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## Über dieses Buch

Through its rapid progress in the last decade, HOOcontrol became an established control technology to achieve desirable performances of con­ trol systems. Several highly developed software packages are now avail­ able to easily compute an HOOcontroller for anybody who wishes to use HOOcontrol. It is questionable, however, that theoretical implications of HOOcontrol are well understood by the majority of its users. It is true that HOOcontrol theory is harder to learn due to its intrinsic mathemat­ ical nature, and it may not be necessary for those who simply want to apply it to understand the whole body of the theory. In general, how­ ever, the more we understand the theory, the better we can use it. It is at least helpful for selecting the design options in reasonable ways to know the theoretical core of HOOcontrol. The question arises: What is the theoretical core of HOO control? I wonder whether the majority of control theorists can answer this ques­ tion with confidence. Some theorists may say that the interpolation theory is the true essence of HOOcontrol, whereas others may assert that unitary dilation is the fundamental underlying idea of HOOcontrol. The J­ spectral factorization is also well known as a framework of HOOcontrol. A substantial number of researchers may take differential game as the most salient feature of HOOcontrol, and others may assert that the Bounded Real Lemma is the most fundamental building block.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
Before 1960, control theory was composed of several classical theorems such as the Routh-Hurwitz stability criterion, Nyquist stability theorem, Bode’s dispersion relations, Wiener’s realizability criterion and factorization theory on the one hand, and a set of design algorithms such as lead-lag compensation, Smith’s prediction method, Ziegler-Nichols ultimate sensitivity method, Evans’ root locus method, and the like on the other hand. Although the preceding theoretical results were highly respected, they were not closely related to each other enough to form a systematic theory of control. However, classical design algorithms such as lead-lag compensation were not clearly formalized as a design problem to be solved, but rather as a tool of practices largely dependent on the cut-and-try process. They were usually explained through a set of examples.
Hidenori Kimura

### Chapter 2. Elements of Linear System Theory

Abstract
A linear system is described in the state space as
$$\dot x = Ax + Bu,$$
(2.1a)
$$y = Cx + Du,$$
(2.1b)
where uR r , yR m , and xR n are the input, the output, and the state, respectively.
Hidenori Kimura

### Chapter 3. Norms and Factorizations

Abstract
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.
Hidenori Kimura

### Chapter 4. Chain-Scattering Representations of the Plant

Abstract
Consider a system ∑ of Figure 4.1 with two kinds of inputs (b1, b2) and two kinds of outputs (a1, a2 ) represented as
$$\left[ {\begin{array}{*{20}{c}} {{{a}_{1}}} \\ {{{a}_{2}}} \\ \end{array} } \right] = \sum {\left[ {\begin{array}{*{20}{c}} {{{b}_{1}}} \\ {{{b}_{2}}} \\ \end{array} } \right]} = \left[ {\begin{array}{*{20}{c}} {{{\sum }_{{11}}}} & {{{\sum }_{{12}}}} \\ {{{\sum }_{{21}}}} & {{{\sum }_{{22}}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} {{{b}_{1}}} \\ {{{b}_{2}}} \\ \end{array} } \right].$$
(4.1)
Hidenori Kimura

### Chapter 5. J-Lossless Conjugation and Interpolation

Abstract
In this section, we introduce the notion of J-lossless conjugation which gives a powerful tool for computing (J, J′)-lossless factorization. The conjugation is a simple operation of replacing a part of the poles of a transfer function by their “conjugates”, that is, the mirror images with respect to the origin, by the multiplication of another transfer function. For instance, if a transfer function
$$G\left( s \right) = \frac{{s + 3}}{{(s + 1)(s - 2)}}$$
(5.1)
Hidenori Kimura

### Chapter 6. J-Lossless Factorizations

Abstract
Assume that G is a stable and invertible transfer function. If G -1 has a stabilizing J-lossless conjugator Θ, then
$$H\text{ : = }G^{ - 1} \Theta$$
(6.1)
is stable. Due to Lemma 5.3, the zeros of H coincide with those of G-1 which are stable from the assumption that G is stable. Hence, H-1 is also stable. Writing the relation (6.1) as
$$G\text{ = }\Theta H^{ - 1} ,$$
we see that G is represented as the product of a J-lossless matrix Θ and a unimodular matrix H-1 . This is a factorization of G which is of fundamental importance in H control theory.
Hidenori Kimura

### Chapter 7. H∞ Control via (J, J′)-Lossless Factorization

Abstract
Now we are in a position to deal with the H control problem based on the results obtained in the previous chapters.
Hidenori Kimura

### Chapter 8. State-Space Solutions to H∞ Control Problems

Abstract
In this chapter, we give a solution to the H control problem in the state space based on the results obtained in the preceding chapters. A state-space realization of the plant (7.1) is given by
$$\dot x = Ax + B_1 w + B_2 u,$$
(8.1a)
$$z = C_1 x + D_{11} w + D_{12} u,$$
(8.1b)
$$y = C_2 x + D_{21} w + D_{22} u$$
(8.1c)
where
$$z \in \text{ }R^m \text{ : errors to be reduced},$$
$$y \in R^q :\text{observation}\,\text{outputs},$$
$$w \in R^r :\text{exogenous}\,\text{inputs},$$
$$u \in R^p :\text{control}\,\text{inputs}.$$
Hidenori Kimura

### Chapter 9. Structure of H∞ Control

Abstract
In this section, we prove the stability of some closed-loop matrices based on the Riccati equations (8.120) and (8.122), under the standing assumption
$$D_{11} = 0.$$
(9.1)
Hidenori Kimura

### Backmatter

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