LINEAR CONTROL SYSTEMSprofsite.um.ac.ir/~karimpor/control/lectures/lec4_lcs.pdf · lecture 4 Dr....

LINEAR CONTROL

SYSTEMS

Ali Karimpour

Associate Professor

Ferdowsi University of Mashhad

lecture 4

Dr. Ali Karimpour Feb 2013

2

Lecture 4

Converting of different representations of

control systems

Topics to be covered include:

Mason’s flow graph loop rule (Converting SD to TF)

Realization (Converting TF to SS model)

Converting high order differential equation to state space

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3

Different representations

ومايشهای مختلف

1

212

21

54

xc

rxxx

xx

SS

SD

TF

45

1

)(

)()(

2

sssr

scsG

HODE

rccc 45

Discussed in the last lecture

Will be discussed in this lecture

Mason’s rule

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4

Mason’s flow graph loop rule

فرمًل بهرٌ ميسًن

SD

TF

45

1

)(

)()(

2

sssr

scsG

Mason’s rule

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5

Mason’s flow graph lop rule

قاوًن گيه ميسًن

m

i

iiM

su

sy 1

)(

)(

outputtoinputfrompathofnumbertheism

.....13

3

2

2

1

1 m

m

m

m

m

m PPP

pathofgain th

i iM

systemofgainloop

pathdeletingaftersystemofgainloop th

i i

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6

Example 1 1مثال

1m

2121 451.......00)45(1 ssss

45

1

451

1.

)(

)(221

2

ssss

s

sr

sc Note

m

i

iiM

sr

sc 1

)(

)(

2

1

sM 11

.....13

3

2

2

1

1

m

m

m

m

m

m PPP

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7

Example 2 2مثال

1411 11

1

HGM

m

262142543226241

262142543226241

1

))(()(1

HGGHGHGGGGHGGGH

HGGHGHGGGGHGGGH

262142543226241

41

1

)1.(1

)(

)(

HGGHGHGGGGHGGGH

GH

sN

sC

?)(

)(

sN

sC

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8

Example 3 3مثال

Find the TF model for the following state diagram.

)(

)(

sr

sc c.1 + bs-1.1

1+as-1

S-1

-a

b r(s) c(s)

1

c

as

bcs

This is a base form for order 1 transfer function.

است 1ايه يک حالت پايٍ برای تابع اوتقال درجٍ

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9

Example 3 3مثال

Find the TF model for following state diagram.

This is a base form for order 2 transfer function.

است 2ايه يک حالت پايٍ برای تابع اوتقال درجٍ

cass

bds

csas

bsds

sr

sc

221

21

1)(

)(

S-1

-a

r(s) c(s) b 1

d

S-1

-c

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10

Realization پيادٌ سازی

1

212

21

54

xc

rxxx

xx

SS TF

45

1)(

2

sssG

Realization

SD

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11

Realization پيادٌ سازی

Some Realization methods چىذ ريش پيادٌ سازی

1- Direct Realization.

2- Series Realization.

3- Parallel Realization.

پياده سازی مستقيم -1

پياده سازی سری -2

پياده سازی موازی -3

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12

Direct Realization مستقيم پيادٌ سازی

254

127

)(

)()(

23

2

sss

ss

sr

scsG

X

X

sss

sss321

321

2541

127

XsXsXssrXsrsssX 321321 254)()()2541(

)(127 321 scXsXsXs

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13

Direct Realization مستقيم پيادٌ سازی

XsXsXssrX 321 254)(

c(s) 12

-4

S-1

r(s) 1 X

-2

S-1 S-1

-5

XsXsXssc 123 712)(

7

1

x1 x2 x3

21 xx

32 xx

rxxxx 3213 452

321 712 xxxc

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14

Series Realization سری پيادٌ سازی

254

127

)(

)()(

23

2

sss

ss

sr

scsG

2

1.

1

4.

1

3

ss

s

s

s

as

bcs

S-1

-a

b r(s) c(s)

1

c

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15

Series Realization

254

127

)(

)()(

23

2

sss

ss

sr

scsG

سری پيادٌ سازی

2

1.

1

4.

1

3

ss

s

s

s

as

bcs

S-1

-a

b 1

c

S-1

-1

3 1

1

S-1

-1

4 1

1

S-1

-2

1 1

r(s) c(s) x1 x2 x3

rxxxxxx 332211 342

rxxxx 3322 3

rxx 33

1xc

3

2

1

3

2

1

3

2

1

]001[

1

1

1

100

210

232

x

x

x

c

r

x

x

x

x

x

x

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16

Parallel Realization مًازی پيادٌ سازی

)3)(2)(1(

17132

6116

17132

)(

)()(

2

23

2

sss

ss

sss

ss

sr

scsG

3

?

2

?

1

?

sss

3 1 -2

as

bcs

S-1

-a

b 1

c

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Parallel Realization مًازی پيادٌ سازی

3

?

2

?

1

?)(

ssssG

3 1 -2

as

bcs

S-1

-a

b 1

c

S-1

-2

1 1 -3

S-1 -2 1

S-1

-1 3 1

r(s) c(s)

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Parallel Realization

rxx 22 3

پيادٌ سازی مًازی

3

?

2

?

1

?)(

ssssG

3 1 -2 rxx 11

rxx 33 2

xc

uxx

]123[

1

1

1

200

030

001

321 23 xxxc x1

x2

x3

S-1

-2

1 1 -3

S-1 -2 1

S-1

-1 3 1

r(s) c(s)

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Parallel Realization

)2()1(

127

254

127

)(

)()(

2

2

23

2

ss

ss

sss

ss

sr

scsG

مًازی پيادٌ سازی

2

?

1

?

)1(

?2

sss

6 2 -1

S-1

-2

2

1

c(s)

x2 x1

x3

211 xxx

rxx 22

rxx 33 2

321 216 xxxc xc

rxx

]216[

1

1

0

200

010

011

S-1

-1 6

1 S-1

-1

1

1

r(s) c(s)

-1

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Converting high order differential equation to state

space

1

212

21

54

xc

rxxx

xx

SS

TF

45

1)(

2

sssG

HODE

rxxx 111 45

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Exercises

4-1 Find Y(s)/U(s) for following system

Ans

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Exercises (Continue)

4-2 Find c1(s)/r2(s) for following system

4-3 Find g such that C(s)/R(s) for following system is

(2s+2)/s+2

Answer

Answer

21

2

2

2

1

kks

k

r

c

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4-4 Find Y1(s)/R(s) for following system

Answer

4-5 Find C1(s)/R1(s) and C2(s)/R2(s) for following system

Answer

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4-6 Find C/R for following system

Answer

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4-7 Find the SS model for following system.

uuyyy 65

It was in some exams.

4-8 Find the TF model for following system without any inverse

manipulation.

xc

uxx

]001[

1

0

0

500

020

131

4-9 Find the SS model for following system.

G(s)

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4-10 Find direct realization, series realization and parallel realization

for following system.

)3)(22(

1)(

2

sss

ssG

4-11 a) Find the transfer function of following system by Masson

formula.

b) Find the state space model of system.

c) Find the transfer function of system from the SS model in part b

d) Compare part “a” and part “c” S-1

-a -b

r(s) c(s) c 1

S-1

It appears in many exams.*****

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27


4-12 a) Find the transfer function of following system by Masson

formula.

b) Find the state space model of system.

c) Find the transfer function of system from the SS model in part b

d) Compare part “a” and part “c”

S-1

-a -b

r(s) c(s) c

1

1 S-1

Regional Electrical engineering Olympiad spring 2008.*****

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Appendix: Example 1 a) Draw the state diagram for the following

differential equation. b) Suppose c(t) as output and r(t) as input and find

transfer function.

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Appendix: Example 2: Express the following set of differential

equations in the form of and draw

corresponding state diagram.

)()()(

)()(3)(2

)(2)(

2213

1312

211

trtxtxx

trtxtxx

txtxx

)()()( tBrtAXtX

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Appendix: Example 3: Determine the transfer function of

following system without using any inverse manipulation.

)(2)(]001[)()(

1

0

0

)(

500

020

131

)( trtXtctrtXtX

)

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Appendix: Example 4: Determine the modal form of this

transfer function. One particular useful canonical form is

called the Modal Form.

It is a diagonal representation of the state-space model.

Assume for now that the transfer function has distinct

real poles pi (but this easily extends to the case with

complex poles.)

Now define a collection of first order systems, each

with state xi

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Which can be written as




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With :

Good representation to use for numerical robustness

reasons.

Avoids some of the large coefficients in the other 4

canonical forms.

LINEAR CONTROL SYSTEMSprofsite.um.ac.ir/~karimpor/control/lectures/lec4_lcs.pdf · lecture 4 Dr....

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