On pole assignment of linear systems by gain output feedback

Systems & Control Letters 9 (1987) 241-247 North-Holland

241

On pole assignment of linear systems by gain output feedback J.-F. M A G N I

a n d C. C H A M P E T I E R

*

C E R T / D E R A , 2 ao. Edouard Belin, 31055 Toulouse, France

Received 3 February 1987 Revised 29 April 1987 In this paper, the Geometric Approach is used to derive in a straightforward way a sufficient condition for pole assignability by gain output feedback. This result leads to a pole assignment procedure which reduces to solving a system of min(n - m, n - p) polynomial equations where n is the number of states, m the number of inputs, p the number of outputs. In the case where m + p > n, this system clearly appears to be linear. The degrees of freedom related to the pole assignment problem are expressed in terms of (right or lef0 eigenvectors.

Abstract:

Keywords:

Linear systems, Output feedback, Pole assignment, Eigenvector assignment, Parametric controller design.

1. Introduction In the last decade, the problem of the pole assignment by constant output feedback has received much attention. Practical considerations have led the designers to improve the classical pole assignment methods. As a matter of fact, in m a n y practical problems, assigning the poles at desired locations is not sufficient to obtain efficient controllers. Now, a question arises: can we do anything else with a constant output feedback? Arithmetic considerations provide a sketchy answer. Let n be the number of states, m the number of inputs, p the number of outputs. At first glance, it seems that if the number of gain coefficients exceeds the number of poles to be ass!gned, that is, if m p >1 n, then the pole assignment is generically possible, and if m p > n , there exist m p - n degrees of freedom in this placement. Unfortunately, the reality is much more intricate (eL WiUems and Hesselink [20]). To get some idea of the involved difficulties, one can try to compute the output gain by equating the coefficients of the closed loop characteristic polynomial with their desired values. This leads to a system of n real polynomial equations in the gain coefficients. Two questions of practical importance m a y be posed: (1) Do there exist real solutions to this set of equations (complex solutions always exist if m p >>.n - of. Hermann and Martin [10])? (2) In the affirmative, what is the domain of the m p - n free parameters for which such solutions exist? In the cases where m + p > n, it appears that Linear Algebra is sufficient to answer these questions. In this case, the problem of genetic pole assignment has been independently solved b y Kimura [12] and Davison and Wang [7] (see Fletcher [8], Mielke and Liberty [16], Roppenecker and O'Reilly [18] for further developments). A necessary and sufficient condition for exact pole assignment is given in [14]. A complete treatment of the second problem can be found in [15]. In the case where m + p ~ n, all seems to be more complicated. Linear Algebra is no longer efficient (except perhaps for some specific systems [13]). Some results related to the first question h a v e been obtained by Brockett and Bymes [5], Giannakopoulos and Karcanias [9] b y using Algebraic Geometry. Anderson et al. [2] have discussed the second question from a different point of view. * This research was supported by the Direction des Reoherches, Etudes, et Techrdques. 0167-6911/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

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J.-F. Magni, C Champetier/ Poleassignmentof linearsystems

In this paper, our objective is to enlighten the transition between the two cases. More precisely, it will be shown that the pole assignment by output feedback can be reduced to solving a set of only m i n ( n - m, n - p ) real polynomial equations in rain(rap- m, m p - p ) unknowns. These equations will clearly appear to be linear in the case where m + p > n. Moreover, the degrees of freedom appearing in the pole placement are directly expressed in terms of (right or left) eigenvector coordinates, which is particularly well adapted to take into account various control design specifications [6]. The plan of this paper is as follows. In Section 2, some classical result of the Geometric Approach of linear systems are recalled. In Section 3, a procedure to obtain the polynomial equations is described. In Section 4, a simple example illustrates the method.

2. Preliminaries

In this section, we recall some well known results of the Geometric Approach from which the procedure described in the next section will be derived. All the notations and concepts used here are those defined by Wonham [22] and Willems and Commault [21]. Let us consider a linear controllable and observable system Yc= A x + Bu,

y = Cx,

(1)

in which x, u, y are functions of time with values in the vector spaces X, U, Y where dim X = n,

dim U = m,

dim Y = p ,

(2)

and B, C are full rank. Let S be a subspace of X. If S is (A, B)-invariant and (C, A)-invariant, it is known (of. Schumacher [19]) that there exists at least one linear mapping K: Y ~ U such that

(A + s r c ) s

c s.

(3)

Let R*S denote the greatest controllability subspace contained in S and Ns* be the smallest complementary observability subspace containing S. It is well known that (3) yields ( A + B K C ) R ~ c R~,

(4)

(A + BKC)Ns* c Ns* ,

(5)

The inclusions (3), (4), (5)justify the following lattice diagram (cf. [21]): 0 ---)R~ AF)S A'F)Ns* ---)X where A F is the fixed part of the spectrum of A + BG for any state feedback G satisfying (A + B G ) S c S and A'E is the fixed part of the spectrum of A + GC for any output injection G satisfying (A + G C ) S c S. Let E be any matrix such that Ker E = S. Since S is the greatest (A, B)-invariant contained in Ker E, it follows from Anderson [1] that AF is the set of the invariant zeros of the tripple (A, B, E). By duality, A'F is the set of the invariant zeros of the triple (A, S, C) where S is any matrix such that Im S = S~ So, the following theorem holds. Theorem. Let S be a (C, A)-inoariant and (A, B)-invariant subspace. Then: (1) the set of the output feedback gains K such that ( A + B K C ) S c S is not empty; (2) any output feedback gain K satisfying the above condition is such that

A~u A'FC a( A + BKC) where A F is the set of inoariant zeros of the triple (A, B, E), E being any matrix such that Ker E = S; wid A'F is the set of inoariant zeros :of the triple (A, B, S), S being any matrix such that Im S = S.

J.-F. Magni, C. Champetier / Pole assignment of linear systems

243

Some well known results contained in [12] concerning the pole placement problem follow from this fundamental result. The geometric approach adopted here permits to release in a natural way some weak hypotheses on the spectrum to be assigned appearing in the above reference. Before stating the specific version of the theorem used in the next section, we need the following definition. Definition. For any ~ ~ C, let S(~) denote the subspace (A - ~ i ) - 1 Im B. For any s ~ S(~), the unique vector w ~ U such that (A - ~ I ) s

+ Bw= 0

is called the input direction related to s. Corollary. Let A be a set of n complex numbers such that A = A 1 U A 2 where A 1 and A 2 are closed under complex conjugation, card A 1 = p and the elements ~1,..., ~p of A1 are distinct (see Remark 2 below). Assume that there exists a subspace S satisfying the following prop_erties: (i) S = S p a n ( s l , . . . , s p } with s i ~ S ( h i ) a n d s i = g j if Xi=Xj; (ii)

c s -- p ;

(iii) the set of the inoariant zeros of the triple (A, S, C) is A 2 where S is a full rank matrix such that Im S = S . Then, there exists a unique real output feedback gain K such that a(A+BKC)=A,

( A + B K C ) s ~ = X i s ~ f o r i = l . . . . . p.

This gain is given by K=[w 1

...

wp](C[Sl

...

$p])-I

(6)

where the w ~'s are the input directions associated to the s~'s. Proof. Clearly, a subspace S satisfying (i) is (A, B)-invariant. Since dim CS = p , then dim S = p and S ca Ker C = 0. So, S is (C, A)-invariant and Ns* = X. From the definition of S, there exists a state feedback K ' such that (A + BK')si=~is

i

for any h i ~ A 1

or equivalently (cf. the definition) [w 1

...

wpl=K'[sl

...

sp].

Since S n Ker C = 0, there exists a unique output feedback K such that KC[sl

...

s.l=K'[s

that is (A + B K C ) s i = ~ i S i

for any h i ~ A 1.

Clearly, K is given by equation (6) and A 1 c tr(A + BKC). Finally, from the theorem A 2 c a(A + BKC). Thus o( A + BKC) = A. Remark 1. The corollary can be dualized in a straightforward way by dealing with left eigenvectors. Remark 2. The elements of A 1 are assumed to be distinct only for notational convenience. This assumption can be released by considering generalized eigenvectors (cf. O'Reilly and Fahrny [17]).

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J.-F. Magni, C. Champetier/ Pole assignment of linear systems

3. A pole assignment procedure In this section, we show how the construction of the space S -- Span(s 1. . . . . sp} can be achieved by solving a set of polynomial equations. The following notations will be used: •for any h ~ C , S(h) denotes a full rank matrix such that Im S ( h ) = S ( k ) ; •Z is a full rank matrix such that Im Z = Ker C; • T(fl) denotes the matrix (A - i l l ) Z ; •~ denotes the vectors in C m such that s i = S(h~)~ i. Lemma. Assume that dim CS =p. Then, the zeros of the triple ( A, S, C) are the roots of the polynomial in fl

det [S(X1)~,

...

S(Xp)~p

T(fl)].

Proof. Let us consider the system matrix

As dim CS = p , the matrix

o

Id

z

s1

0

is full rank. So detM(fl)=O

~

det(M(fl)[ 0

~, det[S

ZO S ] ) = d e t [ o S

T(fl)O

( A - f l I ) S ] =JO c s

T(fl)] =0.

Since det[S

T(fl)] = d e t [ S

Z]fl " - p + . . . ,

it follows that det M(fl) ~ O. Thus, det M(fl), or equivalently det[S T(fl)], is the zero polynomial of the triple (A, S, C). This lemma together with the corollary yield the following result: Proposition. Let A be a set of n complex numbers such that A = A 1 U A 2 where A 1 and A 2 are closed under complex conjugation, card A 1 = p , and the elements of X1,-.-, hp orAl are distinct. A sufficient condition for assigning A by gain output feedback is that there exist vectors ~1. . . . . ~p ~ C m satisfying the following conditions: (i) ~, = ~j if h i =fftj; (i.i) P0(~I . . . . . ~p)_A det[S(hl)~ 1 . . . S(Xp)~p Z] 4= 0; • (iii) P(~I . . . . . $t,, fl)Adet[S(hl)~a "'" S(Xp)~p T ( # ) ] = 0 i f f B e A 2 . This result is similar to Theorem 1 in [12] except for the third condition which is stated as an orthogonality condition An [12]. Remark 3. If multiple eigenvalues (in A1) are to be assigned, it is straightforward to restate the proposition by using generalized eigenvectors with their corresponding parametrization [17] (in this case, to avoid a waste of degrees of freedom, we must choose the corresponding geometric multiplicity equal to one).

J.-F. Magni, C. Champetier/ Poleassignmentof linearsystems

245

Pole Assignment Procedure. It can easily be shown that P(~I . . . . . ~,,/3) is also a polynomial with respect to the entries of the ~i's which can be expressed as

e ( , 1 ..... ~,,/3) = e o ( ~ l , . . . , ~,)/3"-' + e ; ( ~ , . . . . . ~ , ) / 3 , - , - '

+ ... +e-_,(,,

..... ~,)

where the polynomials Pi' are of the form

Pi'(~l .... , ~ , ) = d e t [ S ( h , ) ~ 1

..-

s(x,),, r,.'].

(7)

Let

D(/3) & I-I (/3-/3i) &/3"-" + a1/3 " - ' - 1 + "'" + a . _ , Bl~A2

denote the factor of the desired pole polynomial corresponding to A 2. Condition (iii) of the proposition is equivalent to P(~I .....

~p, f l ) : P o ( ~ x , . . . , ~ p ) D ( f l )

for all B.

Equating the coefficients of these polynomials in/3 leads to n - p equations

ei(~, ..... ~,) = 0,

i=l,...,n-p,

(8)

where the polynomials Pi are defined by Pi(~l

.....

~,)=det[S(A1)~ x

...

S(X,)~,

T/]

(9)

with T, a constant real matrix. Condition (i) of the proposition is essential to obtain real feedback gains. This condition is satisfied iff (1) for any real number h k in A 1, the condition ~k ~ R " is satisfied; (2) for any non-real number.hk in A 1, assuming that hk+ 1 =7~ k, the condition ~k+l = ~k is satisfied. To take into account in the latter condition, it is convenient to express the polynomials P; in terms of the real part and the imaginary part of the ~k's as follows:

[ •

•

I

s,(xk)][[ _,1 ~I' ,,] ~;, .. z]

where(k =" ~kR +J~k • i and S(Xk) a SR(Xk) +jSX(Xk). Therefore, the pole assignment procedure consists in finding the real solutions 71 . . . . . 7, ~ Rm of the polynomial system

P0(Ta ..... 7,) ~ 0, Pl(71,---,Tp)=0,

(10)

fori=l,...,n-p, ¢

where 7,. & ~,. if h i ~ R, (7i, 7,'+1) = ( ~ , ~ ) if ?~; ~ R. Remark 4. From (9), it clearly appears that the polynomials Pi are homogeneous in each ~i- To eliminate the trivial solutions ~i = 0, the vectors ~1. . . . . ~p must be normalized. This can be done by setting one among their components equal to the unit. Hence, the actual number of unknowns appears to be m p - p . This means that m p - n variables among the components of the 7i's - the degrees of freedom of the pole assignment - can be chosen to parametrize the solutions. Remark 5. It is of practical importance to define the domain of this m p - n free parameters for which a real solution always exists. This delicate point will be discussed in the next section.

246

J.-F. Magni, C. Champetier / Pole assignment of linear systems

Remark 6. In the case where m + p > n, we have rap - n >I ( p - 1)(m - 1). This means that p - 1 among the ~i's can be taken as degrees of freedom [16,18]. F r o m (9), it is clear that the polynomial system (10) becomes a linear system of equations. Generic properties related to the existence of solutions in this case may be found in [15]. Remark 7. It is straightforward to show, by duality, that the pole assignment can be achieved by solving a set of n - m polynomial equations in rap - m u n k n o w n s .

4. An illus~ative example

Consider the system is given by:

A=

olo OOLO, 0 0

0 0

B=

0 0

[2o] ix o,

c=

0

o o o o 1 o o 0

0

1

ol .

Assume that A = A 1 U A 2 with A I = { - ½ , - 1 , - 2 } and A 2 = { - 1 , - 1 } . ~1, ~2 and ~3 will be normalized as foUows:

= [x

1]',

= [y

1]

=

lr.

The polynomial system (10) can be easily obtained by using a formal calculus package. It can be written as P o ( x , y , z ) = 8 y z - 8 x y - 48x + 18y - 12z - 147 4= 0,

(11)

y , z ) = 4 0 x y - 2 0 y z - 6 0 x z - 7x + 189y - 224z = 0,

(12)

P2( x , y , z ) = 5 6 x y + 4 y z - 6 0 x z + 137x + 105y - 188z + 441 = 0.

(13)

Pl(x,

The pole assignment reduces to finding the real solutions to this system. From standard elimination theory [4,11], the following polynomial system equivalent to (12)-(13), is obtained: Pl(x,

y , z ) = 4 0 x y + 2 0 y z - 6 0 x z - 7x + 189y - 224z = 0,

Ql(x,

y ) = (10x + 14)y 2 + ( - 1 0 x 2 + 29x + 217)y - 90x 2 - 609x - 1029 = 0,

where Q1 is the resultant of P~ and P2 with respect to z. The resolution of this system is performed upwards. Assume that x is chosen as free parameter. A real solution exists iff the discriminant of Q1 with respect to y is positive. This defines the .domain D of x in which the pole assignment can be achieved: D = {x e R I x4 + 30.2x 3 + 259x 2 + 878x + 1047 >/0}. The calculation of the free parameter domain in which real solutions exist gets more and more difficult as n - p increases. This problem has been theoretically solved by Arnon et al. [3]. Practical algorithms remain to be developed for dealing with the general case.

References [1] B.D.O.Anderson, A note on transmission zeros of a transfer function matrix, IEEE Trans. Automat. Control 21 (1980) 589-591. [2] B.D.O.Anderson, N.K. Boseand E.I. Jury, Output feedbackstabilization and related problems - solution via decisionmethods, IEEE Trans. Automat. Control 20 (1975) 53-66. [3] :D.$, Axnon, G.E. Collins and S. MeCailum, Cylindrical algebraic decomposition(part I and II), S I A M J. Comput. 13 (1984) 865-889,

J.-F. Magni, C. Champetier / Pole assignment of linear systems

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[4] A. Benailou, D.A. Mellichamp and D.E. Seborg, On the number of solutions of multivariate polynomial systems, IEEE Trans. Automat. Control 28 (1983) 224-227. [5] R.W. Brockett and C.I. Byrnes, On the algebraic geometry of output feedback pole placement map, Proc. of the XVIIIth IEEE Conference on Decision and Control (1979) 754-757. [6] C. Champetier and J.-F. Magni, Interpr6tatiou d'un cahier des charges en termes de valeurs-vecteurs propres, Rapport Interne DRET-DERA (1986). [7] E.J. Davison and S.H. Wang, On pole assignment of linear multivariable systems using output feedback, IEEE Trans. Automat. Control 20 (1975) 516-518. [8] L.R. Fletcher, Placement des valeurs propres pour les syst~mes lin~aires multivafiables, C.R. Acaa[ Sci. Paris Set. A 289 (1979) 499-501. [9] C. Giannakopoulos and N. Karcanias, Pole assignment of strictly proper and proper linear systems by constant output feedback, Internat. J. Control 42 (1985) 543-565. [10] R. Hermann and C.F. Martin, Applications of algebraic geometry to system theory Part I, IEEE Trans. Automat. Control 22 (1977) 19-25. [11] P.$. Kamat, Comments on "On the number of solutions of multivariable polynomial systems", IEEE Trans. Automat. Control 31 (1986) 796. [12] H. Kimura, Pole assignment by gain output feedback, IEEE Trans. Automat. Control 20 (1975) 509-516. []3] H. Kimura, A further result in the problem of pole assignment by output feedback, IEEE Trans. Automat. Control 22 (1977) 458-463. [14] J.-F. Magni, Placement de pgles par retour de sorties: une condition n~essaire et suffisante, Rapport Interne DERA (1985). [15] J.-F. Magni and C. Champetier, Commande modale par retour statique de sortie, Rapport Technique DRET-DERA (1987). [16] R.R. Mielke and S.R. Liberty, An eigeuvalue/eigenvector assignment algorithm using output feedback, in: Proc. of IEEE Southeastern Conference, Orlando, FL (1983). [17] J. O'Reily and M.M. Fahmy, The minimum number of degrees of freedom in state feedback control, lnternat. J. Control 41 (1985) 749-768. [18] G. Roppeneeker and J. O'Reilly, Parametric output feedback controller design, Proc. of the Xth World IFAC Congress, Munich (1987). [19] J.M. Schumacher, Compensator synthesis using (C, A, B)-palrs, IEEE Trans. Automat. Control 25 (1980) 1133-1138. [20] J.C. Willems and W.H. Hesselink, Generic properties of the pole placement problem, Proc. of the VIIth World IFAC Congress, Helsinki (1978). [21] J.C. Willems and C. Commault, Disturbance decoupling by measurement feedback with stability or pole placement, S I A M J. Control Optim. 19 (1981) 490-504. [22] W.M. Wonham, Linear Multivariable Control: A Geometric Approach (Springer, Berlin-New York, 1979).