Eigenstructure Assignment by State-derivative and Partial Output-derivative Feedback for Linear Time-invariant Control Systems

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-destate-de-rivative 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-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(t =A x t + B ut x t0 =x0, (1)

wherex(t)ÎRnandu(t)ÎRmare the state and the control input vector, respectively, (m£n), while AÎRn×n and

BÎRn×mare the system and control gain matrices, respec-tively. The fundamental assumption imposed on the system is that, the system is completely controllable and theBmatrix has a full column rankm.

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( )t = -K x&(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(t = In + BK -1Ax t, (3) whereI_nÎRn×nis the identity matrix. In what follows, we assume that (In + BK)has a full rank in order that the closed--loop system is well define. The closedclosed--loop characteristic polynomial is given by

[

]

det lI_n -(I_n + BK)-1A =0 (4) Denote the right eigenvector matrix of the closed-loop matrix (I_n +BK)-1AbyV, and we then have by definition,

(I_n + BK)-1AV = VL (5)

whereL=diag{ ,l₁K,l_n}and det(V)¹0.

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 { ,l1K,ln}ÎC. Select the appropriate state-derivative feedback gain matrix KÎRn×m that will make the closed-loop matrix (In + BK)-1Ahave admissible eigenvalues and associated set

of eigenvectorsV.

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, ,K,An-1B =n and

[

]

rank lI_n -A B, = " În, l C, and matrixAis nonsingular.

Proof: From the condition of that the closed-loop matrix must be defined. This means the matrix (I_n+BK) is of full rank and det(I_n+BK)¹0. Then, from (5) it is easy to rewrite as

AV -VL=BKVL (6)

which can be written as

BK =AVL- -1V 1-In (7)

Then,

det(In +BK)=det(In +AVL- -1V1-In)=

=det(AVL- -1V1)¹0. (8) Since V and L must be nonsingular. This leads to, det ( )A ¹0, and matrix Amust 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 (Ahas full rank). We remark that the requirement that the matrixLis 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 matrixKthat assigns the desired closed-loop poles is derived.

2.2 Case of distinct eigenvalues

In this subsection, all the desired closed-loop eigenvalues

{

l1,K,ln

}

are distinct.

Let (li,v_i) be the closed-loop eigenvalue and the corre-sponding eigenvector, then we can write the right eigen-vectors of the closed-loop system as

(In + BK)-1Avi=li iv, i=1,K,n. (9) The above equation can be rewritten as

Avi=li(In +BK)vi, (10) then

Avi-li iv =liBKvi. (11) Letw_i=Kv_i, then we have

Av_i-l_{i i}v =l_iBw_i. (12) Then (12) can be rewritten in a matrix form as,

[

li n li

]

i i i n I -A B æ 0 è ç ö ø ÷ = = , v , , , w 1K . (13)

Thus the subspace may be calculated in which the closed--loop eigenvector,v_i, and gain-eigenvector product,w_i, must satisfy the null space,null

(

[

l_{i n}I -A,l_iB

]

)

,i=1, …,n.

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

[

li n li

]

i i i i i i I A B 0 K - = = æ è ç ö ø ÷ =æ è ç ö ø ÷ , x , x v v v w . (14)

Thenv_iÎCn andKÎRm×nare unknown and notation-al simplicity is gained by defining the (n+m)×1 vector x_i. The determination ofKconsists of two general steps. First, a sufficient number of solution vectorsxiis found. Then, the internal structure among the components of these vectors is used to find the feedback gain matrixK.

Then, there exists a real feedback gain matrixK 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 {v_iÎCn_{,i=1, …,n}, are linearly independent} and for complex-conjugate poles,l_i=l_jthenv_i=v_j. 3. There exists a set of vectors {wiÎCm,i=1, …,n} satisfying

(13) andw_i=w_j forl_i=l_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(l₁),K, (v l_n) = w(l₁),K, (w l_n) . (15) Finally the gain matrix can be computed as

K =WV-1 (16)

where

[

]

V= v(l₁),K, (v l_n) and W =

[

w(l₁),K, (w l_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 {li},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 {li} are repeated.

LetG =

{

l l_i, _iÎC,i=1,K, ,s 1£ <s n

}

be a set of desired eigenvalues and let us denote the algebraic and geometric multiplicity oflibym_iandq_i, respectively, (q_i£m_i). The length ofq_ichains of generalized eigenvectors withliis denoted by pij, (j=1, …,qi). It is satisfying that _j pij n q i s i = =

å

å

_{1 1} = . Then, the right generalized eigenvector of the closed--loop system with li is denoted by v_ijk ÎC ,n i=1, …, s, j=1, …,q_i,k=1, …,p_ij. Then it is satisfied that

( ) , , , , , , , I_n BK A _ijk _{i ij}k _ijk _ij i s j + = + = = = -1 -1 0 ₀ 1 1 v l v v v K K q_i, k=1,K,p_ij. (17)

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

Av_ijk =l_i(I_n + BK)v_ijk +(I_n +BK)vk_ij-1, v_ij0 =0 . (18) Letw_ijk =Kv_ijk,i=1, …,s,j=1, …, qi,k=1, …, pij. The notations are defined as V =

[

V1,K,Vs

]

,Vi=

[

Vi1,K,Viq_i

]

,

V_{ij = é ij ij}pij

ëêv v ùûú

1_{,K, . The set ofw}

ij

k _{is define in a similar} man-ner to the set ofv_ijk. This leads to

[

li n li

]

ij

[

]

k ij k n ij k ij k I -A B æ I B è ç ç ö ø ÷ ÷= -æ è ç ç ö ø -, v , w v w 1 1÷_÷,vij = ,wij = 0 0 0 0.(19) Finally, the parametric equation to the right generalized eigenvector can be expressed as

[

li n li

]

ijk

[

n

]

ijk ijk ij k ij k I -A B = -I B =æ è ç ç ö ø ÷ ÷ -, x , x 1, x v , x w ij i ij i s j q k p 0 1 1 1 = = = = 0, ,K, , ,K, , ,K, . (20)

Similar to the case of distinct eigenvalues the (n+ m)-di-mentional parameter vectorsx_ijkare chosen arbitrarily, under the condition that the columns of eigenvector matrix Vare linearly independent.

A parametric solutionKto the eigenstructure assignment problem via state-derivative feedback is given by

K =WV-1 (21)

where

[

]

V = V₁(l₁),K,V_s(l_s) and W =

[

W₁(l₁),K,W_s(l_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 matrixK, 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 ÎC ,n i=1,K, ,s j=1,K, ,qi k=1,K,pij

}

are linearly independent and for complex-conjugate poles,

li₁ =li₂, thenvi jk₁ =vki j₂ . 3. There exists a set of vectors

{

wijk ÎC ,m i=1,K, ,s j=1,K, ,qi k=1,K,pij

}

, satisfying (19) andwi jk₁ =wki j₂ forli₁ =li₂.

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

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 statex(t)ÎRn, the output y(t)ÎRrand the con-trol input vector u(t)ÎRm, while AÎRn×n, BÎRn×m and

CÎRr×nare the system, control and output gain matrices, re-spectively. The fundamental assumptions imposed on the sys-tem are that the syssys-tem is completely controllable and observable. Additionally, matricesBandChave 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( )t = -F y&( )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 t = I_n + BFC A x) ( )t (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 lIn -(In + BFC)-1A =0 (25)

Denote the right eigenvector matrix of the matrix (I_n + BFC)-1AbyV, and then we have,

(In + BFC)-1AV V= L. (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 {l₁, …,ln}ÎC. Select the appropriate output-derivative feedback matrix

FÎRm×r that will make the closed-loop matrix (I_n + BFC)-1Ahave admissable eigenvalues and correspond-ing right generalized eigenvectorsV.

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, , ,K An-1B =n and

[

]

rank lI_n -A B, = " În, l C, also, the pair (A,C) is completely observable that is,

[

]

rank C AC, , ,K An-1C =n and

[

]

rank lI_n -A C, = " În, l C, and matrixAis nonsingular.

Proof: From the condition of that the closed-loop matrix must be defined. This means the matrix (I_n+BFC) is of full rank and det(In +BFC)¹0. Then, from (26) it can easily put

in the following form

AV -VL=BFCVL (27)

which can be written as

BFC=AVL- -1V 1-I_n (28) Then,

[

]

[

det I_n + BFC =det I_n +AVL- -1V1-I_n]=

=det [AVL- -1V1]¹0 (29) SinceVandLmust be nonsingular. Thus, matrixAmust 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 (Ahas 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.

LetG =

{

l li, iÎC, i=1,K, ,s 1£ <s n

}

be a set of de-sired eigenvalues and let us denote the algebraic and geo-metric multiplicity oflibym_iandq_i, respectively. The length ofq_ichains of generalized eigenvectors withli, are denoted byp_ij. The right generalized eigenvector of the closed-loop system withliis denoted byv_ijk ÎC ,n i=1, …,s,j=1, …,q_i, k=1, …,p_ij. It is satisfying that p_ij n j q i s i = =

å

å

_{1 1} = . Then we

have the following relation

( , , , , , , In BFC A ijk i ijk ijk ij 0 i s j + = + = = = - -)1 v lv v 1 v0 1K 1K, ,qi k=1,K,pij. (30)

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

(A-l_{i n}I + l_iBFC)vk_ij =(I_n + BFC)v_ijk-1, v_ij0 =0. (31) Letzijk =FCvijk,i=1, …,s,j=1, …,qi,k=1, …,pij. The notations are defined as V =

[

V₁,K,V_s

]

,V_i

[

V_i V_iq

]

i

Vij ij ij pij

= é

ëêv v ùûú

1_{,K, . The set of}

zijk is defined in a similar man-ner to the set ofv_ijk. The above equation can be rewritten as

[

li n li

]

ij

[

]

k ij k n ij k ij k I -A B æ I B è ç ç ö ø ÷ ÷= -æ è ç ç ö ø -, v , z v z 1 1÷_÷, vij0 =0, zij0 =0. (32) The right generalized eigenvector can be expressed as

[

li n li

]

ijk

[

n

]

ijk ijk ij k ij k I -A B = -I B =æ è ç ç ö ø ÷ ÷ -, y , y 1,y v ,y z ij i ij i s j q k p 0 1 1 1 = = = = 0, ,K, , ,K, , ,K, . (33)

Similar to the case of state-derivative feedback, there ex-ists a real matrixFif and only if the (n+m)×1-dimensional parameter vectors,yijk, are arbitrary chosen under the condi-tion that the columns of matrixVare linearly independent.

Finally, a real gain matrixFis expressed as:

F=ZV-1 (34)

where

[

]

V = V1(l1),K,Vs(ls) andZ=

[

Z1(l1),K,Zs(ls)

]

. Then the feedback gain matrixFis 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 ÎC ,n i=1,K, ,s j=1,K, ,q ki =1,K,pij

}

are linearly independent and for complex-conjugate poles,l_i l_i

1 = 2 thenvi j v

k i jk

1 = 2 .

3. There exists a set of vectors

{

zijk ÎC ,m i=1,K, ,s j=1,K, ,q ki =1,K,pij

}

, satisfying (32) andz_{i j}k z_{i j}k

1 = 2 forli1 =li2.

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 t = æ x t ut è ç ö ø ÷ + æ è ç ö ø ÷ 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}. Forl1= -3,

[

l1 l1

]

1 1 4 2 0 0 6 3 In-A B = 0 -æ è ç ö ø ÷ = , x x , arbitrarily selectingx₁= -

[

0 5 1. , ,-2

]

T leads toKæ -è ç ö ø ÷ = -05 1 2 . . Forl₂= -4,

[

l₂ l₂

]

₂ 5 2 0 ₂ 0 7 4 In-A B = 0 -æ è ç ö ø ÷ = , x x , takingx₂= -

[

0 4 1 7 4. , , -

]

T leads toKæ -è ç ö ø ÷ = -0 4 1 7 4 . .

Taking the two equations together, we can write

Kæ- -è ç ö ø ÷ = -0 5 0 4 1 1 2 7 4 . . ( ) .

Finally, the unique state-derivative feedback gain matrix is

[

]

K = 2 5 075. , . .

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

[

lIn-A lB

]

₁₁1 = ₁₁1 0 - -- -æ è ç ö ø ÷ = , x 2 2 0 x 0 4 1 , leads tox₁₁1 = -

[

1 1, ,-4

]

TandKæ -è ç ö ø ÷ = -1 1 4.

The generalized eigenvector equation is,

[

lIn-A,lB

]

x112= -

[

In,B

]

x111, and, - -- -æ è ç ö ø ÷ =æ- _- -è ç ö ø ÷ -æ è ç ç ç ö ø ÷ ÷ 2 2 0 0 4 1 1 0 0 0 1 1 1 1 4 2 x11 ÷= æèç ö ø ÷ 1 3 , thenx₁12= -

[

15 1. , ,-7

]

TandK -æ è ç ö ø ÷ = -15 1 7 . . And the gain equation can be written as

Kæ -è ç ö ø ÷ = -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 t = æ x t u t è ç ö ø ÷ + æ è ç ö ø ÷ 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}. Forl₁= -3,

[

l₁ l₁

]

₁ 4 2 3 0 ₁ 0 6 0 3 I_n -A B = - - - 0 - -æ è ç ö ø ÷ = , x x , takingv₁=

[ ]

a b, T, leads tox₁=

[

a b, ,-4 3 2 3a - b ,-2b

]

T andK a b a b b æ è ç ö ø ÷ =æ- _- -è ç ö ø ÷ 4 3 2 3 2 .

Forl2= -5,

[

l2 l2

]

2 2 6 2 5 0 0 8 0 5 In-A B = 0 - - -- -æ è ç ö ø ÷ = , x x , choosingv2 =

[ ]

c d, T, thenx2 =

[

c d, ,-6 5 2 5 8 5c - d ,-d

]

T, andK c d c d d æ è ç ö ø ÷ =æ- _- -è ç ö ø ÷ 6 5 2 5 8 5 .

Then, the feedback gain matrix equation is

K a c b d a b c d b d æ è ç ö ø ÷ =æ- _-- - _- -è ç ö ø ÷ 4 3 2 3 6 5 2 5 2 8 5 .

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

The gain matrix is

K = - -æ è ç ö ø ÷ 4 3 2 5 0 8 5 .

Case 2:The desired closed-loop poles are, {–1 and –1}. Forl₁= -1,

[

lIn -A lB

]

₁₁1 = ₁₁1 0 - - -- -æ è ç ö ø ÷ = , x 2 2 1 0 x 0 4 0 1 , takingv₁₁1 =

[ ]

a b, T, leads tox₁₁1 =

[

a b, ,-2a-2b,-4b

]

T. The generalized eigenvector equation is,

[

lIn-A,lB

]

x112= -

[

In,B

]

x111 - - -- -æ è ç ö ø ÷ = - -- -æ è ç ö ø ÷ = + 2 2 1 0 0 4 0 1 1 0 1 0 0 1 0 1 2 2 x₁1 x₁₁1 a b b 3 æ è ç ö ø ÷, takingv₁₂1 =

[ ]

c d, T, leads to

[

]

x112= c d, ,- -a 2b-2c-2d,-3b-4d T.

Then, the gain matrix equation can be written as

K = æ è ç ö ø ÷ =æ- -_- - -_{- -}- -è ç ö ø ÷ a c b d a b a b c d b b d 2 2 2 2 2 4 3 4 .

We can seta=d=1 and b=c=0. Then, the gain matrix is

K = - -æ è ç ö ø ÷ 2 3 0 4 .

Example 3:

Consider a controllable and observable, multi-input linear system &( ) ( ) ( ) x t = x t ut -æ è ç ö ø ÷ + æ è ç ö ø ÷ 0 1 3 4 1 0 0 1 , y( )t =(1 1) ( )x t

In the following we consider the first eigenvalue at (l₁= -5),

[

l1 l1

]

1 1 5 1 5 0 3 1 0 5 I_n -A B = - - - 0 - -æ è ç ö ø ÷ = , y y , takingv1=

[ ]

a b, T, leads toy₁=

[

a b, ,- -a 0 2 0 6. , .b a-0 2. b

]

T. This means FC a b a b a b æ è ç ö ø ÷ =æ - - -è ç ö ø ÷ 0 2 0 6 0 2 . . . . Hence, FC a F b a b a b a b æ è ç ö ø ÷ = + = -æ è ç ö ø ÷ ( ) . . . 0 2 0 6 0 2 . The gain matrix is

F= + -æ è ç ö ø ÷ 1 0 2 0 6 0 2 a 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 fora andbare 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 æ è ç ö ø ÷ æ è çç ö_{ø÷÷ - -}æ_èç ö ø ÷ = + - - -- --1 0 1 3 4 1 5 4 4 5 æ è ç ö ø ÷

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

It has been shown, how the eigenstructure assignment approach can be used to design a controller-based state-deriv-ative and partial output-derivstate-deriv-ative 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.

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Eng. Taha H. S. Abdelaziz, Ph.D. e-mail: [email protected] Doc. Michael Valášek, DrSc. e-mail: [email protected] Department of Mechanics

Czech Technical University in Prague Faculty of Mechanical Engineering Karlovo nám. 13