Automatica, Vol. 13, pp. 295-299. Pergamon Press, 1977. Printed in Great Britain

Brief Paper

A General Algorithm for Determining State-Space Representations*

W. A. WOLOVICHt and R. GUIDORZI~

Key Word Index~Canonical forms; multivariable systems; system theory; system analysis; state space methods; matrix algebra; polynomials.

Summary--A new and direct procedure is presented for determining state-space representations of given, time-invariant systems whose dynamical behavior is expressed in a more general, differential operator form. The procedure employs some preliminary polynomial matrix operations, if necessary, in order to "reduce" the given system to an equivalent differential operator form which satisfies four specific conditions. An equivalent state space representation is then determined in a most direct manner; i.e. the algorithm presented requires only a single matrix inversion. An explicit relationship between the partial state and input of the given system and the state of the equivalent state-space system is also obtained.

1. Introduction THE PRIMARY purpose of this paper is to present a new and direct algorithm for determining a state-space system of the form

.~:(t ) = Ax(t ) + Bu(t ); (la)

y(t) =Cx(t)+E(D)u(t), (lb)

which is equivalent[l] to a given differential operator system of the form

P(O)z(t) = O(D)u(t); (2a)

y(t) = R(D)z(t)+ W(D)u(t). (2b)

In the above, x(t) is an n-vector called the state, y(t) is a p-vector called the output, u(t) is a q-vector called the input, and z(t) is an m-vector called the partial state. Furthermore, A, B, and C are appropriately dimensioned real matrices, while E(D), P(D), Q(D), R(D), and W(D) are appropriately dimensioned poly- nomial matrices in the differential operator D =d/dr, with P(D) nonsingular.

The algorithm which we will develop is based on 'appropriate modifications and extensions' of algorithms presented in both [ 1] and [2]. In Section 2 we show how certain state space systems of the form (1) directly imply corresponding and equivalent differential operator representations of the form (2), which satisfy {four) appropriate conditions. In Section 3 the algorithm is then developed, as we essentially reverse the steps employed in Section 2 in order to present a procedure of obtaining state space systems which are equivalent to an appropriate, but restrictive class of differential operator systems. In Section 4, we illustrate how the more general class of differential operator systems can be 'reduced' to equivalent systems which permit the direct employ- ment of our algorithm.

*Received 12 April, 1976; revised 9 November, 1976. The original version of this paper was not presented at any IFAC meeting It was recommended for publication in revised form by associate editor J. Ackermann.

fDivision of Engineering and the Lefschetz Center for Dynamical Systems, Brown University, Providence, RI 02912, U.S.A. Present address: Control Theory Centre, University of Warwick, Coventry CV4 7AL, Warwickshire, England.

++Instituto di Automatica, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy.

295

2. Implications of multi-companion A matrices Consider any state-space system of the form (1). The (n n)

state matrix A will be called a multi-companion matrix if it can be partitioned into m 2 submatrices, Au, such that ifj = i then A, is a companion matrix of dimension d i, i.e., ifd o a---0 and

then

k

ak a__ ~ d~, (3) 0

0 1 ... 0 (4a) Ai i = " : 1

al, a ,+ l+ 1 ai,~r I 1+2 . . . . . . C l ,a

and i f j~ i , A u is a dlxd j matrix whose first d~-I rows are identically zero, i.e.

I 0 0 ... 0 1 Aij= i i 0 0 ... 0

L aG o.j 1+1 . . , ai.aj (4b)

It might be noted that since there are no restrictions on m, virtually every (nxn) real matrix A is multi-companion. However, those of particular importance here are ones which are characterized by the condition, m < n.

We next define (the components of) an m-vector,

z(t) = [z I (t), z 2 (t) .... . zm(t)] T

which we call a partial state, via the relations:

Zi ( t ) ~" Xo , . l+ I ( t ) , (5 )

for i~ m'[', where Xk(t) denotes the k-th component of the state of (1). If the transpose of the n x m polynomial matrix S(D) is now defined as

I I D . Dn,-1 I 0 0 0 [ " ] "'" ! 0 0 ... 0 [ 1 D ... D d2 x [. .

STID)= [ . . . i ' : : : i : 1

0 . . . o [o 0 0 . [

[0 0 ... o

i o o . . . " - [

I I D ...

0

D e . - 1

(6)

then in view of (la), (4), and (5), it can be verified by direct substitution that

S(D)x(t)=x(t)+fl(D)u(t), (7)

tThe notation m is used to denote the integers 1,2 . . . . . m.

296 Br ie f Paper

where fi(D) is defined in terms of the ,-/h rows, h i, oJ B as the following n q polynomial matrix

0

bl

Dbl + b 2 1

Da"~ 2bl +. . . + b% 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

b%+l

fl(D) ~= Db~,+ l + b~ + 2 (8)

Da~- 2b~, + ~ +. . . +b~ _

0

ba 1 1 i

It should be noted that if d k = 1 for any k, then b,, ,+1 {as well as

De" 2b,~ ~ + 1 ) will not appear as an element of fl(D); i.e., only a single, identically zero row of//[D) will correspond to any d k = 1.

We next note that in view of (la),

I DI - A Ix(t ) = Bu(t). (9)

If S(D)z( t ) - f l (D)u( t ) , as given by (7), is now substituted for xit) in (9), we obtain

(D I -A )S(D)z ( t )=[ ID I -A ) f l (D I+B]u l t ) (10)

Let A~ now be defined as the m x n matrix consisting of the (m) ordered ak rows of A, and

P(D ~ x= diag [Dai] - A,,SID ), (11 )

an m x m column proper polynomial matrix with F~[P(D)]* = /,,. If P(D) is row proper as well, it can be verified by direct substitution that the (m) a k rows of(10) imply the relation:?

~diag [ Dai] - .4,,,S1D )'~ :(t ) - P{ D )z(t ) -

=ST(D)MBu( t )=Q{D)u( t ) , (12)

where

Q(D)&Sr (D)MB, (13)

and M is an n x n, nonsingular, real matrix, determined directly from A, which can be partioned as m 2 submatrices, M~j, each of dimensions d i x dj (similar to the partitional structure of A ); i.e. M = [Mii] for iem andjgm, where Mij is d i x d i. I f / - i,

I - -~ l o i+2 --Ui,ai i . 3 . . . --t~l ~ 1 ] " " I

. . "

J - -U .6 - I +3

Mii = : , {14a) a a . 0

1"

*F~[P(D)] (I ',[P(D)]), as defined in [1 ], denotes the m x m real matrix consisting of the coefficients of the highest degree polynomial or polynomials in each column (row) of P(D).

t I f P(D) is not row proper, (12) need not hold: i.e. the row proper condition on P(D) is sufficient to insure that all appropriate derivatives of u(t) in (10) are also contained in Q(D)u(t) of (12).

and if) ~ i, Mii is defined as the firsl d, rows of thc n d, matrix

--(di,aj ~ ~2

--(d o 1~3 Z

-- ~li a " " , ) 0 0

0 0

0 0 . . .

. . - a , . 0

0 "*0

* -O

0

(t4b)

Finally in view of (7), ( I b) clearly implies that

y( t )=CSID)z ( t ) -C f l tD)u( t )+E(D)u{t )=

=R(D)z ( t )+ W(D)u(t) ,

where

and

(15)

R(D)&CS(D) , (16)

W(D) ~=E(D)-Cf l (D). (17)

In summary therefore, we observe that any state-space system of the form (I), with A multi-companion, directly implies a corresponding differential operator system of the form (2), as given by (12) (provided P(D), as given by (11 ), is row proper)and (15), where

and

6)

(ii)

(iii)

(iv)

F, [P(D)]=I , which implies that P(D) i s column proper, [',[/2(D)] is nonsingular, i.e. P(D) i s row proper as well, F,[R(D )] < ,~,.[)~(D 1],*

(w h?,[QID)] < (?,[PID)].*

Finally, it is of interest to note that the matrix M, defined by (14), represents an equivalence transformation which reduces A to a 'dual multi-companion form'; i.e./IT= (MAM 1 )r is multi- companion This. in turn, implies tile ability to obtain a dual state-space realization

1,4, B ,~ ' ,E (D) I=~,MAM-1 ,MB, CM I,F(D)I,

where

(t )= Mx(t )= M S( D )z(t ) - M fl( D )u(t ).

In the next section, we will essentially reverse the steps employed to derive (12) and (15) in order to show that a differential operator system of the form (2), which satisfies the four conditions noted above, directly implies a corresponding state space system of the form (I). with A multi-companion.

3. The algorithm In view of the results presented in the previous section, we will

now present on algorithm which is based largely on the results given in [2] for determining a state-space system of the form (1) which is equivalent to any differential operator system of the form (2), which satisfies conditions (i) thru (iv).

To determine A. Let d i denote the degree of each, i-th column of the m x m nonsingular matrix, P(D)" i.e. (~ci[P(D)] =d i for ie, m.~ In view of condition (i), it follows that

P(D)=diag[D 1 for ie, m, since any d~ =0 would imply irrelevant, non-dynamical equations.

Brief Paper 297

for some real m x n matrix A,., with S(D) given by (6). A,. is therefore directly determined by inspection of

AreS(D) = diag [D d,] - P(D ). (19)

In view of (1) an appropriate A, of the equivalent state-space system, is now given by simply replacing the (m) ordered trk (for kern) rows of the (n x n) companion matrix

I n - 1

0. . .0

by the corresponding k-th ordered rows of A,.. It might be noted that the A, thus obtained, is multi-companion.

In view of conditions (i) and (ii), it follows that

c~,,[P(D)] = Oc,[P(D)] = d,, (20)

for iem, i.e. ifF,[P(D)] =1, and P(D) is now proper as well, then

c~,,[P(D)] >=Oc,[P(D)] =d,, (21)

for iem, since a monic polynomial ofdegree d~ is the diagonal element of each, i-th row of P(D). However, since

m m

c~,,[P(D)] = ~ 0alP(D)] = n, HI

m y] {r'~,,[P(D)]-Oc,EP(D)]} =0 1

which, in view of (21), directly implies (20). To determine B. In view of(20) and condition(iv), it now follows

that

Q(D) = Sr(D)B, (22)

for some real n x q matrix B, which is directly determined by inspection of Q(D). In yiew of (13), an appropriate B, of the equivalent state-space system, is therefore given by the relation:

It is readily verified that conditions (i)-(iv) hold. Since d~ =2 and d2~ l, Ii 0] S(D) = 0

1

and A"S(D)=I--3D+I'2D-3, -~] '

as given by (19), with

[l 3 :] A,. -3 -2 - '

Since a I = dj = 2 and 02 = d~ + d 2 = 3,

L0 0 t 0 -3 I 1 A= . I . I

-3 -2 i -4 I

M is next determined via (14); i.e. (14a) implies that

. and M2z = [1], while (14b) implies that

EOo] andM21 = [2 0]. M is therefore given by

'i-1 3 1 I 0 I 0 I I 0 i I and

B = M- I B, (23)

where M is defined in terms of the A just determined by (14). To determine C. Conditions (i) and (iii) directly imply that

R(D)=CS(D) (24)

for the corresponding, appropriate, real p x n matrix C, which is directly determined by inspection of R (D).

To determine E(D). Iffl(D) is defined, in terms of the B given by (23), via (8), then in view of (17), the appropriate E(D), of the equivalent state-space system, is given by the relation:

E(O ) = W(D ) + Cfl(O). (25)

We finally note that in view of the results presented in the previous section, and (7) in particular, there is a direct and rather simple relationship between the state, x(t), of the equivalent state-space system, which has now been determined, and the partial state and input of the given differential operator system, namely

x(t)=S(D)z(t)-fl(D)u(t). (26)

An example will now be presented in order to illustrate the procedure. In particular, consider a differential operator system (2), with

[-D 2 + 30 - 1 - 1 P(D)=L 2D+3 D+4]' QW)=L[-4D+I]'2

!] I 41 RID)= , and W(D)= . I

M- I~ I

In view of (22),

B = and

0,0] 1 -3 0

0 -2 1

by(23), B=M ~B= [41 1 C is immediately apparent by inspection of R(D); i.e. in view of (24),

and since

as given by (8),

C= [' 'i] 1 0 , 0 -1

[41 Cfl(D) = 0 -4

298 Brief Paper

which implies, in view of (25), that [I, EID)=W(D)+Cfl(D)= 0

0

In summary, therefore, the state space system, .((t)=Ax(t) + Bu(t): y(t)= Cx(t), with

I 0

A= I

-3

B=

and

-3

4

-11

6

I I 1 1 1 C= 1 0 0 0 - 1 2 is equivalent to the given differential operator system, with (see (265)

x2(t) = -~(t)-4u(t) .

X3(I) Z2(/) ,

The reader can now verify that the transfer matrices of the two (equivalent) systems are equal; i.e., for this example,

CIsI-A) ~B=R(s)P t(s)Q(s)+W(s)=

4s2+17s+6 t - s2+30s 14 .j "

s 3 +7s ~ +i3 . ; - i

4. The general case We have now shown how to obtain a state-space system (I)

which is equivalent to a given differential operator system (2) which satisfies conditions (i) (iv). We realize, of course, that in many cases conditions (ik (iv) will not hold, and the purpose of this section is to show how the algorithm of the previous section can be employed to handle the more general case: i.e. when only the nonsingularity of P(D) is assumed.

In particular, if P(D) of (2) is nonsingular, then it is possible to reduce P(D) to simultaneous row proper, column proper form, with F,.[P(D)]=I, via both elementary row and column operations, and references [1] and [3] contain algorithms for performing the appropriate reductions; i.e. for finding a pair of unimodular matrices, Ur(D) and Ua(D). such that P(D) =Ur(O)P(D)Un(D ) of the equivalent[l] differential operator system

P(D)(t)=UIJD)P(D)Un(D)(t)=Ur(D)Q(D)u(t)=O.(D)u(t); (27a)

y(r)=R(D)UR{D)2(t)+ W(D)u(t)=f~(D)'5(t)+ W(D)u(t), (27b)

satisfies conditions (i) and (ii).? In other words, the first step required to employ the algorithm of the previous section is the determination of an equivalent system, (27), with a i6(D5 which

~For example, it is possible to reduce P(D) to Smith form, although it would not usually be necessary to go "so far' in order to obtain an appropriate P(D). In particular, reference [3] shows that only elementary row operations are required.

satisfies (i) and (ii). It should be noted that the partial stales olthc equivalent systems 2) and (27) are related by U~I D t: i.e.

z(t)=URiD)~(t). ~ 28 )

If ~[O~(D)]

Brief Paper 299

To illustrate the results of this section, consider a differential operator system (2), with

I -2D-3 2D2+4D- I ] P(D)= 3DZ+6D_t _3D3_8Da_D j, Q(D)

=F-2 -51 L3o 2 + 12DJ'

2D2+4D+I -2D3-5D2-D ] R(D)=[2D2+7-1_ -2D3-8D2-5D-ID2+D+2 '

and

W(D)=

I _2D2 6D_5]

-2D z- 14D- 1 [ . D+4 ]

The reader can now verify that if

UL(D ' : ID 1 ;1 and UR(D) : [ ; D I I l ,

then

[-D2+3D - - I ] P(D)=Ut(D)P(D)U~(D)=[ 2D+3 l

D + 4 '

and

=[ Dz+TD] O_(D)=ULID)Q(D) k 20+5/'

F 2D2+4D+l D2+4D+I R(D)=R(D)UR(D)= ~ 2D2 +6D-1- D -22

as given by (27). Furthermore, in view of (29),

0,,o,=I;l and

. . [ -4D + 1] O'e(D)=O(D)-P(D)Qt(D)= L 2 J"

Next, the quotient of/~(D)P(D)- ~ is determined; i.e.

/~(D) = 2 0

0 0

which, in view of (35), implies that

f D+I

Re(D) R(D) RI(D)P(D ) 1 -D

,] 0

2

Finally, (38)is employed to obtain

The algorithm of Section 3 can now be employed to deter- mine a state-space system which is equivalent to the given differential operator system with PID), Q_e(D), fCeiD), and ITV(D) of this section substituted for P(D), Q(D), R(D), and W(D) respectively, of the previous section. It should be noted, however, that this example was contrived so that the differential operator system obtained here is identical to the one employed in the earlier example. It therefore follows that the state space system derived in Section 3 is also equivalent to the difl'erential operator system presented here, with the state x(t) related to the partial state and input of this example via (40); i.e.

x,(tl = X3lt)

zl(t)--z2(t)--z2(t)--u(t) ] 2l(t)--'d2(t)-- 2(t)--4u(t)--6(t)

22(t)

It is important to note that the algorithm is most direct and efficient. In particular, the matrices A, M, B, and C are immediately apparent by inspection of the appropriate differen- tial operator representations. Furthermore, the determination of B=M IB requires only a single matrix inversion. Also, B directly implies fl(D) which, in turn, directly implies E(D) = W(D)+ Cfl(D). Finally, an explicit relationship (40) between the partial state and input of the given system and the state of the equivalent state-space system is obtained via the procedure.

5. Conclusions A direct and efficient new algorithm has now been presented

for obtaining state-space representations of the form (1) for given, time-invariant systems whose dynamical behavior is expressed in the more general, differential operator form (2). The algorithm is based on an initial reduction, if necessary, of the given system to an equivalent differential operator form where P(D) is both column and row proper, F[P(D)]=I, and both R(s)P(s) i and P(s)- 1Q(s) are strictly proper transfer matrices. Once such a reduction is accomplished, only a single matrix inversion is required to obtain an equivalent state-space representation. Appropriate algorithms for performing these reductions are presented in Section 4. The actual algorithm for then determining an equivalent state space system is given in Section 3.

Acknowledgements--This work was supported in part by NSF Grant no. ENG-76-10442.

References , [1] W. A. WOLOVlCH: Linear Multivariable Systems. Springer-

Verlag, Berlin (1974). [2] R. GUIDORZI: Canonical structures in the identification of

multivariable systems. Automatica 11,361 374 (1975). [3] S. BEGnEt, LI and R. GOIDORZI: A new input output

canonical form for multivariable systems, to appear.