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Mathematical Problems in Engineering
Volume 2008, Article ID 790530, 19 pages
http://dx.doi.org/10.1155/2008/790530
Research Article

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

Department of Electricity and Electronics, Institute for Research and Development of Processes, Campus of Leioa (Bizkaia), Apartado 644, 48080 Bilbao, Spain

Received 1 June 2007; Revised 22 November 2007; Accepted 28 November 2007

Academic Editor: Giuseppe Rega

Copyright © 2008 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [37]. 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 [922]. 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 [1618, 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, 2331] 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 [14, 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. Section 2 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 matrix 𝐺(𝑠) over a filed K takes values in 𝐊𝑝×𝑚(𝑠), where s is the Laplace transform variable.(2)Any proper rational 𝐺(𝑠)𝐊𝑝×𝑚(𝑠) can be expanded in a formal Laurent series at infinity leading to the formal identity 𝐺(𝑠)=𝑖=0𝐻𝑖𝑠𝑖 with {𝐻𝑖}𝑖0 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 0={0} and 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 𝐴𝑛×𝑛, 𝐵𝑛×𝑚, 𝐶𝑝×𝑛, and 𝐷𝑝×𝑚, of order 𝑛 being minimal, that is, as small as possible, so that, given the infinite sequence {𝐻𝑖𝐊𝑝×𝑚}0, the identity of the proper rational transfer matrix with the formal Laurent series at holds, that is, 𝐺(𝑠)=𝑖=0𝐻𝑖𝑠𝑖=𝐶(𝑠𝐼𝐴)1𝐵+𝐷 and then 𝐻0=𝐷 and 𝐻𝑖=𝐶𝐴𝑖1𝐵, 𝑖. A realization is minimal if and only if it is both controllable and observable.(5)If D = 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 approximation 𝜏 is formulated as follows. Given a finite sequence {𝐻𝑖𝐊𝑝×𝑚}𝜏0 with some 𝜏, satisfying 𝐻0=𝐷, 𝐻𝑖=𝐶𝐴𝑖1𝐵, forall𝑖𝜏={1,2,,𝜏}, there exists a quadruple (𝐴,𝐵,𝐶,𝐷), with 𝐴𝑛×𝑛, 𝐵𝑛×𝑚, 𝐶𝑝×𝑛, and 𝐷𝑝×𝑚, of minimal order 𝑛 (the order of A) such that 𝐶(𝑠𝐼𝐴)1𝐵+𝐷=𝜏𝑖=0𝐻𝑖𝑠𝑖+0(𝑠𝜏1).

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

Consider the linear time-invariant dynamic system in state-space form: ̇𝑥𝑡=𝑞𝑖=0𝐴𝑖𝑥𝑡𝑖+𝑞𝑖=0𝐵𝑖𝑢𝑡𝑖,(2.1)𝑦𝑡=𝐶𝑥𝑡+𝐷𝑢𝑡,(2.2) where 𝑥+×Σ×𝑈Σ𝑛, 𝑢+𝑈𝑚, 𝑦+×Σ×𝑈Y𝑝 are the state, input, and output vector functions in their respective state, input, and output spaces Σ,𝑈, and Y, where +={𝑡𝑡0}, 0=0=0, and 𝑖(>0)+(𝑖𝑞={1,2,,𝑞}), 𝑖(>0)+(𝑖𝑞) are the internal and external point delays. If the input is generated via state feedback, then 𝑢+×Σ𝑈𝑚. If it is generated via output feedback, then 𝑢+×𝑌𝑈𝑚. The internal delays (𝑖), which are assumed to be pairwise distinct, and the external ones (𝑗), which are also assumed to be pairwise distinct, are both incommensurate delays; that is, they are not necessarily equal to 𝑖𝑏 and 𝑗𝑏(𝑖𝑞,𝑗𝑞), with some 𝑏>0, 𝑏>0; 𝐴𝑖𝑛×𝑛(𝑖𝑞{0}), 𝐵𝑖𝑛×𝑚(𝑖𝑞{0}), 𝐶𝑝×𝑛, and 𝐷𝑝×𝑚 are matrices of real entries which parameterize (2.1)-(2.2). The dynamic system is subject to initial conditions, 𝜑[,0]𝑛, where =Max𝑖𝑞(𝑖) 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 𝜑0, a transfer matrix exists, which is defined by 𝐺𝑠=𝑌𝑠𝑈𝑠=[Lap+𝑦𝑡Lap+𝑢𝑡]𝜑0=𝐶(𝑠𝐼𝑛𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠)1(𝑞𝑗=0𝐵𝑖𝑒𝑖𝑠)+𝐷,(2.3) where Lap+(𝑣(𝑡)) is the right Laplace transform of 𝑣+𝐬 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 𝐺𝑖𝑗𝑠=𝑐𝑇𝑖Adj𝑠𝐼𝑛𝑞𝑘=0𝐴𝑖𝑒𝑘𝑠𝑞𝑗=0𝑏𝑗𝑒𝑠Det𝑠𝐼𝑛𝑞𝑘=0𝐴𝑖𝑒𝑘𝑠+𝐷𝑖𝑗,(2.4) where Adj() and Det() stand for the adjoint matrix and the determinant of the ()-matrix, respectively, and 𝑐𝑇𝑖 and 𝑏𝑗 are the ith row of 𝐶(𝑖𝑝) and the jth column of 𝐵(𝑞{0}), respectively. Define complex (𝑞+𝑞) and (̂𝑞+̂𝑞) tuples as follows: 𝑧=(𝑧𝐼,𝑧𝐸)=(𝑧1,,𝑧𝑞,𝑧𝑞+1,𝑧𝑞+𝑞)𝐂𝑞+𝑞,̂𝑧=(̂𝑧𝐼,̂𝑧𝐸)=(𝑧1,,𝑧̂𝑞,𝑧̂𝑞+1,𝑧̂𝑞+̂𝑞)𝐂̂𝑞+̂𝑞,(2.5) where 𝑞̂𝑞̂𝑞0=𝑛(𝑞𝑖=1𝑞𝑖),𝑞̂𝑞̂𝑞0=̂𝑞0𝑞+1(2.6) so that the components 𝑧𝑖 and 𝑧𝑗 correspond to 𝑒𝑖𝑠 and 𝑒𝑗𝑠(𝑖𝑞,𝑗𝑞), respectively, in a multiargument description, where s and all the components 𝑧𝑖 and 𝑧𝑗 are taken as independent variables. The components ̂𝑧𝑖 of the extended ̂𝑧 are associated with combined delays 𝑖 which are formed with all the combinations of sums of the internal delays 𝑖 and their respective integer multiplicities. The components ̂𝑧𝑗 and associated delays 𝑖 are formed with all the above combinations of sums of the internal delays 𝑖 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, 𝑗=𝑗(𝑗𝑛) in a single-input single-output transfer function. Then, 𝑝×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 𝑝×𝑚(𝑠) so that there is a natural bijective mapping between each entry 𝐺𝑖𝑗(𝑠) and 𝐺𝑖𝑗𝑠,̂𝑧=𝑁𝑖𝑗𝑠,̂𝑧𝑀𝑠,̂𝑧=𝑛𝑖𝑗𝑘=0𝑁𝑖𝑗𝑘̂𝑧𝑠𝑘𝑛𝑘=0𝑀𝑘̂𝑧𝑠𝑘=̂𝑞+̂𝑞=0𝑁𝑖𝑗𝑠̂𝑧̂𝑞=0𝑀𝑠̂𝑧=𝑛𝑖𝑗𝑘=0̂𝑞+̂𝑞=0𝑁𝑖𝑗𝑘𝑠𝑘̂𝑧𝑛𝑘=0̂𝑞=0𝑀𝑘𝑠𝑘̂𝑧(2.7) whose numerator 𝑁𝑖𝑗𝑠,̂𝑧 and denominator 𝑀(𝑠,̂𝑧) are, respectively, polynomials in several variables of respective real coefficients 𝑁𝑖𝑗𝑘 and 𝑀𝑘, and 𝑁𝑖𝑗𝑘(𝑍), 𝑀𝑘(̂𝑧), 𝑁𝑖𝑗𝑘(𝑠), and 𝑀𝑘(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 of 𝐺𝑖𝑗𝑠,̂𝑧 are, respectively, in the polynomial additive groups (rings if 𝑝=𝑚) 𝑝×𝑚[𝑠,̂𝑧] and [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=0𝐻𝑖(̂𝑧)𝑠𝑖 with 𝐻𝑖(̂𝑧)𝑝×𝑚[̂𝑧], it follows that 𝑝×𝑚(𝑠,̂𝑧)𝑝×𝑚[[𝑠]][̂𝑧], the additive group of formal Laurent power series with matrices over 𝑝×𝑚 at in the polynomial multiple indeterminate defined by the components of the ̂𝑧-tuple. Note that the formal series additive group 𝑝×𝑚[[𝑠]][̂𝑧] is the completion of the polynomial matrix additive group 𝑝×𝑚[𝑠][̂𝑧] (𝑝×𝑚[𝑠,̂𝑧]) with respect to the I-adic topology, where I is the ideal of the polynomial matrix additive group 𝑝×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)𝑛×𝑛[𝑒𝑖𝑠𝑖𝑞{0}]𝑛×𝑛[𝑧𝐼] if 𝑖+{0} for all 𝑖,𝑗𝑞, and 𝑖𝑗forall𝑖,𝑗(𝑖)𝑞; 𝑛×𝑚[𝑒𝑖𝑠𝑖𝑞{0}]𝑛×𝑚[𝑧𝐸].(ii)𝑝×𝑚(𝑠,̂𝑧)𝑝×𝑚[[𝑠]][̂𝑧]𝑝×𝑚[𝑠,̂𝑧] if 𝑖+{0} for all 𝑖,𝑗𝑞, and 𝑖𝑗 for all 𝑖,𝑗(𝑖)𝑞.(iii)𝑝×𝑚[𝑠][̂𝑧] is a dense subspace of 𝑝×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 𝑝×𝑚[𝑠][̂𝑧] generated by the indeterminate complex (̂𝑞+̂𝑞+1)-tuple (𝑠,̂𝑧).

Note that the isomorphisms of Theorem 3.1(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 0=0=0 and all the remaining internal delays are pairwise distinct while all the remaining external delays are pairwise distinct as well.

From Theorem 3.1(i), the following bijections may be established: 𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠𝐴𝑧𝐼=𝑞𝑖=0𝐴𝑖𝑧𝐼𝑖𝑛×𝑛𝑧𝐼;𝑞𝑖=0𝐵𝑖𝑒𝑖𝑠𝐴𝑧𝐸=𝑞𝑖=0𝐴𝑖𝑧𝐸𝑖𝑛×𝑛𝑧𝐼(3.1) so that via (3.1) the controllability and observability matrices of the nth realization (2.1)-(2.2) result: 𝐶𝑛𝐴𝑧,𝐵𝑧=𝐶𝑛𝐴𝑧𝐼,𝐵𝑧𝐸=𝐵𝑧𝐸,𝐴𝑧𝐼𝐵𝑧𝐸,,𝐴𝑛1𝑧𝐼𝐵𝑧𝐸,𝑂𝑛𝐶,𝐴𝑧=𝑂𝑛𝐶,𝐴𝑧𝐼=𝐶𝑇,𝐴𝑇𝑧𝐼𝐶𝑇,,𝐴𝑛1𝑇𝑧𝐼𝐶𝑇𝑇(3.2) in 𝑛×(𝑛+𝑚)[̂𝑧] and 𝑝×(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 𝑖0(𝑖𝑞), 𝑗0(𝑗𝑞), and 0=0=0 in the indeterminate z does not imply that the rank is lost for some complex indeterminate s satisfying the constraints 𝑧𝑖=𝑒𝑖𝑠, 𝑧𝑞+𝑗=𝑒𝑗𝑠(𝑖𝑞,𝑗𝑞) 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: 𝐡=(1,2,,𝑞,𝑞+1=1,𝑞+2=2,,𝑞+𝑞=𝑞)𝑞+𝑞+,(3.3) which is the closed first orthant in 𝑞+𝑞, and the associated sets of delays are 𝐻𝑢𝑐 and 𝐻𝑢𝑜, where controllability and, respectively, observability are lost: 𝑆𝐶𝑛(𝐡)={𝑧=𝑧1,𝑧2,,𝑧𝑞𝐂𝑞+𝑞𝑧𝑖=|||𝑧𝑖|||𝜃𝑖=𝑒𝑖𝑠,𝜎𝐶=ln|||𝑧𝑖|||𝑖,𝜔𝐶=𝑡𝑔𝜃𝑖𝑖𝑖𝑞+𝑞,rank𝐶𝑛𝐴𝑧𝐼,𝐵𝑧𝐸<𝑛},𝑆𝑂𝑛(𝐡)={𝑧=𝑧1,𝑧2,,𝑧𝑞𝐂𝑞𝑧𝑖=|||𝑧𝑖|||𝜃𝑖=𝑒𝑖𝑠,𝜎𝑂=ln|||𝑧𝑖|||𝑖,𝜔𝑂=𝑡𝑔𝜃𝑖𝑖𝑖𝑞,rank𝑂𝑛𝐶,𝐴𝑧𝐼<𝑛},𝐻𝑢𝑐=𝐡𝑞+𝑞+𝑆𝐶𝑛(𝐡),𝐻𝑢𝑜=𝐡𝑞+𝑞+𝑆𝑂𝑛(𝐡).(3.4) 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 𝐡𝑞+𝑞+, then the system is controllable (resp., observable) in an additive group independent of the delays. Note directly that 𝐡𝐻𝑢𝑐𝑆𝐶𝑛𝐡;𝑆𝐶𝑛𝐡=𝐡𝑞+𝑞+𝐻𝑢𝑐=,(3.5) 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+×𝑛𝑚, fulfilling 𝑢(𝑡)=0forall𝑡=+, such that 𝑈(𝑠)=Lap(𝑢(𝑡)) exists for any given prefixed suited characteristic closed-loop polynomial: 𝑛𝑖=0𝑞+𝑞𝑗=0𝑓𝑖𝑗𝑠𝑖𝑒𝑗𝑠=Det(𝑠𝐼𝑛𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠𝑞𝑖=0𝐵𝑖𝑒𝑖𝑠𝑈𝑠).(3.6) 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+×𝑛𝑝, fulfilling 𝑢(𝑡)=0forall𝑡=+, such that 𝑈(𝑠)=Lap(𝑢(𝑡)) exists for any given prefixed suited characteristic closed-loop polynomial: 𝑛𝑖=0𝑞𝑗=0𝑓𝑖𝑗𝑠𝑖𝑒𝑗𝑠=Det(𝑠𝐼𝑛𝑞𝑗=0𝐴𝑇𝑖𝑒𝑖𝑠𝐶𝑇𝑈𝑠).(3.7) 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 indeterminates 𝑧𝑖 (which replace 𝑒𝑖𝑠 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., [3538]). 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=𝑒𝑠 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 𝑧𝑒𝑠.

Theorem 3.4. The following properties hold.
(i) The state-space realization (2.1)-(2.2) is spectrally controllable for some given 𝐡𝑞+𝑞+ in the first orthant if and only if rank[𝑠𝐼𝑛𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠,𝑞𝑖=0𝐵𝑖𝑒𝑖𝑠]=𝑛forall𝑠𝐂. The state-space realization system (2.1)-(2.2) is spectrally observable if and only if rank[𝑠𝐼𝑛𝑞𝑖=0𝐴𝑇𝑖𝑒𝑖𝑠,𝐶𝑇]=𝑛forall𝑠𝐂. 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, rank[𝑠𝐼𝑛𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠,𝑞𝑖=0𝐵𝑖𝑒𝑖𝑠]=[𝑠𝐼𝑛𝑞𝑖=0𝐴𝑇𝑖𝑒𝑖𝑠,𝐶𝑇]=𝑛𝑠𝐂.(3.8) 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 𝐡𝑞+𝑞+, if rank[𝐶𝑛(𝐴(𝑧),𝐵(𝑧))]=𝑛forall𝑧𝐶𝑞+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))]=𝑛forall𝑧𝐶𝑞+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 rank𝐶𝑛𝐴𝑧,𝐵𝑧=rank𝑂𝑛𝐶,𝐴𝑧=𝑛𝑧𝐶𝑞+𝑞,(3.9) 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 𝐡𝑞+𝑞+ (resp., 𝑆𝑂𝑛(𝐡)= for any 𝐡𝑞+𝑞+). The state-space realization (2.1)-(2.2) is minimal if and only if S𝐶𝑛(𝐡)=𝑆𝑂𝑛(𝐡)= for any 𝐡𝑞+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 𝐡𝑞+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 𝐡𝑞+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 HucHuo=, 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 𝐡𝑞+𝑞+ if and only if 𝐡𝐻𝑢𝑐 (resp., 𝐡Huo) 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 𝜑0. The parallel result for observability follows directly by taking Laplace transforms in (2.1)-(2.2) with 𝑢0 and nonzero point initial conditions at 𝑡=0, namely, 𝑥0=𝑥(0)=𝜑(0)0.
(ii) Note that for any complex matrices A and B and compatible orders, rank[𝑠𝐼𝑛𝐴,𝐵]=𝑛forall𝑠𝐂 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[𝑠𝐼𝑛𝐴(𝑧),𝐵(𝑧)]=𝑛forall𝑠𝐂 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.
If rank[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=𝑒𝑖𝑠 for all 𝑖𝑞+𝑞 and 𝑞+𝑖=𝑖 for all 𝑖𝑞, then the controllability/observability conditions are not necessary.
(iii)–(iv) If 𝑆𝐶𝑛(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=𝑒𝑖𝑠 for all 𝑖𝑞+𝑞 and 𝑞+𝑖=𝑖 for all 𝑖𝑞 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 given 𝐡𝑞+𝑞+ that rank[𝑠𝐼𝑛𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠,𝑞𝑖=0𝐵𝑖𝑒𝑖𝑠]=𝑛𝑠𝐂rank𝐶𝑛𝐴𝑧,𝐵𝑧=𝑛𝑧𝑆𝐶𝑛(𝐡)𝑆𝐶𝑛(𝐡)=,rank[𝑠𝐼𝑛𝑞𝑖=0𝐴𝑇𝑖𝑒𝑖𝑠,𝐶𝑇]=𝑛𝑠𝐂rank𝑂𝑛𝐶,𝐴𝑧=𝑛𝑧𝑆𝑂𝑛(𝐡)𝑆𝑂𝑛(𝐡)=.(3.10)
(vi) It follows directly from properties (iii)–(v), since the sets 𝐻𝑢𝑐 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 of 𝑆𝐶𝑛(𝐡)=, it is only necessary to consider z-(𝑞+𝑞)-tuples with all of their components satisfying simultaneously either |𝑧𝑖|1 or |𝑧𝑖|<1forall𝑖𝑞+𝑞, which satisfy furthermore 𝑔(𝑠,𝑧)=Det(𝑠𝐼𝑛𝐴(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 sequence 𝐻𝜏(̂𝑧)={𝐻𝑖(̂𝑧)}𝜏1 with 𝜏 which defines the 𝜏-finite block complex Hankel matrix 𝐻𝑖,𝜏+1𝑖,̂𝑧=𝐻1̂𝑧𝐻𝜏+1𝑖̂𝑧𝐻𝑖̂𝑧𝐻𝜏̂𝑧=𝐶𝐵𝑧𝐸𝐶𝐴𝜏𝑖𝑧𝐼𝐵𝑧𝐸𝐶𝐴𝑖1𝑧𝐼𝐵𝑧𝐸𝐶𝐴𝜏1𝑧𝐼𝐵𝑧𝐸.(3.11) For 𝜏=, the infinite Hankel block matrix is 𝐻𝐺(̂𝑧)=BlockMatrix(𝐻𝑖+𝑗1(̂𝑧))𝑖,𝑗. 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+𝑞𝐂̂𝑞+̂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)𝐻𝑖,𝜏+1𝑖,̂𝑧=O𝑖𝐶,𝐴𝑧𝐼C𝜏+1𝑖𝐴𝑧𝐼,𝐵𝑧𝐸̂𝑧𝐂̂𝑞+̂𝑞.(ii)rank𝐻𝑖,𝜏+1𝑖,̂𝑧Min(𝑖,𝜏+1𝑖,𝑛)̂𝑧𝐂̂𝑞+̂𝑞.(iii)rank[𝐻(𝑖,𝜏+1𝑖,̂𝑧)]𝑛 for any 𝜏,𝑖 with 𝜏𝑛+𝑖1, 𝑖𝑛forall̂𝑧𝐂̂𝑞+̂𝑞, where n is the order of the state-space realization (2.1)-(2.2).(iv)rank𝐻𝐺̂𝑧𝑔𝑒𝑛𝑟𝑎𝑛𝑘𝜏𝑛+𝑖1,𝑖𝑛,̂𝑧𝐂̂𝑞+̂𝑞𝐻𝐺̂𝑧=𝑔𝑒𝑛𝑟𝑎𝑛𝑘𝜏𝑛+𝑖1,𝑖𝑛,̂𝑧𝐂̂𝑞+̂𝑞𝐻𝑖,𝜏

Lemma 3.6 establishes that the rank of a +1𝑖,̂𝑧𝑛̂𝑧𝐂̂𝑞+̂𝑞.-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): 𝜏 of respective orders n (minimal) and 𝑅=𝐴0,𝐴𝑖𝑖𝑞,𝐵0,𝐵𝑗𝑗𝑞,𝐶,𝐷,𝑅=𝐴0,𝐴𝑖𝑖𝑞,𝐵0,𝐵𝑗𝑗𝑞,𝐶,𝐷𝐷(3.12). Then, the following properties hold independent of the delays.
(i)𝑛>𝑛(ii)𝑛𝑔𝑒𝑛rank𝜏𝑛+𝑖1,𝑖𝑛,̂𝑧𝐂̂𝑞+̂𝑞𝐻𝐺̂𝑧𝑔𝑒𝑛rank𝜏𝑛+𝑖1,𝑖𝑛,̂𝑧𝐂̂𝑞+̂𝑞𝐻𝑖,𝜏+1𝑖,̂𝑧𝑛,Min(𝑔𝑒𝑛rank𝜏𝑛+𝑖1,𝑖𝑛,̂𝑧S𝐶𝑛(𝐡)𝐻𝑖,𝜏+1𝑖,̂𝑧,𝑔𝑒𝑛rank𝜏𝑛+𝑖1,𝑖𝑛,̂𝑧𝑆𝑂𝑛(𝐡)𝐻𝑖,𝜏+1𝑖,̂𝑧)<𝑛,𝑛Min(𝑔𝑒𝑛rank𝜏𝑛,̂𝑧S𝐶𝑛(𝐡)C𝜏𝐴𝑧𝐼,𝐵𝑧𝐸,𝑔𝑒𝑛rank𝜏𝑛,̂𝑧𝑆𝑂𝑛(𝐡)O𝜏𝐴𝑧𝐼,𝐵𝑧𝐸)<𝑛𝐡𝑞+𝑞+.(3.13)(iii)None of the following conditions can hold for a complex function 𝑔𝑒𝑛rank𝜏𝑛+𝑖1,𝑖𝑛𝑁,̂𝑧𝐂̂𝑞+̂𝑞𝐻𝑖,𝜏+1𝑖,̂𝑧=𝑔𝑒𝑛rank𝜏𝑛,̂𝑧𝐂̂𝑞+̂𝑞C𝜏𝐴𝑧𝐼,𝐵𝑧𝐸=𝑔𝑒𝑛rank𝜏𝑛,̂𝑧𝐂̂𝑞+̂𝑞O𝑖𝐶,𝐴𝑧𝐼=𝑛.(3.14) defined by 𝑧𝐂×𝑞+𝑞+𝐂𝑞+𝑞 associated with internal and external delays 𝑧𝑠,𝐡=𝑧𝐼(𝑠,𝐡),𝑧𝐸(𝑠,𝐡)=𝑒1𝑠,,𝑒𝑞𝑠,𝑒1𝑠,,𝑒𝑞𝑠(3.15), 𝑖(𝑖𝑞). Then, 𝑞+𝑗=𝑗(𝑗𝑞)

Proof. Properties (i)-(ii) follow directly from the factorization of Lemma 3.6(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 rank𝐻𝐺̂𝑧<𝑛,rank𝐻𝑖,𝜏+1𝑖,̂𝑧<𝑛forany𝜏𝑛+𝑖1,𝑖𝑛,rankC𝜏𝐴𝑧𝐼,𝐵𝑧𝐸<𝑛𝜏𝑛,rankO𝜏𝐶,𝐴𝑧𝐼<𝑛𝜏𝑛.(3.16) then there would exist 𝑔𝑒𝑛rank𝜏𝑛,𝐂𝑞+𝑞𝐶𝜏𝐴𝑧𝐼,𝐵𝑧𝐸>rank𝜏𝑛,𝑧𝑆𝐶𝑛(𝐡)𝑆𝑂𝑛(𝐡)𝐶𝜏𝐴𝑧𝐼,𝐵𝑧𝐸;(3.17) such that 𝑛(<𝑛) and then the order of the square polynomial matrix 𝑔𝑒𝑛rank𝜏𝑛,𝑧𝐂𝑞+𝑞𝐶𝜏𝐴𝑧𝐼,𝐵𝑧𝐸=rank𝜏𝑛,𝑧𝑆𝐶𝑛(𝐡)𝑆𝑂𝑛(𝐡)𝐶𝜏𝐴𝑧𝐼,𝐵𝑧𝐸=𝑛,(3.18) and, as a result, that of matrices 𝐴(𝑍𝐼) would be 𝐴0,𝐴𝑖(𝑖𝑞). Since n is the order of a minimal realization, 𝑛<𝑛 from Theorem 3.4(iv), which implies and is implied by 𝑆𝐶𝑛(𝐡)𝑆𝑂𝑛(𝐡)=rank[𝐶𝜏(𝐴(𝑧𝐼),𝐵(𝑧𝐸))]=rank[𝑂𝜏(𝐶,𝐴(𝑧𝐼))]=𝑛 and all forall𝜏(𝑛) defined by 𝑧𝐂×𝑞+𝑞+𝐂𝑞+𝑞. 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 𝑧(𝑠,𝐡)=(𝑧𝐼(𝑠,𝐡),𝑧𝐸(𝑠,𝐡))=(𝑒1𝑠,,𝑒𝑞𝑠,𝑒1𝑠,,𝑒𝑞𝑠). Thus, given the sequence with 𝐻𝜏(̂𝑧)={𝐻𝑖(̂𝑧)}𝜏0, find matrices 𝜏(), 𝐴𝑖𝑛×𝑛(𝑖𝑞{0}), and 𝐵𝑖𝑛×𝑚(𝑖𝑞{0}) provided that they exist such that the following matching condition holds independent of the delays either for all 𝐶𝑝×𝑛 (minimal synthesis problem) or for some finite 𝜏 (minimal partial realization problem): 𝜏 where 𝐺𝑠,̂𝑧=𝐶(𝑠𝐼𝐴0𝑞𝑖=1𝐴𝑖𝑧𝐼𝑖)1(𝐵0+𝑞𝑖=1𝐵𝑖𝑧𝐸𝑖)+𝐷=𝑖=0𝐻𝑖𝑠𝑖=𝜏𝑖=0𝐻𝑖𝑠𝑖+0𝑠𝜏1,(4.1) 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 𝐺(𝑠,̂𝑧)𝑝×𝑚(𝑠,̂𝑧)𝑝×𝑚[[𝑠]][̂𝑧]𝜏𝑖=0𝐻𝑖(̂𝑧)𝑠𝑖, 𝑧𝑖=𝑒𝑖𝑠, a state-space realization (2.1)-(2.2) is obtained so that (4.1) holds for 𝑧𝑞+𝑗=𝑒𝑗𝑠(𝑖𝑞,𝑗𝑞) (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 𝜏<) of a rational transfer matrix coincides with that of the rank of the infinite associated block Hankel matrix for 𝜇 (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 degree ̂𝑧𝐂𝑞+𝑞 of the transfer matrix 𝑛=𝜇(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 𝐺(𝑠,𝐡): 𝐡𝑞+𝑞+
(ii) The state-space dimension >𝑛𝐡=𝜇𝐺𝑠,𝐡=Max𝜏(𝜇𝑧SC𝜏𝐡SO𝜏𝐡𝐻𝜏𝑧)(4.2)=Max𝜏(𝑖+𝑗=𝜏+1rank𝐻𝑖,𝑗,𝑧𝑖+𝑗𝜏rank𝐻𝑖,𝑗,𝑧𝑧SC𝜏𝐡S0𝜏𝐡)(4.3)=Max𝑧SC𝐡S0𝐡rank𝐻𝑖,𝑗,𝑧𝑛𝑖,𝑖+𝑛1𝜏.(4.4) of any minimal partial realization satisfies 𝑛𝜏(𝐡)(𝜏) where >𝑛𝜏𝐡=Max𝜏(𝜇𝑧SC𝜏𝐡SO𝜏𝐡𝐻𝑖,𝜏+1𝑖,𝑧𝑖𝜏),(4.5), and then the minimal partial realization is a minimal realization for all 𝑛𝜏(𝐡)=𝑛(𝐡) and sufficiently large finite 𝜏(𝜏0) with 𝜏0𝑁 (iii) Redefine by simplicity the delays according to 𝑛𝜏𝐡=𝑛𝜏0𝐡=𝑛𝐡=Min𝑧SC𝐡S0𝐡rank𝐻𝑖,𝜏+1𝑖,𝑧𝜏0𝑖𝑁,𝑖+𝜏01𝜏𝑁.(4.6). Define 𝑞+𝑖=𝑖(𝑖𝑞) and let 𝐡𝟎𝟎=0 be defined with 𝐡𝐢𝟎 and 𝑖0 for 𝑗=0(𝑗𝑖). Assume that (𝑖𝑞+𝑞) (some constant 𝑛(𝐡𝐢𝟎)=𝑛𝑖0=𝑛0 in 𝑛0) for all , where 𝑖𝑞 with 𝑛𝑖0=Min𝜏𝑖0(𝑖+𝑗=𝜏𝑖0+1rank𝐻𝑖,𝑗,𝛼𝑖𝑖+𝑗=𝜏𝑖0rank𝐻𝑖,𝑗,𝛼𝑖)=Min𝜏0(𝑖+𝑗=𝜏0+1rank𝐻𝑖,𝑗,𝛼𝑖𝑖+𝑗=𝜏0rank𝐻𝑖,𝑗,𝛼𝑖)(4.7) having the ith component distinct from unity and the remaining ones being unity; 𝛼𝑖𝐂𝑞+𝑞 is an associate 𝐡𝐢𝟎-tuple of delays in (𝑞+𝑞) with only the ith component being nonzero and 𝑞+𝑞+, with the remaining ones being zero and 𝐴𝑖=0. Then, the order for any minimal realization independent of the delays is 𝜏0=Max(𝜏𝑖0𝑖𝑞+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=𝑛𝐡=𝑛𝐡𝐢𝟎=𝑛0=𝜏0=Max𝜏(𝜏0)(𝑔𝑒𝑛rank𝑧SC𝜏𝐡SO𝜏𝐡𝐻𝑖+1,𝜏+𝑖,𝑧)𝐡𝑞+𝑞+,(4.8) has to exist such that 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., Max𝜏𝜇𝑧𝑆𝐶𝜏𝐡𝑆𝑂𝜏𝐡𝐻𝜏𝑧=genrank𝜏𝑛+𝑖1,𝑖𝑛,𝑧𝐂𝑞+𝑞𝐻𝐺𝑧=genrank𝜏𝑛+𝑖1,𝑖𝑛,𝑧𝐂𝑞+𝑞𝐻𝑖,𝜏+1𝑖,𝑧,genrank𝜏𝑛,𝑧𝐂𝑞+𝑞𝐶𝜏𝐴𝑧𝐼,𝐵𝑧𝐸=rank𝜏𝑛,𝑧𝑆𝐶𝑛(𝐡)𝑆𝑂𝑛(𝐡)𝐶𝜏𝐴𝑧𝐼,𝐵𝑧𝐸=𝑛𝜏𝑛.(4.9) does not depend on a multidimensional complex tuple z) and by the fact that this rank does not increase for 𝐻(𝑖,𝑗) exceeding a certain finite minimum threshold 𝜏. Property (i) has been proved. Property (ii) follows in the same way by using similar considerations for any given 𝜏0. To prove property (iii), first note that 𝜏 for 𝐡𝐢𝟎𝑞+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 𝑖𝑞+𝑞 being null) as follows: 𝐴()̇𝑧𝑖𝑡=(𝑞𝑗=0𝐴𝑗𝑖𝐴𝑞,𝑖,𝑗)𝑧𝑖𝑡+𝑞𝑗=0𝐵𝑗𝑖𝐵𝑞,𝑞,𝑖,𝑗𝑢𝑡.(4.10) under initial conditions 𝑖𝑞+𝑞, where the binary indicators 𝑧(𝑡)=𝑧0=𝜑(0) have been used for notational simplicity, since 𝑖𝐴𝑞,𝑖,𝑗=1if𝑖,𝑗𝑖𝑞,𝑖𝑞+𝑞,𝑖𝐴𝑞,𝑖=0if𝑖,𝑗=𝑖𝑞,𝑖𝐵𝑞,𝑞,𝑖,𝑗=1if𝑖𝑗𝑞+𝑞/𝑞,𝑖𝑞,𝑖𝐵𝑞,𝑞,𝑖,𝑗=0if𝑖,𝑗=𝑖𝑞+𝑞/𝑞(4.11) and 𝐡𝟎𝟎=(0,0,,0) is defined with components 𝐡𝐢𝟎, for all 𝑗=0, and 𝑗(𝑖)𝑞+𝑞, or 𝐴𝑖=0 if 𝑖= and 𝑖𝑞 or 𝐵𝑖=0 if 𝑞+𝑖=𝑖=. All the delay-free parameterizations (4.10)-(4.11) of (2.1)-(2.2) have a minimal state-space realization of identical dimension 𝑖𝑞+𝑞/𝑞 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 𝑛=𝑛0 algebraic matrix equations to calculate the matrices 𝑞+𝑞, from which 𝐴𝑖=𝑞𝑗(𝑖)=0𝐴𝑗,𝐵=𝑞𝑗()=0𝐵𝑗,𝑖/𝑞{0},𝑞{0}; 𝐴𝑖,𝑖𝑞{0} can be calculated uniquely for some matrix C such that (3.11) holds with 𝐵𝑗,𝑗𝑞{0}. 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 𝐷=𝐻0 parameterized in the sextuple of real scalars 𝐺𝑠,𝛿,𝜌1,𝜌2,𝜆,=𝜌1𝐶(𝑠𝐼𝑛𝛿𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠𝑒𝑠)1(𝑞𝑖=0𝜌2𝐵𝑖𝑒𝑖𝑠𝑒𝑠)+𝜆𝐷(4.12) which models a perturbation of a nominal transfer matrix 𝐩=(𝛿,𝜌1,𝜌2,𝜆,,) parameterized by 𝐺𝑠,𝛿,𝜌1,𝜌2,𝜆,=𝐶(𝑠𝐼𝑛𝑞𝑖=0𝐴𝑖𝑒𝑖𝑠)1(𝑞𝑖=0𝐵𝑖𝑒𝑖𝑠)+𝐷(4.13), 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 𝐩𝟎=(1,1,1,1,0,0), 𝐶𝜌1𝐶, 𝐴𝑖𝛿𝐴𝑖(𝑖𝑞{0}), 𝐵𝑖𝜌2𝐴𝑖 and delay changes 𝐷𝜆𝐷, 𝑖𝑖+ for any finite delay perturbations 𝑗𝑗+(𝑖𝑞{0},𝑗𝑞{0}) and and for any real if and only if 𝜆.

Proof. The block Hankel (i, j)-matrix associated with the p-parameterization is related to that associated with the 𝜌1𝜌2𝛿0-parameterization by 𝐩𝟎 and 𝐻𝐺𝑖,𝑗,𝐩=BlockDiag𝛾𝑠(𝑖1)/2,𝛾𝑠(𝑖1)/2,,𝛾𝑠(𝑖1)/2𝐻𝐺𝑖,𝑗,𝐩𝟎×BlockDiag𝛾𝑠(𝑖1)/2,𝛾𝑠(𝑖1)/2,,𝛾𝑠(𝑖1)/2×BlockDiag(𝜂𝑠,𝜂𝑠,𝑗𝑖+1,𝜂𝑠)(4.14), where 𝐻0=𝜆𝐷 and 𝛾(𝑠,)=𝛿𝑒𝑠. Since the numerator and denominator quasipolynomials have principal terms, they do not have unstable zeros at infinity, that is, zeros with 𝜂(𝑠,)=𝜌1𝜌2𝑒𝑠. Then, Re𝑠 if and only if 𝑛=Max𝑖,𝑗genrank𝐻𝐺𝑖,𝑗,𝐩𝟎=Max𝑖,𝑗𝜇𝐻𝐺𝑖,𝑗,𝐩𝟎=Max𝑖,𝑗𝜇𝐻𝐺𝑖,𝑗,𝐩(4.15) for all complex indeterminate s and any finite delay disturbances 𝛿𝜌1𝜌20𝛾(𝑠,)𝜂(𝑠,)0 and .

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: where 𝐺𝑝𝑠=𝐺𝑠̃𝑛𝑠̃𝑑𝑠,𝐺𝑠=𝑐𝑇𝑠𝐼𝑛𝛿𝐴01𝜌𝑏0+𝜆𝑑0,𝐺0𝑠=𝑐𝑇𝑠𝐼𝑛𝐴01𝑏0+𝑑0,(4.16) and ̃𝑛(𝑠) are polynomials of respective degrees ̃𝑑(𝑠) and ̃𝑛𝑛(𝑠)=deg(̃𝑛(𝑠)) which satisfy the degree constraint ̃𝑛𝑑(𝑠)=deg(̃𝑑(𝑠)) with ̃𝑛𝑑(𝑠)̃𝑛𝑛(𝑠)+𝑚𝑛 Then, the following properties hold.
(i)The state-space realization 𝑛=deg𝑠𝐼𝑛𝛿𝐴0,𝑚=𝑛if𝜆𝑑00,𝑚<𝑛if𝜆𝑑0=0,𝑚=deg𝑐𝑇𝐴𝑑𝑗𝑠𝐼𝑛𝛿𝐴0𝜌𝑏0.(4.17) of (𝑐𝑇,𝐴0,𝑏0,𝑑0) is both controllable and observable, and then minimal of order n, if and only if the associate Hankel matrix satisfies 𝐺0(𝑠). Then, rank𝐻𝐺0=Max𝑖,𝑗(rank𝐻𝐺0(𝑖,𝑗))=rank𝐻𝐺0,𝑗𝑖+𝑛1;𝑖,𝑗(𝑖,𝑗)=𝑛0 is proper (strictly proper if 𝐺0(𝑠) and zero-pole cancellation-free, and 𝑑00) is of order 𝐴0.(ii)Assume that the state-space realization 𝑛0 of (𝑐𝑇,𝐴0,𝑏0,𝑑0) is controllable and observable of minimal order 𝐺0(𝑠). Then, the state-space realization 𝑛0 of (𝑐𝑇,𝛿𝐴0,𝜌𝑏0,𝜆𝑑0) is both controllable and observable, and then minimal of order 𝐺(𝑠), if and only if 𝑛=𝑛0. Then, 𝛿𝜌0 is proper (strictly proper if 𝐺(𝑠) and zero-pole cancellation-free, and 𝜆𝑑00) is of order 𝐴. As a result, 𝑛=𝑛0(iii)rank𝐻𝐺=Max𝑖,𝑗rank𝐻𝐺𝑖,𝑗=rank𝐻𝐺𝑖,𝑗𝑗𝑖+𝑛1;𝑖,𝑗=𝑛.(4.18) is state-space realizable and strictly proper if and only if 𝐺𝑝(𝑠). Assume that ̃𝑛𝑑(𝑠)>̃𝑛𝑛(𝑠)+𝑚𝑛 is zero-pole cancellation-free of order 𝐺0(𝑠). Then, the following hold. (1)𝑛0 is cancellation-free if and only if 𝐺𝑝(𝑠)=𝑛𝐺(𝑠)/𝑑𝐺(𝑠) and the three pairs of polynomials 𝜌𝛿0, (𝑛𝐺(𝑠),̃𝑑(𝑠)), and (𝑑𝐺(𝑠),̃𝑛(𝑠)) are each coprime, where (̃𝑑(𝑠),̃𝑛(𝑠)) and 𝑑𝐺(𝑠)=(𝑠𝐼𝑛𝛿𝐴0) are the denominator and numerator polynomials of 𝑛𝐺(𝑠)=𝑐𝑇𝐴𝑑𝑗(𝑠𝐼𝑛𝛿𝐴0)𝜌𝑏0+𝜆𝑑0𝑑𝐺(𝑠). As a result, 𝐺𝑝(𝑠) has a minimal realization (then, controllable and observable) of order 𝐺𝑝(𝑠).(2)Assume that ̂𝑛=𝑛(=𝑛0)+̃𝑛𝑑 is zero-pole cancellation-free of order 𝐺0(𝑠) and 𝑛0. Assume also that the polynomial pairs 𝜌𝛿0, (𝑛𝐺(𝑠),̃𝑑(𝑠)) are both coprime. Then, (𝑑𝐺(𝑠),̃𝑛(𝑠)) has zero-pole cancellation(s) at the common factors of the pair 𝐺𝑝(𝑠), if any. A minimal realization of (̃𝑑(𝑠),̃𝑛(𝑠)) has order 𝐺𝑝(𝑠), which satisfies ̂𝑛=𝑛0+̃𝑛𝑑̃𝑛𝑑, where 𝑛0+̃𝑛𝑑̂𝑛𝑛0 is the degree of the cancellation polynomial, if any. Also, ̃𝑛𝑑0

Outline of Proof
The proof follows directly from Theorem 4.2 by noting from its definitions of the various transfer functions that 𝑛0+̃𝑛𝑑rank𝐻𝐺𝑝=Max𝑖,𝑗rank𝐻𝐺𝑝𝑖,𝑗=rank𝐻𝐺𝑝𝑖,𝑗𝑗𝑖+̂𝑛1;𝑖,𝑗=̂𝑛𝑛0.(4.19) if 0𝑥0005𝑒𝐺(𝑠)=0𝑥0005𝑒𝐺0(𝑠) and 𝜌=1, and 𝜆=0 if 𝐺𝑝(𝑠)=𝐺(𝑠). As a result, ̃𝑛(𝑠)=̃𝑑(𝑠) is zero-pole cancellation-free if the polynomials 𝐺𝑝(𝑠) and ̃𝑛(𝑠) are nonzero real scalars provided that ̃𝑑(𝑠) is also cancellation-free. The transfer function 𝐺0(𝑠) is also cancellation-free if 𝐺𝑝(𝑠) and 𝐺0(𝑠) are both cancellation-free and, furthermore, ̃𝑛(𝑠)/̃𝑑(𝑠) and (̃𝑛(𝑠),Det(𝑠𝐼𝑛𝐴0)) 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.

Acknowledgments

The author is very grateful to the Spanish Ministry of Education for its partial support of this work through Project no. DPI 2006-00714. He is also grateful to the reviewers for their useful comments.

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