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First-order Representations of Linear Systems by Margreet Kuijper

By Margreet Kuijper

This booklet is ready the speculation of process representations. The structures which are thought of are linear, time-invariant, deterministic and finite­ dimensional. The statement that a few representations are more advantageous for dealing with a specific challenge than others motivates the examine of rep­ resentations. In modeling a procedure, a illustration usually arises certainly from sure legislation that underlie the process. In its so much normal shape the illustration then includes dynamical equations for the process compo­ nents and of constraint equations reflecting the relationship among those elements. looking on the actual challenge that's to be inves­ tigated, it is going to occasionally be precious to rewrite the equations, that's, to rework the illustration. accordingly it really is of unique value to derive alterations that let one to modify from one illustration to a different. a brand new technique of the previous decade has been the so-called "behavioral ap­ proach" brought by means of Willems. one of many major positive aspects of the behavioral method is that it's compatible for modeling the interconnection of sys­ tems. it really is as a result that the behavioral strategy is a typical selection within the context of modeling. during this publication we undertake the behavioral method: we outline a method as a "behavior" , that's, a collection of trajectories whose math­ ematical illustration by way of differential or distinction equations is nonunique. a side of this procedure that's vital within the context of illustration concept is the truth that a average kind of equivalence arises.

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2 that there exists Z2 E v q such that D(U)Z2 = Z1. Defining Z2 = V- 1 (u) [ ~2 ] we have that P(U)Z2 = U(u)D(U)Z2 = U(U)Z1 = Z1 . 5 Let P(s) E Rqxq[s] be nonsingular. Then dim ker P(u) = deg det P(s) . 1 : P(s) = U(s)D(s)V(s) where D(s) = diag (d 1 (s), d2(s) , .. , dq(s)) with dj(s) E R[s] is the Smith form of P(s) over Il[s] and both U(s) and V(s) are R[s]-unimodularmatrices of size q x q. Then det P(s) = d1 (s)d 2(s)· .. (s) . 1. p-l : ker D(u) -+ x 0-+ As a result, ker P(u) V-l(u)x . 3. 6 Let pes) E Rgxq[s] have full column rank for all sEC and let x E V q .

We conclude that (i) holds. 33, the following lemma is immediate from the previous one. 24). Then the following statements are equivalent: (i) A[ ker E] + im E = Z (ii) sE - A has full row rank and <5~ (sE - A) = O. Moreover, if one of the above statements holds, then EV* = im E. 38 Let E and A be constant matrices. Then rank E = <5P (sE - A). 6. Preliminary results (or matrix pencils Proof For all f(s) E Xoo(s) the vector s-2(sE-A)f(s) is strictly proper. 8 that sE - A does not have poles at of order 2: 2.

S-l'n)T(s). e. T(oo) has full row rank. 0 In the above we saw that T(s) can be brought in row reduced form by left multiplication by a Coo(s)-unimodular matrix U_ (s). The next theorem from [71] shows that this can also be accomplished by left multiplication by a C[s]-unimodular matrix U+(s). 11) where U+(s) is a C[s]-unimodular matrix of size n x nand U_ (s) is a Coo(s)-unimodular matrix of size m x m; ~(s) = diag (sl<', ... , sl

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