Integral Effect

If we have a disturbance, $w$, we can add integral effect to our system. We do this by adding a new state, ${\dot{x}}_a={\int }_0^{t}r(\tau )-Cx(\tau )d\tau$ Giving us the augmented system

$$\left[\begin{array}{c}\dot{x}\\ {\dot{x}}_a\end{array}\right]=\left[\begin{array}{cc}A& 0\\ -C& 0\end{array}\right]\left[\begin{array}{c}x\\ x_a\end{array}\right]+\left[\begin{array}{c}B\\ 0\end{array}\right]u+\left[\begin{array}{c}B\\ 0\end{array}\right]w+\left[\begin{array}{c}0\\ \mathbb{I}\end{array}\right]r$$

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With Reference Feed-Forward

$$-\left[\begin{array}{cc}K& -K_a\end{array}\right]\left[\begin{array}{c}x\\ x_a\end{array}\right]+K_{r}r$$

We can achieve the following closed loop system

$$\left[\begin{array}{c}\dot{x}\\ {\dot{x}}_a\end{array}\right]+\left[\begin{array}{cc}A-BK& B{K}_a\\ -C& 0\end{array}\right]\left[\begin{array}{c}x\\ x_a\end{array}\right]+\left[\begin{array}{c}B{K}_r\\ \mathbb{I}\end{array}\right]r+\left[\begin{array}{c}B\\ 0\end{array}\right]w$$