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Review Article
Comparison of Hinf Robust with Mixed Sensitivity and
LQRy Robust with Uncertainty in a Quadcopter Vehicle
Comparación entre el Controlador Hinf con Sensibilidad Mixta y el Controlador LQRy
con Incertidumbre en un Vehículo Cuadricóptero
Huascar M. Montecinos Cortez1. Francisco J. Triveno Vargas2
. Ph.D. Student. Instituto Tecnológico de Aeronáutica. São Paulo. Brasil. Mirko12v@gmail.com
. Consulting. Universidad Privada del Valle. Cochabamba. Bolivia. trivenoj@hotmail.com.
Citar como: Montecinos
Cortez, H.M., Triveño Vargas,
F.J. Comparison of Hinf robust
with mixed sensitivity and
LQRY robust with uncertainty
in a quadcopter vehicle. Revista
Journal Boliviano De Ciencias,
21(57) 94-110. https://doi.
org/10.52428/20758944.
v21i57.1329
Receipt: 08/05/2025
Approval: 16/06/2025
Published: 30/06/2025
Declaración: Derechos
de autor 2025 Montecinos
Cortez, H.M., Triveño Vargas,
F.J. Esta obra está bajo una
licencia internacional Creative
Commons Atribución 4.0.
Los autores/as declaran no tener
en la publicación de este
documento.
ABSTRACT
This article presents a comparative study between two controllers designed for
quadcopter stabilization. The controllers are the mixed-sensitivity Hinf robust
controller and the LQRy robust controller. Both controllers have been designed
considering uncertainties of 10% in the quadcopter mass and inertia. The main
objective of this investigation is to discern which of the two control techniques
model of the quadcopter. Therefore, given that the quadcopter presents a MIMO
more complex when incorporating diagonal uncertainties in MIMO systems, a
facilitates the incorporation of diagonal uncertainties in the quadcopter model.
The simulations were performed in the Matlab-Simulink® environment. The
results indicate that the LQRy controller performs better than the Hinf controller
in stabilizing the quadcopter. The results suggest that the LQRy technique could
Keywords: LQRy, Hinf, Quadcopter, Automatic Control, Robust control.
RESUMEN
Este artículo, presenta un estudio comparativo entre dos controladores, diseñados
para la estabilización de un cuadricóptero. Los controladores son el controlador
robusto Hinf con sensibilidad mixta y el controlador robusto LQRy. Ambos
controladores han sido diseñados teniendo en cuenta incertidumbres del 10% en
la masa e inercias del cuadricóptero. El objetivo principal de esta investigación
es discernir cuál de las dos técnicas de control ofrece un rendimiento óptimo en la
estabilización del cuadricóptero para garantizar la máxima estabilidad de vuelo.
Para este proposito, ambos controladores se diseñaron utilizando el modelo
lineal del cuadricóptero. Por lo tanto, dado que el cuadricóptero presenta una
hace mas complejo al incorporar incertidumbres diagonales en sistemas MIMO,
en el modelo del cuadricóptero. Las simulaciones se realizaron en el entorno
Matlab-Simulink®. Los resultados obtenidos indican que el controlador LQRy
presenta un rendimiento superior al del controlador Hinf en la estabilización del
cuadricóptero. Los resultados obtenidos sugieren que la técnica LQRy podría ser
Keywords: LQRy. Hinf. Cuadricoptero. Control Automático. Control robusto.
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1. INTRODUCTION
The quadcopter, an unmanned aerial vehicle with four propellers and six degrees
sensing and surveillance, among others (Mie, Okuyama, & Saito, 2018). Although
external forces and uncertain parameters of its dynamics.
thrust, and drag forces from the propellers, among others (Praveen & Pillai, 2016).
Furthermore, it presents parametric uncertainties in aspects such as mass and inertia,
which complicates the representation of the mathematical model. Therefore, its
highly nonlinear behavior makes controlling the quadcopter challenging, rendering
it a subject of considerable inter-est in robotics research (Zenkin et al., 2020).
To address these challenges, it is required to design a controller that can hold
the quadcopter stabilized under various real-world conditions (Irfan, Khan, &
Output Control (LQRy), predictive models, Proportional-Integral-Derivative
out (Maaruf, Mahmoud, & Arif, 2022; Peksa & Mamchur, 2024; Khadraoui et al.,
2024; Tomashevich, Borisov, & Gromov, 2017; Brossard, Bensoussan, Landry, &
Hammami, 2019).
This work presents the design of a controller using the Hinf controller with mixed
sensitivity, which will be robust against the uncertainties of the quadcopter. In this
case, the uncertainties of the quadcopter are located in the mass and inertias.
An uncertainty of 10% in the mass has been selected due to potential variability
when altering the quadcopter’s battery or cameras, resulting in a total mass
variation. Additionally, uncertainties of 10% in the inertias Ixx, Iyy, and Izz of the
quadcopter have been considered, acknowledging the potential for human errors in
measuring these parameters.
To assess the performance of this controller, a comparison will be conducted with
the LQRy controller, also designed considering the same selected uncertainty
parameters. This approach aims to facilitate a comprehensive and detailed
control approach concerning uncertainties.
In this work, Hinf control with mixed sensitivity and LQRy control will be
exclusively applied to the quadcopter plant using Matlab-Simulink® software.
Therefore, they will not be implemented in the physical plant as part of the
to address the challenges associated with uncertainties in the quadcopter, thereby
establishing a foundation for future practical implementations (Smith & Shehzad,
2016).
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2. QUADCOPTER MODEL
In this section, the mathematical model of the quadcopter dynamics is presented. It
begins with the concise development of a set of nonlinear equations that describes
the motion of the quadcopter as a rigid body. These nonlinear equations are then
utilized to derive the linearized equations governing the quadcopter’s dynamics.
To initiate the development of the quadcopter’s mathematical model, it is essential
to comprehend the coordinate system used to describe the quadcopter’s body in
Oi, Xi, Yib, Yb, Zb. Here, Oi and Ob are located
at the local reference point and at the center of mass (CM) of the quadcopter,
respectively.
Both frames adhere to the North-East-Down (NED) and Front-Right-Down (FRD)
orientation conventions, as illustrated in Figure N°1.
Figure
Own elaboration, 2025.
Figure N°1 also depicts the thrust forces F1, F2, F3, F4 and the angles (ϕ, θ, ψ)
associated with rotations in each reference system of the body.
Controlling a quadcopter involves adjusting the forces generated by the four
from the quadcopter’s center of gravity.
The equations describe the rigid body dynamics of a six-degree-of-freedom
quadcopter consist of translational and rotational dynamics. These equations refer
to the body’s coordinate system. To simplify the quadcopter model, the following
assumptions are considered:
• The body structure of the quadcopter is rigid.
• The body structure of the quadcopter is symmetrical.
• The propellers are rigid.
• All engines have identical dynamics.
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• Motors with propellers have greater thrust than the weight of the
quadcopter.
• The center of gravity is located in the physical center of the quadcopter.
To develop the mathematical model of the quadcopter, information is needed on
the values of various parameters. Some of these values were calculated by the
author of this work, while others were obtained from the available literature due
to their similarity to the study carried out. These values are presented in Table 1.
Table 1. Quadcopter parameters
Parameter Name Value Reference
Maximum Motor Speed 15330 rpm (Diego, 2015)
Minimum Motor Speed 430 rpm (Diego, 2015)
vMotor Input Voltage 12.5 V (Paiva Peredo,
2016)
i Motor Current 15 A (Diego, 2015)
n 0.93 (Escamilla
Núñez, 2010)
Km Current-Torque Ratio (Diego, 2015)
r Propeller-Engine Ratio 1/3 (Escamilla
Núñez, 2010)
JTP Total Propeller Inertia 0.044 kg·m² (Diego, 2015)
Jr Motor Inertia (Diego, 2015)
R Motor Resistance (Diego, 2015)
q Drag Moment (Paiva Peredo,
2016)
d Drag Factor (Paiva Peredo,
2016)
b Impulse Factor (Paiva Peredo,
2016)
Source: Own elaboration, 2025.
Table 1 presents the parameters of the quadcopter. Some of these parameters were
determined experimentally, and the methodology is explained in [10]. Therefore,
the translation and rotation equations of the quadcopter are described below [8],
[9], [7].
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Rotation Equations:
Translation Equations:
The equations (1), (2), (3), (4), (5), and (6) represent the nonlinear model of
the quadcopter and will be linearized for the Hinf controller project with mixed
sensitivity and LQRy. These equations enable the determination of the position and
orientation of the quadcopter through double integration of its linear and angular
accelerations.
It’s important to note that the variables U1, U2, U3, and U4 represent the rotation
speed command inputs of the quadcopter motors. These rotations are responsible
for the thrust forces resulting in the movement of the quadcopter. These command
inputs are functions of the rotation speed of each motor, , , , , illustrated in Figure
1. The control signal U, also known as the control vector, comprises U1, U2, U3,
and U4, as shown in (7).
With the equations mentioned above (1), (2), (3), (4), (5), (6), and (7), the
development of the linear model of the quadcopter begins. For the control design,
the nonlinear mathematical model has been linearized considering the equilibrium
point, which is presented in (8).
Where:
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For linearization around an equilibrium point, the technique used is based on the
expansion of the Taylor series, with the retention of only the linear term. Higher-
that the values of the variables deviate only slightly from the operating condition
(Ogata, 2010).
Just as the dynamics of the states and inputs of the system are linearized, it is also
important to linearize the outputs of the model. These results are presented in (11).
After performing the Taylor series expansion of the nonlinear model presented
in Equations (1), (2), (3), (4), (5), and (6), the linearized model is derived, as
presented in (12) and (13).
Where is -21.12, b is 0.5, c is 7.938 and d is 5.99. The quadcopter model presented
is in transfer function form (derived using Matlab and the ss2tf function) and can
be described by the following equations:
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the variables are nearly negligible due to the assumptions made in deriving this
quadcopter dynamics.
3. CONTROLLERS DESIGN
This section will focus on two types of controllers: the Hinf controller with mixed
controlling the dynamic behavior of a quadcopter, utilizing the linear model as a
foundation.
3.1. hinf with mixed sensitivity controller
The Hinf controller is a robust linear controller designed for static or dynamic
feedback control (Massé, Gougeon, Nguyen, & Saussié, 2018; Noormohammadi-
Asl et al., 2020). However, in certain cases, it is crucial to adjust the operational
high or low frequencies. Therefore, to accommodate these frequency adjustments,
the Hinf controller with mixed sensitivity is employed. The Hinf control with mixed
sensitivity introduces the capability to assign input and output weighting functions
robust design of the controller (Varghese & Sreekala, 2019; Priya & Kamlu, 2022).
ranges (Madi, Larabi, & Kherief, 2023).
The system’s frequency response is shaped based on its sensitivity function, as
represented by (18):
The equation (18) comprises one or more weight transfer functions that include:
• Minimization of S/KS for the traceback problem.
• Minimization of S/T.
• Minimization of S/T/KS.
In this study, the Hinf controller with mixed sensitivity will be exclusively
utilized for reference tracking. Therefore, within this context, the Complementary
Sensitivity function T is not considered. The primary objective is to minimize and
satisfy the function presented in equation (19).
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Figure N°2 represents equation (19), displaying the weighting functions W1 and
W2, alongside the gain K and the nominal plant Gn. It also illustrates the exogenous
inputs W and the exogenous outputs (Z1, Z2), as well as the control signal u and the
measured signal y, highlighting the representation of the generalized plant P with
blue lines used to solve the Hinf problem with mixed sensitivity.
Figure N° 2. System diagram with weighting functions. Source: Own
elaboration, 2025.
Figure N°2 also depicts the weighting functions W1 and W2. These functions are
following weighting functions have been chosen. For Z, ϕ, θ :
For :
The equations (20) and (21) represent the weighting functions that allow obtaining
the frequency response of the open-loop plant by pre-multiplying and post-
multiplying by the nominal plant Gn. This response can be seen in Figure N° 3.
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Figure N° 3. Frequency response of the nominal plant and the plant with power
functions. Source: Own elaboration, 2025.
The description of the state-space model for the generalized plant P, as depicted in
Figure 2, is as follows (Smith & Shehzad, 2016; Gonzalez & Vargas, 2008):
Now, with P is [16]:
Since this project involves uncertainties in its mathematical model, the decision
was made to utilize the Hinf controller with robust mixed sensitivity. This particular
plant. In this case, uncertainties directly associated with the mass and inertia of
the quadcopter are considered. The strategy adopted to address these uncertainties
involves the incorporation of diagonal uncertainties in the mass parameters and
inertias (Pinheiro & Souza, 2013).
Figure N° 4 illustrates the block diagram representing the system analysis with the
depicted, alongside the diagonal matrix containing the uncertainties.
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Figure N° 4. Generalized plan with uncertainty organized diagonally. Source:
Own elaboration, 2025.
The function present in the equation (24) is the current objective to be minimized
related to Figure N° 4.
To satisfy 24, it is necessary to analyze the functions S and T. Therefore, Figure
N° 5 and Figure N° 6 depict the function S and the function T. It can be observed
that the system is capable of rejecting disturbances and tracking input references.
considering the implemented uncertainties.
Figure N° 5. Sensitivity function and complementary sensitivity function in and
Z. Source: Own elaboration, 2025.
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Figure N° 6. Sensitivity function and complementary sensitivity function in and
Source: Own elaboration, 2025.
Figure N° 6 illustrates that control for θBS
rad/s. Nevertheless, at frequencies higher than WBT
where the working frequency is greater than or equal to WBS but less than or equal
to WBT.
By examining Figure N° 5 and N° 6, which indicate the frequencies where the control
equation (24). This minimization process is visible in Figures 7 to 10.
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(a) 1/W1 vs Si (b) 1/W2 vs KSi
Figure N° 7. Validation of the sensitivity function with Z uncertainties. Source:
Own elaboration, 2025.
(a) 1/W1 vs Si (b) 1/W2 vs KSi
Figure N° 8. Validation of the sensitivity function with uncertainties. Source:
Own elaboration, 2025.
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(a) 1/W1 vs Si (b) 1/W2 vs KSi
Figure N° 9. Validation of the sensitivity function with ϕ uncertainties. Source:
Own elaboration, 2025.
(a) 1/W1 vs Si (b) 1/W2 vs KSi
Figure N° 10. Validation of the sensitivity function with uncertainties. Source:
Own elaboration, 2025.
Figures 7 to 10 illustrate that the function S should be less than the inverse of the
weighting function and similarly, the product of the function (S) and the gain
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3.2. LQRy CONTROLLER
LQRy is a control strategy that aims to optimize system performance based on a
relative ease in handling multi-output systems (Nasir, Ahmad, & Rahmat, 2008).
Figure N° 11 presents a comprehensive overview of the quadcopter feedback
system. The LQRy controller was developed using MATLAB, implemented as a
SISO (Single Input Single Output) system for each movement of the quadcopter.
This approach was chosen to simplify the system while accounting for uncertainties
in the mass and inertias.
Figure N° 11. Complete representation of quadcopter state feedback. Source:
Own elaboration, 2025.
adjusting elements other than R and Q as required, leading to the creation of the
following controllers:
The similarity between K ϕ and K ϕ in the gains K is attributed to the assumptions
made during the derivation of the mathematical model of the quadcopter.
4. RESULTS
In this section, the combined results of two control strategies will be presented:
the Hinf Robust control strategy with mixed sensitivity and the LQRy Robust
control strategy. Both control systems are subject to uncertainties related to mass
and inertia.
Figure N° 12 and N° 13 display the responses of the controllers to a step and
impulse input.
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(a) Z (b) ψ
Figure N° 12. Z and response. Source: Own elaboration, 2025.
(a) θ (b) ϕ
Figure N° 13. θ and ϕ response. Source: Own elaboration, 2025.
The responses of the controllers presented in Figure N° 12 and N° 13 exhibit similar
control signals when subjected to identical inputs. This observation suggests that
It is only necessary to take into account that the degree of the Hinf controller is
quite a higher, for that, sometimes it is necessary to make an order reduction.
5. CONCLUSIONS
This paper focuses on two types of controllers: the Hinf robust controller with
mixed sensitivity and the LQRy robust controller, which are successfully designed
for the quadcopter by considering the uncertainties in the quadcopter’s mass and
inertia.
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The nonlinear model of the quadcopter was presented, along with the linearization
required for the design of the proposed controllers. The main characteristics of the
Hinf and LQRy controllers were also presented.
However, according to simulations, the LQRy robust controller performs better
than the mixed-sensitivity Hinf robust controller. Furthermore, it is essential to
note that the high gains of both controllers could cause control signal saturation in
the physical actuators of the system. Therefore, this project is limited to simulating
only the linearized layout of the quadcopter as presented and not a practical
controllers through simulation.
While there are many simulation and experimental implementation projects around
journal. It is hoped that it will serve as an incentive for research teams in this
country.
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