Mixed H2/H∞ robust controllers in aircraft control problem

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 13, No. 6, December 2023, pp. 6249~6258
ISSN: 2088-8708, DOI: 10.11591/ijece.v13i6.pp6249-6258  6249
Journal homepage: http://ijece.iaescore.com
Mixed H2/H∞ robust controllers in aircraft control problem
Dana Satybaldina, Aida Dabayeva, Nurgul Kissikova, Gulzhan Uskenbayeva, Aliya Shukirova
Department of System Analysis and Control, Faculty of Information Technologies, L.N. Gumilyov Eurаsiаn National University,
Astana, Kazakhstan
Article Info ABSTRACT
Article history:
Received Feb 15, 2023
Revised May 9, 2023
Accepted Jun 4, 2023
A leading cause of accidents during the landing phase of a flight lies in a
considerable altitude loss by an aircraft as a result of the impact of a
microburst of wind. One of the significant factors focuses primarily on the
need to simultaneously satisfy various requirements regarding conditions of
environmental disturbances and a wide range of systemic changes. The
paper presents an algorithm for synthesizing an optimal controller that solves
the mixed H2/H∞ control problem for the stabilization of aircraft in glide-
path landing mode in the presence of uncertainty. Firstly, the principles of
multi-criteria optimization are presented, and the mixed H2/H∞ problem is
interpreted as the synthesis of a system with optimal quadratic performance,
subject to its readiness to operate with the worst disturbance. Then, the
ensuing section expounds upon the mathematical depiction of the vertical
trajectory of aircraft, duly considering the perturbations imposed by wind
phenomena. Subsequently, the effectiveness of mixed H2/H∞ control is
confirmed compared to autonomous H2 or H∞ regulators through simulation
outcomes acquired from the created system. Optimization based on a hybrid
(mixed) criterion allowed combining the strengths of locally optimal systems
based only on H2 or H∞ theory.
Keywords:
Aircraft control
Aircraft landing
Mixed H2/H∞ control
Multi-objective optimization
Robust control
This is an open access article under the CC BY-SA license.
Corresponding Author:
Aida Dabayeva
Department of System Analysis and Control, Faculty of Information Technologies, L.N. Gumilyov Eurаsiаn
National University
11 Pushkin st., Z00T8E0, Astana, Kazakhstan
Email: mashtayeva@mail.ru
1. INTRODUCTION
A high accuracy in determining motion parameters and controlling the aircraft is an essential
requirement for modern control system design [1]–[8]. This emergence necessitates considering various
uncertainty factors during the development phase of appropriate control algorithms. Particular importance is
attached to random uncertainties affecting aircraft flight include the disturbances in the atmosphere, such as
density deviation from the standard value and wind shear, as well as processing errors in control actions,
deviations in the aerodynamic, geometric, and several other factors [9]–[13]. It is important to note that the
vast majority of flight accidents occur due to adverse meteorological conditions. The meteorological
phenomenon of a local disturbance of atmospheric state, known as the vortex ring microburst, poses a
significant threat to aircraft flights, particularly during take-off and landing phases [14]–[18]. In the context
of the examined control algorithms within this domain, the comprehensive review of existing literature
uncovers a multitude of diverse strategies employed for the purpose of aircraft control [19]–[23].
In a comprehensive review of intelligent transforming aircraft, Chu et al. [19] discuss both general
and specific challenges in their development. Ghazali et al. [20] proposes a multinodal hormone regulation of
neuroendocrine proportional-integral-derivative (PID) controller of multiple-input-multiple-output (MIMO)

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Int J Elec & Comp Eng, Vol. 13, No. 6, December 2023: 6249-6258
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systems grounded on adaptive safe experimentation dynamics (ASED). Similarly, Ghazali et al. [21]
investigate the incorporation of controlled sigmoid-based secretion rate neuroendocrine PID in a twin-rotor
MIMO system using ASED algorithm. In reference to the findings presented by Kiselev et al. [22], the
research delves into the examination of flight dynamics exhibited by a hypothetical maneuverable aircraft.
Additionally, it investigates the application of algorithms aimed at augmenting stability and controllability,
thereby compensating for inherent limitations in these characteristics. Notably, a sophisticated boundary
delineating the permissible angle of attack is introduced, contingent upon the specific flight mode under
consideration. Idrissi et al. [23] explores vertical take-off and landing arrangements, presents applicable
modeling tools and control strategies, and applies them to a quadrotor.
The problem of ensuring high-quality landing control is highly relevant, especially in the presence
of atmospheric disturbance. Robust controllers based on H∞ control method is extensively applied
extensively in order to address this problem. The H∞ theory provides a powerful framework for the synthesis
of multivariable robust control systems. The standard (unstructured) and structured H∞ control development
techniques have been effectively used to ensure the establishment of robust controllers. The investigation
in [15] revolves around the examination and formulation of a robust glide-path approach controller of the H∞
structure. The controller is an integral component of automated landing system formulated in response to the
aircraft landing challenge proposed by Airbus. In [16], an integrated control method is considered for the
Autoland system of a civil aircraft, which combined stable inversion swarm intelligence (SI) algorithm and
H∞ synthesis to simultaneously solve the problem of tracking the trajectory and deflection disturbances.
In the realm of linear parameter-varying (LPV) systems, wherein faults in actuators and sensors
occur concurrently, the issue of robust active fault-tolerant control is the focal point of investigation within
Tayari et al. [24]. The assurance of stability for the systems operating in closed-loop configuration is ensured
through the application of H∞ performance measures. Within in [25], an integrated sliding-mode controller
incorporating self-adaptation is devised, aiming to attain finite-time convergence in system control,
regardless of the underlying parameters. The study focuses on the LPV model, which experiences significant
alterations in sweep angle and expansion, encompassing a broad range of parameters. The state-feedback
linear fractional representation (LFR)-H∞ controller is derived through the utilization of constraints based on
linear matrix inequalities. Subsequently, the necessary prerequisites for the existence of sliding mode
characterized by integral action are derived by means of pole assignment.
Yue et al. [26] describes the development of a morphing aircraft engine multi-loop controller, which
ensures the steadiness of the process of wing transition. The offered controller employs a collection of inner
loop gains in order to guarantee stability, leveraging basic methodologies as the foundation for its design.
A self-tuning H∞ controller is formulated for the outer loop gain to attain a satisfactory degree of robust
stability and operational effectiveness, particularly in the presence of non-stationary dynamics.
A comprehensive research in [27] focus on the determination of robust controller parameters for the lateral
control of aircraft, wherein the utilization of auxiliary damping automatic devices (ADAD) plays a pivotal
role. The synthesis of the suggested controller is founded upon the utilization of both H∞ and μ techniques,
serving as the fundamental framework for it is development.
The structured H∞ paradigm has emerged as a versatile approach for implementation of
multi-requirement and multi-variable control systems. In research [14], a structured H∞ method based on a
standard H∞ control structure is examined for a vertical speed controller. Biannic et al. [17] concentrates on
the demanding flare phase in the conditions of high wind and parametric uncertainties based on a structured
principle of H∞ control. The results of the research provide important insights into the problem of aircraft
vertical speed control before landing phase of a flight, minimizing the impact of variations in airspeed, wind
gradient, and ground proximity. Marcos et al. [28] provides an extensive comparative study, centered around
the assessment of two distinct control schemes utilized to actively suppress flutter in a flexible unmanned aerial
vehicle, with thorough analysis and evaluation. The H∞ approach is applied in the development of both
controllers, however, the first is based on a standard (i.e., unstructured) synthesis, and the second is based on
a structured technique. Beisenbi and Basheyeva [29] describes the application of the Lyapunov function to
construct robustly stable aircraft control systems. Karimtaevna and Asylbekkyzy [30] outlines a design
methodology and implementation of robust control using H∞ synthesis tools, which allows to cope more
effectively with parameters and load perturbation. The research conducted in Karimtaevna et al. [31] delves
into a meticulous investigation of the H2 and H∞ synthesis methods, specifically exploring their potential in
the realization systems responsible for controlling the flight of an aircraft during the crucial landing phase,
while effectively mitigating the impact of external disturbances.
A promising approach consists of system optimizing using several criteria, each of which applies
under certain circumstances; consequently, there arises a necessity of considering the problem of robust
controller synthesis in terms of simultaneously satisfying two optimization H2/H∞ robust controller criteria
[32]–[34]. An analysis of scientific publications dedicated to the field of the mixed H2/H∞ robust controller

Int J Elec & Comp Eng ISSN: 2088-8708 
Mixed H2/H∞ robust controllers in aircraft control problem (Dana Satybaldina)
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synthesis indicates that the issue of using the mixed H2/H∞ controller for solving the problem of aircraft control
under conditions of uncertainty has not received sufficient attention. The investigation of the H2/H∞ controller is
carried out only from the perspective of robust stability, and the issue of improving the technical characteristics
therefore remains relevant. The problem of developing mixed H2/H∞ robust controllers for aircraft flight
control under conditions of uncertainty is of relevance to both academic research and industrial applications.
This paper describes the synthesis of the mixed H2/H∞ robust controller for regulating aircraft
motion in the vertical plane throughout the critical landing phase, even in the presence of uncertain
disturbances. This solution effectively enhances the robustness of the system, effectively mitigating the
adverse effects of uncertainties induced by disturbances caused by wind conditions. Section 2, entitled
“research method,” offers an exhaustive assessment of the fundamental principles underlying multi-objective
optimization, interprets the mixed H2/H∞ control approach as the problem of optimal quadratic quality under
the condition of robust stability, and constructs a mathematical model capturing the intricate dynamics of
airplane in the vertical dimension, accounting for the influence of uncertain disturbances. Section 3, entitled
“results and analysis,” presents the findings of the application of the mixed H2/H∞ optimal controller to
aircraft’s flight control mechanisms, specifically addressing the challenges encountered during the critical
landing phase in the face of turbulent wind interferences. The simulation outcomes provide evidence
supporting the effectiveness of the blended H2/H∞ control strategy in terms of its efficiency. The simulation
results provide evidence supporting the effectiveness of the mixed H2/H∞ control strategy in terms of its
efficiency. Finally, section 4 presents the primary findings and imparts recommendations for forthcoming
investigations, thus culminating the study.
2. RESEARCH METHOD
Controller synthesis based on various criteria (i.e., norms) that are related to either to one or
different system outputs is a common aspect of multi-objective optimization. To accurately represent the
output, a quadratic or uniform-frequency index is typically employed. The development of a controller that
optimally represents the first or second indicator is achieved using well-known algorithms described in
literature [35], [36]. Recently, the optimization of the system output based on both frequency-uniform and
quadratic criteria simultaneously, known as mixed H2/H∞-control, has gained significant attention.
Contemplate a stationary linear system depicted in Figure 1, which possesses finite dimensions.
Assume the closed-loop control system exhibits internal stability. The plant 𝐺(𝑠) and controller 𝐾(𝑠) are
described by the state-space equations in (1) and (2) [35], [36].
Figure 1. Scheme of a linear finite-dimensional stationary system
𝑥̇ = 𝐴𝑥 + 𝐵1𝑤 + 𝐵2𝑢;
𝑧0 = 𝐶0𝑥 + 𝐷0𝑢;
𝑧1 = 𝐶1𝑥 + 𝐷1𝑢;
𝑦 = 𝐶2𝑥 + 𝐷2𝑤.
(1)
𝑥̇𝑐 = 𝐴𝑐𝑥𝑐 + 𝐵𝑐𝑦;
𝑢 = 𝐶𝑐𝑥𝑐.
(2)
By substituting expression (2) into (1), the expression (3) is obtained,
𝑥
̃̇ = 𝐴
̃𝑥
̃ + 𝐵
̃𝑤;
𝑧0 = 𝐶
̃0𝑥
̃;
𝑧1 = 𝐶
̃1𝑥
̃,
(3)

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where
𝐴
̃ = [
𝐴 𝐵2𝐶𝑐
𝐵𝑐𝐶2 𝐴𝑐
] , 𝐵
̃ = [
𝐵1
𝐵𝑐𝐷2
] , 𝐶
̃0 = [𝐶0 𝐷0𝐶𝑐], 𝐶
̃1 = [𝐶1 𝐷1𝐶𝑐].
Let 𝑇𝑧𝑤 be the transfer function matrix of a closed-loop control system from input w to z:
𝑇𝑧𝑤 = [
𝑇𝑧0𝑤
𝑇𝑧1𝑤
]. (4)
The synthesized controller must meet the following conditions [36], [37]:
a) A closed-loop system exhibits stability properties, i.e., 𝐴
̃ is a stable matrix.
b) The transfer function 𝑇𝑧1𝑤(𝑠) = 𝐶
̃1(𝑠𝐼 − 𝐴
̃)
−1
𝐵
̃ satisfies the constraint ‖𝑇𝑧1𝑤‖
∞
< 𝛾.
c) The quality functional is minimized: 𝐽(𝑇𝑧0𝑤) = lim
𝑡→∞
∫ {𝑍0
𝑇(𝑡)𝑍0(𝑡)}
𝑡
0
𝑑𝑡 = lim
𝑡→∞
∫ {𝑥
̃𝑇(𝑡)𝑅
̃𝑥
̃(𝑡)}
𝑡
0
𝑑𝑡 =
lim
𝑡→∞
∫ {𝑥𝑇(𝑡)𝑅1𝑥(𝑡) + 𝑢𝑇(𝑡)𝑅2𝑢(𝑡)}
𝑡
0
𝑑𝑡, 𝑅1 = 𝐶0
𝑇
𝐶0, 𝑅2 = 𝐷0
𝑇
𝐷0, 𝑅
̃ = 𝐶
̃0
𝑇
𝐶
̃0 = [
𝐶0
𝑇
𝐶𝑐
𝑇
𝐷0
𝑇] [𝐶0 𝐷0𝐶𝑐] =
[
𝐶0
𝑇
𝐶0 0
0 𝐶𝑐
𝑇
𝐷0
𝑇
𝐷0𝐶𝑐
] = [
𝑅1 0
0 𝐶𝑐
𝑇
𝑅2𝐶𝑐
]; where 𝐽(𝑇𝑧0𝑤) is a special case of the functional of stochastic
linear optimal control task lim
𝑡→∞
1
𝑡
𝐸 {∫ 𝑍0
𝑇(𝑡)𝑍0(𝑡)
𝑡
0
𝑑𝑡} for systems with constant parameters [35].
Minimization of the functional 𝐽(𝑇𝑧0𝑤) is equivalent to the minimization of H2 norm of the transfer
matrix 𝑇𝑧0𝑤, which is regular, and consequently ‖𝑇𝑧0𝑤‖
2
is finite [35].
As the problem formulation includes both H2 and H∞ quality components, similar to the R1 and R2
matrices of the H2, corresponding matrices for the H∞ are introduced. Let 𝑅1∞ = 𝐶1
𝑇
𝐶1, 𝑅2∞ = 𝐷1
𝑇
𝐷1,
𝑅
̃∞ = 𝐶
̃1
𝑇
𝐶
̃1. Similarly, 𝐶1
𝑇
𝐷1 = 0, and let 𝑅2∞ = 𝛽2
𝑅2, where the non-negative scalar β is a design variable.
Let 𝐿𝑐 denote the controllability Gramian for an (𝐴
̃, 𝐵
̃) pair. It satisfies the (5),
𝐴
̃𝐿𝑐 + 𝐿𝑐𝐴
̃𝑇
+ 𝐵
̃𝐵
̃𝑇
= 0 (5)
then [35]:
𝐽(𝑇𝑧0𝑤) = ‖𝑇𝑧0𝑤‖
2
2
= 𝑡𝑟𝑎𝑐𝑒(𝐶
̃0𝐿𝑐𝐶
̃0
𝑇
) = 𝑡𝑟𝑎𝑐𝑒(𝑅
̃𝐿𝑐)
Therefore, solving Riccati equations Y:
𝑅(𝑌) = 𝐴
̃𝑌 + 𝑌𝐴
̃𝑇
+ 𝑌𝑅
̃∞𝑌𝛾−2
+ 𝑉
̃ = 0 (6)
where 𝑉
̃ = 𝐵
̃𝐵
̃𝑇
= [
𝐵1𝐵1
𝑇
0
0 𝐵𝑐𝐷2𝐷2
𝑇
𝐵𝑐
𝑇] = [
𝑉1 0
0 𝐵𝑐𝑉2𝐵𝑐
𝑇] by analogy with (5), the following quality measure
is established:
𝐽(𝑇𝑧𝑤, 𝑌) = 𝑡𝑟𝑎𝑐𝑒(𝐶
̃0𝑌𝐶
̃0
𝑇
) = 𝑡𝑟𝑎𝑐𝑒(𝑌𝑅
̃) (7)
which is a measure consisting of the mixed 𝐻2/𝐻∞ norm, according to the aforementioned property of 𝑌 (6).
As a result, the solution of the Riccati (6) provides the upper bound for the H2 norm criterion subject to the
H∞ norm constraints. According to [35], [36] (𝐴𝑐, 𝐵𝑐, 𝐶𝑐, 𝑌) solve an additional minimization problem.
Therefore, there are non-negative definite matrices 𝑄, 𝑃, 𝑄
̂ such that the (8) equalities hold:
𝐴𝑐 = 𝐴 − 𝑄𝛴
̅ − 𝛴𝑃𝑆 + 𝛾−2
𝑄𝑅1∞;
𝐵𝑐 = 𝑄𝐶2
𝑇
𝑉2
−1
;
𝐶𝑐 = −𝑅2
−1
𝐵2
𝑇
𝑃𝑆,
(8)
while
𝑌 = [
𝑄 + 𝑄
̂ 𝑄
̂
𝑄
̂ 𝑄
̂
] (9)

6253
0 = 𝐴𝑄 + 𝑄𝐴𝑇
+ 𝑉1 + 𝛾−2
𝑄𝑅1∞𝑄 − 𝑄𝛴
̅𝑄 (10)
0 = (𝐴 + 𝛾−2[𝑄 + 𝑄
̂]𝑅1∞)
𝑇
𝑃 + 𝑃(𝐴 + 𝛾−2[𝑄 + 𝑄
̂]𝑅1∞) + 𝑅1 − 𝑆𝑇
𝑃𝛴𝑃𝑆 (11)
0 = (𝐴 − 𝛴𝑃𝑆 + 𝛾−2𝑄𝑅1∞)𝑄
̂ + 𝑄
̂(𝐴 − 𝛴𝑃𝑆 + 𝛾−2𝑄𝑅1∞)𝑇 + 𝛾−2𝑄
̂(𝑅1∞ + 𝛽2𝑆𝑇𝑃𝛴𝑃𝑆)𝑄
̂ + 𝑄𝛴
̅𝑄 (12)
where 𝛴 = 𝐵2𝑅2
−1
𝐵2
𝑇
, 𝛴
̅ = 𝐶2
𝑇
𝑉2
−1
𝐶2, 𝑆 = (𝐼𝑛 + 𝛽2
𝛾−2
𝑄
̂𝑃)
−1
, 𝛽 > 0, and 𝑅2∞ = 𝛽2
𝑅2. In addition, the
auxiliary cost for the system can be represented by the subsequent (13),
𝐽(𝑇𝑧𝑤, 𝑌) = 𝑡𝑟𝑎𝑐𝑒([𝑄 + 𝑄
̂]𝑅1 + 𝑄
̂𝑆𝑇
𝑃𝛴𝑃𝑆) (13)
where 𝑄, 𝑃, and 𝑄
̂ are solutions of modified Riccati (10)-(12). Consequently, the mixed H2/H∞ control
problem can be construed as referring to optimal quadratic quality, provided robust stability. In the instant
case, the upper bound for ‖𝑇𝑧0𝑤‖
2
is minimized under the condition ‖𝑇𝑧1𝑤‖
∞
< 𝛾, and the boundary is
commonly called the mixed H2/H∞ norm. The mixed H2/H∞ optimization algorithm is presented in the
flowchart as shown in Figure 2. The concept of the algorithm assumes that the problem is approximated by
the H2 control theory for sufficiently large 𝛾, what allows to obtain a reliable initial value of the solution. The
parameter 𝛾 is successively reduced until the required value is reached, or further reduction becomes
impossible. The convergence of the algorithm is determined by the number 𝜀.
Figure 2. Flowchart of the mixed 𝐻2/𝐻∞ optimization algorithm
The synthesis of the mixed H2/H∞ controller investigated in this paper is applicable to the problem of
aircraft control. Two crucial control variables of an aircraft, namely engine thrust force 𝑇 and angle of attack
𝛼, are contingent upon the deflection of throttle and elevator, respectively. The equations of flight dynamics
for an aircraft in the vertical dimension, influenced by wind disruption in projection on the coordinate axes,
are defined by a system of nonlinear differential equations [31], [38]:

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{
𝑚𝑉̇ = 𝑇𝑐𝑜𝑠𝛼 − 𝐷 − 𝑚𝑔𝑠𝑖𝑛𝜃 − 𝑚(𝑤
̇ 𝑋𝑐𝑜𝑠𝜃 + 𝑤
̇ 𝑌𝑠𝑖𝑛𝜃),
𝑚𝑉𝜃̇ = 𝑇𝑠𝑖𝑛𝛼 + 𝐿 − 𝑚𝑔𝑐𝑜𝑠𝜃 + 𝑚(𝑤
̇ 𝑋𝑠𝑖𝑛𝜃 − 𝑤
̇ 𝑌𝑐𝑜𝑠𝜃),
𝐽𝑧𝜔̇𝑧 = 𝑀𝑧,
𝜗̇ = 𝜔𝑧.
(14)
M is aircraft weight, 𝐽𝑧 is aircraft moment of inertia about the transverse axis 𝑧, 𝑇 is engine thrust force,
𝑀𝑧 is moment of forces about the 𝑧 axis, 𝜗 = 𝜃в + 𝛼 is pitch angle, 𝜔𝑧 is angular velocity about the 𝑧 axis,
𝑤̇𝑋, 𝑤̇ 𝑌 is derivative of horizontal and vertical components of wind speed. The mentioned equations are valid
in the supposition, that the direction of engine thrust force coincides with the axis of the aircraft, aircraft
weight remains constant, the Earth is flat, and wind flow is stationary. The effect of the earth’s rotation is
neglected. The differential equation for the height of the center of mass ℎ, and the incremental equation
modeling the engine dynamics are formulated as (15) and (16),
ℎ̇ = 𝑉𝑠𝑖𝑛𝜃 + 𝑊ℎ (15)
∆𝑇̇ =
1
𝑇дв
(−∆𝑇 + 𝐾дв∆𝛿𝑡) (16)
where 𝛿𝑡 throttle deflection from the target value. The elevator deflection 𝛿𝑒 is determined by taking into
account the flight contour of the aircraft in its short-term periodic motion, can be summarized as following
equation:
𝛿𝑒 = 𝐾𝜔𝑧
∆𝜔𝑧 + 𝐾𝜗∆𝜗 + 𝐾су∆𝜗су,
where 𝐾𝜔𝑧
, 𝐾𝜗 и 𝐾су numerical coefficients, ∆𝜗су control generated with the assistance of a robust controller.
A significant simplification of the aircraft mathematical model is its linearization. Let linearize the
non-linear aircraft model for system of differential (14) determined by taking into consideration (15), (16).
As a result, the non-linear aircraft model is transformed into a system of linear differential equations in
increments. The matrix representation of linear system takes the form (1), where key vectors:
𝑥 = (∆𝑉, ∆𝜃, ∆𝜔𝑧, ∆𝜗, ∆ℎ, ∆𝑇)𝑇
represents the state, 𝑤 = (𝑤𝑌, 𝑤̇𝑋, 𝑤̇ 𝑌)𝑇
-wind disturbance, 𝑢 = (∆𝜗су, ∆𝛿𝑡)
𝑇
-
control [31], [36].
3. RESULTS AND ANALYSIS
This research is devoted to the analysis of a particular aircraft glide path trajectory, characterized by
a linear trajectory with a defined flight path angle 𝜃𝑔𝑙 (𝜃𝑔𝑙 = 2.7 degrees) in height and range coordinates
[31], [36]. The main purpose of synthesized system is to maintain a consistent airspeed 𝑉0 = 71.375 m/s and
a predetermined height ℎ = 400 m under the influence of wind disturbances, when moving on a glide path.
The model is presented in [31]. Studies have found that the output signal energy is minimized when a
stochastic perturbation model in the form of white noise is served as an input in H2 theory. On the other hand,
the perturbation model is not defined, but its power is restricted in H∞ theory. However, H∞ theory provides
robust control that is appropriate for systems with disturbances having significant power over an arbitrarily
small frequency band. In contrast, H2 theory permits obtaining control for systems with uniform spectral
density of disturbances. Therefore, the H2 controller is well applicable for noise processing, nevertheless, a
potential weak point lies in providing robustness and tracking performance. The H∞ controller offers a
notable advantage in terms of achieving a high level of system robustness. However, it exhibits relative
limitations when it comes to effectively handling noise interference. As a result, this paper contains a
synthesis of robust controllers mainly based mainly on a mixed H2/H∞ approach, which provides an estimate
of all the above-mentioned requirements.
A comparative analysis was conducted to evaluate the transient response characteristics of closed-
loop systems employing the aforementioned H2, H∞ [31], and H2/H∞ controllers. In the process of simulation
an identical input signal was fed to each closed-loop system, imitating the atmospheric disturbance w caused
by wind that affected the aircraft’s motion in the area characterized by microburst-type wind conditions.
Figure 3 [31] illustrates the graphical representation of the vertical component 𝑤𝑦 and horizontal component
𝑤𝑥 of the wind field in relation to the position of the vortex center within the microburst airflow pattern.
Figures 4 and 5 illustrate the deviation graphs of altitude ∆ℎ and speed ∆𝑉 from their nominal values
for H2, H∞ and mixed H2/H∞ controllers, as shown in Tables 1 and 2. An analysis of deviation graphs reveals
that the mixed H2/H∞ controller provides less deviation of flight altitude ℎ and speed 𝑉 than the H2 controller,

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but greater deviation than the H∞ controller. However, a comparison of control signals as shown in Figure 6
and Table 3 demonstrates that the H∞ controller provides a greater deviation than the H2 controller. In
summary: the H∞ controller requires heavy engine loads, whereas the H2 controller requires less loads, but
provides slightly lower quality. As a result, if heavy engine loads are not acceptable, implementing a mixed
H2/H∞ controller would be appropriate.
Figure 3. Vertical component 𝜔𝑦 and horizontal
component 𝜔𝑥 of the wind field
Figure 4. Flight altitude ℎ deviation in cases of 𝐻2,
𝐻∞ and mixed 𝐻2/𝐻∞ controllers using
Figure 5. Speed 𝑉 deviation in cases of 𝐻2, 𝐻∞ and mixed 𝐻2/𝐻∞ controllers using
Table 1. Flight altitude deviation from the nominal
value under the action of wind disturbances
Controller type Flight altitude ℎ deviation (m)
ℎ𝑚𝑖𝑛 ℎ𝑚𝑎𝑥 ℎ𝑚𝑎𝑥 − ℎ𝑚𝑖𝑛
𝐻2 -14.375 4.38 18.75
𝐻∞ -7 0.7 7.7
𝐻2/𝐻∞ -13.125 1.875 15
Table 2. Flight speed deviation from the nominal
value under the action of wind disturbances
Controller type Flight speed 𝑉 deviation (m)
𝑉𝑚𝑖𝑛 𝑉
𝑚𝑎𝑥 𝑉
𝑚𝑎𝑥 − 𝑉𝑚𝑖𝑛
𝐻2 -1.25 2.24 3.49
𝐻∞ -0.125 1.25 1.375
𝐻2/𝐻∞ -1 2 3
Consequently, a mixed H2/H∞ controller can be obtained by manipulating the parameter 𝛾 and the
weighting matrices, possessing almost equivalent qualities of H2 or H∞ control depending on the conditions
of a specific task. It is worth emphasizing that the primary cause of accidents during aircraft landings consist
in a sharp loss of aircraft altitude in conditions of microburst wind action. From this perspective, the results
demonstrate the technical feasibility of the proposed mixed H2/H∞ optimal controller for solving such
problems. Despite the significantly complicated algorithm of calculation, manipulating the level 𝛾 and the

 ISSN: 2088-8708
6256
weighting coefficients provides an opportunity to obtain access to a wide range of transient processes, each
of which is capable of exhibiting high efficiency in certain circumstances, as opposed to optimization by a
single criterion. This article further advances the ongoing exploration of devising and investigating effective
techniques for synthesizing robust controllers to facilitate aircraft flight control during the landing phase,
specifically focusing on the glide path mode. These efforts are conducted in the face of uncertainties arising
from extrinsic and intrinsic disturbances, building upon the foundation established in the previous study [31].
Figure 6. Control signal’s reaction to the assigned wind disturbance
Table 3. Control signals deviation from the nominal value under the action of wind disturbances
Controller type Control signal 𝛿 deviation (degree)
𝛿𝑚𝑖𝑛 𝛿𝑚𝑎𝑥 𝛿𝑚𝑎𝑥 − 𝛿𝑚𝑖𝑛
𝐻2 -1.7 4.25 5.95
𝐻∞ -2.8 5.5 8.3
𝐻2/𝐻∞ -1.95 5 6.95
4. CONCLUSION
The landing phase of aircraft flight embodies the most dangerous flight stage because of the high
risk of an accident. Given the prevalence of substantial external disturbances and uncertainties during this
particular phase of flight, it becomes imperative to employ robust synthesis methods such as H2 and H∞
techniques. These approaches offer a promising foundation for effectively addressing and resolving the
challenges at hand. The H2 controller has the capability of handling and minimizing noise but, on the other
side, plays a weak role in ensuring robustness and tracking performance. The H∞ controller contributes to the
implementation of a high-quality robust system, but is not applicable in noise processing in comparison.
Consequently, this research emphasizes an important aspect of robust controller synthesis by focusing on the
application of a mixed H2/H∞ method that fully complies with the above-mentioned requirements. A mixed
H2/H∞ controller of the required quality, functioning similarly to H∞ or mostly H2 depending on the
conditions, can be developed by applying the technique of manipulating the parameters of 𝛾 and the
weighting matrices. The proposed robust systems exhibit a broad spectrum of applications within the realm
of moving object control, encompassing a wide array of technological challenges that extend beyond the
confines of aircraft flight control. Further research is planned to perform directed towards the development of
robust H2, H∞ and mixed H2/H∞ control in relation to other objects.
ACKNOWLEDGEMENTS
This research is funded by the Science Committee of the Ministry of Science and Higher Education
of the Republic of Kazakhstan (Grant No. AP19680413).
REFERENCES
[1] R. Hess, “Robust flight control design to minimize aircraft loss-of-control incidents,” Aerospace, vol. 1, no. 1, pp. 1–17, Nov.
2013, doi: 10.3390/aerospace1010001.

6257
[2] R. Newman, “Thirty years of airline loss-of-control mishaps,” in AIAA Modeling and Simulation Technologies Conference, 2012.
[3] T. B. Company, “Statistical summary of commercial jet airplane accidents,” Worldwide Operations 1959-2008. 2009. Accessed:
Oct. 23, 2013. [Online]. Available: http://www.airsafe.com/events/models/statsum2008.pdf
[4] C. Wang, I. Gregory, and C. Cao, “L1 adaptive control with output constraints applied to flight envelope limiting,” in AIAA
Guidance, Navigation, and Control Conference, 2012, doi: 10.2514/6.2012-4895.
[5] C. Belcastro, “Validation of safety-critical systems for aircraft loss-of-control prevention and recovery,” in AIAA Guidance,
Navigation, and Control Conference, 2012, doi: 10.2514/6.2012-4987.
[6] J. Wang, H.-L. Pei, and N.-Z. Wang, “Adaptive output feedback control using fault compensation and fault estimation for linear
system with actuator failure,” International Journal of Automation and Computing, vol. 10, no. 5, pp. 463–471, 2013, doi:
10.1007/s11633-013-0743-8.
[7] Y. Wu and J. Dong, “Fault estimation and fault tolerant control for T-S fuzzy systems,” in 2016 3rd International Conference on
Informative and Cybernetics for Computational Social Systems (ICCSS), 2016, pp. 97–102.doi: 10.1109/ICCSS.2016.7586431.
[8] A. Ben Brahim, S. Dhahri, F. Ben Hmida, and A. Sellami, “Multiplicative fault estimation-based adaptive sliding mode fault-
tolerant control design for nonlinear systems,” Complexity, pp. 1–15, Jun. 2018, doi: 10.1155/2018/1462594.
[9] C. Kasnakoğlu, “Investigation of multi-input multi-output robust control methods to handle parametric uncertainties in autopilot
design,” PLOS ONE, vol. 11, no. 10, p. e0165017, Oct. 2016, doi: 10.1371/journal.pone.0165017.
[10] R. Takase, A. Marcos, M. Sato, and S. Suzuki, “Hardware-In-the-loop evaluation of a robust C⁎ control law on MuPAL-α
research aircraft,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 14833–14838, 2020, doi: 10.1016/j.ifacol.2020.12.1927.
[11] H. Alwi and C. Edwards, “Robust fault reconstruction for linear parameter varying systems using sliding mode observers,”
International Journal of Robust and Nonlinear Control, vol. 24, no. 14, pp. 1947–1968, Sep. 2014, doi: 10.1002/rnc.3009.
[12] M. Rodrigues, H. Hamdi, D. Theilliol, C. Mechmeche, and N. BenHadj Braiek, “Actuator fault estimation based adaptive
polytopic observer for a class of LPV descriptor systems,” International Journal of Robust and Nonlinear Control, vol. 25, no. 5,
pp. 673–688, Mar. 2015, doi: 10.1002/rnc.3236.
[13] F.-R. López-Estrada, J.-C. Ponsart, C.-M. Astorga-Zaragoza, J.-L. Camas-Anzueto, and D. Theilliol, “Robust sensor fault
estimation for descriptor-LPV systems with unmeasurable gain scheduling functions: Application to an anaerobic bioreactor,”
International Journal of Applied Mathematics and Computer Science, vol. 25, no. 2, pp. 233–244, Jun. 2015, doi: 10.1515/amcs-
2015-0018.
[14] D. Navarro-Tapia, P. Simplício, A. Iannelli, and A. Marcos, “Robust flare control design using structured H∞ synthesis: a civilian
aircraft landing challenge,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 3971–3976, 2017, doi: 10.1016/j.ifacol.2017.08.769.
[15] P. Simplício, D. Navarro-Tapia, A. Iannelli, and A. Marcos, “From standard to structured robust control design: application to aircraft
automatic glide-slope approach,” IFAC-PapersOnLine, vol. 51, no. 25, pp. 140–145, 2018, doi: 10.1016/j.ifacol.2018.11.095.
[16] X. Wang, Y. Sang, and G. Zhou, “Combining stable inversion and H∞ synthesis for trajectory tracking and disturbance rejection
control of civil aircraft auto landing,” Applied Sciences, vol. 10, no. 4, Feb. 2020, doi: 10.3390/app10041224.
[17] J.-M. Biannic, C. Roos, and A. Knauf, “Design and robustness analysis of fighter aircraft flight control laws,” European Journal
of Control, vol. 12, no. 1, pp. 71–85, Jan. 2006, doi: 10.3166/ejc.12.71-85.
[18] A. Iannelli, P. Simplício, D. Navarro-Tapia, and A. Marcos, “LFT modeling and μ analysis of the aircraft landing benchmark,”
IFAC-PapersOnLine, vol. 50, no. 1, pp. 3965–3970, Jul. 2017, doi: 10.1016/j.ifacol.2017.08.766.
[19] L. Chu, Q. Li, F. Gu, X. Du, Y. He, and Y. Deng, “Design, modeling, and control of morphing aircraft: A review,” Chinese
Journal of Aeronautics, vol. 35, no. 5, pp. 220–246, 2022, doi: 10.1016/j.cja.2021.09.013.
[20] M. R. Ghazali, M. A. Ahmad, and R. M. T. R. Ismail, “A multiple-node hormone regulation of neuroendocrine-PID (MnHR-
NEPID) control for nonlinear MIMO systems,” IETE Journal of Research, vol. 68, no. 6, pp. 4476–4491, Nov. 2022, doi:
10.1080/03772063.2020.1795939.
[21] M. R. Ghazali, M. A. Ahmad, and H. Ishak, “A data-driven sigmoid-based secretion rate of neuroendocrine-PidControl for TRMS
system,” in 2021 IEEE 11th IEEE Symposium on Computer Applications and Industrial Electronics (ISCAIE), Apr. 2021, pp. 1–6,
doi: 10.1109/ISCAIE51753.2021.9431795.
[22] M. Kiselev, S. Levitsky, and M. Shkurin, “Influence of control system algorithms on the maneuvering characteristics of the
aircraft,” Aerospace Systems, vol. 5, no. 1, pp. 123–130, Mar. 2022, doi: 10.1007/s42401-021-00124-8.
[23] M. Idrissi, M. Salami, and F. Annaz, “A review of quadrotor unmanned aerial vehicles: applications, architectural design and
control algorithms,” Journal of Intelligent and Robotic Systems, vol. 104, no. 2, Feb. 2022, doi: 10.1007/s10846-021-01527-7.
[24] R. Tayari, A. Ben Brahim, F. Ben Hmida, and A. Sallami, “Active fault tolerant control design for LPV systems with
simultaneous actuator and sensor faults,” Mathematical Problems in Engineering, vol. 2019, pp. 1–14, Jan. 2019, doi:
10.1155/2019/5820394.
[25] Q. Wu, Z. Liu, F. Liu, and X. Chen, “LPV-based self-adaption integral sliding mode controller with L2 gain performance for a
morphing aircraft,” IEEE Access, vol. 7, pp. 81515–81531, 2019, doi: 10.1109/ACCESS.2019.2923313.
[26] T. Yue, L. Wang, and J. Ai, “Gain self-scheduled H∞ control for morphing aircraft in the wing transition process based on an LPV
model,” Chinese Journal of Aeronautics, vol. 26, no. 4, pp. 909–917, Aug. 2013, doi: 10.1016/j.cja.2013.06.004.
[27] R. Breda, T. Lazar, R. Andoga, and L. Madarasz, “Optimising the aircraft lateral stabilization system in landing mode,” in 2013
IEEE 11th International Symposium on Applied Machine Intelligence and Informatics (SAMI), Jan. 2013, pp. 221–225, doi:
10.1109/SAMI.2013.6480980.
[28] A. Marcos, S. Waitman, and M. Sato, “Fault tolerant linear parameter varying flight control design, verification and validation,”
Journal of the Franklin Institute, vol. 359, no. 2, pp. 653–676, Jan. 2022, doi: 10.1016/j.jfranklin.2021.02.040.
[29] М. А. Beisenbi and Z. O. Basheyeva, “Solving output control problems using Lyapunov gradient-velocity vector function,”
International Journal of Electrical and Computer Engineering (IJECE), vol. 9, no. 4, pp. 2874–2879, Aug. 2019, doi:
10.11591/ijece.v9i4.pp2874-2879.
[30] S. D. Karimtaevna and K. Z. Asylbekkyzy, “Robust control for a tracking electromechanical system,” International Journal of
Electrical and Computer Engineering (IJECE), vol. 12, no. 5, pp. 4883–4891, 2022, doi: 10.11591/ijece.v12i5.pp4883-4891.
[31] S. D. Karimtaevna, A. Z. Bekbolatovna, and M. A. Assilkhanovna, “Robust control of aircraft flight in conditions of
disturbances,” International Journal of Electrical and Computer Engineering (IJECE), vol. 12, no. 4, pp. 3572–3582, 2022, doi:
10.11591/ijece.v12i4.pp3572-3582.
[32] V. Konkov and А. Kiselyov, “Problema smeshannoi H2/H∞ optimizacii,” Vestnik MGTU, Priborostroenie, no. 1, pp. 50–69,
2001.
[33] A. Saberi, B. Chen, P. Sannuti, and U.-L. Ly, “Simultaneous H2/H-infinity optimal control-The state feedback case,” in
Guidance, Navigation and Control Conference, Aug. 1992, doi: 10.2514/6.1992-4476.
[34] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard H2 and H∞ control problems,”
IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831–847, 1989, doi: 10.1109/9.29425.

 ISSN: 2088-8708
6258
[35] N. D. Egupov, Non-stationary automatic control systems: analysis, synthesis and optimization: monograph (in Russian
Nestatsionarnye sistemy avtomaticheskogo upravleniya: analiz, sintez i optimizatsiya: monografiya). Amazon, 2007.
[36] R. S. Ali, “Sintez robastnyh regulyatorov stabilizacii transportnyh sredstv,” Dissertations for a Degree Candidate of Technical
Sciences, St. Petersburg State Polytechnic University, 2002.
[37] S. Skogestad and I. Postlethwaite, Multivariable feedback control: analysis and design, 2nd Edition. John Wiley & Sons, 2005.
[38] D. Schmidt, Modern flight dynamics. New York, USA: McGraw-Hill, 2012.
BIOGRAPHIES OF AUTHORS
Dana Satybaldina is a candidate of Technical Sciences specializing in system
analysis, control, and information processing, currently holds the position of Associate
Professor at DSAC, L.N. Gumilyov ENU. With an extensive publication record of over 90
papers in renowned international journals and conferences, her contributions span a diverse
range of topics in robust control system research and development. Her research interests
primarily revolve around systems analysis and control, modern control theory, and robust
automatic control systems. For further correspondence, she can be contacted via email at:
satybaldina_dk@enu.kz.
Aida Dabayeva successfully completed her undergraduate studies in Automation
and Control, earning a bachelor’s degree in engineering and technology in 2017.
Subsequently, she pursued a master’s degree in engineering in automation and control, which
she attained in 2019, both from L.N. Gumilyov ENU. At present, Aida is actively engaged in
her doctoral studies at the same university. Her research endeavors primarily revolve around
the advancement and implementation of robust aircraft control methodologies. For further
communication, she can be contacted via email at: mashtayeva@mail.ru.
Nurgul Kissikova is a candidate of Physical and mathematical sciences in
differential equations, currently serves as an Associate Professor at DSAC, L.N. Gumilyov
ENU. With a remarkable publication record comprising over 60 scientific articles, her
expertise lies in the domains of control theory models and methods, system modeling, system
analysis, and decision making. Her research pursuits also encompass the exploration of
deterministic chaos in economic systems. For any inquiries or communication, she can be
reached via email at: kissikova_nm@enu.kz.
Gulzhan Uskenbayeva completed a master’s degree in computer sciences at Al-
Farabi Kazakh National University in 2002. Subsequently, she pursued her Ph.D. in
Automation and Control, successfully earning the degree from L.N. Gumilyov ENU in 2016.
Presently, she holds the position of Associate Professor at DSAC within the same university.
Her research interests encompass a wide range of areas, including control engineering, robust
control, fuzzy logic, artificial intelligence, multiservice nodes of telecommunications
networks, deep learning, neural networks, and computer vision. For any inquiries or
correspondence, she can be reached via email at: uskenbayeva_ga_1@enu.kz.
Aliya Shukirova successfully completed her doctoral studies in Automation and
Control at L.N. Gumilyov ENU, where she was awarded a Ph.D. degree. Currently, she holds
the position of Associate Professor at DSAC within the same university. Her research pursuits
encompass various areas of interest, including control engineering, adaptive control, fuzzy
logic, neural networks, and big data. For any inquiries or correspondence, she can be reached
via email at: shukirova_ak_1@enu.kz.

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