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Computation of the structural invariants of linear multivariable systems with an extended version of the program zeros
Syskms 81. Control North-Holland
Letters
6 (1985)
261-266
October
1985
Computation of the structural invariants of linear multivariable systems with an extended version of the program ZEROS F. SVARICEK Mess -, Stewr-
und Regelungstechnik.
Clniuersirtit
-GH - Duisburg.
FB 7. Lotharsrrusse
I. 21. 4100 Duisburg
I, West Gertnan~
Received 22 April 1985 Revised 24 June 1985 Abstract: The slructurel invariants like finite and infinile.zeros play an important role in many problems in the analysis of linear systems. In [l] Emami-Naeini and Van Dooren presented the program ZEROS for compuling the finite zeros of a linear system, which in the aulhor’s opinion is at present the best commonly available program for this problem. Such a program does not exist for computing the other structural invariants, like the infinite zeros and the Kronecker indices. This paper presents an extended version of the program ZEROS, which computes the finite zeros as well as the other structural invariant% Keywords: Feedback
Finite zeros. Infinite block-decoupling.
zeros,
Kronecker
indices,
Numerical
analysis,
Stare-space
methods,
Controllability.
Observability.
1. Introduction Consider the linear time invariant system i(t)=Ax(t)+Bu(t), y(l)=Cx(t)+Du(t),
(1) (4 OX” and A, B, C, D are constant real matrices of appropriate
where x(~)ER”, u(l)~R”, ye dimensions. The system mntrix P(s) [2] of the system (l), (2) is the (II + q)
X
(n‘+ p) pencil
P(s)=[“‘--,” ;I. For the transformations (T, G, R, F, K) representing state, input, output coordinate transformations, state feedback and output injection respectively, a complete set of invariants of the system (l), (2) can be defined from the Kronecker canonical form of the singular pencil P(s) [3,4,5]. Thus, assume that q ap and the normal rank of P(S) = p. p < n +p. Then the set of the structural invariants is [6]: (i) the column minimal indices E,= E, = . . . = en = 0, E,~+, Q . . . $ E, with s = n + p - p; (ii) the row minimal indices g, = q2 = . * . = q,, = 0, q,,+, Q * * * < q, with t = n + q - p; (iii) the infinite elementary divisiors (i.e.d.) of the type &‘I, ~~2,. . . , w”*. with m = p - n; (iv) the finite elementary divisors of the type (s - A)‘. The finite elementary divisors are the zeros of the invariant polynomials of P(S) and are well known as the invariant zeros of the system [7]. These invariant zeros have been studied by many authors in the last decade and can be computed by the numerically stable program ZEROS presented by Emami-Naeini and Van Dooren [l]. The set of the i.e.d. is directly related to the so called infinite zeros of the system (A, B, C, D) [B]. These infinite zeros turn out to be crucially important in solving problems such as decoupling [9], disturbance decoupling [lo], exact model matching [ll], root locus theory [12] and singular optimal control [13]. 0167-6911/85/$3.30
0 1985. Elsevier
Science
Publishers
B.V. (North-Holland)
261
Volume
6. Number
4
SYSTEMS
& CONTROL
LETTERS
October
1985
For the practical application of these results the determination of the orders of the infinite zeros is necessary. At present no commonly available program exists, corresponding to the program ZEROS, for computing these infinite zeros. In this paper it will be shown that both the infinite zeros and the Kronecker indices of the system (A, B, C, D) can be computed in addition to the finite zeros, by a slighly extended version of the program ZEROS. The organization of the paper is as follows. The next section contains a short description of the algorithm ZEROS of Emami-Naeini and Van Dooren. In Section 3 the extended version of the algorithm ZEROS for computing the complete set of structural invariants is presented. In the last section some concluding remarks are given. 2. Brief
description
of the program
ZEROS
The program ZEROS computes the finite zeros of the system (A, B, C, D) in three steps: Step I: reduce the system (A, B, C, D) to a new system (A,, B,, C,, D,) with the same zeros and with D, of full row rank; Step 2: reduce the transposed system (AT, CT, BT, 0,‘) to a new system (A,,, B,,, C,, D,,) with the same zeros and with D,, invertible; Step 3: compress the columns of [C,, D,] to [0 D,] and apply the transformations to the system matrix;
Theorem 1 [l]. The (finite) inoariant ‘finite structure pencil ’ ( sB, - A I).
zeros of the system (A,
B, C, D) are the generalized
eigenvalues
of the
The reductions in step 1 and step 2 are done by the fundamental structure algorithm REDUCE of Emami-Naeini and Van Dooren presented in [l] too. Unitary matrices are used to compress a matrix to either full row rank or full column rank. In the next section it will be shown how the complete set of the structural invariants can be determined by the algorithms REDUCE and ZEROS. 3. Determination
The algorithm
L,P(s)R,
of the structural
invariants
with an extended
version
of the algorithms
REDUCE
and ZEROS
transforms the system matrix P(s) with unitary matrices R,. L, to .. ... * * sI,,r -A, 4 * -C, D, * * 0 0 -s, *
REDUCE
=
(4)
0 * 0
...
..
0
-S1
with D, of full now rank m [l]. The T,x pi matrices Sj have full column rank pi. Therefore the algorithm REDUCE has partitioned the system matrix P(s) in two parts sB, -A, and sB2 -A,:
L,J’(s)R, = [-1. The part
4 SB, -A, = %-A, -c, Dr
1
262
Volume
6, Number
4
SYSTEMS
& CONTROL
LETTERS
October
1985
with D, of full row rank m has no left nullspace, no infinite zeros and contains the finite zeros (the invariant zeros) and the right nullspace of the system (A, B, C, D) [l]. The part sBr -A, contains the left nullspace and Theorem 2 will show that the orders of the i.e.d. of sBz - A 2 are equal to the orders of the infinite zeros of the system. The Kronecker row indices and the i.e.d. of sB2 - A2 can easily be determined from the special structure of sB, -A, with the following lemma. Lemma 1. From sB, - Al can he (i) There are (ii) There are
the structure of sB~ -AZ the infinite elementary divisors atld the directly determined from Corollary 4.6 in [ 141: dj = pi - TV+, i)?fir?ite elementary divisors of degree i (i = 1, . . ~ k ). e, = r, - pi Kroilecker row irldices of size i - 1 (i = 1,. . . , k ).
lleorem 2. The orders of the infinite zeros of the system (A, B, C, D).
elementary
divisors of sB2 -A?
row
minimal
indices of
are equal to the orders of the infinite
Proof. Let EL,,,z IL,,,-, >, . . z FL,> 0 be the orders of the infinite elementary divisors of P(S). It is well known that the orders v,,, k v,,,-, > * . . k V, L 0 of the infinite zeros of P(S) directly relate to the orders of the i.e.d. [8]: (i=l ,-.., n1). v, = P, -1 If the orders of the i.e.d. of sip, -A, are equal to the non-zero orders of the infinite zeros’of P(s) be shown that the i.e.d. pi of P(s) are related to the infinite zeros Y, in the following manner: p,=v,-Fl (i=l,..., m).
it must
Without loss of generality assume that the system (A, B, C. 0) is non-degenerate and that the number of outputs is greater than or equal to the number of inputs (i.e. the system has no column minimal indices). Therefore in a further step (4) can be transformed to . . . . . . * * -sB,-A, * 0 0
I ,,I 0
* -s, 0
* *
*
(7) *
0 Using
...
S, as the pivot,
...
0
(7) can be transformed
-S1 with unitary
matrices
to
(8)
_
0
...
with di=pi-q+,. 263
Volume
6, Number
4
SYSTEMS&CONTROLLETTERS
The i.e.d. of P(s) are the finite elementary divisors at w = 0 of wP(l/o) can be transformed by unimodular row and column transformations to -B+A,
0 0 0
4, 0
0
00
0
0
0 0 I
&I
October
1985
[8]. After this substitution, (8)
0 I N, 0 (9) .. (J"1
_
0
0
I WI,,, 0 ,,I 0 I w”’ 00 *. 1 .Iw’“l
with v,,,> Y,,,-, > * * . > Y, 2 0, where the vi’s > 0 are the i.e.d. of sA, - B, and originate from the integers pi and 7, (see proof of Lemma 1 in [14]). The pencil B, - wAr contains no i.e.d. and from the special structure of 0
(10)
follows directly the proof of Theorem 2. 0
Thus, using Lemma 1 and Theorem 2 the orders of the infinite zeros and the Kronecker row indices can easily be computed by an extended version of the algorithm REDUCE from the integers pi and 7,. The complete set of structural invariants can be determined by the following extended version E-ZEROS of the algorithm ZEROS of Emami-Naeini and Van Dooren. This extended algorithm E-ZEROS uses three new variables namely infz, kronr and kronc with: . infzi=p,-7i+, kronr, = ?; - pi kronc, = Q - pri
infinite zeros of degree i; Kronecker row indices of size i - 1; Kronecker column indices of size i - 1.
Algorithm E-ZEROS (A, B, C, D, n, p, q) result (A,, B,, n,, rank, infz, kronr, kronc) Step 1: comment reduce the system (A, B, C, D) to a new system (A,, B,, C,, 0,) with the same finite zeros and with D, of full row rank and determine the orders of the infinite zeros and the Kronecker row indices; calf E-REDUCE (A, B, C, D, n, p, q) result (A r, B,, C,, D,, n,, p,. q,. infz, kronr); rank := q,; if n, + q, = 0 then begin n r := 0; go to exit end; Step 2: comment reduce the transposed system (AT, CT, BT, 0:) to a new system (A,,, B,. C’,,, D,,) with the same zeros and with D,, invertible and determine the Kronecker column indices; call E-REDUCE (A:, Cr? B,T, D,', n,, pr, qr) result (A,,, B,,, C,, D,,, nrcr prc, qr,, kronc); if n Tc= 0 then begin n r := 0; go to exit end; Step 3: commenr compress the columns of [C,, D,] to [0 D,] and apply the transformations to the system matrix; n, := n,; if rank = 0 then go to exit;
[q-$j:= [*;]w [$]:=[q-+
Exit: stop; Remarks. A)-‘B+D. 264
(i) [l] m =prc = qrc = rank is the normal rank of the transfer function matrix G(S) = C(sl-
SYSTEMS & CONTROL LETTERS
Volume 6. Number 4
October 1985
(ii) [15] The number M qf the i.e.d. of P(s) is equal to the normal rank of G(s). (iii) Note that for p = 0 or q = 0 we have rank = 0 and the eigenvalues of A, are the input or output decoupling zeros of the system [2]. The right and left Kronecker indices are then directly related to the so called controllability or observability indices [16,17] of the pairs (A, B), (A, CT). A comparison of the algorithms ZEROS and E-ZEROS shows that no change of the program-flow is necessary and that the extension of the algorithm REDUCE is limited to the additional determination of the integers infzi, kronr, kronc, from the given integers pi and 7i. Consequently the extended program E-ZEROS has the same properties (for example numerical stability, etc.) as the original program ZEROS. In [l] Emami-Naeini and Van Dooren present also the complete FORTRAN listings ’ of the programs REDUCE and ZEROS. In the subprogram REDUCE the integers 7 are determined at line 164 of the listing and the integers pi at line 178 respectiJely 204. Therefore the simplest extension of the program REDUCE is a printout of the integers pi and ~~after their determination in this subprogram. Then the program ZEROS needs no additional calculations with the result that the execution time of the program ZEROS For computing the finite zeros and the execution time of the program E-ZEROS for computing the complete set of structural invariants are almost the same. A great advantage of the numerically stable program ZEROS of Emami-Naeini and Van Dooren lies in the fact that no assumptions about (A, B, C, D) are made and, moreover, the case p = 0 or q = 0 is also allowed, i.e. the programs ZEROS and E-ZEROS can be used for a numerically stable controllability/observability test [18]. Therefore the presented extended version of the algorithm ZEROS allows a complete numerical analysis of the linear system (A, B, C, D) as well as of the pairs (A, B) and (A, CT).
The research has been supported by the Stiftung Volkswagen under grant number I/38043. The author would like to thank Prof. Dr.-Ing. H. Schwarz for useful discussions and Dipl.-Ing. D. Konik for helpful comments. References [l] A. Emami-Naeini and P. Van Dooren. Computations of zeros of linear multivariable systems, A~romarica 26 (1982) 415-430. [Z] H.H. Rosenbrock, &are Spqce and M~lriuarioble I%ory (William Clowes & Sons, London, 1970). [3] AS. Morse, Structural invariants of linear multivariable systems, Siam J. Cmfrol 11 (1973) 446-465. [4] J.S. Thorp, The singular pencil of a linear dynamical system, I,rrernot. J. Conrrol 18 (1973) 577-596. [S] A.J. van der Weiden and O.H. Bosgra. The determination of structural properties of a linear multivariable system by operations of system similarity: 1. Strictly proper systems, Infernar. J. Conrrol 29 (1979) 835-860. [6] F.R. Gantmacher, Marrizenrechnung Teil2 (VEB Deutscher Verlag der Wissenschaften,Berlin, 1959). [7] A.G.J. MacFarlane and N. Karcanias. Poles and zeros of linear multivariable systems: a survey of the algebraic. geometric and complex-variable theory. Inferna/. J. Control 24 (1976) 33-74. [S] A.I.G. Vardulakis and N. Korcanias, Relations between strict equivalence invariants and structure at infinity of matrix pencils. IEEE Traw. Aulomot. Control 28 (1983) 514-518. [9] J. Descusse,J.F. Lafay and V. Kucera. Decoupling by restricted static-state feedback: The general case, IEEE Trans. Automat. Control
29 (1984)
79-81.
[lo] C. Commault and J.M. Dion, Transfer matrir approach to the disturbance decoupling problem, in: Preprinfs o/the 9th fFAC World Congress, Vol. 8 (1984) 130-133. [ll] M. Malabre and V. Kucera, Infinite structure and exact model matching problem: A geometric.approach, IEEE Trans. Automat. Conrrol 29 (1984) 266-268. ’ The listing contains two errors. The corrected lines are: Line 192 160 IF (Nl .EQ. 0) GOT0 210 DO 170 J=l,Nl Line 218 MAX = NORM(H) 265
Volume
6, Number
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SYSTEMS
& CONTROL
LETTERS
October
1985
[12] D.H. Owens, On structural invariants and the root-loci of linear multivariable systems, Iarentar. J. Cohrrol 28 (1978) 187-196. (131 B.A. Francis. On totally singular linear quadratic optimal control, IEEE Trans. A~trotwl. Cmwol 24 (1979) 616-621. 1141 P. Van Dooren, The computation of Kronecker’s canonical form of a singular pencil, Litrenr Algehru and i/s Applicariom 27 (1979) 103-140. [15] N. Suda and E. Mutsuyoshi, Invariant zeros and input-output structure of linear time-invariant systems, Inrcrnur. J. Cowvl 28 (1978) 525-535. [la] P. Brunovsky, A classification of linear controllable systems, Kvhernericn 3 (1970) 173-187. [17] R.E. Kalman, Kronecker invariants and feedback, in: L. Weiss, Ed.. Proc. oj Con/: oa Ordinoty Di//erenrio/ Equorions, NLR Math. Res. Cent. (1965) 459-471. 1181 F. Svaricek, Eine schnelle - numerisch stabile - Methode zur ijberpriifung der Steuer- bzw. Beobachtbarkeit eines linearen zeitinvarianten Systems, Regehgstechnik 32 (1984) 134-135.
266
Letters
6 (1985)
261-266
October
1985
Computation of the structural invariants of linear multivariable systems with an extended version of the program ZEROS F. SVARICEK Mess -, Stewr-
und Regelungstechnik.
Clniuersirtit
-GH - Duisburg.
FB 7. Lotharsrrusse
I. 21. 4100 Duisburg
I, West Gertnan~
Received 22 April 1985 Revised 24 June 1985 Abstract: The slructurel invariants like finite and infinile.zeros play an important role in many problems in the analysis of linear systems. In [l] Emami-Naeini and Van Dooren presented the program ZEROS for compuling the finite zeros of a linear system, which in the aulhor’s opinion is at present the best commonly available program for this problem. Such a program does not exist for computing the other structural invariants, like the infinite zeros and the Kronecker indices. This paper presents an extended version of the program ZEROS, which computes the finite zeros as well as the other structural invariant% Keywords: Feedback
Finite zeros. Infinite block-decoupling.
zeros,
Kronecker
indices,
Numerical
analysis,
Stare-space
methods,
Controllability.
Observability.
1. Introduction Consider the linear time invariant system i(t)=Ax(t)+Bu(t), y(l)=Cx(t)+Du(t),
(1) (4 OX” and A, B, C, D are constant real matrices of appropriate
where x(~)ER”, u(l)~R”, ye dimensions. The system mntrix P(s) [2] of the system (l), (2) is the (II + q)
X
(n‘+ p) pencil
P(s)=[“‘--,” ;I. For the transformations (T, G, R, F, K) representing state, input, output coordinate transformations, state feedback and output injection respectively, a complete set of invariants of the system (l), (2) can be defined from the Kronecker canonical form of the singular pencil P(s) [3,4,5]. Thus, assume that q ap and the normal rank of P(S) = p. p < n +p. Then the set of the structural invariants is [6]: (i) the column minimal indices E,= E, = . . . = en = 0, E,~+, Q . . . $ E, with s = n + p - p; (ii) the row minimal indices g, = q2 = . * . = q,, = 0, q,,+, Q * * * < q, with t = n + q - p; (iii) the infinite elementary divisiors (i.e.d.) of the type &‘I, ~~2,. . . , w”*. with m = p - n; (iv) the finite elementary divisors of the type (s - A)‘. The finite elementary divisors are the zeros of the invariant polynomials of P(S) and are well known as the invariant zeros of the system [7]. These invariant zeros have been studied by many authors in the last decade and can be computed by the numerically stable program ZEROS presented by Emami-Naeini and Van Dooren [l]. The set of the i.e.d. is directly related to the so called infinite zeros of the system (A, B, C, D) [B]. These infinite zeros turn out to be crucially important in solving problems such as decoupling [9], disturbance decoupling [lo], exact model matching [ll], root locus theory [12] and singular optimal control [13]. 0167-6911/85/$3.30
0 1985. Elsevier
Science
Publishers
B.V. (North-Holland)
261
Volume
6. Number
4
SYSTEMS
& CONTROL
LETTERS
October
1985
For the practical application of these results the determination of the orders of the infinite zeros is necessary. At present no commonly available program exists, corresponding to the program ZEROS, for computing these infinite zeros. In this paper it will be shown that both the infinite zeros and the Kronecker indices of the system (A, B, C, D) can be computed in addition to the finite zeros, by a slighly extended version of the program ZEROS. The organization of the paper is as follows. The next section contains a short description of the algorithm ZEROS of Emami-Naeini and Van Dooren. In Section 3 the extended version of the algorithm ZEROS for computing the complete set of structural invariants is presented. In the last section some concluding remarks are given. 2. Brief
description
of the program
ZEROS
The program ZEROS computes the finite zeros of the system (A, B, C, D) in three steps: Step I: reduce the system (A, B, C, D) to a new system (A,, B,, C,, D,) with the same zeros and with D, of full row rank; Step 2: reduce the transposed system (AT, CT, BT, 0,‘) to a new system (A,,, B,,, C,, D,,) with the same zeros and with D,, invertible; Step 3: compress the columns of [C,, D,] to [0 D,] and apply the transformations to the system matrix;
Theorem 1 [l]. The (finite) inoariant ‘finite structure pencil ’ ( sB, - A I).
zeros of the system (A,
B, C, D) are the generalized
eigenvalues
of the
The reductions in step 1 and step 2 are done by the fundamental structure algorithm REDUCE of Emami-Naeini and Van Dooren presented in [l] too. Unitary matrices are used to compress a matrix to either full row rank or full column rank. In the next section it will be shown how the complete set of the structural invariants can be determined by the algorithms REDUCE and ZEROS. 3. Determination
The algorithm
L,P(s)R,
of the structural
invariants
with an extended
version
of the algorithms
REDUCE
and ZEROS
transforms the system matrix P(s) with unitary matrices R,. L, to .. ... * * sI,,r -A, 4 * -C, D, * * 0 0 -s, *
REDUCE
=
(4)
0 * 0
...
..
0
-S1
with D, of full now rank m [l]. The T,x pi matrices Sj have full column rank pi. Therefore the algorithm REDUCE has partitioned the system matrix P(s) in two parts sB, -A, and sB2 -A,:
L,J’(s)R, = [-1. The part
4 SB, -A, = %-A, -c, Dr
1
262
Volume
6, Number
4
SYSTEMS
& CONTROL
LETTERS
October
1985
with D, of full row rank m has no left nullspace, no infinite zeros and contains the finite zeros (the invariant zeros) and the right nullspace of the system (A, B, C, D) [l]. The part sBr -A, contains the left nullspace and Theorem 2 will show that the orders of the i.e.d. of sBz - A 2 are equal to the orders of the infinite zeros of the system. The Kronecker row indices and the i.e.d. of sB2 - A2 can easily be determined from the special structure of sB, -A, with the following lemma. Lemma 1. From sB, - Al can he (i) There are (ii) There are
the structure of sB~ -AZ the infinite elementary divisors atld the directly determined from Corollary 4.6 in [ 141: dj = pi - TV+, i)?fir?ite elementary divisors of degree i (i = 1, . . ~ k ). e, = r, - pi Kroilecker row irldices of size i - 1 (i = 1,. . . , k ).
lleorem 2. The orders of the infinite zeros of the system (A, B, C, D).
elementary
divisors of sB2 -A?
row
minimal
indices of
are equal to the orders of the infinite
Proof. Let EL,,,z IL,,,-, >, . . z FL,> 0 be the orders of the infinite elementary divisors of P(S). It is well known that the orders v,,, k v,,,-, > * . . k V, L 0 of the infinite zeros of P(S) directly relate to the orders of the i.e.d. [8]: (i=l ,-.., n1). v, = P, -1 If the orders of the i.e.d. of sip, -A, are equal to the non-zero orders of the infinite zeros’of P(s) be shown that the i.e.d. pi of P(s) are related to the infinite zeros Y, in the following manner: p,=v,-Fl (i=l,..., m).
it must
Without loss of generality assume that the system (A, B, C. 0) is non-degenerate and that the number of outputs is greater than or equal to the number of inputs (i.e. the system has no column minimal indices). Therefore in a further step (4) can be transformed to . . . . . . * * -sB,-A, * 0 0
I ,,I 0
* -s, 0
* *
*
(7) *
0 Using
...
S, as the pivot,
...
0
(7) can be transformed
-S1 with unitary
matrices
to
(8)
_
0
...
with di=pi-q+,. 263
Volume
6, Number
4
SYSTEMS&CONTROLLETTERS
The i.e.d. of P(s) are the finite elementary divisors at w = 0 of wP(l/o) can be transformed by unimodular row and column transformations to -B+A,
0 0 0
4, 0
0
00
0
0
0 0 I
&I
October
1985
[8]. After this substitution, (8)
0 I N, 0 (9) .. (J"1
_
0
0
I WI,,, 0 ,,I 0 I w”’ 00 *. 1 .Iw’“l
with v,,,> Y,,,-, > * * . > Y, 2 0, where the vi’s > 0 are the i.e.d. of sA, - B, and originate from the integers pi and 7, (see proof of Lemma 1 in [14]). The pencil B, - wAr contains no i.e.d. and from the special structure of 0
(10)
follows directly the proof of Theorem 2. 0
Thus, using Lemma 1 and Theorem 2 the orders of the infinite zeros and the Kronecker row indices can easily be computed by an extended version of the algorithm REDUCE from the integers pi and 7,. The complete set of structural invariants can be determined by the following extended version E-ZEROS of the algorithm ZEROS of Emami-Naeini and Van Dooren. This extended algorithm E-ZEROS uses three new variables namely infz, kronr and kronc with: . infzi=p,-7i+, kronr, = ?; - pi kronc, = Q - pri
infinite zeros of degree i; Kronecker row indices of size i - 1; Kronecker column indices of size i - 1.
Algorithm E-ZEROS (A, B, C, D, n, p, q) result (A,, B,, n,, rank, infz, kronr, kronc) Step 1: comment reduce the system (A, B, C, D) to a new system (A,, B,, C,, 0,) with the same finite zeros and with D, of full row rank and determine the orders of the infinite zeros and the Kronecker row indices; calf E-REDUCE (A, B, C, D, n, p, q) result (A r, B,, C,, D,, n,, p,. q,. infz, kronr); rank := q,; if n, + q, = 0 then begin n r := 0; go to exit end; Step 2: comment reduce the transposed system (AT, CT, BT, 0:) to a new system (A,,, B,. C’,,, D,,) with the same zeros and with D,, invertible and determine the Kronecker column indices; call E-REDUCE (A:, Cr? B,T, D,', n,, pr, qr) result (A,,, B,,, C,, D,,, nrcr prc, qr,, kronc); if n Tc= 0 then begin n r := 0; go to exit end; Step 3: commenr compress the columns of [C,, D,] to [0 D,] and apply the transformations to the system matrix; n, := n,; if rank = 0 then go to exit;
[q-$j:= [*;]w [$]:=[q-+
Exit: stop; Remarks. A)-‘B+D. 264
(i) [l] m =prc = qrc = rank is the normal rank of the transfer function matrix G(S) = C(sl-
SYSTEMS & CONTROL LETTERS
Volume 6. Number 4
October 1985
(ii) [15] The number M qf the i.e.d. of P(s) is equal to the normal rank of G(s). (iii) Note that for p = 0 or q = 0 we have rank = 0 and the eigenvalues of A, are the input or output decoupling zeros of the system [2]. The right and left Kronecker indices are then directly related to the so called controllability or observability indices [16,17] of the pairs (A, B), (A, CT). A comparison of the algorithms ZEROS and E-ZEROS shows that no change of the program-flow is necessary and that the extension of the algorithm REDUCE is limited to the additional determination of the integers infzi, kronr, kronc, from the given integers pi and 7i. Consequently the extended program E-ZEROS has the same properties (for example numerical stability, etc.) as the original program ZEROS. In [l] Emami-Naeini and Van Dooren present also the complete FORTRAN listings ’ of the programs REDUCE and ZEROS. In the subprogram REDUCE the integers 7 are determined at line 164 of the listing and the integers pi at line 178 respectiJely 204. Therefore the simplest extension of the program REDUCE is a printout of the integers pi and ~~after their determination in this subprogram. Then the program ZEROS needs no additional calculations with the result that the execution time of the program ZEROS For computing the finite zeros and the execution time of the program E-ZEROS for computing the complete set of structural invariants are almost the same. A great advantage of the numerically stable program ZEROS of Emami-Naeini and Van Dooren lies in the fact that no assumptions about (A, B, C, D) are made and, moreover, the case p = 0 or q = 0 is also allowed, i.e. the programs ZEROS and E-ZEROS can be used for a numerically stable controllability/observability test [18]. Therefore the presented extended version of the algorithm ZEROS allows a complete numerical analysis of the linear system (A, B, C, D) as well as of the pairs (A, B) and (A, CT).
The research has been supported by the Stiftung Volkswagen under grant number I/38043. The author would like to thank Prof. Dr.-Ing. H. Schwarz for useful discussions and Dipl.-Ing. D. Konik for helpful comments. References [l] A. Emami-Naeini and P. Van Dooren. Computations of zeros of linear multivariable systems, A~romarica 26 (1982) 415-430. [Z] H.H. Rosenbrock, &are Spqce and M~lriuarioble I%ory (William Clowes & Sons, London, 1970). [3] AS. Morse, Structural invariants of linear multivariable systems, Siam J. Cmfrol 11 (1973) 446-465. [4] J.S. Thorp, The singular pencil of a linear dynamical system, I,rrernot. J. Conrrol 18 (1973) 577-596. [S] A.J. van der Weiden and O.H. Bosgra. The determination of structural properties of a linear multivariable system by operations of system similarity: 1. Strictly proper systems, Infernar. J. Conrrol 29 (1979) 835-860. [6] F.R. Gantmacher, Marrizenrechnung Teil2 (VEB Deutscher Verlag der Wissenschaften,Berlin, 1959). [7] A.G.J. MacFarlane and N. Karcanias. Poles and zeros of linear multivariable systems: a survey of the algebraic. geometric and complex-variable theory. Inferna/. J. Control 24 (1976) 33-74. [S] A.I.G. Vardulakis and N. Korcanias, Relations between strict equivalence invariants and structure at infinity of matrix pencils. IEEE Traw. Aulomot. Control 28 (1983) 514-518. [9] J. Descusse,J.F. Lafay and V. Kucera. Decoupling by restricted static-state feedback: The general case, IEEE Trans. Automat. Control
29 (1984)
79-81.
[lo] C. Commault and J.M. Dion, Transfer matrir approach to the disturbance decoupling problem, in: Preprinfs o/the 9th fFAC World Congress, Vol. 8 (1984) 130-133. [ll] M. Malabre and V. Kucera, Infinite structure and exact model matching problem: A geometric.approach, IEEE Trans. Automat. Conrrol 29 (1984) 266-268. ’ The listing contains two errors. The corrected lines are: Line 192 160 IF (Nl .EQ. 0) GOT0 210 DO 170 J=l,Nl Line 218 MAX = NORM(H) 265
Volume
6, Number
4
SYSTEMS
& CONTROL
LETTERS
October
1985
[12] D.H. Owens, On structural invariants and the root-loci of linear multivariable systems, Iarentar. J. Cohrrol 28 (1978) 187-196. (131 B.A. Francis. On totally singular linear quadratic optimal control, IEEE Trans. A~trotwl. Cmwol 24 (1979) 616-621. 1141 P. Van Dooren, The computation of Kronecker’s canonical form of a singular pencil, Litrenr Algehru and i/s Applicariom 27 (1979) 103-140. [15] N. Suda and E. Mutsuyoshi, Invariant zeros and input-output structure of linear time-invariant systems, Inrcrnur. J. Cowvl 28 (1978) 525-535. [la] P. Brunovsky, A classification of linear controllable systems, Kvhernericn 3 (1970) 173-187. [17] R.E. Kalman, Kronecker invariants and feedback, in: L. Weiss, Ed.. Proc. oj Con/: oa Ordinoty Di//erenrio/ Equorions, NLR Math. Res. Cent. (1965) 459-471. 1181 F. Svaricek, Eine schnelle - numerisch stabile - Methode zur ijberpriifung der Steuer- bzw. Beobachtbarkeit eines linearen zeitinvarianten Systems, Regehgstechnik 32 (1984) 134-135.
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