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Matrices &
Vectors
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:
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:
Complex Conjugate
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
Skew-Hermitian matrixIn 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
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
Find Rank of the Matrix
1.
2.
Solution 1.
2.
Trace of a Matrix can be defined as the sum of the main diagonal elements.
Eigen Values&
Eigen Vectors
• 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
• 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.
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
Determine i)
ii)
iii)
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.
Tacoma Narrows Bridge (1940)
Mode Shape of a Tuning Fork at Eigenfrequency
440.09 Hz
• 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.
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.
• 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.
• 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.
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
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.
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):-
)()(
)(1)(
tvdttdx
tFMdt
tdv
)()()()()(
)()(.
2
1
tFtutxtvtx
txtx
State Equation can be written as:
Output Equation y(t ) is given as :
uMx
xxx
/10
0010
2
1
2
1
2
101)(xx
ty
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”.
M1
t
to
t to
dttFM
dttFM
dttF )(1)(1)(
t
to
dttFtov )()(
t to t
to
dttvdttvdttv )()()(
t
to
t
to
dttFd )(
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
txtx
tx
ty
tyty
ty
tu
tutu
tu
npm
Controlled system(State variables(n))
I/p(u)m variable
o/p (Y)P Variables
State (x)n Variables
• 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
2121111
mnnnn
mn
uuuxxxfxdtdx
uuuxxxfxdtdx
ni
dtuuuxxxftoxtxt
tomniii
....,3,2,1
),...,,,...,()()( 2121
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
ff
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.
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.
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
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
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
aaaaaa
.....
.....
.....
21
.
.
.22221
11211
nmnn
m
m
bbb
bbbbbb
.....
.....
.....
21
.
.22221
11211
• 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
cccccc
.
............
21
22221
11211
For the system shown in figure (a) let us define:
pmpp
m
m
ddd
dddddd
D
........
....
....
21
22221
11211
The state model of linear time invariant systems is thus given by the following equations
)()()()()()(
.
tdutcxtytButAxtx
• 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:-
)()(
)()()(
tiKtT
tBdttdjtT
aTM
M
The voltage loop equation is given by:-
The torque balance equation is given by:-
)()()()( tetiRdttdiLtu baa
aa
• 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
tjBti
jk
dttd
tuL
tLkti
LR
dttdi
a
aT
aa
ba
a
aa
)()(
)(/10
)()(
)()(
1
2
1
2
1
txty
tuLtx
tx
LR
LK
jk
jB
txtx
a
a
a
a
b
T
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
)()()()();()()(
tdutcxtyxtxtbutAxtx o
o
ddcPcbPbAPPA
withtudtxcty
toxPtoxtubtxAtx
ortdutxcPty
tbutxAPtxP
,,,
)()()(
)()(),()()(
)()()(
)()()(
11
1.
.
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
1101
xx
xxx
xx
x
xpx
1101
P
)()(
)()(.
txcty
tuBxAtx
Given below:-
)()()();()(
011101
01
100
100
1101
111110
1101
10111
1101
21211
1
1
toxtoxtoxtoxtox
cPc
bpb
APPA
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.
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).
General State Space form of Physical system BuAxx
DuCxy
xx
y
uA
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
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
First Companion Form(SISO System):If LTI SISO system is described by transfer function of the form;
Decomposition of transfer function:
.
01223
3
012
2
)()(
)()(
asasasabsbsb
sRsC
sUsY
sXbsbsbsCsY 1012
2
101
121
2
2 xbdtdxb
dtxdbty
)2........(..........322110)( xbxbxbty
sXasasasasRsU 1012
23
3
)()()()(10
112
12
231
3
3 txadt
tdxadt
txdadt
txdatu
43322110)( xaxaxaxatu
I.
II.
)1..(..........3
)(3322311304 atuxaaxaaxaax
21 xx 32 xx 1)( xtx 43 xx &Select state variables like :
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
Again from equation (1) & (2) complete state model will be ;
)3).....((
3/100
3
2
1
210
100010
3
1
3
2
1
,
)(
3
13
3
22
3
11
3
043
tuax
x
x
aaaax
x
x
or
tua
xa
ax
a
ax
a
axx
A B
21 xx 32 xx
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
Determine the First Companion form
)(
)(
2024293
3)(
sU
sY
sss
ssG
Ans
3
2
1
3
2
1
3
2
1
]310[
100
92420100010
xxx
y
uxxx
xxx
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
)()........110()()........11( sUnsnsnsYnsnsn
or
0)](0)([..........)](1)(1[1)](0)([ sUsYnsUsYsnsUsYsn
• On dividing by and solving for Y(s), we obtainsn
)]()([1.......)](1)(1[1)(0)( sYnsUnsnsYsUssUsY
Note that is the transfer function of a chain of n integrators.sn1
nsnsnnsnsn
sUsY
......11
.....110
)()(
• 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
αnαn-1 αn-2 α1
u
+
-
x1 x2 xn-1 xn y
Realization of equation (9)
ynun ynun 11
)]()([1.......)](1)(1[1)(0)( sYnsUnsnsYsUssUsY
• .• 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 uẋn-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)
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.
Determine the Second Companion form
)(
)(
2024293
3)(
sU
sY
sss
ssG
Ans
3
3
2
1
3
2
1
013
91024012000
xy
uxxx
xxx
Jordan Canonical Form• In this form the poles of the transfer function form a string along the main diagonal of the
matrix.
nsnnsnnsnsn
sG
......1......1
10)(
• By long division , G(s) can be written as
)('0.....11
'.....1'1
0)( sGnsnsn
nsnsG
or
nsrn
sr
sr
sUsYsG
.....
22
11
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)
• 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
nsrn
sr
sr
sUsYsG
.....
22
11
0)()()(
• Identifying the outputs of integrator with the state variables results in following state
and output equations:
λ1 0 … 00 λ2 … 00 0 … 0: : : :0 0 0 λn
ẋ (t) = x(t) + B u(t)
y (t) = C x(t) + D u(t)
ʌ = ;
111:1
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)
• 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 nnsnsn
sG
• The partial fraction expansion of G(s) is of the form
)()()(.......)(10)(
sUsYsH msHsG
where
)()(
)(.........
)( 12
)(1)(
sUsY i
is
rini
is niri
is nirisH i
• 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)
• 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.
• The corresponding differential equation are
ẋi1 = λi xi1 + xi2
ẋi2 = λi xi2 + xi3
:uxiniixini
.
And the output is given by
xrxrxr ii ininiiiiyi 2211 .........
• If the state vector for the subsystem is defined by
rinixixiT
xi 21
• Then equation can be written in standard form
ẋi = ʌi xi + Bi uyi = Ci xi
where
i
i
i
i
0001000
010001
1
000
Bi; ;
rrrC ini
iii21
(16a)
(16b)
(17)
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
xx
x21
• 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
ẋ1=ʌ1x1+B1uy1=C1x1
ẋ1=ʌ2x2+B2uy2=C2x2
ẋm=ʌmxm+Bmuym=Cmxm
0
y1
y2
ym
yu
m
00
020001
;
BM
BB
B21
C = [ C1 C2 … Cm] ; D = β0
Subsystems in Jordan canonical form combined into overall system
Determine the Jordan canonical form
)(
)(
2024293
3)(
sU
sY
sss
ssG
9/2;1
9/23/1;10
5;20
12
22
11
21
5
9/2
2
9/2
2)2(
3/1)(
)5(2)2(
3)(
cb
cb
ssssG
ss
ssG
3
2
1
.
3
.
2
.
1
]9/29/23/1[
110
500020012
XXX
y
u
X
X
X
CONTROLLABILITY AND
OBSERVABILITY
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.
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.
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)
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”
• 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.
• 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].
CONTROLLABILITY TEST
Sometimes controllability control is not possibleand this can verified by using controllability testmatrix.
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.
• 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.
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
-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.
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.
• 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
• 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.
OBSERVABILITY TEST
• The n×n observability matrix is given by CV= CA .
. CAn-1
V= [ CT : AT :CT : ………. :(AT)n-1 CT] The above matrix is to have a rank of n
QUADRATIC FORMS AND
DEFINITE MATRICES
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.
For example,
consider the matrix
1221
][A
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
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
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
)()()()()()(
sdUscXsysbUsAXxssX o
)]([)()];([)()];([)( tyLsYtuLsUtxLsX
nxnmatrixIsbUxsXAsI o ),()()(
)(])([)()(
)()()()(11
11
sUdbAsIcxAsIcsY
sbUAsIxAsIsXo
o
Inverse of the matrix can be written as:-
So , transfer function is given by:-
For a general nth order matrix given as:-
dbAsIcsGsusy
1)()()()(
AsIAsIAsI
)()( 1
dAsIbAsIcsG
)()(
nnnn
n
n
aaa
aaaaaa
A
...........
...
...
21
22221
11211
• 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
nnn sssAsI ...)( 1
1
nnnn
n
n
asaa
aasaaaas
AsI
...........
...
...
)(
21
22221
11211
Cayley Hamilton Theorem
A matrix satisfies its own characteristics equation.
0....11 IAA n
nn
Thank You
• 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
aaaaaa
A
...........
...
...
21
22221
11211
nna
aa
...00........0...00...0
22
11
• 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...100...01
I
nmnm
n
n
T
aaa
aaaaaa
A
.......
...
...
21
22212
12111
• 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.
ijji
ij mc )1(
n
jkjkjcaA
1
• 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
cccccc
.......
...
...
21
22212
12111
