# Jordan canonical form of a system

I'm given a system with the following state equations:

$$\dot{x}= \begin{bmatrix} 1 & 0 & -4\\ 0 & 3 & 0 \\ -2 & 0 & -1 \\ \end{bmatrix} x+ \begin{bmatrix} 1 \\ 0 \\ -2\\ \end{bmatrix} u$$

$$y = \begin{bmatrix} 1 & 1 & 2\end{bmatrix} x$$

a) Transform the system into its Jordan canonical form, then determine which individual modes are controllable and/or observable

b) Find the transfer function of the system and observe which modes appear as system poles.

The biggest problem I am having is transforming the systems in its Jordan canonical form. I can do that for a single matrix but how do I go about doing that for an entire system?

After that, I think I have to find the eigenvalues (which all have to be unique) to determine is all the individual modes are controllable or observable but I'm not 100% sure.

I have no idea how to even begin to do part c).

Hint:

a) Follow the steps explained here. You will have to determine the matrix $P$ first, then you use the substitution

$$x =Pz$$ for the state equation $\dot{x}=Ax+Bu$ to obtain:

$$P\dot{z}=APz+Bu \implies \dot{z}=P^{-1}APz+P^{-1}Bu.$$

The Matrix $P^{-1}AP=J$ is in Jordan normal form.

b) The transfer function is given by determining the Laplace transform of the state equation. Solve it for $x(s)$

$$sx(s)=Ax(s)+Bu(s) \implies x(s)=\left[sI-A \right]^{-1}Bu(s)$$

and substitute it into the measurement equation for $y$ to obtain:

$$y(s) = C\left[sI-A \right]^{-1}Bu(s)$$

in which the transfer function matrix is given by

$$G(s)= C\left[sI-A \right]^{-1}B.$$