Electrical – Reducing a state space equation to a controllable and observable equation

It would be \$ \left( \begin{array}{cccc} 0 & 0 & 1 & 1 \\ \end{array} \right)^T\$.

The reasoning is as follows. From the output equation, we know that

$$ X_1(s)=\frac{1}{(s+1)^3}U(s) \ \ \ X_2(s)=\frac{1}{(s+1)^2}U(s) \ \ \ X_3(s)=\frac{1}{s+1}U(s) \ \ \ X_4(s)=\frac{1}{s}U(s)$$

This can be simplified as: $$ X_1(s)=\frac{1}{s+1}X_2(s) \ \ \ X_2(s)=\frac{1}{s+1}X_3(s) \ \ \ X_3(s)=\frac{1}{s+1}U(s) \ \ \ X_4(s)=\frac{1}{s}U(s)$$

Now we are beginning to see that the input only appears in the last two equations.

Converting these to the time domain, using the inverse Laplace transform, we get the following equations:

$$ \dot{x}_1+x_1=x_2$$ $$ \dot{x}_2+x_2=x_3$$ $$ \dot{x}_3+x_3=u$$ $$ \dot{x}_4=u$$

which is the desired Jordan form.

The two state variables \$x_1\$ (the inductor current) and \$x_2\$ (the capacitor voltage) do not change directions between the on- and off-states. You can see that during the off-state, as correctly drawn in your lecture notes, the inductor current still splits between \$C\$ and \$R\$. If I may further comment, I would recommend to use the PWM switch model rather than the state-space averaging (SSA) technique for analyzing switching converters. See here for more details: slides 46/47 for the buck converter SSA derivation and the rest of the pages for the PWM switch model analysis.