# Sensor FDI System Based on Discrete-TimeFractional-Order Observers

Authors: R. F. Escobar-Jiménez; M. Alegría-Zamudio; A. C. Téllez-Anguiano; V. H. Olivares-Peregrino
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

The design of sensor fault detection and isolation (FDI) systems has been studied widely by several authors of different fields of science (engineering, mathematics, physics) to ensure the continuous operation of systems or processes. This research presents the design and evaluation of a fault detection and isolation (FDI) system based on discrete fractional-orderhigh-gain observers (DFOHGO) for the outlet temperature sensors of a heat exchanger. The formulation of the discrete fractional-order observers is based on the Grünwald–Letnikov fractional-order derivative. The FDI system is based on a bank of two DFOHGOs to carry out a residual evaluation. The obtained results showed that the DFOHGOs provide accurate estimations of the temperatures. This allows switching the faulty sensor signal to the temperature estimations provided by the DFOHGOs when a fault occurs.


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## Body

## 1. Introduction

In the control engineering area, the use of observers to deal with the problem of estates or parameter estimation has been widely studied by different researchers using different techniques [1, 2] reporting excellent results. On the other hand, fault detection (FD) and fault detection and isolation (FDI) systems have been studied with different purposes using different techniques [3, 4]. In recent years, fractional calculus has become a useful mathematical tool in different fields of science. In control engineering, fractional calculus has been used widely for systems modeling, observers designing, and the design of controllers. Although it is not a new theory, it has been used to characterize real-world systems without complicating their existing models, as shown in [5–10]. These works are focused on using continuous time-fractional definitions. Nevertheless, there are works where discrete derivatives are used; for example, in [11], the Grünwald–Letnikov fractional-order derivative is used for modeling a heating system using an electric heater to heat a thin copper rod. In [12], the authors presented a new approximation for discrete fractional-order LTI SISO systems using the balanced truncation method for obtaining a simplification of the approximation algorithm giving a faster computation without compromising the approximation of the model. Authors in [13] presented a method for modeling an electric circuit using the Grünwald–Letnikov time-discretefractional-order derivative. This model was compared with experimental data.Furthermore, in [14], the authors applied discrete fractional calculus for modeling a Wiener SISO system, and the model presented low prediction errors and managed to reconstruct the characteristics of the systems using the Laguerre time-discrete approximation. In [15], the authors showed an application of the time-discretefractional-order derivative for designing observers. In this work, the fractional Kalman filter was applied to a discrete fractional-order stochastic system using the Grünwald–Letnikov derivative. The authors carried out a comparison with the traditional fractional Kalman filter. The proposed method works adequately with a slower dynamic. The authors in [16] carried out the nonlinear time-discretefractional-order control systems analysis. The authors used the Grünwald–Letnikov definition and presented stability concepts of fractional-ordertwo-dimensional systems.On the other hand, heat exchangers are thermodynamic systems that allow heat transfer between two heat sources. These systems are used widely in the industry as a part of complex processes such as energy generation, water purification, and petrochemical processes, among others. In recent years, several authors have focused their study on these systems using fractional calculus as a mathematical tool to characterize these devices. The authors in [17] presented the estimation of the outlet temperature of the system by fractional-orderhigh-gain observers using the time-variant definition of Atangana–Baleanu–Caputo. The authors in [18] presented the fractional-order model of a helical heat exchanger using the Grünwald–Letnikov derivate. Meanwhile, the authors in [19] presented a fractional-orderfault-tolerant scheme applied to a fractional heat exchanger model using the Riemann–Liouville fractional derivative for the design of the high-gain observers, also in the control area. The authors in [20] developed the design and implementation of a fractional-order PID controller for a heat exchanger (HE), and the authors used particle swarm optimization (PSO) to find the FOPID controller parameters.From the literature review, we conclude that proposing another alternative based on fractional calculus to estimate the states of a heat exchanger contributes to the engineering field. Therefore, the present research work consists of designing a sensor FDI system based on a bank of two discrete fractional-orderhigh-gain observers (DFOHGO). The bank of observers is developed to estimate the outlet cold and hot water temperatures of a heat exchanger. Observer one only requires the cold-water outlet temperature to estimate both cold and hot water temperatures. Meanwhile, observer two only requires the hot water outlet temperature to estimate both cold and hot outlet water temperatures. The challenge of this work is to design a fault detection and isolation (FDI) system based on discrete fractional-order observers using the Grünwald–Letnikov fractional-order derivative.The manuscript is divided as follows: Section2 presents the mathematical preliminaries, Section 3 presents the system description, and Section 4 presents the FDI scheme based on discrete fractional-orderhigh-gain observers. The results and conclusion are given in Sections 5 and 6, respectively.


## 2. Mathematical Preliminaries

### 2.1. The Fractional-Order Grünwald–Letnikov Derivative

Definition 1.
f is a bounded function in a,b and γ>0. The fractional-order derivative of Grünwald–Letnikov (G-L) of order γ is given by(1)GLDγfx=limh⟶0∑s=0n=x/h−1sγsfx−shhγ.
A generalized definition of the factorial has been used through the gamma function to use the previous expression with rational numbers:(2)s!=Γs+1,∀s∈C.
A discrete functionfx defined on an interval a,b is represented by the vector y1,y2,…,yn=fx1,fx2,…,fxn formed by the images of the function at the partition points of the considered interval x1,x2,…,.Definition 2.
Letγ be any real number. The fractional-orderγ Grünwald–Letnikov type difference of xt is given by(3)aΔγxt=∑s=0t−γ−1sγsxt−s,where t∈Na and(4)γs=Γγ+1Γs+1Γγ−s+1,is the binomial coefficient.
According to [21], the binomial term γs is defined as(5)γs=1s=0γγ−1…γ−s+1s!s>1…γs=1s=0Γγ+1Γs+1Γγ−s+1s>1…


## 2.1. The Fractional-Order Grünwald–Letnikov Derivative

Definition 1.
f is a bounded function in a,b and γ>0. The fractional-order derivative of Grünwald–Letnikov (G-L) of order γ is given by(1)GLDγfx=limh⟶0∑s=0n=x/h−1sγsfx−shhγ.
A generalized definition of the factorial has been used through the gamma function to use the previous expression with rational numbers:(2)s!=Γs+1,∀s∈C.
A discrete functionfx defined on an interval a,b is represented by the vector y1,y2,…,yn=fx1,fx2,…,fxn formed by the images of the function at the partition points of the considered interval x1,x2,…,.Definition 2.
Letγ be any real number. The fractional-orderγ Grünwald–Letnikov type difference of xt is given by(3)aΔγxt=∑s=0t−γ−1sγsxt−s,where t∈Na and(4)γs=Γγ+1Γs+1Γγ−s+1,is the binomial coefficient.
According to [21], the binomial term γs is defined as(5)γs=1s=0γγ−1…γ−s+1s!s>1…γs=1s=0Γγ+1Γs+1Γγ−s+1s>1…


## 3. System Description

Experimental data from a heat exchanger were used to demonstrate the effectiveness of the fractional discrete observers’ design. Next, a brief description of the plant and its instrumentation is given. The heat exchanger’s function is to cool water. Figure1 shows the heat exchanger benchmark plant, its configuration is a countercurrent double pipe, where hot water flows in the inner pipe, and cold water flows by the annular side. The heat exchanger instrumentation consists of two analog flow meters, two electro-pneumatic control valves, and four temperature sensors. The flow meters allow us to visualize the flow of hot and cold water inside the heat exchanger. The control valves regulate the water flow, actuator 1 regulates the hot water flow, and actuator 2 regulates the flow of the cold water. And the temperature sensors measure the inlet and outlet temperatures of hot and cold water.Figure 1
Heat exchanger benchmark.


### 3.1. System Modeling

Figure2 shows a schematic representation of the heat exchanger. Where Tci and Tco represent the inlet and outlet cold water temperatures, respectively Thi and Tho represent the inlet and outlet hot water temperatures, respectively. The temperature difference between both fluids is represented by ∆T=Tcot−Thot. Meanwhile, U represents the overall heat transfer coefficient.Figure 2
Heat exchanger configuration scheme.The mathematical model is obtained by an energy balance. The equations of the energy balance are as follows:(6)T˙cot=ωvcVcTcit−Tcot+U1P,T,ωvcAiCpcρcVcThot−Tcot,T˙hot=ωvhVhThit−Thot+U2P,T,ωvhAoCphρhVhTcot−Thot,where U1 and U2 are the heat transfer coefficient of cold and hot water, respectively, ω is the volumetric flow, A is the heat transfer area, V is the volume, and Cp and ρ are the heat capacity and the density (physical properties of water), respectively. The subscripts c and h refer to cold and hot water, respectively. Note that the heat transfer coefficients depend on the temperatures of the fluids (T), flow rates ωv, and pressure (P) in the heat exchanger inner pipe.


## 3.1. System Modeling

Figure2 shows a schematic representation of the heat exchanger. Where Tci and Tco represent the inlet and outlet cold water temperatures, respectively Thi and Tho represent the inlet and outlet hot water temperatures, respectively. The temperature difference between both fluids is represented by ∆T=Tcot−Thot. Meanwhile, U represents the overall heat transfer coefficient.Figure 2
Heat exchanger configuration scheme.The mathematical model is obtained by an energy balance. The equations of the energy balance are as follows:(6)T˙cot=ωvcVcTcit−Tcot+U1P,T,ωvcAiCpcρcVcThot−Tcot,T˙hot=ωvhVhThit−Thot+U2P,T,ωvhAoCphρhVhTcot−Thot,where U1 and U2 are the heat transfer coefficient of cold and hot water, respectively, ω is the volumetric flow, A is the heat transfer area, V is the volume, and Cp and ρ are the heat capacity and the density (physical properties of water), respectively. The subscripts c and h refer to cold and hot water, respectively. Note that the heat transfer coefficients depend on the temperatures of the fluids (T), flow rates ωv, and pressure (P) in the heat exchanger inner pipe.


## 4. FDI Scheme Based on Discrete Fractional-OrderHigh-Gain Observers

### 4.1. FDI Scheme

High-gain observers have been widely studied by several authors to perform the analytical redundancy of a signal in FDI systems. In some cases, the estimated value provided by the observer is used to replace the faulty measured signal in the FDI system. As shown in Figure3, a bank of two observers is required to perform the FDI system in this work. This figure shows that observer 1 and observer 2 use the two inlet water temperatures (Tci, Thi). However, observer uses the outlet cold-water temperature Tco, whilst observer 2 uses the outlet hot water temperature Tho to estimate both states, T^co and T^ho. In the next section, we will show the observers’ equations.Figure 3
Bank of discrete fractional-orderhigh-gain observers (DFOHGO).


### 4.2. Discrete Fractional-OrderHigh-Gain Observers Design

The model of the system is represented in a control-affine nonlinear system to design the observers, as shown in the following equation:(7)x˙t=fxt+∑i=1mgixtuit,yt=hxt,where xt∈Rn, fxt∈Rn, gixt∈Rnuit∈R, i=1,…,m, where m is the number of inputs, yt∈R. If the system is observable, the following coordinate transformation zt=Φxt is considered:(8)Φxt=hxtLfhxt…Lfn−1hxtT,where Lf⋅ represents the Lie derivative of the function h(x(t)) evaluated along f(x(t)), where(9)Lfhxt=∑i=1n∂hxt∂xifixt.The diffeomorphism given by the coordinate transformationΦxt transforms the system in the following form:(10)z˙t=Az+ψzt+∑i=1mϕiztuit,yt=Czt,where ϕxt=htLfhxt…Lfn−1hxtT. The authors in [22] presented the transformation that allows returning to the original coordinates xt=Φ−1zt. As shown in [22], the exponential observer is given by(11)z^˙=Az^t+ψz^t+∑i=1mϕiztuit−sθ−1CTCz^t−yt,where Sθ is a constant matrix of n×n, that is, the solution of the Lyapunov equation(12)θSθ+AθTS+SθA=CTC,where the tuning parameter of the high-gain observer is θ>0, where the Sθ matrix for a second-order system is given by(13)sθ1θ−1θ2−1θ22θ3.Using the system given in equation (11) in the original coordinates, [17, 19, 22] and applying the definition showed in equation (3), the DFOHGOs of the Grünwald–Letnikov type are given in the following equation:(14)Δγx^t−s+1=fx^t+∑i=1mgix^tuit−∂Φx^t∂x^−1Sθ−1CTy^t−yt……−∑s=1t+1−1sγsxt−s+1Δγy^t=Cx^t,where ∂Φx^t/∂x^ is the n×n Jacobian matrix of Φxt, and Φx^t=Φxt|xt=x^t, and γ is the fractional-order difference.


### 4.3. Bank of Discrete Fractional-OrderHigh-Gain Observers

Considering the FDI scheme based on the DFOHGO presented in Figure3 and applying equations (3) and (14), the design of the DFOHGO 1 is carried out using the measured outlet temperature Tco for estimating both state variables, as shown in the following equation:(15)0Δγ1T^co1t+10Δγ2T^ho1t+1=U1A1ρcCpcVcU2A2ρhCphVhT^ho1t−T^co1t+1VcTci−T^co1t0ωVc+……+0+1vhThi−T^ho1tωVh+2θ2θ+θ2gcTcot−T^co1t−∑s=1t+1−1sγ1sT^co1t−s+1∑s=1t+1−1sγ2sT^ho1t−s+1,where gc=U1A1/ρpCpcVc. Meanwhile, DFOHGO 2 equation (16) uses the measured outlet temperature Tho for estimating both state variables, as follows:(16)0Δγ3T^co2t+10Δγ4T^ho2t+1=U1A1ρcCpcVcU2A2ρhCphVhT^ho2t−T^co2t+1VcTci−T^co2t0ωVc+……+01vhThi−T^ho2tωVh+2θ+θ2gh2θThot−T^ho2t−∑s=1t+1−1sγ3sT^co2t−s+1∑s=1t+1−1sγ4sT^ho2t−s+1,where gh=U2A2/ρhCphVh. Both equations (15) and (16) compound the bank of observers proposed in the FDI scheme.


## 4.1. FDI Scheme

High-gain observers have been widely studied by several authors to perform the analytical redundancy of a signal in FDI systems. In some cases, the estimated value provided by the observer is used to replace the faulty measured signal in the FDI system. As shown in Figure3, a bank of two observers is required to perform the FDI system in this work. This figure shows that observer 1 and observer 2 use the two inlet water temperatures (Tci, Thi). However, observer uses the outlet cold-water temperature Tco, whilst observer 2 uses the outlet hot water temperature Tho to estimate both states, T^co and T^ho. In the next section, we will show the observers’ equations.Figure 3
Bank of discrete fractional-orderhigh-gain observers (DFOHGO).


## 4.2. Discrete Fractional-OrderHigh-Gain Observers Design

The model of the system is represented in a control-affine nonlinear system to design the observers, as shown in the following equation:(7)x˙t=fxt+∑i=1mgixtuit,yt=hxt,where xt∈Rn, fxt∈Rn, gixt∈Rnuit∈R, i=1,…,m, where m is the number of inputs, yt∈R. If the system is observable, the following coordinate transformation zt=Φxt is considered:(8)Φxt=hxtLfhxt…Lfn−1hxtT,where Lf⋅ represents the Lie derivative of the function h(x(t)) evaluated along f(x(t)), where(9)Lfhxt=∑i=1n∂hxt∂xifixt.The diffeomorphism given by the coordinate transformationΦxt transforms the system in the following form:(10)z˙t=Az+ψzt+∑i=1mϕiztuit,yt=Czt,where ϕxt=htLfhxt…Lfn−1hxtT. The authors in [22] presented the transformation that allows returning to the original coordinates xt=Φ−1zt. As shown in [22], the exponential observer is given by(11)z^˙=Az^t+ψz^t+∑i=1mϕiztuit−sθ−1CTCz^t−yt,where Sθ is a constant matrix of n×n, that is, the solution of the Lyapunov equation(12)θSθ+AθTS+SθA=CTC,where the tuning parameter of the high-gain observer is θ>0, where the Sθ matrix for a second-order system is given by(13)sθ1θ−1θ2−1θ22θ3.Using the system given in equation (11) in the original coordinates, [17, 19, 22] and applying the definition showed in equation (3), the DFOHGOs of the Grünwald–Letnikov type are given in the following equation:(14)Δγx^t−s+1=fx^t+∑i=1mgix^tuit−∂Φx^t∂x^−1Sθ−1CTy^t−yt……−∑s=1t+1−1sγsxt−s+1Δγy^t=Cx^t,where ∂Φx^t/∂x^ is the n×n Jacobian matrix of Φxt, and Φx^t=Φxt|xt=x^t, and γ is the fractional-order difference.


## 4.3. Bank of Discrete Fractional-OrderHigh-Gain Observers

Considering the FDI scheme based on the DFOHGO presented in Figure3 and applying equations (3) and (14), the design of the DFOHGO 1 is carried out using the measured outlet temperature Tco for estimating both state variables, as shown in the following equation:(15)0Δγ1T^co1t+10Δγ2T^ho1t+1=U1A1ρcCpcVcU2A2ρhCphVhT^ho1t−T^co1t+1VcTci−T^co1t0ωVc+……+0+1vhThi−T^ho1tωVh+2θ2θ+θ2gcTcot−T^co1t−∑s=1t+1−1sγ1sT^co1t−s+1∑s=1t+1−1sγ2sT^ho1t−s+1,where gc=U1A1/ρpCpcVc. Meanwhile, DFOHGO 2 equation (16) uses the measured outlet temperature Tho for estimating both state variables, as follows:(16)0Δγ3T^co2t+10Δγ4T^ho2t+1=U1A1ρcCpcVcU2A2ρhCphVhT^ho2t−T^co2t+1VcTci−T^co2t0ωVc+……+01vhThi−T^ho2tωVh+2θ+θ2gh2θThot−T^ho2t−∑s=1t+1−1sγ3sT^co2t−s+1∑s=1t+1−1sγ4sT^ho2t−s+1,where gh=U2A2/ρhCphVh. Both equations (15) and (16) compound the bank of observers proposed in the FDI scheme.


## 5. Results

### 5.1. DFOHGOs Validation

Once the observers have been developed (DFOHGO 1 equation (15) and DFOHGO 2 equation (16)), the next step is to test their accuracy and performance. Then, experimental values of the heat exchanger are compared with the observers’ estimation, which is shown as follows.The DFOHGO validation is performed under different operating conditions, the cold water inlet flow varies betweenWc,in=1.0875×10−5m3/s and Wc,in=1.1337×10−5m3/s, and the hot water inlet flow is operated constant Wh,in=1.5×10−7m3/s. The pressure in the internal and external pipes remained at atmospheric conditions. The heat exchanger’s hot water inlet temperatures varied between 80.21°C and 81.13°C. Finally, the cooling water was 30°C.Figure4 shows the validation of observer 1. In the figure, the experimental and estimated temperatures are shown in the continuous green line and dashed blue line, respectively. At the upper side of the figure are shown the experimental and estimated cold water temperatures and at the bottom are shown the experimental and calculated hot water temperatures. As can be seen, both temperatures were estimated correctly during the test.Figure 4
DFOHGO validation with experimental heat exchanger data.Table1 presents the quantitative comparison between the experimental and the estimated (DFOHGO 1) temperatures. The error analysis was performed using the mean squared error (MSE). The results showed that the MSE calculated with the cold water temperatures is smaller than the calculated with the hot water temperatures. This is because the observer DFOHGO 1 depends on the measured cold-water temperature Tco.Table 1
MSE between experimental and estimated temperatures using DFOHGO 1.
DFOHGO 1 MSEMSE ofTcoMSE ofTho5.1894×10−040.0039Figure5 shows the validation results of observer 2 (DFOHGO 2) for estimating the cold and hot water temperatures. In the figure, the measured and estimated temperatures are shown in the continuous green line and dashed blue line, respectively. At the upper side of the figure are shown the measured and estimated cold water temperatures, and at the bottom are shown the measured and calculated hot water temperatures. As can be seen, both temperatures were estimated correctly during the test.Figure 5
High-gainfractional-order observers.Table2 presents a quantitative comparison between the experimental and simulated temperatures performed with the DFOHGO 2, and the comparative analysis is performed by using the Mean Squared Error (MSE). The results showed that the MSE between the measured and estimated hot water temperatures is smaller than the MSE of the cold water temperatures. This is because the observer (DFOHGO 2) depends on the measured hot water temperature Tho.Table 2
MSE between experimental and estimated temperatures using DFOHGO 2.
DFOHGO 2 MSEMSE ofTcoMSE ofTho0.02410.0017


### 5.2. FDI System-Based DFOHGOs

Figure6 shows the bank of fractional observers. As shown in the figure, the DFOHGO 1 uses inlet cold and hot water temperatures, and only one of the outlet temperatures Tco to estimate the outlet cold and hot water temperatures T^co1 and T^ho1. In such a way that the outlet hot water temperature Tho always is available even if the physical temperature sensor fails.Figure 6
Fault detection and isolation scheme.On the other hand, the DFOHGO 2 uses inlet cold and hot water temperatures, and only the outlet hot water temperaturesTho to estimate the outlet cold and hot water temperatures T^co2 and T^ho2.The sensor fault detection and isolation depend on the calculus of the residual (equation (17)) and its comparison with a threshold ε. The threshold value depends on the sensor’s accuracy. In this case, the temperature sensor accuracy is ±0.5% − 1°C. In the FDI stage, there will be two states for each sensor: (1) if rT⩽ε, there is no fault in the sensor, and the system keeps operating normally. (2) if rT>ε, there is a fault detected in the sensor. Then, the faulty sensor signal will be switched to the observer estimation.The residuals are given as follows:(17)r1Tho=Thot−T^ho1,r2Tco=Tcot−T^co2.According to equation (17), a fault matrix can be defined by Table 3, which allows to locate and identify faults in the temperature sensors.Table 3
Matrix fault.
Faultr1Thor2TcoTco01Tho10The FDI system works as follows, and ifr1Tco>ε, this indicates a fault in the temperature sensor Tco. Therefore, the system switches to the signal estimated by DFOHGO 2.Ifr2Tho>ε, this indicates a fault in the temperature sensor Tho. Then, the system switches to the signal estimated by DFOHGO 1. However, if r1Tco⩽ε and r2Tho⩽ε, means that there are no faults in the sensors.Figures7 and 8 show the results of the FDI system performance. First, the results of the FDI on the Tho sensor are presented in Figure 7. Second, the results of the FDI on the Tco sensor are presented in Figure 8.Figure 7
Fault in the temperature sensorTho.Figure 8
Fault in the temperature sensorTco.Figure7 shows the performance of observer 1 for estimating the hot water temperature when a total or partial fault occurs in the hot water temperature sensor Tho. First, in sample time 2500, a total fault is induced in the Tho sensor to demonstrate the capability of observer 1 to estimate the hot water temperature. In the middle of Figure 7 are shown the temperature measured signal (continuous green line) and temperature estimated by the observer (dashed blue line). As shown in the figure, the measured signal is lost when the sensor fault occurs, and the temperature estimated by the observer continues estimating the hot water temperature despite the sensor fault. The sensor fault is detected by the residual analysis shown at the bottom of the figure.On the other hand, a partial fault is induced in sample 4400 using white noise. As can be appreciated in Figure7, the hot water measured signal (continuous green line) varies between ±6°C, which is consistent with a partial fault, according to the residual evaluation shown at the bottom of the figure. Once the signal fault is detected, the physical sensor signal is switched by the estimated temperature avoiding stopping the heat exchanger device. In conclusion, the DFOHGO 1 estimates the heat exchanger hot water temperature accurately even if the Tho sensor fails.Figure8 shows the performance of observer 2 (DFOHGO 2) to total and partial faults in the cold water temperature sensor Tco. As shown in Figure 8, in sample 1900, a total fault is induced in the sensor. As shown in the bottom of the figure, in sample 1900, the residual r2Tco overpasses the established threshold, detecting the failure existence. The figure shows that the measured signal (continuous green line) decays due to the sensor failure, switching the faulty signal by the temperature estimated by the observer 2.On the other hand, in sample 3500, a partial fault is induced in the sensorTco. This fault was simulated via software using white noise with oscillations between ±5°C. At the bottom of Figure 8 is shown the instant when the partial fault occurs at sample 3500. At the top of the figure is shown that observer 2 (dashed blue line) estimates the temperature T^co2 accurately, allowing to switch the faulty signal Tco) by the observer estimation T^co2) to keep the heat exchanger operating.


## 5.1. DFOHGOs Validation

Once the observers have been developed (DFOHGO 1 equation (15) and DFOHGO 2 equation (16)), the next step is to test their accuracy and performance. Then, experimental values of the heat exchanger are compared with the observers’ estimation, which is shown as follows.The DFOHGO validation is performed under different operating conditions, the cold water inlet flow varies betweenWc,in=1.0875×10−5m3/s and Wc,in=1.1337×10−5m3/s, and the hot water inlet flow is operated constant Wh,in=1.5×10−7m3/s. The pressure in the internal and external pipes remained at atmospheric conditions. The heat exchanger’s hot water inlet temperatures varied between 80.21°C and 81.13°C. Finally, the cooling water was 30°C.Figure4 shows the validation of observer 1. In the figure, the experimental and estimated temperatures are shown in the continuous green line and dashed blue line, respectively. At the upper side of the figure are shown the experimental and estimated cold water temperatures and at the bottom are shown the experimental and calculated hot water temperatures. As can be seen, both temperatures were estimated correctly during the test.Figure 4
DFOHGO validation with experimental heat exchanger data.Table1 presents the quantitative comparison between the experimental and the estimated (DFOHGO 1) temperatures. The error analysis was performed using the mean squared error (MSE). The results showed that the MSE calculated with the cold water temperatures is smaller than the calculated with the hot water temperatures. This is because the observer DFOHGO 1 depends on the measured cold-water temperature Tco.Table 1
MSE between experimental and estimated temperatures using DFOHGO 1.
DFOHGO 1 MSEMSE ofTcoMSE ofTho5.1894×10−040.0039Figure5 shows the validation results of observer 2 (DFOHGO 2) for estimating the cold and hot water temperatures. In the figure, the measured and estimated temperatures are shown in the continuous green line and dashed blue line, respectively. At the upper side of the figure are shown the measured and estimated cold water temperatures, and at the bottom are shown the measured and calculated hot water temperatures. As can be seen, both temperatures were estimated correctly during the test.Figure 5
High-gainfractional-order observers.Table2 presents a quantitative comparison between the experimental and simulated temperatures performed with the DFOHGO 2, and the comparative analysis is performed by using the Mean Squared Error (MSE). The results showed that the MSE between the measured and estimated hot water temperatures is smaller than the MSE of the cold water temperatures. This is because the observer (DFOHGO 2) depends on the measured hot water temperature Tho.Table 2
MSE between experimental and estimated temperatures using DFOHGO 2.
DFOHGO 2 MSEMSE ofTcoMSE ofTho0.02410.0017


## 5.2. FDI System-Based DFOHGOs

Figure6 shows the bank of fractional observers. As shown in the figure, the DFOHGO 1 uses inlet cold and hot water temperatures, and only one of the outlet temperatures Tco to estimate the outlet cold and hot water temperatures T^co1 and T^ho1. In such a way that the outlet hot water temperature Tho always is available even if the physical temperature sensor fails.Figure 6
Fault detection and isolation scheme.On the other hand, the DFOHGO 2 uses inlet cold and hot water temperatures, and only the outlet hot water temperaturesTho to estimate the outlet cold and hot water temperatures T^co2 and T^ho2.The sensor fault detection and isolation depend on the calculus of the residual (equation (17)) and its comparison with a threshold ε. The threshold value depends on the sensor’s accuracy. In this case, the temperature sensor accuracy is ±0.5% − 1°C. In the FDI stage, there will be two states for each sensor: (1) if rT⩽ε, there is no fault in the sensor, and the system keeps operating normally. (2) if rT>ε, there is a fault detected in the sensor. Then, the faulty sensor signal will be switched to the observer estimation.The residuals are given as follows:(17)r1Tho=Thot−T^ho1,r2Tco=Tcot−T^co2.According to equation (17), a fault matrix can be defined by Table 3, which allows to locate and identify faults in the temperature sensors.Table 3
Matrix fault.
Faultr1Thor2TcoTco01Tho10The FDI system works as follows, and ifr1Tco>ε, this indicates a fault in the temperature sensor Tco. Therefore, the system switches to the signal estimated by DFOHGO 2.Ifr2Tho>ε, this indicates a fault in the temperature sensor Tho. Then, the system switches to the signal estimated by DFOHGO 1. However, if r1Tco⩽ε and r2Tho⩽ε, means that there are no faults in the sensors.Figures7 and 8 show the results of the FDI system performance. First, the results of the FDI on the Tho sensor are presented in Figure 7. Second, the results of the FDI on the Tco sensor are presented in Figure 8.Figure 7
Fault in the temperature sensorTho.Figure 8
Fault in the temperature sensorTco.Figure7 shows the performance of observer 1 for estimating the hot water temperature when a total or partial fault occurs in the hot water temperature sensor Tho. First, in sample time 2500, a total fault is induced in the Tho sensor to demonstrate the capability of observer 1 to estimate the hot water temperature. In the middle of Figure 7 are shown the temperature measured signal (continuous green line) and temperature estimated by the observer (dashed blue line). As shown in the figure, the measured signal is lost when the sensor fault occurs, and the temperature estimated by the observer continues estimating the hot water temperature despite the sensor fault. The sensor fault is detected by the residual analysis shown at the bottom of the figure.On the other hand, a partial fault is induced in sample 4400 using white noise. As can be appreciated in Figure7, the hot water measured signal (continuous green line) varies between ±6°C, which is consistent with a partial fault, according to the residual evaluation shown at the bottom of the figure. Once the signal fault is detected, the physical sensor signal is switched by the estimated temperature avoiding stopping the heat exchanger device. In conclusion, the DFOHGO 1 estimates the heat exchanger hot water temperature accurately even if the Tho sensor fails.Figure8 shows the performance of observer 2 (DFOHGO 2) to total and partial faults in the cold water temperature sensor Tco. As shown in Figure 8, in sample 1900, a total fault is induced in the sensor. As shown in the bottom of the figure, in sample 1900, the residual r2Tco overpasses the established threshold, detecting the failure existence. The figure shows that the measured signal (continuous green line) decays due to the sensor failure, switching the faulty signal by the temperature estimated by the observer 2.On the other hand, in sample 3500, a partial fault is induced in the sensorTco. This fault was simulated via software using white noise with oscillations between ±5°C. At the bottom of Figure 8 is shown the instant when the partial fault occurs at sample 3500. At the top of the figure is shown that observer 2 (dashed blue line) estimates the temperature T^co2 accurately, allowing to switch the faulty signal Tco) by the observer estimation T^co2) to keep the heat exchanger operating.


## 6. Conclusion

This manuscript presented the application of an FDI system based on discrete fractional-order observers to detect and isolate temperature sensor faults of a heat exchanger. Comparisons between experimental data of the heat exchanger and the observers’ estimations were performed and evaluated to validate the DFOHGOs. The results showed the capability of the system to detect and isolate total and partial sensor faults using the DFOHGOs. The calculated mean square errors for observer 1 were5.1894×10−4 and 0.0039 for the cold and hot water temperatures, respectively. Meanwhile, using observer 2, the calculated errors were 0.0241 and 0.0017 for the cold and hot water temperatures, respectively.



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*Source: 9984828-2023-02-17.xml*