Control Theory and Design. An RH₂ and RH∞ Viewpoint by Patrizio Colaneri, José C. Geromel, Arturo Locatelli, José

By Patrizio Colaneri, José C. Geromel, Arturo Locatelli, José C. Geromel and Arturo Locatelli (Auth.)

Content material:
Preface & Acknowledgments

, Page ix
Chapter 1 - Introduction

, Pages 1-2
Chapter 2 - Preliminaries

, Pages 3-68
Chapter three - suggestions structures Stability

, Pages 69-86
Chapter four - RH2 Control

, Pages 87-119
Chapter five - RH Control

, Pages 121-193
Chapter 6 - Nonclassical difficulties in RH2 and RH

, Pages 195-261
Chapter 7 - doubtful structures regulate Design

, Pages 263-300
Appendix A - a few evidence on Polynomials

, Pages 301-302
Appendix B - Singular Values of Matrices

, Pages 303-313
Appendix C - Riccati Equation

, Pages 315-323
Appendix D - Structural Properties

, Pages 325-326
Appendix E - the normal 2-Block Scheme

, Pages 327-336
Appendix F - Loop Shifting

, Pages 337-342
Appendix G - Worst Case Analysis

, Pages 343-347
Appendix H - Convex capabilities and Sets

, Pages 349-358
Appendix I - Convex Programming Numerical Tools

, Pages 359-370

, Pages 371-374

, Pages 375-378

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Extra resources for Control Theory and Design. An RH₂ and RH∞ Viewpoint

Example text

T h e following result is then provided, whose proof is obvious and then omitted. T h e o r e m 2 . 1 1 Let F{s) Then, i) = Fa{s) + Fs{s) with Fa{s) e RH^ and Fs{s) e RH2. =0 E x a m p l e 2 . 1 8 Consider the function ^^'^ = is^l)(s-l) ^ ^^^ Letting Fa{s) := ,rr-^-, , 2(s-l) ' Fs{s):^ ^ ^ • 2(s + l) it follows that F{s) = Fs{s) + Fa{s) and Ga{s) := F^is) = - 1 / 2 ( 5 + 1). 5, Cs = 1, Ca = - 1 and Ds = Da = 0. 21 It is worth noticing that the norm of a generic function F{s) G RLoo coincides with that of a suitable function F{s) G RHoo, which is easily derived from F{s).

Conversely, the subspace RH2 is constituted by the functions RL2 which are analytic in the left half plane. • R e m a r k 2 . 1 8 The given definitions imply that a matrix belongs to RL2 if its elements are strictly proper rational functions without poles on the imaginary axis. It belongs to RH2 (resp. RH2) if its elements are strictly proper rational functions without poles in the closed right (resp. left) half plane. 19 The elements in the spaces RL2, RH2 and RH^ (which are rational functions of complex variable) can be related to the elements of the spaces RL2{—oo 00), RL2[0 00), and RL2{—oo 0], which are functions of the real variable t.

22 Let A and B be two matrices with the same dimensions. 7. 23 Let A and B he two matrices with the same number of rows. 24 Let A he a square matrix. Y^G'1{A) The quantity \\A\\F := ^JiY^ce{A^A) B ]) < V2max[a(A), a(5)] Then, = trace [ A ^ A] is the so called Frobenius norm of A. 25 Let m he the numher of columns of a matrix A and denote hy Aij its element in position {i^j). 7 Basic facts on linear operators In this section some facts on the theory of linear operators are recalled. Since no confusion can arise in the present context, the term linear will be often disregarded.

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