Mechanical Engineering

MEM 355 Final Project, Fall 2020

Due: Friday, December 4, 2020

Preliminaries A rigid aircraft has 6 degrees of freedom comprising three translational coordinates and three rotational coordinates. Thus, it has 12 states: 3 space coordinates X,Y,Z and 3 angle coordinates , ,φ θ ψ roll, pitch and yaw as well as 6 corresponding velocities translational in body coordinates, , ,u v w , and , ,p q r . It is sometimes convenient to replace the translation velocity vector , ,u v w by its polar representation

, ,V α β where V is the speed (magnitude of the velocity vector), α is the angle of attack and β is the sideslip angle.

If a flat earth is assumed and variation of atmospheric density is ignored then the equations are invariant with respect to the X,Y,Z coordinates and heading ψ . So we can ignore them, reducing the number of equations to 8. Furthermore, if the equations are linearized at some trim point they decouple into two smaller sets: a) the longitudinal dynamics consisting of the 4 states , , ,u w q θ (equivalently

, , ,V qα θ ) and b) the lateral dynamics consisting of the remaining 4 states , , ,v p r φ (equivalently, , , ,p rβ φ ). Note that the longitudinal dynamics define the motion in the vertical plane.

Part A Consider the F-18 Longitudinal Dynamics:

0.0239 28.3172 0 32.2 3.8114 0.001 0.0003 0.3621 1 0 0.0515 0

0 2.2115 0.2532 0 2.8791 0 0 0 1 0 0 0

e

T

V V d

q qdt δα α δ

θ θ

− − − −               − − −         = +         − − −                  

0 1 0 1 1 0 0 0

V

V q γ α

θ

   −     =            

1. Compute describe the open loop system modes. 2. Compute the 2x2 system transfer matrix , ,e T Vδ δ γ→ :

( ) ( ) ( ) ( ) ( )

11 12

21 22

G s G s G s

G s G s  

=    

3. Consider the single input – single output system eδ γ→ . Describe the relevant features of its transfer function, ( )11G s . Obtain a state feedback control gain, K, that places the closed loop poles at ( )0.05 0.09, 1.05 1.1j j− ± − ± . Obtain an observer feedback gain L that places observer poles in accordance with ‘Rynaski’ design, e.g., place observer poles at ( )1.149, 8, 8.5, 9− − − − .

4. Determine the equivalent compensator, ( )cG s , obtained in Part 3 and compute the closed loop system transfer function. Compute the closed loop transfer function for compensator ( )cG s and plant

( )11G s .

Part B Consider the lateral dynamics of a Boeing 747 at 0 774 fpsV = and 40,000 feet. The control input is the rudder deflection rδ :

0.0558 0.9968 0.0802 0.0415 0.00729 0.598 0.115 0.0318 0 0.475

, 3.05 0.388 0.4650 0 0.153 0 0.0805 1 0 0

r

r rd y r

p pdt

β β

δ

φ φ

− −               − − −       = + =        − −                

Nomenclature: β sideslip angle, r yaw rate, p roll rate, φ roll angle. We wish to design a compensator to improve yaw rate control performance. The standard

configuration of such a control system is shown the figure below and includes a ‘washout filter1.’ The rudder actuator dynamics need to be considered as well as shown.

Aircraft

Washout Filter

Rudder ActuatorCompensator

- +

( )cG s 10

10s +

1 3

1 3

s s +

rδ rr

Construct a set of state equations that includes the rudder actuator and the washout filter.

1. Choose the following set of closed loop poles: { }0.0051, 0.468, 0.279 0.628 , 0.279 0.628 , 1.106, 9.89j j− − − + − − − −

and find a state feedback controller to achieve those poles. 2. Compare the closed loop and open loop modes. Discuss how they have changed. 3. Design an observer for the system with poles 5 times faster than controller poles:

{ }0.0253, 2.34, 1.39 3.14 , 1.39 3.14 , 5.53, 49.5j j− − − + − − − − 4. Compute the compensator transfer function.

1 Turning an aircraft requires the pilot to coordinate rudder and aileron to establish a desired yaw rate and roll angle. The presence of the washout filter reduces the impact of the rudder on roll dynamics, thereby making the pilot’s job easier.