Page 193 - Servo Motors and Industrial Control Theory -

190 Appendix B

Check the controllability of the system with respect to Q .
i
Suppose that it is required to control the system with state variable feedback
control strategy so that the closed loop system has the following roots of char-
acteristic equation. In this case note that you have to measure two fluids levels
in the tanks and feed them back with deferent gains to the controller and as-
sume it is required to control the level of second tank with the desired value
of h .
2i
s :=−20
1
s :=−80
2
Where numerical values required for some parameters assume some engineer-
ing values for them and insert them in the state equation so that you can solve
the problem. Note that you should use correct units in the equations.
Determine the steady state values of h and h for step input of h and Q . The
2i
1
o
2
large steady state error that may be present in the system poses no problems as
the desired input may be calibrated in terms of output values.
Investigate the possibility of adding an integrator in the controller so that the
system has zero steady state errors. You should note that the order of the system
increases by one and suppose that the system with integrator must have the
eigenvalues as,
s :=−20
1
s :=−50
2
s :=−80
3
13. Consider the following two interacting tanks where the aim is to control the
fluid level in both tanks. Write the governing differential equations and convert
them to state space form. Linearize the nonlinear flow equations and assume
that the two variables to be controlled are the fluid levels in the tanks and the
input variables are the input flow rate in the first tank and the output flow rate
from the second tank. Use the numerical values given in problem 53 in the first
section.
First, assume that the fluid level at the second tank must be controlled by the in-
put voltage to the valve on the inlet side. Find the eigenvalues and eigenvectors
of the system matrix and check the controllability of the two output variables
with respect to the input flow rate. Design state variable control strategy that
the two eigenvalues move to the following position.

s :=− 20
1
s :=− 70
2