Detection, Isolation, and Identification of Multiplicative Faults in a DC Motor and Amplifier Using Parameter Estimation Techniques

Featured Application

The developed laboratory platform and associated algorithms can be applied in educational and research settings for real-time simulation, detection, and isolation of multiplicative faults in electromechanical systems. Its remote-access capability allows engineering students and researchers to conduct fault diagnosis experiments on DC motor systems without physical presence, supporting distance learning and development of robust FDII methods for safety-critical applications.

Abstract

The increasing complexity of modern control systems highlights the need for reliable and robust fault detection, isolation, and identification (FDII) methods, particularly in safety-critical and industrial applications. The study focuses on the FDII of multiplicative faults in a DC motor and its electronic amplifier. To simulate such scenarios, a complete laboratory platform was developed for real-time FDII, using relay-based switching and custom LabVIEW software 2009. This platform enables real-time experimentation and represents an important component of the study. Two estimation-based fault detection (FD) algorithms were implemented: the Sliding Window Algorithm (SWA) for discrete-time models and a modified Sliding Integral Algorithm (SIA) for continuous-time models. The modification introduced to the SIA limits the data length used in least squares estimation, thereby reducing the impact of transient effects on parameter accuracy. Both algorithms achieved high model output-to-measured signal agreement, up to 98.6% under nominal conditions and above 95% during almost all fault scenarios. Moreover, the proposed fault isolation and identification methods, including a decision algorithm and an indirect estimation approach, successfully isolated and identified faults in key components such as amplifier resistors (R₁, R₉, R₁₂), capacitor (C₈), and motor parameters, including armature resistance (R_a), inertia (J), and friction coefficient (B). The decision algorithm, based on continuous-time model coefficients, demonstrated reliable fault isolation and identification, while the reduced Jacobian-based approach in the discrete model enhanced fault magnitude estimation, with deviations typically below 10%. Additionally, the platform supports remote experimentation, offering a valuable resource for advancing model-based FDII research and engineering education.

1. Introduction

As control systems continue to grow in complexity, ensuring their safety and robustness becomes increasingly critical across a wide range of industrial processes. The reliability, robustness, availability, and environmental safety of such systems are often evaluated based on their sensitivity to faults, which may occur randomly during operation. This growing need has resulted in a distinct area within engineering theory and practice–fault detection and isolation (FDI) [1,2,3,4,5].

From a practical engineering standpoint, faults are commonly categorized into additive and multiplicative types. Additive faults are unexpected deterministic inputs (e.g., sudden load change or sensor biases) that affect the system output without altering its internal structure. These faults, which include actuator jamming or sensor drift, are typically absent under normal conditions but produce clear effects on outputs once present [6,7,8,9,10,11,12]. Multiplicative faults, by contrast, involve changes in system parameters, such as resistance, inductance, inertia, or friction coefficients. These faults may appear gradually (due to aging or wear) or abruptly (as in structural damage) and are more challenging to detect due to their parameter-driven, indirect, and often masked effects on system output [13,14,15,16,17].

In DC motors, multiplicative faults include misalignment of brushes, wear of the commutator, armature winding faults, variations in armature resistance, and changes in mechanical friction. They may also occur in the accompanying amplifier—making accurate real-time diagnosis a challenging but essential task [18,19,20,21].

Various FDI methods have been developed to address these issues, which can be classified into four broad categories:

• Signal-based techniques rely on analyzing vibration or current signals, often using spectral, statistical, or wavelet-based methods [19,22,23,24,25]. While model-free and simple to implement, such methods are generally limited to offline analysis, and their performance deteriorates in low-power or noisy systems. They are also less suitable for detecting multiplicative faults, which often do not generate prominent signal features.
• Model-based techniques exploit the system’s dynamic model and include:
  • Parity equation methods, which compare measured and predicted outputs to generate residuals sensitive to specific faults [3,10,11,26]. These methods are simple to implement and effective for additive fault isolation, but they perform poorly for multiplicative faults due to their reliance on accurate models and sensitivity to noise and parameter uncertainties. Moreover, they typically lack fault identification capability, as they do not quantify fault magnitude.
  • Observer and filter-based methods (e.g., Luenberger observers, Kalman filters, sliding-mode observers) that estimate system states and detect deviations due to faults [7,14,27,28]. Such approaches provide good dynamic tracking and are suitable for real-time use, but they often require precise model tuning and can become unstable or inaccurate in the presence of model uncertainty or high noise.
  • Parameter estimation methods (e.g., Least Squares (LS), Recursive Least Squares (RLS), Extended Kalman Filters (EKF), Unscented Kalman Filters (UKF), Algebraic-Geometric Technique (AGT) [18,26,29,30,31,32,33,34,35,36,37,38].
• Knowledge and data-driven approaches such as fuzzy logic, expert systems, and neural networks can learn or encode relationships between faults and observable features [6,12,18,39,40]. While effective for nonlinear and complex systems, these methods require large training datasets and may lack interpretability.
• Optimization-based techniques such as Genetic Algorithms (GA), Particle Swarm Optimization (PSO), and Whale Optimization Algorithm (WOA)) that track deviations in estimated parameters, particularly relevant for multiplicative faults [41,42].

Among model-based approaches, parameter estimation techniques offer significant potential for detecting multiplicative faults, as they enable direct monitoring of physical changes within the system. Compared to traditional parameter estimation-based fault detection methods, the advantages of the proposed Sliding Window Algorithm (SWA) [43] and Sliding Integral Algorithm (SIA) [44] become more apparent.

Classical parameter estimation techniques such as LS [18,32,38] and RLS [30,31] are widely used due to their simplicity and low computational requirements. LS is primarily an offline method, as it processes a fixed batch of data and assumes time-invariant parameters during estimation. In contrast, RLS is designed for real-time application, updating estimates recursively as new data become available and using a forgetting factor to adapt to changing system dynamics. However, despite its real-time capability, RLS often struggles with transient disturbances and abrupt parameter changes caused by faults, especially when the forgetting factor is not optimally tuned. These limitations can lead to delayed or inaccurate fault detection in systems subject to abrupt changes [26,30].

Advanced methods such as EKF [29,34,35] and UKF [36,37] offer improved performance in noisy and nonlinear environments. However, their reliance on accurate system models and careful tuning of noise covariances increases implementation complexity and computational burden, limiting their suitability for embedded or real-time applications.

Similarly, algebraic techniques such as the AGT offer analytical insight and enable fault localization. However, they are generally unsuitable for real-time implementation due to their dependence on signal derivatives, symbolic computations, and high sensitivity to measurement noise [33].

Optimization-based techniques such as GA, PSO, and WOA offer global optimization capabilities for parameter estimation, but their high computational complexity makes them impractical for real-time applications [41,42].

In contrast, SWA and SIA offer robust and low-complexity alternatives that preserve interpretability and can be scaled to more complex systems with relative ease. SWA, applied to discrete-time models with short data windows, enhances responsiveness to abrupt faults, while SIA, using continuous-time integral formulations, reduces noise and transient sensitivity—making it suitable for gradual parameter changes. Both methods are non-intrusive, require minimal tuning, and are computationally efficient, making them ideal for embedded real-time applications. Furthermore, by linking estimated model parameters (e.g., a₁, b₀, b₁) to physical quantities (e.g., resistance, inertia), they enable not only fault detection and isolation but also identification of fault severity.

This paper presents a real-time, parameter-estimation-based framework for detection, isolation, and identification (FDII) of multiplicative faults in a DC motor and its amplifier. The system is implemented on a hardware-in-the-loop experimental platform using LabVIEW 2009 and NI CompactRIO CRIO-9075 (National Instruments, Austin, TX, USA), enabling remote access, real-time fault injection, and online testing.

The key contributions include:

• Development of an interactive laboratory platform for real-time simulation and testing of FDII algorithms, specifically designed to support both practical implementation and engineering education. This is achieved through a hardware-in-the-loop, remotely accessible setup built using LabVIEW and NI CompactRIO, enabling real-time fault injection, detection, isolation, and identification in both the DC motor and its electronic amplifier—representing a significant advancement in applied FDI research and training environments.
• Comprehensive Evaluation and Comparative Analysis of two complementary parameter estimation methods under identical experimental conditions: a modified Sliding Integral Algorithm (SIA) for continuous-time system modeling, which limits the data window to reduce the influence of transients and improves robustness of parameter estimation; and a Sliding Window Algorithm (SWA) for discrete-time modeling, which enhances responsiveness to abrupt parameter changes. This enables a comparative analysis of their performance and limitations, which is rarely addressed in the existing literature [19,20,21,33].
• A modification of the SIA by limiting the data window, which reduces the influence of transients and improves the robustness of parameter estimation.
• Demonstration of two complementary fault identification approaches that enable quantification of the change in affected physical parameters—a capability rarely addressed in the existing literature [13,14,16,18,19]. In addition to detecting and isolating faults, the proposed methods also estimate fault magnitude, which significantly enhances diagnostic resolution. This is achieved through:
  • A novel decision algorithm, based on the model obtained using SIA, which maps deviations in estimated model parameters to specific physical components;
  • An indirect estimation method, applied in the SWA framework, supported by a selective Jacobian matrix approach that improves the accuracy of physical parameter identification and strengthens the overall diagnostic capability of the system, especially when the relation between model and physical parameters is nonlinear or non-negligible. This overcomes limitations of prior indirect approaches that typically rely on the full Jacobian matrix [3,7].

In summary, this work presents a practical and robust methodology for detecting and identifying multiplicative faults using simple yet effective parameter estimation techniques. Its real-time implementation, experimental validation, and educational applicability make it a valuable contribution to both academic research and industrial practice.

2. Materials and Methods

Figure 1 presents the full experimental setup of the measurement equipment, comprising a motor, electronic amplifier, relays, and the CRIO-9075 controller.

2.1. Laboratory Setup

2.1.1. DC Motor

For the planned experiments, a DC motor with permanent magnets A-max 26 A Ø26 mm Precious Metal Brushes CLL (Maxon Motor AG, Sachseln, Switzerland), 4.5 W was used. Figure 2a shows a DC micro-motor (1) equipped with an MR-128-1000 encoder (2), offering a resolution of 1000 impulses per revolution, and a Gear GP-26-B_144032 gearbox (3) with a gear ratio of 3.8:1.

2.1.2. Electronic Amplifier

The developed amplifier is in the form of a linear electronic converter. It represents a linear electronic circuit whose main component is the operational amplifier TDA2030(A) (STMicroelectronics, Geneva, Switzerland), powered by a DC supply via a single-phase transformer with two secondary windings and a diode rectifier. This amplifier can be used either as a voltage-to-current or voltage-to-voltage converter, for operation in two-quadrant or four-quadrant mode.

Figure 2b shows the electronic schematic of the amplifier used to supply the DC motor. The input voltage of the amplifier represents the control voltage U_c. The output of the amplifier is, depending on the mode, either the motor armature current I_a or the reference armature voltage U_m.

The gain values of the amplifier are: in current modes K_UcIa = 1/5 A/V, and in voltage mode K_UcUm = 12/5 V/V. The accuracy of the amplifier operation is 1% of the range. In the experiments, the electronics were set to voltage-to-voltage, two-quadrant mode.

Table 1. Specifications of the A-max 26 A DC motor (Ø26 mm, 4.5 W, with CLL precious metal brushes) and the amplifier.

Nominal armature voltage	Uan = 12 V
Nominal armature current	Ian = 0.629 A
Nominal motor speed	nn = 2750 min−1
Motor moment of inertia	Jm = 1.3·10−6 kgm2 *
Gear reducer moment of inertia	Jgr = 0.6·10−7 kgm2 *
Encoder moment of inertia	Je = 0.7·10−7 kgm2 *
Viscous friction coefficient	B = 1.8·10−6 Nm/rad/s **
Coulomb friction	Mc = 0.48·10−3 Nm **
Motor torque constant	KT = 25.5·10−3 Nm/A
Back electromotive force (EMF) constant	Ke = 25.5·10−3 V/rad/s
Armature resistance	Ra = 7.41 Ω
Armature inductance	La = 0.77·10−3 H
Gear ratio	N = 3.8
Amplifier resistor R1	R1 = 1 Ω
Amplifier resistor R5	R5 = 1 kΩ
Amplifier resistors R9, R10, and R12	R9 = R10 = R12 = 10 kΩ
Amplifier resistors R14 and R15	R14 = R15 = 1 kΩ
Equivalent resistor Re1	Re1 = R9·R10/(R9 + R10) = 5 kΩ
Equivalent resistor Re2	Re2 = R12 + R14 + R15 = 12 kΩ
Amplifier capacitor C8	C8 = 48 pF

* The total moment of inertia of the DC motor is expressed as the sum J = J_m + J_gr + J_e = 1.43·10⁻⁶ kgm². ** This values of B and M_c are not specified in the datasheet for the particular motor but were determined experimentally [45].

presents the data specifications of the DC motor and the amplifier.

2.1.3. CompactRIO 9075

The CompactRIO 9075 integrated system consists of a real-time processor and a reconfigurable FPGA (Field-Programmable Gate Array) on the same chassis, enabling control and monitoring capabilities, Figure 1. The CRIO-9075 integrates a 400 MHz industrial real-time processor with an LX25 FPGA and includes four slots for NI C Series I/O modules.

The modules used in the experimental setup are:

• NI 9402 (LVTTL, bidirectional, 4-channel, 55 ns digital input-output module)—for measuring motor speed after the gearbox, based on the frequency of the pulse train from the optical encoder;
• NI 9403 (5 V/TTL, bidirectional, 32-channel, 7 μs digital input-output module)—for controlling the activation of the corresponding relays;
• NI 9205 (32-channel, ±200 mV to ±10 V, 16-bit, 250 kS/s voltage input module)—for measuring the armature voltage of the motor and the armature current (by measuring the voltage across an additional resistor);
• NI 9263 (4-channel, ±10 V, 16-bit voltage output module)—for supplying the control voltage to the amplifier.

2.1.4. Relays

In order to develop a real-time FDII system, relays were designed to introduce multiplicative faults into the system by abruptly changing one of the amplifier parameters. Figure 3a shows the electronic schematic of the main part of the relay board used for connecting a resistance R_i (i = 1, 9, 12) into the amplifier circuit.

Figure 3b represents an example of a printed circuit board used to switch on or off resistance R₁ in the amplifier circuit by assigning a binary value 0/1 to the corresponding digital outputs.

When the value of the i-th digital output (D_oi)of the NI 9403 module is 0, the 2N3904 transistor is in the conducting state, and the relay’s electromagnet shifts the contact from the NC (normally closed) position to the NO (normally open) position, thereby connecting the desired resistance R_i (i = 1, 9, 12) (or capacitance C₈) into the amplifier circuit.

When the value of the digital output is 1, the bipolar transistor does not conduct, and no current flows through the relay coil. In this case, the NO contacts remain open, while the NC contacts are closed.

There are four printed circuit boards (PCBs), and each can switch between four different resistance values R_i (i = 1, 9, 12), or four capacitance values C₈ of the electronic amplifier, as specified in

Table 2. Amplifier circuit parameter values that can be adjusted using relays.

Doi	R1	Doi	R9
Do17	1 Ω	Do19	10 kΩ
Do16	10 Ω	Do18	33 kΩ
Do1	56 Ω	Do3	56 kΩ
Do0	100 Ω	Do2	100 kΩ
Doi	C8	Doi	R12
Do21	47 pF	Do6	10 kΩ
Do20	1 µF	Do7	5.6 kΩ
Do5	10 µF	Do22	3.3 kΩ
Do4	47 µF	Do23	2 kΩ

.

2.2. System Modeling in the Presence of Multiplicative Faults

The electronic amplifier model, which uses an operational amplifier as its core component (Figure 2b), is influenced by the characteristics of the motor connected to its output. The total motor friction is described by a nonlinear function that combines static, dynamic, and viscous components, as defined by the Karnopp–Reynolds model (1) and illustrated in Figure 4a [46]. Moreover, since viscous friction is temperature-dependent (the value listed in Table 1 corresponds to a measurement at 23 °C), the combined motor–amplifier system exhibits nonlinear behavior [45].

$$M_{\mathrm{f}}(\mathrm{\omega })=\left\{\begin{array}{c}{M}_{\mathrm{s}}\mathrm{sgn}\mathrm{\omega };\mathrm{for}\left|\mathrm{\omega }\right|<\Delta \mathrm{\omega }\\ (M_{\mathrm{c}}+(M_{\mathrm{s}}-M_{\mathrm{c}})e^{-(\mathrm{\omega }/{\mathrm{\omega }}_{\mathrm{k}})\mathrm{\delta }})\mathrm{sgn}\mathrm{\omega }+B\mathrm{\omega };\mathrm{for}\left|\mathrm{\omega }\right|>\Delta \mathrm{\omega }\end{array}\right.$$

(1)

where M_f is total friction in the motor, M_s is static friction, M_c is Coulomb friction, B is viscous friction coefficient, ω is the angular velocity of the motor, and Δω is area around ω = 0. Parameter δ is estimated empirically. Some forms of this model are: Tustin’s (δ = 1) and Gauss’s (δ = 2).

After assigning constant values for B and M_c, provided in Table 1, and after linearizing the friction curve (Figure 4b), a linearized model of the amplifier can be represented in the form:

$$\begin{array}{ll}{U}_{\mathrm{m}}(s)=& {\displaystyle \frac{R_{\mathrm{e}2}(C_8{R}_{5}s+1)[(R_{\mathrm{a}}+s{L}_{\mathrm{a}})(B+Js)+K_{\mathrm{e}}{K}_{\mathrm{T}}]}{R_{\mathrm{e}1}[(R_{\mathrm{e}2}+R_5)C_{8}s+1][(R_{\mathrm{a}}+R_1+s{L}_{\mathrm{a}})(B+Js)+K_{\mathrm{e}}{K}_{\mathrm{T}}]}}{U}_c(s)\\ & -{\displaystyle \frac{R_1{K}_{\mathrm{e}}}{[(R_{\mathrm{a}}+R_1+s{L}_{\mathrm{a}})(B+Js)+K_{\mathrm{e}}{K}_{\mathrm{T}}]}}{M}_{\mathrm{c}}(s).\end{array}$$

(2)

By initially neglecting the inductance L_a of the motor inductor circuit and subsequently reducing the resulting second-order system to a first-order one, a simplified model is obtained:

$$U_{\mathrm{m}}(s)={\displaystyle \frac{b_{11}s+b_{01}}{a_{11}s+1}}{U}_{\mathrm{c}}(s)-{\displaystyle \frac{b_{02}}{a_{12}s+1}}{M}_{\mathrm{c}}(s),$$

(3)

where

$$\begin{array}{c}{b}_{01}={\displaystyle \frac{R_{2\mathrm{e}}}{R_{1\mathrm{e}}}}{\displaystyle \frac{B{R}_{\mathrm{a}}+K_{\mathrm{e}}{K}_{\mathrm{T}}}{B(R_{\mathrm{a}}+R_1)+K_{\mathrm{e}}{K}_{\mathrm{T}}}},\\ b_{11}={\displaystyle \frac{R_{2\mathrm{e}}}{R_{1\mathrm{e}}}}{\displaystyle \frac{[J{R}_{\mathrm{a}}+C_8{R}_5(B{R}_{\mathrm{a}}+K_{\mathrm{e}}{K}_{\mathrm{T}})]}{B(R_{\mathrm{a}}+R_1)+K_{\mathrm{e}}{K}_{\mathrm{T}}}},\\ a_{11}={\displaystyle \frac{J(R_{\mathrm{a}}+R_1)+C_8(R_5+R_{2\mathrm{e}})[B(R_{\mathrm{a}}+R_1)+K_{\mathrm{e}}{K}_{\mathrm{T}}]}{B(R_{\mathrm{a}}+R_1)+K_{\mathrm{e}}{K}_{\mathrm{T}}}},\end{array}$$

(4)

$$\begin{array}{l}{b}_{02}={\displaystyle \frac{R_1{K}_{\mathrm{e}}}{B(R_{\mathrm{a}}+R_1)+K_{\mathrm{e}}{K}_{\mathrm{T}}}},\\ a_{12}={\displaystyle \frac{J(R_{\mathrm{a}}+R_1)}{B(R_{\mathrm{a}}+R_1)+K_{\mathrm{e}}{K}_{\mathrm{T}}}}.\end{array}$$

(5)

By omitting Coulomb friction, simplified continuous model suitable for analyzing multiplicative faults can be derived:

$$U_{\mathrm{m}}(s)={\displaystyle \frac{b_{1}s+b_0}{a_{1}s+1}}{U}_{\mathrm{c}}(s),$$

(6)

$$b_0=b_{01},b_1=b_{11},a_1=a_{11}.$$

(7)

Discretizing the continuous amplifier model with a sampling period T = 10 ms then produces:

$$U_{\mathrm{m}}(z)={\displaystyle \frac{b_{0\mathrm{d}}+b_{1\mathrm{d}}{z}^{-1}}{1+a_{1\mathrm{d}}{z}^{-1}}}{U}_{\mathrm{c}}(z),$$

(8)

where the coefficients in (8) are defined by the discretization process:

$$\begin{array}{c}{b}_{0\mathrm{d}}=b_1/a_1,\\ b_{1\mathrm{d}}=b_0(1-e^{-T/a_1})-b_1/a_1,\\ a_{1\mathrm{d}}=e^{-T/a_1}.\end{array}$$

(9)

The resulting discrete form (8) is directly applicable for implementing discrete multiplicative-fault detection algorithms.

2.3. Sliding Window Algorithm

The SWA is a method for estimating the parameters of a discrete model [47,48]. This algorithm is a variant of the least squares algorithm [38], where the data length K is relatively small.

Consider a system model with a single input and a single output, expressed in the form:

$$y(k)={\displaystyle \frac{b_{0\mathrm{d}}+b_{1\mathrm{d}}{z}^{-1}+b_{2\mathrm{d}}{z}^{-2}+\dots +b_{\upsilon \mathrm{d}}{z}^{-\upsilon }}{1+a_{1\mathrm{d}}{z}^{-1}+a_{2\mathrm{d}}{z}^{-2}+\dots +a_{\upsilon \mathrm{d}}{z}^{-\upsilon }}}u(k),$$

(10)

where:

• $u(k)=U_c(k)$ is the system input,
• $y(k)=U_m(k)$ is the system output,
• $\upsilon$ is the system order.

Defining the regression vector

$${\mathsf{\phi }}^{\prime }(k)=[u(k)u(k-1)\dots u(k-\upsilon );-y(k-1)-y(k-2)\dots -y(k-\upsilon )],$$

(11)

and the parameter vector as

$$\mathsf{\pi }=[b_{0\mathrm{d}}\hspace{1em}{b}_{1\mathrm{d}}\hspace{1em}{b}_{2\mathrm{d}}\hspace{1em}\cdots \hspace{1em}{b}_{\upsilon \mathrm{d}};\hspace{1em}{a}_{1\mathrm{d}}\hspace{1em}{a}_{2\mathrm{d}}\hspace{1em}\cdots \hspace{1em}{a}_{\upsilon \mathrm{d}}{]}^{\prime },$$

(12)

the output can be defined as follows:

$$y(k)={\mathsf{\phi }}^{\prime }(k)\mathsf{\pi }.$$

(13)

The parameter estimation using the least squares algorithm is provided in the form:

$$\hat{\mathsf{\pi }}(k,K)={\left[{\displaystyle \sum _{j=0}^{K-1}\mathsf{\phi }(k-j)}{\mathsf{\phi }}^{\prime }(k-j)\right]}^{-1}\left[{\displaystyle \sum _{j=0}^{K-1}\mathsf{\phi }(k-j)}y(k-j)\right],$$

(14)

$$\hat{\mathsf{\pi }}(k,K)={\left[{\mathsf{\Phi }}^{\prime }(k,K)\mathsf{\Phi }(k,K)\right]}^{-1}{\mathsf{\Phi }}^{\prime }(k,K)\mathit{Y}(k,K),$$

(15)

$$\mathsf{\Phi }=\left[\begin{array}{c}{\mathsf{\phi }}^{\prime }(k)\\ ⋮\\ {\mathsf{\phi }}^{\prime }(k-K+1)\end{array}\right],\mathit{Y}=\left[\begin{array}{c}y(k)\\ ⋮\\ y(k-K+1)\end{array}\right],$$

(16)

where: k denotes current time step, and K represents the length of the data used in the algorithm.

When estimating parameters to detect and analyze multiplicative faults, a transient response occurs following a parameter jump, lasting for K − 1 samples. As a result, the detection of such faults is delayed by at least K − 1 samples. To reduce this delay, the window length K in the least squares algorithm must be limited, prompting a shift to the SWA. A shorter window reduces detection delay but increases sensitivity to noise and variance in parameter estimates. Additionally, successful parameter estimation requires the input signal to be sufficiently informative, satisfying the persistent excitation condition [49].

2.4. Sliding Integral Algorithm

The SIA [3,44,50,51] is a control or signal processing method that computes the integral (i.e., the accumulated sum) of a signal over a moving (sliding) window of time or samples, Figure 5. It can identify the parameters of a continuous-time model directly from sampled data, which is often more complex than discrete-time identification but yields physically meaningful parameters. To reduce sensitivity to noise from derivative approximations, an equivalent integral form of the differential equation is used, typically over a sliding window, to avoid the need for initial conditions, as proposed in [52].

The SIA is explained in detail in [3,51], and a more concise version is presented here.

The sliding integral of a continuous-time variable x(t) is:

$$I_1[x,k]={\displaystyle \underset{t=kT-mT}{\overset{kT}{\int }}x(t)dt},$$

(17)

where $T$ is sampling interval, k is current time step, and $m$ is window length and sample counts representing the number of samples within the integration interval.

With rectangular approximation, this can be written as

$$I_1[x,k]=T{\displaystyle \sum _{i=1}^{m}x(kT-iT)}=TQ(z)x(k),$$

(18)

where

$$Q(z)=({\mathrm{z}}^{-1}+{\mathrm{z}}^{-2}+\dots +{\mathrm{z}}^{-m}).$$

In general, the j-th sliding integral is expressed as:

$$I_j[x,k]=T^j{Q}^j(z)x(k).$$

(19)

The following expression also holds [3,51]

$$I_j[x^{(j)},k]={(1-{\mathrm{z}}^{-m})}^{j}x(k),$$

(20)

where $x^{(j)}=d^{j}x(t)/d{t}^j$.

By combining Equations (19)–(21), this equation can be obtained:

$$I_n[x^{(j)},k]=I_{n-j}[I_j(x^{(j)},k\prime ),k]={\rho }_j(z)x(k),$$

(21)

where

$${\rho }_j(z)=T^{n-j}{Q}^{n-j}{(1-z^{-m})}^j,j=0,1,\dots ,n.$$

(22)

For a single-input single-output (SISO) linear time-invariant (LTI) system, the system dynamics can be described by the following differential equation:

$$y(t)+a_1{\displaystyle \frac{dy(t)}{dt}}+\dots +a_n{\displaystyle \frac{d^{n}y(t)}{d{t}^n}}=b_{0}u(t)+b_1{\displaystyle \frac{du(t)}{dt}}+\dots +b_n{\displaystyle \frac{d^{n}u(t)}{d{t}^n}},$$

(23)

where u(t) is input, y(t) is output of the system.

By applying the sliding integral to the differential Equation (23) n times, it is obtained

$$I_n[y,k]+a_1{I}_n[y^{(1)},k]+\dots +a_n{I}_n[y^{(n)},k]=b_0{I}_n[u,k]+b_1{I}_n[u^{(1)},k]+\dots +b_n{I}_n[u^{(n)},k].$$

(24)

Using (21), (24) can be rewritten as follows:

$${\rho }_0(z)y(k)+a_1{\rho }_1(z)y(k)+\dots +a_n{\rho }_n(z)y(k)=b_0{\rho }_0(z)u(k)+b_1{\rho }_1(z)u(k)+\dots +b_n{\rho }_n(z)u(k).$$

(25)

This is alternatively represented as

$$v(k)={\mathsf{\phi }}^{\prime }(k)\mathsf{\theta },$$

(26)

where

$$v(k)={\rho }_0(z)y(k),$$

(27)

$${\mathsf{\phi }}^{\prime }(k)=\left[{\rho }_0(z)u(k)\dots {\rho }_n(z)u(k);-{\rho }_1(z)y(k)\dots -{\rho }_n(z)y(k)\right],$$

(28)

$$\mathsf{\theta }=[b_0\cdots b_n;a_1\cdots a_n{]}^{\prime }.$$

(29)

Accordingly, the least squares algorithm can be formulated as:

$$\mathsf{\theta }={({\mathsf{\Phi }}^{\prime }\mathsf{\Phi })}^{-1}{\mathsf{\Phi }}^{\prime }\mathit{V},$$

(30)

where

$$\mathsf{\Phi }=\left[\begin{array}{c}{\mathsf{\phi }}^{\prime }(k)\\ ⋮\\ {\mathsf{\phi }}^{\prime }(k-K+1)\end{array}\right],\hspace{1em}\mathit{V}=\left[\begin{array}{c}v(k)\\ ⋮\\ \mathrm{v}(k-K+1)\end{array}\right].$$

(31)

Although the identification procedure still relies on sampled data, the SIA is more complex and computationally intensive than parameter identification in the discrete-time case. Due to the inherent trade-off between computational complexity and noise attenuation, a window length m of up to 20 is recommended [3]. Furthermore, the literature does not impose any specific constraints on the length of the identification dataset K.

2.5. The Main Steps of the Proposed FDII Methodology

A flowchart representing the main steps of the proposed FDII methodology is presented in Figure 6, providing a clear visual overview of the detection, isolation, and identification process.

3. Results

To demonstrate the effectiveness of detecting and isolating individual multiplicative amplifier faults, the amplifier parameters were deliberately and abruptly altered (one at a time) at the midpoint of the recording interval using relay cards (Figure 3). In this experimental setup, two parameter estimation approaches were developed.

The first approach can be described as offline. Signal data were collected in the LabVIEW environment and stored as text files. The estimation process was then carried out in batch mode using MATLAB 2020b, where the estimation algorithm was implemented through dedicated m-files.

Figure 7 presents a portion of the LabVIEW block diagram for generating and recording signals under multiplicative fault conditions. It illustrates the generation of the amplifier’s input signal U_c, using a PRBS (Pseudo-Random Binary Sequence) and the measurement of the amplifier’s output signal U_m. The system also enables monitoring of the armature voltage U_a, the armature current I_a (by monitoring the voltage on the series resistor R_d, Figure 2b and Figure 7), and the motor speed n.

Figure 8 presents a portion of the LabVIEW Block Diagram, which activates specific amplifier parameters by switching the corresponding relays, thereby inducing individual multiplicative faults.

Figure 9 shows the Front Panel of the LabVIEW program used for monitoring input-output signals, featuring controls for the abrupt changing of one of the amplifier’s parameter values in the event of a multiplicative fault. The software is configured so that all parameters are set to their nominal values (slider position 1) prior to the occurrence of a fault. At the fault moment, one of the amplifier parameters is changed. In the example shown, the resistance R₁₂ is in 10 s reduced from its nominal value of 10 kΩ to a lower value of 5.6 kΩ (slider position 3) by automatically activating the corresponding relay.

The second approach to detecting multiplicative faults in the amplifier and DC motor was developed entirely within the LabVIEW environment and can be classified as a real-time method. After acquiring the input–output signals (Figure 7 and Figure 9) during the injection of specific multiplicative faults using relays (Figure 8 and Figure 9), the signals are passed to the next frame of the Flat Sequence structure. Within this structure, the desired estimation algorithm, either discrete (SWA), based on (14)–(16), or continuous (SIA), based on (30) and (31), is implemented using a MathScript Node within the LabVIEW software (Figure 10). The estimated parameters of the amplifier b_0d, b_1d, and a_1d from the discrete model of the amplifier (8) (SWA), or b₀, b₁, and a₁ from the continuous model of the amplifier (6) (SIA), are then displayed in the Chart block of the LabVIEW program. Figure 11 displays section of the LabVIEW Front Panel displaying the estimation of continuous parameters b₀, b₁, and a₁ using the SIA.

The procedure for generating multiplicative motor faults, specifically changes in inductance (L), armature resistance (R_a), moment of inertia (J), and viscous friction coefficient (B), was implemented at the simulation level using the Multisim environment. This setup utilized the amplifier model shown in Figure 12 and the motor model illustrated in Figure 13. In this configuration, the input signal for parameter estimation was a Pseudo-Random Binary Sequence (PRBS) control voltage (U_c), while the output voltage of the amplifier (U_m) was recorded using the Parameter Sweep faults. This approach leverages the mapping of motor parameters onto the electronic amplifier model.

3.1. Multiplicative Fault Detection

3.1.1. Input–Output Signals Under Multiplicative Faults

To ensure accurate estimation of sudden multiplicative faults, the input signal must be persistently exciting, i.e., rich enough to guarantee a unique and identifiable set of parameter estimates [49]. The Pseudo-Random Binary Sequence (PRBS), which is readily available in most software libraries, is, therefore, widely adopted for system identification.

A PRBS signal with values between 2 V and 3 V, as shown in Figure 14, was generated using LabVIEW and applied as the input to the electronic amplifier. Figure 15 and Figure 16 illustrate the amplifier and motor output voltages over a 20 s period, during which a sudden multiplicative fault is introduced at the 10th s. This fault corresponds to an abrupt change in a selected amplifier or motor parameter from its nominal value to a predefined fault value. For the amplifier, faults were introduced in the following components: R₁ (from the nominal value of 1 Ω to fault values 10 Ω, 56 Ω, and 100 Ω), Figure 15a; R₉ (from the nominal value of 10 kΩ to the faults values 33 kΩ, 56 kΩ and 100 kΩ), Figure 15b, C₈ (from the nominal value of 47 pF to the fault values 1 µF, 10 µF, 47 µF), Figure 15c; R₁₂ (from the nominal value of 10 kΩ to the fault values 2 kΩ, 3.3 kΩ and 5.6 kΩ), Figure 15d. For the motor, faults were introduced in the parameters: armature resistance R_a (from the nominal value 7.41 Ω to the fault values 0.741 Ω and 74.1 Ω), Figure 16a,b, viscous friction coefficient B (from the nominal value 1.8·10⁻⁶ Nm/rad/s to the fault values 1.8·10⁻⁴ Nm/rad/s and 1.8·10⁻⁸ Nm/rad/s), Figure 16c, moment of inertia J (from the nominal value 1.43·10⁻⁶ kgm² to the fault values 1.43·10⁻⁴ kgm² and 1.43·10⁻⁸ kgm²), Figure 16d.

3.1.2. Multiplicative Fault Detection Using Sliding Window Algorithm

The following diagrams present selected results from the application of the SWA for estimating amplifier parameters using various window lengths (K). The result of applying the SWA is the set of discrete amplifier parameters, a_1d, b_0d, and b_1d, described in model (8). Figure 17 and Figure 18 show SWA-based parameter estimation in response to a multiplicative fault in the amplifier resistance R₁ and the motor’s viscous friction coefficient B, respectively.

In Figure 17 and Figure 18, the time interval from 0 to 10 s represents the estimated parameters before the fault occurs, while the interval from 10 to 20 s corresponds to the post-fault parameter estimation. A transient interval is noticeable within this post-fault period, lasting approximately (K − 1)·T s, where T = 0.01 s denotes the sampling period.

The graphical estimation results indicated that a sudden change in the amplifier or motor parameters triggers a transient response in the identification algorithm during the measurement period. Consequently, selecting a shorter window length is more advantageous in terms of reducing fault detection delay. However, it is also evident that as the window length decreases, the variance of the estimates increases and the sensitivity to noise becomes more pronounced. In the conducted experiments, K = 50; K = 100; and K = 200, different window lengths were tested. Since the total measurement period was 20 s, the chosen window lengths corresponded to transient durations of approximately 0.49 s, 0.99 s, and 1.99 s. In this particular case, the parameter choice of K = 100 proved to be the most effective.

Table 3. Amplifier parameter estimation using the SWA for multiplicative faults in R₁ and R₉.

R1 = 1 Ω				R9 = 10 kΩ
Fault	b0d	b1d	a1d	Fault	b0d	b1d	a1d
10 Ω	1.421	−0.82	−0.738	33 kΩ	1.488	−0.5	−0.373
56 Ω	0.472	−0.279	−0.896	56 kΩ	1.337	−0.412	−0.35
100 Ω	0.29	−0.18	−0.925	100 kΩ	1.258	−0.359	−0.316

,

Table 4. Amplifier parameter estimation using the SWA for multiplicative faults in R₁₂ and C₈.

R12 = 10 kΩ				C8 = 47 pF
Fault	b0d	b1d	a1d	Fault	b0d	b1d	a1d
2 kΩ	0.757	−0.368	−0.514	1 μF	1.148	−0.013	−0.526
3.3 kΩ	1	−0.458	−0.492	10 μF	0.312	−0.142	−0.929
5.6 kΩ	1.434	−0.784	−0.572	47 μF	0.21	−0.168	−0.982

,

Table 5. Estimation of motor parameters using the SWA for multiplicative faults in R_a and J.

Ra = 7.41 Ω				J = 1.43·10−6 kgm2
Fault	b0d	b1d	a1d	Fault	b0d	b1d	a1d
0.741 Ω	1.9	0.356	−0.06	1.43·10−4 kgm2	2.12	−2.13	−1
74.1 Ω	2.369	−2.227	−0.94	1.43·10−8 kgm2	2.34	−0.192	−0.083

and

Table 6. Estimation of motor parameters using the SWA for multiplicative fault in B.

B = 1.8·10−6 Nm/rad/s
Fault	b0d	b1d	a1d
1.8·10−4 Nm/rad/s	2.162	−0.34	−0.172
1.8·10−5 Nm/rad/s	2.177	−1.062	−0.524

summarize the complete results of applying the SWA for various multiplicative faults in the four amplifier parameters R₁, R₉, R₁₂, and C_8, and the three motor parameters R_a, J, and B. The obtained results pertain to the discrete amplifier parameters b_0d, b_1d, and a_1d from model (8).

3.1.3. Multiplicative Fault Detection Using Sliding Integral Algorithm

The following figures show selected results from applying the SIA to estimate amplifier parameters under different window lengths (m) and data lengths (K). In each case, multiplicative faults were introduced abruptly at 10 s. In Figure 19 and Figure 20, the time interval from 0 to 10 s corresponds to the parameter estimates prior to the fault occurrence, while the interval from 10 to 20 s reflects the parameter estimation after the fault has been introduced. During this post-fault period, a transient response can be observed, lasting approximately (K + m − 1)·T seconds, where T = 0.01 s is the sampling period.

Figure 19 shows amplifier parameter estimation using the SIA in response to a multiplicative fault in the amplifier capacitor C₈, while Figure 20 corresponds to a multiplicative fault in the motor armature resistance R_a.

When applying the estimation algorithm for the detection of abrupt parameter changes, it has been observed that the length of the identification dataset K should be selected to be sufficiently small (yet greater than m, K > m), as the estimation process effectively operates as a “window of length m” nested within a broader window of length K. Such changes induce a transient in the identification process lasting K + m − 1 samples. The used window configurations were: K = 20, m = 5; K = 100, m = 20, and K = 500, m = 20. Since the sampling period T = 10  ms, these configurations correspond to transient durations of approximately 0.24 s, 1.19 s, and 5.19 s, respectively.

Table 7. Amplifier parameter estimation using the SIA for multiplicative faults in R₁ and R₉.

R1 = 1 Ω				R9 = 10 kΩ
Fault	b0	b1	a1	Fault	b0	b1	a1
10 Ω	2.306	0.0546	0.038	33 kΩ	1.331	0.0245	0.0199
56 Ω	1.85	0.047	0.1	56 kΩ	1.425	0.03	0.0221
100 Ω	1.495	0.0352	0.126	100 kΩ	1.575	0.032	0.022

,

Table 8. Amplifier parameter estimation using the SIA for multiplicative faults in R₁₂ and C₈.

R12 = 10 kΩ				C8 = 47 pF
Fault	b0	b1	a1	Fault	b0	b1	a1
2 kΩ	0.8	0.0175	0.023	1 μF	2.39	0.024	0.0209
3.3 kΩ	1.059	0.0228	0.0228	10 μF	2.387	0.044	0.141
5.6 kΩ	1.521	0.032	0.022	47 μF	2.386	0.1142	0.56

,

Table 9. Estimation of motor parameters using the SIA for multiplicative faults in R_a and J.

Ra = 7.41 Ω				J = 1.43·10−6 kgm2
Fault	b0	b1	a1	Fault	b0	b1	a1
0.741 Ω	2.393	0.02	0.011	1.43·10−4 kgm2	2.392	3.794	1.824
74.1 Ω	2.395	0.143	0.06	1.43·10−8 kgm2	2.394	0.018	0.008

and

Table 10. Estimation of motor parameters using the SIA for multiplicative faults in B.

B = 1.8·10−6 Nm/rad/s
Fault	b0	b1	a1
1.8·10−4 Nm/rad/s	2.2	0.026	0.012
1.8·10−5 Nm/rad/s	2.346	0.0462	0.0213

summarize the complete results of applying the SIA for various multiplicative faults in the four amplifier parameters R₁, R₉, R₁₂, and C_8, and the three motor parameters R_a, J, and B. The obtained results pertain to the continuous amplifier parameters b₀, b₁, and a₁ from model (6).

3.2. Multiplicative Fault Isolation and Identification

3.2.1. Decision Algorithm

The applied techniques for detecting multiplicative faults provide information about changes in the system model parameters but do not reveal the values of the specific physical parameters that caused those changes. Hence, they are capable of fault detection, i.e., they confirm that a fault has occurred in the system.

In order to isolate faults (i.e., identify the fault location), a decision algorithm based on the amplifier model, (6), with parameters provided with (4) and (7), has been developed as shown in Figure 21.

This algorithm can be performed in the case of application of continuous fault detection procedure, i.e., SIA. Fault identification, in terms of determining the magnitude of the change in the corresponding physical parameter, is straightforward in this case. The actual faulty physical parameter can be directly determined from the relations provided in (4) and (7), thereby completing the identification.

3.2.2. Indirect Estimation Procedure

An alternative solution to the fault identification problem, which can also be applied in the case of discrete parameter estimation using the SWA, the actual physical parameters can be determined through one of the indirect estimation procedures [3]. This approach is fully applicable when the relationship between the estimated and physical parameters is linear. However, if the relationship is nonlinear (which is more common), the procedure remains applicable only when the change (i.e., the fault) in the physical parameter is relatively small compared to its nominal value.

The starting point is the relationship between the physical parameters $\mathsf{\theta }$, whose number is $L$, and the model parameters $\mathsf{\pi }$, whose number is $N$, expressed through the relation:

$$\mathsf{\pi }=f(\mathsf{\theta }).$$

(32)

When $L=N$, the actual physical parameters can be indirectly estimated according to:

$$\Delta \hat{\mathsf{\theta }}={\mathit{R}}^{-1}\Delta \hat{\mathsf{\pi }},$$

(33)

$$\mathit{R}={\left[{\displaystyle \frac{\mathsf{\partial }\mathsf{\pi }}{\mathsf{\partial }\mathsf{\theta }}}\right]}_{\Delta \mathsf{\theta }=0}={\left[\begin{array}{ccc}{\displaystyle \frac{\mathsf{\partial }{\mathsf{\pi }}_1}{\mathsf{\partial }{\theta }_1}}& \dots & {\displaystyle \frac{\mathsf{\partial }{\mathsf{\pi }}_1}{\mathsf{\partial }{\theta }_L}}\\ ⋮& & ⋮\\ {\displaystyle \frac{\mathsf{\partial }{\mathsf{\pi }}_N}{\mathsf{\partial }{\theta }_1}}& \dots & {\displaystyle \frac{\mathsf{\partial }{\mathsf{\pi }}_N}{\mathsf{\partial }{\theta }_L}}\end{array}\right]}_{\Delta \mathsf{\theta }=0},$$

(34)

where:

• $\Delta \hat{\mathsf{\pi }}=\hat{\mathsf{\pi }}-{\mathsf{\pi }}_{\mathrm{n}}$ is estimation in model parameters variation,
• ${\mathsf{\pi }}_{\mathrm{n}}$ is the nominal model parameters,
• $\hat{\mathsf{\pi }}$ is estimating model parameters,
• $\Delta \hat{\mathsf{\theta }}$ is estimation of actual physical parameters.

The partial derivatives in (34) are calculated using the nominal values of the parameters. If L < N, a slightly modified form of (33) defined in [3] can be used:

$$\Delta \hat{\mathsf{\theta }}=(R^{\prime }R{)}^{-1}{R}^{\prime }\Delta \hat{\mathsf{\pi }}.$$

(35)

In the specific example, the vector of actual physical parameters is defined as:

$$\mathsf{\theta }={\left[\begin{array}{ccccccc}{R}_1& R_9& R_{12}& C_8& R_{\mathrm{a}}& J& B\end{array}\right]}^{\prime },$$

(36)

so, the columns of the Jacobian matrix (matrix of partial derivatives) are:

$${\mathit{R}}_1=\left[\begin{array}{c}{\displaystyle \frac{d{b}_0}{d{R}_1}}\\ {\displaystyle \frac{d{b}_1}{d{R}_1}}\\ {\displaystyle \frac{d{a}_1}{d{R}_1}}\end{array}\right]=\left[\begin{array}{c}-{\displaystyle \frac{R_{\mathrm{e}2}B}{R_{\mathrm{e}1}}}{\displaystyle \frac{R_{\mathrm{a}}B+K_{\mathrm{T}}{K}_{\mathrm{e}}}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ {\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{R_{\mathrm{a}}JB}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ {\displaystyle \frac{J{K}_{\mathrm{T}}{K}_{\mathrm{e}}(R_1+R_{\mathrm{a}})}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\end{array}\right];{\mathit{R}}_2=\left[\begin{array}{c}{\displaystyle \frac{d{b}_0}{d{R}_9}}\\ {\displaystyle \frac{d{b}_1}{d{R}_9}}\\ {\displaystyle \frac{d{a}_1}{d{R}_9}}\end{array}\right]=\left[\begin{array}{c}-{\displaystyle \frac{R_{\mathrm{e}2}}{R_9^2}}{\displaystyle \frac{R_{\mathrm{a}}B+K_{\mathrm{T}}{K}_{\mathrm{e}}}{(R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}}}}\\ -{\displaystyle \frac{R_{\mathrm{e}2}}{R_9^2}}{\displaystyle \frac{J{R}_{\mathrm{a}}}{(R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}}}}\\ 0\end{array}\right]$$

(37)

$${\mathit{R}}_3=\left[\begin{array}{c}{\displaystyle \frac{d{b}_0}{d{R}_{12}}}\\ {\displaystyle \frac{d{b}_1}{d{R}_{12}}}\\ {\displaystyle \frac{d{a}_1}{d{R}_{12}}}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{1}{R_{\mathrm{e}1}}}{\displaystyle \frac{R_{\mathrm{a}}B+K_{\mathrm{T}}{K}_{\mathrm{e}}}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ {\displaystyle \frac{1}{R_{\mathrm{e}1}}}{\displaystyle \frac{R_{\mathrm{a}}J}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ 0\end{array}\right];{\mathit{R}}_4=\left[\begin{array}{c}{\displaystyle \frac{d{b}_0}{d{C}_8}}\\ {\displaystyle \frac{d{b}_1}{d{C}_8}}\\ {\displaystyle \frac{d{a}_1}{d{C}_8}}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{R_5(R_{\mathrm{a}}B+K_{\mathrm{T}}{K}_{\mathrm{e}})}{(R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}}}}\\ 0\\ R_5+R_{2\mathrm{e}}\end{array}\right];$$

(38)

$${\mathit{R}}_5=\left[\begin{array}{c}{\displaystyle \frac{d{b}_0}{d{R}_{\mathrm{a}}}}\\ {\displaystyle \frac{d{b}_1}{d{R}_{\mathrm{a}}}}\\ {\displaystyle \frac{d{a}_1}{d{R}_{\mathrm{a}}}}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{B^2{R}_1}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ {\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{J(R_{\mathrm{a}}B+K_{\mathrm{T}}{K}_{\mathrm{e}})}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ {\displaystyle \frac{J{K}_{\mathrm{T}}{K}_{\mathrm{e}}(R_1+R_a)}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\end{array}\right];{\mathit{R}}_6=\left[\begin{array}{c}{\displaystyle \frac{d{b}_0}{dJ}}\\ {\displaystyle \frac{d{b}_1}{dJ}}\\ {\displaystyle \frac{d{a}_1}{dJ}}\end{array}\right]=\left[\begin{array}{c}0\\ {\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{R_{\mathrm{a}}}{(R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}}}}\\ {\displaystyle \frac{(R_1+R_{\mathrm{a}})}{(R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}}}}\end{array}\right];$$

(39)

$${\mathit{R}}_7=\left[\begin{array}{c}{\displaystyle \frac{d{b}_o}{dB}}\\ {\displaystyle \frac{d{b}_1}{dB}}\\ {\displaystyle \frac{d{a}_1}{dB}}\end{array}\right]=\left[\begin{array}{c}-{\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{K_{\mathrm{T}}{K}_{\mathrm{e}}{R}_1}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ -{\displaystyle \frac{R_{\mathrm{e}2}}{R_{\mathrm{e}1}}}{\displaystyle \frac{J{R}_{\mathrm{a}}(R_1+R_{\mathrm{a}})}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\\ -{\displaystyle \frac{J(R_1+R_{\mathrm{a}})}{{((R_1+R_{\mathrm{a}})B+K_{\mathrm{T}}{K}_{\mathrm{e}})}^2}}\end{array}\right].$$

(40)

After substituting the nominal values of the amplifier and motor model parameters (provided in Table 1), the Jacobian matrix with respect to the actual physical parameters was obtained as:

$$\mathit{R}=\left[\begin{array}{ccccccc}-6.5\cdot {10}^{-3}& -1.197\cdot {10}^{-4}& 1.995\cdot {10}^{-4}& 2.39\cdot {10}^3& 1.76\cdot {10}^{-5}& 0& -3.52\cdot {10}^3\\ -1\cdot {10}^{-4}& -1.9\cdot {10}^{-6}& 3.2\cdot {10}^{-6}& 0& 5.1\cdot {10}^{-3}& 2.67\cdot {10}^4& -4.83\cdot {10}^2\\ 2.1\cdot {10}^{-4}& 0& 0& 1.3\cdot {10}^3& 2.1\cdot {10}^{-3}& 1.26\cdot {10}^4& -2.28\cdot {10}^2\end{array}\right].$$

(41)

It has been concluded that modifying (35) into the form:

$$\Delta {\hat{\mathsf{\theta }}}_i=({R^{\prime }}_i{\mathit{R}}_i{)}^{-1}{\mathit{R}}_i^{\prime }\Delta {\hat{\mathsf{\pi }}}_i,$$

(42)

leads to significantly more accurate parameter estimation, resulting in improved precision and reliability of the results compared to the original approach, where:

• $\Delta {\hat{\mathsf{\theta }}}_i$ is the estimate of the change in the i-th physical parameter,
• $R_i$ is the i-th column of the Jacobian matrix (matrix of partial derivatives)
• $\Delta {\hat{\mathsf{\pi }}}_i={\hat{\mathsf{\pi }}}_i-{\mathsf{\pi }}_{\mathrm{n}}$ is the estimate of the change in model parameters due to the i-th fault,
• ${\mathsf{\pi }}_{\mathrm{n}}$ is nominal model parameters,
• ${\hat{\mathsf{\pi }}}_i$ is estimated model parameters in the presence of a fault in the i-th physical parameter.

Table 11. Estimation of amplifier physical parameters under multiplicative faults.

Parameter	R1 = 1 Ω			R9 = 10 kΩ			R12 = 10 kΩ			C8 = 47 pF
Fault	10 Ω	56 Ω	100 Ω	33 kΩ	56 kΩ	100 kΩ	2 kΩ	3.3 kΩ	5.6 kΩ	1 μF	10 μF	47 μF
Estimated fault	9.8 Ω	49 Ω	79 Ω	17 kΩ	18.2 kΩ	19 kΩ	2 kΩ	3.3 kΩ	5.6 kΩ	0.97 μF	9.7 μF	46 μF

and

Table 12. Estimation of motor physical parameters under multiplicative faults.

Parameter	${\mathit{R}}_{\mathbf{a}}=7.41 \mathbf{\Omega }$		J = 1.43·10−6 kgm2		$\mathit{B}=1.8\cdot {10}^{-6} \mathbf{N}\mathbf{m}/\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{s}$
Fault	0.741 Ω	74.1 Ω	1.43·10−4 kgm2	1.43·10−8 kgm2	1.8·10−4 Nm/rad/s	1.8·10−5 Nm/rad/s
Estimated fault	0.62 Ω	64.4 Ω	1.43·10−4 kgm2	1.45·10−8 kgm2	0.57·10−4 Nm/rad/s	1.53·10−5 Nm/rad/s

present the estimates of the actual physical parameters of the amplifier and DC motor in the presence of individual multiplicative faults, using the described indirect method provided in (42). Thereby, the parameter estimates were calculated using:

$${\hat{\mathsf{\theta }}}_i=\Delta {\hat{\mathsf{\theta }}}_i+{\mathsf{\theta }}_{\mathrm{n}},$$

(43)

where ${\mathsf{\theta }}_{\mathrm{n}}$ represent nominal values of the actual physical parameters of the system.

Based on the values presented in Table 11 and Table 12, it can be concluded that the indirect estimation of physical parameters R₁₂, C₈, and J is most accurate in the presence of multiplicative faults. This accuracy arises from the linear relationship with parameters b₀, b₁, and a₁ of the continuous amplifier model. During the estimation of R₁, R₉, R_a, and B, due to the nonlinear relationship with the amplifier parameters and the large deviation of these parameters from their nominal values, the estimation results were less accurate. Nevertheless, the deviations obtained using the proposed solution (42) are significantly smaller than the completely unacceptable results produced by applying the approach available in the literature (35).

4. Discussion

4.1. Validation of Fault Detection Measurement Results

To evaluate the effectiveness of different methods for detecting multiplicative faults in the motor and amplifier, the outputs of the amplifier model (3), obtained by substituting system parameters into Equations (4) and (5) for three different values of Coulomb friction [45] (M_c = 0, M_c = 0.48 mNm and M_c = 0.33 mNm), with the outputs of model (6), which incorporates parameters estimated using the SIA, and model (8), based on the SWA results, have been compared. These outputs were compared with the measured output voltage U_m, and the percentage agreement was calculated using MATLAB’s System Identification Toolbox, as shown in

Table 13. Percentage match of amplifier output before fault occurrence.

SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
98.6%	98.59%	92.27%	91.92% (92%)

,

Table 14. Percentage match of system output during parametric fault in R₁.

R1 = 1 Ω	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
10 Ω	95.71%	96.42%	70.78%	69% (72.11%)
56 Ω	81.65%	86.5%	−30.61%	35.13% (76.74%)
100 Ω	74.44%	76.75%	−150.2%	−17.67% (79%)

,

Table 15. Percentage match of system output during parametric fault in R₉.

R9 = 10 kΩ	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
33 kΩ	97.7%	97.14%	90.54%	89.01% (89.52%)
56 kΩ	97.28%	96.72%	91.07%	89.26% (89.87%)
100 kΩ	93.09%	97.57%	90.25%	88.27% (88.93%)

,

Table 16. Percentage match of system output during parametric fault in R₁₂.

R12 = 10 kΩ	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
2 kΩ	98.32%	98.57%	92.14%	90% (90.89%)
3.3 kΩ	95.89%	98.59%	92%	91.4% (91.6%)
5.6 kΩ	98.77%	97.96%	92.48%	91.52% (91.89)

,

Table 17. Percentage match of system output during parametric fault in C₈.

C8 = 47 pF	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
1 μF	97.85%	90.6%	75.41%	75.4% (75.43%)
10 μF	95.07%	97.57%	88.53%	89.04% (88.98%)
47 μF	87.96%	96.72%	96.3%	96.69% (96.7%9

,

Table 18. Percentage match of system output during parametric fault in R_a.

Ra = 7.41 Ω	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
0.741 Ω	99.68%	89.53%	95.43%	95.01% (95.25%)
74.1 Ω	95.84%	99.31%	99.75%	98.53% (98.93%)

,

Table 19. Percentage match of system output during parametric fault in J.

J = 1.43·10−6 kgm2	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
1.43·10−4 kgm2	85.84%	96.63%	99.59%	98.72% (99.12%)
1.43·10−8 kgm2	82.76%	82.69%	75.73%	75.71% (75.73%)

and

Table 20. Percentage match of system output during parametric fault in B.

B = 1.8·10−6 Nm/rad/s	SWA	SIA	Mc = 0	Mc = 0.48 mNm (Mc = 0.33 mNm)
1.8·10−4 Nm/rad/s	99.54%	99.21%	96.88%	96.84% (96.86%)
1.8·10−5 Nm/rad/s	99.45%	99.1%	96.5%	96.2% (96.35%)

.

An example of the Simulink models and the comparison interface for model outputs with the measured output voltage, in the case of a fault R₁ = 100 Ω, is shown in Figure 22. The corresponding output signals are presented in Figure 23.

Based on the results, it is clear that a multiplicative fault in resistor R₁ = 100 Ω (Table 14) significantly amplifies the effect of Coulomb friction (Figure 23). In this scenario, the model incorporating M_c = 0.33  mNm achieves the closest match to the measured output (79%), compared with the SIA-based model (76.8%) and the SWA-based model (74.4%). By contrast, for faults in C₈, R_9, or R₁₂, including Coulomb friction makes virtually no difference: the percentage match to the measured output is effectively the same whether friction is modeled or not.

A comparison of the fault-detection algorithms and the linearized models (with and without Coulomb friction) yields the following insights:

• Although the SIA is computationally more complex and more sensitive to noise than the discrete-time approach, it proved almost as effective at detecting multiplicative faults. As it has been observed, it is important to ensure that the length of the identification dataset K is chosen to be sufficiently small yet greater than m (i.e., K > m).
• The SWA is computationally simpler and more robust to measurement noise, making it better suited for real-time applications. Its parameter estimates also showed strong agreement with the actual system response.
• Obtained results confirm that, at the nominal resistance R₁ = 1 Ω, Coulomb friction is negligible. When R₁ increases abruptly, however, Coulomb friction can no longer be ignored (which is consistent with Equation (2)). Hence, simplified SIA and SWA models based on (6) and (8) that omit Coulomb friction lose accuracy. Also, the Coulomb friction value M_c = 0.33  mNm, which yields the best fit, appears to be more accurate than the value reported in [45]. However, the agreement between the measured system output and the output of the linearized model (2), with M_c = 0.33  mNm, does not exceed 80% for any fault magnitude in R₁, indicating that the system can only be linearized approximately.

4.2. Discussion of Multiplicative Fault Isolation and Identification

4.2.1. Discussion of the Decision Algorithm

After the fault isolation procedure is carried out using the decision algorithm, fault identification can be performed (when using one of the continuous estimation procedures, such as the SIA) based on a known, generally relatively simple relationship with the model parameters.

4.2.2. Discussion of the Indirect Estimation Procedure

When fault detection is performed through parameter estimation of a discrete model, such as the SWA, faults can only be identified using indirect approaches. However, due to the higher complexity and nonlinearity of the relationship with the model parameters in this case, the accuracy of the method is directly affected by how much the fault magnitude deviates from nominal values.

Here, too, the advantage of the continuous estimation method over the discrete one becomes evident, particularly in terms of the complexity of diagnosing multiplicative faults.

Also, more accurate parameter estimation in the indirect identification procedure has been achieved by extracting only selected columns from the Jacobian matrix (matrix of partial derivatives), instead of applying the full matrix as suggested in the literature. This is supported by the analyzed case involving large multiplicative faults that significantly deviate from the nominal operating point, where the literature-proposed method proved inapplicable. In such cases of large faults, when the relationship between model parameters and actual physical parameters is linear, the proposed method provides highly accurate identification. In the case of a nonlinear relationship, the result remains approximate but is still more acceptable than the entirely inaccurate solution obtained using the literature-based approach.

4.3. Conparison of the Results

A summary of key parameter estimation, knowledge-based, and optimization methods used in fault detection, isolation, and identification (FDII) is presented in

Table 21. Comparison of Proposed FDII Framework with Existing Literature.

Method	Real-Time Capability	Effectiveness	Computational Complexity	Characteristics	Fault Detection/Isolation/Identification
Least Squares (LS) [18,32,38]	Offline	Effective for static faults	Low	Not suitable for real time; sensitive to noise; poor performance under transients	Detection only
Recursive Least Squares (RLS) RLS [30,31]	Real time	Effective for gradual parameter changes	Moderate	Sensitive to noise and abrupt faults; requires forgetting factor tuning	Detection + Identification
Extended Kalman Filter (EKF) [29,34,35]	Real time (with tuning)	Suitable for nonlinear systems	High	Requires accurate model; sensitive to noise covariance; tuning required	Full (Detection, Isolation, Identification)
Unscented Kalman Filter (UKF) UKF [36,37]	Real time (with tuning)	High estimation accuracy for nonlinear systems	Very High	Computationally intensive; complex tuning process	Full (Detection, Isolation, Identification)
Algebraic-Geometric Technique (AGT) [33]	Offline	High analytical accuracy	High	Requires symbolic derivation; not robust to noise	Full (Detection, Isolation, Identification)
Genetic Algorithm (GA) [42].	Offline (training/optimisation)	Effective in feature selection, parameter tuning	High	Not suitable for real time; used primarily for offline training/optimisation	Detection, Optimisation (offline)
Particle Swarm Optimisation (PSO) [41].	Offline (training/optimisation)	Effective for weight tuning in ML models	Moderate–High	Not real time; applied offline for tuning; performance depends on swarm settings	Detection, Optimisation (offline)
Neural Networks (NN) [6,18,41]	Possible (inference only)	High accuracy with nonlinear and complex patterns	High (training), Low–Mod (inference)	Training is offline; real time only feasible if model is simple and pretrained	Full (if combined with logic/rules)
Proposed method: Sliding Window Algorithm (SWA)+ selected Jacobian	Real-time	Robust to noise; suited for abrupt faults;	Low	Performance depends on window size;	Full (Detection, Isolation, Identification)
Proposed method: Sliding Integral Algorithm (SIA)+ Decision algorithm	Real-time	Robust to noise; suited for abrupt faults; clear model-parameter link	Moderate	Performance depends on window lengths and data lengths	Full (Detection, Isolation, Identification)

, highlighting their real-time applicability, computational complexity, robustness to noise and transients, and overall FDII capability.

As shown in Table 21, each method presents a trade-off between real-time capability, complexity, and fault coverage. The proposed SWA and SIA approaches demonstrate a favorable balance, offering full diagnostic functionality with low-to-moderate complexity and strong robustness to noise.

5. Conclusions

The paper discusses diagnosis of multiplicative faults caused by sudden changes in one of the parameters of the converter or motor, which significantly deviate from the nominal operating point. Four amplifier faults were considered (changes in resistances R₁, R₉, R₁₂, and capacitance C₈), along with three motor faults (changes in moment of inertia J, viscous friction coefficient B, and armature resistance R_a). The method leverages the advantageous fact that the electromechanical parameters of the motor are reflected in the parameters of the amplifier model, which is realized as a linear electronic circuit with an operational amplifier. Therefore, the fault detection algorithms could be based solely on parameter estimation of the discrete or continuous model of the amplifier, which is a unique approach to the problem.

Diagnosing faults caused by abrupt parameter changes is particularly demanding. Classical parameter estimation techniques yield unsatisfactory results because a sudden change in one system parameter initiates a transient process during which the effect of the change spreads to other parameters as well. To address this, more advanced estimation techniques were employed for detecting multiplicative faults induced by sudden parameter changes in the amplifier or motor. These included: the SWA for estimating parameters of the discrete model and the SIA for estimating parameters of the continuous model.

An additional requirement for the application of these techniques is the necessity of a sufficiently informative input signal (i.e., persistent excitation) of adequate order to enable parameter identification. To satisfy this, a PRBS (Pseudo-Random Binary Sequence) signal was used in the experiments.

This paper implements a modified version of the SIA, which introduces a constraint on the data length in the least squares algorithm, within which the sliding integral is applied. This modification aims to reduce the impact of transients on parameter estimation.

To compare the two approaches to multiplicative fault detection and to assess estimation quality, as well as the influence of Coulomb friction disturbances on system behavior under fault conditions, the responses of the respective models were compared with the measured system response. It was concluded that although the continuous parameter estimation method is more demanding to implement (particularly in real-time applications), it proved equally effective in detecting multiplicative faults in the specific case studied. Furthermore, the influence of Coulomb friction as a disturbance was most pronounced in cases involving a fault in resistance R₁, where both the SWA and SIA algorithms yielded the best results.

To support fault isolation in case of SIA, a new decision algorithm was developed to enable accurate fault localization. In the case of continuous parameter estimation, fault identification—defined as the determination of both the faulty physical parameter and the magnitude of its change—is straightforward. The faulty parameter can be directly identified using the analytical relationships that define the amplifier model.

An alternative approach, applicable when fault detection is performed using the SWA method, involves indirect estimation of changes in physical parameters. This technique is based on evaluating the partial derivatives (i.e., sensitivity coefficients) of the model parameters with respect to the physical parameters at the nominal operating point. The method yields accurate results when the relationship between estimated model parameters and physical parameters is linear. However, in the more common case of nonlinear dependencies, the estimation becomes an approximation, valid primarily for small deviations around the nominal point. Consequently, its accuracy diminishes as the magnitude of the physical fault increases.

It was observed that better estimation results are obtained by selecting specific columns of the Jacobian matrix (i.e., the matrix of partial derivatives), rather than using the full matrix, as suggested in the literature. This approach proved especially effective in cases of nonlinear dependencies, even when the multiplicative faults are large and significantly deviate from the nominal operating point—representing one of the key contributions of this paper.

For the considered multiplicative faults, LabVIEW codes with partial real-time implementation were developed, using the Flat Sequence Frame structure. After signal acquisition, they are processed within LabVIEW MathScript Nodes for implementing the required parameter estimation algorithms. This results in practical solutions suitable for remote laboratories and training in the FDII.

Based on the above, it can be concluded that FDII techniques originally intended for LTI models, with appropriate modifications, can be successfully applied to certain classes of nonlinear dynamic systems, such as the analyzed permanent magnet DC motor with the corresponding amplifier.

Supported by the experimental results, which include precise fault detection, reliable fault isolation, and identification, it can be concluded that the presented findings are fully supported by the obtained data and the observed system behavior.

Feature work will be based on development of a fault-tolerant control (FTC) system that is robust against faults by incorporating a compensation mechanism between the faulty plant and the nominal controller. This compensation mechanism should allow the fault to be accommodated by the controller, ensuring that the faulty plant, under nominal control, remains globally stable and delivers satisfactory performance in both the transient and steady-state regimes.