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My main goal in writing this monograph is to provide a detailed treatment of uncertainty analysis for sampled-data systems in the context of sys­ tems control theory. Here, sampled-data system refers to the hybrid sys­ tem formed when continuous time and discrete time systems are intercon­ nected; by uncertainty analysis I mean achievable performance in the pres­ ence of worst -case uncertainty and disturbances. The focus of the book is sampled-data systems; however the approach presented is applicable to both standard and sampled-data systems. The past few years has seen a large surge in research activity centered around creating systematic methods for sampled-data design. The aim of this activity has been to deepen and broaden the, by now, sophisticated viewpoint developed for design of purely continuous time or discrete time systems (e.g. J{oo or -I!l optimal synthesis, J1 theory) so that it can be ap­ plied to the design of sampled-data systems. This research effort has been largely successful, producing both interesting new mathematical tools for control theory, and new methodologies for practical engineering design. Analysis of structured uncertainty is an important objective in control design, because it is a flexible and non-conservative way of analyzing sys­ tem performance, which is suitable in many engineering design scenarios.

My main goal in writing this monograph is to provide a detailed treatment of uncertainty analysis for sampled-data systems in the context of sys­ tems control theory. Here, sampled-data system refers to the hybrid sys­ tem formed when continuous time and discrete time systems are intercon­ nected; by uncertainty analysis I mean achievable performance in the pres­ ence of worst -case uncertainty and disturbances. The focus of the book is sampled-data systems; however the approach presented is applicable to both standard and sampled-data systems. The past few years has seen a large surge in research activity centered around creating systematic methods for sampled-data design. The aim of this activity has been to deepen and broaden the, by now, sophisticated viewpoint developed for design of purely continuous time or discrete time systems (e.g. J{oo or -I!l optimal synthesis, J1 theory) so that it can be ap­ plied to the design of sampled-data systems. This research effort has been largely successful, producing both interesting new mathematical tools for control theory, and new methodologies for practical engineering design. Analysis of structured uncertainty is an important objective in control design, because it is a flexible and non-conservative way of analyzing sys­ tem performance, which is suitable in many engineering design scenarios.

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1 Introduction.- 1.1 Modelling and Uncertainty.- 1.2 Summary of Contents.- 2 Preliminaries.- 2.1 Hilbert Space and Banach Algebras.- 2.2 Operator Theory.- 2.3 Analytic Functions.- 2.4 Time Domain Spaces and Lifting.- 2.5 Frequency Domain Function Spaces.- 2.6 The Structured Singular Value.- 3 Uncertain Sampled-data Systems.- 3.1 Summary.- 4 Analysis of LTI Uncertainty.- 4.1 Converting to Frequency Domain.- 4.2 Destabilizing Perturbations.- 4.3 Robustness Test.- 4.4 Sampled-data Frequency Response.- 4.5 Summary.- 5 A Computational Framework.- 5.1 Lower Bounds.- 5.2 Upper Bounds.- 5.2.1 Convergence.- 5.2.2 Characterization in Finite Dimensions.- 5.2.3 Evaluating ?Mn.- 5.3 An Algorithm.- 5.4 Example.- 5.5 Summary.- 6 Robust Performance.- 6.1 Robust Performance Conditions.- 6.1.1 Periodic Perturbations.- 6.1.2 Quasi-Periodic Perturbations.- 6.1.3 Arbitrary Time-Varying Uncertainty.- 6.2 Computational Tools.- 6.2.1 Definition and Properties.- 6.2.2 Reduction to Finite Dimensions.- 6.3 Example Algorithm.- 6.3.1 A Cutting Plane Approach.- 6.3.2 Numerical Example.- 6.4 Minimizing the Scaled Hilbert-Schmidt Norm.- 6.4.1 The Hilbert-Schmidt Norm.- 6.4.2 Scaling the Hilbert-Schmidt Norm and Osborne’s Method.- 6.5 Summary.- A State space for ?M.- B Proof of Proposition 5.4.- D State Space for Section 6.2.- E Proof of Lemma 6.10.- G The S-Procedure.

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