Riasat Morshed Khan

Riasat Morshed Khan

Water Power Technology | Mechanical Engineering PhD Student

06.4 Augmented System - 2 with Control Command

Once we complete 06.3 Observer (State Estimator) and Augmented System, we would incorporate the control law into the augmented system.

This is without control law

\(\begin{bmatrix}\dot{x} \\ \dot{\hat{x}}\end{bmatrix} = \begin{bmatrix} A & 0 \\ KC & \hat{A}-K\hat{C}\end{bmatrix} \begin{bmatrix} x \\ \hat{x} \end{bmatrix}+ \begin{bmatrix}B \\ \hat{B}+K(D-\hat{D}) \end{bmatrix} u\)

\(\begin{bmatrix} y \\ \hat{y} \end{bmatrix} = \begin{bmatrix} C & 0 \\ 0 & \hat{C} \end{bmatrix} \begin{bmatrix}x \\ \hat{x} \end{bmatrix} + \begin{bmatrix} D \\ \hat{D} \end{bmatrix} u\)

If we add control law $u=-Gx$ to the augmented system above we will end up with the following

\[\begin{bmatrix}\dot{x} \\ \dot{\hat{x}}\end{bmatrix} = \begin{bmatrix} A & -BG \\ KC & \hat{A}-K\hat{C}-\hat{B}G-K(D-\hat{D})G\end{bmatrix} \begin{bmatrix} x \\ \hat{x} \end{bmatrix}\] \[\begin{bmatrix} y \\ \hat{y} \end{bmatrix} = \begin{bmatrix} C & -DG \\ 0 & \hat{C}-\hat{D}G \end{bmatrix} \begin{bmatrix}x \\ \hat{x} \end{bmatrix}\]

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