Eigenvalue Assignment by Static Output Feedback

Copyright © IFAC System Structure and Control, Bucharest, Romania, 1997

EIGENVALUE ASSIGNMENT BY STATIC OUTPUT FEEDBACK M. Dandache J. Lottin Laboratoire d' Automatique et de Microlnformatique Industrielle LAMII-CESALP-ESIA 41 , avenue de la plaine B.P. 806 74016 ANNECY cedex . FRANCE Phone: (33)-450-66-60-70 Fax: (33)-450-66-60-20 E-mail: {dandache.lottin}@esia.univ-savoie.fr

Abstract. This paper investigates a new method for the search of a solution to the problem of eigenvalue assignment by means of static output feedback or, when there is no solution to this problem, for the synthesis of a minimal order controller that allows arbitrary pole placement. The aim of this work is an attempt to give an interpretation of the dynamical part of the control scheme in terms of unavailable information reconstruction such as minimal order observer design in single input system control problem . This study is based on a signal flow graph representation of the system. A first result concerns indications about redundant parameters in the output feedback gain matrix by inspecting independent subsystems. A second result gives a necessary condition for arbitrary eigenvalue assignment by static output feedback. More than a boolean verdict, this result provides with indications about the location of missing measured information in the system. Once the characteristic polynomial coefficients are expressed in terms of output feedback gains, it is possible to analyze the existence of a solution , at least lo cally, for the arbitrary eigenvalue assignment . Copyright © 1998 IFAC

Keywords: Linear systems, multivariable systems, pole assignment, static output feedback .

1

Introduction

(Kimura, 1978) , (Champetier and Magni , 1989) , (Champetier and Magni, 1991), (Magni), (Alexandridis and Paraskevopoulos, 1996), (Rosenthal and Wang , 1996), (Wang, 1992) . Critical cases appear when m .p=n where m , p, n are respectively the number of input, output and state. The aim of this work is an attempt to give an interpretation of the dynamical part of the control scheme in terms of unavailable information re con-

This paper investigates a new method for the search and design of a solution to the problem of eigenvalue assignment by means of static output feedback. Several necessary and/or sufficient conditions to the existence of a solution have been given in the literature (Kimura, 1975), (Kimura, 1977),

279

struction such as minimal order observer design in single input system control problem. However, this paper is limited to the presentation of the framework of this analysis. This study is based on a signal flow graph representation of the system. A first result concerns indications about redundant parameters in the output feedback gain matrix by inspecting independent subsystems. A second result gives a necessary condition for arbitrary eigenvalue assignment by static output feedback . It implies that it is possible to find loop, or association of loops, of length P that include output feedback gains, for every P between 1 and

and the problem for arbitrary pole assignment by static output feedback involves the selection of a constant feedback gain matrix K, since when a solution exists, it is not necessarily unique. Kimura (Kimura, 1975), (Kimura, 1977), (Kimura, 1978), showed that, under some conditions, min(n,m+p-l) poles are assignable generically. This translates into the sufficient condition for generic poles assignability that m+p ~ n+1. If this condition fails to hold then m+p-1 eigenvalues can be assigned (Champetier and Magni, 1989), (Champetier and Magni, 1991), (Magni). Recently Wang (Wang, 1992) has shown that n< m.p is sufficient for arbitrary pole placement. In the case of n=m .p, it is sufficient if dm,p is th d h cl m,p = l!m!........(p-l)!(mp)!. (m+p-l)! IS e egree o dd ,were of Grassc(m, m + p) . When for example m=p=2 and n=4, d m,p=2 is even, therefore we can't conclude. We propose to examine the case of m . p~n using the paths and loops extracted from the signal flow graph as defined in the next section . For sake of simplicity we consider the case of a triangular state matrix A.

n.

More than a boolean verdict, possibility or not, this result provides with indications about the location of missing measured information in the system, and as a consequence the place for a reduced observer if an additional sensor cannot be set up . Once the characteristic polynomial coefficients are expressed in terms of output feedback gains, which can be done by inspection of loops in the signal flow graph for example, and provided that the necessary condition is satisfied, it is possible to analyze the existence of a solution, at least locally, for the arbitrary eigenvalue assignment. The paper is organized as follows. Section 2 deals with problem formulation while section 3 establishes several definitions that are used in the sequel in order to simplify the presentation. Section 4 gives the main results that can be derived from the signal flow graph analysis, and presents the outlines of the analysis of a local solution by means of characteristic polynomial. All along the paper many simple examples are given in order to improve the understanding of the procedure.

2

3

Definitions: • Definition 1: We call a loop of order p a curly path which includes a dynamic of order p.

For example, the graphs below are two examples of a same first order loop:

pt 't

or

Problem Formulation Fig 1: Canonical form of a 1rst order loop

This study concerns a linear multi variable system described by: x(t) = A.x + B.u { y(t) = C.x

• Definition 2: We call a gain of a loop, the coefficient of its feedback, when put in the canonical form. The gain of the above example is not Q' but

(1)

Q' -

T.

where A, Band C are matrices of appropriate dimensions, and the state x(t)E nn, the input u(t) E nm and the output y(t) E n p . Under the influence of static output feedback of the form :

• Definition 3: Two loops are called independent if they haven't any common dynamic .

u(t) = K.y(t) + v(t)

• Definition 4: We call a E-loop of order p a set of independent loops of order Pi such that

the closed loop system becomes: x(t) = (A + B.K.C).x + B.v

280

1--------------------------------1 , ,

Let us consider the fourth order two input two output system described by the following diagram:

U

L ________________________________ I

1

p+'t 1

--------------------------------1,

p

Fig 2: Two input two output example One possible state space representation by:

1

(

given

o o o o

-Tl

A=

IS

Fig 4:

o 1 o o

C-_ (00

o o

~-loops

of ord er 2.

Figure 5 shows three realizations of ~-loops of order 3. Here also dashed loops mustn't be taken into account . One situation involves two independent loops of order 1 and 2 respectively, while the two other cases concern unique loops of order three. These situations make appearing the coefficient / , already met in ~-loops of order 2, and coeffi cients Q and 6 of the output feedback gain matrix .

1

0

where K is the static output feedback gain matrix. It is easy to list the various ~-loop of order p, p 1..4 , as shown below.

=

Let us first consider ~-loops of order 1. It can be seen that two loops can be built using output feedback matrix together with bloc diagram in figure 2. ,---------------,

,---------------1

1

1

1

1

1

1

1

1

1

I 1

1 1

I

I

1

I I

1 1

.--------------~

Fig 3:

~-loops

1

u~

.--------------~

of order 1.

p

It can be noticed that none of the loop gains involves any coefficient of the static output feedback gain matrix.

p

-+;

:

p

: G: ··: · :...... .....

:

- 't 3

..: ..... . :

________________________________ J

---------------------------------,

Concerning ~-loop of order 2, two different situations occur. One involves the two first order ~>loops previously mentionned, but they must be considered together . Another possibility is given by one simple ~-loop of order 2 as shown in the second part of the figure 4. Note that the feedback loop with gain -T3 is dashed since it doesn't account for this second order loop. The parameter / is the sole coefficient of the output feedback matrix that appears in the set of second order loops.

,----------la.~-----_,

,

::......... I,;l: ~ ... :

!:........... r::l : ~ .. ' ,

I ________________________________ J

Fig 5: I;-loops of order 3.

281

Maximum length of ~-loops is equal to four for this example. Two realizations of simple loops are given in figure 6. In these cases, parameters 0 and (3 appear explicitly, and it is the unique situation for parameter (3.

described in the previous section:

It's easy to find that

,,,-----------------------------------------------,

+ T3

- for

~-loops

of order 1:

a3

=

T1

for

~-loops

of order 2:

a2

=

T1 T3 -

- for

~-loops

of order 3: a1 =

-T1 i

- for

~-loops

of order 4: ao =

-TI

i - 0- a

0 - (3

• Result 2: A necessary condition for pole placement by a static output feedback is that there exists a ~-loop of order p , with p=l..n , which contains at least one element of the feed back matrix K= {kij } .

,

In the previous example the coefficient a3 1S 1Udependent of parameters a, (3, i, and O. So arbitrary pole assignment is not feasible.

----5]--:

r--------------------------------I I

I

Let us consider another example with the same number of input, output and state as before, described in figure 7.

I

,,

: G: ~---

: ,

,

-

1 I ---,

5]:

'

,---- - 13 -'

,

U

, ________________________________ J

I

p Fig 6: I:-loops of order 4.

4

Fig 7: Second example

Results:

A possible state space representation is given by:

• Result 1: The characteristic polynomial of the closed loop system can be deduced from independent loops of the closed loop system diagram . More precisely let us write the characteristic polynomial as:

Then

ai

=L

II( _l)k

A_ -

(-;1 0

o o

0

1

-T3

o o

c -_ (

0 1

1 000 0 001

Following the same procedure, it is possible to derive the coefficients of the characteristic polynomial:

pgj .

p=1 j=1

where: - Sn-i is the number of realizations of ~-loops of order n-i

a3 a2

{

- kp is the number of independent loops appearing in the pth realization

0

a1 =ao-i(3-T10

ao

- gj is the feedback gain of the /h loop.

= 71 + T3 - a = 7173 - aT3 = -(3

It is easily seen that the necessary condition is satisfied, which means it is worth to look after arbitrary pole assignment.

For example let us derive the coefficients of the characteristic polynomial of the close loop system

282

• Result 3: The poles are locally assignable if the rank of the first derivative matrix of M(kij) is equal to n, where M(k ij ) E 'Rn is the vector function which maps the set of gain matrix parameters to the set of characteristic polynomial coefficients aj, j=O .. n-1.

C

=

(

1 0

o

0 0 0

0 0 0

0 1 0

0) 0 1

K

=

(

kll k21 k31

The characteristic polynomial of closed loop system becomes:

In the second example we can give the first derivative matrix J of M(a ,,8, I, 8): with : 0

-r

(

-1

J=

0

0 0

-"(

-(3

-1

0

0 -1 -Tl

+ 0

0'

)

a4 = T3 - k33 + Tl - k 11 a3 = Tl T3 + kl1 k33 - kl1 T3 - Tlk33 - k31 k 13 -T3 k33 a2 = -Tl T3k33 - k3lk13T3 + kl1 T3k33 - k22 al = k22k33 - k12 + k11k n - Tlk22 - k23k32 -k21k12 aD = Tlk22 k 33 - Tlk23k32 + k12k33 - k13k32 -k21k13k32 - k31 k 12k23 - kl1k22 k33 +kl1 k 23 k32 + k31 k 13 k 22 + k21k12 k33

The jacobian determinant is: det(J)=-,8 When ,8 :f. 0 which means aD :f. 0 there exists a solution to obtain a characteristic polynomial

peA ):

In such a form the problem becomes redundant with respect to gain matrix parameters since five constraints are to be satisfied by means of nine parameters. One could reduce the size of the problem with a proper choice of some particular parameters. However, the study of various I:-loops allows elimination of some particular solutions that would not satisfy result 2. Final validation of a simplified case is done by checking the rank of first derivative matrix as presented in result 3.

It can be noticed that a D :f. 0 is not a constraint since otherwise closed loop is unstable.

In the first example the jacobian determinant of M(kij) is obviously null. Then a compensator is necessary for arbitrary pole placement. It can be shown that a first order controller is sufficient . Then the extented system becomes:

Making the a prIOrI choice for four parameters, namely:

~::: ~

------------u;--------------------

{

I I

=

k l3 1 k12 = 1

U

I

One obtains a square problem which, in this case, provides the following solution:

p

U3~-

k33 = -~a4 + ~T3 - ~Ja~ + 2T3a4 - 3Tj - 4 - 4a3 2 2 k 32 -- - (k 33 ( al - T34 - T3a3 - a4 a 2 - T32 a4+ a4 T3a 3 + T3 a 2 + 2Tla4) + Tl + aD - a3 a 2 +T3a~ - Tja2 - 2Tia4 + 2Tra3 + Tra~+ T3a4a2 - 2Tja4a3)/(T34 - T3a2 - a4 T3a3 +a4 a 2 + T322+2 a 4 T3 a3 - 23 T3 a4 - al ) k22 = -a2 + Tf - Tja4 + T3 a 3 k21 = -1 - T3a2 + Tt - 2Tfa4 + Tja3 + a4a2 +Tj a~ - a4 T3a3 - al kl1 = T3 - k33 + Tl - a4

y 3

[}

P

I

l _________________________________

J

Fig 8: Extented system for first example

The corresponding state space representation given by:

i

-Tl

h

(

o o

o o

1

-T3

o

o

1

o

IS

o 0 o 0 o 0 o 0 o 0

Let us remark that one of the parameters that are a priori fixed, namely k 12 , doesn't belong to the set of parameters introduced by the links between initial system and the controller .

283

5

Conclusion

Champetier, C. and J.F. Magni (1991). On eigenstructure assignment by gain output feedback. Siam J. Control and Optimisation, 29:4, pp. 848-865. Garcia, G., J. Bernussou and P. Camozi (1994) . Placement de poles robustes dans un disque pour les systemes lineaires. APII, 28:6, pp. 687-713. Garcia, G. and J. Bernussou (1995). Pole assignment for uncertain systems in a specified disk by state feedback. IEEE Transaction on Automatic Control, 40:1, pp. 184-190. Kimura, H. (1975). Pole assignment by output feedback. IEEE Transaction on Automatic Control, 20, pp. 509-516. Kimura, H. (1977). A further result on the problem of pole assignment by output feedback. IEEE Transaction on Automatic Control, 22, pp. 458-463. Kimura, H. (1978). On pole assignment by output feedback. Int. J. Control and Optimisation, 28:1, pp. 11-22. Magni, J.F. (1993). Robustesse et performances par la commande modale. Ecole d'ite. d 'A utomatique de Grenoble , pp. 1-37. Rosenthal, J. and X. Wang (1996). Output feedback pole placement with dynamic compensator.JEEE Transaction on Automatic Control, 41:6, pp. 830-843 Wang, X. (1992). Pole placement by static output feedback. Journal of Mathematical Systems, Estimation, and Control, 2:2, pp. 205-218

One main interest of the procedure which is presented in this paper is that it gives important information about the system structure and the way to exploit it when arbitrary eigenvalue assignment is not feasible by means of static output feedback with original sensor location. Examples that are given illustrate the importance of structural properties of the system as well as the local vanishing of solution in some particular numerical situations. The work currently in progress in this field, concerns the comparison between multiple solutions when there are redundant parameters in the feedback gain matrix, namely m.p > n, and the search of a criterion which would give a comparative measure of the robustness of each solution. Another aspect. of this work is related to the interpretation of the variables appearing in the dynamic controller when it is impossible to find a solution by means of static output feedback.

References Alexandridis , A.T., and P.N. Paraskevopoulos (1996). A new approach to eigenstructure assignment by output feedback . IEEE Transaction on Automatic Control, 41:7, pp. 1046- 1050. Champetier, C . and J .F. Magni (1989). Analyse et synthese de lois de commande modales. La Recherche Aerospatiale, 6, pp . 17- 35.

284