# Predictive and robust control strategies applied to ground antenna systems

Rodrigo Negri de Azeredo, Mohamed Hajjem, Lara Thomas, Stéphane Victor, Patrick Lanusse, Pierre Melchior

► To cite this version:

Rodrigo Negri de Azeredo, Mohamed Hajjem, Lara Thomas, Stéphane Victor, Patrick Lanusse, et al. Predictive and robust control strategies applied to ground antenna systems. Annual Reviews in Control, 2025, 60, pp.100995. 10.1016/j.arcontrol.2025.100995. hal-05152801

HAL Id: hal-05152801

https://hal.science/hal-05152801v1

Submitted on 7 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Full Length Article

Predictive and robust control strategies applied to ground antenna systems

Rodrigo Negri de Azeredo a,b, Mohamed Hajjem b, Lara Thomas b, Stéphane Victor a,b,*,
Patrick Lanusse a, Pierre Melchior a,b

ARTICLE INFO

Keywords:

CRONE
H-infinity ($H_\infty$)

Predictive control

Robust control

Tracking antenna

Uncertain models

A B S T R A C T

A mobile antenna system requires precise control that combines good trajectory tracking and disturbance rejection. The system has several operating scenarios, demanding a robust control design that ensures good performance and stability for a set of uncertain models. Moreover, ground stations are mainly outdoor operating systems, and they are subject to wind gusts that impact the accuracy of the position control. This paper presents a study using predictive control and robust controllers (CRONE and $H_\infty$ strategies) to assess the application feasibility of these methodologies in tracking antenna systems. The control system design of both CRONE and $H_\infty$ strategies are carried out in frequency domain. Time domain analysis demonstrates that both robust controllers yield very similar results, ensuring excellent stability and outstanding performances in tracking trajectory operation. The choice of control strategy ultimately depends on the designer's expertise, software/hardware requirements, and the complexity of the system model, with an advantage of the CRONE controller in terms of robustness and practical implementation. Simulation results are validated on tracking satellite trajectories by comparing MPC, CRONE and $H_\infty$ controls with various robustness tests.

1. Introduction

In the telemetry and satellite communications fields, Safran Data Systems is a leading expert in receiving antennas for ground stations. Parabolic antennas communicate with satellites or track targets during flight tests (aircraft or missiles). The system operates in the [1, 30 GHz] frequency band, and the antenna reflector size has a diameter ranging from 1 to 13 m. The antenna comprises two mechanical axes that enable motion in both vertical and horizontal planes to direct the radio-frequency (RF) source towards a target. These axes are mechanically coupled, i.e. a movement on one axis affects the dynamics of the other one and vice-versa.

To improve their dynamical performances with new tracking sys-
tem technologies, and aiming to ensure precise motion and maximum
signal reception, antennas require high performance levels in accuracy,
stability, and speed specifications. For this reason, the synthesis of an
effective control system is an essential part of the final product. It
calls for a dynamic model that faithfully describes the behavior of the
antenna operational conditions, to simulate their motions and repro-
duce the different mechanical phenomena to which they are subjected,
e.g. noise, aging, wind turbulence, friction, nonlinearities, ...

Ground stations have a wide range of working capacities, in dif-
ferent environmental, operational, and functional conditions. Thus, the
system may experience considerable variations throughout its useful
life. The control system must guarantee the project specifications for
all operating conditions, ensuring stability degree of system robustness
and providing the required performances and stability requirements.
For these reasons, different control techniques are applied for better
performances of the global antenna control and tracking.

Predictive control is a control strategy where a dynamic model of the process is applied to predict the future behavior of the real system (Clarke et al., 1987). This ability is appreciated when ground stations work with a reference signal defined *a priori*, i.e. the trajectory to be followed is known in advance. In this case, this control strategy presents a good performance. However, the estimation of the system behavior is based on a mathematical model that describes the model dynamics. The variations in the real plant model, differentiating itself from the predictive controller model, cause performance and stability robustness issues to the predictive control law. Hence, it is essential to consider them in the control system design of ground antenna systems.

<footnote>ª Univ. Bordeaux, CNRS, IMS UMR 5218, Bordeaux INP/ENSEIRB-MATMECA, Talence, 33405, France</footnote>

<footnote>ᵇ Safran Data Systems, La Teste-de-Buch, 33260, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: rodrigo.negri-de-azeredo@safrangroup.com (R.N. de Azeredo), mohamed.hajjem@safrangroup.com (M. Hajjem), lara.thomas@safrangroup.com (L. Thomas), stephane.victor@ims-bordeaux.fr (S. Victor), patrick.lanusse@ims-bordeaux.fr (P. Lanusse), pierre.melchior@ims-bordeaux.fr (P. Melchior).</footnote>
Control system design that cope with such uncertainties, can be
carried out through robust control methodologies. Several applications
can be found in the literature, such as in the work of Sedhom et al.
(2019), which presented a comparison between H<sub>∞</sub> and predictive
controllers applied to micro-grid systems. In Yang et al. (2021), the
authors exposed the same control strategies for trajectory tracking
control in automated vehicles. Ravasco et al. (2020) applied CRONE
control for wind turbine. Lanusse et al. (2023) tackled robust control
problems with nonlinear models by using the CRONE robust control.

The CRONE (French acronym for “Non-Integer Order Robust Con-
trol”) control strategy (Lanusse et al., 2013; Oustaloup, 1991) is a
frequency-based approach based on the closed-loop configuration with
unitary feedback, where the nominal open-loop frequency response
is defined by using fractional-order integration and differentiation.
The resulting controllers ensure the stability degree robustness and
guarantees the required performances as given in the specifications and
control constraints, despite model uncertainties and disturbances. The
other one is the H<sub>∞</sub> control design (Francis & Doyle, 1987), which
formulates a feedback system as an optimization problem with an
operator norm, particularly, the infinity norm. Such a methodology
uses an alternate approach as the CRONE control, aiming to synthesize
stable controllers that ensure the specified performance for a set of
uncertain models.

Due to the robustness issues of the predictive model, the two control strategies, namely CRONE and H∞, are applied to the position control of a real 2-axis antenna developed by Safran Data Systems and compared in terms of stability and performances in trajectory tracking, considering a set of uncertain models. The controller designs are performed with the same constraints and specifications. Additionally, wind gusts are estimated by the authors using a fractional order model. Such disturbances are added to the simulation model and allow the assessment of the control law capability in terms of disturbance rejections. Thus, this article analyzes the advantages and limitations of predictive, CRONE, and H∞ control strategies for ground antenna system applications.

The paper is organized as follows. Section 2 describes the antenna system and the parametric uncertainties to be considered in the control design. Section 3 presents the wind turbulence disturbance signal, which is based on non-integer system identification from frequency analysis of real data spectral density. Afterwards, Section 4 details the predictive control methodology applied to the tracking antenna system. Section 5 explains the control system requirements for a robust controller synthesis and exposes the two robust control system design. All the control system strategies are implemented and the predictive, CRONE, and H∞ controllers are compared and evaluated through simulation results in Section 6. Finally, a conclusion is drawn in Section 7.

## 2. System modeling

Ground stations with 2-axis antenna type are equipped with an RF
source and a reflector with a diameter ranging from 1.8 to 3.0 m. The
system is composed of two mechanical axes, namely the azimuth (Az)
and the elevation (El). A multiple-input multiple-output (MIMO) system
that takes these angles as inputs to point at a target which is located
in a spherical coordinate system and whose origin is the antenna. The
system outputs are the angular positions of the antenna mechanical
axes.

### 2.1. Antenna nominal model

The dynamic equation between the flexible joints of each axis de-
scribes the antenna mechanical model. The antenna system is modeled
as a flexible two degrees of freedom (DOF) robot, as presented in Ha-
jjem et al. (2022). Due to high loads and flexible antenna gearboxes,
position deviations of the different components of the mechanical axes

<figure><img src="image_2.png" /><figcaption>Fig. 1. Illustration of 2-axis antenna in a geometric rigid model form.</figcaption></figure>

may occur. In this case, these uncertainties must be incorporated in the
control system design, as they definitely will hurt the system stability
and precision. An additional flexibility is considered between the motor
and the gearbox of each axis, which adds another degree of freedom per
axis. Consequently, the antenna model has 4 DOF and Fig. 1 illustrates
the 2-axis antenna in its simplified rigid geometric model form.

The model is linearized around the operating point with azimuth Az = 0° and elevation angle El = 45°. Such a model defines the nominal system for the control system design. Its multivariate frequency responses are plotted in Fig. 2. The non-diagonal frequency responses (red dotted lines), corresponding to the transfer functions between the inputs and outputs of different axes, are non-zero, thus indicating the presence of coupling between the antenna axes. However, these frequency responses have quite low gains compared to the direct transfer ones: the axes can be considered independent in the control system design.

### 2.2. Uncertainty analysis

Model uncertainties can be classified into dynamic perturbations
and disturbance signals (Gu et al., 2013). Dynamic perturbations come
from the difference between the mathematical model and the actual
dynamics of the system. The mathematical model of a real system
normally represents an approximation of its dynamics, caused by un-
modeled dynamics, neglected non-linearities, and parametric variations
due to environmental conditions and time-related wear of physical
components. Disturbance signals are input/output disturbances and
sensor/actuator noises. These errors in system modeling can affect the
stability and performance of the controlled system.

The nominal model has already been defined in the previous Sec-
tion 2.1. Nevertheless, it is remarked that this model almost always
differs from reality, mainly because of the uncertainties in system
identification and simplifications adopted to obtain its mathematical
representation. Concerning the robustness of a control system, it is
necessary to clearly define the class of modeling errors to which this
robustness refers. Consequently, the closed-loop system stability is
guaranteed for all the processes whose modeling errors belong to the set
of models, ensuring stability degree robustness. The same holds for the
performance specifications of all these uncertain processes, ensuring
performance robustness.

The first important factor is the working point regarding the el-
evation axis position. Considering the antenna elevation, there is a
variation of the azimuth axis inertia, and consequently, of the equiv-
alent linear model. The nominal condition is considered at 45°, and
the variation range is in [0°, 90°]. Frictional parameter variations
will affect the amplitude of the resonance peak. The main causes are
variations in temperature, lubrication, wear, and estimation errors. A
variation of 20% of these nominal values is considered, as well as
20% of the variation of the inertia of the antenna due to external
<figure><img src="image_3.png" /><figcaption>Fig. 2. Bode diagrams of the linearized models: direct-axis transfer functions (blue solid lines) and cross-axis ones (red dotted lines).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Frequency responses of all uncertain models for the azimuth (left) and elevation (right) axes.</figcaption></figure>

disturbances (rain and snow) or modifications in the structure (onboard components). The frequency responses of all uncertain models are illustrated in Fig. 3 for both azimuth and elevation axes.

The azimuth axis is more impacted because the load applied to its motor depends on the antenna elevation position. Hence, the resonant point of the frequency response varies considerably in frequency and peak value for all uncertain models. These uncertainties will affect the stability and the performances of the system.

## 3. Wind turbulence modeling

Antennas are mainly outdoor systems and they are subject to inclement weather conditions. In the case of position control of these systems, the wind is an important disturbance that continuously affects the antenna operation. Wind turbulence generates disturbances that increase position errors. For this reason, these disturbances must be rejected during the system operation.
Some antennas are installed under radomes to protect them from wind, but this represents a significant cost in the final project. Therefore, it is essential to minimize wind effects with system control to guarantee good performance without additional costs. Wind effects must be integrated into the global antenna dynamic system, allowing the development and evaluation of control laws that can minimize them.

The aerodynamic coefficients used for the study of the impact of wind on antennas are coefficients measured and published in Association and Institute (1996) by the Telecommunications Industry Association (TIA). Antenna manufacturers widely use these coefficients. The studies on wind modeling available in the literature focus mainly on statistical and probabilistic aspects. These studies do not reproduce the impact of wind on the fast dynamics of systems, which depends primarily on turbulence and the short-term component. The latter one is a non-stationary stochastic process and it is hard to model it properly in time domain. For that reason, the modeling method considered is based on the spectral characteristics of the wind, as explained in Hajjem et al. (2023b). The frequency analysis of the power spectral density of wind turbulence from real measurements have enabled designing non-integer order filters which are very accurate to generate random wind turbulence speeds, by specifying the wind turbulence intensity, its mean component and a length scale that is linked with the landscape topology.

### 3.1. Wind forces and moments

Wind exerts dynamic pressure on the antenna, which results from its kinetic energy and generates forces applied to the structure. The forces applied to the reflector disrupt the behavior of the antenna different axes. The forces on the shaft and the frame remain negligible compared to those exerted on the reflector. For this reason, the torques requested from the actuators of each axis must be adapted to compensate for the external disruptive forces.

The expression for dynamic pressure $p$ is:

$$p = \frac{1}{2} \rho V^2, \qquad (1)$$

with $\rho$ being the density of the air and $V$ the relative wind speed. From the pressure and the shape of the obstacle encountered, the aerodynamic force $F$ exerted by the wind is expressed by:

$$F = p S_{ref} C = \frac{1}{2} \rho V^2 S_{ref} C, \qquad (2)$$

where $S_{ref}$ is the reference surface and $C$ is the aerodynamic coefficient. In the case of this study, the reference surface is a circle corresponding to the projection of the reflector in the plane perpendicular to the wind direction. In addition to the force, the wind exerts a moment $M$ on the obstacle encountered and can create deformations on it. The moment is calculated from an aerodynamic coefficient $C_m$ as:

$$M = \frac{1}{2} \rho V^3 S_{ref} D C_m, \qquad (3)$$

with $D$ being the reference length, i.e. the diameter of the reflector. The torque applied to each joint is calculated from the forces and moments applied to each antenna body.

### 3.2. Wind speed modeling

At the origin, wind turbulence models are usually generated from von Karman's model (von Kármán, 1948), but such a model is suited for fast moving aircrafts in high altitude, which is definitely not the case for antennas operating near the ground. An analysis of the power spectral density (PSD) of wind turbulence, as explained in Hajjem et al. (2023b), have shown that there are two dynamics: one in middle frequency and on in high frequency. Based on the works of Cole and Cole (1941)), an improved fractional turbulence model has been designed by using two

Cole-Cole (CC) cell models, one for each dynamics. The improved PSD is expressed as follows Hajjem et al. (2023b):

$$\hat{S}_{CC \times 2}(f) = \frac{K}{D_1(f) D_2(f)}, \qquad (4)$$

where

$$D_1(f) = 1 + 2 \cos(v\pi/2)(\tau_1 f)^v + (\tau_1 f)^{2v}, \qquad (5)$$

$$D_2(f) = 1 + 2 \cos(v\pi)(\tau_2 f)^{2v} + (\tau_2 f)^{4v}. \qquad (6)$$

The resulting improved Cole-Cole model (CC×2) is then defined by:

$$H_{CC \times 2}(s) = \frac{\sqrt{K}}{\left(1 + \left(\frac{\tau_1}{2\pi} s\right)^v\right) \left(1 + \left(\frac{\tau_2}{2\pi} s\right)^{2v}\right)}, \qquad (7)$$

where the static gain $K$ has the same expression as the von Kármán one:

$$K = \frac{4\sigma^2 L}{\bar{V}}, \qquad (8)$$

with $\sigma$ being the turbulence standard deviation, $L$ is the length scale and $\bar{V}$ is the mean speed. The time constants $\tau_1$ and $\tau_2$ are calculated as:

$$\tau_1 = 8.9 \frac{L}{\bar{V}}, \qquad (9)$$

$$\tau_2 = \frac{\tau_1}{3.6}. \qquad (10)$$

Hajjem et al. (2023b) show that the parameter $v_2$ is extremely close to $2v_1$. Hence, the model has only three parameters to be estimated. From the defined variables $\bar{V}$ and $\sigma$, and considering fixed values of $L$ and $v$ estimated from the database (Hajjem et al., 2023b), the obtained fractional transfer function of the CC×2 model is given by:

$$H_{CC \times 2}(s) = \frac{16.43}{40.7s^{1.548} + 7.62s^{1.032} + 5.34s^{0.516} + 1}. \qquad (11)$$

The rational transfer function to be implemented is obtained from Oustaloup's synthesis (Oustaloup, Melchior, et al., 2000). The rational synthesis of a fractional transfer function is recalled in Appendix. This function allows real-time simulations to be performed by taking random white noise as input. The real-time wind speed simulator, as explained in Hajjem et al. (2023b), is built considering the CC×2 model specified by (11) and user-defined inputs (Fig. 4): mean speed $\bar{V}$, the length scale $L$, and the turbulence intensity $I_{ref}$.

### 4. Predictive control system

Predictive control techniques are powerful tools for dealing with constrained control problems. These methods use a dynamic model of the system to predict the actual system future behavior within a finite time window, called the prediction horizon. Given these predictions and the current states of the system, an optimized control signal is calculated and applied to the system. This signal may be subject to input/output constraints and one or more objectives. In other words, the optimal control actions are calculated with one or multiple objectives in the cost function, and which may include restrictions on the process variables.

The advantage of this control methodology is that it inherently has a delay and feedforward compensation of measurable disturbances. In addition, it includes processing restrictions during the controller design and implementation. Nevertheless, solving real-time algorithms requires more computational time and cost. It is also necessary to have an appropriate mathematical model of the process. The predictive control approach is based on the knowledge of a physical model. Thus, the performance obtained will depend on the reliability of the model and the variations between the real plant and the mathematical model considered in the algorithm.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Scheme of the wind speed generator.</figcaption></figure>

The predictive control does not consider parametric variations of the model or couplings between the antenna axes. As shown in Section 2, the system presents uncertainties and variations depending on the operating point. In this case, the predictive control is synthesized from a nominal plant model, at 45° of elevation. The controller is synthesized independently for each axis. Consequently, it does not deal with the system coupling.

## 4.1. Predictive control law design

The implemented controller model must consider an integral action in its control law. The interest lies in the fact that the integrator component can completely cancel the static errors concerning a constant setpoint. For this, the synthesis is based on the controlled auto-regressive integrated moving average (CARIMA) model (Camacho & Bordons, 2007):

$$ \Delta \cdot A(q^{-1}) y(t) = q^{-k} B(q^{-1}) \cdot \Delta \cdot u(t - 1) + C(q^{-1}) \xi(t) \quad (12) $$

with:

$$ \Delta = (1 - q^{-1}) $$

$$ A(q^{-1}) = 1 + a_1 q^{-1} + a_2 q^{-2} + \dots + a_{na} q^{-na} \quad (13) $$

$$ B(q^{-1}) = b_0 + b_1 q^{-1} + b_2 q^{-2} + \dots + a_{nb} q^{-nb} $$

$$ C(q^{-1}) = 1 + c_1 q^{-1} + c_2 q^{-2} + \dots + c_{nc} q^{-nc}, $$

where $q^{-1}$ is a delay operator (or backward-shift operator), k the number of time delay intervals or dead time system, $y(t)$ is the system output at time t, $u(t)$ the control signal at time t, and $\xi(t)$ zero mean white noise at time t.

## 4.2. Optimization criterion

The calculation of the control using the predictive method results from the minimization of a quadratic criterion proposed in Clarke et al. (1987). The criterion J, defined in (14), includes on the one hand a term coming from the error between the setpoint and the predicted outputs, and on the other hand, a term linked to the control energy.

$$ J = \mathbb{E} \left[ \sum_{j=N_1}^{N_2} [w(t+j) - y(t+j)]^2 + \sum_{j=1}^{N_u} \lambda(j) [\Delta u(t+j-1)]^2 \right] \quad (14) $$

with $N_1$ and $N_2$ being the minimum and maximum costing horizons, $N_u$ the control horizon, $\lambda$ a weighting factor, $w(t+j)$ the setpoint at time $t+j$, $y(t+j)$ the predicted output at time $t+j$ and $\Delta u(t+j-1)$ the command increments at time $t+j-1$, where $\Delta u(t) = u(t) - u(t-1)$. The optimization is performed under constraints of axes speed and acceleration. The output prediction is done by solving two Diophantine equations from the model.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Real and estimated open-loop system responses to an impulse input in azimuth axis.</figcaption></figure>

## 4.3. Predictive control synthesis

The implemented predictive control is synthesized from an auto-regressive moving average (ARMA) model identified at an elevation of 45°, the plant nominal model. The estimated parameters for polynomials A, B and C are not presented here for confidentiality reasons.

Two input signals are applied to the real plant and the estimated model to validate the system identification. For an impulse signal applied to the azimuth axis, the estimated model presents a response equivalent to the real plant, as illustrated in Fig. 5. The error between the models is extremely low and within the accepted values for the system accuracy. For a sinusoidal signal to the azimuth axis, the estimated model also presents the same dynamics as the real open-loop system (Fig. 6), which validates the model used by this control law.

## 5. Robust control system

This section compares two robust control system designs (CSD): the CRONE and $H_\infty$ strategies. Only the control system design of azimuth is presented here, as CSD for the elevation axis follows exactly the same methodology.
<figure><img src="image_7.png" /><figcaption>Fig. 6. Real and estimated open-loop system responses to a sinus input in azimuth axis.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Diagram of the position feedback loop.</figcaption></figure>

## 5.1. Specifications and loop structure

The objective of applying a robust control law consists of the need to increase the crossover frequency for the control system and to guarantee stability degree for all uncertain models (Hajjem et al., 2022). In this project, it is desired to implement faster controllers with $\omega_{cg} = 10$ rad/s and a phase margin of $M_{\phi} \ge 50^{\circ}$ for the nominal model, which guarantees a good degree of stability and an overshoot of less than 20%. It is also desirable to ensure zero static error in the steady state and reject wind gusts, besides the lowest possible response time.

As explained in Section 2.1 and plotted in Fig. 3, the axes can be considered independent for control system design given the small coupling. Hence, a SISO controller can be designed by considering axis uncertainties in the frequency domain. Fig. 7 depicts the block diagram of the antenna position closed-loop control. The maximum frequency in this type of antenna is 20 Hz, which is approximately 125 rad/s. The sampling period is 50 ms, and because of Shannon's theorem, the maximum frequency is set to 62.5 rad/s. All higher frequencies from the original model can be neglected, simplifying the model used in the CSD.

## 5.2. CRONE CSD

The 3rd CRONE controller (Lanusse et al., 2013; Oustaloup, 1991) is used in the problem of mobile antennas because the system has several types of uncertainties in the frequency domain (not only gain-like) and presents under-damped modes.

<figure><img src="image_9.png" /><figcaption>Fig. 8. Open-loop Nichols locus (left) and its optimal shape (right) defined by the 3rd gen. CRONE methodology.</figcaption></figure>

### 5.2.1. Methodology

The third-generation CRONE controller has its nominal open-loop transfer function defined by an ideal integrator of complex fractional order whose Nichols locus must touch a contour of necessary magnitude ($M_{r0}$) around the resonance frequency ($\omega_r$) of the nominal closed-loop transfer function ($T_{nom}$).

The variations of $M_r$ for the other parametric states of the plant ($M_{rmax2} \le M_{rmax1}$) are minimized by the optimal open-loop function, as shown in Fig. 8. It is obtained by minimizing the objective function $J$ defined by $J = (M_{rmax} - M_{r0})^2$. The CRONE controller is derived from the open-loop transfer function.

The nominal fractional open-loop transfer function contains integral and low-pass filters, and could include a notch filter to control under-damped modes present in the nominal plant $G_{nom}(s)$:

$$ \beta_{nom}(s) = K \left( \frac{\omega_l}{s} + 1 \right)^{n_l} \frac{1}{\left(1 + \frac{s}{\omega_h}\right)^{n_h}} \left( \frac{1 + \frac{s}{\omega_h}}{1 + \frac{s}{\omega_l}} \right)^a \times \mathfrak{R}_{/i} \left[ \left( C_0 \frac{1 + \frac{s}{\omega_h}}{1 + \frac{s}{\omega_l}} \right) \right]^{-\text{sign}(b)} G_{und}(s) N(s) \quad (15) $$

with

$$ C_0 = \left( \frac{1 + \left( \frac{\omega_r}{\omega_l} \right)^2}{1 + \left( \frac{\omega_r}{\omega_h} \right)^2} \right)^{1/2}, $$

$$ N(s) = \frac{1 + 2z'_{1} \frac{s}{\omega_{l}} + \left( \frac{s}{\omega_{l}} \right)^{2}}{1 + 2z_{1} \frac{s}{\omega_{l}} + \left( \frac{s}{\omega_{l}} \right)^{2}} $$

is a notch filter, $G_{und}(s)$ the under-damped part of the nominal plant, $\omega_l, \omega_h$ and $K \in \mathbb{R}^+$, and $n_l$ and $n_h \in \mathbb{N}^+$.

### 5.2.2. Constraints

The specifications are translated into the frequency domain by inequality constraints on the sensitivity functions. Robustness is guaranteed for all possible models that are included within the domain of model uncertainties. The highest resonant peak is bounded by:

$$ |T(j\omega)| \le T_{\max}, \quad (16) $$

with $T(j\omega) = \beta(j\omega)/(1 + \beta(j\omega))$, with is called the complementary sensitivity function, and $\beta(j\omega) = G(j\omega)C(j\omega)$ the open-loop transfer function. The stability degree requirement is respected with $T_{max}$ equal to 2 dB for all model uncertainties, which limits an overshoot around 20% for small step responses. The lowest modulus margin is restricted by:

$$ |S(j\omega)| \le S_{\max} = 6 \text{ dB}, \quad (17) $$

where $S(j\omega) = \frac{1}{1 + \beta(j\omega)}$ is the sensitivity function.

To improve the input disturbance rejection:

$$ |G(j\omega)S(j\omega)| \le G S_{\max} = -8 \text{ dB}, \quad (18) $$
<table><caption>**Table 1**
Open-loop parameters for the CRONE CSD.</caption><tr><td>M<sub>r0</sub> = 1.7 dB</td><td>ω<sub>r</sub> = 6.51 rad/s</td><td>a = 0.93</td></tr><tr><td>n<sub>l</sub> = 2</td><td>ω<sub>l</sub> = 2.96 rad/s</td><td>b = -0.51</td></tr><tr><td>n<sub>h</sub> = 5</td><td>ω<sub>h</sub> = 99.58 rad/s</td><td>K = 2.60</td></tr><tr><td></td><td>Y<sub>r</sub> = 3.95 dB</td><td></td></tr></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 9.** General configuration for the control system.</figcaption></figure>

with $G(jω)S(jω)$ being the input disturbance sensitivity function. To limit the control effort:

$$|C(j\omega)S(j\omega)| \le \left| \frac{3\%U_{\max}}{N_{\max}} \right| = 36 \text{ dB}, \qquad (19)$$

with $C(jω)S(jω)$ being the control sensitivity function. The greatest admissible magnitude of the control-effort noise is 3%$U_{\max}$ and the greatest measurement noise is defined as $N_{\max}$.

As previously observed in Fig. 3, the resonance peak of a 2-axis antenna is not high, so there is no under-damped mode to include in (15). Since the $H_{\infty}$ controller does not consider this filter in its synthesis by default, for reasons of reliable comparison, the synthesized CRONE controller excludes a notch filter.

### 5.2.3. Optimization

The optimization step consists of the definition of open-loop parameters, the orders for the integral action, $n_l$ in low frequency and for better measurement noise filtering, $n_h$, and the resonant peak ($M_{r0}$) of $T_{nom}$. The open-loop parameters to optimize are the resonant frequency $\omega_r$, $\omega_l$, $\omega_h$ and $Y_r = |\beta_{nom}(j\omega_r)|$. The optimized open-loop parameters are obtained by using the MATLAB®/CRONE toolbox (Lanusse et al., 2013) and the results are presented in Table 1.

## 5.3. $H_{\infty}$ CSD

The $H_{\infty}$ (or H-infinity) optimization approach has shown interesting results over the last decades. This control theory was introduced by Zames (1976, 1981), where the author formulates a basic feedback system as an optimization problem with an operator norm. Francis and Doyle (1987) present the main results of $H_{\infty}$ control theory in the context of continuous-time linear systems, focusing on the mathematical development of the problem.

### 5.3.1. Methodology

The general configuration for the controlled system is shown in Fig. 9.

The control system can be described in terms of inputs/outputs as

$$ \begin{gathered} \begin{bmatrix} z \\ v \end{bmatrix} = P(s) \cdot \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11}(s) & P_{12}(s) \\ P_{21}(s) & P_{22}(s) \end{bmatrix} \cdot \begin{bmatrix} w \\ u \end{bmatrix}, \qquad (20) \\ u = K(s) \cdot v \end{gathered} $$

where signals are defined as control variables $u$, exogenous inputs $w$, measured variables $v$, and error signals $z$ to be minimized in order to satisfy the specifications. The generalized process is expressed by $P$ and

<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Block diagram for the mixed sensitivity approach.</figcaption></figure>

the desired controller by $K$. The state space realization for the process $P$ is given by

$$ P = \begin{bmatrix} A & B_1 & B_2 \\ C_1 & D_{11} & D_{12} \\ C_2 & D_{21} & D_{22} \end{bmatrix}. \qquad (21) $$

Considering the relations (20), the closed-loop transfer function from $w$ to $z$ can be rewritten by the linear fractional transformation (LFT):

$$ z = F_l(P, K) w, \qquad (22) $$

where $F_l(P, K) = P_{11} + P_{12}K(I - P_{22}K)^{-1}P_{21}$. The objective of the $H_{\infty}$ synthesis is to find the regulator $K$ such that the $H_{\infty}$ norm of the LFT is minimal, according to (23), which is called the $H_{\infty}$ optimization problem:

$$ \min_{K \text{ stabilizing}} \|F_l(P, K)\|_{\infty} \qquad (23) $$

with

$$ \|F_l(P, K)\|_{\infty} = \sup \bar{\sigma}(F_l(P, K)(j\omega)), \qquad (24) $$

the sigma function, $\sigma$, represents the singular values of the frequency response, where $\bar{\sigma}$ are the upper values.

As shown by Skogestad and Postlethwaite (2005), maximum singular values are forced to stay below user-defined limits, ensuring desirable bandwidths and roll-off. The mixed sensitivity approach uses the shaping of the sensitivity function $S$, the control sensitivity function $KS$ – also called $R$ – and the complementary sensitivity function $T$. The mixed sensitivity problem (also called $S$ over $KS$) is stated as

$$ \min_{K \text{ stabilizing}} \left\| (I + PK)^{-1} K(I + PK)^{-1} \right\|_{\infty}. \qquad (25) $$

The output disturbance is a low-frequency signal and will be rejected by reducing the maximum singular value of $S$ at low frequencies. A low-pass filter $W_1(s)$ with the same bandwidth as the noise is applied and a stabilizing $K$ is sought that minimizes $\|W_1 S\|_{\infty}$. However, this consideration alone can lead to practical problems. For a non-minimum phase system, the stability requirement will indirectly limit the controller gains. In practice, a high pass filter $W_2(s)$ with a cut-off frequency close to the desired closed-loop bandwidth is used to limit the gains. The ability to shape the $T$ function is also desirable for tracking problems and noise attenuation, in addition to robust stability against multiplicative disturbances at the process output. Therefore, the minimization problem is defined as

$$ \left\| \begin{matrix} W_1 S \\ W_2 K S \\ W_3 T \end{matrix} \right\|_{\infty} < 1. \qquad (26) $$
<figure><img src="image_12.png" /><figcaption>Fig. 11. Optimal solution with performance (top) and control (down) constraints.</figcaption></figure>

The exogenous input is a reference command $w$, and the error signals are $z_1 = W_1 \cdot e = W_1 \cdot (w - y)$, $z_2 = W_2 \cdot u$ and $z_3 = W_3 \cdot y$. The weighting functions reflect the specified steady-state error and disturbance attenuation, control input constraints, and stability condition. The $W_1$ weights $S$ against the performance specifications; the $W_2$ weights $KS$ to limit control effort; and the $W_3$ weights $T$ to reflect the stability condition and RF noise rejection. The block diagram of the standard mixed-sensitivity optimization problem is defined in Fig. 10.

From the relation $S + T = I$, as explained by Henry (2020), causes in $|W_1^{-1}(j\omega)| + |W_3^{-1}(j\omega)| > 1$. Hence, $W_1$ and $W_3$ are coupled, and if $S$ and $KS$ have correct constraints, the synthesis of $W_3$ is facultative.

### 5.3.2. Constraints

The choice of the weighting functions is fundamental for the controller synthesis and depends on the design problem and its constraints. According to the performance specifications in the time domain, one can determine the corresponding requirements in the frequency domain, taking into account the bandwidth $\omega_b$, the peak of the sensitivity function $M_s$, and the steady-state error $\epsilon_s$. By choosing a weighting function $W_1$ that satisfies the condition $|W_1(0)| \ge 1/\epsilon_s$, the condition $\|W_1S\|_{\infty} \le 1$ can be satisfied. If a steeper transition between low frequencies and high frequencies is desired, to improve the performance of the controller, the roll-off can be modified by the filter order $k$:

$$W_1(s) = \left( \frac{s + \frac{\kappa}{\sqrt{M_s}} + \omega_b}{s + \omega_b + \frac{\kappa}{\sqrt{\epsilon_s}}} \right)^k, \quad k \ge 1. \qquad (27)$$

The weighting function for the control signal ($W_2$) is given in the same way. The $|KS|$ gain at low frequencies is essentially limited by the control energy and the actuator saturation limits. Therefore, the maximum gain ($M_u$) of $KS$ can be quite large, while the high-frequency gain is limited by the bandwidth of the controller $\omega_{bc}$ and the sensor noise:

$$W_2(s) = \left( \frac{s + \omega_{bc} / \frac{\kappa}{\sqrt{M_u}}}{\frac{\kappa}{\sqrt{\epsilon_u}} s + \omega_{bc}} \right)^k, \quad k \ge 1. \qquad (28)$$

### 5.3.3. Optimization

The filter $W_1$ has been synthesized to have the crossover frequency $\omega_b = 4.5$ rad/s, the gain margin $M_s = 2.0$ and $\epsilon_s = 10^{-6}$ to ensure static error according to the specifications. To improve performance, the roll-off has been set equal to -40 dB/dec with a second-order filter ($k = 2$). The condition to be satisfied is given by $\|W_1S\|_{\infty} < 1 \Rightarrow \|S\|_{\infty} < \|W_1^{-1}\|_{\infty}$. The filter $W_2$ has been synthesized with a crossover frequency $\omega_{bc} = 360$ rad/s so that the cut-off frequency is close to 10 rad/s, $M_u = 500$ and $\epsilon_u = 10^{-6}$. It also meets the condition $\|W_2KS\|_{\infty} < 1 \Rightarrow \|KS\|_{\infty} < \|W_2^{-1}\|_{\infty}$.

<figure><img src="image_13.png" /><figcaption>Fig. 12. Frequency responses of the nominal (-) and extreme open-loops considering CRONE controller (left) and $H_{\infty}$ controller (right).</figcaption></figure>

The controller synthesis was performed on MATLAB® with the `hinfsyn` function of the Robust Control toolbox. Given the weighting functions presented earlier, the controller $K_{H_{\infty}}$ minimizes the infinite norm to 0.6925, respecting criterion (26). Fig. 11 shows the constraints and desired closed-loop sensitivity functions of the controlled system. This solution guarantees that the constraints are above the sensitivity functions, respecting the imposed performance and stability conditions for nominal (nom) and uncertain (unc) models.

### 5.4. Frequency domain comparison

Note that the $H_{\infty}$ controller is a system of order 9 whereas the CRONE controller is only a system of order 7, namely 4 fewer parameters. After synthesizing the control laws under the same constraints, a stability analysis is performed with the nominal and extreme cases of the system frequency response. Fig. 12 presents the open-loop Nichols diagrams considering both controllers, with the nominal (blue) and extreme (green and red) cases for the azimuth axis. The extremes cases are the worst system frequency responses presented in Fig. 3, which corresponds to the extreme elevation angles of 0° and 90°.

The CRONE and $H_{\infty}$ controllers have essentially the same gain margin $M_g$, around 9.1 dB. This parameter gives the smallest variation of the block of uncertainty $\Delta(s)$, which destabilizes the closed-loop system. Thus, these two laws have the same robustness to the pure gain variation of the model uncertainties. The resonance peak $M_r$ of the complementary sensitivity function $T$ is indicated in Nichols' diagram iso-contour lines. The CRONE peaks at 1.93 dB and the $H_{\infty}$ peaks at 2.8 dB. The resonant peak is strongly related to the overshoot of the closed-loop system. In this case, the CRONE controller should show less overshoot when the response is analyzed in the time domain. The $H_{\infty}$
<figure><img src="image_14.png" /><figcaption>Fig. 13. Robustness test with step responses for $H_\infty$ (up), CRONE (middle), MPC (down) controllers considering nominal (-) and extreme (-/-) cases.</figcaption></figure>

controller exhibits a slightly faster integral action than the CRONE at low frequency, which can be verified by the distance between the $\beta$ curves and the central phase axis at -180° (right side of the diagram). This results in a faster trajectory tracking recovery for the $H_\infty$, but also makes it more sensitive to friction at low speeds, resulting in a larger amplitude oscillation relative to the CRONE.

## 6. Implementation and results

The predictive control is implemented using a RST controller, where R, S, and T are polynomials in the backward shift operator. This control law is composed of a feedforward term and a feedback part. The approach performs the prediction using the Diophantine equations computed offline, which diminishes the cost and computational time of the controller algorithm.

The results presented in this sections are obtained through simulations, thus allowing a complete evaluation of the control laws under specific and similar operational scenarios. In this way, the plant model can be forced to have worse behavior in relation to the frequency response gain and is evaluated together with external disturbances (e.g. wind gusts). Such cases allow an objective assessment of the different control strategies under the most critical conditions in which the real system can operate and consequently ensure a more reliable performance validation. Due to confidentiality reasons from Safran Data Systems, the MPC feedforward design cannot be detailed. True experiments on ground antennas have been validated and published with the CRONE CSD (Hajjem et al., 2023) with a comparison with a PI controller. In the latter reference, the tests do not necessarily present such severe conditions, and demonstrate the effectiveness of the CRONE control laws. The objectives of the present work are to evaluate the system under the worst operational conditions and scenarios in order to guarantee the extreme effectiveness of the applied control system strategies.

The transfer function of the CRONE controller can be obtained in two ways. The Oustaloup's synthesis calculates the equivalent rational open-loop function, which allows the controller function to be derived (see Appendix for a recall of Oustaloup's synthesis). Nonetheless, this technique sometimes leads to complex controllers with superabundant

orders. The alternative is to derive the rational CRONE controller function directly from its fractional frequency response, approximating the ideal fractional-order frequency response to a low-order integer one. Thus, the transfer function coefficients are an approximation and do not need a significant amount of digits to the right of the decimal point. This simplifies the implementation of the controller in the antenna computer system.

The synthesis approach for the $H_\infty$ controller was the mixed sensitivity, where the controller transfer function is computed by matrix algebra. Therefore, the coefficients are relevant for its performance. In practical tests with antennas, an accuracy of at least 8 digits to the right of the decimal point was required to ensure system convergence and steady-state error elimination. In the CRONE controller, 4 digits provided the same margin of error.

Time-domain analyses evaluate the performance of the control laws, assessing their ability to track a reference signal, reject disturbances at the process output, and attenuate sensor measurement noise.

### 6.1. Step responses

The step responses considering nominal and extreme cases are shown in Fig. 13, checking for oscillations and performance degradation to model uncertainties. The CRONE and $H_\infty$ controllers exhibit stable responses despite pattern variations, with low impact on settling time and overshoot. The step responses for the azimuth axis are less stable because its system frequency responses are more impacted by parametric uncertainties and working points, presenting a greater variation in resonance frequencies and peaks compared to the elevation axis. It reduces the robustness of the stability margin and generates oscillations for certain cases, as shown in the red curves (right side). The CRONE control demonstrates better stability robustness concerning $H_\infty$ strategy, especially for the azimuth axis (left side).

The predictive control ignores parametric variations in its synthesis. Consequently, such a control system will only perform well around the nominal operating point for which the model was identified. It also presents a slower response since its bandwidth is lower than the other robust control methods. For the nominal case and one of the extreme cases (red line), the controller is stable and has almost
<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Azimuth and elevation control system output (top) and control signal (down) for a real tracking satellite trajectory.</figcaption></figure>

no overshoot. For the other extreme case (green line), the response produces a large oscillation, confirming that a greater plant variation respecting the mathematical model used to controller synthesis causes a lack of robustness in terms of stability and performance.

## 6.2. Tracking satellite trajectory

### 6.2.1. Independent control system responses

The controllers are compared in a real tracking satellite overpass. The reference signals is shown in Fig. 14, with the system output for each control law. The satellite pass has a maximum elevation of 85°, which results in high dynamics for the azimuth axis when the antenna rotates to obtain the maximum elevation position. Hence, the azimuth axis control signal ($U_{Azimuth}$) is higher than the elevation ($U_{Elevation}$) one. The results for all control systems are similar, especially when the antenna turns around, between 300 and 400 s.

The system output position errors are more relevant at the moment of antenna rotation when the elevation axis reaches its maximum value (Fig. 15). The predictive controller has greater accuracy in tracking the azimuth trajectory, with a maximum position error of about $4 \cdot 10^{-3}$°. Even with a smaller frequency bandwidth regarding robust control laws, this error is six times lower than CRONE and $H_\infty$ controlled systems.

### 6.2.2. Control system response with MPC feedforward

As explained previously, the structure used by this control law has a feedforward term, which allows anticipation of the system output. It avoids motor saturation in acceleration and causes a smaller trajectory tracking error when a high dynamic is requested. The robust controllers have a larger bandwidth, and consequently, a faster response, as observed in the elevation axis position error (zoom plot on the left side of Fig. 15). Their maximum error is three times lower than the predictive controller ones. However, both robust controllers saturate in acceleration in the azimuth axis, presenting a higher error than the predictive controller.

A feedforward control was incorporated into the CRONE and $H_\infty$ models to allow a fairer comparison with the predictive controller. In this instance, robust control systems do not saturate in acceleration and the maximum error was approximately $6 \cdot 10^{-4}$° in azimuth and $2.5 \cdot 10^{-3}$° in elevation, values significantly lower than those achieved by the predictive controller. The results are illustrated in Fig. 16. Table 2 provides the maximum values of position error ($e_{max}$) and control signal ($u_{max}$) for the cases where robust control strategies are used with and without a feedforward controller. This controller allows anticipating the reference signal, as MPC does, resulting in an extremely low position error for robust control laws, with a slight advantage for the CRONE controller as compared to $H_\infty$ controller.

<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Output position errors and control signals for a satellite tracking for predictive control (-), $H_\infty$ (...) and CRONE (-) controllers.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 16.** Output position errors and control signals for a satellite tracking with predictive control (-) and with $H_\infty$ (...) and CRONE (-) controllers using a feedforward compensation.</figcaption></figure>

### 6.2.3. Control system response with MPC feedforward and rejection of wind turbulence

Previous simulations show that a fast controller can increase the accuracy of antennas by reducing position errors when tracking target trajectories. However, by coupling a slow regulator with feedforward control, the position errors obtained could be satisfactory to meet customer specifications, particularly in cases where the model used in the feedforward control is well calibrated and close to the real system. The question of whether there is an interest in developing faster regulators could then arise. In addition to refining trajectory tracking, the interest of fast feedback regulators lies in the ability to reject external disturbances more effectively.

The wind speed generator is integrated into the plant model to evaluate the ability of the control law to reject output disturbances. The
<caption>**Table 2**</caption>

Performance metrics for each control strategy in the trajectory tracking of a satellite target with/without feedforward controller, $e_{max}$ being the maximum absolute position error (°) and $u_{max}$ being the maximum absolute control signal (°/s).

<table><caption>**Table 3**</caption><thead><tr><th rowspan="2">Control strategy</th><th rowspan="2"></th><th colspan="2">Azimuth axis</th><th colspan="2">Elevation axis</th></tr><tr><th>e<sub>max</sub></th><th>u<sub>max</sub></th><th>e<sub>max</sub></th><th>u<sub>max</sub></th></tr></thead><tbody><tr><td rowspan="3">Control system<br>without feedforward<br>(Section 6.2.1)</td><td>MPC</td><td><strong>0.0040</strong></td><td><strong>7.0632</strong></td><td>0.0112</td><td><strong>0.5498</strong></td></tr><tr><td>H<sub>∞</sub></td><td>0.0250</td><td>7.0738</td><td><strong>0.0052</strong></td><td>0.5499</td></tr><tr><td>CRONE</td><td>0.0235</td><td>7.0738</td><td>0.0053</td><td>0.5499</td></tr><tr><td rowspan="3">Control system<br>with feedforward<br>(Section 6.2.2)</td><td>MPC</td><td>0.0042</td><td><strong>7.0632</strong></td><td>0.0112</td><td><strong>0.5498</strong></td></tr><tr><td>H<sub>∞</sub></td><td><strong>0.0006</strong></td><td>7.0691</td><td>0.0027</td><td>0.5501</td></tr><tr><td>CRONE</td><td><strong>0.0006</strong></td><td>7.0691</td><td><strong>0.0024</strong></td><td>0.5501</td></tr></tbody></table>

Performance metrics for each control strategy in the trajectory tracking of a satellite target with feedforward controller and in presence of wind disturbances, $e_{max}$ being the absolute maximum of the position error (°), $\Delta e_{max}$ the one of the error variation (°), $u_{max}$ the one of the control signal (°/s), and $\Delta u_{max}$ the one of the control signal variation (°/s).

<table><thead><tr><th rowspan="2">Control strategy</th><th colspan="4">Azimuth axis</th><th colspan="4">Elevation axis</th></tr><tr><th>e<sub>max</sub></th><th>Δe<sub>max</sub></th><th>u<sub>max</sub></th><th>Δu<sub>max</sub></th><th>e<sub>max</sub></th><th>Δe<sub>max</sub></th><th>u<sub>max</sub></th><th>Δu<sub>max</sub></th></tr></thead><tbody><tr><td>MPC</td><td>0.0191</td><td>0.0272</td><td><strong>7.0620</strong></td><td>0.1717</td><td>0.0324</td><td>0.0667</td><td>0.6275</td><td>0.2395</td></tr><tr><td>H<sub>∞</sub></td><td>0.0197</td><td>0.0165</td><td>7.0679</td><td><strong>0.1455</strong></td><td>0.0157</td><td>0.0322</td><td><strong>0.6254</strong></td><td><strong>0.2046</strong></td></tr><tr><td>CRONE</td><td><strong>0.0187</strong></td><td><strong>0.0135</strong></td><td>7.0679</td><td><strong>0.1455</strong></td><td><strong>0.0154</strong></td><td><strong>0.0321</strong></td><td><strong>0.6254</strong></td><td><strong>0.2046</strong></td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>**Fig. 17.** Time domain signal generated by wind speed generator.</figcaption></figure>

generator calculates the instantaneous wind speed along the satellite trajectory, as explained in Section 3. A resulting torque is applied to the antenna axes, adding disturbances to the control system output. With the view to simulate real operating conditions, the applied wind has an average speed of 80 km/h and a direction of 20° in the azimuth axis reference frame. The temporal signal of wind speed generated has values varying between 50 and 100 km/h and is represented in Fig. 17.

The results presented for the satellite tracking simulation with the effect of wind gust disturbance are illustrated in Fig. 18. The position error is higher than the values presented previously, as expected. The first half of the tracking presents a smaller error than the second half since the wind direction causes a higher impact on the system when in the opposite direction to the reflector. In both axes, the error presented by the predictive control is mostly twice the amplitude of the robust controls. The shorter response time of the robust controllers allows a quick recovery to the trajectory tracking point. Thus, they are more efficient in disturbance rejection caused by wind gusts. Moreover, the CRONE controller presents a small advantage regarding the $H_\infty$ control performance, resulting in a lower amplitude position error.

The results presented in the case of satellite tracking considering wind disturbances are summarized in Table 3. The performance metrics used to evaluate the control strategies are the absolute maximum of the position error ($e_{max}$), the error variation ($\Delta e_{max}$), the control signal ($u_{max}$) and the control signal variation ($\Delta u_{max}$). The CRONE controller presents better results for rejecting wind disturbances, with lower error and oscillation values, and requires less control effort. The results

<figure><img src="image_19.png" /><figcaption>**Fig. 18.** Output position errors and control signals for a satellite tracking with predictive control (-), $H_\infty$ (···) and CRONE (-) controllers and wind gust disturbance.</figcaption></figure>

presented for the $H_\infty$ controller are similar, but slightly lower in performance levels on disturbance oscillations.

## 7. Conclusion

This paper presents a comparative study of predictive control and robust control strategies (CRONE and $H_\infty$) applied to a two-axis antenna. The closed-loop system is subject to parametric uncertainties and various operating points, providing a set of uncertain models applied to robust control system design. Verifications in time and frequency domains demonstrate that the robust controllers yield nearly equivalent performance and stability, with a slight advantage for the CRONE strategy over the $H_\infty$ controller. Indeed, the controller order is lesser for the CRONE controller in comparison with the $H_\infty$ one, the digital precision can be lesser for the CRONE controller in comparison with the $H_\infty$ one, and there are a bit lesser oscillations with the CRONE controller. Therefore, in practice, the chosen methodology relies on the designer's expertise, accessibility to the controller toolbox, software and
hardware limitations, and complexity of the system model for controller synthesis. It can be noted that the CRONE CSD allows adding a notch filter contrary to the $H_\infty$ CSD.

Antennas are generally outdoor operating systems. They are constantly subjected to wind, which exerts aerodynamic forces (forces and moments) on the surface of the parabolic reflector of the antenna. These forces are then reflected on the axes as torques disturbing their movements, demanding compensation to maintain the correct position of the system. In this case, wind modeling becomes important to add its effects to the simulation models of the controlled system. In addition, the calculated torques also allow improved dimensioning of the electrical components of the ground station by estimating the maximum wind torque to be compensated by the electro-mechanical system.

Time-domain simulations demonstrate superior performance for both robust controllers, with a slight advantage for the CRONE controller, which offers better stability robustness and greater simplicity in its implementation compared to the predictive control law.

Predictive control is designed only for a nominal system model, therefore it ignores system parametric variations in its synthesis. Hence, this control strategy is efficient around the operating point for which the plant model has been identified. Since the real system presents variations in its model, predictive control does not guarantee performance and stability similar to those performed for the nominal model. Consequently, the specifications and operating requirements are disrespected and the closed-loop system loses pointing accuracy. Predictive control works autonomously if the trajectory is known over the prediction horizon. Otherwise, for operating modes where the antenna receives the position error calculated from the received RF signal, a structure to generate the reference trajectory is required.

The CRONE controller provides robust, stable, and fast control as required by specifications and achieves better performance and rejection of wind disturbances compared to predictive and $H_\infty$ controllers. The robust control laws are also improved with the help of other control tools, such as the anticipation of the reference signal and the treatment of non-linearities existing in the ground station systems. These topics are not presented in this document because they are not part of the main objective of this study and for confidentiality reasons.

Further work could focus on the treatment of uncertain trajectories, evaluating the robustness of control strategies in operating modes where the target trajectory is not known in advance, but defined by the power of the RF signal received by the antenna. In this case, other disturbing phenomena affect the system operation, generating different control dynamics.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix. Rational filter design

Fractional operators are usually approximated by high-order rational functions in a limited frequency band. The fractional model and its rational approximation must have the same dynamical behavior in this limited frequency band. The most commonly used approximation of the fractional integro-differentiator $s^\nu$ (with $\nu$ a non-integer order) in the limited frequency band [$\omega_A, \omega_B$] is the recursive distribution of zeros and poles or the Oustaloup's approximation (Oustaloup, 1995; Oustaloup, Lebron, et al., 2000). However, the rational approximation of the fractional operators in [$\omega_A, \omega_B$] induces deterioration around the limit frequencies $\omega_A$ and $\omega_B$. This deterioration is usually reduced by extending the frequency band over which the approximation is performed from [$\omega_A, \omega_B$] to [$\omega_B, \omega_H$]. In the case of wind turbulence, the frequency band [$\omega_A, \omega_B$] will be $2\pi[10^{-4}, 10^{-1}]$. Therefore, the rational

<figure><img src="image_20.png" /><figcaption>Fig. A.19. Fractional integrator $s^{-0.5}$ and its approximation $s^{0.5}_{[0.1,10]}$ based on Oustaloup's recursive poles and zeros approximation.</figcaption></figure>

approximation will be carried out on $2\pi[10^{-6}, 10^1]$. $s^\nu$ is approximated by (see Fig. A.19):

$$s^\nu \rightarrow s^\nu_{[\omega_A, \omega_B]} = C_0 \left( \frac{1 + \frac{s}{\omega_B}}{1 + \frac{s}{\omega_H}} \right)^\nu \approx C_0 \prod_{k=1}^{N} \frac{1 + \frac{s}{\omega_k}}{1 + \frac{s}{\omega_k}} \quad (A.1)$$

where:

$$\omega_1' = \eta^{1/2} \omega_B, \quad \omega_1 = \alpha \omega_1', \quad (A.2)$$

$$\omega_{k+1}' = \alpha\eta\omega_k', \quad \omega_{k+1} = \alpha\eta\omega_k, \quad (A.3)$$

$$\nu = \frac{\log \alpha}{\log(\alpha\eta)}, \quad \alpha = \left(\frac{\omega_h}{\omega_B}\right)^{\frac{\nu}{N}}, \quad \eta = \left(\frac{\omega_h}{\omega_B}\right)^{\frac{1-\nu}{N}} \quad (A.4)$$

and

$$C_0 = \left| \frac{1 + j \frac{1}{\omega_h}}{1 + j \frac{1}{\omega_B}} \right|^\nu = \left( \frac{\omega_h}{\omega_B} \right)^\nu \left( \frac{1 + \omega_B^2}{1 + \omega_h^2} \right)^{\frac{1}{2}} \quad (A.5)$$

## References

Association, T. I., & Institute, A. N. S. (1996). *Structural standards for steel antenna towers and antenna supporting structures*. Telecommunications Industry Association.

Camacho, E., & Bordons, C. (2007). Model predictive control. In *Number 2 in advanced textbooks in control and signal processing*. Springer London, http://dx.doi.org/10.1007/978-0-85729-398-5.

Clarke, D., Mohtadi, C., & Tuffs, P. (1987). Generalized predictive control—part i. the basic algorithm. *Automatica*, **23**, 137–148. http://dx.doi.org/10.1016/0005-1098(87)90087-2.

Cole, K., & Cole, R. (1941). Dispersion and absorption in dielectrics i. alternating current characteristics. *Journal of Chemical Physics*, **9**, 341–351. http://dx.doi.org/10.1063/1.1750906.

Francis, B., & Doyle, J. (1987). Linear control theory with an $h_\infty$ optimality criterion. *SIAM Journal on Control and Optimization*, **25**, 815–844. http://dx.doi.org/10.1137/0325046.

Gu, D. W., Petkov, P. H., & Konstantinov, M. M. (2013). Robust control design with matlab®. In *Advanced textbooks in control and signal processing* (2nd ed.). Springer London, http://dx.doi.org/10.1007/978-1-4471-4682-7.

Hajjem, M., Lanusse, P., Victor, S., Melchior, P., & Thomas, L. (2022). Robust control of 2-axis tracking antennas using the crone approach. In *1st IFAC workshop on control of complex systems - COSY 2022 Bologna, Italy*, (pp. 169–174). http://dx.doi.org/10.1016/j.ifacol.2023.01.067.

Hajjem, M., Lanusse, P., Victor, S., Melchior, P., & Thomas, L. (2023). Crone mutli-siso control of a 2-axis tracking antenna. In *22nd IFAC world congress Yokohama, Japan*, (pp. 4319–4324). http://dx.doi.org/10.1016/j.ifacol.2023.10.1802.
Hajjem, M., Victor, S., Melchior, P., Lanusse, P., & Thomas, L. (2023b). Wind turbulence modeling for real-time simulation. *Fractional Calculus and Applied Analysis, 26*, http://dx.doi.org/10.1007/s13540-023-00165-0.

Henry, D. (2020). Application of the $h_\infty$ control theory to space missions in engineering education. *IFAC-PapersOnLine, 53*, 17132-17137. http://dx.doi.org/10.1016/j.ifacol.2020.12.1663.

Lanusse, P., Airimitoaie, T., Shulga, E., & Maurel, S. (2023). Principles of a safe and high performing crone control of nonlinear systems. In *2023 international conference on fractional differentiation and its applications* (pp. 1-6). http://dx.doi.org/10.1109/icfda58234.2023.10153309.

Lanusse, P., Malti, R., & Melchior, P. (2013). Crone control system design toolbox for the control engineering community: tutorial and case study. *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371*, http://dx.doi.org/10.1098/rsta.2012.0149.

Oustaloup, A. (1991). *La commande CRONE: commande robuste d'ordre non entier*. Paris: Hermès.

Oustaloup, A. (1995). *La dérivation non-entière : théorie, synthèse et applications*. Paris: Hermès.

Oustaloup, A., Levron, F., Mathieu, B., & Nanot, F. (2000). Frequency-band complex noninteger differentiator: characterization and synthesis. *IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47*, 25-39. http://dx.doi.org/10.1109/81.817385.

Oustaloup, A., Melchior, P., Lanusse, P., Cois, O., & Dancla, F. (2000). The crone toolbox for matlab. *CACSD. Conference proceedings. In IEEE international symposium on computer-aided control system design (cat. no. 00TH8537)* (pp. 190-195). http://dx.doi.org/10.1109/CACSD.2000.900210.

Ravasco, F., Melicio, R., Batista, N., & Valério, D. (2020). A wind turbine and its robust control using the crone method. *Renewable Energy, 160*, 483-497. http://dx.doi.org/10.1016/j.renene.2020.05.180.

Sedhom, B., El-Saadawi, M., Hatata, A., & Abd-Raboh, E. (2019). H-infinity versus model predictive control methods for seamless transition between islanded- and grid-connected modes of microgrids. *IET Renewable Power Generation, 14*, 856-870. http://dx.doi.org/10.1049/iet-rpg.2019.0018.

Skogestad, S., & Postlethwaite, I. (2005). *Multivariable feedback control: analysis and design*. Wiley.

von Kármán, T. (1948). Progress in the Statistical Theory of Turbulence. *Proceedings of the National Academy of Sciences, 34*, 530 LP - 539.

Yang, K., Tang, X., Qin, Y., Huang, Y., Wang, H., & Pu, H. (2021). Comparative study of trajectory tracking control for automated vehicles via model predictive control and robust h-infinity state feedback control. *Chinese Journal of Mechanical Engineering, 34*, http://dx.doi.org/10.1186/s10033-021-00590-3.

Zames, G. (1976). Feedback and complexity, special plenary lecture addendum. In *Proceedings of the IEEE conference on decision and control*.

Zames, G. (1981). Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. *IEEE Transactions on Automatic Control, 26*, 301-320. http://dx.doi.org/10.1109/TAC.1981.1102603.