The  problem of  optimal  state feedback  controller design in  terms of quadratic performance 
index  is  considered.  The  problem  is  formulated  in  the  form  of  linear  matrix  inequalities  (LMIs).  The 
obtained solution guarantees stabilization of the aircraft during flight mission. During flight envelope the 
aircraft is subjected to the external stochastic disturbances. The efficiency of the proposed approach is 
illustrated by a case study of airplane longitudinal motion

Komnatska, M. M.

Electronics and Control Systems

English

Інституційний репозитарій erNAU (Electronic Repository of the National Aviation University)

46                                                                   ISSN 1990-5548   Electronics and Control Systems  2015. N 2(44): 46-50  ___________________________________________________________________________________________________________ © National Aviation University, 2015 http://ecs.in.ua UDC 629.735.051'622 (045) M. M. Komnatska OPTIMAL GUARANTEED COST CONTROL OF AIRCRAFT MOTION Aircraft Control Systems Department, National Aviation University, Kyiv, Ukraine Е-mail: martakomnatska@gmail.com  Abstract—The problem of optimal state feedback controller design in terms of quadratic performance index is considered. The problem is formulated in the form of linear matrix inequalities (LMIs). The obtained solution guarantees stabilization of the aircraft during flight mission. During flight envelope the aircraft is subjected to the external stochastic disturbances. The efficiency of the proposed approach is illustrated by a case study of airplane longitudinal motion. Index Terms—Aircraft motion control, external disturbances, linear matrix inequality, performance index, robustness, state feedback. I.  INTRODUCTION The problem of optimal control design has been considered in a number of publications [1]–[3]. Especially, it is very crucial question in the area of aircraft control, where it is necessary to satisfy the manifold requirements imposed on the aircraft during flight envelope. A great number of control approaches have been proposed to solve the problem autopilot design. Among them, it is possible to enumerate some works related to the combination of observer and linear quadratic regulator [3], [4]. Furthermore, to preserve the required level of performance without losing the robustness of the flight control system, the mixed 2H H – robust optimization procedure is used. The main idea behind this technique is to seek a trade-off between the performance and the robustness of the overall closed loop system [3], [4]. The autopilot design is also may be performed basing on the available information about the output variables. This circumstance leads to the problem of static output feedback (SOF) controller design. The main advantage of SOF design is that it requires only available signals from the plant to be controlled. The SOF problem concerns finding a static or feedback gain to achieve certain desired closed-loop characteristics. It is necessary to admit that the output feedback problem is much more difficult to solve in comparison to state feedback control problem. A survey devoted to this problem is presented in [5]. This paper deals with static state feedback controller design in terms of LMIs [6], [7] for aircraft control during flight envelope. The main feature of this paper is that the obtained state feedback controller stabilizes the set of autonomous systems, simultaneously. Moreover, the designed controller possesses with robustness properties. To prove the efficiency of the proposed technique, the longitudinal motion of the aircraft is considered as a case study.  II. PROBLEM STATEMENT Let us consider procedure of state feedback design for an aircraft control whose dynamics is described by the following differential equation      i it t t x A x B u ,   00 x x , (1) where  nx R  is the state space vector,  mu R is the control vector. The uncertainties of the model are represented by the set of matrices i i,A B  that satisfy the following requirement:  1 1 N N[ ] [ ],...,[ ] , i 1,..., NCo A B A B A B , where Co  is a convex set; N is the set of models associated with certain operating conditions within the flight envelope. The main problem is to find the state feedback of the following form    t tu K x ,               (2) where K  is a constant state feedback gain matrix that assures that the system is asymptotically stable. Thus, the closed-loop system taking into account (1) and (2) takes the well known description in form of differential equation as:      i it t x A B K x ,   00 .x x  The obtained solution minimizes performances index given by              T T0T T0                 ,J t t t t dtt t dt  x Q x u Rux Q K R K x   (3) where Q  and R are diagonal matrices, weighting each state and control variables, respectively.  This cost depends on the trajectory, of  tx , taken, such that the worst trajectory will correspond M.M. Komnatska  Optimal guaranteed cost control of aircraft motion                                                                          47  to the worst cost. The control problem is to find the state feedback gain K and a quadratic Lyapunov function P  that minimizes the bound T0 0x P x  on the worst cost of J . The problem can be translated into an optimization problem as follows: minimize T0 0 ;x P x  subject to 0P ,    T T 0.i i i i     A B K P P A B K Q K RK  (4) The goal is to find state feedback controller (2) that simultaneously stabilizes the set of autonomous systems. The obtained solution is optimal and brings to the minimum performance index given by (3). The linear matrix inequality (LMI) technique permits to solve this problem [6], [7]. Thus, it is possible to transform the non-linear inequality (4) into the LMI form. This procedure reduces to the defining new matrices X  and M  such that 1,X P  1M K P  and 1P X , 1 K M P M X . By substituting new variables X  and M  instead of P  and K  into Lyapunov inequality (4), and then pre-multiplying and post-multiplying the left and the right hand sides by X , the inequality (4) becomes T T T T 0.i i i i i i i i      XA A X M B M XQX M RM  (5) Basing on Schur’s complement the matrix inequality (5) can be expressed as the LMI T T T 1 2 T 1 21 21 20 0.0i i i i i ii          XA AX B M M XQ M RQ X IR M I(6) By applying Schur’s complement, the cost  T T 10 0 0 0  x Px x X x , is expressed as the LMI T000    xx X.   (7) The following optimization problem can be represented in terms of LMI (6) and (7) minimizes γ subject to T T T 1 2 T 1 2i i i i i i1 21 2i0 00          XA AX B M M XQ M RQ X IR M I, T000.    xx X III. CASE STUDY To demonstrate the efficiency of the proposed approach the longitudinal channel of regional jet is used as a case study. The longitudinal dynamics of regional jet in the state space is represented by the phase and control vectors, respectively. The phase and control vector have the following form:  T, , , ,u w q h x  and  Te u , where u is a longitudinal component of true airspeed; w is a vertical component of true airspeed; q is the aircraft pitch rate;   is a pitch angle; h is the aircraft altitude. The control vector  Te u  is represented by the elevator defection [8]. It is considered two operating modes of the Boeing 737-100 aircraft with true airspeed at V1 = 240.88 m/sec and V2 = 237.26 m/sec. Thus, we have two mathematical models that correspond to these airspeeds. These linear models in the state space are represented by the matrices  ,A B . The set of matrices that correspond to nominal and parametrically perturbed models are given below: – nominal model 0.674 -2 0.498 -1 0.250 2 0.322 2 00.122 -1 0.738 0.788 3 0.102 1 00.245 -2 0.625 -2 0.846 0.104 -13 00 0 1 0 00.317 -1 0.1 1 0 0.790 3 0ne e e ee e ee e ee e e                    A ;    0.191 10.6010.802 100nee          B ; – perturbed model 48                                                                   ISSN 1990-5548   Electronics and Control Systems  2015. N 2(44): 46-50  0.595 -2 0.525 -1 0.290 +2 0.321 +2 00.156 -2 0.688 0.776 +3 0.120 +1 00.231 -2 0.602 -2 0.771 0 00 0 1 0 00.373 -1 0.999 0 0.799 +3 0pe e e ee e ee ee e              A ;    0.206 -10.5530.736 -100pee         B .where the subscript “n” corresponds to the nominal model and perturbed model is designated by the subscript “p”. Disturbance,   affecting the longitudinal motion of the aircraft involves the following components: horizontal and vertical components of true airspeed, gu  and gw , and pitch rate, gq  such that that T, ,g g gu w q     .  In order to simulate the atmospheric turbulence a Dryden filter is used [8]. The aircraft is considered to fly in a moderate turbulence. The transfer functions of forming filter according to standard MIL–F–8785C [13], [14] used in simulation to account external disturbances have the following structure: 2;(s)11uu uuLLVsVH      23(1;L(1 s)s)www wwLsL VVVH     s(s) (s).4(1 s)q wVHbVH    . To solve the problem of state feedback design with the control law (2) via LMIs as given by (6) it is necessary to define weighting matrices Q  and R . In our case these matrices are defined as follows   814 10 diag 1 1 1 1 1 ; Q   0.011R . By solving the LMIs given by (6) and (7) the gain matrix K  for the state feedback has the following structure  0.0090 0.1683 27.7311 195.6772 0.0374  K . The static state feedback gain matrix obtained with the help of the proposed approach assures simultaneous stabilization the set of autonomous systems that testifies the robust properties of the controller. Performance indices of closed loop nominal and parametrically perturbed systems with the state feedback are given in Table 1. Standard deviations of the Boeing 737-100 regional jet with state feedback control for nominal and parametrically perturbed models are given in Table 2. .TABLE 1 STANDARD DEVIATIONS OF THE BOEING 737-100 OUTPUTS IN A STOCHASTIC CASE  Plant Standard deviation  V , m/sec q , 0/sec  , 0 h ,m el , 0 V1= 240.88 m/s 0.3439 0.0094 0.0341 4.5747 0.3439 V2= 237.26 m/s 0.3087 0.0076 0.0315 4.3585 0.3087 TABLE2 PERFORMANCE INDICES OF CLOSED-LOOP SYSTEMS  Plant Performance Index Н2-norm Н∞-norm V1= 240.88 m/s 0.1090 0.3037 V2= 237.26 m/s 0.0739 0.2096 The simulation results of the closed loop systems operation taking into account the influence of the random wind, simulated according to the standard Dryden model of turbulence confirm the efficiency of proposed approach. Results of the simulation are shown in Figure. M.M. Komnatska  Optimal guaranteed cost control of aircraft motion                                                                          49  Simulation results of Boeing 737-100 aircraft longitudinal motion in the presence of external disturbances: a is the pitch rate of nominal and perturbed models, deg/s; b is the pitch angle of nominal and perturbed models, deg; c is the altitude of nominal and perturbed models, m  CONCLUSIONS Simulation results prove the efficiency of the proposed approach. It can be seen that the handling quality of the nominal and the perturbed models are satisfied.  REFERENCES [1] Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems, Wiey Interscience: New York, 650 p.  [2] Klipa, A. and Sydorenko, A. (2014). “Performance and Robustness Estimation of UAV Flight Control Systems using Different Methods of Flight Parameters Measurements”, Polit –2014: XIV International scientific-technical conference, 152 p., 2-3 April. 2014 (in Ukrainian). [3] Tunik, A. A. and Galaguz, T. A. (2004). “Robust Stabilization and Nominal Performance of the Flight Control System for Small UAV”, Applied and Computation Mathematics, vol. 3, no. 1, pp. 34–45, 2004.  [4] Tunik, A.; Galaguz, T. and Dong Xu Guo (2005). “Design of suboptimal robust flight control system for small UAV”, Aviation in the XXI-st Century. Safety in Aviation and Space Technologies: The Second World Congress, pp. 1.70–1.79, 22-24 September, 2005. [5] Syrmos, V. L.; Abdallah, C.; Dorato, P. and Grigoriadis, K. (1997). “Static Output Feedback – A Survey”, Automatica, vol. 33, pp. 125–137.  [6] Boyd, S.; Ghaoui, L. E.; Feron, E. and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, Philadelphia, 416 p. [7] Nesterov, Yu. and Nemirovskii, A. (1994). Interior-point polynomial algorithms in convex programming. Philadelphia: SIAM, 407 p. [8] McLean D. (1990). Automatic Flight Control Systems, Prentice Hall, NY, 593 p. Received 27 April 2015. a  b  c 50                                                                   ISSN 1990-5548   Electronics and Control Systems  2015. N 2(44): 46-50  Komnatska Marta. Ph. D. Associate Professor.  Aircraft control systems Department, National Aviation University, Kyiv, Ukraine. Education: National Aviation University, Kyiv, Ukraine (2007). Research interests: control systems and data processing.  Publications: 38. E-mail: martakomnatska@gmail.com  М. М. Комнацька. Оптимальне керування рухом літака з гарантованими мінімальними затратами Представлено процедуру синтезу оптимального регулятора, що мінімізує обраний квадратичний критерій. Задача розв’язується в термінах лінійних матричних нерівностей. Ефективність запропонованого підходу ілюструється на прикладі керування повздовжнім рухом літака. Ключові слова: керування рухом літака; зовнішні збурення; лінійна матрична нерівність; показник якості; робастність; зворотній зв’язок за станом.  Комнацька Марта Миколаївна. Кандидат технічних наук. Доцент. Кафедрa систем управління літальних апаратів, Національний авіаційний університет, Київ, Україна. Освіта: Національний авіаційний університет, Київ, Україна (2007). Напрямок наукової діяльності: системи управління та обробка інформації. Кількість публікацій: 38. E-mail: martakomnatska@gmail.com  М. Н. Комнацкая. Оптимальное управление движением самолета с гарантированными минимальными затратами Представлена процедура синтеза оптимального регулятора, который минимизирует выбранный показатель качества. Задача решается с использованием аппарата линейных матричных неравенств. Эффективность предложенного подхода иллюстрируется на примере управления продольным движением самолета.  Ключевые слова: управление движением самолета; внешние возмущения; линейное матричное неравенство; показатель качества; робастность; обратная связь по состоянию.  Комнацкая Марта Николаевна. Кандидат технических наук, доцент. Кафедра систем управления летательных аппаратов, Национальный авиационный университет, Киев, Украина. Образование: Национальный авиационный университет, Киев, Украина (2007). Направление научной деятельности: системы управления и обработка информации. Количество публикаций: 38.  E-mail: martakomnatska@gmail.com  

OPTIMAL GUARANTEED COST CONTROL OF AIRCRAFT MOTION

Scientific and technical library of the National Aviation University

46                                                                   ISSN 1990-5548   Electronics and Control Systems  2015. N 2(44): 46-50 
 
___________________________________________________________________________________________________________ 
© National Aviation University, 2015 
http://ecs.in.ua 
UDC 629.735.051'622 (045) 
M. M. Komnatska 
OPTIMAL GUARANTEED COST CONTROL OF AIRCRAFT MOTION 
Aircraft Control Systems Department, National Aviation University, Kyiv, Ukraine 
Е-mail: martakomnatska@gmail.com  
Abstract—The problem of optimal state feedback controller design in terms of quadratic performance 
index is considered. The problem is formulated in the form of linear matrix inequalities (LMIs). The 
obtained solution guarantees stabilization of the aircraft during flight mission. During flight envelope the 
aircraft is subjected to the external stochastic disturbances. The efficiency of the proposed approach is 
illustrated by a case study of airplane longitudinal motion. 
Index Terms—Aircraft motion control, external disturbances, linear matrix inequality, performance 
index, robustness, state feedback. 
I.  INTRODUCTION 
The problem of optimal control design has been 
considered in a number of publications [1]–[3]. 
Especially, it is very crucial question in the area of 
aircraft control, where it is necessary to satisfy the 
manifold requirements imposed on the aircraft 
during flight envelope. A great number of control 
approaches have been proposed to solve the problem 
autopilot design. Among them, it is possible to 
enumerate some works related to the combination of 
observer and linear quadratic regulator [3], [4]. 
Furthermore, to preserve the required level of 
performance without losing the robustness of the 
flight control system, the mixed 2H H – robust 
optimization procedure is used. The main idea 
behind this technique is to seek a trade-off between 
the performance and the robustness of the overall 
closed loop system [3], [4]. The autopilot design is 
also may be performed basing on the available 
information about the output variables. This 
circumstance leads to the problem of static output 
feedback (SOF) controller design. The main 
advantage of SOF design is that it requires only 
available signals from the plant to be controlled. The 
SOF problem concerns finding a static or feedback 
gain to achieve certain desired closed-loop 
characteristics. It is necessary to admit that the 
output feedback problem is much more difficult to 
solve in comparison to state feedback control 
problem. A survey devoted to this problem is 
presented in [5]. 
This paper deals with static state feedback 
controller design in terms of LMIs [6], [7] for 
aircraft control during flight envelope. The main 
feature of this paper is that the obtained state 
feedback controller stabilizes the set of autonomous 
systems, simultaneously. Moreover, the designed 
controller possesses with robustness properties. To 
prove the efficiency of the proposed technique, the 
longitudinal motion of the aircraft is considered as a 
case study.  
II. PROBLEM STATEMENT 
Let us consider procedure of state feedback 
design for an aircraft control whose dynamics is 
described by the following differential equation 
     i it t t x A x B u ,   00 x x , (1) 
where  nx R  is the state space vector,  mu R is the 
control vector. The uncertainties of the model are 
represented by the set of matrices i i,A B  that 
satisfy the following requirement: 
 1 1 N N[ ] [ ],...,[ ] , i 1,..., NCo A B A B A B , 
where Co  is a convex set; N is the set of models 
associated with certain operating conditions within 
the flight envelope. The main problem is to find the 
state feedback of the following form 
   t tu K x ,               (2) 
where K  is a constant state feedback gain matrix 
that assures that the system is asymptotically stable. 
Thus, the closed-loop system taking into account (1) 
and (2) takes the well known description in form of 
differential equation as: 
     i it t x A B K x ,   00 .x x  
The obtained solution minimizes performances 
index given by 
        
     
T T
0
T T
0
                 ,
J t t t t dt
t t dt


 
 


x Q x u Ru
x Q K R K x
   (3) 
where Q  and R are diagonal matrices, weighting 
each state and control variables, respectively.  
This cost depends on the trajectory, of  tx , 
taken, such that the worst trajectory will correspond 
M.M. Komnatska  Optimal guaranteed cost control of aircraft motion                                                                          47 
 
to the worst cost. The control problem is to find the 
state feedback gain K and a quadratic Lyapunov 
function P  that minimizes the bound T0 0x P x  on the 
worst cost of J . The problem can be translated into 
an optimization problem as follows: 
minimize T0 0 ;x P x  
subject to 0P , 
   
T T 0.i i i i     A B K P P A B K Q K RK  (4) 
The goal is to find state feedback controller (2) 
that simultaneously stabilizes the set of autonomous 
systems. The obtained solution is optimal and brings 
to the minimum performance index given by (3). 
The linear matrix inequality (LMI) technique 
permits to solve this problem [6], [7]. Thus, it is 
possible to transform the non-linear inequality (4) 
into the LMI form. This procedure reduces to the 
defining new matrices X  and M  such that 
1,X P  1M K P  and 1P X , 
1 K M P M X . 
By substituting new variables X  and M  instead 
of P  and K  into Lyapunov inequality (4), and then 
pre-multiplying and post-multiplying the left and the 
right hand sides by X , the inequality (4) becomes 
T T T T 0.i i i i i i i i      XA A X M B M XQX M RM  (5) 
Basing on Schur’s complement the matrix inequality 
(5) can be expressed as the LMI 
T T T 1 2 T 1 2
1 2
1 2
0 0.
0
i i i i i i
i
    
 
  
 
  
XA AX B M M XQ M R
Q X I
R M I
(6) 
By applying Schur’s complement, the cost  
T T 1
0 0 0 0
  x Px x X x , 
is expressed as the LMI 
T
0
0
0
 
 
  
x
x X
.   (7) 
The following optimization problem can be 
represented in terms of LMI (6) and (7) minimizes γ 
subject to 
T T T 1 2 T 1 2
i i i i i i
1 2
1 2
i
0 0
0
    
 
  
 
  
XA AX B M M XQ M R
Q X I
R M I
, 
T
0
0
0.
 
 
  
x
x X
 
III. CASE STUDY 
To demonstrate the efficiency of the proposed 
approach the longitudinal channel of regional jet is 
used as a case study. The longitudinal dynamics of 
regional jet in the state space is represented by the 
phase and control vectors, respectively. The phase 
and control vector have the following form: 
 T, , , ,u w q h x  and  Te u , where u is 
a longitudinal component of true airspeed; w is a 
vertical component of true airspeed; q is the aircraft 
pitch rate;   is a pitch angle; h is the aircraft 
altitude. The control vector  Te u  is represented 
by the elevator defection [8]. It is considered two 
operating modes of the Boeing 737-100 aircraft with 
true airspeed at V1 = 240.88 m/sec and V2 = 237.26 
m/sec. Thus, we have two mathematical models that 
correspond to these airspeeds. These linear models 
in the state space are represented by the 
matrices  ,A B . The set of matrices that 
correspond to nominal and parametrically perturbed 
models are given below: 
– nominal model 
0.674 -2 0.498 -1 0.250 2 0.322 2 0
0.122 -1 0.738 0.788 3 0.102 1 0
0.245 -2 0.625 -2 0.846 0.104 -13 0
0 0 1 0 0
0.317 -1 0.1 1 0 0.790 3 0
n
e e e e
e e e
e e e
e e e
     
      
   
 
 
    
A ;    
0.191 1
0.601
0.802 1
0
0
n
e
e
 
  
   
 
 
  
B ; 
– perturbed model 
48                                                                   ISSN 1990-5548   Electronics and Control Systems  2015. N 2(44): 46-50 
 
0.595 -2 0.525 -1 0.290 +2 0.321 +2 0
0.156 -2 0.688 0.776 +3 0.120 +1 0
0.231 -2 0.602 -2 0.771 0 0
0 0 1 0 0
0.373 -1 0.999 0 0.799 +3 0
p
e e e e
e e e
e e
e e
   
    
   
 
 
  
A ;    
0.206 -1
0.553
0.736 -1
0
0
p
e
e
 
  
  
 
 
  
B .
where the subscript “n” corresponds to the nominal 
model and perturbed model is designated by the 
subscript “p”. 
Disturbance,   affecting the longitudinal motion 
of the aircraft involves the following components: 
horizontal and vertical components of true airspeed, 
gu  and gw , and pitch rate, gq  such that that 
T
, ,g g gu w q     .  
In order to simulate the atmospheric turbulence a 
Dryden filter is used [8]. The aircraft is considered 
to fly in a moderate turbulence. The transfer 
functions of forming filter according to standard 
MIL–F–8785C [13], [14] used in simulation to 
account external disturbances have the following 
structure: 
2
;(s)
1
1
u
u u
u
L
LV
s
V
H  


    
2
3
(
1
;
L
(1 s)
s)
w
w
w w
w
L
s
L V
V
V
H





    
 
s
(s) (s).
4
(1 s)
q w
VH
b
V
H


 
  
 
. 
To solve the problem of state feedback design 
with the control law (2) via LMIs as given by (6) it 
is necessary to define weighting matrices Q  and R . 
In our case these matrices are defined as follows 
  814 10 diag 1 1 1 1 1 ; Q   0.011R . 
By solving the LMIs given by (6) and (7) the 
gain matrix K  for the state feedback has the 
following structure 
 0.0090 0.1683 27.7311 195.6772 0.0374  K . 
The static state feedback gain matrix obtained with 
the help of the proposed approach assures 
simultaneous stabilization the set of autonomous 
systems that testifies the robust properties of the 
controller. 
Performance indices of closed loop nominal and 
parametrically perturbed systems with the state 
feedback are given in Table 1. 
Standard deviations of the Boeing 737-100 
regional jet with state feedback control for nominal 
and parametrically perturbed models are given in 
Table 2. 
.
TABLE 1 
STANDARD DEVIATIONS OF THE BOEING 737-100 OUTPUTS IN A STOCHASTIC CASE 
 
Plant 
Standard deviation  
V , m/sec q , 
0/sec  , 
0 h ,m el , 
0 
V1= 240.88 m/s 0.3439 0.0094 0.0341 4.5747 0.3439 
V2= 237.26 m/s 0.3087 0.0076 0.0315 4.3585 0.3087 
TABLE2 
PERFORMANCE INDICES OF CLOSED-LOOP SYSTEMS 
 
Plant 
Performance Index 
Н2-norm Н∞-norm 
V1= 240.88 m/s 0.1090 0.3037 
V2= 237.26 m/s 0.0739 0.2096 
The simulation results of the closed loop systems 
operation taking into account the influence of the 
random wind, simulated according to the standard 
Dryden model of turbulence confirm the efficiency 
of proposed approach. Results of the simulation are 
shown in Figure. 
M.M. Komnatska  Optimal guaranteed cost control of aircraft motion                                                                          49 
 
Simulation results of Boeing 737-100 aircraft longitudinal motion in the presence of external disturbances: a is the pitch 
rate of nominal and perturbed models, deg/s; b is the pitch angle of nominal and perturbed models, deg; 
c is the altitude of nominal and perturbed models, m 
 
CONCLUSIONS 
Simulation results prove the efficiency of the 
proposed approach. It can be seen that the handling 
quality of the nominal and the perturbed models are 
satisfied.  
REFERENCES 
[1] Kwakernaak, H. and Sivan, R. (1972). Linear 
Optimal Control Systems, Wiey Interscience: New York, 
650 p.  
[2] Klipa, A. and Sydorenko, A. (2014). 
“Performance and Robustness Estimation of UAV Flight 
Control Systems using Different Methods of Flight 
Parameters Measurements”, Polit –2014: XIV 
International scientific-technical conference, 152 p., 2-
3 April. 2014 (in Ukrainian). 
[3] Tunik, A. A. and Galaguz, T. A. (2004). “Robust 
Stabilization and Nominal Performance of the Flight 
Control System for Small UAV”, Applied and 
Computation Mathematics, vol. 3, no. 1, pp. 34–45, 2004.  
[4] Tunik, A.; Galaguz, T. and Dong Xu Guo 
(2005). “Design of suboptimal robust flight control 
system for small UAV”, Aviation in the XXI-st Century. 
Safety in Aviation and Space Technologies: The Second 
World Congress, pp. 1.70–1.79, 22-24 September, 2005. 
[5] Syrmos, V. L.; Abdallah, C.; Dorato, P. and 
Grigoriadis, K. (1997). “Static Output Feedback – 
A Survey”, Automatica, vol. 33, pp. 125–137.  
[6] Boyd, S.; Ghaoui, L. E.; Feron, E. and 
Balakrishnan, V. (1994). Linear Matrix Inequalities in 
System and Control Theory, SIAM Studies in Applied 
Mathematics, Philadelphia, 416 p. 
[7] Nesterov, Yu. and Nemirovskii, A. (1994). 
Interior-point polynomial algorithms in convex 
programming. Philadelphia: SIAM, 407 p. 
[8] McLean D. (1990). Automatic Flight Control 
Systems, Prentice Hall, NY, 593 p. 
Received 27 April 2015.
 
a 
 
b 
 
c 
50                                                                   ISSN 1990-5548   Electronics and Control Systems  2015. N 2(44): 46-50 
 
Komnatska Marta. Ph. D. Associate Professor.  
Aircraft control systems Department, National Aviation University, Kyiv, Ukraine. 
Education: National Aviation University, Kyiv, Ukraine (2007). 
Research interests: control systems and data processing.  
Publications: 38. 
E-mail: martakomnatska@gmail.com 
 
М. М. Комнацька. Оптимальне керування рухом літака з гарантованими мінімальними затратами 
Представлено процедуру синтезу оптимального регулятора, що мінімізує обраний квадратичний критерій. 
Задача розв’язується в термінах лінійних матричних нерівностей. Ефективність запропонованого підходу 
ілюструється на прикладі керування повздовжнім рухом літака. 
Ключові слова: керування рухом літака; зовнішні збурення; лінійна матрична нерівність; показник якості; 
робастність; зворотній зв’язок за станом. 
 
Комнацька Марта Миколаївна. Кандидат технічних наук. Доцент. 
Кафедрa систем управління літальних апаратів, Національний авіаційний університет, Київ, Україна. 
Освіта: Національний авіаційний університет, Київ, Україна (2007). 
Напрямок наукової діяльності: системи управління та обробка інформації. 
Кількість публікацій: 38. 
E-mail: martakomnatska@gmail.com 
 
М. Н. Комнацкая. Оптимальное управление движением самолета с гарантированными минимальными 
затратами 
Представлена процедура синтеза оптимального регулятора, который минимизирует выбранный показатель 
качества. Задача решается с использованием аппарата линейных матричных неравенств. Эффективность 
предложенного подхода иллюстрируется на примере управления продольным движением самолета.  
Ключевые слова: управление движением самолета; внешние возмущения; линейное матричное неравенство; 
показатель качества; робастность; обратная связь по состоянию. 
 
Комнацкая Марта Николаевна. Кандидат технических наук, доцент. 
Кафедра систем управления летательных аппаратов, Национальный авиационный университет, Киев, Украина. 
Образование: Национальный авиационный университет, Киев, Украина (2007). 
Направление научной деятельности: системы управления и обработка информации. 
Количество публикаций: 38.  
E-mail: martakomnatska@gmail.com 
 


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OPTIMAL GUARANTEED COST CONTROL OF AIRCRAFT MOTION

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