Disturbance Decoupling for a Single-Phase Pulse Width Modulation Rectifier Based on an Extended H-Infinity Filter

Abstract

The growing utilization of single-phase pulse width modulation (PWM) rectifiers in various applications has spurred interest in detecting and monitoring faults in these devices. In particular, voltage and current sensors play a crucial role in the control loop of these rectifiers. However, sensor faults can significantly affect the converter’s performance and availability. This paper introduces a novel and efficient method for detecting and decoupling sensor faults in single-phase PWM rectifiers. The proposed method utilizes residual generation and incorporates an extended filter within the rectifier. Unlike conventional filters, the presented fault detection and isolation (FDI) method effectively eliminates the influence of disturbances on the residual signal. This feature helps prevent false alarms in the monitored system, ensuring reliable fault detection. To evaluate the effectiveness of the approach, hardware-in-the-loop and simulation tests were conducted. The results from these tests provide substantial evidence supporting the efficacy of the proposed method. The hardware-in-the-loop experiments involved real-world implementation, validating the practicality and reliability of the approach. Meanwhile, simulation tests allowed for a comprehensive analysis of system behavior and performance under various fault scenarios. The findings demonstrate the rapid and dependable nature of the proposed method for detecting and decoupling sensor faults in single-phase PWM rectifiers. By effectively mitigating the impact of disturbances on the residual signal, false alarms are minimized, ensuring accurate fault detection. The experimental validation highlights the practical applicability and effectiveness of the proposed approach, making it a valuable contribution to fault detection in single-phase PWM rectifiers.

1. Introduction

Single-phase pulse width modulation (PWM) rectifiers are a popular choice for various industrial applications such as grid-connected solar modules, electric train traction, active power filters, and cascaded H-bridge multilevel converters [1,2,3,4]. These critical industrial applications require high system availability and operational reliability. Full- bridge AC/DC converters are favored for their ability to regulate voltage on the DC-link side and maintain a high power factor by sinusoidally absorbing grid-side current. This is achieved through a closed-loop control mechanism, which necessitates one current sensor and two voltage sensors. However, if any of these sensors malfunction, this can lead to control system instability and problems with power semiconductors. Consequently, the converter’s overall performance relies heavily on sensor availability and the control system’s sensitivity to accurate measurements. To ensure the stability and optimal operation of the monitored plant, even in the presence of sensor defects, it is essential to develop and implement a sensor fault diagnostic algorithm and a fault-tolerant control mechanism into the converter controller. A review of defect detection and fault tolerance methods is available in [5,6], which discuss various approaches used to address the challenges of detecting and identifying failures.

The literature offers a wide range of fault detection and diagnosis (FDD) techniques, including model-based, structural graph, data-driven, and pattern recognition methods [7,8].

Several techniques have been proposed to detect faults in converters. A survey of fault diagnosis techniques is presented in [5], and [9,10,11,12] introduces Park’s Vector as a method for electric current–based fault identification and diagnosis, while other methods based on current are presented in [13,14,15,16]. An unknown input observer was utilized in [17] to detect faults in single-phase PWM (pulse width modulation) rectifiers. The literature on fault detection and isolation (FDI) has mostly focused on three-phase adjustable speed drive systems. This includes single-phase PWM rectifiers under fault-tolerant control and state-based sensor FDI for electrical railway traction, as described in [2]. In [18], a hybrid system model-based current residual vector fault diagnostic approach for inverters in permanent magnet synchronous motor drive systems is presented. This method uses sensors already present in drive systems to identify open-circuit (OC) issues in inverters. In [19], a fault detection method based on signal normalization using a basic trigonometric function is described for a fault-tolerant application under non-sinusoidal asymmetrical current waveforms.

The primary objective of FDI is to identify faults and locate faulty components. However, disturbances can often cause false alarms in the residuals, leading to confusion regarding the need for maintenance. Methods based on robust observers are recommended to address this problem. There are various approaches to achieving robust defect detection, such as FDI based on unknown input observers, FDI based on eigenstructure assignment, and FDI based on the linear matrix inequality (LMI) approach. To be considered robust, each method must satisfy three conditions:

• enhancing the effects of faults on the residual;
• decreasing the effects of disturbances on the residual;
• stabilizing the gain observer [20].

Therefore, disturbance decoupling is crucial in achieving robust FDI. The H-infinity filter is a type of Kalman filter that minimizes estimation energy error for all potential disturbances with fixed energy, based on the H-infinity optimum estimation [21]. Due to its robustness against modeling uncertainty, H-infinity is often used for FDI problems. However, the H-infinity filter only minimizes the impact of disturbances on the residual signal and does not eliminate it.

This paper’s primary contribution is:

(1)

computation of a novel $H_{\infty }$ filter that eliminates the impact of perturbations on the residual signal, contrary to the standard $H_{\infty }$ filter which minimizes only the impacts of disturbances;

(2)

application of the developed strategy on a single-phase PWM rectifier for sensor fault detection;

(3)

implementation of the proposed algorithm in an experimental test with the HIL platform to validate the effectiveness of this method.

This paper has the following structure: in the first part, an introduction is presented; in the second part, the state space representation for a single-phase PWM rectifier is developed; and in the third part, a filter design via the linear matrix inequality approach is introduced. In the fourth and fifth sections, the results discussion and a conclusion are presented, respectively.

2. Materials and Methods

2.1. State Space Representation of a Single-Phase Pulse Width Modulation Rectifier

Applying the mesh flow to Figure 1, the state space of the single-phase rectifier is defined as follows:

$$\{\begin{array}{l}{U}_n=-L\frac{d{i}_c}{dt}-R{i}_c+U_{ab}\\ i_{cd}=C_{cd}\frac{d{V}_{dc}}{dt}\\ i_d=i_{cd}+i_2+i_{ld}\end{array}$$

(1)

where $U_n$, $i_c$, $i_{cd}$, $U_{ab}$, $V_{dc}$ and $i_d$ are input source AC-voltage, sensor current, capacitor current in the output filter, rectifier input voltage, output DC-link voltage, and output DC-link current, while $L$, $R$, $C_{dc}$ and $i_{ld}$ are inductance filter, resistance filter, capacitor, and output load current. This is as in [22], while moving from a stationary location to a synchronously spinning coordinate as seen in Figure 2.

From Equations (1) and (2), we get:

$$\{\begin{array}{l}{u}_{sd}=L\frac{d{i}_{sd}}{dt}+R{i}_{sd}-wL{i}_{sq}+u_{ab}\\ u_{sq}=L\frac{d{i}_{sq}}{dt}+R{i}_{sq}+wL{i}_{sd}+u_{ab}\end{array}$$

(2)

After some simplification, we get:

$$\{\begin{array}{l}\left[\begin{array}{c}d\\ q\end{array}\right]=\left[\begin{array}{cc}\cos (\mathrm{wt})& \sin (\mathrm{wt})\\ -\sin (\mathrm{wt})& \cos (\mathrm{wt})\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]\\ \left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\left[\begin{array}{cc}\cos (\mathrm{wt})& -\sin (\mathrm{wt})\\ \sin (\mathrm{wt})& \cos (\mathrm{wt})\end{array}\right]\left[\begin{array}{c}d\\ q\end{array}\right]\end{array}$$

(3)

$$\{\begin{array}{l}L\frac{d{i}_{sd}}{dt}=v_d-R{i}_{sd}+wL{i}_{sq}\\ L\frac{d{i}_{sq}}{dt}=v_q-R{i}_{sq}+wL{i}_{sd}\end{array}$$

(4)

where: $wt=\theta$ in Figure 2.

$$\begin{array}{l}x=\left[\begin{array}{c}{i}_{sd}\\ i_{sq}\end{array}\right]\\ u=\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{c}{u}_n-u_{abd}\\ -u_{abq}\end{array}\right]\end{array}$$

The following equation provides the state space of the model under uncertainties:

$$\{\begin{array}{l}\dot{x}(t)=[A(t)+\Delta A(\upsilon ,t)]x(t)+[B(t)+\Delta B(v,t)]u(t)+E_1(t)d(t)\\ y(t)=C(t)x(t){+\mathrm{E}}_2\mathrm{d}\left(\mathrm{t}\right)\end{array}$$

(5)

The model (4) without parameter disturbances can be presented in matrix form as follows:

$$\begin{array}{l}A=\left[\begin{array}{cc}-\frac{R}{L}& w\\ -w& 0\end{array}\right],B=\left[\begin{array}{cc}\frac{1}{L}& 0\\ 0& \frac{1}{L}\end{array}\right]\\ C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\mathrm{D}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}\}$$

(6)

Assuming that only the vector control is impacted by the perturbations, the matrix representing the distribution of the unknown input can be addressed using matching criteria. We can make this assumption using the theory of relativity form [23], as follows:

Assumption 1.

Consider that the function $\{A(\mathrm{t}),B(\mathrm{t})\}$ is consistently and fully controllable.

Assumption 2.

for each $(x,t)\in {\Re }^n\times \Re$, constant matrix functions $\delta (\upsilon ,\mathrm{t}),\vartheta (\mathrm{v},\mathrm{t}),\varphi (\mathrm{t})$ of the appropriate size exist, and they are such that:

$$\begin{array}{l}a)\Delta A(\upsilon ,t)=B(t)\delta (\upsilon ,t)\\ b)\Delta B(v,t)=B(t)\vartheta (v,t)\\ c)E(t)=B(t)\varphi (t)\end{array}$$

(7)

where the matrices of system uncertainties $\Delta A(\upsilon ,\mathrm{t}),\Delta B(\nu ,\mathrm{t})$ remain constant across all their arguments. $\upsilon$ and $v$ are the uncertain vectors.

First observation: The matching requirements are unique to system architecture. Be careful to make sure that the control vector $u$ does not have a greater impact on dynamics state than the uncertainty vector.

In the event that Assumptions 1 and 2 are true, (5) takes the next structure:

$$\{\begin{array}{l}\dot{x}(t)=A(t)x(t)+B(t)[(u(t)+\eta (\upsilon ,v,t]\\ y(t)=C(t)x(t)\end{array}$$

(8)

where $\eta =\delta (\upsilon ,\mathrm{t})x(t)+\vartheta (\nu ,\mathrm{t})u(t)+\varphi (\mathrm{t})d(t)$.

This illustration suggests that the dynamics are not more affected by unpredictability $\eta$ than the control vector $u$, as mentioned in remark one.

If the matching condition is not satisfied, methods based on estimating the distribution matrix can be used. This paper only considers sensor faults. The next section will present the linear matrix inequality (LMI) approach and robust fault detection observer designing. The duality of $H_{\infty }$ filter estimation will also be established in a similar way to what was developed in [24], and we propose a way to eliminate the effect of perturbation on the residual signal.

2.2. Filter Design via LMI Approach

Designing a diagnosis that delivers good FDI performance requires considering two factors: robustness against perturbations and sensitivity to defects. This is according to [25]. This section describes the LMI strategy for robust FDI, which is based on the work of Patton & Hou [26]. The linear time-invariant system is presented below (5):

$$\{\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+R_{1}f(t)+E_{1}d(t)\\ y(t)=Cx(t)+Du(t)+R_{2}f(t)+E_{2}d(t)\end{array}$$

(9)

With $f\in {ℝ}^m$ describing the faults, the vector $f$ can be a component, an actuator, or a sensor fault.

Assumption 3.

$\{C,A\}$are the detectable pair, and the matrix $\left[\begin{array}{cc}C& E_2\end{array}\right]$ has full rank.

Assumption 4.

$\{A,R_1,C,R_2\}$containing no transmission zeros, i.e., for any $s\in ℂ$, rank $\left[\begin{array}{cc}-sI+A& R_1\\ C& R_2\end{array}\right]=n+m$ to guarantee detectability in (9).

From Equation (9), it is possible to construct a failure detection observer as follows:

$$\{\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+Bu+K(y+C\widehat{x}-Du)\\ \widehat{y}=C\widehat{x}-Du\\ \mathrm{r}=\mathrm{Q}[\mathrm{y}-\widehat{\mathrm{y}}]\end{array}$$

(10)

where $\widehat{x}\in {ℝ}^n$, $\widehat{y}\in {ℝ}^p$, $r\in {ℝ}^p$, $u$ are the observer state, output estimate, residual signal, and input control signal, respectively. $K$ is the filter gain, and $Q$ is the constant estimation weight to the design. Their design must satisfy that the residual $r$ doesn’t depend on $u$ and $x(0)$ for $t\to \infty$ when the state error is defined as $x(t)-\widehat{x}(t)$. The following equation provides the filter assessment dynamic error:

$$\{\begin{array}{l}\dot{\tilde{x}}=(A-KC)\tilde{x}(t)+(R_1+K{R}_2)f(t)+(E_1-K{E}_2)d(t)\\ r(t)=H\tilde{x}(t)+\mathrm{Q}{\mathrm{R}}_{2}f(t)\end{array}$$

(11)

with $H=QC$.

When designing a robust observer, it is important not only to ensure the stability of the observer gain but also to ensure the sensitivity of the residual to the faults. This can be achieved by reducing the impact of disturbances on the residual when it is not possible to remove them, in order to avoid false alarms.

A standard observer should be selected such that the transfer function from the disturbance to the observer residual has an H-infinity norm, resulting in a specific small real number $\gamma >0$. Consider a real number $\gamma >0$; observer (10) is an $H_{\infty }$ observer if and only if:

$$A_0\mathrm{is}\mathrm{asymptotically}\mathrm{stable}$$

(12)

$${\Vert G_{rd}(jw)\Vert }_{\infty }<\gamma$$

(13)

where $\{\begin{array}{l}{G}_{rd}(s)=C{(sI-A_0)}^{-1}(E_1-K{E}_2)+E_2;\\ A_0=A-KC\end{array}$

From (11), one can see that the dynamic error is governed by two terms:

$$r(s)=G_{rf}(s)+G_{rd}(s)$$

(14)

The purpose of fault detection based on the filter is not to completely decouple disturbances from the residual but to minimize the sensitivity of the residual to disturbances and increase its sensitivity to faults by designing an appropriate filter gain.

In this paper, we propose an alternative approach to achieving the decoupling of disturbances instead of minimizing the sensitivity of the residual to disturbances using an H-infinity filter. The proposed approach involves manipulating the weight matrix $Q$ and the output signal $y(t)$ to completely decouple the disturbances. Two techniques are suggested to achieve this, since the unknown input term appears in both the state system equation and the output equation.

Step 1: nulling of disturbance in the output

Let’s rewrite the output equation of (9) as follows:

$$y(t)=y_E(t)=\psi Cx(t)+\psi Du(t)+\psi R_{2}f(t)+\psi E_{2}d(t)$$

(15)

where $\psi$ is designed so that $\psi E_2=0$. This situation is equal to the one where there are no output disturbances and (9) becomes as:

$$\{\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+R_{1}f(t)+E_{1}d(t)\\ y_E(t)=\psi Cx(t)+\psi Du(t)+\psi R_{2}f(t)\end{array}$$

(16)

Step 2: decoupling of $E_1$

Now to deal with one disturbance, let’s rewrite (10) based on the modification made in (16)

$$\{\begin{array}{l}\dot{\widehat{x}}(t)=A\widehat{x}(t)+Bu(t)+K(y(t)+C\widehat{x}(t)-Du(t))\\ y(t)=C\widehat{x}(t)-Du(t)\\ \mathrm{r}(\mathrm{t})=\mathrm{Q}[{\mathrm{y}}_E(\mathrm{t})-{\widehat{\mathrm{y}}}_E(\mathrm{t})]\end{array}$$

(17)

From (11) and (17), the residual response to faults and disturbances after the Laplace transformation is as follows:

$$\begin{array}{ll}r(s)=& H_E{(sI-A+K{C}_E)}^{-1}(R_1-K\psi R_2)\mathrm{f}(\mathrm{s})\\ & +H_E{(sI-A+K{C}_E)}^{-1}{E}_{1}d(s)+Q\psi R_{2}f(s)\end{array}$$

(18)

where $C_E=\psi C$.

One can see that the one condition to get $r(s)$ independent of $d(t)$ is to design $Q$ so that $H_E{E}_1=0$. As the filter gain will be generated on the basis of the $H_{\infty }$ theory, the designing of $Q$ depends only on $C_E$ and $E_1$. When this condition is satisfied and $K$ is generated, then our residual is independent of disturbance and (19) becomes:

$$r(s)=H_E{(sI-A+K{C}_E)}^{-1}(R_1-K\psi R_2)\mathrm{f}(\mathrm{s})+Q\psi R_{2}f(s)$$

(19)

The goal of this paper is to compute $\psi$ and $Q$ so that the residual becomes independent of disturbances. Those two steps are successfully implemented in this paper.

The H-infinity filter gain is computed using a linear matrix inequality as described below. The resulting theorem is presented in the following:

Theorem 1.

Provided 2 scalars $\beta >\gamma >0$, there is an H-infinity filter in the form of Equation (10) with the criteria (12)–(13) only if there exists $Y=Y^T>0$ gives solution to the LMI.

$${\left[\begin{array}{c}{C}_E^T\\ E_2{}^T\end{array}\right]}^{\perp }\left[\begin{array}{cc}{A}^{T}Y+YA+C_E^T{C}_E& Y{E}_1+C_E^T{E}_2\\ E_1{}^{T}Y+E_2^T{C}_E^T& E_2^T{E}_2-{\gamma }^{2}I\end{array}\right]{\left[\begin{array}{c}{C}_E^T\\ E_2^T\end{array}\right]}^{\perp T}<0$$

(20)

When a quadratic matrix inequality restriction is presented as:

$$\begin{array}{l}{\left[C_E{(Y{C}_E^T{C}_E-A)}^{-1}(R_1-Y{C}_E^T{R}_2)+R_2\right]}^T\\ \left[C_E{(Y{C}_E^T{C}_E-A)}^{-1}(R_1-Y{C}_E^T{R}_2)+R_2\right]>{\beta }^{2}I\end{array}$$

(21)

$(Y{C}_E^T{C}_E-A)$ must be no singular to find a solution of (21). According to this, the $H_{\infty }$ filtering performance metric modeled as a quadratic cost function is as follows:

$$J(K,k)=\sup \frac{{\Vert x-\widehat{x}\Vert }_2}{\Vert k\Vert }={\Vert H_{\epsilon k}(s)\Vert }_{\infty }$$

(22)

Substituting the decision variable $Y$ (which is a positive definite matrix) into the observer equation, we can express it as follows:

$$\{\begin{array}{l}\widehat{x}(t)=(A-Y{C}_E^T)\widehat{x}(t)+Bu(t)+Y{C}_E^{T}y(t)\\ {\widehat{\mathrm{y}}}_E(t)=C_E\widehat{x}(t)\end{array}$$

(23)

We have ${\Vert H_{\epsilon k}\Vert }_{\infty }$ if and only if $Y\ge 0$ exists. From [27], for a resilient FD, it is desired that the failure only influences the residual vector. To achieve this, $K$ and $H_E$ should be created in a way that the FDF is stable and satisfies the following condition:

$$G_{rd}(s)=Q(C_E{(sI-A_0)}^{-1}{E}_1+E_2)=0$$

(24)

$$G_{rf}(s)=Q(C_E{(sI-A_0)}^{-1}{R}_1+R_2)\ne 0$$

(25)

For (24) and (25) to hold,

$$rank\left[\begin{array}{cc}{A}_o-sI& E_1\\ C_E& E_2\end{array}\right]<rank\left[\begin{array}{ccc}{A}_o-sI& R_1& E_1\\ C_E& R_2& E_2\end{array}\right]\le n+m$$

The transfer function of our disturbance system can be presented as

$$G(s)=C_E{(sI-A+K{C}_E)}^{-1}(E_1-K{E}_2)+E_2$$

(26)

The requirement for robustness is given by the following inequality:

$$\Vert G(s)=C_E{(sI-A+K{C}_E)}^{-1}(E_1-K{E}_2)+E_2\Vert <\gamma$$

(27)

This can be simplified as: ${\Vert G\Vert }_{\infty }<\gamma$ and is true if and only if a symmetric matrix $Y>0$ and a matrix $K$ exist, such that LMI

$$\left[\begin{array}{ccc}{A}_o^{T}Y+Y{A}_o& Y{E}_o& C_E^T\\ E_o^{T}Y& -\gamma I& E_2^T\\ C_E& E_2& -\gamma I\end{array}\right]<0$$

(28)

where $A_o=(A-K{C}_E)$ and we express $R=\gamma I-E_2^T{E}_2$, $E_o=E_1-K{E}_2$. The LMI is the transformation of the following Ricatti Equation:

$$\{\begin{array}{l}Y(A_o+E_o{R}^{-1}{E}_2^T{C}_E)+{(A_o+E_o{R}^{-1}{E}_2^T{C}_E)}^{T}Y\\ +Y{E}_o{R}^{-1}{E}_o^{T}Y+C_E^T(I+E_2^T{R}^{-1}{E}_2^T)C_E=0\end{array}$$

(29)

If this is true, the systems will be asymptotically stable, which will satisfy the inequality stated by:

$${\Vert r\Vert }_2<\gamma {\Vert d\Vert }_2$$

(30)

The FDF dynamics given in Equation (11) will be as follows, since we are only reducing the impact of disturbances on the residual signal:

$$\{\begin{array}{l}\dot{\tilde{x}}=(A-K{C}_E)\tilde{x}+E_{o}d\\ r=H_E\tilde{x}+E_{2}d)\end{array}$$

(31)

Since

$${\Vert G_{rd}\Vert }_{\infty }=\frac{\Vert r(t)\Vert }{\Vert d(t)\Vert }<\gamma$$

(32)

$$r^{T}r<{\gamma }^2{d}^{T}d$$

The Lyapunov candidate function is defined as follows:

$$v=e^{T}Ye$$

By using Equation (29) and the time derivative of the proposed Lyapunov function, the following outcomes are achieved:

$$\dot{v}=e^{T}Y(A_{o}e+E_{o}d)+{(A_{o}e+E_{o}d)}^{T}Ye$$

(33)

The objective function defined in Equation (30) is designed to satisfy the H-infinity performance requirement stated in Equation (29).

$$J=\int (r^{T}r-{\gamma }^2{d}^{T}d)<0$$

(34)

We must show that $J<0$ for every non-zero $d$ and $\dot{v}$

$$\begin{array}{l}{(C_{E}e+E_{2}d)}^T(C_{E}e+E_{2}d)-{\gamma }^2{d}^{T}d+e^{T}Y(A_{o}e+E_{o}d)\\ +{(A_{o}e+E_{o}d)}^{T}Ye\end{array}$$

(35)

After performing some simple calculations, we obtain:

$$\left[\begin{array}{cc}{e}^T& d^T\end{array}\right]\left[\begin{array}{cc}{A}_o^{T}Y+Y{A}_o+C_E^T{C}_E& Y{E}_o+C_E^T{E}_2\\ E_o^{T}Y+E_2^T{C}_E& E_2^T{E}_2-{\gamma }^{2}I\end{array}\right]\left[\begin{array}{c}e\\ d\end{array}\right]<0$$

(36)

So

$$J\le \int (r^{T}r-{\gamma }^2{d}^{T}d+\dot{v}(x))<0=\int \left[\begin{array}{cc}{e}^T& d^T\end{array}\right]\Phi \left[\begin{array}{c}e\\ d\end{array}\right]$$

(37)

with

$$\Phi =\left[\begin{array}{cc}{A}_o^{T}Y+Y{A}_o+C_E^T{C}_E& Y{E}_o+C_E^T{E}_2\\ E_o^{T}Y+E_2^T{C}_E& E_2^T{E}_2-{\gamma }^{2}I\end{array}\right]<0$$

(38)

After applying the Schur complement and performing some additional processing, we obtain:

$$\left[\begin{array}{ccc}{A}_o^{T}Y+Y{A}_o& Y{E}_o& C_E^T\\ E_o^{2}Y& -\gamma I& E_2^T\\ C_E& E_2& -\gamma I\end{array}\right]<0$$

(39)

Equation (39) represents a nonlinear matrix inequality. To obtain a solvable solution, we need to convert it into LMI form by changing the variables. Let us define:

$$YK=Z\hspace{1em}{Z}^T=K^{T}Y$$

(40)

Using (38) to (40) we get:

$$\left[\begin{array}{ccc}{A}^{T}Y+YA-C_E^T{Z}^T-Z{C}_E& Y{E}_1-Z{E}_2& C_E^T\\ E_1^{T}Y-E_2^{T}Z& -\gamma I& E_2^T\\ C_E& E_2& -\gamma I\end{array}\right]<0$$

(41)

If Equation (41) has a solution, then the filter gain can be computed as follows:

$$K=Y^{-1}Z$$

(42)

By applying the parameter values from

Table 1. Single-phase pulse width modulation rectifier parameters.

Parameters	Variables Values	Unity
Filter resistor (R)	$0.530$	$\Omega$
Filter inductor (L)	$2.30\times {10}^{-3}$	$H$
DC-link Capacitor (Cd)	$8.0\times {10}^{-3}$	$F$
Grid voltage ($U_n$)	$1550$	$V$
DC-link Voltage ($V_{dc}$)	$3000$	$V$
Rectifier switching frequency ($f_s$)	$1250$	$Hz$

in Equation (6) and by selecting the appropriate values of the distributed matrices of the faults and disturbances signals, the following matrices are provided to validate the numerical parameters of the single-phase PWM rectifier stand:

$$A=\left[\begin{array}{cc}-100& 314.15\\ -314.15& -100\end{array}\right]$$

$$B=\left[\begin{array}{cc}200& 0\\ 0& 200\end{array}\right];C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];D=0;E_1=\left[\begin{array}{cc}11& 0\\ 0& 0\end{array}\right],R_2=\left[\begin{array}{l}1\\ 0\end{array}\right]R_1=\left[\begin{array}{c}0.5\\ 0\end{array}\right];E_2=\left[\begin{array}{cc}0& 0\\ 0& 0.5\end{array}\right]$$

We are interested in the effectiveness and dependability of the best filter gain solution through the analysis of fault detection filtering. The value of $\gamma =1$ is adopted in this paper, and the numerical results are computed as follows:

$$Q=\left[\begin{array}{cc}0& 1\\ 0& 1\end{array}\right];\psi =\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right];C_E=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$$

$$Z=\left[\begin{array}{cc}758784493.82& 0.00064\\ 0.00064& 0.00023\end{array}\right]$$

$$Y=\left[\begin{array}{cc}1022& 421.70\\ 421.70& 1996410.30\end{array}\right]$$

$$K=\left[\begin{array}{cc}742508.16& 6.33\times {10}^{-7}\\ -156.84& -1.80\times {10}^{-11}\end{array}\right]$$

$$\lambda =\left[\begin{array}{l}742508.16\\ 1.15\times {10}^{-10}\end{array}\right]$$

where the matrices $Q,\psi$ and $C_E$ are selected to satisfy the requirements described in step 1 and step 2.

3. Experimental Setup

The effectiveness of the proposed fault detection and isolation (FDI) mechanism was assessed using Simulink/MATLAB simulations. To ensure the stability of the system, various strategies for instantaneous current control were investigated.

To further evaluate the proposed FDI mechanism, experiments were conducted on a hardware-in-the-loop platform, as illustrated in Figure 3. The experimental setup comprised two personal computers (PCs), a DSP control board named TMS320F28335, a real-time simulator provided by dSPACE, and an oscilloscope for visualizing signal waveforms. In this configuration, the DSP board received analog signals from the dSPACE simulator, which underwent analog-to-digital conversion and computation processes. Subsequently, the DSP board generated control signals to regulate the switching devices.

The control algorithm for the single-phase pulse width modulation (PWM) rectifier was implemented using the Code Composer Studio and subsequently loaded onto the DSP board. By executing these experiments on the hardware-in-the-loop platform, the effectiveness and validity of the proposed H-infinity filter technique were thoroughly assessed and validated.

4. Results and Discussion

4.1. Simulation Result

In order to achieve a power factor of unity and effectively control the DC-link voltage in a single-phase PWM rectifier, a control strategy was implemented. This strategy utilizes a proportional integrator (PI) controller in the external control loop and a proportional resonant (PR) controller in the inner loop control. The specific details regarding the design of this controller can be found in reference [15].

The role of the PI controller in the external control loop is to regulate the DC-link voltage by adjusting the rectifier’s output voltage. This is accomplished by comparing the actual DC-link voltage to a reference voltage set at 3000 V. The resulting error signal is then inputted into the PI controller, which calculates the necessary adjustment to the output voltage. This adjustment is utilized to modify the pulse width of the switching signals.

Within the inner loop control, a PR controller is employed to regulate the rectifier’s output current. The PR controller is designed to track the reference current signal while simultaneously suppressing any existing harmonics in the system. This is achieved by decomposing the current signal into its fundamental and harmonic components and applying the proportional resonant controller to eliminate the harmonic component.

The effectiveness of the adopted controller in normal operation is depicted in Figure 4. The reference of the DC-link set at $3000\mathrm{V}$ is reached at $t\simeq 0.25\mathrm{s}$, and from this, the system is maintained stable for the rest of the operation time.

Additionally, random signals with evenly distributed magnitudes were introduced as disturbances. Specifically, disturbances with a magnitude of 5 were injected at $t=0.8\mathrm{s}$ to $t=1\mathrm{s}$. The input voltage is significantly affected as it is shown in Figure 5, but it does not affect the detection decision, since it has been effectively decoupled from the residual.

In order to evaluate the performance of the proposed fault detection and isolation methods, an intentional fault was introduced into the system at t = 5 using a signal generator. The specific fault introduced in this experiment was a sensor fault, which aimed to simulate a scenario where the sensor providing measurements for the catenary current was compromised.

To observe the impact of the fault on the estimated catenary current, the experimental results were presented in both Figure 6 and Figure 7. These figures visually illustrate the effect of the fault on the estimated current.

Upon introducing the fault, it was observed that the error in the estimated current increased. This indicates that the fault had a significant influence on the accuracy of the estimated current value. The magnitude of this error was found to be dependent on the direction of the fault, as demonstrated in Figure 7. “Direction of the fault” refers to whether the fault caused an overestimation or underestimation of the catenary current.

Figure 7 provides a clear representation of how the fault affected the estimated current in different fault directions. It showcases the variation in the error magnitude, thereby highlighting the sensitivity of the estimation process to the direction of the introduced sensor fault.

These experimental findings emphasize the importance of the proposed fault detection and isolation methods in identifying and addressing sensor faults in the system. By accurately detecting and isolating faults, the system can mitigate the adverse effects on the estimated catenary current and ensure the reliability and effectiveness of the overall system operation.

The experimental results demonstrate that the system’s error remains close to zero before the fault occurs but rapidly increases once the fault is introduced. This behavior enhances the system’s fault detection capabilities by significantly increasing the sensitivity of the residual to the presence of a fault. The evolution of the residual signal, generated using Equation (19), and the proposed disturbance decoupling technique are depicted in Figure 8. In the absence of a fault, the residual signal remains relatively stable. However, when a fault is present, the residual signal experiences a noticeable increase, providing a clear indication of the fault. This amplified residual signal serves as an effective indicator for fault detection, enabling timely actions for fault isolation and mitigation.

The experimental findings underscore the effectiveness of the proposed disturbance decoupling technique in enhancing fault detection by amplifying the residual signal in the presence of a fault. Incorporating this technique into the fault detection framework improves the system’s reliability and allows for prompt detection and response to faults, ensuring smooth and dependable operation.

The effectiveness of the suggested technique in detecting faults is evident, as the residual signal generated using the proposed filter provides sufficient information on the sensor fault in a reasonable time and is decoupled from the unknown input. This guarantees the achievement of the goal of this study.

4.2. Experiment Results

In order to replicate the simulation in the experimental setup, a fault was intentionally introduced into the electric current sensor signal. The fault had a specific amplitude of 30 and a frequency of 50 Hz, and it was introduced at a predetermined time. This fault simulation aimed to mimic a realistic scenario where a fault occurs in the electric current measurement.

The experimental results provide valuable insights into the behavior of the system under the influence of the fault. Prior to the fault occurrence, the measured and estimated electric current signals are observed to be merged and aligned with each other. This alignment indicates that the designed control strategy effectively maintains consistency between the measured and estimated signals.

Figure 9 visually represents the alignment between the measured and estimated electric current signals before the fault affects the plant. This alignment signifies the accurate estimation and reliable measurement of the electric current, which demonstrates the efficacy of the implemented control strategy.

By introducing the fault and observing the subsequent changes in the measured and estimated signals, the experimental results shed light on the impact of the fault on the system’s behavior. These findings not only validate the effectiveness of the control strategy in normal operation but also emphasize the importance of fault detection and isolation mechanisms to mitigate the effects of faults on the system’s performance.

The experimental results confirm that the designed control strategy effectively maintains alignment between the measured and estimated electric current signals in the absence of faults. This aligns with the objective of reliable estimation and measurement, enhancing the overall performance and robustness of the system.

In the experimental study, it was observed that the introduction of a fault into the plant resulted in a significant discrepancy between the estimated and measured electric current signals. This discrepancy led to a noticeable gap between the two signals, as depicted in Figure 10. Prior to the fault, the estimated and measured signals closely matched each other. However, once the fault occurred, the signals diverged, indicating a deviation from the expected behavior of the system.

The observed gap between the estimated and measured signals serves as a clear indicator of the fault’s impact on the system’s dynamics. Analyzing these differences is crucial for effective fault detection, enabling operators to identify and diagnose faults accurately. By monitoring and understanding the gaps between estimated and measured signals, appropriate actions can be taken for fault isolation and mitigation, ensuring the continued reliable operation of the system.

The difference between the measured and estimated current is utilized to generate an estimated fault (as seen in Figure 11), which aids in making informed maintenance decisions. The filter developed in this study is highly responsive and sensitive to faults, which is crucial for effective fault detection and identification. The estimated fault (f_{_est} in Figure 11) is three times larger than the introduced fault, and the same trend is observed in the simulation results.

This is because the filter uses a factor gain to address the disturbance issue, and the error output is multiplied by this gain’s parameters. Judging from the experimental findings, it is evident that the proposed method is effective in detecting faults, decoupling the disturbance from the fault and increasing residual sensitivity to faults by eliminating the disturbance’s impact.

5. Discussion

The provided results demonstrate the effectiveness of the proposed method in disturbance decoupling and fault detection and isolation in a system. The experiment involved introducing disturbances with a magnitude of 5 at a specific time interval, and the results showed that these disturbances significantly affected the input voltage. However, it was noted that the detection decision was not affected by these disturbances because they were effectively decoupled from the residual signal. This indicates that the proposed method successfully separated the disturbances from the fault-related signals, allowing for accurate fault detection.

Furthermore, an intentional sensor fault was introduced to simulate a realistic scenario in which the measurement of the catenary current was compromised. The experimental results indicated that the fault had a significant impact on the accuracy of the estimated current value. The error in the estimated current increased upon the introduction of the fault, and the magnitude of the error depended on the direction of the fault.

Figure 6 and Figure 7 visually depict the effect of the fault on the estimated current in different fault directions. These figures highlighted the sensitivity of the estimation process to the direction of the introduced sensor fault. By accurately detecting and isolating faults, the proposed fault detection and isolation methods mitigated the adverse effects on the estimated catenary current, ensuring the reliability and effectiveness of the overall system operation.

The evolution of the residual signal, generated using the proposed disturbance decoupling technique, is depicted in Figure 8. In the absence of a fault, the residual signal remained relatively stable. However, in the presence of a fault, the residual signal experienced a noticeable increase, providing a clear indication of the fault. This amplified residual signal served as an effective indicator for fault detection, enabling timely actions for fault isolation and mitigation. The experimental findings underscored the effectiveness of the proposed disturbance decoupling technique in enhancing fault detection by amplifying the residual signal in the presence of a fault. Incorporating this technique into the fault detection framework improved the system’s reliability and allowed for prompt detection and response to faults, ensuring smooth and dependable operation.

The experimental results further validated the effectiveness of the suggested technique in detecting faults. The residual signal generated using the proposed filter provided sufficient information on the sensor fault in a reasonable time and was decoupled from the unknown input. This demonstrated the achievement of the goal of the work, which was to accurately detect and isolate faults while eliminating the impact of disturbances.

The experiment also introduced a fault in the electric current sensor signal to mimic a realistic scenario. Prior to the fault occurrence, the measured and estimated electric current signals were observed to be merged and aligned with each other, indicating the accurate estimation and reliable measurement of the electric current. Figure 9 visually represented this alignment, highlighting the efficacy of the implemented control strategy in maintaining consistency between the measured and estimated signals.

However, upon introducing the fault, the experimental results showed a significant discrepancy between the estimated and measured electric current signals. This discrepancy led to a noticeable gap between the two signals, as depicted in Figure 10. The observed gap served as a clear indicator of the fault’s impact on the system’s dynamics. Analyzing these differences was crucial for effective fault detection and identification, enabling operators to identify and diagnose faults accurately. By monitoring and understanding the gaps between estimated and measured signals, appropriate actions could be taken for fault isolation and mitigation, ensuring the continued reliable operation of the system.

The difference between the measured and estimated current was utilized to generate an estimated fault, as seen in Figure 11. The experimental findings showed that the filter developed in this study was highly responsive and sensitive to faults, which was crucial for effective fault detection and identification. The estimated fault was found to be three times larger than the introduced fault, and this trend was consistent with the simulation results. This behavior was attributed to the factor gain used in the filter to address the disturbance issue, where the error output was multiplied by the residual weighting function.

6. Conclusions

In conclusion, this study has successfully developed a reliable extended H-infinity filter for fault detection in single-phase pulse width modulation (PWM) rectifiers. The filter effectively addresses the challenges posed by disturbances, ensuring accurate fault detection while minimizing false alarms.

The developed filter employs a robust design approach, effectively isolating the influence of disturbances from the residual signal. This isolation eliminates the impact of disturbances on the fault detection process, enhancing the reliability and accuracy of the system.

The proposed method has been applied to a single-phase PWM rectifier and experimentally validated on a hardware-in-the-loop platform. The experimental tests conducted confirmed the effectiveness and efficiency of the proposed approach. The results demonstrated the filter’s ability to accurately detect faults while maintaining a low false alarm rate, highlighting its suitability for practical fault detection applications.

By successfully addressing the challenges associated with disturbances and providing accurate fault detection capabilities, the developed extended H-infinity filter offers a valuable contribution to the field of fault detection in single-phase PWM rectifiers. The research findings pave the way for improved fault detection techniques in similar systems, enhancing their reliability, efficiency, and overall performance in practical applications.

Research Limitation:

The scope of the present study is limited to addressing current sensor faults while disregarding the investigation of DC-link voltage sensor faults. The study focuses on utilizing an extended H-infinity filter to detect faults and mitigate disturbances. Although the proposed method successfully tackles one aspect of sensor faults, a more comprehensive analysis incorporating both current and DC-link voltage sensor faults would provide a deeper understanding of the system’s fault diagnosis capabilities.

Potential Improvement for Future Studies:

For future studies, it is recommended to extend the investigation to encompass both current and DC-link voltage sensor faults. This expansion will enhance the robustness and effectiveness of the proposed method in fault diagnosis. By incorporating fault detection and isolation techniques for both sensors, a more comprehensive and reliable fault diagnosis framework can be established. Additionally, to ensure system resilience in the presence of sensor faults, future studies should explore fault-tolerant control strategies. The integration of fault diagnosis and fault-tolerant control will enable the system to maintain stable operation and mitigate the impact of sensor faults on overall performance. Conducting such studies will contribute to the advancement of more sophisticated and dependable fault diagnosis techniques for practical applications.