Module 8 p1 - Nonlinear Adaptive Control Module 8-Part I...

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1 Module 8-Part I Differential Geometry Concepts
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2 Topics 1. Differential Geometry concepts 2. Ergodicity and Chaos 3. Embedding Ideas and Relationships to Observability
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3 A m-dimensional manifold is a set of points that locally is similar to m , i.e., it is an object which we can imagine of being constructed out of a small and possibly overlapping pieces of m . Examples: 1) m itself is an m -dimensional manifold. 2) The surface of a sphere or torus are examples of 2-dimensional manifold . Say, the state space manifold is the surface of a sphere. Then, the trajectories will be curves on this surface, and the vector field will define velocity vectors that are tangent to each trajectory at each point.
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4 Now, the vectors which are tangent to the curve on the sphere are also tangent to the sphere. That is the reason, the vector field of the dynamical system defines a set of vectors, called tangent vectors Æ tangent to the state space manifold at each point of the state space. Consider another dynamical system on the same state space manifold. Consider a point which is common to both trajectories. At this point, velocity vectors will be different, but still tangent to the manifold. This two tangent vectors then form a surface which is tangent to the manifold. In the most general situation, the set of all possible tangent vectors at a given point x on an m -dimensional manifold spans an m -dimensional linear space x which is called the tangent space of the manifold at x .
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5 M S 1 S 2 x T 1 T 2 Tangent Vectors and Tangent Spaces
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6 For dynamical system with dynamics f , the space derivative (or Jacobian) D x f defines the tangent space of the state space manifold. = m x m f x m f m x m f x f J x f x D " # % # " 1 1 1 ) ( ) ( ) ( c x z b dt dz ay x dt dy z y dt dx + = + = + = Rösler Flow c x z a 0 0 1 1 1 0 Jacobian Jacobians
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7 Over small regions of the state space manifold M , M and x are nearly coincident at the point x . If d is a vector sufficiently close to x , then d is also a tangent vector in x . Now the dynamics can be linearized as: f ( x + δ ) - f ( x ) D x f . Æ 1 D x f . 0 The eigenspaces of the linearized dynamics, D x f , form a decomposition of x . The action of the dynamics on vectors in these eigenspaces produce contraction, expansion or no effect. Manifolds and Vector Spaces
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8 Eigenspaces Define eigenspaces corresponding to the property of expansion or contraction of tangent vectors. E
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This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

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Module 8 p1 - Nonlinear Adaptive Control Module 8-Part I...

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