By Hidenori Kimura
The creation of H-infinity-control was once a very extraordinary innovation in multivariable concept. It eradicated the classical/modern dichotomy that have been a massive resource of the long-standing skepticism concerning the applicability of contemporary keep watch over conception, by means of amalgamating the "philosophy" of classical layout with "computation" in response to the state-space challenge atmosphere. It better the applying by means of deepening the idea mathematically and logically, now not by means of weakening it as was once performed via the reformers of contemporary keep an eye on conception within the early 1970s.
However, only a few functional layout engineers are accustomed to the idea, even supposing a number of theoretical frameworks were proposed, particularly interpolation idea, matrix dilation, differential video games, approximation conception, linear matrix inequalities, and so forth. yet none of those frameworks have proved to be a common, easy, and entire exposition of H-infinity-control idea that's available to sensible engineers and demonstrably the main traditional keep watch over technique to in achieving the keep watch over objectives.
The objective of this e-book is to supply this sort of traditional theoretical framework that's comprehensible with little mathematical heritage. The idea of chain-scattering, popular in classical circuit idea, yet new to regulate theorists, performs a primary function during this booklet. It captures an important function of the keep an eye on structures layout, decreasing it to a J-lossless factorization, which leads us evidently to the belief of H-infinity-control. The J-lossless conjugation, an primarily new inspiration in linear procedure concept, then offers a robust device for computing this factorization. hence the chain-scattering illustration, the J-lossless factorization, and the J-lossless conjugation are the 3 key notions that offer the thread of improvement during this booklet. The e-book is conpletely self contained and calls for little mathematical historical past except a few familiarity with linear algebra. it will likely be invaluable to praciticing engineers up to speed process layout and as a textual content for a graduate path in H-infinity-control and its applications.
The reader is meant to be accustomed to linear structures basically at an straight forward point and, even if complete proofs are given, the exposition is cautious in order that it can be available to engineers. H. Kimura's textbook is an invaluable resource of knowledge for everyone who desires to study this a part of the trendy regulate conception in a radical demeanour. —Mathematica Bohemica
The e-book comes in handy to training engineers up to speed procedure layout and as a textbook for a graduate direction in H∞ keep an eye on and its functions. —Zentralblatt MATH
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Extra resources for Chain-Scattering Approach to H∞ Control
4. Stability of Linear Systems Proof. 56) is given by p = fooo eATtCTCeAtdt 2: o. If Px = 0 for some x, we see that Hence, CeAtx = 0, Vt. 5(vi), x = O. Hence, P > o. 56) is positive definite. Let A be an eigenvalue of A with corresponding eigenvector f" that is, Af, = Af,. 13, we have Since (A, C) is observable, Cx assertion. =1= O. \ < 0, which verifies the • We can state the converse of the preceding lemma. 56} has a positive definite solution for a stable A, then (A, C) is observable. Proof.
1 Norms of Signals and Systems In the recent development of control theory, various kinds of norms of signals and systems play important roles. The norm of signals is a nonnegative number assigned to each signal which quantifies the "length" of the signal. Some of the norms of signals can induce a norm of systems through the input/output relation generated by the system. The notion of induced norm is the key idea of the contemporary design theory of control systems. Let I (t) be a signal represented as a time function.
50) implies X An + Af1 X + XWllX - Q = O. This implies X = Ric( fI). 31) is of particular interest. 2. 53) only if A has no eigenvalue on the imaginary axis. The rank of X is equal to the number of unstable eigenvalues of A and the eigenspace of A corresponding to the stable eigenvalues is equal to KerX. Proof. Assume that A has an eigenvalue jW, that is, A~ = jw~, ~ # O. 52), (jwI + AT + XW)X~ = O. Since A + W X is stable, we conclude that X~ = O. This implies that (A + W X)~ = jw~. This contradicts the assumption that A + W X is stable.