Control Theory and Applications 10.1 System Theory 233
Part A | 10.1
fractions of the form shown in (10.20)
H.s/ D
n
X
v D1
k v
s v
:
(10.22)
In the above constants (coefficients) k v ; v D 1; : : : ; n
can be calculated by using the following relationship
k v D lim
s!v
Œ.s v /H.s/ ; v D 1; : : : ; n : (10.23)
Based on the fact (10.20) and the expansion (10.22)
the following expression is derived for the system’s impulse response
h.t/ D
n
X
v D1
k v exp.. v t/ ; t 0 :
(10.24)
The partial fraction expansion can also be employed
in cases for which the characteristic polynomial obtains multiple real or complex roots. First, the case of
a purely real pole, 0 , with multiplicity l > 1, is examined. Then, the factorized form of the characteristic
polynomial of the system includes factor .s 0 /
l . Consequently, in the system’s transfer function this factor
appears in the denominator. Therefore, the partial function expansion of the following general form is needed
p c .s/ D .s 0 /
l
n
Y
v D1
.s v /
) H.s/ D
n
X
v D1
k v
s v
C
l
X
D1
k 0
.s 0 /
: (10.25)
In the above, additional terms augment the expansion of
H.s/ in (10.22) in order to encompass factor .s 0 /
l
of the characteristic polynomial. To obtain the values of
constants k 0 of the expansion, the following formula is
used
k 0 D lim
s!0
(
1
.l
d
.l/
ds .l/
.s 0 /
l H.s/
)
;
D 1; : : : ; l :
(10.26)
The inverse Laplace transform of each additional term
in the expansion (10.25) is given by the following
L
1
(
1
.s 0 /
)
D
1
.. 1/Š
t
..1/ e
0t
:
(10.27)
The above comes as a generalization of (10.20). In conclusion, each additional term (10.25) corresponds to
a term of the form (10.27) in the impulse response expression (10.24).
In the case that the characteristic polynomial obtains a pair of complex conjugate roots, C and
C , the
factorized form includes the factor
.s C /.s
C / D s
2
2Re.. C /s C j C j
2 :
In this case, (10.20), (10.22), and (10.23) are still valid
for each one of the poles C and
C , separately. Furthermore, the following fact for the combined partial
fraction expansion of this factor can be shown
1
s 2 2Re.. C /s C j C j
2
D
k C
s C
C
k C
s
C
) k C D .k C /
:
(10.28)
Therefore, for calculating the constant k C corresponding to factor .s
C / it suffices to calculate constant k C
corresponding to factor .s C /.
10.1.4 Multivariable Systems
and State Space
State Equations
State equations constitute a description in the time
domain encompassing, besides LTI systems, other categories of systems such as time-varying or nonlinear
systems [10.1–4]. A system described by state equations may in general accept more than one scalar input
signals and produce more than one scalar output signals, also referred to as measurements. If the number of
scalar input signals is m and that of scalar output signals is p, the following column vectors are introduced
for convenience in notation
u D u.t/ D
u 1 u 2 : : : u m
T
and
y D y.t/ D
y 1 y 2 : : : y p
T :
(10.29)
In the disciplines of dynamic systems and control,
the term state or state vector is used for a set of n 1
variables (state variables) conveniently packed in a column vector as follow
x D x.t/ D
x 1 x 2 : : : x n
T
(10.30)
If the value of x is known at some initial time instant t 0 ,
then it must be possible to determine the values of both
x and y for times t t 0 provided that (a) the values of u
are given and (b) the mathematical law connecting the
input, output, and state vectors is known.
Part A | 10.1
fractions of the form shown in (10.20)
H.s/ D
n
X
v D1
k v
s v
:
(10.22)
In the above constants (coefficients) k v ; v D 1; : : : ; n
can be calculated by using the following relationship
k v D lim
s!v
Œ.s v /H.s/ ; v D 1; : : : ; n : (10.23)
Based on the fact (10.20) and the expansion (10.22)
the following expression is derived for the system’s impulse response
h.t/ D
n
X
v D1
k v exp.. v t/ ; t 0 :
(10.24)
The partial fraction expansion can also be employed
in cases for which the characteristic polynomial obtains multiple real or complex roots. First, the case of
a purely real pole, 0 , with multiplicity l > 1, is examined. Then, the factorized form of the characteristic
polynomial of the system includes factor .s 0 /
l . Consequently, in the system’s transfer function this factor
appears in the denominator. Therefore, the partial function expansion of the following general form is needed
p c .s/ D .s 0 /
l
n
Y
v D1
.s v /
) H.s/ D
n
X
v D1
k v
s v
C
l
X
D1
k 0
.s 0 /
: (10.25)
In the above, additional terms augment the expansion of
H.s/ in (10.22) in order to encompass factor .s 0 /
l
of the characteristic polynomial. To obtain the values of
constants k 0 of the expansion, the following formula is
used
k 0 D lim
s!0
(
1
.l
d
.l/
ds .l/
.s 0 /
l H.s/
)
;
D 1; : : : ; l :
(10.26)
The inverse Laplace transform of each additional term
in the expansion (10.25) is given by the following
L
1
(
1
.s 0 /
)
D
1
.. 1/Š
t
..1/ e
0t
:
(10.27)
The above comes as a generalization of (10.20). In conclusion, each additional term (10.25) corresponds to
a term of the form (10.27) in the impulse response expression (10.24).
In the case that the characteristic polynomial obtains a pair of complex conjugate roots, C and
C , the
factorized form includes the factor
.s C /.s
C / D s
2
2Re.. C /s C j C j
2 :
In this case, (10.20), (10.22), and (10.23) are still valid
for each one of the poles C and
C , separately. Furthermore, the following fact for the combined partial
fraction expansion of this factor can be shown
1
s 2 2Re.. C /s C j C j
2
D
k C
s C
C
k C
s
C
) k C D .k C /
:
(10.28)
Therefore, for calculating the constant k C corresponding to factor .s
C / it suffices to calculate constant k C
corresponding to factor .s C /.
10.1.4 Multivariable Systems
and State Space
State Equations
State equations constitute a description in the time
domain encompassing, besides LTI systems, other categories of systems such as time-varying or nonlinear
systems [10.1–4]. A system described by state equations may in general accept more than one scalar input
signals and produce more than one scalar output signals, also referred to as measurements. If the number of
scalar input signals is m and that of scalar output signals is p, the following column vectors are introduced
for convenience in notation
u D u.t/ D
u 1 u 2 : : : u m
T
and
y D y.t/ D
y 1 y 2 : : : y p
T :
(10.29)
In the disciplines of dynamic systems and control,
the term state or state vector is used for a set of n 1
variables (state variables) conveniently packed in a column vector as follow
x D x.t/ D
x 1 x 2 : : : x n
T
(10.30)
If the value of x is known at some initial time instant t 0 ,
then it must be possible to determine the values of both
x and y for times t t 0 provided that (a) the values of u
are given and (b) the mathematical law connecting the
input, output, and state vectors is known.