Let us discuss the design of an observer when a system is in observer canonical form defined by the system matrices , and . In observer canonical form, these matrices are represented in the following way:

(26.5)

Whenever, the desired eigen-values related to the error-dynamics are specified, one can construct the desired characteristic equation identical to eqn. (26.7). The observer gain matrix for such cases may be obtained from the simple relationship

(26.6)

Now, if a system is not in observer canonical form, then one needs to transform the system matrices first into canonical form. The transformation matrix required for such cases has been derived as

(26.7)

where, and are the observability matrices related to the non-canonical and canonical forms respectively. After obtaining the observer gains in observer canonical form using eqn. (26.6), one can transform the gain vector to the original non-canonical form as

(26.8)

Let us illustrate the process of estimator design with the help of an example.