Steen Toffner-Clausen, Palle Andersen and Jakob Stoustrup U96-4153· April 18,2001 4th edition Dep artment of Control Engineering, Institute of Electronic Systems Aalborg University, Fredrik Bajers Vej 7, DK-9220 Aalborg O, Denmark
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Robus t (b This note has been written for a b asic course in robust and optimal control at 9th term of the ystem Construction Line, nstitute of Electronic Sy stems, Aalborg University. Originally the note was intended for a course consisting of six modules of four hours each. Currently, the courses at Aalborg University has been reduced to five modules, and hence, the note cont ains material, that can not be included in the course. The note has been adapted to a level, whid can be expected of 9th term students at the System Construction Line. The students are expected to be acquainted with classical feedback control theory The purposes of the note is to provide an introduction to mo dern robust and optimal control especially in Hoo and i theory. n Chapter 1, a short introduction to the concept of robust control is given; m Chapter 2 nominal and robust stability for single variable(SoSO)systems is described. n Chapter 3 nominal and robust performance for sso sy stems is analyzed, and the concepts Hoo and H2 optimal control are introduced. Chapter 4 gives an introduction to the analysis of multi variable systems, and in Chapter 5, st ability and performance of multi variable systems are studied. n Chapter 6, a solution to the Hoo control problem presented. Finally, in Chapter 7, an introduction to the structured singular value I is given, and controller design with i is treated. Robust optimal control; robust st ability; robust performance; Hoo optimal control; I ana I synthes afigution of thb authors The authors are with Grundfos A/S, and with the Department of Control Engineering, an tute of Electronic Systems, Aalborg University, DK-9220 Aalborg o, Denmark. The depart lenthasthefollowinghomepagehttp://www.conTroL.auc.dk.Theemailsofthelasttwo authors are: ipa, js]@control. auc.dk
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Contents 1 Robust feedback Conti 1. 1 Feedback and Uncertainty 1. 2 Classical Com pensator De sign Metho ds 1. 3 Mo del Uncertainty. 1. 4 Feedback Systems with Model Uncertainties 1.5 Robust Control Nominal and Robust Stability 1357799 2.1 A Model of the proces 2.2 Mo del Uncertainty 2.3 Nominal Stability 2.4 Robust Stability. 3 Nominal and robust performance 13 3.1 Signal No 3.2 Norms for Systems and Transfer Functions 3.3 Specification of inputs 3.4 Requirements for Perform ance 3.5 H2 Optimal Control(LQ) 3.6H。 Optim al 3.7 Robust Perform an ce 3.8 H2 Robust Performance 3.9 Hoo Robust P n ce 3.10 Loop Shaping 4 An Introduction to Multivariable Systems 26 4.1 Poles and zeros of multi ariable Sy stems 4.1.1 Smith-McMillan Form of a Transfer matrix 4.2 Nominal Stability for Multivariable Systems 4.2.1 Internal Stability 4.2.2 The Generalized Ny quist Theorem 4.3 Frequency Responses for Multiv ariable Systems 3 4.3. 2 Induced Norms 4.3.3 Singular values
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5 Robustness Analysis for Multi ble Systems 5. 1 Nominal perfo 44 5. 1.1 Norms of Signals for Multivariable Sy stem 45 5.1 5.2 Robust Stability 5. 2.1 The Sm all Gain Theorem 5.3 Robust performan 5.3. 1 Specifications with Mixed Sensitivity Functions 5.3.2 The Significance of Zeros and Poles in the Right Half Plane Robust Design for Multivariable Systems 6.1 Loop Shaping 6.2 Mo deling Individual Channels 6.3 H Control 6.3. 1 Remarks to the Hoo solution 6.3.2 The MAtLaB Toolboxes 7 7 Design of Robust Comp ensat ors based on A Theory 7.1 7.1.1 Robust Stability 7.1 7. 1.3 Computing p esIs 7.2.1 omplex u Sy nthesis - D-K iter ation
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Chapter 1 Robust feedback Control Chapl er n conl ains a general presenl al ion of classical feedback conl rol, wilh an emphasis on robusl ness l o mo del uncerl ainl y of feedb ack sysl ems The need lo develop design mel hods, hal explicil ly can handle mo del uncerl ainl y is demons ral ed 1.1 Feedback and Uncertainty Conl rol of a dy namical process by feedb ack of a measured oul pul is a well(known principle pically wil h i he primary ob jeclive of keeping I he oul pul of I he process close lo a given e(s K(s) G(s Standard feedback configuration Fh Figure nn, a sl andard feedback configur al ion is shown, consis ing of a process G(s)and a compensa or K(s). For sy sl ems wil h feedb ack I he following imp orl anl properl ies can ofl en oys ems I hal are unsl able from nal ure, can be sl abilized fecl of exl ernal disl t The I ime cons anl s or chara eris ic frequencies of I he sysl em can be shifl ed Finally, ofl en Ihe above properlies can be obl ained even wil h an in complel e knowledge
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