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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' $• ecessary condition for a system to
completely output controlla(le is thatoutput controlla(ility matri'
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• 6rom the state transition equation' we have:
• The a(ove equation can modied as:
• 1riting the ),S of a(ove equation in thematri form:
)()()( - Du - Cx - y +=
)()()0()(1
0
1 - Du ! Hu xC - y
-
!
! - - +
+= ∑
−
=
−−
)()()0()(1
0
1 - Du ! HuC xC - y
-
!
! - - +
=− ∑
−
=
−−
=− )0()( xC-y -
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• *utput controlla(ility matri'
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• State transition equation of the system isgiven as:
• *utput equation is given as:
• &f we consider the instant as forinitial state:
• Then state transition equation (ecomes:
)()()( k Duk Cxk y +=
∑
−
=
−−
+=
1
0
1
)()0()(
k
!
!k k
! Hu xk x
th /
∑−=
−−+=+1
0
1)()()(
k
!
!k k ! Hu / xk / x
• And output equation (ecomes
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p q
• Su(stituting the epression for !M0/"from state transition equation into outputequation:
• 1hen / assumes values from . to !-."'we get p!-." equation$
•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
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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
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−+
+
+
×
+
−−−
)1(
)2(
)1(
)(
00
000
0000
432
- / u
/ u
/ u
/ u
DCH H C H C H C
DCH CH
DCH
- - -
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2ontrolla(le?uncontrolla(le decomposition
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p
• &f a system is not completely controlla(le'
it can (e decomposed into a controlla(leand a completely uncontrolla(lesu(system$
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• *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
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)oss of controlla(ility and
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)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$
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• 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$• ecause for eample: the output has to (e ero
over a time interval' while the sampled timesystem output has to (e ero only at the sampling
instants$
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• oth controlla(ility and o(serva(ility is21T
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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
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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
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• 6or any / C n (y repeatedly applying the
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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
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.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
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.)()(
:,)(
).()()()(
:,.,,,,
,,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
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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
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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−−
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for given 7' can (e simplied (y thefollowing method:
)et:
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•
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• 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
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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 $ot%m&ti!iedis1)(n'&ation
**
:ase&ation1)(ncreate+e
a+
=
+
Th di d ! ." i i
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I " " " " + + "
I "a "a "a + a + "a
I "a "a "a + a + "a
I "a "a + a + "a
"I a+ a "+ a + a + a
nnnnnn
n
n
n
n
n
n
n
n
n
n
+++++=
++++=
+++=
++=
+==
−−−
−−
−−
−−
−−
−−
1221
1
1
2
1
2
1
1
1
1
1
3
3
2
3
2
3
3
3
3
3
2
22
2
2
2
2
111
00
• Thus' modied !n0." equations as writtenas:
• The a(ove equations can (e summed onthe (oth sides to give:
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the (oth sides to give:
• The a(ove equation can (e representedas:
1
11
1
22
2
1
100
=
++
+++=−+−
−=
−
=
−
===
∑∑∑∑∑
n
nnnin
ni
i
in
ii
in
ii
in
ii
in
ii
awhere
I " " a
" a " a + a + " a
∑ ∑∑∑= =
−
==
+
=
n
!
n
!i
!i
i
!n
i
i
i
in
i
i a " + a + " a100
• Due to 2ayley-,amilton theorem' The
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y yrst term on the right-hand side of theequation (ecome a null matri' whichlead to epression of 6 as:
"I
a " +
" a
a "
+
n
!
n
!i
!i
i
!
in
i
i
n
!
n
!i
!i
i
!
−
=
=
∑ ∑
∑
∑ ∑
= =
−
=
= =
−
1
0
1
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( ) ( ) +=++=)3()1()3)(1(
)4( B A "I I +
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( ) ( )
( ) ( )
−+−−−−−
−−−−−−=
−
+
−−
+
=
+
+−
+
+=
+−=
+=
++++
+++
+
111
1
)3()1()3()1(3
)3()1()3(3)1(
2
1
)3(2)1(2
3
:transorm-inerse)3(2)1(2
3,
2
)( ;2
3
)3()1()3)(1(
k k k k
k k k k
k k k I I
takin0 "
" I
"
" I + so
I B
I A
" " " " "
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6eed(ac/• &f system is controlla(le and o(serva(le' the
poles of the closed-loop system (e placed atany desired locations (y means of statefeed(ac/ through an appropriate state-feed(ac/ gain matri$
• 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 =
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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"
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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
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• 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
=
= −
•
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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
∧
==
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• The closed loop system characteristicsequation (ecomes:
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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
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• 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
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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 ,
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• System matri in the new domain: 0010
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• )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
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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
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• 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:
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( )
...(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 −=
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)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+
=
−−−
−+−−=+++
+−−−−=
−−−−=
−−−=
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= nn I I α α
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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
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)(
)(
)(
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
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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 )()(
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[ ]
++++
+++×−
=
−−−
−−−
−−−
−
+
+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
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y 'of ran/ J n K and so its inverse eist$ So
we can write:
•
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• 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 −+−=
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−
−−
=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"
8/15/2019 Updated Module 4
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)(
8/15/2019 Updated Module 4
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