Investigating the Performance of Adaptive Methods in application to Autopilot of General aviation Aircraft

Transcription

1 I J C T A, 8(5), 2015, pp International Science Press Investigating the Performance of Adaptive Methods in application to Autopilot of General aviation Aircraft V Rajesari 1 and L Padma Suresh 2 Abstract: This ork emphasizes an integrated investigation, revie, design and performance tracking of ne generation autopilot system by controlling the longitudinal and lateral axes of Navion using the Variable Structure Control and Linear Quadratic Regulator methods The major contribution made in this ork is to design and develop adaptive controllers for the autopilot system to control the longitudinal and lateral axes of the general aviation aircraft in the MATLAB environment For this, the aircraft dynamic equations of motion for both longitudinal and lateral directions are modeled and the stability derivatives are used to get the state space representation The sliding mode controller design is based on designing the sitching surface so that the plant restrained in this surface has the dynamics inclined to the desired one and also designing a sitched control ill drive the plant state to the sitching surface In this ork, the sitching surface is designed by using the quadratic minimization approach The controllers ere designed and the responses are analyzed and the performance characteristics of both controllers are verified in time domain Key Words: Autopilot, Longitudinal Dynamics, Lateral Dynamics, Pitch angle, Roll angle, LQR, SMC 1 INTRODUCTION An aircraft control system is an assemblage of mechanical and electronic equipment that let on an aircraft to sail ith remarkable accuracy and fidelity This system comprises cockpit controls, sensors, actuators (hydraulic, mechanical or electrical) and computers The introduction of control systems is to establish an additional role for stabilization devices to play since if a device is coupled to control devices then it could correct any departure from a stabilized condition The earlier flight control system designs are mechanical based ones As technology gros, the system designs are influenced not only by the advances in aerodynamics and aircraft controllability characteristics but also by the advances taking place in other technological fields The diversity of present day automatic flight control system arises principally to suit the aerodynamic and flight handling characteristics of individual types of aircraft The main parts of the aircraft consist of the primary and secondary systems The primary control systems of flight are the ailerons, elevator, and rudder These are required for the safe control of an aircraft during flight such as landing, take-off, attitude changes etc The ing flaps, leading edge devices, spoilers, and trim systems consist the secondary control devices By using these devices, the performance characteristics of the aircraft are improved and also the pilot ill be relieved from carrying out enormous control forces [1] Commonly, the aircraft is free to revolve about the pitch, roll and ya axes hich are perpendicular to each other In this research ork, to robust control methods: the Linear Quadratic and the Sliding mode controllers ere designed and the pitch and roll axes of the aircraft are controlled The automatic control system using the above methods ere established and the performance characteristics of these controllers ere studied by simulating in MATLAB environment 1 Assistant Professor, Department of ICE, G Narayanamma Institute of Technology & Science, Hyderabad, India 2 HOD, Department of EEE, Noorul Islam Centre for Higher Education, Tamilnadu, India * rajivisanath28@gmailcom

2 2424 V Rajesari and L Padma Suresh 2 MATHEMATICAL FORMULATIONS 21 Modeling of Longitudinal Equations of motion of aircraft The aircraft comprises to dynamical equations: The lateral axis is represented by the Lateral dynamic equations of motion and longitudinal axis is represented by the longitudinal equations of motion The equations governing the motion of aircraft are a set of six non-linear differential equations hich can be decoupled and linearized into longitudinal and lateral equations The aerodynamic forces, moments and velocity components in body axis system are depicted in figure 1 In this X B, Y B and Z B denotes a body fixed axis system L, M, N shos the aerodynamic moments and p, q, r shos the angular velocities roll, pitch and ya respectively [2, 3] The equations of motion of aircraft can be established in terms of the translational and angular accelerations by applying Neton s 2 nd la to the aircraft model The aircraft is assumed to be a steady state cruise at constant altitude and velocity so that the thrust and drag cancel out each other and lift and eight balance out each other The change in angle of pitching does not change the speed of aircraft The angular orientation and velocities of gravity vector relative to body axis is illustrated in figure 2 ith respect to the angular velocity of the body axes about the vector mg With respect to fig2, the equation of motion of aircraft is expressed in equations (1) to (3) x mg sin m( u q rv) (1) z mg cos m( pv qu) (2) Figure 1: Aerodynamic forces, moments and velocity components Figure 2: Angular orientation and velocities of gravity vector

3 Investigating the Performance of Adaptive Methods in application to Autopilot 2425 M I q qr I I I p r (3) 2 2 yy ( xx zz ) xz ( ) The equations (1)-(3) are considered to be nonlinear and simplified by assuming that the aircraft may be comprising to factors: a mean motion that represents the equilibrium or trim conditions and a dynamic motion hich accounts for the perturbations about the mean motion Thus every motion variable is considered to have to components U U u, Q q, R R r, M M m, Y Y y, o o o o o P P p, L L l, V V v, (4) o o o o It is assumed that the reference flight condition are symmetric and the propulsive forces as constants v0 q0 u0 r (5) The complete linearised equations of motion are obtained as in eqn, (6)-(8) belo d dt X u u X u g cos 0 X e E (6) d 1 d Zuu Z Z u0 Zq g sin 0 Z e e dt dt (7) 2 d d d M uu M M M 2 q M e e dt dt dt (8) Using the equations (6), (7), (8) the state space model for the pitch control problem can be formulated u X X u X u 0 0 e Zu Z uo 0 Z e q M u M u Zu M u M Z M q M uo 0 q M e M Z e e (9) The change in elevator deflection angle is the input and change in pitch angle is the output of the pitch control problem The General Aviation airplane Navion is considered [1] in this ork The longitudinal stability derivatives are listed in table1 [4] from hich the state space representation can be attained [5] Table 1 Longitudinal stability derivative Longitudinal Derivatives Components X-Force Derivatives Z-Force Derivatives Pitching moment Yaing velocities X = 0254 X 0 Z = 2 Z 0 M =- 005 M Rolling velocities X u = 004 Z u = 037 M u = 0 Angle of attack X = 0 X 0 Z = 355 Z 0 M = M 0898 Pitching rate X q = 0 Z q = 0 M q = 205 ElevatorDeflection X e =0 Z e = 2815 M e = 1188

4 2426 V Rajesari and L Padma Suresh The equation (10) shos the state space representation for longitudinal control problem u u e q q (10) 22 Modeling of Lateral Equations of motion of Aircraft Considering fig2, the equation for the lateral axis is expressed as follos: Y mg cos sin m[ v ru p] (11) L I p I r qr( I I ) (12) xx xz zz yy N I p I r pq( I I ) I qr (13) xz zz yy xx xz With the assumptions made before, the equations for roll axis are given in eqn (14)-(16) as follos: d Yv v Ypp ( u0 Yr ) r ( g cos 0 ) Y r r dt d Lv v Lp p Lr r L r r L a a dt (14) (15) d Nvv Nr r N pp N r r N a a dt Using the equations (14),(15), (16) the state space model for the roll control problem can be formulated The roll control problem [5] has the input as the aileron deflection angle and output as change in roll angle of the aircraft On to these equations, the lateral stability derivatives are substituted to achieve the state space representation of the roll control problem The table 2 shos the lateral stability derivatives hich are substituted to get the state space representation of roll control problem Lateral Derivatives Table 2 Lateral stability derivative Components (16) X-Force Derivatives Rolling moment Derivatives Yaing moment Derivative Pitching velocities Y v = 0254 L v = 0091 N v = 025 Sideslipangle Y = 446 L = 1584 N = 43 Rolling rate Y p = 0 L p = 8349 N p = 342 Yaing rate Y r = 0 L r = 2086 N r = 076 Rudder Deflection Y r = 1243 L r = 267 N r = 479 Aileron Deflection Y a = 0 L a = 2868 N a = 216 The eqn (17 & 18) shos the state space representation [5] of the roll control problem

5 Investigating the Performance of Adaptive Methods in application to Autopilot 2427 y y p y g cos r 0 1 y r 0 u 0 u 0 u 0 u 0 p u 0 p a L L 0 p L r r L a L r r r N N 0 N p N r a N r (17) p p 2868 a r r (18) 3 DESIGN OF CONTROL METHODOLOGIES The ork is aimed to develop a pitch control scheme for controlling the pitch and roll angle of aircraft For this to control methodologies (ie) Linear Quadratic Regulator (LQR) and Sliding Mode Control methodologies ere proposed The performance of both the control strategies ere investigated and compared 31 Design of Linear Quadratic Regulator (LQR) Linear quadratic optimization [6], [7] is a basic method for designing controllers of dynamical systems LQR is a poerful method for designing flight control systems This method is based on the manipulation of the equations of motion in state space and the system can be stabilized using full state feedback system Consider the state and output equations describing the longitudinal equations of motion X ( t) Ax( t) Bu( t) (19) y( t) Cx( t) Du( t) (20) In the LQR design [8], the lqr function in Matlab can be used to determine the value of the vector K hich is used to find the feedback control la This control la has to minimize the performance index, u( t) kx( t) e N (21) ( T T J X QX u Ru ) dt 0 here, Q- state cost matrix, R- performance index matrix Here R = 1 and Q = C T C Fig3 shos the full state feedback controller ith reference input For the present study, the value of K is to be determined The controller is tuned by varying the elements in Q matrix hich is done in m-file code With the belo values, R = 1 & Q = [ ; ; ; ], the values of K ith N can be obtained

6 2428 V Rajesari and L Padma Suresh 32 Design of Sliding Mode Control (SMC) The SMC is a non-linear method [10] that alters the dynamics of the non-linear system by applying a high frequency sitching control This is a variable structure control The aim of VSS is to ride the state of the system from an original condition x(0) to the origin of state space as t The state feedback control vector U(x) has a j th component U j (j = 1, 2, m) hich has a discontinuity on the j th sitching surface hich is a hyperplane M j passing through the state origin The hyper plane is defined by Consider the system From equation(14), the sliding mode satisfies the condition M j = {x: C j x = 0}, j = 1, 2, m (22) x (t) = Ax(t) + Bu(t) (23) S = Cx(t) = 0, t t s (24) C = mxn matrix, t s is the time of reaching the sliding subspace Differentiating eqn(22) ith respect to time and substituting for eqn(24), e get C is the hyperplane matrix so that CB 0 Figure 3: Full state feedback controller The equivalent control is obtained by rearranging Eqn(26), C x (t) = CAx(t) + CBu(t) = 0, (25) CBu(t) = CAx(t) (26) U eqt = (CB) 1 CAx(t) U eqt = Kx(t) (27) x (t) = [A Bk]x(t) (28) Eqn(26) gives the system equation for the closed loop system dynamics during sliding The choice of C determines the matrix K The plant dynamics are changed to a regular form and are expressed as: y ( t) A y A y y ( t) A y A y B u( t) (29) This regular form is used to get the solution for the reachability problem and the sliding condition is equivalent to C y ( t) C y ( t) C y ( t) y ( t) (30) C2

7 Investigating the Performance of Adaptive Methods in application to Autopilot 2429 y ( t) Fy ( t) Eqn (29) is the reduced order equivalent system a) Design of Sliding hyperplane [9],[10],[11] To design the hyperplane, the quadratic performance y ( t) [ A FA ] y ( t) (31) 1 T J x Qx dt can be minimised, here Q > 0 is positive definite symmetric matrix and t s is the time of reaching the sliding mode The matrix F is calculated after solving the Riccati equation b) Design of feedback control la Once the sliding surface is attained, then it is easy to solve the reachability problem For this the state feedback control function is selected to drive the state x in to the sliding subspace and to maintain that in it This control la comprises to components: 1 A linear control la (U l ) 2 Discontinuous component (U n ) And the control la, 2 ts Where, U ( t) U ( t) U ( t) (32) l n U t U t CB CAx t CB CA C x t (33) The non-linear component is defined to as 1 1 l ( ) eq( ) ( ) ( ) ( ) [ ] ( ) S Cx( t) U n( t), 0 s Cx( t) (34) C is the symmetric positive definite matrix satisfying the Lyapunov equation, T C C I, is stable design matrix The control la is 4 SIMULATION AND RESULTS 1 Cx( t) U ( t) ( CB) [ CA C] x( t) Cx( t) (35) A system for controlling the pitch and roll axes ere developed and simulated using LQR and SMC and the results of simulation are investigated and compared To analyze the performance of the controllers, fe limitations from the time domain specifications are met The values of Q and R are selected for the pitch control problem as belo and illustrated in figure 4 For Pitch Control using LQR, Q = [ ; ; ; ], R = 1 and the value of state feedback matrix K [ ] & N For pitch control using SMC,

8 2430 V Rajesari and L Padma Suresh Q and R The value of the sitching function F = [ ] Optimal control la, U(t) = [ ] Figure 4: Response of Pitch Control problem using LQR&SMC The values of Q and R are selected for the roll problem as belo and illustrated in figure 5 For Roll Control using LQR, Q = [ ; ; ; ], R = 1 and the value of state feedback matrix, K = [ ], & N = For Roll Control using SMC, Q and R The value of the sitching function F = [ ] Optimal control la, U(t) = [ ] The results of both the Linear Quadratic Regulator and the Sliding Mode Control methodologies for the pitch and roll control problems are presented for comparison as in table 3 By comparing the values of the performances of control strategies in table3, it is observed that the SMC shos fast and good response and gave an optimal performance than the LQR controller in controlling the longitudinal and lateral axes of the aircraft

9 Investigating the Performance of Adaptive Methods in application to Autopilot 2431 Figure 5: Response of Roll Control problem using LQR & SMC Table 3 Performance of control strategies for comparison Controller Rise Time tr Settling Time ts Percentage Over- (secs) (secs) shoot Mp(%) Pitch Roll Pitch Roll Pitch Roll LQR SMC CONCLUSION This paper shos the design of an automatic control system for controlling the longitudinal and lateral axes of aircraft hich as executed using LQR and SMC on the MATLAB environment The controllers ere designed and the responses ere analyzed and verified in time domain From the result of simulation it is observed that the sliding mode controller gives an optimal performance in controlling the pitch and roll axes of the aircraft efficiently than the LQR by handling the effect of disturbances in the system References [1] RC Nelson, 1998, Flight Stability and Automatic Control, McGra Hill, Second Edition [2] Donald Mclean, 1990, Automatic Flight Control Systems, Prentice Hall, International Series in Systems and Control Engineering [3] nasagov( ) [4] E Seckel and JJ Moris, 1971, The Stability Derivatives of the Navion Aircraft Estimated by Various Methods and Derived from Flight Test Data, Federal Aviation Administration, Systems Research and Development Service [5] Ryan C Struett, June 2012, Empennage Sizing and Aircraft Stability using MATLAB, American Institute of Aeronautics and Astronautics [6] Wahid N, Hassan N, Rahmat MF, Mansor, Application of Intelligent Controllers in Feedback Control Loop For Aircraft Pitch Control, Australian Journal of Basic and Applied Sciences, 2011 [7] Donald E Kirk, 1970, Optimal Control Theory- An Introduction, Prentice Hall [8] Hespanha P, 2007, Undergraduate Lecture Notes on LQR/LQG Controller Design, [9] CEdards, Sarah K Spurgeon, A Sliding Mode Control MATLAB Toolbox [10] C Edards, Sarah K Spurgeon, Sliding Mode Control Theory and Applications, Taylor& Francis [11] John YHung, WeibingGao, james C Hung, Variable Structure Control: A Survey, IEEE Transactions on Industrial Electronics, Vol 40, No1, February 1993