By Margreet Kuijper

This booklet is ready the idea of approach representations. The structures which are thought of are linear, time-invariant, deterministic and finite dimensional. The remark that a few representations are improved for dealing with a selected challenge than others motivates the research of rep resentations. In modeling a approach, a illustration usually arises clearly from yes legislation that underlie the method. In its such a lot common shape the illustration then contains dynamical equations for the process compo nents and of constraint equations reflecting the relationship among those elements. reckoning on the actual challenge that's to be inves tigated, it is going to occasionally be necessary to rewrite the equations, that's, to rework the illustration. accordingly it truly is of particular value to derive changes that permit one to modify from one illustration to a different. a brand new technique of the prior decade has been the so-called "behavioral ap proach" brought by way of Willems. one of many major gains of the behavioral technique is that it's like minded for modeling the interconnection of sys tems. it truly is therefore that the behavioral procedure is a normal selection within the context of modeling. during this publication we undertake the behavioral process: we outline a approach as a "behavior" , that's, a suite of trajectories whose math ematical illustration through differential or distinction equations is nonunique. a facet of this procedure that's very important within the context of illustration idea is the truth that a typical kind of equivalence arises.

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**Example text**

16) follows. Let us now assume that T(s) is row reduced, and let us denote the row contents at 00 of T(s) by /Li (i = 1, ... , n). 18. Then T(s) = diag (SI'I, ... 3) . This implies the other statements in the lemma. 19; the right Wiener-Hopf indices of T(s) and the right Wiener-Hopf order of T(s), denoted by Ier(T), are defined in the obvious way. 23 Every matrix T(s) E cnxm(s) of rank r can be factorized as where U_ (s) is a Coo (s )-unimodular matrix of size n x nand U+(s) is a C[s]-unimodular matrix of size m x m; ~(s) = diag (s"", ...

N). 18. Then T(s) = diag (SI'I, ... 3) . This implies the other statements in the lemma. 19; the right Wiener-Hopf indices of T(s) and the right Wiener-Hopf order of T(s), denoted by Ier(T), are defined in the obvious way. 23 Every matrix T(s) E cnxm(s) of rank r can be factorized as where U_ (s) is a Coo (s )-unimodular matrix of size n x nand U+(s) is a C[s]-unimodular matrix of size m x m; ~(s) = diag (s"", ... , s"'r) with leI 2: 1e2 2: . 2: le r · Moreover, the integers lei are unique (i = 1, ...

27 if [V(s) R(s)] is row reduced. 58 3. Representations of linear time-invariant systems We now introduce a different type of first-order representation that we call the dual pencil representation (DP representation): (IKe = Le + Mw. 30) Here K and L are mappings from X to Z, and M is a mapping from W to Zj we denote the representation by (/(, L, M). Representations of this type have been considered before in [1, 67, 68]. 22), respectively. 22)). 5 Systems with split external variables As mentioned in Chapter 1 there are situations in which the external variables are split into two parts.