
The steady-state error due to a unit-ramp command and due to a unit-ramp disturbance of I controller with an internal feedback loop of first order plant for the given cases.

Answer to Problem 10.28P
The steady-state error values for all the cases are:
Case 1. For
Error due to unit-ramp command:
Error due to unit-ramp disturbance:
Case 2. For
Error due to unit-ramp command:
Error due to unit-ramp disturbance:
Case 3. For a root separation factor of 10,
Error due to unit-ramp command:
Error due to unit-ramp disturbance:
Explanation of Solution
Given:
The I controller with an internal feedback loop of first order plant is as shown below:
Where, the parameter values are as given:
Also, the performance specifications require the time constant of the system to be
The values of gains for the I controller are as follows:
Case 1. For
Case 2. For
Case 3. For a root separation factor of 10,
Concept Used:
- The transfer functions for the block diagram are as shown below:
- The steady-state error of a system using final value theorem is:
Calculation:
From the block diagram as shown, the transfer functions are as:
Therefore, the response
And from the block diagram shown in figure, we have
On keeping the values of the parameters such that
Case 1. When
Since,
Therefore, for unit-ramp command response
Thus, the steady-state error for this design is:
Therefore, for zero command response
Thus, the steady-state error for this design is:
Case 2. When
Since,
Therefore, for unit-ramp command response
Thus, the steady-state error for this design is:
Therefore, for zero command response
Thus, the steady-state error for this design is:
Case 3. For a root separation factor of 10,
Since,
Therefore, for unit-ramp command response
Thus, the steady-state error for this design is:
Therefore, for zero command response
Thus, the steady-state error for this design is:
Conclusion:
The steady-state error values for all the cases are:
Case 1. For
Error due to unit-ramp command:
Error due to unit-ramp disturbance:
Case 2. For
Error due to unit-ramp command:
Error due to unit-ramp disturbance:
Case 3. For a root separation factor of 10,
Error due to unit-ramp command:
Error due to unit-ramp disturbance:
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Chapter 10 Solutions
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