4.1 ControllerHessenberg 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 controllerHessenberg form of the pair (, ) or, simply, the controllerHessenberg form.
Note that in the singleinput case, is just an upper Hessenberg matrix and is a multiple of .
The subdiagonal blocks , , ..., in the matrix are of particular importance: the statespace system is controllable if and only if they all have full rank, in which case the controllerHessenberg form is called unreduced. When the system is uncontrollable, the subdiagonal blocks have full rank and the last block is zero.
The lower controllerHessenberg form is similarly defined:
,
where is an rectangular lower triangular matrix.
Both upper and lower controllerHessenberg forms are constructed by using the staircase algorithm (Paige (1981), Boley (1981), and Van Dooren and Verhaegen (1985)). By default, upper controllerHessenberg form is used.
The controllerHessenberg realizations.
Pivoting, obtained with the option value PivotingTrue, 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 statespace systems.
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This is a statespace model of a drum boiler.
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This controllerHessenberg 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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This is the lower controllerHessenberg form of the drum boiler system .
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