The document describes modeling and control of aircraft pitch dynamics. It presents the longitudinal equations of motion, transfer function and state-space models. PID and root locus controllers are designed in MATLAB to meet requirements like overshoot <10% and settling time <10 seconds. A Simulink model is built containing the physical setup, state-space model and open/closed loop responses. PID control achieves the design goals while root locus analysis places poles for natural frequency >0.9 rad/sec and zero steady-state error.

1. Topic
Aircraft pitch

2. Contents:
 Aircraft Pitch System Modeling
 Aircraft Pitch System Analysis
 Aircraft Pitch PID Controller Design
 Aircraft Pitch Root Locus Controller Design
 Aircraft Pitch Simulink Modeling

3. System Modeling
 Design Requirements
1. Overshoot less than 10%
2. Rise time less than 2 seconds
3. Settling time less than 10 seconds
4. Steady-state error less than 2%

4. Physical setup and system equations
The 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.

5. The basic coordinate axes and forces
of an aircraft

6. Axis of flight
3. Pitch is controlled by
the air flow across the
elevators.
2. Yaw is controlled by
the air flow across the
rudder
1. Roll is controlled by
the air flow across the
ailerons

7. Conti…
Longitudinal equations of motion for the
aircraft can be written as follows.

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

9. Transfer Function
To find the transfer function of the above system, we need to
take the Laplace transform of the above modeling
equations.

10. State Model
Recognizing the fact that the modeling
equations above are already in the state-
variable form, we can rewrite them as
matrices as shown below

11. Conti…
Since our output is pitch angle, the output
equation is the following

12. MATLAB
Representation:

13. Output:

14. System Analysis
 Open loop response
 Close loop response

15. Open-loop response
MATLAB Code:

16. Pole(P_pitch)
One of the poles of the open-loop transfer
function is on the imaginary axis
 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

17. 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.

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

19. MATLAB code

20. Output

21. PID Controller Design
Proportional Control
PI Control
PID Control

22. Conti…
From the main problem, the open-loop
transfer function for the aircraft pitch
dynamics is

23. For a step reference of 0.2 radians, the design
criteria are the following.
Overshoot less than 10%
Rise time less than 2 seconds
Settling time less than 10 seconds
Steady-state error less than 2%

24. PID
We will implement combinations of proportional (Kp),
integral (Ki), and derivative (Kd) control in the unity
feedback to achieve the desired system behavior.
In PID controller design we use SISO TOOL

25. 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

26. The Control and Estimation Tools Manager window
displays the architecture of the control system being
designed as shown below. This default agrees with the
architecture we are employing.

27. 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 by Compensator
Editor

28. .
 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.

29. Output

30. Conti…
Examination of the above shows that aside from
steady-state error, the given design requirements
have not been met. The gain chosen for Kp can be
adjusted in an attempt to modify the resulting
performance through the Compensator
Editor tab.
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

31. PI Control
Integral control is often helpful in reducing steady-state error.
In our case, the steady-state error requirement is already
being met. We will again use automated tuning to choose our
controller gains. Under the Automated Tuning tab change
the Controller type to PI.
 This transfer function is a PI compensator with Ki =
0.56 and Kp = 1.00.

32. The resulting closed-loop step
response

33. Conti…
From inspection of the above, notice that the
addition of integral control helped reduce the
average error in the signal more quickly.
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

34. PID
 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
 This transfer function is a PID compensator with Ki = 4.45, Kp =
0.98, and Kd = 4.90.
 This response meets all of the requirements except for the settle
time which at 12.6 seconds is a little larger than the given
requirement of 10 seconds. We will attempt to increase the
proportional gain in order to reduce the system's settle time.

35. Output

36. 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.

37. Root Locus Controller Design
A root locus plot shows all possible closed-
loop pole locations as a parameter (usually a
proportional gain K) is varied from 0 to infinity.
 We will employ the root locus to design our
controller to place our system's closed-loop
poles in locations that will result in behavior
that satisfies our given requirements.
We will specifically use MATLAB's SISO Design
Tool to modify the system

38. MATLAB (M_file) Code
 Two windows will initially open, one is the SISO Design
Task which will open with the root locus of the plant with
gain K
 Other is the Control and Estimation Tool
Manager which allows you to design compensators and
analyze plots

39. Root Locus Plot

40. Root Locus plot:
 Our requirement that rise time be less than 2 seconds
corresponds approximately to a natural frequency of greater
than 0.9 rad/sec Adding this requirement to the root locus
plot in addition to the settle time and overshoot requirements
generates the following figure

41. Conti…
 From examination of the above figure, since none of the three
branches of the root locus enter the unshaded region, we
cannot place the system's closed-loop poles in the desired
region by varying the proportional gain K.
 Therefore, we must attempt a dynamic compensator with
poles and/or zeros in order to reshape the root locus
 We specifically need to shift the root locus more to the left in
the complex plane to get it inside our desired region

42. Conti…
 we can see that for the current value of gain K the
settling time and rise time are both too large. Let's
attempt to modify the loop gain graphically by clicking
on one of the pink boxes on the root locus and
dragging the box along the locus in the direction of
increasing K.
 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.

43. Required Output
when K=200

44. Simmulation Model
Simulink model contains,
1. Physical setup and system equations
2. Building the state-space model
3. Generating the open-loop and closed-loop
response

45. Physical setup
We will now build a Simulink model of the
equations already described.

46. State-Space Model
 By double-click on the State-Space block we can enter the
system parameters as shown in the figure below.

47. Complete model
 When finished, the completed model should appear
as shown below.

48. Open loop response
 Next we will generate the open-loop step response
by running the simulation
This response is unstable

49. Closed loop responce
In order to view a stable response, we will
now add the state-feedback control gain K

50. Output response
This response is stable