Page 167 - Servo Motors and Industrial Control Theory -

Appendix A 163

6. Prove the nonlinear equation of problem 3. If you have difficulties, refer to a
fluid mechanics book.
7. Derive the Laplace Transform for the following step, velocity, acceleration, and
exponential inputs that often are used as test functions in control systems. Use
the definition of Laplace Transform and for the second and third functions, use
integral by parts. Compare your results with those stated in the table of Laplace
Transform.
x: = A
x: = At
x: = At 2
x: = Ae − t τ
where A and τ are constants.
8. A simple but complicated control system is the driver and his car. Identify sev-
eral input and output variables that have to be controlled by the driver. Show
every control loop in descriptive block diagram form. Show the driver as the
controller and discuss what functions he performs. Identify what kind of time
delay is present in every block.
9. There are several control systems in an airplane. Identify several input and out-
put variables that have to be controlled by the pilot. Show each control system
by descriptive block diagram. And discuss what methods can be used to control
this system in the autopilot position. There are various possibilities of measur-
ing the output variables needed for control in the feedback loop. Discuss these
possibilities.
10. The diagram below shows schematic cross-section view of an electric oven.
The purpose is to keep the temperature of the oven at constant value. The dia-
gram shows the open loop control. The closed loop properties of the oven will
be discussed later in the book.

Qo
Qi To Ta

Where Q is the input heat to the heater (watt/sec)
i
T is the oven temperature (centigrade)
o
T is the outside temperature (centigrade)
a
Q is the heat loss through the heaters walls
o
Show that the transfer function of the oven relating the oven temperature to the
input heat and the outside temperature is,