Question

A system is represented in state-space is \(\dot X = Ax + Bu,where\;{\rm{ }}A = \left[ {\begin{array}{*{20}{c}} 1&2\\ \alpha &6 \end{array}} \right]and{\rm{ }}\;B = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right]\). The value of α for which the system is not controllable is _______.

Answer 1

\(A = \left[ {\begin{array}{*{20}{c}} 1&2\\ \alpha &6 \end{array}} \right]B = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right]\)

For controllability, |Q_c| ≠ 0<!--[endif]--><!--[endif]-->

In the questions given that, the system is not controllable.

⇒ |Q_c| = 0

\({\left[ {{Q_c}} \right]_{\min }} = \left[ {\begin{array}{*{20}{c}} B&{AB}&{{A^2}B}&{ - }&{{A^{n-1}}B} \end{array}} \right]\)

\(\begin{array}{l} {\left[ {{Q_c}} \right]_{2 \times 2}} = \left[ {\begin{array}{*{20}{c}} 1&3\\ 1&{\alpha + 6} \end{array}} \right]\\ AB = \left[ {\begin{array}{*{20}{c}} 1&2\\ \alpha &6 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3\\ {\alpha + 6} \end{array}} \right] \end{array}\)

\(\left| {{Q_c}} \right| = \left| {\begin{array}{*{20}{c}} 1&3\\ 1&{\alpha + 6} \end{array}} \right| = 0\)<!--[endif]--><!--[endif]-->

⇒ α + 6 – 3 = 0 ⇒ α = -3