1. E(s): Laplace transform of the error e(t)
6. Steady-State Errors
6.1 Steady-state errors of single-loop systems with unity feedback ) s ( R
E(s): Laplace transform of the error e(t)
Steady-state error:
DEU-MEE 5017 Advanced Automatic Control

2. 6.2 System type:
ess depends on the characteristics of G(s). We can show that ess depends on the number of poles that G(s) has at s = 0. This number is known as system type, which explains the type of the control system.
Example 6.1 :
Determine the system type of the forward path transfer functions.
System type: 1
System type: 2
System type: 0
System type: 1

3. 6.3(a) Step error : ) s ( R Step-errror constant System type: ess 1
Step Input ) s ( R
Step-errror constant
System type:
ess 1

4. 6.3(b) Ramp error : ) s ( R Ramp-errror constant System type: ess 1 2
Ramp input
Ramp-errror constant
System type:
ess 1 2

5. Parabolic-errror constant
6.3(c) Parabolic error :
Parabolic input ) s ( R
Parabolic-errror constant
System type:
ess 1 2 3

6. Example 6.2: ) s ( R System type: 0 Test stability ess: Step Ramp
Parabolic
Test stability

7. I-control improves steady-state error.
Example 6.3 : ) s ( R
System type: 1
I-control improves steady-state error.
ess:
Step
Ramp
Parabolic

8. D-control does not affect and improve steady-state error.
Example 6.4 : ) s ( R
System type: 1
D-control does not affect and improve steady-state error.
ess:
Step
Ramp
Parabolic

9. 6.4 The effect of PID control on the performance parametersOV
P-action : As Kp increases, the steady-state error decreases, the overshoot increases, the settling time does not change, the rise time decreases.
I-action: When the integral control is added, the steady-state error eliminates, the overshoot increases, the settling time increases, the rise time decreases.
D-action: When the derivative control is added, the steady-state error does not change, the overshoot and settling time decreases, the rise time does not change.

10. 6.5. Other parameters at steady-state) s ( R
Final value theorem
Css: Steady state value of the ouput.
CNss: Sensitivity.

11. ) s ( R Steady-state error:
6.6 Steady-state errors of single-loop systems with nonunity feedback ) s ( R
Steady-state error:

12. Example 6.5 :
(Example 7-5, Kuo:P-381) Consider a single-loop system with nonunity feedback.
Find the steady-state errors for step, ramp, and parabolic inputs.
Observe the result by Matlab/Simulink

13. Example 6.6 : ) s ( R Answer: K=14.7
(Problem 7-7, Kuo:P-458) Consider the closed-loop system with nonunity feedback.
Find K so that the steady-state error for the step input is % 2.
Model the system by Matlab/Simulink and observe the result . ) s ( R
Answer: K=14.7