Ch05Solns - 5000 Solutions Manual . Feedback Control of...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
5000 Solutions Manual . Feedback Control of Dynamic Systems . . Gene F. Franklin . J. David Powell . Abbas Emami-Naeini . . . . Assisted by: H.K. Aghajan H. Al-Rahmani P. Coulot P. Dankoski S. Everett R. Fuller T. Iwata V. Jones F. Safai L. Kobayashi H-T. Lee E. Thuriyasena M. Matsuoka
Background image of page 1

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

View Full DocumentRight Arrow Icon
Chapter 5 The Root-Locus Design Method Problems and solutions for Section 5.1 1. Set up the following characteristic equations in the form suited to Evans’s root-locus method. Give L ( s ) ,a ( s ) , and b ( s ) and the parameter, K, in terms of the original parameters in each case. Be sure to select K so that a ( s ) and b ( s ) are monic in each case and the degree of b ( s ) is not greater than that of a ( s ) . (a) s +(1 )=0 versus parameter τ (b) s 2 + cs + c +1=0 versus parameter c (c) ( s + c ) 3 + A ( Ts +1)=0 i. versus parameter A , ii. versus parameter T , iii. versus the parameter c , if possible. Say why you can or can not. Can a plot of the roots be drawn versus c for given constant values of A and T by any means at all (d) 1+[ k p + k I s + k D s τs +1 ] G ( s . Assume that G ( s )= A c ( s ) d ( s ) where c ( s ) and d ( s ) are monic polynomials with the degree of d ( s ) greater than that of c ( s ) . i. versus k p ii. versus k I iii. versus k D iv. versus τ 5001
Background image of page 2
5002 CHAPTER 5. THE ROOT-LOCUS DESIGN METHOD Solution: (a) K =1 ; a = s ; b (b) K = c ; a = s 2 +1; b = s +1 (c) Part (c) i. K = AT ; a =( s + c ) 3 ; b = s /T ii. K = AT ; a s + c ) 3 + A ; b = s iii. The parameter c enters the equation in a nonlinear way and a standard root locus does not apply. However, using a polynomial solver, the roots can be plotted versus c. (d) Part (d) i. K = k p ; a = s ( s ) d ( s )+ k I ( s ) c ( s k D τ s 2 Ac ( s ); b = s ( s ) c ( s ) ii. K = Ak I ; a = s ( s ) d ( s Ak p s ( s k D τ s 2 Ac ( s ); b = s ( s ) c ( s ) iii. K = Ak D τ ; a = s ( s ) d ( s Ak p s ( s ) c ( s Ak I ( s + 1 ) c ( s ); b = s 2 c ( s ) iv. K ; a = s 2 d ( s k p As 2 c ( s k I Asc ( s ); b = sd ( s k p sAc ( s k I Ac ( s k D s 2 Ac ( s )
Background image of page 3

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

View Full DocumentRight Arrow Icon
5003 Problems and solutions for Section 5.2 2. Roughly sketch the root loci for the pole-zero maps as shown in Fig. 5.48. Show your estimates of the center and angles of the asymptotes, a rough evaluation of arrival and departure angles for complex poles and zeros, and the loci for positive values of the parameter K . Each pole-zero map is from a characteristic equation of the form 1+ K b ( s ) a ( s ) =0 , where the roots of the numerator b ( s ) are shown as small circles o and the roots of the denominator a ( s ) are shown as × 0 s on the s -plane. Note that in Fig. 5.48(c), there are two poles at the origin. Solution: Root Locus Real Axis Imag Axis -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 Root Locus Real Axis -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 Root Locus Real Axis -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 Root Locus Real Axis -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 Root Locus Real Axis -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 Root Locus Real Axis -6 -4 -2 0 2 -3 -2 -1 0 1 2 3 Figure 5.48: Pole-zero maps (a) a ( s )= s 2 + s ; b ( s s +1 Breakin(s) -3.43; Breakaway(s) -0.586 (b) a ( s s 2 +0 . 2 s +1; b ( s s
Background image of page 4
5004 CHAPTER 5. THE ROOT-LOCUS DESIGN METHOD Angle of departure: 135.7 Breakin(s) -4.97 (c) a ( s )= s 2 ; b ( s )=( s +1) Breakin(s) -2 (d) a ( s s 2 +5 s +6; b ( s s 2 + s Breakin(s) -2.37 Breakaway(s) -0.634 (e) a ( s s 3 +3 s 2 +4 s 8 Center of asymptotes -1 Angles of asymptotes ± 60 , 180 Angle of departure: -56.3 (f) a ( s s 3 s 2 + s 5; b ( s s +1 Center of asymptotes -.667 Angles of asymptotes ± 60 , 180
Background image of page 5

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

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 78

Ch05Solns - 5000 Solutions Manual . Feedback Control of...

This preview shows document pages 1 - 6. Sign up to view the full document.

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