This document discusses state space analysis and the conversion of transfer functions to state space models. It covers:
1. The need to convert transfer functions to state space form in order to apply modern time domain techniques for system analysis and design.
2. Three possible representations for realizing a transfer function as a state space model: first companion form, second companion form, and Jordan canonical form.
3. The concepts of eigenvalues and eigenvectors, and how they relate to state space models.
4. Worked examples of converting transfer functions to state space models in first and second companion forms, as well as the Jordan canonical form for systems with repeated and non-repeated roots.
The document provides an overview
The Most Attractive Pune Call Girls Budhwar Peth 8250192130 Will You Miss Thi...
State space analysis, eign values and eign vectors
1. By:
Shilpa Mishra
M.E., NITTTR Chandigarh
.
STATE SPACE ANALYSIS
Conversion of T.F. models to canonical state variable models and
concept of Eignvalues & Eignvectors.
2. Contents_______________
1. Introduction
2. Need of realization of transfer function into state variable models
3. Realization of transfer function into a state space model or
mathematical model in following possible representation:
a. First companion form (controllable form)
b. Second companion form (observable form)
c. Jordan canonical form
4. Eigenvalues and Eigen vectors
08/05/142 Shilpa Mishra ME IC 122509
3. Plant
Mathematical Model :
Differential equation
Linear, time invariant
Frequency Domain
Technique
Time Domain
Technique
Two approaches for analysis and design of control system:
1.Classical Technique or Frequency Domain Technique.(T.F. + graphical plots like
root locus, bode etc.)
2.Modern Technique or Time Domain Technique (State variable approach).
08/05/143
Shilpa Mishra ME IC 122509
4. Transfer Function form
Need of conversion of transfer function form into state space form:
1.A transfer function can be easily fitted to the determined experimental data in best
possible manner. In state variable we have so many design techniques available for system.
Hence in order to apply these techniques T.F. must be realized into state variable model.
08/05/144 Shilpa Mishra ME IC 122509
6. BuAxx +=
DuCxy +=
x
x
y
u
A
B
C
D
= state vector
= derivative of the state vector with respect to time
= output vector
= input or control vector
= system matrix
= input matrix
= output matrix
= Feed forward matrix
State equation
output equation
General State Space form of Physical System
08/05/146 Shilpa Mishra ME IC 122509
7. Deriving State Space Model from Transfer Function Model
The process of converting transfer function to state space form is NOT
unique. There are various “realizations” possible.
All realizations are “equivalent” (i.e. properties do not change). However,
one representation may have some advantages over others for a particular
task.
Possible representations:
1. First companion form
2. Second companion form
3. Jordan canonical form
08/05/147 Shilpa Mishra ME IC 122509
8. 1. First Companion Form (SISO System)
If LTI SISO system is described by transfer function of the form;
Decomposition of transfer function:
.
012
23
3
01
2
2
)(
)(
)(
)(
asasasa
bsbsb
sR
sC
sU
sY
+++
++
==
08/05/148 Shilpa Mishra ME IC 122509
9. ( ) ( ) ( ) ( )sXbsbsbsCsY 101
2
2 ++==
( ) 10
1
12
1
2
2 xb
dt
dx
b
dt
xd
bty ++=
)2........(..........322110)( xbxbxbty ++=
( ) ( ) ( ) ( )sXasasasasRsU 101
2
2
3
3 +++==
( ) )(
)()()(
10
1
12
1
2
23
1
3
3 txa
dt
tdx
a
dt
txd
a
dt
txd
atu +++=
43322110)( xaxaxaxatu +++=
I.
II.
)1...().........(32211043 tuxaxaxaxa +−−−=
21 xx = 32 xx =1)( xtx = 43 xx =&
Select state
variables like :
08/05/149 Shilpa Mishra ME IC 122509
10. 21 xx = 32 xx =1)( xtx =
from equation (1) & (2) and state equation, block diagram realization in first
companion form of TF will be
43 xx =
08/05/1410 Shilpa Mishra ME IC 122509
11. Again from equation (1) & (2) complete state model will be ;
)3).....((
3
/1
0
0
3
2
1
210
100
010
3
1
3
2
1
,
)(
3
1
3
3
2
2
3
1
1
3
0
43
)(32211043
tu
a
x
x
x
aaa
a
x
x
x
or
tu
a
x
a
a
x
a
a
x
a
a
xx
tuxaxaxaxa
+
−−−
=
+−−−==
+−−−=
A
B
08/05/1411 Shilpa Mishra ME IC 122509
12. Equation (3)&(4) combining together gives the complete realization of the given
transfer function.
Matrix A has coefficients of the denominator of the TF preceded by minus sign in its
bottom row and rest of the matrix is zero except for the superdiagonol terms which are
all unity.
In matrix theory matrix with this structure is said to be in companion form therefore
this realization is called first companion form of realizing a TF.
[ ] )4.......(
3
2
1
210
)(
,
322110)(
=
++=
x
x
x
bbbty
or
xbxbxbty
C
08/05/1412 Shilpa Mishra ME IC 122509
13. Example :TF to State Space (constant term in numerator)
rcccc 2424269 =+++
cx =1 cx =2 cx =3
1. Inverse Laplace
2. Select state variables
21 xx =
32 xx =
rxxxx 2492624 3213 +−−−=
1xcy ==
( ) ( )
( )sD
sN
sG =
( )sN
( )sD
numerator
denominator
08/05/1413 Shilpa Mishra ME IC 122509
16. 2. Second Companion Form (SISO System)
In second companion form coefficient of the denominator of the transfer
function appear in one of the column of the A matrix.
This form can be obtained by the following steps:
Let the transfer function is,
n
nnn
n
nnn
asasas
bsbsbsb
sU
sY
sH
++++
++++
== −−
−−
......
........
)(
)(
)( 2
2
1
1
2
2
1
10
)5)].......(()([
1
....)]()([
1
)()(
0)]()([...)]()([)]()([,
)()....()().....(
110
11
1
0
1
10
1
1
sYasUb
s
sYasUb
s
sUbsY
sUbsYasUbsYassubsYsor
sUbsbsbsYasas
nnn
nn
nn
n
nn
n
nn
−++−+=
=−++−+−
+++=+++
−
−−
On dividing by and solving for Y(s);n
s
08/05/1416 Shilpa Mishra ME IC 122509
17. By equation (5) block diagram can be drawn as follows which is called second companion
form of realization:-
08/05/1417 Shilpa Mishra ME IC 122509
18. To get state variable model, output of each integrator is identified as
state variables starting at the left and preceding to the right.
The corresponding differential equations are,
ub
n
xy
u
n
bub
n
x
n
ax
u
n
bub
n
x
n
axx
ubub
n
xa
n
x
n
x
ubub
n
xa
n
x
n
x
0
isequationoutputtheand
)
0
(
1
1
)
0
(
112
2
)
0
(
221
1
)
0
(
11
+=
++−=
−
++
−
−=
++−
−
=
−
++−
−
=
08/05/1418 Shilpa Mishra ME IC 122509
19. Now from here state and output equations organized in matrix form are given
below:
BuAxx +=
DuCxy +=
[ ] 0
011
011
0
1
2
1
;1000
B;
100
010
001
000
bDC
bab
bab
bab
a
a
a
a
A nn
nn
n
n
n
==
−
−
−
=
−
−
−
−
= −−−
−
08/05/1419 Shilpa Mishra ME IC 122509
20. •In this form of realizing a TF the poles of the transfer function form a string
along the main diagonal of the matrix A.
•In Jordan canonical form state space model will be like:-
3. Jordan canonical form (Non-repeated roots)
08/05/1420 Shilpa Mishra ME IC 122509
28. Eignvalues and Eignvector
Definition :
Given a linear transformation A, a non-zero vector x is defined to be an
eigenvector of the transformation if it satisfies the eigenvalue equation
for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A
corresponding to the eigenvector x.
It indicates that vector x has the property that its direction is not changed by the
transformation A, but that it is only scaled by a factor of λ.
Only certain special vectors x are eigenvectors, and only certain special scalars λ
are eigenvalues.
The eigenvector must be non-zero because the equation A0 = λ0 holds for every A
and every λ.
λxAx =
08/05/1428 Shilpa Mishra ME IC 122509
29. A acts to stretch the vector x, not change its direction, so x is an eigenvector of A.
The eigenvalue λ is simply the amount of "stretch" or "shrink" to which a vector
is subjected when transformed by A.
If λ = 1, the vector remains unchanged (unaffected by the transformation). A
transformation I under which a vector x remains unchanged, Ix = x, is defined as
identity transformation
If λ = −1, the vector flips to the opposite direction; this is defined as reflection.
08/05/1429 Shilpa Mishra ME IC 122509
30. Computation of eigenvalues & the characteristic
equation
When a transformation is represented by a square matrix A, the eigenvalue equation
can be expressed as;
Ax = λx
(A − λI)x = 0
As x must not be zero, this can be rearranged to;
det(A − λI) = 0.
which is defined to be the characteristic equation of the n × n matrix A.
Expansion of the determinant generates a polynomial of degree n in λ and may be
written as;
here λ1, λ2……λn are called eignvalues. For each eign value there exist eign vector x.
08/05/1430 Shilpa Mishra ME IC 122509
31. Example 1.
find the eigenvalues and eigenvectors of the matrix
Solution
So the eigenvalues are 3.421 and 0.3288.
08/05/1431 Shilpa Mishra ME IC 122509
32. Let
be the eigenvector corresponding to
Hence
.421x1 +1.5x2 = 0
.75x1 +2.671x2 = 0
08/05/1432 Shilpa Mishra ME IC 122509
33. If
then
The eigenvector corresponding to then is
The eigenvector corresponding to
is
Similarly, the eigenvector corresponding to
is
08/05/1433
Shilpa Mishra ME IC 122509
34. Example 2.
Find the eigenvalues and eigenvectors of
Solution
The characteristic equation is given by
The roots of the above equation are
Note that there are eigenvalues that are repeated. Since there are only two distinct
eigenvalues, there are only two eigenspaces. But, corresponding to λ = 0.5 there should be
two eigenvectors that form a basis for the eigenspace
08/05/1434
Shilpa Mishra ME IC 122509
35. To find the eigenspaces, let
Given
then
For ,
Solving this system gives
08/05/1435 Shilpa Mishra ME IC 122509
36. So
So the vectors and form a basis for the eigenspace for the eigenvalue .
For ,
Solving this system gives
The eigenvector corresponding to is ;
Hence the vector is a basis for the eigenspace for the eigenvalue of .
08/05/1436
Shilpa Mishra ME IC 122509