On an algebraic theory of systems defined by convolution operators

On an Algebraic Theory of Systems Defined by Convolution Operators* by E. W. KAMEN School of Electrical Engineering Georgia Institute of Technology Atlanta, Georgia ABSTRACT

For a large class of linear continuous-time systems including delay-differential systems, an algebraic theory is presented in terms of Noetherian operator rings generated from a finite number of elements belonging to a convolution algebra of distributions. The external behavior of these systems is given by a finite set of input/output (convolution) operator equations which are solved in a novel manner by constructing an operational transfer function matrix and then applying an extension of the Mikusifiski operational calculus. After the formulation of an internal representation consisting of a finite set of scalar operational-differential equations, the problem of realizing an operational transfer function matrix by such an internal description is considered. Results on the existence and construction of realizations are given.

1. Introduction. The practicality of existing theories on finite-dimensional time-invariant systems is mainly a result of "finiteness" in the structural aspects of the mathematical representation; for example, the finite degree of polynomials in the transfer function matrix or the finite size of matrices in the classical state space description. Unfortunately, these nice characteristics are lost when the theory is extended to include infinite-dimensional systems. As a result, in the infinite-dimensional case the usual transform and state space techniques do not yield practical computational procedures unless an approximation theory is implemented. However, in many cases the infinite elements (or "devices") within a system play an integral role in system behavior such that approximations (by lumped elements, say) cannot be made without losing the direct relationships between system properties and the characteristics of the infinite elements. A very common example is a continuous-time system containing ideal time delays with important system properties depending on the magnitudes of the time delays. In the study of infinite-dimensional systems there is a need for an algebraic theory whose structural properties are "finite" so that computations within * This work was supported by the NSF under Grant No. GK-32697. 57 MATHEMATICALSYSTEMSTHEORY,Vo|. 9, NO. 1. © 1975 by Springor-Vorlag N~w York Inc.

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E . W . KAMEN

this framework can be performed. Algebraic techniques have been applied to various special situations, but a sufficiently general algebraic framework is still lacking even in the time-invariant case. The purpose of this paper is to establish the foundations of such a theory for a large class of infinite-dimensional timeinvariant continuous-time systems including delay-differential systems. The basic idea of the approach given here is to construct a one-to-one correspondence between the "primitive" elements of a system and the generators of the mathematical representation of the system. More precisely, given a class of systems consisting of an interconnection of elements having impulse responses 0j, 02,- .-, 0q belonging to a convolution algebra V of distributions, the systems in this class are represented by (convolution) operators belonging to the subring of V generated by 0~,..., 04. The study of the input/output representation in terms of these operators is given in Sections 2, 3, and 4. The finiteness of this algebraic framework is primarily a result of the fact that the ring of operators is a Noetherian domain which means that every ideal is finitely generated. Furthermore, in many cases the elements 0 j , . . . , 0q are algebraically independent (as defined in Section 2) with the result that the operator ring is isomorphic to the polynomial ring in q symbols. In these cases the rich theory of polynomials in several symbols can be utilized to study the given class of systems. In Sections 5 and 6, it is shown that the description of operational-differential systems in terms of polynomials over Noetherian domains makes it possible to compute internal representations from an operational transfer function matrix. An example of the realization procedure is given in Section 7. 2. Input/OutputDescription. Let R denote the field of real numbers and let V denote the linear space of R-valued Schwartz distributions (generalized functions) defined on R with supports bounded on the left. As proved by Schwartz [1], with addition ( + ) and convolution (,) V is also a commutative ring with no divisors of zero. (In other words, V is an integral domain.) Further, the linear structure on V is compatible with the ring structure in that V is a convolution algebra over R. The identity of V is the Dirac distribution ~o. Note that R can be viewed as a subring of V under the embedding R--~ V: a -+ a3 o. Letting U denote a fixed linear subspace of V, we shall be concerned mainly with systems consisting of an interconnection of devices having inputs belonging to U and outputs belonging to V. In particular, given a single-input single-output device, the input/output behavior is specified by the linear operator U ~ V: u -+ 0 • u where 0 ~ V is the impulse response of the device. Common examples of such devices are integrators (0 = Heaviside function) and scalors for which 0=ab0, a~R. As is well known, operators of the form u ~ 0 • u, 0 e V, can also represent devices that are distributed in an axial direction. For example, 0 = 3a, a > 0, can be viewed as the impulse response of an LC transmission line (or ideal delay line) with time delay a. Elements of V are also utilized to specify lossy and dispersive delay lines of various types. An example of the latter is an RC transmission line with 0 = .5(~t3) -~ a exp ( - a2/4t) h (t), h(t) = Heaviside function.

On an Algebraic Theory of Systems Defined by Convolution Operators

59

Generalizing, it is true that many devices of practical interest can be specified by an input/output operator U ~ V: u --->0 • u with 0 ~ V. However, V may be unnecessarily "large" for some applications, in which case a theory corresponding to that given below could be constructed in terms of a proper subalgebra of V or some other convolution algebra. Also we could consider convolution algebras defined over the field of complex numbers. The systems under consideration here are described in terms of (convolution) operator equations that are generated in the following manner. Given a finite list of fixed elements 0~, 0 2 , ' ' ", Oq belonging to V, let R[01,. •., Oq] denote the smallest subring of V containing 0~,..., 0q and R (viewed as a subring of V), An element ~(01,. •., 04) in R[Ox,..., 04] can be written as a finite sum

4) =

Z

.....

0fo,

Jl ,"" ,Jq

where the j~ are non-negative integers, ajl ..... s~ e R, O~ = jth-fold convolution of 0i, 0° = 3o. The ring R[01,. • -, 04] is an integral domain since it is a subring of the integral domain V. Any fixed ~(0) E R[Oa,..., 0q] (0 denotes the list 0a,-.., 04) defines a linear operator V ~ V: v ~ ~(0) • v. With the usual addition and composition, the set of all operators on V of the form v ~ ~(0), v, ~(0) ~ RIOt,..., 0j, is a ring which is isomorphic to R[OD'", 0q]. For this reason, we shall usually refer to R[01,..., 04] as a ring of operators. For any fixed operator ring R[O~,..., Oq], we consider the class of m-input terminal k-output terminal systems whose external description is given by the following finite set of (convolution) operator equations (1)

~, o:ij(O) * yj =

j=l

j=l

flij(O) * uj,

i = 1, 2 , ' " , k,

where **ij(0), /3~j(0) ~ R[O~,..., 0j, the yj ~ V are the outputs, and the uj e U are the inputs, U = fixed linear subspace of V. If q = 1 and 0 = 3~o1) = first derivative of 30, then since (3~ol))", v = 3~o") • v = nth derivative of v e V, (1) is a set of ordinary linear constant differential equations which is often taken as the input/output representation of a finite-dimensional time-invariant system. If q = 1 and 0 is any fixed element of V, then (1) could be the external representation of a system consisting of an interconnection of a finite number of adders, scalors, and devices having impulse response 0 (or 0- ~ if 0 is invertible in V). More generally, if 0 = 01,... , 04, then (1) could represent a system consisting of an interconnection of adders, scalors, and finite combinations of devices having impulse responses 01,..., 04 (or 0~-~ if 01 is invertible in V). This latter case includes a large class of infinitedimensional systems which are defined in terms of the following notions. A device with impulse response 0 ~ V is said to be finite (or lumped) if there exist elements ~,/3 e R[p],p = 3~o~), such that/3 * 0 = ~. A device is infinite if it is not finite. Via standard constructions in realization theory, it can be shown that a device admits a finite-dimensional state space representation if and only if it is finite, hence the motivation for the term finite. Integrators and

60

E . W . KAMEN

scalors are two common examples of finite devices. Examples of" infinite devices are ideal delay lines and dispersive delay lines. If we then let 0 = p, 01,. •., 0r where p = 8to1), the set of operator equations (1) could represent a system consisting of an interconnection of finite devices and finite combinations of infinite devices having impulse responses 01,..., 0r (or 0~-1). These systems will be referred to as operational-differential systems. Common examples are delay-differential systems in which the infinite devices are ideal delay lines. An obvious but important point is that the properties of a system specified by (1) depend on the algebraic properties of the operator ring R [ O l , . . . , Oq]. To determine the structure of this ring, in the remainder of this section we relate it to the ring of polynomials over R in q symbols. Let R[s 1, s 2 , . . . , Sq] denote the ring of polynomials in the symbols sl, s2," " , s~ with coefficients in R, and define the map

p: R[sl,. •., sq] ~ R[0~,..., 0q]: ~(s) ~ ~(0) The map p is a surjective ring homomorphism, and thus R [ 0 1 , . . . , 0q] is isomorphic to the factor ring R[sl,...,s~]/ker p where ker p = {~(s): p(~(s)) = 0}. Then since R [ O I , . . . , 0~] is a homomorphic image of the ring R [ s ~ , . . . , Sq] which is Noetherian (by the Hilbert Basis Theorem), it follows that R[O~,..., 0~] is also a Noetherian ring. Summing up these results, we have P R O P O S I T I O N 1. Given any finite list 0~,. •., Oq o f elements belonging to V, the operator ring R [ 0 1 , . . . , OJ is a Noetherian (integral) domain which is isomorphic to R[sl,. . ., sq]/ker p. The elements 0~,..-, 0q are said to be algebraically independent over R (viewed as a subring of V) if the map ~(s) --~ ~(0) is an isomorphism in which case R [ 0 1 , . . . , 0q] is isomorphic to the polynomial ring R [ s l , . . . , sq]. Hence, 0~,..., 0~ are algebraically independent over R if and only if there does not exist a nonzero polynomial ~(s) such that ~(0) = 0. For q = 1, an element 0 e 11, algebraically independent over R, is said to be transcendental over R. Examples of transcendental elements are given in the following result. P R O P O S I T I O N 2. For any a e R, a # O, 8 a is transcendental over R, and p = 8~o1) is transcendental over R. Proof. For a # 0, 8a is transcendental over R since the supports of 8° = 80, 8~, 82 = 82~,-.., 8~" = 8~ do not intersect for any positive integer n. If a(s) = n ~.~=0 ai si # O, a, # O, then ~(p) * h" = a ~ o + ~ =n -o1 ai h~-~ # 0, h = Heaviside function, since supp (E,=o ,-1 a, h " - ' ) # {0}. Examples of algebraically independent elements are given in the following result) T H E O R E M 1. Given al," " ~, ar ~ R, the elements p, 8a~~• . ~, 8at are algebraically independent over R if and only if given m l , . . . , mr ~ Z = integers, such that mlax + • " • +mrar = O, then m i = O, all i. Proof. The necessity of the condition on the a~ is clear, for suppose that 1 The condition on the a~ was given by one of the referees.

On an Algebraic Theory of SystemsDefined by Convolution Operators

61

there exist m~ e Z with m~ ~ 0 for at least one i, such that ~ m~a~ -- 0. Then 87: . . . . , 8~:-8o = 0 which shows that 3 , : . . . , bo, are not algebraically independent. To prove sufficiency we shall use the following result from ring theory. LEMMA. 01," " , 04, q > 1, are algebraically independent over R i f and only (f each 0 i, i = I, 2 , . . . , q, is transcendental over R[01,. . ., 0 i_ 1]. P r o o f o f Sufficiency in Theorem 1. When r = 0, p is transcendental over R by Proposition 2. Now let

0 # =(s) = rr.s"+'" + , r : + %

ER[p, 3a,,''', 3a~_1] [S],

r > 1,

and let a l , ' " , ar satisfy the hypothesis of the theorem. Then a(3,,) ~ 0 since the supports of the rq3,~ do not intersect for all i. Thus 8~, is transcendental over R[p, 3a,,. •., 3. . . . ], and by the above lemma the proof of the theorem is complete. In many cases of interest, the elements 0x, 02," "', 04 generating the operator ring R [ 0 1 , . . . , 0q] are algebraically independent over R. For example, it can be shown that any delay-differential system can be specified by the set of equations (1) where 0 = p, 3~,,. •., 8~ with the a~ as in the above theorem.

3. Quotient Field Operations. Given the finite set of equations (1), in this section we consider the existence and construction of solutions by utilizing operations in quotient fields. Since the ring V of distributions is an integral domain, the smallest field in which V can be embedded is its quotient field, denoted by Q. The elements of Q are equivalence classes whose representatives are denoted by u/v where u, v ~ V, v # 0. The ring V is embedded in its quotient field by the map 0: V ~ Q: v -+ v/3 o. Usually, oa(v) will be denoted by v. Now let 01, 02,- • -, 0q be a finite list of fixed elements in V as before. Then since the ring R[01,'" ", 0j is an integral domain, the smallest field in which R [ 0 1 , . . - , 0q] can be embedded is its quotient field, denoted by R ( 0 1 , . . . , Oq). Clearly, R ( 0 1 , . . . , Oq) is a subfield of Q, and in fact it is the smallest subfield of Q containing 01,..., 04 and R (viewed as a subfield of Q). For positive integers m and k, let R [ 0 1 , . . ' , 0~]k×m denote the R [ 0 1 , . . . , 04]module of k × m matrices over R [ 0 1 , . . . , 0j, and let Vm denote the free Vmodule of m-element column vectors over V. With respect to this notation, the set of equations (1) can be written in the following matrix form: (2)

A(O) • y = B(O) • u,

where A(O) = (~ij(0)) e R[01,. . . , Oq] k × k , B(O) = (flij(O)) E R [ O I , . . . , Oq]k×ra, y = (Yl," " "' YR)ra ~ vR, and u = (ua," " ", urn)rR ~ U m ( T R = transpose). Let R ( O ~ , . . . , Oq)k×k denote the ring of k x k matrices over the quotient field R ( 0 1 , . . . , 04). By a well-known result in matrix algebra, A(O) ~ R[01,.. ~, oq]k× k has a (unique) inverse in the matrix ring R ( 0 1 , . . . , 0~)k× k if and only if the determinant of A(O), denoted by det A(O), is not zero. If det A(O) ~ 0, we denote the inverse of A(O) by A(O)-1. Finally, letting V k ×m denote the V-module of k x k matrices over V, we have the following results on the existence of solutions of (2).

62

E . W . KAMEN

P R O P O S I T I O N 3. I f det A(O) # O, f o r any u E V m, (2) has the unique solution y -- A(O)- i. B(O) , u ~ V k i f and only i f A(O)- t. B(O) ~ V k ×", where • denotes that componentwise multiplications are in Q. P r o o f If: Viewing (2) as a set of equations over Q, we obtain the solution y = A(O)- 1. B(O). u ~ Qk. Then since A(O)- 1. B(O) ~ V k ×m, y = A(O)- 1. B(O)'u E Vk, all u ~ V ' . Only if: Let .W(O) = (wij(O)) = A(O) -1.B(O), and for every j = 1, 2 , . - . , m, let ej be the element of V m all of whose components are zero except for the jth which is equal to 3o. Then since y = W(O).ej = ( w l j , . . . , Wkj) rR E V k for all j, we have that W(O) ~ V k ×m. C O R O L L A R Y 1. I f det A(O) has an inverse (det A(O))-1 ~ V, then f o r any u E V", (2) has a unique solution y in V k, given by (3)

y = (det A(O))- 1 . .~(0) • B(O) • u,

where A(O) is the transpose o f the matrix o f cofactors o f A(O). P r o o f Let V k ×k denote the ring of k x k matrices over the ring V. Then R [ O l , . . . , Oq]k×k is a subring of V k×k and by standard results in matrix algebra, A(O) e R[01,. . . , Oq]k×k has a (unique) inverse in V k ×k if and only if det A(O) is a unit (invertible element) in V. If det A(O) is a unit in V, then A(O)- 1 = (det A ( O ) ) - 1 . ~(0) and A(O)-1 . B(O) e V k ×m. Hence, by the above proposition, the

corollary is proved. C O R O L L A R Y 2. I f B(O) has an m x k right inverse over V, then f o r any u E V ~, (2) has a unique solution y e V k i f and o n l y / f d e t A(O) is a unit in V. P r o o f The if part follows from Proposition 3. Now suppose that (2) has a unique solution y e V k for any u E V m, and for every i = I, 2,. •., k, let wij denote the ith component of the solution when u = ej, j = 1, 2, .o., m. Then A(O) • W = B(O), where W = (wij), and if B(O) has a right inverse M over V, we have that A(O) • ( W . M ) = lk, where I k is the k x k identity matrix. Thus, A(O) is a unit in V k ×k which implies that det A(O) is a unit in V. Note that if A(O) -1.B(O) e V k ×~, the system given by (2) can be represented by the restriction to U" of the input/output operator f : V ~ ~ v k : V -+ A(O) -~" B(O) * v. With respect to the free V-module structure on V" and V k, f is a Vmodule homomorphism and A(O)-1. B(O) can be viewed as the matrix representation of f relative to the standard bases in V" and V k.

As seen from (3), solutions of (2) can be computed by first finding the inverse in V of det A(O) if it exists. To simplify this computation, we can utilize the property that the operator ring R [ O , , . . . , 0q] is Noetherian. As a consequence, every non-unit of the ring R[01,.-., 0q] can be written as a finite product of factors that are irreducible in R[01,. • -, 0q]. Further, if 01,. •., 0q are algebraically independent over R, then R [ 0 1 , . . . , 0q] is a unique factorization domain, and hence factorizations into irreducible elements are unique. Now as a consequence of the commutative ring structure on V, we have the following LEMMA. Let det A(O) be a non-unit in R[O~,..., OJ and let det A(O) = ~r1 * ~r2 *" • "* 7rt be a decomposition o f det A(O) into irreducible factors. Then det A(O) is a unit in V i f and only i f each rri has an inverse ~ri- 1 in V, in which ease (det A(O))- 1 = ~r-Z1. ~rz 1 . . . . , ~r-71

On an Algebraic Theory of Systems Defined by Convolution Operators

63

Combining Corollary 1 of Proposition 3 and the above/_,emma, we have T H E O R E M 2 . / f det A(O) is a non-unit in R[OD..., O~] and if det A(O) = ~ • rr2 *'" "* ~rt is a decomposition o f d e t A(O) into irreducible factors with each ~ri having inverse rr:{ l in V, then for any u e V m, (2) has the unique solution (4)

y = (rr i- ~ * ~r2 ~ * " " * ~r~-1 ) , .4(0) * B(O) * u e V k.

A fundamental point here is that in determining the existence of solutions of (2) assuming det A(O) # O, it is sufficient to consider the invertibility in V of the irreducible elements of the operator ring R[OD..., 0q]. In particular, combining the above results, we obtain T H E O R E M 3. I f every irreducible element of R[01,. •., 0~] is a unit in l/~ then the quotient field R(01,. . ., 0~) is contained in V, and for any A(O) e R[01,. . ., Oq]k×k and B(O) e R[01,. . ., 0~]k×m with det A(O) ~ O, (2) has the unique solution (3) f o r all u ~ V ' . As an application of Theorem 3, let q -- 1 and let 0 e V be any fixed element transcendental over R. Then the only irreducible elements of RIO] are linear elements aO+b and quadratic elements aO2 +bO+c with negative discriminant b 2 - 4 a c < 0. Hence if these elements are units in V, we have that R(O) ~ V. For example, it is easy to show (and already known) that these elements are units in V when 0 = p or 0 = ~ , and thus both R(p) and R(8o) are contained in V. When 0 = 0~,..., Oq, q > l, the problem of determining the irreducible elements of R[O~,..., 0~] is very difficult in general. However, when 0 i is transcendental over R[O- 0~] for some fixed i, where 0 - 0~ denotes the list 0~,. •., 0~_ 1, 0~+ 1," " ", 0~, we could consider a splitting field over R(O-0~). We leave this for further studies. An example in which R(O1,.. -, 0q), q > 1, is contained in V is given in the following result.

P R O P O S I T I O N 4. For any a l , ' " , a, e R, the field R(p, 8a~,"., 8~,) is contained in V. Proof. Let 0 # ~-e R[p, 8~,,..., 8a] and consider ~r-1 which belongs to R(p, 8 ~ , . . . , 8 J . It will be shown that ~r-1 e V. Multiply ~-1 by ~/o with o consisting of Dirac distributions so that ~r-~ E R(p, 8a~,..., 8a,), where d~ _> 0 for all i. Now ~r-1 e R(p) (8~,," --, 8a.) and we can expand this into a formal series in the elements 8~,,..., 8~, with coefficients in R(p); that is

jl,,,o,Jp

where Ji > Ni with - ov < N~ < 0. Reorder the terms of this sum so that "IT

~

n--al

~a

• •

n>O

and ( j ; , - - . , j ; ) ~ n' < n ~ (Jr,"" ",J,) ¢~'.~'=1 dij'i < ~,'~=x d j , . Let { A , } denote the sequence of partial sums obtained from this series. For any 9~ e ~ =

64

E.W. KAMEN

Schwartz space of infinitely continuously differentiable functions R - + R with compact support, consider the sequence {A,(~0)}. Since for any ,X~ R(p) the support of A as an element of V is contained in [0, oo), and since 3~', is concentrated at the point d~ji, only a finite number of the A,(9) differ in value. Hence { A,(9)} converges in R, and thus ,r has an inverse in V. Note that if R(Ol,..., Oq) c V, then V is a linear space over R(SI,- .., 0q) with the multiplication R(01,.. ., Oq)x V ~ V: (~/{3, v ) - - > 3 - 1 , o,. v. In this case, to solve (2) we first can simplify the set of equations by applying the process of Gauss elimination. For the case q = 1 and 0 = derivative operator, this approach is similar to that given by Blomberg et al. [2]. Unfortunately, even if det A(O) can be decomposed into irreducible factors, the actual computation of solutions via (4) is usually quite difficult as a result of the complexity of convolution operations. To simplify the problem of computation, in the next section we present an algebraic procedure that extends the operational calculus of Mikusifiski [3]. 4. Computation of Solutions. Given a system specified by the input/output equations (5)

A(O) • y = B(O) • u,

where A(O) ~ R[01," " , Oq]k× k with det A(O) ¢ O, and B(O) e R[OD. . . , OJk×m, the expression of A(O)-1.B(O) as an element of R(01,..., Oq)k×'' (i.e., as a matrix of rational functions) is referred to as the operational transfer function matrix of the system. In correspondence with the standard terminology, the expression of A(O)- 1. B(O) as an element of V k ×" (assuming that A(O)- 1. B(O) is contained in V k ×m) is called the impulse response function matrix of the system. Since R ( 0 1 , ' " , Oq) is a field, the inverse of A(O) over R(01,..., Oq) can be computed by the usual techniques of inverting matrices over a field. Hence the operational transfer function matrix A(0)-I.B(0) can be determined from (5) by using standard procedures. Furthermore, if the input u can be expressed as an element of R(Ol," • ", 0~)", then as an element of R(01,..., 0~)k, the output y is readily computed. The main difficulty in obtaining solutions of (5) via this procedure is expressing y as an element of V k (when possible). Hence the central problem is computing the inverse image, under the embedding ~9: V - + Q D R(01,... , 0~), of elements in R(01,..., 0~). We now consider an algebraic method which simplifies this problem. First, let q = 1 and let 0 ~ V be any fixed transcendental element over R. Since R[O] is a principal ideal domain (pid) by applying the method of partial fraction expansions, we can reduce the problem of finding thee inverse image under ~ of elements in R(O) to computing the inverses in V of linear and quadratic elements in R[O] (or linear elements in C[O], C = field of complex numbers). If 0 = p = derivative operator, this procedure yields the operational calculus of Mikusifiski [3]. The main point here is that operational techniques apply to any class of systems described by operators in R[O] with 0 e V transcendental over R. Now let us consider the case when q > I and 0~,-.., 0q are algebraically independent. For any fixed i, the elements of R(O~,..., Oq) can be viewed as elements in the quotient field of the ring R(O- 0~) [0~]. Then since R(O- 0~) [0~] is

On an Algebraic Theory of SystemsDefined by Convolution Operators

65

a pid, any element of R(01," ", 0q), viewed as an element of R(O- 0i) (0i), can be decomposed via a repeated partial fraction expansion as follows. Let f~ denote the set of monic irreducible polynomials in R(O- 0~) [0i]. After multiplication by elements in R[O-0i] if necessary, the elements of f~ actually belong to R[O- 01] [Oi]since if ~ is an irreducible polynomial in R[O- 0i] [0i], it is also an irreducible polynomial in R(O- 0i) [0d (see Zariski and Samuel [4, p. 102]). Then given any element ~/fl ~ R(01,'" ", Oq), ~/fl has a unique decomposition

(6)

+

where 7r~,, ~, e R(O-Oi) [0il, j(oJ) are non-negative integers, rr~o = 0 if j(co) = 0, ~r~ is relatively prime to co ifj(~) > 0, and deg rr~ < deg coJ(°~) ifj(w) > 0. Note: The expression (6) can be decomposed further by viewing the ~r,~ as polynomials in R(O-Oj) (Oi), i ¢ j, and then applying the partial expansion to each ~r,~, and so on. This procedure can greatly simplify the problem of finding the inverse image of elements in R(01," • ", Oq). For illustrative purposes, we present the following Example. Consider the delay-differential system given by the following input/output equations

d2yl(t) + dy,(t-1) + y2(t_ 1)+y2(t) _ dul(t-1) + 2ul(t) + 2du2(t-2) dt 2 dt dt dt dyx(t) dt

y l ( t - 1)+y2(t) = - u l ( t - 1 ) - u 2 ( t - 2 ) .

If we let d = 3x and p = b(o1), then the system can be specified in terms of operators belonging to the ring R[d, p] with d and p algebraically independent over R (by Theorem 1). In matrix form, we have A(p, d) • y = B(p, d) • u, where

A ( p , d ) = ( p2+dp \-p-d

dlX )

'

B(p,d)=(dPd2 -

2d2p'~ -d2] "

Computing the inverse A(p, d)-a of A(p, d) over R(p, d) and multiplying by

B(p, d), we obtain 1 (dp+d2+d+2 A(p'd)-X'B(p'd)=p2+(Ed+l)p+dZ+d\ 2p+Ed

2d2p+da+dZ'~ d2pZ+dap ] '

which is the operational transfer function of the system (we are omitting the * from convolution). Now let u = (-e-th(t), h(t)) rR where h(t)= Heaviside function. As an element of R(p, d) 2, we have that u = ( - ( p + l ) -~, p-~)rR. Viewing det A(p, d) = p2+(2d+l)p+d2+d as a polynomial in R(d)[p] and applying the quadratic formula, we obtain det A(p, d) = (p+d) (p+d+ 1). Then / ( 2 d 2 -d)p 2 + (d 3 + 2d 2 - d - 2)p + d 3 + d 2 \

Y

A(p, d)-l.B(p, d).u \

d2p+d2-2 -(p+l)(p+d)(p+d+l)

l" ]

66

E.W.

KAMEN

Since det A(p, d) ~ O, by Theorem 3 and Proposition 4, the solution y belongs to V z. To compute the inverse image under t~ of y, we shall view the components of y as elements in the quotient field of R(d) [p] and expand by partial fractions. This gives d2+2 da-2d2+2d+2 d3+2 (7)

Yl =

d

d(d- 1) - p+l

p

d- 1

+

p+d

2 Y2

=

p+d+l

-d3+d2-2

d(d- 1) (8)

d -d3-2

d- 1

-

d +

p+d

p+l

'

-

-

p+d+l

"

Now ya and Y2 can be decomposed further by performing the following expansions d2+2 2 3 (9)

-

-

=

d(d- 1)

-

d

2

(10)

1 -

d(d-1)

+

- -

d- 1 '

-2 2 -- + d d- 1

-

From (7)-(10), it is seen that finding the inverse image of y~ and Y2 reduces to the problem of inverting p + l , p+d, d - l , and p+d÷l. Via power series expansion, we obtain ( p + l ) -~ =

e-'h(t)

( d - l ) - 1 = _ ~. ~. n=0

(p+d) -1 = ~ (n-t)"h(t-n) A=f(t) n=O

n!

(p+d+ 1) -1 = ~ (n-t)" e_(,_.)h(t_n) & g(t). n=O

Hence, the solution

y = (Yl, Y2) TR ~

n!

V2 is given by

Yl = h(t- 1)-e-th(t)+ 2e-(t+X)h(t+ l)+ 3 ~ e-(t-")h(t-n) n=0

- ~ [ f ( t - n - 3 ) - Z f ( t - n - 2 ) + Z f ( t - n - 1)+2f(t-n)] n=0 -

-

g(t-2)-g(t+ 1),

Y2 = 2e-~t+~)h(t+l)- 2 ~ e-~t-")h(t-n) n=0

+ ~ [ f ( t - n - 3 ) - f ( t - n - 2 ) + 2 f ( t - n ) ] - g ( t - 2 ) - 2 g ( t + 1). n=0

It is clear from this example that the success of the above algebraic procedure in simplifying the computation of solutions depends on the decom-

On an Algebraic Theory of SystemsDefined by Convolution Operators

67

posability of det A(O). When det A(O) is decomposable into factors of low degree, this technique of computing solutions compares quite favorably with classical procedures for solving operational-differential equations (see Bellman and Cooke [5]). Furthermore, this approach is an extension of Mikusifiski's operational calculus to operator rings in several variables. We also mention that when the generators of the operator ring are specified, the algebraic framework could be applied to equations with initial conditions. 5. Internal Description. Consider a system specified by the input/output operator f : U m ~ vk: U --~ W * u where W ~ V k×m n R(p, 01,. . ., Or)k×". If the inputs and resulting outputs are regular distributions (generated by locally integrable functions), the classical state representation of the system (if one exists) is given by dx(t) = Fx(t) + Gu(t) dt

(11)

y(t) = Hx(t), where F, G, H are linear maps, u(t) E R ~, y(t) ~ R k, and the state x(t) at time t belongs to some locally convex R-linear topological space X, called the state space. If the system contains infinite elements, then X is an infinite-dimensional linear space, and thus the matrix representation of the linear maps F, G, H have infinite size. To circumvent this infinite dimensionality, we consider the class of "hereditary systems" in which X = R", n < 0% and the derivative of x(t) at time t depends on x(t) and u(t) over a past interval ( t - r , t] for some fixed r, 0 < r + oo. (Here x(t) is no longer the state in the classical sense.) An example of a finite hereditary system (r < oo) is a delay-differential system of the form (dx( (12)

/)=

] dt

t

' E F,x(t-b,) + E G,u(t-c,) ~=1 i=1

y(O =

Z

q

I-rix(t - d,),

i=1

where b~, c~, d~ > 0 and F~, Gi, Hi are matrices over R of size n '< n, n x m, and k x n, respectively. Representations similar to (12) have been used extensively to study the internal properties, such as control, of delay-differential systems. (For example, see OgfiztiSreli [6].) Usually, in the literature y(t) = x(t), but it is reasonable to consider situations in which there are also time delays between x(t) and the output y(t). Our objective here is to extend representations of the form (12) to a general class of operational-differential systems and to do this in terms of operators belonging to the ring Rip, 01,..., Or]. First note that we can write (12) in the form (13)

I dx(t) = (F * x) (t)+ (G * u) (t) /&y(t)

(H • x) (t),

68

E. W, KAMEN

where F, G, H are matrices of size n × n, n × m, k x n over the operator ring R[8,~,. •., 8,r] for some a i e R. From (13) we obtain the desired generalization as follows. Let ~ denote the space o f "testing functions" associated with the space of distributions V. Given any positive integer n and v = (v~,..., v,) TM e V", we define v(9), ~ E ~ , by v(~0)= (vl(~),... , V,(~0))TRe R". Given x E V", we let p * x -- dx/dt denote the operation o f p = 8(01)on x in the V-module structure on V". We then have the following Definition. A n m-input terminal k-output terminal operational-differentia! system over R[p, 01," • ", 0,] is a triple (F, G, H ) of n × n, n × m, k × n matrices over R[O,,. • -, Or] such that ( p I - F ) - 1. G z V" ×m, together with the following set of operational-differential equations in the sense of distributions

l (p * x) "tq~) " - -dx(rp) ~ - - (F * x) (qO + (G * u) (q~) (14) y(9) = ( H * x) (~o),

where u ~ U m, y E V k, and x e S", S = linear subspace of V. The integer n is called the size of the system. The set of equations (14) represents a hereditary system in a generalized sense if the components of F, G, H have their supports contained in [0, oo). Furthermore, (14) is a finite hereditary system if the elements of F, G, H have compact support contained in [0, m), which will be the case if the supports of 01," • ", 0r are compact and contained in [0, oe). This latter condition implies that the infinite devices comprising the system have impulse responses with compact support c [0, oo), such as ideal delay lines. Note that by placing suitable constraints on U and F, G, H, we could restrict our attention to operational-differential equations defined in the ordinary sense (that is, we can replace ~oby t). We could then consider extending the theory of hereditary systems by using the framework given by (14). However, the fundamental problem of interest here is constructing representations of the form (14) from operational transfer functions W e R(p, 0 ~ , . . . , Or)k×m. First, solving (14) over the quotient field Q of V, we have that x = ( p I - F ) - 1. G * u e V" since ( p I - F ) -1. G e V "×". Then y = H ' ( p I - F ) -1" G * u e V k since H is over R[01,..., Or] c V. Hence, H . ( p I - F ) - I ' G ~ R(p, 0 1 , . ' . , 0,) k×" n V k ×" is the operational transfer function matrix of the system. We then have the following Definition o f Realization. Given an input/output operator f: u m - ~ vR: U ~ W * u, W E R(p, 0 1 , ' " , Or) k x m ('~ V kxm, a realization of f over R[01,..., Or] is a system of the form (14) with W = H . ( p I - F ) - I ' G . In the next section we pursue the problem of constructing realizations by considering the decomposability of the operational transfer function matrix. 6. Decomposition of Transfer Functions. Again, let 0 = 01,. •., 0q be a finite list of elements (not necessarily algebraically independent) belonging to 1I. Given W e R ( 0 1 , . . . , Oq)k×m, we say that W is decomposable over R(O-Oi)

On an Algebraic Theory of Systems Defined by Convolution Operators

69

(respectively, over R[O-Oi]) if there exist matrices F, G, H over R(O-Oi) (respectively R[O- 0~]), where F is n x n, G is n x m, and H is k x n, such that (15)

W = H.(O,I-F)-I.G.

The integer n is called the size of the decomposition. A decomposition (F, G, H ) , over R(O-0~) (RIO-0~]) is said to be minimal if n is minimal among all possible decompositions over R(O-Oi) (RIO-Oil ). A decomposition (F, G, H), over R(O-0~) or R[O-0i] yields the following set of operational equations (16)

(Oi'x = F . x + G . u y = H'x,

(

where u e U", and in general x e Q", y e Qk, where Q is the quotient field of V. If 0~ is a unit in V, (16) corresponds to a system with the following "wiring diagram".

This diagram illustrates that in a decomposition over R(O-0~) or R[O-0i] all devices with impulse response 0~-~ are "extracted". However, in a decomposition over R(O-0~) the elements of F, G, H may not be distributions (i.e., they belong to Q) or if they are, their supports may not be contained in [0, oe) even if the supports of the elements in the list 0-0~ are contained in [0, oe). The main interest here in decompositions over R(O-0~) is that this problem can be viewed as a first step in determining decompositions over R[O- 0~] which, in turn, for the case 0~ = p = ~(oI) leads to the construction of realizations as defined in the preceding section. In particular, if (F, G, H), is a decomposition of W e R(p, 01,..., 0,)k×m over R[O-p], then (F, G, H), defines a realization of the operator f : um---~ vk: U---~ IV* u as given by (14) if ( p I - F ) - I . G e V "×m. This latter condition is satisfied if det ( p I - F) is a unit in V, which is always the case if R(p, 0~,..., 0,) is contained in K Although we are primarily interested in the case 0~ = 3~~), we consider the construction of decompositions over R(O- 0~) and R[O- 0~] for any 0~ such that R(O-0~) [0d is a pid. The main result on decompositions over R(O-0~) is given in the following T H E O R E M 4. Let O~be transcendental over R[O-Oi] .for some fixed i. Then W = (~j/fllj)e R(01,..- , Oq)k×r" is decomposable over R(O-0~) if the degree of o~ij is less than the degree of flij fo r any fixed i, j when ~j and flij are viewed as elements in R[O-0,.] [0i].

70

E . W . KAMEN

Several constructive proofs of this theorem can be given. The first one t h a t we consider is based on the invariant factor theorem for pids. Let W satisfy the hypothesis of the theorem. Since R[O-0;] [0i] is contained in R(O-03 [0~], the elements of W can be viewed as elements in the quotient field of the ring R(O-03 [01], which is a principal ideal domain since 0 i is transcendental over R[O-O~]. Let ~" be the least common denominator of W as a matrix over R(O-Oi) (03. Then since deg ~ j < deg/~j, ~ ' W is a matrix over R(O-O3 [0i] whose entries have degree less than deg ~'. Since R(O-03 [0,.] is a pid, by the invariant factor theorem we can reduce ~FW to diagonal form which yields a Smith-McMillan-type form for W. From this the matrices F, G, H of a minimal decomposition can be computed by using Kalman's procedure [7]. For the details along with an example, see Kamen [8]. Other proofs of Theorem 4 are based on a Hankel matrix sequence which is generated in the following manner. Again viewing W as a matrix over the quotient field of the pid R(O-03 [0i], since deg ~ j < deg/3ij by long division we can expand W into a formal power series in 071 of the form

W= ~ At07~1,

AzER(O-OI)k×",

l=

1,2,3,'".

I=1

We then define a sequence of Hankel matrices for W by

Fi, i =

A 2t

A23 A

A3 A,

Aj 1 f Aj+

A3

A4

A 5

Aj+2 | = .

Ai+l

Ai+2

li i

Ai+j-i

Now if a decomposition of W over R(O- 03 exists, then W = H. ( O / - F ) - ~. G. Expanding (OiI-F)- 1, we obtain

H.(OtI-F)-I.G = ~,

H'FI-t.G.07~ t.

/=1

Hence W is decomposable over R(O-Oi) if there exist matrices F, G, H such that A t = H. F ~- 1. G, l = 1, 2, 3,-. -. Again let W = least common denominator of W. Then it follows from the results of Ho [9] that W has a minimal decomposition over R(O-03 of size equal to the rank of I' . . . . where cr is the degree of ~F as an element of R[O- 0i] [01]. The matrices F, G, H can be computed from I',+ a,,+ 1 by using Ho's algorithm. The details have been carried by Newcomb [10] for the case in which W is a rational function in several complex variables. Another procedure for computing F, G, H is to use Silverman's formulas as derived by Rouchaleau [11]: Let J be a submatrix of P , , , having maximal rank n. Let K be the n x m submatrix of the first block column (the first m elementary columns) of I',,~ corresponding to the rows of J. Finally, let L be the k x n submatrix of the first block row (the first k elementary rows) of I ' , . , corresponding to the columns of J. Then F = j-1M, G = J-1K, H = L is a minimal decomposition where M is the n xn submatrix of I ' , . , + i , for i

On an Algebraic Theory of Systems Defined by Convolution Operators

71

suitably large, sitting to the right of J. An example of this procedure is given in the following section. Any two minimal decompositions over R(O-0i) with 0 i transcendental over R[O-0i] are unique in the following sense. T H E O R E M 5. If(F, G, H), and (/0, ~, ~ ) , are two minimal decompositions of W over R(O- Oi), then there exists an n x n invertible matrix T over R(O- 0i) such that F = TFT-L, d = TG, ffl = H T - 1 . Proof. Let K denote the field R(O-O~), write f2 = K[Oi], and let Y' = K[[Oi- 1]] = ring of formal power series over K in 0~-1. Define A: f~"-+ Fk: ~o -+ W o oJ, where W o/,J is the usual multiplication of a matrix of power series in 0i" 1 by a vector of polynomials in 0 i with all terms containing nonpositive powers of 0~-1 omitted. Using the constructions given by Kalman [7] in his algebraic theory of discrete-time systems, we have that A is a K-linear homomorphism and that each minimal decomposition defines a canonical factorization of A through K n. The existence of T then follows from Theorem 6.9 and Proposition 6.10 in [7, pp. 258-259]. Comment. The approach used in the proof of Theorem 5 shows that algebraic results on discrete-time systems can be carried over directly to operational systems in continuous-time, as will be demonstrated again shortly. However, it is very interesting to note that some of the system-theoretic interpretations of the algebraic constructions do not carry over. For example, in the discrete-time theory K" is the state space, whereas here K " = R(O-Oi)" bears no direct relationship to the state space of the continuous-time system. We now consider decompositions over R[O- 0~]. Here we utilize the algebraic theory of linear discrete-time systems over commutative rings as developed by Rouchaleau [11] and Rouchaleau, Wyman and Kalman [12]. One of the main contributions of this work is the application of this algebraic theory to operational systems in continuous-time. In the remainder of this section, we assume that 0i is transcendental over R[O-Oi] for some fixed i. T H E O R E M 6. I f W is decomposable over R(O-Oi) and W = ~°= i Aft:,', A z ~ R[O-Oi] k×m, I = 1, 2 , ' " , then W i s decomposable over R[O-Oi]. Proof. Since R[O-0~] is a Noetherian domain, it follows directly from the results of Rouchaleau, Wyman and Kalman [12] that W is decomposable over R[O-0i]. This theorem can also be proved by using the fact that R[O-0i] is a Fatou ring as discussed in Cahen and Chabert [13]. Details of this approach in the discrete-time setting are given by Rouchaleau and Wyman [14]. COROLLARY. I f W is decomposable over R(O-0i) and W has a common denominator which is monic when viewed as an element of R[O- 0i] [0i], then W is decomposable over R[O-Oi]. Proof. If W has a monic common denominator, then the A~ in the expansion W = ~ = 1 AlO:, t belong to R[0-0~] k×m. The proof given in [12] of the existence of the matrices F, G, H is fairly constructive. However, in general the decomposition obtained in this manner is not minimal, and as yet there are no practical procedures for computing minimal decompositions over an arbitrary Noetherian ring R[O-0i]. However, when q = 2 and 01 and 02 are algebraically independent over R, R[O-0~] is a

72

E . W . KAMEN

pid and we can apply Rouchaleau's algorithm [11] to compute minimal decompositions over RIO- 0~]. The procedure is as follows. Let (F, G, H), be a minimal decomposition of W e R(O1, 02)k ×" over R(O- Oi), computed from the matrix J in the Silverman procedure. Since R[O-0~] is a pid, it follows that there exists a n x n invertible matrix T over R(O-Oi) such that F = T-1FT, G = T - 1 G , H = H T i s a minimal decomposition of W over R[O- 0i]. To compute T, let N be the n × am submatrix of the Hankel matrix F , , , containing the same rows as J. We then find a basis for the columns of N over the pid R[O-Oi]. Let ~ be the greatest common divisor of the elements in the first row of N. There is a linear combination Yl over R[O-Oi] of the columns of N having ~r1 as first element, and for each column ~ of N, there exists an ~ e R[O- 8i] such that the first element of 7 - ~ 7 1 is zero. Doing this for each column of N, we obtain a matrix N1 such that ~,~ and the columns of N1 generate the columns of N over R[O- Oi]. Applying this procedure to N I, and so on, we obtain a matrix (Yl,'" ", 7,) such that ~'1,"" ", 7, generate the columns of N over R[O-O~] and T = J - a ( 7 1 , . ' . , 7,). An example of this construction is given in the next section. In the general case, the question of the uniqueness of minimal decompositions over R[O-0~] appears to be difficult to answer; we leave this as an open problem. However, we do have the following result. T H E O R E M 7. I f q = 2 and 01 and 02 are algebraically independent over R, then given any two minimal decompositions (F, G, H), and (F, (i, ffI), over R[O-Oi] of W e R(01, 02)k×" there exists an n x n invertibIe matrix A over R(O-Oi) such that ff = AFA-1, ~ = AG, A = HA-1. Proof. Let (F, G, H ) , be a minimal decomposition over R[O-0~]. We claim that (F, G, H ) , is also a minimal decomposition over R(O-0~), for suppose it is not. Then there exists (F, G, B)~, m < n, which is a minimal decomposition over R(O-Oi). By Rouchaleau's results [11], from (F, G, H),, we can construct a decomposition over R[O- 0~] of size m, a contradiction. Now given two minimal decompositions over R[O-Oi], since they are also minimal over R(O-O~), by Theorem 5 we have the desired result. 7. An ExampLe. Consider a delay-differential system whose input/output operator f is given by f : U 2 -+ V2: u -+ W . u, where

W-

1 ( 2dEp p2+pd\-2dap

-6

d)

- 2 p + 4 _ _ eR(p,

d)2×2'

p = 8(o~), d = 3~.

It is seen that W satisfies the hypothesis of the corollary to Theorem 6 with 0~ = p, and thus, it has a minimal decomposition over R[d] which we now compute. Since the degree o f p 2 +pd, viewed as an element of R[d] [p], is two, we need to consider the Handel matrix F2.2. Expanding the elements of W, we obtain: // 2d 2 0 -2d a -6 \ -2d 3 -2 2d 4 6d P2'2 = l - 2 d 3 -6 2d 4 6d " \ 2d 4 6d -2d 5 -6d 2

)

On an Algebraic Theory of Systems Defined by Convolution Operators

0)

73

(1 o)

The rank of 1'2, 2 (as a matrix over R(d)) is two. We then pick j =

,

-2d 3

j-I

=

2d 2

-d/2

-2

-1/2

We then have that K = J, L = J, and

(-2d 3

-6) 6d '

M = \ 2d 4

which gives the following decomposition over R(d):

_~ =

,

C,=

,

~ =

.

0

-2d 3

-2

Now ( 2d 2 N = \_2d a

0 -2

-2d a 2d 4

-6) ' 6d

and via the procedure given above, we find that y, =

(:) J

and Y2 =

(o) 2

generate the columns of N over R[d]. Hence,

T = J - l ( y l , Y2) =

Then

~=~-=(0 ~ 6o) 0=~0 (~0 H=HT=(dl ?)'2 Since R(p, d) c V by Proposition 4, the decomposition (F, G, H ) . over R[d] yields a realization of minimal size of the input/output operatorf. In component form the realization is given by

dxl(t) dt

xl(t- 1)+6x2-2ul(t-2)

dx2(O = .2(t) dt

yl(t) = - x l ( t ) y2(t) = x l ( t - 1)-2Xz(t ).

74

E.W. KAMEN

REFERENCES

[1] L. SCHWARTZ, Thdorie des Distributions, Hermann, Paris, 1966. [2] H. BLOMaERG,J. SINERVO,A. HALMEand R. YLINEN, On algebraic methods in systems theory, ACTA Poly. Scandinav. Ser. 19, Helsinki, 1969. [3] J. MIKOSlrqSKI,Operational Calculus, Pergamon Press, New York, 1959. [41 O. ZARISKtand P. SAMUEL,Commutative Algebra, Vol. 1, Van Nostrand, Princeton, 1958. [5] R. BELLMANand K. COOKE,Differential-Difference Equations, Academic Press, New York, 1963. [6] M. OG0ZT6RELI, Time-Lag Control Systems, Academic Press, New York, 1966. [7] R. KALMAN,P. FALa and M. ARBIS, Topics in Mathematical System Theory, McGrawHill, New York, 1969. [8] E. KAMEN, "A Distributional-Module Theoretic Representation of Linear ContinuousTime Systems", Rept. SEL-71-044 (TR. No. 6560-24), Stanford Electronics Lab., Stanford, Calif., 1971. [9] B. Ho, "On Effective Construction of Realizations from Input-Output Description", Ph.D. Dissertation, Stanford Univ., 1966. [10] R. NEWCOMB, "On the Realization of Multivariable Transfer Functions", Research Report EERL 58, Cornell Univ., 1966. [I1] Y. ROUCHALEAO,"Linear, Discrete-Time, Finite-Dimensional,Dynamical Systems Over Some Classes of Commutative Rings", Ph.D. Dissertation, Stanford Univ., 1972. [12] Y. ROUCHALEAU,n. WYMANand R. KALMAN,Algebraic structure of linear dynamical systems, III, realization theory over a commutative ring, Proc. Nat. Acad. Sci. USA 69 (1972), 3404-3406. [131 P. CAHEN and J. CHAaERT, "El6ments quasi-entiers et extensions de Fatou", Queen's Math. Preprint No. 1972-22, Queen's University, Ontario, Canada, 1972. [14] Y. ROUCHALEAUand B. WYMAN,"Linear Dynamical Systems over Integral Domains", to appear.

(Received 6 May 1973 and in revised form 7 December 1973)