Chapter 8 State Space Analysis

8/22/2019 Chapter 8 State Space Analysis

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Chapter 8 (page# 535)

State Space Analysis

Chapter 8 (page# 535)

State Space Analysis

Reading assignment:

Section 8.2 (page 536 to 545)

8.2.5 (page 551 to 554)


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Why State-Variable ModelsWhy State-Variable Models

Computer-aided analysis works better forstate models than

the transfer function approach.

State variable model provide more internal information about

the plant, allowing more complete control.

Optimal (best) design procedures are mostly based on theuse of state-variable models.

The state variable models are required fordigital simulation.

System State DefinitionSystem State DefinitionThe state of a system is defined as

The state of a system at any time t0 is the amount of information

at t0 that, together with all inputs for t t0, uniquely defines the

behavior of the system for all t

t0.


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State Equations Standard Formtate Equations tandard Form

)()()()()()(

tttttt

DuCxyBuAxx

+=

+=

State equationOutput equation

State vectorSystem matrix

u(t) = input vector = (n 1) vector of input functions

y(t) = output vector = (p 1) vector of defined outputs.

is the time derivative ofx(t) x(t) is the state vector, an (n 1) vector of the states A is the system matrix, an (n n) matrix of the coefficients

Bis an (n r) input matrix where ris the number of inputs C= (p n) output matrix D= (p r) matrix representing direct coupling between the

input and output. (Neglected)

)t(x


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Cxy

BuAxx

=

+=

=s1B CU

x

A

Y

X

+

+

responseCxy

solutionBuAxx

=

+=

State Equation

InitialC

onditions


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Example 2 RLC CircuitExample 2 RLC Circuit

+

vOUT

_

v1

i1

Reference node

v2

+v

_

i+vIN_

R L

C

v1 v2

The system has two state variables, the inductor current (i) andthe capacitor voltage (v).

We can obtain the system equations by use of 1) mesh

analysis, 2) nodal analysis, or3) mixed (both) analysis. We would

nor-mally choose mesh analysis since there is only one mesh;

how-ever, we have a new constraint avoiding integrals.

The results of each approach are given on the following page.


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System EquationsSystem Equations

=

=+++

t

OUT

t

IN

idtC

v

idtCdt

diLiRv

Mixed

0

0

1

01

01

01

21

02

021

1

=+

=+

dt

dvCdt)vv(

L

dt)vv(LR

vv

AnalysisNodal

t

tIN

0

0

=+

=+++

dt

dvCi

vdt

diLiRv

AnalysisMesh

IN

All three results are valid models but only the equations obtainedfrom the mixed analysis can be represented as state equations in

the standard form. The state equations are given below:

C

i

dt

dv

vLvLiL

R

dt

diIN

=

+=

11

This form is frequently used in digital simulation.


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State EquationsState Equations

vv

vLvi

C

LL

R

vi

OUT

IN

CC

=

+

=

0

1

01

1

The state equations, in standard form, for the series RLC circuit

are:


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Solution Using TransformsSolution Using Transforms

)866.05.0)(866.05.0(

1

1

)(

)( 2 jsjssLC

sL

Rs LC

sV

sV

IN

OUT+++=++=

The solution forvOUT

(t) will have the form

tjtj

OUT eVeVVtv)866.05.0(

2

)866.05.0(

10)(+

++=The constants V

0, V

1, and V

2are evaluated using partial fractions:

1)866.05.0)(866.05.0(

10 =

+=

jjV

,

2887.05.0)732.1)(866.05.0(

11 j

jjV +=

=

2887.05.0)732.1)(866.05.0(

12 j

jjV ==

)866.0sin(5774.0)866.0cos(1)( 5.05.0 tetetv ttOUT =

vOUT(t) can be obtained by taking the inverse:


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IN

CCvLv

i

C

LL

R

v

i

+

=

0

1

01

1

+=

=

sC

LL

Rs

C

LL

R

1

1

01

1

10

01AI

Formal SolutionFormal SolutionWe may use the standard form directly:

The characteristic equation for a

set of state equations is given by|I A| where A is the system

matrix, I is the identity matrix,

and the values are the

eigenvalues. The matrix |I

A|is given by

Taking the determinant of this matrix yields 012 =++=LCL

RAI

This agrees with the previous results (the denominator of the

trans-form).


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)(2

2

tfKydt

dyB

dt

ydM =++

)(2

2

tfKydt

dyB

dt

ydM +=

KBsMs

sG

++

=2

1)(

ytxLet =)(1

)()()(

)(12

txdt

tdx

dt

tdytx ===

2

2

22

)()(

dt

tdy

dt

tdxx ==

21 xx =

fKxBxxM += 122

M

fx

M

Kx

M

Bx += 122

[ ]xy

fM

x

M

B

M

kx

x

01

1

010

2

1

=

+

=

State variable Modeling


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23112111 ukuykyky +=++

1615242

ukykyky =++

Example: Consider the system described by the coupled differential equations

u1 & u2 are inputs, y1 & y2 are outputs. ki ;i=1,.6 are system parameters.

23

112

11

yx

xyx

yxLet

=

==

=

23121122 ukuxkxkx ++=

1634253 ukxkxkx +=

32

11

xy

xy

equationoutput

=

=

xy

f

k

kx

kk

kk

x

x

x

=

+

=

100

001

0

1

00

0

0

010

6

3

45

12

3

2

1


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Simulation Diagram

[ ] ;x)t(y

)t(u)t(x)t(xGiven

14

1

1

28

10

=

+

=


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Control canonical (Controllable)

&

Observer Canonical (Observable)

Matrix A,B & C are constructed from the transfer

function.

Next - Examples


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Next slides consider general standard Transfer Function


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2

2 1 0

3 2

2 1 0

( )

( )

Y s b s b s b

U s s a s a s a

+ +=

+ + +

2 1 0

2 3

2 1 0

2 3

( )

( ) 1

b b bY s s s s

a a aU s

s s s

+ +=

+ + +

Dividing each term by the highest order ofs yields

Control canonical form block diagram.

[ ] [ ]

1 1

2 2

3 0 1 2 3

1

0 1 2 2

3

0 1 0 0

0 0 1 01

0

x x

x x ux a a a x

x

y b b b x u

x

= +

= +

&

&&

GH

G

U

Y

+==

1

is canonical form

E l


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2 1 02 2 3 2 3

3 2

2 1 02 3 2 3

2 8 6( ) 2 8 6

8 26 6( ) 8 26 6 1 1

b b bC s s s s s s s s s

a a aR s s s s

s s s s s s

+ + + ++ +

= = =

+ + + + + + + + +The form is:

[ ]

1 1

2 2

3 3

1

2

3

0 1 0 0

0 0 1 0

6 26 8 1

6 8 2 [0] .

x x

x x r

x x

x

y x r

x

= +

= +

&

&

&

System controllable but not observable. Why?

Ans: State model is based on controllable canonical form, this

can be confirmed by another method later in the next lecture.

is controllable canonical form

Example


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Observer canonical form block diagram.

2

2 1 0

3 2

2 1 0

( )

( )

Y s b s b s b

U s s a s a s a

+ +=

+ + +

2 1 0

2 3

2 1 0

2 3

( )

( ) 1

b b bY s s s s

a a aU s

s s s

+ +=

+ + + GH

G

U

Y

+==

1

[ ]001

00

10

01

2

1

0

3

2

1

0

1

2

3

2

1

=

+

=

Y

u

b

b

b

X

X

X

a

a

a

X

X

X

called observable canonical


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[ ]001

00

10

01

2

1

0

3

2

1

0

1

2

3

2

1

=

+

=

Y

u

b

b

b

X

X

X

a

a

a

X

X

X

[ ]210

3

2

1

2103

2

1

1

0

0

100

010

bbbY

u

X

X

X

aaaX

X

X

=

+

=

observable but not controllableControllable but not observable

called observable canonicalcalled controllable canonical


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3

1

34

1

31

12 +

=++

+=

++

+=

sss

s

)s)(s(

s)s(G

Controllable canonical

[ ]

=

+

=

2

111

1

0

43

10

x

xy

uxx

Observable canonical

[ ]

=

+

=

2

110

1

1

41

30

x

xy

uxx

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Chapter 8 State Space Analysis