feedback control systems (fcs)
DESCRIPTION
Feedback Control Systems (FCS). Lecture-26-27-28-29 State Space Canonical forms. Dr. Imtiaz Hussain email: [email protected] URL : http://imtiazhussainkalwar.weebly.com/. Lecture Outline. Canonical forms of State Space Models Phase Variable Canonical Form - PowerPoint PPT PresentationTRANSCRIPT
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Feedback Control Systems (FCS)
Dr. Imtiaz Hussainemail: [email protected]
URL :http://imtiazhussainkalwar.weebly.com/
Lecture-26-27-28-29State Space Canonical forms
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Lecture Outline– Canonical forms of State Space Models
• Phase Variable Canonical Form
• Controllable Canonical form
• Observable Canonical form
– Similarity Transformations
• Transformation of coordinates
– Transformation to CCF
– Transformation OCF
![Page 3: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/3.jpg)
Canonical Forms• Canonical forms are the standard forms of state space models.
• Each of these canonical form has specific advantages which makes it convenient for use in particular design technique.
• There are four canonical forms of state space models– Phase variable canonical form– Controllable Canonical form– Observable Canonical form– Diagonal Canonical form– Jordan Canonical Form
• It is interesting to note that the dynamics properties of system remain unchanged whichever the type of representation is used.
Companion forms
Modal forms
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Phase Variable Canonical form
• The method of phase variables possess mathematical advantage over other representations.
• This type of representation can be obtained directly from differential equations.
• Decomposition of transfer function also yields Phase variable form.
![Page 5: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/5.jpg)
Phase Variable Canonical form• Consider an nth order linear plant model described by the
differential equation
• Where y(t) is the plant output and u(t) is the plant input.
• A state model for this system is not unique but depends on the choice of a set of state variables.
• A useful set of state variables, referred to as phase variables, is defined as:
𝑑𝑛 𝑦𝑑𝑡𝑛
+𝑎1𝑑𝑛− 1𝑦𝑑𝑡𝑛− 1 +⋯+𝑎𝑛−1
𝑑𝑦𝑑𝑡 +𝑎𝑛 𝑦=𝑢(𝑡)
𝑥1=𝑦 , 𝑥2=�̇� , 𝑥3=𝑦 ,⋯ , 𝑥𝑛=𝑑𝑛−1 𝑦𝑑𝑡𝑛−1
![Page 6: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/6.jpg)
Phase Variable Canonical form
• Taking derivatives of the first n-1 state variables, we have
𝑥1=𝑦 , 𝑥2= �̇� , 𝑥3=𝑦 ,⋯ , 𝑥𝑛=𝑑𝑛−1 𝑦𝑑𝑡𝑛−1
�̇�1=𝑥2 , �̇�2=𝑥3 , �̇�3=𝑥4⋯ , �̇�𝑛−1=𝑥𝑛
�̇�𝑛=−𝑎𝑛 𝑥1−𝑎𝑛−1𝑥2−⋯−𝑎1𝑥𝑛+𝑢(𝑡)
u
xx
xx
aaaaxx
xx
n
n
nnnn
n
10
00
1000
01000010
1
2
1
131
1
2
1
![Page 7: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/7.jpg)
Phase Variable Canonical form
• Output equation is simply
𝑥1=𝑦 , 𝑥2= �̇� , 𝑥3=𝑦 ,⋯ , 𝑥𝑛=𝑑𝑛−1 𝑦𝑑𝑡𝑛−1
n
n
xx
xx
y
1
2
1
0001
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8
∫ ∫ ∫ ∫
1a
2a
na
1xy 2xy
nn xy )1(
)(ny
1)2(
nn xy
…
)(tu
+ +
Phase Variable Canonical form
![Page 9: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/9.jpg)
9
Phase Variable Canonical form
yu s1
s1
s1
s1
1 1
1
2
3
1 n
n
1x
21 xx nx 1nx2nx
![Page 10: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/10.jpg)
• Obtain the state equation in phase variable form for the following differential equation, where u(t) is input and y(t) is output.
• The differential equation is third order, thus there are three state variables:
• And their derivatives are (i.e state equations)
2 𝑑3 𝑦𝑑𝑡3 +4 𝑑2 𝑦
𝑑𝑡2 +6 𝑑𝑦𝑑𝑡 +8 𝑦=10𝑢 (𝑡)
𝑥1=𝑦 𝑥2=�̇� 𝑥3= �̈�
�̇�1=𝑥2
�̇�2=𝑥3
�̇�3=−4 𝑥1−3𝑥2−2𝑥3+5𝑢 (𝑡)
Phase Variable Canonical form (Example-1)
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Phase Variable Canonical form (Example-1)
• In vector matrix form
𝑥1=𝑦 𝑥2=�̇� 𝑥3= �̈��̇�1=𝑥2
�̇�2=𝑥3
�̇�3=−4 𝑥1−3𝑥2−2𝑥3+5𝑢 (𝑡)
3
2
1
3
2
1
3
2
1
001)(
)(500
234100010
xxx
ty
tuxxx
xxx
Home Work: Draw Sate diagram
![Page 12: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/12.jpg)
• Consider the transfer function of a third-order system where the numerator degree is lower than that of the denominator.
• Transfer function can be decomposed into cascade form
• Denoting the output of the first block as W(s), we have the following input/output relationships:
Phase Variable Canonical form (Example-2)
𝑌 (𝑠)𝑈 (𝑠 )
=𝑏𝑜𝑠2+𝑏1𝑠+𝑏2
𝑠3+𝑎1𝑠2+𝑎2𝑠+𝑎3
1𝑠3+𝑎1𝑠2+𝑎2𝑠+𝑎3
𝑏𝑜𝑠2+𝑏1𝑠+𝑏2𝑈 (𝑠) 𝑌 (𝑠)𝑊 (𝑠)
𝑊 (𝑠)𝑈 (𝑠)
= 1𝑠3+𝑎1𝑠
2+𝑎2𝑠+𝑎3
𝑌 (𝑠)𝑊 (𝑠)
=𝑏𝑜𝑠2+𝑏1𝑠+𝑏2
![Page 13: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/13.jpg)
• Re-arranging above equation yields
• Taking inverse Laplace transform of above equations.
• Choosing the state variables in phase variable form
Phase Variable Canonical form (Example-2)
𝑊 (𝑠)𝑈 (𝑠)
= 1𝑠3+𝑎1𝑠
2+𝑎2𝑠+𝑎3
𝑌 (𝑠)𝑊 (𝑠)
=𝑏𝑜𝑠2+𝑏1𝑠+𝑏2
+
𝑌 (𝑠)=𝑏𝑜𝑠2𝑊 (𝑠 )+𝑏1𝑠𝑊 (𝑠)+𝑏2𝑊 (𝑠)
+
𝑦 (𝑡)=𝑏𝑜�̈� (𝑡 )+𝑏1 �̇� (𝑡 )+𝑏2𝑤(𝑡 )
𝑥1=𝑤𝑥2=�̇�𝑥3=�̈�
![Page 14: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/14.jpg)
• State Equations are given as
• And the output equation is
�̇�1=𝑥2 �̇�2=𝑥3 �̇�3=−𝑎3𝑥1−𝑎2𝑥2−𝑎1𝑥3+𝑢(𝑡 )
Phase Variable Canonical form (Example-1)
𝑦 (𝑡 )=𝑏2𝑥1+𝑏1𝑥2+𝑏𝑜 𝑥3
𝑏𝑜
𝑏2
𝑏1
𝑎1𝑎2
𝑎3
![Page 15: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/15.jpg)
• State Equations are given as
• And the output equation is
�̇�1=𝑥2 �̇�2=𝑥3 �̇�3=−𝑎3𝑥1−𝑎2𝑥2−𝑎1𝑥3+𝑢(𝑡 )
Phase Variable Canonical form (Example-1)
𝑦 (𝑡 )=𝑏2𝑥1+𝑏1𝑥2+𝑏𝑜 𝑥3
𝑏𝑜
𝑏2
𝑏1
−𝑎1
−𝑎2
−𝑎3
![Page 16: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/16.jpg)
• State Equations are given as
• And the output equation is
• In vector matrix form
�̇�1=𝑥2 �̇�2=𝑥3 �̇�3=−𝑎3𝑥1−𝑎2𝑥2−𝑎1𝑥3+𝑢(𝑡 )
3
2
1
12
3
2
1
1233
2
1
)(
)(100
100010
xxx
bbbty
tuxxx
aaaxxx
o
Phase Variable Canonical form (Example-1)
𝑦 (𝑡 )=𝑏2𝑥1+𝑏1𝑥2+𝑏𝑜 𝑥3
![Page 17: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/17.jpg)
Companion Forms
• Consider a system defined by
• where u is the input and y is the output. • This equation can also be written as
• We will present state-space representations of the system defined by above equations in controllable canonical form and observable canonical form.
ububububyayayay nn
nn
onn
nn
1
1
11
1
1
𝑌 (𝑠)𝑈 (𝑠 )
=𝑏𝑜𝑠𝑛+𝑏1𝑠𝑛−1+⋯+𝑏𝑛−1𝑠+𝑏𝑛
𝑠𝑛+𝑎1𝑠𝑛−1+⋯+𝑎𝑛−1𝑠+𝑎𝑛
![Page 18: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/18.jpg)
Controllable Canonical Form
• The following state-space representation is called a controllable canonical form:
𝑌 (𝑠)𝑈 (𝑠 )
=𝑏𝑜𝑠𝑛+𝑏1𝑠𝑛−1+⋯+𝑏𝑛−1𝑠+𝑏𝑛
𝑠𝑛+𝑎1𝑠𝑛−1+⋯+𝑎𝑛−1𝑠+𝑎𝑛
u
xx
xx
aaaaxx
xx
n
n
nnnn
n
10
00
1000
01000010
1
2
1
121
1
2
1
![Page 19: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/19.jpg)
Controllable Canonical Form
𝑌 (𝑠)𝑈 (𝑠 )
=𝑏𝑜𝑠𝑛+𝑏1𝑠𝑛−1+⋯+𝑏𝑛−1𝑠+𝑏𝑛
𝑠𝑛+𝑎1𝑠𝑛−1+⋯+𝑎𝑛−1𝑠+𝑎𝑛
ub
xx
xx
babbabbabbaby o
n
n
ooonnonn
1
2
1
112211
![Page 20: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/20.jpg)
Controllable Canonical Form
∫ ∫ ∫ ∫
1
2
n
1xv 2xv )(nv…
)(tu
+ + n
1n
1dtd
dtd
dtd
…
+
![Page 21: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/21.jpg)
Controllable Canonical Form (Example)𝑌 (𝑠)𝑈 (𝑠 )
= 𝑠+3𝑠2+3𝑠+2
0 1 3 3 2 1212 obbbaa
𝑌 (𝑠)𝑈 (𝑠 )
=0𝑠2+𝑠+3𝑠2+3𝑠+2
• Let us Rewrite the given transfer function in following form
uxx
aaxx
1010
2
1
122
1
uxx
xx
10
3210
2
1
2
1
![Page 22: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/22.jpg)
Controllable Canonical Form (Example)
0 1 3 3 2 1212 obbbaa
𝑌 (𝑠)𝑈 (𝑠 )
=0𝑠2+𝑠+3𝑠2+3𝑠+2
ubxx
babbaby ooo
2
11122
2
113xx
y
![Page 23: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/23.jpg)
Controllable Canonical Form (Example)𝑌 (𝑠)𝑈 (𝑠 )
= 𝑠+3𝑠2+3𝑠+2
• By direct decomposition of transfer function
)()(
233
)()(
2
2
2 sPssPs
sss
sUsY
)(2)(3)()(3)(
)()(
21
21
sPssPssPsPssPs
sUsY
• Equating Y(s) with numerator on the right hand side and U(s) with denominator on right hand side.
)1.......().........(3)()( 21 sPssPssY
)2.......().........(2)(3)()( 21 sPssPssPsU
![Page 24: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/24.jpg)
Controllable Canonical Form (Example)• Rearranging equation-2 yields
)3.......().........(2)(3)()( 21 sPssPssUsP
• Draw a simulation diagram using equations (1) and (3)
)(3)()( 21 sPssPssY )(2)(3)()( 21 sPssPssUsP
1/s 1/sU(s) Y(s)
-2
-3
P(s)
2x
12 xx 1x3
1
![Page 25: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/25.jpg)
Controllable Canonical Form (Example)
• State equations and output equation are obtained from simulation diagram.
213)( xxsY
122 23)( xxsUx
1/s 1/sU(s) Y(s)
-2
-3
P(s)
2x
12 xx 1x3
1
21 xx
![Page 26: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/26.jpg)
Controllable Canonical Form (Example)
• In vector Matrix form
213)( xxsY 122 23)( xxsUx 21 xx
)(10
3210
2
1
2
1 tfxx
xx
2
113xx
y
![Page 27: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/27.jpg)
Observable Canonical Form
• The following state-space representation is called an observable canonical form:
𝑌 (𝑠)𝑈 (𝑠 )
=𝑏𝑜𝑠𝑛+𝑏1𝑠𝑛−1+⋯+𝑏𝑛−1𝑠+𝑏𝑛
𝑠𝑛+𝑎1𝑠𝑛−1+⋯+𝑎𝑛−1𝑠+𝑎𝑛
u
babbab
babbab
xx
xx
aa
aa
xx
xx
o
o
onn
onn
n
n
n
n
n
n
11
22
11
1
2
1
1
2
1
1
2
1
100000
001000
![Page 28: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/28.jpg)
Observable Canonical Form
𝑌 (𝑠)𝑈 (𝑠 )
=𝑏𝑜𝑠𝑛+𝑏1𝑠𝑛−1+⋯+𝑏𝑛−1𝑠+𝑏𝑛
𝑠𝑛+𝑎1𝑠𝑛−1+⋯+𝑎𝑛−1𝑠+𝑎𝑛
ub
xx
xx
y o
n
n
1
2
1
1000
![Page 29: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/29.jpg)
Observable Canonical Form (Example)𝑌 (𝑠)𝑈 (𝑠 )
= 𝑠+3𝑠2+3𝑠+2
0 1 3 3 2 1212 obbbaa
𝑌 (𝑠)𝑈 (𝑠 )
=0𝑠2+𝑠+3𝑠2+3𝑠+2
• Let us Rewrite the given transfer function in following form
ubabbab
xx
aa
xx
o
o
11
22
2
1
1
2
2
1
10
uxx
xx
13
3120
2
1
2
1
![Page 30: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/30.jpg)
Observable Canonical Form (Example)
0 1 3 3 2 1212 obbbaa
𝑌 (𝑠)𝑈 (𝑠 )
=0𝑠2+𝑠+3𝑠2+3𝑠+2
ubxx
y o
2
110
2
110xx
y
![Page 31: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/31.jpg)
Similarity Transformations• It is desirable to have a means of transforming one state-space
representation into another.
• This is achieved using so-called similarity transformations.• Consider state space model
• Along with this, consider another state space model of the same plant
• Here the state vector , say, represents the physical state relative to some other reference, or even a mathematical coordinate vector.
)()()( tButAxtx
)()()( tDutCxty
)()()( tuBtxAtx
)()()( tuDtxCty
![Page 32: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/32.jpg)
Similarity Transformations• When one set of coordinates are transformed into another
set of coordinates of the same dimension using an algebraic coordinate transformation, such transformation is known as similarity transformation.
• In mathematical form the change of variables is written as,
• Where T is a nonsingular nxn transformation matrix.
• The transformed state is written as
)( )( txTtx
)( )( 1 txTtx
![Page 33: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/33.jpg)
Similarity Transformations• The transformed state is written as
• Taking time derivative of above equation )( )( 1 txTtx
(t) )( 1 xTtx
)()( )( 1 tButAxTtx
)( )( txTtx
)()()( tButAxtx
)()( )( 1 tButxATTtx
)()()( 11 tBuTtxATTtx )()()( tuBtxAtx
ATTA 1 BTB 1
![Page 34: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/34.jpg)
Similarity Transformations• Consider transformed output equation
• Substituting in above equation
• Since output of the system remain unchanged [i.e. ] therefore above equation is compared with that yields
)()()( tuDtxCty
)()()( 1 tuDtxTCty
CTC DD
![Page 35: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/35.jpg)
Similarity Transformations
• Following relations are used to preform transformation of coordinates algebraically
CTC DD
ATTA 1 BTB 1
![Page 36: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/36.jpg)
Similarity Transformations• Invariance of Eigen Values
ATTsIAsI 1
ITTATTTsT 111
TAsIT 1
AsI
AsIAsI
![Page 37: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/37.jpg)
Transformation to CCF• Transformation to CCf is done by means of transformation matrix
P.
• Where CM is controllability Matrix and is given as
and W is coefficient matrix
Where the ai’s are coefficients of the characteristic polynomial
WCMP
𝐶𝑀=[𝐵 𝐴𝐵 ⋯ 𝐴𝑛−1 𝐵 ]
0001001
011
1
32
121
a
aaaaa
Wnn
nn
s+
![Page 38: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/38.jpg)
Transformation to CCF• Once the transformation matrix P is computed following
relations are used to calculate transformed matrices.
CPC DD APPA 1 BPB 1
![Page 39: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/39.jpg)
Transformation to CCF (Example)• Consider the state space system given below.
• Transform the given system in CCF.
[𝑥1
𝑥2
𝑥3]=[1 2 1
0 1 31 1 1] [𝑥1
𝑥2
𝑥3]+[101 ]𝑢(𝑡 )
![Page 40: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/40.jpg)
Transformation to CCF (Example)
• The characteristic equation of the system is
[𝑥1
𝑥2
𝑥3]=[1 2 1
0 1 31 1 1] [𝑥1
𝑥2
𝑥3]+[101 ]𝑢(𝑡 )
|𝑠𝐼− 𝐴|=|𝑠−1 −2 −10 𝑠−1 −3−1 −1 𝑠−1|=𝑠3−3𝑠2−𝑠−3
𝑎1=−3 ,𝑎2=−1 ,𝑎3=−1
001013131
001011
1
12
aaa
W
![Page 41: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/41.jpg)
Transformation to CCF (Example)
• Now the controllability matrix CM is calculated as
• Transformation matrix P is now obtained as
[𝑥1
𝑥2
𝑥3]=[1 2 1
0 1 31 1 1] [𝑥1
𝑥2
𝑥3]+[101 ]𝑢(𝑡 )
𝐶𝑀=[𝐵 𝐴𝐵 𝐴2 𝐵 ]
𝐶𝑀=[ 1 2 100 3 91 2 7 ]
𝑃=𝐶𝑀×𝑊=[1 2 100 3 91 2 7 ] [−1 −3 1
−3 1 01 0 0 ]
𝑃=[3 −1 10 3 00 −1 1]
![Page 42: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/42.jpg)
Transformation to CCF (Example)• Using the following relationships given state space
representation is transformed into CCf as
APPA 1 BPB 1
313100010
1APPA
100
1BPB
|𝑠𝐼− 𝐴|=𝑠3−3𝑠2−𝑠−3
![Page 43: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/43.jpg)
Transformation to OCF• Transformation to CCf is done by means of transformation matrix
Q.
• Where OM is observability Matrix and is given as
and W is coefficient matrix
Where the ai’s are coefficients of the characteristic polynomial
1)( OMWQ
𝑂𝑀=[𝐶 𝐶𝐴 ⋯ 𝐶𝐴𝑛−1 ]𝑇
0001001
011
1
32
121
a
aaaaa
Wnn
nn
s+
![Page 44: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/44.jpg)
Transformation to OCF• Once the transformation matrix Q is computed following
relations are used to calculate transformed matrices.
CQC DD AQQA 1 BQB 1
![Page 45: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/45.jpg)
Transformation to OCF (Example)• Consider the state space system given below.
• Transform the given system in OCF.
[𝑥1
𝑥2
𝑥3]=[1 2 1
0 1 31 1 1] [𝑥1
𝑥2
𝑥3]+[101 ]𝑢(𝑡 )
𝑦 (𝑡)= [1 1 0 ] [𝑥1
𝑥2
𝑥3]
![Page 46: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/46.jpg)
Transformation to OCF (Example)
• The characteristic equation of the system is
[𝑥1
𝑥2
𝑥3]=[1 2 1
0 1 31 1 1] [𝑥1
𝑥2
𝑥3]+[101 ]𝑢(𝑡 )
|𝑠𝐼− 𝐴|=|𝑠−1 −2 −10 𝑠−1 −3−1 −1 𝑠−1|=𝑠3−3𝑠2−𝑠−3
𝑎1=−3 ,𝑎2=−1 ,𝑎3=−1
001013131
001011
1
12
aaa
W
![Page 47: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/47.jpg)
Transformation to OCF (Example)
• Now the observability matrix OM is calculated as
• Transformation matrix Q is now obtained as
[𝑥1
𝑥2
𝑥3]=[1 2 1
0 1 31 1 1] [𝑥1
𝑥2
𝑥3]+[101 ]𝑢(𝑡 )
𝑂𝑀=[𝐶 𝐶𝐴 𝐶𝐴2 ]𝑇
𝑂𝑀=[1 1 01 3 45 6 10]
𝑄=(𝑊 ×𝑂𝑀 )− 1=[ 0 .333 −0.166 0.333−0.333 0.166 0.6660.166 0.166 0.16 6]
𝑦 (𝑡)= [1 1 0 ] [𝑥1
𝑥2
𝑥3]
![Page 48: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/48.jpg)
Transformation to CCF (Example)• Using the following relationships given state space
representation is transformed into CCf as
310101300
1AQQA
123
1BQB
CQC DD AQQA 1 BQB 1
100CQC
![Page 49: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/49.jpg)
Home Work
• Obtain state space representation of following transfer function in Phase variable canonical form, OCF and CCF by – Direct Decomposition of Transfer Function– Similarity Transformation– Direct Approach
𝑌 (𝑠)𝑈 (𝑠 )
=𝑠2+2 𝑠+3
𝑠3+5𝑠2+3 𝑠+2
![Page 50: Feedback Control Systems (FCS)](https://reader031.vdocuments.mx/reader031/viewer/2022020919/56815f11550346895dcdd377/html5/thumbnails/50.jpg)
END OF LECTURES-26-27-28-29
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