# On Minimal Realizations and Minimal Partial Realizations of Linear Time-Invariant Systems Subject to Point Incommensurate Delays

Authors: M. De la Sen
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

This paper investigates key aspects of realization and partial realization theories for linear time-invariant systems being subject to a set of incommensurate internal and external point delays. The results are obtained based on the use of formal Laurent expansions whose coefficients are polynomial matrices of appropriate orders and which are also appropriately related to truncated and infinite block Hankel matrices. The above-mentioned polynomial matrices arise in a natural way from the transcendent equations associated with the delayed dynamics. The results are linked to the properties of controllability and observability of dynamic systems. Some related overview is given related to robustness concerned with keeping the realization properties under mismatching between a current transfer matrix and a nominal one.


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## Body

## 1. Introduction

The realization theory is a
very basic issue in theory of linear dynamic systems. It basically consists of
a state-space realization from available external description as, for instance,
a transfer matrix model or models based on measured sets of input-output data.
The minimal realization problem of dynamic linear time-invariant delay-free systems is to find a linear state-space description of the minimal possible
dimension whose associate transfer matrix exactly matches a proper predefined
rational matrix with entries over a field. Any proper, that is, realizable, rational
transfer matrix can be expanded in a formal Laurent series at infinity resulting
in a formal identity of the Laurent series and the transfer matrix. The
coefficients of the Laurent series form an infinite sequence of matrices which
are the block matrices of the infinite block Hankel matrix. The minimal realization
problem in the delay-free case may be focused on finding a state-space
realization on minimal order, that is, as small as possible, so that the above-mentioned
identity holds. The classical related
problem was firstly formulated by Kalman [1, 2] for the single-input
single-output case. The minimal partial realization problem of any
approximation is formulated in terms of a certain finite-order truncation of
the identity of the Laurent series expansion with the transfer matrix [3–7]. In this paper, this
formalism is extended to linear time-invariant systems subject to any finite
number of, in general, incommensurate internal
(i.e., in the state) and external (i.e.,
in the input and/or output) point delays; namely, they are not all an integer
multiple of a base delay, contrary to commensurate delays. These systems are
very common in the real life, like, for instance, in biological problems,
transportation problems, signal transmission, war/peace models, and so forth [8].
There is a wide recent interest in studying the properties of time-delay
systems as associated to either linear dynamics or nonlinear dynamics or even
to dynamics described by differential equations in partial derivatives [9–22].
Impulsive time-delay systems have been studied recently in [9, 12, 16, 19]. In particular, impulses may be associated with the singularity of the
dynamics and the loss of uniqueness of the solution as a result [16]
or to the fact that the forcing terms are impulsive. The robust stability
problem has been studied in [11] via linear matrix inequalities and Lyapunov's
functions. The oscillatory behavior under delays and possible unmodeled
dynamics is investigated in
[10, 21]. Different aspects and conditions of
positivity of the solutions and equilibrium points have been recently described
in [16–18, 22], either in the first orthant or in generic cones. The
central purpose of this paper is concerned with the realization theory and
associated properties of controllability and observability of dynamic systems
under linear delayed dynamics. An infinite polynomial block Hankel matrix, as
well as its associate τ-finite polynomial block Hankel matrices, is defined
in order to relate the spectral controllability and observability properties of
minimal realizations [3, 4, 8, 23–31]  with the minimum feasible finite rank of
such a Hankel matrix. Then, such a rank is proved to equalize that of its
associate finite polynomial block matrices whose orders exceed a minimum lower bound
related to the minimal realization to be synthesized. Potential extensions of the obtained results might
be addressed to investigate controllability and observability and then minimal
and partial minimal realizations of expanded composite systems [23, 24] and hybrid
systems [25, 31, 32], including systems subject to switches between multiple
parameterizations. Therefore, problems related to properties of dynamic systems
like, for instance, controllability, observability, or realizability have received
important attention from years up till now. An important point of view for
modeling dynamic systems is to synthesize both nonparametric and parametric
models which describe the mappings relating collections of measured
input/output data as closely as possible. The Hankel matrix-based models are
appropriate to describe linear input/output mappings by infinitely many
parameters, in general, since they might be obtained directly from available
input/output data on the system. In a second step, those models may be mapped
either exactly or approximately into finitely parameterized models, for
instance, via transfer matrices and associate state-space realizations [1–4, 6, 7]. In particular, the point of view of starting from Hankel matrices-based models formulated
in rings, in general, has been discussed in [3, 4]. Note that realizability
results and related properties formulated over rings lead only to
sufficiency-type conditions when applied to time-delay systems. The reason is
that those systems are modeled by transfer matrices involving rational entries
being quotients of quasipolynomials or their associate matrix impulse responses,
infinite series, and so forth. It is known that bijective mappings exist
between such quasipolynomials and their corresponding polynomials of several
variables depending on the number of delays. However, quasipolynomials are in
fact transcendent functions which depend only on one variable since there is a
precise functional dependence on the remaining independent variables in the
polynomial context and that one. Such a variable is, roughly speaking, the
argument of the Laplace transform. In this paper, stronger results are obtained
since the fact that only one independent variable exists is taken into account.The paper is organized as
follows. Section2 deals with proper transfer matrix descriptions of linear
time-invariant time-delay systems from their state-space realizations in the
general case where delays are incommensurate. Section 3 establishes
connections to the level of appropriate isomorphisms between formal Laurent
expansions at infinity as well as their finite truncations of a certain finite order,
rational transfer matrices, and rings of polynomials. It turns out that Laurent
expansions at infinity are equivalent to transfer matrices while their
truncations of a certain finite order are related to Hankel matrices. Such formalism
is applied to linear time-invariant systems subject to internal incommensurate
point delays. In particular, the properties of spectral controllability and
observability are investigated together with the associated problems of minimal
realizations for simultaneously controllable and observable systems and minimal
partial state-space realizations. Related results are obtained for the independence
of the delay case as well as for the dependence on the delay case. The related
problems of synthesis of minimal state-space realizations and minimal partial
realizations are dealt with in Section 4 with special emphasis on the
single-input single-output case. The formulation is made in terms of finding a
state-space realization such that it matches a certain transfer matrix which is
formally identical to a series Laurent expansion at infinity. Finally, a
section of concluding remarks ends the paper.


### 1.1. Some Basic State-Space Realization Concepts and Related Notations

The following concepts about
state-space realizations, minimal realizations, and minimal partial
realizations are dealt with through the manuscript.(1)
A proper rational transfer matrixG(s) over a filed K takes values in Kp×m(s),
where s is the Laplace transform
variable.(2)
Any proper rationalG(s)∈Kp×m(s) can be expanded in a formal Laurent series at
infinity leading to the formal identity G(s)=∑i=0∞His−i with {Hi}i∈ℕ0 being an infinite sequence of matrices which
are the block matrices of the infinite block Hankel matrix.(3)
The infinite block Hankel matrix is often
denoted as{Hi}0∞, with ℕ0=ℕ∪{0} and ℕ being the set of the natural numbers.(4)
The minimal realization
problem in the delay-free system consists in finding a state-space
realization(A,B,C,D),
with A∈ℝn×n, B∈ℝn×m, C∈ℝp×n,
and D∈ℝp×m,
of order n∈ℕ being minimal, that is, as small as
possible, so that, given the infinite
sequence {Hi∈Kp×m}0∞,
the identity of the proper rational transfer matrix with the formal Laurent
series at ∞ holds, that is, G(s)=∑i=0∞His−i=C(sI−A)−1B+D and then H0=D and Hi=CAi−1B, i∈ℕ.
A realization is minimal if and only if it is both controllable and observable.(5)
IfD = 0, then G(s) is strictly proper; that is, the number of poles exceeds strictly the
number of zeros at any entry of the transfer matrix.(6)
The so-called minimal
partial realization problem of any approximationτ is formulated as follows. Given a finite
sequence {Hi∈Kp×m}0τ
with some τ∈ℕ,
satisfying H0=D, Hi=CAi−1B, foralli∈τ¯:={1,2,…,τ},
there exists a quadruple (A,B,C,D),
with A∈ℝn×n, B∈ℝn×m, C∈ℝp×n,
and D∈ℝp×m,
of minimal order n∈ℕ (the order of A) such that C(sI−A)−1B+D=∑i=0τHis−i+0(s−τ−1).


## 1.1. Some Basic State-Space Realization Concepts and Related Notations

The following concepts about
state-space realizations, minimal realizations, and minimal partial
realizations are dealt with through the manuscript.(1)
A proper rational transfer matrixG(s) over a filed K takes values in Kp×m(s),
where s is the Laplace transform
variable.(2)
Any proper rationalG(s)∈Kp×m(s) can be expanded in a formal Laurent series at
infinity leading to the formal identity G(s)=∑i=0∞His−i with {Hi}i∈ℕ0 being an infinite sequence of matrices which
are the block matrices of the infinite block Hankel matrix.(3)
The infinite block Hankel matrix is often
denoted as{Hi}0∞, with ℕ0=ℕ∪{0} and ℕ being the set of the natural numbers.(4)
The minimal realization
problem in the delay-free system consists in finding a state-space
realization(A,B,C,D),
with A∈ℝn×n, B∈ℝn×m, C∈ℝp×n,
and D∈ℝp×m,
of order n∈ℕ being minimal, that is, as small as
possible, so that, given the infinite
sequence {Hi∈Kp×m}0∞,
the identity of the proper rational transfer matrix with the formal Laurent
series at ∞ holds, that is, G(s)=∑i=0∞His−i=C(sI−A)−1B+D and then H0=D and Hi=CAi−1B, i∈ℕ.
A realization is minimal if and only if it is both controllable and observable.(5)
IfD = 0, then G(s) is strictly proper; that is, the number of poles exceeds strictly the
number of zeros at any entry of the transfer matrix.(6)
The so-called minimal
partial realization problem of any approximationτ is formulated as follows. Given a finite
sequence {Hi∈Kp×m}0τ
with some τ∈ℕ,
satisfying H0=D, Hi=CAi−1B, foralli∈τ¯:={1,2,…,τ},
there exists a quadruple (A,B,C,D),
with A∈ℝn×n, B∈ℝn×m, C∈ℝp×n,
and D∈ℝp×m,
of minimal order n∈ℕ (the order of A) such that C(sI−A)−1B+D=∑i=0τHis−i+0(s−τ−1).


## 2. State-Space and Transfer Matrix Descriptions of Time-Delay Systems

Consider the linear
time-invariant dynamic system in state-space form:(2.1)x˙(t)=∑i=0qAix(t−hi)+∑i=0q′Biu(t−hi′),(2.2)y(t)=Cx(t)+Du(t), where x:ℝ+×Σ×U→Σ⊂ℝn, u:ℝ+→U⊂ℝm, y:ℝ+×Σ×U→Y⊂ℝp are the state, input, and output vector
functions in their respective state, input, and output spaces Σ,U,
and Y, where ℝ+:={t∈ℝ:t≥0}, h0=h0′=0,
and hi(>0)∈ℝ+(i∈q¯:={1,2,…,q}), hi′(>0)∈ℝ+(i∈q¯′) are the internal and external
point delays. If the input is generated via state feedback, then u:ℝ+×Σ→U⊂ℝm.
If it is generated via output feedback, then u:ℝ+×Y→U⊂ℝm.
The internal delays (hi), which are assumed to be pairwise distinct, and the external ones (hj′), which are also assumed to be pairwise
distinct, are both incommensurate delays;
that is, they are not necessarily equal to ihb and jhb′(i∈q¯,j∈q¯′),
with some hb>0, hb′>0; Ai∈ℝn×n(i∈q¯∪{0}), Bi∈ℝn×m(i∈q¯′∪{0}), C∈ℝp×n,
and D∈ℝp×m are matrices of real entries which
parameterize (2.1)-(2.2). The dynamic system is subject to initial conditions, φ:[−h,0]→ℝn,
where h:=Maxi∈q¯(hi) is piecewise continuous possibly with bounded
discontinuities on a subset of zero measures of its definition domain. Closed formulas for the unique
state and output trajectory solutions of (2.1)-(2.2) are provided in [33, 34], although
the dynamic system is infinite-dimensional by nature. One of these formulas is based
on defining a C0-semigroup generated by the infinitesimal
generator of the delay—free matrix A0—which is
trivially valid for the time-invariant case only. An alternative second formula
is based on an evolution operator which satisfies the unforced differential
system (2.1) which may be generalized to the linear time-varying case. Both formulas are equivalent since the solutions
are unique.By taking right Laplace transforms in the state-space description (2.1)-(2.2)
with φ≡0,
a transfer matrix exists, which is defined by
(2.3)G(s):=Y(s)U(s)=[Lap+(y(t))Lap+(u(t))]φ≡0=C(sIn−∑i=0qAie−his)−1(∑j=0q′Bie−hi′s)+D, where Lap+(v(t)) is the right Laplace transform of v:ℝ+→ℝs provided that it exists. Note that G(s) is a
complex matrix function in Cp×m in the complex indeterminate s whose (i,j)th entry is
(2.4)Gij(s)=ciTAdj(sIn−∑k=0qAie−hks)(∑j=0qbℓje−hℓ′s)Det(sIn−∑k=0qAie−hks)+Dij, where Adj(⋅) and Det(⋅) stand
for the adjoint matrix and the determinant of the (⋅)-matrix, respectively, and ciT and bℓj are the ith row of C(i∈p¯) and the jth column of Bℓ(ℓ∈q¯′∪{0}),
respectively. Define complex (q+q′) and (q^+q^′) tuples as follows:
(2.5)z:=(zI,zE)=(z1,…,zq,zq+1,zq+q′)∈Cq+q′,z^:=(z^I,z^E)=(z1,…,zq^,zq^+1,zq^+q^′)∈Cq^+q^′, where
(2.6)q≤q^≤q^0:=n(∑i=1q(qi)),q′≤q^′≤q^0′:=q^0(q′+1) so that the components zi and zj correspond to e−his and e−hj′s(i∈q¯,j∈q¯′), respectively, in
a multiargument description, where s and all the components zi and zj are taken as independent variables. The
components z^i of the extended z^ are associated with combined delays h^i which are formed with all the
combinations of sums of the internal delays hi and their respective integer multiplicities. The components z^j and associated delays h^i′ are formed with all the above combinations of sums of the internal
delays hi and their respective integer multiplicities with the various external
delays. The appearance of these delays arises in a natural way in the transfer
function and then, roughly speaking, in the input/output data model via direct
calculation in the numerator and denominator of (2.4). Intuitively, that means
there are usually much more delays in the external system representation than
in the internal one due to the evaluation of the adjoint matrix and the determinant
and the products in the numerator of (2.4). This feature leads to inequalities
with upper bounds in (2.6) so that if identical internal/external delays appear
as a result of calculations in the transfer functions (see (2.4)), the resulting coefficients
are regrouped so that each of the identical delays appears only once. As
intuitive related example, one single internal delay h in the
state-space representation generates, up till n commensurate delays, hj=jh(j∈n¯) in a single-input single-output transfer
function. Then, ℝp×m(s,z),
the space of realizable rational transfer p×m matrices of real coefficients in the complex (q^+q^′+1)-tuple (s,z^) of numerator and denominator being,
respectively, a quasipolynomial matrix and a quasipolynomial, is isomorphic (in
the sequel denoted by the symbol “≈”) to ℝp×m(s) so that there is a natural bijective mapping
between each entry Gij(s) and
(2.7)Gij(s,z^)=Nij(s,z^)M(s,z^)=∑k=0nijNijk(z^)sk∑k=0nMk(z^)sk=∑ℓ=0q^+q^′N¯ijℓ(s)z^ℓ∑ℓ=0q^M¯ℓ(s)z^ℓ=∑k=0nij∑ℓ=0q^+q^′Nijkℓskz^ℓ∑k=0n∑ℓ=0q^Mkℓskz^ℓ whose numerator Nij(s,z^) and denominator M(s,z^) are, respectively, polynomials in several variables of respective
real coefficients Nijkℓ and Mkℓ, and Nijk(Z^), Mk(z^), N¯ijk(s), and M¯k(s) are also polynomials in their
respective single or multiple arguments.


## 3. Analysis of Minimal Realizations and Formal Series Descriptions

Note that the numerator and
denominator ofGij(s,z^) are, respectively, in the polynomial additive groups (rings if p=m) ℝp×m[s,z^] and ℝ[s,z^] generated by (s,z^).
By using a formal Laurent series expansion at ∞ in the variable s of the form G(s,z^)=∑i=0∞Hi(z^)s−i with Hi(z^)∈ℝp×m[z^],
it follows that ℝp×m(s,z^)≈ℝp×m[[s]][z^], the additive group of formal
Laurent power series with matrices over ℝp×m at ∞ in the polynomial multiple indeterminate
defined by the components of the z^-tuple. Note that the formal series additive group ℝp×m[[s]][z^] is
the completion of the polynomial matrix additive group ℝp×m[s][z^] (≈ℝp×m[s,z^]) with respect to the I-adic topology, where I is the ideal of the
polynomial matrix additive group ℝp×m[s][z^] generated by the indeterminate complex (q^+q^′+1)-tuple (s,z^). The above discussion is formalized as follows.Theorem 3.1.
The following properties hold for any positive integers p, m, and n.
(i)
ℝn×n[e−his:i∈q¯∪{0}]≈ℝn×n[zI] if hi∈ℝ+∖{0} for all i,j∈q¯,
and hi≠hjforalli,j(≠i)∈q¯; ℝn×m[e−hi′s:i∈q¯′∪{0}]≈ℝn×m[zE].(ii)
ℝp×m(s,z^)≈ℝp×m[[s]][z^]≈ℝp×m[s,z^] if hi′∈ℝ+∖{0} for all i,j∈q¯′, and hi≠hj for all i,j(≠i)∈q¯′.(iii)
ℝp×m[s][z^] is a dense subspace of ℝp×m[[s]][z^], which is a
complete topological additive group with respect to the I-adic topology, where I is the ideal of the
additive group ℝp×m[s][z^] generated by the indeterminate complex (q^+q^′+1)-tuple (s,z^).Note that the isomorphisms of
Theorem3.1(i) are useful to formulate controllability/observability and minimal realizations for the dynamic system (2.1)-(2.2) since
the only delays which are zero are h0=h0′=0 and all the remaining internal delays are
pairwise distinct while all the remaining external delays are pairwise distinct
as well.From Theorem3.1(i), the following bijections may be established:
(3.1)∑i=0qAie−his⟷A(zI):=∑i=0qAiziI∈ℝn×n[zI];∑i=0q′Bie−hi′s⟷A(zE):=∑i=0qAiziE∈ℝn×n[zI] so that via (3.1) the
controllability and observability matrices of the nth realization (2.1)-(2.2)
result:
(3.2)Cn(A(z),B(z))=Cn(A(zI),B(zE)):=(B(zE),A(zI)B(zE),…,An−1(zI)B(zE)),On(C,A(z))=On(C,A(zI)):=(CT,AT(zI)CT,…,An−1T(zI)CT)T
in ℝn×(n+m)[z^] and ℝp×(p+m)[z^], respectively. Note
that if some of the above matrices are full rank, then the state-space
realization (2.1)-(2.2) is controllable (resp., observable) in an additive group. However, the
respective full-rank conditions are not necessary for controllability/observability
since the additive group isomorphism defined by (3.1) does not preserve the metric and
topologic properties. In
particular, the loss of rank of any of the polynomial matrices (see (3.2)) for
some delays hi≥0(i∈q¯), hj≥0(j∈q¯′),
and h0=h0′=0 in the indeterminate z does not imply that the rank
is lost for some complex indeterminate s satisfying the constraints zi=e−his, zq+j=e−hj′s(i∈q¯,j∈q¯′) for some
predefined delays. Define the following controllability and
observability testing sets SCn(h) and SOn(h),
respectively, depending  on the real (q+q′)-tuple of delays:
(3.3)h=(h1,h2,…,hq,hq+1=h1′,hq+2=h2′,…,hq+q′=hq′′)∈ℝ+q+q′, which is the closed first orthant
in ℝq+q′,
and
the associated sets of delays are
Huc and Huo,
where controllability and, respectively,
observability are lost:
(3.4)SCn(h):={z=(z1,z2,…,zq)∈Cq+q′:zi=|zi|≺θi=e−his,σC=ln|zi|hi∈ℝ,ωC=tg(θi)hi∈ℝ∀i∈q+q′¯,rank[Cn(A(zI),B(zE))]<n},SOn(h):={z=(z1,z2,…,zq)∈Cq:zi=|zi|≺θi=e−his,σO=ln|zi|hi∈ℝ,ωO=tg(θi)hi∈ℝ∀i∈q¯,rank[On(C,A(zI))]<n},Huc:={h∈ℝ+q+q′:SCn(h)≠∅},Huo:={h∈ℝ+q+q′:SOn(h)≠∅}. Note that if the full ranks in
the polynomial matrices (see (3.2)) are lost for some h such that the respective testing set in (3.4) is nonempty, then
controllability (resp., observability) in an additive group becomes lost for the corresponding set of delays.
If the full-rank property in the polynomial matrices (see (3.2)) is lost for some h such that the respective testing set
in (3.4) is empty, then controllability (resp., observability) in an additive group holds for the corresponding set of
delays. If the sets SCn(h) and SOn(h),
respectively, are empty for any h∈ℝ+q+q′,
then the system is controllable (resp., observable) in an additive group independent of
the delays. Note directly that
(3.5)h∈Huc⟺SCn(h)≠∅;SCn(h)=∅∀h∈ℝ+q+q′⟺Huc=∅, and similar assertions are
applicable to the sets SOn(h) and Huo.
Then, the following definitions on spectral controllability and observability
are provided. Then, a related result is given as a formal statement of the
above informal discussions, which states formally the equivalences between
spectral controllability (observability) and controllability (observability).Definition 3.2.
The dynamic system (2.1)-(2.2) is spectrally controllable if there exists a state-feedback
control law u:ℝ+×ℝn→ℝm, fulfilling u(t)=0forallt∈ℝ−:=ℝ∖ℝ+, such that U(s)=Lap(u(t)) exists for any given prefixed suited characteristic
closed-loop polynomial:
(3.6)∑i=0n∑j=0q+q′fijsie−hjs=Det(sIn−∑i=0qAie−his−∑i=0q′Bie−hi′sU(s)). If the above property holds
for any given vector of delays h,
then the system is spectrally controllable independent of the delays.Definition 3.3.
The dynamic
system (2.1)-(2.2) is spectrally observable
if its dual is spectrally controllable, that
is, if there exists a state-feedback control law u:ℝ+×ℝn→ℝp,
fulfilling u(t)=0forallt∈ℝ−:=ℝ∖ℝ+,
such that U(s)=Lap(u(t)) exists for any given prefixed suited
characteristic closed-loop polynomial:
(3.7)∑i=0n∑j=0q′fijsie−hjs=Det(sIn−∑j=0qAiTe−his−CTU(s)). If the above property holds
for any given vector of internal delays, then the system is spectrally
observable independent of the delays.Controllability and
observability in rings are defined in parallel to their above spectral versions
in the complex indeterminateszi (which replace e−his in Definitions 3.2 and 3.3) which are considered
to be mutually independent; this is not true since they are related by the Laplace transform indeterminate s. Therefore,
controllability and observability in rings are sufficient (but not necessary)
for their corresponding spectral versions to hold in the context of time-delay
systems. Note that observability is defined by duality in order to simplify the
description. On the other hand, note that the spectral definitions of
controllability and observability are established in terms of the ability of
arbitrary coefficient assignment of the characteristic closed-loop polynomial
(or that of the dual system) through some realizable control law. These
definitions are equivalent to the classical spectral definitions for time-delay
time-invariant systems which were stated in equivalent terms via Popov-Belevitch-Hautus
controllability/observability tests (see, e.g., [35–38]). Such tests are used
in Theorem 3.4 in terms of being
necessary and sufficient to guarantee both properties. Finally, note that the real
controllability/observability (in terms of necessary and sufficient conditions
for prefixed assignment of closed-loop modes) has to be stated in the spectral
context. Alternative classical formulations in rings provide only sufficient
conditions for controllability/observability of the dynamic system since
characteristic quasipolynomial s is treated as if it were polynomial of
several independent variables, that is, as if exponential terms of the form z=e−hs were independent of s. In other words,
conditions implying loss of controllability/observability appear by considering
the arguments s and z as independent variables. Such conditions
are spurious and have to be removed in the cases where z≠e−hs.Theorem 3.4.
The following properties
hold.
(i) The state-space realization (2.1)-(2.2)
is spectrally controllable for some given h∈ℝ+q+q′ in the first orthant if and only if rank[sIn−∑i=0qAie−his,∑i=0q′Bie−hi′s]=nforalls∈C. The state-space realization system (2.1)-(2.2) is spectrally observable
if and only if rank[sIn−∑i=0qAiTe−hi′s,CT]=nforalls∈C. The state-space realization (2.1)-(2.2) is minimal of order n if
and only if it is spectrally controllable and spectrally observable [27], that
is,
(3.8)rank[sIn−∑i=0qAie−his,∑i=0q′Bie−hi′s]=[sIn−∑i=0qAiTe−hi′s,CT]=n∀s∈C. Both full-rank conditions hold
simultaneously; then the state-space realization (2.1)-(2.2) is minimal and the
converse  is also true.
(ii) The state-space
realization (2.1)-(2.2) is controllable in a ring independent of the delays, that
is, for any h∈ℝ+q+q′,
if rank[Cn(A(z),B(z))]=nforallz∈Cq+q′,
while the converse is not true in general. The state-space realization (2.1)-(2.2)
is observable in a ring independent of the delays if rank[On(C,A(z))]=nforallz∈Cq+q′,
and the converse is not true. The state-space realization (2.1)-(2.2) is minimal of
order n independent of the delays if
(3.9)rank[Cn(A(z),B(z))]=rank[On(C,A(z))]=n∀z∈Cq+q′, while the converse is not true
in general.
(iii) The state-space
realization (2.1)-(2.2) is controllable (resp., observable) in a ring independent
of the delays if and only if SCn(h)=∅ for any h∈ℝ+q+q′ (resp., SOn(h)=∅ for any h∈ℝ+q+q′). The state-space realization (2.1)-(2.2) is
minimal if and only if SCn(h)=SOn(h)=∅ for any h∈ℝ+q+q′,
that is, if and only if it is both controllable and
observable in a ring independent of the delays.
(iv) The state-space
realization (2.1)-(2.2) is controllable (resp., observable) in a ring for any given h∈ℝ+q+q′ if and only if SCn(h)=∅ (resp., SOn(h)=∅). The state-space realization (2.1)-(2.2) is minimal if
and only if SCn(h)=SOn(h)=∅, that is, if and only if it is both controllable and
observable in a ring.
(v) The state-space
realization (2.1)-(2.2) is controllable (resp., observable) in a ring being either
dependent on h∈ℝ+q+q′ or independent of the delays if and only if it
is spectrally controllable (resp., spectrally observable) being either
dependent on or independent of the delays.
(vi) The state-space
realization (2.1)-(2.2) is controllable (resp., observable) in a ring independent
of the delays, and equivalently spectrally controllable (resp., spectrally
observable) independent of the delays if and only if Huc=∅ (resp., Huo=∅) with Huc and Huo defined in (3.4). The state-space realization (2.1)-(2.2)
is minimal independent of the delays of order n if and only if Huc∪Huo=∅, that is, if and only if it is both controllable and observable
in a ring independent of the delays.
The state-space realization (2.1)-(2.2) is spectrally uncontrollable (resp., spectrally unobservable) for a
given h∈ℝ+q+q′ if and only if h∈Huc (resp., h∈Huo) and equivalently if and only if SCn(h)≠∅ (resp., SOn(h)≠∅).Proof.
(i) It is a direct generalization of
the Popov-Belevitch-Hautus rank controllability/observability tests [39] to the
case of point time delays. The result for controllability follows directly by
taking Laplace transforms in (2.1) with initial condition
φ≡0.
The parallel result for observability follows directly by taking Laplace
transforms in (2.1)-(2.2) with u≡0 and
nonzero point initial conditions at t=0, namely, x0=x(0)=φ(0)≠0.
(ii) Note that for any complex
matricesA and B and compatible orders, rank[sIn−A,B]=nforalls∈C if and only if
rank[Cn(A,B)]=n from Popov-Belevitch-Hautus rank
controllability test for the linear time-invariant delay-free case. Thus, for
polynomial matrices A(z) and B(z) in
several complex variables, rank[sIn−A(z),B(z)]=nforalls∈C for some given complex (q+q′)-tuple z if and only if rank[Cn(A(z),B(z))]=n for some given complex (q+q′)-tuple z, since
for each z, A(z) and B(z) are complex matrices.
Ifrank[Cn(A(z),B(z))]=n for any complex (q+q′)-tuple z, then the
property (ii) follows from the ring isomorphisms of Theorem 3.1, made
explicit in (3.1). A similar proof follows for observability. Since a loss of
full rank at some z does not necessarily imply that all of its components
satisfy zi=e−his for all i∈q+q′¯ and hq+i=hi′ for all i∈q′¯,
then the controllability/observability conditions are not necessary.
(iii)–(iv) IfSCn(h),
defined in (3.4), is empty, then rank[Cn(A(z),B(z))]=n for all complex (q+q′+1)-tuple (s,z) such
that the constraints zi=e−his for all i∈q+q′¯ and hq+i=hi′ for all i∈q¯′ are satisfied. This proves necessity.
Sufficiency follows directly from (ii). A similar reasoning applies to
observability with SCn(h) being empty and rank[On(C,A(z))]=n.
(v) The results
(i)–(v) imply for any givenh∈ℝ+q+q′ that
(3.10)rank[sIn−∑i=0qAie−his,∑i=0q′Bie−hi′s]=n∀s∈C⟺rank[Cn(A(z),B(z))]=n∀z∉SCn(h)⟺SCn(h)=∅,rank[sIn−∑i=0qAiTe−his,CT]=n∀s∈C⟺rank[On(C,A(z))]=n∀z∉SOn(h)⟺SOn(h)=∅.
(vi) It follows
directly from properties (iii)–(v), since the setsHuc and Huo are empty, that there is no
vector of delays such that the respective spectral controllability and
observability tests fail resulting in the corresponding matrices being
rank-defective. As a result, the system is controllable (resp., observable)
independent of the delays.Remark 3.5.
Note that from the definition ofSCn(h)=∅,
it is only necessary to consider z-(q+q′)-tuples with all of their components satisfying
simultaneously either |zi|≥1 or |zi|<1foralli∈q+q′¯,
which satisfy furthermore g(s,z):=Det(sIn−A(zI))=0 (since rank[Cn(A(z),B(z))]=n for all z such that g(s,z)≠0) in order to test SCn(h)=∅. Similar considerations apply for testing SOn(h)=∅. This
facilitates the way of performing the controllability/observability tests in
practice.Now, consider the sequenceHτ(z^):={Hi(z^)}1τ with τ∈ℕ which defines the τ-finite block complex Hankel matrix
(3.11)H(i,τ+1−i,z^):=⌊H1(z^)⋯Hτ+1−i(z^)⋮⋮⋮Hi(z^)⋯Hτ(z^)⌋=⌊CB(zE)⋯CAτ−i(zI)B(zE)⋮⋮⋮CAi−1(zI)B(zE)⋯CAτ−1(zI)B(zE)⌋. For τ=∞,
the infinite Hankel block matrix is HG(z^):=BlockMatrix(Hi+j−1(z^))i,j∈ℕ.
From (3.2) and (3.11), the subsequent technical result holds directly, where the
generic rank (denoted as gen rank) of the (⋅)-polynomial matrix (⋅) is its
maximum rank reached on the overall set of values of its argument. Note that
there is a natural surjective mapping Cq+q′→Cq^+q^′ which maps each argument z into one
corresponding z^(z); it is irrelevant to replace the argument z^ by its preimage z in all of the subsequent notations
and related discussions about controllability/observability in the appropriate
rings of polynomials, quasipolynomials, or series. Therefore, both arguments z
and z^ are used indistinctly where appropriate according
to convenience for clarity.Lemma 3.6.
The following properties hold independent of the delays.
(i)
H(i,τ+1−i,z^)=Oi(C,A(zI))Cτ+1−i(A(zI),B(zE))∀z^∈Cq^+q^′.(ii)
rank[H(i,τ+1−i,z^)]≤Min(i,τ+1−i,n)∀z^∈Cq^+q^′.(iii)
rank[H(i,τ+1−i,z^)]≤n for any τ,i∈ℕ with τ≥n+i−1, i≥nforallz^∈Cq^+q^′, wheren is the order of the state-space realization (2.1)-(2.2).(iv)
rank[HG(z^)]≤genrankτ≥n+i−1,i≥n,z^∈Cq^+q^′[HG(z^)]=genrankτ≥n+i−1,i≥n,z^∈Cq^+q^′[H(i,τ+1−i,z^)]≤n∀z^∈Cq^+q^′.Lemma3.6 establishes that the
rank of a τ-finite or infinite block Hankel matrix is
always finite and it cannot exceed the order of given state-space
realization.Theorem 3.7.
Consider two state-space realizations of the transfer matrix of (2.1)-(2.2):
(3.12)R:=(A0,Ai(i∈q¯),B0,Bj(j∈q¯′),C,D),R¯:=(A¯0,A¯i(i∈q¯),B¯0,B¯j(j∈q¯′),C¯,D¯≡D) of respective orders n (minimal) and n−>n.
Then, the following properties hold independent of the delays.
(i)
(3.13)n≤genrankτ≥n¯+i−1,i≥n¯,z^∈Cq^+q^′[H¯G(z^)]≤genrankτ≥n¯+i−1,i≥n¯,z^∈Cq^+q^′[H¯(i,τ+1−i,z^)]≤n¯,Min(genrankτ≥n+i−1,i≥n,z^∈SCn¯(h)[H¯(i,τ+1−i,z^)],genrankτ≥n+i−1,i≥n,z^∈SOn¯(h)[H¯(i,τ+1−i,z^)])<n¯,n≤Min(genrankτ≥n,z^∈SCn¯(h)[Cτ(A¯(zI),B¯(zE))],genrankτ≥n,z^∈SOn¯(h)[Oτ(A¯(zI),B¯(zE))])<n¯∀h∈ℝ+q+q′.(ii)
(3.14)genrankτ≥n+i−1,i≥n∈N,z^∈Cq^+q^′[H(i,τ+1−i,z^)]=genrankτ≥n,z^∈Cq^+q^′[Cτ(A(zI),B(zE))]=genrankτ≥n,z^∈Cq^+q^′[Oi(C,A(zI))]=n.(iii)
None of the following
conditions can hold for a complex functionz:C×ℝ+q+q→Cq+q′ defined by
(3.15)z(s,h)=(zI(s,h),zE(s,h))=(e−h1s,…,e−hqs,e−h1′s,…,e−hq′s) associated with internal and
external delays hi(i∈q¯), hq+j=hj′(j∈q¯′).
Then,
(3.16)rank[HG(z^)]<n,rank[H(i,τ+1−i,z^)]<nforanyτ(≥n+i−1),i(≥n)∈ℕ,rank[Cτ(A(zI),B(zE))]<n∀τ(≥n)∈ℕ,rank[Oτ(C,A(zI))]<n∀τ(≥n)∈ℕ.Proof.
Properties (i)-(ii) follow directly from
the factorization of Lemma3.6(i) and the rank constraints in Lemmas 3.6(ii) and 3.6(iii)
since the ranks of the controllability and observability matrices never exceed
the order of a minimal realization, and on the other hand, the generic ranks of
the observability and controllability matrices equalize the order of any minimal
realization. This proves that the generic rank is upper-bounded by n.
The fact that it is identical to a minimum n follows from the
contradiction which would arise if
(3.17)genrankτ≥n,∈Cq+q′[Cτ(A(zI),B(zE))]>rankτ≥n,z∈SCn(h)∪SOn(h)[Cτ(A(zI),B(zE))]; then there would exist n¯(<n)∈ℕ such that
(3.18)genrankτ≥n¯,z∈Cq+q′[Cτ(A(zI),B(zE))]=rankτ≥n¯,z∈SCn¯(h)∪SOn¯(h)[Cτ(A(zI),B(zE))]=n¯, and then the order of the square polynomial matrix A(ZI) and, as a result, that of matrices A0,Ai(i∈q¯) would be n¯<n. Since n is the order of a minimal
realization, SCn(h)∪SOn(h)=∅ from Theorem 3.4(iv), which implies and is
implied by rank[Cτ(A(zI),B(zE))]=rank[Oτ(C,A(zI))]=nforallτ(≥n)∈ℕ and all z:C×ℝ+q+q→Cq+q′ defined by z(s,h)=(zI(s,h),zE(s,h))=(e−h1s,…,e−hqs,e−h1′s,…,e−hq′s). This proves property
(iii).


## 4. Synthesis of Minimal Realizations

The problems of synthesis of a
minimal realization or a minimal partial realization are formulated in terms of
finding a state-space realization such that it matches a certain transfer
matrix which is formally identical to a series Laurent expansion at∞.
Thus, given the sequence Hτ(z^):={Hi(z^)}0τ with τ(≤∞)∈ℕ,
find matrices Ai∈ℝn×n(i∈q¯∪{0}), Bi∈ℝn×m(i∈q¯′∪{0}),
and C∈ℝp×n provided that they exist such that the following matching condition holds independent
of the delays either for all τ∈ℕ (minimal synthesis problem) or for some finite τ∈ℕ (minimal partial realization problem):
(4.1)G(s,z^):=C(sI−A0−∑i=1qAiziI)−1(B0+∑i=1q′BiziE)+D=∑i=0∞His−i=∑i=0τHis−i+0(s−τ−1), where G(s,z^)∈ℝp×m(s,z^)≈ℝp×m[[s]][z^]∋∑i=0τ≤∞Hi(z^)s−i such that n is as small as possible. If
the minimal (resp., partial minimal) realization synthesis problem is solvable
(i.e., it has a solution), then by making the changes zi=e−his, zq+j=e−hj′s(i∈q¯,j∈q¯′),
a state-space realization (2.1)-(2.2) is obtained so that (4.1) holds for τ∈ℕ (resp., for some natural number τ<∞). If the problem is solvable, then there are
infinitely many minimal (resp., partial minimal) realizations satisfying it,
since any nonsingular state transformation preserves the transfer matrix. In
what follows, the result, where the McMillan degree (denoted by μ) of a rational transfer matrix coincides with
that of the rank of the infinite associated block Hankel matrix for z^∈Cq+q′ (which is also called the McMillan degree of
this one), is extended from the delay-free case. The block Hankel matrices are
now polynomial matrices. The idea is extended also to truncated finite block
Hankel matrices, and it concludes that such a degree equalizes the order of
minimal (or partial minimal) state-space realizations.Theorem 4.1.
The following properties hold.
(i) The McMillan degreen=μ(G(s,h)) of the transfer matrix G(s,h) is the unique order of any minimal realization
of G(s,h) and satisfies the following constraints for
any set of delays being components of some given h∈ℝ+q+q′:
(4.2)∞>n(h)=μ(G(s,h))=Maxτ∈ℕ(μz∈SCτ(h)∪SOτ(h)(Hτ(z)))(4.3)=Maxτ∈ℕ(∑i+j=τ+1rankH(i,j,z)−∑i+jτrankH(i,j,z):z∈SCτ(h)∪S0τ(h))(4.4)=Maxz∈SC∞(h)∪S0∞(h)(rankH(i,j,z):n≤i∈ℕ,i+n−1≤τ∈ℕ).
(ii) The state-space dimensionnτ(h)(τ∈ℕ) of any minimal partial realization satisfies
(4.5)∞>nτ(h)=Maxτ∈ℕ(μz∈SCτ(h)∪SOτ(h)(H(i,τ+1−i,z)):i∈τ¯), where nτ(h)=n(h),
and then the minimal partial realization is a minimal realization for all τ(≥τ0)∈ℕ and sufficiently large finite τ0∈N with
(4.6)nτ(h)=nτ0(h)=n(h)=Minz∈SC∞(h)∪S0∞(h)(rankH(i,τ+1−i,z):τ0≤i∈N,i+τ0−1≤τ∈N).
(iii) Redefine
by simplicity the delays according to hq+i=hi′(i∈q¯′).
Define h00=0 and let hi0 be defined with hi≠0 and hj=0(j≠i) for (i∈q+q′). Assume that n(hi0)=ni0=n0 (some constant n0 in ℕ) for all i∈q¯,
where
(4.7)ni0:=Minτi0∈ℕ(∑i+j=τi0+1rankH(i,j,αi)−∑i+j=τi0rankH(i,j,αi))=Minτ0∈ℕ(∑i+j=τ0+1rankH(i,j,αi)−∑i+j=τ0rankH(i,j,αi)) with αi∈Cq+q′ having the ith component distinct from unity and the
remaining ones being unity; hi0 is an associate (q+q′)-tuple of delays in ℝ+q+q′ with only the ith component being nonzero
and Ai=0, with the remaining ones being zero and τ0:=Max(τi0:i∈q+q′¯).
Then, the order for any minimal realization independent of the delays is
(4.8)n=n(h)=n(hi0)=n0=τ0=Maxτ(≥τ0)∈ℕ(genrankz∈SCτ(h)∪SOτ(h)H(i+ℓ−1,τ+ℓ−i,z))∀h∈ℝ+q+q′, and
all the matrices defining the state-space realization (2.1)-(2.2) are independent
of the delays.Proof.
Equation (4.4) of property (i) follows directly by using a close reasoning to that in Theorem 3.7(ii) since n∈ℕ has to exist such that
(4.9)Maxτ∈ℕ(μz∈SCτ(h)∪SOτ(h)(Hτ(z)))=genrankτ≥n+i−1,i≥n,z∈Cq+q′[HG(z)]=genrankτ≥n+i−1,i≥n,z∈Cq+q′[H(i,τ+1−i,z)],genrankτ≥n,z∈Cq+q′[Cτ(A(zI),B(zE))]=rankτ≥n,z∈SCn(h)∪SOn(h)[Cτ(A(zI),B(zE))]=n∀τ(≥n)∈ℕ. Equation
(4.4) is identical to (4.2) from the definition of McMillan degrees of minimal
realizations, namely, of partial
minimal realizations [4, 7] and transfer matrices and the fact that the
generic ranks of infinite block Hankel matrices [4, 5] are constant and
equalize that of those exceeding appropriate sizes under certain thresholds. Equation
(4.3) equalizes (4.2) and (4.4) by extending a parallel result given in [4] for
real block Hankel matrices describing the realization problem of the delay-free
case (i.e., H(i,j) does not depend on a multidimensional complex tuple z) and
by the fact that this rank does not increase for τ∈ℕ exceeding a certain finite minimum threshold τ0∈ℕ.
Property (i) has been proved. Property (ii) follows in the same way by using
similar considerations for any given τ∈ℕ.
To prove property (iii), first note that hi0∈ℝ+q+q′ for i∈q+q′¯ defines each particular delay-free parameterization
of (2.1)-(2.2) with some zero delays and the remaining ones being infiniy (or, equivalently, with their associated
matrices of dynamics A(⋅) being null) as follows:
(4.10)z˙i(t)=(∑j=0qAjiA(q,i,j))zi(t)+∑j=0q′(BjiB(q,q′,i,j))u(t).i∈q+q′¯ under initial conditions z(t)=z0=φ(0),
where the binary indicators
(4.11)iA(q,i,j)=1ifi,j(≠i)∈q¯,i∈q+q′¯,iA(q,i)=0ifi,j(=i)∈q¯,iB(q,q′,i,j)=1ifi(≠j)∈q+q′¯/q¯,i∈q¯,iB(q,q′,i,j)=0ifi,j(=i)∈q+q′¯/q¯ have been used for notational
simplicity, since h00=(0,0,…,0) and hi0 is defined with components hj=0, for all j(≠i)∈q+q′¯,
and Ai=0,
or hi=∞ if i∈q¯ and Bi=0 or hq+i=hi′=∞ if i∈q+q′¯/q¯.
All the delay-free parameterizations (4.10)-(4.11) of (2.1)-(2.2) have a minimal state-space
realization of identical dimension n=n0 from (4.7). Using (3.11), now independent of the
complex indeterminate z, for each of the delay-free state-space realizations,
Lemma 3.6(i), and the fact that all the above delay-free realizations are
spectrally controllable and observable, one can construct q+q′ algebraic matrix equations to calculate the matrices A^i=∑j(≠i)=0qAj,B^ℓ=∑j(≠ℓ)=0qBj,i∈/q¯∪{0},ℓ∈q¯′∪{0},
from which Ai,i∈q¯∪{0}; Bj,j∈q¯′∪{0} can be calculated uniquely for some matrix C
such that (3.11) holds with D=H0.
Since all the matrices of parameters of (2.1)-(2.2) may be
calculated, then the minimal order is identical, independent of the delays, so
that (4.8) holds.It is of interest to provide some
result concerning the case when controllability and observability are maintained,
and then the order of the minimal realization is not modified, under some
parametrical and delay disturbances. In what follows, parametrical
perturbations consisting of matrix scaling and constant perturbations of all
delays are discussed.Theorem 4.2.
Consider the
transfer matrix(4.12)G^(s,δ,ρ1,ρ2,λ,h˜)=ρ1C(sIn−δ∑i=0qAie−hise−h˜s)−1(∑i=0q′ρ2Bie−hise−h˜′s)+λD parameterized
in the sextuple of real scalars p:=(δ,ρ1,ρ2,λ,h˜,h˜′) which models a perturbation of a nominal
transfer matrix
(4.13)G^(s,δ,ρ1,ρ2,λ,h˜)=C(sIn−∑i=0qAie−his)−1(∑i=0q′Bie−his)+D parameterized by p0:=(1,1,1,1,0,0),
and assume that the denominator quasipolynomial and all the numerator quasipolynomials
of (4.12) possess principal terms
[8]. Assume that (2.1)-(2.2) is a minimal realization of (minimal) order n of (4.13). Then, a minimal realization of the same order n of (4.12) is given
by (2.1)-(2.2) with the parametrical changes C→ρ1C, Ai→δAi(i∈q¯∪{0}), Bi→ρ2Ai, D→λD and delay changes hi→hi+h˜, hj′→hj′+h˜′(i∈q¯∪{0},j∈q¯′∪{0}) for any finite delay perturbations h˜ and h˜′ and for any real λ if and only if ρ1ρ2δ≠0.Proof.
The block Hankel (i, j)-matrix associated with the p-parameterization is related to that associated with the p0-parameterization by
(4.14)HG^(i,j,p)=BlockDiag(γ(s)(i−1)/2,γ(s)(i−1)/2,…,γ(s)(i−1)/2)HG^(i,j,p0)×BlockDiag(γ(s)(i−1)/2,γ(s)(i−1)/2,…,γ(s)(i−1)/2)×BlockDiag(η(s),η(s),…j−i+1︸,η(s)) and H0=λD,
where γ(s,h˜)=δe−h˜s and η(s,h˜′)=ρ1ρ2e−h˜′s.
Since the numerator and denominator quasipolynomials have principal terms, they
do not have unstable zeros at infinity, that is, zeros with Res→∞.
Then,
(4.15)n=Maxi,j∈ℕ(genrankHG^(i,j,p0))=Maxi,j∈ℕ(μ(HG^(i,j,p0)))=Maxi,j∈ℕ(μ(HG^(i,j,p))) if and only if δρ1ρ2≠0⇔γ(s,h˜)η(s,h˜′)≠0 for all complex indeterminate
s and any finite delay
disturbances h˜ and h˜′.The
above result establishes that scalar nonzero scaling of the matrices which
parameterize (2.1)-(2.2) preserves the spectral controllability/observability, and then the
degree of any minimal realization for any finite constant change of the
internal delays and a finite constant change of the external point delays. For these purposes, it is assumed with no loss of generality that the system transfer function is
defined by numerator and denominator quasipolynomials which possess principal
terms. It has been also proved that those
properties hold by adding any zero or nonzero interconnection constant matrix
to a strictly proper transfer matrix whose nominal state-space realization
possesses them. An immediate consequent robustness result for minimal
state-space realizations of very easy
testing and interpretation applicable to the delay-free single-input single-output
case is as follows.Corollary 4.3.
Consider the following three transfer functions:(4.16)G^p(s)=G^(s)n˜(s)d˜(s),G^(s)=(cT(sIn−δA0)−1ρb0+λd0),G^0(s)=(cT(sIn−A0)−1b0+d0), where n˜(s) and d˜(s) are polynomials of respective degrees n˜n(s)=deg(n˜(s)) and n˜d(s)=deg(d˜(s)) which satisfy the degree constraint n˜d(s)≥n˜n(s)+m−n with
(4.17)n=deg(sIn−δA0),m={nifλd0≠0,m′<nifλd0=0,m′=deg(cTAdj(sIn−δA0)ρb0). Then, the following properties hold.
(i)
The state-space
realization(cT,A0,b0,d0) of G^0(s) is both controllable and observable, and then
minimal of order n, if and only if the associate Hankel matrix satisfies rankHG^0=Maxi,j∈ℕ(rankHG^0(i,j))=rankHG^0,j≥i+n−1;i,j∈ℕ(i,j)=n0.
Then, G^0(s) is proper (strictly proper if d0≠0) and zero-pole cancellation-free, and A0 is of order n0.(ii)
Assume that the state-space realization(cT,A0,b0,d0) of G^0(s) is controllable and observable of minimal
order n0.
Then, the state-space realization (cT,δA0,ρb0,λd0) of G^(s) is both controllable and observable, and then
minimal of order n=n0,
if and only if δρ≠0.
Then, G^(s) is proper (strictly proper if λd0≠0) and zero-pole cancellation-free, and A is of order n=n0.
As a result,
(4.18)rankHG^=Maxi,j∈ℕ(rankHG^(i,j))=rankHG^(i,j)j≥i+n−1;i,j∈ℕ=n.(iii)
G^p(s) is state-space realizable and strictly proper
if and only if n˜d(s)>n˜n(s)+m−n.
Assume that G^0(s) is zero-pole cancellation-free of order n0.
Then, the following hold.
(1)
G^p(s)=nG^(s)/dG^(s) is cancellation-free if and only if ρδ≠0 and the three pairs of polynomials (nG^(s),d˜(s)), (dG^(s),n˜(s)),
and (d˜(s),n˜(s)) are each coprime, where dG^(s)=(sIn−δA0) and nG^(s)=cTAdj(sIn−δA0)ρb0+λd0dG^(s) are the denominator and numerator polynomials
of G^p(s).
As a result, G^p(s) has a minimal realization (then, controllable
and observable) of order n^=n(=n0)+n˜d.(2)
Assume thatG^0(s) is zero-pole cancellation-free of order n0 and ρδ≠0.
Assume also that the polynomial pairs (nG^(s),d˜(s)), (dG^(s),n˜(s)) are both coprime. Then, G^p(s) has zero-pole cancellation(s) at the common
factors of the pair (d˜(s),n˜(s)),
if any. A minimal realization of G^p(s) has order n^=n0+n˜d−n˜d′,
which satisfies n0+n˜d≥n^≥n0,
where n˜d′≥0 is the degree of the cancellation polynomial,
if any. Also,
(4.19)n0+n˜d≥rankHG^p=Maxi,j∈ℕ(rankHG^p(i,j))=rankHG^p(i,j)j≥i+n^−1;i,j∈ℕ=n^≥n0.Outline of Proof
The proof follows directly from Theorem4.2 by noting
from its definitions
of the various transfer functions that G^(s)=G^0(s) if ρ=1 and λ=0,
and G^p(s)=G^(s) if n˜(s)=d˜(s).
As a result, G^p(s) is zero-pole cancellation-free if the
polynomials n˜(s) and d˜(s) are nonzero real scalars provided that G^0(s) is also cancellation-free. The transfer function G^p(s) is also cancellation-free if G^0(s) and n˜(s)/d˜(s) are both cancellation-free and, furthermore, (n˜(s),Det(sIn−A0)) and (d˜(s),cTAdj(sIn−A0)b0+d0Det(sIn−A0)) are both coprime pairs of polynomials.
Then,
proceed as follows to complete the proof of the various properties.
(1)
Remove the delays and consider the single-input single-output case by relating ranks of infinite or
partial block Hankel matrices with orders being minimal, then being controllable
and observable.(2)
Note those minimal state-space realizations which cannot have zero-pole
cancellations in their transfer function and vice versa.(3)
If there are cancellations, then the
associate realization is never minimal.
The
above result can be extended very easily to the multivariable case and to the
presence of delays.


## 5. Concluding Remarks

This
paper addresses the problem of synthesizing minimal realizations and partial
minimal realizations of linear time-invariant systems with (in general, incommensurate)
multiple constant internal and external point delays. The main body of the
formalism of the properties of controllability, observability,
minimal realizations, and minimal partial realizations is discussed
through a formulation over appropriate rings of polynomials and corresponding (roughly
speaking, isomorphic) truncations of formal Laurent expansions of rational
transfer matrices. However, the spectral versions of controllability and
observability are used to remove spurious conditions which lead to apparent loss
of those properties. In this sense, the presented results are stronger than
those previous parallel ones derived in a full formalism over rings. Some
particular results are also obtained for the single-input single-output case by
mutually relating realizations of transfer functions with given basic control
and output vectors, input-output interconnection gain, and dynamics matrix, but
being on the other hand dependent at most on three potentially freely chosen
real parameters. The minimal state-space realizations are interpreted in terms of absence of zero-pole
cancellation in the transfer function by giving some direct relationships among
those parameters. Since one starting point for the analysis is the Hankel
matrix, the formalism is appropriate to be applied for obtaining transfer
matrices, minimal realizations, and minimal partial realizations collected
input-output data.



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*Source: 790530-2008-04-06.xml*