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 , ,? ? ? 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 ? ? where V is the speed (magnitude of the velocity vector), ? is the angle of attack and ? 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 ? . 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 ? (equivalently
, , ,V q? ? ) and b) the lateral dynamics consisting of the remaining 4 states , , ,v p r ? (equivalently, , , ,p r? ? ). 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 ?? ? ?
? ?
? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ? ? ?= + ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
0 1 0 1 1 0 0 0
V
V q ? ?
?
? ? ? ??? ? ? ? ? ?=? ? ? ? ? ?? ? ? ? ? ? ? ?
1. Compute describe the open loop system modes. 2. Compute the 2×2 system transfer matrix , ,e T V? ? ?? :
( ) ( ) ( ) ( ) ( )
11 12
21 22
G s G s G s
G s G s ? ?
= ? ? ? ?
3. Consider the single input single output system e? ?? . 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 r? :
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
? ?
?
? ?
? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ? ?= + = ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Nomenclature: ? sideslip angle, r yaw rate, p roll rate, ? 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 +
r? 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 pilots job easier.
