Page 165 - Servo Motors and Industrial Control Theory -

Appendix A: Exercise Problems on Classical
Feedback Control Theory (Chaps. 1 and 2)

For all calculations in this book, you can use the MathCad software or any other
mathematical software that you are familiar with.

1. Determine which of the following differential equations represents linear and
which one represents nonlinear systems.

d 5 d 4 d 3 d 2
a. dt 5 y 5+ dt 4 y 10+ dt 3 y 25+ dt 2 y ⋅= 6x

d 3 d 2 d 1 d
+
b. dt 3 y () +10 dt 2 y () + 50 dt 1 y () +100 y () = 60 dt X 200x

d 2 d 2
c. dt 2 y () +125 dt y () +1250 y () = 1 − xx()

d 2 d 1
d. dt 2 (y) + dt 1 (y)⋅ y 100y:+ = x

2. Find a linearized model for the following function around the point x = 5
y: 120x= 2

3. Find a linearized equation for the following equation which may represent a flow
equation in a valve. Find the linearized coefficients for a valve for four extreme
positions of around fully closed and fully open positions, and at valve opening of
6 mm and extremely low and extremely high pressure. Assume that the fluid is
at pressure of 150 bars and the valve displacement is maximum of 6 mm.

(P − P)
Q: C A X= d ⋅⋅⋅ 2· s ρ

Where Q is the flow rate and ρ = 0.95 kg/l and C are the fluid density and valve
d
shape factor and P is the back pressure. A is the cross-section area of the valve.
Assume that C is 0.85.
d
R. Firoozian, Servo Motors and Industrial Control Theory, Mechanical Engineering Series, 161
DOI 10.1007/978-3-319-07275-3, © Springer International Publishing Switzerland 2014