A State Space Approach to Canonical Factorization with by Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M.

By Harm Bart, Israel Gohberg, Marinus A. Kaashoek, André C.M. Ran

The current publication offers with canonical factorization difficulties for di?erent sessions of matrix and operator services. Such difficulties seem in a variety of components of ma- ematics and its functions. The capabilities we examine havein universal that they seem within the country area shape or might be represented in this type of shape. the most effects are all expressed when it comes to the matrices or operators showing within the country house illustration. This comprises priceless and su?cient stipulations for canonical factorizations to exist and particular formulation for the corresponding f- tors. additionally, within the purposes the entries within the nation area illustration play a vital function. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. one of many preliminary steps are available in mathematical platforms concept and electric community conception, the place a cascade decomposition of an input-output process or a community is said to a factorization of the linked move functionality. Canonical factorization has an extended and fascinating historical past which begins within the idea of convolution equations. fixing Wiener-Hopf indispensable equations is heavily relating to canonical factorization. the matter of canonical factorization additionally appears to be like in different branches of utilized research and in mathematical structures concept, in H -control concept in particular.

Show description

Read or Download A State Space Approach to Canonical Factorization with Applications PDF

Best linear programming books

Integer Programming: Theory and Practice

Integer Programming: concept and perform comprises refereed articles that discover either theoretical points of integer programming in addition to significant purposes. This quantity starts off with an outline of latest confident and iterative seek tools for fixing the Boolean optimization challenge (BOOP).

Extrema of Smooth Functions: With Examples from Economic Theory

It isn't an exaggeration to kingdom that almost all difficulties handled in financial conception will be formulated as difficulties in optimization idea. This holds precise for the paradigm of "behavioral" optimization within the pursuit of person self pursuits and societally effective source allocation, in addition to for equilibrium paradigms the place lifestyles and balance difficulties in dynamics can usually be said as "potential" difficulties in optimization.

Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems

This e-book displays an important a part of authors' examine task dur­ ing the final ten years. the current monograph is developed at the effects acquired by way of the authors via their direct cooperation or as a result authors individually or in cooperation with different mathematicians. a lot of these effects slot in a unitary scheme giving the constitution of this paintings.

Optimization on Low Rank Nonconvex Structures

Worldwide optimization is among the quickest constructing fields in mathematical optimization. actually, more and more remarkably effective deterministic algorithms were proposed within the final ten years for fixing a number of sessions of enormous scale specifically established difficulties encountered in such parts as chemical engineering, monetary engineering, place and community optimization, creation and stock regulate, engineering layout, computational geometry, and multi-objective and multi-level optimization.

Additional info for A State Space Approach to Canonical Factorization with Applications

Sample text

0 0 . I −q0 Ir −q1 Ir .. −qk−1 Ir ⎤ ⎥ ⎥ ⎥, ⎥ ⎦ ⎡ ⎢ ⎢ B=⎢ ⎢ ⎣ H0 H1 .. Hk−1 ⎤ ⎥ ⎥ ⎥, ⎥ ⎦ C = 0 . . 0 Ir . Then the resolvent set ρ(A) of A coincides with ΩW , the subset of C on which q takes non-zero values. For λ ∈ ρ(A), define C1 (λ), . . , Ck (λ) by [ C1 (λ) C2 (λ) . . Ck (λ) ] = C(λ − A)−1 . 3. Realization of analytic operator functions 23 From the special form of the matrix A (second companion type) we see that Cj+1 (λ) = λCj (λ), j = 0, . . , k − 1, and C1 (λ) = q(λ)−1 I. Hence C(λ − A) −1 k−1 B = Cj+1 (λ)Hj = j=0 1 H(λ).

6) are matrix-valued realizations. 6), in that order. The counterpart of taking products is factorization. In the next section this topic will be discussed for functions given by a biproper realization. We close the present section with a remark preparing for this discussion. The main operator A in the product realization is given in the form of a 2 × 2 upper triangular operator matrix: A= A1 B1 C2 0 A2 ˙ 2 → X1 +X ˙ 2. 6), respectively. Note that M = X1 + ˙ 2 is an invariant subspace for A× , and that M subspace for A, that M × = {0} +X and M × match in the sense that the state space of the product realization is the direct sum of M and M × .

The chapter consists of six sections. 1 presents preliminaries on realization, including the relevant definitions and the connection with systems theory. In the next two sections the realization problem is discussed. 3. 6). 1 Preliminaries on realization Let W be a rational matrix function which is also proper, that is, W has no pole at infinity. As is well-known such a function can always be represented (see the next section for an explicit construction) in the form W (λ) = D + C(λI − A)−1 B. 1) Here λ is a complex variable, A is a square matrix, I is the identity matrix of the same size as A, and B and C are matrices of appropriate sizes.

Download PDF sample

Rated 4.17 of 5 – based on 47 votes