Systems & Control Letters 5 (1984) 213-215 December 1984 North-Holland

On the minimum delay problem

G. C O N T E a n d A. P E R D O N

lstituto di Matematica, Universitit di Genova, Via L.B. Alberti 4, 16132 Genova. Italy

Received 19 July 1984

In this note we solve the following problem: given a full column rank p × m transfer function matrix, find the mini- mum non-negative integers L i, i = l . . . . . m. such that the equation G(z)F(z) = diag{z -Lt ..... z-t..,} has a proper solu- tion G(z).

Keywords: Linear systems, Inversion problems, Zero structure at infinity.

I. Introduction

By the minimum delay problem (m.d.p.) we mean the following:

1.1. Problem. Given a full column rank p × m transfer function matrix F(z ) with coefficients in a field K, find the minimum non-negative integer L such that the equation

G ( z ) F ( z ) : z- l ' l , , (1)

has a proper solution G(z).

L is the overall delay of the system represented by F(z) . Clearly, L = 0 means that F(z ) has a proper left inverse and it is known [4] that a necessary and sufficient condit ion for the existence of a proper inverse is that F(z) has no zeros at oo. When the zero-structure of F(z ) at oo is non-triv- ial, the solution L of the m.d.p, is the following: let ooo = ( p ! . . . . . Pn, } be the zero-structure of F(z ) at oo [3,6.5], then L = vn,, the maximum order of zero at oo (see [4]). Any proper G(z) satisfying (1) is called, in this case, a pseudoinverse of F(z) .

In this note we consider a modified version of the m.d.p., namely:

1.2. Problem. Given a full column rank p × m transfer function matrix F(z) , find the minimum non-negative integers L i, i = 1 . . . . . m, such that

the equation

G ( z ) F ( z ) = diag( z -L, . . . . . z -L-, ) (2)

has a proper solution G(z).

Essentially, 1.2 is a m.d.p, in which different delays are allowed on the various channels. It is clear that, if L is the solution of 1.1, then L i ~< L for i = 1 . . . . . m. We remark that, in dealing with any inversion problem, it is possible to use a proper solution G(z) of (2) instead of a pseudoin- verse. The advantage in doing this is given by the fact that the MacMillan degree of the minimal proper solutions of (2) may be less than the Mac- Millan degree of the minimal proper solutions of (1) (see the example below). Since it is possible to give, as we will point out, a practical procedure to solve 1.2 using essentially the same tools needed to solve 1.1, the above remark gives a sufficient motivation to consider 1.2.

1.3. Example. Let F(z) have the following pur- posely simple form:

F ( z ) = 0 1 / z 2 ,

0 0

with zero-structure at oo given by %0 = {1, 2}. The solution of the m.d.p, is L - - 2 and a minimal pseudoinverse is

G ( z ) = ( lo/ Z 0 0 0 ) 1

whose MacMillan degree is 1. The solution of 1.2 is clearly L 1 = 1, L 2 = 2 (the fact that it coincides with the zero-structure is due to the simple form of F(z) , the point, obviously, is that at least one of the L, ' s is different f rom L) and a minimal solu- tion of

G ( z ) F ( z ) = diag( z - t , z -2 )

is

0°) whose MacMillan degree is 0.

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Volume 5, Number 3 SYSTEMS & CONTROL LETTERS December 1984

In the following section we will solve 1.2 using the notion of infinite zero module introduced in [11.

2. Solution of the modified m.d.p.

Denot ing by K the field of coefficients, we take U = K " and Y = K v and we write

U(z) = U ~ # ¢ ( z ) , r ( z ) = Y ® f l ¢ ( z ) .

Then, F(z) can be viewed as an injective (by the rank assumption) K(z)- l inear map F(z) : U(z) --+ Y(z).

By the already mentioned result of [4], it is clear that an equivalent formulation of 1.2 is the follow- ing: given F(z ) , find the min imum non-negative integers L i, i = 1 . . . . , m, such that, for

A ( z ) = diag{ z L' . . . . . z L - } ,

the transfer function F(z )A( z ) has no zeros at oo. Let us recall now that the zero-structure at oo

of a transfer function T(z) can be described by means of the infinite zero module Zoo(T ) intro- duced in [1].

Denot ing respectively by 12.oU and by I2ooY the submodules (over the ring of proper rational func- tions) of all the proper elements of U(z) and Y(z), the infinite zero module ZOO(FA) of F ( z ) A ( z ) is given by

( FA ) - ' ( [2=Y) + ~ U ZOO( FA ) = ~=U

(see [I] for details). In particular, F ( z ) A ( z ) has no zeros at 0o iff ZOO(FA)= O, and to solve 1.2 it is therefore sufficient to find L~, i = 1 . . . . . m, such that

( FA ) -'(g2ooY) c laooV.

To this aim, let us write

F ( z ) = S x ( z ) S ( z ) S 2 ( z )

where Bl(z), B2(z ) are bicausal and

with

D ( z ) = diag{ z - ' , . . . . . z - ' - },

is the Smi th-MacMil lan form of F(z) at oo. Then

( FA )-'(g2OOY) = A - l ( z ) ( F- ' (~ooY)) c 9ooU

iff

A - I ( z ) B~ ' ( z )( S - ' ( a2~Y) )

= A - l ( z ) B 2 1 ( z ) D - l ( z ) ( 9 o o U )

a I2,~U.

The last condit ion is equivalent to the transfer function

A - l ( z ) B ~ l ( z ) D - l ( z ) (3)

being proper and, al though in the Smi th -MacMi l - lan factorization at oo of F(z) the bicausal matrices Bl(Z ) and B2(z ) are not uniquely determined, it depends only on F(z). In fact, if

F ( z ) =

is a different factorization, we have from [2] that

B , ( z ) = B , ( z ) B L I ( z ) , B 2 ( z ) = B R ( z ) B 2 ( z )

where Ba(z ) and BR(z ) are bicausal matrices,

l o 1

with B(z) and B'(z) bicausal, P(z) proper and B( z )D( z ) = D( z )BR ( z ). This implies that

A - l ( z )B;~( z ) D - l ( z)( ~2ooU)

= A - ' ( z ) ~ l ( z ) B { t l ( z ) D - l ( z ) ( I 2 o o U)

= A - ' ( z ) B ; l ( z ) D - ' ( z ) B - l ( z ) ( ~ 2 . o U )

= n - ' ( z ) i ~ { l ( z ) D - l ( z ) ( f a ~ U ) .

Therefore, denot ing by d;, i = 1 . . . . . m, the maxi- m u m non-negative degree of z in the i-th row of B { l ( z ) D - l ( z ) , we have that (3) is proper iff L i >1 d i for i = 1 . . . . . m.

This proves the following:

2.2 Proposition. The solution of Problem 1.2 is given by L i = d i for i = 1 . . . . . m where d i is the maximum non-negative degree of z in the i-th row of B~I ( z )D- I ( z ) ,

being a Smith-MacMillan factorization at o0 of F(z).

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2.3. Remark. Since B2(z ) is bicausal, (3) is surely proper if L~ = p,,, for all i. In this way one reob- tains immediately the known solution of the m.d.p. However, d i may be less than 1,,, for some i as it happens in 1.3.

2.4. Remark. It is clear that to solve Problem 1.2 practically one needs a Smith-MacMillan factori- zation at oo. This can be obtained using, for instance, the nested algorithm described in [6] or the invariant factors algorithm (see [5]) over the ring of propel rational functions.

References

of S.vstems, Part 1, Lecture Notes in Control and Informa- tion Science (Springer, Berlin-New York, 1984).

[2] J.M. Dion and C. Commauit, Smith-MacMillan factoriza- tions at infinity of rational matrix functions and their control interpretation, S.vstems Control Lett. 1 (1982) 312-320.

[3] T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, NJ, 1980).

[4] S. Kung and T. Kailath, Some notes on valuation theory in linear systems, Proc. IEEE Conference on Decision and Control, San Diego (1978).

[5] M. Newman, Integral Matrices (Academic Press, New York, 1972).

[6] L.M. Silverman and A. K.itapci, Structure at infinity of a rational matrix, in: Outils et Modbles Mathbmatiques (CNRS, 1983).

[1] G. Conte and A. Perdon, Infinite zero module and infinite pole module, Proc. VII Int. Conf Anal.vsis and Optimization

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