2.1 Modeling of RF components: a case study on the LNA

The X-band chain of the monopulse system at the Oran station incorporates three identical amplifiers [Reference Nasri, Saiah, Meghabber and Bouchiba22] of reference AMFW-6S-07700850-50-N-R [23]. Their modeling is based on the manufacturer’s specifications in order to faithfully reproduce their performance (see Table 1).

To anticipate the effects of potential failures, it is essential to define a tolerance strategy for the LNA, particularly by considering the impact of a fault on the input impedance. The effective gain of the LNA depends on the impedance mismatch and can be expressed using the reflection coefficient ${{\Gamma }}$ , defined as

(1) \begin{align}{\boldsymbol{\Gamma }} = \frac{{{Z_{in}} - {Z_s}}}{{{Z_{in}} + {Z_s}}}\end{align}

with ${Z_{in}}$ being the input impedance of the LNA and ${Z_s}$ the source impedance (typically 50Ω). The corresponding gain loss in decibels is given by [Reference Pozar24]:

(2) \begin{align}\Delta \boldsymbol{G}({{{dB}}}) = 10{\textrm{log}}_{10}({1 - {{|{\boldsymbol{\Gamma }}|}^2}})\end{align}

Table 1.

Main specifications of the AMFW-6S-07700850-50-N-R amplifier

Table 1 Long description

A table with three columns and eight rows detailing the specifications of the AMFW-6S-07700850-50-N-R amplifier. The columns are labeled Parameter, Value, and Significance. Row 1: Parameter, Frequency band; Value, 7.70 – 8.50 GHz; Significance, Amplifier bandwidth. Row 2: Parameter, Gain; Value, ≥ 50 dB; Significance, Nominal gain. Row 3: Parameter, Gain flatness; Value, ± 1 dB; Significance, Frequency variation of gain. Row 4: Parameter, Gain vs temperature; Value, ± 5 dB; Significance, Thermal dependence. Row 5: Parameter, Noise factor; Value, 45 K max (40 K typ.); Significance, Added noise. Row 6: Parameter, P1dB output; Value, + 10 dBm min; Significance, Linearity limit. Row 7: Parameter, VSWR in/out; Value, ≤ 1.5:1; Significance, Impedance matching. Row 8: Parameter, Temperature; Value, -40 to +60 °C; Significance, Temperature rating.

For a given voltage standing wave ratio (VSWR), the reflection coefficient can be calculated as follows:

(3) \begin{align}\left| {\boldsymbol{\Gamma }} \right| = \frac{{{\textrm{VSWR}} - 1}}{{{\textrm{VSWR}} + 1}}\end{align}

Table 2 illustrates the impact of the reflection coefficient $\left| {{\Gamma }} \right|$ on the LNA gain.

Table 2.

Impact of impedance matching on LNA gain

Table 2 Long description

A table with four columns and nine rows. The columns are labeled as follows: |Γ|, VSWR (approx.), ΔG (dB), and Interpretation. The rows present the following data: Row 1: |Γ|, 0.00; VSWR (approx.), 1.00:1; ΔG (dB), 0.00; Interpretation, Perfect match, nominal gain 50 dB. Row 2: |Γ|, 0.05; VSWR (approx.), 1.11:1; ΔG (dB), 0.01; Interpretation, Very good match, minimal loss. Row 3: |Γ|, 0.10; VSWR (approx.), 1.22:1; ΔG (dB), 0.04; Interpretation, No noticeable loss. Row 4: |Γ|, 0.15; VSWR (approx.), 1.35:1; ΔG (dB), 0.11; Interpretation, Slight reduction, <0.2 dB. Row 5: |Γ|, 0.20; VSWR (approx.), 1.50:1; ΔG (dB), 0.18; Interpretation, Maximum datasheet limit, loss <0.2 dB. Row 6: |Γ|, 0.25; VSWR (approx.), 1.67:1; ΔG (dB), 0.28; Interpretation, Acceptable but outside LNA specs. Row 7: |Γ|, 0.30; VSWR (approx.), 1.86:1; ΔG (dB), 0.92; Interpretation, Visible impact (1 dB). Row 8: |Γ|, 0.40; VSWR (approx.), 2.33:1; ΔG (dB), 1.92; Interpretation, Poor match, strong impact. Row 9: |Γ|, 0.50; VSWR (approx.), 3.00:1; ΔG (dB), 3.01; Interpretation, Significant loss. Row 10: |Γ|, 0.70; VSWR (approx.), 5.67:1; ΔG (dB), 5.55; Interpretation, Critical, very poor match. Row 11: |Γ|, 0.90; VSWR (approx.), 19.00:1; ΔG (dB), 7.20; Interpretation, Almost total reflection.

Assumption 1. Gain failure due to LNA mismatch (critical mission). A relevant interval for the failure assumption is defined by:

\begin{align*}\left\{\! \begin{array}{l@{\quad}l}|\unicode{x1D71E} | \in[{0.25,{{\;}}0.40} [(\Delta \boldsymbol{G} \in [-\!1.92, - 0.28\,dB])\qquad\quad \textrm{Potential risk out of specification}.\\| \unicode{x1D71E} | \ge 0.40\;\;\Delta G\; \le - 1.92\;dB \qquad \qquad \qquad \qquad \textrm{Strict and valid failure assumption.}\end{array} \right.\end{align*}

2.2 The probabilistic decision algorithm

A probabilistic decision algorithm is employed to estimate the failure probability of the LNA component. The algorithm uses the outputs of the TSR as observations $\unicode{x1D71F}{\boldsymbol{G}_k}$ to compute the likelihood of a fault. The likelihood function is based on the measured reflection coefficient of magnitude ${\boldsymbol{x}_k} = \left| {{{\boldsymbol{\Gamma }}_k}} \right|$ . In the event of failure or degradation of one of the LNAs, the likelihood function automatically transfers control to the ML-TSR module, which relies on ML to maintain system reliability.

The gain loss can be expressed in terms of the reflection coefficient as:

(4) \begin{align}\unicode{x1D71F} {\boldsymbol{G}_k} = 10\,{\textrm{log}}_{10}\big({1 - {\boldsymbol{x}_k}^2}\big),\,{\boldsymbol{x}_k} = |{\boldsymbol{\Gamma}}|\end{align}

The likelihood of the measured gain variation $\unicode{x1D6E5} {G_k}$ under state ${H_{s(k)}}$ is modeled as a normal (Gaussian) distribution [Reference Casella and Berger25]:

(5) \begin{align}L({\unicode{x1D71F} {\boldsymbol{G}_k}|{H_{s(k)}}}) = \frac{1}{{\sqrt {2\pi \boldsymbol\sigma _{s(k)}^2} }}\exp\left({-\frac{{{{({\unicode{x1D71F} {\boldsymbol{G}_k} - {\boldsymbol\mu _{s(k)}}})}^2}}}{{2\boldsymbol\sigma _{s( k )}^2}}}\right)\end{align}

where ${\boldsymbol\mu _{s(k)}}$ and ${\boldsymbol\sigma _{s(k)}}$ are the mean and standard deviations assigned according to the state of the observation.

Assignment to observations. For each of the ${{\;}}{N_s}$ observation $\unicode{x1D71F}{\boldsymbol{G}_k}({k = 1, \ldots ,\,{N_s}})$ the corresponding parameters ${\boldsymbol\mu _{s(k)}}$ and ${\boldsymbol\sigma _{s(k)}}$ are assigned based on the associated state $s(k)$ , (0 for normal, 1 for faulty):

Note (for reference). If estimated classically from the measurements, the mean and variance for each state $i \in \left\{{0,1}\right\}$ would be:

(6) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\boldsymbol\mu _i} = \;\dfrac{1}{{{N_i}}}\mathop \sum \limits_{k = 1}^{{N_s}} {\textbf{1}_{\left\{ {s(k) = i} \right\}}}\unicode{x1D71F} {\boldsymbol{G}_k}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\[9pt]{\boldsymbol\sigma _i^2 = \dfrac{1}{{{N_i} - 1}}\mathop \sum \limits_{k = 1}^{{N_s}} {\textbf{1}_{\left\{ {s(k) = i} \right\}}}{{(\unicode{x1D71F} {\boldsymbol{G}_k} - {\boldsymbol\mu _i})}^2}\;\;\;\;}\end{array}} \right.\end{align}

where:

• • ${N_i}$ is the number of measurements corresponding to state ${{\;}}i$ .

• • ${\textbf{1}_{\left\{ {s(k) = i} \right\}}}$ is the indicator function ( $1$ if measurement $k$ belongs to state $i$ , $0$ otherwise).

The likelihood ratio for each observation is computed as:

(7) \begin{align}\boldsymbol\varLambda({{\unicode{x1D71F}}{\boldsymbol{G}_{\boldsymbol{k}}}}) = \frac{{{{L}}( {{\unicode{x1D71F}}{\boldsymbol{G}_{{k}}}\,{{{H}}_1}} )}}{{{{L}}( {{\unicode{x1D71F}}{\boldsymbol{G}_{{k}}}\,{{{H}}_0}} )}} \lessgtr {{\eta }}\end{align}

2.3 Grey wolf optimisation

GWO is a nature-inspired metaheuristic mimicking grey wolf hunting behaviour and hierarchy [Reference Boualem26]. The three best solutions $\;\left( {\alpha ,{{\;}}\beta ,{{\;}}\delta } \right)$ guide the population. Wolf positions are updated iteratively using:

1. 1. Coefficient vectors:

   (8) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{\vec{A}} = 2.{\vec{a}}.\,{{{\vec{r}}}_1} - {\vec{a}}\;}\\{{\vec{C}} = 2.{{{\vec{r}}}_2}\;\;\;\;\;\;\;\;\;\;\;\;}\end{array}} \right.\end{align}
   such as
   (9) \begin{align}{\vec{a}} = 2 - 2\;.\,{{n}}/{Max\_iter}\end{align}
   where ${\vec r_1}$ , ${\vec r_2}$ $ \in \left[ {0,1} \right]$ are random vectors, $n$ is the actual number of iterations and $Max\_{iter}$ is the maximum number of iterations.

2. 2. Distance to leaders:

   (10) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{{{\vec{D}}}_{{\alpha }}} = \left| {{\vec{C}}.\,{{{\vec{X}}}_{{\alpha }}} - {{{\vec{X}}}_{{j}}}} \right|}\\[5pt]{{{{\vec{D}}}_{{\beta }}} = \left| {{\vec{C}}.\,{{{\vec{X}}}_{{\beta }}} - {{{\vec{X}}}_{{j}}}} \right|}\\[5pt]{{{{\vec{D}}}_{{\delta }}} = \left| {{\vec{C}}.\,{{{\vec{X}}}_{{\delta }}} - {{{\vec{X}}}_{{j}}}} \right|}\end{array}} \right.\end{align}

3. 3. Position update:

   (11) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{{{\vec{X}}}_1} = \left| {{{{\vec{X}}}_{{\alpha }}} - {{A}}.\,{{{\vec{D}}}_{{\alpha }}}} \right|}\\[5pt]{{{{\vec{X}}}_2} = \left| {{{{\vec{X}}}_{{\beta }}} - {{A}}.\,{{{\vec{D}}}_{{\beta }}}} \right|}\\[5pt]{{{{\vec{X}}}_3} = \left| {{{{\vec{X}}}_{{\delta }}} - {{A}}.\,{{{\vec{D}}}_{{\delta }}}} \right|}\end{array}} \right.\end{align}
   (12) \begin{align}{{\vec{X}}_{{j}}}\left( {{{n}} + 1} \right) = \;\frac{{{{{\vec{X}}}_1} + {{{\vec{X}}}_2} + {{{\vec{X}}}_3}}}{3}\end{align}

The fitness function measures how well a candidate solution ${\vec{\boldsymbol{X}}_j} = ({{\boldsymbol\mu _{0,\;j}},\,{\boldsymbol\sigma _{0,\;j}},\,{\boldsymbol\mu _{1,\;j}},\,{\boldsymbol\sigma _{1,\;j}}})$ explains the observed gain variations ${\boldsymbol{\Delta }}{\boldsymbol{G}_k}$ . It is defined as the negative log-likelihood over all observations:

(13) \begin{align}{{f}}({{\vec{\boldsymbol{X}}_{{j}}}}) = - \mathop \sum \limits_{{{k}} = 1}^{{{{N}}_{{s}}}} {\textrm{log}\,L}({\boldsymbol{\Delta }}{\boldsymbol{G}_{{k}}}| {{{{H}}_{{{s}}({{k}})}},\,{\vec{\boldsymbol{X}}_{{j}}}}) .\end{align}

• • Here, ${{\;}}{\vec{\boldsymbol{X}}_j}$ contains the parameters of the $j$ -th wolf in the GWO population.

• • This function is evaluated individually for each wolf $j = 1, \ldots ,N$ .

Equivalently, in expanded form:

(14) \begin{align}{{f}}({{\vec{\boldsymbol{X}}_{{j}}}}) = \mathop \sum \limits_{{{k}} = 1}^{{{{N}}_{{s}}}} \left[ {\frac{{{{({\boldsymbol{\Delta }}{\boldsymbol{G}_{{k}}} - {{\boldsymbol{\mu }}_{{{s}}\left( {{k}} \right),{{j}}}})}^2}}}{{2{\boldsymbol{\sigma }}_{{{s}}\left( {{k}} \right),{{j}}}^2}} + \frac{1}{2}{\textrm{log}}( {2{{\unicode{x03C0} \boldsymbol\sigma }}_{{{s}}({{k}}),{{j}}}^2})} \right]\end{align}

The optimisation algorithm proceeds through the following sequential steps:

Step 1: Initialisation

1. 1. Define the GWO control parameters:

   • • Population size $N$ (number of wolves),

   • • Maximum number of iterations $Max\_{iter}$ ,

   • • Convergence tolerance $\varepsilon $ .

2. 2. Define problem-specific parameters:

   • • Number of observations ${N_{{s}}}$ (number of TSR measurements ${{\Delta }}{G_k})$

3. 3. Define the search bounds for the decision variables:

   \begin{align*}{\vec X_j} = ({{\mu _{0,j}},\,{\sigma _{0,j}},\,{\mu _{1,j}},\,{{\;}}{\sigma _{1,j}}}),{{\;\;\;\;\;\;\;\;\;\;\;\;}}j = 1, \ldots ,{{\;}}N\end{align*}

with

\begin{align*}{\mu _{0,\,j}} \in {{\;}}\left[ {\mu _0^{min},{{\;}}\mu _0^{max}} \right],\,{\sigma _{0,j}} \in {{\;}}\left[ {\sigma _0^{min},{{\;}}\sigma _0^{max}} \right],\,\,{\mu _{1,j}} \in {{\;}}\left[ {\mu _1^{min},{{\;}}\mu _1^{max}} \right],\,{\sigma _{1,j}} \in {{\;}}\left[ {\sigma _1^{min},{{\;}}\sigma _1^{max}} \right]\end{align*}

1. 4. Randomly initialise the position of each wolf ${\vec X_j}\left( 0 \right)$ within the defined bounds.

2. 5. Evaluate the fitness of each wolf using the objective function defined in Step 2.

3. 6. Identify the three best wolves as ${\vec X_\alpha },{{\;}}{\vec X_\beta },{{\;}}{\vec X_\delta }$

Step 2: Definition of the fitness function

Compute the fitness of each wolf using Equation (14)

• • This evaluates how well is a candidate solution ${\vec{\boldsymbol{X}}_j} = \left( {{\boldsymbol\mu _{0,j}},\,{\boldsymbol\sigma _{0,j}},\,{\boldsymbol\mu _{1,j}},\,{\boldsymbol\sigma _{1,j}}} \right)$

• • The optimisation goal is to minimise this fitness.

Step 3: Main GWO optimisation loop

For iteration $n = 1$ to $Max\_{iter}$

1. 1. Update the coefficient vectors $\vec A$ and $\vec C$ according to Equation (8).

2. 2. For each wolf ${\vec X_j}\left( n \right),{{\;}}j = 1, \ldots ,{{\;}}N:$

   • • Compute the distances to the leaders $\alpha ,{{\;}}\beta ,{{\;}}\delta $ using Equation (10).

   • • Compute the candidate positions using Equation (10).

   • • Update the wolf position using the GWO update rule (Equation (12)).

   • • Enforce the search bounds on ${\vec X_j}\left( {n + 1} \right)$ .

3. 3. Re-evaluate the fitness of all wolves using Step 2 over all $k = 1, \ldots ,{{\;}}{N_s}$ .

4. 4. Update ${\vec X_\alpha }$ , ${\vec X_\beta }$ , ${\vec X_\delta }$ as the three best solutions.

Step 4: Termination criteria

1. 1. Let ${f_\alpha }\left( n \right) = f\left( {{{\vec X}_\alpha }\left( n \right)} \right)$

2. 2. Convergence check:

   • • If $\left|\, {{f_\alpha }\left( n \right) - {f_\alpha }\left( {n - 1} \right)} \right| \lt $ $\varepsilon $ or $n = {{\;}}Max\_{iter}$ is reached

     1. ° Report the best solution ${\vec X_\alpha }$ (position of α).

     2. ° Terminate the algorithm.

   • • Else: return to Step 3.

Notes: ${\vec{\boldsymbol{X}}_{\boldsymbol{j}}} \boldsymbol\in {{\boldsymbol{\mathbb{R}}}^4},\;{\vec{\boldsymbol{X}}_{\boldsymbol\alpha}},$ ${\vec{\boldsymbol{X}}_{\boldsymbol\beta}}$ , ${\vec{\boldsymbol{X}}_{\boldsymbol\delta}}$ $ \boldsymbol\in {{\boldsymbol{\mathbb{R}}}^{\textbf{4}}},\;{\boldsymbol{N}_{\textbf{s}}} \boldsymbol\in {{\boldsymbol{\mathbb{R}}}^ + }$ .

Table 3 reports the results obtained by the learning process based on the GWO algorithm, together with the corresponding three-dimensional fitness landscapes (see Figs 2 and 3). The statistical parameters of the gain ${\boldsymbol{\Delta }}{\boldsymbol{G}_k}$ were estimated as ${\mu _0}$ $ = {{\;}} - 0.0655{{\;}}dB$ , ${\sigma _0}$ $ = {{\;}}0.0564{{\;}}dB$ for the normal state, and ${\mu _1}$ = $ - 3.6050{{\;}}dB$ , ${\sigma _1}$ $ = $ $2.2194{{\;}}dB$ for the faulty state, with a minimum fitness value of $f({{{\vec X}_j}}) = 8.1910 \times {10^{ - 2}}$ .

Table 3.

Optimisation of LNA fault detection parameters using GWO

Table 3 Long description

The table presents the optimization of LNA fault detection parameters using GWO. It has 10 rows and 4 columns. The columns are labeled Parameters, Range (min, max), Optimised values, and Coefficients. The rows list the parameters with their respective ranges, optimized values, and coefficients. Row 1: μ0, [-0.1, 0.1 dB], -0.0655 dB, N/A. Row 2: σ0, [0.01, 0.1 dB], 0.0564 dB, N/A. Row 3: μ1, [-5, 0 dB], -3.6050 dB, N/A. Row 4: σ1, [0.1, 5 dB], 2.2194 dB, N/A. Row 5: Ns, N/A, N/A, 400. Row 6: f(X⃗j), N/A, 1.2623.10^-2, N/A. Row 7: N, N/A, N/A, 200. Row 8: Max_iter, N/A, N/A, 100. Row 9: ε, N/A, N/A, 1e-6.

Figure 2.

Fitness function surface over ( ${\mu _0}$ , ${\sigma _0}$ ) for the normal operating state.

Figure 3.

Fitness function surface over ( ${\mu _1}$ , ${\sigma _1}$ ) for the faulty operating state.

Figure 3 Long description

A 3D surface plot represents the fitness function over the variables mu1 and sigma1. The x-axis is labeled mu1 in decibels, ranging from approximately -4.5 to 3. The y-axis is labeled sigma1 in decibels, ranging from approximately 1.8 to 3. The z-axis represents the fitness function f(Xi), ranging from 0.015 to 0.06. The color gradient ranges from blue (lower values) to red (higher values), indicating the magnitude of the fitness function. The surface shows multiple peaks and valleys, suggesting varying levels of fitness across different combinations of mu1 and sigma1. Higher fitness values are indicated by red peaks, while lower values are shown in blue.

The analysis of the convergence surfaces shows that, for the normal state (Fig. 2), the fitness function is smooth and convex, exhibiting a well-defined global minimum. This topology indicates that the normal operating regime is statistically stable and that the algorithm converges easily toward a robust estimation of ${\mu _0}$ and ${\sigma _0}$ . In contrast, the surface associated with the faulty state (Fig. 3) is more irregular and multimodal, reflecting the increased dispersion of the observations in this regime. Despite this complexity, the global minimum remains identifiable, which justifies the use of a global optimisation strategy to avoid entrapment in local minima. These results confirm that the system exhibits stable behaviour under normal operating conditions and more dispersed behaviour in the faulty regime, and they demonstrate the effectiveness of the proposed approach for the robust estimation of statistical parameters in both states.

2.4 ML models for autotrack prediction

Three ML algorithms were selected to predict the elevation and azimuth parameters of the auto-track system, namely XGBoost, decision trees and KNN. These models were chosen for their ability to model nonlinear relationships, their robustness to noise and their established use in nonlinear regression problems [Reference Chen and Guestrin27–Reference Cover and Hart29].

1. 1. XGBoost: A gradient boosting approach based on a set of decision trees optimised to minimise a loss function. It is well known for its robustness against overfitting and its high efficiency when handling large-scale datasets.

2. 2. Decision trees: A hierarchical partitioning of the feature space, offering high interpretability while effectively capturing feature interactions and nonlinear patterns.

3. 3. KNN: A non-parametric method based on proximity between observations, particularly suitable for modeling complex local structures in the data.

The study is based on a dataset comprising 336,000 measurements collected between 2017 and 2024 from the ALSAT-1B mission, including the actual antenna azimuth and elevation values, the received signal power, as well as the azimuth and elevation measurements provided by the auto-track system. After preprocessing and formatting, the data were used to train the models and to develop an accurate predictor for elevation and azimuth angles. Table 4 summarises the optimised hyperparameters of each algorithm, ensuring an appropriate trade-off between prediction accuracy and generalisation capability.

Table 4.

Tuning of KNN, XGBoost and decision tree algorithms

Table 4 Long description

A table comparing optimal parameters for three algorithms: XGBoost, Decision Tree, and KNN. The table has three rows and two columns. The first column lists the algorithms: XGBoost, Decision Tree, and KNN. The second column lists the optimal parameters for each algorithm. For XGBoost, the optimal parameters are n_estimators = 300, learning_rate = 0.3, max_depth = 9, and subsample = 1. For Decision Tree, the optimal parameters are criterion = squared_error, max_depth = None, min_samples_split = 2, min_samples_leaf = 1, and max_features = None. For KNN, the optimal parameters are algorithm = ball_tree, n_neighbors = 10, leaf_size = 30, metric = manhattan, and weights = distance.

Algorithm 1: Baseline KNN weighted multi-output regression (optimised with ball tree)

Step 1: Input definition

1. 1. Query point: ${x_q}$ : [ ${{A}}{{{z}}_{{{mesured}}}}$ , ${{E}}{{{l}}_{{{mesured}}}}$ , Signal Power]

2. 2. Historical database $D = \left\{ {\left( {{x_i},{y_i}} \right)} \right\}_{i = 1}^N$ (here $N = {{\;}}336{{\;}}000{{\;}}measurements$ )

3. 3. Parameters:

   • • $v$ = 10 (number of neighbours)

   • • Distance metric $d$ (Manhattan)

   • • $lea{f_{size}} = 30$

Step 2: Initialisation phase (baseline setup)

1. 1. Build the ball tree spatial index over the dataset $D$ by recursively partitioning the points into hyperspheres until each leaf node contains at most 30 points.

Step 3: Prediction phase (search for nearest neighbours)

1. 1. Go through the ball tree using the triangle inequality to find the nearest neighbours according to the Manhattan distance defined in Equation (15):

   (15) \begin{align}{d_{Manhattan}}({{x_q},{x_i}}) = \mathop \sum \limits_{l = 1}^m \left| {{x_{q,\,f}} - {x_{i,\,f}}} \right|\;\end{align}

where $l = 1, \ldots ,{{\;}}m$ indexes the features (azimuth, elevation, signal power).

Step 4: Prediction phase (weight computation and aggregation)

1. 1. Compute distance-based weights with numerical stabilisation:

   (16) \begin{align}{{{\omega }}_{{i}}} = \frac{1}{{{{d}}\left( {{{{x}}_{{q}}},\,{{{x}}_{{i}}}} \right) + {{\varepsilon }}}}{{\;}},\,{{\;\;\varepsilon }} \gt 0\;\end{align}

   • • Normalise the weights:

     (17) \begin{align}{{{\omega }}_{{i}}} = \frac{{{{{\omega }}_{{i}}}}}{{\mathop \sum \nolimits_{{{n}} = 1}^{{\nu }} {{{\omega }}_{{n}}}}},\,{{\;}}\mathop \sum \limits_{{{i}} = 1}^{{v}} {{{\omega }}_{{i}}} = 1{{\;}}\end{align}

2. 2. Compute the multi-output prediction:

   (18) \begin{align}\hat y = \mathop \sum \limits_{i = 1}^\nu {\omega _i}{y_i}\end{align}

   • • Here, ${y_i}$ is the output vector of neighbour $i$ (azimuth, elevation).

Step 5: Output

• • Return the predicted vector $\hat y$ , containing azimuth and elevation.

Notes:

${x_q} = ({x_{q,1}}$ , …, ${x_{q,m}}) \in {R^m}$ , (query point, $m = 3$ ), ${{\;}}{x_i} = ({x_{i,1}}$ , …, ${x_{i,m}}) \in {{\mathbb R}^m}$ , (historical input samples), ${{\;}}{y_i} = ({y_{i,1}}$ , …, ${y_{i,r}}) \in {{\mathbb R}^r}$ , (output vectors, $r = 2$ ), ${{\;\hat y}} = ({\hat y_1}$ , …, ${\hat y_{{r}}}) \in {\mathbb{R}^r}$ , $d\left( {{x_q},{{\;}}{x_i}} \right) \in {\mathbb{R}^ + },{{\;}}{\omega _i} \in {\mathbb{R}^ + }$ .

3.1 Results of phase 1: dataset-based algorithms evaluation

Each algorithm was analysed by graphically representing the azimuth and elevation data from the satellite pass in 2017 (AOS: 10:11:39, LOS: 10:16:22). Figures 5 and 6, respectively, show the response of the XGBoost algorithm compared to the reference azimuth and elevation and that of the automatic tracking system upon activation.

Figure 5.

Azimuth auto-tracking predicted with XGBoost.

Figure 5 Long description

A line graph showing azimuth auto-tracking predictions using XGBoost compared to ground reference and ground solution data. The horizontal axis represents time in Coordinated Universal Time (UTC) ranging from 10:11:45 to 10:16:15. The vertical axis represents the angle in degrees ranging from 198 to 208 degrees. The graph includes three lines: a green line for ground reference, a red line for auto-track ground solution, and a blue line for XGBoost prediction. The lines show a general downward trend in the angle over time. An inset zooms in on a specific time range from 10:11:40 to 10:11:45, highlighting detailed variations in the angle measurements.

Figure 6.

Elevation auto-tracking predicted with XGBoost.

First, it should be noted that at the beginning of the pass, azimuth tracking remained generally acceptable until time 10:45:00, with only slight fluctuations, as shown in the enlarged area of Fig. 5 (auto-track vs REF: $MSE{{\;}} = {{\;}}0.003290^\circ $ ; XGBoost vs auto-track: $MSE{{\;}} = {{\;}}0.114913^\circ $ ). After this time, a marked discrepancy appeared and persisted until the end of the pass, indicating a significant degradation in azimuth tracking accuracy.

Regarding elevation (Fig. 6), tracking remains generally stable, with only minor fluctuations. However, a slight degradation is observed between 10:15:40 and 10:16:01 (XGBoost vs. auto-track: $MSE{{\;}} = {{\;}}0.035245^\circ $ ), indicating significantly better performance compared to azimuth. This degradation, particularly pronounced in azimuth, is primarily attributable to a decrease in signal quality, which can lead to telemetry losses and impair the operation of the automatic tracking system.

For Figs 7 and 8, illustrating the azimuth and elevation estimates obtained with the decision tree algorithm, we initially observe more accurate tracking than that provided by the XGBoost algorithm. At the beginning of the pass, the azimuth calculated by the decision tree closely follows the ground reference trajectory as well as that measured by the auto-track system. However, slight oscillations appear at times 10:12:16, 10:13:12, 10:13:15 and 10:13:26, with respective angles of 204.2°, 201.6°, 202.0° and 201.8° showing deviations from nominal.

Figure 7.

Azimuth auto-tracking predicted with decision tree.

Figure 8.

Elevation auto-tracking predicted with decision tree.

Figure 8 Long description

Panel: A line graph with the horizontal axis labeled Time in U T C and the vertical axis labeled Angle in degrees. The graph shows three lines representing Ground Reference, Auto-track Ground Solution, and Auto-track Decision Tree Prediction. The Ground Reference line is green, the Auto-track Ground Solution line is red, and the Auto-track Decision Tree Prediction line is blue. The graph includes two zoomed-in insets highlighting specific time intervals. The main trend shows a general decrease in angle over time for all three lines, with the blue line deviating more significantly from the green and red lines as time progresses.

Thereafter, the algorithm very accurately follows the auto-track system, particularly between 10:15:30 and 10:16:22, with a remarkably low RMS error ( $RMS{{\;}} = {{\;}}0.000136^\circ $ ). For elevation, accuracy remains high throughout the pass, as illustrated by the zoom between 10:15:28 and 10:16:01. The RMS error calculated over this period (10:15:28–10:16:22) is $0.000772^\circ $ , confirming stability and performance consistency of the algorithm for elevation tracking.

The final algorithm considered for this phase is KNN. Figure 9 shows that the azimuth estimated by KNN very accurately follows the auto-track system throughout the pass. The RMS error calculated over the entire pass is $0.000190^\circ $ , confirming high azimuth tracking accuracy.

Figure 9.

Azimuth auto-tracking predicted with KNN.

In terms of elevation (Fig. 10), the tracking performance also remains high. Although a brief peak of $1.73^\circ $ is observed at 10:16:08, the RMS error over the entire pass remains low, with a value of $0.026375^\circ $ . Overall, the KNN-based algorithm demonstrates high and stable accuracy. Compared to the decision tree, which remains competitive, the KNN offers better overall performance thanks to its ability to adapt to local signal variations, ensuring more accurate tracking in both azimuth and elevation. These observations are consistent with the results presented in Table 6.

Figure 10.

Elevation auto-tracking predicted with KNN.

At the end of this section, Table 7 provides a summary of the performance indicators (azimuth and elevation) for all models, in accordance with Remark 4, without multiplying the graphs.

3.2 Results of phase 2: KNN implementation on real satellite passes

Based on the performance observed in Phase 1, KNN was chosen as the implementation algorithm, with the likelihood function used to trigger a switch in the event of an LNA failure in the auto-track chain.

The satellite pass of October 10, 2025 (AOS: 06:22:13, LOS: 06:34:11) was selected as a case study, corresponding to a cumulative volume of downloaded multispectral images of 2,526.8 MB. Table 8 summarises the details related to the image acquisitions.

Figure 11 highlights a clear transition from a stationary nominal regime ( ${H_0}$ ) to an unstable degraded regime ( ${H_1}$ ) around 06:27:26, characterised by a significant drop and an increase in its variance associated with an increase in the reflection coefficient $\left| {{\Gamma }} \right|$ of the LNA ( $\left| {{\Gamma }} \right|{{{\;}}_{{{mean\;}}}} \approx 0.35$ ).

As shown in Fig. 12, this transition is detected when the log-likelihood ratio abruptly changes from a stable value of approximately −4.3 under normal operating conditions to positive values greater than 10 after 06:27:26, reflecting a statistically significant change in $\Delta G$ that renders the observations inconsistent with the nominal model and triggers the fault decision. This separation is quantitatively confirmed by the estimated parameters of the distributions, with ${\mu _0} \approx - 0.0655{{\;dB}},\,{\sigma _0} \approx - 0.0564{{\;dB}},$ ${\mu _1} \approx - 3.6050{{\;dB}},\,{\sigma _1} \approx 2.2194{{\;dB}}$ .

Figures 13 and 14 show the temporal evolution of the satellite’s azimuth ( $AZ{{'}}$ ) and elevation ( $EL{{'}}$ ), comparing the measured values with the predictions provided by the ML-TSR algorithm (see Table 9). Until 06:27:25, the LNA operates in nominal mode ( ${{\boldsymbol{H}}_0}$ ), and the predictions remain consistent with the measurements. At 06:27:26, the likelihood function detects a fault in the LNA, signaling the transition to degraded mode ( ${{\boldsymbol{H}}_1}$ ). From this point onward, the ML-TSR algorithm takes over and initially provides $AZ{{'\;}} \approx {{\;}}49.5^\circ $ and $EL{{'\;}} \approx {{\;}}32.8^\circ ,$ then values that evolve according to the orbital dynamics until the end of the pass, thus ensuring continuous tracking.

Table 6.

Performance metrics – $A{Z^{{'}}}$ & $E{L^{{'}}}$ (comparison of models, 21 April 2017, AOS: 10 :11 :39, LOS: 10 :16 :22)

Table 6 Long description

A table comparing performance metrics of different models across two datasets, AZ and EL. The table has 4 rows and 6 columns. Column headers are Dataset, Model pair, Metric (deg), XGBoost, Decision tree, and KNN. Row labels include Ground REF vs prediction and Auto-track vs prediction for each dataset. The table lists values for Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) for each model pair and dataset. For dataset AZ, Ground REF vs prediction shows MSE values of 0.084537 for XGBoost, 0.006351 for Decision tree, and 0.004653 for KNN. RMSE values are 0.290753 for XGBoost, 0.079693 for Decision tree, and 0.068211 for KNN. Auto-track vs prediction shows MSE values of 0.098286 for XGBoost, 0.001871 for Decision tree, and 0.000190 for KNN. RMSE values are 0.313507 for XGBoost, 0.043259 for Decision tree, and 0.013788 for KNN. For dataset EL, Ground REF vs prediction shows MSE values of 0.008999 for XGBoost, 0.006752 for Decision tree, and 0.005139 for KNN. RMSE values are 0.094862 for XGBoost, 0.082171 for Decision tree, and 0.071687 for KNN. Auto-track vs prediction shows MSE values of 0.010398 for XGBoost, 0.002419 for Decision tree, and 0.000696 for KNN. RMSE values are 0.101971 for XGBoost, 0.049178 for Decision tree, and 0.026375 for KNN.

Table 7.

Performance metrics – $A{Z^{{'}}}$ & $E{L^{{'}}}$ (comparison of models, 19 Nov 2024, AOS: 19 :32 :03, LOS: 19 :40 :50)

Table 7 Long description

A table comparing performance metrics of different models. The table has 2 groups of rows labeled AZ and EL, 3 columns labeled Model pair, Metric (deg), and XGBoost, Decision tree, KNN. Row 1: Model pair, Ground REF vs prediction; Metric (deg), MSE; XGBoost, 0.259704; Decision tree, 0.157743; KNN, 0.155526. Row 2: Model pair, Auto-track vs prediction; Metric (deg), MSE; XGBoost, 0.144817; Decision tree, 0.009556; KNN, 0.006925. Row 3: Model pair, Ground REF vs prediction; Metric (deg), RMSE; XGBoost, 0.509612; Decision tree, 0.397169; KNN, 0.394368. Row 4: Model pair, Auto-track vs prediction; Metric (deg), RMSE; XGBoost, 0.380549; Decision tree, 0.097757; KNN, 0.083216. Row 5: Model pair, Ground REF vs prediction; Metric (deg), MSE; XGBoost, 0.024203; Decision tree, 0.013129; KNN, 0.014371. Row 6: Model pair, Auto-track vs prediction; Metric (deg), MSE; XGBoost, 0.013377; Decision tree, 0.005348; KNN, 0.004509. Row 7: Model pair, Ground REF vs prediction; Metric (deg), RMSE; XGBoost, 0.155574; Decision tree, 0.114583; KNN, 0.119879. Row 8: Model pair, Auto-track vs prediction; Metric (deg), RMSE; XGBoost, 0.115658; Decision tree, 0.073130; KNN, 0.067149.

Table 8.

Scheduled ALSAT-1B image acquisitions and related operations

Table 8 Long description

The table has five rows and five columns. The columns are labeled Download start (UTC), Download end (UTC), Image area (km^2), Size (MB), and Session 1 (06:22:13-06:34:11). The rows provide details for each image acquisition. Row 1: Download start, 06:24:00; Download end, 06:26:30; Image area, 22,486.6; Size, 631.7; Session 1, Download. Row 2: Download start, 06:26:30; Download end, 06:29:00; Image area, 22,486.6; Size, 631.7; Session 1, Download. Row 3: Download start, 06:29:00; Download end, 06:31:30; Image area, 22,486.6; Size, 631.7; Session 1, Download. Row 4: Download start, 06:31:30; Download end, 06:34:00; Image area, 22,486.6; Size, 631.7; Session 1, Download.

Figure 11.

Transition from nominal to degraded LNA regime ( ${H_0}$ → ${H_1}$ ): output ΔG dynamics.

Figure 12.

Statistical detection of LNA defects.

Figure 13.

ML-TSR azimuth auto-tracking predictions profile during phase 2.

Figure 14.

ML-TSR elevation auto-tracking predictions profile during phase 2.

Although a fault in the LNA was observed for approximately 4 s around 06:27:26, it did not cause an interruption of the transmission, as illustrated in Fig. 15. This event resulted in a transient degradation of the RSL, which briefly dropped to approximately $ - 58{{\;}}dBm$ , accompanied by an increase in the LNA’s reflection coefficient, momentarily rendering the observations incompatible with the nominal regime. The likelihood function decision triggered the activation of the ML-TSR controller, which took over the auto-tracking chain and allowed the tracking to stabilise again from 06:27:30. The RSL then recovered to values approximately between $ - 55$ and $ - 42{{\;}}dBm$ , with the link lock maintained, ensuring continuity of the transmission until the Loss of Signal (LOS).

Table 9 illustrates the values distinguishing the nominal state of the LNA from the period affected by a fault. In the absence of a fault, the average $\Delta G{{\;}}\left( {dB} \right)$ corresponds to the component specifications and remains within the normal operating range. In the presence of a fault, according to assumption 1, the average ΔG is in the range $\left[ { - 1.92,{{\;}} - 0.28{{\;}}dB} \right]$ , indicating a potential out-of-specification risk requiring careful monitoring.