International Journal of Control, Automation, and Systems (2011) 9(1):15-22
DOI 10.1007/s12555-011-0103-9
http://www.springer.com/12555
Parametric Eigenstructure Assignment for Descriptor Systems via
Proportional plus Derivative State Feedback
Biao Zhang
Abstract: Eigenstructure assignment for descriptor systems with proportional plus derivative state
feedback is studied. Based on a simple complete explicit parametric solution to a group of recursive
equations, a parametric approach for eigenstrucure assignment in descriptor systems via proportional
plus derivative state feedback is proposed. The proposed approach possesses the following features: 1)
it does not impose any condition on the closed-loop eigenvalues, simultaneously assigns arbitrary n finite and infinite eigenvalues to the closed-loop system and guarantees the closed-loop regularity; 2) it
is simple and needs less computational work; 3) it gives general complete parametric expressions for
the closed-loop eigenvectors, the proportional state feedback gain matrix and the derivative state feedback gain matrix.
Keywords: Closed-loop regularity, descriptor linear systems, eigenstructure assignment, proportional
plus derivative state feedback.
1. INTRODUCTION
In this paper, we consider the control of the following
linear descriptor system
Ex  Ax  Bu
(1)
with proportional plus derivative (PD) state feedback
u  K1 x  K2 x,
(2)
where in (1), x  R n , u  R r are respectively, the state
vector and the input vector; E , A  R nn and B Rnr
are real matrices with rank( E )  m  n, rank( B)  r;
where in (2), K1 , K2  Rrn are respectively, the
proportional (P) state feedback gain matrix and the
derivative (D) state feedback gain matrix.
It is well known from classical control theory that
derivative feedback is very essential for improving the
stability and the performance of a control system (see
e.g., [1-3]). In [1], PD controllers are employed to
provide anticipatory action for overshoot reduction in the
responses. In [2] and [3], PD controllers are used to
achieve decoupling of linear systems. For descriptor
systems, derivative feedback is even more important
since it can alter many properties of a descriptor system,
which a pure proportional state feedback can not.
Because of this reason, the use of PD control law in
descriptor systems has been intensively and widely
__________
Manuscript received August 13, 2009; revised February 22,
2010 and July 9, 2010; accepted July 21, 2010. Recommended by
Editor Jae Weon Choi. This work was supported by the Chinese
National Natural Science Foundation under Grant No. 10671046.
Biao Zhang is with the Department of Mathematics, Harbin
Institute of Technology, Harbin, P. R. China (e-mail: zhangb@
hit.edu.cn).
© ICROS, KIEE and Springer 2011
studied since the early 1980s [3-17].
Eigenstructure assignment in descriptor linear systems
is a very important problem in descriptor systems theory
and has been studied during the past three decades. In
particular, the topic of eigenstructure assignment for
descriptor systems via PD state feedback has been
studied by a number of researchers [13-17]. Chen and
Chang [13] and Jin [14] considered the problem of
eigenstructure assignment in descriptor systems using the
special case of PD state feedback where K1   K2 , i.e.,
the constant-ratio proportional plus derivative (CRPD)
state feedback, while Duan and Patton [15], Owens and
Askarpour [16], and Wang and Lin [17] considered the
problem of eigenstructure assignment in descriptor
systems via PD state feedback. The work in [13] depends
on the properties of the standard form descriptor systems,
but the computed gain matrix and assigned eigenvectors
are in the original coordinates, and no transformations
are needed. However, the result requires the solution of
(sE−A)−1 or (i E  A)1, i  1, 2, , n, and thus requires
more computational work and may subject to numerical
problems. The works in [14] and [15] are based on the
right coprime factorization of the open-loop system. The
work in [14] removes the restriction that the closed-loop
finite eigenvalues are different from the open-loop
eigenvalues, which was required in [13], while the work
in [15] removes the restriction required in [13] that the
assigned closed-loop finite eigenvalues are different
from the open-loop eigenvalues and also releases the
open-loop regularity assumption required in both [13]
and [14]. However, the results in [14] and [15] require
right coprime matrix polynomials to be determined, and
thus are not desirable to use in high dimension cases
because the determination of right coprime matrix
polynomials is computationally expensive and not in
general numerically reliable [18]. The works in [16] and
Biao Zhang
16
[17] are based on solving a recursive eigenvector chains
of the matrix pair ( E  BK2 , A  BK1 ). Unlike [14] and
[15], the works in [16] and [17] do not require right
coprime matrix polynomials to be determined, and thus
overcome the defect of [14] and [15]. However, the
results in [16] and [17] contain a series of iterative
computations with parameter vectors involved, and thus
are complex and need more computational work [19,20].
Moreover, all the reported works for eigenstructure
assignment by PD state feedback except the work [17]
cannot assign infinite eigenvalues to the closed-loop
system.
This paper considers eigenstructure assignment in the
descriptor system (1) via PD state feedback (2). We
relate the problem to the following recursive equations
Lzk  Mzk 1 , z0  0, k  1, 2,
, l,
(3)
where L, M  Cst (s  t ) with L of full row-rank are
known matrices; zk , k  1, 2, , l , are to be determined.
Based on a presented simple complete explicit
parametric solution to (3), a simple complete parametric
approach for eigenstructure assignment in the descriptor
system (1) via PD state feedback (2) is proposed. The
proposed approach possesses the following features: 1) it
does not impose any condition on the closed-loop
eigenvalues, simultaneously assigns arbitrary n finite and
infinite eigenvalues to the closed-loop system and
guarantees the closed-loop regularity; 2) it is very simple
and needs less computational work; 3) it gives general
complete parametric expressions for the closed-loop
eigenvectors, and the P and D state feedback gain
matrices.
2. FORMULATION OF THE PROBLEM
Assume that the descriptor system (1) is complete
controllable (C-controllable), i.e., system (1) satisfies the
following C-controllability assumption.
Assumption 1: rank[sE  A B]  n for all s C
and rank[ E B]  n.
If the PD feedback control law (2) is applied to (1), a
closed-loop system is obtained in the form
Ec x  Ac x
(4)
with
Ec  E  BK 2 , Ac  A  BK1.
(5)
Let   {i , i  1, 2, , , }, where i , i  1, 2, , ,
are a group of distinct self-conjugate complex numbers
and   , be the set of eigenvalues of the matrix pair
( Ec , Ac ), and denote the algebraic and geometric
multiplicity of i by mi and qi, respectively, then there
are qi chains of generalized eigenvectors of ( Ec , Ac )
associated with i . Denote the lengths of those qi
chains by pij , j  1, 2, , qi , then the following
relations hold:
pi1  pi 2 
 piqi  mi ,
(6)
m1  m2 
 m  m  n.
(7)
Let the right eigenvector chains of the matrix pair
( Ec , Ac ) associated with finite eigenvalue i be
denoted by vijk  Cn , k  1, 2,
, pij , j  1.2,
, qi . Then
they satisfy
[ A  BK1  i ( E  BK2 )]vijk  ( E  BK2 )vijk 1 , vij0  0,
k  1, 2,
, pij ,
j  1, 2,
, qi , i  1, 2,
, .
(8)
Let s  1   0, then s is the zero eigenvalue of
the matrix pair (Ac, Ec). Denote the right eigenvector
chains of the matrix pair (Ac, Ec) associated with s by
vk j  Cn , k  1, 2,
, p , j  1, 2,
, q . Then we have
the following equations by definition
[ E  BK 2  s ( A  BK1 )]vk j  ( A  BK1 )vk j 1 ,
v0 j  0, k  1, 2,
, p j ,
j  1, 2,
(9)
, q .
Now the problem of eigenstructure assignment (EA)
via PD feedback controller (2) for the descriptor system
(1) can be stated as follows: Determine a pair of real
matrices K1 , K2  Rrn , and a group of vectors vijk 
C n , k  1, 2,
, pij , j  1, 2,
, qi , i  1, 2,
, , , such
that the following three requirements are simultaneously
satisfied.
1) all the equations in (8) and (9) hold;
2) vectors vijk  C n , k  1, 2, , pij , j  1, 2, , qi , i 
1, 2, , , , are linearly independent;
3) the matrix pair (Ec, Ac) is regular, i.e., det( sEc −Ac)
is not identically zero.
3. GENERAL SOLUTION OF EQUATION (3)
Using matrix elementary transformation and in view
of the assumption that L is of full row-rank, a pair of
matrices P  Css and Q Ctt can be obtained such
that
PLQ  [I 0].
(10)
Partition the matrix Q as follows:
Q  [Q1 Q2 ], Q1  Cts .
(11)
From (10) and (11), we obtain
LQ1  P1 , LQ2  0.
(12)
Denote
H k  (Q1PM )k 1Q2 , k  1, 2,
, l.
(13)
Then the general solution of (3) is given in the following
theorem.
Theorem 1: All solutions of (3) are given by
Parametric Eigenstructure Assignment for Descriptor Systems via Proportional plus Derivative State Feedback
zk  H1 f k  H 2 f k 1 
 H k f1 , k  1, 2,
(14)
, l,
where f k Ct  s , k  1, 2, , l , are a group of arbitrarily
chosen free parameter vectors; H k , k  1, 2, , l , are
determined by (10)-(13).
Proof: First, let us show that the vectors given by (14)
satisfy the equations in (3). Using (12) and (13), we have
Lzk  L( H1 f k  H 2 f k 1 
 H k f1 )
 (Q1 PM ) k  2 Q2 f1 )
 0  P 1 PM ( H1 f k 1  H 2 f k  2 
f  [ f1T
H l 1
, pij ,
j  1, 2,
 In
Pi  i , Qi   0
 0
(18)
, .
, , satisfying
, .
0nr ], i  1, 2,
(19)
0
Ir
0
0 

i I r   i
0
I r  
0
, i  1, 2,
I r 
, .
(20)
It can be easily verify that the following holds
Pi Li Qi  [ I n
, .
0n2r ], i  1, 2,
Partition the matrix Qi , i  1, 2,
(21)
, , as follows:
Qi  [Qi1 Qi 2 ], Qi1  C(n2r )n , i  1, 2,
H ik  [ NikT
k  1, 2,
Since the matrix Q is nonsingular, the matrix H1 (=Q2) is
of full column-rank. Thus, the matrix H is of full
column-rank. Therefore, all elements in f contribute
independently to  . With this we complete the proof. 
4. SOLUTION OF THE PROBLEM EA
Let
wijk  K1vijk , sijk  K 2vijk ,
j  1, 2,
, qi , i  1, 2,
Denote
matrix H ik , k  1, 2,
  Hf .
, pij ,
k  1, 2,
where di  max1 j qi { pij }, i  1, 2,
then the equations in (3) can be written in the compact
matrix form
k  1, 2,
Li zijk  M i zijk 1 , zij0  0,
k 1
Hik  (Qi1PM
Qi 2 , k  1, 2,
i i)



,


H1 
H1
(17)
, .
, . (22)
Denote
flT ]T
and
 H1
H
 2
H 


 H l
, qi , i  1, 2,
Then the equations in (8) can be equivalently written as
i [ A  i E B]i  [ I n
zlT ]T ,
f2T
j  1, 2,
and i C(nr )(nr ) , i  1, 2,
 H k 1 f1 )
which holds for all k  1, 2, , l. Therefore, the vectors
given by (14) satisfy the equations in (3).
Next, let us show that the solution (14) is complete,
that is, it contains the maximum degrees of freedom, and
therefore forms a complete parametric solution to (3).
It is obvious that the maximum degrees of freedom
involved in the general solution to the group of equations
in (3) is l(t − s), while the solution (14) happen to contain
l(t − s) parameters. Thus, we need only to show that
these parameters involved in the solution (14) all
contributes to the vectors zk independently.
Let
z2T
, pij ,
(sijk )T ]T ,
Assumption 1, we obtain two sets of matrices i Cnn
 Mzk 1 , z0  0,
  [ z1T
k  1, 2,
(wijk )T
Applying matrix elementary transformations to the
matrices [ A  i E B], i  1, 2, , , and in view of
 LQ2 f k  L(Q1 PM )(Q2 f k 1  (Q1 PM )Q2 f k  2

zijk  [(vijk )T
17
, qi , i  1, 2,
, .
(15)
Denote
Li   A  i E B i B , Mi   E 0nr
B , (16)
DikT
, di , i  1, 2,
, ,
(23)
, . Partition the
, di , i  1, 2,
, , as follows:
SikT ]T , Dik , Sik  Cr 2r ,
, di , i  1, 2,
(24)
, .
By Theorem 1, the general complete parametric
expressions for the closed-loop eigenvectors associated
with the finite closed-loop eigenvalues, together with the
corresponding vectors wijk , sijk , k  1, 2, , pij , j  1, 2,
, qi , i  1, 2,
, , are obtained as
 vijk 
 Ni 2 
 Nik 
   Ni1 




k
k
k

1
 wij   Di1 fij  Di 2 fij    Dik  fij1 ,




   
 Si 2 
 Sik 
 sijk   Si1 
 
k  1, 2, , pij , j  1, 2, , qi , i  1, 2, , ,
where fijk  C 2 r , k  1, 2,
, pij , j  1, 2,
(25)
, qi , i  1, 2,
, , are a group of arbitrarily chosen free parameter
vectors; Nik , Dik , Sik , k  1, 2, , pij , j  1, 2, , qi , i
 1, 2,
Let
, , are determined by (19)-(24).
Biao Zhang
18
wk j  K1vk j , sk j  K 2 vk j ,
k  1, 2
, pj ,
j  1, 2,
(26)
, q .
termined by (30)-(35).
Define the matrix Vf as follows:
Denote
L  [ E  s A s B B], M   [ A B 0nr ], (27)
T T
zk j  (vk j )T ( wk j )T ( sk j )  ,
k  1, 2 , pj , j  1, 2, , q .
(28)
Then the equations in (9) can be equivalently written as
L zk j  M  zk j 1 , z0 j  0,
k  1, 2
, pj ,
j  1, 2,
(29)
, q .
Applying matrix elementary transformation to the matrix
[E B] and in view of Assumption 1, we obtain a pair of
matrices   R nn and   R(nr )(nr ) , satisfying
 [ E
B]  [ I n
(30)
0nr ].
Denote
In
P   , Q   0
 0
0
0
Ir
0

I r   
0
0  
0
.
I r 
(31)
(32)
0n2r ].
Q  [Q1 Q2 ], Q1 R(n2r )n .
(33)
Denote
, d ,
(34)
where d  max1 j q { pj }. Partition the matrix H k ,
k  1, 2,
, d , as follows:
H  k   N T k DT k
k  1, 2, , d .
T
ST k  , Dk , Sk  R r 2r ,
(35)
, q , are obtained as
 vk j 
 N 2 

  N 1 
 wk j    D1  f kj   D 2  f kj1 




 
 S 2 
 sk j   S1 


k  1, 2 , pj , j  1, 2, , q ,
where f kj  R 2 r , k  1, 2,
Vi  [Vi1 Vi 2
Viqi ],
Vij  [vij1
vij ij ].
vij2
p
Similarly, columns of matrices Wf and Sf are also
composed of wijk , sijk , k  1, 2, , pij , j  1, 2, , qi , i =1,
2, , . Then the equations in (15) can be equivalently
written in the unified matrix forms
W f  K1V f , S f  K2V f .
(37)
Again, define the matrix V∞ as
V  [V1 V2
V q ],
Vj  [v1j
vj j ],
p
v2 j
and similarly matrices W∞ and S∞ have the same form.
Then the equations in (26) can be equivalently written in
the unified matrix forms
(38)
By combining (37) and (38) we have
In order that real matrices K1 and K2 to be solved from
(39), we choose vijk  R n , wijk , sijk  R r for a real finite
eigenvalue i , whereas vljk  vijk  Cn , wljk  wijk , sljk 
sijk  C r
for a complex conjugate pair of finite
eigenvalues i , l  i . From (25), it is easy to see that
this condition can be equivalently converted into the
following constraint on the group of parameter vectors
f ijk , k  1, 2, , pij , j  1, 2, , qi , i  1, 2, , .
Constraint 1: fijk  R 2 r for a real finite eigenvalue
By Theorem 1, the general complete parametric
expressions for the closed-loop eigenvectors associated
with the infinite closed-loop eigenvalues, together with
the corresponding vectors wk j , sk j , k  1, 2, , pj , j 
1, 2,
V ],
[Wf W ]=K1[V f V ], [S f S ]  K2[V f V ]. (39)
Partition the matrix Q∞ as follows:
H  k  (Q1 P M  ) k 1 Q 2 , k  1, 2,
V f  [V1 V2
W  K1V , S  K2V .
It can be easily verify that the following holds
P LQ  [ I n
group of arbitrarily chosen free parameter vectors;
Nk , Dk , Sk , k  1, 2, , pj , j  1, 2, , q , are de-
 N k 
  Dk  f 1j ,
(36)
 Sk 
, pj , j  1, 2,
, q , are a
i , whereas fljk  fijk  C2r for a complex conjugate
pair of finite eigenvalues i , l  i .
To ensure the requirement 2) in the problem EA, we
need to supply the following constraint on the group of
parameter vectors f ijk , k  1, 2, , pij , j  1, 2, , qi , i 
1, 2, , , .
Constraint 2: det[V f
V ]  0.
When this constraint is met, the gain matrices K1 and K2
are given by
K1  [W f
W ][V f
V ]1 ,
K 2  [S f
S ][V f
V ]1.
(40)
Parametric Eigenstructure Assignment for Descriptor Systems via Proportional plus Derivative State Feedback
Constraint 3: det[ EV f  BS f
Let
J  diag( J1, J 2 ,
i


J ij  



1
i
1
, Jν ), Ji  diag( Ji1, Ji 2 ,
, Jiqi ),



p p
  C ij ij

1
i 
0 1
 0

Nj  



1
, , , satisfying Con-
2, , qi , i  1, 2, , ,  (or the closed-loop eigenvector
matrix [V f V ]), is given by (25) and (36), and the
, Nq ),



p p
  R j j .

1
0 
( A  BK1 )V f  ( E  BK2 )V f J ,
(41)
( E  BK2 )V  ( A  BK1 )V N .
(42)
Concerning the regularity of system (4), we have the
following lemma.
Lemma 1: Let Vf, V∞ and K1, K2 be matrices satisfying
equations (41) and (42) and Constraint 2. Then the
matrices K1 and K2 make the matrix pencil [ s ( E 
BK2 )  ( A  BK1 )] regular if and only if
( A  BK1 )V ]  0.
(43)
Proof: Using (41) and (42), we obtain
[ s ( E  BK 2 )  ( A  BK1 )][V f
, qi , i  1, 2,
straints 1-3. When this condition is met, the group of
closed-loop eigenvectors vijk  C n , k  1, 2, , pij , j  1,
Then all the equations in (8) and (9) can be written
respectively in the following matrix forms
det[( E  BK2 )V f
AV  BW ]  0.
To summarize, we have the following general result
for the solution to the problem EA.
Theorem 2: The problem EA has solutions if and only
if there exists a group of parameter vectors f ijk , k  1, 2,
, pij , j  1, 2,
and
N  diag(N1, N2 ,
19
V ]
 sI  J
 [( E  BK 2 )V f ( A  BK1 )V ] 
 0
 (44)
.
sN  I 
0
feedback gain matrices K1, K2 are given by (40) with
Constraints 1-3 satisfied.
Remark 1: The above approach for eigenstructure
assignment in descriptor systems by PD state feedback
has several advantages over the approaches of previously
published works in the following.
1) The approach does not impose any condition on the
closed-loop eigenvalues, and simultaneously assigns
arbitrary n finite and infinite eigenvalues to the closedloop system. Consequently, the approach automatically
removes the restriction required in [13] that the assigned
closed-loop finite eigenvalues are different from the
open-loop eigenvalues and the restriction required in [1316] that all the assigned closed-loop eigenvalues are
finite. The approach also releases the open-loop
regularity assumption required in [13] and [14].
2) The approach involves mainly matrix elementary
transformations (19) and (30). Unlike [13], it does not
require the solution of (sE  A)1 or (i E  A)1, i  1,
2, , n, to be determined. Unlike [14] and [15], it does
not require right coprime matrix polynomials to be
determined. Unlike [16] and [17], it does not contain
iterative computations. Thus, comparing with the
approaches in [13-17], the approach is much simpler and
needs less computational work.
3) Unlike [16] and [17], the approach gives the direct,
explicit parametric solution to the problem EA. It is
known that a direct, explicit solution usually provides
much convenience in some system design problems [19].
Therefore, we have
det[ s( E  BK 2 )  ( A  BK1 )]det[V f
 det[( E  BK 2 )V f
( A  BK1 )V ]
5. EXAMPLES
V ]
(45)
 det( sI  J ) det( sN  I ).
Since det[V f V ]  0, det(sN  I )  0 and det( sI  J )
is not identically zero, it follows from (45) that det[ s( E
 BK2 )  ( A  BK1 )] is not identically zero if and only if
condition (43) holds.

By using (25) and (36)-(38), condition (43), which
ensures the closed-loop regularity, can be turned into the
following constraint also on the group of parameter
vectors f ijk , k  1, 2, , pij , j  1, 2, , qi , i  1, 2, , ,
, as follows.
Two examples are given in this section. The first one
is to demonstrate the effect of the proposed approach; the
second one is an application of the proposed approach to
model refinement.
Example 1: Consider a system of the form (1) with
the following matrix parameters [13,15]
1
0

0
E
0
0

1
0 0 0 0 0
1 0 0 0 0 
0 1 0 0 0
,
0 0 0 1 0
0 0 0 0 0

0 0 0 0 0 
Biao Zhang
20
0
1

0
A
0
0

1
0 1 0 0 0
1

0
0 0 0 0 0

0
1 0 1 0 0
, B  
0 0 1 0 0
0
1
0 0 0 1 0


0 0 0 0 1 
 0
0
0 
0
,
0
1

0 
where n  6, m  4, r  2. It is easy to verify that
Assumption 1 is satisfied.
In the following, we consider the assignment of the
following closed-loop eigenstructure:
  {1, 0, }, m1  q1  1,
p11  3, m2  q2  1,
p21  2, m  q  p1  1.
1 0 0
2
 1
1 0 0 

0
1 0 0
 , N12  
0 0 0
1

0
0 0 0


2 0 0 
 5
1 0 0
0

0
0 0 0

0
0 0 0
 , N 21  
0 0 0
0
1
0 0 0


4 0 0 
0
0 0 0
1 0 0 
0 0 0
1
 , D11  
0 0 0
 2

0 0 0

0 0 0 
2 0 0 
1 0 0 
0 0 0
,
0 0 0
0 0 0

5 0 0 
0 0 0
0 0 0 
1 0 0
,
0 0 0
0 0 0

0 0 0 
2 1 0 
,
2 0 1
 3 3
D12  
 3 3
 0 1
D21  
 1 1
S11  S21
1 0
 3 3 0 0 
, D13  

,
0 1
 3 3 0 0 
0 0
0 0 1 0 
, D22  
,
0 0 
0 0 0 1 
0 0 1 0 

 , S12  S13  S22  024 ,
0 0 0 1 
From (30)-(35), we have
N 1
0
0

0

0
0

1
Therefore the closed-loop eigenvectors are given by
1
2
1
v11
 N11 f111 , v11
 N11 f112  N12 f11
,
3
1
1
v11
 N11 f113  N12 f112  N13 f11
, v121  N21 f 21
,
2
2
1
v21
 N21 f21
 N22 f21
, v11  N1 f11
and the corresponding vectors are given by
1
2
1
w11
 D11 f111 , w11
 D11 f112  D12 f11
,
3
1
1
w11
 D11 f113  D12 f112  D13 f11
, w121  D21 f21
,
2
2
1
w21
 D21 f21
 D22 f21
, w11  D1 f11
and
1
1
2
s11
 S11 f11
, s11
 S11 f112  S12 f111 ,
In this case, from (19)-(24), we have
 1
1

0
N11  
 1
1

 2
 1
0

0
N13  
0
0

 4
0
 1

0
N 22  
1
0

 0
S1  024.
0 0 0
0 0 0 
0 0 0
0 0 1 0 
 , D1  
,
1 0 0
0 0 0 1 
0 0 0

0 0 0 
3
1
s11
 S11 f113  S12 f112  S13 f111 , s121  S21 f 21
,
2
2
1
s21
 S21 f21
 S22 f 21
, s11  S1 f11.
By specially choosing
1 
0 
0 
1 
1
f111  f 21
 f 11    , f112    ,
0 
0 
 
 
 0 
0 
0 
1 
0 
0 
f113    , f 212    ,
1 
0 
 
 
 0 
1 
which satisfy Constraints 1-3, we obtain
[V f
[W f
[S f
 1
1

0
V ]  
 1
1

 2
1
W ]  
 2
3 0
0
2 1 0 1 0 
1 0 0 0 0
,
1 0 0 1 0
0 0 1 1 0

7 9 0 0 1 
5 5 0 0 0 
,
5 6 1 1 0 
3
0
0 0 1 0 0 0 
S ]  
.
0 0 0 0 1 0 
Then, according to (40), the P and D state feedback gain
matrices are given by
 1 2 0 2
K1  
 1 3 1 3
0 1 1 1
K2  
 1 3 1 2
0 0
,
1 0 
0 0
.
0 0 
Example 2 (Model refinement [21]): Given a secondorder dynamical model of the form
M x  Dx  Kx  Hf ,
(46)
Parametric Eigenstructure Assignment for Descriptor Systems via Proportional plus Derivative State Feedback
where M is the positive definite mass matrix, D is the
positive definite (semidefinite) damping matrix, K is the
positive definite (semidefinite) stiffness matrix, H is the
disturbance input influence matrix, x is a n  l vector of
displacements, and f is a l  l vector of disturbances to
the system. Find position, velocity, and acceleration gain
matrix that reassign a desired subset of the eigenvalues
of the model, along with partial mode shapes.
It is known [21] that the dynamics of the refined
system may be written as
(M  M ) x  ( D  D) x  ( K  K ) x  Hf ,
(47)
where M , D and K are symmetric matrices
satisfying (M  M )  0, ( D  D)  0 and ( K  K )
> 0 respectively. Clearly, the first-order descriptor
representation of this system is
  In

 0
0   0    x

K2 
M   I n    x 
 0
 
  K
given by
v7  N7 f7 , w7  D7 f7 , s7  S7 f7 ,
where f7  g  hi, g , h  R 8 , N7 , D7 and S7 are determined by (19)-(24). Assume that
Re(v7 ) Im(v7 )])  0.
det([V 0
(49)
Then the gain matrices K1 and K2 are given by
K1  [086
Re(w7 ) Im(w7 )][V 0 Re(v7 ) Im(v7 )]1,
K2  [086
Re(s7 ) Im(s7 )][V 0 Re(v7 ) Im(v7 )]1.
Since K1  [K D], K2  [0 M ], we have
0
0
I 
M  K 2   , D   K1   , K   K1  4 
I
I
0
 4
 4
with
(48)
In   0    x  0 

K1 

f,
 D   I n    x   H 
where K1  [K D], K2  [0 M ].
We consider the case n =4. The mass, damping, and
stiffness matrices are, respectively, chosen as [22]
1 0 0
 2 1 0 0 
 1 2 1 0 
4 1 0 
,
, K  1000 
 0 1 2 1
1 4 1



0 1 2 
 0 0 1 1 
0
0
 1.0898 0.7071
 0.7071 1.6310 0.9239 0 
1 
.
D
0.9239 1.9239 1
100  0


0
1
1 
 0
4
1
M 
0

 0
0
1,2
 8.1947e  3  4.2282e  1i,
0
3,4
 3.2590e  3  2.9239e  1i,
Assume that
M  M  0, D  D  0, K  K  0.
(51)
We focus on finding low gain matrices M , D and
K . Then we may define an objective as
J ( g , h)   M
2
  D 2   K 2 ,
where  ,  ,  are positive scalars representing the
weighting factors,
2
represents the spectral norm.
Therefore the model refinement problem can be
converted into the following minimization problem
min
s.t.(49) (51)
J ( g , h).
 0.0390 0.0721 0.0942 0.0510]T ,
0
7,8
 7.6814e  5  5.1024e  0i.
h  [6.7674 12.5174 16.3667 9.1360
0
7,8
to their
target values ˆ7,8  5.0000e  4  3.0000e  0i while
ensuring that the remaining eigenvalues
(50)
g  [380.8006 703.6246 919.3225 497.8529
0
5,6
 9.6437e  4  1.6096e  1i,
It is desired that assigning the eigenvalues
I 
K 2  4   0.
0
Taking       1 and using the Matlab command
fmincon, the solution to this minimization problem is
obtained as
The eigenvalues of this system are
i0 , i  1, 2,
vi0 , i  1, 2,
…, 6, and the corresponding eigenvectors
…, 6, remain invariant (a property known as the no spillover phenomenon [23]). Let
V 0  [Re(v10 ) Im(v10 )
21
Re(v50 ) Im(v50 )].
From (19)-(25), the general parametric solution of the
eigenvector and the corresponding vectors of the
extended first-order system (48) associated with ̂7 are
2.2064 4.0769 5.3267 2.8828]T .
With this group of parameters, we have
 0.0141
0.0260
M  
0.0340

0.0184
0.0260 0.0340 0.0184
0.0481 0.0629 0.0340
,
0.0629 0.0822 0.0445

0.0340 0.0445 0.0241
 0.0072
 0.0132
D  1000 
 0.0173

 0.0094
0.0132 0.0173 0.0094 
0.0245 0.0320 0.0173
,
0.0320 0.0418 0.0226 

0.0173 0.0226 0.0122 
Biao Zhang
22
 0.0351

1 0.0664
K 
100  0.0883

 0.0481
0.0664 0.0883 0.0481
0.1256 0.1669 0.0909 
.
0.1669 0.2218 0.1207 

0.0909 0.1207 0.0657 
6. CONCLUSIONS
This paper deals with eigenstructure assignment in
descriptor systems via PD state feedback. Based on a
simple complete explicit parametric solution to a group
of recursive equations, a parametric approach for
eigenstrucure assignment in descriptor systems via PD
state feedback is proposed. The proposed approach
possesses the following features: 1) it does not impose
any condition on the closed-loop eigenvalues,
simultaneously assigns arbitrary n finite and infinite
eigenvalues to the closed-loop system and guarantees the
closed-loop regularity; 2) it is simple and needs less
computational work; 3) it gives general complete
parametric expressions for the closed-loop eigenvectors,
and the P and D state feedback gain matrices.
[12]
[13]
[14]
[15]
[16]
[17]
REFERENCES
B. C. Kuo, Digital Control Systems, Holt, Rinehart
and Winston, York, 1980.
[2] M. B. Estrada and M. Malabre, “Proportional and
derivative state-feedback decoupling of linear systems,” IEEE Trans. on Automatic Control, vol. 45,
vol. 4, pp. 730-733, 2000.
[3] M. A. Shayman and Z. Zhou, “Feedback control
and classification of generalized singular systems,”
IEEE Trans. on Automatic Control, vol. 32, no. 6,
pp. 483-494, 1987.
[4] R. Mukundan and W. Dayawansa, “Feedback control of singular systems—proportional and derivative feedback of the state,” Int. J. System Sci., vol.
14, no. 6, pp. 615-632, 1983.
[5] Z. Zhou, M. A. Shayman, and T. J. Tarn, “Singular
systems: a new approach in the time domain,”
IEEE Trans. on Automatic Control, vol. 32, no. 1,
pp. 42-50, 1987.
[6] L. Dai, Singular Control Systems, Springer-Verlag,
Berlin, 1989.
[7] M. A. Shayman, “Homogeneous indices, feedback
invariants, and control structure theorem for generalized linear systems,” SIAM J. Contr. Opt., vol. 26,
pp. 387-400, 1988.
[8] J. J. Loiseau, “Pole placement and connected problems,” Prep. IFAC Workshop on System Structure
and Control, pp.193-196, 1989.
[9] F. L. Lewis and V. S. Syrmos, “A geometric theory
for derivatives feedback,” IEEE Trans. on Automatic Control, vol. 36, no. 9, pp. 1111-1116, 1991.
[10] V. S. Syrmos and F. L. Lewis, “A geometric approach to proportional plus derivative feedback using quotient and partitioned subspaces,” Automatica, vol. 27, pp. 349-369, 1991.
[11] P. Zagalak and V. Kucera, “Eigenstructure assignment by PD state feedback in linear systems,” Proc.
[1]
[18]
[19]
[20]
[21]
[22]
[23]
of the 30th Conf. Decision and Control, pp. 12841286, 1991.
V. X. Le, “Synthesis of proportional-plus-derivative
feedbacks for descriptor systems,” IEEE Trans. on
Automatic Control, vol. 37, no. 5, pp. 672-675,
1992.
H. C. Chen and F. R. Chang, “Chained eigenstructure assignment for constraint-ratio proportional
and derivative (CRPD) control law in controllable
singular systems,” Syst. Control Lett., vol. 21, pp.
405-411, 1993.
H. Y. Jin, “Eigenstructure assignment by proportional-derivative state-feedback in singular systems,” Syst. Control Lett., vol. 22, pp. 47-52, 1994.
G. R. Duan and R. J. Patton, “Eigenstructure assignment in descriptor systems via proportional
plus derivative state feedback,” Int. J. Control, vol.
68, no. 5, pp. 1147-1162, 1997.
T. J. Owens and S. Askarpour, “Integrated approach
to eigenstructure assignment in descriptor systems
by PD and P state feedback,” IEE Proc. Control
Theory Appl., vol. 147, no. 4, pp. 407-415, 2000.
A. P. Wang and S. F. Lin, “The parametric solutions
of eigenstructure assignment for controllable and
uncontrollable singular systems,” Journal of Mathematical Analysis and Applications, vol. 248, pp.
549-571, 2000.
G. R. Duan, “Parametric eigenstructure assignment
via state feedback: a simple numerically reliable
approach,” Proc. of the 4th World Congress on Intelligent Control and Automation, pp. 165-173,
2002.
B. Zhang, “Parametric eigenstructure assignment
by state feedback in descriptor systems,” IET Control Theory Appl., vol. 2, no. 4, pp. 303-309, 2008.
B. Zhang, “Infinite eigenvalue assignment in descriptor systems via state feedback,” Int. J. Syst.
Sci., vol. 41, no. 9, pp. 1075-1084, Sep. 2010.
P. G. Maghami, “Model refinement using eigensystem assignment,” Journal of Guidance, Control,
and Dynamics, vol. 23, no. 4, pp. 683-692, 2000.
B. N. Datta, S. Elhay, and Y. M. Ram, “Orthogonality and partial pole assignment for the symmetric
definite quadratic pencil,” Linear Alg. Appl., vol.
257, pp. 29-48, 1997.
M. T. Chu, B. N. Datta, W. W. Lin, and S. F. Xu,
“The spill-over phenomenon in quadratic model
updating,” AIAA J., vol. 46, pp. 420-428, 2008.
Biao Zhang received his B.S. degree in
Mathematics from Peking University,
Beijing, an M.S. degree in Applied
Mathematics, and a Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, in
1984, 1989, and 2007, respectively. He is
currently an associate professor in the
Department of Mathematics at Harbin
Institute of Technology, Harbin. His research interests include
eigenstructure assignment, robust control, and descriptor sys-
Parametric Eigenstructure Assignment for Descriptor Systems via Proportional plus Derivative State Feedback
tems.
23