Mrinal K Pal. Power system stability
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REVIEW OF MATRICES
A-7
Useful properties of determinants
1. If A and B are both n × n, then BAAB = .
2.
AA
′
=
.
3.
If all the elements in any row or in any column are zero, then 0=A .
4.
If any two rows of A are proportional, 0=A . The same holds for columns.
5.
Interchanging any two rows (or any two columns) of a matrix changes the sign of its
determinant.
6.
Multiplying all elements of any one row (or column) of a matrix A by a scalar k yields a
matrix whose determinant is k
A
7.
Any multiple of a row (column) can be added to any other row (column) without changing
the value of the determinant.
Problem
Verify the above statements.
Inverse of a matrix
The inverse of a matrix A is defined such that
IAAAA ==
−− 11
(A.24)
where
A
−1
is the inverse of A.
It is evident that only square matrices (number of rows equal to the number of columns) can
possess inverses.
The inverse of a matrix
A can be obtained as
A
A
A
det
ofadjoint
1
=
−
(A.25)
when det
A is not equal to zero.
Product of inverse matirces
Consider the product BAC = . Pre-multiplying both sides of the equation by
11 −−
AB and post-
multiplying both sides by
1−
C results in the relationship
111 −−−
= CAB (A.26)
Derivative of the inverse matrix
If A(t) is differentiable and possesses an inverse, the derivative of )(
1
t
−
A is given by
[]
)()()()(
111
tttt
dt
d
−−−
−= AAAA
&
(A.27)
REVIEW OF MATRICES
A-8
Problem
Derive equation (A.27).
Consider the following vector-matrix equation in partitioned form as indicated by the vertical
and horizontal dotted lines.
=
5
4
3
2
1
5554535251
4544434241
3534333231
2524232221
1514131211
5
4
3
2
1
x
x
x
x
x
aaaa
aaaaa
aaaaa
aaaaa
aaaaa
y
y
y
y
y
(A.28)
Equation (A.28) can be written in terms of submatrices as
=
2
1
43
21
2
1
x
x
AA
AA
y
y
(A.29)
Equation (A.29) can be expanded as
22111
xAxAy
+
= (A.30)
24132
xAxAy
+
= (A.31)
From (A.31)
13
1
42
1
42
xAAyAx
−−
−= (A.32)
Substituting (A.32) into (A.30),
[
]
2
1
4213
1
4211
yAAxAAAAy
−−
+−=
(A.33)
Equations (A.33) and (A.32) can be arranged as
−
−
=
−−
−−
2
1
1
43
1
4
1
423
1
421
2
1
y
x
AAA
AAAAAA
x
y
(A.34)
The above procedure demonstrates how a group of variables can be transferred from the right-
hand side of equation (A.28) to the left-hand side. If the variables are transferred one at a time
and the procedure is repeated n times for the n variables, the resulting equation would be, in
matrix form,
yBx
=
where
1−
= AB
The above illustrates an efficient technique for inverting matrices.
In equation (A.29) if
2
y is a null vector (i.e., 0
54
=
=
yy ) then, from (A.33),
[
]
13
1
4211
xAAAAy
−
−= (A.35)
Equation (A.35) is useful in the elimination of passive nodes in network reduction.
REVIEW OF MATRICES
A-9
Characteristic values (eigenvalues) and characteristic vectors (eigenvectors)
Consider the vector-matrix equation
xAy
=
(A.36)
where
y and x are column vectors, and A is a square n × n matrix. This equation can be viewed
as a transformation of the vector
x into the vector y. The question arises whether there exists a
vector
x, such that the transformation A produces a vector y, which has the same direction in
vector space as the vector
x. If such a vector exists, then y is proportional to x, or
xxAy
λ
=
= (A.37)
where λ is a scalar of proportionality. This is known as the characteristic value problem, and a
value of λ, e.g.,
i
λ
for which equation (A.37) has a solution )0(
≠
i
x , is called a characteristic
value or eigenvalue of
A. The corresponding vector solution )0(
≠
i
x is called a characteristic
vector or eigenvector of
A associated with the characteristic value
i
λ
.
Equation (A.37) can be written in the form
[]
0xIA
=
−
λ
(A.38)
This system of homogeneous equations has a nontrivial solution if, and only if, the determinant
of the coefficients vanishes, i.e., if
0=− IA
λ
(A.39)
The nth order polynomial in λ, given by equation (A.39), is called the characteristic equation
corresponding to the matrix
A. The general form of the equation is
0)(
1
2
2
1
1
=+++++=
−
−−
nn
nnn
aaaaP
λλλλλ
L
(A.40)
The roots of the characteristic equation are precisely the characteristic values or eigenvalues of
A. When all the eigenvalues of A are different, A is said to have distinct roots. When an
eigenvalue occurs m times, the eigenvalue is said to be a repeated root of order m. When a
characteristic root is of the form α + jβ, the root is said to be complex. Complex roots must occur
in conjugate pairs, assuming that the elements of
A are real.
Modal matrix
For each of the n eigenvalues
i
λ
(i = 1, 2, ... n) of A, a solution of equation (A.38) for x can be
obtained, provided that the roots of equations (A.39) are distinct. The vectors
i
x , which are the
solutions of
[]
ni
ii
L,2,1
=
=
− 0xIA
λ
(A.41)
are the characteristic vectors or eigenvectors of
A. Since equation (A.41) is homogeneous,
ii
k x ,
where
i
k is any scalar, is also a solution. Thus only the directions of each of the
i
x are uniquely
determined by equation (A.41). The matrix formed by the column vectors
ii
k x is called the
modal matrix.
REVIEW OF MATRICES
A-10
Diagonalizing a square matrix
Consider the case in which the modal matrix M is non-singular, so that its inverse exists (this is
always the case if the eigenvalues of
A are distinct). The solution to equation (A.41) can be
combined to form the single equation
=
nnnn
n
n
nnnn
n
n
nnnnn
nn
nn
xxx
xxx
xxx
aaa
aaa
aaa
xxx
xxx
xxx
L
MOMM
L
L
L
MOMM
L
L
L
MOMM
L
L
21
22221
11211
21
22221
11211
2211
2222211
1122111
λλλ
λλλ
λλλ
(A.42)
or
MAΛM
=
(A.43)
where
=
n
λ
λ
λ
O
2
1
Λ
is a diagonal matrix composed of the eigenvalues
n
λ
λ
λ
L,,
21
. Since
1−
M exists, the diagonal
matrix Λ can be found by pre-multiplying both sides of equation (A.43) by
1−
M , yielding
MAMΛ
1−
= (A.44)
Higher powers of
A can be diagonalized in the same manner. For example,
[
]
[
]
MAMMAMMAMΛ
21112 −−−
== (A.45)
A transformation of the type
QAQB
1−
= , where A and B are square matrices and Q is a non-
singular square matrix, is called a collineatory or similarity transformation.
Problem
Show that the eigenvalues of B are the same as the eigenvalues of A.
Sensitivities of eigenvalues and eigenvectors
Sensitivities of the eigenvalues and eigenvectors to system parameters or the elements of the
system matrix can provide useful information about the system performance.
From the definitions of eigenvalues and eigenvectors we have
Ax = λx
and
A′y = λy
where y is a column vector which is the transpose of the left eigenvector of A
y
′A = y′λ
REVIEW OF MATRICES
A-11
α
λ
α
λ
αα
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
∴
x
x
x
Ax
A
where α is any parameter of the system. Pre-multiplying both sides by
y′
α
λ
α
λ
α
α
∂
∂
′
+
∂
∂
′
=
∂
∂
′
+
∂
∂
′
x
yxy
x
Ayx
A
y
or
α
λ
α
λ
α
λ
α
∂
∂
′
+
∂
∂
′
=
∂
∂
′
+
∂
∂
′
x
yxy
x
yx
A
y
(A.46)
From (A.46) it follows that
xy
x
A
y
′
∂
∂
′
=
∂
∂
α
α
λ
(A.47)
When the parameter α is an element a
ij
of the matrix A, all the elements of the matrix
α
∂
∂
A
are
zero except the one in the ijth position which is unity. Hence
xy
′
=
∂
∂
ji
ij
xy
a
λ
(A.48)
where y
i
is the ith element of the vector y and x
j
is the jth element of the vector x. The sensitivity
matrix
S
i
, which is the matrix of the sensitivities to each element of A, is thus given by
ii
ii
i
xy
xy
S
′
′
=
(A.49)
for each eigenvalue λ
i
.
To evaluate
α
∂
∂
i
x
we first show that the left and right eigenvectors corresponding to distinct
eigenvalues are orthogonal.
We start with
iii
xAx
λ
=
(A.50)
and
)(
jjjjjj
λ
λ
yAyyyA
′
=
′
=
′
(A.51)
From (A.50) we can write
ijiij
xyAxy
′
=
′
λ
(A.52)
Similarly, from (A.51)
ijjij
xyAxy
′
=
′
λ
(A.53)
Subtracting (A.53) from (A.52)
ijji
xy
′
−
= )(0
λ
λ
Since
ji
λ
λ
≠ , it follows that
0
=
′
ij
xy
(A.54)
REVIEW OF MATRICES
A-12
Now, as before, for the ith eigenvalue and eigenvector
α
λ
α
λ
αα
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
i
ii
ii
i
x
x
x
Ax
A
Pre-multiplying both sides by
j
y
′
α
λ
α
λ
αα
∂
∂
′
+
∂
∂
′
=
∂
∂
′
+
∂
∂
′
i
iji
i
j
i
jij
x
yxy
x
Ayx
A
y
which yields
α
λλ
α
∂
∂
′
−=
∂
∂
′
i
jjiij
x
yx
A
y
)( (A.55)
Let us set
∑
≠
=
=
∂
∂
n
ij
j
jij
i
1
x
x
µ
α
, which is a valid assumption as the n linearly independent eigenvectors
form a basis for the generation of any n vector, and since eigenvectors are only determined to
within a multiplicative constant we are not interested in any perturbation in the direction of
x
i
so
that we can allow µ
ii
to be zero. Thus
∑
≠
=
′
=
′
=
∂
∂
′
n
ij
j
jjijjijj
i
j
1
xyxy
x
y
µµ
α
which yields, from (A.55),
jjji
ij
ij
xy
x
A
y
′
−
∂
∂
′
=
)(
λλ
α
µ
and thus
=
∂
∂
∑
≠
=
n
ij
j
jij
i
1
x
x
µ
α
is determined.
To obtain the sensitivity of the eigenvector to an element a
pq
of A, we set
pq
a∂
∂
=
∂
∂
AA
α
to obtain
jjji
ij
ij
pq
xy
xy
′
−
′
=
)(
ofelement th
λλ
µ
References
1. P.M. DeRusso, R.J. Roy and C.M. Close, State Variables for Engineers, John Wiley and Sons, Inc., 1965.
2. W.L. Brogan, Modern Control Theory - 2
nd
Ed., Prentice-Hall, Inc., 1985.
3. D.K. Fadeev and V.N. Fadeeva, Computational Methods of Linear Algebra, W.H. Freeman & Co., 1963.
B-1
APPENDIX B
REVIEW OF SELECTED TOPICS FROM CONTROL THEORY
The Laplace Transform
The Laplace transformation is a functional transformation in that it changes the expression for a
given quantity from a function of time, f(t), to a function of the complex operator, F(s), where s
is the complex operator.
A real function f(t) has a Laplace transform if it is defined and single-valued almost everywhere
for 0 ≤ t, and if f(t) is such that
∫
∞
−
∞<
0
)( dttf
t
σ
ε
(B.1)
for some real number σ. Then f(t) is said to be L transformable. The functions of time that
describe the performance of actual control systems are in this category.
The Laplace transformation for a function of time, f(t), is defined as
∫
∞
−
==
0
)()( dttfsF
st
ε
L
)(tf
(B.2)
where s is a complex variable such that its real part is greater than σ, the number described in the
definition for L
transformability.
The inverse transformation is generally defined implicitly by the definition
=)(tfL tsF ≤
−
0,)(
1
(B.3)
Some important Laplace transform pairs are given in Table B.1.
Table B.1 Important Laplace Transform pairs
______________________________________________________________
f(t) F(s)
______________________________________________________________
Unit step
s
1
at−
ε
as +
1
t
ω
sin
22
ω
ω
+
s
t
ω
cos
22
ω
+
s
s
)(tf
at−
ε
)( asF
+
______________________________________________________________
SELECTED TOPICS FROM CONTROL THEORY
B-2
Table B.1 (continued)
______________________________________________________________
f(t) F(s)
______________________________________________________________
n
t
1
!
+n
s
n
k
k
k
dt
tfd
tf
)(
)( = )0()0()0()(
121 +−+−+−
−
′
−−
kkkk
ffsfssFs L
∫
∞−
t
dttf )(
s
dttf
s
sF
∫
∞−
+
0
)(
)(
Unit impulse 1
_______________________________________________________________
In the solution of differential equations by the Laplace transform method, the differential
equations are transformed into algebraic equations and the resulting algebraic equations are
solved for the transform of the variable of interest. The time response solution is then obtained
by an inverse Laplace transformation.
In the context of differential equations, the Laplace variable s can be considered to be the
differential operator, so that
dt
d
s ≡ (B.4)
Then, we also have the integral operator
∫
+
≡
t
dt
s
0
1
(B.5)
In order to illustrate the use of Laplace transformation consider the spring-mass-damper system
described in Chapter 2, which is
)(
2
2
tuyK
dt
dy
f
dt
yd
M =++ (B.6)
Taking the Laplace transform of (B.6),
[]
)()()0()(
)0(
)0()(
2
sUsYKysYsf
dt
dy
yssYsM =+−+
−−
+
+
+
(B.7)
When
0,)0(,0)(
0
o
===
=
+
t
dt
dy
yytu
we have
0)()()(
oo
2
=+−+− sYKyfsYsfysMsYsM (B.8)
Solving for Y(s), we obtain
SELECTED TOPICS FROM CONTROL THEORY
B-3
)(
)(
)(
)(
2
o
sq
sp
KsfsM
yfsM
sY =
++
+
= (B.9)
The denominator polynomial q(s) is the characteristic equation, and the roots of this equation
determine the character of the time response. These roots are also called the poles or singularities
of the system. The roots of the numerator polynomial p(s) are called the zeros of the system. At
the poles the function Y(s) becomes infinite; while at the zeros, the function becomes zero. The
complex s-plane plot of the poles and zeros graphically portrays the character of the natural
transient response of the system.
The time response solution y(t) is obtained by taking the inverse Laplace transform of equation
(B.9).
The steady-state or final value of the response y(t) is obtained from the relation
0
)(lim)(lim
→
∞→
=
s
t
sYsty (B.10)
where a simple pole of Y(s) at the origin is permitted, but poles on the imaginary axis and in the
right-half plane and higher-order poles at the origin are excluded.
The initial value of the response y(t) is obtained from
∞→
→
=
s
t
sYstf )(lim)(lim
0
(B.11)
The transfer function of linear systems
The transfer function of a linear system is defined as the ratio of the Laplace transform of the
output variable to the Laplace transform of the input variable, with all initial conditions assumed
to be zero. The transfer function of a system (or element) represents the relationship describing
the dynamics of the system under consideration.
A transfer function may only be defined for a linear, stationary (constant parameter) system.
Also, the transfer function description does not include any information concerning the internal
structure of the system and its behavior.
The transfer function of the spring-mass-damper system is obtained from the original describing
equation (B.7), rewritten with zero initial conditions as follows:
)()()()(
2
sUsYKsYsfsYsM =++ (B.12)
Then the transfer function is
KsfsM
sU
sY
sG
++
===
2
1
)(
)(
)(
input
output
(B.13)
The transfer function of the RC network shown in Figure B.1 is obtained by writing the voltage
equations, in transformed form, yielding
)(
1
)(
1
sI
Cs
RsV
+= (B.14)
)(
1
)(
2
sI
sC
sV = (B.15)
SELECTED TOPICS FROM CONTROL THEORY
B-4
Fig. B.1 An RC network.
From (B.14) and (B.15) we obtain the transfer function as
τ
τ
τ
/1
/1
1
1
1
1
)(
)(
)(
1
2
+
=
+
=
+
==
sssCRsV
sV
sG
(B.16)
where τ = RC, the time constant of the network.
Block diagrams
The dynamic systems that comprise automatic control systems are represented mathematically
by a set of simultaneous differential equations. The introduction of the Laplace transformation
reduces the problem to the solution of a set of linear algebraic equations. Since control systems
are concerned with the control of specific variables, the interrelationship of the controlled
variables to the controlling variables is required. This relationship is typically represented by the
transfer functions of the subsystems relating the input and output variables in the form of block
diagrams. Block diagrams consist of unidirectional, operational blocks that represent the transfer
function of the variables of interest. A block diagram of the spring-mass-damper system is
shown in Figure B.2.
Fig. B.2 Block diagram of the spring-mass-damper system.
In order to represent a system with several variables under control, an interconnection of blocks
is utilized.
The block diagram representation of a given system may often be reduced by block diagram
reduction techniques to a simplified block diagram with fewer blocks than in the original
diagram.
Block diagram transformation and reduction techniques are derived by considering the algebra of
the diagram variables. For example, consider the block diagram shown in Figure B.3.