Modelling the Longitudinal Motion of an Aircraft

Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 Valentina E. BĂLAù !Aurel Vlaicu" University of Arad, Engineering Faculty Bd. RevoluĠiei nr. 77, 310130, Arad, Romania E-mail: [email protected] Marius M. BĂLAù !Aurel Vlaicu" University of Arad, Engineering Faculty Bd. RevoluĠiei nr. 77, 310130, Arad, Romania E-mail: [email protected] MODELLING THE LONGITUDINAL MOTION OF AN AIRCRAFT NOTE: This paper was presented at the International Symposium •Research and Education in an Innovation Era•, Engineering Sciences, November 20-21, 2008, "Aurel Vlaicu" University of Arad, Romania 7 Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 ABSTRACT: The paper is making a short introduction into the field of the aircraft modeling. A basic aircraft model is build, aiming to obtain a simulation platform for different related control algorithms and for further studies on the switching controllers effects in avionics. A Matlab-Simulink implementation is provided. KEYWORDS: switching controllers, mathematical model, state space. 8 Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 INTRODUCTION The Stability and the Control of the airplanes was a key issue from the very beginning of the aviation. A great step forward in the field was the introduction of the automatic pilot (robot pilot, autopilot). The initial purpose of an autopilot was to replace the human pilot during cruise modes. They are expected to perform more rapidly and with greater precision than the human pilot and to make the aircraft fly in the same manner as a highly trained pilot: smoothly and with no sudden and erratic maneuvers. The modern autopilots are implemented by complex digital computers and they are able to stabilize the aircraft, protect the aircraft from undesirable maneuvers, and realize automatic landings. Although at the first glance the reliability of the digital computers seem to be indubitable, in particular circumstances, the perturbations produced when switching between automate pilot and manual pilot may cause sudden and erratic instabilities that can cause fatal airplane crashes. Official and reliable reports on such accidents are not easy to find, but it is unanimously accepted that the on-line switching of two different controllers may produce uncontrollable transient regimes and even destabilizations. In some previous papers we investigated the Switching Controllers Instability (SCI) for the case of some second order plants. The objective of this paper is to choose an appropriate mathematical model of an aircraft, that could stand for a simulation support, in further studies on control algorithms and on the switching controllers effects in avionics. 9 Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 MATHEMATICAL MODELS FOR AIRCRAFTS The first mathematical model of an aircraft was proposed by G.H. Bryan, in a fundamental early book: Stability in Aviation, 1911. Bryan#s model is a system of 6-degrees-offreedom equations that are still in use for the computer simulation of the most advanced of today#s aircrafts, with some supplementary developments needed for the airplane control [1]. An interesting fact about the Brian#s model is that his simplifying assumptions, which are affecting the model#s accuracy for the subsonic aircrafts, are more suitable for the supersonic aircraft models. Figure 1. The Brian#s aircraft parametrical model Starting from this model, that offers a structural view of the aircraft#s dynamics, off-line or on-line accurate experimental models can be obtained. 10 Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 An on-line identification was communicated in Ref. [2]. The identified aircraft is an L-410 Turbolet, manufactured by the Czech aircraft manufacturer LET. L-410 is a twin engine shortrange transport aircraft (see Fig. 2). The state vector that was used for the aircraft longitudinal motion modeling contains four state variables: aircraft velocity v, angle of attack Į, pitch angle ij and derivative pitch angle ij#. The control vector contains only one input variable: the elevator angle į. Figure 2. The L-410 Turbolet It was used a linear model: X' = Ɏ(X) + Ƚ(U) (1) where Ɏ ҏ is the plant matrix (n x n), Ƚ the control matrix (n x r), X the state vector (n x 1) and U the control vector (r x 1). The state vector and the control vector are the following: X = [ v Į ij ij# ] T U=[į]T (2) (3) The final result of the identification, after n = 21 data measurements (t = 2.0 s), using a Matlab implemented version of the classic least squares method [3] is the following: 11 Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 9.5079e-001 3.4779e+001 -1.7931e+001 -1.4134e+001 -1.3064e+001 [Ɏ21 , Ƚ21] = 3.8000e-004 8.2254e-001 3.3844e-002 9.5337e-002 -1.4376e-002 (4) -1.5513e-004 9.8492e-002 9.3600e-001 6.8752e-002 -2.7441e-002 3.3045e-004 1.5789e-001 -2.5812e-001 6.4224e-001 -5.3852e-001 A SIMULINK IMPLEMENTATION Gain20 0.6422 Gain19 0.2581 Gain8 Gain18 0.1579 Gain17 Gain7 0.00033 0.5385 Gain15 -0.02744 0.09849 Gain13 Gain6 0.000155 0.01437 Gain14 0.936 Gain12 12 Figure 3. The aircraft longitudinal motion Simulink model 4 f' [rad/s] Saturation Integrator3 1 xo s 0 f'0 [rad] [fd] Goto3 3 f [rad] Integrator2 Saturation1 0 f0 [rad]1 1 xo s 0.06875 Gain16 [f] Goto2 2 Saturation2 Integrator1 a0 [rad] 0 0.8225 Gain9 0.00038 d 1 From1 Gain10 0.03384 Gain 13.064 From3 0.09534 Gain11 1 xo s [a] Goto1 220 V0 [km/h] Gain5 Fcn1 14.134 [fd] Gain3 17.931 [f] From2 Gain4 34.779 Gain2 0.9508 [a] [v] From a [rad] Test v [km/h] 3.6 1 xo s u/3.6 Saturation3 Integrator [v] Gain1 Goto 1 The previous mathematical model is implemented in Simulink-Matlab as shown in Fig. 3. This deployed version is more complicate that the state-space model, but has the advantage of a transparent and complete control of the initial values of the state variables. Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 360 v [km/h] V [km/h] 0.05246 a [rad] A [rad] 0 PID Imposed pitch PID Controller d -0.01735 f [rad] Pitch [rad] 0.04166 f ' [rad/s] F [rad/s] Aircraft Figure 4. The control of the pitch by the elevator angle 0.01 input 0.005 0 -0.005 output - elevator angle [rad] -0.01 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 time [s] 6 7 8 9 10 0.01 0.005 0 -0.005 -0.01 Figure 5. A simulation result 13 Scientific and Technical Bulletin Series: Electrotechnics, Electronics, Automatic Control and Computer Science, Vol. 5, No. 4, 2008, ISSN 1584-9198 This model can be used now for testing different control algorithms. A simulation result is shown in Fig. 5. The main advantages of this model are the simplicity, the linearity and the accuracy for the given identification conditions. It is adapted for the real-time identification of the specific steady flight regimes. On the other side its nature is synthetic: no information about the physical structure of the airplane system is included. Although the state variables are physical parameters, they are not able to catch the nonlinear functionality of the system. That is why this model can be hardly used outside of its context. CONCLUSIONS The paper is presenting a deployed continuous time version of a state-space mathematical model of the longitudinal motion of an L-410 airplane. REFERENCES [1] M.J. Abzug and E.E. Larabee. Airplane Stability and Control. A History of the Technologies That Made Aviation Possible. Second edition. Cambridge University Press, 2002. [2] M. Dub and R. Jalovecký. Experimental online identification of aircraft longitudinal motion. Cybernetic Letters ! Informatics [online], Cybernetics and Robotics, 15.12.2006. <http://www. cybletter.com>. [3] N.E. Leonard and W.S. Levine. Using Matlab to Analyse and Design Control Systems, ed. II, Addison-Wesley Publishing Company, 1995. 14