Control Theory and Applications 10.4 Pole Placement of LTI Systems 261
Part A | 10.4
pass through the same point on the real axis. This
point is given as
0 D D
a 1 b 1
n m
D D
P n
iD1 p i
P m
iD1 z i
n m
:
(10.149)
7. a) A real-axis segment is part of the root locus provided that an odd cumulative number of poles
and zeros of G.s/ are located on its right-hand
side.
b) A real-axis segment is part of the root locus provided that an even cumulative number of poles
and zeros of G.s/ are located on its right-hand
side.
8. The breakpoints (corner points) of the locus are the
repeated (multiple) roots of the closed-loop characteristic polynomial p c .s/; those can be determined
as the roots of
d
ds
ŒKG.s/ D 0 )
d
ds
G.s/ D 0 :
(10.150)
9. a) The root locus angle of departure # j from the jth pole of G.s/, (p j ), is given by employing the
following condition for s !
p j
,
1 C KG.s/ D 0
)
8
ˆ <
ˆ :
jKG.s/j D 1
†KG.s/ D ˙ .2v C 1/ /;
v D 0; 1; 2; : : :
:
(10.151)
The second (phase-related) of the above yields
for s !
p j
lim
s!.pj/
ΠKG.s/
D
m
X
iD1
†
z i p j
n
X
iD1
i¤j
†
p i p j
# j
D ˙ .2v C 1/ / :
(10.152)
b) The root locus angle of arrival j at the j-th zero
of G.s/, (z j ) is given by the same condition as
the one used in 9(a) but applied for s !
z j
.
lim
s!.zj/
ΠKG.s/
D j C
m
X
iD1
i¤j
†
z i z j
n
X
iD1
†
p i z j
D ˙ .2v C 1/ / ; v D 0; 1; 2; : : :
(10.153)
The properties presented previously are now applied
to the construction of the root locus of various examples in Table 10.4 of process transfer function, G.s/, as
in (10.145).
10.4 Pole Placement of LTI Systems
Modern control design techniques rely heavily on the
system’s description in state space. The reason for this
is the need to develop control systems for SISO as well
as MIMO systems.
In the remaining of this chapter, LTI systems are
considered. For the open-loop system (referred to as
plant or process) a state-space description must be provided of the general form
P
x D Ax C Bu ;
y D Cx :
(10.154)
In (10.154), u stands for the m-dimensional input vector
(excitations), x for the n-dimensional state vector, and y
for the p-dimensional output vector (measurements) of
the LTI system at hand. By manipulating (10.154) the
following open-loop transfer function is obtained
Y.s/ D H.s/U.s/ D C .sI n A/
1 BU.s/ (10.155)
10.4.1 Input–Output Decoupling
Consider the case below where the open-loop transfer
function matrix H.s/ is square and diagonal, that is, the
process has the same number of inputs and outputs (p D
m).
H.s/ D
2
6
6
6
4
H 11 .s/
0
: : :
0
0
H 22 .s/ : : :
0
: : :
: : :
: : :
: : :
0
0
: : : H mm .s/ :
3
7
7
7
5
(10.156)
It is straightforward that the LTI–MIMO system is
equivalent to m in number LTI–SISO systems with
scalar transfer functions those appearing as the diagonal, nonzero elements of the transfer function matrix,
H.s/. This case is referred to by the term input–output
decoupling.
Part A | 10.4
pass through the same point on the real axis. This
point is given as
0 D D
a 1 b 1
n m
D D
P n
iD1 p i
P m
iD1 z i
n m
:
(10.149)
7. a) A real-axis segment is part of the root locus provided that an odd cumulative number of poles
and zeros of G.s/ are located on its right-hand
side.
b) A real-axis segment is part of the root locus provided that an even cumulative number of poles
and zeros of G.s/ are located on its right-hand
side.
8. The breakpoints (corner points) of the locus are the
repeated (multiple) roots of the closed-loop characteristic polynomial p c .s/; those can be determined
as the roots of
d
ds
ŒKG.s/ D 0 )
d
ds
G.s/ D 0 :
(10.150)
9. a) The root locus angle of departure # j from the jth pole of G.s/, (p j ), is given by employing the
following condition for s !
p j
,
1 C KG.s/ D 0
)
8
ˆ <
ˆ :
jKG.s/j D 1
†KG.s/ D ˙ .2v C 1/ /;
v D 0; 1; 2; : : :
:
(10.151)
The second (phase-related) of the above yields
for s !
p j
lim
s!.pj/
ΠKG.s/
D
m
X
iD1
†
z i p j
n
X
iD1
i¤j
†
p i p j
# j
D ˙ .2v C 1/ / :
(10.152)
b) The root locus angle of arrival j at the j-th zero
of G.s/, (z j ) is given by the same condition as
the one used in 9(a) but applied for s !
z j
.
lim
s!.zj/
ΠKG.s/
D j C
m
X
iD1
i¤j
†
z i z j
n
X
iD1
†
p i z j
D ˙ .2v C 1/ / ; v D 0; 1; 2; : : :
(10.153)
The properties presented previously are now applied
to the construction of the root locus of various examples in Table 10.4 of process transfer function, G.s/, as
in (10.145).
10.4 Pole Placement of LTI Systems
Modern control design techniques rely heavily on the
system’s description in state space. The reason for this
is the need to develop control systems for SISO as well
as MIMO systems.
In the remaining of this chapter, LTI systems are
considered. For the open-loop system (referred to as
plant or process) a state-space description must be provided of the general form
P
x D Ax C Bu ;
y D Cx :
(10.154)
In (10.154), u stands for the m-dimensional input vector
(excitations), x for the n-dimensional state vector, and y
for the p-dimensional output vector (measurements) of
the LTI system at hand. By manipulating (10.154) the
following open-loop transfer function is obtained
Y.s/ D H.s/U.s/ D C .sI n A/
1 BU.s/ (10.155)
10.4.1 Input–Output Decoupling
Consider the case below where the open-loop transfer
function matrix H.s/ is square and diagonal, that is, the
process has the same number of inputs and outputs (p D
m).
H.s/ D
2
6
6
6
4
H 11 .s/
0
: : :
0
0
H 22 .s/ : : :
0
: : :
: : :
: : :
: : :
0
0
: : : H mm .s/ :
3
7
7
7
5
(10.156)
It is straightforward that the LTI–MIMO system is
equivalent to m in number LTI–SISO systems with
scalar transfer functions those appearing as the diagonal, nonzero elements of the transfer function matrix,
H.s/. This case is referred to by the term input–output
decoupling.