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Eigenstructure Decoupling in State
Feedback Control Design
Pavol Kocsis and Róbert Fónod
Abstract
The presented design method aim is to synthesize a state feedback control law in
such way that with respect to the prescribed eigenvalues of the closed-loop system
matrix the corresponding eigenvectors are as close as possible to a decoupled sys-
tem eigenvectors. It is demonstrated that some degree of freedom existing in the
control design, representing by the parametric vectors set as well as by the set of
closed-loop eigenvalues, can be properly used to meet some desired specification
requirement. An illustrative example and the simulation results show that the pro-
posed design principle is effective and simple.
Keywords: State feedback, eigenstructure assignment, mode decoupling, singular
value decomposition, orthogonal complement.
Introduction
The static and the dynamic pole placement belongs to the
prominent design problems of modern control theory, and,
although its practical usefulness has been continuously in
dispute, it is one of the most intensively investigated in con-
trol system design. It seems that the state-feedback pole
assignment in control system design is one from the pre-
ferred techniques. In the single-input case the solution to
this problem, when it exists, is unique. In the multi-input
multi output (MIMO) case various solutions may exist [6],
[10], and to obtain a specific solution some additional condi-
tions have to be supplied in order to eliminate the extra
degrees of freedom in design strategy.
In the last decade significant progress has been achieved in
this field, coming in its formulation closest to the algebraic
geometric nature of the pole placement problem [11], [17].
The reason for the discrepancy in opinions about the condi-
tioning of the pole assignment problem is that one has to
distinguish among three aspects of the pole placement
problem, the computation of the memory-less feedback
control law matrix gain, the computation of the closed loop
system matrix eigenvalues spectrum and the suppressing of
the cross-coupling effect [16], where one manipulated input
variable cause change in more outputs variables.
Thus, eigenstructure assignment seems to be a powerful
technique concerned with the placing of eigenvalues and
their associated eigenvectors via feedback control laws, to
meet closed-loop design specifications. The eigenvalues are
the principal factors that govern the stability and the rates of
decay or rise of the system dynamic response. The right
and left eigenvectors, on the other hand, are dual factors
that together determine the relative shape of the system
dynamic response [9], [12], [15].
The general problem of assigning the system matrix eigen-
structure using the state feedback control is considered in
this paper. Based on the classic algebraic methods [3], [4],
[14], as well as on the algorithms for pole assignment using
Singular Value Decomposition (SVD) [5], [13] the exposition
of the pole eigenstructure assignment problem is genera-
lized here to handle the specified structure of the left eigen-
vector set in state feedback control design for MIMO linear
systems. Extra freedom, which makes dependent the
closed-loop eigenvalues spectrum, is used for closed-loop
state variables mode decoupling.
The integrated procedure provides a straightforward metho-
dology usable in linear control system design techniques
when the memory-free controller in the state-space control
structures takes the standard form. Presented application
for closed-loop state variables mode decoupling is relative
simple and its worth can help to disclose the continuity be-
tween eigenstructure assignment and system variable do-
minant dynamic specification.
1. Problem Statement
Linear dynamic system with
n
degree of freedom can be
modelled by the state-space equations
)()()( ttt BuAqq +=
&
(1)
)()( tt Cqy =
(2)
where
n
tRq ∈
∈∈
∈)( ,
r
tRu ∈
∈∈
∈)( and
m
tRy ∈
∈∈
∈)( are vectors of the
state, input and measurable output variables, respectively,
and the system matrix parameters
nn×
∈RA
,
rn×
∈RB
, and
nm×
∈RC are constant and finite valued.
Generally, to the controllable time-invariant linear MIMO
system (1) a linear state-feedback controller can be defined
by a control policy
)()()( ttt LwKqu +−=
(3)
nr×
∈RK ,
mr×
∈RL to give rise to the closed-loop system
description
)()()( ttt
c
BLwqAq +=
&
(4)
in such way that roots of the closed loop characteristic
polynomial are eigenvalues of
(
((
(
)
))
)
BKAA −
−−
−=
==
=
c
,
nn
c×
∈RA .
Throughout the paper it is assumed that the pair
(
)
BA,
is
controllable.
2. Basic Preliminaries
2.1 Orthogonal Complement
Definition 1. (Null space) Let
hh×
∈REE,,
hk <=)(Erank
be a rank deficient matrix. Then the null space
E
N
of
E
is
the orthogonal complement of the row space of
.E
Proposition 1.
Let
hh×
∈REE,
, hk <=
)(E
rank be a rank
deficient matrix. Then an orthogonal complement
⊥
E
of
E
is
T
2
DUE =
⊥
(5)
where
T
2
U
is the null space of
E
and
D
is an arbitrary
matrix of appropriate dimension.
Proof.
(see e.g. [8]) The SVD of
hh×
∈REE,
,
hk
<=
)(
E
rank gives
[
[[
[ ]
]]
]
∑
∑∑
∑
=
==
=
=
==
=
2221
121
21
2
1
00 0
VVE
U
U
VEU
T
T
T
(6)
where
hh×
××
×
∈
∈∈
∈
RU
is the orthogonal matrix of the left singular
vectors,
hh×
∈
RV
is the orthogonal matrix of the right singu-
lar vectors of
E
and
kk×
××
×
∈
∈∈
∈∑
∑∑
∑R
1
is the diagonal positive
definite matrix
[
[[
[
]
]]
]
0,
1
>
>>
>≥
≥≥
≥≥
≥≥
≥=
==
=∑
∑∑
∑
k11
diag
σσσσ
LL
k
(7)
which diagonal elements are the singular values of
.
E
Using
orthogonal properties of
U
and
V
, i.e.
h
T
IUU =
==
=
,
h
T
IVV =
==
=
,
0UU =
==
=
12
T
, then
[
[[
[ ]
]]
]
=
==
=
∑
∑∑
∑
=
==
=
∑
∑∑
∑
=
==
=
T
T
T
2
1
2221
121
21
V
V
00 0
UUVUE
(8)
[
[[
[ ]
]]
]
11
2
1
21
SU
0
S
UU =
==
=
=
==
=
where
.
111
T
VS ∑
∑∑
∑=
==
=
Thus (8) implies
[
[[
[ ]
]]
]
0
0
S
UUUEU =
==
=
=
==
=
2
1
2122 TT
(9)
It is evident that for an arbitrary matrix
D
is
0EEEUD =
==
==
==
=
⊥
⊥⊥
⊥T
2
(10)
respectively, which implies (5).
■
2.2 System Model Canonical Form
Preposition 2.
If m
=
==
=
)(
CB
rank then there exists a coordi-
nates change in which
),,(
ooo
CBA
takes the structure
[
[[
[ ]
]]
]
m
I0C
B
0
B
AA AA
A
=
==
=
=
==
=
=
==
=
o
o
o
oo
oo
o
,,
2
2221
1211
(11)
where
,
)()(
11
mnmn −
−−
−×
××
×−
−−
−
∈
∈∈
∈RA
omm×
××
×
∈
∈∈
∈RB
o
2
is a non-singular
matrix, and
mm
m×
××
×
∈
∈∈
∈RI
is identity matrix.
Proof.
(see e.g. [7]) Considering the state-space description
of the system (1), (2) with
m
r
=
==
=
and defining the transform
matrix
1
1
−
−−
−
T
such that
[
[[
[ ]
]]
]
=
==
==
==
==
==
=
−
−−
−
−
−−
−
C0I
TI0CTC
mn
m1
111
,
(12)
then
=
==
=
=
==
=
=
==
==
==
=
−
−−
−−
−−
−
12
111
2
1
1
1
1
11
B
B
CB
B
B
B
TBTB
(13)
If
12
BCB =
==
=
is a regular matrix (in opposite case the pseudo
inverse of
12
B
is possible to use), then the second trans-
form matrix
1
2
−
−−
−
T
can be defined as follows
−
−−
−
=
==
=
−
−−
−
−
−−
−
−
−−
−
m
mn
I0 BBI
T
1
1211
1
2
(14)
=
==
=
−
−−
−
−
−−
−
m
mn
I0 BBI
T
1
1211
2
(15)
This application results in
BTBTT
B
0
BTB
11
1
1
2
2
1
1
2
−
−−
−−
−−
−−
−−
−−
−−
−
=
==
==
==
=
=
==
==
==
=
c
o
o
(16)
where
1
2
1
1
1
122111
,,
−
−−
−−
−−
−−
−−
−
=
==
==
==
==
==
==
==
=TTTCBBBBB
c
o
(17)
and analogously
[
[[
[
]
]]
]
[
[[
[
]
]]
]
mmc
I0TI0CTTCC =
==
==
==
==
==
==
==
=
221
o
(18)
Finally, it yields
21
1
1
1
2
1
TATTTATTA
−
−−
−−
−−
−−
−−
−
=
==
==
==
=
cc
o
(19)
Thus, (16), (18), and (19) implies (11). This concludes the
proof.
■
Note, the structure of
1
1
−
−−
−
T
is not unique and others can be
obtained by permutations of the first
m
n
−
−−
−
rows in the struc-
ture defined in (12).
2.3 System Modes Properties
Proposition 3.
Given system eigenstructure with distinct
eigenvalues then for
},,2,1{},,,2,1{, mlnkj LL ∈
∈∈
∈∈
∈∈
∈
,
r
m
=
==
=
i.
the
k-th
mode
)(
k
ss
−
−−
−
is unobservable from the
l-th
sys-
tem output if the
l-th
row of matrix
C
is orthogonal to the
k-th
eigenvector of the closed-loop system matrix
,
c
A
i.e. with
kj
≠
≠≠
≠
[
[[
[
]
]]
]
m
T
k
T
jk
T
l
ccCnnnc
L
1
,0 =
==
==
==
==
==
=
(20)
ii.
the
k-th
mode
)(
k
ss
−
−−
−
is uncontrollable from the
l-th
col-
umn of matrix
B
is orthogonal to the
k-th
eigenvector of the
closed-loop system matrix
,
c
A
i.e. with
kj
≠
≠≠
≠
[
[[
[
]
]]
]
rj
T
kl
T
k
bbBnnbn
L
1
,0 =
==
==
==
==
==
=
(21)
Proof.
(see e.g. [13]) Let
k
n
is the
k-th
right eigenvector
corresponding to the eigenvalue
,
k
s
i.e.
(
((
(
)
))
)
kkkkc
nsnBKAnA =
==
=−
−−
−=
==
=
(22)
By definition, the closed-loop system resolvent kernel is
(
((
(
)
))
)
1−
−−
−
−
−−
−=
==
=
cn
sAI
Υ
ΥΥ
Υ
(23)
If the closed-loop system matrix is with distinct eigenvalues,
(22) can be written in the compact form
[
[[
[ ]
]]
] [
[[
[ ]
]]
]
=
==
=
n
nnc
s
s
nnnnA OLL
1
11
(24)
T
c
NNNSNA =
==
==
==
=
−
−−
−
1
,
(25)
respectively, where
[
[[
[
]
]]
]
[
[[
[
]
]]
]
nn
ss nnNS LL
11
diag =
==
==
==
=,
(26)
Using the orthogonal property given in (25), the resolvent
kernel of the system takes the next form
(
((
(
)
))
)
(
((
( )
))
)
T
ss NSINNSNNN
1
1
11 −
−−
−
−
−−
−
−
−−
−−
−−
−
−
−−
−=
==
=−
−−
−=
==
=
Υ
ΥΥ
Υ
(27)
[
[[
[ ]
]]
]
=
==
=
T
n
T
n
n
n
n
s
s
nn
MOL
11
1
Υ
ΥΥ
Υ
(28)
∑
∑∑
∑
=
==
=
−
−−
−
=
==
=
n
hh
T
hh
ss
1
nn
Υ
ΥΥ
Υ
(29)
respectively. Thus, the closed loop transfer functions matrix
takes form
(
((
( )
))
)
∑
∑∑
∑
=
==
=
−
−−
−
−
−−
−
=
==
=−
−−
−=
==
=
n
hh
T
hh
c
ss
s(s)
1
1
L
BnCn
BLAICG
(30)
It is obvious that (30) implies (20), (21). This concludes the
proof. ■
3. Eigenstructure Assignment
In the pole assignment problem, a feedback gain matrix K
is sought so that the closed-loop system has a prescribed
eigenvalues spectrum
{
{{
{
}
}}
}
nhss
hhc
,...,2,1,0)(:)( =
==
=<
<<
<ℜ
ℜℜ
ℜ=
==
=AΩ
.
Note, the spectrum
)(
c
AΩ
is closed under complex conju-
gation, and the observability and controllability of modes is
determined by the closed-loop eigenstructure.
Considering the same assumptions as above then (22) can
be rewritten as
[
[[
[ ]
]]
]
0
Kn
n
L
Kn
n
ABI =
==
=
=
==
=
−
−−
−
h
h
h
h
h
h
s
(31)
where
)( rnn
h+
++
+×
××
×
∈
∈∈
∈RL ,
[
[[
[
]
]]
]
ABIL −
−−
−=
==
=
hh
s
(32)
Subsequently, the singular value decomposition (SVD) of
h
L
gives
[
[[
[ ]
]]
]
=
==
=
+
++
++
++
+
)(,)1(,1
1
rnhnhhnhh
T
hn
T
h
vvvvL
u
u
LLM
(33)
=
==
=
+
++
++
++
+
hn
rnn
h
σ
σ
00 LO
1
1
where },,2,1,{},,,2,1,{ rnknl
hk
T
hl
+
++
+=
==
==
==
=LL vu are sets of
the left and right singular vectors of
h
L associated with the
set of singular values
}.,,2,1,{
nl
hl
L
=
==
=
σ
It is evident that vectors },...,2,1,{ rnnnj
hj
+
++
++
++
++
++
+=
==
=v satisfy
(31), i.e.
[
[[
[
]
]]
]
0vABIvL =
==
=−
−−
−=
==
=
hjhhjh
s
(34)
The set of vectors },...,2,1,{
rnnnj
hj
+
++
++
++
++
++
+=
==
=v
is a non-
trivial solution of (32), and results the null space of
nh
h
,...,2,1,
=
==
=L
[
[[
[ ]
]]
]
0ABI
Kn
n=
==
=−
−−
−∈
∈∈
∈
h
h
h
sN
(35)
The null space (35) consists of the normalized orthogonal
set of vectors. Any combination of these vectors (the span
of null space) will provide a vector
h
n
which used as an
eigenvector produces the desired eigenvalue
h
s
in the
closed-loop system matrix.
Proposition 4. The canonical form eigenstructure optimiza-
tion provides optimal eigenstructure also for that model from
which the canonical form was derived.
Proof. Using (16), (18), (19) and (22) it can be simply writ-
ten
(
((
(
)
))
)
(
((
(
)
))
)
(
((
( )
))
)
hhhcc
hccccck
snnTKBAT
nTKTBTTATnBKA
=
==
=−
−−
−=
==
=
=
==
=−
−−
−=
==
=−
−−
−
−
−−
−
−
−−
−−
−−
−
1
11
ooo
oo
(36)
(
((
(
)
))
)
hchhc
snTnTKBA
11 −
−−
−−
−−
−
=
==
=−
−−
−
ooo
(37)
(
((
(
)
))
)
ooooooo
hhhch
snnAnKBA =
==
==
==
=−
−−
−
(38)
respectively, where .,
oo hchc
nTnKTK =
==
==
==
=
Writing compactly the set },,2,1,{ nh
hch
L
o
=
==
==
==
=nTn as fol-
lows
11
,
−
−−
−−
−−
−
=
==
==
==
=
c
T
c
TNNNTN
oo
(39)
then using (27), (30), (39) it yields
(
((
(
)
))
)
(
((
( )
))
)
(
((
( )
))
)
LBNSINC
BLTNSINCT
BLNSICNBLCG
oooo
oo
Tc
T
c
s
s
s(s)
1
1
1
1
1
−
−−
−
−
−−
−
−
−−
−
−
−−
−
−
−−
−
−
−−
−=
==
=
=
==
=−
−−
−=
==
=
=
==
=−
−−
−=
==
==
==
=
Υ
ΥΥ
Υ
(40)
(
((
( )
))
)
(
((
(
)
))
)
LBAICBLAICG
ooo 1
1−
−−
−
−
−−
−
−
−−
−=
==
=−
−−
−=
==
=
cc
ss(s)
(41)
respectively, and evidently
∑
∑∑
∑∑
∑∑
∑
=
==
==
==
=
−
−−
−
=
==
=
−
−−
−
=
==
=
n
hh
T
hh
n
hh
T
hh
ssss
(s)
11
L
BnnC
L
BnCn
G
oooo
(42)
It is obvious that optimizing product
oo h
nC then it is opti-
mized also .
h
nC
This concludes the proof. ■
0 1 2 3 4 5 6 7 8 9 10
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time [s]
y(t)
Fig. 1 System output response
4. Parameter Design
Using eigenvector orthogonal properties, (22) can be rewrit-
ten for
nh ,,2,1 L=
==
=
as follows
(
((
(
)
))
)
ooooooo hhhh
srBnKBnAI −
−−
−=
==
=−
−−
−=
==
=−
−−
−
(43)
(
((
(
)
))
)
oooooo hhhhh
srVrBAIn =
==
=−
−−
−−
−−
−=
==
=
−
−−
−1
(44)
respectively, where
(
((
(
)
))
)
oooooo
BAIVnKr
1
,
−
−−
−
−
−−
−−
−−
−=
==
==
==
=
hhhh
s
(45)
Subsequently, it can be obtained
ooo h
h
h
nVr
†
=
==
=
(46)
where
(
((
(
)
))
)
TT† oooo hhh
h
VVVV
1−
−−
−
=
==
=
(47)
is Moore-Penrose pseudo inverse of .
o
h
V
Of interest are the eigenvectors of the closed-loop system
having minimal orthogonal projection to rows of the ortho-
gonal complement
⊥
⊥⊥
⊥To
C of the output matrix
To
C and as-
sociated with
m
element eigenvalues subset
))()(),(()(
oooo
AACA Ω⊂
⊂⊂
⊂=
==
=
ρρ
rankm of the desired closed-
loop eigenvalues set },...,2,1,0)(,{)( nhss
hh
=
==
=<
<<
<ℜ
ℜℜ
ℜ=
==
=
o
AΩ
).()( AA ΩΩ =
==
=
o
The rest
)( mn −
−−
−
eigenvalues can be asso-
ciated with rows of the complement matrix
•
••
•
C obtained in
such way that all zero elements in
o
C be changed to ones,
and all ones to zeros. Note, direct use of
o
C maximize ma-
trix weights of modes.
Let
},,...,2,1,0)(,{)( nhss
hh
=
==
=<
<<
<ℜ
ℜℜ
ℜ=
==
=
o
A
ρ
then
mh
TT
h
h
h
,...,2,1,
=
==
==
==
=
⊥
⊥⊥
⊥∗
∗∗
∗oo
cVr
†
(48)
nmh
T
h
h
h
,...,1,
+
++
+=
==
==
==
=
•
••
••
••
•
cVr
†o
(49)
Thus, computing
•
••
••
••
•∗
∗∗
∗∗
∗∗
∗
=
==
==
==
=
hhhhhh
rVnrVn
oo
,
(50)
0 1 2 3 4 5 6 7 8 9 10
−30
−20
−10
0
10
20
30
time [s]
u(t)
Fig. 2 Control actions
it is possible to construct and to separate the matrix
o
Q of
the form
[
[[
[
]
]]
]
=
==
==
==
=
•
••
••
••
•+
++
+
∗
∗∗
∗∗
∗∗
∗o
o
o
LL R
P
vvvvQ
mmm 11
(51)
with
nn×
××
×
∈
∈∈
∈
RP
o
,
nr×
××
×
∈
∈∈
∈
RR
o
such that
11
,
−
−−
−−
−−
−
=
==
==
==
=
c
TKKPRK
ooo
(52)
4. Illustrative Example
The system under consideration was described by (1), (2),
where
=
==
=
=
==
=
−
−−
−−
−−
−−
−−
−
=
==
=01
12
11
,
52
12
31
,
595
100
010
T
CBA
Constructing the transformation matrices
−
−−
−
−
−−
−−
−−
−−
−−
−
=
==
=
−
−−
−
=
==
=
−
−−
−
5.75.30.1
5.65.20.1
5.55.20.1
,
0.00.10.1
0.10.20.1
5.25.00.4
1cc
TT
the system model canonical form parameters were com-
puted as
[
[[
[ ]
]]
]
2
,
43
107
00
,
10.10
20.30
65.101
I0CBA =
==
=
=
==
=
−
−−
−
−
−−
−−
−−
−
−
−−
−
=
==
=
ooo
Thus, considering
}6,2.1,5.0{)( −
−−
−−
−−
−−
−−
−=
==
=
o
AΩ
it is
−
−−
−−
−−
−
−
−−
−−
−−
−
=
==
=1538.64615.4
9231.07692.0
4615.543846.37
1
o
V
−
−−
−−
−−
−
−
−−
−−
−−
−
=
==
=4878.105610.7
0976.65122.4
4815.50610.10
2
o
V
−
−−
−−
−−
−
=
==
=0176.01176.0
4118.34118.2
3059.72059.5
3
o
V
and with
[
[[
[
]
]]
]
001=
==
=
⊥
⊥⊥
⊥To
c,
[
[[
[
]
]]
]
101
1
=
==
=
•
••
•T
c it yields
−
−−
−
=
==
=
−
−−
−
=
==
=
−
−−
−
=
==
=
•
••
•
2737.13
4978.18
,
1194.0
1645.0
,
2854.0
3891.0
321
rrr
oo
[
[[
[
]
]]
]
0205.00358.09983.0
1
=
==
=
To
n
[
[[
[
]
]]
]
0082.00144.09997.0
2
−
−−
−−
−−
−=
==
=
To
n
[
[[
[
]
]]
]
0205.00358.09983.0
3
=
==
=
•
••
•T
n
Constructing the matrix
o
Q
=
==
=
−
−−
−−
−−
−−
−−
−
−
−−
−−
−−
−
=
==
=
o
o
o
R
P
Q
2737.131194.02854.0
4978.181645.03891.0
6146.00082.00205.0
6745.00144.00358.0
6788.09997.09983.0
the control law parameters satisfying (52) are
−
−−
−
−
−−
−−
−−
−
=
==
=7151.186301.20036.0
9402.257944.30062.0
o
K
−
−−
−−
−−
−−
−−
−
=
==
=6211.24532.130707.16
7990.33483.181212.22
K
It is possible to verify that closed-loop system matrix eigen-
values belongs to the desired one.
In the presented Fig. 1, 2 the example is shown of the un-
forced closed-loop system output response, as well as con-
trol actions, where nonzero initial state was considered.
5.0
1
−
−−
−←
←←
←
•
••
•T
c 2.1
1
−
−−
−←
←←
←
•
••
•T
c 6
1
−
−−
−←
←←
←
•
••
•T
c
o
K
17.2524 15.9878 32.3181
K
46.2521 43.7478 35.8664
ISE 1.8746 0.8183 0.1712
IAE 3.7033 1.6238 0.5651
ITAE 7.0822 1.8607 0.7077
Tab.1 Comparison of performance indicators for differ-
ent combination of eigenvalue assignment
Concluding remarks
This paper provides a design method for memory-free con-
trollers where the general problem of assigning the eigen-
structure for state variable mode decoupling in state feed-
back control design is considered. The method exploits
standard numerical optimization procedures to manipulate
the system feedback gain matrix as a direct design variable.
The manipulation is accomplished in a manner that produc-
es desired system global performance by pole placement
and output dynamics by modification of the mode observa-
bility.
With generalization of the known algorithms for pole as-
signment the modified exposition of the problem is pre-
sented here to handle the optimized structure of the left
eigenvector set in state feedback control design. Presented
method makes full use of the freedom provided by eigen-
structure assignment to find a controller which stabilizes the
closed-loop system. Therefore, the feedback control law has
a clear physical meaning and provides a valid design me-
thod of the controller for real systems. It is shown by appro-
priately assigning closed-loop eigenstructure in state feed-
back control the overall stability is achieved. Finally the
design methodology is illustrated by an example.
Acknowledgment
The work presented in this paper was supported by VEGA,
Grant Agency of Ministry of Education and Academy of
Science of Slovak Republic under Grant No. 1/0256/11, as
well by Research & Development Operational Programme
Grant No. 26220120030 realized in Development of Centre
of Information and Communication Technologies for Know-
ledge Systems. These supports are very gratefully acknowl-
edged.
References
[1] BACHELIER, O., BOSCHE, J., and MEHDI, D.: On pole
placement via eigenstructure assignment Approach, IEEE
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[2] BURNS, R.S.: Advanced Control Engineering, Butter-
worth-Heinmann, Oxford, 2001.
[3] DATTA, B.N.: Numerical Methods for Linear Control
Systems: Design and Analysis, Elsevier, London, 2004.
[4] GOLUB, G.H. and LOAN Van C.F.: Matrix Computations,
The John Hopkins University Press, Baltimore, 1989.
[5] FILASOVÁ, A.: Robust control design for large-scale
uncertain dynamic system, In New Trends in Design of
Control Systems, STU, Bratislava, 247-432, 1997.
[6] FILASOVÁ, A.: Robust control design: An optimal con-
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ference on Intelligent Engineering Systems INES’99, Stará
Lesná, Slovakia, 515-518, 1999.
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Abstrakt
Cieľom návrhu je syntetizovať stavové riadenie tak, aby
vzhľadom na predpísané vlastné hodnoty matice dynamiky
uzavretého obvodu boli odpovedajúce vlastné vektory čo
najviac podobné vlastným vektorom systému s rozviazaným
vstupom a výstupom. V príspevku je ukázané, že existujúci
stupeň voľnosti v návrhu, reprezentovaný množinou para-
metrických vektorov a množinou pólov uyavretého obvodu,
možno využiť na dosiahnutie vopred zadaných charakteris-
tík uzavretého obvodu. Príklad a číselné simulácie ukazujú,
že takýto spôsob návrhu je efektívny a jednoduchý.
Ing. Pavol Kocsis
Ing. Róbert Fónod
Technical University of Košice
Faculty of Electrical Engineering and Informatics
Department of Cybernetics and Artificial Intelligence
Letná 9
042 00 Košice
Tel.: ++421 55 602 2749
Fax: ++421 55 625 3574
E-mail: robert.fonod@student.tuke.sk, pavol.kocsis@tuke.sk
