On-line switched robust fault detection framework for switched systems with unknown inputs

Abstract

This paper concerns an on-line robust fault detection (RFD) for continuous-time switched linear systems with unknown disturbances and noises. Owing to a novel event-triggered mechanism, the current mode-dependent event detector receives the sampled information of the unknown synchronous switching signal at each sampling instant; the active mode is updated. Thus, the on-line switched unknown input observer immediately estimates the system states. Using the average dwell-time (ADT) approach, new linear matrix inequalities (LMIs) are determined through an online switched quadratic Lyapunov function ensuring the globally uniformly asymptotically system stability (GUAS). So, we establish less conservative stability criteria, while the ADT switching mechanism is formulated to ensure that the estimation error converges by determining a minimum average dwell time for the switching signal. The designed robust fault detection framework is proper to find out all kinds of unknown faults (affected by each current mode), in real-time, via online residual indicators with adaptive thresholds that are robust to the deterministic input disturbances, as well as the impact of the stochastic disturbances is reduced regarding an $H\infty$ performance with on-line switched index. Simulation results on a DC-DC buck converter are given to underline the online applicability of the suggested approach. A comparative study of the RFD scheme concerning existing works is provided, where our results are shown to be less conservative and more valid.

Introduction

In the present work, our research direction is concentrated on the switched systems, which account for a broad class of the HDS and have brought about increasing interest because of the huge amount of practices in engineering systems, including traffic control, industrial systems, power conversion systems, chemical production processes, and so on. ¹ It is significant to note that the switched system is formed of subsystems where the transitions between these operating modes are governed by a switching law that identifies the switching sequence between them; thus, every mode is associated with an independent submodel. Autonomous (or arbitrary) switching and controlled (or constrained) switching are likewise two categories that depend on the characterization of switched systems. ² Thereby, a large number of researchers has explored the properties of this class,^2,3 and some attempts have been spent on fault monitoring and diagnosis analysis and that over the last decade.^1,4

The stability and stabilization issues associated with switched systems are complex and intriguing. Notably, even if all subsystems are stable, the switching signal can introduce instability to the overall system. The concept of dwell time is crucial in the context of switching. By leveraging this concept, stability conditions can be formulated based on the dynamics of the switching signals and each mode, mainly by determining the shortest dwell time allowed by the switching law. ⁵ However, the classic dwell-time switching approach tends to be more conservative, as real-world switching signals typically have frequency limitations. The average dwell-time (ADT) approach mitigates this conservatism when designing the switching law, providing greater flexibility. Significantly, the ADT has been further developed into a novel concept known as the mode-dependent average dwell time (MDADT) approach. This new framework is less restrictive in conservatism than the traditional average dwell time method, enabling the association of an average dwell time to each specific mode. The MDADT approach specifies a minimum average dwell time tailored for each subsystem.^6,7

As is well known, the common Lyapunov function (CLF) and multiple Lyapunov function (MLF) are well-established methods for addressing the stability of switched systems. The CLF approach is typically employed for switched systems subject to arbitrary switching by requiring that all subsystems possess a CLF. ⁸ However, the necessity for all subsystems to share a single CLF is often unrealistic in practical scenarios. Consequently, the analysis of switched systems with constrained switching becomes particularly significant. It is important to highlight that the MLF approach offers greater flexibility for addressing various issues related to switched systems with constrained switching, allowing for a reduction in the conservatism of theoretical outcomes. Nonetheless, the MLF constructed in these studies is continuous throughout each mode-running interval, which still introduces conservatism in the analysis and synthesis of switched systems. Thus, a switched multiple Lyapunov function (SMLF) is proposed for switched systems, where each MLF can be discontinuous during the operation of any subsystem. Thus, the first question is how to obtain a less conservative stability criterion according to the ADT concept and exploit the SCLF based on LMI synthesis.

Up to now, the fundamental issue is that the dynamics of a switched system belong both to the subsystems behaviors and the switching signal, which renders it extra challenging to look into the fault detectability, isolability and diagnosability of such systems, as pointed out in El Harabi et al. ⁹ The preliminary knowledge of the system switches that specifies the change of modes is vital to theoretical primary outcomes in these present-day topics. ¹⁰ From fault diagnosis scenarios, a surprising passage from a fault-free mode to a faulty mode may be regarded as an unrecognized switching (i.e. unknown). Hence, one must identify the active mode since the Fault Detection (FD) procedure gives rise to a delay between the active mode and its involved fault generator.

In light of the literature reviewed below, the main contribution of this paper revolves around an online robust fault detection scheme able to estimate both the unknown active mode and continuous system state of synchronously switched linear systems with both unknown inputs and unknown switching sequences simultaneously in real-time. There is currently no work to consider this important issue. In comparison with the existing results,^1–10 the guarantees related to system reliability and safety improve from online algorithms, leading to the current active mode estimation and the robust fault detection scheme. Therefore, the proposed on-line robust fault detection framework is involved in Figure 1 and includes: (i) New event-triggering current active mode estimator is designed to update the active mode at each instant and identify the current operating mode of the addressed system in unknown synchronous switching situations. (ii) online switched robust unknown input fault detection observer is proposed. The less conservative stability criterion uses an online switched quadratic Lyapunov function, exploiting the average dwell-time concept. This approach yields novel and less restrictive linear matrix inequalities that guarantee the system’s globally uniformly asymptotic stability. As a result, the ADT mechanism facilitates the convergence of the estimation error towards optimal values, while the observer gain matrices ensure the system’s overall stability. While the robustness against noises is achieved by using the $H\infty$ performance with an online switched index. The online residual signal generator is sensitive to actuator faults and insensitive to disturbances and other system factors. (iii) Adaptive thresholds are determined for online fault detection. Ultimately, the results of the analysis are significant and also lead to an important step in monitoring and supervisor switched-mode power converters. A synchronous buck converter case study can improve the proposed online robust fault detection scheme and validate the derived results.

Figure 1.

An overview of the proposed approach.

The rest of this article proceeds as follows. Section “Motivation and overview” depicts the related work. In Section “General model and preliminaries,” the general model is briefly outlined, and the preliminaries are stated. The main results are presented in the following sections. First, the design of a new event-triggering current active mode estimator is explained in Section “Current active mode-dependent event-triggering estimator.” Second, online switched robust unknown input fault detection observer synthesis is established in Sections “On-line switched unknown input fault detection observer design,”“On-line switched robust unknown input fault detection observer design,” and “On-line fault detection adaptive threshold design,” respectively. The online adaptive thresholds for fault detection are later provided. A simulation synchronous buck converter case study is given. Compared to existing results, we use linear matrix inequalities (LMIs) to overcome some of the main sources of conservatism and establish more relaxed stability conditions. These will be delineated in detail in Section “Synchronous buck converter case study.” At last, the conclusions end the paper in Sections “Discussion and future scope” and “Conclusions.”

Motivation and overview

The past decade has seen many upshots in the existing literature focusing on the FD problem survey of switched systems. The observer-based technique is one of the most favored approaches in fault diagnosis. ¹ According to the residual signal deducing from the difference between the system output and the observer output, the alarm generation and the fault shapes estimation can be performed. So far, many switched Luenberger-type observers dedicated to fault diagnosis targets have been designed, in recent years, for example, fuzzy observer,^11,12 adaptive observer, ¹³ sliding-mode observer,^14,15 etc.

In this regard, active mode identification and continuous state estimation have been the subject matter of several existing studies, which can be divided into two kinds, taking into account assumptions related to the active mode; some research works reflect only on continuous states with known operating modes, whereas others assume that both the operating modes and the continuous states are unknown.

Within the same context, the data-driven quality linked to the FD issue has been considered in Han et al. ¹¹ A new observer-based fuzzy filter has been conceived to find out faults for nonlinear switched stochastic systems through Takagi-Sugeno fuzzy modeling techniques. ¹² Nevertheless, the FD performances of these frameworks founded on experimental design and data analysis investigations rely on the availability of a specified fault database to acquire the faulty modes used after the learning stage. In addition, learning all such modes for industrial processes is experimentally impossible. For this reason, analytical model-based FD may be used as an alternative.

Furthermore, some authors^14,15 suggested an FD scheme for switched systems based on sliding mode observers, which enables them to detect faults, discern the root causes, and identify the fault forms quickly and in real-time. In Ali et al., ¹⁶ $H\infty$ performance index has been used to minimize disturbance impact from residual indicators. Switched system stability analysis has been introduced through a Lyapunov function, and average dwell time (ADT) constraints have also been envisaged. Note that the ADT technique has been extensively utilized to treat fault diagnosis and fault tolerant control for such systems.^6,7 Whereas, Yin et al. ¹⁷ synthesized a descriptor sliding mode observer to handle the fault estimation task devoted to linear switched systems.

On the other hand, even if diverse studies look at FD for HDS, it seems that merely little research deals with the case of HDS described through modeling errors, noise measurements, and external disturbances. In Belkhiat et al., ¹⁸ a robust hybrid observer-based fault detection for a switched linear system has been deemed. Afterwards, an optimal algorithm for robustness to external disturbances and sensitivity to prior known faults was developed using the Linear Matrix Inequality (LMI) technique. While a novel robust FD strategy for switched fuzzy systems together with unknown input has been presented in Han et al. ¹⁹ The latter holds the benefit of utilizing an observer enable to getting residual signals that are robust vis-a-vis disturbances and sensitive against faults. Besides, the Fault Detection and Isolation (FDI) scheme has been equally extended to switched systems with parameter uncertainties in Du et al. ²⁰ through profiting from the main results cited in Wang et al., ²¹ Xiang et al., ²² and Benzaouia and Eddoukali. ⁸ More details related to fault estimation based on Unknown Input Observers (UIO) are provided in Li et al., ²³ Du et al., ²⁴ and Su et al. ²⁵

Nonetheless, to identify the current mode and determine the faults that may take place, the co-authors in Hakem et al. ²⁶ have considered a set of multi-mode observers conceived with regard to a kind of slow-switching linear systems governed by synchronous and asynchronous switching signals. Recently, the article ²⁷ has proposed an exact finite optimization-based scheme consisting of a bank of filters and an algorithm of diagnostic rules allowing the generation of fault indicators. This FD framework affords an optimal bank of filters in which the measurement noise effect on the residual signal is minimized. From the diagnostic point of view, a bank of residual signals is often necessitated to look into synchronous multi-mode systems dominated by a switching signal. Additionally, the systems should accomplish particular classification conditions to ensure that two partial models could be differentiated. This capability is called the distinguishability or discernibility of switched systems. ²⁸

However, as a significant research topic with application potentials to practical engineering, the robust fault detection problem of switched systems has not received much research attention yet, not to mention the case where current active mode and state estimation are considered simultaneously in real time.

To overcome these limits, our present objective is predicated on the advantage of an event-triggered sampling mechanism. This means that the event-triggered technology has been extensively investigated in recent years. The key principle of event-triggered technology is to elaborate an event detector to transmit useful data, which differs significantly from the previously transmitted one. ²⁹ Many outcomes have been obtained for different systems in multiple domains.^30–32 To mention a few, event-triggered control has been looked into for diverse non-switched systems. Newly, the event-trigger control and fault diagnosis of switched systems has too captured some attention. For instance, the authors ³³ put forward an event-triggered sampling mechanism and synthesized an observer-based controller for switched neutral systems. About previous works,^34,35 the event-triggered control of networked switched linear systems and uncertain switched linear systems have been addressed. As well, the event-triggered containment control for second-order nonlinear multi-agent systems and the consensus tracking control of switched stochastic nonlinear multi-agent systems via event-triggered strategy have been respectively discussed in Liu et al. ³⁶ and Zou et al. ³⁷ In Hajshirmohamadi et al., ³⁸ an event-triggered filter to detect and isolate the fault for discrete-time systems has been designed. However, discrete-time model analysis has investigated most results for event-triggered fault-detection problems. Few related results for stochastic switched linear systems have been obtained, which motivates the current study.

On the other hand, communication network applications induce the appearance of damaging phenomena like packet losses, network attacks, … etc.^39,40 To decrease useless communication transmissions for switched networked systems, an event-triggered mechanism with dynamical behavior is recently taken into account in the designing of finite-time filter dedicated to fault detection purposes for the class of networked systems with time-delays. ⁴¹ A nonlinear fault detection filter is proposed to generate the residual signal and detect system faults, and an event-triggered strategy is applied to limit signal transmission. ⁴²

Unlike the offline FDI and FTC techniques, online fault detection for switched systems received less attention, and only a few results are available.

General model and preliminaries

General model

An unwanted modification in the system that leads global to degrade the system performance is known as a “fault.” ⁴³ For the hybrid dynamical systems, faults may assign the active mode behavior or the trajectory of the discrete evolution and take place whenever:

• When the system switches, a new mode is activated; however, the latter is not considered the possible successor mode. This type of fault has a rapport with the system switching signal that compels the trajectory of the discrete state in ordinary cases;

• The system passes to an eligible successor following a result of both controlled and uncontrolled external events;

• The system stays in the same mode even if the switching conditions are satisfied.

During this research, we will look into actuator faults that may act on the continuous part of a switched linear system. Considering a class of switched systems subject to unknown bounded disturbances and noises. The dynamic of such continuous-time linear system $\sum _{\sigma }$ is described as

$$\{\begin{array}{c}\stackrel{·}{x}\left(t\right)=A_{\sigma }x\left(t\right)+B_{\sigma }u\left(t\right)+D_{\sigma }d\left(t\right)+F_{\sigma }f\left(t\right)\hfill \\ +W_{\sigma }w\left(t\right);x\left(t_0\right)=x_0\hfill \\ y\left(t\right)=C_{\sigma }x\left(t\right)\hfill \end{array}$$

(1)

where $x\left(t\right)\in {\Re }^n$ , $y\left(t\right)\in {\Re }^m$ , and $u\left(t\right)\in {\Re }^r$ are the state vector, the measurement output, and the known input signal, respectively. $d\left(t\right)\in {\Re }^q$ is the bounded unknown input vector given rise to either disturbances or modeling errors, and $w\left(t\right)\in {\Re }^w$ is the independent Gaussian white noise disturbance. Whereas $f\left(t\right)\in {\Re }^l$ stands for the unknown actuator fault vector needed to be detected. Note that $d\left(t\right)$ , $w\left(t\right)$ and $f\left(t\right)$ are assumed to belong to $L_2[0,+\infty )$ without loss of generalities. Besides, $A_{\sigma }$ , $B_{\sigma }$ , $C_{\sigma }$ , $F_{\sigma }$ , $D_{\sigma }$ , and $W_{\sigma }$ are known constant real matrices with appropriate dimensions which vary as a function of $\sigma \left(t\right)$ .

The following assumptions related to the considered system (1) are performed:

• A.1: The function $\sigma \left(t\right)$ defining the switching sequence is assumed to be unknown and synchronous at each sampling instant.

• A.2: Matrices $A_{\sigma }$ are all stables.

Moreover, $\sigma \left(t\right)$ is a system switching signal and assumed to be an unknown piecewise constant function of time which takes its values in the finite set $I_m=\{1,2,3....m\}(m\in \mathbb{N})$ where $m$ is the number of subsystems or modes. For $\sigma \left(t\right)\in I_m$ , the switching sequence $\mathrm{\Lambda }$ is specified as

$$\mathrm{\Lambda }=\{x_0:(i_0,t_0),(i_1,t_1),...(i_k,t_k),...|i_k=1,2....,m,i,j\in \mathbb{N}\}$$

(2)

in which $t_k,k\in \mathbb{N}$ is the switching instant. While the switching sequence indicates that the $i_k\mathrm{th}$ mode is activated for any $t\in [t_k,t_{k+1})$ .

Let us consider a simple notation $\sigma \left(t_k\right)$ by ${\sigma }_k$ , $k\in \mathbb{N}$ , and ${\sigma }_0=\sigma \left(t_0\right)$ be any element of $I_m$ , for the given switching instant $t_k$ , and the next switching instant $t_{k+1}$ , the system switching signal $\sigma \left(t\right)$ can be defined as

$$\begin{array}{c}\sigma \left(t\right)=\sigma \left(t_k\right)={\sigma }_k,t\in [t_k,t_{k+1})\hfill \\ \sigma \left(t_{k+1}\right)={\sigma }_{k+1}\in \mathrm{\Phi }(.,{\sigma }_k)\hfill \end{array}$$

(3)

where the function $\mathrm{\Phi }(.,{\sigma }_k)$ denotes a state and/or time-dependent switching strategy for both autonomous or controlled switched systems.

Note that the next switching instant $t_{k+1}$ can be choose as

$$t_{k+1}=\min \{t:t\ge {\tilde{t}}_{k+1},\mathrm{\Phi }(.,{\sigma }_k)|\mathrm{existe}\mathrm{some}j\in I_m\}$$

(4)

where ${\tilde{t}}_{k+1}=t_k+{\tau }_{d_{{\sigma }_k}}$ , ${\tau }_{d_{{\sigma }_k}}$ designates the minimum dwell time during $[t_k,t_{k+1})$ .

Consequently, the $i_k^{\mathrm{th}}$ mode remains active within the dwell-time ${\tau }_{d_{{\sigma }_k}}$ from each considered switching instant $t_k$ . It means that the switching function $\mathrm{\Phi }(.,{\sigma }_k)$ is maintained fixed at ${\sigma }_k$ over $[t_k,t_{k+1})$ whose duration is au minimum equal to minimum dwell time ${\tau }_{d_{{\sigma }_k}}$ . Once this latter has been attained, the succeeding mode is taken founded on the switching function stated from equation (3). The infinitely fast switching describing a chattering phenomenon is prevented, and inherent catastrophic situations are avoided.

• Remark 1. It is to be observed that there exists an infinite number of switches whether $t_{k+1}$ is finite for all $k\in \mathbb{N}$ .

• Remark 2. Next, in this work, time will be used to select the operating mode. In DC-DC buck converters, the Pulse Width Modulation (PWM) circuit usually dictates how long it will stay in each subsystem.

Preliminaries

Let us shortly recall the following assumptions, definitions and lemmas which are helpful hereinafter.

• Assumption 1. Note that the dimension of the unknown input is always less than the dimension of the measured output, that is to say, $d\le m$ .

• Assumption 2. The matrix $D_{\hat{\sigma }}$ is full column rank.

• Assumption 3. Following Assumptions 1 and 2, one may obtain that $\mathrm{rank}(C_{\hat{\sigma }},D_{\hat{\sigma }})=\mathrm{rank}\left(D_{\hat{\sigma }}\right)\forall \hat{\sigma }\in I_m,d\le m$ .

Assumption 3 serves as a fundamental criterion for the existence of an unknown input observer. These observers rely on fulfilling the “relative degree condition” concerning the measured output, essential for effectively estimating the unknown input.

• Assumption 4. All the pairs $(A_{\hat{\sigma }},C_{\hat{\sigma }})$ are detectable for all $\hat{\sigma }\in I_m$ .

It is important to note that Assumption 4 does not inherently ensure the presence of an observer structure for the system. Consequently, a consideration of an average dwell time is introduced later in the paper to address this gap.

• Definition 1. ⁵ The system switching signal $\sigma \left(t\right)$ satisfies an average dwell-time ${\tau }_{\mathrm{ad}}$ , that implies for a given chattering bound $N_0\ge 0$ and ${\tau }_{\mathrm{ad}}>0$ verifying

$$N_{\sigma \left(t\right)}(t_1,t_2)\le N_0+\frac{t_2-t_1}{{\tau }_{\mathrm{ad}}},\forall t_2>t_1>0$$

(5)

where $N_{\sigma \left(t\right)}(t_1,t_2)$ indicates the number of discontinuities of $\sigma \left(t\right)$ in the interval $(t_1,t_2)$ .

Definition 1 states that a positive number ${\tau }_{\mathrm{ad}}$ ensures a switching signal has the average dwell-time property. This means the average time interval between consecutive switchings is at least ${\tau }_{\mathrm{ad}}$ . This characterizes a slow switching signal, which is less conservative than a traditional dwell-time switching signal. As ordinarily taken on in the literature, the chattering bound $N_0$ is selected to be null along this research work.

• Lemma 1. ⁴⁴ Take into account the continuous-time switched system defined as $\stackrel{·}{x}\left(t\right)=f(\sigma \left(t\right),x\left(t\right))$ . Assume that there exist scalars $\alpha >0,\mu >1,C^1$ Lyapunov function $V_{\sigma \left(t\right)}$ and $K_{\infty }$ functions $k_1$ and $k_2$ suchthat $\forall x\in {\Re }^n$ the following equations hold

$$\begin{array}{c}{k}_1\left(x\left(t\right)\right)\le V_{\sigma \left(t\right)}\left(x\left(t\right)\right)\le k_2\left(x\left(t\right)\right)\hfill \\ {\stackrel{·}{V}}_{\sigma \left(t\right)}\left(x\left(t\right)\right)\le -\alpha V_{\sigma \left(t\right)}\left(x\left(t\right)\right)\hfill \\ \mathrm{and}\forall (\sigma \left(t_{k+1}\right)=i,\sigma \left(t_k\right)=j)\in I_m\times I_m,i\ne j\hfill \\ V_i\left(x\left(t_{k+1}\right)\right)\le \mu V_j\left(x\left(t_{k+1}\right)\right)\hfill \end{array}$$

If the average dwell time ${\tau }_{\mathrm{ad}}$ satisfies

$${\tau }_{\mathrm{ad}}>{\tau }_{\mathrm{ad}}^{*}=\frac{\ln \mu }{\alpha }$$

(6)

thus the switched system $\stackrel{·}{x}\left(t\right)=f(\sigma \left(t\right),x\left(t\right))$ is globally uniformly asymptotically stable $\left(\mathrm{GUAS}\right)$ for any system switching signal along with (6).

• Lemma 2. ⁴⁵ For a matrix $D$ satisfying Assumption 2 and a given matrix C. $A$ is a general solution of the consistent equation $\mathrm{ACD}=D$ verifying this condition $\mathrm{rank}\left(\mathrm{CD}\right)=\mathrm{rank}\left(D\right)$ and is given by

$$A=D{\left(\mathrm{CD}\right)}^{+}+X(I-\left(\mathrm{CD}\right){\left(\mathrm{CD}\right)}^{+})$$

(7)

in which ${\left(\mathrm{CD}\right)}^{+}$ is the Moore-Penrose inverse of $\mathrm{CD}$ , while, $X$ is an arbitrary matrix along with proper dimension.

Notations

In this article, the subsequent notations are involved. $\mathbb{N}$ is a non-negative natural number set. The n-dimensional Euclidean space is denoted by ${\mathbb{R}}^n$ . ∥.∥ stand for the Euclidean vector norm. $L_2[0,+\infty )$ indicates space of square-integrable vector functions over time interval $[0,+\infty )$ . Meanwhile, $I$ represents the identity matrix along with suitable dimensions. By the side of $A^T$ signifies the transpose of whatever real matrix $A$ . Besides, $A^{-1}$ denotes the inverse of any non-singular matrix $A$ . $P<0(P>0)$ implies that $P$ is a symmetric negative definite matrix (respectively a symmetric positive definite matrix). $\parallel .{\parallel }_{\infty }$ denotes the standard $H_{\infty }$ norm.

The details that are related to the new proposed robust fault detection scheme dedicated to the switched linear system $\sum _{\sigma }$ are cited below.

Current active mode-dependent event-triggering estimator

In the previous research works,^2–4 the switching signal (or discrete state) has been assumed to be known when the continuous state estimation has been done. Nevertheless, the switching sequence law is mainly seen as an unknown signal for any actual system.^10,11

Throughout this work, we propose to adapt an event-triggered sampling mechanism to identify the current operating mode of the investigated system (1) in an unknown synchronous switching situation. By this discrete state detector block, the active mode is updated in real-time at each instant. For that, the mechanism of event-triggered is straightforwardly designed and can be considered a very easy implementation with a more flexible and practical sampling strategy.

The core of the proposed event-triggered estimator (or detector), taking into account the system switching signal, is to compare the currently active mode with the last mode, and the mechanism also permits information about whether the current mode change/modification has been done or not. In other words, the current sampled signal is compared with the last released signal, and the sampled measurement will be merely brought (i.e. transmitted) to the online switched robust UI fault detection Observer block if the condition triggered by a predefined event is satisfied.

Henceforth, the suggested event-triggered active mode estimator is conceived. Firstly, $\sigma \left(t\right)$ has been sampled under a well-chosen sampling period corresponding to the sampler, which checks $0<h\le {\tau }_d<{\tau }_{\mathrm{ad}}$ . Secondly, the trigger condition is violated, therefore, the on-line switched robust UIO receives the latest operating mode data (active mode data). So, the sampling instant sequence of the relative event-triggered mechanism is presumed to be

$$\mathrm{\Theta }=\{[{\hat{t}}_0,{\hat{t}}_1),[{\hat{t}}_1,{\hat{t}}_2),\dots ,[{\hat{t}}_e,{\hat{t}}_{e+1}),...|e\in \mathbb{N}\}$$

(8)

where ${\hat{t}}_e<{\hat{t}}_{e+1}$ and ${\hat{t}}_0=t_0=0$ represent the first sampling instant of the event-triggered detector. Also, $t={\hat{t}}_e$ is supposed to be an updated sampling instant of the associated estimator. Regarding this sampling adjustment, the discrete behavior (known as switching law) inducing various operational modes can be sampled as

$$\sigma \left(t\right)=\sigma \left({\hat{t}}_e\right),t\in [{\hat{t}}_e,{\hat{t}}_{e+1})$$

(9)

For the notational simplicity, let $\sigma \left({\hat{t}}_e\right)={\sigma }_{{\hat{t}}_e}$ without loss of generality. This latter can be determined as

$$\hat{\sigma }\left(t\right)=\{\begin{array}{c}{\sigma }_{{\hat{t}}_e},\mathrm{when}{\sigma }_{{\hat{t}}_e}\ne {\hat{\sigma }}_{{\hat{t}}_e}\hfill \\ \multicolumn{1}{c}{{\hat{\sigma }}_{{\hat{t}}_e},\mathrm{otherwise}}\end{array}$$

(10)

where ${\hat{\sigma }}_{{\hat{t}}_e}$ denotes the previous estimated mode switching signal. Thereby, the sampled switching signal definitions (9) and (10) mean the switches among an arbitrary number of modes are done following the switching decision just made at sampling instant ${\hat{t}}_e$ . At each sampling instant ${\hat{t}}_e$ if ${\sigma }_{{\hat{t}}_e}\ne {\hat{\sigma }}_{{\hat{t}}_e}$ , the switching law $\sigma \left(t\right)$ values are modified if and only if ${\sigma }_{{\hat{t}}_e}\ne {\hat{\sigma }}_{{\hat{t}}_e}$ , in other ways, it alternatively keeps its further the latest active value. The mode is updated when an event is triggered.

For any $t\in [{\hat{t}}_e,{\hat{t}}_{e+1})$ , the estimated system switching law $\hat{\sigma }\left(t\right)$ is

$$\hat{\sigma }\left(t\right)=\sigma \left(t\right)$$

(11)

Moreover, the online switched UIO module receives the latest estimated active mode once the event triggering mechanism is switched on and stays steady at a fixed constant value up to the next event. Here, it is undertaken that the mode is altered just whenever the Euclidean norm of variance between the current sampled switching value $\sigma \left(t_k\right)$ and the previously one $\hat{\sigma }\left(t_k\right)$ is upper than a given threshold value ${\epsilon }_{{\sigma }_k}$ , that is,

$$\Vert \sigma \left(t\right)-\sigma \left({\hat{t}}_e\right)\Vert >{\epsilon }_{{\sigma }_k}\left(t\right)$$

(12)

In this regard, the following sampling instant ${\hat{t}}_{e+1}$ can be described by

$${\hat{t}}_{e+1}=\inf \{t:t\ge {\hat{t}}_e,\parallel \sigma \left(t\right)-\sigma \left({\hat{t}}_e\right)\parallel \le {\epsilon }_{{\sigma }_k}\left(t\right)\}$$

(13)

where ${\epsilon }_{{\sigma }_k}\left(t\right)=n_{{\sigma }_k}\parallel \sigma \left(t\right)\parallel$ in which $n_{{\sigma }_k}$ is a predetermined constant linked to the ${\sigma }_k$ th mode and is strictly greater than zero $(n_{{\sigma }_k}>0)$ .

So to avoid the mismatch between current active modes and the associated event estimator, let’s note that at most one switching is permitted within each event-triggered sampling interval $[{\hat{t}}_e,{\hat{t}}_{e+1})$ . For that, the length of time interval may be selected less than $\min \left\{{\tau }_{d_{{\sigma }_k}}\right\}$ .

• Remark 3. The parameters ${\epsilon }_{{\sigma }_k}\left(t\right)$ may be calculated from the x-fold of the greatest possible amplitude of $\sigma \left(t\right)$ , that is, ${\epsilon }_{{\sigma }_k}\left(t\right)=n_{{\sigma }_k}\underset{k}{max}\parallel \sigma \left(t\right)\parallel$ with $n_{{\sigma }_k}>0$ is the event detection parameter for the system discrete behavior signal; ⁴⁶ in other words, this latter can be system mode dependent. In addition, it should be mentioned that ${\epsilon }_{{\sigma }_k}\left(t\right)$ has significantly impact the length of inter-event intervals and they indirectly specify the rapidity of current active mode estimation.

• Remark 4. Based on the abovementioned condition, an adequately small sampling period $h$ considering the dwell time restriction hinders the switching actions from the Zeno phenomenon. The triggering time sequence related to the event detector scheme is likewise a subsequence of the sampling time sequence; in other words, it signifies that $\{{\hat{t}}_{0}h,{\hat{t}}_{1}h,{\hat{t}}_{2}h,...\}\subseteq \{0h,1h,2h,...\}$ . Whence, $\underset{e}{min}\{{\hat{t}}_{e+1}-{\hat{t}}_e\}$ represents the minimum time-sequence and $h$ is its lower bound. Therefore, the undesired chattering causes are avoid and are never happened in the proposed event triggering strategy herein.

In the sequel of this research work, note that the index time $t$ will be omitted for the estimated variable $\hat{\sigma }\left(t\right)$ when there is no ambiguity.

On-line switched unknown input fault detection observer design

To detect the possible actuator faults in a considered continuous-time switched linear system (1), an online switched unknown input observer is designed with respect to each mode and constructed as

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}\stackrel{·}{z}\left(t\right)=N_{\hat{\sigma }}z\left(t\right)+K_{\hat{\sigma }}y\left(t\right)+G_{\hat{\sigma }}u\left(t\right)\hfill \\ \multicolumn{1}{c}{\hat{x}\left(t\right)=z\left(t\right)+E_{\hat{\sigma }}y\left(t\right)}\end{array};z\left(t_0\right)=z_0}\end{array}$$

(14)

where $z\left(t\right)\in {\Re }^n$ stands for the observer state. While, $\hat{x}\left(t\right)\in {\Re }^n$ designates the state estimation of the investigated system (1). Parameter matrices $N_{\hat{\sigma }}$ , $G_{\hat{\sigma }}$ , $E_{\hat{\sigma }}$ and $K_{\hat{\sigma }}$ ( $K_{\hat{\sigma }}=K_{1\hat{\sigma }}+K_{2\hat{\sigma }}$ ) depict the UIO gains to be afterward computed so that $\underset{t\to \infty }{\lim }(x\left(t\right)-\hat{x}\left(t\right))=0$ for any initial functions. Meanwhile, some specific conception criteria will be handled and might lead to the decoupling of the unknown inputs from other inputs. ⁴⁷

The related state estimated error $e\left(t\right)$ and the residual signal $r\left(t\right)$ are respectively defined as

$$e\left(t\right)=x\left(t\right)-\hat{x}\left(t\right),r\left(t\right)=y\left(t\right)-\hat{y}\left(t\right)$$

(15)

The time derivative of the state error may be easily written as

$$\stackrel{·}{e}\left(t\right)={\mathrm{\Delta }}_{\hat{\sigma }}\stackrel{·}{x}\left(t\right)-\stackrel{·}{z}\left(t\right)$$

(16)

in which a matrix ${\mathrm{\Delta }}_{\hat{\sigma }}$ is equal to $I-E_{\hat{\sigma }}{C}_{\hat{\sigma }}$ .

Substituting both equations (1) and (14) into the derivative of the state error (16), one can then obtained

$$\{\begin{array}{c}\stackrel{·}{e}\left(t\right)=A_{\hat{\sigma }}^{\mathrm{obs}}e\left(t\right)+G_{\hat{\sigma }}^{\mathrm{obs}}z\left(t\right)+B_{\hat{\sigma }}^{\mathrm{obs}}u\left(t\right)+C_{\hat{\sigma }}^{\mathrm{obs}}y\left(t\right)+D_{\hat{\sigma }}^{\mathrm{obs}}d\left(t\right)\hfill \\ +F_{\hat{\sigma }}^{\mathrm{obs}}f\left(t\right)+W_{\hat{\sigma }}^{\mathrm{obs}}w\left(t\right)\hfill \\ r\left(t\right)=C_{\hat{\sigma }}e\left(t\right)\hfill \end{array}$$

(17)

where

$$\begin{array}{c}{A}_{\hat{\sigma }}^{\mathrm{obs}}={\mathrm{\Delta }}_{\hat{\sigma }}{A}_{\hat{\sigma }}-K_{1\hat{\sigma }}{C}_{\hat{\sigma }},\hfill \\ G_{\hat{\sigma }}^{\mathrm{obs}}={\mathrm{\Delta }}_{\hat{\sigma }}{A}_{\hat{\sigma }}-K_{1\hat{\sigma }}{C}_{\hat{\sigma }}-N_{\hat{\sigma }},\hfill \\ B_{\hat{\sigma }}^{\mathrm{obs}}={\mathrm{\Delta }}_{\hat{\sigma }}{B}_{\hat{\sigma }}-G_{\hat{\sigma }},\hfill \\ C_{\hat{\sigma }}^{\mathrm{obs}}=N_{\hat{\sigma }}{E}_{\hat{\sigma }}-K_{1\hat{\sigma }},\hfill \\ D_{\hat{\sigma }}^{\mathrm{obs}}={\mathrm{\Delta }}_{\hat{\sigma }}{D}_{\hat{\sigma }},F_{\hat{\sigma }}^{\mathrm{obs}}={\mathrm{\Delta }}_{\hat{\sigma }}{F}_{\hat{\sigma }},W_{\hat{\sigma }}^{\mathrm{obs}}={\mathrm{\Delta }}_{\hat{\sigma }}{W}_{\hat{\sigma }}\hfill \end{array}.$$

In this subsection, let us assume that the noise vector $w\left(t\right)=0$ . The existence of an on-line switched UIO is derived by providing some necessary and sufficient conditions.

• Theorem 1. Let Assumptions 1–4 be satisfied and let $\mu >1$ be freely chosen positive scalar. The observer having the structure of (14) is an online switched UIO for system (1) if and only if these subsequent conditions are verified

• (i) There exist two symmetric matrices ${\mathit{P}}_{1\hat{\sigma }}>0$ and ${\mathit{S}}_{\hat{\sigma }}>0$ with an on-line switched observer gain ${\mathit{K}}_{1\hat{\sigma }}$ , which satisfy

$${({\mathrm{\Delta }}_{\hat{\sigma }}-K_{1\hat{\sigma }}{C}_{\hat{\sigma }})}^T{P}_{1\hat{\sigma }}+P_{1\hat{\sigma }}({\mathrm{\Delta }}_{\hat{\sigma }}-K_{1\hat{\sigma }}{C}_{\hat{\sigma }})=-S_{\hat{\sigma }}$$

(18)

 and there exist positive definite matrices $P_{{\hat{\sigma }}_{k+1}}$ and $P_{{\hat{\sigma }}_k}$ , $\forall {\hat{\sigma }}_{k+1},{\hat{\sigma }}_k\in I_m$ which satisfy

$$P_{{\hat{\sigma }}_{k+1}}<\mu P_{{\hat{\sigma }}_k},\forall {\hat{\sigma }}_{k+1}\ne {\hat{\sigma }}_k$$

(19)

 thus, for the system switching law $\sigma \left(t\right)$ together with average dwell time restriction (6), the error system $\stackrel{·}{e}\left(t\right)=({\mathrm{\Delta }}_{\hat{\sigma }}{A}_{\hat{\sigma }}-K_{1\hat{\sigma }}{C}_{\hat{\sigma }})e\left(t\right)$ is asymptotically stable.

• (ii)

$$\begin{array}{c}{N}_{\hat{\sigma }}={\mathrm{\Delta }}_{\hat{\sigma }}{A}_{\hat{\sigma }}-K_{1\hat{\sigma }}{C}_{\hat{\sigma }}\hfill \end{array}$$

(20)

$$K_{2\hat{\sigma }}=N_{\hat{\sigma }}{E}_{\hat{\sigma }}$$

(21)

$$D_{\hat{\sigma }}=E_{\hat{\sigma }}{C}_{\hat{\sigma }}{D}_{\hat{\sigma }}$$

(22)

$$G_{\hat{\sigma }}={\mathrm{\Delta }}_{\hat{\sigma }}{B}_{\hat{\sigma }}$$

(23)

$$E_{\hat{\sigma }}=D_{\hat{\sigma }}{\left(C_{\hat{\sigma }}{D}_{\hat{\sigma }}\right)}^{+}+{\mathrm{\Theta }}_{\hat{\sigma }}(I-\left(C_{\hat{\sigma }}{D}_{\hat{\sigma }}\right){\left(C_{\hat{\sigma }}{D}_{\hat{\sigma }}\right)}^{+})$$

(24)

• Proof. From equations (20) and (17), we have

$$N_{\hat{\sigma }}=A_{\hat{\sigma }}^{\mathrm{obs}}$$

(25)

Substituting equations (21)–(23) into (17), we get the error derivative $\stackrel{·}{e}\left(t\right)$ as

$$\Delta \stackrel{·}{e}\left(t\right)=A_{\hat{\sigma }}^{\mathrm{obs}}e\left(t\right)=N_{\hat{\sigma }}e\left(t\right)$$

(26)

The state estimation can be fully separated (i.e. decoupled) from the unknown input if and only if the following sufficient condition should hold:

$${\mathrm{\Delta }}_{\hat{\sigma }}{D}_{\hat{\sigma }}=0$$

(27)

Referring to Lemma 2, a solution for this condition exists when Assumption 3 is verified. Then, the matrix $E_{\hat{\sigma }}$ of the on-line switched UIO can be immediately calculated from equation (24) where ${\left(C_{\hat{\sigma }}{D}_{\hat{\sigma }}\right)}^{+}$ is the Moore ′Penrose of $C_{\hat{\sigma }}{D}_{\hat{\sigma }}$ , and an arbitrary matrix ${\mathrm{\Theta }}_{\hat{\sigma }}$ has an appropriate dimension which leads to a lot of further design freedom.

After that, we must check if the system (14) is stable. By setting $w\left(t\right)$ to $0$ and assuming that each subsystem of the SLC (1) is stable (Assumption A.2), we can construct a switched Lyapunov function to examine the error dynamics (17) as

$$V_{\hat{\sigma }}(t,e\left(t\right))=e^T\left(t\right)P_{1\hat{\sigma }}e\left(t\right)$$

(28)

where $P_{1\hat{\sigma }}$ let be a positive define matrix. Now, using the Leibniz’s formula, one has

$$\begin{array}{c}{\stackrel{·}{V}}_{\hat{\sigma }}(t,e\left(t\right))=e^T\left(t\right)(N_{\hat{\sigma }}^T{P}_{1\hat{\sigma }}+P_{1\hat{\sigma }}{N}_{\hat{\sigma }})e\left(t\right)<0\hfill \\ \iff N_{\hat{\sigma }}^T{P}_{1\hat{\sigma }}+P_{1\hat{\sigma }}{N}_{\hat{\sigma }}^{<}0\hfill \end{array}$$

(29)

Then, there exists also $S_{\hat{\sigma }}>0$ such that

$$N_{\hat{\sigma }}^T{P}_{1\hat{\sigma }}+P_{1\hat{\sigma }}{N}_{\hat{\sigma }}^{+}{S}_{\hat{\sigma }}=0$$

(30)

which is the on-line switched quadratic Lyapunov equation.

On the other hand, more precisely, the stability criterion is ensured by the fulfillment of an ADT condition. Consider $t_k$ as the $\mathrm{kth}$ switching time instant and $t_{k+1}$ as the next switching instant, the ${\sigma }_{\mathrm{kth}}$ subsystem is said to be active when $\sigma \left(t\right)=\sigma \left(t_k\right)={\sigma }_k\in I_m$ and $t\in [t_k,t_{k+1})$ .

To formalize the third condition from Lemma 1 as inequality (19), one has

$$\begin{array}{c}{V}_{{\hat{\sigma }}_{k+1}}(t_k,e\left(t_k\right))-\mu V_{{\hat{\sigma }}_k}(t_k,e\left(t_k\right))=e^T\left(t_k\right)P_{{\hat{\sigma }}_{k+1}}e\left(t_k\right)\hfill \\ -\mu e^T\left(t_k\right)P_{{\hat{\sigma }}_k}e\left(t_k\right)=e^T\left(t_k\right)(P_{{\hat{\sigma }}_{k+1}}-\mu P_{{\hat{\sigma }}_k})e\left(t_k\right)\hfill \end{array}$$

(31)

Therefore, condition (6) is satisfied if it holds, $\mathrm{for}\mathrm{all}{\hat{\sigma }}_{k+1}\ne {\hat{\sigma }}_k$

$$P_{{\hat{\sigma }}_{k+1}}-\mu P_{{\hat{\sigma }}_k}<0$$

(32)

Afterward, the online switched unknown input fault detection observer (14) is asymptotically stable for any discrete behavior law and average dwell time switching.

• Remark 5. In this sequel, the perfect decoupling UIO-based FD condition is satisfied. Whereas, Condition (27) may be a hard constraint in some situations. If it is beyond the bounds of possibility to reach a feasible solution that satisfies precondition (27), from which one can partially separate the unknown input $d\left(t\right)$ from the switched linear system 1 and incorporate the non-decoupled disturbances into the vector $w\left(t\right)$ .

Next, a new robust UIO dedicated to fault detection is offered in the sequel hereinafter.

On-line switched robust unknown input fault detection observer design

In practice, it is common that the disturbance vector $w\left(t\right)$ is different from zero. The effects from the $w\left(t\right)$ to the residual signal $r\left(t\right)$ might be minimized. That is to say, the impact of the other unknown inputs on the residual indicator will be restrained using the $H_{\infty }$ performance index. Accordingly, this ensures that the residual signal is robust against the external unknown inputs $w\left(t\right)$ and sensitive to actuator faults $f\left(t\right)$ .

For analyzing the robustness of the state estimation to $w\left(t\right)$ , take into consideration a switched linear system (1) with $f\left(t\right)=0$ and equation (17) became

$$\{\begin{array}{c}\stackrel{·}{e}\left(t\right)=N_{\hat{\sigma }}e\left(t\right)+{\mathrm{\Delta }}_{\hat{\sigma }}{W}_{\hat{\sigma }}w\left(t\right)\hfill \\ \multicolumn{1}{c}{r\left(t\right)=C_{\hat{\sigma }}e\left(t\right)}\end{array}$$

(33)

Hence, the next theorem is devoted to outline the on-line Switched Robust Fault Detection Observer with Unknown Inputs.

• Theorem 2. Let Assumptions 1–4 be satisfied and let $\mu >1$ and $\alpha >0$ be freely chosen positive scalars. The observer in the form of (14) can ensure robustness to $w\left(t\right)$ owing to the $H_{\infty }$ performance with on-line switched index ${\lambda }_{\hat{\sigma }}$ , that is,

$$\{\begin{array}{c}\underset{0}{\overset{\infty }{\int }}{r}^T\left(t\right)r\left(t\right)\mathrm{dt}<{{\lambda }_{\hat{\sigma }}}^2\underset{0}{\overset{\infty }{\int }}{w}^T\left(t\right)w\left(t\right)\mathrm{dt},\mathrm{if}w\left(t\right)\ne 0\hfill \\ \underset{t\to \infty }{\lim }e\left(t\right)=0,\mathrm{if}w\left(t\right)=0\hfill \end{array}$$

(34)

$$\Vert C_{\hat{\sigma }}{\left[s(I-A_{\hat{\sigma }})\right]}^{-1}{B}_{\hat{\sigma }}{\Vert }_{\infty }<{\lambda }_{\hat{\sigma }}$$

(35)

and if the conditions from (20) to (24) hold, there exist on-line switched matrices $P_{\hat{\sigma }}=P_{\hat{\sigma }}^T>0$ , $P_{{\hat{\sigma }}_{+1}}=P_{{\hat{\sigma }}_{+1}}^T>0$ , $\forall \hat{\sigma },{\hat{\sigma }}_{+1}\in I_m$ and ${\mathrm{\Psi }}_{\hat{\sigma }}$ satisfying the following LMIs

$$P_{\hat{\sigma }}<\mu P_{{\hat{\sigma }}_{+1}},\forall \hat{\sigma }\ne {\hat{\sigma }}_{+1}$$

(36)

$$\left[\begin{array}{cc}\mathrm{\Upsilon }+C_{\hat{\sigma }}^T{C}_{\hat{\sigma }}\hfill & P_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}{W}_{\hat{\sigma }}\hfill \\ \multicolumn{1}{c}{*}& -{\lambda }_{\hat{\sigma }}^{2}I\hfill \end{array}\right]<0$$

(37)

in which $\mathrm{\Upsilon }=A_{\hat{\sigma }}^T{\mathrm{\Delta }}_{\hat{\sigma }}^T{P}_{\hat{\sigma }}+P_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}{A}_{\hat{\sigma }}-{\Psi }_{\hat{\sigma }}{C}_{\hat{\sigma }}-C_{\hat{\sigma }}^T{\mathrm{\Psi }}_{\hat{\sigma }}^T+\alpha P_{\hat{\sigma }}+I$

and the symbol * generically designates every of its symmetrical blocks.

Whence system (33) under arbitrary estimated switching signal $\hat{\sigma }\left(t\right)$ is (GUAS) asymptotically stable, and ensures the robust $H_{\infty }$ performance (34) with on-line switched index ${\lambda }_{\hat{\sigma }}$ . The robust UI FD observer gains in form of (14) can be given by ${\Psi }_{\hat{\sigma }}=P_{\hat{\sigma }}{K}_{1\hat{\sigma }}$ .

• Proof. Consider the stability for system (33) in which the state error dynamic is written for $f\left(t\right)=0$ . To guarantee stability, an on-line switched quadratic Lyapunov function (SQLF) is selected as

$$V_{\hat{\sigma }}\left(t\right)=e^T\left(t\right)P_{\hat{\sigma }}e\left(t\right)$$

(38)

The time derivative of the SQLF along the trajectory of the state error dynamic (33) yields as follows:

$${\stackrel{·}{V}}_{\hat{\sigma }}\left(t\right)={\stackrel{·}{e}}^T\left(t\right)P_{\hat{\sigma }}e\left(t\right)+e^T\left(t\right)P_{\hat{\sigma }}\stackrel{·}{e}\left(t\right)$$

(39)

$$\begin{array}{c}{\stackrel{·}{V}}_{\hat{\sigma }}\left(t\right)=e^T\left(t\right)(N_{\hat{\sigma }}^T{P}_{\hat{\sigma }}+N_{\hat{\sigma }}^T{P}_{\hat{\sigma }})e\left(t\right)+w^T\left(t\right)W_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}^T{P}_{\hat{\sigma }}e\left(t\right)\hfill \\ +e^T\left(t\right)P_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}{W}_{\hat{\sigma }}w\left(t\right)\hfill \end{array}$$

(40)

Thus, one can develop this relation as:

$$\begin{array}{c}{\stackrel{·}{V}}_{\hat{\sigma }}\left(e\left(t\right)\right)-\alpha V_{\hat{\sigma }}\left(e\left(t\right)\right)\hfill \\ =e^T\left(t\right)(N_{\hat{\sigma }}^T{P}_{\hat{\sigma }}+N_{\hat{\sigma }}^T{P}_{\hat{\sigma }}-\alpha P_{\hat{\sigma }})e\left(t\right)\hfill \\ +w^T\left(t\right)W_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}^T{P}_{\hat{\sigma }}e\left(t\right)+e^T\left(t\right)P_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}{W}_{\hat{\sigma }}w\left(t\right)\hfill \end{array}$$

(41)

Therefore, the inequality ${\stackrel{·}{V}}_{\hat{\sigma }}\left(e\left(t\right)\right)<\alpha V_{\hat{\sigma }}\left(e\left(t\right)\right)$ is satisfied based on condition (20) and online switched unknown input observer (14).

After that, let $V_{\hat{\sigma }}\left(e\left(t_k\right)\right)=e^T\left(t_k\right)P_{\hat{\sigma }}e\left(t_k\right)$ be the Lyapunov function given in the mode preceding this instant $\mathrm{tk}$ , and $V_{{\hat{\sigma }}_{+1}}\left(e\left(t_k\right)\right)=e^T\left(t_k\right)P_{{\hat{\sigma }}_{+1}}e\left(t_k\right)$ is the Lyapunov function in the following mode. Then, for this Lyapunov function to decrease after each switching instant, $V_{\hat{\sigma }}\left(e\left(t_k\right)\right)\le \mu V_{{\hat{\sigma }}_{+1}}\left(e\left(t_k\right)\right)$ , with $\mu >1$ , will be necessary. So equation (36) is proven.

Then, from (34), let us define the performance index $\chi \left(t\right)$ through

$${\chi }_{\hat{\sigma }}\left(t\right)={\stackrel{·}{V}}_{\hat{\sigma }}\left(t\right)+r^T\left(t\right)r\left(t\right)-{{\lambda }_{\hat{\sigma }}}^2{w}^T\left(t\right)w\left(t\right)$$

(42)

Assume that ${\chi }_{\hat{\sigma }}\left(t\right)<0$ , for $w\left(t\right)=0$ , equation (42) can be reduced to ${\chi }_{\hat{\sigma }}\left(t\right)={\stackrel{·}{V}}_{\hat{\sigma }}\left(t\right)+r^T\left(t\right)r\left(t\right)\mathrm{dt}$ . Noting that conditions (34) are verified, if ${\chi }_{\hat{\sigma }}\left(t\right)<0$ . So, ${\stackrel{·}{V}}_{\hat{\sigma }}\left(t\right)<-r^T\left(t\right)r\left(t\right)\mathrm{dt}\le 0$ is gotten and $\underset{t\to \infty }{\lim }e\left(t\right)=0$ is deduced from the Lyapunov theory.

Besides, under the initial condition e (0) = 0 and for $w\left(t\right)\ne 0$ , one has

$$\int {\chi }_{\hat{\sigma }}\left(t\right)=V_{\hat{\sigma }}\left(t\right)+\int r^T\left(t\right)r\left(t\right)\mathrm{dt}-{{\lambda }_{\hat{\sigma }}}^2\int w^T\left(t\right)w\left(t\right)\mathrm{dt}$$

(43)

Therefore, according to $V_{\hat{\sigma }}\left(t\right)>0$ , condition ${\chi }_{\hat{\sigma }}\left(t\right)<0$ implies that $\underset{0}{\overset{\infty }{\int }}{r}^T\left(t\right)r\left(t\right)\mathrm{dt}<{{\lambda }_{\hat{\sigma }}}^2\underset{0}{\overset{\infty }{\int }}{w}^T\left(t\right)w\left(t\right)\mathrm{dt}$ .

By substituting the residual vector (33) and (41) in (42), the above inequality can be reformulated as

$$\left[\begin{array}{cc}\mathrm{\Upsilon }+C_{\hat{\sigma }}^T{C}_{\hat{\sigma }}\hfill & P_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}{W}_{\hat{\sigma }}\hfill \\ \multicolumn{1}{c}{*}& -{\lambda }_{\hat{\sigma }}^{2}I\hfill \end{array}\right]<0$$

(44)

with regard to the sake of clarity, setting $\mathrm{\Upsilon }=A_{\hat{\sigma }}^T{\mathrm{\Delta }}_{\hat{\sigma }}^T{P}_{\hat{\sigma }}+P_{\hat{\sigma }}{\mathrm{\Delta }}_{\hat{\sigma }}{A}_{\hat{\sigma }}-{\Psi }_{\hat{\sigma }}{C}_{\hat{\sigma }}-C_{\hat{\sigma }}^T{\Psi }_{\hat{\sigma }}^T+\alpha P_{\hat{\sigma }}+I$ Note that ${\Psi }_{\hat{\sigma }}=P_{\hat{\sigma }}{K}_{1\hat{\sigma }}$ , the LMI holds.

It is clear now that the observer (14) is globally uniformly asymptotically stable (GUAS) and can asymptotically estimate the state with the switched $H\infty$ performance index ${\lambda }_{\hat{\sigma }}$ .

Based on the results obtained in Theorem 2, the Switched FD UI Observer design procedure can be summarized in the following algorithm to compute the parameter matrices $N_{\hat{\sigma }}$ , $K_{\hat{\sigma }}$ , and $E_{\hat{\sigma }}$ in observer (14):

1. Establish the augmented system (17);

2. Compute the UIO gains $E_{\hat{\sigma }}$ as given in equation (24);

3. Resolve $P_{\hat{\sigma }}$ , $P_{{\hat{\sigma }}_{+1}}$ and $Q_{\hat{\sigma }}$ matrices through conditions (36) and (37) in the Theorem 2, and determine the Switched FD UIO gains $K_{1\hat{\sigma }}$ ;

4. $N_{\hat{\sigma }}$ , $G_{\hat{\sigma }}$ and $K_{2\hat{\sigma }}$ gains are computed by using conditions (20)–(22);

5. Implement the Switched FD UIO (14) to estimate the state estimation $\hat{x}\left(t\right)$ .

The feasibility of the proposed linear matrix inequalities (LMIs) is verified using Yalmip.

This section consists of a design methodology for fault detection observer and fault detection logic.

On-line fault detection adaptive threshold design

Now, the fault detection framework can be formulated as follows: for the switched linear system in faulty case, the actuator fault affecting all modes given as (14) can be detected through dynamic observer (1) in which observer gains ( $K_{1\hat{\sigma }}$ and $K_{2\hat{\sigma }}$ ) are computed on-line for each active current mode estimated from the proposed event-triggered detector as stated above. If (18) and (20)–(24) in Theorem 1 are satisfied and for a static or an adaptive threshold $J_r$ , $J_r\le \parallel r\left(t\right)\parallel$ is satisfied, where $r\left(t\right)$ stands for the residual.

After generating the residual signal, as seen in the previous sections, the remaining step is designing the fault detection logic to conclude the presence or absence of a fault. Here, the adaptive threshold ${\Im }_{\mathrm{th}}$ is determined to find out the tolerant limit for the maximal influence of unknown/known inputs under fault-free operation conditions, that is

$${\Im }_{\mathrm{th}}=\underset{u\ne 0,d\ne 0,w\ne 0,f=0}{\sup }{\Im }_r$$

(45)

where ${\Im }_r$ is a residual evaluation function. The root mean square norm is often adopted and determined by

$${\Im }_r=\parallel r\left(t\right){\parallel }_2={\left(\frac{1}{\mathrm{T}}\underset{0}{\overset{T}{\int }}r{\left(\tau \right)}^{T}r\left(\tau \right)d\tau \right)}^{1/2}$$

(46)

where $T$ is the evaluation time step and $0$ is the initial evaluation time instant. Thus, the below criterion is used to detection decision-making by comparing ${\Im }_r$ and ${\Im }_{\mathrm{th}}$ as this simple form

$$\{\begin{array}{c}{\Im }_r\le {\Im }_{\mathrm{th}},\mathrm{faulty}\mathrm{case}\hfill \\ \multicolumn{1}{c}{{\Im }_r>{\Im }_{\mathrm{th}},\mathrm{fault}-\mathrm{free}\mathrm{case}}\end{array}$$

(47)

As a result, the fault detection stage is combined with low computational on-line effort.

Synchronous buck converter case study

This part illustrates the effectiveness, utility, and flexibility of the proffered robust fault detection framework. To be precise, the simulation results using MATLAB LMI toolbox verify the aforementioned theoretical outcomes.

To this end, let us consider a DC-DC buck converter circuit with load resistor $\left(R_{\mathrm{Load}}\right)$ as pointed out in Corradini et al. ⁴⁸ and depicted through Figure 2 where the inductor current $i_L\left(t\right)$ is controlled through the switch network composed of two main insulated-gate transistor-diode switches $Q1$ and $Q2$ and driven by a converter control.

Figure 2.

DC-DC buck converter topology.

From this sketch 2, the switching signal $\sigma \left(t\right)$ is generated via the Pulse Width Modulation (PWM) block; that is to say, the DC-DC voltage step-down ratio relies on the PWM duty cycle $D$ . Thereby, the switching component positions have two possible discrete states, that is, open (off) or close (on) cases as shown in Table 1. For that, the Boolean variable $d_k=1$ means that the switch $Q_k$ is closed, while, $d_k=0$ indicates that the switch $Q_k$ is open, where $k=[1,2]$ .

Table 1.

Possible on/off switch positions for $Q_k$ .

$\sigma \left(t\right)$	Mode 1	Mode 2
${\sigma }_1$	0	1
${\sigma }_2$	1	0

For the simulation purpose, Table 2 indicates variables’ description and parameter values related to the DC-DC buck converter.

Table 2.

Parameter values for the DC-DC buck converter.

Parameter description	Symbol	Numerical value
Inductance of the inductor	$L$	$4.1\mathrm{\mu }\mathrm{H}$
Resistance of the resistor	$R_L$	$80\mathrm{m}\mathrm{\Omega }$
Capacitance of the capacitor	$C$	$376\mathrm{\mu }\mathrm{F}$
Switching frequency	$f_s$	$100\mathrm{KHz}$
Voltage source	$V_g$	12 V
Duty cycle	D	$0.42$
Load resistance	$R_{\mathrm{load}}$	$1\mathrm{\Omega }$

Moreover, the studied power electronic converter dynamic can be characterized as a linear-switched state-space model subject to both disturbances and Gaussian white noises in the presence of actuator faults (refer to general model (1)) with:

$$\begin{array}{c}{A}_{\sigma }=\left[\begin{array}{cc}\frac{-R_C}{L}\hfill & \frac{1}{C}-\frac{R_L{R}_C}{L}\hfill \\ \multicolumn{1}{c}{\frac{-1}{L}}& \frac{-R_L}{L}\hfill \end{array}\right],B_{\sigma }=\left[\begin{array}{c}d\frac{R_C}{L}\hfill \\ \multicolumn{1}{c}{d\frac{1}{L}}\end{array}\right],C_{\sigma }=\left[\begin{array}{c}1,0\hfill \end{array}\right],\hfill \\ F_{\sigma }=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\right],D_{\sigma }=\left[\begin{array}{c}\frac{1}{2}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right],\mathrm{and}{W}_{\sigma }=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \end{array}$$

$V_g\left(t\right)$ is the voltage source input having a constant known value. $i_L\left(t\right)$ denotes the inductor current and $v_0\left(t\right)$ is equal to $v_C\left(t\right)+R_C{i}_L\left(t\right)$ where $v_C\left(t\right)$ is the capacitor voltage.

Here, we consider $x\left(t\right)=\left[\begin{array}{cc}{i}_L\left(t\right)\hfill & v_0\left(t\right)\hfill \end{array}\right]$ as the continuous state vector, $u\left(t\right)=V_g\left(t\right)$ and both modes having a common output $y\left(t\right)=i_L\left(t\right)$ .

Here, let us explain the switching mechanism. An unknown synchronous time-dependent switching signal for the controlled switched linear system is adapted for this showcase as given by Figure 3 where the first active mode is 2. The binary signal evolves by switching between two functioning modes and obeys the average dwell time constraint (6). The switches herein introduced are only for simulation viewing. They are as well used to be compared with the estimated results. Our mechanism aims to estimate in real time the current active mode and the corresponding switching time sequence.

Figure 3.

Evolution of the switching signal $\sigma \left(t\right)$ .

The event-triggered scheme is able to detect the current active mode, thus, a direct on-line estimation of the system switching law $\sigma \left(t\right)$ from PWM block is accomplished. We presume the switching structure and event-triggered estimation (or transmission) are synchronous. Whence, any transmission or switching-induced delays (e.g. by sampling feature) are omitted from this research work.

Meanwhile, from inequality (6), one obtains ${\tau }_{\mathrm{ad}}>{\tau }_{\mathrm{ad}}^{*}=2.{8010}^{-4}s$ for given scalars $\mu =1.0002$ and $\alpha =0.81$ . Then, let the constant sampling period value $h={10}^{-6}s$ . Using Remark 3, the parameter of the event-triggered condition is calculated following $1/9-\mathrm{fold}$ of the maximal magnitude of the system switching law along with $\epsilon =1.2{10}^{-3}$ . It can be noticed that the active mode estimation rapidity is reduced when $n_{{\sigma }_k}$ rises. Besides, Figure 4 describes the inter-event intervals in which the Zeno phenomenon is precluded.

Figure 4.

Inter-event intervals.

The actual switching law $\sigma \left(t\right)$ and its estimation $\hat{\sigma }\left(t\right)$ are illustrated in Figure 5.

Figure 5.

Actual switching signal and its estimation via the current active mode event-triggered estimator.

It is easy to see that the state of the designed event-triggering estimator approximates that of the actual system switching signal. This reveals the effectiveness of the proposed method.

To tackle the robust fault detection issue devoted to this class of hybrid dynamical systems, the designed online switched unknown input observer schemes mentioned here before are considered. Assumption 4 holds since all the pairs $(A_{\hat{\sigma }},C_{\hat{\sigma }})$ are observable for all $\hat{\sigma }\in \{1,2\}$ . For that, the disturbance $d\left(t\right)$ is some type of Gaussian distributed random number and is taken as $d\in \left[\begin{array}{cc}-0.5,\hfill & 0.5\hfill \end{array}\right]$ . While, the input signal is corrupted by an additive band-limited Gaussian white noise $w\left(t\right)$ chosen such that the Signal to Noise Ratio is equal to $10\mathrm{dB}$ with respect to the input signal.

Let’s suppose that the actuator fault scenario corresponds to an unknown additive fault on the actuator signal, that is, a decrease of the voltage source performance (e.g. for DC-DC solar PV system, increment or decrement of the PV panel voltage). This latter (biased fault) is defined as follows

$$f\left(t\right)=\{\begin{array}{cc}10\%u\left(t\right)& ift>0.004s\\ 0& \mathrm{otherwise}\end{array}$$

(48)

In what follows, the initial state conditions are set as $x_0={\left[\begin{array}{cc}0.001\hfill & 0.001\hfill \end{array}\right]}^T$ and $z_0={\left[\begin{array}{cc}0\hfill & 0\hfill \end{array}\right]}^T$ .

Now, assume that both subsystems and the proposed observer (14) are switching in a synchronous manner, that is, under the actual estimated switching sequence, any actuator faults may be immediately detected via the online switched UIO (14) with respect to some conditions. It is easy to find $\mathrm{rank}\left(C_{\hat{\sigma }}{D}_{\hat{\sigma }}\right)=\mathrm{rank}\left(D_{\hat{\sigma }}\right)=1$ , that is, the switched UIO decoupling condition is satisfied, subsequently, the suggested Theorems 1 and 2 can be utilized.

In accordance with (35), the $H\infty$ performance online switched index ${\lambda }_1$ and ${\lambda }_2$ are respectively equal to $4.0817{10}^{-10}$ and $4.0817{10}^{-10}$ . Calculating the parameters $E_{\hat{\sigma }},\hat{\sigma }=1,2$ in (24) under the conditions in Lemma 2 yields

$$E_1=E_2=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{-0.5}\end{array}\right]$$

Further, when conditions of Theorem 1 (from (20) to (24)) are fulfilled, solving the LMIs in Theorem 2 gives the parameter matrices of the on-line switched observer (14) as

$$\begin{array}{c}{P}_1=P_2={10}^8\left[\begin{array}{cc}5.6333\hfill & 0.1348\hfill \\ \multicolumn{1}{c}{0.1348}& 0.0101\hfill \end{array}\right]\hfill \\ {\psi }_1={\psi }_2={10}^8\left[\begin{array}{c}0.3586\hfill \\ \multicolumn{1}{c}{-4.8007}\end{array}\right]\hfill \\ K_{11}=K_{12}=\left[\begin{array}{c}16.8197\hfill \\ \multicolumn{1}{c}{-700.1644}\end{array}\right]\hfill \end{array}$$

According to Theorem 1 (ii), the other UIO parameters may be gotten

$$\begin{array}{c}{N}_1=N_2={10}^5\left[\begin{array}{cc}-0.0002\hfill & 0\hfill \\ \multicolumn{1}{c}{-2.4381}& -0.1823\hfill \end{array}\right]\hfill \\ G_1={10}^5\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{2.4451}\end{array}\right],G_2=0\hfill \\ K_{21}=K_{22}={10}^5\left[\begin{array}{c}-0.0002\hfill \\ \multicolumn{1}{c}{-2.3470}\end{array}\right]\hfill \end{array}$$

The state responses of the inductor current $x_1\left(t\right)$ and the voltage $x_2\left(t\right)$ along with its estimate $z_1\left(t\right)$ and $z_2\left(t\right)$ are depicted in Figures 6 and 7. Owing to the above estimated active modes (see Figure 5), the continuous states estimations of the buck converter are accurately fulfilled. Since the attenuation of the unknown input (disturbance and noise) effects is derived from Theorems 1 and 2, the designed on-line switched UIO shows the expected convergence performances regardless of the unknown inputs and switching instants.

Figure 6.

Continuous state responses of the system and its estimation in normal case.

Figure 7.

Continuous state responses of the system and its estimation in abnormal case.

As can be seen from Figures 8 and 9, the residual signal responses $r\left(t\right)$ are deduced from the proposed online switched UIO framework devoted to fault detection purposes. Obviously, the generated fault indicator is only sensitive to additive actuator faults. This means that the robustness vis-a-vis disturbance and noise is insured. That is to say, the unknown input attenuation is achieved. At $t=0.004s$ , the occurrence of a bias fault is immediately alerted and perfectly detected. As known, $r\left(t\right)$ is supposed to be nonzero under a faulty condition and null otherwise (see Figure 9).

Figure 8.

Residual response curves in fault-free situation (without actuator faults).

Figure 9.

Residual response curves in additive actuator-faulty situation.

The state estimations show remarkable accuracy, tracking actual states with deviations of less than $0.01\%$ and achieving rapid convergence speeds of under $3.33{10}^{-4}$ s. Furthermore, the residuals produced are both bounded and stable, indicating that the proposed observer effectively eliminates errors caused by noise and control switching actions. Unlike Ali et al.’s ¹⁶ research work which is based on $H\infty$ performance index to minimizing disturbance impact from residual indicators generated offline.

Figure 10 plots the evaluation function of $J\left(r\right)$ under the fault (solid line) and fault-free conditions (dash-dot line). With a threshold $\mathrm{Jth}=0.001$ , it can be concluded from the results that $J\left(r\right)>\mathrm{Jth}$ at time $t=0.004s$ . It is worth pointing out that fault detection tasks are readily guaranteed.

Figure 10.

Evaluation function.

Here, the fault detection time is faster than the results shown in both Ali et al. ¹⁶ and Su et al., ²⁵ which means that, regardless of the initial state chosen (as long as it remains within a reasonable range), the estimation consistently converges to the target value and maintains accurate tracking thereafter. As a result, the parameters established for the proposed observer will influence the time required for the system to achieve convergence and the variations observed. This dependence underscores the importance of carefully selecting and tuning these parameters to optimize performance and accuracy in the given context.

Discussion and future scope

From the simulation results, further discussions can be pointed out. The innovative interest of the proposed framework about the existing schemes is compared herein. In this work, we suggest a straightforward design perspective in the FD framework that relies on an online fault detection observer-based event-triggered mode estimation scheme for systems under synchronous unknown switching law.

Earlier research works undertake switched systems. Both the operating modes and the continuous states are unknown, as taken in this work, whereby the current switching mode is here estimated online (unlike others) using the proposed event-triggered estimator without any necessity for numerical calculation, and it is generally used for real-time online mode identification.

Based on the aforementioned simulation, Theorem 2 enhances fault detection robustness by effectively addressing disturbances and noise. Consequently, Theorem 2 demonstrates superior performance compared to Theorem 1 in terms of both accuracy and reliability. Notably, Theorem 2 reduces the error rate through the minimization of process noise, a critical factor for achieving optimal outcomes. In order to test the robustness of the estimation performance, two quantitative performance metrics are used to rigorously evaluate the suggested fault detection framework’s accuracy. The first criterion, the Integral Squared Error (ISE), quantifies the average squared deviation between the estimated states and the actual values over time. The second metric, the Variance Accounted For (VAF), measures the correlation between the estimated states and the system states, indicating the similarity between the two, as given here:

$$f_{\mathrm{ISE}}=\frac{1}{T}\underset{0}{\overset{T}{\int }}{(x\left(t\right)-\hat{x}\left(t\right))}^2\mathrm{dt}$$

(49)

$$f_{VAF}=\max \left(1-\frac{\mathrm{var}\left(x\left(t\right)-\widehat{x}\left(t\right)\right)}{\mathrm{var}\left(\widehat{x}\left(t\right)\right)},0\right)100\%$$

(50)

The proposed strategy demonstrates intense matches between the estimated states and the system ones, underscored by an ISE of $0.0012$ and a VAF of $99.105\%$ . The elevated VAF indicates that the proposed FD scheme delivers a highly accurate depiction of the desired system dynamics, even in the presence of faults and noise. In Cen and Stewart, ⁴⁹ a fault diagnosis scheme utilizing adaptive model observers is proposed and implemented within a switched hybrid system. This scheme addresses the fault effects caused by common power-loss malfunctions in solar photovoltaic buck converters, even in the presence of faults and noise. The Variance-Accounted-for (VAF) metric was used to evaluate its effectiveness, achieving a VAF of $98\%$ .

On the other hand, it is obviously that the computational difficulty of such fault indicator deduction approaches^26–28 rises crucially as the system dimension and the number of partial models (i.e. modes) enhance. ⁵⁰ The online FD observer modifies with each sampling time. So, we avoid a bank of filters whose dimension does not necessarily scale up with the dimension of the system. This feature enables a possibility of low-ordered filters compared to the existing literature. This way simplifies the residual signal generation and avoid sophisticated algorithms.

The fault detection framework presented in this study effectively formalizes the online design of a switched unknown input observer using Linear Matrix Inequalities, eliminating the need for an ensemble of filters tailored offline for associated residual signals, as proposed in Kazemiand Montazeri. ⁵¹ Our findings indicate that a singular residual signal generated in real time maintains high efficiency. A comparative analysis of the minimum average dwell time ${\tau }_{\mathrm{ad}}^{*}$ , as outlined in Table 3, demonstrates the feasibility of the LMIs derived in Theorem 2 for lower values of ${\tau }_{\mathrm{ad}}^{*}$ , highlighting that this novel formalization is less conservative. As a result, the switched system can achieve stabilization across a broader class of switching signals.

Table 3.

Comparison with the reference Kazemi and Montazeri. ⁵¹

Approach	${\tau }_{\mathrm{ad}}^{*}$
Reference Kazemi and Montazeri. 51	$3.2{10}^{-4}s$
Theorem 2	$2.80{10}^{-4}s$

Today actual industrial applications (hybrid renewable energy processes, vehicles, … etc.) become greatly sophisticated.^9,52 The proposed approach constitutes an advantageous alternative for such systems described by switching dynamics. It represents not only a unified fault detection procedure but also a multitasking method that permits attaining complex hybrid model-based issues like robust model-based diagnosis.

This method offers a flexible and adaptable solution for complex systems where traditional modeling may be intricate. This paper addresses two critical fault types, as outlined in Qiao et al., ⁵³ emphasizing their significance in real-world systems. We focus on abrupt faults, which can lead to catastrophic system failures if not detected promptly. Implementing early fault detection mechanisms or proactive system configurations can effectively mitigate such risks related to the fault’s nature or severity, as mentioned in Yahia et al. ⁵⁴ The time of fault detection may be longer for slowly developing or incipient faults, but these faults can often be detected more efficiently, as seen in Qiao and Shu. ⁵⁵

Qiao et al. ⁵³ introduce an innovative technique aimed at enhancing the detection of weak fault signatures in complex nonlinear systems characterized by partially unknown dynamics. This method uses an adaptive coupled neuron architecture integrated with multi-objective optimization to identify unforeseen or incipient fault signatures effectively. Thus, both the signal-to-noise ratio (SNR) and the residence-time distribution ratio are seen as the multi-objective function that optimizes the adjusting parameters of the coupled neurons and the rescaling factor simultaneously by using genetic algorithms. Therefore, this adaptive framework is particularly adept at swiftly pinpointing potential issues, making it highly suitable for diagnosing early-stage failures in rotating machinery. Additionally, it addresses the challenges posed by complex systems where the underlying model may be inadequately defined. Consequently, our new formalization may be less computationally efficient than Qiao et al.’s^53,55 works and may have limitations in particular faults. Our results present sufficient conditions with reducing the conservativeness of stability results for the studied system whose all modes are stable. Moreover, the online state estimation capability and the rapid fault detection can minimize sharp errors caused by noise and switching actions.

Conclusions

This paper has been investigated an online robust switching fault detection observer for continuous-time switched linear systems influenced by unknown inputs such as disturbances noise and additive faults. We have introduced an online switched unknown input observer-based robust fault detection algorithm utilizing a mode-dependent event-triggered estimator block. This approach effectively identifies various types of faults while mitigating the impact of unknown inputs on the generation of online residual signals, employing adaptive thresholds. The theorems developed in this research have been derived using the online switched Lyapunov function method and the ADT technique. Additionally, observer gains are computed online through the resolution of linear matrix inequalities. Consequently, this research facilitates full automation in two critical stages: current mode estimation and robust fault diagnosis, achieving validation and reduced conservatism that surpasses existing literature.

Nevertheless, some limitations appear; the system partial models are usually identified by taking more time and putting on the appropriate observer. This induces a delay between the event-triggered estimator and the robust online switched observer block. Hence, it is possible to extend the suggested framework to perform state estimation and fault detection schemes by thinking of the delayed observer synthesis.