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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 PivotingRuleTrue, 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.


Load the collection of state-space systems.


This is a state-space model of a drum boiler.



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.



This is the lower controller-Hessenberg form of the drum boiler system .