4.2 ObserverHessenberg Forms
The duals of the controllerHessenberg forms are the observerHessenberg 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 observerHessenberg form of the pair (, ) or simply the observerHessenberg form.
Note that in the singleinput case, is just a lower Hessenberg matrix and is a multiple of .
The superdiagonal blocks , , ..., in the matrix are of particular importance: the statespace system is observable if and only if they all have full rank, in which case the observerHessenberg form is called unreduced. When the system is unobservable, the superdiagonal blocks have full rank and the last block is zero.
The upper observerHessenberg form is given by
where is a rectangular upper triangular matrix.
Both the lower and upper observerHessenberg forms are constructed by duality of the staircase algorithm for the controllerHessenberg forms. By default, lower observerHessenberg form is used.
The observerHessenberg realizations.
Pivoting, obtained with the option value PivotingTrue, 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.
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Load the collection of statespace systems.
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Consider the same drum boiler system in the previous example.
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This is the lower observerHessenberg form of the drum boiler system.
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Out[4]//NumberForm=
This is the upper observerHessenberg form of the drum boiler system.
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