MA30061 Control Theory

1. Consider the state-output system \(\dot {x}=Ax\), \(y=Cx\) with

\[ A=\bbm {1&0\\0&-1}\qquad C=\bbm {1&0}. \]
- (a) Use the Hautus observability matrix to show that this system is detectable and not observable.
- (b) Show using only the definitions of detectability and observability that this system is detectable and not observable.
2. Consider
\(\seteqnumber{0}{H.}{0}\)
\begin{align*} \ddot {q}_1+2\dot {q}_1+q_1&=0,\\ \ddot {q}_2+q_2&=u. \end{align*} The equations can be written in first order form \(\dot {x}=Ax+Bu\) with

\[ x=\bbm {q_1\\\dot {q}_1\\q_2\\\dot {q}_2},\qquad A=\bbm {0&1&0&0\\-1&-2\zeta &0&0\\ 0&0&0&1\\0&0&-1&0},\qquad B=\bbm {0\\0\\0\\1}. \]
- (a) Use the Hautus controllability matrix to show that the system is stabilizable and not controllable.
- (b) Show using only the definitions of stabilizability and controllability that this system is stabilizable and not controllable.
3. Consider the input-state-output system \(\dot {x}=Ax+Bu\), \(y=Cx\) where

\[ A=\bbm {-1&0\\1&3},\qquad B=\bbm {0\\1},\qquad C=\bbm {2&1}. \]

Show that an observer-based stabilizing controller exists without explicitly calculating such a controller.