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4.1 Controller-Hessenberg Forms
Given the  state matrix  and  input matrix  , there exists an orthogonal matrix  , such that
 , 
where each  is an  matrix, the matrix  is  and  is the transpose of  . The rectangular subdiagonal block matrices  ,  , ...,  and  are upper triangular matrices. The matrices  and  are transformed to  and  as  and . The pair ( ,  ) is called the upper controller-Hessenberg form of the pair ( ,  ) or, simply, the controller-Hessenberg form.
Note that in the single-input case,  is just an upper Hessenberg matrix and  is a multiple of  .
The subdiagonal blocks  ,  , ...,  in the matrix  are of particular importance: the state-space system is controllable if and only if they all have full rank, in which case the controller-Hessenberg form is called unreduced. When the system is uncontrollable, the subdiagonal blocks  have full rank and the last block  is zero.
The lower controller-Hessenberg form is similarly defined:
 , 
where  is an  rectangular lower triangular matrix.
Both upper and lower controller-Hessenberg forms are constructed by using the staircase algorithm (Paige (1981), Boley (1981), and Van Dooren and Verhaegen (1985)). By default, upper controller-Hessenberg form is used.
The controller-Hessenberg realizations.
Pivoting, obtained with the option value Pivoting True, improves the computational accuracy. Note, however, that the triangular structure of subdiagonal blocks may be lost when pivoting is used.
Make sure the application is loaded.
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Load the collection of state-space systems.
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This is a state-space model of a drum boiler.
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This controller-Hessenberg form of the drum boiler system has a  subdiagonal block  and a  subdiagonal block  . The system is close to an uncontrollable system as indicated by the near rank deficiency of the last subdiagonal block.
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Out[4]//NumberForm=
This is the lower controller-Hessenberg form of the drum boiler system .
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