Adaptive Systems: An Introduction by Iven Mareels

By Iven Mareels

Loosely talking, adaptive platforms are designed to accommodate, to evolve to, chang­ ing environmental stipulations while preserving functionality pursuits. through the years, the idea of adaptive platforms advanced from particularly easy and intuitive suggestions to a fancy multifaceted thought facing stochastic, nonlinear and limitless dimensional structures. This ebook offers a primary creation to the idea of adaptive platforms. The ebook grew out of a graduate direction that the authors taught numerous instances in Australia, Belgium, and The Netherlands for college kids with an engineering and/or mathemat­ ics historical past. after we taught the path for the 1st time, we felt that there has been a necessity for a textbook that will introduce the reader to the most facets of variation with emphasis on readability of presentation and precision instead of on comprehensiveness. the current ebook attempts to serve this want. we think that the reader may have taken a simple direction in linear algebra and mul­ tivariable calculus. except the fundamental techniques borrowed from those components of arithmetic, the publication is meant to be self contained.

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Assume moreover that the high frequency gain, indicative of the initial reaction of the system, is positive. Such systems can be stabilized by positive high (arbitrarily large) gain output feedback. This may be understood from the observation that the root loci converge towards the open loop zeros or to negative infinity along the real axis for positive output feedback gain. This example is studied in detail in Chapter 6. For such systems we propose an adaptive scalar gain output feedback strategy in which we allow the gain to increase as long as the output is non-zero.

For polynomials in lR[~, r-I] a similar technique exists. Let PI (~, ~-I), p2(~, ~-I) E lR[~, ~-I] be such that deg PI (~, ~-I) ~ deg P2(~' ~-I). Long division can now be done as follows. Write both PI (~, ~-I) and P2 (~, ~-I ) such that all terms appear in descending powers of ~. Take the term of P2(~, ~-I) with highest power of ~ as leading term for the division. Alternatively, determine kl' k2 such that ~kl Pi = Pi(~) E lR[~] and such that Pi(O) =1= O. Notice that the second requirement implies that the degrees of Pi(~' ~-I) equal the normal degrees.

Although an obvious statement, it took more than 30 years from the birth of adaptive control/signal pro• cessing before the importance of this statement was realized. 4. 9 The MIT rule can be formulated in more general circumstances in _ which several parameters of the plant model need to be estimated. 10 The above control scheme has been implemented with success in the position control loop for the antennae in the Australia Telescope. There the _ gain variation is due to the change in antenna position.

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