ONT Re: Differential Geometry for Engineers
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Note 7
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| 2. Manifolds And Their Maps
|
| 2.2. Examples
|
| Another trivial but important example of a class of manifolds is
| afforded by any open subset of R^n. There the atlas may consist
| of the set itself, together with the identity map. Thus, the
| notion that manifolds are spaces that locally look like open
| subsets of R^n is at least self-consistent. This example
| is important because the whole idea of the definition
| of manifolds is to be able to see how calculations
| valid in R^n carry over into any other manifold.
|
| Another example of a manifold, which is
| an open set of Euclidean space and which
| is important in systems theory, follows.
| Let:
|
| x` = Ax + bu
|
| be a single-input controllable system.
| Recall that controllability is equivalent
| to having the rank of the matrix:
|
| [b, A b, A^2 b, ..., A^(n-1) b]
|
| equal to n, where A is an n x n matrix.
| Now let M be the set of pairs (A, b)
| such that a system is controllable:
|
| M = {(A, b) : x` = Ax + bu is controllable}.
|
| The complement of this set is the
| set that satisfies the condition:
|
| det [b, A b, A^2 b, ..., A^(n-1) b] = 0.
|
| Since this is a closed set in R^(n^2 + n), the set M
| is open in R^(n^2 + n), and therefore is a manifold.
|
| The system, being of single input, is a special case.
| In general, when the control distribution function B
| is an n x m matrix, M* is also a manifold where M* is
| the set:
|
| M* = {(A, B) : x` = Ax + Bu is controllable}.
|
| Although the conditions are more involved and less
| easy to describe than the determinant condition above,
| a similar argument shows that the controllable pairs are
| an open subset of R^n(n+m).
|
| A more general example along these same lines is the set
| of triples of matrices (A, B, C) representing the system:
|
| x` = Ax + Bu
|
| y = Cx
|
| If the system is controllable and observable, it can be shown
| that this set of triples is also an open subset of a suitable
| Euclidean space.
|
| Related to this manifold is a set of matrix transfer functions
| T(s). These are matrices of rational functions that arise as the
| Laplace transforms of the above systems. Whether this set {T(s)} is
| a manifold is a deep question in systems theory. It has been answered
| affirmatively by Martin Clark [5], Roger Brockett [6], and independently
| by Michiel Hazewinkel [7], and by Christopher Byrnes and N.E. Hurt [8].
| Much of the study in linear systems is involved with various properties
| of this manifold.
|
| Doolin & Martin, DGFE, pages 17-18.
|
| 5. J.M.C. Clark,
| "The Consistent Selection of Parameterizations in System Identification",
| 'Proceedings of the Joint Automatic Control Conference', July 27-30, 1976,
| Purdue University, West Lafayette, IN, pages 576-585.
| American Society of Mechanical Engineers, New York, NY.
|
| 6. R.W. Brockett,
| "Some Geometric Questions in the Theory of Linear Systems",
| 'IEEE Transactions on Automatic Control', vol. AC-21:4,
| August 1976, pages 449-455.
|
| 7. M. Hazewinkel & R.E. Kalman,
| "On Invariance, Canonical Forms, and Moduli for Linear
| Constant, Finite-Dimensional, Dynamical Systems", in:
| 'Lecture Notes on Economics & Mathematical System Theory',
| vol. 131, pages 440-454, Springer-Verlag, Berlin, 1976.
|
| 8. C.I. Byrnes & N.E. Hurt,
| "On the Moduli of Linear Dynamical Systems", 'Advances in Mathematics',
| Suppl. Series, vol. 4, 1978, pages 83-122. Academic Press, New York, NY.
|
| Brian F. Doolin & Clyde F. Martin,
|'Introduction to Differential Geometry for Engineers',
| Marcel Dekker, New York, NY, 1990.
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