- Email: [email protected]
On the zeros of transfer functions of delayed systems
Volume
I. Number
SYSTEMS
3
& CONTROL
LETTERS
November
19X I
On the zeros of transfer functions of delayed systems *
Received 7 August I98 I Revised 13 September 1981
Let T(z) be the transfer function of a system u-irh input. SIBIC and outpul delays. The zeros of T(z) MacMillan-Smith form for meromorphic marrices. and some of their propertics arc investigated.
Kqw~ords;
Delayed
System.
Transfer
function.
Zeros.
arc dcfincd
using
a (local)
Poles.
1. Preliminaries
The zeros of linear autonomous systemsof ordinary differential equations have been widely studied (see [1,2] and the references therein). In spite of this, the zeros of delayed systems have been rarely dealt with (for example see[3]). In this paper we consider the system (S) which is governed by the equation i=Ax,(.)+Bu,(.),
fro,
(1)
with output Y= cx,(*)
(2)
where x E R”, y E R”o, u E R”l. We assumethat u(l) is a continuous function. Let h be a fixed positive number. If x(r) E C( -h, cc), x,( .) is the (continuous) function from (0, + CL))in C,, = C( -h, 0) defined by x,( *) = x(f + s), s E [--A, 01.y,( .), u,( .) are defined in a similar way. If cpE C,,, the operator A is defined by AT(.)
= ; 49(-h,) t=o
+I0
A(s)v(s)
- It
ds,
O=h,
G-h.
A, are constant n X n matrices, A(s) is a square integrable n X n matrix. The operators B and C are defined in a similar way: Bu(.)=
ds,
f: 13,0(-h,)+j-~,fB(s)o(s)
r=O
Cd.)
= i
Cd-h,)
+/_u,,Wd4
ds,
t==O
where the matrices Bi, B(s), C,, C(s) have suitable dimensions, and B(s), C(S) have square integrable elements. It is well known that for every initial data x(r) = cp(r), u(r) = u(r), r E [-h, 01, x(O) = q”, and every control u(t), f 3 0, (1) has a unique solution 40
=x(c(~“d~),u(~)),
2.4)
(the initial time will always be zero). * Paper written 204
under
the auspices
of the GNAFA
group
of the Consiglio
Narionale
delle Ricerchc
0167-691l/81/0000-0000/$02.75
0 1981 North-Holland
Volume
1. Number
3
SYSTEMS
& CONTROL
LETTERS
If the initial data is set to be zero and if we take the Laplace transform 3(z) (whenever
7-(z)=C(z)A-‘(z)B(z).
= T(z)4z),
it appears, L is the identity
Novcmbcr
1981
of (I), (2) we find that
A(z)=zl-A(z)
matrix of suitable dimension).
A(z)
= $ A, exp( -zh,) i=o
+/:,?A(s)
e” ds,
B(z)
= i B, exp( -z/r,) t=o
+/:,,E(s)
e” ds,
C(z)
= i C, exp( -A,) r=O
+/:,C(s) I
e” ds.
We put d(z) = det A( z). The equation d( 2) = 0 is called the characteristic equation of (1). In [4] the following definition was used: Let z. be a pole of T(z). Let U= (U’(Z),..., u”(z)) be a set of entire functions such that T(z)u(z)=u,/(i-z,)”
+f;(z),
r, >O,
whereL(z)(z - z~)~I is bounded, and the vectors u, are independent. If n?(U) is the sum of the exponents r,, the order of z. as pole of the transfer function T(z) is mr( zo) = max,{m(U)}. In this paper we study the zeros of r(r). Several (not equivalent) definitions can be given, exactly as in the case of rational transfer functions (see [l]). We start with the following: Let z. be a complex number such that T(z) is bounded near zo. Let m be the greatest number such that at least one m X m minor of T(z) is not identically zero. Let p > 0 be the greatest number such that (z - zo)P divides all the m X m minors of r(z). We say that z. is a zero of T(z) when p is positive. The number p is the multiplicity of zo, and is denoted m,( zo). Remark 1.1. When T(z) is a rational of all its minors of maximum order.
function,
many authors say that z0 is a zero of T(z) when it is a zero
If T(z) is a rational matrix, a point z. can be both a pole and a zero of T(z). This situation is studied using the MacMillan-Smith form for rational matrices. In our case T(z) is a matrix of meromorphic functions. In the next section we study its poles and zeros using an extension of the MacMillan-Smith form which holds locally. near a given point. This form is sufficient for our needs.
2. A local block MacMillan-Smith
form for T(z)
We recall the following local Smith form for a matrix M(z) which is holomorphic near zo. We assume that M(r) is an m X p matrix, with m >p. The case m Gp can be treated with obvious modifications. It is possible to prove the existence of two holomorphic matrices F,(z), F,(z) such that det( e(z)) = 1, i = 1.2, and F,(z)M(z)F,(z)=[diag((z-zo)“M,(z),(z-zo)”LM,(z),...,(z-zo)”M,(r))~~] 1. If det(M,(t,))=O, then where Mi(z) are d, X d, holomorphic matrices. det(M,( ro)) # 0 if 1 GiGs-M,(z) c 0. We assume that the indices have been ordered so that Y, < v2 < . . .
A(z)]B(z))/dz) 205
SYSTEMS & CONTROL LETTERS
Volume I. Number 3
November I98 I
so that (near r,,) we can write F,(r)T(z)F,(~)=[diag((z-z,)~‘~,(z),...,(z-z~)~~~~(z)))O]
=S(z)
then ns =O, otherwise n,-, ~7~. where as(z) are holomorphic, and 9, C n2 < * * * c;ql,-,. If Ms(z)=O, The numbers vi are entire (and may be negative). We call the matrix S(z) the local block MacMillan-Smith form of 2”(z). Let r be such that 7, < 0, n,+ , 2 0. We associate with z,, the two matrices d r+I d,+2 . . . d, 77,+1 ?,+2 **. 7, (the first matrix is not defined if 0, 2 0, and in this case we assume r = 0, while the second one is not defined if 9s C 0, and in this case we assume r = s). Now observe that if all the exponents vi are nonnegative, z0 is a zero according to the definition of Section 1 when at least one of the exponents is positive. This justifies the following definition: Definition 2.1. A complex number z0 is a zero of r(z)
when in the local MacMillan-Smith
form of T(z)
near z0 one of the exponents 17,is strictly positive. We have the following result: Theorem 2.1. I’ zOis a pole of T(z), then
If
z,, is a zero, but ndt a pole, of T(z), then fin
=
2 diqi* i:q,=-0
Proof. We prove the first assertion. Let T(z)u,(z) zero in to, and r, > 0 (i = 1,. . . , k). Then
= (z - zo)-‘tcpi(z),
qi(z) holomorphic, different from
F,(z)T(z)F,(z)F,-i(z)~;(z)~=S(z)~2-’(z)u(z)=(z-z,)-“F,(z)rp;(z) and of course Ft( zo)qi( zc) Z 0. Moreover, if { cpi(zo)} are independent are independent, and conversely. Hence, 4-b,)
=msbo>
=
2 t=l
vectors, also the vectors F,( zo)‘pi( zo)
d,v;.
Now we prove the second assertion of the theorem. It is clear that fff~(Z~)
=
2 i:q,
diqi* >O
So we must prove that iEr( zO) = ms( zo). We can see that mr( zo) < m,( zo). We remember that the normal rank of the matrix T(z) is the number max,(rank T(z)). Let it be k. Then also k = (normal rank S(z)). Hence we must show that if (z -to)? divides all the minors of T(z) of order k, (z - zo)’ divides all the minors of S(z) of order k. This is obvious, since the minors of S(z) of order k are sums of products one factor of which is a minor of order k of T(z). Hence mr( to) G #?,( zo). Interchanging the role of T(z) and S(z) we have the converse inequality. This finishes the proof. q The second assertion of the above theorem shows that we can define the multiplicity T(z) as
of z. as a zero of
Volume
I. Number
3
SYSTEMS
Remark 2.1. For some applications ~r(~o)=min(q,;
& CONTROL
LETTERS
November
19X1
it could be useful to define
qi >O).
3. An application As an application of the above material, we can consider the series connection of canonical systems. The question is wether the tandem system is canonical or not. A sufficient condition, which is well known in the ordinary case [7], can easily be generalized. The results that we obtain should be compared also with the theorem in [8] for the series connection of shift canonical realizations in Hilbert spaces. In [4] we called ‘canonical’ a system which is fully detectable and fully stabilizable, i.e. a system (S) such that rank[A*(r), C*(Z)] =h, Vz, (3) rank[A(z),
B(Z)] =n,
VZ.
(4) Let (S,) be two canonical systems (j = 1,2). The number i will denote any quantity that is referred to system (Si). Assume that nb = nf, so that the two systems can be connected in series, to form the new system (S). The transfer function of (S) is T(z) = T’( z)T,( z). Assume that ni z nf, fl: an b. We can show that under some conditions on the poles and zeros of the two systems, the tandem system (S) is canonical. For this we need a lemma which is the converse of Theorem 4.2 in [4]. Let mO be the order of zO as zero of d(z). Lemma 3.1. If mT( zO) = m, for euery pole zO oj T(z), then (S) is canonicul. Proof. Assume that (S) is not canonical. There exists a point zO such that (3) and/or (4) does not hold. We proved in [4] that near z0 T(z)u(z) may be written T(z)u=[Co(z)(zI-A,)-‘l&(z)
+ F(z)]u(z)
where F(z) is bounded near zc, and the characteristic polynomial of A, is (z - zO)“lO. To obtain this expression for T(z) we simply project (S) on the generalized eigenspace of zc. If, for example, (4) does not hold, then rank[zl-A,,
B,(Z)]
cm,,
forz=zO.
This means [5] that the system A,, B,(z) may be decomposed in a reachable part and in a part that is not effected by the control, so that the order of the pole zO of (zl- A,)-‘&( z) is less then m,. against our assumption, as we can see if we calculate (zl - A,)-’ using the Jordan form of A,, and if we note that B,(z) and C,(z) are bounded near zO. 0 Hence we must prove that 443)
=m,,(zo)
+%+o>.
We need the following lemma: Lemma 3.2. If z,, is not a zero of T(z), then the order of to as pole of T(z) is the greatest of the orders of z0 as pole of the minors of order k of T(z), k = normal rank T(z). Proof. The assertion is clear for S(z). Pre- and post-multiplying T(z) for regular invertible matrices does not change the order of a pole. Let r be the maximum order of z. as pole of the minors of order k of T(Z). From the Bin&t-Cauchy theorem, r =Gm,(z,) (compare with the proof of Theorem 3.1). Exchanging the Cl role of T(z) and of S(z) we find the equality. 207
Volume
I, Number
SYSTEMS
3
& CONTROL
LETTERS
November
I98 I
3.1. L.et T(r) = T,(z)T,(z). Assume that ni an:, n) 2 nb. Assume that when zO is a pole of T,(z), z,, is not a zero of q.(z) (j # k) and that not all the minors of maximum order of T(z) are identically zero. Then Theorem
4zd
= w,(z0)
+wJzd.
We start considering the case that z0 is a pole and a zero of T,(z), so that rank T,(z,) = nf. In this
Proof.
case w44
= T,w+M4.
T,(z) is analytic in z,,. If r< m,,(z,),
consider that
(z-z,)‘T(z)u(z)=T,(z)[T,(z)u(z)(z-z,)r]
f-0
forz-z,,
since ker Tz( z,,) = { 0). Hence m.(zo)~mT,(zg)=m.,(z,)+nr,~(zo). The converse inequality is obvious. If z0 is a pole and a zero of T’(z), then T,(z) is analytic in zO, and rank T,(z,) = nf. Hence for every u(z) there exists an analytic function o(z) such that u(z) = T,(z)o(z) near z,,, and we can see directly by the definition that mA+>
=mT2(zO) =mTl(zo> +mT,(zo).
The last casethat we must examine is the casein which z0 is a pole of both T,(z) and T,(z). We use Lemma 3.2. Any minor of maximum order of T(z) is a sum of products of minors of maximum order respectively of T’(z) and of r,(z), so that
Obviously, by the very definition of m,(z,),
the inequality cannot be strict. This finishes the proof.
Cl
As a consequencewe have the result: 3.2. Assumethat the conditions of the above theorem hold. If the systems(S,), i = 1, 2, are canonical, then the tandem system is canonical. 0 Theorem
4. A local fraction
decomposition
of T(z)
Let US consider again the local Smith-MacMillan negative. Then F,(z)T(z)&(z)= - ([diag(M,(z),...,
form. As above, r is the last index such that nlr is
([diag((z-z,)“Z,...,(z-zO)“‘Z,Z,...,Z)]) M,(z),
(z-z,)““~M,,,(z),...,(z-z,)%,(z))10])
=d,‘(z)NL(z)
= ([diag(M,(r),...,M,(z),(L.-z~)~~+~~~+,(z),...,(z-z~)~~~~(z))~~]) * ([diag((z-z,)“Z,...,(z-z,)“‘Z,Z,...,Z)])
=fi,(z)d;‘(z)
at least in a neighboorhood of zc. Hence T(z) =[oL(z)F,(z)]-‘[~,(z)F2-I(z)] and, analogously, T(z) =N,(z)D,‘(z). 208
=D;‘(z)N,(z)
(6)
Volume
I. Number
3
SYSTEMS
Hence, the Smith-MacMillan The following ‘coprimeness’ rank[Q(z,), rank[Nf:(z,),
& CONTROL
form of T(z) provides conditions hold:
LETTERS
November
a local right or left fraction decomposition
198 I
of T(Z).
= normal rank[ DL( z), N~( z)],
K.(z~)]
= normal rank[N;(z),
Do]
D;(z)],
(7) (8) of T(z) satisfies (7) or (8)
for every choice of z,,. In general, if any local right or left fraction decomposition for every z, we say that it is a comprime decomposition. In the following we consider a local left coprime factorization of T(z). The results that we find can be easily translated to the case of right coprime factorizations. We drop the index L, because no confusion can arise. 4.1. If (7) holds, then there are matrices P(z), Q(z) (holomorphic near zO) such that the equality
Lemma
D(z)P(z)+N(z)Q(z)=I
(9)
holds in a neighbourhoodof zo, and conuersely. Proof. It is sufficient to show that [D(z), N(z)]X(z) = u has a holomorphic solution for every vector u. Observe that the normal rank of [D(z), N(z)] is n,, because D(z) is invertible near zO. Let us choose n, independent columns of [D(i), N(z)]. Let P(z) be their matrix. The rows of X(z) which do not correspond to columns of [D(z), N(z)] appearing in P(z) may be put zero, while the others can be (locally) determined by the rows of P-‘(z). The converse part of the lemma is obvious. 0 Now in the spirit of [9], we can investigate the relationships of the zeros of T(z) with the condition rank iV( za) < normal rank N(z).
(10)
We have the following result: Theorem
4.1. (10) holds if and only if m,( zO) > 0.
Proof. If N(z) is that special matrix which is provided by the local MacMillan-Smith form of T(z) near za, the assertion of the theorem is seen to hold by inspection. Now we prove that if (10) holds for one coprime local decomposition of T(z), it holds for any other coprime decomposition. Let T(z)=D-‘(z)iv(z)=D-‘(z)rn(z), so that F(z)
=D(z)DP(z)N(z).
Premultiplying (9) by D( z)D-‘(z) &z)D-‘(z)
we have
=D(z)D-‘(z)D(z)P(z)
+D(z)D-‘(z)N(;)Q(z)
=D(i)P(z)
+F(z)Q(z)
which is a holomorphic function. Now we assumethat (10) holds. Then rank m( za) G min(rank D( z)D-‘(
z) IZZZ,,,rank N( z,))
< min(norma1 rank D( z)D-‘(
I), normal rank N(z))
= normal rank iV( z) = normal rank m( z ). This finishes the proof.
q
Remark 4.1. See [lo] for the corresponding result in the case that Yf(z) is a rational function. We finish this paper with an observation about the decoupling properties of the zeros of T(Z). 209
Volume
I, Number
3
SYSTEMS
& CONTROL
LETTERS
Novcmhcr
19X I
Let z0 be a zero (not a pole) of T(z), and consider the system i=A(t,)x+B(z,)u, which is is a zero T&z) in u 0 E R”l
y = C(z,)x,
described by an ordinary differential equation. Let T,(z) be its transfer function. It is clear that z0 of T,(z). If mr,,(zO) is positive and finite, from the above theorem we know that z0 is a zero of the sense of [9], and that T,(z) is not identically zero. Hence there exist vectors x0 E R” and such that
(zol-A(r,))x,
=B(zo)uo,
C(z,)x,
=o.
) -h. Then x(t) = x0 exp( zof ) is the solution Letx(t)=xoexp(zor),rE[-Izh,0],andu(~)=uoexp(zof),t of (1) with the given initial condition and control. The corresponding output is given by v(f)
= C(zo)xo
exp(zot)
which is identically zero. Under the above conditions, assume that, moreover, no G n,. We know from [9] that we can find a vector c E R”II such that for every u. there exists a vector x0 which satisfies the relations (zol-A(zo))xo
=B(z,)u,,
c*c(zo)xo
=o.
Hence, if we refer to the system (S), for every control U(I) = u. exp( z,t), there exists a solution x(t) such that the output given by (2) satisfies c*r(t) G 0.
of (1)
References [I] F. Fallside. Editor. Co~rrol .S,r~slern Design h,, Pole Zero Assrgwmwr (Academic Press. London. 1977). [2] A.G.J. MacFarlane and N. Karcanias. Poles and zeros of linear multivariable systems: A survey of the algebraic and complex variable theory, fnrernur. J. Co11rrol24 (1976) 33-74. [3] K.M. Prz@ski, Zeros of linear distributed parameter systems with application to the theory of delayed systems. Report. Inst. 01 Elect. Found.. Warsaw Technical University (1979). [4] L. Pandolfi. Canonical realizations of systems with delays. Rapport0 interno. 1st. Matcmatico del Polit.. Torino. Ser. II, No. I3 ( 1980). (51 L. Pandolfi. Canonical realizations of systems with delayed controls. Rmw/~e A~rrom. IO ( 19X0) 27-37. [6J F. Kappel and H.K. Wimmer. An elementary divisor theory for autonomous linear lunctional diffcrcntial equations. J. Di//eretrrtul Equurrom 2 I ( 1976) I34- 147. [7] H.H. Rosenbrock. Srore Spuce uptd Mu/ricunoh/e T/tear), (Thomas Nelson and Sons, London, 1970). [8] P.A. Fuhrman, On series and parallel coupling of a class of discrete time infinite-dimensional systems, SIAM J. ~onrrol Oprim. I4 (1976) 339-358. [9] C.H. Desoer and J.D. Schulman. Zeros and poles of matrix transfer functions and their dynamical intcrprctation. f,5’EE Trurrs. Circuits u~7d .Qsrenrr. 2 I ( 1974) 3-8. [IO] W.A. Wolovich. Multivariable systems zeros. in: F. Fallside. Ed.. Co,rrro/ ~~..~rerrts Des/,g,t ht. /‘o/e Zero Assrgrmte,t( (Academic Press. London, 1977) pp. 226-236.
210
I. Number
SYSTEMS
3
& CONTROL
LETTERS
November
19X I
On the zeros of transfer functions of delayed systems *
Received 7 August I98 I Revised 13 September 1981
Let T(z) be the transfer function of a system u-irh input. SIBIC and outpul delays. The zeros of T(z) MacMillan-Smith form for meromorphic marrices. and some of their propertics arc investigated.
Kqw~ords;
Delayed
System.
Transfer
function.
Zeros.
arc dcfincd
using
a (local)
Poles.
1. Preliminaries
The zeros of linear autonomous systemsof ordinary differential equations have been widely studied (see [1,2] and the references therein). In spite of this, the zeros of delayed systems have been rarely dealt with (for example see[3]). In this paper we consider the system (S) which is governed by the equation i=Ax,(.)+Bu,(.),
fro,
(1)
with output Y= cx,(*)
(2)
where x E R”, y E R”o, u E R”l. We assumethat u(l) is a continuous function. Let h be a fixed positive number. If x(r) E C( -h, cc), x,( .) is the (continuous) function from (0, + CL))in C,, = C( -h, 0) defined by x,( *) = x(f + s), s E [--A, 01.y,( .), u,( .) are defined in a similar way. If cpE C,,, the operator A is defined by AT(.)
= ; 49(-h,) t=o
+I0
A(s)v(s)
- It
ds,
O=h,
G-h.
A, are constant n X n matrices, A(s) is a square integrable n X n matrix. The operators B and C are defined in a similar way: Bu(.)=
ds,
f: 13,0(-h,)+j-~,fB(s)o(s)
r=O
Cd.)
= i
Cd-h,)
+/_u,,Wd4
ds,
t==O
where the matrices Bi, B(s), C,, C(s) have suitable dimensions, and B(s), C(S) have square integrable elements. It is well known that for every initial data x(r) = cp(r), u(r) = u(r), r E [-h, 01, x(O) = q”, and every control u(t), f 3 0, (1) has a unique solution 40
=x(c(~“d~),u(~)),
2.4)
(the initial time will always be zero). * Paper written 204
under
the auspices
of the GNAFA
group
of the Consiglio
Narionale
delle Ricerchc
0167-691l/81/0000-0000/$02.75
0 1981 North-Holland
Volume
1. Number
3
SYSTEMS
& CONTROL
LETTERS
If the initial data is set to be zero and if we take the Laplace transform 3(z) (whenever
7-(z)=C(z)A-‘(z)B(z).
= T(z)4z),
it appears, L is the identity
Novcmbcr
1981
of (I), (2) we find that
A(z)=zl-A(z)
matrix of suitable dimension).
A(z)
= $ A, exp( -zh,) i=o
+/:,?A(s)
e” ds,
B(z)
= i B, exp( -z/r,) t=o
+/:,,E(s)
e” ds,
C(z)
= i C, exp( -A,) r=O
+/:,C(s) I
e” ds.
We put d(z) = det A( z). The equation d( 2) = 0 is called the characteristic equation of (1). In [4] the following definition was used: Let z. be a pole of T(z). Let U= (U’(Z),..., u”(z)) be a set of entire functions such that T(z)u(z)=u,/(i-z,)”
+f;(z),
r, >O,
whereL(z)(z - z~)~I is bounded, and the vectors u, are independent. If n?(U) is the sum of the exponents r,, the order of z. as pole of the transfer function T(z) is mr( zo) = max,{m(U)}. In this paper we study the zeros of r(r). Several (not equivalent) definitions can be given, exactly as in the case of rational transfer functions (see [l]). We start with the following: Let z. be a complex number such that T(z) is bounded near zo. Let m be the greatest number such that at least one m X m minor of T(z) is not identically zero. Let p > 0 be the greatest number such that (z - zo)P divides all the m X m minors of r(z). We say that z. is a zero of T(z) when p is positive. The number p is the multiplicity of zo, and is denoted m,( zo). Remark 1.1. When T(z) is a rational of all its minors of maximum order.
function,
many authors say that z0 is a zero of T(z) when it is a zero
If T(z) is a rational matrix, a point z. can be both a pole and a zero of T(z). This situation is studied using the MacMillan-Smith form for rational matrices. In our case T(z) is a matrix of meromorphic functions. In the next section we study its poles and zeros using an extension of the MacMillan-Smith form which holds locally. near a given point. This form is sufficient for our needs.
2. A local block MacMillan-Smith
form for T(z)
We recall the following local Smith form for a matrix M(z) which is holomorphic near zo. We assume that M(r) is an m X p matrix, with m >p. The case m Gp can be treated with obvious modifications. It is possible to prove the existence of two holomorphic matrices F,(z), F,(z) such that det( e(z)) = 1, i = 1.2, and F,(z)M(z)F,(z)=[diag((z-zo)“M,(z),(z-zo)”LM,(z),...,(z-zo)”M,(r))~~] 1. If det(M,(t,))=O, then where Mi(z) are d, X d, holomorphic matrices. det(M,( ro)) # 0 if 1 GiGs-M,(z) c 0. We assume that the indices have been ordered so that Y, < v2 < . . .
A(z)]B(z))/dz) 205
SYSTEMS & CONTROL LETTERS
Volume I. Number 3
November I98 I
so that (near r,,) we can write F,(r)T(z)F,(~)=[diag((z-z,)~‘~,(z),...,(z-z~)~~~~(z)))O]
=S(z)
then ns =O, otherwise n,-, ~7~. where as(z) are holomorphic, and 9, C n2 < * * * c;ql,-,. If Ms(z)=O, The numbers vi are entire (and may be negative). We call the matrix S(z) the local block MacMillan-Smith form of 2”(z). Let r be such that 7, < 0, n,+ , 2 0. We associate with z,, the two matrices d r+I d,+2 . . . d, 77,+1 ?,+2 **. 7, (the first matrix is not defined if 0, 2 0, and in this case we assume r = 0, while the second one is not defined if 9s C 0, and in this case we assume r = s). Now observe that if all the exponents vi are nonnegative, z0 is a zero according to the definition of Section 1 when at least one of the exponents is positive. This justifies the following definition: Definition 2.1. A complex number z0 is a zero of r(z)
when in the local MacMillan-Smith
form of T(z)
near z0 one of the exponents 17,is strictly positive. We have the following result: Theorem 2.1. I’ zOis a pole of T(z), then
If
z,, is a zero, but ndt a pole, of T(z), then fin
=
2 diqi* i:q,=-0
Proof. We prove the first assertion. Let T(z)u,(z) zero in to, and r, > 0 (i = 1,. . . , k). Then
= (z - zo)-‘tcpi(z),
qi(z) holomorphic, different from
F,(z)T(z)F,(z)F,-i(z)~;(z)~=S(z)~2-’(z)u(z)=(z-z,)-“F,(z)rp;(z) and of course Ft( zo)qi( zc) Z 0. Moreover, if { cpi(zo)} are independent are independent, and conversely. Hence, 4-b,)
=msbo>
=
2 t=l
vectors, also the vectors F,( zo)‘pi( zo)
d,v;.
Now we prove the second assertion of the theorem. It is clear that fff~(Z~)
=
2 i:q,
diqi* >O
So we must prove that iEr( zO) = ms( zo). We can see that mr( zo) < m,( zo). We remember that the normal rank of the matrix T(z) is the number max,(rank T(z)). Let it be k. Then also k = (normal rank S(z)). Hence we must show that if (z -to)? divides all the minors of T(z) of order k, (z - zo)’ divides all the minors of S(z) of order k. This is obvious, since the minors of S(z) of order k are sums of products one factor of which is a minor of order k of T(z). Hence mr( to) G #?,( zo). Interchanging the role of T(z) and S(z) we have the converse inequality. This finishes the proof. q The second assertion of the above theorem shows that we can define the multiplicity T(z) as
of z. as a zero of
Volume
I. Number
3
SYSTEMS
Remark 2.1. For some applications ~r(~o)=min(q,;
& CONTROL
LETTERS
November
19X1
it could be useful to define
qi >O).
3. An application As an application of the above material, we can consider the series connection of canonical systems. The question is wether the tandem system is canonical or not. A sufficient condition, which is well known in the ordinary case [7], can easily be generalized. The results that we obtain should be compared also with the theorem in [8] for the series connection of shift canonical realizations in Hilbert spaces. In [4] we called ‘canonical’ a system which is fully detectable and fully stabilizable, i.e. a system (S) such that rank[A*(r), C*(Z)] =h, Vz, (3) rank[A(z),
B(Z)] =n,
VZ.
(4) Let (S,) be two canonical systems (j = 1,2). The number i will denote any quantity that is referred to system (Si). Assume that nb = nf, so that the two systems can be connected in series, to form the new system (S). The transfer function of (S) is T(z) = T’( z)T,( z). Assume that ni z nf, fl: an b. We can show that under some conditions on the poles and zeros of the two systems, the tandem system (S) is canonical. For this we need a lemma which is the converse of Theorem 4.2 in [4]. Let mO be the order of zO as zero of d(z). Lemma 3.1. If mT( zO) = m, for euery pole zO oj T(z), then (S) is canonicul. Proof. Assume that (S) is not canonical. There exists a point zO such that (3) and/or (4) does not hold. We proved in [4] that near z0 T(z)u(z) may be written T(z)u=[Co(z)(zI-A,)-‘l&(z)
+ F(z)]u(z)
where F(z) is bounded near zc, and the characteristic polynomial of A, is (z - zO)“lO. To obtain this expression for T(z) we simply project (S) on the generalized eigenspace of zc. If, for example, (4) does not hold, then rank[zl-A,,
B,(Z)]
cm,,
forz=zO.
This means [5] that the system A,, B,(z) may be decomposed in a reachable part and in a part that is not effected by the control, so that the order of the pole zO of (zl- A,)-‘&( z) is less then m,. against our assumption, as we can see if we calculate (zl - A,)-’ using the Jordan form of A,, and if we note that B,(z) and C,(z) are bounded near zO. 0 Hence we must prove that 443)
=m,,(zo)
+%+o>.
We need the following lemma: Lemma 3.2. If z,, is not a zero of T(z), then the order of to as pole of T(z) is the greatest of the orders of z0 as pole of the minors of order k of T(z), k = normal rank T(z). Proof. The assertion is clear for S(z). Pre- and post-multiplying T(z) for regular invertible matrices does not change the order of a pole. Let r be the maximum order of z. as pole of the minors of order k of T(Z). From the Bin&t-Cauchy theorem, r =Gm,(z,) (compare with the proof of Theorem 3.1). Exchanging the Cl role of T(z) and of S(z) we find the equality. 207
Volume
I, Number
SYSTEMS
3
& CONTROL
LETTERS
November
I98 I
3.1. L.et T(r) = T,(z)T,(z). Assume that ni an:, n) 2 nb. Assume that when zO is a pole of T,(z), z,, is not a zero of q.(z) (j # k) and that not all the minors of maximum order of T(z) are identically zero. Then Theorem
4zd
= w,(z0)
+wJzd.
We start considering the case that z0 is a pole and a zero of T,(z), so that rank T,(z,) = nf. In this
Proof.
case w44
= T,w+M4.
T,(z) is analytic in z,,. If r< m,,(z,),
consider that
(z-z,)‘T(z)u(z)=T,(z)[T,(z)u(z)(z-z,)r]
f-0
forz-z,,
since ker Tz( z,,) = { 0). Hence m.(zo)~mT,(zg)=m.,(z,)+nr,~(zo). The converse inequality is obvious. If z0 is a pole and a zero of T’(z), then T,(z) is analytic in zO, and rank T,(z,) = nf. Hence for every u(z) there exists an analytic function o(z) such that u(z) = T,(z)o(z) near z,,, and we can see directly by the definition that mA+>
=mT2(zO) =mTl(zo> +mT,(zo).
The last casethat we must examine is the casein which z0 is a pole of both T,(z) and T,(z). We use Lemma 3.2. Any minor of maximum order of T(z) is a sum of products of minors of maximum order respectively of T’(z) and of r,(z), so that
Obviously, by the very definition of m,(z,),
the inequality cannot be strict. This finishes the proof.
Cl
As a consequencewe have the result: 3.2. Assumethat the conditions of the above theorem hold. If the systems(S,), i = 1, 2, are canonical, then the tandem system is canonical. 0 Theorem
4. A local fraction
decomposition
of T(z)
Let US consider again the local Smith-MacMillan negative. Then F,(z)T(z)&(z)= - ([diag(M,(z),...,
form. As above, r is the last index such that nlr is
([diag((z-z,)“Z,...,(z-zO)“‘Z,Z,...,Z)]) M,(z),
(z-z,)““~M,,,(z),...,(z-z,)%,(z))10])
=d,‘(z)NL(z)
= ([diag(M,(r),...,M,(z),(L.-z~)~~+~~~+,(z),...,(z-z~)~~~~(z))~~]) * ([diag((z-z,)“Z,...,(z-z,)“‘Z,Z,...,Z)])
=fi,(z)d;‘(z)
at least in a neighboorhood of zc. Hence T(z) =[oL(z)F,(z)]-‘[~,(z)F2-I(z)] and, analogously, T(z) =N,(z)D,‘(z). 208
=D;‘(z)N,(z)
(6)
Volume
I. Number
3
SYSTEMS
Hence, the Smith-MacMillan The following ‘coprimeness’ rank[Q(z,), rank[Nf:(z,),
& CONTROL
form of T(z) provides conditions hold:
LETTERS
November
a local right or left fraction decomposition
198 I
of T(Z).
= normal rank[ DL( z), N~( z)],
K.(z~)]
= normal rank[N;(z),
Do]
D;(z)],
(7) (8) of T(z) satisfies (7) or (8)
for every choice of z,,. In general, if any local right or left fraction decomposition for every z, we say that it is a comprime decomposition. In the following we consider a local left coprime factorization of T(z). The results that we find can be easily translated to the case of right coprime factorizations. We drop the index L, because no confusion can arise. 4.1. If (7) holds, then there are matrices P(z), Q(z) (holomorphic near zO) such that the equality
Lemma
D(z)P(z)+N(z)Q(z)=I
(9)
holds in a neighbourhoodof zo, and conuersely. Proof. It is sufficient to show that [D(z), N(z)]X(z) = u has a holomorphic solution for every vector u. Observe that the normal rank of [D(z), N(z)] is n,, because D(z) is invertible near zO. Let us choose n, independent columns of [D(i), N(z)]. Let P(z) be their matrix. The rows of X(z) which do not correspond to columns of [D(z), N(z)] appearing in P(z) may be put zero, while the others can be (locally) determined by the rows of P-‘(z). The converse part of the lemma is obvious. 0 Now in the spirit of [9], we can investigate the relationships of the zeros of T(z) with the condition rank iV( za) < normal rank N(z).
(10)
We have the following result: Theorem
4.1. (10) holds if and only if m,( zO) > 0.
Proof. If N(z) is that special matrix which is provided by the local MacMillan-Smith form of T(z) near za, the assertion of the theorem is seen to hold by inspection. Now we prove that if (10) holds for one coprime local decomposition of T(z), it holds for any other coprime decomposition. Let T(z)=D-‘(z)iv(z)=D-‘(z)rn(z), so that F(z)
=D(z)DP(z)N(z).
Premultiplying (9) by D( z)D-‘(z) &z)D-‘(z)
we have
=D(z)D-‘(z)D(z)P(z)
+D(z)D-‘(z)N(;)Q(z)
=D(i)P(z)
+F(z)Q(z)
which is a holomorphic function. Now we assumethat (10) holds. Then rank m( za) G min(rank D( z)D-‘(
z) IZZZ,,,rank N( z,))
< min(norma1 rank D( z)D-‘(
I), normal rank N(z))
= normal rank iV( z) = normal rank m( z ). This finishes the proof.
q
Remark 4.1. See [lo] for the corresponding result in the case that Yf(z) is a rational function. We finish this paper with an observation about the decoupling properties of the zeros of T(Z). 209
Volume
I, Number
3
SYSTEMS
& CONTROL
LETTERS
Novcmhcr
19X I
Let z0 be a zero (not a pole) of T(z), and consider the system i=A(t,)x+B(z,)u, which is is a zero T&z) in u 0 E R”l
y = C(z,)x,
described by an ordinary differential equation. Let T,(z) be its transfer function. It is clear that z0 of T,(z). If mr,,(zO) is positive and finite, from the above theorem we know that z0 is a zero of the sense of [9], and that T,(z) is not identically zero. Hence there exist vectors x0 E R” and such that
(zol-A(r,))x,
=B(zo)uo,
C(z,)x,
=o.
) -h. Then x(t) = x0 exp( zof ) is the solution Letx(t)=xoexp(zor),rE[-Izh,0],andu(~)=uoexp(zof),t of (1) with the given initial condition and control. The corresponding output is given by v(f)
= C(zo)xo
exp(zot)
which is identically zero. Under the above conditions, assume that, moreover, no G n,. We know from [9] that we can find a vector c E R”II such that for every u. there exists a vector x0 which satisfies the relations (zol-A(zo))xo
=B(z,)u,,
c*c(zo)xo
=o.
Hence, if we refer to the system (S), for every control U(I) = u. exp( z,t), there exists a solution x(t) such that the output given by (2) satisfies c*r(t) G 0.
of (1)
References [I] F. Fallside. Editor. Co~rrol .S,r~slern Design h,, Pole Zero Assrgwmwr (Academic Press. London. 1977). [2] A.G.J. MacFarlane and N. Karcanias. Poles and zeros of linear multivariable systems: A survey of the algebraic and complex variable theory, fnrernur. J. Co11rrol24 (1976) 33-74. [3] K.M. Prz@ski, Zeros of linear distributed parameter systems with application to the theory of delayed systems. Report. Inst. 01 Elect. Found.. Warsaw Technical University (1979). [4] L. Pandolfi. Canonical realizations of systems with delays. Rapport0 interno. 1st. Matcmatico del Polit.. Torino. Ser. II, No. I3 ( 1980). (51 L. Pandolfi. Canonical realizations of systems with delayed controls. Rmw/~e A~rrom. IO ( 19X0) 27-37. [6J F. Kappel and H.K. Wimmer. An elementary divisor theory for autonomous linear lunctional diffcrcntial equations. J. Di//eretrrtul Equurrom 2 I ( 1976) I34- 147. [7] H.H. Rosenbrock. Srore Spuce uptd Mu/ricunoh/e T/tear), (Thomas Nelson and Sons, London, 1970). [8] P.A. Fuhrman, On series and parallel coupling of a class of discrete time infinite-dimensional systems, SIAM J. ~onrrol Oprim. I4 (1976) 339-358. [9] C.H. Desoer and J.D. Schulman. Zeros and poles of matrix transfer functions and their dynamical intcrprctation. f,5’EE Trurrs. Circuits u~7d .Qsrenrr. 2 I ( 1974) 3-8. [IO] W.A. Wolovich. Multivariable systems zeros. in: F. Fallside. Ed.. Co,rrro/ ~~..~rerrts Des/,g,t ht. /‘o/e Zero Assrgrmte,t( (Academic Press. London, 1977) pp. 226-236.
210