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Structural properties of linear dynamical systemsH. H. ROSENBROCK aa Control Systems Centre, University of Manchester Institute of Science and Technology,Manchester, U.K.Version of record first published: 27 Mar 2007.

To cite this article: H. H. ROSENBROCK (1974): Structural properties of linear dynamical systems, International Journal ofControl, 20:2, 191-202

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INT. J. CONTROL, 1974, YOLo 20, No.2, 191-202

Structural properties of linear dynamical systems

H. H. ROSENBROCKt

The system considered is Ex = Ax + Bu, y =ex, with E singular. It is shown thatinfinite decoupling zeros can be defined, and these Induce a fourfold decomposition, ofthat part of the system giving rise to the polynomial part of the transfer functionmatrix, analogous to Kalman's canonical decomposition. Minimal indices are alsodefined, and are shown to be equal to dynamical indices based on the work of Forneyand studied recently by Roaenbrock and Hayton.

1. IntroductionThe (Laplace-transformed) system

SX=AX+BU}

y,=OX(1)

has been taken as the starting-point for much recent work III control andsystems theory. It gives rise to a transfer function matrix

G(s) =0(s1 - A )-1B (2)

which is strictly proper (that is, tends to zero as s..... co) but this restriction caneasily be removed by adding a term D(s)u to the right-hand side of the secondof eqns. (1), and hence also to G(s), where D(s) is a polynomial matrix.

When 'we wish to consider a system S as an assembly of subsystems Si(Rosenbrock and Pugh 1974) eqns. (1) are no longer adequate. If the sub­systems Si are described by such equations, then these, with the equations ofinterconnection, are a description of S which is not in the form (1). It can bebrought to this form, but in the process some important properties may be lost.

A second defect of eqns. (1) is that whereas the matrices A, B, 0 containstructural information not contained in G(s), no such information about thesystem is preserved with respect to D(s) when this is added. As will be shownbelow, the system S will in general have properties entirely analogous, withrespect to D(s), to the canonical decomposition (Kalman 1962) of (1) withrespect to G(s).

For these reasons we are led to study the more general linear system

(3)y=Ox

sEx=Ax+Bu }

where E, A, B, 0 are real matrices, respectively r x r, r x r, r x l, m x r. Theinteresting case occurs when E is singular, for otherwise (3) reduces to (1) when

Received 16 October 1973.t Control Systems Centre, University of Manchester Institute of Science and

Technology, Manchester, U.K.

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192 H. H. Rosenbrock

the first equation is premultiplied on both sides by E-I. The order of thesystem (3) is (Rosenbrock 1970) the degree n of IsE-AI, which will be lessthan.r if E is singular. There is no need to add a term D(s)u in the second ofeqns. (3), because it will be shown that (3) can already give rise to a transferfunction matrix which has a polynomial part. In the following sections of thepaper the structural properties of (3) will be studi~d.

2. Restricted system equivalence

The system matrix (Rosenbrock 1970) corresponding to (3) is

[

SE - AP(s)=

-0(4)

in which we require that IsE - A I$ O. We shall say that two system matricesP, P', in this form are related by restricted system equivalence (or are r.s.e.) if

o ][SE - A B][N 0] = [SE' - A'

i; -0 0 0 I, -0'B']o = P'(s) (5)

(6)

where 111, N are real r x r non-singular matrices. This transformation is aspociulization of strict system equivalence (s.s.e.) and therefore leaves invariantthe transfer function matrix, the order n, the decoupling zeros, the systemzeros and the system poles (Rosenbrock 1970, 1973).

Theorem 1

The system matrix (4) is r .s.e. to the matrix

Pl(S)=[Sln~AI 1r_n

O

+ S'/ I::]····.·.·.·--·--------------·.····.·------1·.·.·.

-01 -02 i 0

where J is l.l real matrix having all its entries zero except perhaps those in thefirst superdiagonal, where the entries are either 0 or I.

Proof

Because /sE - A I$ 0, sE - A is a regular pencil of matrices (Gantmacher19M)). Hence there exist real non-singular matrices M, N such that

Then we write

[

SI - AlM(sE-A)N= 0

MB=[BI

]

B2

ON = (01 O2 )

(7)

(8)

(9)

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Structural properties of linear dynamical systems

Let the transfer function matrix G(s) arising from (4) be written

G(s) =O(sE - A)-lB= O(s) + D(s)

193

(10)

where O(s) is strictly proper and D(s) is polynomial. Then it follows at oncefrom (6) that

0(s)=01(s1 -A1)-lB1 (11)

D(s) =02(1 +sJ)-lB2 (12)

Theorem 2The proper matrix D(S-l) is

(13)

ProofWe have

(14)

and the matrix 1 +s-V is block diagonal, with the blocks on the principaldiagonal having the form

1 S-l 0 ... 0 0

0 1 S-1 ... 0 0jxj (15)

0 0 0 ... 1 S-l

0 0 0 ... 0 1

the inverse of which is

1 _S-l 8-2 ... (-s)2-i ( -s)1-1

0 1 _8-1 ... (_8)3-; ( - 8)2-i(16)

0 0 0 1 _8-1

0 0 0 0 1

On the other hand, the matrix 81+J is also block diagonal (it is in fact inJordan form) with its blocks corresponding to those of 1 +s-V and having theform

s 1 0 ... 0 0

0 s 1 ... 0 0jxj (17)

0 0 0 ... 8 1

0 0 0 ... 0 S

CON. 20

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194 H. H. Rosenbrock

with inverse

S-1 _S-2 S-3 ... (-s)l-} ( -s)i

0 S-1 _S-2 ... (_S)2-i (-s)l-} ,

. , (18)0 0 0 S-1 _S-2

0 0 0 0 S-1

Comparison of (14), (16) and (18) gives the first result.For the second result we notice that an operation of s.s.e. gives

[SI+J SB2 ] [I-02 0 0

(19)

The first system matrix gives rise to D(S-I) by the preceding result, and there­fore the second system matrix also gives D(S-I). This completes the proof.

3. Infinite decoupllng zerosThe decoupling zeros of (4), namely the i.d., o.d. and i.o.d. zeros, have been

defined elsewhere (Rosenbrock 1970). We now call these the finite decouplingzeros of (4). We define the infinite i.d. zeros of (4) to be equal in number tothose finite i.d.zeros of

'[E-SA B]-0 0

(20)

which are equal to O. When (4) is brought by r.s.e. to the form (6), the sameoperation brings (20) to the form

l~:~:;·~I:~:1~;·1and the only finite i.d. zeros of (21) which are equal to 0 are those of

(21)

(22)

Since finite decoupling zeros are invariant under s.s.e., and therefore also underr.s.e., the finite i.d. zeros of (22) are the infinite i.d. zeros of (4). Similardefinitions and remarks apply to the infinite o.d. and i.o.d. zeros of (4).

It follows from what has just been said that the infinite decoupling zeros of(4) are invariant under r.s.e. They are not invariant under s.s.e., because thesame operation of s.s.e. on (4) and on (20) may produce results which are notrelated in the way that (6) is related to (21).

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Structural properties oj linear dynamical systems 195

The finite decoupling zeros of (4), which are those of (6), are also those of

(23)

because IIr-n + sJ I = 1. These finite decoupling zeros are related to thecanonical decomposition of (23) introduced by Kalman (1962). Explicitly

(i) the finite i.d. zeros are the eigenvalues corresponding to the uncon­trollable part in any canonical decomposition;

(ii) the finite o.d. zeros are the eigenvalues corresponding to the unobserv­able part;

(iii) the finite i.o.d. zeros are the eigenvalues corresponding to that partwhich is uncontrollable and unobservable.

In a similar way the infinite decoupling zeros of (4), which are the finite de­coupling zeros of (22), induce a fourfold decomposition of (22) and therefore ofthat part of (6) giving rise to D(s).

As an example, if (6) has the form

s ..n 0 0

o s+2 0

o 0 s+3

o

o

o

o 0 0 0

o 0 0 1

o 0 0 0

o o o s+4 0 0 0 1

o

o

o

o

o

o

o

o

1 s 0 0

o 1 0 1

o o o o 0 0 1 1.....~ ~ __ _--:::-~-_ .._---:::--~--_ .. __ ..I ..__ ~ ., ~r..~..

then the decoupling zeros are

i.d. zeros {-I, - 3, co}o.d. zeros {-I, - 2, co}i.o.d. zeros {- 1, co}

and the set of all decoupling zeros is {-I, - 2, - 3, co}. The transfer functionis s + 1+ 1/(s +4). Notice that an infinite decoupling zero may cancel a poleat infinity in the usual sense, or it may cancel a constant term.

Infinite decoupling zeros can be removed from (6) by the following pro­cedure. If (22) has, for example, an i.d. zero, then there- exists a real non­singular matrix K such that K(sI +J B 2 ) has one row divisible by s; say row i(Rosenbrock 1970, p. 61). Hence row i of KB2 is zero, and row i of K(I +sJ)

282

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196 H. H. Rosenbrock

(26)

does not contain s. Moreover, K has its entry in position (i, i) equal to 1. Bythe row and column operations of s.s.e. we may bring the matrix

[

K +sKJ KB2](25)

-02 0

tothe form

[:--~~:~~~-~.]Finally we delete the first row and first column of the matrix. In this way rin (.6) is reduced by 1, while G(s) is unchanged. Removal of infinite o.d. zeroscan be ,accomplished in the analogous way.

As an example, subtract row 6 from row 7 in (24), then add the new row 7to row 8, then add column 8 to column 7. There is now a 1 in position (7, 7),and zeros elsewhere in row 7 and column 7. Eliminating row 7 and column 7completes the reduction.

The operation of deleting the first row and column from (26) is permitted inthe definition given earlier (Rosenbrock 1970, p. 59) of system equivalence.This operation can now be interpreted as the removal-of an infinite i.o.d. zero.Removal of finite decoupling zeros is also possible by system equivalence, so'that a pleasing symmetry is achieved.

Theorem 3

Let P, PI be two system matrices having the form (4), neither of which has a(finite or infinite) decoupling zero. Then P, PI are r.s.e. if and only if theygive the same G(8).

Proof

If P and PI are r.s.e. they are s.s.e., and therefore give the same G(s). Toprove the converse, bring P and PI by r.s.e, to the form (6) and call the resultingmatrices P', PI'. Since P' and PI' give the same G, they give the same G.The two matrices of the form (23), obtained from P' and PI' respectively, haveno decoupling zero, give the same G, and hence are system similar (Rosenbrock1970, p. 106). In the same way P' and PI' give the same D(s) and thereforethe same D(s-I). The two matrices of the form (22), formed respectively fromP' and PI" have no decoupling zero, give the same transfer function matrixs-ID(S-l), and hence are system similar. We thus obtain

and the proof is complete.

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Structural properties of linear dynamical systems 197

This theorem extends in a satisfying way the celebrated theorem first givenby Kalman (1963) and proved by Ho and Kalman (1966) and Youla and Tissi(1966). That theorem related to (1), and it extends without difficulty when aterm D(s)u "is added on the right-hand side of the second equation (as inRosenbrock 1970): essentially Kalman's theorem is then being applied tcO(s) = G(s) - D(s). The generalization in Theorem 3, on the other hand,demands a lack of redundancy in the way the system generates D(s) which isentirely analogous to the lack of redundancy, demanded by Kalman's theorem,in the way that the system generatesO(s).

4. Minimal indicesThe matrix (sE - A B) which forms the first (block) row of P(s) in (4) is a

singular pencil of matrices (Gantmacher 1959). The invariants of such apencil under the transformation

[

N ll

M(sE-A B)

N'1

(28)

where the transforming matrices are real and non-singular, were investigatedby Kronecker. A complete set of invariants comprises

(i) the finite elementary divisors,

(ii) the infinite elementary divisors,(iii) the minimal indices for the rows,

(iv) the minimal indices for the columns.

A particularly interesting case is the one in which P(s) has no (finite orinfinite) input decoupling zero. We then have the following result.

Theorem 4

If P(s) has no (finite or infinite) i.d. zero, then the matrix (sE - A B) has nofinite elementary divisor and no minimal index for the rows. It has infiniteelementary divisors, each of degree 1, equal in number-to the rank defect of E.It has I minimal indices for the columns, which when P(s) has no finite orinfinite o.d. zero are the dynamical indices for G(s) = C(sE - A )-1Bstudied inan earlier paper (Rosenbrock and Hayton 1974).

Proof

The action of r.s.e. upon (sE-A B) is represented by (28) with N1.=0,

N 21=0, N 22=I. Consequently we may choose P(s) in the form (6), and havethen to find the invariants of

o(29)

If this pencil had a finite elementary divisor, then study of Kronecker'scanonical form (Gantmacher 1959, especially p. 48) shows that the rows of

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198 H. H. Rosenbrock

(29) would be linearly dependent for some .finite (real or complex) s =sQ'But then So would be a finite i.d. zero of P(s), contradicting the assumptionsof the theorem. If the pencil had a minimal index for the rows, these wouldbe linearly dependent for all s, contradicting the assumption that IsE - A I $ O.

The infinite elementary divisors of (29) are those finite elementary divisorsof the pencil

(30)

which have the form ,.,P. Among the minors of (30) of order t > n is one atleast having the form

(31 )

which is divisible by ,.,.k if and only if IQ2("") I is divisible by ,.,.k. Hence the finiteelementary divisors of (30) of the form ,.,." are those of

(32)

Correspondingto each block of ,.,.1 +J there is a set of rows in the matrix(,.,.1 +J B 2 ) having the form

0 ... ,.,. 1 0 ... 0 0 0 ... b1

0 ... 0 ,.,. I ... 0 0 0 ... b2

0 ... 0 0 0 ... ,.,. I o ... bj _ 1

0 ... 0 0 0 ... 0 ,.,. 0 ... bj

The corresponding rows of (,.,.1 +J JB2 ) are

0 ... ,.,. 1 0 ... 0 0 0 ... b2

0 ... 0 ,.,. 1 .: 0 0 0 ... b3

0 ... 0 0 0 ... fL .1 0 ... b j

0 ... 0 0 0 ... 0 fL 0 ... 0

(33)

(34)

The vector bj in (34) contains a non-zero entry, for otherwise (33) would not havefull rank when u = 0, and P(s) would have an infinite i.d. zero, By columnoperations we can eliminate all constant entries in that column of (,.,.1 +J JB 2 )

which has a I in ,the same ~ow as bj in (34), leaving only multiples of fL. By row

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Structural properties of linear dynamical systems . 199

operations we can eliminate all but one of these multiples to leave other rows of(/11 +J JB2 ) unchanged and those under consideration of the form

o ... ~ 1 0 ... 0 0 0 ... b2

0 ... 0 ~ 1 ... 0 0 0 ... ba

0 ... 0 0 0 ... ~ 0 0 ... b j

(35)

0 ... 0 0 0 ... 0 I-' 0 ... 0

The first j -1 rows of (35) are the last j - 1 rows of (33) and these and similarrows from other blocks are linearly independent when ~ = 0, because P(s) has noinfinite i.d. zero. From this it follows that the Smith form of (~l +J JB2 ) is

[lT~n_q ~:J (36)

where q is the number of blocks in ~l+J. But the highest degree of anyminor in (29) is easily seen to be r - q, and this degree is invariant under r.s.e.Consequently q is the rank defect of E. This proves the assertion regardinginfinite elementary divisors. It also shows that the system matrix on theright-hand side of (19) has q finite i.d. zeros.

The Kronecker canonical form for (29) is now seen to be

(37)

where the blocks Q;(s) are -\ x (-\ + 1) matrices] corresponding to the minimalindices ,l.; for the columns. Because (29) is r x (r + I), it follows that j = 1 in(37). That is, there are exactly 1minimal indices for the columns of (sE - A B).

When P(s) has no finite i.d. or o.d. zero, the McMillan degree o(G) of G(s)=O(sE-A)-IB is (Rosenbrock 1970, p. 137) n+v[D(s-I)]. We have shown

. above that the system matrix on the right-hand side of (19) has q finite i.d.zeros, and when P(s) has no infinite o.d. zero, the system matrix in (19) has nofinite o.d. zero. Consequently

o(G)=n+ v[D(s-I)]=n+r-n-q=r-q (38)

On comparing this with (37) we see thatI

L .:\;=o(G)i= 1

(39)

Now the minimal indices -\ are also defined as the degrees of certainpolynomial solution vectors of the equation

o

l+sJ(40)

t When ,\;=0, the corresponding Q; is 0 1 I' and (37) is interpreted as in Gant-macher (1959). ' .

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200 H. H. Rosenbrock

The output y(i)(S) corresponding to such a solution vector is, from (6),

y(i)(s) =C,x(i)(s)+ C2g(i)(S) (41)

(40)

The maximum degree '\ of the solution vector may occur in u(i)(s) or in g(i)(s) :it cannot occur in x(i)(s) = (s1 - A,)-'B,u(i)(s). If this maximum degree occursin U(i)(S) and not in g(i)(s), then (41) shows that S(y(i») ~ -\ -1. If it occurs ing(i)(S) (and perhaps also in u(i)(s)), then Styli») ~ -\.

Furthermore, there are I solution vectors of (40), defining the I minimalindices, and the corresponding u(i)(s) are linearly independent considered asvectors over the rational functions. For if the contrary were true there would(Rosenbrock 1970, p. 20) be polynomials "'i(S) such that

I

L "'i(S)U(i)(s) = 0i "" 1

Then (40) gives

I [ X(i)(S)] I [(S1 - A,)-'B,U(i)(S)].L "'i(S) g(i)(s) =.L "'i(S) (1+&1)-1B2U(~)(S).=1 -u(t)(s) t e I -u(i)(s) ,

I

(s1 - A 1).,-IB1 L "'i(S)u(i)(s)i=l

I

(1+sJ)-'B2 L''''i(S)U(i)(s) =0 .(43)i=l

I

- L "'i(S)U(i)(s)i=l

which is impossible because the solution vectors of (40) which define the Ai arelinearly independent. .

Now assemble the vectors u(i)(s) and the corresponding y(i)(s) to formmatrices •

T(s) = (u(l)(s) U(2)(S) ... u(l)(s)), I x I

Vis) = (y(l)(s) y(2)(S) .. , y(l)(s)), m xl

where IT(s) 1;$ 0 because' the u(i)(s) are linearly independent.y(i)(s) = G(s)u(i)(s),

Vis) = G(s)T(s)

G(s) = V(S)T-l(S)

and G(s) arises from the system matrix

[ ~ T~S) 1~].~._-----------~_.._---_._- \

o - Vis) 0

(44)

(45)

Then since

(46)

(47)

(48)

The complexity c of this system matrix is by definition (Rosenbrock 1974)the sum of the column degrees. If for some i the highest degree Ai in the

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Structural properties of linear dynamical systems 201

(49)

solution vector to (40) occurs only in u(i)(s), and therefore in the ith column ofT(s), then it was shown above that the ith column of V(s) has degree o(y(i»)';:;\ - 1. The corresponding column of (48) then has degree ,\;. If the highestdegree in the solution vector occurs in g(i)(S), then in the same way the corres­pendingcolumn degree in (48) does not exceed Ai' We therefore have

I

c,;:; L \= o(G)i= 1

by (39). But it is known (Rosenbrock 1974) that c~o(G), whence equalityholds in (49) and the column degrees in the second (block) column of (48) areprecisely ,1.1> ,1.2' ... , ,1./.

Now because c =o(G), the system matrix (48) is minimal, and the columns of

[

T(s) ]

- V(s)(50)

are a minimal basis in the sense of Forney (1974; see Rosenbrock 1974).The column degrees \ of (50) are therefore the dynamical indices 0i studied inRosenbrock and Hayton (1974). This completes the proof.

5. RemarksThe above theory has been given for the field 8£ of real numbers. With

small changes (as in Forney 1974) it can be extended to other fields fF. Asthere is no apparent technical application to fields other than 8£ (Rosenbrockand Hayton 1974) the generalization has not been made here.

The theory of infinite decoupling zeros given above is an application ofKronecker's theory of infinite elementary divisors. The chief problem in thisapplication is that the infinite elementary divisors which occur most naturally,as in (32),' are the finite elementary divisors of (sI +J JB2) , not those of(sI +J B 2 ) . The first of these matrices always has finite elementary divisorswhen E is singular, whereas in generalizing the idea of decoupling zeros-we wishto exhibit systems which have no infinite decoupling zero. We are thereforeled to define the decoupling zeros by way of the second matrix. Theorem 3shows that this leads to results 'of the type we are seeking.

Having adopted this course, we find as in Theorem 2 that we are led toconsider the finite decoupling zeros of the system (22), which gives S-ID(s-I)rather than D(S-I). An infinite decoupling zero may therefore cancel a con­stant term which would otherwise occur in D(s), as for example the infinitei.o.d. zero in (26). Such a constant term in D(s) does not affect the McMillandegree, so that the connection between infinite decoupling zeros and McMillandegree becomes awkward.

Some of these difficulties, but not all, can be overcome by replacing (4) by

[SE-A B]

-0 Do(51 )

where Do is the constant term in D(s). This, however, does not fit well withthe requirements of compound matrices (Rosenbrock and Pugh 1974).

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202 Structural properties-o] linear dynamical systerruJ

In the theory as developed, the following quantities appear to have aspecial significance.

(i) The number r (the dimension of A) appears as a generalization of theorder n. .

(ii) The least value of r which can occur among all systems (4) giving rise toG(s)=G(s)+D(s) is e=v(G)+v[s-lD(s-l)], where v denotes the least ordercorresponding to the strictly proper rational matrices. This number e appearsas a generalization of least order. We have e ~ 1>( G), where 1>( G) is the McMillandegree of G, and equality holds if and only if D(s) == O. Also r =e if and only ifthe system has no finite or infinite decoupling zero.

(iii) The degree d defined earlier (Rosenbrock 1974) is d=n+ v[D(S-l)],where v[D(S-l)] means the least order corresponding to the strictly proper partof the proper rational matrix D(S-l). We have d ~ I>(G), with equality holdingif and only if the system has no finite decoupling zero. The degree d appearsas a partial generalization of order, taking no account of infinite decouplingzeros in (4).

(iv) The McMillan degree is I>(G) = v(G) + v[D(S-l)]. It appears as a partialgeneralization of least order, taKing no account of that part of the system whichgenerates the constant term in D(s). It is known to have special significancein defining the least number of reactive elements needed to realize G in electricalnetwork theory.

(v) The complexity c (defined above and in Rosenbrock 1974) is thenumber of initial conditions, not necessarily independent, used in obtaining theLaplace transformed equations represented by (4). It is also the number ofdynamical elements in appropriately defined compound systems (Rosenbrockand Pugh 1973). We have c ~ d; and if E is diagonal in (4), as in certaincompound system matrices, then c=r-q, the rank of E.

(vi) The order n, which has its usual significance.

In developing the theory the course taken was the one which seemed mostilluminating from the viewpoint of systems theory. So for example, Theorem 3was proved by two applications of Kalman's theorem. A more coherent andlogically appealing account could no doubt be given by redeveloping the theoryin a purely algebraic way.

REFERENCESFORNEY, G.·D., 1974, SIAM J. Control (to be published).GANTMACHER, F. R., 1959, Applications o{ the Theory o{ Matrices (Interscience).Ho, B. L., and KALMAN, R. E., 1966, Regelungstechnik, 14,545.KALMAN, R. E., 1962, Proc. natn. Acad. Sci., Wash., 48,596; 1963, SIAM J. Control,

A, 1, 152.ROSENBROCK, H. H., 1970, State-space and Multivariable Theory (Nelson-Wiley);

1973, Int. J. Control, 18, 297, see also a correction to appear; 1974, Int. J.Control, 19, 323.

ROSENBROCK, H. H., and HAYTON, G. E., 1974, Int. J. Control, 20,177.ROSENBROCK, H. H., and PUOH, A. C., 1974, Int. J. Control, 19,845.YOULA, D. C., and TISSI, P., 1966, I.E.E.E. Int. convent. Rec., Part 7, p. 183.

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