Applications Linear Control Design Techniques in Aircraft Control I

Transcription

1 Lecture 29 Applications Linear Control Design Techniques in Aircraft Control I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

2 Topics Brief Review of Aircraft Flight Dynamics An Overview of Automatic Flight Control Systems Automatic Flight Control Systems: Classical (Frequency Domain) Designs 2

3 Brief Review of Aircraft Flight Dynamics Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

4 Geometry of Conventional Aircrafts 4

5 Basic Force Balance Weight Lift Drag Thrust 5

6 Basic Moment Balance Rolling Y Pitching Yawing X Z 6

7 Aileron Roll 7

8 Elevator Pitch 8

9 Rudder Yaw 9

10 Airplane Dynamics: Six Degree-of-Freedom Model Ref: Roskam J., Airplane Flight Dynamics and Automatic Controls, U = VR WQ gsin Θ+ ( X + XT ) m 1 V = WP UR+ gsin Φcos Θ+ Y + YT m 1 W = UQ VP+ g cos Φcos Θ+ Z + Z m ( ) ( ) ( ) ( ) P = c1qr+ c2pq+c3 L + LT +c4 N + NT 2 2 Q = c5pr c6( P R ) + c7( M + MT ) R = c PQ c QR + c L + L + c N + N ( ) ( ) T 9 Φ = P+ Qsin Φtan Θ+ Rcos Φtan Θ Θ = Qcos Φ Rsin Φ x I y I Ψ = sin Φ+ cos Φ sec Θ ( Q R ) T T cos ψ sinψ 0 cosθ 0 sinθ U sinψ cosψ cosφ sin φ V = z I sinθ 0 cosθ 0 sinφ cos φ W h = z = UsinΘ V cosθsinφ W cosθcosφ I 10

11 Representation of Longitudinal Dynamics in Small Perturbation State space form: X = AX + BU c A B XU XW 0 g Z Z U 0 U W 0 = MU + M ZU MW M ZW MQ M U0 0 W + W + W Xδ X E δt Zδ Zδ M + M Z M + M Z 0 0 E T = δ W δ δt W δ E E T ΔU ΔW X = Δ Q Δθ U Δδ E c = Δδ T 1 X 1 X XU =, XW = etc. m U m W 11

12 Short Period Mode Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Heavily damped Short time period Constant velocity 12

13 Short Period Dynamics State Space Equation: Δ α a 1 Δα b Δq 11 1 = a21 a + Δ 22 q b Δ 2 δ E Transfer Function Equations: Δα() s As+ B = Δ δ () + + E α α 2 s As Bs C Δqs () q q = 2 s As Bs C Δ δ () + + E As + B 13

14 Long Period (Phugoid) Dynamics Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Lightly damped Changes in pitch attitude, altitude, velocity Constant angle of attack 14

15 Long Period (Phugoid) Dynamics State Space Equation: Δu a g Δu b b Δδ = + Δ θ Transfer Function Equations: Assumption: Δ δ T = E a21 0 θ b21 b 22 δ Δ Δ T Δus () As u + Bu = 2 Δ δ () s As + Bs+ C E Δθ () s As+ B = Δ δ () + + E θ θ 2 s As Bs C 15

16 Representation of Lateral Dynamics in Small Perturbation State space form: X = AX + BU c YV YP ( U0 YR) gcosθ0 * IXZ IXZ I XZ LV + NV LP + NP LR + NR 0 IX IX IX A = IXZ IXZ IXZ NV + LV NP + LP NR + LR 0 IZ IZ IZ Yδ R * IXZ IXZ Lδ + N A δ L A δ + N R δr IX I X B = IXZ IXZ Nδ + L N L A δa δ + R δ R IZ I Z 0 0 X U ΔV ΔP = Δ R Δφ Δδ A c = Δδ R 16

17 Lateral Dynamic Instabilities Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Directional divergence Do not possess directional stability Tend towards ever-increasing angle of sideslip Largely controlled by rudder Spiral divergence Spiral divergence tends to gradual spiraling motion & leads to high speed spiral dive Non oscillatory divergent motion Largely controlled by ailerons 17

18 Lateral Dynamic Instabilities Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Dutch roll oscillation Coupled directional-spiral oscillation Combination of rolling and yawing oscillation of same frequency but out of phase each other Controlled by using both ailerons and rudders 18

19 Dutch Roll Dynamics State Space Equation: Δ β a11 a12 Δβ b11 b12 Δδ A = r a21 a + 22 Δr b21 b 22 Δδ Δ R Transfer Function Equations: Δβ() s As B () s As ˆ Bˆ β + β Δβ β + β = = Δ δ () s As + Bs + C Δ δ () s As + Bs + C A 2 2 R Δ rs () As+ B Δ rs () As ˆ + Bˆ = = Δ δ () s As + Bs + C Δ () s As + Bs + C A r r r r 2 2 δr 19

20 Automatic Flight Control Systems: An Overview Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

21 ADVANCEDCONTROLSYSTEMDESIGN 21

22 Sensors Altimeter: Height above sea level Air Data System: Airspeed, Angle of Attack, Mach No., Air Temperature etc. Magnetometer: Heading Inertial Navigation System (INS) Accelerometers: Translational motion of the aircraft in the three axes Gyroscopes: Rotational motion of the aircraft in the three axes GPS: Accurate position, ground speed The transfer function for most sensors can be approximated by a gain k 22

23 Actuators Electrical actuators: Its a second order system in general It can be approximated to a first order system with small angle displacements Hydraulic / Pneumatic actuators First order system Combination of the above 23

24 Applications of Automatic Flight Control Systems Cruise Control Systems Attitude control (to maintain pitch, roll and heading) Altitude hold (to maintain a desired altitude) Speed control (to maintain constant speed or Mach no.) Stability Augmentation Systems Stability enhancement Handling quality enhancement Landing Aids Alignment control (to align wrt. runway centre line) Glideslope control Flare control 24

25 Techniques for Autopilot Design Frequency domain techniques: Root locus Bode plot Nyquist plot PID design etc. Time domain techniques: Pole placement design Lyapunov design Optimal control design etc. 25

26 Automatic Flight Control Systems: Frequency Domain Designs Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

27 Velocity Hold Control System Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, The forward speed of an airplane can be controlled by changing the thrust produced by propulsion system. The function of the speed control system is to maintain the some desired flight speed. 27

28 Attitude Control System Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Sense the angular position Compare the angular position with the desired angular position Generate commands proportional to the error signal 28

29 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Components: Model for roll dynamics (transfer function) Attitude gyro Comparator Aileron actuator Reference: R. C. Nelson: Flight Stability and Automatic Control, McGraw-Hill,

30 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Problem: Design a roll attitude control system to maintain a wings level attitude for a vehicle having the following characteristics. L = 0.5 rad / s L = 2.0 / s p δ a The system performance is to have damping ratio ζ = and an undamped natural frequency ω n = 10 rad/s. 2 Assume the aileron actuator and the sensor (gyro) can be represented by the gains k a and k s 30

31 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, The system transfer function Δφ() s Lδ a = Δδ () s s( s L ) a p Forward path transfer function Δδa() s Δφ() s Lδ a Gs () = = ka es () Δδ () s ss ( L) a H() s = k = 1 (unity feedback assumption) s p The loop transfer function GsHs () () = k, where ss ( + 0.5) k = kal δ a 31

32 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, The desired damping needed is ζ = We know ζ = cos θ. Hence, 0 draw a line of 45 from the origin. Any root intersecting this line will have ζ = Gain is determined from: k s s+ 0.5 = 1, This leads to k = However, ω = 0.35 rad / s (much lower than desired!) n 32

33 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, desired achieved Moving pole P to P is impractical, unless there is an increase in wingspan of the aircraft..! 1 2 Hence, the cure is to have a stability augmentation system. 33

34 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Compensator is added in form of rate feedback loop to meet desired damping and natural frequency. The inner loop transfer function can be expressed as follows: Transfer function of inner loop Δps () Lδ a = Δδ () s ( s L ) a kil Gs () IL =, Hs () IL = 1( krg), kil = kaslδ a s Gs () IL M(s) IL = 1 + Gs ( ) ILHs ( ) kil = s k p IL IL 34

35 Roll Attitude Autopilot Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Inner loop gain is selected to move the augmented root farther on the negative real axis. If the inner loop is located at s = 14.14, then the root locus will shift to the left and desired ξ and ω can be achieved. n 35

36 Stability Augmentation System (SAS) Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Inherent stability of an airplane depends on the aerodynamic stability derivatives. Magnitude of derivatives affects both damping and frequency of the longitudinal and lateral motion of an airplane. Derivatives are function of the flying characteristics which change during the entire flight envelope. Control systems which provide artificial stability to an airplane having undesirable flying characteristics are commonly called as stability augmentation systems. 36

37 Example: SAS Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Consider an aircraft with poor short period dynamic characteristics. Assume one degree of freedom (only pitching motion about CG) to demonstrate the SAS. Short period dynamics: ( q ) θ M + M θ + M θ = M δ α α δ Substituting the numerical values θ θ θ = 6.71δ e This leads to ζ = ω = 2.34 rad / s sp nsp It is seen that the airplane has poor damping (flying quality). 37

38 Example: SAS Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, One way to improve damping is to provide rate feedback. Artificial damping is provided by producing an elevator deflection in proportion to pitch rate and then adding it to the pilot's input; i.e. δe = δ ep + k θ Pilot input Artificial command Rate gyro measures the θ and creates an electrical signal to provide k θ in addtion to δe. With this, the modified dynamics becomes: θ θ θ = 6.71δ ( k ) 2 2ζωn = k, ωn = 5.49 Hence, by varying k, the desired damping is achieved. e 38

39 Landing System Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, Key Components: Alignment control (to align wrt. runway centre line) Glideslope control Flare control 39

40 Alignment Control Through Electronic Aid and Pilot Input Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill,

41 Glide Slope: Pitch Control Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill,

42 Glide Slope: Speed Control Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill,

43 Flare: Sink Rate Control Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill,

44 44