Algebraic and Geometric Methods in Nonlinear Control Theory

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point"of a Pin'. van GuIik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; ihe Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras ·are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

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Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point"of a Pin'. van GuIik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non­ trivially) in regional and theoretical economics; algebraic geometry interacts with physics; ihe Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras ·are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Résumé
**... this volume has already become one of the reference titles in nonlinear control theory. Moreover, the volume shows how modern sophisticated elements of mathematics can lead not only to new theoretical results but also to direct immediate industrial applications. This way, the volume has many samples ofThe art of applying mathematics' as stated in the `manifest' written by Hazewinkel in the first number of Acta Applicandae Mathematicae.'
Acta Applicandae Mathematicae, Vol. 11, 1988.

Contenu
Controllability, Observability, Realization and other Structural Properties.- Realization Theory for Nonlinear Systems; Three Approaches.- The Local Realization of Generating Series of Finite Lie Rank.- Realizations of Polynomial Systems.- Symmetries and Local Controllability.- The Intrinsic Geometry of Dynamic Observations.- Design of Nonlinear Observers by a Two-Step-Transformation.- Feedback Synthesis and Linearization Techniques.- On the Input-Output Decoupling of Nonlinear Systems.- Control of Nonlinear Systems Via Dynamic State-Feedback.- A Classification of Nonlinear Systems Based on the Invariant Subdistribution Algorithm.- Asymptotic Expansions, Root-Loci and the Global Stability of Nonlinear Feedback Systems.- Everything You Always Wanted to Know About Linearization.- Feedback Linearization and Simultaneous Output Block Decoupling of Nonlinear Systems.- Global Feedback Linearizability of Locally Linearizable Systems.- Global Aspects of Linearization, Equivalence to Polynomial Forms and Decomposition of Nonlinear Control Systems.- The Extended-Linearization Approach for Nonlinear Systems Problems.- About the Local Linearization of Nonlinear Systems.- Optimal Control.- Envelopes, Conjugate Points, and Optimal Bang-Bang Extremals.- Geometry of the Optimal Control.- Volterra Series and Optimal Control.- Optimal Control and Hamiltonian Input-Output Systems.- Discrete-Time Systems.- Nonlinear Systems in Discrete Time.- Local Input-Output Decoupling of Discrete Time Nonlinear Systems.- Orbit Theorems and Sampling.- Various other Theoretical Aspects.- An Infinite Dimensional Variational Problem Arising in Estimation Theory.- Iterated Stochastic Integrals in Nonlinear Control Theory.- Approximation of Nonlinear Systems by Bilinear Ones.- Applications.- Feedback Linearization Techniques in Robotics and Power Systems.- C.A.D. for Nonlinear Systems Decoupling, Perturbations Rejection and Feedback Linearization with Applications to the Dynamic Control of a Robot Arm.- A Nonlinear Feedback Control Law for Attitude Control.- Identification of Different Discrete Models of Continuous Non-linear Systems. Application to Two Industrial Pilot Plants.- Bang-Bang Solutions for a Class of Problems Arising in Thermal Control.