5. Symmetric matrix
A symmetric matrix is a square matrix that is
equal to its transpose. Formally, matrix A is
symmetric if
The following 3×3 matrix is symmetric:
6. Skew Symmetric Matrix
A skew symmetric (or antisymmetric or antimetric)
matrix is a square matrix whose transpose is also its
negative; that is, it satisfies the condition -A = AT.
For example, the following matrix is skew-symmetric:
8. Hermitian matrix
A Hermitian matrix (or self-adjoint matrix) is
a square matrix with complex entries that is
equal to its own conjugate transpose
matrix A is Hermitian if it satisfies the relation
9. Skew-Hermitian matrix
In linear algebra, a square matrix with complex
entries is said to be Skew-Hermitian
or antihermitian if its conjugate transpose is
equal to its negative.
matrix A is skew-Hermitian if it satisfies the
relation
10.
11.
12.
13. Rank of a Matrix
The maximum number of linearly independent
rows in a matrix A is called the row rank of A,
and the maximum number of linearly
independent columns in A is called
the column rank of A.
Rank (A m*n) <=min(m,n)
r3=2r1-r2
r4=-3r1+2r2
18. • In linear algebra, an eigenvector or characteristic
vector of a square matrix is a vector that does not
change its direction under the associated linear
transformation.
In other words—if v is a vector that is not zero,
then it is an eigenvector of a square
matrix A if Av is a scalar multiple of v. This
condition could be written as the equation:
Av = λv
19. • where λ is a scalar known as
the eigenvalue or characteristic
value associated with the eigenvector v.
Geometrically, an eigenvector corresponding
to a real, nonzero eigenvalue points in a
direction that is stretched by the
transformation and the eigenvalue is the
factor by which it is stretched. If the
eigenvalue is negative, the direction is
reversed.
20. In this shear mapping the red arrow changes
direction but the blue arrow does not. The blue
arrow is an eigenvector of this shear mapping
because it doesn't change direction, and since its
length is unchanged, its eigenvalue is 1
27. Vibration Analysis
The eigenvalues are used to determine the natural frequencies
(or eigenfrequencies) of vibration, and the eigenvectors determine
the shapes of these vibrational modes.
Practical Applications in Structural Engineering
Most structures from buildings to bridges have a natural frequency
of vibration. It means all these structures have their own system of
eigenvibrations and eigenfrequencies. Now external forces like wind
and earthquake may cause these structures to undergo vibrations.
In case the frequency of these vibrations becomes equal to the
natural frequencies of these structures, vibrations with large
amplitudes are set up. It is a phenomena called Resonance. This can
lead to the collapse of the structure by a process called aeroelastic
flutter. One very famous example of the collapse of a structure due
to this phenomena is the Tacoma Narrows Bridge (1940) in which
the wind provided an external periodic frequency that matched the
bridge's natural structural frequency. So vibration analysis of these
structures are done at the time of their design using eigenvalues
and eigenvectors.
29. Mode Shape of a Tuning
Fork at Eigenfrequency
440.09 Hz
30. In image processing, processed
images of faces can be seen as
vectors whose components are
the brightnesses of each
pixel. The dimension of this
vector space is the number of
pixels. The eigenvectors of
the covariance matrix associated
with a large set of normalized
pictures of faces are
called eigenfaces; this is an
example of principal components
analysis. In the facial
recognition branch of biometrics,
eigenfaces provide a means of
applying data compression to
faces for identification purposes.
31. Research related to Eigen vision systems
determining hand gestures has also been
made.
eigenvoices represent the general direction of
variability in human pronunciations of a
particular utterance, such as a word in a
language. Based on a linear combination of
such eigenvoices, a new voice pronunciation
of the word can be constructed. These
concepts have been found useful in automatic
speech recognition systems for speaker
adaptation.
32. • Eigenvalues can also be used to test for cracks or
deformities in structural components used for construction.
When a beam is struck, its natural frequencies
(eigenvalues) can be heard or measured. If the beam
"rings," then it is not flawed. A dull sound will result from a
flawed beam because the flaw causes the eigenvalues to
change. Sensitive machines can be used to "see" and
"hear" eigenvalues more precisely.
• The eigenvalues can also be used to determine if a
structure has deformed under the application of a
particular force. Eigenvalues for the structure are measured
before and after the application of force. If a change in the
eigenvalues is observed, it means the structure has
undergone deformation.
33. • Eigenvalues were used by Claude Shannon to determine the
theoretical limit to how much information can be transmitted
through a communication medium like your telephone line or
through the air. This is done by calculating the eigenvectors and
eigenvalues of the communication channel (expressed a matrix),
and then waterfilling on the eigenvalues.
The eigenvalues are then, in essence, the gains of the fundamental
modes of the channel, which themselves are captured by the
eigenvectors.
Google uses the eigenvector corresponding to the maximal
eigenvalue of the Google matrix to determine the rank of a page
for search.
Eigenvectors are fundamental to principal components analysis
which is commonly used for dimensionality reduction in face
recognition and other machine learning applications.
Eigenvectors can also be used for latent semantic analysis, a NLP
technique for extracting topics and concepts from text documents.
34. Contents
• Need of State Variable Approach
• Concept of State, State Variable, State Vector,
State Space
• State Variable Modeling
• Transformation of State Variable
• Conversion of State Variable Models to Transfer
functions
• Cayley-Hamilton Theorem
35. Need of State variable Approach
Earliest methods modeled the physical systems in the form of transfer function.
It suffered from certain drawbacks:-
1) defined only under zero initial conditions.
2) Applicable only to LTI systems & are generally restricted to SISO systems.
3) reveals system o/p only for a given i/p & gives no information regarding the
internal states of the systems.
4) Classical design methods based on transfer function model are essentially trial &
error procedures which are difficult to visualize & organize in complex systems.
we needed a more general mathematical representation of a system which gives
information about the state of the system variables. State Variable
Approach(time domain approach) is a very powerful technique for design &
analysis of linear & non-linear ,time invariant & time varying MIMO
system.
36. Concept of state ,state variable, state space
A mathematical abstraction to represent the dynamics of a system
utilizes three types of variables called the
1) i/p
2) o/p
3) state variables.
Consider the mechanical system shown in figure below wherein
mass M is acted upon by force F(t). fig (a):-
38. State Equation can be written as:
Output Equation y(t ) is given as :
u
Mx
x
x
x
/1
0
00
10
2
1
2
1
2
1
01)(
x
x
ty
39. From these relations we get:-
v(t) =
=
x(t) =
= x(to) + [t-to]v(to) +
“The state of a dynamical system is a minimum set of
variables(known as state variable) such that knowledge of these
variables at t = to with the knowledge of the inputs for t ≥ to
completely determine the behavior of the system for t ≥ to”.
M
1
t
to
t to
dttF
M
dttF
M
dttF )(
1
)(
1
)(
t
to
dttFtov )()(
t to t
to
dttvdttvdttv )()()(
t
to
t
to
dttFd )(
40. The different variables may be presented by i/p vector u(t), o/p
vector y(t) & state vector x(t).
The state space representation may be visualized in block
diagram form as shown below:
)(
.
.
)(
)(
)(,
)(
.
.
)(
)(
)(,
)(
.
.
)(
)(
)(
2
1
2
1
2
1
tx
tx
tx
tx
ty
ty
ty
ty
tu
tu
tu
tu
npm
Controlled system
(State variables(n))
I/p(u)
m variable
o/p (Y)
P Variables
State (x)
n Variables
41. • For a general system of fig. above the state variable
representation can be arranged in the form of n first-order
differential equations(state equations):
• Integration of equation gives:-
thus the n state variables & hence the state of the system can be
determined uniquely at any t > to if each state variable is
known at t = to and all the m control forces are known
throughout the interval to to t.
),..,,,..,,(
.
.
),..,,,,..,,(
2121
212111
1
mnnn
n
mn
uuuxxxfx
dt
dx
uuuxxxfx
dt
dx
ni
dtuuuxxxftoxtx
t
to
mniii
....,3,2,1
),...,,,...,()()( 2121
42. The above ‘n’ differential equations may be written in vector
notation as
where x is n x 1 state vector, u is m x 1 is a input vector &
f(.) =
is n x 1 function vector.
))(),(()( tutxftx
(.)
.
.
(.)
(.)
2
1
nf
f
f
43. State The state of a dynamic system is the smallest set of variables
(state variables) such that the knowledge of these variables at t=t0,
together with the knowledge of the input for t>=to, completely
determine the behaviour of the system for any time t>=to
State Variables
In mechanical systems, the position coordinates and velocities of
mechanical parts are typical state variables; knowing these, it is
possible to determine the future state of the objects in the system.
In thermodynamics, a state variable is also called a state function.
Examples include temperature, pressure, volume, internal
energy, enthalpy, and entropy. In contrast heat and work are not
state functions, but process functions.
In electronic circuits, the voltages of the nodes and
the currents through components in the circuit are usually the state
variables.
44. In ecosystem models, population sizes (or
concentrations) of plants, animals and resources
(nutrients, organic material) are typical state variables.
In electric circuits, the number of state variables is
often, though not always, the same as the number of
energy storage elements in the circuit such
as capacitors and inductors.
State Vector If n state variables x1,x2,…..xn are needed
to completely describe the behaviour of a given system,
then these n state variables can be considered the n
components of a vector x. Such a vector is called a state
vector.
45. At any time to the state vector x determines a point (called the
state point) in an n-dimentional space (x1 axis, x2 axis……..xn
axis) called state space.
As the time progresses and the system state changes, a set of
points will be defined. This set of points, locus of the above tip
of the state vector as time progresses, is called the state
trajectory of the system.
The o/p y(t) in general form can be expressed as:-
y(t) = g(x(t),u(t))
The state equations and output equations constitute the
state model of any system.
y(t) = g(x(t),u(t))
))(),(()( tutxftx
46. State Variable Modeling
State model of a linear time invariant system is a special case
of the general time invariant model. Derivative of each state
variable now becomes a linear combination of system states &
outputs, i.e.
mmnn ubububxaxaxax 121211112121111
mmnn ubububxaxaxax 22222121222221212
mnmnnnnnnnn ubububxaxaxax 22112211
.
.
Written as:-
)()()( tButAxtx
47. Where x(t) is a nx1 state vector, u(t) is mx1 input vector . A is n x
n system matrix defined by:-
A =
B is n x m input matrix defined by:-
B =
nnnn
n
n
aaa
aaa
aaa
.....
.....
.....
21
.
.
.
22221
11211
nmnn
m
m
bbb
bbb
bbb
.....
.....
.....
21
.
.
22221
11211
48. • Similarly o/p variables at time t are linear combinations of
values of input and state variables at time t i,e,:
where coefficients cij & dij are constant. In matrix form :
y(t) = C x(t) + D u(t)
where y(t) is p x 1 o/p vector. C is p x n o/p matrix defined by:
C =
D is transmission matrix .
)(...)()(....)()(
.
.
)(...)()(....)()(
111
111111111
tudtudtxctxcty
tudtudtxctxcty
mpmpnpnpp
mmnn
pnnp
n
n
ccc
ccc
ccc
.
....
....
....
21
22221
11211
49. For the system shown in figure (a) let us define:
pmpp
m
m
ddd
ddd
ddd
D
....
....
....
....
21
22221
11211
The state model of linear time invariant systems is thus
given by the following equations
)()()(
)()()(
.
tdutcxty
tButAxtx
50. • Use of DC motor in speed control systems:-
• Separately excited DC motor drives the load. A DC tachometer is
attached to the motor shaft . Speed signal is feedback & error signal
is used to control the armature voltage of motor.
• To drive the plant model we have the following diagram:-
52. • The counter electromotive force eb which is proportional to ø & ω
is given as:-
where kb is back emf cont. (volts /rad/sec)
so we can write:-
are the state variable & o/p variable is y(t) = ω (t). Then plant model
can be written as:-
)()( tkte bb
)()(&)()(
)()(
)(
)(
1
)()(
)(
21 titxttx
t
j
B
ti
j
k
dt
td
tu
L
t
L
k
ti
L
R
dt
tdi
a
a
T
aa
b
a
a
aa
)()(
)(
/1
0
)(
)(
)(
)(
1
2
1
2
1
txty
tu
Ltx
tx
L
R
L
K
j
k
j
B
tx
tx
a
a
a
a
b
T
53. Transformation of state variable
2) The change of variables is represented by a linear
transformation:- x = P ----------(1)
Transformation matrix p is a nonsingular constant matrix of n
x n order. The original dynamics are presented by:-
Substitution of eq.(1) gives:-
1) The state variables used in the original formulation of the
dynamics of a system ate not as convenient as another set of
state variables.
x
)()()(
)();()()(
tdutcxty
xtxtbutAxtx o
o
ddcPcbPbAPPA
with
tudtxcty
toxPtoxtubtxAtx
or
tdutxcPty
tbutxAPtxP
,,,
)()()(
)()(),()()(
)()()(
)()()(
11
1
.
.
54. For the speed control system we have angular velocity ω(t) & armature
current as state variable. S o
We define new state variables:-
With transformation
We can write:-
)(tia
aixx 21 ,
2
1
21
1
2
1
11
01
x
x
xx
x
x
x
x
xpx
11
01
P
)()(
)()(
.
txcty
tuBxAtx
56. Transfer Function form:
Need of conversion of transfer function form into state space form:
1. Often the system dynamics is determined using standard test signals like a step, impulse
or sinusoidal signal. 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.
57. 2. For transient response simulation frequency domain design methods are not
helpful. For that It is must to invert design from s-domain to t-domain but
there is not much software for this. Hence it is better to convert transfer
function to state variable model and numerically integrating the resulting
differential equations rather than attempting to compute the inverse laplace
transform by numerical methods.
Note: There are certain limitations in classical design techniques
which can be overcome by time domain technique or state variable
approach. (discussed earlier).
58. General State Space form of Physical system
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
= feedforward matrix
State equation
output equation
59. 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
60. 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
61. 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..(..........
3
)(3322311304 atuxaaxaaxaax
21 xx 32 xx 1)( xtx 43 xx &
Select state
variables like :
62. 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
63. 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
tu
a
x
x
x
aaa
a
x
x
x
or
tu
a
x
a
a
x
a
a
x
a
a
xx
A B
21 xx 32 xx
64. 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
65. Determine the First Companion form
)(
)(
2024293
3
)(
sU
sY
sss
s
sG
67. Second Companion Form
• In this form the coefficient appear in a column of the A matrix.
• This can be obtain by writing equation (1) as
)()........1
10()()........1
1( sUnsnsnsYnsnsn
or
0)](0)([..........)](1)(1[1)](0)([ sUsYnsUsYsnsUsYsn
• On dividing by and solving for Y(s), we obtainsn
)]()([1.......)](1)(1[1)(0)( sYnsUn
sn
sYsU
s
sUsY
Note that is the transfer function of a chain of n integrators.
sn
1
nsnsn
nsnsn
sU
sY
......1
1
.....1
10
)(
)(
68. • The signal passes through n integrators ; the signal
passes through n-1 integrators and so forth to complete the realization of equation
βn βn-1 βn-2 β1
β0
αn
αn-1 αn-2 α1
u
+
-
x1
x2
xn-1 xn
y
Realization of equation (9)
ynun ynun 11
)]()([1.......)](1)(1[1)(0)( sYnsUn
sn
sYsU
s
sUsY
69. •.• To write the differential equation for the realization identify the output of each
integrator with a state variable starting at the left and proceeding to the right
ẋn = xn-1 -α1 ( xn + β0 u ) + β1 u
ẋn-1 = xn-2 - α2 ( xn + β0 u ) + β2 u
:
ẋ2 = x1 - αn-1 ( xn + β0 u ) + βn-1 u
ẋ1 = - αn ( xn + β0 u ) + βn u
and the output equation is
y = xn + β0 u
• The state and output equation organized in vector matrix form are given below
ẋ (t) = A x(t) + B u(t)
y (t) = C x(t) + D u(t)
(10)
70. 0 0 … 0 -αn
1 0 … 0 -αn-1
0 1 … 0 -αn-2
: : : : :
0 0 0 1 -α1
A =
; B =
βn – αn β0
βn-1 – αn-1 β0
βn -2 – αn-2 β0
:
- β1 – α1 β0
C = [ 0 0 … 0 1 ]
;
D = β0
A , B , C or D matrix of second companion form correspond ot the transpose of
the A , B , C or D respectively to the first one.
• This state-space realization is also called observable canonical form because the
resulting model is guaranteed to be observable (i.e., because the output exits from
a chain of integrators, every state has an effect on the output).
• These form also play an important role in pole placement design through state
feedback.
73. Jordan Canonical Form
• In this form the poles of the transfer function form a string along the main diagonal of the
matrix.
nsn
nsn
nsnsn
sG
......1
......1
10)(
• By long division , G(s) can be written as
)('0
.....1
1
'.....1'
1
0)( sG
nsnsn
nsn
sG
or
ns
rn
s
r
s
r
sU
sY
sG
.....
2
2
1
1
0)(
)(
)(
• The coefficient (i = 1,2,…….,n ) are the residue of the transfer function G’(s)
at the poles at s = ( i = 1,2,…..,n).
ri
i
(11)
74. • The transfer function consists of a direct path with gain , and first order transfer
function in parallel.
0
λ1
λ2
λn
β0
r1
r2
rn
+
u y
Realization of G(s) in equation (11)
x1
x2
xn
ns
rn
s
r
s
r
sU
sY
sG
.....
2
2
1
1
0)(
)(
)(
75. • Identifying the outputs of integrator with the state variables results in following state
and output equations:
λ1 0 … 0
0 λ2 … 0
0 0 … 0
: : : :
0 0 0 λn
ẋ (t) = x(t) + B u(t)
y (t) = C x(t) + D u(t)
ʌ = ;
1
1
1
:
1
B =
C = [ r1 r2 ….. rn ] ; D = β0
• It is observed that for this canonical state variable model , the matrix A is a diagonal
with the poles of G(s) as its diagonal elements.
• The unique decoupled nature of the canonical model is obvious from eqn (12); the
n first order differential equation are independent of each other.
ẋ (t) = λi xi(t) + u(t) ; i = 1 , 2 , 3 …….,n
(12)
(13)
76. • Assume that G(s) has m distinct poles at s = λ1 , λ2 , ……… , λm of multiplicity
n1 , n2 , ……… , nm respectively: s = n1 + n2 + ……… + nm i.e. G(s) is of the form
)(.........)2( 2)1( 1
'.........2'
2
1'
1
0)(
ms nms ns n
nsnsn
sG
• The partial fraction expansion of G(s) is of the form
)(
)(
)(.......)(10)(
sU
sY
sH msHsG
where
)(
)(
)(
.........
)( 1
2
)(
1)(
sU
sYi
is
rini
is ni
ri
is ni
risHi
• The first term in Hi(s) can be synthesized as a chain of ni identical, first order
systems , each having transfer function 1/(s-λi).
• The second order term can be synthesized by a chain of (ni-1) first order system ,
and so forth.
(14)
(15)
77. • The entire Hi(s) can be synthesized by the system having block diagram shown in
figure.
rin1 ri2 ri1
λi λi λi
u +
+
yi
xini xi2 xi1
Realization of Hi(s) in equation (15)
• Now to get state variable we identify the output of each integrator with a state variable
starting at the right and proceeding to the left.
78. • The corresponding differential equation are
ẋi1 = λi xi1 + xi2
ẋi2 = λi xi2 + xi3
:
uxiniixini
.
And the output is given by
xrxrxr ii ininiiii
yi 2211 .........
• If the state vector for the subsystem is defined by
rinixixi
T
xi 21
• Then equation can be written in standard form
ẋi = ʌi xi + Bi u
yi = Ci xi
where
i
i
i
i
000
1000
010
001
1
0
0
0
Bi; ;
rrrC ini
iii
21
(16a)
(16b)
(17)
79. Note that the matrix has two diagonals- the principle diagonal has the corresponding
characteristic root (pole) and the super diagonal has all 1’s.
i
• In matrix theory , a matrix having this structure is said to be in Jordan form. That’s
why this realization is identified as Jordan Canonical Form.
• The state vector of the overall system consists of the concatenation of state vector
of each of the Jordan blocks:
xm
x
x
x
2
1
• Since there is no coupling between any of the subsystem , the matrix of the
overall system is ‘block diagonal’: where each of the sub matrices is in the
Jordan canonical form.
i
84. INTRODUCTION
• Controllability is an important property of a
control system,and the controllability property
plays crucial role in many control problems,such
as stabilization of unstable systems by feedback
or optimal control.
• The conditions of controllability and observability
may govern the existence of a complete solution
to the control system design problem.
85. DEFINITIONS
• Controllability In order to be able to whatever
we want with the given dynamic system under
control input,the system must be controllable.
• Observability The method of determining the
state of a system by observing its output
concerns observability.In order to see what is
going on inside the system under observation,the
system must be observable.
86. CONTROLLABILITY
Controllability is in relation to transfer of a system
from one state to another by appropriate input
controls in a finite time.
Consider the continous linear time-invariant
system.
ẋ(t)=Ax(t)+Bu(t) state equation(a)
y(t)=Cx(t)+Du(t) output equation(b)
87. where,
A is the n×n “state matrix”
B is the n×1 “input matrix”
C is the 1×n “output matrix”
D is the 1×1 “feed forward matrix”
x(t) is the n×1 “state vector”
y(t) is the “output variables”
u(t) is the “input variables”
88. • For the linear system given by equation
(a),if there exists an input u[0,t1] which
transfers the initial state x(0)=x0 to the
state x¹ in a finite time t1,the state x0 is
said to be controllable.If all initial states
are controllable,the system is said to be
completely controllable,or simply
controllable.Otherwise , the system is said
to be uncontrollable.
89. • The solution of equation (a) is
If the system is controllable , there exists an input
u[0,t1] such that
From this equation we observe that complete
controllability of a system depends on A and B and
is independent of output matrix C. The
controllability of the system is frequently referred
to as the controllability of the pair [A,B].
90. CONTROLLABILITY TEST
Sometimes controllability control is not possible
and this can verified by using controllability test
matrix.
The n×n controllability matrix is given by
U=[B AB A2B.....An-1B]
This test allows the controllability of a system
to be easily checked.
91. • The matrix U is known as controllability test
matrix.
• The controllability condition of a system
depends on the coefficient matrices A and B,
and it is said that the pair A,B is controllable
indicating that the rank of the test matrix is n.
• However if controllability matrix U is n x n i.e.
square matrix, then the condition for state
controllability is I U I ≠ 0 i.e. matrix be non
singular.
92. Example 1 - Verify whether the following system
is controllable :
• ẋ1 -2 0 x1 1
= + u
ẋ2 0 -1 x2 1
Soln.
1 -2 0
B = and A =
1 0 -1
93. -2 0 1 -2
AB= =
0 -1 1 -1
U= [B : AB ]= 1 -2
1 -1
I U I= [1x (-1)- 1×(-2)]=1
The test matrix U is found to non-singular,
hence the rank of the test matrix U is equal to
n(n=2) and the system is controllable.
94. OBSERVABILITY
• For the linear system given by equation (a) ,if
the knowledge of the output y and the input u
over a finite time (0,t1) suffices to determine
the state x(0)= x0 the state x0 is said to be
observable. If all initial states are observable,
the system is said to be completely observable
,or simply observable. Otherwise , the system
is said to be unobservable.
95. • The output of the system is given by
The output and the input can be measured and
used so that following signal ŋ.
Multiplying by and integrating from 0 to t1,
gives
96. • When the signal ŋ(t) is available over a time
interval [0, t1] and the system is observable
then the initial state x0 can be uniquely
determined from above equation.
• From above equation we see that complete
observability of a system depends on A and C
and is independent of B. The observability of
the system is frequently referred to as the
observability of the pair A,C.
97. OBSERVABILITY TEST
• The n×n observability matrix is given by
C
V= CA
.
.
CAn-1
V= [ CT : AT :CT : ………. :(AT)n-1 CT]
The above matrix is to have a rank of n
99. QUADRATIC FORM
Let A denote an n x n symmetric matrix with real entries
and x denote an n x 1 column matrix.
Then,
Q = x’Ax
is said to be a quadratic form.
101. Classification of the quadratic form Q
a: positive definite:
Q > 0 when x ≠ 0
b: positive semi-definite:
Q ≥ 0 for all x
c: negative definite:
Q < 0 when x ≠ 0
d: negative semi-definite:
Q ≤ 0 for all x
e: indefinite:
Q > 0 for some x and Q < 0 for some other x
102. Testing for Definiteness
Eigen values of A Nature of
quadratic form Q
All λi > 0 positive definite
All λi ≥ 0 positive semi-definite
All λi < 0 negative definite
All λi ≤ 0 negative semi-definite
some λi ≥ 0 and
some λi ≤ 0
indefinite
103. Consider the state variable model:-
Taking the Laplace:
Where
After manipulation we get:-
Or,
In case of zero initial conditions ,we get i/p, o/p relation by transfer
function :-
Conversion of state variables to Transfer
function
)()(
)()(
.
txcty
tBuxAtx
0
)( xtox
)()()(
)()()(
sdUscXsy
sbUsAXxssX o
)]([)()];([)()];([)( tyLsYtuLsUtxLsX
nxnmatrixIsbUxsXAsI o
),()()(
)(])([)()(
)()()()(
11
11
sUdbAsIcxAsIcsY
sbUAsIxAsIsX
o
o
104. Inverse of the matrix can be written as:-
So , transfer function is given by:-
For a general nth order matrix given as:-
dbAsIcsG
su
sy
1
)()(
)(
)(
AsI
AsI
AsI
)(
)( 1
d
AsI
bAsIc
sG
)(
)(
nnnn
n
n
aaa
aaa
aaa
A
...
....
....
...
...
21
22221
11211
105. • Matrix ISI-AI has the following matrix:-
ISI-AI will be of following form:-
where are the constants scalars.
This is known as characteristic polynomial of matrix A. it plays a vital role
in the dynamic behavior of the system. The roots of this characteristics
equation are called the characteristics roots or eigen values of the matrix
A.
i
n
nn
sssAsI
...)( 1
1
nnnn
n
n
asaa
aasa
aaas
AsI
...
....
....
...
...
)(
21
22221
11211
106. Cayley Hamilton Theorem
A matrix satisfies its own characteristics equation.
0....1
1
IAA n
nn
109. • An rectangular array of nm elements. n= rows, m=no. of columns.
aij= (I,j)th element.
• Diagonal Matrix :- all elements are zero except diagonal
elements.
nnnn
n
n
aaa
aaa
aaa
A
...
....
....
...
...
21
22221
11211
nna
a
a
...00
....
....
0...0
0...0
22
11
110. • Unity Matrix :- A diagonal matrix whose all elements are unity.
• Transpose Matrix:- If row & column of a matrix A are interchanged
then we get transposed matrix of A.
1...00
....
0...10
0...01
I
nmnm
n
n
T
aaa
aaa
aaa
A
...
....
...
...
21
22212
12111
111. • Determinant of a matrix:- defined only for square matrix.
Represented as IAI or detA is a scalar valued function of A. found
with the help of minors & cofactors.
a) Minors:- minor mij of any element aij is determinant of a matrix of
order of (n-1)x(n-1) obtained from A removing row & column
containing aij.
b) Cofactors:- cofactors cij of the element aij is defined by the
equation:-
So determinant is given by:-
K = any arbitrary row.
ij
ji
ij mc
)1(
n
j
kjkjcaA
1
112. • Singular Matrix:- A matrix whose associated determinant is zero.
• Non –singular Matrix:- A matrix whose associated determinant is
not equal to zero.
• Adjoint Matrix:- found by replacing each element of a matrix A by its
cofactor & then transposing.
adjA =
• Inverse Matrix:- inverse of a matrix is given by:-
A-1 = adjA/ IAI
also we have:- A A-1 = I = A-1A
ji
nnnn
n
n
c
ccc
ccc
ccc
...
....
...
...
21
22212
12111
