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4.2 Observer-Hessenberg Forms

The duals of the controller-Hessenberg forms are the observer-Hessenberg forms. Given the state matrix A and the output matrix , there exists an orthogonal matrix T, such that

where each is an matrix, the matrix is and is the transpose of . The rectangular superdiagonal block matrices , , ..., are lower triangular and the matrix is also rectangular and lower triangular. The matrices and are transformed to and as and. The pair (, ) is called the lower observer-Hessenberg form of the pair (, ) or simply the observer-Hessenberg form.

Note that in the single-input case, is just a lower Hessenberg matrix and is a multiple of .

The superdiagonal blocks , , ..., in the matrix are of particular importance: the state-space system is observable if and only if they all have full rank, in which case the observer-Hessenberg form is called unreduced. When the system is unobservable, the superdiagonal blocks have full rank and the last block is zero.

The upper observer-Hessenberg form is given by

where is a rectangular upper triangular matrix.

Both the lower and upper observer-Hessenberg forms are constructed by duality of the staircase algorithm for the controller-Hessenberg forms. By default, lower observer-Hessenberg form is used.

The observer-Hessenberg realizations.

Pivoting, obtained with the option value PivotingRuleTrue, improves the computational accuracy. Note that the triangular structure of superdiagonal blocks may be lost when pivoting is used.

Make sure the application is loaded.


Load the collection of state-space systems.


Consider the same drum boiler system in the previous example.



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



This is the upper observer-Hessenberg form of the drum boiler system.