Feedback Control Systems (FCS)

Dr. Imtiaz Hussainemail: imtiaz.hussain@faculty.muet.edu.pk

URL :http://imtiazhussainkalwar.weebly.com/

Lecture-36-37Transfer Matrix and solution of state equations

Transfer Matrix (State Space to T.F)• Now Let us convert a space model to a transfer function model.

• Taking Laplace transform of equation (1) and (2) considering initial conditions to zero.

• From equation (3)

)()()( tButAxtx (1)

)()()( tDutCxty (2)

)()()( sBUsAXssX (3)

)()()( sDUsCXsY (4)

)()()( sBUsAXssX

)()()( sBUsXAsI

)()()( 1 sBUAsIsX (5)

Transfer Matrix (State Space to T.F)• Substituting equation (5) into equation (4) yields

)()()()( 1 sDUsBUAsICsY

)()()( 1 sUDBAsICsY

DBAsICsUsY

1)()()(

Example#1

• Convert the following State Space Model to Transfer Function Model if K=3, B=1 and M=10;

)(tfMv

x

MB

MK

vx

1010

vx

ty 10)(

Example#1

• Substitute the given values and obtain A, B, C and D matrices.

)(1010

101

103

10tf

vx

vx

vx

ty 10)(

Example#1

101

103

10A

10C

1010

B

0D

DBAsICsUsY

1)()()(

Example#1

101

103

10A

10C

1010

B

0D

1010

101

103

10

00

10)()(

1

ss

sUsY

Example#1

1010

101

103

10

00

10)()(

1

ss

sUsY

1010

101

103

110

)()(

1

ss

sUsY

1010

103

1101

103)

101(

110)()(

s

s

sssUsY

Example#1

1010

103

1101

103)

101(

110)()(

s

s

sssUsY

1010

103

103)

101(

1)()( s

sssUsY

10103)

101(

1)()( s

sssUsY

Example#1

10103)

101(

1)()( s

sssUsY

3)110()()(

sss

sUsY

Example#2

• Obtain the transfer function T(s) from following state space representation.

Answer

Forced and Unforced Response

• Forced Response, with u(t) as forcing function

• Unforced Response (response due to initial conditions)

)(tubb

xx

aaaa

xx

2

1

2

1

2221

1211

2

1

)()(00

2

1

2221

1211

2

1

xx

aaaa

xx

Solution of State Equations• Consider the state equation given below

• Taking Laplace transform of the equation (1))()( tAxtx (1)

)()0()( sAXxssX

)0()()( xsAXssX

)0()( xsXAsI

)0()( 1xAsIsX

)0(1)( xAsI

sX

Solution of State Equations

• Taking inverse Laplace

)0(1)( xAsI

sX

)0()( xetx At

Atet )( State Transition Matrix

Example-3• Consider RLC Circuit obtain the state transition matrix ɸ(t).

Vc

+

-

+

-

VoiL

)(0

1

1

10tuCi

v

LR

L

Civ

L

c

L

c

5013 ., CandLR

)(tuiv

iv

L

c

L

c

02

3120

Example-3 (cont...)

)(tuiv

iv

L

c

L

c

02

3120

1

111

3120

00

])[()(S

SASIt

))(())((

))(())(()(2121

121

221

31

SSS

SS

SSSSS

t

• State transition matrix can be obtained as

• Which is further simplified as

Example-3 (cont...)

))(())((

))(())(()(2121

121

221

31

SSS

SS

SSSSS

t

• Taking the inverse Laplace transform of each element

)()()()()(

tttt

tttt

eeeeeeee

t22

22

2222

Example#4• Compute the state transition matrix if

300020001

A

])[()( 11 ASIt

Solution

State Space Trajectories• The unforced response of a system released from any initial

point x(to) traces a curve or trajectory in state space, with time t as an implicit function along the trajectory.

• Unforced system’s response depend upon initial conditions.

• Response due to initial conditions can be obtained as

)()( tAxtx

)()()( 0xttx

State Transition• Any point P in state space represents the state of the system

at a specific time t.

• State transitions provide complete picture of the system

1x

2xP(x1, x2)

1x

2xt0

t1

t2

t3

t4t5

t6

Example-5• For the RLC circuit of example-3 draw the state space trajectory

with following initial conditions.

• Solution

21

00)()(

L

c

iv

21

)2()()22()2(

22

22

tttt

tttt

L

c

eeeeeeee

iv

)()()( 0xttx

tt

tt

L

c

eeee

iv

2

2

333

ttL

ttc

eei

eev2

2

3

33

Example-5 (cont...)• Following trajectory is obtained

-1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

Vc

iLState Space Trajectory of RLC Circuit

t-------->inf

Example-5 (cont...)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vc

iLState Space Trajectories of RLC Circuit

01

01

10

10

Equilibrium Point• The equilibrium or stationary state of the system

is when0)(tx

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vc

iL

State Space Trajectories of RLC Circuit

Solution of State Equations• Consider the state equation with u(t) as forcing function

• Taking Laplace transform of the equation (1))()()( tButAxtx (1)

)()()0()( sBUsAXxssX

)()0()()( sBUxsAXssX

)()0()( sBUxsXAsI

AsIsBUxsX

)()0()(

Solution of State Equations

• Taking the inverse Laplace transform of above equation.

AsIsBUxsX

)()0()(

AsIsBU

AsIxsX

)()0()(

dtutxttxt

)()()0()()(0

Natural ResponseForced Response

Example#6• Obtain the time response of the following system:

• Where u(t) is unit step function occurring at t=0. consider x(0)=0.

)(10

3210

2

1

2

1 tuxx

xx

Solution

• Calculate the state transition matrix])[()( 11 ASIt

Example#6• Obtain the state transition equation of the system

dtutxttxt

)()()0()()(0

END OF LECTURES-36-37

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