Page 190 - Servo Motors and Industrial Control Theory -

Appendix B 187

a. Derive the second order governing differential equation for the above system.
Assume that the x is measured from the static equilibrium position and hence
you can ignore the weight of the mass (M).
ForK = 10000 (N/m), M = 50(kg),and C10(N/(m/sec))=

Solve both problems 5 and 6 by hand because the order of system is two and
it is convenient to understand the principle of state variable control theory.
b. Write the governing differential equation in state space form.
c. Determine the eigenvalues of the system from the determinant of the dynamic
matrix. Calculate the natural frequency and damping ratio of the system.
d. Determine the eigenvectors of the system and discuss the meaning of the
eigenvectors.
e. Find the steady state value of x for a given step input of force from the state
equation.

6. In problem 5, assume that it is required to control the position of the mass very
fast with over damped transient response characteristic. Assume that the desired
roots of the characteristic equation should be,

s := 50 s := 100
2
1
Also assume that the system is controllable. Design state variable feedback so
that the closed loop system has the above mentioned roots. Note that the closed
loop system is quite fast with the dominant time constant 0.02 s. Determine the
two gains of state variable feedback. Assume that the demand position is x . De-
i
termine the error for step input of the position.
7. The open loop a system has the following transfer function
θ 100
o :=
θ i s + 40s + 2100s 34000+
3
2
Write the transfer function in state space form and using MathCAD or any other
mathematical software determine the eigenvalues of the system matrix. Suppose
that the system must have an over damped characteristic behavior with the fol-
lowing roots,

s :=−50
1
s :=−100
2
s :=−150
3
Check the controllability of the system and if controllable design a state variable
feedback controls so that all roots move to the required position. Find the steady
state value of the output for a unit step input.