3.4 Quotient manifolds 3.4.1 Theory of quotient manifolds 3.4.2 Functions on quotient manifolds 3.4.3 The real projective space RP n−1 3.4.4 The Grassmann manifold Grass(p, n) 3 3.5 Tangent vectors and differential maps CONTENTS ix 5.3 Riemannian connection 5.3.1 Symmetric connections 5.3.2 Definition of the Riemannian connection 5.3.3 Riemannian connection on Riemannian submanifolds 5.3.4 Riemannian connection on quotient manifolds 5.4 Geodesics, exponential mapping, and parallel translation 5.5 Riemannian Hessian operator 5.6 Second covariant derivative* 5.7 Notes and references 6. Newton's Method 6.1 Newton's method on manifolds 6.2 Riemannian Newton method for real-valued functions 6.3 Local convergence 6.3.1 Calculus approach to local convergence analysis 6.4 Rayleigh quotient algorithms 6.4.1 Rayleigh quotient on the sphere 6.4.2 Rayleigh quotient on the Grassmann manifold 6.4.3 Generalized eigenvalue problem 6.4.4 The nonsymmetric eigenvalue problem 6.4.5 Newton with subspace acceleration: Jacobi-Davidson 6.5 Analysis of Rayleigh quotient algorithms 6.5.1 Convergence analysis 6.5.2 Numerical implementation 6.6 Notes and references 7. Trust-Region Methods 7.1 Models 7.1.1 Models in R n 7.1.2 Models in general Euclidean spaces 7.1.3 Models on Riemannian manifolds 7.

In this book several streams of nonlinear control theory are merged and directed towards a constructive solution of the feedback stabilization problem. Analytic, geometric and asymptotic concepts are assembled as design tools for a wide variety of nonlinear phenomena and structures. Differential-geometric concepts reveal important structural properties of nonlinear systems, but allow no margin for modeling errors. To overcome this deficiency, we combine them with analytic concepts of passivity, optimality and Lyapunov stability. In this way geometry serves as a guide for construction of design procedures, while analysis provides robustness tools which geometry lacks.Our main tool is passivity. As a common thread, it connects all the chapters of the book. Passivity properties are induced by feedback passivation designs. Until recently, these designs were restricted to weakly minimum phase systems with relative degree one. Our recursive designs remove these restrictions. They are applicable to wider classes of nonlinear systems characterized by feedback, feedforward, and interlaced structures.After the introductory chapter, the presentation is organized in two major parts. The basic nonlinear system concepts -passivity, optimality, and stability margins -are presented in Chapters 2 and 3 in a novel way as design tools. Most of the new results appear in Chapters 4, 5, and 6. For cascade systems, and then, recursively, for larger classes of nonlinear systems, we construct design procedures which result in feedback systems with optimality properties and stability margins.The book differs from other books on nonlinear control. It is more designoriented than the differential-geometric texts by Isidori [43] and Nijmeijer and Van der Schaft [84]. It complements the books by Krstić, Kanellakopoulos and Kokotović [61] and Freeman and Kokotović [26], by broadening the class of systems and design tools. The book is written for an audience of graduate students, control engineers, and applied mathematicians interested in control theory. It is self-contained and accessible with a basic knowledge of control theory as in Anderson and Moore [1], and nonlinear systems as in Khalil [56]. vii viii For clarity, most of the concepts are introduced through and explained by examples. Design applications are illustrated on several physical models of practical interest.The book can be used for a first level graduate course on nonlinear control, or as a collateral reading for a broader control theory course. Chapters 2, 3, and 4 are suitable for a first course on nonlinear control, while Chapters 5 and 6 can be incorporated in a more advanced course on nonlinear feedback design.

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Abstract-This paper addresses the design of mobile sensor networks for optimal data collection. The development is strongly motivated by the application to adaptive ocean sampling for an autonomous ocean observing and prediction system. A performance metric, used to derive optimal paths for the network of mobile sensors, defines the optimal data set as one which minimizes error in a model estimate of the sampled field. Feedback control laws are presented that stably coordinate sensors on structured tracks that have been optimized over a minimal set of parameters. Optimal, closed-loop solutions are computed in a number of low-dimensional cases to illustrate the methodology. Robustness of the performance to the influence of a steady flow field on relatively slow-moving mobile sensors is also explored.

Abstract-This paper proposes a design methodology to stabilize isolated relative equilibria in a model of all-to-all coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a low-order parametric family of stabilizable collectives that offer a set of primitives for the design of higher-level tasks at the group level.