Control System Toolbox™

canon - State-space canonical realization

Syntax

csys = canon(sys,'modal') csys = canon(sys,'modal',CONDT) csys = canon(sys,'companion')

Description

canon computes a canonical state-space model for the continuous or discrete LTI system sys. Two types of canonical forms are supported.

Modal Form

csys = canon(sys,'modal') returns a realization csys in modal form. If A has no repeated eigenvalues, the real eigenvalues appear on the diagonal of the A matrix and the complex conjugate eigenvalues appear in 2-by-2 blocks on the diagonal of A. For a system with eigenvalues , the modal A matrix is of the form

csys = canon(sys,'modal',CONDT) specifies an upper bound CONDT on the condition number of the block-diagonalizing transformation T. The default value is CONDT=1e8. Increase CONDT to reduce the size of the eigenvalue clusters (setting CONDT=Inf amounts to diagonalizing A).

Companion Form

csys = canon(sys,'companion') produces a companion realization of sys where the characteristic polynomial of the system appears explicitly in the rightmost column of the A matrix. For a system with characteristic polynomial

the corresponding companion matrix is

For state-space models sys,

[csys,T] = canon(sys,'type')

also returns the state coordinate transformation T relating the original state vector and the canonical state vector , where

This syntax is meaningful only when sys is a state-space model.

Algorithm

Transfer functions or zero-pole-gain models are first converted to state space using ss.

The transformation to modal form uses the matrix P of eigenvectors of the A matrix. The modal form is then obtained as

The state transformation returned is the inverse of .

The reduction to companion form uses a state similarity transformation based on the controllability matrix [1].

Limitations

The companion transformation requires that the system be controllable from the first input. The companion form is often poorly conditioned for most state-space computations; avoid using it when possible.

References

[1] Kailath, T. Linear Systems, Prentice-Hall, 1980.