Module 8 p2 - Nonlinear & Adaptive Control Module...

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1 Nonlinear & Adaptive Control Module 8-Part II Differential Geometry Concepts
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2 Topics 1. Differential Geometry concepts This material is in Chapter 13 of Khalil and in 1. Nonlinear Control Systems - Isidori 3 rd Edition. 2. Nonlinear Dynamical Systems - Nijmeier and van der Schaft The first book in particular is a very well developed discussion of the ideas of feedback-linearization and differential geometric based designs.
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3 New Mathematics & Formalism To properly develop feedback linearization, we need to learn about differential geometry. We start with some definitions, examples, and work our way to the linearization theory.
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4 Differential Geometric Concepts Vector field: ctor) column ve dim - ( field. vector a be to said is domain, a is where : mapping A n R D R D f n n Covector field: A transpose of a vector field is said to be a covector field. ( -dim row vector) n Inner Product: field. vector a is and field covector a is where ) ( ) ( ) ( ) ( , 1 f w x f x w x f x w f w n i i i = >= < = Differential (gradient): ] [ i.e., field, covector a is of al differenti The . : Let 1 n x h x h x h dh h R D h = = " ) ( h
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5 Example of a Vector field θ (x, y) Y b Z b
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6 Smooth Vector fields A smooth vector field is a smooth map It takes to
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7 Manifolds & Tangent Spaces 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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8 Tangent Vectors & Spaces 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.
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Module 8 p2 - Nonlinear &amp; Adaptive Control Module...

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