Adaptive Systems: An Introduction by Iven Mareels

By Iven Mareels

Loosely conversing, adaptive platforms are designed to house, to conform to, chang­ ing environmental stipulations while keeping functionality ambitions. through the years, the idea of adaptive structures advanced from quite easy and intuitive strategies to a fancy multifaceted conception facing stochastic, nonlinear and limitless dimensional structures. This ebook presents a primary creation to the idea of adaptive structures. The e-book grew out of a graduate direction that the authors taught numerous occasions in Australia, Belgium, and The Netherlands for college students with an engineering and/or mathemat­ ics history. once we taught the direction for the 1st time, we felt that there has been a necessity for a textbook that might introduce the reader to the most elements of edition with emphasis on readability of presentation and precision instead of on comprehensiveness. the current booklet attempts to serve this desire. we predict that the reader can have taken a easy direction in linear algebra and mul­ tivariable calculus. except the fundamental innovations borrowed from those components of arithmetic, the ebook is meant to be self contained.

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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. 3 (2) that every behavior described by equations of the form R(a, a-I)w = 0, admits a representation that does not involve negative powers of~: simply multiply all rows with a sufficiently high power of~.

Notice that e == 0 does not imply that the plant parameter estimates equal the plant parameters. , the control objective is achieved whether or not the parameters are correctly estimated. This phenomenon is analyzed extensively in Chapters 4 and 5. Here we see again the dual nature of the adaptation error. It is indicative of the control objective, and it is the driving force in the adaptation mechanism. ", is nontrivial. It is one of the key (theoretical) difficulties in indirect adaptive control to be able to deal with these singular points.

8), but it does not belong to ~2 since it obviously 1The degree of a polynomial P(~. ~-l) is defined as the difference of the largest and the smallest exponent in P(~, ~-l). So the degree of ~2 + 3 - ~-l is three. 30 Chapter 2. 8). This is, of course, due to the common factor of Al (~, ~-I) and BI (~, ~-I) which disappears in the transfer function. The conclusion is that the transfer function of a system does not necessarily represent the whole system. This is one of the main reasons why we prefer the behavioral approach over the transfer function approach.

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