Fault-Tolerant Tracking Control for Linear Parameter-Varying Systems under Actuator and Sensor Faults

Abstract

In this study, we delve into the intricacies of addressing the challenge posed by simultaneous external disturbances and ever-changing actuator and sensor faults in the context of linear parameter-varying (LPV) systems. Our focus is on fault estimation (FE) and the pursuit of fault-tolerant tracking control (FTTC). LPV systems are described through a polytopic LPV representation with measurable gain scheduling functions. An adaptive LPV sliding mode observer (ASMO) is developed for the purpose of simultaneously estimating the system states and faults despite external disturbances. Compared with other conventional ASMO designs, the proposed observer has the capability to reconstruct the actuator faults by exploiting the equivalent output error injection signal required to maintain sliding motion and to directly estimate sensor faults using an adaptive algorithm. Based on online FE information, an FTTC is synthesized to compensate for the fault effect and to force closed-loop system states to track their desired reference signals. Sufficient conditions to ensure the stability of the state estimation errors and closed-loop system are established using Lyapunov stability theory together with $H_{\infty }$ techniques. These requirements are articulated using linear matrix inequalities (LMIs), which can be effortlessly addressed through optimization problem-solving methods. To illustrate the potency of the proposed approaches, an illustrative example is provided. To illustrate the potency of the proposed approaches and to validate their practical effectiveness, we offer an illustrative example featuring a vertical takeoff and landing aircraft. This real-world case study serves as a practical application of our theoretical contributions, demonstrating the adaptability and robustness of our approach in the face of complex, real-world challenges.

1. Introduction

As the demand for heightened performance in contemporary industrial systems grows, particularly in terms of system safety, dependability, ease of maintenance, and the ability to withstand adverse conditions, the likelihood of system failures also rises. Disruptions and defects in control elements, such as actuators and sensors, can lead to subpar performance or even instability, constraining the applicability of conventional control techniques that were once well-established. To overcome these limitations, novel control strategies have surfaced, aiming to ensure the stability of the feedback system even in the face of potential component malfunctions. A novel approach known as fault-tolerant control (FTC) has gained traction, involving the dynamic adjustment of the control law guided by fault estimations impacting the system. The problems of fault estimation (FE) and FE-based FTC have been widely investigated and fruitful results have been reported in several books [1,2,3], survey papers [4,5,6], and the references therein.

At another research forefront, in the last two decades, linear parameter-varying (LPV) systems have attracted much attention owing to their wide practical applications. The main reason arises from the fact the LPV approach provides efficient ways to deal with some nonlinearities. In the realm of LPV systems, the transformation of a nonlinear system into an array of interpolated, localized models governed by convex weighting functions is a fundamental approach. In a broader context, LPV systems are considered as nonlinear systems that experience a momentary linearization along a path defined by a parameter vector. This parameter vector is commonly assumed to be continuously and dynamically measurable in real-time, evolving within a polytopic region. Hence, scholarly focus in the academic sphere has significantly gravitated towards the creation of FE and FTC techniques based on LPV models, as indicated by references [7,8,9].

Within this landscape of diverse approaches, the sliding mode observer (SMO) emerges as a notably effective technique for addressing FE challenges, particularly in the presence of disturbances. This method has been demonstrated to have the capability of concurrently estimating faults and states by capitalizing on the injection of an equivalent output error signal, vital for sustaining sliding motion within the state space of estimation errors, as evidenced by references [10,11,12,13,14].

For instance, in [10], a novel FE scheme predicated on an innovative SMO design method was introduced for LPV systems afflicted by sensor faults. A filtering mechanism for the discontinuous term was introduced to counteract chattering phenomena. The authors in [11] tackled the intertwined issues of FE and FTC for LPV descriptor systems grappling with actuator faults and time delays, engineering a synthesized SMO to simultaneously estimate system states and actuator faults.

In ref. [13], a pair of polytopic SMOs was ingeniously designed for an LPV system, offering the ability to estimate actuator and sensor faults simultaneously through the injection of two equivalent output error signals. However, it is important to note that this approach is primarily tailored for incipient fault signals.

Furthermore, in [14], an integrated sensor fault-tolerant control scheme was proposed for LPV systems, where sensor faults were reconstructed using the SMO framework. This approach involved the simultaneous synthesis of gains in the LPV observer and the LPV static feedback controller to optimize the closed-loop system’s performance.

Nevertheless, there remains a conspicuous research gap pertaining to the challenge of FE and FTTC rooted in dynamic output feedback controllers for LPV systems subjected to simultaneous external disturbances, actuator faults, and sensor faults. It is precisely this void that serves as the impetus for the present study reported in this paper.

In this paper, we tackle the challenge of simultaneously dealing with disturbances, actuator faults, and sensor faults in LPV systems. We begin by employing a clever coordinate transformation to distinguish between actuator and sensor faults. We then proceed to devise an LPV adaptive sliding mode observer (ASMO) that accomplishes the concurrent estimation of system state, actuator faults, and sensor faults. Notably, this design allows us to reconstruct actuator faults through equivalent output error injection, while sensor FE is carried out through an adaptive algorithm driven by a proportional and integral component of the output estimation error. This innovative FE algorithm significantly enhances both the speed and accuracy of fault detection.

In the contemporary industrial landscape, where the demand for enhanced system performance and robustness continues to grow, the specter of system failures looms ever larger. Factors such as system safety, dependability, ease of maintenance, and the ability to withstand adverse conditions have become paramount. These factors underscore the critical need for control systems capable of functioning reliably in the presence of component malfunctions. Traditional control techniques, once the cornerstone of industrial control, are rendered less effective in the face of disruptions or defects in control elements, such as actuators and sensors. It is in response to this challenge that novel control strategies, with a particular emphasis on fault-tolerant control, have emerged.

Fault-tolerant control involves dynamically adjusting the control law in response to fault estimations, thus safeguarding the stability of the entire closed-loop system, even in the presence of faults. While substantial research has been conducted in the field of FE and FE-based FTC, the development of such techniques has primarily been centered on LPV systems. LPV systems have garnered significant attention in the past two decades due to their wide range of practical applications and their inherent ability to address some nonlinearities efficiently.

In the realm of LPV systems, a fundamental approach involves transforming nonlinear systems into a collection of interpolated, localized models defined by convex weighting functions. These models are governed by a continuously and dynamically measurable parameter vector, which evolves within a polytopic region. As a result, research in the academic sphere has seen a notable shift towards the development of FE and FTC techniques based on LPV models [15,16,17].

Within this landscape of diverse approaches, the SMO has emerged as a particularly effective technique for addressing FE challenges, especially in the presence of disturbances. The SMO method is capable of concurrently estimating faults and system states through the injection of an equivalent output error signal, a vital component for sustaining sliding motion within the state space of estimation errors. Previous research has explored the application of SMO in various scenarios, from sensor fault estimation to the simultaneous estimation of actuator faults and system states [18,19,20].

However, a notable research gap remains when it comes to dynamic output feedback controllers for LPV systems subjected to simultaneous external disturbances, actuator faults, and sensor faults. This specific research void serves as the driving force behind the study presented in this paper.

Our research endeavors to bridge this gap by addressing the simultaneous challenges posed by external disturbances, actuator faults, and sensor faults in LPV systems. We begin by employing a clever coordinate transformation to differentiate between actuator and sensor faults, a crucial step in the fault-tolerant process. Subsequently, we introduce a novel LPV ASMO, capable of simultaneously estimating system states, actuator faults, and sensor faults. This innovative design enables the reconstruction of actuator faults through the injection of equivalent output error, while sensor fault estimation is carried out through an adaptive algorithm driven by the proportional and integral components of the output estimation error. This novel FE algorithm enhances the speed and accuracy of fault detection. In summary, our study addresses the pressing need for a comprehensive solution to simultaneous disturbances and multiple fault scenarios in LPV systems, contributing to the growing body of research in the field of fault-tolerant control for complex industrial systems.

Following this, we formulate a tracking controller resilient to faults, with the goal of mitigating their effects and guaranteeing that the states of the closed-loop system accurately track their intended reference signals. The primary contributions of this paper can be succinctly summarized as follows:

• In the case of LPV systems encountering concurrent external disturbances, actuator and sensor faults, we delve into the novel challenge of addressing FE and FTTC using a dynamic output feedback controller.
• Compared with the results in [7,11,20], the proposed ASMO can not only reconstruct the actuator faults by exploiting the equivalent output error injection, but can also estimate the sensor faults using an adaptive algorithm of the output estimation error.
• Unlike the results reported in reference [13], the proposed ASMO presents a more forgiving perspective on the structural prerequisites necessary for creating an observer-based estimation system that can effectively manage both actuator and sensor faults concurrently.
• By employing $H_{\infty }$ performance standards, fresh adequate requirements for achieving the desired LPV ASMO and FTTC are extracted and formulated as a convex optimization challenge utilizing linear matrix inequalities (LMIs).
• It is worth emphasizing that the design of the ASMO and FTTC is carried out separately—a design method that conveniently lessens computational intricacy.

2. System Description

Let us contemplate a continuous LPV system, influenced by actuator faults, sensor anomalies, and external disturbances.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=A\left(\rho \right)x+B\left(\rho \right)u+M\left(\rho \right)\xi +F\left(\rho \right)f_a\hfill \\ y=Cx+G{f}_s\hfill \end{array}\right.\end{array}$$

(1)

In this context, we have several variables at play: x representing the system state in ${\mathbb{R}}^n$, u denoting the control input in ${\mathbb{R}}^m$, and y indicating the measured output in ${\mathbb{R}}^p$. Additionally, we have signals like $f_a$ residing in ${\mathbb{R}}^q$ to represent the actuator faults, $f_s$ in ${\mathbb{R}}^s$ for the sensor faults, and $\xi$ in ${\mathbb{R}}^l$ representing the disturbances.

The dynamics of the system are characterized by parameter-varying matrices denoted as $A\left(\rho \right)$, $B\left(\rho \right)$, $M\left(\rho \right)$, and $F\left(\rho \right)$, which depend affinely on the varying parameters vector $\rho \in \Im \subset {\mathbb{R}}^r$. It is worth noting that $\rho$ is measured in real-time, facilitating dynamic adaptation.

Furthermore, we have fixed matrices in play: G has full column rank, C has full row rank, and $n>p\ge s$.

In this paper, it is assumed that the varying parameters vector $\rho$ is bounded and varies in a hypercube ℑ such that

$$\begin{array}{c}\hfill \rho \in \Im =\left\{{\rho }_i\left|{\rho }_{di}\le {\rho }_i\le {\rho }_{mi}\right.\right\},\forall i=\left[1,\dots ,r\right]\end{array}$$

(2)

where ${\rho }_{di}$ and ${\rho }_{mi}$ represent, respectively, the minimum and maximum values of ${\rho }_i$. Under the assumption of the affine dependence of the parameter vector $\rho$, the system (1) can be represented by a convex combination of k vertices where each vertex $S_i$ is defined as

$$\begin{array}{c}\hfill S_i=\left[\begin{array}{cccccc}{A}_i\hfill & B_i& M_i& F_i& C& G\end{array}\right],\forall i\in \left[1,\dots ,k\right],k=2^r\end{array}$$

(3)

The polytopic coordinates are denoted by ${\alpha }_i\left(\rho \right)$ and vary within the convex set $\Omega$.

$$\begin{array}{c}\hfill \Omega =\left\{\alpha \left(\rho \right)\in R^k,{\alpha }_i\left(\rho \right)\ge 0,\sum _{i=1}^k{\alpha }_i\left(\rho \right)=1,\forall i\in \left[1,\dots ,k\right]\right\}\end{array}$$

(4)

To unveil the core findings, we lay the groundwork by introducing specific assumptions and lemmas.

Assumption 1.

We presuppose that the varying matrix $F\left(\rho \right)$ can be decomposed into

$$\begin{array}{c}\hfill F\left(\rho \right)=FE\left(\rho \right)\end{array}$$

(5)

where $F\in {\mathbb{R}}^{n\times q}$ is fixed and $E\left(\rho \right)\in {\mathbb{R}}^{q\times q}$ is a varying matrix that is assumed to be invertible for all $\rho \in \Im$.

Remark 1.

The utilization of factorization (5) will contribute to the advancement of the intended observer. While this constraint is evident, it is noteworthy that this ideal factorization is frequently encountered in over-actuated systems or in fault-tolerant systems equipped with inherent redundancy, such as large civil aircraft. An illustration of a corresponding factorization for a large civil aircraft can be found in references [12,21,22].

Applying Equation (5), we can express the system from Equation (1) as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=A\left(\rho \right)x+B\left(\rho \right)u+M\left(\rho \right)\xi +F\underset{f_{\nu }\left(\rho \right)}{\underbrace{E\left(\rho \right)f_a}}\hfill \\ y=Cx+G{f}_s\hfill \end{array}\right.\end{array}$$

(6)

where $f_{\nu }\left(\rho \right):{\mathbb{R}}^{{}_r}\to {\mathbb{R}}^q$ is the virtual actuator faults. In the following, an observer will be designed to firstly estimate $f_{\nu }$ and then $f_a$ will be recuperated by exploiting the fact that $det\left(E\right(\rho \left)\right)\ne 0$. Assume that

$$\begin{array}{c}\hfill ∥f_{\nu }\left(\rho \right)∥<\phi (\rho ,u)\end{array}$$

(7)

where $\phi (.)$ is a known scalar function.

Assumption 2.

We posit that the distribution matrix for virtual actuator faults, denoted as F, possesses a complete column rank, signifying:

$$rank\left(CF\right)=rank\left(F\right)=q$$

(8)

Assumption 3.

For all $\rho \in \Im$, $A\left(\rho \right)$, F and C in (6) satisfy the rank condition

$$rank\left[\begin{array}{cc}s{I}_n-A\left(\rho \right)& F\\ C& 0\end{array}\right]=n+q$$

(9)

for every complex number s with a non-negative real part.

Assumption 4.

The sensor fault $f_s$, its derivative, and the disturbance ξ satisfy

$$∥f_s∥\le {\gamma }_s,∥{\dot{f}}_s∥\le {\gamma }_{ss},∥\xi ∥\le {\gamma }_{\xi }$$

(10)

where ${\gamma }_s$, ${\gamma }_{ss}$, and ${\gamma }_{\xi }$ are known positive constants. Moreover, the disturbance ξ belongs to $L_2\left[0,\infty \right)$.

Remark 2.

Remark 2 asserts that Assumptions 2 and 3 are both essential and comprehensive conditions for formulating a stable sliding-mode, particularly in scenarios involving disturbances, as documented in references [3,13]. Assumption 3 posits that all invariant zeros of the triple $\left(A\right(\rho ),F,C)$ must reside in the left half-plane, or alternatively, the triple $\left(A\right(\rho ),F,C)$ must exhibit minimum phase characteristics. Additionally, Assumption 4 mandates that the derivative of the sensor fault, as well as of the disturbances, remains bounded, which aligns with a general assumption in fault estimation (FE) and fault-tolerant control (FTC) methodologies, as elucidated in reference [13].

Lemma 1

([23]). Suppose we have U and V, two matrices with suitable dimensions, $\exists X>0$, such that

$$U^{T}V+V^{T}U\le U^{T}XU+V^T{X}^{-1}V$$

(11)

Lemma 2.

According to reference [24], for a square matrix $\mathcal{A}\in {\mathbb{R}}^{n\times n}$, it is possible to ascertain that the eigenvalues of $\mathcal{A}$ lie within a circular region denoted as $\mathcal{D}(\kappa ,\beta )$. This circular region is centered at the coordinates $(\kappa ,0)$ and exhibits a radius of β. This significant property remains valid, provided that there exists a positive-definite matrix $\mathcal{P}\in {\mathbb{R}}^{n\times n}$ such that the subsequent condition is satisfied:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\kappa \mathcal{P}& \beta \mathcal{P}+\mathcal{P}\mathcal{A}\\ \beta \mathcal{P}+{\left(\mathcal{P}\mathcal{A}\right)}^T& -\kappa \mathcal{P}\end{array}\right]<0\end{array}$$

(12)

3. ASMO Design

3.1. State Transformation

Lemma 3

([25]). With Assumption 2 in place, we can uncover coordinate transformations denoted as $\upsilon =T_{c}x$ and $\eta =S_{c}y$, where $T_c\in {\mathbb{R}}^{n\times n}$ and $S_c\in {\mathbb{R}}^{p\times p}$. Consequently, the system outlined in Equation (6) can be elegantly reformulated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\upsilon }=T_{c}A\left(\rho \right)T_c^{-1}\upsilon +T_{c}B\left(\rho \right)u+T_{c}M\left(\rho \right)\xi +T_{c}F{f}_{\nu }\left(\rho \right)\hfill \\ \eta =S_{c}C{T}_c^{-1}\upsilon +S_{c}G{f}_s\hfill \end{array}\right.\end{array}$$

(13)

where

$$\begin{array}{ccc}\hfill T_{c}A\left(\rho \right)T_c^{-1}& =& \left[\begin{array}{cc}{A}_{11}\left(\rho \right)& A_{12}\left(\rho \right)\\ A_{21}\left(\rho \right)& A_{22}\left(\rho \right)\end{array}\right],T_{c}B\left(\rho \right)=\left[\begin{array}{c}{B}_1\left(\rho \right)\\ B_2\left(\rho \right)\end{array}\right],T_{c}M\left(\rho \right)=\left[\begin{array}{c}{M}_1\left(\rho \right)\\ M_2\left(\rho \right)\end{array}\right]\hfill \\ \hfill T_{c}F& =& \left[\begin{array}{c}{F}_1\\ 0\end{array}\right],S_{c}C{T}_c^{-1}=\left[\begin{array}{cc}{C}_1& 0\\ 0& C_2\end{array}\right],S_{c}G=\left[\begin{array}{c}0\\ G_2\end{array}\right]\hfill \end{array}$$

(14)

where $A_{11}\left(\rho \right)\in {\mathbb{R}}^{q\times q}$, $A_{22}\left(\rho \right)\in {\mathbb{R}}^{(n-q)\times (n-q)}$, $B_1\left(\rho \right)\in {\mathbb{R}}^{q\times m}$, $M_1\left(\rho \right)\in {\mathbb{R}}^{q\times l}$, $F_1\in {\mathbb{R}}^{q\times q}$, $C_1\in {\mathbb{R}}^{q\times q}$, $C_2\in {\mathbb{R}}^{(p-q)\times (n-q)}$, $G_2\in {\mathbb{R}}^{(p-q)\times s}$, and $C_1$ is a nonsingular matrix while $F_1$ has full column rank.

Considering Lemma 3, the system (13) is divided into the following two subsystems:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\upsilon }}_1=A_{11}\left(\rho \right){\upsilon }_1+A_{12}\left(\rho \right){\upsilon }_2+B_1\left(\rho \right)u+M_1\left(\rho \right)\xi +F_1{f}_{\nu }\left(\rho \right)\hfill \\ {\eta }_1=C_1{\upsilon }_1\hfill \end{array}\right.\end{array}$$

(15)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\upsilon }}_2=A_{21}\left(\rho \right){\upsilon }_1+A_{22}\left(\rho \right){\upsilon }_2+B_2\left(\rho \right)u+M_2\left(\rho \right)\xi \hfill \\ {\eta }_2=C_2{\upsilon }_2+G_2{f}_s\hfill \end{array}\right.\end{array}$$

(16)

where $\upsilon ={\left({\upsilon }_1^T,{\upsilon }_2^T\right)}^T$ with ${\upsilon }_1\in {\mathbb{R}}^q$, ${\upsilon }_2\in {\mathbb{R}}^{(n-q)}$, $\eta ={\left({\eta }_1^T,{\eta }_2^T\right)}^T$ with ${\eta }_1\in {\mathbb{R}}^q$ and ${\eta }_2\in {\mathbb{R}}^{(p-q)}$.

For subsystem (16), define a new state

$$\begin{array}{c}\hfill {\dot{\upsilon }}_3=C_2{\upsilon }_2+G_2{f}_s\end{array}$$

(17)

Then, the subsystem (16) can be expanded into an augmented system described by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\upsilon }}_f=\overline{A}\left(\rho \right){\upsilon }_f+{\overline{A}}_{21}\left(\rho \right){\eta }_1+\overline{B}\left(\rho \right)u+\overline{M}\left(\rho \right)\xi +\overline{G}{f}_s\hfill \\ {\eta }_f=\overline{C}{\upsilon }_f\hfill \end{array}\right.\end{array}$$

(18)

where

$$\begin{array}{ccc}\hfill {\upsilon }_f& =& \left[\begin{array}{c}{\upsilon }_2\\ {\upsilon }_3\end{array}\right],\overline{A}\left(\rho \right)=\left[\begin{array}{cc}{A}_{22}\left(\rho \right)& 0\\ C_2& 0\end{array}\right],{\overline{A}}_{21}\left(\rho \right)=\left[\begin{array}{c}{A}_{21}\left(\rho \right)C_1^{-1}\\ 0\end{array}\right]\hfill \\ \hfill \overline{B}\left(\rho \right)& =& \left[\begin{array}{c}{B}_2\left(\rho \right)\\ 0\end{array}\right],\overline{M}\left(\rho \right)=\left[\begin{array}{c}{M}_2\left(\rho \right)\\ 0\end{array}\right],\overline{G}=\left[\begin{array}{c}0\\ G_2\end{array}\right],\overline{C}=\left[\begin{array}{cc}0& I_{p-q}\end{array}\right]\hfill \end{array}$$

Lemma 4

([25]). The observability of the pair ($\overline{A}\left(\rho \right)$, $\overline{C}$) is confirmed when Assumption 3 is satisfied.

Lemma 5.

If there exist matrices $P_f\in {\mathbb{R}}^{(n+p-2q)\times (n+p-2q)}>0$ and $L_f\in {\mathbb{R}}^{(n+p-2q)\times (p-q)}$ such that for some $Q_f\in {\mathbb{R}}^{(n+p-2q)\times (n+p-2q)}>0$, the following matrix condition holds:

$$\begin{array}{c}\hfill {(\overline{A}\left(\rho \right)-L_f\overline{C})}^T{P}_f+P_f(\overline{A}\left(\rho \right)-L_f\overline{C})=-Q_f\end{array}$$

(19)

where $L_f=\left[\begin{array}{c}{L}_{f1}\\ L_{f2}\end{array}\right]$, $L_{f1}\in {\mathbb{R}}^{n\times p}$, $L_{f2}\in {\mathbb{R}}^{p\times p}$, $P_f=\left[\begin{array}{cc}{P}_{f1}& P_{f2}\\ P_{f2}^T& P_{f3}\end{array}\right]$, $P_{f1}\in {\mathbb{R}}^{(n-q)\times (n-q)}$, $P_{f3}\in {\mathbb{R}}^{(p-q)\times (p-q)}$, $Q_f=\left[\begin{array}{cc}{Q}_{f1}& Q_{f2}\\ Q_{f2}^T& Q_{f3}\end{array}\right]$, $Q_{f1}\in {\mathbb{R}}^{(n-q)\times (n-q)}$, and $Q_{f3}\in {\mathbb{R}}^{(p-q)\times (p-q)}$, then the matrix $A_{22}\left(\rho \right)+P_{f1}^{-1}{P}_{f2}{C}_2$ is stable $\forall \rho \in \Im$.

Proof. 

According to Lemma 4, we have $(\overline{A}\left(\rho \right),\overline{C})$ is observable $\forall \rho \in \Im$, and we know that there exists a matrix $L_f$ such that the condition (19) holds. We consider the structure of $\overline{A}\left(\rho \right)$, $\overline{C}$, $L_f$, $P_f$, and $Q_f$, so, the matrix condition (19) has the following form:

$$\begin{array}{ccc}\hfill & & {\left(\left[\begin{array}{cc}{A}_{22}\left(\rho \right)& 0\\ C_2& 0\end{array}\right]-\left[\begin{array}{c}{L}_{f1}\\ L_{f2}\end{array}\right]\left[\begin{array}{cc}0& I_{p-q}\end{array}\right]\right)}^T\left[\begin{array}{cc}{P}_{f1}& P_{f2}\\ P_{f2}^T& P_{f3}\end{array}\right]\hfill \\ & & +\left[\begin{array}{cc}{P}_{f1}& P_{f2}\\ P_{f2}^T& P_{f3}\end{array}\right]\left(\left[\begin{array}{cc}{A}_{22}\left(\rho \right)& 0\\ C_2& 0\end{array}\right]-\left[\begin{array}{c}{L}_{f1}\\ L_{f2}\end{array}\right]\left[\begin{array}{cc}0& I_{p-q}\end{array}\right]\right)=\left[\begin{array}{cc}{Q}_{f1}& Q_{f2}\\ Q_{f2}^T& Q_{f3}\end{array}\right]\hfill \end{array}$$

(20)

It is easy to obtain

$$\begin{array}{c}\hfill A_{22}^T\left(\rho \right)P_{f1}+C_2^T{P}_{f2}^T+P_{f1}{A}_{22}\left(\rho \right)+P_{f2}{C}_2=-Q_{f1}\end{array}$$

(21)

We obtain

$$\begin{array}{c}\hfill {\left(A_{22}\left(\rho \right)+P_{f1}^{-1}{P}_{f2}{C}_2\right)}^T{P}_{f1}+P_{f1}\left(A_{22}\left(\rho \right)+P_{f1}^{-1}{P}_{f2}{C}_2\right)=-Q_{f1}\end{array}$$

(22)

Given that $P_{f1}$ and $Q_{f1}$ are both greater than zero, we can conclude, based on Lyapunov theory, that $A_{22}\left(\rho \right)+P_{f1}^{-1}{P}_{f2}{C}_2$ remains stable for all $\rho \in \Im$. □

If we apply the coordinate transformation $z=T_f{\upsilon }_f$, where

$$\begin{array}{c}\hfill T_f=\left[\begin{array}{cc}{I}_{n+h-p}& P_{f1}^{-1}{P}_{f2}\\ 0& I_p\end{array}\right]\end{array}$$

(23)

then, subsystem (18) can be transformed into

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{z}=\tilde{A}\left(\rho \right)z+{\tilde{A}}_{21}\left(\rho \right){\eta }_1+\tilde{B}\left(\rho \right)u+\tilde{M}\left(\rho \right)\xi +\tilde{G}{f}_s\hfill \\ {\eta }_f=\tilde{C}z\hfill \end{array}\right.\end{array}$$

(24)

where

$$\begin{array}{ccc}\hfill \tilde{A}\left(\rho \right)& =& \left[\begin{array}{cc}{\tilde{A}}_{11}\left(\rho \right)& {\tilde{A}}_{12}\left(\rho \right)\\ {\tilde{A}}_{21}\left(\rho \right)& {\tilde{A}}_{22}\left(\rho \right)\end{array}\right]\hfill \\ \hfill & =& \left[\begin{array}{cc}{A}_{22}\left(\rho \right)+P_{f1}^{-1}{P}_{f2}{C}_2& -\left(A_{22}\left(\rho \right)+P_{f1}^{-1}{P}_{f2}{C}_2\right)P_{f1}^{-1}{P}_{f2}\\ C_2& -C_2{P}_{f1}^{-1}{P}_{f2}\end{array}\right]\hfill \\ \hfill {\tilde{A}}_{21}\left(\rho \right)& =& \left[\begin{array}{c}{A}_{21}\left(\rho \right)C_1^{-1}\\ 0\end{array}\right],\tilde{B}\left(\rho \right)=\left[\begin{array}{c}{B}_2\left(\rho \right)\\ 0\end{array}\right],\tilde{M}\left(\rho \right)=\left[\begin{array}{c}{M}_2\left(\rho \right)\\ 0\end{array}\right]\hfill \\ \hfill \tilde{G}& =& \left[\begin{array}{c}{P}_{f1}^{-1}{P}_{f2}{G}_2\\ G_2\end{array}\right],\tilde{C}=\left[\begin{array}{cc}0& I_{p-q}\end{array}\right]\hfill \end{array}$$

(25)

In the transformed coordinate system, it can be demonstrated that the Lyapunov matrix $P_z$ assumes a quadratic form as follows:

$$\begin{array}{c}\hfill P_z={\left(T_f^T\right)}^{-1}{P}_f{T}_f^{-1}=\left[\begin{array}{cc}{P}_{f1}& 0\\ 0& P_{f0}\end{array}\right]\end{array}$$

(26)

where $P_{f0}=-P_{f2}^T{P}_{f1}^{-T}{P}_{f2}+P_{f3}$.

Denoting $K_f=P_{f1}^{-1}{P}_{f2}$, subsystem (24) becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{z}}_1=\left(A_{22}\left(\rho \right)+K_f{C}_2\right)z_1-\left(A_{22}\left(\rho \right)+K_f{C}_2\right)K_f{z}_2+A_{21}\left(\rho \right)C_1^{-1}{\eta }_1+B_2\left(\rho \right)u\hfill \\ +M_2\left(\rho \right)\xi +K_f{G}_2{f}_s\hfill \\ {\dot{z}}_2=C_2{z}_1-C_2{K}_f{z}_2+G_2{f}_s\hfill \\ {\eta }_f=z_2\hfill \end{array}\right.\end{array}$$

(27)

where $z={\left(z_1^T,z_2^T\right)}^T$.

Subsystem (15), in the new coordinate z, can be rewritten as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\upsilon }}_1=A_{11}\left(\rho \right){\upsilon }_1+A_{12}\left(\rho \right)z_1-A_{12}\left(\rho \right)K_f{\eta }_f+B_1\left(\rho \right)u+M_1\left(\rho \right)\xi +F_1{f}_{\nu }\left(\rho \right)\hfill \\ {\eta }_1=C_1{\upsilon }_1\hfill \end{array}\right.\end{array}$$

(28)

For subsystem (28), a sliding mode observer is constructed by the following:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\widehat{\upsilon }}}_1=A_{11}\left(\rho \right){\widehat{\upsilon }}_1+A_{12}\left(\rho \right){\widehat{z}}_1-A_{12}\left(\rho \right)K_f{\eta }_f+B_1\left(\rho \right)u+L_{11}\left(\rho \right)({\eta }_1-{\widehat{\eta }}_1)+\vartheta \hfill \\ {\widehat{\eta }}_1=C_1{\widehat{\upsilon }}_1\hfill \end{array}\right.\end{array}$$

(29)

where ${\widehat{\upsilon }}_1$, ${\widehat{z}}_1$ and ${\widehat{\eta }}_1$ denote, respectively, the estimated ${\upsilon }_1$, $z_1$ and ${\eta }_1$. $L_{11}\left(\rho \right)$ is the observer gain matrix to be determined. The discontinuous vector $\vartheta$ is designed as

$$\begin{array}{c}\hfill \vartheta =\left\{\begin{array}{cc}\mathcal{K}(\rho ,y,u)\frac{P_1{C}_1^{-1}({\eta }_1-{\widehat{\eta }}_1)}{∥P_1{C}_1^{-1}({\eta }_1-{\widehat{\eta }}_1)∥},\hfill & ({\eta }_1-{\widehat{\eta }}_1)\ne 0\\ 0,\hfill & \mathrm{otherwise}\end{array}\right.\end{array}$$

(30)

where the matrix $P_1\in {\mathbb{R}}^{q\times q}>0$ and the scalar $\mathcal{K}(y,u,\rho )$ satisfying

$$\begin{array}{c}\hfill \mathcal{K}(\rho ,y,u)>∥F_1∥\phi (\rho ,u)+{\varrho }_0\end{array}$$

(31)

where ${\varrho }_0>0$.

Furthermore, in the context of subsystem (27), we establish the subsequent adaptive observer:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\widehat{z}}}_1=\left(A_{22}\left(\rho \right)+K_f{C}_2\right){\widehat{z}}_1-\left(A_{22}\left(\rho \right)+K_f{C}_2\right)K_f{\widehat{z}}_2+A_{21}\left(\rho \right)C_1^{-1}{\eta }_1+B_2\left(\rho \right)u\hfill \\ +K_f{G}_2{\widehat{f}}_s+L_{21}\left(\rho \right)({\eta }_f-{\widehat{\eta }}_f)\hfill \\ {\dot{\widehat{z}}}_2=C_2{\widehat{z}}_1-C_2{K}_f{\widehat{z}}_2+G_2{\widehat{f}}_s+L_{22}\left(\rho \right)({\eta }_f-{\widehat{\eta }}_f)\hfill \\ {\widehat{\eta }}_f={\widehat{z}}_2\hfill \\ {\dot{\widehat{f}}}_s=\Gamma G_2^T{P}_{f0}\left({\dot{\tilde{\eta }}}_f+\sigma {\tilde{\eta }}_f\right)\hfill \end{array}\right.\end{array}$$

(32)

where ${\widehat{z}}_2$, ${\widehat{\eta }}_f$ and ${\widehat{f}}_s$ denote, respectively, the estimated $z_2$, ${\eta }_f$ and $f_s$. $L_{21}\left(\rho \right)$ and $L_{22}\left(\rho \right)$ are the observer gain matrices to be determined and the parameter $\Gamma \in {\mathbb{R}}^{q\times q}>0$ represents the learning rate.

Let us define the discrepancies in the state estimation, output estimation, and fault estimation as follows: ${\tilde{\upsilon }}_{{}_1}={\upsilon }_1-{\widehat{\upsilon }}_1$, ${\tilde{z}}_1=z_1-{\widehat{z}}_1$, ${\tilde{\eta }}_f={\eta }_f-{\widehat{\eta }}_f=z_2-{\widehat{z}}_2$, and ${\tilde{f}}_s=f_s-{\widehat{f}}_s$. To construct the matrices $L_{11}\left(\rho \right)$, $L_{21}\left(\rho \right)$, and $L_{22}\left(\rho \right)$, we can opt for the following expressions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{L}_{11}\left(\rho \right)=\left(A_{11}\left(\rho \right)-A_{11}^s\right)C_1^{-1}\hfill \\ L_{21}\left(\rho \right)=-\left(A_{22}\left(\rho \right)+K_f{C}_2\right)K_f\hfill \\ L_{22}\left(\rho \right)=-C_2{K}_f+H\hfill \end{array}\right.\end{array}$$

(33)

where $A_{11}^s\in {\mathbb{R}}^{q\times q}$ is a stable matrix and $H\in {\mathbb{R}}^{(p-q)\times (p-q)}$ will be determined. Then, the state estimation errors dynamics can be obtained by

$$\begin{array}{ccc}\hfill {\dot{\tilde{\upsilon }}}_1& =& A_{11}^s{\tilde{\upsilon }}_1+A_{12}\left(\rho \right){\tilde{z}}_1+M_1\left(\rho \right)\xi +F_1{f}_{\nu }\left(\rho \right)-\vartheta \hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill {\dot{\tilde{z}}}_1& =& \left(A_{22}\left(\rho \right)+K_f{C}_2\right){\tilde{z}}_1+M_2\left(\rho \right)\xi +K_f{G}_2{\tilde{f}}_s\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\dot{\tilde{\eta }}}_f& =& C_2{\tilde{z}}_1+H{\tilde{\eta }}_f+G_2{\tilde{f}}_s\hfill \end{array}$$

(36)

From (36), we have

$$\begin{array}{ccc}\hfill {\dot{\tilde{f}}}_s& =& {\dot{f}}_s-\Gamma G_2^T{P}_{f0}\left({\dot{\tilde{\eta }}}_f+\sigma {\tilde{\eta }}_f\right)\hfill \\ & =& {\dot{f}}_s-\Gamma G_2^T{P}_{f0}\left(C_2{\tilde{z}}_1+H{\tilde{\eta }}_f+G_2{\tilde{f}}_s+\sigma {\tilde{\eta }}_f\right)\hfill \end{array}$$

(37)

Introducing the concept of a weighted estimation error vector, denoted as m, we can define it as follows:

$$m=\chi \left(\begin{array}{c}{\tilde{\upsilon }}_1\\ {\tilde{z}}_1\\ {\tilde{\eta }}_f\\ {\tilde{f}}_s\end{array}\right)$$

(38)

where $\chi =diag({\chi }_1,{\chi }_2,{\chi }_3,{\chi }_4)$ is a full-rank design matrix.

Let us now introduce Theorem 1, which reveals the essential requirement for the presence of the suggested observers (29) and (32), all the while guaranteeing the prescribed $H_{\infty }$ performance

$${∥\mathcal{H}∥}_{\infty }=\underset{{∥\xi ∥}_{L_2}\ne 0}{sup}\frac{{∥m∥}_{L_2}^2}{{∥\xi ∥}_{L_2}^2}\le \mu$$

(39)

where $\mu >0$.

Theorem 1.

Given the scalars $\sigma >0$ and $\epsilon >0$ along with a matrix $N>0$, if we can find the matrices $P_1>0$, $P_{f1}>0$, $P_{f2}>0$, $P_{f0}>0$, $X<0$, and Y, and a scalar $\mu >0$ such that the following inequalities hold:

$$\mathit{min}\mu \mathit{subject}\mathit{to}$$

$$\begin{array}{c}\hfill \left(\begin{array}{ccccc}{\Phi }_{11}& P_1{A}_{12}\left(\rho \right)& 0& 0& P_1{M}_1\left(\rho \right)\\ *& {\Phi }_{22}& C_2^T{P}_{f0}& {\Phi }_{24}& P_{f1}{M}_2\left(\rho \right)\\ *& *& {\Phi }_{33}& -\frac{1}{\sigma }{H}^T{P}_{f0}{G}_2& 0\\ *& *& *& {\Phi }_{44}& 0\\ *& *& *& *& -\mu I\end{array}\right)<0\end{array}$$

(40)

where

$$\begin{array}{ccc}\hfill {\Phi }_{11}& =& X+X^T+{\chi }_1^T{\chi }_1\hfill \\ \hfill {\Phi }_{22}& =& P_{f1}{A}_{22}\left(\rho \right)+P_{f2}{C}_2+A_{22}^T\left(\rho \right)P_{f1}+C_2^T{P}_{f2}+{\chi }_2^T{\chi }_2\hfill \\ \hfill {\Phi }_{24}& =& P_{f2}{G}_2-\frac{1}{\sigma }{C}_2^T{P}_{f0}{G}_2\hfill \\ \hfill {\Phi }_{33}& =& Y+Y^T+{\chi }_3^T{\chi }_3\hfill \\ \hfill {\Phi }_{44}& =& -\frac{2}{\sigma }{G}_2^T{P}_{f0}{G}_2+\frac{1}{\epsilon \sigma }N+{\chi }_4^T{\chi }_4\hfill \end{array}$$

then, the estimation error vectors ${\tilde{\upsilon }}_1$, ${\tilde{z}}_1$, ${\tilde{\eta }}_f$ and ${\tilde{f}}_s$ are bounded and satisfy (39). Moreover, the matrices $A_{11}^s$ and H are determined by:

$$\begin{array}{ccc}\hfill A_{11}^s& =& P_1^{-1}X\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill H& =& P_{f0}^{-1}Y\hfill \end{array}$$

(42)

Proof. 

We opt for the Lyapunov function in the following manner:

$$\begin{array}{c}\hfill V=V_1+V_2+V_3+V_4\end{array}$$

where $V_1={\tilde{\upsilon }}_1^T{P}_1{\tilde{\upsilon }}_1$, $V_2={\tilde{z}}_1^T{P}_{f1}{\tilde{z}}_1$, $V_3={\tilde{\eta }}_f^T{P}_{f0}{\tilde{\eta }}_f$, and $V_4=\frac{1}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}{\tilde{f}}_s$. The derivation of $V_1$ can be shown as

$$\begin{array}{ccc}\hfill {\dot{V}}_1& =& {\tilde{\upsilon }}_1^T\left(A_{11}^{sT}{P}_1+P_1{A}_{11}^s\right){\tilde{\upsilon }}_1+2{\tilde{\upsilon }}_1^T{P}_1{A}_{12}\left(\rho \right){\tilde{z}}_1+2{\tilde{\upsilon }}_1^T{P}_1{M}_1\left(\rho \right)\xi +2{\tilde{\upsilon }}_1^T{P}_1{F}_1{f}_{\nu }\left(\rho \right)\hfill \\ & & -2{\tilde{\upsilon }}_1^T{P}_1\vartheta \hfill \end{array}$$

(43)

Using (7), (30) and (31), we have

$$\begin{array}{ccc}\hfill 2{\tilde{\upsilon }}_1^T{P}_1{F}_1{f}_{\nu }\left(\rho \right)-2{\tilde{\upsilon }}_1^T{P}_1\vartheta & =& 2{\tilde{\upsilon }}_1^T{P}_1{F}_1{f}_{\nu }\left(\rho \right)-2\mathcal{K}(y,u,\rho ){\tilde{\upsilon }}_1^T{P}_1\frac{P_1{\tilde{\upsilon }}_1}{∥P_1{\tilde{\upsilon }}_1∥}\hfill \\ \hfill & \le & 2∥P_1{\tilde{\upsilon }}_1∥\left(∥F_1∥f_{\nu }\left(\rho \right)-\mathcal{K}(y,u,\rho )\right)\hfill \\ & \le & -2{\varrho }_0∥P_1{\tilde{\upsilon }}_1∥<0\hfill \end{array}$$

(44)

Therefore,

$$\begin{array}{ccc}\hfill {\dot{V}}_1& \le & {\tilde{\upsilon }}_1^T\left(A_{11}^{sT}{P}_1+P_1{A}_{11}^s\right){\tilde{\upsilon }}_1+2{\tilde{\upsilon }}_1^T{P}_1{A}_{12}\left(\rho \right){\tilde{z}}_1+2{\tilde{\upsilon }}_1^T{P}_1{M}_1\left(\rho \right)\xi \hfill \end{array}$$

(45)

Similarly, the derivatives of $V_2$, $V_3$ and $V_4$ can be obtained as

$$\begin{array}{ccc}{\dot{V}}_2& =& {\tilde{z}}_1^T\left({\left(A_{22}\left(\rho \right)+K_f{C}_2\right)}^T{P}_{f1}+P_{f1}\left(A_{22}\left(\rho \right)+K_f{C}_2\right)\right){\tilde{z}}_1+2{\tilde{z}}_1^T{P}_{f1}{M}_2\left(\rho \right)\xi \hfill \\ & & +2{\tilde{z}}_1^T{P}_{f1}{K}_f{G}_2{\tilde{f}}_s\hfill \end{array}$$

(46)

$${\dot{V}}_3={\tilde{\eta }}_f^T\left(H^T{P}_{f0}+P_{f0}H\right){\tilde{\eta }}_f+2{\tilde{\eta }}_f^T{P}_{f0}{C}_2{\tilde{z}}_1+2{\tilde{\eta }}_f^T{P}_{f0}{G}_2{\tilde{f}}_s$$

(47)

$$\begin{array}{ccc}\hfill {\dot{V}}_4& =& \frac{1}{\sigma }{\dot{\tilde{f}}}_s^T{\Gamma }^{-1}{\tilde{f}}_s+\frac{1}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}{\dot{\tilde{f}}}_s\hfill \\ \hfill & =& \frac{1}{\sigma }{\left({\dot{f}}_s-{\dot{\widehat{}}f}_s\right)}^T{\Gamma }^{-1}{\tilde{f}}_s+\frac{1}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}\left({\dot{f}}_s-{\dot{\widehat{}}f}_s\right)\hfill \\ \hfill & =& \frac{2}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}{\dot{f}}_s-\frac{2}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}{\dot{\widehat{f}}}_s\hfill \\ \hfill & =& \frac{2}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}{\dot{f}}_s-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}{C}_2{\tilde{z}}_1-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}H{\tilde{\eta }}_f-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}{G}_2{\tilde{f}}_s\hfill \\ & & -2{\tilde{f}}_s^T{G}_2^T{P}_{f0}{\tilde{\eta }}_f\hfill \end{array}$$

(48)

Given a positive-definite matrix N and making use of Lemma 1, we can acquire the following:

$$\begin{array}{ccc}\hfill \frac{2}{\sigma }{\tilde{f}}_s^T{\Gamma }^{-1}{\dot{f}}_s& \le & \frac{1}{\epsilon \sigma }{\tilde{f}}_s^{T}N{\tilde{f}}_s+\epsilon \sigma {\dot{f}}_s^T{\Gamma }^{-1}{N}^{-1}{\Gamma }^{-1}{\dot{f}}_s\hfill \\ & \le & \frac{1}{\epsilon \sigma }{\tilde{f}}_s^{T}N{\tilde{f}}_s+\delta \hfill \end{array}$$

(49)

where

$$\begin{array}{c}\hfill \delta =\epsilon \sigma {\gamma }_{ss}^2{\lambda }_{max}\left({\Gamma }^{-1}{N}^{-1}{\Gamma }^{-1}\right)>0\end{array}$$

(50)

Substituting (49) into (48) gives

$$\begin{array}{ccc}\hfill {\dot{V}}_4& \le & -\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}{C}_2{\tilde{z}}_1-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}H{\tilde{\eta }}_f-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}{G}_2{\tilde{f}}_s-2{\tilde{f}}_s^T{G}_2^T{P}_{f0}{\tilde{\eta }}_f\hfill \\ & & +\frac{1}{\epsilon \sigma }{\tilde{f}}_s^{T}N{\tilde{f}}_s+\delta \hfill \end{array}$$

(51)

By considering (45)–(47) and (51), $\dot{V}$ will be bounded as follows:

$$\begin{array}{ccc}\hfill \dot{V}& \le & {\tilde{\upsilon }}_1^T\left(A_{11}^{sT}{P}_1+P_1{A}_{11}^s\right){\tilde{\upsilon }}_1+2{\tilde{\upsilon }}_1^T{P}_1{A}_{12}\left(\rho \right){\tilde{z}}_1+2{\tilde{\upsilon }}_1^T{P}_1{M}_1\left(\rho \right)\xi \hfill \\ \hfill & & +{\tilde{z}}_1^T\left({\left(A_{22}\left(\rho \right)+K_f{C}_2\right)}^T{P}_{f1}+P_{f1}\left(A_{22}\left(\rho \right)+K_f{C}_2\right)\right){\tilde{z}}_1+2{\tilde{z}}_1^T{P}_{f1}{M}_2\left(\rho \right)\xi \hfill \\ \hfill & & +2{\tilde{z}}_1^T{P}_{f1}{K}_f{G}_2{\tilde{f}}_s+{\tilde{\eta }}_f^T\left(H^T{P}_{f0}+P_{f0}H\right){\tilde{\eta }}_f+2{\tilde{\eta }}_f^T{P}_{f0}{C}_2{\tilde{z}}_1-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}{C}_2{\tilde{z}}_1\hfill \\ \hfill & & -\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}H{\tilde{\eta }}_f-\frac{2}{\sigma }{\tilde{f}}_s^T{G}_2^T{P}_{f0}{G}_2{\tilde{f}}_s+\frac{1}{\epsilon \sigma }{\tilde{f}}_s^{T}N{\tilde{f}}_s+\delta \hfill \\ \hfill & =& {\left(\begin{array}{c}{\tilde{\upsilon }}_1\\ {\tilde{z}}_1\\ {\tilde{\eta }}_f\\ {\tilde{f}}_s\end{array}\right)}^T\left(\begin{array}{cccc}{{\rm Y}}_{11}& P_1{A}_{12}\left(\rho \right)& 0& 0\\ *& {{\rm Y}}_{22}& C_2^T{P}_{f0}& {{\rm Y}}_{24}\\ *& *& {{\rm Y}}_{33}& -\frac{1}{\sigma }{H}^T{P}_{f0}{G}_2\\ *& *& *& {{\rm Y}}_{44}\end{array}\right)\left(\begin{array}{c}{\tilde{\upsilon }}_1\\ {\tilde{z}}_1\\ {\tilde{\eta }}_f\\ {\tilde{f}}_s\end{array}\right)\hfill \\ & & 2{\tilde{\upsilon }}_1^T{P}_1{M}_1\left(\rho \right)\xi +2{\tilde{z}}_1^T{P}_{f1}{M}_2\left(\rho \right)\xi +\delta \hfill \end{array}$$

(52)

where

$$\begin{array}{ccc}\hfill {{\rm Y}}_{11}& =& A_{11}^{sT}{P}_1+P_1{A}_{11}^s\hfill \\ \hfill {{\rm Y}}_{22}& =& {\left(A_{22}\left(\rho \right)+K_f{C}_2\right)}^T{P}_{f1}+P_{f1}\left(A_{22}\left(\rho \right)+K_f{C}_2\right)\hfill \\ \hfill {{\rm Y}}_{24}& =& P_{f2}{G}_2-\frac{1}{\sigma }{C}_2^T{P}_{f0}{G}_2\hfill \\ \hfill {{\rm Y}}_{33}& =& H^T{P}_{f0}+P_{f0}H\hfill \\ \hfill {{\rm Y}}_{44}& =& -\frac{2}{\sigma }{G}_2^T{P}_{f2}{G}_2+\frac{1}{\epsilon \sigma }N\hfill \end{array}$$

When $\xi \ne 0$, we define

$$\begin{array}{c}\hfill J\left(\rho \right)=\dot{V}+m^{T}m-\mu {\xi }^T\xi \end{array}$$

(53)

Substituting (52) into (53) and using the Schur Lemma yields

$$\begin{array}{c}\hfill J\left(\rho \right)\le {\varsigma }^T\Phi \left(\rho \right)\varsigma +\delta \end{array}$$

(54)

where $\varsigma ={\left[\begin{array}{ccccc}{\tilde{\upsilon }}_1^T& {\tilde{z}}_1^T& {\tilde{\eta }}_f^T& {\tilde{f}}_s^T& {\xi }^T\end{array}\right]}^T$ and $\Phi \left(\rho \right)$ is a matrix defined as follows:

$$\begin{array}{c}\hfill \Phi \left(\rho \right)=\left(\begin{array}{ccccc}{\Phi }_{11}& P_1{A}_{12}\left(\rho \right)& 0& 0& P_1{M}_1\left(\rho \right)\\ *& {\Phi }_{22}& C_2^T{P}_{f0}& {\Phi }_{24}& P_{f1}{M}_2\left(\rho \right)\\ *& *& {\Phi }_{33}& -\frac{1}{\sigma }{H}^T{P}_{f0}{F}_2& 0\\ *& *& *& {\Phi }_{44}& 0\\ *& *& *& *& -\mu I\end{array}\right)\end{array}$$

(55)

where

$$\begin{array}{ccc}\hfill {\Phi }_{11}& =& A_{11}^{sT}{P}_1+P_1{A}_{11}^s+{\chi }_1^T{\chi }_1\hfill \\ \hfill {\Phi }_{22}& =& P_{f1}{A}_{22}\left(\rho \right)+P_{f2}{C}_2+A_{22}^T\left(\rho \right)P_{f1}+C_2^T{P}_{f2}+{\chi }_2^T{\chi }_2\hfill \\ \hfill {\Phi }_{24}& =& P_{f2}{G}_2-\frac{1}{\sigma }{C}_2^T{P}_{f0}{G}_2\hfill \\ \hfill {\Phi }_{33}& =& H^T{P}_{f0}+P_{f0}H+{\chi }_3^T{\chi }_3\hfill \\ \hfill {\Phi }_{44}& =& -\frac{2}{\sigma }{G}_2^T{P}_{f0}{G}_2+\frac{1}{\epsilon \sigma }N+{\chi }_4^T{\chi }_4\hfill \end{array}$$

Hence, if $\Phi \left(\rho \right)<0$ holds for all $\rho \in \Im$, then there exists a positive $\tau$ such that $\tau ={\lambda }_{min}(-\Phi \left(\rho \right))$, satisfying the condition ${\tau \parallel \varsigma \parallel }^2>\delta$ for all $t\ge 0$. As a consequence, the performance index $J\left(\rho \right)$ is constrained within the bound $J\left(\rho \right)\le -{\tau \parallel \varsigma \parallel }^2+\delta$. This unequivocally establishes the negativity of $J\left(\rho \right)$, indicating that every constituent of the state vector $\varsigma$, encompassing the error vectors for state estimation $\left({\tilde{v}}_1,{\tilde{z}}_1\right.$, and $\left.{\tilde{\eta }}_f\right)$, as well as the fault estimation error vector ${\tilde{f}}_s$, remains bounded. □

Remark 3.

Moreover, in accordance with Theorem 1, the availability of a solution to the linear matrix inequality (LMI) optimization problem indicates a constraint on ς, implying the existence of a positive parameter $\tilde{\chi }$ for which $\parallel \varsigma \parallel <\tilde{\chi }$. Additionally, it can be deduced that $∥{\tilde{z}}_1∥<\tilde{\chi }$.

3.2. Sliding Motion Reachability

Moving forward, our attention shifts to the formulation of the scalar function $\mathcal{K}(\rho ,y,u)$ as delineated in Equation (30). The objective is to guide the error dynamics, within a finite time window, toward the attainment of a sliding surface while ensuring a seamless and continuous sliding motion along it. In the context of the system represented by Equations (34)–(36), let us delve into the notion of a sliding surface.

$$\begin{array}{c}\hfill \mathcal{S}=\left\{({\tilde{\upsilon }}_1,{\tilde{z}}_1,{\tilde{\eta }}_f)\left|s=C_1^{-1}{\eta }_1-{\widehat{\upsilon }}_1=0\right.\right\}\end{array}$$

(56)

The following theorem delineates the prerequisites for attainability that guarantee an impeccable sliding motion along the hyperplane labeled as $\mathcal{S}$.

Theorem 2.

The error dynamics are directed towards reaching the sliding surface $\mathcal{S}$ for all times $t\ge t_s$, and they sustain sliding motion on this surface provided that the LMI optimization problem presented in (40) has a solution, and the scalar function $\mathcal{K}(\rho ,y,u)$ adheres to the condition

$$\mathcal{K}(\rho ,y,u)\ge a_{12}\tilde{\chi }+m_1{\gamma }_{\xi }+∥F_1∥\phi (\rho ,u)+{\varrho }_0$$

(57)

where ${\varrho }_0$ is a positive scalar.

Proof. 

Let us examine the subsequent Lyapunov candidate, denoted as $V_s=\frac{1}{2}{s}^T{P}_{1}s$, which is associated with the sliding mode surface $\mathcal{S}$. The rate of change of $V_s$ can be calculated as follows:

$$\begin{array}{ccc}\hfill {\dot{V}}_s& =& s^T{P}_1\left(A_{11}^{s}s+A_{12}\left(\rho \right){\tilde{z}}_1+M_1\left(\rho \right)\xi +F_1{f}_{\nu }\left(\rho \right)-\vartheta \right)\hfill \end{array}$$

Since $A_{11}^s$ is a stable design matrix, it is easy to see that

$$\begin{array}{c}\hfill s^T{P}_1{A}_{11}^{s}s=\frac{1}{2}{s}^T\left(A_{11}^{sT}{P}_1+P_1{A}_{11}^s\right)s\le 0\end{array}$$

(58)

Hence, we have

$$\begin{array}{c}\hfill {\dot{V}}_s\le ∥P_{1}s∥\left(∥A_{12}\left(\rho \right)∥∥{\tilde{z}}_1∥+∥M_1\left(\rho \right)∥∥\xi ∥+∥F_1∥∥f_{\nu }\left(\rho \right)∥-\mathcal{K}(\rho ,y,u)\right)\end{array}$$

(59)

Let us define

$$\begin{array}{c}\hfill a_{12}=\underset{\rho \in \Im }{max}\left(∥A_{12}\left(\rho \right)∥\right),m_1=\underset{\rho \in \Im }{max}\left(∥M_1\left(\rho \right)∥\right)\end{array}$$

(60)

By utilizing the fact that $∥f_{\nu }\left(\rho \right)∥<\phi (\rho ,u)$, $∥{\tilde{z}}_1∥<\tilde{\chi }$, and $∥\xi ∥\le {\gamma }_{\xi }$, if the gain $\mathcal{K}(\rho ,y,u)$ is chosen as

$$\begin{array}{c}\hfill \mathcal{K}(\rho ,y,u)\ge a_{12}\tilde{\chi }+m_1{\gamma }_{\xi }+∥F_1∥\phi (\rho ,u)+{\varrho }_0\end{array}$$

(61)

then,

$$\begin{array}{c}\hfill {\dot{V}}_s\le -{\varrho }_0∥P_{1}s∥\le -{\varrho }_0\sqrt{{\lambda }_{min}\left(P_1\right)}{V}_s^{1/2}\end{array}$$

(62)

This suggests that the conditions for reachability, as detailed in [26], are satisfied, leading to the achievement of an ideal sliding motion on the surface $\mathcal{S}$ within a finite time duration $t_s$. □

Remark 4.

It follows that the definition of the scalar function $\mathcal{K}(y,u,\rho )$ in (57) is consistent with the assumption on its size in Equation (31).

4. Faults Estimation

During the sliding motion on $\mathcal{S}$, ${\dot{\tilde{\upsilon }}}_1={\tilde{\upsilon }}_1=0$ [19], as a result, Equation (34) is reduced to

$$\begin{array}{c}\hfill 0=A_{12}\left(\rho \right){\tilde{z}}_1+M_1\left(\rho \right)\xi +F_1{f}_{\nu }\left(\rho \right)-{\vartheta }_{eq}\end{array}$$

(63)

In this context, ${\vartheta }_{eq}$ represents the injection signal equivalent to the output error required for sustaining the sliding mode, as explained in reference [19]. An approximate expression for this signal can be derived as follows:

$$\begin{array}{c}\hfill {\vartheta }_{eq}=\mathcal{K}(\rho ,y,u)\frac{P_1{C}_1^{-1}{\tilde{\eta }}_1}{∥P_1{C}_1^{-1}{\tilde{\eta }}_1∥+\phi }\end{array}$$

(64)

Using $\phi$, which is a minute positive scalar, we can consequently establish the following:

$$\begin{array}{ccc}\hfill {∥F_1{f}_{\nu }\left(\rho \right)-{\vartheta }_{eq}∥}_{L_2}& =& {∥A_{12}\left(\rho \right){\tilde{z}}_1+M_1\left(\rho \right)\xi ∥}_{L_2}\hfill \\ \hfill & \le & a_{12}{∥{\tilde{z}}_1∥}_{L_2}+m_1{∥\xi ∥}_{L_2}\hfill \\ & \le & \left(\mu a_{12}{\sigma }_{max}\left({\chi }^{-1}\right)+m_1\right){∥\xi ∥}_{L_2}\hfill \end{array}$$

(65)

Here, ${\sigma }_{max}\left(A\right)$ represents the largest singular value of a matrix A. The outcome given in Equation (65) arises from the consideration that $∥{\tilde{z}}_1∥\le {\sigma }_{max}\left({\chi }^{-1}\right)\mu ∥\xi ∥$. Consequently, we can deduce that:

$$\underset{{∥\xi ∥}_{L_2}\ne 0}{sup}=\frac{{∥F_1{f}_{\nu }\left(\rho \right)-{\vartheta }_{eq}∥}_{L_2}}{{∥\xi ∥}_{L_2}}={\lambda }_1\mu +{\lambda }_2$$

(66)

where ${\lambda }_1=a_{12}{\sigma }_{max}\left({\chi }^{-1}\right)$ and ${\lambda }_2=m_1$. Thus, for a small ${\lambda }_1\mu +{\lambda }_2$, it can be seen from (66) that $F_1{f}_{\nu }\left(\rho \right)\approx {\vartheta }_{eq}$. Therefore, the virtual actuator fault $f_{\nu }\left(\rho \right)$ can be estimated as

$${\widehat{f}}_{\nu }\left(\rho \right)\approx F_1^{+}\mathcal{K}(y,u,\rho )\frac{P_1{C}_1^{-1}{\tilde{\eta }}_1}{∥P_1{C}_1^{-1}{\tilde{\eta }}_1∥+\phi }$$

(67)

The actual actuator fault can be estimated as

$${\widehat{f}}_a\approx E{\left(\rho \right)}^{-1}{F}_1^{+}\mathcal{K}(y,u,\rho )\frac{P_1{C}_1^{-1}{\tilde{\eta }}_1}{∥P_1{C}_1^{-1}{\tilde{\eta }}_1∥+\phi }$$

(68)

From (32), we have

$${\widehat{f}}_s\approx \Gamma G_2^T{P}_{f0}{\tilde{\eta }}_f+\sigma \Gamma G_2^T{P}_{f0}{\int }_{t_f}^t{\tilde{\eta }}_f\left(\tau \right)d\tau$$

(69)

Here, $t_f$ denotes the moment when a sensor fault is detected.

5. Fault-Tolerant Tracking Controller Design

Employing the obtained FE data, we will proceed to formulate a dynamic output feedback controller tailored for fault-tolerant tracking. The objective is to guarantee that the specified output of the compromised LPV system faithfully follows the prescribed reference signal.

To counteract the repercussions stemming from sensor faults, we will employ the subsequent sensor compensation output:

$$\begin{array}{c}\hfill y_c=y-G{\widehat{f}}_s=Cx+G{f}_s-G{\widehat{f}}_s=Cx+G{\tilde{f}}_s\end{array}$$

(70)

where $y_c$ is called the reliable output. The objective is to revamp the LPV system using an extended model, primarily encompassing the state vector x and the state representing the tracking error, denoted as $x_t$. This error state is defined as the integral of the difference between the desired reference and the rectified output, with the objective of accomplishing the following:

$$\begin{array}{c}\hfill x_t={\int }_0^t{e}_t\left(\tau \right)d\tau ={\int }_0^t\left(y_r-S_r{y}_c\right)d\tau \end{array}$$

(71)

where $e_t$ is the tracking error, $y_r$ is the desired reference, and $S_r\in {\mathbb{R}}^{l\times p}$ is used to define which output variable is considered to track the desired reference.

Considering $x_t$ and $y_c$ to be the outputs, the LPV augmented tracking system can be constructed in the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_f=A_f\left(\rho \right)x_f+B_f\left(\rho \right)u+M_f\left(\rho \right)\xi +F_f\left(\rho \right)f_a+I_f{y}_r+S_f{\tilde{f}}_s\hfill \\ y_f=C_f{x}_f+G_f{\tilde{f}}_s\hfill \end{array}\right.\end{array}$$

(72)

$$\begin{array}{c}\hfill x_f=\left[\begin{array}{c}{x}_t\\ x\end{array}\right],y_f=\left[\begin{array}{c}{x}_t\\ y_c\end{array}\right]\end{array}$$

$$\begin{array}{ccc}\hfill A_f\left(\rho \right)& =& \left[\begin{array}{cc}0& -S_{r}C\\ 0& A\left(\rho \right)\end{array}\right],B_f\left(\rho \right)=\left[\begin{array}{c}0\\ B\left(\rho \right)\end{array}\right],F_f\left(\rho \right)=\left[\begin{array}{c}0\\ F\left(\rho \right)\end{array}\right],M_f\left(\rho \right)=\left[\begin{array}{c}0\\ M\left(\rho \right)\end{array}\right]\hfill \\ \hfill I_f& =& \left[\begin{array}{c}I\\ 0\end{array}\right],S_f=\left[\begin{array}{c}-S_{r}G\\ 0\end{array}\right],C_f=\left[\begin{array}{cc}I& 0\\ 0& C\end{array}\right],G_f=\left[\begin{array}{c}0\\ G\end{array}\right]\hfill \end{array}$$

For system (72), we propose the following structure of the FTTC law:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\zeta }=A_k\left(\rho \right)\zeta +B_k\left(\rho \right)e_f\hfill \\ u=C_k\left(\rho \right)\zeta +D_k\left(\rho \right)e_f-F_k\left(\rho \right){\widehat{f}}_a\hfill \end{array}\right.\end{array}$$

(73)

Here, we have several components at play:

• $\zeta$ is the state of the controller, residing in ${\mathbb{R}}^{n\times n}$.
• $e_f$ represents the closed-loop error signal, which is defined as $R_r{y}_r-y_f$. We introduce the matrix $R_r$ in ${\mathbb{R}}^{(p+l)\times l}$ to ensure that the dimensions of $y_r$ and $y_f$ align appropriately.
• $A_k\left(\rho \right)$, $B_k\left(\rho \right)$, $C_k\left(\rho \right)$, $D_k\left(\rho \right)$, and $F_k\left(\rho \right)$ denote the controller matrices, which we will derive subsequently. These matrices have dimensions in ${\mathbb{R}}^{n\times n}$, ${\mathbb{R}}^{n\times p}$, ${\mathbb{R}}^{m\times n}$, ${\mathbb{R}}^{m\times p}$, and ${\mathbb{R}}^{m\times q}$, respectively.

Substituting (72) into (73), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\zeta }=A_k\left(\rho \right)\zeta +B_k\left(\rho \right)R_r{y}_r-B_k\left(\rho \right)C_f{x}_f-B_k\left(\rho \right)G_f{\tilde{f}}_s\hfill \\ u=C_k\left(\rho \right)\zeta +D_k\left(\rho \right)R_r{y}_r-D_k\left(\rho \right)C_f{x}_f-D_k\left(\rho \right)G_f{\tilde{f}}_s-F_k\left(\rho \right){\widehat{f}}_a\hfill \end{array}\right.\end{array}$$

(74)

Then, substituting (74) into (72), we further obtain

$$\begin{array}{ccc}\hfill {\dot{x}}_f& =& \left(A_f\left(\rho \right)-B_f\left(\rho \right)D_k\left(\rho \right)C_f\right)x_f+B_f\left(\rho \right)C_k\left(\rho \right)\zeta +\left(I_f+B_f\left(\rho \right)D_k\left(\rho \right)R_r\right)y_r\hfill \\ & & +\left(S_f-B_f\left(\rho \right)D_k\left(\rho \right)G_f\right){\tilde{f}}_s+F_f\left(\rho \right){\tilde{f}}_a+M_f\left(\rho \right)\xi \hfill \end{array}$$

(75)

The result (75) follows by selecting $F_k\left(\rho \right)=B_f^{+}\left(\rho \right)F_f\left(\rho \right)$, where $B_f^{+}\left(\rho \right)$ is the pseudo-inverse of $B_f\left(\rho \right)$ for all $\rho \in \Im$. Hence, $F_k\left(\rho \right)$ is considered as a known gain in the derivation of the LMI-based FTTC design.

We derive the dynamic equation governing the closed-loop system as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\eta }=A_{\eta }\left(\rho \right)\eta +E_{\eta }\left(\rho \right)\varpi \hfill \\ y_f=C_{\eta }\eta +G_{\eta }\varpi \hfill \end{array}\right.\end{array}$$

(76)

where $\eta =\left[\begin{array}{c}{x}_f\\ x_c\end{array}\right]$, $\varpi ={\left[\begin{array}{cccc}{\tilde{f}}_a^T& {\tilde{f}}_s^T& y_r^T& {\xi }^T\end{array}\right]}^T$, and

$$\begin{array}{ccc}\hfill A_{\eta }\left(\rho \right)& =& \left[\begin{array}{cc}{A}_f\left(\rho \right)-B_f\left(\rho \right)D_k\left(\rho \right)C_f& B_f\left(\rho \right)C_k\left(\rho \right)\\ -B_k\left(\rho \right)C_f& A_k\left(\rho \right)\end{array}\right]\hfill \\ \hfill E_{\eta }\left(\rho \right)& =& \left[\begin{array}{cccc}{F}_f\left(\rho \right)& S_f-B_f\left(\rho \right)D_k\left(\rho \right)G_f& I_f+B_f\left(\rho \right)D_k\left(\rho \right)R_r& M_f\left(\rho \right)\\ 0& -B_k\left(\rho \right)G_f& B_k\left(\rho \right)R_r& 0\end{array}\right]\hfill \\ \hfill C_{\eta }& =& \left[\begin{array}{cc}{C}_f& 0\end{array}\right],G_{\eta }=\left[\begin{array}{cccc}0& G_f& 0& 0\end{array}\right]\hfill \end{array}$$

(77)

We define the output performance $y_L$ as

$$y_L=C_L{y}_f$$

(78)

where $C_L$ is part of the design.

The next step is to design the controller gains to make the closed loop system (75) stable, and to satisfy the following performance index:

$$\underset{{∥\varpi ∥}_{L_2}\ne 0}{sup}\frac{{∥y_L∥}_{L_2}^2}{{∥\varpi ∥}_{L_2}^2}\le {\mu }_k$$

(79)

where ${\mu }_k$ is a positive scalar.

Theorem 3.

Given the circular region $\mathcal{D}(\kappa ,\beta )$, if there exist matrices $\mathbb{X}>0$, $\mathbb{Y}>0$, $\widehat{A}\left(\rho \right)$, $\widehat{B}\left(\rho \right)$, $\widehat{C}\left(\rho \right)$, and $\widehat{D}\left(\rho \right)$ such that the following inequalities hold:

$$\mathit{min}{\mu }_k\mathit{subject}\mathit{to}$$

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}{\Psi }_{11}& {\Psi }_{12}& G_f& {\Psi }_{14}& {\Psi }_{15}& {\Psi }_{16}& M_f\left(\rho \right)& \mathbb{X}{C}_f^T\\ *& {\Psi }_{22}& \mathbb{Y}{G}_f& {\Psi }_{24}& {\Psi }_{25}& {\Psi }_{26}& \mathbb{Y}{M}_f\left(\rho \right)& C_f^T\\ *& *& -{\mu }_{k}I& 0& 0& 0& 0& 0\\ *& *& *& -{\mu }_{k}I& 0& 0& 0& 0\\ *& *& *& *& -{\mu }_{k}I& 0& 0& I_f^T\\ *& *& *& *& *& -{\mu }_{k}I& 0& 0\\ *& *& *& *& *& *& -{\mu }_{k}I& 0\\ *& *& *& *& *& *& *& -I\end{array}\right)<0\end{array}$$

(80)

and

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}-\kappa \mathbb{X}& -\kappa I& A_f\left(\rho \right)+B_f\left(\rho \right)\widehat{C}\left(\rho \right)+\beta & A_f\left(\rho \right)-B\left(\rho \right)\widehat{D}\left(\rho \right)C_f+\beta I\\ -\kappa I& -\kappa \mathbb{Y}& \widehat{A}\left(\rho \right)+\beta I& A\left(\rho \right)+\widehat{B}\left(\rho \right)C_f+\beta \\ *& *& -\kappa \mathbb{X}& -\kappa I\\ *& *& *& -\kappa \mathbb{Y}\end{array}\right)<0\end{array}$$

(81)

with

$$\begin{array}{ccc}\hfill {\Psi }_{11}& =& A_f\left(\rho \right)\mathbb{X}+\mathbb{X}{A}_f^T\left(\rho \right)+B_f\left(\rho \right)\widehat{C}\left(\rho \right)+{\widehat{C}}^T\left(\rho \right)B_f^T\left(\rho \right)\hfill \\ \hfill {\Psi }_{12}& =& {\widehat{A}}^T\left(\rho \right)+A_f\left(\rho \right)-B_f\left(\rho \right)\widehat{D}\left(\rho \right)C_f\hfill \\ \hfill {\Psi }_{14}& =& S_f-B_f\left(\rho \right)\widehat{D}\left(\rho \right)G_f\hfill \\ \hfill {\Psi }_{15}& =& I_f+B_f\left(\rho \right)\widehat{D}\left(\rho \right)R_r\hfill \\ \hfill {\Psi }_{16}& =& F_f\left(\rho \right)-B_f\left(\rho \right)F_k\left(\rho \right)\hfill \\ \hfill {\Psi }_{22}& =& \mathbb{Y}{A}_f\left(\rho \right)+A_f^T\left(\rho \right){\mathbb{Y}}^T+\widehat{B}\left(\rho \right)C_f+C_f^T{\widehat{B}}^T\left(\rho \right)\hfill \\ \hfill {\Psi }_{24}& =& \mathbb{Y}{S}_f+\widehat{B}\left(\rho \right)G_f\hfill \\ \hfill {\Psi }_{25}& =& \mathbb{Y}{I}_f+\widehat{B}\left(\rho \right)R_r\hfill \\ \hfill {\Psi }_{26}& =& \mathbb{Y}\left(F_f\left(\rho \right)-B_f\left(\rho \right)F_k\left(\rho \right)\right)\hfill \end{array}$$

Following this, the closed-loop system, as described by Equation (76), exhibits robust stability at the ${\mathcal{H}}_{\infty }$ performance level denoted as ${\mu }_k$, as indicated in (79). Additionally, the eigenvalues of the matrix $A_{\eta }\left(\rho \right)$ are situated within the set $\mathcal{D}(\kappa ,\beta )$. The determination of the controller gains is subsequently carried out using the following methodology:

$$\begin{array}{ccc}\hfill D_k\left(\rho \right)& =& \widehat{D}\left(\rho \right)\hfill \\ \hfill C_k\left(\rho \right)& =& \left(\widehat{C}\left(\rho \right)-D_k\left(\rho \right)C_f\right)U^{-T}\hfill \\ \hfill B_k\left(\rho \right)& =& V^{-1}\left(\widehat{B}\left(\rho \right)-\mathbb{Y}{B}_f\left(\rho \right)D_k\left(\rho \right)\right)\hfill \\ \hfill A_k\left(\rho \right)& =& V^{-1}\left(\widehat{A}\left(\rho \right)-\mathbb{Y}\left(A_f\left(\rho \right)-B_f\left(\rho \right)\widehat{D}\left(\rho \right)C_f\right)\mathbb{X}-\mathbb{Y}{B}_f\left(\rho \right)C_k\left(\rho \right)U^T\right.\hfill \\ \hfill & & \left.+V{B}_k\left(\rho \right)C_f\mathbb{X}\right)U^{-T}\hfill \end{array}$$

(82)

where U, $V\in {\mathbb{R}}^{n\times n}$ satisfy $U{V}^T=I-\mathbb{X}\mathbb{Y}$.

Proof. 

Supposing that $V_{\eta }={\eta }^T{P}_{\eta }\eta$. We have

$$\begin{array}{c}\hfill {\dot{V}}_{\eta }={\eta }^T\left(A_{\eta }^T\left(\rho \right)P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)\right)\eta +2{\eta }^T{P}_{\eta }{E}_{\eta }\left(\rho \right)\varpi \end{array}$$

(83)

Let

$$\begin{array}{c}\hfill J_{\eta }\left(\rho \right)={\dot{V}}_{\eta }+y_L^T{y}_L-{\mu }_k{\varpi }^T\varpi \end{array}$$

(84)

Substituting (83) into (84), we have

$$\begin{array}{c}\hfill J_{\eta }\left(\rho \right)={\eta }^T\left(A_{\eta }^T\left(\rho \right)P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)\right)\eta +2{\eta }^T{P}_{\eta }{E}_{\eta }\left(\rho \right)\varpi +y_L^T{y}_L-{\mu }_k{\varpi }^T\varpi \end{array}$$

(85)

Let us define

$$\begin{array}{c}\hfill y_L=R_r{y}_r-y_f\end{array}$$

(86)

We have

$$\begin{array}{ccc}\hfill y_L^T{y}_L& =& {\left(R_r{y}_r-y_f\right)}^T\left(R_r{y}_r-y_f\right)\hfill \\ \hfill & =& y_r^T{R}_r^T{R}_r{y}_r-2y_r^T{R}_r^T{y}_f+y_f^T{y}_f\hfill \\ & =& {\varpi }^T{R}_{\eta }^T{R}_{\eta }\varpi -2{\varpi }^T{R}_{\eta }^T{C}_{\eta }\eta +{\eta }^T{C}_{\eta }^T{C}_{\eta }\eta \hfill \end{array}$$

(87)

where $R_{\eta }=\left[\begin{array}{cccc}0\hfill & 0& R_r& 0\end{array}\right]$. Substituting (87) into (85), we have

$$\begin{array}{ccc}\hfill J_{\eta }\left(\rho \right)& =& {\eta }^T\left(A_{\eta }^T\left(\rho \right)P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)+C_{\eta }^T{C}_{\eta }\right)\eta +2{\eta }^T\left(P_{\eta }{E}_{\eta }\left(\rho \right)-C_{\eta }^T{R}_{\eta }\right)\varpi \hfill \\ \hfill & & +{\varpi }^T{R}_{\eta }^T{R}_{\eta }\varpi -{\mu }_k{\varpi }^T\varpi \hfill \\ & =& {\left(\begin{array}{c}\eta \\ \varpi \end{array}\right)}^T\left(\begin{array}{cc}{A}_{\eta }^T\left(\rho \right)P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)+C_{\eta }^T{C}_{\eta }& P_{\eta }{E}_{\eta }\left(\rho \right)-C_{\eta }^T{R}_{\eta }\\ *& R_{\eta }^T{R}_{\eta }-{\mu }_{k}I\end{array}\right)\left(\begin{array}{c}\eta \\ \varpi \end{array}\right)\hfill \end{array}$$

(88)

Using the Schur complement, it is easy to find that $J_{\eta }\left(\rho \right)<0$ if

$$\begin{array}{c}\left(\begin{array}{ccc}{A}_{\eta }^T\left(\rho \right)P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)& P_{\eta }{E}_{\eta }\left(\rho \right)& C_{\eta }^T\\ *& -{\mu }_{k}I& R_{\eta }^T\\ *& *& -I\end{array}\right)<0\end{array}$$

(89)

We can find that the form (89) is not an LMI. As a result, we can change (89) to make it solvable.

Let us define

$$\begin{array}{ccc}\hfill P_{\eta }& =& \left(\begin{array}{cc}\mathbb{Y}& V\\ V^T& W\end{array}\right),P_{\eta }^{-1}=\left(\begin{array}{cc}\mathbb{X}& U\\ U^T& Z\end{array}\right)\hfill \\ \hfill {\Pi }_1& =& \left(\begin{array}{cc}\mathbb{X}& I_n\\ U^T& 0\end{array}\right),{\Pi }_2=\left(\begin{array}{cc}{I}_n& \mathbb{Y}\\ 0& V^T\end{array}\right)\hfill \end{array}$$

(90)

Since $P_{\eta }{P}_{\eta }^{-1}=I_{2n}$, we can deduce that $P_{\eta }{\Pi }_1={\Pi }_2$. By performing matrix multiplication on both sides of Equation (89) with $diag\left({\Pi }_1^T,I,I\right)$ and its transpose, we can derive the following:

$$\begin{array}{c}\left(\begin{array}{ccc}{\Pi }_1^T\left(A_{\eta }^T\left(\rho \right)P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)\right){\Pi }_1& {\Pi }_1^T{P}_{\eta }{E}_{\eta }\left(\rho \right)& {\Pi }_1^T{C}_{\eta }^T\\ *& -{\mu }_{k}I& R_{\eta }^T\\ *& *& -I\end{array}\right)<0\end{array}$$

(91)

Taking into consideration the formulations of $A_{\eta }\left(\rho \right)$, $E_{\eta }\left(\rho \right)$, and $C_{\eta }$ as provided in (77), and incorporating the subsequent change of variables:

$$\begin{array}{ccc}\hfill \widehat{A}\left(\rho \right)& =& \mathbb{Y}(A_f\left(\rho \right)-B_f\left(\rho \right)\widehat{D}\left(\rho \right)C_f)\mathbb{X}+\mathbb{Y}{B}_f\left(\rho \right)C_k\left(\rho \right)U^T-V{B}_k\left(\rho \right)C_f\mathbb{X}\hfill \\ \hfill & & +V{A}_k\left(\rho \right)U^T\hfill \\ \hfill \widehat{B}\left(\rho \right)& =& -\mathbb{Y}{B}_f\left(\rho \right)\widehat{D}\left(\rho \right)-V{B}_k\left(\rho \right)\hfill \\ \hfill \widehat{C}\left(\rho \right)& =& C_k\left(\rho \right)U^T-\widehat{D}\left(\rho \right)C_f\mathbb{X}\hfill \\ \hfill \widehat{D}\left(\rho \right)& =& D_k\left(\rho \right)\hfill \end{array}$$

(92)

Inequality (80) can, thus, be easily obtained.

Our subsequent objective is to formulate the conditions for regional pole constraints. When we substitute $\mathcal{A}={\mathcal{A}}_{\eta }\left(\rho \right)$ and $\mathcal{P}=P_{\eta }$ into Lemma 2, it becomes evident that the eigenvalues of $A_{\eta }\left(\rho \right)$ lie within the sector shaped like a cone, denoted as $D(\kappa ,\beta )$, if:

$$\begin{array}{c}\hfill \left(\begin{array}{cc}-\kappa P_{\eta }& \beta P_{\eta }+P_{\eta }{A}_{\eta }\left(\rho \right)\\ & -\kappa P_{\eta }\end{array}\right)<0\end{array}$$

(93)

Next, by left- and right-multiplying (93) with $diag({\Pi }_1^T,{\Pi }_1^T)$ and its transpose, and subsequently employing the definitions of $\widehat{A}\left(\rho \right)$, $\widehat{B}\left(\rho \right)$, $\widehat{C}\left(\rho \right)$, and $\widehat{D}\left(\rho \right)$ as presented in (92), we can readily derive the inequality (81). This concludes the proof. □

The structure of the proposed FE and FTTC strategy is shown in Figure 1.

6. Simulation Results

In this section, two examples are presented to validate the developed FTTC and FE schemes.

6.1. Example 1

Let us consider an LPV system in the form of (1) with the following system matrices [27]:

$$\begin{array}{ccc}\hfill A\left(\rho \right)& =& \left[\begin{array}{cccc}-1.75+{\rho }_2& 1& 0& 0\\ 1& -1+{\rho }_1& 0& 0\\ -1.8& -1& -0.75+{\rho }_1& 0\\ -1& 0& 0& -1+{\rho }_2\end{array}\right],B\left(\rho \right)=\left[\begin{array}{cc}1+{\rho }_1& 1\\ 1& 0.5+{\rho }_2\\ 1& 0\\ {\rho }_2& 0\end{array}\right]\hfill \\ \hfill M\left(\rho \right)& =& \left[\begin{array}{c}0\\ 0.6+{\rho }_1\\ 0\\ 1\end{array}\right],F\left(\rho \right)=F=\left[\begin{array}{c}1\\ 0.6\\ 0\\ 1\end{array}\right],C=\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\end{array}\right],G=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\hfill \end{array}$$

Let us consider the varying parameters vector $\rho ={\left[\begin{array}{cc}{\rho }_1& {\rho }_2\end{array}\right]}^T$, where ${\rho }_1$ lies in the interval $[-0.05,0.05]$ and ${\rho }_2$ ranges between $[-0.1,0.1]$. Consequently, the system operates within a hypercube characterized by four vertices. Each vertex corresponds to the extreme values of ${\rho }_i$, where i can be either 1 or 2. In this context, the polytopic LPV representation (1) can be written as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}={\displaystyle \sum _{i=1}^4}{\alpha }_i\left(\rho \right)\left(A_{i}x+B_{i}u+M_i\xi +F{f}_a\right)\hfill \\ y=Cx+G{f}_s\hfill \end{array}\right.\end{array}$$

(94)

The matrices of the system can be determined at the vertices of the polytope for extreme values of the parameters ${\alpha }_i$. So, the matrices $A_i$, $B_i$, $M_i$ are defined as follows:

$$\begin{array}{c}\hfill A_1=\left[\begin{array}{cccc}-1.85& 1& 0& 0\\ -1& -1.05& 0& 0\\ -1.8& -1& -0.8& 0\\ -1& 0& 0& -1.1\end{array}\right],A_2=\left[\begin{array}{cccc}-1.65& 1& 0& 0\\ -1& -1.05& 0& 0\\ -1.8& -1& -0.8& 0\\ -1& 0& 0& -0.9\end{array}\right]\end{array}$$

$$\begin{array}{c}\hfill A_3=\left[\begin{array}{cccc}-1.85& 1& 0& 0\\ -1& -0.95& 0& 0\\ -1.8& -1& -0.7& 0\\ -1& 0& 0& -1.1\end{array}\right],A_4=\left[\begin{array}{cccc}-1.65& 1& 0& 0\\ -1& -0.95& 0& 0\\ -1.8& -1& -0.7& 0\\ -1& 0& 0& -0.9\end{array}\right]\end{array}$$

$$\begin{array}{c}\hfill B_1=\left[\begin{array}{cc}0.95& 1\\ 1& 0.4\\ 1& 0\\ -0.1& 0\end{array}\right],B_2=\left[\begin{array}{cc}0.95& 1\\ 1& 0.6\\ 1& 0\\ 0.1& 0\end{array}\right],B_3=\left[\begin{array}{cc}1.05& 1\\ 1& 0.4\\ 1& 0\\ -0.1& 0\end{array}\right]\end{array}$$

$$\begin{array}{ccc}\hfill B_4& =& \left[\begin{array}{cc}1.05& 1\\ 1& 0.6\\ 0& 0\\ 0.1& 0\end{array}\right],M_1=M_2=\left[\begin{array}{c}0.1\\ -0.15\\ 0\\ 0\end{array}\right],M_3=M_4=\left[\begin{array}{c}0.1\\ -0.05\\ 0\\ 0\end{array}\right]\hfill \end{array}$$

The expressions defining the weighting functions ${\alpha }_i\left(\rho \right)$ are given as follows:

$$\begin{array}{ccc}\hfill {\alpha }_1\left(\rho \right)& =& \frac{{\rho }_1-{\underline{\rho }}_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\rho }_2-{\underline{\rho }}_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{({\rho }_1+0.05)({\rho }_2+0.1)}{0.02}\hfill \\ \hfill {\alpha }_2\left(\rho \right)& =& \frac{{\rho }_1-{\underline{\rho }}_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\overline{\rho }}_2-{\rho }_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{({\rho }_1+0.05)(0.1-{\rho }_2)}{0.02}\hfill \\ \hfill {\alpha }_3\left(\rho \right)& =& \frac{{\overline{\rho }}_1-{\rho }_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\rho }_2-{\underline{\rho }}_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{(0.05-{\rho }_1)({\rho }_2+0.1)}{0.02}\hfill \\ \hfill {\alpha }_4\left(\rho \right)& =& \frac{{\overline{\rho }}_1-{\rho }_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\overline{\rho }}_2-{\rho }_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{(0.05-{\rho }_1)(0.1-{\rho }_2)}{0.02}\hfill \end{array}$$

It is worth noting that the parameters ${\rho }_1$ and ${\rho }_2$ are governed by the functions ${\rho }_1=0.05sin\left(t\right)$ and ${\rho }_2=0.1sin\left(t\right)$, respectively. The weighting functions ${\alpha }_i\left(\rho \right)$ for each vertex (where i ranges from 1 to 4) are illustrated in Figure 2.

Since Assumption 2 holds, the nonsingular transformation matrices $T_c$ and $S_c$ can be found as:

$$\begin{array}{c}\hfill T_c=\left[\begin{array}{cccc}1& 0& 1& 0\\ -0.6& 1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 1\end{array}\right],S_c=\left[\begin{array}{ccc}1& 0& 0\\ -1.6& 1& 0\\ -1& 0& 1\end{array}\right]\end{array}$$

In this investigation, the following parameter values are selected: $\sigma$ is fixed at 1, $\epsilon$ takes the value of 1, $\Gamma$ is assigned a value of 100, ${\chi }_1$ is set to 2, ${\chi }_2$ is represented as a $3\times 3$ identity matrix scaled by 2, ${\chi }_3$ is described as a $2\times 2$ identity matrix scaled by 2, and ${\chi }_4$ is defined as a $1\times 1$ identity matrix scaled by 2. Additionally, we define the matrices $S_r$ and $R_r$ as:

$$\begin{array}{c}\hfill S_r=\left[\begin{array}{ccc}0& 0& 1\end{array}\right],R_r={\left[\begin{array}{cccc}0& 0& 0& 1\end{array}\right]}^T\end{array}$$

We employ the MATLAB YALMIP toolbox solver for the convenient identification of solutions to the linear matrix inequality (LMI) optimization problems stated in (40) and (80)–(81).

(a) ASMO: For this part, we obtain the optimum value $\mu =1.2307$, and

$$\begin{array}{ccc}\hfill P_1& =& 0.2746,P_{f1}=\left[\begin{array}{ccc}98.4322& 3.7770& -153.1626\\ 3.7770& 16.3044& -76.0368\\ -153.1626& -76.0368& 545.3477\end{array}\right]\hfill \\ \hfill P_{f2}& =& \left[\begin{array}{cc}1.1945& 2.0638\\ 2.2850& -1.4075\\ 0.7579& 0.9907\end{array}\right],P_{f0}=\left[\begin{array}{cc}1.0316& 1.9049\\ 1.9049& 8.0249\end{array}\right]\hfill \\ \hfill X& =& -2.3822,Y=\left[\begin{array}{cc}2.4601& -0.0468\\ 0.0468& 2.4601\end{array}\right]\hfill \end{array}$$

Then, $A_{11}^s$ and H can be computed as

$$\begin{array}{ccc}\hfill A_{11}^s& =& -8.6762,H=\left[\begin{array}{cc}4.2266& -1.0886\\ -0.9974& 0.5650\end{array}\right]\hfill \end{array}$$

The gain matrices of the ASMO $L_{11}\left(\rho \right)$, $L_{21}\left(\rho \right)$, and $L_{22}\left(\rho \right)$ are given by

$$\begin{array}{c}\hfill L_{11}\left(\rho \right)=\sum _{i=1}^4{\alpha }_i{L}_{11,i},L_{21}\left(\rho \right)=\sum _{i=1}^4{\alpha }_i{L}_{21,i},L_{22}\left(\rho \right)=\sum _{i=1}^4{\alpha }_i{L}_{22,i}\end{array}$$

where

$$\begin{array}{ccc}\hfill L_{11,1}& =& L_{11,3}=5.0262,L_{11,2}=L_{11,4}=5.2262\hfill \\ \hfill L_{21,1}& =& \left[\begin{array}{cc}0.4307& -3.7390\\ 2.4979& -6.4150\\ 0.6263& -0.6535\end{array}\right],L_{21,2}=\left[\begin{array}{cc}0.1565& -3.5701\\ 2.4979& -6.4150\\ 0.4747& -0.8516\end{array}\right]\hfill \\ \hfill L_{21,3}& =& \left[\begin{array}{cc}0.4483& -4.0299\\ 2.2694& -6.2743\\ 0.6263& -0.6535\end{array}\right],L_{21,4}=\left[\begin{array}{cc}0.1741& -3.8610\\ 2.2694& -6.2743\\ 0.4747& -0.8516\end{array}\right]\hfill \\ \hfill L_{22,1}& =& L_{22,2}=L_{22,3}=L_{22,4}=\left[\begin{array}{cc}5.9302& -6.3951\\ -1.7554& -0.4257\end{array}\right]\hfill \end{array}$$

(b) FTTC: We attain the optimum value of ${\mu }_k$ as 0.45644 and calculate the matrices for the controller gains using Theorem 3 within the specified domain of $D(-30,30)$. For brevity, the controller parameters are provided in Appendix A due to space constraints.

In the simulation, we apply the input signal $u=sin\left(t\right)$ to the LPV system (1), while the external disturbance is modeled as $\xi =0.1sin\left(0.2t\right)$. Initially, the system states are set to $x_0={\left[0000\right]}^T$, and the estimated states begin at $\widehat{x_0}={[-2-0.52-1.5]}^T$. The parameters related to the equivalent output error injection, denoted as ${\vartheta }_{eq}$, are selected as $\mathcal{K}(\rho ,y,u)=10$ and $\phi =0.01$.

The left of Figure 3 shows the state estimation errors, whereas the right of Figure 3 illustrates the actuator fault signal $f_a$ along with its estimate ${\widehat{f}}_a$. The left of Figure 4 shows the sensor FE for the case of a constant sensor fault, whereas the right of Figure 4 provides the case for a time-varying sensor fault. These figures collectively demonstrate that the proposed ASMO can effectively estimate both states and faults with a high degree of accuracy, effectively mitigating the impact of disturbances.

For a more comprehensive assessment of the system’s performance, we present the simulation results for the tracking error of the output $y_3$ in Figure 5. It is evident that the actual output $y_3$ from the LPV system under the influence of the developed FTTC closely follows its reference trajectory with remarkable accuracy, even in the presence of actuator and sensor faults and disruptive external disturbance. This underscores the effectiveness of our proposed controller in compensating for fault effects, enabling precise tracking of the desired trajectory.

6.2. Example 2

In this example, consider a real-world model of an aircraft provided by [28]. As is pointed in [28], the dynamic model describing the motion of vertical takeoff and landing (VTOL) aircraft in the vertical plane can be represented in the following polytopic LPV form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}=A\left(\rho \right)x+B\left(\rho \right)u+M\left(\rho \right)\xi +F\left(\rho \right)f_a\hfill \\ y=Cx+G{f}_s\hfill \end{array}\right.\end{array}$$

(95)

Here, the state vector is denoted as $x={\left[\begin{array}{cccc}{V}_h\hfill & V_v\hfill & q\hfill & \theta \hfill \end{array}\right]}^T$, where $x_1=V_h$ represents the horizontal velocity (in knots), $x_2=V_v$ signifies the vertical velocity (in knots), $x_3=q$ denotes the pitch rate (in degrees per second), and $x_4=h$ stands for the pitch angle (in degrees). Figure 6 illustrates the standard coordinate system for a VTOL aircraft in the vertical plane. The control input vector is represented as $u={\left[\begin{array}{cc}{u}_1\hfill & u_2\hfill \end{array}\right]}^T$, where $u_1$ corresponds to the collective pitch control, adjusting the pitch angle (the angle of attack with respect to the air) of the main rotor blades collectively to facilitate vertical movement. Additionally, $u_2$ denotes the longitudinal cyclic pitch control, tilting the main rotor disc by adjusting the pitch of the main rotor blades individually to enable horizontal movement. The system matrices, incorporating the variable parameters, can be expressed as:

$$\begin{array}{ccc}\hfill A\left(\rho \right)& =& \left[\begin{array}{cccc}-9.9477& -0.7478& 0.2632& 5.0337\\ 52.1659& 2.7452& 5.5532& -24.4221\\ 26.0922& 2.6361+{\rho }_1& -4.1975& -19.2774+{\rho }_2\\ 0& 0& 1& 0\end{array}\right]\hfill \\ \hfill B\left(\rho \right)& =& F\left(\rho \right)=\left[\begin{array}{cc}0.4422& 0.1761\\ 3.5446+{\rho }_2& -7.5922\\ -5.5200& 4.4900\\ 0& 0\end{array}\right]\hfill \\ \hfill M\left(\rho \right)& =& \left[\begin{array}{c}1\\ 0\\ 1\\ 1\end{array}\right],C=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right],G=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\hfill \end{array}$$

Let us assume that the varying parameters vector $\rho ={\left[\begin{array}{cc}{\rho }_1& {\rho }_2\end{array}\right]}^T$, where ${\rho }_1\in [-0.5,0.5]$ and ${\rho }_2\in [-2,2]$. Consequently, the VTOL aircraft system operates within a hypercube characterized by four vertices. Consequently, the LPV model can be written as the following polytopic representation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}={\displaystyle \sum _{i=1}^4}{\alpha }_i\left(\rho \right)\left(A_{i}x+B_{i}u+M\xi +F_i{f}_a\right)\hfill \\ y=Cx+G{f}_s\hfill \end{array}\right.\end{array}$$

(96)

where

$$\begin{array}{ccc}\hfill A_1& =& \left[\begin{array}{cccc}-9.9477& -0.7478& 0.2632& 5.0337\\ 52.1659& 2.7452& 5.5532& -24.4221\\ 26.0922& 2.1361& -4.1975& -21.2774\\ 0& 0& 1& 0\end{array}\right]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill A_2& =& \left[\begin{array}{cccc}-9.9477& -0.7478& 0.2632& 5.0337\\ 52.1659& 2.7452& 5.5532& -24.4221\\ 26.0922& 2.1361& -4.1975& -17.2774\\ 0& 0& 1& 0\end{array}\right]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill A_3& =& \left[\begin{array}{cccc}-9.9477& -0.7478& 0.2632& 5.0337\\ 52.1659& 2.7452& 5.5532& -24.4221\\ 26.0922& 3.1361& -4.1975& -21.2774\\ 0& 0& 1& 0\end{array}\right]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill A_4& =& \left[\begin{array}{cccc}-9.9477& -0.7478& 0.2632& 5.0337\\ 52.1659& 2.7452& 5.5532& -24.4221\\ 26.0922& 3.1361& -4.1975& -17.2774\\ 0& 0& 1& 0\end{array}\right]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill B_1& =& F_1=\left[\begin{array}{cc}0.4422& 0.1761\\ 1.5446& -7.5922\\ -5.5200& 4.4900\\ 0& 0\end{array}\right],B_2=F_2=\left[\begin{array}{cc}0.4422& 0.1761\\ 5.5446& -7.5922\\ -5.5200& 4.4900\\ 0& 0\end{array}\right]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill B_3& =& F_3=\left[\begin{array}{cc}0.4422& 0.1761\\ 1.5446& -7.5922\\ -5.5200& 4.4900\\ 0& 0\end{array}\right],B_4=F_4=\left[\begin{array}{cc}0.4422& 0.1761\\ 5.5446& -7.5922\\ -5.5200& 4.4900\\ 0& 0\end{array}\right]\hfill \end{array}$$

The local weighting functions are as follows:

$$\begin{array}{ccc}\hfill {\alpha }_1\left(\rho \right)& =& \frac{{\rho }_1-{\underline{\rho }}_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\rho }_2-{\underline{\rho }}_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{({\rho }_1+0.5)({\rho }_2+2)}{4}\hfill \\ \hfill {\alpha }_2\left(\rho \right)& =& \frac{{\rho }_1-{\underline{\rho }}_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\overline{\rho }}_2-{\rho }_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{({\rho }_1+0.5)(2-{\rho }_2)}{4}\hfill \\ \hfill {\alpha }_3\left(\rho \right)& =& \frac{{\overline{\rho }}_1-{\rho }_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\rho }_2-{\underline{\rho }}_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{(0.5-{\rho }_1)({\rho }_2+2)}{4}\hfill \\ \hfill {\alpha }_4\left(\rho \right)& =& \frac{{\overline{\rho }}_1-{\rho }_1}{{\overline{\rho }}_1-{\underline{\rho }}_1}\frac{{\overline{\rho }}_2-{\rho }_2}{{\overline{\rho }}_2-{\underline{\rho }}_2}=\frac{(0.5-{\rho }_1)(2-{\rho }_2)}{4}\hfill \end{array}$$

We assumed that an actuator fault occurs in the second input channel. Hence, the actuator fault vector $f_a$ can be expressed as ${\left[\begin{array}{cc}0& f_{a2}\end{array}\right]}^T$, and

$$\begin{array}{c}\hfill F\left(\rho \right)=F=\left[\begin{array}{c}0.1761\\ -7.5922\\ 4.4900\\ 0\end{array}\right]\end{array}$$

(97)

We checked that Assumptions 2–4 hold. We select the gain matrices $T_c$ and $S_c$ as

$$\begin{array}{c}\hfill T_c=\left[\begin{array}{cccc}1& 0& 0& 0\\ 43.1130& 1& 0& 0\\ -25.4969& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],S_c=\left[\begin{array}{ccc}1& 0& 0\\ 43.1130& 1& 0\\ -25.4969& 0& 1\end{array}\right]\end{array}$$

Choosing $\sigma =1$, $\epsilon =1$, $\Gamma =100$, ${\chi }_1=1$, ${\chi }_2=I_3$, ${\chi }_3=I_2$, ${\chi }_4=1$. By solving the LMI (40) in Theorem 1, we obtain the minimal optimization index $\mu =5.7689$, and

$$\begin{array}{ccc}\hfill P_1& =& 0.9929,A_1^s=-0.8164\hfill \\ \hfill P_{f_1}& =& \left[\begin{array}{ccc}1.5534& 2.0879& -12.4614\\ 2.0879& 3.4835& -16.0862\\ -12.4614& -16.0862& 128.9241\end{array}\right],P_{f_2}=\left[\begin{array}{cc}-1.3195& 0.1886\\ 0.2040& 1.3132\\ -0.1040& 0.1831\end{array}\right]\hfill \\ \hfill P_{f_0}& =& \left[\begin{array}{cc}0.7304& 0.1302\\ 0.1302& 1.2834\end{array}\right],H=\left[\begin{array}{cc}1.3338& -0.2343\\ -0.0805& 0.7747\end{array}\right]\hfill \end{array}$$

Choosing $S_r=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]$, $R_r={\left[\begin{array}{cccc}0& 0& 0& 1\end{array}\right]}^T$, and LMI region $\mathcal{D}=(30,30)$. By solving the LMI (80) in Theorem 3, we obtain the $H_{\infty }$ performance index ${\mu }_k=5.9141$. Thus, according to Theorem 3, we can obtain the controller gain matrices.

The outcomes of the simulation for example 2 are depicted in Figure 7 and Figure 8. Figure 7 illustrates the actuator and sensor faults alongside their estimations. Meanwhile, Figure 8 displays the tracking error of the output $y_3$. Analysis of these figures reveals that the devised active sliding mode observer (ASMO) successfully estimates the faults, and the fault-tolerant tracking controller (FTTC) effectively stabilizes the systems.

7. Conclusions

In this work, we explore the complex issues surrounding the domains of FE and FTTC within the framework of LPV systems. These systems confront the daunting task of coping with concurrent disturbances, actuator faults that vary over time, and sensor faults.

The LPV systems we are examining are depicted within a polytopic LPV framework, encompassing measurable gain scheduling functions. We introduce an adaptive LPV SMO designed to provide robust state and fault estimations, even in the presence of external disturbances. What sets our proposed approach apart from traditional SMO designs is its dual capability: it can directly reconstruct actuator faults by harnessing the equivalent output error injection required to maintain sliding motion, while also employing an adaptive algorithm to estimate time-varying sensor faults.

To address the FE and FTTC challenges in LPV systems plagued by simultaneous external disturbances, actuator faults, and sensor faults, we propose a novel approach based on dynamic output feedback controllers. Our contributions can be summarized as follows:

• We venture into uncharted territory by investigating FE and FTTC problems specifically for LPV systems grappling with concurrent external disturbances, actuator faults, and sensor faults.
• Our proposed SMO outperforms existing approaches by not only enabling direct actuator fault reconstruction through equivalent output error injection but also by effectively estimating time-varying sensor faults using an adaptive algorithm driven by the output estimation error.
• Differing from prior research, our SMO takes a more flexible approach by alleviating specific structural constraints necessary for the design of an observer-based system capable of simultaneously estimating actuator and sensor faults.
• We present novel adequate prerequisites for the establishment of the sought-after adaptive LPV SMO and FTTC, making use of $H_{\infty }$ performance standards. These prerequisites are reformulated as convex optimization challenges rooted in LMIs.
• Notably, our SMO and FTTC designs are independent, reducing computational complexity, and enhancing their practical applicability.

In conclusion, our comprehensive investigation and innovative solutions pave the way for enhanced FE and FTTC in LPV systems, fostering greater reliability and performance in the face of complex real-world challenges.