AP04_4web.vp


1 Introduction

Eigenstructure assignment is one of the basic techniques
for designing linear control systems. The eigenstructure as-
signment problem is the problem of assigning both a given
self-conjugate set of eigenvalues and the corresponding
eigenvectors. Assigning the eigenvalues allows one to alter the
stability characteristics of the system, while assigning
eigenvectors alters the transient response of the system.
Eigenstructure assignment via state feedback has developed
the design methods for a wide class of linear systems under
full-state feedback with the objective of stabilizing control
systems. Parametric solutions of eigenstructure assignment
for state feedback have been studied by many researchers
[4–9].

However, this paper focuses on a special feedback using
only state derivatives instead of full-state feedback. Therefore
this feedback is called state-derivative feedback. The prob-
lem of arbitrary eigenstructure assignment using full-state
derivative feedback naturally arises. The motivation for state
derivative feedback comes from controlled vibration suppres-
sion of mechanical systems. The main sensors of vibration
are accelerometers. From accelerations it is possible to
reconstruct velocities with reasonable accuracy, but not dis-
placements. Therefore the available signals for feedback are
accelerations and velocities only, and these are exactly the
derivatives of the states of mechanical systems, which are
velocities and displacements. Then, direct measurement of a
state is difficult to achieve. One necessary condition for a
control strategy to be implementable is that it must use
the available measured responses to determine the control
action. All of the previous research work in control has as-
sumed that all of the states can be directly measured (i.e., full
state feedback).

To control this class of systems, many papers (e.g. [10–15])
have been published describing the acceleration feedback
for controlled vibration suppression. However, the eigen-
structure assignment approach for feedback gain determina-
tion has not been used at all or has not been solved generally.
Other papers dealing with acceleration feedback for mechan-
ical systems are [16–17], but here the feedback uses all states
(positions, velocities) and accelerations additionally.

Only recently, Abdelaziz and Valášek [1–3] have presented
a pole placement technique by state-derivative feedback for
single-input and multi-input time-invariant and time-varying
linear systems. This paper proposes a parametric approach
for eigenstructure assignment in linear systems via state-de-
rivative and partial output-derivative feedback. The necessary
and sufficient conditions for the existence of eigenstructure
assignment problem are described. The proposed controller
is based on measurement and feedback of the state derivatives
of the system. This work has successfully extended previous
techniques by state feedback and modified to state-derivative
feedback. Finally, numerical examples are included to dem-
onstrate the effectiveness of this approach. The main contri-
bution of this work is the efficient technique that solves the
eigenstructure assignment problem via state-derivative feed-
back systems. The procedure defined here represents a
unique treatment for the extension of the eigenstructure as-
signment technique using the derivative feedback in the
literature.

The paper is organized as follows. The next section in-
troduces the eigenstructure assignment problem formulation
via state-derivative feedback. The necessary and sufficient
conditions that ensure solvability are described. Additionally,
the parametric solution to the eigenstructure assignment
problem via state-derivative feedback for distinct and
repeated right eigenvectors is presented. Section 3 introduces
the eigenstructure assignment problem by partial output-

54 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 4/2004

Eigenstructure Assignment by
State-derivative and Partial
Output-derivative Feedback for Linear
Time-invariant Control Systems
T. H. S. Abdelaziz, M. Valášek

This paper introduces a parametric approach for solving the problem of eigenstructure assignment via state-derivative feedback for linear
control time-invariant systems. This problem is always solvable for any controllable systems if the open-loop system matrix is nonsingular. In
this work, the parametric solution to the feedback gain matrix is introduced that describes the available degrees of freedom offered by the
state-derivative feedback in selecting the associated eigenvectors from an admissible class. These freedoms can be utilized to improve the
robustness of the closed-loop system. Finally, the eigenstructure assignment problem via partial output-derivative feedback is introduced.
Numerical examples are included to show the effectiveness of the proposed approach.

Keywords: Eigenstructure assignment, state-derivative feedback, output-derivative feedback, linear systems, feedback stabilization,
parametrization.



-derivative feedback. Section 4 presents illustrative examples.
Finally, conclusions are discussed in section 5.

2 Eigenstructure assignment by
state-derivative feedback for
time-invariant systems
In this section, we present a parametric approach for

solving the eigenstructure assignment problem via state-deri-
vative feedback for linear time-invariant systems.

2.1 Eigenstructure assignment problem
formulation

Consider a linear, time-invariant, completely controllable
system

� ) ( ) ( ), ( )x A x B u x x(t t t t� � �0 0 , (1)

where x (t) � Rn and u (t) � Rm are the state and the control
input vector, respectively, (m n� ), while A � Rn×n and
B � Rn×m are the system and control gain matrices, respec-
tively. The fundamental assumption imposed on the system is
that, the system is completely controllable and the B matrix
has a full column rank m.

The objective is to stabilize the system by means of a linear
feedback that enforces a desired characteristic behavior for
the states. The eigenstructure assignment problem is to find
the state-derivative feedback control law

u K x( ) � )t t� � ( (2)

which assigns the prescribed closed-loop eigenvalues and
the corresponding eigenvectors that stabilize the system and
achieve the desired performance. Then, the closed-loop
system dynamics becomes:

� ) ( ) ( )x I BK A x(t tn� �
�1 , (3)

where In � R
n×n is the identity matrix. In what follows, we

assume that ( )I BKn � has a full rank in order that the closed-
-loop system is well define. The closed-loop characteristic
polynomial is given by

� �det ( )� I I BK An n� � ��1 0 (4)
Denote the right eigenvector matrix of the closed-loop

matrix ( )I BK An �
�1 by V, and we then have by definition,

( )I BK AV = Vn �
�1

� (5)

where � � diag{ , , }� �1 � n and det (V) � 0.

We now formulate the eigenstructure assignment problem
via state-derivative feedback as follows.

Eigenstructure Assignment Problem 1: Given the controllable
pair ( A, B ) and the desired self-conjugate set { , , }� �1 � n �C.
Select the appropriate state-derivative feedback gain
matrix K � Rn×m that will make the closed-loop matrix
( )I BK An �

�1 have admissible eigenvalues and associated set
of eigenvectors V.

Now, we will prove the necessary and sufficient conditions
for the existence of the eigenstructure assignment problem
via state-derivative feedback. The necessary conditions to
ensure solvability are presented in the following theorem.

Theorem 1:
The eigenstructure assignment problem for the real pair

(A, B) is solvable for any arbitrary self-conjugate closed-loop
poles, if and only if (A, B) is completely controllable, that is,

� �rank , ,B AB A B�, n- n1 �
and

� �rank , C� �I A Bn � � 	 �n, ,
and matrix A is nonsingular.
Proof: From the condition of that the closed-loop matrix
must be defined. This means the matrix (In � BK) is of full
rank and det(In � BK) � 0. Then, from (5) it is easy to rewrite
as

AV V BKV� �� � (6)
which can be written as

BK AV V I� �� �� 1 1 n (7)

Then,

det( ) det( )I BK I AV V In n n� � � � �
� �

�
1 1

� 
� �det( )AV V� 1 1 0. (8)

Since V and � must be nonsingular. This leads to,
det ( )A 
 0, and matrix A must be of full rank in order for
the matrix to be defined.

Thus, the necessary and sufficient conditions for the exis-
tence of the solution to the eigenstructure assignment prob-
lem via state-derivative feedback is that the system is com-
pletely controllable and all eigenvalues of the original system
are nonzero (A has full rank). We remark that the requirement
that the matrix � is diagonal, together with the invertibility of
V, ensures that the closed-loop system is non-defective.

Based on the above necessary and sufficient conditions,
the parametric formula for the state-derivative feedback gain
matrix K that assigns the desired closed-loop poles is derived.

2.2 Case of distinct eigenvalues
In this subsection, all the desired closed-loop eigenvalues

� �� �1, ,� n are distinct.
Let (�i, vi) be the closed-loop eigenvalue and the corre-

sponding eigenvector, then we can write the right eigen-
vectors of the closed-loop system as

( ) , , ,I BK An i i i i n� � �
�1 1v v� � . (9)

The above equation can be rewritten as

A I BKv vi i n i� �� ( ) , (10)
then

A BKv v vi i i i i� �� � . (11)

Let w vi i� K , then we have

A Bv v wi i i i i� �� � . (12)

Then (12) can be rewritten in a matrix form as,

� �� �i n i
i

i
i nI A B 0�



�
�

�

�
� � �, , , ,

v
w

1 � . (13)

Thus the subspace may be calculated in which the closed-
-loop eigenvector, vi, and gain-eigenvector product, wi, must
satisfy the null space, � �� �null � �i n iI A B� , , i � 1, …, n.

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Acta Polytechnica Vol. 44  No. 4/2004



Finally, the parametric equation to the right eigenvector
can be expressed as

� �� �i n i i i
i

i

i

i
I A B 0

K
� � �



�
�

�

�
� �



�
�

�

�
�, ,� �

v
v

v
w

. (14)

Then vi � C
n and K � Rm×n are unknown and notation-

al simplicity is gained by defining the (n � m)×1 vector �i.
The determination of K consists of two general steps. First,
a sufficient number of solution vectors �i is found. Then, the
internal structure among the components of these vectors is
used to find the feedback gain matrix K.

Then, there exists a real feedback gain matrix K if and
only if the following three conditions are satisfied:

1. The assigned eigenvalues are symmetric with respect to
the real axis.

2. The vectors {vi � C
n, i � 1, …, n}, are linearly independent

and for complex-conjugate poles, �i � �j then v vi j� .

3. There exists a set of vectors {wi � C
m, i � 1, …, n} satisfying

(13) and w wi j� for �i � �j.

The parametric formula for the state-derivative feedback
gain matrix K that assigns the desired closed-loop eigen-
values and eigenvectors is derived as

� � � �K v v w w( ), , ( ) ( ), , ( )� � � �1 1� �n n� . (15)
Finally the gain matrix can be computed as

K WV� �1 (16)

where

� �V � v v( ), , ( )� �1 � n and � �W � w w( ), , ( )� �1 � n .
This is a parametric solution to the eigenstructure assign-

ment problem via state-derivative feedback. It is clear that,
for the case of multi-input, there are infinite solutions to the
feedback gain. Eigenstructure assignment uses the extra de-
grees of freedom in the undetermined solution to specify
the closed-loop right-eigenvectors, V, corresponding to the
desired self-conjugate set {�i}, i � 1, …, n.

2.3 Case of repeated eigenvalues
Our main work is to find a parametric solution in the case

that some or all of the desired closed-loop eigenvalues {�i}
are repeated.

Let � �� � � � � �� �i i i s s n, , , , ,C 1 1� be a set of desired
eigenvalues and let us denote the algebraic and geometric
multiplicity of �i by mi and qi, respectively, (qi � mi). The length
of qi chains of generalized eigenvectors with �i is denoted by
pij, (j � 1, …, qi). It is satisfying that p nijj

q

i

s i
�� �� �11 .

Then, the right generalized eigenvector of the closed-
-loop system with �i is denoted by vij

k n�C , i � 1, …, s,

j � 1, …, qi, k � 1, …, pij. Then it is satisfied that

( ) , ,

, , , , ,

I BK An ij
k

i ij
k

ij
k

ij

i s j

� � � �

� �

� �1 1 0 0

1 1

v v v v�

� � q k pi ij, , , .�1 �
(17)

This equation demonstrates the relation of assignable
right generalized eigenvectors with the corresponding
eigenvalue, and can be rewritten as,

A I BK I BKv v v vij
k

i n ij
k

n ij
k

ij� � � � �
�

� ( ) ( ) ,1 0 0 . (18)

Let w vij
k

ij
k� K , i � 1, …, s, j � 1, …, qi, k � 1, …, pij. The

notations are defined as � �V V V� 1, ,� s , � �V V Vi i iq i� 1, ,� ,
Vij ij ij

p ij� �
��

�
��

v v1 , ,� . The set of w ij
k is define in a similar man-

ner to the set of vij
k . This leads to

� � � �� �i n i
ij
k

ij
k n

ij
k

ij
kI A B I B�



�

�
�

�

�

�
�
� �



�

�
�

�

�

�

�, ,
v

w

v

w

1

1
�
�

� �, ,v wij ij
0 00 0 .(19)

Finally, the parametric equation to the right generalized
eigenvector can be expressed as

� � � �� �i n i ij
k

n ij
k

ij
k ij

k

ij
kI A B I B� � � �



�

�
�

�

�

�
�

�, , , ,� � � �1
v

w
ij

i iji s j q k p

0

1 1 1

�

� � �

0,

, , , , , , , , .� � �

(20)

Similar to the case of distinct eigenvalues the (n�m)-di-
mentional parameter vectors �ij

k are chosen arbitrarily, under

the condition that the columns of eigenvector matrix V are
linearly independent.

A parametric solution K to the eigenstructure assignment
problem via state-derivative feedback is given by

K WV� �1 (21)

where

� �V V V� 1 1( ), , ( )� �� s s and � �W W W� 1 1( ), , ( )� �� s s .
Then the feedback gain matrix is parameterized directly

in terms of the eigenstructure of the closed-loop system,
which can be selected to ensure robustness by exploiting the
freedom of these parameters.

Then, there exists a real feedback gain matrix K, such that
(19) holds, if and only if the following three conditions are
satisfied:
1. The assigned eigenvalues are symmetric with respect to

the real axis.
2. The vectors
� �vijk n i iji s j q k p� � � �C , , , , , , , , ,1 1 1� � � are
linearly independent and for complex-conjugate poles,
� �i i1 2

� , then v vi j
k

i j
k

1 2
� .

3. There exists a set of vectors
� �w ijk m i iji s j q k p� � � �C , , , , , , , , ,1 1 1� � � ,
satisfying (19) and w wi j

k
i j
k

1 2
� for � �i i1 2� .

From the above results we observe that the system poles
can always be assigned by a state-derivative feedback for
any controllable system if and only if all eigenvalues of the
original system are nonzero. In the case of single-input,
m � 1, there is only at most one solution. In the case of
multi-input, m > 1, the solution is generally non-unique, and
extra conditions must be imposed to specify a solution. The
extra freedom can be used to give the closed-loop system
other desirable properties. The extra freedom can be used
in different ways, for example to decrease the norm of the
feedback gain matrix or to improve the condition of the
eigenvalues of the closed-loop matrix. Additionally, the ro-
bustness of the closed-loop system against system parameter

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Acta Polytechnica Vol. 44  No. 4/2004



perturbation is increased. This issue becomes very important
when the system model is not sufficently precise or the system
is subject to parameter uncertainty.

3 Eigenstructure assignment by
partial output-derivative feedback
for time-invariant systems

Let us consider eigenstructure assignment via output-de-
rivative feedback for linear time-invariant systems. In prac-
tice, it may be expensive to measure all the state-derivative
variables, or they may not all be available for measurement.
We then feedback some of the derivative-outputs via a con-
troller. Output-derivative feedback is a more difficult problem
than state-derivative feedback.

Consider a linear, time-invariant, completely controllable
and observable system

�( ) ( ) ( ), ( ) ,
( ) ( ),

x A x B u x x
y C x

t t t t
t t
� � �

�
0 0 (22)

where the state x(t) � Rn, the output y(t) � Rr and the con-
trol input vector u(t) � Rm, while A � Rn×n, B � Rn×m and
C � Rr×n are the system, control and output gain matrices, re-
spectively. The fundamental assumptions imposed on the sys-
tem are that the system is completely controllable and
observable. Additionally, matrices B and C have a full rank.

The objective is to stabilize the system by means of a linear
feedback that enforces a desired characteristic behavior for
the state. The eigenstructure assignment problem is to find
the output-derivative feedback control law

u F y( ) �( )t t� � , (23)

which assigns prescribed closed-loop eigenvalues that stabi-
lize the system and achieve the desired performance, and the
closed-loop system dynamics becomes

�( ) (x I BFC A xt tn� � ) ( ) (24)

In what follows, we assume that (In � BFC) has a full rank
in order that the closed-loop system is well defined. The
closed-loop characteristic equation of this system is given by

� �det (�I I BFC An n� � ��) 1 0 (25)
Denote the right eigenvector matrix of the matrix

(I BFC An �
�) 1 by V, and then we have,

(I BFC AV Vn =�
�) 1 � . (26)

Now, the eigenstructure assignment problem for output-
-derivative feedback is as follows.

Eigenstructure Assignment Problem 2: Given the reat triple
(A, B, C) and the desired self-conjugate set {�1, …, �n} � C.
Select the appropriate output-derivative feedback matrix
F � Rm×r that will make the closed-loop matrix
(I BFC An �

�) 1 have admissable eigenvalues and correspond-
ing right generalized eigenvectors V.

Now, the necessary and sufficient conditions for the exis-
tence of the eigenstructure assignment problem via output-
-derivative feedback are presented in the following theorem.

Theorem 2:
The eigenstructure assignment problem for the triple

(A, B, C) is solvable for any arbitrary self-conjugate closed-
-loop poles, if and only if (A, B) is completely controllable,
that is,

� �rank B AB A B, , ,� n n� �1 and
� �rank C� �I A Bn n� � 	 �, , ,

also, the pair (A, C) is completely observable that is,

� �rank C AC A C, , ,� n n� �1 and
� �rank C� �I A Cn n� � 	 �, , ,

and matrix A is nonsingular.

Proof: From the condition of that the closed-loop matrix
must be defined. This means the matrix (In � BFC) is of full
rank and det(I BFCn � 
) 0. Then, from (26) it can easily put
in the following form

AV V BFCV� �� � (27)

which can be written as

BFC AV V I� �� �� 1 1 n (28)

Then,

� � �det det ]I BFC I AV V In n n� � � � �
� �

�
1 1

� 
� �det [ ]AV V� 1 1 0 (29)

Since V and � must be nonsingular. Thus, matrix A must
be of full rank in order for the closed-loop matrix to be
defined.

From the above results we can observe that, the neces-
sary and sufficient conditions for existence the solution to
the eigenstructure assignment problem via output-derivative
feedback are the system is completely controllable and
observable and all eigenvalues of the original system are non-
zero (A has full rank).

Our main work in this section is to find a parametric solu-
tion to the gain matrix in the case of right generalized
eigenvectors.

Let � �� � � � � �� �i i i s s n, , , , ,C 1 1� be a set of de-
sired eigenvalues and let us denote the algebraic and geo-
metric multiplicity of �i by mi and qi, respectively. The length
of qi chains of generalized eigenvectors with �i, are denoted
by pij. The right generalized eigenvector of the closed-loop
system with �i is denoted by vij

k n�C , i � 1, …, s, j � 1, …, qi,

k � 1, …, pij. It is satisfying that p nijj
q

i

s i
�� �� �11 . Then we

have the following relation

( , ,

, , , ,

I BFC A 0n ij
k

i ij
k

ij
k

ij

i s j

� � � �

� �

� �) 1 v v v v� 1 0

1 1� �, , , , .q k pi ij�1 �
(30)

This equation demonstrates the relation of assignable
right generalized eigenvectors with the eigenvalue, and can
be rewritten,

( ) ( ,A I BFC I BFC 0� � � � ��� �i n i ij
k

n ij
k

ijv v v)
1 0 . (31)

Let z vij
k

ij
k� FC , i � 1, …, s, j � 1, …, qi, k � 1, …, pij. The

notations are defined as � �V V V� 1, ,� s , � �V V Vi i iq i� 1, ,� ,

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Acta Polytechnica Vol. 44  No. 4/2004



Vij ij ij
p ij� �

��
�
��

v v1 , ,� . The set of zij
k is defined in a similar man-

ner to the set of vij
k . The above equation can be rewritten as

� � � �� �i n i
ij
k

ij
k n

ij
k

ij
kI A B I B�



�

�
�

�

�

�
�
� �



�

�
�

�

�

�

�, ,
v

z

v

z

1

1
�
�

� �, ,v zij ij
0 00 0 . (32)

The right generalized eigenvector can be expressed as

� � � �� �i n i ij
k

n ij
k

ij
k ij

k

ij
kI A B I B� � � �



�

�
�

�

�

�
�

�, , , ,� � � �1
v

z
ij

i iji s j q k p

0

1 1 1

�

� � �

0,

, , , , , , , , .� � �

(33)

Similar to the case of state-derivative feedback, there ex-
ists a real matrix F if and only if the (n � m) ×1-dimensional
parameter vectors, �ij

k , are arbitrary chosen under the condi-

tion that the columns of matrix V are linearly independent.
Finally, a real gain matrix F is expressed as:

F ZV� �1 (34)

where

� �V V V� 1 1( ), , ( )� �� s s and � �Z Z Z� 1 1( ), , ( )� �� s s .

Then the feedback gain matrix F is parameterized directly
in terms of the eigenstructure of the closed-loop system.

There exists a real feedback gain matrix F, such that
(32) holds, if and only if the following three conditions are
satisfied:
1. The assigned eigenvalues are symmetric with respect to

the real axis.
2. The vectors � �vijk n i iji s j q k p� � � �C , , , , , , , , ,1 1 1� � �

are linearly independent and for complex-conjugate
poles, � �i i1 2� then v vi j

k
i j
k

1 2
� .

3. There exists a set of vectors
� �zijk m i iji s j q k p� � � �C , , , , , , , , ,1 1 1� � � ,
satisfying (32) and z zi j

k
i j
k

1 2
� for � �i i1 2� .

4 Illustrative examples
In this section, we present numerical examples to

illustrate the feasibility and effectiveness of the proposed
eigenstructure assignment technique.

Example 1:
Consider a single-input linear system described in the

state-space form

�( ) ( ) ( )x x ut t t�


�
�

�

�
� �



�
�

�

�
�

1 2
0 3

0
1

In the following, we consider the assignment of two differ-
ent cases:
Case 1: The desired closed-loop eigenvalues are selected as,
{�3 and �4}.

For �1 � �3, � �� �1 1 1 1
4 2 0
0 6 3

I A B 0n � �
� �

� �


�
�

�

�
� �, � � ,

arbitrarily selecting � ��1 0 5 1 2� � �. , ,
T

leads to K
�

�
�

�

�
� � �

0 5
1

2
.

.

For �2 � �4, � �� �2 2 2 2
5 2 0
0 7 4

I A B 0n � �
� �

� �


�
�

�

�
� �, � � ,

taking � ��2 0 4 1 7 4� � �. , ,
T

leads to K
�

�
�

�

�
� � �

0 4
1

7 4
.

.

Taking the two equations together, we can write

K
� �

�
�

�

�
� � � �

0 5 0 4
1 1

2 7 4
. .

( ) .

Finally, the unique state-derivative feedback gain matrix is

� �K � 2 5 0 75. , . .

Case 2: The desired eigenvalues are, {–1 and –1}.

For � � �1, � �� � ��
�

��
�I A B 0n � �

� �

� �


�
�

�

�
� �, � �

2 2 0
0 4 1

,

leads to � ����
� � � �1 1 4, , T and K

�

�
�

�

�
� � �

1
1

4.

The generalized eigenvector equation is,

� � � �� � �
�

�
�I A B I Bn n� � �, ,� �2 1 ,

and,
� �

� �


�
�

�

�
� �

�

� �


�
�

�

�
�

�

�



�

�
�
�

�

�

�
�

2 2 0
0 4 1

1 0 0
0 1 1

1
1
4

2��
�

�
�


�
�

�

�
�

1
3

,

then � ���
�
2 15 1 7� � �. , ,

T and K
�

�
�

�

�
� � �

15
1

7
.

.

And the gain equation can be written as

K
� �

�
�

�

�
� � � �

1 15
1 1

4 7
.

( ).

Finally, the unique feedback gain is

� �K � 6 2, .

Example 2:
Consider a multi-input controllable linear system

�( ) ( ) ( )x x ut t t�


�
�

�

�
� �



�
�

�

�
�

1 2
0 3

1 0
0 1

In the following, we consider the assignment of two differ-
ent cases:

Case 1: The desired closed-loop eigenvalues are selected as
{–3 and –5}.

For �1 � �3, � �� �1 1 1 1
4 2 3 0

0 6 0 3
I A B 0n � �

� � �

� �


�
�

�

�
� �, � � ,

taking � �v1 � a b,
T,

leads to � ��1 4 3 2 3 2� � � �a b a b b, , ,
T

and K
a
b

a b
b



�
�

�

�
� �

� �

�


�
�

�

�
�

4 3 2 3
2

.

58 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 4/2004



For �2 � �5, � �� �2 2 2 2
6 2 5 0

0 8 0 5
I A B 0n � �

� � �

� �


�
�

�

�
� �, � � ,

choosing � �v2 � c d,
T,

then � ��2 6 5 2 5 8 5� � � �c d c d d, , ,
T,

and K
c
d

c d
d



�
�

�

�
� �

� �

�


�
�

�

�
�

6 5 2 5
8 5

.

Then, the feedback gain matrix equation is

K
a c
b d

a b c d
b d



�
�

�

�
� �

� � � �

� �


�
�

�

�
�

4 3 2 3 6 5 2 5
2 8 5

.

Then any values of a, b, c and d will give a valid gain matrix
as long as the required inverse exists. To give the orthogo-
nal set of the closed-loop eigenvector, we set a � d � 1 and
b � c � 0.

The gain matrix is

K �
� �

�


�
�

�

�
�

4 3 2 5
0 8 5

.

Case 2: The desired closed-loop poles are, {–1 and –1}.

For �1 � �1, � �� � ��
�

��
�I A B 0n � �

� � �

� �


�
�

�

�
� �, � �

2 2 1 0
0 4 0 1

,

taking � �v11
1 � a b, T,

leads to � ����
� � � � �a b a b b, , ,2 2 4 T.

The generalized eigenvector equation is,

� � � �� � �
�

��
�I A B I Bn n� � �, ,� �2

� � �

� �


�
�

�

�
� �

� �

� �


�
�

�

�
� �

�2 2 1 0
0 4 0 1

1 0 1 0
0 1 0 1

2
2� ��

�
��
�

a b
b3



�
�

�

�
�,

taking � �v12
1 � c d, T, leads to

� ���
�
2 2 2 2 3 4� � � � � � �c d a b c d b d, , ,

T.

Then, the gain matrix equation can be written as

K �


�
�

�

�
� �

� � � � � �

� � �


�
�

�

�
�

a c
b d

a b a b c d
b b d

2 2 2 2 2
4 3 4

.

We can set a � d � 1 and b � c � 0.
Then, the gain matrix is

K �
� �

�


�
�

�

�
�

2 3
0 4

.

Example 3:
Consider a controllable and observable, multi-input linear

system

�( ) ( ) ( )x x ut t t�
� �



�
�

�

�
� �



�
�

�

�
�

0 1
3 4

1 0
0 1

,

y x( ) ( ) ( )t t� 1 1

In the following we consider the first eigenvalue at (�1 � �5),

� �� �1 1 1 1
5 1 5 0

3 1 0 5
I A B 0n � �

� � �

� �


�
�

�

�
� �, � � ,

taking � �v1 � a b,
T,

leads to � ��1 0 2 0 6 0 2� � � �a b a b a b, , . , . .
T.

This means FC
a
b

a b
a b



�
�

�

�
� �

� �

�


�
�

�

�
�

0 2
0 6 0 2

.
. .

.

Hence, FC F
a
b

a b
a b
a b



�
�

�

�
� � � �

� �

�


�
�

�

�
�( )

.
. .

0 2
0 6 0 2

.

The gain matrix is

F �
�

� �

�


�
�

�

�
�

1 0 2
0 6 0 2a b

a b
a b

.
. .

.

Clearly indicating that the real output-derivative gain ma-
trix is not unique and a finite number of valid choices for a
and b are possible.

The closed-loop system equation can be computed as

( )

. .
. . . .

I BFC A

I

n

n
a b

a b a b
a b a

� �

�
�

� � � �

� �

�1

1 0 2 0 2
0 6 18 0 6 0 2b

a b

a b b
b a b



�
�

�

�
�



�
��

�

�
�� � �



�
�

�

�
� �

�

� � �

� � �

�1
0 1
3 4

1 5 4
4 5



�
�

�

�
�

then it is easily to shown that the closed-loop system has
eigenvalues at –5 and –1, regardless of the a and b values.

It has been shown, how the eigenstructure assignment
approach can be used to design a controller-based state-deriv-
ative and partial output-derivative feedback control, which
yields a closed-loop system with specified characteristics. The
approach is relevant for design with preservation of stability
when some necessary and sufficient conditions are provided.
Compared to state feedback, the state-derivative feedback
controller in some cases achieves the same performance with
lower gain elements. From the practical point of view, it is de-
sirable to determine feedback with small gains. Intuitively,
this must be advantageous, since small gains lead to smaller
control signals, and thus to less energy consumption.

5 Conclusions
This paper proposes a parametric approach for solving

the eigenstructure assignment problem via state-derivative
feedback. The necessary conditions to ensure solvability are
that the system is controllable and the open-loop system ma-
trix is nonsingular. The main result of this work is an efficient
algorithm for solving the eigenstructure assignment problem
of linear systems via state-derivative feedback. The extra
degrees of freedom on the choice of feedback gains can be
exploited to further to improve the closed-loop robustness
against perturbation. Additionally, the eigenstructure assign-
ment problem via partial output-derivative feedback is intro-
duced. The numerical examples prove the feasibility and
effectiveness of the proposed technique.

References
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for SISO linear systems by state-derivative feedback.”
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[2] Abdelaziz T. H. S., Valášek M.: “A direct algorithm for
pole placement by state-derivative feedback for single-

©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 59

Acta Polytechnica Vol. 44  No. 4/2004



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Eng. Taha H. S. Abdelaziz, Ph.D.
e-mail: tahahelmy@yahoo.com

Doc. Michael Valášek, DrSc.
e-mail: valasek@fsik.cvut.cz

Department of Mechanics

Czech Technical University in Prague
Faculty of Mechanical Engineering
Karlovo nám. 13
121 35 Praha 2, Czech Republic

60 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 44  No. 4/2004