State-space models can be realized in the following standard forms:
Modal Canonical Form
Companion Canonical Form
Observable Canonical Form
Controllable Canonical Form
In modal form, A is a block-diagonal matrix. The block size is typically 1-by-1 for real eigenvalues and 2-by-2 for complex eigenvalues. However, if there are repeated eigenvalues or clusters of nearby eigenvalues, the block size can be larger.
For example, for a system with eigenvalues , the modal A matrix is of the form
In the companion realization, the characteristic polynomial of the system appears
explicitly in the rightmost column of the A matrix. You can obtain
the companion canonical form of your system by using the canon
canon
(System Identification Toolbox) command in the following
way:
csys = canon(sys,'companion')
For a system with characteristic polynomial
the corresponding companion A matrix is
The companion transformation requires that the system is controllable from the first input. The transformation to companion form is based on the controllability matrix which is almost always numerically singular for mid-range orders. Hence, avoid using it for computation when possible. The companion canonical form is that same as the observable canonical form.
The observable canonical form is the same as the companion canonical form where the
characteristic polynomial of the system appears explicitly in the rightmost column of
the A
matrix. You can obtain the observable canonical form of your
system by using the canon
command in the following
way:
csys = canon(sys,'companion')
For a system with defined by the transfer function
the corresponding matrices are:
The observable canonical form which is the same as the companion form is poorly conditioned for most state-space computation. The transformation of the system to companion form is based on the controllability matrix which is almost always numerically singular for mid-range orders. Hence, avoid using it for computation when possible.
The controllable canonical form of a system is the transpose of its observable
canonical form where the characteristic polynomial of the system appears explicitly in
the last row of the A
matrix.
For a system with defined by the transfer function
the corresponding matrices are:
The relationship between the observable and controllable canonical realizations are as follows:
The controllable canonical form is useful for controller design using pole placement method. However, the transformation of the system to companion form is based on the controllability matrix which is almost always numerically singular for mid-range orders. Hence, avoid using the controllable form for computation when possible.