- state equations can be converted to transfer functions. The derivation follows.


- state equation coefficient matrices can be transformed to another equivalent for, if the state vector is rearranged.



• The transfer function form can be put into a matrix form. In this case the denominator is the characteristic equation.


• The free (homogeneous) response of a system can be used to find the state transition matrix.


• The forced response (particular) response of the system can be found using convolution,


• As an example the homogeneous/free response of the system is shown below.


• The forced/particular solution is shown below,


• If a matrix is diagonizable, the diagonal matrix can be found with the following technique. This can be used for more advanced analysis techniques to create diagonal (and separable) system matrices.


• example,


• If there are repeated Eigenvalues in the system the Jordan Form can be used.





• The Eigenvectors can be used to calculate the system response.


• zeros of state space functions can be found using the state matrices.






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