On the complete controllability indexes assignment ...

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Proof of Theorem 3.3: The proof of the first part of the theorem is similar to that of Theorem 3.1, and is omitted. Next, we prove the second part of the theorem. Suppose that the state estimator (5) with G given by (6) is a nonfragile quadratic guaranteed cost estimator with cost matrix Q0 , then from Lemma 3E, (31) and (44) hold for some  > and

1

0

Qe

Q0 QT12

=

Q12 : Q2

+

From (44), direct algebraic manipulations yield AQa Qa AT 01 G2 CQa V1 < , where Qa Qa C T GT2 Rm Q0 Q12 T Q12 Q2 > . From [13, Lemma A.2.6] and V1 > , it follows has a stabilizing solution that the Riccati equation H1m P;  P > . By (44), we have T I 0 I Q12 Q201 Tm Qe0 ;  I 0 I Q12 Q201 01 G2 C Qb A Gb I V2 GT2 Rm 01 G2 C T Qb A Gb I V2 GT2 Rm 01 G2 CQb V1 Qb C T GT2 Rm 01 G2 V2 m G1 GT Gb < (45) Gb V2 V2 GT2 Rm 1  0 1 0 1 T where Gb 0 I Q12 Q2 G, Qb Q0 0 Q12 Q2 Q12 > . Denote 1 AQ QAT V 0 QC T CQ (46) H2m0 Q;  1 m where 01 G2 T I V2 GT2 Rm m 01 1 V2 V2 GT2 Rm01G2 V2 m G1 GT1 1 I V2 GT2 Rm01G2 0 G2T Rm01G2 : By (45) and completing the square, it follows H2m0 Qb ;  < , which further implies from [13, Lemma A.2.6] that H2m0 Q;  has a stabilizing solution Q > and Q < Qb  Q0 . By Lemma 3B, (27) and (46), it follows that H2m Q;  H2m0 Q;  . Q  P follows from (17), (24), and (27). Thus, the proof is complete.

+ +

+ 0 ( + = + + + + =

0

)

( + + ( +

+ ( +

(

)=

+ (

0 )=0

(

)

+

+

+

)

)

0

(

+

0

)

)

)

+ ( +

( +

)

+ =

1 =( +

=

0

0

1

+

)=

(

(

(

)

[10] A. Jadbabaie, T. Chaouki, D. Famularo, and P. Dorato, “Robust, nonfragile and optimal controller design via linear matrix inequalities,” in Proc. Amer. Control Conf., Philadelphia, PA, 1998, pp. 2842–2846. [11] B. N. Jain, “Guaranteed error estimation in uncertain systems,” IEEE Trans. Automat. Contr., vol. 20, pp. 230–232, 1975. [12] L. H. Keel and S. P. Bhattacharyya, “Robust, fragile, or optimal?,” IEEE Trans. Automat. Contr., vol. 42, pp. 1098–105, 1997. [13] H. W. Knobloch, A. Isidori, and D. Flockerzi, Topics in Control Theory. Basel, Switzerland: Birkhauser Verlag, 1993. [14] L. El Ghaoui and G. Calafiore, “Worst-case prediction under structured uncertainty,” in Proc. Amer. Control Conf., San Diego, CA, 1999, pp. 3402–3406. [15] I. R. Petersen, “A stabilization algorithm for a class of uncertain systems,” Syst. Contr. Lett., vol. 8, pp. 181–188, 1987. [16] I. R. Petersen and D. C. McFarlane, “Optimal guaranteed cost control and filtering for uncertain linear systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 1971–1977, 1994. [17] U. Shaked and C. E. de Souza, “Robust minimum variance filtering,” IEEE Trans. Signal Processing, vol. 43, pp. 2474–2483, 1995. [18] L. Xie and Y. C. Soh, “Robust Kalman filtering for uncertain systems,” Syst. Contr. Lett., vol. 2, pp. 123–129, 1994. [19] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1996.

0 )=0

)

On the Complete Controllability Indexes Assignment Problem Angel Herrera and Sabine Mondié Abstract—New solvability conditions for the simultaneous assignment by static-state feedback of both the controllability indexes of ( ) and the invariant factors of ( ) are given. These conditions enhance the fact that this problem is a generalization of the control structure problem of Rosenbrock and of the controllability indexes assignment problem of Heymann. The choice of a polynomial framework allows to obtain simpler proofs of the results than those previously published. Index Terms—Completion problems, controllability indexes, invariant factors, state feedback.

REFERENCES [1] J. Ackerman, Sampled-Data Control Systems. Berlin, Germany: Springer-Verlag, 1985. [2] B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice-Hall, 1990. [3] D. P. Bertsekas and I. B. Rhodes, “Recursive state estimation for a setmembership description of uncertainty,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 17–123, 1971. [4] F. Blanchini, R. Lo Cigno, and R. Tempo, “Control of ATM networks: fragility and robustness issues,” in Proc. Amer. Control Conf., Philadelphia, PA, 1998, pp. 2947–2851. [5] P. Bolzern, P. Colaneri, and G. De Nicolao, “Optimal robust filtering with time-varying parameter uncertainty,” Int. J. Contr., vol. 63, pp. 557–576, 1996. [6] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [7] D. Famularo, C. T. Abdallah, A. Jadbabais, P. Dorato, and W. M. Haddad, “Robust nonfragile LQ controllers: The static state feedback case,” in Proc. Amer. Control Conf., Philadelphia, PA, 1998, pp. 109–113. [8] W. M. Haddad and J. R. Corrado, “Non-fragile controller design via quadratic Lyapunov bounds,” in Proc. IEEE Conf. Dec. Contr., San Diego, CA, 1997, pp. 2678–2683. [9] W. M. Haddad and D. S. Bernstein, “Robust reduced-order, nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds,” IEEE Trans. Automat. Contr., vol. 33, pp. 591–595, 1988.

I. INTRODUCTION Let the time invariant linear system described by xt Ax t Bu t ; x 2 n ; u 2 m ; rankB = m

_( ) = ( ) + ( )

2 r

(1)

u(t) = F x(t) + Gv (t); v ; r  m (2) be given, where v (t) is the new input of the closed-loop system. Let (n+m)2m (N; D ) 2 [s]1 be a right coprime factorization of (sI 0 0 1 A) B , where D is column reduced, so that the column degrees of D

Manuscript received November 22, 1996; revised May 26, 1999. Recommended by Associate Editor K. Zhou. The authors are with the Departamento de Control Automático Departamento de Ingeniería Eléctrica, CINVESTAV-IPN. A.P. 14-740. 07000, México D.F. (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0018-9286(01)01019-4. 1 [s] denotes the ring of polynomials over , the field of reals. (s) stands for the field of rational functions over , while the rings of proper and strictly proper functions are denoted, respectively, by (s) and (s). Further, and [s]; . . ., denote the sets of m 2 n matrices having elements in , [s]; . . ., respectively, while f1g denotes a set having n elements. For M 2 , d M (d M ) denotes the degree of its ith column (row). A nonsingular m 2 m polynomial matrix is said to be column (row) reduced if its column (row) coefficient leading matrix is nonsingular. Units of the ring [s] are called unimodulars and those of the ring (s) bipropers.

0018–9286/01$10.00 © 2001 IEEE

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are the controllability indexes fci : ci  ci+1 gm of the pair (A; B ). The static-state feedback (2) is called regular if G is square and invertible, and nonregular if not. The complete controllability indexes assignment problem (CCIAP) consists of finding a nonregular static-state feedback (F; G), where G is only monic, such that (A + BF; BG) has a desired set of controllability indexes fc0i : ci0  ci0 +1 gr , r < m, and its noncontrollable part has a set of invariant factors fi (s): i ji+1 gm0r , where j denotes polynomial division. The controllability indexes and the invariant factors form a set of complete structural invariants of system (1). They are the skeleton over which the designer can build in order to achieve given control objectives. The limits of their modifications under the action of nonregular control laws are described by the solvability conditions obtained in this paper. They are crucial in understanding open problems such as model matching and input–output decoupling [12]. They describe the structure that can be obtained in the squaring down problem [6]. They are also a solid tool in the so-called integral design approach [7] that consist in selecting properly the sensors and actuators of the system at an early stage of the design process, according to a given control strategy. The CCIAP is indeed a generalization of the seminal problem solved in Rosenbrock Control Structure Theorem [11] (G = 0, i.e., r = 0) and of the Controllability Indexes Assignment Problem (CIAP) solved by Heymann [4] [the invariant factors fi (s)gm0r are not specified] and revisited in a polynomial framework in [8]. The problem under consideration has been previously studied in the framework of matrix pencil completion problems [1] and within a polynomial approach [2], [5], and [9] that allowed to eliminate redundant conditions. However these works do not exploit the fact that the problem is a generalization of the above mentioned seminal results, resulting in lengthy proofs and obscure solvability conditions. The polynomial formulation, recalled in Section II, shows the link between the problem considered here and the CIAP. This formulation leads to a new proof of the result of Heymann in Section II, and to new conditions and proofs for the CCIAP in Section III. These proofs are much simpler than those previously given, due to a wide and elegant use of the properties of unimodular and biproper matrices, and the solvability conditions show clearly the link with Rosenbrock and Heymann results.

diag

349

Proof—Necessity: Without any loss of generality, consider D = fsc gm . Then, from (3) diag

Vi DUi =

fsc gi Xi Yi

0

;

i = 1; 2; . . . ; r

(5)

where Vi is biproper, Ui is unimodular, Xi 2 i2(m0i) [s], and Yi 2 (m0i)2(m0i) [s]. Biproperness of Vi implies dc DUi = dc Vi DUi . In this way, dc DUi = cj0 ; j = 1; 2; . . . ; i; i = 1; 2; . . . ; r . Then, uji = 0 8 cj > ci0 , j 2 f1; 2; . . . ; mg. So, Ui has the following form i i U11 U12 i 0 U22 i 2 l 2i [s] and U i 2 (m0l )2(m0i) [s] is full row rank where U11 22 since i  li . So, there exists a unimodular (m 0 i) 2 (m 0 i) matrix Li such that U i11 U i12 I 0 Vi DUi i = Vi D 0 Li 0 Im0l c diag fs gi X i = 0 Yi i where the li 2 li matrix U 11 is unimodular, and (X Ti Y Ti )T = T T T T [Li (Xi Yi )] . In this way c i ~i D11 U i11 diag fs gi X Vi = 0 0 Y~i

Ui =

f c ; . . . ; cl g

i D11

= diag c1 ; T T T T [(Il 0i 0) (X i Y i )] .

where

2

~T and (X i

Y~iT )T

=

The rank of the right-hand side is equal to li , therefore, rank Y~i = li 0 i and the (m 0 i) 2 (li 0 i) matrix Y~i is full column rank since m 0 i  li 0 i. So, Lemma 2 implies that there exist a biproper matrix Wi 2 p(m0i)2(m0i) (s), a unimodular matrix Zi 2 (pl 0i)2(l 0i) [s], and a list of positive integers f i : i  i+1 gl 0i such that Wi Y~i Zi =

T (diag fs gl 0i 0) . Thus, we have

V^i

diag

D11 U^11

=

0

fsc gi 0

X^ i

diag fs gl 0i

0

0

where II. PREVIOUS RESULTS In this section, the polynomial formulation of the problem is recalled, and a new proof of the CIAP is presented. Lemma 1 [10]: There exists a nonregular state feedback (F; G) 2 m2(n+r) , rankG = r , which solves the CIAP if and only if there exist a unimodular matrix U 2 m2m [s] and a biproper matrix V 2 m2m (s) such that p

V DU

=

diag

fsci gr X 0

(3)

Y (m0r )2(m0r ) [s] has invariant

where X 2 r2(m0r) [s], and Y 2 factors fi (s)gm0r . This polynomial formulation and the extended use of unimodular and biproper matrices properties allow us to derive a new proof of the result of Heymann. Theorem 1: The CIAP has a solution if and only if i

where li = maxj fj jcj

j =1

cj0



l j =1

cj

 ci0 g, i = 1; 1 1 1 ; r.

(4)

V^i =

Im+i0l

0

Wi

0

i i = U 11 Zi U^11

Vi ;

and

X^ i = X~ i Zi : Then, partitioning V^ i as V^ i =

V^11i V^12i V^21i V^22i

i is a l 2 l matrix, we have that V ^ i Di U ^i where V^11 i i 21 11 11 = 0. In consei ^ i is biproper. In this way, quence V^21 = 0, and so V 11 i ^i V^11i D11 U11 =

diag

fsc gi 0

X~ i :

diag fs gl 0i

Taking determinants on both sides leads to 1 c 0 c +

2s (6) ki i is a biproper function whose relative degree is where bi (s) = det V^11 i ^ zero, and ki = det U11 is a non zero constant. Finally, computing the

bi (s) =

degrees on both sides of (6) gives (4).

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Sufficiency: By hypothesis c01

 k c g. Then, c0 = c 0 h + ~ j 1 1 l j =1 diagfsc

sc c s +c

111

U

g~l 

s

0 111 111 111 111 0 111 111 111 111 111 1 1 1 1 1 1 sc 01 0 0 1 1 1 0 sc 111 111 111 111 0 111 1 1 1 c1 1 01 1 1 1 1 0sch 111 s 111 0 s

sc cj

sc

0 01 0 sc B  111 111 0 111 sc 0 U

l c . Let ~l = min fkjc0  j 1 k 1 ~lj =1 01 cj , where h1  c~ . So l j =1

k j =1

k

III. MAIN RESULT This section is devoted to the obtention of solvability conditions for the solution of the CCIAP as well as of an explicit construction procedure. These conditions clearly show the role of the CIAP derived in the previous section, and give a good insight of how things work. Theorem 2: There exists a nonregular state feedback (F; G) which solves the CCIAP if and only if

j =1 where li

cj0 

j =1

cj ;

i = 1; 2; . . . ; r

hj +

j =1

B

l

hj +

i0k j =1

cj0



i0k

j =1

j



i j =1

j =1

(8)

Remark 1: Solvability conditions (7) and (8) reduce in a straightforward manner to well known particular cases. If r = 0, condition (7) vanishes and we recognize in (8) the conditions of Rosenbrock Control Structure Theorem. If the list fi (s)gm0r is not specified, condition (8) is less restrictive than (7), the conditions of Heymann for the CIAP. Proof—Necessity: The inequalities (7) necessarily hold because the problem solved in Theorem 1 is a subproblem of the one considered here. Now, it follows from (3) and Lemma 2 of the Appendix that there exist a biproper matrix V1 , a unimodular matrix U1 , and a proper matrix V12 such that

=

i j =1

i = 1; 2; . . . ; m

cj ;

X I 0 Y 0 U1 c ~ diag fs gr X 0 diag fsh gm0r

i j =1

i 

k j =1

hj +

i0k

deg(j );

j =1

i = 1; 2; . . . ; m

i = ci + di01 0 (ci+1 + di ) + ci+1 i = 1; 2; . . . ; m with d0 = 0: (12) If ci + di01  (ci+1 + di ) and i 2 = fci0 g, then i is constructed as

follows:

0

sc

0

c U sc +sd 0d 0 B

sc

U

0

sc i sc

(13)

B

where  and  stand for postmultiplication and premultiplication by appropriate unimodular and biproper matrices, respectively. Now, if ci + di01 > (ci+1 + di ) and i 2 fdeg i (s)gm0r , then such a term is constructed as a controllability index, since the biproper matrix B used in (13) does not exist. Note that in this case (12) implies that i > ci+1 , and so i > cj , j = 1; 2; . . . ; i. Then from (8)

j =1  2fc

j + i 

g

i j =1

i01

cj 0 

j =1

j



2fdeg  g

i j =1

cj :

This relation has the form (7). Now, define

f^ci0 : c^i0  c^i0 +1 g^rr = fci0 gr [ fi : i 2 fdeg i g&i > ci+1 g f^i (s): ^i j^i+1 gm0^r = fi : i 2 fdeg i g&t  ct+1 g

0

~ are diviswhere all the elements of column i = 1; 2; . . . ; m 0 r of X ible by sh . So, the degrees of the invariant factors of the polynomial matrices on the right hand side of (9) are f~i : ~i  ~i+1 gm obtained by reordering fci0 gr [ fhi gm0r . Therefore, it follows from Lemma 2

(11)

with equality for i = m, holds and the result follows from (10) and (11). Sufficiency: Define di = ij =1 j  ij =1 cj ; i = 1; 2; . . . ; m. Then

diag fsc gr

(9)

(10)

with equality for i = m. The list fhi gk [ fdeg(j )gi0k is a sublist of size i of the ordered list fi gm , therefore, the inequality

(7)

with equality for i = m, where fi : i  i+1 gm is the reordered list fci0 gr [ fdeg (i )gm0r .

I V12 0 V1

i = 1; 2; . . . ; m

cj ;

sc +d 0d

i = 1; 2; . . . ; m

cj ;

j =1

deg (j ) 

= maxj fci0  cj g, and i

i

with equality for i = m. Note also that because of Lemma 2 of the Appendix, we have that ij =1 deg(j )  ij =1 hj ; i = 1; 2; . . . ; m 0 r with equality for i = m 0 r . Therefore,

where  and  stand for postmultiplication and premultiplication by obvious unimodular and biproper matrices, respectively. Now, c20 is obl2 = mink fkjc10 + tained from fh1 ; c~l +1 ; c~l +2 ; 1 1 1 ; c~l ; g where ~ k 0 c2  j =1 cj g in the same way as we used fc1 ; 1 1 1 ; c~l g when we constructed c10 . The result follows by repeating r 0 1 times this process.

i

i c ; i = 1; 2; 1 1 1 ; m with of the Appendix that ij =1 ~j  j =1 j equality for i = m. Now, the sublist of the first i elements of f~i gm includes k  m 0 r elements of the list fhi gm0r and i 0 k elements of fci0 gr , i.e.,

(14) (15)

therefore,

i j =1 where ^ li

c^j0 

^l j =1

cj ;

i = 1; 2; . . . ; r^

= maxj fj : cj  c^i0 g, i = 1; 2; . . . ; r^.

(16)

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Construction of c^01 . Let t = minj fj : c^10  cj +1 + dj g; 1  t  ^ l1 , and c^10 = k , k 2 f1; 2; . . . ; mg. Then, since c10 > ct + dt01 , we have that c^10 = ct+1 + dt 0 "1 , 0  "1  ci+1 0 i . In this way

11 0 111 111 111 0 11 sc

sc

111 0 0 111 0 0 111 111 111 111 0 1 1 1 sc 0 0 1 1 1 0 sc

sc c s +d

U

BDU

sc

sc +d s^c

0 111 0 0 s 1 1 1 0 B 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0s " c s 0 0 1 1 1 sc

sc

0

where Y~ has invariant factors fsdeg(~ ) g~r . Note that some of the terms s~c s are invariant factors. Therefore, additional left and right permutations give

0

111

0s

0

351

:

+

(17) IV. CONCLUSION

Remark 2: The minor 1; 2; . . . ;

of

the

right

hand

t

111; t + 1

matrix in (17) has invariant factors g, since j  cj , j = 1; 2; . . . ; t + 1.

fs ; s ; . . . ; s 0; s Construction of c^2 Define

i1 = and

ci1 =

t+i01 ; i = 1; 2; . . . ; k 0 t i = k 0 t + 1; . . . ; m 0 t t+i ; t + "1 ; i = 1 : i = 2; . . . ; m 0 t ct+i

j =1

j1 

i j =1

cj1 ;

j = 1; 2; . . . ; m 0 t

In this note, new necessary and sufficient conditions for the solutions of the controllability indexes assignment and complete controllability indexes assignment problems are obtained. The interest of these conditions is that they reflect the fact that the problem is a generalization of the well known results of Rosenbrock and Heymann. The way these two results are combined is not so simple, and previous works on CCIAP did not clarified these seemingly evident facts. The use of the polynomial approach was decisive in obtaining an elegant and simple proof. These results can be generalized to the noncontrollable case as in [10]. APPENDIX

Let us show now that

i

0

X Y

where Y has invariant factors fsdeg( ) gr . Finally, [5, Lem. 7.2.3] implies that there exist a biproper matrix V and a unimodular matrix W , both of appropriate dimensions, such that V Y W has invariant factors fi gr . Properly collecting all the bipropers and unimodular matrices in this procedure leads to (3).

^

2; 3;

sc

diag

=

(18)

with equality for i = m 0 t. It follows from (14)–(15) that t+i  ct+i+1 , i = 1; 2; . . . ; k 0 t 0 1. This implies that ij =1 j1  i c1 ; i = 1; 2; . . . ; k 0 t. From (8) let i   i c j =1 j j =1 j j =1 j 1 t+1 c , where k  i  m. Then, from k + jt0  + (  + " ) = j t 1 =1 j =1 j where k = c~10 , we have

The following technical result links the Wiener–Hopf canonical form of a polynomial matrix to the conditions of Rosenbrock Control Structure Theorem. m2m [s] with Lemma 2: Given a nonsingular matrix D 2 invariant factors fi (s): i ji+1 gm , there exist a biproper matrix B 2 m2m (s), a unimodular matrix U 2 m2m [s], and a unique list fki : ki  ki+1 gm , named left Wiener–Hopf indexes, such that BDU = diag fsk gm . Moreover, the lists fi (s)gm and fki gm are related as follows:

i j =1

deg(i )



i j =1

kj ;

i = 1; m

(20)

i c1 ; i = k 0 t + This relation is equivalent to ij =1 j1  j =1 j i c1 0 i  1 ; 1 1; k 0 t + 2; . . . ; m 0 t. Define now di = j =1 j j =1 j i = 1; 2; . . . ; m 0 t. Then

with equality for i = m. Proof: There always exists a unimodular matrix U such that DU is column reduced with column degrees fki gm . Matrix DU ^ , with D0 can be written uniquely as DU = D0 diag fsk gr + D ^ ^ < ki nonsingular and D a polynomial matrix such that dc D ^ diag fs0k gr )diag fsk gr and the result therefore DU = (D0 + D ^ diag fs0k gr ): In addition, note that follows with B 01 := (D0 + D 0 1 k DU = B diag fs gr is a column reduced polynomial matrix with column degrees fki gm and invariant factors fi (s)gm . Then (20) follows from [5, Lem. 7.2.2].

i1 = ci1 + di101 0 (ci1+1 + di1 ) + ci1+1

REFERENCES

i j =t j 6=k

j

 (t + " ) + 1

i j =t+2

cj :

(19)

with d01 = 0. Now, apply the same procedure used to construct c~10 , with relations (18)–(19) along with the following definitions: t1 = 1 1 0 ^1 g; 1  t1  ^ l2 , and c^20 = k ; k1 2 minj fj : cj +1 + dj  c 0 1 1 f1; 2; . . . ; m 0 tg. So, c^2 = ct+1 + dt 0 "2 , 0  "2  ct1+1 0 t1 . The rest of the procedure is similar and we can find a biproper matrix B and a unimodular matrix U such that ~ U ~ BD

=

s~c

diag 0

X~ Y~

[1] I. Baragaña and I. Zaballa, “Column completion of a pair of matrices,” Linear Multilinear Alg., vol. 27, no. 4, pp. 243–273, 1990. [2] I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials. New York: Academic, 1982. [3] M. L. J. Hautus and M. Heymann, “Linear feedback: An algebraic approach,” SIAM J. Control Optimiz., vol. 7, no. 1, pp. 50–63, 1978. [4] M. Heymann, “Feedback simulation,” Int. J. Syst. Sci., vol. 12, no. 12, pp. 1477–1484, 1981. [5] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. [6] N. Karcanias and C. Giannakopoulos, “Necessary and sufficient conditions for zero assignment by constant squaring down,” Lin. Alg. Appl., vol. 122–124, no. 3, pp. 415–446, 1989.

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[7] N. Karcanias, “The selection of input and output schemes for a system and the model projection problems,” Kybernetika, vol. 39, pp. 585–596, 1994. [8] J. J. Loiseau and P. Zagalak, “Feedback simulation,” in Second IEEE Mediteranean Symp. New Directions Control Theory Applications, Chania, Greece, 1994. [9] S. Mondié, “Contribution à la modification de la structure des systèmes linéaires,” Ph.D. dissertation, Laboratoire d’automatique, Universitè de Nantes, France, Mexico, 1996.

[10] S. Mondié and J. J. Loiseau, “Simultaneous zeros and controllability indexes assignment through nonregular static state feedback,” in 36th Conf. Decision Control, San Diego, CA, USA, 1997. [11] H. H. Rosenbrock, State Space and Multivariable Theory. London, U.K.: Nelson, 1970. [12] P. Zagalak, V. Eldem, and K. Özcaldiran, “On a special case of the Morgan problem,” in IFAC Conf. System Structure Control, Nantes, France, July 8–10, 1998, p. 181.