By Iven Mareels

Loosely conversing, adaptive structures are designed to house, to conform to, chang ing environmental stipulations while conserving functionality ambitions. through the years, the speculation of adaptive platforms developed from quite uncomplicated and intuitive innovations to a fancy multifaceted idea facing stochastic, nonlinear and limitless dimensional structures. This e-book offers a primary creation to the idea of adaptive platforms. The e-book grew out of a graduate path that the authors taught numerous occasions in Australia, Belgium, and The Netherlands for college kids with an engineering and/or mathemat ics heritage. after we taught the path for the 1st time, we felt that there has been a necessity for a textbook that may introduce the reader to the most elements of edition with emphasis on readability of presentation and precision instead of on comprehensiveness. the current e-book attempts to serve this want. we think that the reader can have taken a simple path in linear algebra and mul tivariable calculus. except the elemental options borrowed from those parts of arithmetic, the ebook is meant to be self contained.

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**Example text**

We say that R(t ~-l) has full row rank if R(~, ~-l) has a g x g submatrix of which the determinant is a non-zero polynomial in ~, ~-l . 3 Let R(~, ~-l) E lRgxq[~, ~-l]. 31) and R' (~, ~-l ) has full row rank. We are now ready to describe how to eliminate latent variables. 33) with M'(~, ~-l) of full row rank. 9. Elimination of Latent variables 41 where, of course, the partition is according to the partition of U(~, ~-l )M(t ~-l). We claim that the manifest behavior is represented by R~ (~, ~-l).

57). 2 we conclude that the system is not controllable. 65) is defined by the matrix triple: A~[! ~ ~2]b~[-2]c~[O 0 ']. 5. 65) itself is not controllable. 12 A word about the notation In this chapter we have tried to be rather precise about the underlying algebraic structure of the difference equations describing the behaviors. In particular we consistently use the notation R(~, ~-I), even if terms involving ~-I do not appear. This complication is natural as we consider behaviors defined as the collection of solutions of linear time invariant difference equations defined on Z.

2) The pair (A, c) is detectable if and only if there exists a vector I E jRn such that A + Ie has all its eigenvalues in the open unit disk. 8 Stability Stability is a minimal requirement of all adaptive systems that we will consider. In this section we confine ourselves to linear systems for which we shall define marginal stability and asymptotic stability. 3. First we define stability for behaviors with only outputs and no inputs. 1 Let P(~, ~-I) E jRPXP[~, ~-I] such that det P(~, ~-I) =1= 0 (by 0 we mean the zero polynomial).