State Space Analysis Hany Ferdinando Dept. of Electrical Engineering Petra Christian University

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State Space Analysis

Hany FerdinandoDept. of Electrical Engineering

Petra Christian University

State Space 2 - Hany Ferdinando 2

OverviewState Transition MatrixTime ResponseDiscrete-time evaluation

State Space 2 - Hany Ferdinando 3

State Transition Matrix

The solution of

t

o

tt deet )()0()( )( Buxx AA

t

o

dttt )()()0()()( Buxxis

If the initial condition x(0), input u() and the state transition matrix (t) are known the time response of x(t) can be evaluated

State Space 2 - Hany Ferdinando 4

State Transition Matrix

The (t) is inverse Laplace Transform of (s) and 1)( AIss

When the input u(t) is zero, then

)0()()( xX ss

State Space 2 - Hany Ferdinando 5

State Transition Matrix

)0()()( xX ss

From the equation above we can expand the matrix into (for example, two elements)

)0(

)0(

)()(

)()(

)(

)(

2

1

2221

1211

2

1

x

x

ss

ss

sX

sX

State Space 2 - Hany Ferdinando 6

State Transition Matrix

The 11(s) can be evaluated from the relation between X1(s) and x1(0), the 12(s), 21(s) and

22(s) can be evaluated with the same procedure

)0(

)0(

)()(

)()(

)(

)(

2

1

2221

1211

2

1

x

x

ss

ss

sX

sX

State Space 2 - Hany Ferdinando 7

Time Response

It is the time response of X(t).

First, find (t) from (s). It is simply the inverse Laplace Transform of (s). Do the inverse Laplace Transform for each element of (s).

)0()()( xX tt

State Space 2 - Hany Ferdinando 8

Example

i(t)

LC

C itidt

dvCi )( LC

LL Riv

dt

diLu

State Space 2 - Hany Ferdinando 9

Example

LC

LC

iC

tiC

v

iC

tiCdt

dv

1)(

1

1)(

1

LCL

LCL

iL

RvL

i

iL

RvLdt

di

1

1

If x1 = vC and x2 = iL then

21

1)(

1x

Cti

Cx 212

1x

L

Rx

Lx

State Space 2 - Hany Ferdinando 10

Example

)(0

1

1

10

2

1

2

1 tiCx

x

L

R

L

Cx

x

)(0

2

31

20

2

1

2

1 tix

x

x

x

For R = 3, L = 1 and C = 0.5,

State Space 2 - Hany Ferdinando 11

Example

I(s) V(s)s-1 s-11/C

-1/C

1/L

-R/LX1(s) X2(s)

R

x1(0)/s x2(0)/s

State Space 2 - Hany Ferdinando 12

Example

s-1 s-1

-1/C

1/L

-R/LX1(s) X2(s)

x1(0)/s

x2(0)/s

When U(s) = 0

State Space 2 - Hany Ferdinando 13

Example11(s) is transfer function of X1(s)/x1(0). Here, use the Mason Gain Formula to get 11(s)

s

xs

sX

)0()(.1

)(

11

1

1

111

1

1 )(.1)(

)0(

)( sss

x

sX

State Space 2 - Hany Ferdinando 14

Example

1(s) is path cofactor of , is 1 + 3s-1 + 2s-2

1(s) = 1 + 3s-1

1

111

)(.1)(

sss

23

3

231

)31()(

221

11

11

ss

s

ss

sss

State Space 2 - Hany Ferdinando 15

Example

With the same procedures, find the 12(s), 21(s) and 22(s)!

2323

123

2

23

3

)(

22

22

ss

s

ss

ssss

s

s

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Example

31

2

s

ss AI

s

s

ssss

1

23

2)3(

1)( 1AI

2323

123

2

23

3

)(

22

22

ss

s

ss

ssss

s

s

State Space 2 - Hany Ferdinando 17

Example

tttt

tttt

eeee

eeeet

22

22

2

222)(

Then the X(t) can be calculated with

)0()()( xX tt

State Space 2 - Hany Ferdinando 18

Discrete-time Evaluation

For discrete-time, use the approximation

T

tTt )()( xxx

)()()()(

)()()()(

)()()()(

tTtTTt

tTttTTt

ttT

tTt

BuxIAx

BuxAxx

BuAxxx

State Space 2 - Hany Ferdinando 19

Discrete-time Evaluation

)()()()1(

)()()()1(

kTktk

kTTkTTTk

Buxx

BuxIAx

)()( IA Tt

State Space 2 - Hany Ferdinando 20

ExampleWith the same example above and T = 0.2s,

)(0

2

31

20

2

1

2

1 tix

x

x

x

)(2.0)()2.0()1( kkk BuxIAx

)(0

4.0

4.02.0

4.01

2

1

2

1 tix

x

x

x

State Space 2 - Hany Ferdinando 21

Matlab

Use function expm to calculate the (t)

A = [0 -2; 1 -3]; T = 0.2

psy = expm(A*T)