Steady-state tracking & sys. typesG(s) C(s)+

-r(s) e y(s)

plant

1 01 1

1 1

0 0 0

0 1 0 1

0 1 2

TF from e to for r is

System type #factors of s in d(s) Type 0: 0, 0, /

Type 1: 0, 0, , 0, /

Type 1: , 0, 0,

e r ol

mm

n n N Nn N N

v a p

p a v

p v

G G

b s b s bs a s a s a s

Na K K K b a

a a K K K b a

a a a K K

0 2, /aK b a

controller0

0

2

0

1lim ( ); to step1

1lim ( ); to ramp

1lim ( ); to acc

p e r sssp

v e r sssv

a e r sssa

K G s eK

K sG s eK

K s G s eK

1 2

20

20 0

TF from to : ( ) ( ); TF from to : ( ) ( );

( ) 1; if is step: 1 / 2 ( ) 1 / (0) (0)

1 1if is ramp: ; if is acc: (0) | (0) |

e d d e

sse B e Bs

e B s e B s

e d G s G s d e G s G s

sd se dG G s G G

d dsG s G

G1(s)+

-r(s) e

G2(s)

d(s)

AB

y(s)Type w.r.t. d

1# factor in ( ).e dG ss

Type 0 rejects nothing, has finite to step d, to ramp or acc dType 1 rejects const d, has finite to ramp d, has to acc dType 2 rejects step or ramp d, has finite

ss ss

ss ss

ss

e ee e

e

to acc d

The root locus technique1. Obtain closed-loop TF and char eq d(s) = 02. Re-arrange to get 3. Mark zeros with “o” and poles with “x”4. High light segments of x-axis and put arrows5. Decide #asymptotes, their angles, and x-axis meeting

place:

6. Determine jw-axis crossing using Routh table7. Compute breakaway:8. Departure/arrival angle:

01)()(

1

1 sdsnK

mnzerospoles

)(/)();()()()( 11'11

'11 sdsnKsdsnsnsd

k

kk

kp ppanglezpanglem )()(

k

kk

kz pzanglezzanglem )()(

-8 -6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

Root Locus

Real Axis

Imag

inar

y Ax

is

rlocus([1 3], conv([1 2 2 0],[1 11 30]))

3 2 2

3 1 0( 2 2 ) 11 30

sKs s s s s

Example: motor control

The closed-loop T.F. from θr to θ is:

PD

Plc KsKss

KsG22410

223..

What is the open-loop T.F.?The o.l. T.F. of the system is:

But for root locus, it depends on which parameter we are varying.

1. If KP varies, KD fixed, from char. poly.

sKssK

D

P

24102

23

PD KsKsssd 22410 23

012410

12 23 sKss

KD

P

The o.l. T.F. for KP-root-locus is the system o.l. T.F.

In general, this is the case whenever the parameter is multiplicative in the forward loop.

2. If KD is parameter, KP is fixedFrom

3 22 1 0

10 4 2DPK

G s

sKs s s K

022410 23 PD KsKsssd

What if neither is fixed?• Multi-parameter root locus?

– Some books do this• Additional specs to satisfy?

– Yes, typically– Then use this to reduce freedom

• The o.l. T.F. of the system is:

• It is type 1, tracks step with 0 error• Suppose ess to ramp must be <=1

sKssK

D

P

24102

23

• Since ess to ramp = 1/Kv• Kv = lim_s->0 {sGol(s)} =2KP/(4+2KD)• Thus, ess=(4+2KD)/2KP• Design to just barely meet specs: ess=1• (4+2KD)/2KP = 1• 4+2KD=2KP

0110

220)22(10

02210

022410

23

23

23

23

sssK

Ksss

KsKsssd

KsKsssd

P

P

PP

PD

>> rlocus([2 2], [1 10 0 0]);>> grid;>> axis equal;

-20 -15 -10 -5 0 5 10-15

-10

-5

0

5

10

150.220.420.60.740.84

0.91

0.96

0.99

0.220.420.60.740.84

0.91

0.96

0.99

5101520

Root Locus

Real Axis (seconds -1)

Imag

inar

y Ax

is (s

econ

ds-1

)

The grid line rays correspond to zThe semi circles corresponds to wn

If there is a design specification: Mp <= 10 %,then we need z >= 0.6

Select data point tool, and click at a point on the root locus with z >= 0.6

-5 -4 -3 -2 -1 0

-1.5

-1

-0.5

0

0.5

1

1.50.250.50.680.80.880.94

0.975

0.994

0.250.50.680.80.880.94

0.975

0.994

12345

System: sysGain: 7.84Pole: -0.827 + 1.09iDamping: 0.603Overshoot (%): 9.29Frequency (rad/s): 1.37

Root Locus

Real Axis (seconds -1)

Imag

inar

y Ax

is (s

econ

ds-1)

Desired KP = 7.84Closed loop dominant pole at -0.827+-1.09iPole damping ratio is 0.603Undamped frequency is 1.37Estimated overshoot is 9.29%

• With KP = 7.84• Design specs for ess=1 required

4+2KD=2KP• KD=KP-2 = 5.84

3 2

3 2

10 4 2 2

10 15.68 15.68D Pd s s s K s K

s s s

>> roots([1 10 15.68 15.68])ans = -8.3464 + 0.0000i -0.8268 + 1.0932i -0.8268 - 1.0932i

• Closed loop TF is:

3 2

3 2

210 4 2 2

15.6810 15.68 15.68

Pcl

D P

cl

KG ss s K s K

G ss s s

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

System: sysTime (seconds): 3.01Amplitude: 1.09

Step Response

Time (seconds)

Ampl

itude

>>max(y)

ans =

1.0915

Actual Mp=9.15%

More examples1.

No finite zeros, o.l. poles: 0,-1,-2Real axis: are on R.L.Asymp: #: 3

021 Ksss

0121

1

sssK

3 21 11, 3 2n s d s s s s

0,1&2,

13

021

3,: e

-axis crossing:char. poly: Ksss 23 23

wj

21:3sKs 3:2

361 : Ks

Ks :0

60set 36 KK

633: 222 sKssAs 223 2,1

2 jss

2at axis- cross R.L. ,6at jjK w

-8 -6 -4 -2 0 2 4-5

-4

-3

-2

-1

0

1

2

3

4

50.220.420.60.740.84

0.91

0.96

0.99

0.220.420.60.740.84

0.91

0.96

0.99

2468

Root Locus

Real Axis

Imag

inar

y A

xis

>> rlocus(1,[1 3 2 0])>> grid>> axis equal

Example:

Real axis:(-2,0) seg. is on R.L.

0222 2 Kssss

43,

41,4Asymp# e

01222

12

ssssK

zeros no ,11 sn

22s2 21 ssssd

1120:poles j,-,-

2or 0for compute toneed No dep

For ,1 j

iidep

43

24

2from-

j-

1from

0from

2

jj

1 from up vert.going R.L. symmetry,By 1 fromdown vert.going R.L.

Break away point: sssssd 464 234

1

0 so 1 '11 sdsn

412124' 231 ssssd

1334 23 sss

314 s

poly. char.for root quadruple isit

0'for root tripleis 1 1 sds

out.split and 1at meet branches R.L. 4 s

each. 45 into 360 dividing

out, 4 andin branches 4

-axis crossing:char. poly: Kssss 464 234

wj

113 44:s

21

2 2: KKs

21:1 Ks

2:0 Ks

202

1set KK

012

: 222 sKssAs js

jjK at axis- cross R.L. ,2at w

Ks 61:4

-4 -3 -2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.96

0.99

0.220.420.60.740.84

0.91

0.96

0.99

0.511.522.533.54

0.220.420.60.740.84

0.91

Root Locus

Real Axis

Imag

inar

y A

xis

>> rlocus(1, conv([1 2 0], [1 2 2]))>> axis equal>> sgrid

Example: in prev. ex., change s+2 to s+3

01223

12

ssssK

45

4113

43,

4: e

j 1,3,0:poleszeros no

R.L.on is seg 1,3:axis Real

4 :Asymptotes #

need. no :3,0for depjdep 1for

i-dep

43

221tan 1-

21tan

41-

symmetry.by of neg is 1for jdep

0 :pointsaway Break sd

0616154 23 ssssd

R.L.on 6.2at solution One?

s

-axis crossing:char. poly:

Kssss 685 234

wj

65:3s

Ks5

34:2

534

534

1 56:

Ks

Ks :0

55634

534 056set

KK

56

556342

5342 0: jsss

56

25634 at axis- cross R.L. ,at jjK w

Ks 81:4

-10 -8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

80.220.420.60.740.84

0.91

0.96

0.99

0.220.420.60.740.84

0.91

0.96

0.99

246810

Root Locus

Real Axis

Imag

inar

y A

xis

>> rlocus(1, conv([1 3 0], [1 2 2]))>> axis equal>> sgrid