A contemporary approach of flight control system design via static output feedback design is proposed. The static output feedback is formulated in terms of linear matrix inequalities. The obtained solution guarantees stabilization of unmanned aerial vehicle during flight mission. During flight envelope the  unmanned  aerial  vehicle  is  subjected  to  the  external  stochastic  disturbances.  The  efficiency  of  the proposed approach is illustrated by a case study of unmanned aerial vehicle longitudinal motion.Запропоновано нову процедуру синтезу статичного зворотного зв’язку за виходом на основі апарату лінійних матричних нерівностей. Показано, що управління забезпечує стабілізацію безпілотного літального апарата під час виконання льотного завдання в умовах дії на нього зовнішніх неструктурованих збурень. Дослідження проведено на прикладі управління поздовжнім рухом безпілотного літального апарата

M. M. Komnatska

Electronics and Control Systems

Scientific and technical library of the National Aviation University

FLIGHT CONTROL SYSTEM DESIGN VIA LMI-APPROACH

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

M.M. Komnatska  Flight control system design VIA LMI-approach                                                                                 49  ___________________________________________________________________________________________________________ © National Aviation University, 2013 http://ecs.org.ua UDC 629.735.33-519 (045) M. M. Komnatska FLIGHT CONTROL SYSTEM DESIGN VIA LMI-APPROACH Aircraft Control Systems Department, National Aviation University, Kyiv, Ukraine E-mail: martakomnatska@gmail.com Abstract. A contemporary approach of flight control system design via static output feedback design is proposed. The static output feedback is formulated in terms of linear matrix inequalities. The obtained solution guarantees stabilization of unmanned aerial vehicle during flight mission. During flight envelope the unmanned aerial vehicle is subjected to the external stochastic disturbances. The efficiency of the proposed approach is illustrated by a case study of unmanned aerial vehicle longitudinal motion.  Keywords: aircraft control; disturbance attenuation; external disturbances; linear matrix inequality; ro-bustness; static output feedback; unmanned aerial vehicle.Introduction During the last years, the problem of robust con-troller design has attracted considerable attention from the automatic control society, especially in the area of unmanned aerial vehicles (UAVs) [1]. The wide UAV application is explained by the fact that such vehicles are able to perform various tasks. Preferably, they are used in dangerous and inacces-sible regions to avoid physical injuries in case of manned vehicles usage. These conditions lead to the robust flight control system design which possesses with ability to meet the contradictory requirements imposed on the UAV during the flight. Furthermore, one should care about various problems connected with law cost design and power consumption in order to be implemented onboard computer with restricted abilities. In turn, it leads to the limited number of navigation sensors, their size and weight. These circumstances lead to the problem of a 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 feed-back 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. Nevertheless, the obtained control law is simpler and easier to be realized. Moreover, in contrast to the ob-server based controllers, SOF controller does not need to solve differential equations that results in a decreas-ing of power consumption and computational cost. A survey devoted to this problem is presented in [2]. This paper deals with the static output feedback (SOF) controller design in terms of linear matrix inequalities (LMIs) [3] – [7] for UAV control during flight envelope. The main feature of this paper is that the obtained SOF controller stabilizes the set of autonomous systems, simultaneously. To prove the efficiency of the proposed technique, the longitudinal motion of the aircraft control is considered as a case study. Problem statement  Consider a linear time invariant system described by the following differential equation           ;,t t t tt t   x Ax Bu By Cx where  nx R  is the state space vector,  mu R is the control vector,  py R is the output vector and  nR  is a disturbance vector. Besides that, the state space matrices of the controlled plant have the following dimensions , ,  nЧn mЧn pЧnA R B R C R . It could be seen that number of measuring variables p  is less than number of all phase coordinates, n . Therefore, the control law is designed taking into account only variables that are available for measurement.  The control law is given by      t t t   u Ky K C x ,  (1) where K is a constant output feedback gain, that minimizes performance index:                2020 0,T T TJ z t dtt t t t dt t t dt       Kx Q x u R u  0t  , where 0Q and 0R  are diagonal matrices, weighting each state and control variables, respectively. Output signal  tz  used for performance evaluation is defined as follows: 00Q xzuR          . Bounded 2L  gain design problem The system 2L  gain is said to be bounded or attenu-ated by   if [2] – [5]: 50                                                                    ISSN 1990-5548   Electronics and Control Systems  2013. N 3(37): 49-52       220 020 0T TTz t dt x Q x u Ru dtd t dt d d dt      . Therefore, it is necessary to find constant output feedback gain matrix K  that stabilizes the control plant such that the infinity norm of the transfer function referring exogenous input to performance output  tz  approaches minimum. The minimum gain is denoted by * .  The output feedback gain matrix K  (1) could be found by solving the following iterative LMI 20 0 00 00 0iiT Tn i i n n i n nTiTn          P A A P Q P B P B LB P RB P IL R,    (2) where Ni ,,1  in (2) denotes the set of models associated with certain operating conditions within the flight envelope.  The main advantages of the proposed approach: there is no necessity to define the initial matrix K ; an opportunity to find solution to the set of matrices that proves the robustness properties of the system.  The matrices K  and L  for each n th iteration are updated as follows:     11 ;T T Tn+1 n n  K R B P L C CC  1 1Tn n n  L RK C B P . On the last stage a convergence is checked, namely if 1n n  K K  (if 1nK  and nK  are close enough to each other) than terminate and set 1 nKK , otherwise set 1 nn  and solve the ine-quality (2). Case study To demonstrate the efficiency of the proposed approach a longitudinal channel of the UAV is used as a case study. The state space vector of the longi-tudinal channel is  , , Ttx V q h   , where Vt is the true airspeed of UAV, α is the angle of attack, θ is the pitch angle, q is the pitch rate and h is the alti-tude. The control input vector  , Tthl e u  is represented by the throttle and elevator deflections, respectively.  It is considered two operating modes with true airspeed at Vt = 33.9 m/s and Vt = 38.8 m/s. Thus, we have two mathematical models that correspond to these airspeeds. The linear models in the state space are represented by the matrices  ,A B : – nominal model 0.2123 25.6346 9.81 0 00.0442 3.0958 0.2875 0.9734 00 0 0 1,0 00.5404 114.9272 0.5970 10.1163 00 33.9910 33.9910 0 0n            A ; 1045.1375 0.98560 0.51710 00 258.70560 0n        B ; – perturbed model 0.2428 33.5390 9.81 0 00.0442 3.5420 0.2513 0.9734 00 0 0 1,0 00.6184 150.5485 0.5983 11.5816 00 38.880 38.880 0 0p            A; 1045.1375 1.28950 0.63120 00 338.12540 0p        B , where the subscript “n” corresponds to the nominal model and perturbed model is designated by the subscript “p”. The actuator dynamics is described by quadruple matrix given by 1/ 1/1 0a a a aa aA B T TC D            , where Та = 0,23 s is an actuator time constant. The output vector of measured variables is given as fol-lows  , , , Test ty V q h  . Disturbance, υ affecting the longitudinal motion of the aircraft involves the following components: the true airspeed, Vt, angle of attack, α and pitch rate, q, so that  , , TtV q  . In order to simulate the atmospheric turbulence the Dryden filter is used [8]. The attenuation level   is found to be equal to 1.04. By solving LMI (2) the stabilizable static output feedback gain matrix is obtained. The obtained con-M.M. Komnatska  Flight control system design VIA LMI-approach                                                                                  51  troller guarantees disturbance attenuation with pre-defined value of γ. The gain matrix has the following form 0.0125 0.2963 0.0067 0.01140.0170 1.3775 0.0292 0.0510       K . Performance indices for a set of nominal and para-metrically perturbed of the closed loop systems are given in table 1. Table 1 Estimated performance indices for the set of nominal and parametrically perturbed closed loop systems Performance index Plant Nominal Perturbed  Н2 –norm deterministic case  0.3830  0.3795 stochastic case 0.5936 0.6181 Н∞ – norm 0.5534 0.6917 Transient processes in nominal and parametrically perturbed system, which were simulated taking into account all nonlinear functions inherent to the real autopilot as well as the influence of the random wind, simulated according to the standard Dryden model of turbulence [8]. Simulation results are shown in Fig. 1.  a  b  c Fig. 1. Simulation results for longitudinal channel of UAV in the presence of external disturbances: a is velocity; b is altitude of UAV nominal and perturbed models; c is pitch angle of UAV nominal and perturbed models 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. Table 2 reflects standard deviations of the UAV outputs in a stochastic case of nominal and parame-trically perturbed model with static output feedback controller in a control loop  Table 2 Standard deviations of the UAV outputs in a stochastic case  Plant Standard deviation σV, m/s σα, o σq, deg/sec σθ, o σh, m σel, o σth, % Nominal 0.1681 1.1954 3.5785 0.9677 0.4547 0.0579 0.0010 Perturbed 0.1798 1.2056 3.9755 0.9838 0.4657 0.0601 0.0010  Conclusions The paper presents procedure of static output feedback controller design in terms of linear matrix inequalities. The main advantages of static controller application are their simplicity; the control law directly forms basing on available information about measuring output vector. The simulation results of longitudinal motion  control with static controller in the loop including performance indices prove the efficiency of the proposed approach of flight control system design.  52                                                                    ISSN 1990-5548   Electronics and Control Systems  2013. N 3(37): 49-52   The maximum deflection of pitch angle is enclosed within acceptable interval: 2 20    deg. The altitude h  and velocity V  are also held at their reference signals 50mrefh   and 4m srefV   respectively with acceptable deflections.  Simulation results along with numerical results, represented in Table1 and Table 2 show that the controller possesses with robustness property. References [1] Magni; J.-F.; Bernani, S.; Terlouw; J. “Robust flight control. A design challenge.” London. Springer. 1997. 649 p. [2] Syrmos, V. L.; Abdallah, C. and Dorato, P. “Static output feedback: a survey.” Proceedings of 33rd IEEE Conference on Decision and Control, Orlando, FL. 1997. pp. 837–842.  [3] Boyd, S.; Ghaoui, L. El; Feron, E.; Balakrishnan, V. “Linear matrix inequalities insystem and control theory.” Philadelphia: PA SIAM. 1994. 205 p. [4] Gadewadikar, J.; Lewis, F.; Abu-Khalaf, M. “Necessary and sufficient conditions for H-infinity static output feedback control.” Journal of Guidance, Control and Dynamics. 2006. vol. 29. pp. 915–921. [5] Gadewadikar, J.; Lewis, F. “Aircraft flight controller tracking design using H-infinity static output-feedback.” Transactions of the Institute of Measurement and Control. 2006. vol. 28. no. 8. pp. 429–440. [6] Basanets, O. P.; Tunik, A. A.; Komnatska, M. M. “LMI-based static output feedback design for rotating solid body.” ‘I-st Intern.Conf. Methods and Systems of Navigation and Motion Control’ Kyiv. 2010. pp. 88–90.  [7] Komnatska, M. M. “LMI based design of flight control system combined with fuzzy tuning.” 2010. Proceedings of NAU. vol. 3 (44). pp. 25–34. [8] McLean, D. “Automatic flight control systems.” 1990. Prentice Hall Inc., Englewood Cliffs. 593 p. Received 18 October 2013  Komnaska Marta M. Associate Professor.  Aerospace Control Systems Department, National Aviation University, Kyiv, Ukraine. Education: National Aviation University, Kyiv, Ukraine (2006). Research interests: control theory and its application.  Publications: 27. E-mail: martakomnatska@gmail.com М. М. Комнацька. Синтез системи управління польотом за допомогою апарату лінійних матричних не-рівностей Запропоновано нову процедуру синтезу статичного зворотного зв’язку за виходом на основі апарату лінійних матричних нерівностей. Показано, що управління забезпечує стабілізацію безпілотного літального апарата під час виконання льотного завдання в умовах дії на нього зовнішніх неструктурованих збурень. Дослідження про-ведено на прикладі управління поздовжнім рухом безпілотного літального апарата. Ключові слова: безпілотний літальний апарат; гасіння збурень; неструктуровані зовнішні збурення; робаст-ність; стабілізація літального апарата; статичний зворотний зв’язок за виходом. Комнацька Марта Миколаївна. Кандидат технічних наук, доцент. Кафедрa систем управління літальних апаратів, Національний авіаційний університет, Київ, Україна. робочий Освіта: Національний авіаційний університет, Київ, Україна (2006). Напрямок наукової діяльності: системи управління. Кількість публікацій: 27. E-mail: martakomnatska@gmail.com М. Н. Комнацкая. Синтез системы управления полетом с помощью аппарата линейных матричных не-равенств Предложена новая процедура синтеза статической обратной связи по выходу на основе аппарата линейных матричных неравенств. Показано, что управление обеспечивает стабилизацию беспилотного летательного аппарата во время выполнения летного задания в условиях действия на него внешних неструктурированных возмущений. Исследование проведено на примере управления продольным движением беспилотного летательного аппарата. Ключевые слова: беспилотный летательный аппарат; гашение возмущений; неструктурированные внешние возмущения; робастность; стабилизация летательного аппарата; статическая обратная связь по выходу. Комнацкая Марта Николаевна. Кандидат технических наук, доцент. Кафедра систем управления летательных аппаратов, Национальный авиационный университет, Киев, Украина. Образование: Национальный авиационный университет, Киев, Украина (2006). Направление научной деятельности: системы управления. Количество публикаций: 27.  E-mail: martakomnatska@gmail.com 

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FLIGHT CONTROL SYSTEM DESIGN VIA LMI-APPROACH

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Scientific and technical library of the National Aviation University

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