Minimal realization of transfer function matrices†

The concepts of weak and strong transfer equivalence of constant (time-invariant) linear dynamical systems are defined and analyzed. The analysis leads to a simple new algorithm for constructing minimal realizations of transfer function matrices. In addition, it provides new information on the significance of the polynomial invariants which appear in the Smith–McMillan canonical form.

Full probabilistic solution of a finite dimensional linear control system with random initial and final conditions

The state‐space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time‐invariant case was solved, particular attention was paid to the case of linear periodic systems. Recently, such systems have regained importance, for instance, in the context of coding theory where periodic convolutional encoders play an important role. The majority of the contributions within this area only concern the realization of transfer functions (or impulse responses), thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. In previous research conducted by the authors, some results on the realization of periodic behavioral systems were obtained, which, despite their importance, are conservative with respect to the dimension of the obtained realizations. In this paper, we revisit the behavioral periodic realization problem and give further insight into this question. This allows to derive new results and set up an algorithm to compute low‐dimensional state‐space realizations.

Previously, we posted the following question concerning the array realization problem: Assume that one or more (mathematical) model realizes (faithfully describes) an arbitrary (two dimensional) array; if two such models exist that realize the array, then are these models isomorphic? The solution to this question is the array realization theorem: Canonical commutative linear representative systems necessarily exist that realize an arbitrary array, and these systems are identical excluding isomorphism. An investigation of finite dimensional canonical commutative linear representation systems that can be dealt with on a computer is performed. An algorithm is provided that realizes any given finite dimensional array. In this paper, an efficient coding (communications channel coding) is described for arrays that result from the structure problem of commutative linear representative systems based on these results and its solution. This structure problem is an extension of the structure problem occurring in linear system theory to a structure theorem for commutative linear representative systems. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(3): 58–68, 1999

Minimal realizations for discrete-time linear periodic systems

It is shown that two properties of rational transfer functions with real coefficients are realized by means of a single complex IIR digital filter. Equivalently, the real and imaginary parts of output from a complex filter for a real input sequence correspond to real coefficient transfer function respectively. In a real signal processing, a delay operator z/sup -1/ corresponds to a real operation. In this case, a complex transfer function is formed by connecting two real transfer functions with an imaginary unit. These real functions are approximated to the desired properties and are expressed by a complex coefficient transfer function.

This note investigates the canonical realization of the single-input single-output transfer function. An algorithm for generating all canonical realizations of the transfer function is presented and the alternative formulas for the total number of realizations of the denominator are derived.

In this paper we consider a linear autonomous system with delays and we prove that we can associate a mantainability subspace to each zero of the system matrix. This space is the generalized eigenspace of the closed loop system obtained by using a special feedback. This observation provides a representation of the system which generalizes a known result on finite dimensional linear systems.

In this paper we attempt to describe a circle of ideas which makes possible a unified exposition of a large part of linear algezbra, operator theory and both finite and infinite dimensional multivariable linear systems. As the title suggests the unifying conzcepts will be those of functional models, module theory and various factorizations of polynomial and analytic matrix valued functions. The full exposition of the ideas presented here will be the theme of a forthcoming monograph [20].

The well-known Richards' Theorem of the continuous-time filter theory is reformulated in the digital domain in a convenient manner, leading to a simple derivation of cascaded lattice digital filter structures, realizing lossless bounded transfer functions. The theorem is also extended to the matrix case, leading to a derivation of m -input p -output cascaded lattice filter structures with lossless building blocks, that realize an arbitrary p \times m digital Lossless Bounded Real (LBR) transfer matrix. Extensions to the synthesis of arbitrary, stable p \times m transfer matrices in the form of such cascaded lattices is also outlined. The derivation also places in evidence a means of testing the stability of an arbitrary p \times m transfer matrix of a discrete-time linear system.

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

A new method for computation of positive realizations of given transfer matrices of linear discrete-time linear systems is proposed. Sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by an example.

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

Consistent estimation of real NMP zeros in stable LTI systems of arbitrary complexity

The main result of this paper establishes that if a controller $C$ (comprising of a linear feedback of the output and its derivatives) globally stabilizes a (nonlinear) plant $P$, then global stabilization of $P$ can also be achieved by an output feedback controller $C[h]$, where the output derivatives in $C$ are replaced by an Euler approximation with sufficiently small delay $h>0$. This is proved within the conceptual framework of the nonlinear gap metric approach to robust stability. The main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain. Results are also given for systems with nonzero initial conditions.