{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Module 8 p1

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

This preview shows pages 1–9. Sign up to view the full content.

1 Module 8-Part I Differential Geometry Concepts

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Topics 1. Differential Geometry concepts 2. Ergodicity and Chaos 3. Embedding Ideas and Relationships to Observability
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
5 M S 1 S 2 x T 1 T 2 Tangent Vectors and Tangent Spaces

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
8 Eigenspaces Define eigenspaces corresponding to the property of expansion or contraction of tangent vectors. E
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 36

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

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online