A Bruhat type decomposition of the set of conditioned invariant - PowerPoint PPT Presentation

A Bruhat type decomposition of the set of conditioned invariant subspaces Jochen Trumpf Jochen.Trumpf@anu.edu.au Department of Systems Engineering Research School of Information Sciences and Engineering The Australian National University and National ICT Australia Ltd. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.1/22

overview output injection A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

overview output injection Brunovsky form A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

overview output injection Brunovsky form restrictions A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

overview output injection Brunovsky form restrictions restriction indices A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

overview output injection Brunovsky form restrictions restriction indices image representations A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

overview output injection Brunovsky form restrictions restriction indices image representations tightness A Bruhat type decomposition of the set of conditioned invariant subspaces – p.2/22

overview restricted output injection A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

overview restricted output injection Kronecker form A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

overview restricted output injection Kronecker form restrictions A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

overview restricted output injection Kronecker form restrictions image representations A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

overview restricted output injection Kronecker form restrictions image representations generalised flag manifolds A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

overview restricted output injection Kronecker form restrictions image representations generalised flag manifolds the retraction map A Bruhat type decomposition of the set of conditioned invariant subspaces – p.3/22

output injection Given a pair of matrices ( C, A ) ∈ R p × n × R n × n the group action ( T, J, S ) , ( C, A ) �→ ( SCT − 1 , T ( A − JC ) T − 1 ) where J ∈ R n × p is arbitrary and T ∈ R n × n and S ∈ R p × p are invertible is called the output injection equivalence action. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.4/22

output injection Given a pair of matrices ( C, A ) ∈ R p × n × R n × n the group action ( T, J, S ) , ( C, A ) �→ ( SCT − 1 , T ( A − JC ) T − 1 ) where J ∈ R n × p is arbitrary and T ∈ R n × n and S ∈ R p × p are invertible is called the output injection equivalence action. Its corresponding normal form is the so-called Brunovsky normal form . In the case where ( C, A ) is observable it looks like this. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.4/22

Brunovsky form   1 ... 0   ......         0 ... 01 1 0     ... ... C = A =  and              1 ... 0 ... 01 0   µ 1 . . . � �� � ......   � �� �   µ p 1 0 . . . � � �� � �� � µ 1 µ p where µ 1 ≥ ... ≥ µ p ≥ 0 are the observability indices of ther pair ( C, A ) which form a complete set of invariants. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.5/22

restrictions A conditioned invariant or ( C, A ) -invariant subspace is a subspace V ⊂ R n for which there exists an output injection J such that ( A − JC ) V ⊂ V Note that this concept is invariant under output injection equivalence. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.6/22

restrictions A conditioned invariant or ( C, A ) -invariant subspace is a subspace V ⊂ R n for which there exists an output injection J such that ( A − JC ) V ⊂ V Note that this concept is invariant under output injection equivalence. There is a natural notion of restriction of the pair ( C, A ) to V as follows. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.6/22

� � �� � � � �� � � �� � � � � � � � � restrictions ˜ ˜ A C R n / V R n / V R p /C ( V ) R n A − JC C R n R p ¯ ¯ A C C ( V ) V V A Bruhat type decomposition of the set of conditioned invariant subspaces – p.7/22

restriction indices It can be shown that all matrix representations ( ¯ C, ¯ A ) of restrictions are output injection equivalent and that all pairs that are output injection equivalent to a given matrix representation of a restriction are in fact such. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.8/22

restriction indices It can be shown that all matrix representations ( ¯ C, ¯ A ) of restrictions are output injection equivalent and that all pairs that are output injection equivalent to a given matrix representation of a restriction are in fact such. Hence there is a matrix representation of a restriction in Brunovsky normal form. Its indices ( λ 1 , . . . , λ p ) are called the restriction indices of ( C, A ) with respect to V . It is λ i ≤ µ i for i = 1 , . . . , p . A Bruhat type decomposition of the set of conditioned invariant subspaces – p.8/22

image representations Theorem [FPP]: The codimension q ( C , A ) -invariant subspaces are precisely the images of full rank matrices Z ∈ R n × ( n − q ) which fullfill C ⊤ ¯ (1) A Z = Z ¯ A + A Z ¯ C C ⊤ ¯ (2) C Z = C Z ¯ C C ⊤ has full rank (3) C Z ¯ where ( ¯ C, ¯ A ) is the matrix representation of a restriction in Brunovsky form. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.9/22

image representations The set of all these Z s is called M ( λ, µ ) , we denote Γ( λ ) = M ( λ, λ ) which is a group. In fact Im Z 1 = Im Z 2 for Z 1 , Z 2 ∈ M ( λ, µ ) if and only if Z 2 = Z 1 S − 1 for S ∈ Γ( λ ) . A Bruhat type decomposition of the set of conditioned invariant subspaces – p.10/22

image representations The set of all these Z s is called M ( λ, µ ) , we denote Γ( λ ) = M ( λ, λ ) which is a group. In fact Im Z 1 = Im Z 2 for Z 1 , Z 2 ∈ M ( λ, µ ) if and only if Z 2 = Z 1 S − 1 for S ∈ Γ( λ ) . Hence the Brunovsky stratum of all codimension q conditioned invariant subspaces with fixed restriction indices λ is a smooth manifold diffeomorphic to M ( λ, µ ) / Γ( λ ) . A Bruhat type decomposition of the set of conditioned invariant subspaces – p.10/22

image representations The set of all these Z s is called M ( λ, µ ) , we denote Γ( λ ) = M ( λ, λ ) which is a group. In fact Im Z 1 = Im Z 2 for Z 1 , Z 2 ∈ M ( λ, µ ) if and only if Z 2 = Z 1 S − 1 for S ∈ Γ( λ ) . Hence the Brunovsky stratum of all codimension q conditioned invariant subspaces with fixed restriction indices λ is a smooth manifold diffeomorphic to M ( λ, µ ) / Γ( λ ) . Equations (1) and (2) can be solved explicitely. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.10/22

image representations    z 1 0 ... 0 .  ... ...  .  .     z 2 ... ...     .    .  . 0    ...         zµi − λj z 1 , λ j ≤ µ i  zµi − λj +1 ...         z 2 ... ...     .   Z = ( Z ij ) , Z ij = .  .  0 . ... ... .   .     zµi − λj     0 ... 0 zµi − λj +1        � 0 ... 0  �  . .   . .  . . , λ j > µ i    0 ... 0 A Bruhat type decomposition of the set of conditioned invariant subspaces – p.11/22

tightness A ( C , A ) -invariant subspace V is called tight if V + Ker C = R n or, equivalently, if rk ¯ C = p , i.e. λ p > 0 . A Bruhat type decomposition of the set of conditioned invariant subspaces – p.12/22

tightness A ( C , A ) -invariant subspace V is called tight if V + Ker C = R n or, equivalently, if rk ¯ C = p , i.e. λ p > 0 . C ⊤ is invertible. We will consider only Then C Z ¯ this case. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.12/22

restricted output injection The same construction can be repeated starting from a slightly different group action ( T, J, U ) , ( C, A ) �→ ( UCT − 1 , T ( A − JC ) T − 1 ) where U is unipotent lower triangular. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.13/22

restricted output injection The same construction can be repeated starting from a slightly different group action ( T, J, U ) , ( C, A ) �→ ( UCT − 1 , T ( A − JC ) T − 1 ) where U is unipotent lower triangular. The normal form ( Kronecker form ) looks very similar. A Bruhat type decomposition of the set of conditioned invariant subspaces – p.13/22