Geometric structure of single/combined equivalence classes of a controllable pair

Given a pair of matrices representing a controllable linear system, its equivalence classes by the single or combined action of feedbacks, change of state and input variables, as well as their intersection are studied. In particular, it is proved that they are differentiable manifolds and their dimensions are computed. Some remarks concerning the effect of different kinds of feedbacks are derived.Postprint (published version

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)\in\w C^n\times\w C^{n+m} without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary)

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)\in\w C^n\times\w C^{n+m} without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary)

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)\in\w C^n\times\w C^{n+m} without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary)

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)\in\w C^n\times\w C^{n+m} without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary)

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)\in\w C^n\times\w C^{n+m} without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary)

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)\in\w C^n\times\w C^{n+m} without any assumption on the observability. More precisely we prove that the set of $(C,A)$-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all $(C,A)$-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of $(C,A)$ has at most one eigenvalue (this condition is in general necessary)

Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

We extend some known results about the smooth stratification of the set of conditioned invariant subspaces for a general pair (C,A)€C^nXC^(n+m) without any assumption on the observability. More precisely we prove that the set of (C,A)-conditioned invariant subspaces having a fixed Brunovsky-Kronecker structure is a submanifold of the corresponding Grassman manifold, with a vector bundle structure relating the observable and nonobservable part, and we compute its dimension. We also prove that the set of all (C,A)-conditioned invariant subspaces having a fixed dimension is connected, provided that the nonobservable part of (C,A) has at most one eigenvalue (this condition is in general necessary)