Updated Module 4

8/15/2019 Updated Module 4

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Module -4

State Space Modeling andAnalysis of Discrete Time System


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Review of modeling of continuous-timesystem in state-space

• State and output equation are given as:

• State transition equation is given as:

• where state transition matri !STM" isgiven as:

)()()(

)()()(

t u Dt Cxt y

t But Axt xdt

d

+=

+=

∫ −+=

t

d But xt t x0

)()()0()()( τ τ τ φ φ

1

)()(

−

−== A sI of Laplaceinverseet At

φ


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!a" State Space representation of Discretetime system with sampled signal$

• )et us consider M&M* discrete-timecomprising of +*, devices andcontinuous-time process:

• *utputs of ero-order holds are: T k t kT for kT ekT ut u iii )1()()()( +


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• &nput vector in the state transitionequation of !." are constant (etween twoconsecutive sampling instants:

• So the can (e placed outside theintegral as u!/T":

• As range of time starts from t % /T'

setting initial time

T k t kT for kT uu i )1()()( +


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• State transition equation (ecomes:

• The a(ove equation gives state vector

!t" for all time (etween the samplinginstants' /T and !/0."T$

• )et us consider:

• A(ove state transition equation (ecomes:

T k t kT for

kT u Bd t kT xkT t t x

t

kT

)1(

)()()()()(

+


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• To descri(e the !t" only at samplinginstants' put t % !/0."T in a(ovestate transition equation:

• State transition equation (ecomes:

• *r

• 1e have state transition matri !STM" ofgiven continuous-time system as:

• So'

At et =)(φ

AT

eT =)(φ

)()()()())1(( kT ukT T kT kT xkT T kT T k x −++−+=+ θ φ

)()()()())1(( kT uT kT xT T k x θ φ +=+


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• 1e have considered:

• at t % !/0."T' it (ecomes:

• which on simplifying (ecomes:

• 2onsider change of varia(le asin integral$ Then

∫ −=−

t

kT

Bd t kT t τ τ φ θ )()(

∫

+

−+=−+T k

kT

Bd T kT kT T kT

)1(

)()( τ τ φ θ

∫

+

−+=

T k

kT Bd T kT T

)1(

)()( τ τ φ θ τ +−= kT m

dmd =τ


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• 2hange in limits

• 3pression for (ecomes:

• *utput equation for continuous-timesystem was given as:

• At t % !/0."T' the output equation(ecomes:

• which can (e generalied as:

T T kT kT mT kT when

kT kT mkT when

=++−=+=

=+−==

;

0;

τ

τ

∫ −=

T

BdmmT T 0

)()( φ θ

)(T θ

)()()( t Dut Cxt y +=

))1(())1(())1(( T k DuT k CxT k y +++=+

)()()( kT DukT CxkT y +=


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• Thus'

• represents set of rst order di5erenceequations' referred as discrete stateequation of sampled data system$

• where system matri and input matri iso(tained as:

• and the output equation is representedas:

)()()()())1(( kT uT kT xT T k x θ φ +=+

AT eT =)(φ ∫ −=T

BdmmT T

0

)()( φ θ

)()()( kT DukT CxkT y +=


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• 3ample: The discrete-time system'represented (y (loc/ diagram given in 6ig' has a continuous time linear process

7p!s" which is descri(ed (y followingdynamic equation$

• and its output equation is given (y :

• 6ind the state-space representation of theover-all system

sec1=T

ZOH )( s p)(t r )(* t r )(t y)(t u

processlinear

timeContinuous−

)(1

0

)(

)(

56

10

)(

)(

2

1

2

1t u

t x

t x

t x

t x

dt

d

+

−−

=

[ ]

=

)(

)(10)(

2

1

t x

t xt y


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S l ti f St t 3 ti


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Solution of State 3quation

• Solution of state equation is called as state transitionequation

• 1e consider two methods for o(taining state transitionequation:

!." Recursive Method

!9" sing +-transform approach$

!."Recursive Method(•) 6or a nth order system' 7iven the /nowledge of !#"

and past inputs u!#"'u!."';' u!/-."' we can nd the!/"$

(•) State equation:(•)


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•


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)(k


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• &n terms of state transition matri'' the state equation can (e written as:

!9" sing the +-transform approach

• 1e have the state equation as:

• Ta/ing the -transform on (oth sides:

• *n simplifying:

)(kT ϕ

[ ]∑−

=−−+=

1

0)()1()0()()(

k

! ! HuT !k xkT kT x ϕ ϕ

[ ] )()()1( kT HukT xT k x +=+

[ ] )()()0()( " H# " $ x " $ " +=−

[ ] [ ] )()0()( 11 " H# "I x "I " " $ −−

−+−=


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• Ta/ing the inverse -transform:

• 2omparing the two methods' we get:

[ ]{ } [ ]{ })()0()( 1111

" H# "I Z x "I " Z kT x −−−−

−+−=

[ ]{ }

[ ]{ })()(

)(

111

0

1

11

" H# "I Z ! Hu

"I " Z kT

k

!

!k

k

−−−

=

−−

−−

−=

−==

∑

ϕ

l i ( f d l d


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!a"6rom state-space to Transfer function model

()&f the discrete-data system is given as:

() Ta/ing the -transformation

()*n simplication:

( )1 ( ) ( )

( ) ( ) ( )

x k T x kT H u kT

y kT C x kT Du kT

+ = + = +

&nter-relation (etween -transform model andstate ?varia(le model

( ) (0) ( ) ( )

( ) ( ) ( )

"$ " "x $ " H# "

% " C $ " D# "

− = +

= +

( ) ( ) ( )1 1

(0) ( )

( ) ( ) ( )

$ " " "I x "I H# "

% " C $ " D# "

− −= − + −

= +

( i f f i ! " d ! "


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• To o(tain transfer function !" and @!"'!#" is made to (e null matri:

• So the output equation (ecome:

• The pulse transfer function is given (y:

• &f the system contains ero-order hold 'them pulse transfer can (e given as:

( ) ( ) 1

( ) $ " "I H# " −

= −

( ) 1

( ) ( )% " C "I H D # " − = − +

( ) 1

C "I H D− = − +

( )

1

( ) ( )C "I T T Dφ θ

−

= − +


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i f


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• or in form :

• or in form of di5erence equation:

• Aim is to o(tain the state space equation:

1 2

1 2

( ) ( 1) ( 2) ( )

( ) ( 1) ( 2) ( )

n

o n

y k a y k a y k a y k n

& u k &u k & u k & u k n

+ − + − + − =

+ − + − + −

)

)

)()()(

)()()1(

k Duk Cxk y

k Huk xk x

+=

+=+

n

nnn

n

nnn

a " a " a "

& " & " & " &

" #

" %

++++++++

= −−−−

2

2

1

1

2

2

1

10

)(

)(

. 2 t ll (l i l f ! l ll d


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.$ 2ontrolla(le canonical form:!also calledas phase canonical form"$ The Stateequation is given as:

and output equation:

)(

10

0

0

)(

)(

)(

1000

0100

0010

)1(

)1(

)1(

2

1

121

2

1

k u

k x

k x

k x

aaaak x

k x

k x

nnnnn

+

−−−−

=

+

++

−−

( ) ( ) ( )[ ] )(

)(

)(

)(

)( 02

1

0110110 k u&

k x

k x

k x

&a&&a&&a&k y

n

nnnn +

−−−= −−

3 l 643)( 123


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• 3ample:

• The given transfer function can (emodied as:

427

6543

)(

)(23

123

++++++

= " " "

" " "

" #

" %

321

321

4271

6543

)(

)(−−−

−−−

++++++

= " " "

" " "

" #

" %

321

321

4271)4*36()2*35()7*34(3

)()( −−−

−−−

+++ −+−+−+= " " " " " "

" # " %

)(4271

)6()1()17(

)(3)( 321

321

" # " " "

" " "

" # " % −−−

−−−

+++

−+−+−

+=

)()(3)(^

" % " # " % +=

)6()1()17( 321^


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• Rewriting the a(ove equation as:

• 6ollowing two equations can (e writtenas:

• Dene the state varia(les as:

)(4271

)6()1()17()(

321

321^

" # " " "

" " " " % −−−

−−−

+++−+−+−

=

)(4271

)(

)6()1()17(

)(321321

^

" ' " " "

" #

" " "

" % =

+++=

−+−+− −−−−−−

)()6()()1()()17()(

)()(4)(2)(7)(

321^

321

" ' " " ' " " ' " " %

" # " ' " " ' " " ' " " '

−−−

−−−

−+−+−=

+−−−=

)()();()();()( 3

1

2

2

1

3

" ' " " $ " ' " " $ " ' " " $ −−− ===

&t ( h th t


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• &t can (e shown that:

• &n term of di5erence equation:

• Su(stituting the state varia(les inequation a(ove:

);()(

)()(

32

21

" $ " "$

" $ " "$

=

=

)()1(

)()1(

32

21

k xk x

k xk x

=+

=+

)()6()()1()()17()(

)()(4)(2)(7)(

123

^

1233

" $ " $ " $ " %

" # " $ " $ " $ " "$

−+−+−=+−−−=

T /i th i t f d


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• Ta/ing the inverse -transform ando(taining the di5erence equation' we get:

• 1riting in matri form' we getcontrolla(le or phase canonical form:

)()6()()1()()17()(

)()(4)(2)(7)1(

123

^

1233

k x " xk xk y

k uk xk xk xk x

−+−+−=

+−−−=+

[ ] )(3

)(

)(

)(

1716)(

)(

1

0

0

)(

)(

)(

724

100

010

)1(

)1(

)1(

3

2

1

3

2

1

3

2

1

k u

k x

k x

k x

k y

k u

k x

k x

k x

k x

k x

k x

+

−−−=

+

−−−

=

+

++


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3ample 6543)( 123%


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• 3ample:

• The given transfer function can (emodied as:

• 2ross Multiplying ' we gets:

• Rearranging in powers of :

427

6543

)(

)(23

123

++++++

= " " "

" " "

" #

" %

( ) ( )321321 6543)(4271)( −−−−−− +++=+++ " " " " # " " " " %

( ) ( ) ( )

( ) 0)(6)(4

)(5)(2)(4)(7)(3)(

3

21

=−+

−+−+−−

−−

" " # " %

" " # " % " " # " % " # " %

321

321

4271

6543

)(

)(−−−

−−−

++++++

= " " "

" " "

" #

" %

( ) ( ) 21 )(5)(2)(4)(7)(3)( −− ++++ ""#"%""#"%"#"%


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•Rewriting the a(ove equation as:

• Dening the state varia(les as:

• y putting the state varia(le' a(ove

equation (ecomes:

( ) ( ) ( ){ }{ })(6)(4)(5)(2)(4)(7)(3)(

111 "# "% " "# "% " "# "% "

"# "%

+−++−++−+

+=−−−

( ){ }

( ){ }

( ))(6)(4)(

)()(5)(2)(

)()(4)(7)(

1

1

1

1

2

2

1

3

" # " % " " $

" $ " # " % " " $

" $ " # " % " " $

+−=

++−=

++−=

−

−

−

)()(3)( 3 " $ " # " % +=

( ) ( )

( ) 3)(6)(4

)(5)(2)(4)(7)(3)(

−+−++−++−+=

" " # " %

" " # " % " " # " % " # " %

• 6rom the epression of state varia(les:


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• 6rom the epression of state varia(les:

•


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• Ta/ing the inverse -transform:

• and epression

• 1riting the a(ove equations in the matriform:

)(6)(4)1(

)()(2)()1(

)(17)(7)()1(

31

312

323

k uk xk x

k uk xk xk x

k uk xk xk x

−−=+

−−=+

−−=+

)(3)()( 3 k uk xk y +=

[ ] )(3

)(

)(

)(

100)(

)(

17

1

6

)(

)(

)(

710

201

400

)1(

)1(

)1(

3

2

1

3

2

1

3

2

1

k u

k x

k x

k x

k y

k u

k x

k x

k x

k x

k x

k x

+

=

−

−−

+

−

−−

=

+

++

• ote that: oftranspose =


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• ote that:

=" Diagonal 2anonical 6orm:

&f the poles of -transfer function are alldistinct' then state-space representationcan (e put in the diagonal form:

&f are poles of the systemand

are correspondingresidues$ !o(tained (y partial fraction

-

n p p p ,,, 21

oc

oc

oc

oc

D D

H of transposeC

C of transpose H

of transpose

=

=

==

n ( ( ( ,,, 21

• State equation can (e given as:


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• State equation can (e given as:

• And output equation:

)(

1

1

1

1

)(

)(

)(

0000

0000

0000

0000

)1(

)1(

)1(

2

1

1

2

1

2

1

k u

k x

k x

k x

p

p

p

p

k x

k x

k x

nn

n

n

+

=

+

+

+

−

[ ] )(

)(

)(

)(

)( 02

1

21 k u&

k x

k x

k x

( ( (k y

n

n +

=

4" Bordan 2anonical 6orm


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4" Bordan 2anonical 6orm

&f the pulse transfer function involves amultiple pole of order m at % p. andother poles are distinct' the stateequation can (e given as:

)(

1

11

0

0

0

)(

)(

)(

000000

0000000000

00000

00010

00001

)1(

)1()1(

)1(

)1(

)1(

2

1

2

1

1

1

1

1

3

2

1

k u

k x

k x

k x

p

p p

p

p

p

k x

k xk x

k x

k x

k x

n

nn

m

m

+

=

+

++

+

+

+

+

• And output equation is given (y:


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• And output equation is given (y:

STAT3 D&A7RAMS *6 A D&7&TA) S@ST3M1hen the a digital system is represented

(y di5erence equation' a state diagram

can (e drawn to represent therelationships (etween the discrete statevaria(les$


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asic linear operation on a digitalcomputer are multiplication (y aconstant' addition of varia(les and time

delay or storage$Mathematical epression of these

operation and its corresponding -

domain representation:." Multiplication (y a constant

9" Summing operation

)()(

)()(

12

12

" $ a " $

k xak x

=

=

)()()(

)()()(

123

123

" $ " $ " $

k xk xk x

+=

+=


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• 9"


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• 9"

• ="

• 3planation wor/ing of delay mechanism:


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• 3planation wor/ing of delay mechanism:

• )et;;;;;;;;;$!."

• So its -transform will (e:

•So to o(tain .!/":

( )[ ]

( ))0()()(

1)(

112

12

x " $ " " $

T k xkT x

−=

+=

)0()()( 121

1 x " $ " " $ += −

[ ]

[ ]

[ ]

[ ]

[ ]T xT xk if

T xT xk if

T xT xk if

T xT xk if

T x xk if

5)4(;4

4)3(;3

3)2(;2

2)(;1

)0(;0

12

12

12

12

12

==

==

==

==

==

• 7etting 9!/" from


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7etting 9!/" from.!/" is the left shift of.!/"

• ut if one has o(tained.!/" from the given9!/"'

• Then'

• 1ay is to delay thesignal 9!/" ! Right shift

the signal 9!/""• And Then add .!#" tothe shifted 9!/"$

• State diagram can (e used to o(tained the


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State diagram can (e used to o(tained thefollowing from a given di5erence equation of asystem$

!a" State equation and *utput equation !(" *verall Transfer function

!c" State transition equation ! in the -domain"

*ne advantage of using the state diagram' isthat state transition equation and overalltransfer function can (e o(tained (y using theMasonEs 7ain 6ormula

Saves the e5ort of performing the inverse ofmatri !& - 7"$

State transition equation in time-domain can (eo(tained (y ta/ing inverse -transform of -

domain state transition equation$


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• 3ample:

• 6or a discrete-time system whosedynamics is represented (y followingdi5erence equation$

with initial conditions:

Draw the state diagram for the system',ence using it ' 6ind

!a" State equation and *utput equation

!(" State transition equation in -domain

!c" *verall -transfer function (etween @!"

)()(2.0)1(2.1)2( k uk yk yk y =++−+

7.0)1(5.0)0( == yand y


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• Solution:

• Arrange the di5erence equation as:

• Drawing the state diagram as:

)()(2.0)1(2.1)2( k uk yk yk y +−+=+


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• !(" state Transition equation in -domain:

• sing the Mason 7ain 6ormula'

can (e found as:• 2onsidering

)()(

)(

)0(

)0(

)()(

)()(

)(

)(

2

1

2

1

2221

1211

2

1 " # " L

" L

x

x

" + " +

" + " +

" $

" $

+=

)(11 " +

:)(1)0( 11 output as " $ and nput as x

)2.02.1(1

1

)0(

)()(

31

1

111

−− −−=∆

∆==

" " where

x

" $ " +


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• &n same way:

∆

−

==

∆==

−

−

1

1

2

21

1

2

112

2.0

)0(

)(

)(

)0(

)()(

"

x

" $

" +

"

x

" $ " +


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• And

∆

== 1

)0(

)()(

2

222

x

" $ " +

∆==

∆==

−

−

1

22

21

1

)(

)()(

)(

)()(

"

"#

" $ " L

"

"#

" $ " L

Response (etween sampling instants


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Response (etween sampling instantsusing state varia(le approach

• evaluation of system responses (etweenthe sampling instants of discrete-datacontrol systems$

• represents a modern alternative to the

modied -transform$

• State transition equation:

• where

)()()()()( 0000 t ut t t xt t t x −+−= θ φ

∫ −=−t

t o

Bd t t t τ τ φ θ )()( 0


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• &f one wants response (etween samplinginstant' then put and

• and where / % #' .' 9'; and

• Then state equation (ecomes:

• Farying the value of (etween # and .'all information on !t" for all t can (e

o(tained$

[ ] )()()()()( kT uT kT xT T k x ∆+∆=∆+ θ φ

kT t =0 T k t )( ∆+=

10 ≤∆≤

∆

2ontrolla(ility


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2ontrolla(ility• 2onsider a discrete-time system as shown in

gure:

• A controlled process is said to (e controlla(le ifevery state of system can (e a5ected orcontrolled in nite time (y same un-constrainedcontrol signal u!/"$

• &f any one state is not accessi(le from control'

u!/"' there is no way of driving that particularstate to a desired state in nite time (y means ofsame control e5ort$

• Such a state varia(le is said to (e uncontrolla(le

and system is said to (e uncontrolla(le

• 1e consider two type of controlla(ility:


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1e consider two type of controlla(ility:

!." 2omplete state controlla(ility$

!9" 2omplete output controlla(ility$

!."2omplete state controlla(ility$

(•)System is said to (e completely statecontrolla(le if for any initial time !/ % #"'there eist a set unconstrained controlu!/"' / % #'.'9';'!-."' which transferseach initial state !#" to any nal state

!" for a nite $(•) Theorem: System is completely statecontrolla(le if only if the matri'

is of ran/ GnH

[ ] H H H H , - 12 −=

•


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• writing a(ove equation in a condensedform:

• *(>ective is nd such that a initial

vector !#" is driven to !"$

[ ]

−

= −

)0(

)1(

)2(

)1(

)( 12

u

u

- u

- u

H H H H - + -

,# - + =)(

•


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psimultaneous linear equations given (ya(ove equations for a given S' !"' !#"$

• 6or solution to eit' equations mustlinearly independent

• 6or this necessary and suIcient condition

is that matri S has a ran/ of GnH or inother word' S must have least G n Gindependent columns$

• &f state equation is given as:

• Then State controlla(ility matri is given

(y:

)()()()())1(( kT uT kT xT T k x θ φ +=+

[ ] [ ][ ])()()()()()()( 12

T T T T T T T ,

-

θ φ θ φ θ φ θ

−

= • 9" 2omplete *utput controlla(ility


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" p p y

• System is said to (e completely outputcontrolla(le if for any initial stage !time"' /

% #' there eit a set unconstrained controlsu!/"' / % #' .' 9' $$$' !-."' such that anynal output y!" can (e reached from

ar(itrary initial states in nite time' $• ecessary condition for a system to

completely output controlla(le is thatoutput controlla(ility matri'


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• 6rom the state transition equation' we have:

• The a(ove equation can modied as:

• 1riting the ),S of a(ove equation in thematri form:

)()()( - Du - Cx - y +=

)()()0()(1

0

1 - Du ! Hu xC - y

-

!

! - - +

+= ∑

−

=

−−

)()()0()(1

0

1 - Du ! HuC xC - y

-

!

! - - +

=− ∑

−

=

−−

=− )0()( xC-y -


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• *utput controlla(ility matri'


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61/117

• State transition equation of the system isgiven as:

• *utput equation is given as:

• &f we consider the instant as forinitial state:

• Then state transition equation (ecomes:

)()()( k Duk Cxk y +=

∑

−

=

−−

+=

1

0

1

)()0()(

k

!

!k k

! Hu xk x

th /

∑−=

−−+=+1

0

1)()()(

k

!

!k k ! Hu / xk / x

• And output equation (ecomes


62/117

p q

• Su(stituting the epression for !M0/"from state transition equation into outputequation:

• 1hen / assumes values from . to !-."'we get p!-." equation$

•1e have from output equation:

)()()( k / Duk / Cxk / y +++=+

.1,,1,0

)()()()(1

0

1

−=

++

+=+

∑

−

=

−−

- k for

k / Du ! Hu / xC k / yk

!

!k k

)()()( / Du / Cx / y += • Altogether we have p linear alge(raic


63/117

g p gequations that can (e put in a matriform as follows:

+

=

−+

+

+

−

)(

)1(

)2(

)1(

)(

1

2 / x

C

C

C

C

- / y

/ y

/ y

/ y

-

0000 D


64/117

−+

+

+

×

+

−−−

)1(

)2(

)1(

)(

00

000

0000

432

- / u

/ u

/ u

/ u

DCH H C H C H C

DCH CH

DCH

- - -


65/117


66/117

2ontrolla(le?uncontrolla(le decomposition


67/117

p

• &f a system is not completely controlla(le'

it can (e decomposed into a controlla(leand a completely uncontrolla(lesu(system$


68/117


69/117

• *utput has a componentthat does not depend on manipulated

input' u!/"$

• 2aution must (e eercised when

controlling a system which is notcompletely controlla(le$

• The controllable subspace of a state-space model is composed of all statesgenerated through every possi(le

com(ination of the states in $

)(k xC ncnc

c x

c A


70/117

)oss of controlla(ility and


71/117

)oss of controlla(ility ando(serva(ility due to sampling

• Sampling of a continuous time system gives adiscrete-time system with system matrices thatdepend on the sampling period

• 1hen a continuous-time control system with

comple poles is discretied' the introduction ofsampling may impair the controlla(ility ando(serva(ility of the resulting discrete system$


72/117

• To get a controlla(le sampled system' it isnecessary that the continuous-time system is also(e controlla(le

• ecause it allowa(le control signals for the samplesystem !piecewise constant signals" are a su(setof allowa(le control signals for the continuous-time system$

• ,owever it may happen that controlla(ility is lostfor some sampling period$

• The condition for uno(serva(ility are more

restricted in the continuous time system$• ecause for eample: the output has to (e ero

over a time interval' while the sampled timesystem output has to (e ero only at the sampling

instants$


73/117


74/117

• oth controlla(ility and o(serva(ility is21T


75/117

lost for:

• That is when sampling interval is half theperiod of the oscillation of the harmonicoscillator or an integer multiple of that

period$

• )oss of controlla(ility and o(serva(ilitydue to sampling only when the

continuous time s stem has oscillator

,2,1,2

,2,1,2

,2,1,

==

==

==

nT nT

nnT T

nnT

os

os

π π

π ω

A system' that was completely statell (l d l l ( (l i


76/117

controlla(le and completely o(serva(le inthe a(sence of sampling' remains

completely state controlla(le andcompletely o(serva(le after introductionof sampling'

if and only if' for every eigen-value !rootof the characteristic equation" for thecontinuous-time control system' the

relation

implies

!i λ λ ReRe =

( )T

n !i

π λ λ 2

Im ≠−

,3,2,1 ±±±=n


77/117


78/117

• 6or any / C n (y repeatedly applying the


79/117

6or any / C n' (y repeatedly applying thetheorem'

can (e eventually (e epressed interms of 7$

Second Method

1e are concerned a(out ndingepression or value of functions which arerepresented as a series of the powers of a

matri2onsider a matri polynomial as:

• This can (e computed (y consideration of

k

++++++= ++1

1

2

210)(n

n

n

n aaaa I a f

++++++= ++1

1

2

210)(n

n

n

n aaaaa f λ λ λ λ λ

• 2haracteristic equation is given (y


80/117

.0)(

0)(

1

2

2

1

1 =+++++=∆

=−=∆

−

−−

nn

nnn

I

α λ α λ α λ α λ λ

λ λ

).()()()(

:.)(

: !o"nomina)(

)(

)(

)()(

)(

,)()(

1

1

2

210

λ λ λ λ

λ β λ β λ β β λ

λ

λ

λ

λ λ

λ

λ λ

0 * f

aswritten&ecane*autiona&ove 0

formof remainder is 0 where

0

*

f

0et we&y f Dividin0

n

n

+∆=

++++=

∆+=∆

∆

−−

ofsei0envaluetheAt λλλ


81/117

.)()(

:,)(

).()()()(

:,.,,,,

,,2,1);()(

:

,,2,1;0)(

,,,,

1

1

2

210

1210

21

−−

−

++++==∆

+∆=

==

==∆

n

n

n

ii

i

n

I 0 f

followsit "ero yidenticall is A,ince

0 * f

0et we for n0 ,u&stituticomputed &ecant CoefficienThe

ni 0 f

havewethen

ni

of sei0envaluethe At

β β β β

λ β β β β

λ λ

λ

λ λ λ

• &f matri 7 has an 3igen-value ofpλ


82/117

gmultiplicity of m ' then only oneindependent equation can (e o(tained (ysu(stituting $

• The remaining !m-." equations can (eo(tained (y di5erentiating (oth sides ofthe equation$

• Since

• &t follows that

pi λ λ =

p

)1(,2,1,0;0)( −==

∆

=

m !d d

p

!

!

λ λ

λ λ

)1(,2,1,0;)()( −=

=

==

m ! 0 d

d f

d

d

p p

!

!

!

!

λ λ λ λ

λ λ

λ λ


83/117

9$ sing the -transform method of ndingSTM

C STM of matri 7 can (e epressed as:

) &t involves the nding of inverse of !&-7"' then applying inverse -transform$

)6or the second order systems' thesesteps can (e carried out with ease$

)6or higher order systems' it (ecometedious (y hand calculation$

( ){ }1−−−= "I " of transform " inversek

Tas/ of nding inverse -transform of( ) " "I 1−−


84/117

for given 7' can (e simplied (y thefollowing method:

)et:


85/117

•


86/117

• Thus' result of repeated pre-multiplying(y !& 0 7"' can (e o(tained (y putting >% 4' ' ;' n to form to equation !4" toequation !n":

I " " " " + + "

I " " " " " + + "

I " " " " + + "

nnnnnn +++++=

+++++=

++++=

−−− 1221

542332455

3322344

,,4,3,2,1

:),3(),2(),1(

1221

n ! for

I " " " " + + "

as 0enerali"ecanonee*uationat Lookin0

! ! ! ! ! !

=

+++++= −−−

• )et the characteristics equation of 7 (eas: 021 −− nnn


87/117

as:

• 1e modify the GnH equations o(tained (yusing the coeIcients of the characteristicequation as given:

0012

2

1

1 =+++++ −

−−

− a " a " a " a " n

n

n

n

n

1sides $ot%m&ti!iedis(n)'&ation

$"sides $ot%m&ti!iedis1)(n'&ation

$"sides $ot%m&ti!iedis(2)'&ation

$"sides $ot%m&ti!iedis(1)'&ation

1

2

1

&y

a

a

a

n−

0

t%

t%

$"sides $ot%m&ti!iedis1)(n'&ation

**

:ase&ation1)(ncreate+e

a+

=

+

Th di d ! ." i i


88/117

I " " " " + + "

I "a "a "a + a + "a

I "a "a "a + a + "a

I "a "a + a + "a

"I a+ a "+ a + a + a

nnnnnn

n

n

n

n

n

n

n

n

n

n

+++++=

++++=

+++=

++=

+==

−−−

−−

−−

−−

−−

−−

1221

1

1

2

1

2

1

1

1

1

1

3

3

2

3

2

3

3

3

3

3

2

22

2

2

2

2

111

00

• Thus' modied !n0." equations as writtenas:

• The a(ove equations can (e summed onthe (oth sides to give:


89/117

the (oth sides to give:

• The a(ove equation can (e representedas:

1

11

1

22

2

1

100

=

++

+++=−+−

−=

−

=

−

===

∑∑∑∑∑

n

nnnin

ni

i

in

ii

in

ii

in

ii

in

ii

awhere

I " " a

" a " a + a + " a

∑ ∑∑∑= =

−

==

+

=

n

!

n

!i

!i

i

!n

i

i

i

in

i

i a " + a + " a100

• Due to 2ayley-,amilton theorem' The


90/117

y yrst term on the right-hand side of theequation (ecome a null matri' whichlead to epression of 6 as:

"I

a " +

" a

a "

+

n

!

n

!i

!i

i

!

in

i

i

n

!

n

!i

!i

i

!

−

=

=

∑ ∑

∑

∑ ∑

= =

−

=

= =

−

1

0

1


91/117

( ) ( ) +=++=)3()1()3)(1(

)4( B A "I I +


92/117

( ) ( )

( ) ( )

−+−−−−−

−−−−−−=

−

+

−−

+

=

+

+−

+

+=

+−=

+=

++++

+++

+

111

1

)3()1()3()1(3

)3()1()3(3)1(

2

1

)3(2)1(2

3

:transorm-inerse)3(2)1(2

3,

2

)( ;2

3

)3()1()3)(1(

k k k k

k k k k

k k k I I

takin0 "

" I

"

" I + so

I B

I A

" " " " "


93/117

6eed(ac/• &f system is controlla(le and o(serva(le' the

poles of the closed-loop system (e placed atany desired locations (y means of statefeed(ac/ through an appropriate state-feed(ac/ gain matri$

• 2onsider an open-loop system whose stateequation is given as:

• where 7 is !n 8 n" system matri' , is !n 8 ."input matri and !/" is state vector of !n 8

."$

)()()1( k Huk xk x

+=+

• Aim is to design a control law:

)()( k+xku =


94/117

where 6 is !. 8 n" state-feed(ac/ gain matri

such that it can place eigen-values of theclosed-loop system at desired location in -plane$

• )et the desired location of the closed-looppoles (e at:

• The characteristic equation of open-loopsystem is given (y:

• 6ollowing are 6our Methods to nd the state

feed(ac/ 7ain matri 6:

;,;; 21 n " " " µ µ µ ===

nn

nnn

a " a " a " a " "I +++++=− −−−

1

2

2

1

1

)()( k +xk u −=

!."Method-. !2ontrolla(le 2anonical 6ormmethod"


95/117

method"

[ ]

=

=

=

−−

−−

−

0001

001

011

:

,:matrition/ransormaeine

1

32

121

12

a

aaaaa

/

&y 0ivenis / and

H H H H , where

,/ ' x

nn

nn

n

• Ran/ of Matri M should (e equal to J n K$

sing the Transformation matri we


96/117

• sing the Transformation matri ' wetransform the representation of the

system from given state space to anotherstate space domain dened (y:

• State equation and output equation of the

system (ecomes:

)()()(

)()()1(

k Duk C'vk y

k Huk 'vk 'v

+=+=+

)()(

)()( 1

k 'vk xor

k x'k v

=

= −

•


97/117

inverse of :

• Dening

• Modied state-space representation ofthe system (ecomes:

• y this transformation' will (e incontrolla(le canonical form$

• The 2hosen control law is change to:

H ' H and '' 11 −

∧−

∧

==)()()(

)()()1( 11

k Duk C'vk y

k Hu'k 'v'k v

+=

+=+ −−

)()()1( k u H k vk v∧∧

+=+∧∧

H and

)()()( kv+k+'vku

∧

==


98/117

• The closed loop system characteristicsequation (ecomes:


99/117

equation (ecomes:

• 1hich on simplication (ecomes:

• The desired closed loop systemcharacteristics equation can (e o(tainedas:

0

)()()(

0

00

01

1111

=

++++

−

−− δ δ δ a " aa

"

"

nnnn

0)()(

1

11 =+++++

−

nn

nn

a " a " δ δ

0)())(( 21 =n""" µµµ • 1hich when epended' we gets:

01 ++++ −nn ααα


100/117

• 2omparing the coeIcients of li/e-powers

and ndingas:

• sing the inverse transformation' o(tainfeed(ac/ gain matri in the -domain$

011

1 =++++ − nnnn

" " " α α α

1221 ,,, δ δ δ δ δ −− nnn

111

111 ;

;

a

a

a

nnn

nnn

−=

−=

−=

−−−

α δ

α δ

α δ

1−∧

'++ 3ample: Determine a state feed(ac/ gain

matri Q such that system will have the closed


101/117

matri Q such that system will have the closedloop poles at:

7iven the open loop system matrices as:

Solution:

2ontrolla(ility matri

! " 3.02.0,.0 ±=

=

=

1

0

1

;

4.01.00

013.0

2.005.0

H

3)(;

10401

51.03.0043.07.01

=

= , rank ,


102/117

• System matri in the new domain: 0010


103/117

• )et

• 2losed loop system matri will (e

• 2losed loop system characteristicsequation :

==

−

== −∧

−∧

1

0;

.11.1206.0

100 11 H ' H ''

[ ];321 f f f + =∧

−−−−

=− ∧∧∧

).1()1.1()206.0(

100

010

321 f f f

+ H

0)206.0()1.1().1( 122

3

3 =−−−−−−− f " f " f "

• 2losed loop system characteristicsequation can also made with desired


104/117

equation can also made with desiredclosed loop 3igen-value:

• 2omparing the coeIcient of li/e-powers

in two equation:

• 2alculating un/nown parameters f.'f9'f=:

• 6 in original domain

0117.04.03.1

0)3.02.0)(3.02.0)(.0(

23 =−+−

=+−−−−

" " "

! " ! " "

117.0)206.0(;4.0)1.1(;3.1).1( 123 −=−−=−−−−=−− f f f

[ ] [ ]6.061.00.0321 −===∧

f f f +

[ ]033.0333.06333.01 −== −∧

' + +

Method 9: Ac/ermannHs 6ormula• The desired closed loop system


105/117

• The desired closed loop systemcharacteristic equation is given:

• Dene

• 2ayely-,amilton Theorem states that 1satises its own characteristic equation$

• Then

( )0

0)())((

1

1

1

21

=++++==−−−=+−

−−

nn

nn

n

" " "

" " " H+ "I

α α α

µ µ µ

1 H+ =−

0)( 11

1 =++++= −− I 1 1 1 1 f nn

nn α α α

• 2onsider the following identities:


106/117

( )

...(3)

)2(

...(2a) ))((

)(

..(2).

(1)...

22

2

2

22

H+1 H+ 1

from

H+ H+ H+

H+H+ H+H+ H+ H+

H+ 1

H+ 1

I I

−−=

−−−=

−−−=−−=

−=−=

=

)(33

H+ 1 −=


107/117

)4(

:),2()2()

....()(

...

...

))((

))()((

223

223

223

2

H+1 H+1 H+

0et weaand from H+H+ H+H+

H+ H+ H+ H+

H+H+H+ H+H+ H+H+

H+H+H+ H+ H+

H+H+ H+H+ H+

H+ H+ H+

=

−−−

−+−−=+++

+−−−−=

−−−−=

−−−=


108/117

= nn I I α α


109/117

and adding the ),S and R,S of a(ove!n0." equation' we get:

)(

)(

)(

)(

121

00

223

3

3

3

2

2

2

2

11

−−−

−−

−−

−−

−−−−=

−−−=

−−=

−=

nnnnn

nn

nn

nn

nn

H+1 H+1 H+ 1

H+1 H+1 H+ 1

H+1 H+ 1

H+ 1

α α

α α

α α

α α

0

2

21 −− =++++n

nnn 1 1 1 I α α α α


110/117

)(

)(

)(

121

0

22

3

2

1

0

2

21

021

−−−

−

−

−

−−

−−−−+

−−−+−−+

− ++++

nnn

n

n

n

n

nnn

nnn

H+1 H+1 H+

H+1 H+1 H+

H+1 H+

H+ I

α

α

α

α α α α α

• 1riting in terms of closed loop systemcharacteristics equation and epress the


111/117

characteristics equation and epress therest term as factor of ,' 7,' 79,:

)(

)(

)(

)(

)()(

0

1

3

043

2

2

032

1

021

+H

+1 +1 + H

+1 +1 + H

+1 +1 + H

f 1 f

n

n

nn

n

nn

n

nn

α

α α α

α α α

α α α

−

−

−−

−−−

−−−

++−

++−

+++−=

• 1riting the a(ove equation in matriform: = f1f )()(


112/117

[ ]

++++

+++×−

=

−−−

−−−

−−−

−

+

+1 +1 +

+1 +1 +

+1 +1 +

H H H H

f 1 f

n

nn

nnn

n

nn

n

0

3

043

2032

1

021

12

)(

)(

)(

)()(

α

α α α

α α α

α α α

• Since we have f!1" % # and since systemis controlla(le controlla(ility matri' S is


113/117

y 'of ran/ J n K and so its inverse eist$ So

we can write:

•


114/117

• This epression for nding the matri'6 is

commonly called as A/ermannHs formula$• 3ample: Ta/ing previous pro(lem:

• 2losed loop system characteristics

equation can also made with desiredclosed loop 3igen-value:

• 6inding the value of closed loop

[ ] )(1000 f , +

117.04.03.1)(

)3.02.0)(3.02.0)(.0()(

23

−+−=

+−−−−=

" " " " f

! " ! " " " f

I f 117.04.03.1)( 23 −+−=


115/117

−

−−

=05.00230.001.0

0360.007.007.0

014.0012.0066.0

)( f

• &nverse of controlla(ility matri:

• Then (y Ac/ermannHs formula

−−−−=−

7037.37037.37037.3

263.6630.2263.614.2415.014.1

1,

[ ] [ ]033.0333.06333.0)(100 1 −== − f , +

Method-= !3igen-vector method"


116/117

Method = !3igen vector method"

• if desired 3igen-values

are distinct' then desired state feed(ac/gain matri ' Q can (e found (y:

• is the 3igen-vectors of the matri !7 ?,6" that is satisfy the equation:

n µ µ µ ,,, 21

[ ][ ]

.,,2,1;)(

:

1111

1

1

121

ni H I

e*uation satisfywhere

+

ii

i

nn

=−=

=

−

−

−

µ ξ

ξ

ξ ξ ξ ξ

iξ

iiiH+ ξµξ)(


117/117

Method-4 !Direct 2alculation method"

• &f the order of system is low' su(stitute:

• &nto the characteristic equation:

• Match coeIcients of powers of in a(ovecharacteristic equation with li/e power of

[ ]n + + + + 21=

0=+− H+ "I

Updated Module 4

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Transcript of Updated Module 4