Damien Koenig — Associate Professor (HDR)

In a classic control loop, a sensor fault is copied by the state feedback: the controller attempts to cancel a flawed error and drives the real system toward a dangerous drift. To detect these failures, two paradigms clash:

In dependability, it is essential to distinguish execution-level functions from supervision-level functions:

To isolate (localize) which sensor or actuator is faulty among $N$ outputs $y_1, y_2, \dots, y_N$, a single observer is generally not enough. We build a bank of $N$ state estimators (DOS / GOS scheme):

• Estimator $i$: Receives the command $u$ and a subset of measurements (e.g., all measurements except $y_i$) to estimate $\hat{y}_{e_i}$.
• Structured residual $r_i(t)$: Compares the real output $y_i$ to its estimation $\hat{y}_{e_i}$: $$r_i(t) = y_i(t) - \hat{y}_{e_i}(t)$$
• Adaptive thresholds: In the presence of model uncertainties, residuals fluctuate within a bounded envelope. As soon as a fault occurs, $r_1(t)$ crosses its adaptive threshold!

Let $x_i(t)$ be the real physical quantity and $y_i(t)$ the measurement provided by sensor $i$ from the time of fault occurrence $t_{F_i}$. There are 5 classic profiles of sensor faults:

Diagnostic analysis systematically decomposes the physical chain into three subsystems subject to three classes of faults:

1. Actuators: Subject to actuator faults $f_A(t)$ (providing $u_a(t)$ from the command $u_c(t)$).
2. Process Dynamics: Subject to component faults $f_c(t)$ and state disturbances $w(t)$.
3. Sensors: Subject to sensor faults $f_s(t)$ and measurement noise $\nu(t)$.

The core of model-based analytical diagnosis relies on the generation of a discrepancy signal called a residual $r(t)$. The methods in the literature are divided into 3 main families:

The most intuitive idea to detect an anomaly consists in running a simulator or output estimator in parallel with the real system:

• Open-Loop Simulator: Receives only $u(t)$. Very sensitive to initialization errors $x(0) \neq \hat{x}(0)$ and unusable if the system is unstable!
• Closed-Loop Estimator (Observer): Uses both $u(t)$ and the measurement $y(t)$ to make $\hat{y}(t) \to y(t)$ converge in healthy conditions.

In discrete time (operator $z$) or continuous time ($p$), the system output subjected to known inputs $u^*(z)$ and faults $f(z)$ is written:

Any linear residual generator filters the known signals $u^*(z)$ and $y(z)$ via two transfer matrices $H_{u^*}(z)$ and $H_y(z)$:

By substituting $y(z)$, we obtain: $r(z) = \big[H_{u^*}(z) + H_y(z)G_{yu^*}(z)\big]\,u^*(z) + H_y(z)G_{yf}(z)\,f(z)$.

In the presence of measurement noise, the residual $r(t)$ is never strictly zero even without a fault. We define an evaluation function (e.g., $J(r(t)) \equiv |r(t)|$) compared to a detection threshold $\varepsilon(t)$:

Statistical threshold setting (Gaussian process):
We estimate in healthy conditions the mean $\bar{r}_i$ and variance $\bar{\sigma}_i^2$ of each residual $r_i(t)$:

For a normal distribution, $\delta = 3$ guarantees that $99.73\%$ of healthy samples remain within the corridor $[\bar{r}_i - 3\bar{\sigma}_i, \bar{r}_i + 3\bar{\sigma}_i]$, thereby limiting the false alarm rate to less than $0.27\%$.

To generate a residual $r(t) = y(t) - \hat{y}(t)$, why not just run a simple open-loop simulator $\dot{\hat{x}}(t) = A\hat{x}(t) + Bu(t)$ in parallel with the real system?

First, let us consider the linear time-invariant state-space model of a process in the absence of any fault and any disturbance:

By defining the state estimation error $e_x(t) = x(t) - \hat{x}(t)$, its dynamics are obtained by subtraction ($\dot{e}_x = \dot{x} - \dot{\hat{x}}$):

Now, let's introduce a fault vector $f(t)$ acting both on the state equation via the matrix $F_x$ and on the measurement via the matrix $F_s$:

Since the observer keeps the same nominal structure, the dynamics of the estimation error $e_x = x - \hat{x}$ and the output error $\tilde{y} = y - \hat{y}$ become:

To analyze in detail the effect of a specific fault on a BFDF (Beard-Jones Fault Detection Filter), we explicitly express the column of the faulty actuator $i$ and the row of the faulty sensor $j$:

• Fault on the $i$^th actuator ($i = 1, \dots, r$): $b_i \in \mathbb{R}^n$ is the $i$^th column of the control matrix $B$, and $f_{ai}(t)$ is the unknown scalar signal describing the evolution of the actuator fault.
• Fault on the $j$^th sensor ($j = 1, \dots, m$): $I_j \in \mathbb{R}^m$ is the $j$^th standard basis vector ($1$ at row $j$, $0$ elsewhere), and $f_{sj}(t)$ is the fault on sensor $j$.

Consider an electromechanical system of order $n = 2$ with a scalar measurement $y(t) = x_1(t)$ and an actuator fault $f_a(t)$:

1 Observability: $\mathcal{O} = \begin{bmatrix} C \\ CA \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix} \implies \text{rank}(\mathcal{O}) = 2 = n$.

2 Calculation of the gain $L = [l_1 \;\; l_2]^T$ to place the poles at $\{-3, -4\}$:

3 Static response of the residual to a step fault $f_a(t) = f_{a0}$:

Consider a system with no actuator fault but subject to potentially simultaneous sensor faults described by the vector $f_s(t) = [f_{s1}(t), \dots, f_{sm}(t)]^T$:

To isolate each sensor independently of the others, we build a bank of $m$ observers where the $i$^th observer is driven solely by the scalar measurement $y_i(t)$:

By assembling the $m$ dedicated observers in parallel (one per output sensor), we obtain the DOS architecture (Dedicated Observer Scheme, introduced by R.N. Clark):

• Number of estimators: Equal to the number $m$ of measured outputs of the system.
• Signal Ungrouping: The measurement vector $y \in \mathbb{R}^m$ is split into $m$ scalar channels $y_1, \dots, y_j, \dots, y_m$.
• Inputs of each observer: The $j$^th identity observer (or Kalman Filter $\text{KF}_j$) receives all control inputs $u$ and a single scalar output $y_j$.
• Total decoupling: A fault on the $j$^th output affects only the residual $r_j(t)$!

Thanks to the direct decoupling between measurement channels, elementary threshold logic allows instant localization of each sensor fault:

Incidence table (Signature matrix) of the DOS scheme:
A "1" at the intersection of row $i$ and column $j$ indicates that residual $r_j$ is sensitive to fault $f_{si}$, while a "0" indicates total insensitivity:

When observability from a single sensor is impossible and we assume that only one sensor fails at a time (no simultaneous faults), we use the GOS scheme (Generalized Observer Scheme, P.M. Frank):

• Signal Grouping: For each channel $j \in \{1, \dots, m\}$, we form a reduced vector $y^j \in \mathbb{R}^{m-1}$ containing all measurements EXCEPT the $j$^th measurement $y_j$: $$y^j = \begin{bmatrix} y_1 & \dots & y_{j-1} & y_{j+1} & \dots & y_m \end{bmatrix}^T \in \mathbb{R}^{m-1}$$
• Observer $j$: Driven by $u$ and by the sub-vector $y^j$ (observation matrix $C^j$ without row $C_j$). It generates the vectorial residual: $$r^j(t) = y^j(t) - C^j\,\hat{x}^j(t) \in \mathbb{R}^{m-1}$$
• Key property: Since observer $j$ never uses sensor $j$, the residual $r^j(t)$ is completely insensitive to fault $f_{sj}$, but sensitive to all other faults $f_{sk}$ ($k \neq j$)!

In the GOS scheme, the incidence table is the exact logical complement of that of the DOS scheme: the diagonal is composed of "0"s and all off-diagonal terms are "1":

Let $L(\|r^i\|)$ be the binary logic function equal to $1$ if $\|r^i\| \ge \varepsilon^i$ and $0$ otherwise. The alarm for sensor $1$ (denoted $a_1$) is triggered by the Boolean equation:

Consider a process of order $n = 2$ equipped with $m = 3$ redundant sensors (assuming only one sensor can fail at a time):

1 Construction of the 3 GOS sub-models ($C^j \in \mathbb{R}^{2\times 2}$ without row $c_j$):

2 Pole placement for each observer $j \in \{1, 2, 3\}$ at $\{-5, -6\}$:
Since each pair $(A, C^j)$ is of rank 2 (observable), we calculate the 3 gain matrices $L^j \in \mathbb{R}^{2\times 2}$ such that $\lambda(A - L^j C^j) = \{-5, -6\}$. The residual $r^j(t) = y^j(t) - C^j \hat{x}^j(t) \in \mathbb{R}^2$ is evaluated by its norm $\|r^j(t)\|_2$.

What happens if the real process is subjected to an external disturbance or model uncertainty $E\,d(t)$ (wind, unknown load torque, nonlinearity) while no fault is present ($f = 0$)?

Critical consequence: Whatever the choice of gain $L$, the term $+E\,d(t)$ remains intact in the equation of $\dot{e}_x(t)$ and causes permanent false alarms!

In the presence of model uncertainties or unknown external disturbances $d(t) \in \mathbb{R}^q$, the goal of robust FDI is to construct a residual totally independent of $d(t)$ but sensitive to faults.

The matrices $F, T, G, N$ are the degrees of freedom to be synthesized to exactly cancel the effect of $d(t)$ on $e_x(t)$.

Let us express the state estimation error $e_x(t) = x(t) - \hat{x}(t)$ by substituting $\hat{x}(t) = z(t) + N\big(Cx(t) + f_s(t)\big)$:

By differentiating $e_x(t)$ with respect to time ($\dot{e}_x = T\dot{x} - \dot{z} - N\dot{f}_s$) and substituting $z(t) = Tx(t) - Nf_s(t) - e_x(t)$, we get:

Step 1 Verify the unknown input decoupling condition:

Step 2 Calculate the matrices $N$ and $T$ via the pseudo-inverse $(CE)^+$:
Since $TE = 0 \iff (I - NC)E = 0 \iff N(CE) = E$, an explicit solution is given by the left Moore-Penrose pseudo-inverse:

Step 3 Solve the Sylvester equation & place the poles:
By setting $K = G - FN$, the equation $TA - GC - FT = 0$ with $T = I - NC$ can be rewritten as:

If the pair $(TA, C)$ is observable (or detectable), we calculate $K$ by pole placement or the LQ method to stabilize $F$, and then deduce $G$.

Once the UIO observer is synthesized ($TE = 0$ and $F = TA - KC$), the residual is governed by the reduced system:

We can see that the disturbance $d(t)$ has completely disappeared. But under what condition does an actuator fault $f_a(t)$ remain visible on the residual $r(t)$?

How do we combine robustness against disturbances $d(t)$ AND fault localization (isolation)?

• Isolation problem: To localize with certainty which sensor or actuator is faulty despite the permanent presence of the unknown input $d(t)$.
• Sensitivity / insensitivity properties: Ideally, each residual $r^j(t)$ should be sensitive to a single fault and insensitive to all others (DOS scheme), but the rank constraints $\text{rank}(C_j E) = \text{rank}(E)$ often make this impossible with a single sensor.
• Standard industrial solution (GOS Bank of UIOs):
  • To isolate sensor faults: each $\text{UIO}_j$ is driven by $u$ and by $y^j$ (all measurements except $y_j$) and decouples $d(t)$.
  • To isolate actuator faults: the $i$^th observer $\text{UIO}_i$ groups the $i$^th actuator fault $b_i f_{ai}(t)$ with the unknown input by defining an augmented matrix $E_i = [E \;\; b_i]$.

At the end of Sequence 2, we obtained the error equations for a single UIO observer decoupling the disturbance $E\,d(t)$ ($T E = 0$):

To design a robust sensor fault isolation scheme, we assume that all actuators are healthy ($f_a(t) = 0$). For each index $j \in \{1, 2, \dots, m\}$, we partition the measurement vector $y(t) \in \mathbb{R}^m$ by isolating the $j$^th component:

• $c_j \in \mathbb{R}^{1 \times n}$ denotes the $j$^th row of the observation matrix $C$.
• $C^j \in \mathbb{R}^{(m-1) \times n}$ is obtained from $C$ by deleting the $j$^th row $c_j$.
• $y_j(t) \in \mathbb{R}$ is the $j$^th scalar component of the measurement vector $y(t)$.
• $y^j(t) \in \mathbb{R}^{m-1}$ (and $f_s^j(t) \in \mathbb{R}^{m-1}$) is obtained from $y(t)$ (resp. $f_s(t)$) by deleting its $j$^th component $y_j(t)$ (resp. $f_{sj}(t)$).

Based on the previous reduced sub-model, we build a bank of $m$ UIO-based residual generators (for $j = 1, 2, \dots, m$):

For each observer $j$ to be an Unknown Input Observer (UIO) decoupling $E\,d(t)$, its parameter matrices must satisfy:

When all actuators are healthy ($f_a = 0$) and a fault occurs on the $j$^th sensor ($f_{sj}(t) \neq 0$), the $m$ residuals satisfy the following isolation logic:

Physical explanation of the mechanism:

• The Signal Grouping block forms the $m$ sub-vectors $y^1, \dots, y^j, \dots, y^m \in \mathbb{R}^{m-1}$.
• The observer UIO $j$ does not receive the measurement $y_j(t)$ corrupted by $f_{sj}(t)$ and decouples $d(t)$: its state error tends to $0$ and $\|r^j(t)\| < \varepsilon^j$.
• All other observers UIO $k$ ($k \neq j$) use the faulty measurement $y_j(t)$ in their vector $y^k(t)$: their residuals $\|r^k(t)\|$ cross their thresholds $\varepsilon^k$!

The bank of $m$ UIO observers realizes a Generalized Observer Scheme (GOS) that is robust against $d(t)$. Its isolability table (Table 1) presents "0"s on the diagonal and "1"s everywhere else:

By defining the binary logic function $L(\|r^i\|) = 1$ if $\|r^i\| \ge \varepsilon^i$ (and $0$ otherwise), the sensor alarm $a_j$ is generated by the Boolean intersection:

For each of the $m$ UIO observers ($j = 1, \dots, m$) to simultaneously satisfy the decoupling constraint $T^j E = 0$ and the stability of $F^j = T^j A - K^j C^j$, two rank conditions must be met for all $j \in \{1, \dots, m\}$:

Proof of condition $c_1$ (Rao & Mitra, 1971):

To isolate sensor $j$, it was enough not to connect the wire $y_j(t)$ to observer $\text{UIO } j$. Why is it not enough to disconnect the control wire $u_i(t)$ to isolate actuator $i$?

To design a robust actuator fault isolation scheme, we now assume that all sensors are healthy ($f_s(t) = 0$). We want to isolate a fault $f_{ai}(t)$ on the $i$^th actuator ($i = 1, \dots, r$):

• $b_i \in \mathbb{R}^n$ is the $i$^th column of $B$, and $B^i \in \mathbb{R}^{n \times (r-1)}$ is obtained by deleting column $b_i$ from $B$.
• $u_i(t)$ is the $i$^th component of $u(t)$, and $u^i(t) \in \mathbb{R}^{r-1}$ is obtained by deleting $u_i(t)$ from $u(t)$.
• Fundamental trick: We merge the path of actuator $i$ with the vector of unknown disturbances $d(t)$ by defining:

Starting from the augmented representation $E^i = [E \;\; b_i]$, we build a bank of $r$ UIO-based residual generators (one per actuator, for $i = 1, 2, \dots, r$):

For the $i$^th observer to exactly decouple $E^i d^i(t)$ (meaning $T^i E^i = 0$), its parameter matrices must satisfy:

When all sensors are healthy ($f_s = 0$) and a fault occurs on the $i$^th actuator ($f_{ai}(t) \neq 0$), the estimation error dynamics of the $k$^th observer is written as:

Since $f_a^i(t)$ does not contain $f_{ai}(t)$ (because $T^i b_i = 0$), the $r$ residuals satisfy the isolation logic:

The bank of $r$ UIO observers realizes a Generalized Observer Scheme (GOS) for the $r$ actuators. A "1" at row $i$ and column $k$ means that the residual $r^k$ is sensitive to the actuator fault $f_{ui}$ ($f_{ai}$), while a "0" indicates total insensitivity:

With the binary logic function $L(\|r^i\|) = 1 \iff \|r^i\| \ge \varepsilon^i$, the failure alarm of the 1^st actuator (or the $i$^th actuator) obeys the Boolean law:

For each of the $r$ UIO observers ($i = 1, \dots, r$) to satisfy the decoupling constraints $T^i E^i = 0$ and the stability of $F^i = T^i A - K^i C$, the following two rank conditions must be met for each $i \in \{1, \dots, r\}$:

The analysis of the two robust isolation schemes (sensors and actuators) presented previously leads to three essential methodological remarks:

Let's consider a linear system of order $n = 3$, equipped with $m = 3$ sensors ($C = I_3$), $r = 2$ actuators ($B = [b_1 \;\; b_2]$), and subjected to $q = 1$ scalar disturbance $d(t)$:

Question 1 Can the 3 sensor faults be isolated by a GOS bank of 3 UIOs?

Question 2 Can the 2 actuator faults ($b_1$ and $b_2$) be isolated?

Let's continue the previous example by explicitly calculating the static matrices $N^1$ and $T^1$ of the observer UIO 1, whose task is to simultaneously decouple the disturbance $E$ and the first actuator $b_1$:

Step 1 Calculation of the left pseudo-inverse $(C E^1)^+ = [(E^1)^T E^1]^{-1}(E^1)^T$:

Step 2 Deduction of $N^1 = E^1 (C E^1)^+$ and the projector $T^1 = I_3 - N^1 C$:

Let's complete the example with the following state matrix $A$ and the reconfigured actuator $\tilde{b}_2 = [0, 0, 1]^T$. We calculate the dynamic matrices $F^1, K^1, G^1$ of UIO 1:

Step 3 Hautus Verification ($c_2$) & Pole Placement of $F^1 = T^1 A - K^1 C$:
Since $C = I_3$, the pair $(T^1 A, I_3)$ is trivially observable ($\text{rank}[pI_3 - T^1 A ; I_3] = 3$). We choose $K^1$ to place the 3 eigenvalues of $F^1$ at $\lambda_1 = \lambda_2 = \lambda_3 = -4$:

Step 4 Calculation of gain $G^1 = K^1 + F^1 N^1$ and input matrix $T^1 B^1$:

To simulate a UIO isolation bank in MATLAB / Simulink, each residual generator $j$ is implemented using a standard State-Space block receiving the concatenated vector $[u(t) ; y^j(t)]$:

What is the practical advantage of the parity relations approach compared to the state observers studied in Sequences 2 and 3?

Consider the general problem of measuring an unknown state vector $x_k \in \mathbb{R}^n$ using $m$ sensors described by the algebraic equation:

Depending on the number of sensors $m$ relative to the dimension $n$ of the state, two fundamental situations arise:

• Case 1: $m < n$ (or $m = n$ with $C$ invertible — No direct spatial redundancy): There are fewer sensors than state unknowns. At a given instant $k$, any variation in $y_k$ can be explained by a legitimate variation in $x_k$.
• Case 2: $m > n$ with $\text{rank}(C) = n$ (With direct spatial redundancy): The rows of $C$ are linearly dependent; we can eliminate $x_k$ at instant $k$.

When $m < n$ with $y_k = C\,x_k$ and $\text{rank}(C) = m$, no direct spatial redundancy relation exists at an isolated instant $k$ (the left null space of $C$ is reduced to $\{0\}$ because the $m$ rows of $C$ are linearly independent).

On the other hand, if $x_k$ evolves according to a known dynamic law, we can artificially create redundancy by stacking the sensor measurements over a time horizon of $s+1$ samples (sliding window from $k-s$ to $k$):

When $m > n$ and $\text{rank}(C) = n$, the rows of $C$ are linked by $m - n$ static algebraic relations. Consider a system with $n = 2$ states measured by $m = 4$ sensors:

1 Elimination of $X$: Since $C_1$ is invertible, $X = C_1^{-1} Y_1$, which we substitute into $Y_2 = C_2 X = C_2 C_1^{-1} Y_1$:

2 3^rd relation (by eliminating $y_2$): $r_3 = r_1 - 0.25\,r_2 = 0.5\,y_1 - y_3 + 0.25\,y_4 = 0$.

Let's generalize the previous example (Example 2): let $y_k = C\,x_k + f_k$ with $m > n$ and $\text{rank}(C) = n$. The number of measurements is strictly greater than the number of state variables $x_k$.

For diagnosis (FDI), we linearly combine the components of $y_k$ via a parity matrix $V \in \mathbb{R}^{(m-n) \times m}$ to generate the parity vector (residual) $r_k$:

For the residual $r_k$ to be zero in the absence of a fault and completely insensitive to the unknown state $x_k$ ($\forall x_k \in \mathbb{R}^n$), the matrix $V$ must belong to the left null space of $C$:

When this orthogonality condition is met, the residual contains only the information about the sensor faults:

Let's consider the noisy static system (Example 3) where the fault vector $d_k \in \mathbb{R}^p$ is decomposed into $d_k^- \in \mathbb{R}^{p-1}$ (faults or disturbances to make undetectable) and $d_k^+ \in \mathbb{R}^1$ (the target fault to detect):

1 Ideal solution (Exact decoupling): Find a row vector $\Omega \in \mathbb{R}^{1 \times m}$ such that $P_k = \Omega\,y_k = \Omega\,F^+\,d_k^+ + \Omega\,\epsilon_k$, i.e.:

2 Optimal projection (Approximate decoupling): In practice, if the number of columns of $[C \;\; F^-]$ is equal to or greater than $m$, the condition $\Omega\,[C \;\; F^-] = 0$ with $\Omega F^+ \neq 0$ is impossible to achieve. We then look for an optimal vector $\Omega$ that provides the best compromise:

How to simultaneously solve the two antagonistic objectives $J_1 = \min_{\Omega} \|\Omega\,[C \;\; F^-]\|$ (robustness) and $J_2 = \max_{\Omega} \|\Omega\,F^+\|$ (sensitivity to fault $d^+$)?

• A solution minimizing $J_1$ alone ($\Omega = 0$) obviously cannot maximize $J_2$ at the same time.
• To obtain a mathematically well-posed and scale-invariant compromise (homogeneous of degree 0 in $\Omega$), we define a single performance index $J$ equal to the quadratic ratio of the two criteria:

We are trying to solve the optimization problem under an equality constraint:

Since $C \in \mathbb{R}^{m \times n}$ has full column rank ($\text{rank}(C) = n$), we can partition $C$ and $\Omega = \begin{pmatrix} \Omega_1^T & \Omega_2^T \end{pmatrix}$ with a regular invertible block $C_1 \in \mathbb{R}^{n \times n}$:

By substituting $\Omega = \Omega_2^T P$, the constrained problem in $\Omega \in \mathbb{R}^{1 \times m}$ is reduced to an unconstrained problem on the free vector $\Omega_2 \in \mathbb{R}^{m-n}$:

By defining the two symmetric positive semi-definite matrices $A = P F^-(F^-)^T P^T$ and $B = P F^+(F^+)^T P^T$, the optimization problem is written:

This problem is solved by computing the smallest generalized eigenvalue $\underline{\gamma}$ of the pair $(A, B)$ and its associated generalized eigenvector $X^* = \Omega_2^*$.
Rigorous proof: Let $F_1(X) = \dfrac{X^T A X}{X^T B X}$. At the minimum $X = X^*$, where $F_1(X^*) = \underline{\gamma}$, the gradient vanishes:

The roots of $\det(A - \gamma B) = 0$, ordered as $\underline{\gamma} < \gamma_j < \dots < \bar{\gamma}$, are the generalized eigenvalues: the global minimum is $\underline{\gamma}$!

Let's now move to the general case of a linear time-invariant discrete dynamic system described by the following state representation:

• $u_k \in \mathbb{R}^r$ is the vector of known control inputs.
• $y_k \in \mathbb{R}^m$ is the vector of known measured outputs.
• $x_k \in \mathbb{R}^n$ is the unknown state vector of the system.
• $f_k \in \mathbb{R}^g$ denotes the vector of unknown faults (which can group actuator, component, or sensor faults).
• $\{A, B, C, D, R_1, R_2\}$ are known real matrices of appropriate dimensions.

By stacking the output equations from instant $k-s$ to instant $k$, we obtain the temporal redundancy matrix equation:

And the fault transfer matrix $M \in \mathbb{R}^{(s+1)m \times (s+1)g}$ is built exactly like the Toeplitz matrix $H$ by replacing the pair $\{D, B\}$ with $\{R_2, R_1\}$!

From the data relation $Y_k - H\,U_k = W\,x_{k-s} + M\,F_k$, we define the residual signal $r_k \in \mathbb{R}^p$ calculated online by:

To analyze the internal behavior of the residual, let's replace $Y_k - H U_k$ with its expression as a function of the state and faults:

For this parity vector to be a valid FDI residual, the projection matrix $V$ must imperatively satisfy two fundamental conditions:

Consider a discrete system of order $n = 2$ (period $T_e = 0.1\text{ s}$) equipped with a single sensor ($m = 1 < n$), subjected to an actuator fault $f_{a,k}$ and a sensor fault $f_{s,k}$ ($f_k = [f_{a,k} \;\; f_{s,k}]^T$):

1 Choice of order $s$: With $m=1, n=2$, the condition $(s+1)m > n$ requires $\mathbf{s = 2}$ ($(2+1)\times 1 = 3 > 2$). The horizon matrices $W \in \mathbb{R}^{3\times 2}$ and $H \in \mathbb{R}^{3\times 3}$ are:

2 Parity vector $V = [v_0 \;\; v_1 \;\; v_2] \in \text{Ker}(W^T)$:

The MATLAB script below automatically constructs the horizon matrices $W, H, M$, calculates the parity vector $V$ via the null(W')' command, and simulates the residual $r_k$:

Let's analyze the time response obtained in Simulink ($T_e = 0.1\text{ s}$, non-zero initial condition $x_0 = [2 \;\; -1]^T$) for two successive fault scenarios:

Let us consider the dynamic model subject only to sensor faults $f_k = [f_{1k}, \dots, f_{mk}]^T$:

The auto-redundancy approach consists of building a time parity relation sensor by sensor (using only the history $Y_{ik}$ of the $i$^th scalar measurement $y_{ik} = C_i x_k + f_{ik}$):

Finally, let's consider the general discrete dynamic system subject simultaneously to faults $f_k \in \mathbb{R}^g$ and unknown inputs (disturbances) $d_k \in \mathbb{R}^q$:

Over the sliding horizon $[k-s, k]$, the stacked data equation becomes:

To detect and isolate the faults $f_k$, a DOS or GOS scheme is then deployed in the parity space (exactly as for UIO banks).

Why are observers and parity space not always sufficient when an internal component of the process degrades (heating of a resistor $\Delta R$, leak, wear of a spring $\Delta k$)?

Parametric estimation for fault detection relies on the following postulates and methodological steps:

• Parameter ignorance or drift: The process parameters are not known at all, or not known with sufficient precision. They can be determined by parametric identification methods.
• Structural knowledge: The basic structure of the model (order of differential equations, physical laws) must be known a priori.
• Physical hypothesis: It is assumed that component faults (wear, leak, short circuit, fouling) are directly reflected in the physical parameters of the system.
• Online estimation: The parameters of the real process are estimated in real-time from the measurements $(u(t), y(t))$.
• Comparison with reference model: The estimates $\hat{\theta}(t)$ are compared to the nominal parameters $\theta_0$ initially obtained under healthy conditions (fault-free). Any significant deviation ($\Delta\theta \neq 0$) indicates the occurrence of a fault.

Let us consider a linear-in-its-parameters model (SISO) linking the sensor measurements vector $Z$ to the unknown parameters vector $\underline{\theta}$:

Example of a polynomial calorimeter: $Y = \beta_0 + \beta_1 U + \beta_3 U^3 + \epsilon$ is written in linear matrix form with respect to $\underline{\theta}$:

We minimize the quadratic criterion weighted by $V^{-1}$:

Consider a continuous linear dynamic system described around its operating point $(U_{00}, Y_{00})$ by the centered signals $u(t) = U(t) - U_{00}$ and $y(t) = Y(t) - Y_{00}$:

By isolating $y(t)$, this differential equation can be written as a linear regression:

• Additive faults: $f_u(t)$ (input fault) and $f_y(t)$ (output fault).
• Multiplicative (parametric) faults: Variations $\Delta a_i$ and $\Delta b_j$ of the coefficients of the transfer function $G_p(s) = \dfrac{B(s)}{A(s)}$.

The previous differential equation can also be written in the form of a canonical state representation (companion form):

In this state representation, the two categories of faults are clearly distinguished:

• Additive faults: $f_l(t)$ (input or state variable fault) and $f_m(t)$ (fault on the output measurement).
• Multiplicative (parametric) faults: Matrix perturbations $\Delta A$, $\Delta b$ and $\Delta c^T$ caused by variations $\Delta a_i$, $\Delta b_i$, $\Delta c_j$.

Parametric identification minimizes the quadratic cost function $V = \sum_k e^2(k)$ relating to the equation error $e(k) = y(k) - h^T(k)\hat{\Theta}$ according to two variants:

From the estimate $\hat{\Theta}$, two levels of symptoms (residuals) are generated:

Consider a discrete-time system described by an ARX (AutoRegressive with eXogenous input) model using the backward shift operator $q^{-1}$ ($q^{-1} y(t) = y(t-1)$):

By expanding the polynomials in $q^{-1}$, the output at instant $t$ is expressed in the standard linear regression form:

By collecting measurements for $k = 1, \dots, t$, we stack the equations row by row into a large matrix relation.

Using the stacked vector notations up to instant $t$:

By minimizing the Ordinary Least Squares (OLS) criterion with $V = \sigma^2 I_t$:

We obtain the non-recursive analytical solution (OLS) at instant $t$:

At instant $t+1$, a new scalar measurement $y(t+1)$ and a new regression vector $h^T(t+1)$ are appended to the bottom of the matrix system:

Let's apply the Least Squares formula at instant $t+1$:

By performing the block matrix multiplications, we obtain the fundamental starting relation:

To invert the sum $H^T(t)H(t) + h(t+1)h^T(t+1)$ without recalculating a matrix inverse, we apply the Matrix Inversion Lemma:

Let's identify the 4 terms of our problem with the general formula:

Since $C = 1$ and $D A^{-1} B = h^T(t+1) P(t) h(t+1)$ are scalars, the inverse $(C^{-1} + D A^{-1} B)^{-1}$ reduces to a simple division by the scalar $1 + h^T(t+1)P(t)h(t+1)$! This leads to the RLS equations:

By expanding the product of $P(t+1)$ and $\big(H_t^T Z(t) + h_{t+1} y(t+1)\big)$ and using the identity $\hat{\theta}(t) = (H_t^T H_t)^{-1} H_t^T Z(t)$, we obtain:

By bringing the bracketed term before $y(t+1)$ to a common denominator:

We then factor out the gain vector $K_{t+1}$ from the a priori prediction error:

Why does standard RLS fall asleep after a long period of healthy operation?
In classical RLS, the matrix $P^{-1}(t) = \sum_{k=1}^t h(k)h^T(k)$ grows indefinitely with $t$. Consequently, $\mathbf{P(t) \to 0}$ and the adaptation gain $\mathbf{K_{t+1} \to 0}$: if a component fault occurs at $t = 5000$, the algorithm barely reacts!

The solution for FDI diagnosis: the Forgetting Factor $\lambda \in ]0, 1[$
We exponentially weight past measurements in the criterion in order to favor recent data:

The division by $\lambda < 1$ keeps $P(t)$ and $K_{t+1}$ constantly active (effective memory of $N_{\text{eq}} \approx \frac{1}{1-\lambda}$ samples).

Consider the series $RC$ electrical circuit supplied by voltage $u_1(t)$, with current $i(t) = C\frac{du_2(t)}{dt}$ flowing through it, and capacitor voltage $u_2(t)$. Let's set the model parameters $b_0 = 1$, $b_1 = C$, and $a_1 = R C$:

How to implement on a digital computer the estimation of Case 2 starting from the continuous differential equation:

1 Approximation of derivatives by the implicit Euler scheme (step $\Delta t$):

2 Putting into linear regression form $y_{n+1} = h_{n+1}^T\,\theta$:
By isolating the current measurement $i^{n+1}$ on the left, we directly obtain:

In industrial practice, the number of sensors $m$ is limited and does not allow for exact decoupling of all disturbances. The disturbances are then broken down into two classes:

• Structured dominant disturbances $d(t)$: Decoupled exactly by the algebraic UIO structure within a GOS bank ($\mathbf{T^j E_d = 0}$ and decoupling of the $j$^th fault).
• Residual noises and uncertainties $w(t) \in \mathcal{L}_2$: Impossible to cancel exactly. The remaining degrees of freedom in the observer gain $K^j$ ($F^j = T^j A - K^j C^j$) are then used to perform a robust $H_\infty$ / $H_-$ synthesis via Linear Matrix Inequalities (LMI):

The ultimate goal of diagnosis (FDD) is to allow Fault Tolerant Control (FTC) to maintain the stability and performance of the closed-loop system despite the occurrence of a fault:

• 1. Passive FTC via Robust $H_\infty$ Control: We synthesize a single controller $u(t) = -K_c \hat{x}(t)$ guaranteeing $\|T_{z w}\|_\infty < \gamma$ for a whole family of bounded faults $(\Delta A, \Delta B)$ using the same LMI / Riccati tools as the $H_\infty$ observer.
• 2. Active FTC (Fault Accommodation / Compensation): Thanks to the fault estimate $\hat{f}_a(t)$ provided by the UIO / $H_\infty$ observer bank, we add a direct compensating term in the control law:

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