This ppt has the main part of aircraft pitch and MATLAB simulation results

2. Group members
Muhammad Bilal (11-EE-074)
Waheed Ashfaq (11-EE-140)
Farhaj Ishtiaq ch. (11-EE-031)
HITEC UNIVERSITY TAXILA CANTT PAKISTAN
(DEPARTMENT OF ELECTRICAL ENGINEERING)

3. TOPIC
Aircraft Pitch

4. CONTENTS:
 Aircraft Pitch System Modeling
 Aircraft Pitch System Analysis
 Aircraft Pitch PID Controller Design
 Aircraft Pitch Root Locus Controller Design
 Aircraft Pitch Simulink Modeling

5. What is pitch angle and
angle of attack in
air-craft

6.  Pitch angle: is the angle between the center
line of the roll axis of the aircraft (nose to tail,
the longitudinal axis) and the horizon.
 Angle of attack is the angle between the
chord line of an airfoil and the relative wind
through which it is moving.
 The angle of attack and the pitch angle are
usually different in flights
 Sometimes the pitch angle may be positive or
negative in normal flight, but the angle of
attack is always positive.

7. Principle of flight

8. There are 4 forces involved with flight:
i. Lift
ii. Weight
iii. Thrust
iv. Drag

9. AXIS OF FLIGHT

10. Three axis’ of flight
include:
Pitch, Roll, and Yaw
1. Roll is controlled by
the air flow across the
ailerons
1. Yaw is controlled by
the air flow across the
rudder
1. Pitch is controlled by
the air flow across the
elevators.

11. Design Requirements
Of Air-craft
a) Overshoot less than 10%
b) Rise time less than 2 seconds
c) Settling time less than 10 seconds
d) Steady-state error less than 2%

12. Aircraft Pitch: System
Modeling

13. System Modeling
• The of equations governing the motion of an aircraft
are a very complicated set of six nonlinear coupled
differential equations.
• However, under certain assumptions, they can be
decoupled and linearized into longitudinal and lateral
equations. Aircraft pitch is governed by the
longitudinal dynamics. In this example we will design
an autopilot that controls the pitch of an aircraft.

14. Conti……
 We will assume that the aircraft is in steady-state at
constant altitude and velocity.
 Thus, the thrust, drag, weight and lift forces balance each
other in the x- and y-directions.
 We will also assume that a change in pitch angle will not
change the speed of the aircraft under any circumstance
(unrealistic but simplifies the problem a bit).
 Under these assumptions, the longitudinal equations of
motion for the aircraft can be written as follows:

16. Transfer Function:
 Before finding the transfer function and state-space models,
let's plug in some numerical values to simplify the modeling
equations shown above:
 Above equation become:

17. Transfer Function:
 To find the transfer function of the above system, we need to
take the Laplace transform of the above modeling equations.
 The Laplace transform of the above equations are shown
below.

18. MATLAB
Representation:

19. Output:

20. Aircraft Pitch: System
Analysis

21. System analysis contain:
Open loop response.
Close loop response.

22. Open loop response:
MATLAB (M-FILE CODE)

23. Output:
 From the above plot, we see that the open-loop response does not satisfy the design
criteria at all. In fact, the open-loop response is unstable.
 Stability of a system can be determined by examining the poles of the transfer
function where the poles can be identified using the MATLAB command pole as
shown below.

24. Pole(P_pitch)
 As indicated by this function, one of the poles of the open-loop
transfer function is on the imaginary axis while the other two poles are
in the left-half of the complex s-plane.
 A pole on the imaginary axis indicates that the free response of the
system will not grow unbounded, but also will not decay to zero. Even
though the free response will not grow unbounded.

25. Closed-loop response:
 In order to stabilize this system and eventually
meet our given design requirements, we will add
a feedback controller.

26. Closed loop code:

27. OUTPUT

28. conti…..
 The above plot again demonstrates that this
closed-loop system does not meet the given
design requirements.

29. Aircraft Pitch:
PID Controller Design

30. PID Controller Design
In PID controller design we cover:
• Proportional control
• PI control
• PID Control
1. In PID controller designing we use SISO tool.
2. The main advantage of using the SISO tool
Design within MATLAB is that it have the
automated tuning capabilities which design
the PID controller

32. Design Criteria For
PID:

33. Proportional control
Let's begin by designing a proportional controller
of the form C(s) = Kp. The SISO Design Tool we will
use for design can be opened by typing SISOtool
(P_pitch) at the command line.
This will open both the SISO Design Task window
as well as the Control and Estimation Tools
Manager window.

34.  Since our reference is a step function of 0.2 radians, we can
set the precompensator block F(s) equal to 0.2 to scale a unit
step input to our system.
 This can be accomplished from the Compensator Editor tab of
the open window. Specifically, choose F from the drop-down
menu in the Compensator portion of the window and set the
compensator equal to 0.2 as shown in the figure below

35.  To see the performance of our system with this
controller, go to the Analysis Plots tab of
the Control and Estimation Tools
Manager window.
 Then choose a Plot Type of Step for Plot 1 in
the Analysis Plots section of the window as
shown below. Then choose a response of Closed
loop r to y for Plot 1 as shown in the figure below:

37.  The resulting performance is improved, though the settle
time is still much too large. We will likely need to add integral
and/or derivative terms to our controller in order to meet the
given requirements.

38. PI control

39.  From inspection of the above, notice that the addition of integral
control helped reduce the average error in the signal more quickly.
 Unfortunately, the integral control also made the response more
oscillatory, therefore, the settle time requirement is still not met.
 the overshoot requirement is no longer met either. Let's try also
adding a derivative term to our controller

40. PID Control
Increasing the derivative gain Kd in a PID controller
can often help reduce overshoot.
Therefore, by adding derivative control we may be
able to reduce the oscillation in the response a
sufficient amount that we can then increase the
other gains to reduce the settling time.
Let's test our hypothesis by changing
the Controller type to PID and again clicking
the Update Compensator button. The generated
controller is shown below.

41. Output of PID:

42. Conti…….
The response shown above meets all of the given
requirements as summarized below.
Overshoot = 5% < 10%
Rise time = 1.2 seconds < 2 seconds
Settling time = 5 seconds < 10 seconds
Steady-state error = 0% < 2%
Therefore, this PID controller will provide the desired
performance of the aircraft's pitch.

43. Aircraft Pitch: Root Locus Controller
Design

44. M-File Code For
Root Locus:

45. Root Locus Plot:

46. Root-Locus plot:

47. Output Of Root Locus:
 This diagram meets all the requirements which we needed
except settling time which we required 10sec but here it is
35.1sec

48. Conti…..
In order to get the required criteria we move the
pink boxes on the root locus and dragging the box
along the locus in the direction of increasing K.
The step response plot will automatically update
The effect of increasing K to a value of 200 will be
that the two slowest closed-loop poles will
approach closed-loop zeros thereby making their
effect minimal

49. Conti….
A loop gain of K = 200 keeps all of the poles
on the real-axis, leading to no overshoot and
the presence of the integrator in the plant
guarantees zero steady-state error.
Therefore, this controller meets all of the
given requirements as shown in the figure
below.

50. Our Required Output
when K=200

51. Aircraft Pitch: Simulink
Modeling

52. In Simulink
Model contains:
Building the state-space model
Generating the open-loop and closed-loop
response

53. State-Space Model

54. Conti….

55. Simulink models:
Open-loop response:

56. Open-Loop Response Output

57. Conti…..
This response is unstable and identical to
that obtained in the Aircraft Pitch: System
Analysis.
 In order to view a stable response, we will
now quickly add the state-feedback control
gain K.
When gain of K is added our new Simulink
diagram of open-loop response becomes…….

58. Simulink model
closed-loop response:

59. Closed-Loop Response Output

60. Conti…
The output of closed-loop response is similar
to Aircraft Pitch: State-Space Methods for
Controller Design.
Where the state-feedback controller was
designed.