By Hidenori Kimura
The introduction of H-infinity-control used to be a very amazing innovation in multivariable idea. It eradicated the classical/modern dichotomy that were a big resource of the long-standing skepticism in regards to the applicability of recent keep watch over idea, by means of amalgamating the "philosophy" of classical layout with "computation" according to the state-space challenge environment. It greater the appliance through deepening the idea mathematically and logically, now not by way of weakening it as was once performed via the reformers of recent regulate thought within the early 1970s.
However, only a few sensible layout engineers are acquainted with the speculation, even if a number of theoretical frameworks were proposed, particularly interpolation concept, matrix dilation, differential video games, approximation conception, linear matrix inequalities, and so forth. yet none of those frameworks have proved to be a typical, basic, and complete exposition of H-infinity-control conception that's obtainable to sensible engineers and demonstrably the main common keep watch over technique to in achieving the keep watch over objectives.
The function of this ebook is to supply this type of typical theoretical framework that's comprehensible with little mathematical historical past. The concept of chain-scattering, renowned in classical circuit concept, yet new to manage theorists, performs a primary position during this ebook. It captures a necessary function of the regulate platforms layout, lowering it to a J-lossless factorization, which leads us clearly to the belief of H-infinity-control. The J-lossless conjugation, an basically new thought in linear method thought, then offers a strong software for computing this factorization. therefore the chain-scattering illustration, the J-lossless factorization, and the J-lossless conjugation are the 3 key notions that supply the thread of improvement during this booklet. The publication is conpletely self contained and calls for little mathematical history except a few familiarity with linear algebra. it will likely be helpful to praciticing engineers on top of things procedure layout and as a textual content for a graduate path in H-infinity-control and its applications.
The reader is meant to be familiar with linear structures in simple terms at an straightforward point and, even though complete proofs are given, the exposition is cautious in order that it can be obtainable to engineers. H. Kimura's textbook is an invaluable resource of knowledge for everyone who desires to research this a part of the trendy keep watch over conception in a radical demeanour. —Mathematica Bohemica
The booklet turns out to be useful to working towards engineers up to the mark approach layout and as a textbook for a graduate direction in H∞ regulate and its functions. —Zentralblatt MATH
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Additional info for Chain-Scattering Approach to H∞ Control
9 shows that the eigenvalues of A + LC can be chosen arbitrarily by choosing L, if and only if (A, C) is observable. 6 Block-Diagram of Output Insertion. Through the output insertion, we can define the dual of stabilizable pair. 11 The pair (A, C) is said to be detectable if there exists a matrix L such that A + LC is stable. Obviously, an observable pair is detectable, but the converse is not true. 1). 1) is minimal, z is a transmission zero of G(s) iff rank [ B A -C zI D 1< n + min(m, r). 46) Chapter 2.
However, a special class of transformations preserves the Hamiltonian properties. An example of such transformations is given by 0] ' T = [I L I L = LT. 3 Let Ho be a Hamiltonian matrix with Xo = Ric(Ho). 42} and X = Ric(H) exists which is given by X = Xo + L. Proof. 34). 42), it follows that TETT Since Ho is a Hamiltonian, we have = E. HE+EHT = -T(HoT-lET-T+T-lET-THl)rr = -T(HoE + EHl)TT = o. This implies that H is also a Hamiltonian. Let Xo = Ric(Ho). Then it is clear that ;0 ]- [;0 ] Ho [ A for a stable matrix A.
The following theorem is one of the most important results in linear system theory. 71) has a stabilizing solution X = Ric(H). 73) where M(s) F is unimodular. 75) _R-l(C - BTX), 56 Chapter 3. Norms and Factorizations Proof. 72) holds, r(s) has no zeros on the imaginary axis. Hence, r(s)-l has no poles on the jw-axis, which implies that H has no eigenvalues on the jw-axis. 2 implies the existence of X = Ric(H). 71). 73). 75). 76), A + BF is stable. Direct manipulations yield M~(s)RM(s) [ = 1 ° _AT _FTRF FTR A B.
Chain-Scattering Approach to H∞ Control by Hidenori Kimura