Homework3solutions_mae143b

Homework3solutions_mae143b - MAE143B Homework Set 3 -...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAE143B Homework Set 3 - Solutions Question 1 FPE-N Question 5.30: Assume that the closed-loop system has plant transfer function P ( s ) = 1 s ( s + 2) . Design a lag compensator so that the dominant poles of the closed-loop system are located at s =- 1 j and the steady-state error to a unit ramp input is less than 0.2. Without compensation, the root locus of the system is as shown in the following figure. By choosing 2.5 2 1.5 1 0.5 0.5 1.5 1 0.5 0.5 1 1.5 Root Locus Real Axis Imaginary Axis a constant feedback gain of K = 2 , we can place the closed loop poles at s =- 1 j . However, this leaves the steady-state error to a ramp as lim t e t = lim s sE ( s ) = lim s s s 2 + 2 s s 2 + 2 s + 1 1 s 2 = 1 . We need to increase the low-frequency gain via a lag compensator without affecting the closed-loop poles very much. Since we need an increase in gain by a factor of five to get our error down to 0.2, lets try C ( s ) = s +0 . 1 s +0 . 02 . This has the low-frequency gain of five and has a pole and zero which effectively cancel out when s is as big as- 1 j . With this new choice of compensator, we plot the root locus of C ( s ) P ( s ) = s + 0 . 1 s + 0 . 02 1 s ( s + 2) = s + 0 . 1 s 3 + 2 . 02 s 2 + 0 . 04 s . This yields Note that the gain for poles close to s =- 1 + j is given by K =- s 3 + 2 . 02 s 2 + 0 . 04 s s + 0 . 1 s =- 1+ j = 1 . 9900- . 0101 j 2 . 2.5 2 1.5 1 0.5 0.5 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Root Locus Real Axis Imaginary Axis Figure 1: Root locus with lag compensator. Question 2....
View Full Document

Page1 / 7

Homework3solutions_mae143b - MAE143B Homework Set 3 -...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online