6.4.2 The Luenberger Observable Canonical Form

The transformation to Luenberger observable canonical form, introduced here, is used in the design of the Luenberger reduced-order state-estimator discussed in Section 6.5.

Given an n^"th"-order linear system in state-space form, as described by Eq. (6.7), then using a transformation matrix T formed from the rows of the observability matrix for the pair {A, C}, the resulting matrices and have special block structured forms, namely, for a system with q outputs,

where the diagonal blocks are in the observable companion form

for i = 1, ..., q, with dimensions _i×_i, and the symbol * stands for possible nonzero entries. The off-diagonal blocks have zero entries, except perhaps in the last column, that is, these look like the of the Luenberger controllable canonical form (see Section 6.4.1) transposed. The transformed output matrix is

where the q ×_i blocks have the special form

and the unit entry in the first row occurs in the last column. The entries below the unit entry in the last column of are not, in general, equal to zero. These nonzero entries in the matrix can be readily removed, if desired, by a simple nonsingular transformation of the system outputs.

Here, the matrix has no special form.

The parameters _k that appear in the formulation are the observability indices of the system, and are determined in a similar way to the controllability indices, defined in Section 6.4.1, but using the rows of the observability matrix for the system. The observability index for the system is defined as _o= Max[_i]. The function ObservabilityIndices can be used to determine these.

Observability indices.

The observable canonical form of the system and the required transformation matrix T are both generated via elementary row and column operations on the original system A and C matrices, using a version of Daly's algorithm (Daly (1976)). If a state-space system is completely observable, and the rows of its matrix C are linearly independent, then the function LuenbergerObservableForm can be used to transform the system into this canonical form.

Luenberger observable canonical form.

Make sure the application is loaded.

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Consider a 5^th-order state-space system with two inputs and two outputs.

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These are its observability indices.

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These indices indicate that in the corresponding Luenberger observable canonical form, the transformed matrix A will have two observable companion form diagonal blocks of size 3 × 3 and 2 × 2, with a two-block structured transformed matrix C.

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The Control System Professional functions TransformationMatrix and SimilarityTransform are both applicable to the LuenbergerObservableForm.

This gives the transformation matrix between the two realizations.

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You can confirm the similarity transformation of the original system with this matrix.

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