Question

Gilbert's test for controllability and observability

Answer 1

State space Representation in Normal form.

i) Diagonal Canonical Form (DCF): For non-repeated Eigen value of matrix ‘A’. DCF is used provided A must be a scalar matrix.

Ex:

ii) Jordan Canonical Form (JCF): For repeated Eigen values of matrix ‘A’, JCF is used.

Ex:

Ex:

Controllability and Observability 

By using Gilbert’s test we can directly tell about controllability and observability of a system.

This test can be applied to the systems which are in Normal form (DCF / JCF).

Controllability

– The rows of a ‘B’ matrix that correspond to the distinct Eigen value should not have all zero elements.

– The rows of a ‘B’ matrix that correspond to the last row of Jordan block should not have all zero elements.

Observability

– The columns of a ‘C’ matrix that correspond to the distinct Eigen Values should not have all zero elements.

– The columns of a ‘C’ matrix that corresponds to the first row of a Jordan block should not have all zero elements.

Problem

Comment on Controllability and observability for the given state space representation.

Solution:

The above system is in JCF form, So we can apply gilbert’s test. First check for controllability the last row of Jordan block is [-2] and the corresponding row in B matrix is [0,0].

Which is zero matrix hence the system is not controllable.

Now check for observability.

The first row of Jordan block is [-2] and the corresponding column in C matrix is , which is non–zero matrix.

Hence the system is observable.

The given system is not controllable but observable.