Chapter 6: Root Locus

Algebra/calculus review items

• Rational functions

• Limits of rational functions

• Polynomial long division

• Binomial theorem

Simple example 1• Unity Gain negative feedback system

• Gp(s) =1/(s+2)

• Gc(s)=K

• OL TF: L(s)=Gp(s)Gc(s)=K/(s+2)

– Plot OL poles and zeros in s-plane

• CL TF: T(s)=K/(s+2+K)– Plot CL poles and zeros in s-plane as K changes

• Make observations

Simple example 2• Unity Gain negative feedback system

• Gp(s) =(s+1)/[s(s+2)]

• Gc(s)=K

• OL TF: L(s)=Gp(s)Gc(s)=K(s+1)/[s(s+2)]

– Plot OL poles and zeros in s-plane

• CL TF: T(s)=K(s-1)/(s2+(2+K)s+K)– Plot CL poles and zeros in s-plane as K changes

• Make observations

Simple example 3• Unity Gain negative feedback system

• Gp(s) =1/[s(s+5)]

• Gc(s)=K

• OL TF: L(s)=Gp(s)Gc(s)=K/[s(s+5)]

– Plot OL poles and zeros in s-plane

• CL TF: T(s)=K/(s2+5s+K)– Plot CL poles and zeros in s-plane as K changes

• Make observations

• Learning from simple examples.

• Going beyond simple examples.

Basic RL Facts:Consider standard negative gain unity feedback systemTR(s) = L(s)/[1+L(s)], S(s) = 1/[1+L(s)], L=GCG, L=GH, etcCharacteristic equation 1+L(s) = 0

For any point s on the root locusL(s) = -1=1e+/-j(2k+1)180°

|L(s)|=1 magnitude criterionarg(L(s)) = +/- (2k+1)180° angle criterionAngle and magnitude criterion useful in constructing RL

RL is set of all roots (= locus of roots) of the characteristic equation (= poles of closed loop system)OL poles (zeros) are poles (zeros) of L(s)CL poles are poles of TR(s), or S(s), …Closed loop poles start at OL poles (=poles of L(s)) when K=0Closed loop poles end at OL zeros (=zeros of L(s)) when K infinityStable CL systems have all poles in LHP (no poles in RHP)

Outline

•Graphical RL construction•Mathematical common knowledge•Motivational Examples•Summary of RL construction Rules•Matlab/Sysquake & RL•Assignments

Pole-Zero Form of L(s)

2

1

31

( )( )

( )( )( )s p

K s

sss

p

zL

p

Examples?

1 2

1

3

( )1

( ( ))1

)3

(2

( )( )

( )( )( )

j s z

j j sp p ps j s

K s z eL s

s p ep s es e p

For any point s in the s-Plane, (s+z) or (s+p) can be expressed in polar form (magnitude and angle, Euler identity)

2

1

31

( )( )

( )( )( )s p

K s

sss

p

zL

p

211 3( ) ( ( ( ))) ( )s p s pL s s z s p

For use with magnitude criterion

For use with angle criterion

Graphical representation/determination.

Mathematical Common Knowledge2

4 3 25 5

3

3

( )lim lims s

K s bs cs d K

s es fs gs hs k s

11( ) n nn n ns a na ss nas a

Binomial theorem

Polynomial long division 2

2

2

2

3

2

3 2

2

2 4

( )

2 4 2

2 2

3 2

8

3

3 6

s s s s

s

s

s s

s

s

s

s

s s

Final remainder is lower order than divisor.

Remainder = 0 => you found a factor.

Example 1 ( )( 1)( 2)

KL s

s s s

1. RL on real axis. Apply angle criterion (AC) to various test pts on real axis.

2. RL asymptotes. 1. Angles. Apply AC to test point very far from origin,

approximate L(s) = K/sm-n

2. Center. Approximate L(s) = K/(s+c)m-n, c center3. RL Breakaway points. Find values of s on real axis so that

K = -1/L(s) is a maximum or minimum.4. RL intersects imaginary axis. R-H criterion, auxiliary equation.5. Complete RL plot (see Fig. 6-6, pg. 346).6. Design. Use RL plot to set damping ratio to .5.

Example 22

( 2)( )

2 3

K sL s

s s

1. Plot OL poles and zeros. Standard beginning.2. RL on real axis. Apply angle criterion (AC) to various test pts

on real axis.3. RL asymptotes.

1. Angles. Apply AC to test point very far from origin, approximate L(s) = K/sm-n

2. Center. Approximate L(s) = K/(s+c)m-n, c center4. RL Break-in points. Find values of s on real axis so that K =

-1/L(s) is a maximum or minimum.5. RL intersects imaginary axis. R-H criterion, auxiliary equation.6. Complete RL plot (see Fig. 6-6, pg. 346).7. Design. Use RL plot to set damping ratio to .5.

New featurs: Complex roots, break-in points, departure angles.

Root Locus Construction Rules

1. RL on real axis. To the left of an odd number of poles & zeros2. RL asymptotes.

1. Angles. +/- 180(2k+1)/(#poles - #zeros)2. Center. C = -[(sum of poles)-(sum of zeros)]/(#poles - # zeros)

3. RL Break-in points. K=-1/L(s), dK/ds =0, s’s on real axis portion of RL

4. RL intersects imaginary axis. R-H criterion, auxiliary equation.5. Other rules. We will use MatLab for details.

Matlab/Sysquake and RL

Chapter 6 Assignments

B 1, 2, 3, 4, 5, 10, 11

Root locus Big Picture• Characteristic equation: 1+KL(s)=0; L(s)=N(s)/D(s)

– N(s)=sm+bsm-1+…=(s+z1)…(s+zm); b=z1+…+zm

– D(s)=sn+asn-1+…=(s+p1)…(s+pn); a=p1+…+pn

• RL For various values of K>0 solve D(s)+KN(s)=0• Small K RL

– Real axis portion of RL– Choose a K, find the roots

• Large K RL– Poly long division yields approximation L(s)=1/(s+c)n-m

– binomial theorem relates c to a and b;– a determined by zeros; b determined by poles