E(s): Laplace transform of the error e(t)

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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

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

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

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

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

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

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

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

6.4 The effect of PID control on the performance parameters OV 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.

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

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

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

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