UNIT IVUNIT IV

State Variable Models.

OutlineOutline

• Linking state space representation and transferLinking state space representation and transfer function

• Phase variable canonical form• Phase variable canonical form • Input feedforward canonical form• Physical state variable model• Diagonal canonical formg• Jordan canonical form

Consider the following RLC circuitg

We can choose state variables to be ),(),( 21 tixtvx Lc ==We can choose state variables to beAlternatively, we may choose ).(ˆ),(ˆ 21 tvxtvx Lc ==

This will yield two different sets of state space equations, but both of them have the identical input-output relationship, expressed by )(0 RsVp yCan you derive this TF?

.1)(

)(2

0

++=

RCsLCssU

Linking state space representation and f f itransfer function

Given a transfer function there exist infinitelyGiven a transfer function, there exist infinitely many input-output equivalent state space models.We are interested in special formats of state spaceWe are interested in special formats of state space representation, known as canonical forms.It is useful to develop a graphical model thatIt is useful to develop a graphical model that relates the state space representation to the corresponding transfer function The graphicalcorresponding transfer function. The graphical model can be constructed in the form of signal-flow graph or block diagram.g p g

We recall Mason’s gain formula when all feedback loops are t hi d l t h ll f d thtouching and also touch all forward paths,

Δ ∑∑ kkk PP

gain loopfeedback of sum1gainpathforwardofSum

11

−=

−=

Δ

Δ=

∑

∑∑

=

N

qq

kk

kkk

L

PPT

Consider a 4th-order TF

40

012

23

34

0

)()()(

−

++++==

sb

asasasasb

sUsYsG

We notice the similarity between this TF and Mason’s gain f l b T t th t 4 t t

40

31

22

13

0

1 −−−− ++++=

sasasasasb

formula above. To represent the system, we use 4 state variables Why?.,,, 4321 xxxx

Signal‐flow graph modelSignal flow graph model

This 4th‐order system can be represented bys o de syste ca be ep ese ted by

4321

40

1)()()( −−−−

−

++++==

sasasasasb

sUsYsG

01231)( ++++ sasasasasU

How do you verify this signal‐flow graph by Mason’s gain formula?gain formula?

Block diagram modelBlock diagram modelAgain, this 4th‐order TF 

0)( bsY

4321

40

012

23

34

0

1

)()()(

−−−−

−

++++=

++++==

sasasasasb

asasasasb

sUsYsG

can be represented by the block diagram as 

01231 ++++ sasasasa

shown

With either the signal‐flow graph or block diagram of th i 4th d tthe previous 4th‐order system,

we define state variables as ,,,, 3423120

1 xxxxxxbyx &&& ====

then the state space representation is 21

xxxx

==

&

&

433221104

43

32

uxaxaxaxaxxxxx

+−−−−===

&

&

10

433221104

xby =

Writing in matrix formg

)()()()()()(tttttt

DuCxyBuAxx

+=+=&

we have

)()()(y

00010⎥⎤

⎢⎡

⎥⎤

⎢⎡

100

,10000100

3210⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ −−−−

=

aaaa

BA

[ ] 0,0000

3210

==⎦⎣⎦⎣

DbC

Let us consider a more general 4th‐order systemg y

23)( +++ bsbsbsbsY

40

31

22

13

012

23

34

0123

)()()(

−−−− +++=

+++++++

==

sbsbsbsb

asasasasbsbsbsb

sUsYsG

How do we construct the signal‐flow graph and block 

40

31

22

131 −−−− ++++ sasasasa

g g pdiagram using Mason’s gain formula?• forward paths (they have to touch all the loops)• feedback loops (all of them are touching)• integrators

For the 4th‐order TF 

One form of the signal flow graph and block

40

31

22

13

40

31

22

13

1)()()( −−−−

−−−−

+++++++

==sasasasa

sbsbsbsbsUsYsG

One form of the signal‐flow graph and block diagram is Phase variable canonical form

Phase variable canonical formPhase variable canonical form

23)( +++ bsbsbsbsY

4321

40

31

22

13

012

23

34

0123

1

)()()(

−−−−

−−−−

+++++++

=

+++++++

==

sasasasasbsbsbsb

asasasasbsbsbsb

sUsYsG

The state space equation developed from the above graph is

01231 ++++ sasasasa

p q p g p

with 43322110

433221104433221 ,,,xbxbxbxby

uxaxaxaxaxxxxxxx+++=

+−−−−==== &&&&

00010 ⎤⎡⎤⎡x1, x2, x3, x4 are called phase

1000

,100001000010

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

= BA

x1, x2, x3, x4 are called phase variables.

[ ] 0,1

3210

3210

==

⎥⎦

⎢⎣

⎥⎦

⎢⎣ −−−−

Dbbbbaaaa

C

012

23

34

012

23

3

)()()(

+++++++

==asasasas

bsbsbsbsUsYsG

There is an alternative state space representation by feeding forward

40

31

22

13

40

31

22

13

0123

1

)(

−−−−

−−−−

+++++++

=

++++

sasasasasbsbsbsb

asasasassUp y ginput signal.

Input feedforward canonical formca o ca o

Input feedforward canonical formInput feedforward canonical form

23)( +++ bsbsbsbsY

4321

40

31

22

13

012

23

34

0123

1

)()()(

−−−−

−−−−

+++++++

=

+++++++

==

sasasasasbsbsbsb

asasasasbsbsbsb

sUsYsG

The state space equation representing the above graph is 

01231 ++++ sasasasa

p q p g g p

i h1

0104141132312232131 ,,,xy

ubxaxubxxaxubxxaxubxxax=

+−=++−=++−=++−= &&&&

with

,100010001

1

2

3

1

2

3

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−

=bbb

aaa

BA

[ ] 0,0001

000 0

1

0

==

⎥⎦

⎢⎣

⎥⎦

⎢⎣−

D

ba

C

When studying an actual control system block diagram, we wish to select the physical variables as state variables.  For example, the block p y p

diagram of an open loop DC motor is

1

1

5155

−

−

++

ss

1

1

21 −

−

+ ss

1

1

316

−

−

+ ss

We draw the signal‐flow diagraph of each block separately and then connect them. We select x1=y(t), x2=i(t) and x3=(1/4)r(t)‐(1/20)u(t) to form the state space 1 y 2 3 prepresentation.

Physical state variable modelPhysical state variable model

The corresponding state space equation is

xx )(150

5002020063

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−= tr&

x]001[1500

=

⎥⎦⎢⎣⎥⎦⎢⎣y

We revisit the block diagram model of the open loop DC motor.   

325)3)(2)(5()1(30

)()( 321

++

++

+=

++++

=sk

sk

sk

ssss

sRsYThe overall TF is

Distinct poleswhere k1=‐20, k2=‐10, k3=30.  If we choose state variables associated with distinct poles, we can build a ‘decoupled’ form of state space model.

Distinct poles

o o state space ode

Diagonal canonical formDiagonal canonical form

330

210

520

)3)(2)(5()1(30

)()(

++

+−

++

−=

++++

=ssssss

ssRsY

325)3)(2)(5()( ++++++ sssssssR

Distinct l

The state space equation for the above model is 1005 ⎤⎡⎤⎡−

poles

xx

]301020[

)(111

300020005

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−= tr&

x]301020[ −−=y

Jordan canonical formJordan canonical form

If t h lti l l th t t t tiIf a system has multiple poles, the state space representation can be written in a block diagonal form, known as Jordan canonical form.  For example,

Three poles are equal