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STATE FEEDBACK CONTROL DESIGN USING
EIGENSTRUCTURE DECOUPLING
R. F´onod and P. Kocsis
Department of Cybernetics and Artificial Intelligence
Faculty of Electrical Engineering and Informatics
Technical University of Koˇsice, Koˇsice, Slovakia
fax: ++ 421 55 625 3574
e-mail: robert.fonod@student.tuke.sk; pavol.kocsis@tuke.sk
Abstract: In this paper the design of controlling a class of linear systems via state
feedback eigenstructure assignment is investigated. The design aim is to synthesize
a state feedback control law such that for prescribed eigenvalues of the closed-loop
control system corresponding eigenvectors are as close to decoupled ones as possible.
The set of parametric vectors and the set of closed-loop eigenvalues represent the
degrees of freedom existing in the control design, and can be further properly chosen to
meet some desired specification requirement, such as mode decoupling and robustness.
An illustrative example and the simulation results show that the proposed parametric
method is effective and simple.
Keywords: Mode decoupling, singular value decomposition, state feedback, linear
control systems, eigenstructure assignment.
1. INTRODUCTION
The static and the dynamic pole placement be-
longs to the prominent design problems of modern
control theory, and, although its practical useful-
ness has been continuously in dispute, it is one
of the most intensively investigated in control
system design. It seems that the state-feedback
pole assignment in control system design is one
from the preferred 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 (Filasov´a (1999),
Ipsen (2009)), and to determine a specific solution
additional conditions have to be supplied in order
to eliminate the extra degrees of freedom in design
strategy.
In last significant progress has been achieved in
this field, coming in its formulation closest to the
algebraic geometric nature of the pole placement
problem (Kautsky et al. (1985), Wonham (1985)).
The reason for the discrepancy in opinions about
the conditioning of the pole assignment problem
is that one has to distinguish among three aspects
of the pole placement problem, the computation
of the memoryless feedback control law matrix
gain, the computation of the closed loop system
matrix eigenvalues spectrum and the suppressing
of the cross-coupling effect (Wang (2003)), 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 (Kocsis and
Krokavec (2008), Sobel and Lallman (1989)).
The general problem of assigning the system ma-
trix eigenstructure using the state feedback con-
trol is considered in this paper. Based on the
classic algebraic methods (Golub and Van Loan
(1989), Datta (2004), Poznyak (2008)), as well as
on the algorithms for pole assignment using Singu-
lar Value Decomposition (SVD) (Filasov´a (1997),
Krokavec and A. Filasov´a (2006)) the exposition
of the pole eigenstructure assignment problem is
generalized here to handle the specified struc-
ture of the left eigenvector 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 straightfor-
ward methodology usable in linear control sys-
tem 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 rel-
ative simple and its worth can help to disclose the
continuity between eigenstructure assignment and
system variable dominant dynamic specification.
2. PROBLEM STATEMENT
Linear dynamic systems with ndegree of freedom
can be modelled by the state-space equations
˙
q(t)=Aq(t)+Bu(t)(1)
y(t)=Cq(t)(2)
with constant matrices A∈IRn×n,B∈IRn×r,
and C∈IRm×n. Generally, to the controllable
time-invariant linear MIMO system (1) a linear
state feedback regulator control law, defined gen-
erally as
u(t)=−Kq(t)+Lw(t)(3)
with K∈IRr×n,L∈IRr×mgives rise to the
closed-loop system
˙
q(t)=Acq(t)+BLw(t)(4)
which closed loop poles are eigenvalues of matrix
Ac=(A−BK)andAc∈IRn×n.
Throughout the paper it is assumed the pair
(A,B) is controllable.
3. BASIS PRELIMINARIES
3.1 Orthogonal Complement
Definition 1. (Null space) Let E,E∈IRh×h,
rank(E)=k<hbe a rank deficient matrix.
Then the null space NEof Eis the orthogonal
complement of the row space of E.
Proposition 1. Let E,E∈IRh×h,rank(E)=k<
hbe a rank deficient matrix. Then an orthogonal
complement E⊥of Eis
E⊥=DUT
2(5)
where UT
2is the null space of Eand Dis an
arbitrary matrix of appropriate dimension.
Proof. (Filasov´a and Krokavec (2010b)) The
SVD of E,E∈IRh×h,rank(E)=k<hgives
UTEV =UT
1
UT
2EV1V2=Σ1012
021 022(6)
where UT∈IRh×his the orthogonal matrix of the
left singular vectors, V∈IRh×his the orthogonal
matrix of the right singular vectors of Eand Σ1∈
IRk×kis the diagonal positive definite matrix
Σ1=diagσ1··· σk,σ
1≥···≥σk>0(7)
which diagonal elements are the singular values of
E. Using orthogonal properties of Uand V, i.e.
UTU=Ih,VTV=Ih,UT
2U1=0,then
E=UΣVT=U1U2Σ1012
021 022VT
1
VT
2=
=U1U2S1
02=U1S1
(8)
where S1=Σ1VT
1. Thus, (8) implies
UT
2E=UT
2U1U2S1
02=0(9)
It is evident that for an arbitrary matrix Dis
DUT
2E=E⊥E=0(10)
respectively, which implies (5).
3.2 System Model Canonical Form
Proposition 2. If rank(CB)=mthen there ex-
ists a coordinates change in which (A◦,B◦,C◦)
takes the structure
A◦=A◦
11 A◦
12
A◦
21 A◦
22,B◦=0
B◦
2,C◦=0Im(11)
where A◦
11 ∈IR(n−m)×(n−m),B◦
2∈IRm×mis a
non-singular matrix, and Im∈IRm×mis identity
matrix.
Proof. (Filasov´a and Krokavec (2010a)) Consid-
ering the state-space description of the system (1),
(2) with r=mand defining the transform matrix
T−1
1such that
C1=CT1=0Im,T−1
1=In−m0
C(12)
then
B1=T−1
1B=T−1
1B1
B2=B1
CB=B11
B12(13)
If CB =B12 is a regular matrix (in opposite case
the pseudoinverse of B12 is possible to use), then
the second transform matrix T−1
2can be defined
as follows
T−1
2=In−m−B11
B−1
12
0Im(14)
T2=In−mB11 B−1
12
0Im(15)
This results in
B◦=T−1
2B1=0
B◦
2(16)
where
B11 =B1,B◦
2=B12 =CB (17)
and
C◦=C1T2=0ImT2=0Im(18)
Finally, with T−1
c=T−1
2T−1
1it yields
A◦=T−1
cATc=T−1
2T−1
1AT 1T2(19)
Thus, (16), (18), and (19) implies (11). This
concludes the proof.
Note, the structure of T−1
1is not unique and
others can be obtained by permutations of the first
n−mrows in the structure defined in (12).
3.3 System Modes Properties
Proposition 3. Given system eigenstructure with
distinct eigenvalues then for j, k ∈{1,2,...n},l ∈
{1,2,...m},m=r
i.thek-thmode(s−sk) is unobservable from the
l-th system output if the l-th row of matrix Cis
orthogonal to the k-th eigenvector of the closed-
loop system matrix A
c, i.e. with j=k
cT
lnk=nT
jnk=0,CT=c1··· cm(20)
ii.thek-thmode(s−sk) is uncontrollable from the
l-th system input if the l-th column of matrix B
is orthogonal to the k-th eigenvector of the closed-
loop system matrix A
c, i.e. with j=k
nT
kbl=nT
knj=0,B=b1··· br(21)
Proof. (Krokavec and A. Filasov´a (2006)) Let nk
is the k-th right eigenvector corresponding to the
eigenvalue sk, i.e.
A
cnk=(A−BK)nk=sknk(22)
By definition, the closed-loop system resolvent
kernel is
Υ=(sIn−A
c)−1(23)
If the closed-loop system matrix is with distinct
eigenvalues, (22) can be written in the compact
form
A
cn1··· nn=
n1··· nn⎡
⎢
⎣
s1
...
sn
⎤
⎥
⎦
(24)
A
cN=NS,N−1=NT(25)
respectively, where
S=diags1···sn,N=n1···nn(26)
Using the property of orthogonality given in (25),
the resolvent kernel of the system takes form
Υ=sNN−1−NSN−1−1=
=N(sI−S)−1NT(27)
Υ=n1···nn
⎡
⎢
⎢
⎢
⎢
⎣
1
s−s1...
1
s−sn
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
nT
1
.
.
.
nT
n
⎤
⎥
⎦(28)
Υ=
n
h=1
nhnT
h
s−sh
(29)
respectively. Thus, the closed loop transfer func-
tions matrix takes form
G(s)=
=C(sI−A
c)−1BL =
n
h=1
CnhnT
hB
s−sh
L(30)
It is obvious that (30) implies (20), (21). This
concludes the proof.
4. EIGENSTRUCTURE ASSIGNMENT
In the pole assignment problem, a feedback gain
matrix Kis sought so that the closed-loop system
has a prescribed eigenvalues spectrum Ω(A
c)=
{sh:(sh)<0,h=1,2, ..., n}. Note, the spec-
trum Ω(A
c) is closed under complex conjugation,
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
shI−AB
nh
Knh=Lhnh
Knh=0(31)
where Lh∈IRn×(n+r),
Lh=shI−AB
(32)
Subsequently, the singular value decomposition
(SVD) of Lhgives
⎡
⎢
⎣
uT
h1
.
.
.
uT
hn
⎤
⎥
⎦Lhvh1
···vhn
vh,(n+1)
···vh,(n+r)=
=⎡
⎢
⎣
σh1
...
σhn
0n+1
···0n+r⎤
⎥
⎦
(33)
{uT
hl,l =1,2,...,n},{vhk,k =1,2,...,n+r}are
sets of the left and right singular vectors of Lh
associated with the set of singular values {σhl,
l=1,2,...,n}
It is evident that vectors {vhj ,j =n+1,n+
2,...,n+r}satisfy (31), i.e.
Lvhj =shI−AB
vhj =0(34)
The set of vectors {vhj ,j =n+1,n+2,...,n+r}
is a non-trivial solution of (32), and results the
null space of Lh,h=1,2,...,n
nh
Knh∈NshI−AB
(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 nhwhich used as an eigenvector produces
the desired eigenvalue shin the closed-loop system
matrix.
Proposition 4. The canonical form eigenstructure
optimization provides optimal eigenstructure also
for that model from which the canonical form was
derived.
Proof. Using (16), (18), (19) and (22) it can be
written
(A−BK)nh=
=(TcA◦T−1
c−TcB◦
KTcT−1
c)nh=
=Tc(A◦−B◦K◦)T−1
cnh=shnh
(36)
shT−1
cnh=shn◦
h=(A◦−B◦K◦)n◦
h(37)
respectively, where
K◦=KTc,nh=Tcn◦
h,(38)
N=TcN◦,N−1=N◦TT−1
c(39)
and subsequently using (27) it yields
G(s)=CN(sI−S)−1N−1BL =
=CTcN◦(sI−S)−1N◦TT−1
cBL =
=C◦N◦(sI−S)−1N◦TB◦L
(40)
G(s)=C(sI−A
c)−1BL =
=C◦(sI−A◦)−1B◦L(41)
G(s)=
=
n
h=1
CnhnT
hB
s−sh
L=
n
h=1
C◦
n◦
hn◦T
hB◦
s−sh
L(42)
respectively. It is obvious that optimizing C◦n◦
h
is optimized Cnh. This concludes the proof.
5. PARAMETER DESIGN
Using eigenvector orthogonal properties, (22) can
be rewritten for h=1,2,...,n as follows
(shI−A◦)n◦
h=−B◦
K◦nh=−B◦r◦
h(43)
n◦
h=−(shI−A◦)−1B◦r◦
h=V◦
hr◦
h(44)
respectively, where
r◦
h=K◦n◦
h,V◦
h=−(shI−A◦)−1B◦(45)
Subsequently, it can be obtained
r◦
h=V◦1
hn◦
h(46)
where
V◦1
h=(V◦T
hV◦
h)−1V◦T
h(47)
is Moore-Penrose pseudoinverse of V◦
h.
Of interest are the eigenvectors of the closed-loop
system which are as orthogonal as possible to rows
of the orthogonal complement C◦T⊥of the output
matrix C◦Tand associated with the prescribed
m=rank(C◦) elements subset ρ(A◦)⊂Ω(A◦)
of the desired closed-loop eigenvalues set Ω(A◦)=
{sh,(sh)<0,h=1,2,...,n},Ω(A◦)=Ω(A).
The rest (n−m) eigenvalues can be associated with
rows of the complement matrix C•obtained in
such way that all zero elements in C◦be changed
to ones, and all ones to zeros. Note, direct use of
C◦maximize matrix weights of modes.
Let ρ(A◦)={sh,(sh)<0,h=1,2,...,n},
r
h=V◦1
hc◦T⊥T
h,h=1,2,...,m (48)
r•
h=V◦1
hc•T
h,h=m+1,...n (49)
Then, computing
n
h=V◦
hr
h,n•
h=V◦
hr•
h(50)
it is possible to construct and to separate the
matrix Q◦of the form
Q◦=v
1···v
mv•
m+1 ···w•
m=P◦
R◦(51)
with P◦∈IRn×n,R◦∈IRr×nsuch that
K◦=R◦P◦−1,K=K◦T−1
c(52)
6. ILLUSTRATIVE EXAMPLE
The system under consideration was described by
(1), (2), where
A=⎡
⎣
010
001
−5−9−5
⎤
⎦,B=⎡
⎣
13
21
25
⎤
⎦,CT=⎡
⎣
11
21
10
⎤
⎦
Constructing the transform matrices
T−1
c=⎡
⎣
4.00.5−2.5
1.02.01.0
1.01.00.0
⎤
⎦,Tc=⎡
⎣
1.02.5−5.5
−1.0−2.56.5
1.03.5−7.5
⎤
⎦
the system model canonical form parameters were
computed as C◦=[0I2]
A◦=⎡
⎣
−1 10.56
0−3.0−2
01.0−1⎤
⎦B◦=⎡
⎣
00
710
34
⎤
⎦
Thus, considering Ω(A◦)={−0.5,−1.2,−6}be
V◦
1=⎡
⎣
−37.3846 −54.4615
0.7692 0.9231
−4.4615 −6.1538
⎤
⎦
V◦
2=⎡
⎣
−10.0610 −5.4878
4.5122 6.0976
−7.5610 −10.4878⎤
⎦
V◦
3=⎡
⎣
−5.2059 −7.3059
2.4118 3.4118
0.1176 0.0176⎤
⎦
and with c◦T⊥=100
,c•T
1=101
yields
r◦
1=0.3891
−.0.2854,r◦
2=−0.1645
0.1194
r•
3=18.4978
−13.2737
n◦T
1=0.9983 0.0358 0.0205
n◦T
2=0.9997 −0.0144 −0.0082
n•T
3=0.6788 −0.6745 0.6146
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
Constructing the matrix Q◦
Q◦=⎡
⎢
⎢
⎢
⎢
⎣
0.9983 0.9997 0.6788
0.0358 0.0144 −0.6745
0.0205 −0.0082 0.6146
0.3891 −0.1645 18.4978
−0.2854 0.1194 −13.2737
⎤
⎥
⎥
⎥
⎥
⎦
=P◦
R◦
the control law parameters satisfying (52) are
K◦=−0.0062 −3.7944 25.9402
0.0036 2.6301 −18.7151
K=22.1212 18.3483 −3.7990
−16.0707 −13.4532 2.6211
It is possible to verify that closed-loop system
matrix eigenvalues belongs to the desired one.
In the presented Fig. 1 the example is shown of
the unforced closed-loop system output response,
where nonzero initial state was considered.
7. CONCLUDING REMARKS
This paper provides a design method for memory-
free controllers where the general problem of as-
signing the eigenstructure for state variable mode
decoupling in state feedback control design is con-
sidered. 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 produces desired system global performance
by pole placement and output dynamics by mod-
ification of the mode observability.
With generalization of the known algorithms for
pole assignment the modified exposition of the
problem is presented here to handle the optimized
structure of the left eigenvector set in state feed-
back control design. Presented method makes full
use of the freedom provided by eigenstructure
assignment to find a controller which stabilizes the
closed-loop system. Therefore, the feedback con-
trol law has a clear physical meaning and provides
a valid design method of the controller for real
systems. It is shown by appropriately assigning
closed-loop eigenstructure in state feedback con-
trol the overall stability is achieved. Finally the
design methodology is illustrated by an example.
ACKNOWLEDGMENT
The work presented in this paper was supported
by Grant Agency of Ministry of Education and
Academy of Science of Slovak Republic VEGA
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 Knowledge Systems. These supports are very
gratefully acknowledged.
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