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Systems & Control Letters 14 (1990) 267-273 267 North-Holland A polynomial matrix method for computing stable rational doubly coprime factorizations * Kenji SUGIMOTO Department of Information Technology, Faculty of Engineering, Okayama University, Okayama 700, Japan Yutaka YAMAMOTO Department of Applied Systems Science, Faculty of Engineering, Kyoto University, Kyoto 606, Japan Received 14 August 1989 Revised 4 December 1989 Abstract: This paper proposes a new method for computing stable rational doubly coprime factorizations from a given transfer matrix. In contrast to the well-known method which requires a state space representation, the proposed method makes full use of polynomial matrices, and the whole operation is carried out directly in the frequency domain. Furthermore, the paper clarifies the meaning of the obtained factorization as a controller by showing that this factorization coincides with the one by the state space method for a suitable choice of feedback and observer gains. Keywords: Polynomial matrix method; fractional representa- tion; proper stable factorization; Bezout identity; linear system theory. 1. Introduction Fractional representation by proper stable ra- tional matrices has been of great interest as a powerful tool for the analysis and synthesis of linear control systems (see, e.g., [6,13], and the references therein). In this approach we first need to compute the so-called doubly coprime factori- zation of a given system. As is well known, Nett et al. [11] gave this factorization by means of a state * An original version of this paper was presented at the IFAC Workshop on System Structure and Control: State-space and polynomial methods, which was held in Prague, Czechoslovakia during 25-27 September 1989. The first author is grateful to the C&C Foundation for their travel support. The second author is also supported in part by the Inamori Foundation. space representation of the system. Their method is based on a standard design procedure of control systems (i.e., a state feedback and a full order observer), and hence is easy to understand. In actual calculation, however, their method requires some redundant operations when the sys- tem is given in terms of a transfer matrix: we must find its realization, determine stabilizing feedback and observer gains, and then compute the eight transfer matrices which give a doubly coprime factorization. This process is indirect because find- ing the realization is not essential to the computa- tion. In this sense, it is more desirable to compute such factorizations directly in the frequency do- main. In this paper we derive a doubly coprime fac- torization via a certain set of polynomial matrices instead of state space representation. These oper- ations are carried out entirely in the frequency domain, and hence the algorithm is transparent. Furthermore, it turns out that the result coincides with the one by Nett et al. for a suitable choice of feedback and observer gains. Consequently our method makes it possible to compute a doubly coprime factorization equivalent to that of Nett et al. without recourse to the state space method. 2. Problem formulation Let us start by introducing a few notions on polynomial matrices [9,14]. Let Q be a square polynomial matrix. (We usually omit the inde- terminate s of rational functions in this paper.) Denote by [Q]c the matrix whose columns consist of the highest-degree coefficients of the respective columns of Q; Q is said to be column reduced (column proper) if [Q]e is invertible. [Q]r and row reducedness are defined similarly. Let W be a rational matrix which is not neces- sarily proper. We denote by [W]poly the poly- nomial part of W and by [W]s.p the strictly proper 0167-6911/90/$3.50 © 1990, Elsevier Science Publishers B.V. (North-Holland)

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268 K. Sugimoto, E Yamamoto /Computing stable doubly coprime factorizattons part of it. Naturally we always have W= [W]poly + [Wkp Now let Z be a transfer matrix of a given system; i.e., Z is proper rational. Our goal is to obtain a doubly coprime factorization of Z; i.e., to obtain a set of proper stable rational matrices N, D, X, Y, N, D, X, and 17 such that Z = ND -1 = D-'57, (1) (2) An existing method for computing these matrices in the frequency domain is that of resorting to solutions of the following equations, similar to (1) and (2), with respect to polynomial matrices P, Q, U, V, /3, Q, u, and 17: Z = PQ -1 = 0-1/3 , (3) t(0 f)=i ,4, -/30P Q is column reduced and 0 is row reduced. (5) The method is as follows: (i) change the variable s of Z into X by s = (1- X)/h; (ii) compute the polynomial matrices (3)-(5); and then (iii) return to s. Here, (4) is called the generalized Bezout identity and plays an important role in the poly- nomial matrix approach [9]. The matrices satisfy- ing (3)-(5) can be obtained via, for example, ele- mentary transformations [9]. Furthermore, due to the recent work by Beelen et al., it has become feasible to compute them in a numerically reliable way [2,3,4]. Although the above method is conceptually simple, it is not attractive in practice because changing variables requires much calculation. Another drawback is that the meaning of the obtained factorization is not clearly understood. Recall that, in contrast, the factorization by Nett et al. corresponds to a concrete controller com- posed of a state feedback and a full order ob- server. In this paper, we present another effective method for deriving a doubly coprime factoriza- tion from the polynomial matrices satisfying (3)- (5). The method requires only a few number of multiplications and divisions of the polynomial matrices. Also, it is shown that the obtained fac- torization has a clear meaning as a controller. 3. Main result Consider the polynomial matrices (3)-(5). It is well known in the pole-shifting theory [14, §7.2] that, for the matrix Q, there exists a polynomial matrix Qo of the same size such that: (C1) The column degrees of Q0 are equal to the respective ones of Q; (C2) [Qo],. = [Q],; (C3) det Qo(s) 4:0 for Re s >_ O. Similarly, for 0 there exists a polynomial ma- trix 0o of the same size such that: (R1) The row degrees of Q0 are equal to the respective ones of Q; (R~) [O0]r = [0]r; (R3) det Q0(s) ~ 0 for Re s >__ 0. It is not difficult at all to find such Q0 and 0o. We will give concrete examples of Q0 and Q0 later in an algorithm. With these notions, we are able to show the following simple fact. Lemma 3.1. Suppose that the polynomial matrices in (3)-(5) are obtained. Then, for any polynomial matrices Qo and 00 satisfying (C1)-(C3) and (R1)-(R3) respectively, the set of matrices N := PQo 1 D :: QQo 1 , (6) := 0 1b, b ..= 0o10 (7) gives a proper stable factorization (1) of Z. Proof. These matrices clearly satisfy (1). They are stable by (C3) and (R3). Also we have D(s) ~ I, U(s) --* Z(oc) < oc, (8) b(s)--,I, 57(s) --, Z(oo) < o~ (9) as s --, oo, which means they are proper. [] Note that Lemma 3.1 is a generalization of the well-known method for scalar systems, where the denominator and the numerator of a transfer function are divided by (s + 1) n so as to obtain a coprime factorization. It is more difficult, how- ever, to derive a doubly coprime factorization corresponding to (6), (7). To this end, let us pre- and postmultiply (4) respectively by diag(Q0, Qo I ) and diag(Qo a, 00)- Then, in view of (6), (7) we have =I. (10) -57 VPo ]

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K. Sugimoto, Y. Yamamoto / Computing stable doubly coprime factorizations 269 Although this identity contains N, D, ~7, and /3, there still remain four polynomial matrices in it. We want to replace the latter with proper stable ones by some elementary transformations. This is done in the following theorem. Theorem 3.2. Under the same hypotheses as in Lemma 3.1, there exists a polynomial matrix R such that X := Qo U - RD (11) is strictly proper. Furthermore, the matrices X and Y := Qo V + R~[, I7 := l~00 + NR, (12) 2.'= OO.o - DR, together with (6), (7) give a doubly coprime factori- zation (1), (2) of Z. Proof. We starty by showing the existence of R. Define QoUD- 1 = ~ + F, ~:= [QoS/~-llpoly, F:= [Qog/~-llsp (13) Then we have Qo U = ~b + F[?. (14) Here, FD is strictly proper, since F is strictly proper and b is proper. Therefore, in view of (14), it is enough to take R .'= ~ (and then X = F/)). Next we have (2) by pre- and postmultiplying (10) respectively by (0 /R) and (/ R). So the proof will be complete if the matrices in (11) and (12) are proper stable. The poles of these are the zeros of det Q0 and det Q0, and hence they are stable by (C1)-(C3) and (R1)-(R3). So we have only to show r(00) = I, 17(oo) = I, (15) x(00) =0, 2(00) =0. We have already shown X(00) = 0. In view of (2), we have YD + XN = I. Taking s ~ 00 in this iden- tity then yields Y(00) = I. Therefore we obtain -N(00) 6(00) -Z(00) I " Then we have .g(00)= 0 and 17(00)= I again by (2). We have thus shown (15). [] Choosing in particular a simple form of Q0 and Q0 in Theorem 3.2, we have an algorithm for computing a doubly coprime factorization as fol- lows. Algorithm. Step 1. Denote by o 1 ..... o m the column de- grees of Q, and by ~1 ..... ~-p the row degrees of Q. Assigning poles at, say -1, take Oo:=[O]cdiag((s+l) a', .... (s+l)°m), (17) Q0 := diag((s + 1)",..., (s + 1)'P)[0] r . (18) Step 2. Compute Qo I and Qo 1. These are ob- tained simply by inverting numerical matrices and then multiplying them by diagonal rational matrices. Substitute them into (6), (7) and obtain N, D, N, and /). Step 3. Define I" := max(r/). Then (s + 1)~/) is a polynomial matrix by (7) and (18). Since/5(00) = I, we can write (s+ l)~D=Is~ +Als~-l + ... +A~. (19) Therefore, by means of division of polynomial matrices (see [7,5], and the references therein), we can readily compute polynomial matrices R and E such that (s + 1)'QoU = R(s + 1)'D + E, (20) and each entry of E is of degree less than T. Let X.'= (s + 1)-'E. Step 4. Compute Y, ,g, and I 7 by (12). 4. Connection with the state space method In this section we discuss a connection between the method of Nett et al. [11] and ours. To begin with, recall the following: if a minimal realization is fixed, the method of Nett et al. does not cover all possible doubly coprime factorizations of Z by adjusting feedback and observer gains. For exam- ple, suppose that we have obtained a right coprime factorization Z = ND -1 by their method. Then the pair s+3 D' s+3 N':=~N and :=~D

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270 K. Sugimoto, Y, Yamamoto / Computing stable doubly coprime factorizations apparently gives another right coprime factoriza- tion of Z. As is already known, however, there exist no feedback and observer gains which give this factorization, since N' and D' are of degrees larger than N and D. It is also known that there exist doubly coprime factorizations of degrees less than N and D [12]. In view of this fact, we may ask the following: How wide is the class of doubly coprime factoriza- tions obtained by our method, compared to that of Nett et al.? In this section we answer this question by showing that the factorizations by these two different methods are in one-to-one correspondence, and hence the two sets are identi- cal. Lemma 4.1 (Nett et al. [11]). Let Z be a proper rational matrix, and let (F, G, H) be a minimal realization of Z - Z(og). Select a feedback gain K and an observer gain L such that the eigenvalues of F- GK and F- LH all have negative real parts. Then the following matrices give a doubly coprime factorization of Z: D'=I-K(sI-F+GK) 1G, (21a) N'= H(sI- F+ GK)-1G + Z(oe)D', X' = K(sI- F+ LH) 1L, Y'=I+K(sI-F+LH) 'G (21b) D' = I- H(sI- F+ LH)-IL, (21c) N'= H(sI- F+ LH)-'G + D'Z(~), 2' = K(sl- F+ GK)-IL, Y' = I + H(sI- F+ GK)-IL (21d) - z(oo)2'. Theorem 4.2. Suppose that, by taking Qo and Qo in (C1)-(C3) and (R1)-(R3), a doubly coprime factorization is computed according to Theorem 3.2. Then there exist unique K and L such that the obtained matrices (6), (7), (11) and (12) are equal to those given by (21). Conversely, for any stabiliz- ing K and L of compatible sizes there exist unique Qo and Qo such that the matrices in (21) are equal to those in (6), (7), (11) and (12). Proof. Take Q0 in (C1)-(C3). Then it is known [14] that these exists a unique feedback gain K such that QoQ l = l + K(sl_ F) 1G. (22) The right-hand side is known as the return dif- ference matrix. As is also known, its inverse is the sensitivity matrix and QQol=I-K(sI-F+GK) 1G. But the left-hand side is D by the definition (6), and the right-hand side is D' in (21). Thus we obtain D = D'. Then it follows that N = ZD = ZD'= N'. We can show similarly that b =/)' and 37 = N' for some L. To show X = X', we claim that F defined by (13) satisfies F=K(sI-F) 'L. (23) This is shown by using the following three identi- ties. First, we have from the generalized Bezout identity (4). In view of its (1, 2) block, we have QU= UQ. (24) Secondly, it is known [9] that there exists a poly- nomial matrix S such that (sl-F) 1G=SQ -~. (25) Finally we also have H(sl-F) -I=Q_ 1~ (26) for some polynomial matrix S ~. Now we are ready to show (23) as follows. We have QoUD- ' = QoQ-1QU [ I + H( sI - F )- ' L ] =[I+K(sl-F)-tG]QU [I+H(sl-F)-'L] (by (22)) =QU+K(sI-F) 'GQU + QUH(sI- F) 1L + K(sl- F) 1GQUH(sI- F) IL. (27)

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K. Sugimoto, Y. Yamamoto / Computing stable doubly coprime factorizations 271 The first term of the right-hand side is a poly- nomial matrix. And so is the second term by (25). The third term is QUH(sI- F)-aL = O0.O-1SL = OSL by (24) and (26), and hence it is also a polynomial matrix. As a result we obtain r= [eoUb-lls p = [r(~I- F)-'GQUH(sI- V)-'L]s.p. (by (27)) =[KSUH(sI-F)-aL]s.p. (by (25)) = KA(sI- F)-aL, (28) where A is the remainder of S UH divided by (sI - F): sun = A(sI - F) + A, A being a polynomial matrix. Therefore (23) holds if A = I. To show this, pre- and postmultiply the Bezout identity VQ + UP = I (derived from (4)) by S and Q-1, respec- tively. Then we have SV + SUH(sl- F)-'G = SQ -1 = (s[-- F)-IG In view of the strictly proper part of this, we obtain A(sI- F)-IG = (sI-- F)-'G. Hence A = I holds, since (F, G) is reachable. In view of (28), we obtain (23). Now we are able to show the following. We compute X = F/) (by (11) and (14)) =K(sI-F)-'L [I-H(sI-F+LH)-aL] (by(23)) = K(sI- F+ LH)-aL S t ' Y= [I- XN]D-' = [I- X'N']D '-1 = Y'. Finally, we have -1 -1 (o ,) --N' D' N' I 7' ' and hence 2 = X' and i7 = i7,. The converse is also shown similarly. [] 5. An example Let us compute a doubly coprime factorization of the following transfer matrix [10] by our al- gorithm: 1 (s(s + 1) 2 s(s- 1) 2 ) Z=s(s_l)(s+l) I s+l (s+l)(s-1) " First, the matrices in (3)-(5) are computed as e= 's(s+ a) 1 O = s(s- 1) 0 0.5 U= -0.5 V-- ( -0.5 0.5 /3= (s+ 31 -(s-l)], s+l ] -s+l),s s-5) s--1 ' 4s) 0 s(s- 1) ' 0=(0.5s-0.5 -s+2) -0.5 1 ' 1~= ( -0.5s s-l) 0 0 " Then the computation goes as follows: Step 1. Take the polynomial matrices Q°:=[Q]c( (s+l)20 s+10 ) and 00,=((s+l) 0 ) 0 (s+l) a [Qlr, where we have 1 -1)and I ]r=(1 -4) [Ql~=(o 1

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272 K. Sugimoto, E Yamamoto / Computing stable doubly coprime factorizations Step 2. Compute the coprime factorization (6), (7) as _ 1 (s(s+ 1)(s + 1)(s- 2)) N = PQo 1 (s + 1) ------5 1 s + 2 ' D=QQol 1 {s(s- 1) -(s-l) (s + 1) 2 0 (s + 1) 2 A7 = 0o'P __ 1 (s2+4s+7 (s + 3)(s- 3)) (s + 1) 2 1 s -- 1 ' 1 (s+l) 2 -8s ) b=0o10= (s+l) 2 0 s(s-1) " Step 3. By division of polynomial matrices, we obtain /0.5(s + 1)(s + 2) --S3--5S2--6S+5) R = 1 -0.5(s + 1) (s + 1)(s + 3) " Step 4. Finally we have 1 (0 15s+ 1) X- (s + 1) 2 0 0 ' 1 (s2+5s+11 s-13 t, Y- (s+ 1) -----~k 0 (s+ 1)2J 1 (0 15s+ 1) = (s + 1) 2 0 0 ' = __ 1 ((s+1)2 -7(s+l) ) (s + 1) 2 0 S 2 + 5s + 11 " 6. Conclusion The authors wish to thank the reviewers who suggested various improvements. Notes After the submission of this paper the authors became aware of the existence of the paper by Hippe [8] which studies a similar problem. Let us compare these two papers here. Hippe [8] studies the structure of (full order) observer-based controllers in the frequency do- main, and reaches a nice parametrization of all such controllers. A main goal there is to clarify the degree of freedom in this design. On the other hand, the present paper aims at deriving doubly coprime factorizations from the polynomial matrices in (3)-(5) in a natural way, and then computing one such factorization as easily as pos- sible. This has been done by taking the simplest denominators Q0 and Q0 in the Algorithm of Section 3. From a technical viewpoint, the both methods have several similarities. These methods will give the same factorization if we start with the same polynomial matrices. In its derivation, however, there is one remarkable difference. Hippe [8] uses the matrices V 1, V 2 obtained from solutions of the two independent Diophantine equations (8.1) and (9.1). On the contrary, in the present paper we can compute the same doubly coprime factorization by using only the single polynomial matrix R; see (11) and (12). This is because we have started with the generalized Bezout identity (3), (4) instead of the individual Diophantine equations. As a result, the present approach reduces both theoretical and algorithmic complexity. We have proposed a new method for comput- ing a doubly coprime factorization of a given transfer matrix. It should be noted that Antsaklis [1] has also studied a connection between (right or left) polynomial factorizations and proper stable ones (but not doubly coprime). Our method can be regarded as a generalization of his result to the doubly coprime factorization. Furthermore, the factorization by our method has an advantage in that it corresponds uniquely to that of Nett et al, [11]. References [1] P.J. Antsaklis, Proper stable transfer matrix factorizations and internal systems descriptions, IEEE Trans. Automat. Control 31 (1986) 634-638. [2] Th. G.J. Beelen, New algorithms for computing the Kronecker structure of a pencil with applications to sys- tems and control theory, Ph.D. Thesis, Eindhoven Univer- sity of Technology, Netherlands (1987). [3] Th. G.J. Beelen and G.W. Veltkamp, Numerical computa- tion of a coprime factorization of a transfer function matrix, Systems Control Lett. 9 (1987) 281-288.

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K. Sugimoto, Y. Yamamoto / Computing stable doubly coprime factorizations 273 [4] Th. G.J. Beelen and P. van Dooren, A pencil approach for embedding a polynomial matrix into a unimodular matrix, SlAM J. Matrix Anal. Appl. 9 (1988) 77-89. [5] B. Codenotti and G. Lotti, A fast algorithm for the division of two polynomial matrices, IEEE Trans. Auto- mat. Control 34 (1989) 446-448. [6] B.A. Francis, A Course in H~ Control Theory (Springer, New York, 1987). [7] F.R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959). [8] P. Hippe, Parametrization of the full-order compensator in the frequency domain, lnternat. J. Control 48 (1988) 1583-1603. [9] T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, N J, 1980). [10] H. Maeda, Design theory for linear control systems II (in Japanese), Systems and Control 31 (1987) 728-734. [11] C.N. Nett, C.A. Jacobson, and M.J. Balas, A connection between state-space and doubly coprime fractional repre- sentations, IEEE Trans. Automat. Control 29 (1984) 831- 832. [12] T. Sugie and T. Ono, On doubly coprime factorizations, Linear Algebra Appl. 122/123/124 (1989) 681-696. [13] M. Vidyasagar, Control System Synthesis: A Factorization Approach (MIT Press, Cambridge, MA, 19850. [14] W.A. Wolovich, Linear Multivariable Systems (Springer, New York, 1974).

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Systems & Control Letters 14 (1990) 267-273 267 North-Holland A polynomial matrix method for computing stable rational doubly coprime factorizations * Kenji SUGIMOTO Department of Information Technology, Faculty of Engineering, Okayama University, Okayama 700, Japan Yutaka YAMAMOTO Department of Applied Systems Science, Faculty of Engineering, Kyoto University, Kyoto 606, Japan Received 14 August 1989 Revised 4 December 1989 Abstract: This paper proposes a new method for computing stable rational doubly coprime factorizations from a given transfer matrix. In contrast to the well-known method which requires a state space representation, the proposed method makes full use of polynomial matrices, and the whole operation is carried out directly in the frequency domain. Furthermore, the paper clarifies the meaning of the obtained factorization as a controller by showing that this factorization coincides with the one by the state space method for a suitable choice of feedback and observer gains. Keywords: Polynomial matrix method; fractional representa- tion; proper stable factorization; Bezout identity; linear system theory. 1. Introduction Fractional representation by proper stable ra- tional matrices has been of great interest as a powerful tool for the analysis and synthesis of linear control systems (see, e.g., [6,13], and the references therein). In this approach we first need to compute the so-called doubly coprime factori- zation of a given system. As is well known, Nett et al. [11] gave this factorization by means of a state * An original version of this paper was presented at the IFAC Workshop on System Structure and Control: State-space and polynomial methods, which was held in Prague, Czechoslovakia during 25-27 September 1989. The first author is grateful to the C&C Foundation for their travel support. The second author is also supported in part by the Inamori Foundation. space representation of the system. Their method is based on a standard design procedure of control systems (i.e., a state feedback and a full order observer), and hence is easy to understand. In actual calculation, however, their method requires some redundant operations when the sys- tem is given in terms of a transfer matrix: we must find its realization, determine stabilizing feedback and observer gains, and then compute the eight transfer matrices which give a doubly coprime factorization. This process is indirect because find- ing the realization is not essential to the computa- tion. In this sense, it is more desirable to compute such factorizations directly in the frequency do- main. In this paper we derive a doubly coprime fac- torization via a certain set of polynomial matrices instead of state space representation. These oper- ations are carried out entirely in the frequency domain, and hence the algorithm is transparent. Furthermore, it turns out that the result coincides with the one by Nett et al. for a suitable choice of feedback and observer gains. Consequently our method makes it possible to compute a doubly coprime factorization equivalent to that of Nett et al. without recourse to the state space method. 2. Problem formulation Let us start by introducing a few notions on polynomial matrices [9,14]. Let Q be a square polynomial matrix. (We usually omit the inde- terminate s of rational functions in this paper.) Denote by [Q]c the matrix whose columns consist of the highest-degree coefficients of the respective columns of Q; Q is said to be column reduced (column proper) if [Q]e is invertible. [Q]r and row reducedness are defined similarly. Let W be a rational matrix which is not neces- sarily proper. We denote by [W]poly the poly- nomial part of W and by [W]s.p the strictly proper 0167-6911/90/$3.50 © 1990, Elsevier Science Publishers B.V. (North-Holland)

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268 K. Sugimoto, E Yamamoto /Computing stable doubly coprime factorizattons part of it. Naturally we always have W= [W]poly + [Wkp Now let Z be a transfer matrix of a given system; i.e., Z is proper rational. Our goal is to obtain a doubly coprime factorization of Z; i.e., to obtain a set of proper stable rational matrices N, D, X, Y, N, D, X, and 17 such that Z = ND -1 = D-'57, (1) (2) An existing method for computing these matrices in the frequency domain is that of resorting to solutions of the following equations, similar to (1) and (2), with respect to polynomial matrices P, Q, U, V, /3, Q, u, and 17: Z = PQ -1 = 0-1/3 , (3) t(0 f)=i ,4, -/30P Q is column reduced and 0 is row reduced. (5) The method is as follows: (i) change the variable s of Z into X by s = (1- X)/h; (ii) compute the polynomial matrices (3)-(5); and then (iii) return to s. Here, (4) is called the generalized Bezout identity and plays an important role in the poly- nomial matrix approach [9]. The matrices satisfy- ing (3)-(5) can be obtained via, for example, ele- mentary transformations [9]. Furthermore, due to the recent work by Beelen et al., it has become feasible to compute them in a numerically reliable way [2,3,4]. Although the above method is conceptually simple, it is not attractive in practice because changing variables requires much calculation. Another drawback is that the meaning of the obtained factorization is not clearly understood. Recall that, in contrast, the factorization by Nett et al. corresponds to a concrete controller com- posed of a state feedback and a full order ob- server. In this paper, we present another effective method for deriving a doubly coprime factoriza- tion from the polynomial matrices satisfying (3)- (5). The method requires only a few number of multiplications and divisions of the polynomial matrices. Also, it is shown that the obtained fac- torization has a clear meaning as a controller. 3. Main result Consider the polynomial matrices (3)-(5). It is well known in the pole-shifting theory [14, §7.2] that, for the matrix Q, there exists a polynomial matrix Qo of the same size such that: (C1) The column degrees of Q0 are equal to the respective ones of Q; (C2) [Qo],. = [Q],; (C3) det Qo(s) 4:0 for Re s >_ O. Similarly, for 0 there exists a polynomial ma- trix 0o of the same size such that: (R1) The row degrees of Q0 are equal to the respective ones of Q; (R~) [O0]r = [0]r; (R3) det Q0(s) ~ 0 for Re s >__ 0. It is not difficult at all to find such Q0 and 0o. We will give concrete examples of Q0 and Q0 later in an algorithm. With these notions, we are able to show the following simple fact. Lemma 3.1. Suppose that the polynomial matrices in (3)-(5) are obtained. Then, for any polynomial matrices Qo and 00 satisfying (C1)-(C3) and (R1)-(R3) respectively, the set of matrices N := PQo 1 D :: QQo 1 , (6) := 0 1b, b ..= 0o10 (7) gives a proper stable factorization (1) of Z. Proof. These matrices clearly satisfy (1). They are stable by (C3) and (R3). Also we have D(s) ~ I, U(s) --* Z(oc) < oc, (8) b(s)--,I, 57(s) --, Z(oo) < o~ (9) as s --, oo, which means they are proper. [] Note that Lemma 3.1 is a generalization of the well-known method for scalar systems, where the denominator and the numerator of a transfer function are divided by (s + 1) n so as to obtain a coprime factorization. It is more difficult, how- ever, to derive a doubly coprime factorization corresponding to (6), (7). To this end, let us pre- and postmultiply (4) respectively by diag(Q0, Qo I ) and diag(Qo a, 00)- Then, in view of (6), (7) we have =I. (10) -57 VPo ]

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K. Sugimoto, Y. Yamamoto / Computing stable doubly coprime factorizations 269 Although this identity contains N, D, ~7, and /3, there still remain four polynomial matrices in it. We want to replace the latter with proper stable ones by some elementary transformations. This is done in the following theorem. Theorem 3.2. Under the same hypotheses as in Lemma 3.1, there exists a polynomial matrix R such that X := Qo U - RD (11) is strictly proper. Furthermore, the matrices X and Y := Qo V + R~[, I7 := l~00 + NR, (12) 2.'= OO.o - DR, together with (6), (7) give a doubly coprime factori- zation (1), (2) of Z. Proof. We starty by showing the existence of R. Define QoUD- 1 = ~ + F, ~:= [QoS/~-llpoly, F:= [Qog/~-llsp (13) Then we have Qo U = ~b + F[?. (14) Here, FD is strictly proper, since F is strictly proper and b is proper. Therefore, in view of (14), it is enough to take R .'= ~ (and then X = F/)). Next we have (2) by pre- and postmultiplying (10) respectively by (0 /R) and (/ R). So the proof will be complete if the matrices in (11) and (12) are proper stable. The poles of these are the zeros of det Q0 and det Q0, and hence they are stable by (C1)-(C3) and (R1)-(R3). So we have only to show r(00) = I, 17(oo) = I, (15) x(00) =0, 2(00) =0. We have already shown X(00) = 0. In view of (2), we have YD + XN = I. Taking s ~ 00 in this iden- tity then yields Y(00) = I. Therefore we obtain -N(00) 6(00) -Z(00) I " Then we have .g(00)= 0 and 17(00)= I again by (2). We have thus shown (15). [] Choosing in particular a simple form of Q0 and Q0 in Theorem 3.2, we have an algorithm for computing a doubly coprime factorization as fol- lows. Algorithm. Step 1. Denote by o 1 ..... o m the column de- grees of Q, and by ~1 ..... ~-p the row degrees of Q. Assigning poles at, say -1, take Oo:=[O]cdiag((s+l) a', .... (s+l)°m), (17) Q0 := diag((s + 1)",..., (s + 1)'P)[0] r . (18) Step 2. Compute Qo I and Qo 1. These are ob- tained simply by inverting numerical matrices and then multiplying them by diagonal rational matrices. Substitute them into (6), (7) and obtain N, D, N, and /). Step 3. Define I" := max(r/). Then (s + 1)~/) is a polynomial matrix by (7) and (18). Since/5(00) = I, we can write (s+ l)~D=Is~ +Als~-l + ... +A~. (19) Therefore, by means of division of polynomial matrices (see [7,5], and the references therein), we can readily compute polynomial matrices R and E such that (s + 1)'QoU = R(s + 1)'D + E, (20) and each entry of E is of degree less than T. Let X.'= (s + 1)-'E. Step 4. Compute Y, ,g, and I 7 by (12). 4. Connection with the state space method In this section we discuss a connection between the method of Nett et al. [11] and ours. To begin with, recall the following: if a minimal realization is fixed, the method of Nett et al. does not cover all possible doubly coprime factorizations of Z by adjusting feedback and observer gains. For exam- ple, suppose that we have obtained a right coprime factorization Z = ND -1 by their method. Then the pair s+3 D' s+3 N':=~N and :=~D

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270 K. Sugimoto, Y, Yamamoto / Computing stable doubly coprime factorizations apparently gives another right coprime factoriza- tion of Z. As is already known, however, there exist no feedback and observer gains which give this factorization, since N' and D' are of degrees larger than N and D. It is also known that there exist doubly coprime factorizations of degrees less than N and D [12]. In view of this fact, we may ask the following: How wide is the class of doubly coprime factoriza- tions obtained by our method, compared to that of Nett et al.? In this section we answer this question by showing that the factorizations by these two different methods are in one-to-one correspondence, and hence the two sets are identi- cal. Lemma 4.1 (Nett et al. [11]). Let Z be a proper rational matrix, and let (F, G, H) be a minimal realization of Z - Z(og). Select a feedback gain K and an observer gain L such that the eigenvalues of F- GK and F- LH all have negative real parts. Then the following matrices give a doubly coprime factorization of Z: D'=I-K(sI-F+GK) 1G, (21a) N'= H(sI- F+ GK)-1G + Z(oe)D', X' = K(sI- F+ LH) 1L, Y'=I+K(sI-F+LH) 'G (21b) D' = I- H(sI- F+ LH)-IL, (21c) N'= H(sI- F+ LH)-'G + D'Z(~), 2' = K(sl- F+ GK)-IL, Y' = I + H(sI- F+ GK)-IL (21d) - z(oo)2'. Theorem 4.2. Suppose that, by taking Qo and Qo in (C1)-(C3) and (R1)-(R3), a doubly coprime factorization is computed according to Theorem 3.2. Then there exist unique K and L such that the obtained matrices (6), (7), (11) and (12) are equal to those given by (21). Conversely, for any stabiliz- ing K and L of compatible sizes there exist unique Qo and Qo such that the matrices in (21) are equal to those in (6), (7), (11) and (12). Proof. Take Q0 in (C1)-(C3). Then it is known [14] that these exists a unique feedback gain K such that QoQ l = l + K(sl_ F) 1G. (22) The right-hand side is known as the return dif- ference matrix. As is also known, its inverse is the sensitivity matrix and QQol=I-K(sI-F+GK) 1G. But the left-hand side is D by the definition (6), and the right-hand side is D' in (21). Thus we obtain D = D'. Then it follows that N = ZD = ZD'= N'. We can show similarly that b =/)' and 37 = N' for some L. To show X = X', we claim that F defined by (13) satisfies F=K(sI-F) 'L. (23) This is shown by using the following three identi- ties. First, we have from the generalized Bezout identity (4). In view of its (1, 2) block, we have QU= UQ. (24) Secondly, it is known [9] that there exists a poly- nomial matrix S such that (sl-F) 1G=SQ -~. (25) Finally we also have H(sl-F) -I=Q_ 1~ (26) for some polynomial matrix S ~. Now we are ready to show (23) as follows. We have QoUD- ' = QoQ-1QU [ I + H( sI - F )- ' L ] =[I+K(sl-F)-tG]QU [I+H(sl-F)-'L] (by (22)) =QU+K(sI-F) 'GQU + QUH(sI- F) 1L + K(sl- F) 1GQUH(sI- F) IL. (27)

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K. Sugimoto, Y. Yamamoto / Computing stable doubly coprime factorizations 271 The first term of the right-hand side is a poly- nomial matrix. And so is the second term by (25). The third term is QUH(sI- F)-aL = O0.O-1SL = OSL by (24) and (26), and hence it is also a polynomial matrix. As a result we obtain r= [eoUb-lls p = [r(~I- F)-'GQUH(sI- V)-'L]s.p. (by (27)) =[KSUH(sI-F)-aL]s.p. (by (25)) = KA(sI- F)-aL, (28) where A is the remainder of S UH divided by (sI - F): sun = A(sI - F) + A, A being a polynomial matrix. Therefore (23) holds if A = I. To show this, pre- and postmultiply the Bezout identity VQ + UP = I (derived from (4)) by S and Q-1, respec- tively. Then we have SV + SUH(sl- F)-'G = SQ -1 = (s[-- F)-IG In view of the strictly proper part of this, we obtain A(sI- F)-IG = (sI-- F)-'G. Hence A = I holds, since (F, G) is reachable. In view of (28), we obtain (23). Now we are able to show the following. We compute X = F/) (by (11) and (14)) =K(sI-F)-'L [I-H(sI-F+LH)-aL] (by(23)) = K(sI- F+ LH)-aL S t ' Y= [I- XN]D-' = [I- X'N']D '-1 = Y'. Finally, we have -1 -1 (o ,) --N' D' N' I 7' ' and hence 2 = X' and i7 = i7,. The converse is also shown similarly. [] 5. An example Let us compute a doubly coprime factorization of the following transfer matrix [10] by our al- gorithm: 1 (s(s + 1) 2 s(s- 1) 2 ) Z=s(s_l)(s+l) I s+l (s+l)(s-1) " First, the matrices in (3)-(5) are computed as e= 's(s+ a) 1 O = s(s- 1) 0 0.5 U= -0.5 V-- ( -0.5 0.5 /3= (s+ 31 -(s-l)], s+l ] -s+l),s s-5) s--1 ' 4s) 0 s(s- 1) ' 0=(0.5s-0.5 -s+2) -0.5 1 ' 1~= ( -0.5s s-l) 0 0 " Then the computation goes as follows: Step 1. Take the polynomial matrices Q°:=[Q]c( (s+l)20 s+10 ) and 00,=((s+l) 0 ) 0 (s+l) a [Qlr, where we have 1 -1)and I ]r=(1 -4) [Ql~=(o 1

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272 K. Sugimoto, E Yamamoto / Computing stable doubly coprime factorizations Step 2. Compute the coprime factorization (6), (7) as _ 1 (s(s+ 1)(s + 1)(s- 2)) N = PQo 1 (s + 1) ------5 1 s + 2 ' D=QQol 1 {s(s- 1) -(s-l) (s + 1) 2 0 (s + 1) 2 A7 = 0o'P __ 1 (s2+4s+7 (s + 3)(s- 3)) (s + 1) 2 1 s -- 1 ' 1 (s+l) 2 -8s ) b=0o10= (s+l) 2 0 s(s-1) " Step 3. By division of polynomial matrices, we obtain /0.5(s + 1)(s + 2) --S3--5S2--6S+5) R = 1 -0.5(s + 1) (s + 1)(s + 3) " Step 4. Finally we have 1 (0 15s+ 1) X- (s + 1) 2 0 0 ' 1 (s2+5s+11 s-13 t, Y- (s+ 1) -----~k 0 (s+ 1)2J 1 (0 15s+ 1) = (s + 1) 2 0 0 ' = __ 1 ((s+1)2 -7(s+l) ) (s + 1) 2 0 S 2 + 5s + 11 " 6. Conclusion The authors wish to thank the reviewers who suggested various improvements. Notes After the submission of this paper the authors became aware of the existence of the paper by Hippe [8] which studies a similar problem. Let us compare these two papers here. Hippe [8] studies the structure of (full order) observer-based controllers in the frequency do- main, and reaches a nice parametrization of all such controllers. A main goal there is to clarify the degree of freedom in this design. On the other hand, the present paper aims at deriving doubly coprime factorizations from the polynomial matrices in (3)-(5) in a natural way, and then computing one such factorization as easily as pos- sible. This has been done by taking the simplest denominators Q0 and Q0 in the Algorithm of Section 3. From a technical viewpoint, the both methods have several similarities. These methods will give the same factorization if we start with the same polynomial matrices. In its derivation, however, there is one remarkable difference. Hippe [8] uses the matrices V 1, V 2 obtained from solutions of the two independent Diophantine equations (8.1) and (9.1). On the contrary, in the present paper we can compute the same doubly coprime factorization by using only the single polynomial matrix R; see (11) and (12). This is because we have started with the generalized Bezout identity (3), (4) instead of the individual Diophantine equations. As a result, the present approach reduces both theoretical and algorithmic complexity. We have proposed a new method for comput- ing a doubly coprime factorization of a given transfer matrix. It should be noted that Antsaklis [1] has also studied a connection between (right or left) polynomial factorizations and proper stable ones (but not doubly coprime). Our method can be regarded as a generalization of his result to the doubly coprime factorization. Furthermore, the factorization by our method has an advantage in that it corresponds uniquely to that of Nett et al, [11]. References [1] P.J. Antsaklis, Proper stable transfer matrix factorizations and internal systems descriptions, IEEE Trans. Automat. Control 31 (1986) 634-638. [2] Th. G.J. Beelen, New algorithms for computing the Kronecker structure of a pencil with applications to sys- tems and control theory, Ph.D. Thesis, Eindhoven Univer- sity of Technology, Netherlands (1987). [3] Th. G.J. Beelen and G.W. Veltkamp, Numerical computa- tion of a coprime factorization of a transfer function matrix, Systems Control Lett. 9 (1987) 281-288.

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K. Sugimoto, Y. Yamamoto / Computing stable doubly coprime factorizations 273 [4] Th. G.J. Beelen and P. van Dooren, A pencil approach for embedding a polynomial matrix into a unimodular matrix, SlAM J. Matrix Anal. Appl. 9 (1988) 77-89. [5] B. Codenotti and G. Lotti, A fast algorithm for the division of two polynomial matrices, IEEE Trans. Auto- mat. Control 34 (1989) 446-448. [6] B.A. Francis, A Course in H~ Control Theory (Springer, New York, 1987). [7] F.R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959). [8] P. Hippe, Parametrization of the full-order compensator in the frequency domain, lnternat. J. Control 48 (1988) 1583-1603. [9] T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, N J, 1980). [10] H. Maeda, Design theory for linear control systems II (in Japanese), Systems and Control 31 (1987) 728-734. [11] C.N. Nett, C.A. Jacobson, and M.J. Balas, A connection between state-space and doubly coprime fractional repre- sentations, IEEE Trans. Automat. Control 29 (1984) 831- 832. [12] T. Sugie and T. Ono, On doubly coprime factorizations, Linear Algebra Appl. 122/123/124 (1989) 681-696. [13] M. Vidyasagar, Control System Synthesis: A Factorization Approach (MIT Press, Cambridge, MA, 19850. [14] W.A. Wolovich, Linear Multivariable Systems (Springer, New York, 1974).

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