Example 26.1:

The system matrices for a plant (A, B and C) are as follows:

Design an observer for the plant, where the desired characteristic polynomial is given by

Solution:

The system is evidently in non-canonical form. Let us first check the observability of the system by obtaining the observability matrix as

One may check the rank of the above matrix to be of 3 indicating that the system is completely observable.

To obtain the transformation matrix T, we first need to find out the observability matrix corresponding to the canonical form. The characteristic equation corresponding to the open loop system may be obtained by using Eqn. (26.2) as

Accordingly, following Eqn. (29.5) the system matrices in observer canonical form could be written as

Based on the canonical system matrices, one can obtain the observability matrix as

Now, the transformation matrix may be obtained using Eqn. (26.7) as:

Using equation (26.6), the observer gain matrix in the canonical form may be obtained as

Finally, the observer gain matrix L, in the non-canonical form may be obtained by using eqn. (26.8) ;

Congratulations! You have finished Lecture 26.