Search

In this chapter we review the factorization theory for the case of real matrix functions with respect to real divisors. As in the complex case the minimal factorizations are completely determined by the supporting projections of a given realization, but in this case one has the additional requirement that all linear transformations must be representable by matrices with real entries. Due to the difference between the real and complex Jordan canonical form the structure of the stable real minimal factorizations is somewhat more complicated than in the complex case. This phenomenon is also reflected by the fact that for real matrixes there is a difference between the stable and isolated invariant subspaces. [ABSTRACT FROM AUTHOR]

In this chapter we shall prove that there exist stable factorizations which are not spectral factorizations. In fact, for the finite-dimensional case we shall give a complete description of all possible stable minimal factorizations. It will also be shown that stability amounts to the same as the property of being isolated provided the underlying field is complex (which will be the case in this chapter). [ABSTRACT FROM AUTHOR]

In numerical computations of minimal factors of a given transfer function questions concerning the conditioning of the factors turn up naturally. According to the division theory developed in the previous chapters, all minimal factorizations may be obtained in an explicit way in terms of supporting projections of minimal systems. This fact allows one to reduce questions concerning the conditioning of minimal factorizations to questions concerning the stability of divisors of a system. In the present chapter we study the matter of stability of spectral divisors mainly. In this case the investigation can be carried out for finite- as well as for infinite-dimensional state spaces. The invariant subspace method employed in this chapter will also be used to prove that "spectral" solutions of an operator Riccati equation are stable. The case of minimal non-spectral factorizations will be considered in the next chapter. [ABSTRACT FROM AUTHOR]

In this chapter a connection is made between the issue of quasicomplete factorization discussed in Section 10.4 and a problem from the theory of combinatorial job scheduling. The problem in question is the so-called two machine flow shop problem (2MFSP for short) where one wants to find optimal schedules for processing jobs on two machines, given certain precedence constraints. It turns out that such problems are in correspondence (one-to-one, essentially) with the companion based rational matrix functions considered in the previous chapter. We show that the number of factors in a quasicomplete factorization of a companion based matrix function is directly related to the minimum makespan (i.e., the time needed for carrying out a optimal schedule) of the associated instance of 2MFSP. Illustrative examples are given. In one of them the (computationally fast) algorithm called Johnson's rule for 2MFSP is used to compute the quasidegree of a companion based function. [ABSTRACT FROM AUTHOR]

In this chapter results of the previous chapter are specified further for rational matrix functions of a special type, namely for the so-called companion based functions. These are characterized by the fact that they are rational matrix function having a minimal realization in which both the main matrix and the associate main matrix are (first) companions. A description of such functions is presented for the 2 × 2 case, and necessary and sufficient conditions are given for such functions to admit a complete factorization. The factorization results in this chapter are based on a detailed analysis of simultaneous reduction to complementary forms of pairs of companion matrices. [ABSTRACT FROM AUTHOR]

In this chapter we study the factorization of a proper rational m × m matrix function having the value Im at infinity into elementary factors satisfying the same constraints. These elementary factors are of McMillan degree one by definition. It turns out, by using realization, that the problem of factorizing a function in such degree one factors is intimately connected with the issue of simultaneous reduction to complementary triangular forms of pairs of matrices. We prove that factorization into elementary factors is always possible. [ABSTRACT FROM AUTHOR]

In this chapter the notion of minimal factorization of rational matrix functions, which has its origin in mathematical system theory, is introduced and analyzed. In Section 9.1 minimal factorizations are identified as those factorizations that do not admit pole zero cancellation. Canonical factorization is an example of minimal factorization but the converse is not true. In Section 9.2 we use minimal factorization to extend the notion of canonical factorization to rational matrix functions that are allowed to have poles and zeros on the curve. In Section 9.3 (the final section of this chapter) the concept of a supporting projection is extended to finite-dimensional systems that are not necessarily biproper. This allows us to prove one of the main theorems of the first section also for proper rational matrix functions of which the value at infinity is singular. [ABSTRACT FROM AUTHOR]

In this chapter finite-dimensional systems are studied in terms of the zero or pole data of their transfer functions. In the first two sections we describe the local zero and pole data, and related Jordan chains, of a meromorphic m×m matrix function of which the determinant does not vanish identically. In the third section these results are used to construct minimal realizations of rational matrix functions in terms of the zero or pole data of the function. The fourth section deals with the notions of local degree of a transfer function and local minimality of a finite-dimensional system. The global versions of these notions are studied in the final section. [ABSTRACT FROM AUTHOR]

In this chapter the notion of a minimal system is considered. If two systems are similar, then they have the same transfer function. The converse statement is not true. In fact, systems with rather different state spaces may have the same transfer function. For minimal systems this phenomenon does not occur. In Section 7.1 minimal systems are defined as systems that are controllable and observable. The latter two notions are explained in more detail for finite-dimensional systems in Section 7.2. In the finite-dimensional case the connection between a minimal system Θ and its transfer function WΘ is very close. For example in that case Θ is uniquely determined up to similarity by WΘ. This result, which is known as the state space similarity theorem, will be proved in Section 7.3. Several examples, presented in Section 7.4, show that a generalization of the finite-dimensional theory to an infinite-dimensional setting is not possible in a straightforward way. An appropriate generalization requires a further refinement of the state space theory. In Section 7.5 the notion of minimality is considered for Brodskii systems, Kreîn systems, unitary systems, monic systems, and polynomial systems. [ABSTRACT FROM AUTHOR]

As we have seen in Chapter 1 canonical factorization serves as tool to solve Wiener-Hopf integral equations and their discrete analogues, the block Toeplitz equations. In this chapter the state space factorization method developed in Chapter 2 is used to solve the problem of canonical factorization (necessary and sufficient conditions for its existence) and to derive explicit formulas for its factors. This is done in Section 6.1 for rational matrix functions. The results are applied to invert Wiener- Hopf integral equations (Section 6.2) and block Toeplitz operators (Section 6.3) with a rational matrix symbol. [ABSTRACT FROM AUTHOR]

In this chapter the state space factorization theory from Section 2.4 is presented using a different terminology. Here it will be based on the notion of an angular operator and the algebraic Riccati equation. [ABSTRACT FROM AUTHOR]

The main problem addressed in this chapter is the realization problem for operatorvalued functions. Given such a function the problem is to find a system for which the transfer function coincides with the given function. In the first section we consider rational operator functions, and in the second analytic ones. In Section 4.3 it is shown that, in a certain sense, the transfer function of a system with an invertible external operator can be reduced to a linear function, and we use this reduction to describe the singularities of the transfer function. In the final section a connection between Schur complements and linearization is described. [ABSTRACT FROM AUTHOR]

In this chapter we review the notions and results from the previous chapter for various classes of systems. Included are Brodskii systems (Section 3.1), Kreîn systems (Section 3.2), unitary systems (Section 3.3), monic systems (Section 3.4) and polynomial systems (Section 3.5). The final section (Section 3.6) concerns a change of variable in the transfer function defined by a Möbius transform. [ABSTRACT FROM AUTHOR]

In this chapter the concepts of an operator node (abstract system) and its transfer function are introduced and developed systematically. Important operations on operator nodes (abstract systems) and the corresponding operations on the associated transfer functions are studied in detail: inversion (Section 2.2), products (Section 2.3) and factorization (Section 2.4). With an eye on future applications, a detailed analysis of the relationships between the various results is given in the final section. [ABSTRACT FROM AUTHOR]