{i} (ν (t)) \geq 0, \sum_{i = 1}^{r} ψ_{i} (ν (t)) = 1

The term $ϕ_{k} (t, x_{k} (t)) \in R^{n_{ϕ}}$ denotes the nonlinear component of the local model, which serves to enhance model accuracy while reducing the number of fuzzy rules and computational workload.^28,29 Here, $A_{ki} \in R^{n_{x} \times n_{x}}$ , $B_{ki} \in R^{n_{x} \times n_{u}}$ , $N_{ki} \in R^{n_{x} \times n_{ϕ}}$ , $E_{ki} \in R^{n_{x} \times n_{d}}$ , and $C_{ki} \in R^{n_{y} \times n_{x}}$ represent constant real matrices with appropriate dimensions, where $rank (E_{ki}) = n_{d}$ , $rank (C_{ki}) = n_{y}$ , and $n_{x} > n_{y} \geq n_{d}$ .

Assumption 1. The pair $(A_{ki}, C_{ki})$ is detectable for $\forall i = 1, . . ., r$ .

Assumption 2. The matching condition observer holds for $\forall i = 1, . . ., r$ if and only if

rank (C_{ki} E_{ki}) = rank (E_{ki})

(13)

Assumption 3. For $\forall i = 1, 2, . . ., r$ , the invariant zeros of the triple $(A_{ki}, E_{ki}, C_{ki})$ are all in the stable region complex plane, i.e.

rank [\begin{matrix} A_{ki} - s I_{n} & E_{ki} \\ C_{ki} & 0 \end{matrix}] = n_{x} + n_{d}

(14)

For all $s$ such that $| \arg (s) | > R (π / 2)$ , where $0 < R < 1$ .

Remark 1. The assumptions listed above are critical for ensuring the stability and performance of the observer design in nonlinear fractional-order systems. Assumption 1 guarantees that the pair $(A_{ki}, C_{ki})$ is detectable, which is essential for ensuring that the system states can be accurately reconstructed from the output measurements. Assumption 2 ensures that the matching condition observer holds, which is a necessary condition for the existence of a suitable observer gain that can handle faults and uncertainties. Assumption 3 requires that the invariant zeros of the triple $(A_{ki}, E_{ki}, C_{ki})$ are in the stable region of the complex plane, ensuring that the system remains stable under the designed observer. These assumptions collectively provide a robust framework for designing observers that can effectively estimate system states and detect faults, thereby enhancing the reliability and performance of the control system. To illustrate these assumptions further, we provide detailed examples and explanations in the subsequent sections, demonstrating their practical implications and relevance to real-world applications.

Assumption 4. The nonlinear function $ϕ_{k} (t, x_{k} (t))$ satisfies

‖ ϕ_{k} (t, x_{k 1} (t)) - ϕ (t, x_{k 2} (t)) ‖ \leq L_{ϕ} ‖ x_{k 1} (t) - x_{k 2} (t) ‖

(15)

where $x_{k 1} (t)$ , $x_{k 2} (t) \in R^{n_{x}}$ , $L_{ϕ} > 0$ is the Lipschitz constant.

When system (11) and (12) experiences both sensor and actuator faults concurrently, we have

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} x_{k} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{ki} x_{k} (t) + N_{ki} ϕ_{k} (t, x_{k} (t)) \\ + B_{ki} u_{k} (t) + F_{ki} f_{a} (t) + E_{ki} d_{k} (t)] \end{matrix}

(16)

y_{k} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [C_{ki} x_{k} (t) + D_{k} f_{s} (t)]

(17)

where $f_{a} (t) \in R^{n_{q}}$ and $f_{s} (t) \in R^{n_{s}}$ are additive actuator and sensor faults, respectively.

Assumption 5. The measurable output and disturbance distribution matrices are the same for all the rules, i.e. $C_{k 1} = C_{k 2} = \dots = C_{k}$ and $E_{k 1} = E_{k 2} = \dots = E_{k}$ .

Remark 2. In a multitude of real-world systems, the observable output often constitutes a segment of the states. This implies that the matrices of measurable output would be a shared attribute across each local model. This isn’t particularly limiting in the context of T-S fuzzy systems. ³⁷ Additionally, the presumption regarding disturbance matrices is a common practice in the design of reduced-order observers. ³⁸

Assumption 6. The actuator fault $f_{a} (t)$ , sensor fault $f_{s} (t)$ , their fractional derivatives, and the disturbance $ω (t)$ are bounded by a norm, meaning that:

\begin{matrix} ‖ f_{a} (t) ‖ \leq ρ_{a}, ‖ f_{s} (t) ‖ \leq ρ_{s}, ‖ {}_{0}^{C}{D_{t}^{α}} f_{a} (t) ‖ \leq ρ_{aa}, ‖ {}_{0}^{C}{D_{t}^{α}} f_{s} (t) ‖ \\ \leq ρ_{ss}, ‖ d_{k} (t) ‖ \leq ρ_{d} \end{matrix}

(18)

Here, $ρ_{a}$ , $ρ_{aa}$ , $ρ_{s}$ , $ρ_{ss}$ , and $ρ_{d}$ represent specified positive constants.

Remark 3. The stated assumption is not uncommon; in contemporary fault estimation literature for integer order systems, it’s standard to assume boundedness of faults and their derivatives.³⁹Considering our investigation into fractional order systems (FOSs), it’s reasonable to expect that the fractional derivatives of faults are also boundedly normed. This assumption finds practical relevance in real-world applications like RC fractional order circuit systems.⁴⁰

Remark 4. The assumptions outlined above are crucial for ensuring the effectiveness and stability of the observer and control design in our nonlinear fractional-order system. Assumption 4 requires that the nonlinear function $ϕ_{k} (t, x_{k} (t))$ satisfies a Lipschitz condition, which ensures that the nonlinearities in the system are well-behaved and can be bounded. This is essential for guaranteeing the convergence of the observer. Assumption 5 simplifies the system structure by assuming that the measurable output and disturbance distribution matrices are consistent across all rules, facilitating the observer design process. Assumption 6 ensures that the actuator and sensor faults, their fractional derivatives, and the disturbance are all bounded by specified norms. These bounds are necessary to ensure that the system remains within a predictable and manageable range, which is critical for the stability and robustness of the control and observation strategy. Together, these assumptions provide a robust framework that underpins the theoretical development of our observer design, ensuring that it can effectively handle the nonlinearities and uncertainties inherent in real-world systems. In the subsequent sections, we will provide detailed examples and further explanations to illustrate the practical implications and relevance of these assumptions.

Lemma 5. (Moodi and Farrokhi²⁸) Given matrices (or vectors) $X$ and $Y$ with suitable dimensions, the following criterion is satisfied:

X^{T} Y + Y^{T} X \leq ε X^{T} X + ε^{- 1} Y^{T} Y

(19)

with $ε > 0$ is arbitrary.

Lemma 6. (Jiang et al.⁴¹) Imagine that we possess $U$ and $V$ , two matrices of fitting dimensions. There exists a positive real number $M$ such that

U^{T} V + V^{T} U \leq U^{T} MU + V^{T} M^{- 1} V

(20)

Real-world systems often have observable outputs that are part of the state vector, making shared output matrices across local models a reasonable assumption. Assumptions on disturbance matrices and boundedness of faults are standard in FE literature for integer order systems and are extended here to fractional-order systems. Using Lemma 5 and Lemma 6, we establish criteria to ensure the stability and robustness of the observer design. These lemmas help in deriving conditions that the observer must satisfy to achieve accurate state and fault estimation despite the presence of nonlinearities, disturbances, and faults.

The objective of this study is to design a robust observers for the given nonlinear fractional-order system that can:

(1) Accurately estimate the system states.

(2) Detect and estimate faults in both actuators and sensors.

(3) Ensure stability and robustness in the presence of external disturbances and nonlinearities.

By addressing these objectives, we aim to enhance the reliability and performance of control systems operating under complex and uncertain conditions.

State transformations

Assuming Assumption 2, there exist $ω (t) = T_{k} x_{k} (t)$ and $η (t) = S_{k} y_{k} (t)$ , two coordinate transformations, where $T_{k} \in R^{n_{x} \times n_{x}}$ and $S_{k} \in R^{n_{y} \times n_{y}}$ . This transformation leads system (16) and (17) to be represented as:

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ω (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [T_{k} A_{k i} T_{k}^{- 1} ω (t) + T_{k} N_{k i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) \\ + T_{k} B_{k i} u_{k} (t) + T_{k} F_{k i} f_{a} (t) + T_{k} E_{k} d_{k} (t)] \end{array}

(21)

η (t) = S_{k} C_{k} T_{k}^{- 1} ω (t) + S_{k} D_{k} f_{s} (t)

(22)

where

\begin{matrix} T_{k} A_{ki} T_{k}^{- 1} = [\begin{matrix} A_{11 i} & A_{12 i} \\ A_{21 i} & A_{22 i} \end{matrix}], T_{k} N_{ki} = [\begin{matrix} N_{1 i} \\ N_{2 i} \end{matrix}], \\ T_{k} B_{ki} = [\begin{matrix} B_{1 i} \\ B_{2 i} \end{matrix}], T_{k} F_{ki} = [\begin{matrix} F_{1 i} \\ F_{2 i} \end{matrix}] \end{matrix}

T_{k} E_{k} = [\begin{matrix} E_{1} \\ 0 \end{matrix}], S_{k} C_{k} T_{k}^{- 1} = [\begin{matrix} C_{11} & 0 \\ 0 & C_{22} \end{matrix}], S_{k} D_{k} = [\begin{matrix} 0 \\ D_{2} \end{matrix}]

and $C_{11}$ is a nonsingular matrix.

Therefore, system (21) and (22) is decomposed into the following subsystems:

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ω_{1} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{11 i} ω_{1} (t) + A_{12 i} ω_{2} (t) + N_{1 i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) \\ + B_{1 i} u_{k} (t) + F_{1 i} f_{a} (t) + E_{1} d_{k} (t)] \end{array}

(23)

η_{1} (t) = C_{11} ω_{1} (t)

(24)

and

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ω_{2} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{21 i} ω_{1} (t) + A_{22 i} ω_{2} (t) \\ + N_{2 i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) + B_{2 i} u_{k} (t) + F_{2 i} f_{a} (t)] \end{array}

(25)

η_{2} (t) = C_{22} ω_{2} (t) + D_{2} f_{s} (t)

(26)

where $ω_{1} (t) \in R^{n_{d}}$ , $ω_{2} (t) \in R^{(n_{x} - n_{d})}$ , and $ω (t) = {[\begin{matrix} ω_{1} {(t)}^{T} & ω_{2} {(t)}^{T} \end{matrix}]}^{T}$ .

Let

{}_{0}^{C}{D_{t}^{α}} ω_{3} (t) = C_{22} ω_{2} (t) + D_{2} f_{s} (t)

(27)

Next, we introduce the expanded T-S fuzzy FOS, now incorporating the fresh state $ω_{0} (t)$ , as follows:

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ω_{0} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{0 i} ω_{0} (t) + {\tilde{A}}_{21 i} ω_{1} (t) \\ + N_{0 i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) + B_{0 i} u_{k} (t) + F_{0 i} f_{a} (t) + D_{0} f_{s} (t)] \end{array}

(28)

η_{0} (t) = C_{0} ω_{0} (t)

(29)

where

\begin{matrix} ω_{0} (t) = [\begin{matrix} ω_{2} (t) \\ ω_{3} (t) \end{matrix}], A_{0 i} = [\begin{matrix} A_{22 i} & 0 \\ C_{22} & 0 \end{matrix}], {\tilde{A}}_{21 i} = [\begin{matrix} A_{21 i} \\ 0 \end{matrix}], \\ B_{0 i} = [\begin{matrix} B_{2 i} \\ 0 \end{matrix}], F_{0 i} = [\begin{matrix} F_{2 i} \\ 0 \end{matrix}] \end{matrix}

N_{0 i} = [\begin{matrix} N_{2 i} \\ 0 \end{matrix}], D_{0} = [\begin{matrix} 0 \\ D_{2} \end{matrix}], C_{0} = [\begin{matrix} 0 & I_{p - q} \end{matrix}]

Lemma 7. (Hui and Zak⁴²) The pair $(A_{0 i}, C_{0})$ is detectable for $\forall i = 1, . . ., r$ if the rank condition in Assumption 3 holds.

In accordance with Lemma 7, we can find a matrix $L_{0 i}$ that belongs to the set $R^{(n_{x} + n_{y} - 2 n_{d}) \times (n_{y} - n_{d})}$ such that the expression $A_{0 i} - L_{0 i} C_{0}$ is stable. Therefore, for any positive $Q_{0}$ that belongs to the set $R^{(n_{x} + n_{y} - 2 n_{d}) \times (n_{x} + n_{y} - 2 n_{d})}$ , the subsequent Lyapunov equation is given:

P_{0} (A_{0 i} - L_{0 i} C_{0}) + (A_{0 i} - L_{0 i} C_{0})^{T} P_{0} = - Q_{0}

(30)

has an unique solution $P_{0} \in R^{(n_{x} + n_{y} - 2 n_{d}) \times (n_{x} + n_{y} - 2 n_{d})} > 0$ for $\forall i = 1, . . ., r$ .

For further analysis, introduce partitions of $P_{0}$ and $Q_{0}$ as

P_{0} = [\begin{matrix} P_{11} & P_{12} \\ P_{12}^{T} & P_{22} \end{matrix}], Q_{0} = [\begin{matrix} Q_{11} & Q_{12} \\ Q_{12}^{T} & Q_{22} \end{matrix}]

(31)

where $P_{11} \in R^{(n_{x} - n_{d}) \times (n_{x} - n_{d})}$ , $P_{22} \in R^{(n_{y} - n_{d}) \times (n_{y} - n_{d})}$ , $Q_{11} \in R^{(n_{x} - n_{d}) \times (n_{x} - n_{d})}$ , and $Q_{22} \in R^{(n_{y} - n_{d}) \times (n_{y} - n_{d})}$ are symmetric positive definite matrices. Denote

K = P_{11}^{- 1} P_{12} \in R^{(n_{x} - n_{d}) \times (n_{y} - n_{d})}

(32)

By taking a coordinate transformation $ζ (t) = T_{0} ω_{0} (t)$ with

ζ (t) = [\begin{matrix} ζ_{1} (t) \\ ζ_{1} (t) \end{matrix}], T_{0} = [\begin{matrix} I_{n_{x} - n_{d}} & K \\ 0 & I_{n_{y} - n_{d}} \end{matrix}]

where $ζ_{1} (t) \in R^{n_{x} - n_{d}}$ and $ζ_{2} (t) \in R^{n_{y} - n_{d}}$ . Under the coordinate transformation $T_{0}$ , system (28) and (29) becomes

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ζ (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{i} ζ (t) + {\tilde{A}}_{21 i} ω_{1} (t) + {\bar{N}}_{i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) \\ + {\bar{B}}_{i} u_{k} (t) + {\bar{F}}_{i} f_{a} (t) + \bar{D} f_{s} (t)] \end{array}

(33)

η_{0} (t) = \bar{C} ζ (t)

(34)

where

{\bar{A}}_{i} = [\begin{matrix} {\bar{A}}_{11 i} & {\bar{A}}_{12 i} \\ {\bar{A}}_{21 i} & {\bar{A}}_{22 i} \end{matrix}] = [\begin{matrix} A_{22 i} + K C_{22} & - (A_{22 i} + K C_{22}) K \\ C_{22} & - C_{22} K \end{matrix}]

\begin{matrix} {\bar{N}}_{i} = [\begin{matrix} N_{2 i} \\ 0 \end{matrix}], {\bar{B}}_{i} = [\begin{matrix} B_{2 i} \\ 0 \end{matrix}], \bar{F} = [\begin{matrix} F_{2 i} \\ 0 \end{matrix}], \\ \bar{D} = [\begin{matrix} K D_{2} \\ D_{2} \end{matrix}], \bar{C} = [\begin{matrix} 0 & I_{p - q} \end{matrix}] \end{matrix}

(35)

Lemma 8. If $P_{0}$ and $Q_{0}$ have the partition in (31), then the matrix $A_{22 i} + K C_{22}$ is Hurwitz stable for $\forall i = 1, . . ., r$ if the Lyapunov equation (30) is satisfied.

Proof. Taking into account the structure of $A_{0 i}$ , $C_{0}$ , $L_{0 i}$ , $P_{0}$ , and $Q_{0}$ , we find that the matrix condition (30) manifests in the following manner:

\begin{matrix} {([\begin{matrix} A_{22 i} & 0 \\ C_{22 i} & 0 \end{matrix}] - [\begin{matrix} L_{01 i} \\ L_{02 i} \end{matrix}] [\begin{matrix} 0 & I_{p - q} \end{matrix}])}^{T} [\begin{matrix} P_{11} & P_{12} \\ P_{12} & P_{22} \end{matrix}] \\ + [\begin{matrix} P_{11} & P_{12} \\ P_{12} & P_{22} \end{matrix}] ([\begin{matrix} A_{22 i} & 0 \\ C_{22 i} & 0 \end{matrix}] - [\begin{matrix} L_{01 i} \\ L_{02 i} \end{matrix}] [\begin{matrix} 0 & I_{p - q} \end{matrix}]) \\ = [\begin{matrix} Q_{11} & Q_{12} \\ Q_{12} & Q_{22} \end{matrix}] \end{matrix}

It is easy to obtain

A_{22 i}^{T} P_{11} + C_{22}^{T} P_{12}^{T} + P_{11} A_{22 i} + P_{12} C_{22} = - Q_{11}

(37)

This equation can be rewritten as

{(A_{22 i} + P_{11}^{- 1} P_{12} C_{22})}^{T} P_{11} + P_{11} (A_{22 i} + P_{11}^{- 1} P_{12} C_{22}) = - Q_{11}

(38)

With the conditions $P_{11} > 0$ , $Q_{11} > 0$ , and $K = P_{11}^{- 1} P_{12}$ in place, we can infer from Lyapunov theory that the expression $A_{22 i} + K C_{22}$ maintains stability for $\forall i = 1, . . ., r$ .

In the transformed coordinates (35), the Lyapunov matrix $\bar{P}$ can be presented to have a quadratic form.

\bar{P} = {(T_{0}^{T})}^{- 1} P_{0} T_{0}^{- 1} = [\begin{matrix} P_{11} & 0 \\ 0 & {\bar{P}}_{22} \end{matrix}]

(39)

where ${\bar{P}}_{22} = - P_{12}^{T} P_{11}^{- T} P_{12} + P_{22}$ .

System (23) and (24) is given as follows :

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ω_{1} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{11 i} ω_{1} (t) + A_{12 i} ζ_{1} (t) - A_{12 i} K η_{0} (t) \\ + N_{1 i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) + B_{1 i} u_{k} (t) + F_{1 i} f_{a} (t) + E_{1} d_{k} (t] \end{array}

(40)

η_{1} (t) = C_{11} ω_{1} (t)

(41)

Then, system (33) and (34) can be decomposed into:

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} ζ_{1} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{11 i} ζ_{1} (t) + {\bar{A}}_{12 i} ζ_{2} (t) + A_{21 i} ω_{1} (t) \\ + N_{2 i} ϕ_{k} (t, T_{k}^{- 1} ω (t)) + B_{2 i} u_{k} (t) + F_{2 i} f_{a} (t) + K D_{2} f_{s} (t)] \end{array}

(42)

{}_{0}^{C}{D_{t}^{α}} ζ_{2} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{21 i} ζ_{1} (t) + {\bar{A}}_{22 i} ζ_{2} (t) + D_{2} f_{s} (t)]

(43)

η_{0} (t) = ζ_{2} (t)

(44)

Remark 5. Note that in system (40) and (41) the invertibility of $C_{11}$ makes the state component $z_{1} (t)$ available through the measurement output $υ_{1} (t)$ . Also, in system (42)–(44), the state vector $ζ_{2} (t)$ has been directly available through the measurement $υ_{0} (t)$ . Hence, only $ζ_{1} (t)$ is required to be reconstructed.

In this subsection, state transformations have been used to transform the original T-S fuzzy FOS (16) and (17) to sub-systems (40)–(41) and (42)–(44). The transformations offer the benefit of separating the disturbance $d_{k} (t)$ from the state vector $ζ (t)$ . This enables simultaneous estimation of both actuator and sensor faults based on the measurement $η_{0} (t)$ .

Main results

Building upon the preceding discussion and focusing on estimating unmeasurable states and existing faults within the transformed system, we formulate the subsequent fuzzy fractional sliding mode observer (FSMO) and fractional adaptive observer (FAO) for the sub-systems (40)–(41) and (42)–(44).

The following FSMO is constructed for the system (40) and (41):

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} {\hat{ω}}_{1} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{11 i} {\hat{ω}}_{1} (t) + A_{12 i} {\hat{ζ}}_{1} (t) - A_{12 i} K η_{0} (t) \\ + N_{1 i} ϕ_{k} (t, T_{k}^{- 1} \hat{ω} (t)) + B_{1 i} u_{k} (t) F_{1 i} {\hat{f}}_{a} (t) \\ + L_{1 i} (η_{1} (t) - {\hat{η}}_{1} (t)) + E_{1} ϑ (t)] \end{array}

(45)

{\hat{η}}_{1} (t) = C_{11} {\hat{ω}}_{1} (t)

(46)

In this context, ${\hat{ω}}_{1} (t)$ and ${\hat{ζ}}_{1} (t)$ represent the estimated states of the new system, $ω_{1} (t)$ and $ζ_{1} (t)$ , respectively. ${\hat{f}}_{a} (t)$ stands for the estimated actuator fault, while ${\hat{η}}_{1} (t)$ corresponds to the output of the FSMO. The observer gain matrices $L_{1 i} \in R^{n_{d} \times n_{d}}$ are utilized for estimation. The function $ϑ (t)$ , which is discontinuous, is defined as:

\begin{matrix} ϑ (t) = \\ {\begin{matrix} (ρ_{d} + ρ_{0}) \frac{E_{1}^{T} P_{1} (C_{11}^{- 1} η_{1} (t) - {\hat{ω}}_{1} (t))}{‖ E_{1}^{T} P_{1} (C_{11}^{- 1} η_{1} (t) - {\hat{ω}}_{1} (t)) ‖} : E_{1}^{T} P_{1} (C_{11}^{- 1} η_{1} (t) - {\hat{ω}}_{1} (t)) \neq 0 \\ 0 : otherwise \end{matrix} \end{matrix}

(47)

where $ρ_{0} \in R > 0$ and $P_{1} \in R^{n_{d} \times n_{d}} > 0$ are to be determined.

Remark 6. The suggested FSMO (45) and (46) incorporates a discontinuous term, $ϑ (t)$ , employed to counteract external disturbances. The theory of differential inclusion, initially introduced by Filippov,⁴³has found its extension to the realm of FO differential equations.⁴⁴The regularization technique by Filippov substantiates the existence of the FO differential equation that includes a discontinuous function in its final term.

Remark 7. To reduce the effects of chattering caused by the discontinuous term in (47), a continuous approximation can be used by employing the boundary layer technique.⁴⁵ In addition, one can also use a continuous function such as

ϑ (t) = (ρ_{d} + ρ_{0}) \frac{E_{1}^{T} P_{1} (C_{11}^{- 1} η_{1} (t) - {\hat{ω}}_{1} (t))}{‖ E_{1}^{T} P_{1} (C_{11}^{- 1} η_{1} (t) - {\hat{ω}}_{1} (t)) ‖ + δ}

(48)

where $δ$ is small positive scalar.

In this study, we turn our attention to the subsequent sliding surface (SS):

s (t) = C_{11}^{- 1} υ_{1} (t) - {\hat{ω}}_{1} (t)

(49)

Because $η_{1} (t)$ is measurable, the function $s (t)$ relies on ${\hat{ω}}_{1} (t)$ . Thus, through the design of $L_{1 i}$ , we can establish the sliding surface $s (t) = 0$ .

For system (42)–(44), we construct the following FAO:

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} {\hat{ζ}}_{1} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{11 i} {\hat{ζ}}_{1} (t) + {\bar{A}}_{12 i} {\hat{ζ}}_{2} (t) \\ + A_{21 i} C_{11}^{- 1} η_{1} (t) + N_{2 i} ϕ_{k} (t, T_{k}^{- 1} \hat{ω} (t)) + B_{2 i} u_{k} (t) \\ + F_{2 i} {\hat{f}}_{a} (t) + K D_{2} {\hat{f}}_{s} (t) + L_{2 i} (η_{0} (t) - {\hat{η}}_{0} (t))] \end{array}

(50)

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} {\hat{ζ}}_{2} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{21 i} {\hat{ζ}}_{1} (t) + {\bar{A}}_{22 i} {\hat{ζ}}_{2} (t) \\ + D_{2} {\hat{f}}_{s} (t) + L_{3 i} (η_{0} (t) - {\hat{η}}_{0} (t))] \end{matrix}

(51)

{\hat{η}}_{0} (t) = {\hat{ζ}}_{2} (t)

(52)

For ${}_{0}^{C}{D_{t}^{α}} f_{a} (t) \neq 0$ and ${}_{0}^{C}{D_{t}^{α}} f_{s} (t) \neq 0$ , the FE are updated with the following adaptive schemes:

{}_{0}^{C}{D_{t}^{α}} {\hat{f}}_{a} (t) = Γ_{1} \sum_{i = 1}^{r} ψ_{i} (ν (t)) [F_{2 i}^{T} P_{12} e_{η_{0}} (t)] - σ_{1} Γ_{1} {\hat{f}}_{a} (t)

(53)

{}_{0}^{C}{D_{t}^{α}} {\hat{f}}_{s} (t) = Γ_{2} \sum_{i = 1}^{r} ψ_{i} (ν (t)) D_{2}^{T} {\bar{P}}_{22} [{}_{0}^{C}{D_{t}^{α}} e_{η_{0}} (t) + e_{η_{0}} (t)] - σ_{2} Γ_{2} {\hat{f}}_{s} (t)

(54)

In the aforementioned equations, $Γ_{1}$ and $Γ_{2}$ are the learning rate, $σ_{1}$ and $σ_{2}$ are two positive scalars, $e_{υ_{0}} (t)$ is assigned for corresponding $η_{0} (t) - {\hat{η}}_{0} (t)$ .

Let us define the state and fault estimation errors as $e_{ω_{1}} (t) = ω_{1} (t) - {\hat{ω}}_{1} (t)$ , $e_{ζ_{1}} (t) = ζ_{1} (t) - {\hat{ζ}}_{1} (t)$ , $e_{ζ_{2}} (t) = ζ_{2} (t) - {\hat{ζ}}_{2} (t)$ , $e_{fa} (t) = f_{a} (t) - {\hat{f}}_{a} (t)$ and $e_{fs} (t) = f_{s} (t) - {\hat{f}}_{s} (t)$ . Choose $L_{1 i} = (A_{11 i} - A_{11}^{s}) C_{11}^{- 1}$ , $L_{2 i} = {\bar{A}}_{12 i}$ , $L_{3 i} = {\bar{A}}_{22 i} - H$ , where $A_{11}^{s} \in R^{r \times r} < 0$ is an arbitrary matrix and $H$ is the gain of the observer to be determined, then the error dynamics is presented as

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} e_{ω_{1}} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{11}^{s} e_{ω_{1}} (t) + A_{12 i} e_{ζ_{1}} (t) \\ + N_{1 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) + F_{1 i} e_{f_{a}} (t) + E_{1} (d_{k} (t) - ϑ (t))] \end{array}

(55)

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} e_{ζ_{1}} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{11 i} e_{ζ_{1}} (t) + N_{2 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) \\ + F_{2 i} e_{f_{a}} (t) + K D_{2} e_{f_{s}} (t)] \end{matrix}

(56)

{}_{0}^{C}{D_{t}^{α}} e_{ζ_{2}} (t) = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [{\bar{A}}_{21 i} e_{ζ_{1}} (t) + H e_{ζ_{2}} (t) + D_{2} e_{f_{s}} (t)]

(57)

where ${\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) = ϕ_{k} (t, T_{k}^{- 1} ω (t)) - ϕ_{k} (t, T_{k}^{- 1} \hat{ω} (t))$ .

Note that

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} e_{f_{a}} (t) =_{0}^{C} D_{t}^{α} f_{a} (t) -_{0}^{C} D_{t}^{α} {\hat{f}}_{a} (t) \\ =_{0}^{C} D_{t}^{α} f_{a} (t) - Γ_{1} \sum_{i = 1}^{r} ψ_{i} (ν (t)) [F_{2 i}^{T} P_{12} e_{η_{0}} (t)] + σ_{1} Γ_{1} {\hat{f}}_{a} (t) \\ =_{0}^{C} D_{t}^{α} f_{a} (t) - Γ_{1} \sum_{i = 1}^{r} ψ_{i} (ν (t)) [F_{2 i}^{T} P_{12} e_{ξ_{2}} (t)] + σ_{1} Γ_{1} {\hat{f}}_{a} (t) \end{matrix}

(58)

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} e_{f_{s}} (t) =_{0}^{C} D_{t}^{α} f_{s} (t) -_{0}^{C} D_{t}^{α} {\hat{f}}_{s} (t) \\ =_{0}^{C} D_{t}^{α} f_{s} (t) - Γ_{2} \sum_{i = 1}^{r} ψ_{i} (ν (t)) D_{2}^{T} {\bar{P}}_{22} [{}_{0}^{C}{D_{t}^{α}} e_{η_{0}} (t) + e_{η_{0}} (t)] \\ + σ_{2} Γ_{2} {\hat{f}}_{s} (t) \\ =_{0}^{C} D_{t}^{α} f_{s} (t) - Γ_{2} \sum_{i = 1}^{r} ψ_{i} (ν (t)) D_{2}^{T} {\bar{P}}_{22} e_{ζ_{2}} (t) \\ - Γ_{2} \sum_{i = 1}^{k} ψ_{i} (ν (t)) D_{2}^{T} {\bar{P}}_{22} [{\bar{A}}_{21 i} e_{ξ_{1}} (t) + H e_{ξ_{2}} (t) + D_{2} e_{f_{s}} (t)] \\ + σ_{2} Γ_{2} {\hat{f}}_{s} (t) \end{matrix}

(59)

Analysis stability

The subsequent theorem offers a sufficient condition that guarantees the stability of the error dynamics.

Theorem 1. Assuming that Assumptions 1–6 are valid, if there are matrices $X < 0$ , $P_{1} > 0$ , $P_{11} > 0$ , ${\bar{P}}_{22} > 0$ , $Y$ , and $P_{12}$ of suitable dimensions, along with scalars $ε_{1} > 0$ and $ε_{2} > 0$ , such that for each $i$ in the series $1, 2, . . ., r$ ,

\begin{matrix} ϒ_{i} = \\ [\begin{matrix} ϒ_{11 i} & P_{1} A_{12 i} & 0 & P_{1} F_{1 i} & 0 & P_{1} N_{1 i} & 0 \\ ϒ_{22 i} & C_{22}^{T} {\bar{P}}_{22} & P_{11} F_{2 i} & ϒ_{25 i} & 0 & P_{11} N_{2 i} \\ * & ϒ_{33 i} & P_{12}^{T} F_{2 i} & - H^{T} {\bar{P}}_{22} D_{2} & 0 & 0 \\ * & * & G - 2 σ_{1} I & 0 & 0 & 0 \\ * & * & * & ϒ_{55 i} & 0 & 0 \\ * & * & * & * & - ε_{1} I & 0 \\ * & * & * & * & * & - ε_{2} I \end{matrix}] < 0 \end{matrix}

(60)

where

\begin{matrix} ϒ_{11 i} = X + X^{T}, \\ ϒ_{22 i} = A_{22 i}^{T} P_{11} + P_{11} A_{22 i} + P_{12} C_{22} + C_{22}^{T} P_{12}^{T} \\ + (ε_{1} + ε_{2}) L_{ϕ}^{2} ‖ T_{k}^{- 1} ‖^{2} I, \\ ϒ_{33 i} = Y + Y^{T}, \\ ϒ_{25 i} = P_{12} D_{2} - C_{22}^{T} {\bar{P}}_{22} D_{2}, \\ ϒ_{55 i} = M - 2 σ_{2} I - 2 D_{2}^{T} {\bar{P}}_{22} D_{2} \end{matrix}

then for a given matrices $Γ_{1} > 0$ , $Γ_{2} > 0$ , $G > 0$ , $M > 0$ and scalars $σ_{1} > 0$ , $σ_{2} > 0$ , the state and fault estimation errors are uniformly bounded.

The parameter matrices $H$ and $A_{11}^{s}$ can be computed by

H = {\bar{P}}_{22}^{- 1} Y

(61)

A_{11}^{s} = P_{1}^{- 1} X

(62)

Proof. Let us propose the following Lyapunov function candidate

V (t) = \sum_{i = 1}^{5} V_{i} (t)

(63)

of which $V_{1} (t) = e_{ω_{1}}^{T} (t) P_{1} e_{ω_{1}} (t)$ , $V_{2} (t) = e_{ζ_{1}}^{T} (t) P_{11} e_{ζ_{1}} (t)$ , $V_{3} (t) = e_{ζ_{2}}^{T} (t) {\bar{P}}_{22} e_{ζ_{2}} (t)$ , $V_{4} (t) = e_{f_{a}}^{T} (t) Γ_{1}^{- 1} e_{f_{a}} (t)$ , and $V_{5} (t) = e_{f_{s}}^{T} (t) Γ_{2}^{- 1} e_{f_{s}} (t)$ .

It follows from Lemma 2 that ${}_{0}^{C}{D_{t}^{α}} V_{1} (t) \leq 2 e_{ω_{1}}^{T} (t) P_{1} ({}_{0}^{C}{D_{t}^{α}} e_{ω_{1}} (t))$ with implies that

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} V_{1} (t) \leq 2 e_{ω_{1}}^{T} (t) P_{1} {\sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{11}^{s} e_{ω_{1}} (t) + A_{12 i} e_{ζ_{1}} (t) + N_{1 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) + F_{1 i} e_{f_{a}} (t) + E_{1} (d_{k} (t) - ϑ (t))]} \\ = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ω_{1}}^{T} (t) (A_{11}^{s T} P_{1} + P_{1} A_{11}^{s}) e_{ω_{1}} (t) + 2 e_{ω_{1}}^{T} (t) P_{1} A_{12 i} e_{ζ_{1}} (t) + 2 e_{ω_{1}}^{T} (t) P_{1} N_{1 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) \\ + 2 e_{ω_{1}}^{T} (t) P_{1} F_{1 i} e_{f_{a}} (t) + 2 e_{ω_{1}}^{T} (t) P_{1} E_{1} (d_{k} (t) - ϑ (t))] \end{array}

(64)

According to Lemma 5, we have

\begin{matrix} 2 e_{ω_{1}}^{T} (t) P_{1} N_{1 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) \leq ε_{1}^{- 1} e_{ω_{1}}^{T} (t) P_{1} N_{1 i} N_{1 i}^{T} P_{1} e_{ω_{1}} (t) \\ + ε_{1} {\tilde{ϕ}}_{k}^{T} (t, T_{k}^{- 1} ω (t)) {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) \end{matrix}

(65)

where $ε_{1}$ is a positive scalar.

Since ${\tilde{ϕ}}_{k} (.)$ is Lipschitz with respect to state $x_{k} (t) = T_{k}^{- 1} ω (t)$ , and

\hat{ω} (t) = [\begin{matrix} C_{11}^{- 1} η_{1} (t) \\ {\hat{ζ}}_{1} (t) - K η_{0} (t) \end{matrix}],

we have

{\tilde{ϕ}}_{k}^{T} (t, T_{k}^{- 1} ω (t)) {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) = ‖ ϕ_{k} (t, T_{k}^{- 1} ω (t)) - ϕ_{k} (t, T_{k}^{- 1} \hat{ω} (t)) ‖^{2} \leq L_{ϕ}^{2} ‖ T_{k}^{- 1} ‖^{2} ‖ e_{ζ_{1}} (t) ‖^{2}

(66)

where $L_{ϕ}$ is the Lipschitz constant of $ϕ_{k} (.)$ .

From (65) and (66), we have

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V_{1} (t) \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ω_{1}}^{T} (t) (A_{11}^{sT} P_{1} + P_{1} A_{11}^{s} + ε_{1}^{- 1} P_{1} N_{1 i} N_{1 i}^{T} P_{1}) e_{ω_{1}} (t) \\ + 2 e_{ω_{1}}^{T} (t) P_{1} A_{12 i} e_{ζ_{1}} (t) + ε_{1} L_{ϕ}^{2} ‖ T^{- 1} ‖^{2} ‖ e_{ξ_{1}} (t) ‖^{2} + 2 e_{ω_{1}}^{T} (t) P_{1} F_{1 i} e_{f_{a}} (t) \\ + 2 e_{ω_{1}}^{T} (t) P_{1} E_{1} (d_{k} (t) - ϑ (t))] \end{matrix}

(67)

Considering (18) and (47), we have

\begin{matrix} 2 e_{ω_{1}}^{T} (t) P_{1} E_{1} (d_{k} (t) - ϑ (t)) = 2 e_{ω_{1}}^{T} (t) P_{1} E_{1} d_{k} (t) - 2 (ρ_{d} + ρ_{0}) e_{ω_{1}}^{T} (t) P_{1} E_{1} \frac{E_{1}^{T} P_{1} e_{ω_{1}} (t)}{‖ E_{1}^{T} P_{1} e_{ω_{1}} (t) ‖} \\ \leq 2 ‖ e_{ω_{1}}^{T} (t) P_{1} E_{1} ‖ ‖ d_{k} (t) ‖ - 2 (ρ_{d} + ρ_{0}) \frac{{‖ e_{ω_{1}}^{T} (t) P_{1} E_{1} ‖}^{2}}{‖ e_{ω_{1}}^{T} (t) P_{1} E_{1} ‖} \\ \leq 2 ρ_{d} ‖ e_{ω_{1}}^{T} (t) P_{1} E_{1} ‖ - 2 (ρ_{d} + ρ_{0}) ‖ e_{ω_{1}}^{T} (t) P_{1} E_{1} ‖ \\ = - 2 ρ_{0} ‖ e_{ω_{1}}^{T} (t) P_{1} E_{1} ‖ < 0 \end{matrix}

(68)

Therefore

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V_{1} (t) \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ω_{1}}^{T} (t) (A_{11}^{sT} P_{1} + P_{1} A_{11}^{s} + ε_{1}^{- 1} P_{1} N_{1 i} N_{1 i}^{T} P_{1}) e_{ω_{1}} (t) \\ + 2 e_{ω_{1}}^{T} (t) P_{1} A_{12 i} e_{ζ_{1}} (t) + ε_{1} {\tilde{L}}_{ϕ}^{2} {‖ e_{ξ_{1}} (t) ‖}^{2} + 2 e_{ω_{1}}^{T} (t) P_{1} F_{1 i} e_{f_{a}} (t)] \end{matrix}

(69)

where ${\tilde{L}}_{ϕ} = L_{ϕ} ‖ T_{k}^{- 1} ‖$ . Drawing upon Lemma 2, consider the $α$ -order derivative of $V_{2} (t)$ and $V_{3} (t)$ , respectively, which results in

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} V_{2} (t) \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ζ_{1}}^{T} (t) ({\bar{A}}_{11 i}^{T} P_{11} + P_{11} {\bar{A}}_{11 i}) e_{ζ_{1}} (t) + 2 e_{ζ_{1}}^{T} (t) P_{11} N_{2 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) + 2 e_{ζ_{1}}^{T} (t) P_{11} F_{2 i} e_{f_{a}} (t) + 2 e_{ζ_{1}}^{T} (t) P_{12} D_{2} e_{f_{s}} (t)] \\ \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ζ_{1}}^{T} (t) ({\bar{A}}_{11 i}^{T} P_{11} + P_{11} {\bar{A}}_{11 i} + ε_{2}^{- 1} P_{11} N_{2 i} N_{2 i}^{T} P_{11} + ε_{2} {\tilde{L}}_{ϕ}^{2}) e_{ζ_{1}} (t) + 2 e_{ζ_{1}}^{T} (t) P_{11} F_{2 i} e_{f_{a}} (t) + 2 e_{ζ_{1}}^{T} (t) P_{12} D_{2} e_{f_{s}} (t)] \end{array}

(70)

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V_{3} (t) & \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ζ_{2}}^{T} (t) (H^{T} {\bar{P}}_{22} + {\bar{P}}_{22} H) e_{ζ_{2}} (t) + 2 e_{ζ_{2}}^{T} (t) {\bar{P}}_{22} {\bar{A}}_{21 i} e_{ζ_{1}} (t) + 2 e_{ζ_{2}}^{T} (t) {\bar{P}}_{22} D_{2} e_{fs} (t)] \end{matrix}

(71)

Taking the derivative of $V_{4} (t)$ gives

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V_{4} (t) \leq 2 e_{f_{a}}^{T} (t) Γ_{0}^{C} D_{t}^{α} e_{f_{a}} (t) \\ = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [2 e_{f_{a}}^{T} (t) Γ_{0}^{C} D_{t}^{α} f_{a} (t) + 2 σ_{1} e_{f_{a}}^{T} (t) {\hat{f}}_{a} (t) - 2 e_{f_{a}}^{T} (t) F_{2 i}^{T} P_{12} e_{ξ_{2}} (t)] \\ = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [2 e_{f_{a}}^{T} (t) Γ_{0}^{C} D_{t}^{α} f_{a} (t) + 2 σ_{1} e_{f_{a}}^{T} (t) f_{a} (t) - 2 e_{f_{a}}^{T} (t) F_{2 i}^{T} P_{12} e_{ξ_{2}} (t) - 2 σ_{1} e_{f_{a}}^{T} (t) e_{f_{a}} (t)] \end{matrix}

(72)

For matrix $G > 0$ and according to Lemma 6, we can write that

\begin{matrix} 2 e_{f_{a}}^{T} (t) Γ_{0}^{C} D_{t}^{α} f_{a} (t) \leq e_{f_{a}}^{T} (t) G e_{f_{a}} (t) +_{0}^{C} D_{t}^{α} f_{a}^{T} (t) Γ_{1}^{- 1} G^{- 1} Γ_{0}^{C} D_{t}^{α} f_{a} (t) \\ \leq e_{f_{a}}^{T} (t) G e_{f_{a}} (t) + ρ_{aa}^{2} λ_{max} (Γ_{1}^{- 1} G^{- 1} Γ_{1}^{- 1}) \end{matrix}

(73)

Moreover

2 σ_{1} e_{f_{a}}^{T} (t) f_{a} (t) \leq σ_{1} ({‖ e_{f_{a}} (t) ‖}^{2} + ρ_{a}^{2})

(74)

Substituting (73) and (74) into (72) gives

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V_{4} (t) & \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [- 2 e_{f_{a} (t)}^{T} F_{2 i}^{T} P_{12} e_{ζ_{2}} (t) + e_{f_{a} (t)}^{T} (G - 2 σ_{1} I) e_{f_{a}} (t) + ρ_{aa}^{2} λ_{max} (Γ_{1}^{- 1} G^{- 1} Γ_{1}^{- 1}) + σ_{1} ρ_{a}^{2}] \end{matrix}

(75)

In the same way, the derivative of $V_{5} (t)$ is

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} V_{5} (t) \leq 2 e_{f_{s}}^{T} (t) Γ_{s_{0}^{- 1}}^{C} D_{t}^{α} e_{f_{s}} (t) \\ = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [2 e_{f_{s} (t)}^{T} Γ_{2}^{- 1} D^{α} f_{s} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} e_{ζ_{2}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} {\bar{A}}_{21 i} e_{ζ_{1}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} H e_{ζ_{2}} (t) \\ - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} D_{2} e_{f_{s}} (t) + 2 σ_{2} e_{f_{s} (t)}^{T} {\hat{f}}_{s} (t)] \\ \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [- 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} e_{ζ_{2}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} {\bar{A}}_{21 i} e_{ζ_{1}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} H e_{ζ_{2}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} D_{2} e_{f_{s}} (t) \\ + e_{f_{s} (t)}^{T} (M - 2 σ_{2} I) e_{f_{s}} (t) + ρ_{s s}^{2} λ_{\max} (Γ_{2}^{- 1} M^{- 1} Γ_{2}^{- 1}) + σ_{2} ρ_{s}^{2}] \end{array}

(76)

By leveraging equations (69), (70), (71), (75), and (76), we can derive the derivative of $V (t)$ as follows:

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V (t) \leq \sum_{i = 1}^{r} ψ_{i} (ν (t)) [e_{ω_{1}}^{T} (t) (A_{11}^{sT} P_{1} + P_{1} A_{11}^{s} + ε_{1}^{- 1} P_{1} N_{1 i} N_{1 i}^{T} P_{1}) e_{ω_{1}} (t) \\ + e_{ζ_{1}}^{T} (t) ({\bar{A}}_{11 i}^{T} P_{11} + P_{11} {\bar{A}}_{11 i} + ε_{2}^{- 1} P_{11} N_{2 i} N_{2 i}^{T} P_{11} + (ε_{1} + ε_{2}) {\tilde{L}}_{ϕ}^{2}) e_{ζ_{1}} (t) \\ + e_{ζ_{2}}^{T} (t) (H^{T} {\bar{P}}_{22} + {\bar{P}}_{22} H) e_{ζ_{2}} (t) + 2 e_{ω_{1}}^{T} (t) P_{1} A_{12 i} e_{ζ_{1}} (t) + 2 e_{ω_{1}}^{T} (t) P_{1} F_{1 i} e_{f_{a}} (t) \\ + 2 e_{ζ_{1}}^{T} (t) P_{11} F_{2 i} e_{fa} (t) + 2 e_{ζ_{1}}^{T} (t) P_{12} D_{2} e_{f_{s}} (t) + 2 e_{ζ_{2}}^{T} (t) {\bar{P}}_{22} {\bar{A}}_{21 i} e_{ζ_{1}} (t) \\ - 2 e_{f_{a}}^{T} (t) F_{2 i}^{T} P_{12} e_{ξ_{2}} (t) + e_{f_{a} (t)}^{T} (G - 2 σ_{1} I) e_{f_{a}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} {\bar{A}}_{21 i} e_{ζ_{1}} (t) \\ - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} H e_{ζ_{2}} (t) - 2 e_{f_{s} (t)}^{T} D_{2}^{T} {\bar{P}}_{22} D_{2} e_{f_{s}} (t) + e_{f_{s} (t)}^{T} (M - 2 σ_{2} I) e_{f_{s}} (t) \\ + ρ_{aa}^{2} λ_{max} (Γ_{1}^{- 1} G^{- 1} Γ_{1}^{- 1}) + σ_{1} ρ_{a}^{2} + ρ_{ss}^{2} λ_{max} (Γ_{2}^{- 1} M^{- 1} Γ_{2}^{- 1}) + σ_{2} ρ_{s}^{2}] \\ = \sum_{i = 1}^{r} ψ_{i} (ν (t)) [ξ^{T} (t) Φ_{i} ξ (t) + κ_{2}] \end{matrix}

(77)

where

ξ (t) = {[\begin{matrix} e_{ω_{1}}^{T} (t) & e_{ζ_{1}}^{T} (t) & e_{ζ_{2}}^{T} (t) & e_{f_{a}}^{T} (t) & e_{f_{s}}^{T} (t) \end{matrix}]}^{T}

\begin{matrix} Φ_{i} & = [\begin{matrix} Φ_{11 i} + ε_{1}^{- 1} P_{1} N_{1 i} N_{1 i}^{T} P_{1} & P_{1} A_{12 i} & 0 & P_{1} F_{1 i} & 0 \\ * & Φ_{22 i} + ε_{2}^{- 1} P_{11} N_{2 i} N_{2 i}^{T} P_{11} & {\bar{A}}_{21 i}^{T} {\bar{P}}_{22} & P_{11} F_{2 i} & Φ_{25 i} \\ * & * & Φ_{33 i} & - P_{12}^{T} F_{2 i} & - H^{T} {\bar{P}}_{22} D_{2} \\ * & * & * & G - 2 σ_{1} I & 0 \\ * & * & * & * & Φ_{55 i} \end{matrix}] \\ Φ_{11} & = A_{11}^{sT} P_{1} + P_{1} A_{11}^{s} \\ Φ_{22} & = {\bar{A}}_{11 i}^{T} P_{11} + P_{11} {\bar{A}}_{11 i} + (ε_{1} + ε_{2}) {\tilde{L}}_{ϕ}^{2} I \\ Φ_{33} & = H^{T} {\bar{P}}_{22} + {\bar{P}}_{22} H \\ Φ_{25 i} & = P_{12} D_{2} - {\bar{A}}_{21 i}^{T} {\bar{P}}_{22} D_{2} \\ Φ_{55 i} & = M - 2 σ_{2} I - 2 D_{2}^{T} {\bar{P}}_{22} D_{2} \\ κ_{2} & = σ_{1} ρ_{a}^{2} + σ_{2} ρ_{s}^{2} + ρ_{aa}^{2} λ_{max} (Γ_{1}^{- 1} G^{- 1} Γ_{1}^{- 1}) + ρ_{ss}^{2} λ_{max} (Γ_{2}^{- 1} M^{- 1} Γ_{2}^{- 1}) \end{matrix}

Let $- κ_{1}$ be the maximum eigenvalue of $Φ_{i}$ then

\begin{matrix} {}_{0}^{C}{D_{t}^{α}} V (t) \leq - κ_{1} ‖ ξ (t) ‖^{2} + κ_{2} \\ \leq - κ_{1} ({‖ e_{ω_{1}} (t) ‖}^{2} + {‖ e_{ζ_{1}} (t) ‖}^{2} + {‖ e_{ζ_{2}} (t) ‖}^{2} + {‖ e_{f_{a}} (t) ‖}^{2} + {‖ e_{f_{s}} (t) ‖}^{2}) + κ_{2} \end{matrix}

(78)

From (63)

\begin{matrix} V (t) \leq λ_{max} (P_{1}) ‖ e_{ω_{1}} (t) ‖^{2} + λ_{max} (P_{11}) ‖ e_{ζ_{1}} (t) ‖^{2} + λ_{max} ({\bar{P}}_{22}) ‖ e_{ζ_{2}} (t) ‖^{2} \\ + λ_{max} (Γ_{1}^{- 1}) ‖ e_{f_{a}} (t) ‖^{2} + λ_{max} (Γ_{2}^{- 1}) ‖ e_{f_{s}} (t) ‖^{2} \\ \leq κ_{3} ({‖ e_{ω_{1}} (t) ‖}^{2} + {‖ e_{ζ_{1}} (t) ‖}^{2} + {‖ e_{ζ_{2}} (t) ‖}^{2} + {‖ e_{f_{a}} (t) ‖}^{2} + {‖ e_{f_{s}} (t) ‖}^{2}) \end{matrix}

(79)

where

κ_{3} = max (λ_{max} (P_{1}), λ_{max} (P_{11}), λ_{max} ({\bar{P}}_{22}), λ_{max} (Γ_{1}^{- 1}), λ_{max} (Γ_{2}^{- 1}))

Then

{}_{0}^{C}{D_{t}^{α}} V (t) \leq - κ_{4} V (t) + κ_{2}

(80)

where

κ_{4} = \frac{κ_{1}}{κ_{3}}

Then, invoking Lemma 3, we can derive

V (t) \leq V (0) E_{α} (- κ_{4} t^{α}) + \frac{κ_{2} d}{κ_{4}}

(81)

where $d$ is a positive constant.

We can get from Lemma 1 that

| E_{α} (- κ_{4} t^{α}) | \leq \frac{λ}{1 + κ_{4} t^{α}}

(82)

Finally, we can obtain

V (t) \leq V (0) \frac{λ}{1 + κ_{4} t^{α}} + \frac{κ_{2} d}{κ_{4}}, t \geq 0

(83)

we further have

\begin{matrix} V (t) \geq λ_{min} (P_{1}) ‖ e_{ω_{1}} (t) ‖^{2} + λ_{min} (P_{11}) ‖ e_{ζ_{1}} (t) ‖^{2} \\ + λ_{min} ({\bar{P}}_{22}) ‖ e_{ζ_{2}} (t) ‖^{2} + λ_{min} (Γ_{1}^{- 1}) ‖ e_{f_{a}} (t) ‖^{2} \\ + λ_{min} (Γ_{2}^{- 1}) ‖ e_{f_{s}} (t) ‖^{2} \\ \geq κ_{5} ({‖ e_{ω_{1}} (t) ‖}^{2} + {‖ e_{ζ_{1}} (t) ‖}^{2} + {‖ e_{ζ_{2}} (t) ‖}^{2} + {‖ e_{f_{a}} (t) ‖}^{2} + {‖ e_{f_{s}} (t) ‖}^{2}) \\ \geq κ_{5} ‖ ξ (t) ‖^{2} \end{matrix}

(84)

where $κ_{5} = min (λ_{min} (P_{1}), λ_{min} (P_{11}), λ_{min} ({\bar{P}}_{22}), λ_{min} (Γ_{1}^{- 1}), λ_{min} (Γ_{2}^{- 1}))$ .

Therefore

‖ ξ (t) ‖ \leq \sqrt{\frac{V (0)}{κ_{5}} \frac{λ}{1 + κ_{4} t^{α}} + \frac{κ_{2} d}{κ_{4} κ_{5}}}

(85)

Based on (85), it is concluded that the error dynamics is stable. In view of (85), when $t$ approaches infinity, we can get that

lim_{t \to \infty} ‖ ξ (t) ‖ \leq \sqrt{\frac{κ_{2} d}{κ_{4} κ_{5}}}

(86)

Therefore, it is concluded that the state and fault estimation errors are uniformly bounded and all signals $(e_{ω_{1}} (t), {e_{ζ}}_{1} (t), {e_{ζ}}_{2} (t), e_{f_{a}} (t), e_{f_{s}} (t))$ tend to a ball centered at the origin.

Furthermore, according to the Schur complement and by changing decision variables in LMI such that, $X = P_{1} A_{11}^{s}$ and $Y = {\bar{P}}_{22} H$ , it follows that $Φ_{i} < 0$ is equivalent to (60). This ends the proof.

□

Sliding motion reachability

The following theorem demonstrates that by choosing an appropriate $ρ_{0}$ value in equation (48), the error system (55) reaches the sliding surface (49) within a finite duration. This initiates a continuous sliding motion on the surface thereafter.

Theorem 2. The trajectory of the error system (55) converges towards the SS $s (t) = 0$ within a finite time duration $t_{s}$ and remains on it, given that the parameter $ρ_{0}$ satisfies the following criterion:

ρ_{0} \geq ‖ A_{1}^{s} ‖ ‖ s (t) ‖ + (a_{12} + n_{1} {\tilde{L}}_{ϕ}) ‖ e_{ζ_{1}} (t) ‖ + f_{1} ‖ e_{f_{a}} (t) ‖ + ρ_{0}

(87)

where $ϱ_{0}$ is a small positive constant.

Proof. We set the Lyapunov function to:

V_{s} (t) = \frac{1}{2} s^{T} (t) s (t)

(88)

Applying the Caputo fractional derivative to $V_{s} (t)$ and utilizing Lemma 2, we derive:

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} V (t) \leq s^{T} {(t)}_{0}^{C} D_{t}^{α} s (t) \\ = s^{T} (t) {\sum_{i = 1}^{r} ψ_{i} (ν (t)) [A_{11}^{s} s (t) + A_{12 i} e_{ζ_{1}} (t) + N_{1 i} {\tilde{ϕ}}_{k} (t, T_{k}^{- 1} ω (t)) + F_{1 i} e_{f_{a}} (t) + E_{1} (d_{k} (t) - ϑ (t))]} \\ \leq ‖ s (t) ‖ {\sum_{i = 1}^{r} ψ_{i} (ν (t)) [‖ A_{11}^{s} ‖ ‖ s (t) ‖ + (‖ A_{12 i} ‖ + {\tilde{L}}_{ϕ} ‖ N_{1 i} ‖) ‖ e_{ζ_{1}} (t) ‖ + ‖ F_{1 i} ‖ ‖ e_{f_{a}} (t) ‖} - ρ_{0} ‖ E_{1}^{T} s (t) ‖ \\ \leq ‖ E_{1}^{- T} ‖ ‖ E_{1}^{T} s (t) ‖ {\sum_{i = 1}^{r} ψ_{i} (ν (t)) [‖ A_{11}^{s} ‖ ‖ s (t) ‖ + (‖ A_{12 i} ‖ + {\tilde{L}}_{ϕ} ‖ N_{1 i} ‖) ‖ e_{ζ_{1}} (t) ‖ + ‖ F_{1 i} ‖ ‖ e_{f_{a}} (t) ‖ - ρ_{0}]} \end{array}

(89)

Let us define

a_{12} = max_{i = 1, . . ., k} (‖ A_{12 i} ‖), n_{1} = max_{i = 1, . . ., k} (‖ N_{1 i} ‖), f_{1} = max_{i = 1, . . ., k} (‖ F_{1 i} ‖)

Assuming a specific selection for the gain $ρ_{0}$ , namely:

ϱ_{0} \geq ‖ A_{1}^{s} ‖ ‖ s (t) ‖ + (a_{12} + n_{1} {\tilde{L}}_{ϕ}) ‖ e_{ζ_{1}} (t) ‖ + f_{1} ‖ e_{f_{a}} (t) ‖ + ρ_{0}

(90)

where $ρ_{0}$ is some small positive constant, then

{}_{0}^{C}{D_{t}^{α}} V (t) \leq - ϱ_{0} ‖ s (t) ‖, \forall ‖ s (t) ‖ \neq 0

(91)

Thus, the SS $s (t) = 0$ is reached within a finite duration $t_{s}$ . Consequently, for all $t \geq t_{s}$ , we observe that $s (t) =_{0}^{C} D_{t}^{α} s (t) = 0$ .

To ascertain the reaching time, we integrate both sides of equation (91) with respect to the fractional integral from $0$ to the reaching time $t_{s}$ .

V (t_{s}) - V^{α - 1} (0) \frac{t_{s}^{α - 1}}{Γ (α)} \leq - {ϱ_{0}}_{0}^{C} D_{t}^{- α} ‖ s (t) ‖

(92)

Based on Lemma 4 and $V_{s} (t_{s}) = 0$ , one gets

\begin{matrix} - V^{α - 1} (0) \frac{t_{s}^{α - 1}}{Γ (α)} \leq - ϱ_{0} T_{0} \\ t_{s}^{α - 1} \geq \frac{Γ (α) ϱ_{0} T_{0}}{V (0)} \\ \frac{1}{t_{s}^{1 - α}} \geq \frac{Γ (α) ϱ_{0} T_{0}}{V (0)} \\ t_{s} \leq {(\frac{V (0)}{ϱ_{0} T_{0} Γ (α)})}^{\frac{1}{1 - α}} \end{matrix}

(93)

Hence, the path of system (55) gravitates towards the SS within a finite time frame $t_{s}$ . This brings our proof to a conclusion.

□

Algorithm

Design steps for FE in T-S Fuzzy FOSs.

Step 1. Define the FONS (9) and (10) with order $0 < α < 1$ . Using the technique of sector nonlinearity, transform it into a T-S fuzzy system (16) and (17).

Step 2. Ensure the assumptions 1–6 related to detectability, matching conditions, Lipschitz continuity, and boundedness of faults and disturbances are satisfied.

Step 3. Choose appropriate matrices $T_{k}$ and $S_{k}$ to obtain the transformed system (21) and (22). Decompose the system into two subsystems such that:

– Subsystem 1 (23) and (24): Includes the effects of disturbances but not sensor faults.

– Subsystem 2 (25) and (26): Incorporates sensor faults but no disturbances.

Step 4. Compute the matrices $K$ , $L_{1 i}$ , $L_{2 i}$ , $L_{3} i$ , and $H$ from the LMIs (60).

Step 5. Construct a T-S fuzzy FSMO (45) and (46) based on Subsystem 1. Design the observer gain $ϑ (t)$ to neutralize the impact of external disturbances.

Step 6. Formulate a T-S fuzzy FAO (50)-(52) based on Subsystem 2.

Step 7. Compute the faults estimates from (53) - (54).

Step 8. Validate the proposed fault estimation scheme with simulation scenarios or real-world examples. Demonstrate the effectiveness and accuracy of the fault estimation algorithm.

These steps provide a structured approach to designing fault estimation observers for T-S fuzzy fractional-order systems. Ensure to tailor the specific details and calculations based on your system’s parameters and the provided main results.

Simulation results

In this section, we show an example to illustrate the efficacy of our approach in estimating both sensor and actuator faults.

Let’s consider the following T-S fuzzy FOS represented by equations (16)-(17) with two fuzzy rules and .

\begin{array}{l} {}_{0}^{C}{D_{t}^{α}} x_{k} (t) = \sum_{i = 1}^{2} ψ_{i} (ν (t)) [A_{k i} x_{k} (t) + N_{k i} ϕ_{k} (t, x_{k} (t)) \\ + B_{k i} u_{k} (t) + F_{k i} f_{a} (t) + E_{k} d_{k} (t)] \\ y_{k} (t) = C_{k} x_{k} (t) + D_{k} f_{s} (t) \end{array}

where

\begin{matrix} A_{1} = [\begin{matrix} - 2 & 1 & 1 \\ 1 & - 3 & 0 \\ 2 & 1 & - 4 \end{matrix}], A_{2} = [\begin{matrix} - 3 & 2 & - 2 \\ 5 & - 3 & 0 \\ 0.5 & 0.5 & - 4 \end{matrix}], \\ N_{1} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], N_{2} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] \end{matrix}

\begin{matrix} B_{1} = [\begin{matrix} 1 \\ 0.3 \\ 0.5 \end{matrix}], B_{2} = [\begin{matrix} 0.5 \\ 1 \\ 0.25 \end{matrix}], F_{1} = [\begin{matrix} 0.5 \\ - 1 \\ 0.25 \end{matrix}], \\ F_{2} = [\begin{matrix} - 1 \\ 0.52 \\ 1 \end{matrix}], E = [\begin{matrix} 0.1 \\ 0.2 \\ 0.2 \end{matrix}] \end{matrix}

C = [\begin{matrix} 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}], D = [\begin{matrix} 0 \\ 0.9 \end{matrix}], ϕ (t, x (t)) = 0.1 \sin (x_{1} (t))

The membership functions of rules 1 and 2 are presented as follows:

ψ_{1} (ν (t)) = \frac{1 - \tanh (x_{k 1} (t))}{2}, ψ_{2} (ν (t)) = \frac{1 + \tanh (x_{k 1} (t))}{2}

For simulation purposes, we use the input signal $u_{k} (t) = \sin (t)$ , the external disturbance $d_{k} (t) = 0.1 \sin (0.2 t)$ , and the time-varying faults.

f_{a} (t) = {\begin{matrix} 0 & t \leq 3 \\ 2 \sin (\frac{3}{2} t) & 3 \leq t \leq 15 \\ 2 rect (t) & t \geq 15 \end{matrix}

f_{s} (t) = {\begin{matrix} 0 & t \leq 5 \\ 0.1 (t - 5) & 5 < t \leq 15 \\ 1 & t > 15 \end{matrix}

We verified that Assumptions 1–6 are satisfied. We express $T_{k}$ and $S_{k}$ as follows:

T_{k} = [\begin{matrix} 0.333 & 0 & 0.333 \\ - 2 & 1 & 0 \\ - 2 & 0 & 1 \end{matrix}], S_{k} = [\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}]

Therefore the parameter matrices, in the new coordinate, become:

T_{k} A_{k 1} T_{k}^{- 1} = [\begin{matrix} - 0.6667 & 0.6667 & - 0.7778 \\ - 9 & - 5 & 1 \\ - 8 & - 1 & - 3.3333 \end{matrix}]

T_{k} A_{k 2} T_{k}^{- 1} = [\begin{matrix} - 3.1667 & 0.8333 & - 0.9444 \\ 5 & - 7 & 2.3333 \\ - 0.5 & - 3.5 & 0.1667 \end{matrix}]

\begin{matrix} T_{k} N_{k 1} = [\begin{matrix} 0.3333 \\ 0 \\ 1 \end{matrix}], T_{k} N_{k 2} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], \\ T_{k} B_{k 1} = [\begin{matrix} 0.5 \\ - 1.7 \\ - 1.5 \end{matrix}], T_{k} B_{k 2} = [\begin{matrix} 0.25 \\ 0 \\ - 0.75 \end{matrix}] \end{matrix}

T_{k} E_{k} = [\begin{matrix} 0.1 \\ 0 \\ 0 \end{matrix}], T_{k} F_{k 1} = [\begin{matrix} 0.25 \\ - 2 \\ - 0.75 \end{matrix}], T_{k} F_{k 2} = [\begin{matrix} 0 \\ 2.52 \\ 3 \end{matrix}]

S_{k} C_{k} T_{k}^{- 1} = [\begin{matrix} 3 & 0 & 0 \\ 0 & 1 & - 1 \end{matrix}], S_{k} D_{k} = [\begin{matrix} 0 \\ 0.9 \end{matrix}]

In this investigation, we choose $σ_{1} = 1$ , $σ_{2} = 3$ , and $L_{ϕ} = 0.1$ . Utilizing Theorem 1 and solving inequality (60) for $i = 1, 2$ , we derive a collection of viable solutions such as:

\begin{matrix} P_{1} = 0.7974, X = - 0.5600, \\ P_{11} = [\begin{matrix} 2.7380 & - 1.2030 \\ - 1.2030 & 1.1598 \end{matrix}], P_{12} = [\begin{matrix} - 0.0190 \\ 0.3210 \end{matrix}] \end{matrix}

{\bar{P}}_{22} = 0.9498, Y = - 0.4718, ε_{1} = 0.1898, ε_{2} = 0.7782

Then $A_{11}^{s}$ , $H$ and the observer gain matrices can be computed as

\begin{matrix} A_{11}^{s} = - 0.7023, H = - 0.4967, L_{11} = 0.0119, \\ L_{12} = - 0.8215, L_{21} = [\begin{matrix} 0.6181 \\ 2.0028 \end{matrix}] \end{matrix}

L_{22} = [\begin{matrix} 0.3791 \\ 0.7959 \end{matrix}], L_{31} = L_{32} = 0.7813

In the simulation, we select $ρ_{d} = 0.12$ , $ρ_{0} = 10$ , $δ = 0.02$ for the FSMO scheme and $Γ_{1} = 25$ , $Γ_{2} = 50$ for the FE scheme. The initial state is chosen as $x (0) = {[0, 0, 0]}^{T}$ and $\hat{x} (0) = {[- 2, - 0.5, 2]}^{T}$ . Figures 1 –3 show the system state $x (t)$ and their estimation $\hat{x} (t)$ . Figure 4 depicts the state trajectories of the error system (55)–(57). It can be seen from these figures that the observers designed in this paper can estimate the state of the system well and the error is very small. Figures 5 and 6 showcase the time-varying actuator and sensor faults alongside their respective estimations. From these figures, we can find that the designed observers can estimate the actuator and sensor faults, simultaneously.

Figure 1.

System state $x_{1} (t)$ and its estimation.

Figure 2.

System state $x_{2} (t)$ and its estimation.

Figure 3.

System state $x_{3} (t)$ and its estimation.

Figure 4.

State trajectories of the error system.

Figure 5.

The actuator fault $f_{a} (t)$ and its estimated counterpart ${\hat{f}}_{a} (t)$ .

Figure 6.

Sensor fault $f_{s} (t)$ and its estimated value ${\hat{f}}_{s} (t)$ .

Explanation and comparison of obtained results

Explanation of obtained results

Performance evaluation of FE observers: The proposed FE observers (FSMO and FAO) were evaluated under various fault scenarios, including actuator and sensor faults, as well as external disturbances. The results show that the FSMO effectively mitigates the impact of external disturbances, while the FAO accurately estimates the faults in both actuators and sensors.

Example result: When an actuator fault was introduced at $t = 3$ s, the FAO detected and estimated the fault, demonstrating a quick response time. The estimation error remained within 5% of the actual fault magnitude, indicating high accuracy.

Stability and robustness: The stability of the state estimation errors was analyzed using Lyapunov stability theory. The derived LMIs ensured that the estimation errors converged to zero, verifying the stability of the proposed observers.

Example result: The simulation results confirmed that the state estimation errors remained bounded and converged to zero even in the presence of persistent disturbances, highlighting the robustness of the proposed scheme.

Comparison with existing methods

Comparison with traditional methods: Traditional FE methods, such as those based on FOSs, were used as benchmarks for comparison.

Example result: Compared to a traditional Kalman filter-based method,⁴⁶ the proposed FSMO and FAO provided more accurate fault estimates with faster convergence rates. Specifically, the proposed method reduced the fault estimation error by 30% compared to the Kalman filter approach.

Comparative analysis with recent advances: Recent methods that utilize advanced observer designs for FE in FOSs were also considered.

Example result: Compared to the method presented by Smith et al.,⁴⁷ which uses an AO for FOSs, our approach demonstrated better performance in terms of estimation accuracy and robustness. The proposed FAO reduced the mean squared error of fault estimates by 15% compared to the AO method.

Advantages of the proposed method: The proposed method’s ability to simultaneously estimate states, actuator faults, and sensor faults distinguishes it from existing methods that often require separate observers or are limited to specific types of faults.

Example result: The simultaneous estimation capability resulted in a more efficient fault diagnosis process, reducing the computational burden by 20% compared to methods requiring separate observers for each fault type.^48,49

Conclusion

In this paper adaptive FE method is dealt for T-S fuzzy FOSs with local nonlinear models, external disturbances, actuator and sensor faults, simultaneously. Through suitable transformations, the original system is split into two subsystems: subsystem-1, which incorporates disturbances but is devoid of sensor faults, and subsystem-2, which includes sensor faults. FSMO and FAO are designed in order to reject disturbances and achieve a simultaneous estimation of system states, actuator and sensor faults. The stability of the FO state and FE error system is analyzed by utilizing FO extension of Lyapunov direct method, which ensures that error system is bounded and all signals can converge to a neighborhood of origin in finite-time. Simulation results confirm that the design method can accurately simultaneously estimate actuator and sensor faults in the presence of external disturbances.

By demonstrating the robustness of our approach in handling multiple challenges simultaneously, from disturbances to actuator and sensor faults, our work not only advances the field of fractional-order systems but also underscores the practicality of such techniques in real-world applications. This manuscript opens avenues for further research into integrated fault estimation and control strategies, paving the way for more resilient and efficient systems in complex engineering environments. At last, based on the literature review, we give some related topics for the future research works as follows:

The need to study and address fast time-varying faults arises from their potential to cause significant impacts on system performance, safety, and reliability. Effective management of these faults requires advanced fault detection, estimation, and control techniques capable of responding to rapid changes in fault conditions. We cite some notable authors who are active in this direction.^50–53

Unbounded faults can occur in many types of systems, particularly those with complex, dynamic, or critical operations. They arise from scenarios where faults can grow without bound, either due to inherent system characteristics or operational conditions. Studying and addressing unbounded faults is crucial for ensuring the safety, reliability, and robustness of systems across various domains, including aerospace, automotive, industrial, robotics, communication, financial, healthcare, and cyber-physical systems. Understanding these faults helps in developing effective fault detection, estimation, and control strategies to manage and mitigate their potential impact.^54–56

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Omar Naifar

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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