Chapter

# Combined Sensitivity–Complementary Sensitivity Min–Max Approach for Load Disturbance–Setpoint Trade-off Design

DOI: 10.1007/978-0-387-74905-1_24

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**ABSTRACT:**In this paper, a simple PI controller design method is proposed which can achieve user-specified gain and phase margins if they are suitable, or otherwise the automatically adjusted alternatives for performance enhancement. Unlike reduced-order model-based tuning methods, exact margins, once specified or adjusted, can be accomplished regardless of the process order and damping nature. The response of the closed-loop system using the proposed design is hence more predictable than those using model-based methods when performance specifications are given in gain and phase margins. The proposed method is based on finding the intersections between two graphs that are plotted using the frequency response of the process. The given gain and phase specifications can be achieved if intersections can be found, and each intersection corresponds to one solution. Suggestions are provided on how to modify the specifications for achieving satisfactory responses for cases where there are no solution or when the specifications can be met but poor responses result. Simulation examples are given to show the usefulness of the method.Automatica 09/1998; · 3.13 Impact Factor -
##### Article: A course in H 1 control theory

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**ABSTRACT:**H ∞ denotes the Hardy space of complex-valued functions F(s) of a complex variable s which are analytic and bounded in the open right half-plane, Re s>0; bounded means that there is a real number b such that |F(s)|≤b, Re s>0. The least such bound b is the H ∞ -norm of F, denoted ∥F∥ ∞ . The subset of H ∞ consisting of real-rational functions is denoted by RH ∞ · This book is devoted to the solution of a model matching problem: given three matrices T i in RH ∞ , find a matrix Q in RH ∞ to minimize ∥T 1 -T 2 QT 3 ∥ ∞ . If α denotes the mf {T 1 -T 2 QT 3 ∥ ∞ : Q∈RH ∞ }, the problems concerned with are: (1) find α ; (2) find Q (when it exists) for which α is attained. Many problems in modern control synthesis and design can be stated as special cases of this quite general problem. The first problem of this type was proposed by Zames (author’s reference 1979) and since then, over the past seven or eight years this subject has become quite popular both for control theorists and operator theorists. The reason for this is that problems of this sort have their classical roots in the interpolation theory of Carathéodory, Nevanlinna, Pick and their modern operator theoretic ones in the work of Nehari, Adamjan, Arov and Krein, and Sarason. The author was uniquely qualified in writing this text. He was actively involved in all phases of the research on this problem. I would say he is probably the only one that mastered all the techniques and directions that this research involved. He also has an extremely clear and concise writing style that allows him to express non-trivial ideas in a very clear elementary fashion. This book is a significant contribution to the growing field of control theory and should be read by everyone interested in this area. The book begins with three introductionary chapters where the basic terminology and mathematical framework of the problem are described. Chapter four is concerned with deriving the Youla parametrization of all stabilizing compensators for a given plant. All results are given with corresponding state-space formulas for the actual computation of the compensators. This is a unique feature that is continued throughout the book. All solutions are given in state space form which allows straightforward computational algorithms. In Chapter 5 the author defines the notion 0+a Hankel operator, relating it to the system-theoretic notions of controllability and observability Grammians. In Chapter 6 the main results of the book begin to appear. Here a sufficient condition for the existence of an optimal Q is given and a complete solution of the problem is given for the scalar case. An actual computational example is worked out in complete detail. The rest of the book is dedicated to the much more complicated matrix version of the problem. This uses such tools as canonical factorizations of the Wiener-Hopf and Inner-Outer type, the introduction of Krein spaces and the Ball-Helton theory of invariant subspaces Krein spaces. While these notions are quite technically complicated they are presented in a style which is so clear and compelling that it leaves one with the feeling of having undergone a great intellectual experience. The last chapter is dedicated to the analysis of some qualitative properties of the problem. To summarize, I highly recommend this book strongly as the best way to enter into an area of research which has been central in control theory during the last ten years. - Industrial & Engineering Chemistry Process Design and Development 04/2002; 25(1).

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