{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Quiz_6_sol

# Quiz_6_sol - Solution The part between-2 and 0(2.5 pts Find...

This preview shows pages 1–2. Sign up to view the full content.

ECE 483 Quiz # 6 Solution Suppose a feedback sampled-data control system has the following characteristic equation: 1 + Kz ( z + 2)( z 2 + 0 . 25) = 0 , where K 0 is an adjustable parameter. (10 pts) Find the range of K 0 so that the closed-loop system is stable. Solution: We use the Jury’s test. The characteristic equation is equivalent to Q ( z ) = ( z + 2)( z 2 + 0 . 25) + Kz = z 3 + 2 z 2 + ( K + 0 . 25) z + 0 . 5 = 0 . We apply the Jury’s test by constructing the array: z 0 z 1 z 2 z 3 0.5 K + 0 . 25 2 1 1 2 K + 0 . 25 0.5 -0.75 0.5K-1.875 0.75-K For the system to be stable, we must have: Q (1) = K + 3 . 75 > 0 K > - 3 . 75 ( - 1) 3 Q ( - 1) = K - 1 . 25 > 0 K > 1 . 25 | a 0 | = 0 . 5 < a 3 = 1 | b 0 | = 0 . 75 > | b 2 | = | 0 . 75 - K | 0 < K < 1 . 5 . To sum up, for the closed-loop system to be stable, we must have 1 . 25 < K < 1 . 5. (2.5 pts) Find the open-loop zeros and poles of the system and plot them on the complex plane. Solution: The only open-loop zero is 0 and the open-loop poles are - 2 and ± j 0 . 5. (2.5 pts) Find the parts of the real axis on the root locus, and mark them on the above plot.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solution: The part between-2 and 0. (2.5 pts) Find the asymptotes of the root locus (their center and angles) and mark them on the above plot. Solution: There are two asymptotes diverging to in±nity along ± 90 ◦ angles, with the center [(-2 + j . 5-j . 5)-0] / (3-1) =-1. (2.5 pts) Determine if z = j 2 is on the root locus. If not, what is the angle of de±ciency φ ? Solution: The angle of di±ciency is φ = 180 ◦-n G ( j 2) =-45 ◦ , which is not 0. Therefore, j 2 is not on the root locus. See the plot next page for the root locus by Matlab. 1-3-2-1 1 2 3-2-1.5-1-0.5 0.5 1 1.5 2 2.5 Root Locus Real Axis Imaginary Axis 2...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Quiz_6_sol - Solution The part between-2 and 0(2.5 pts Find...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online