Chapter G Problem Sheet 7 (Lectures 14–15)

  • 1. Consider the undamped second order scalar differential equation

    \[ \ddot {q}+q=0,\qquad y=q. \]

    • (a) Write this in the standard form \(\dot {x}=Ax\), \(y=Cx\) with \(x=\sbm {q\\\dot {q}}\).

    • (b) Use the Kalman observability matrix to show that this system is observable.

    • (c) Use the Hautus observability matrix to show that this system is observable.

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

    \[ A=\bbm {1&0&0\\1&1&1\\0&0&1},\qquad C=\bbm {0&1&0}. \]

    Determine whether or not this system is observable.

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

    \[ A=\bbm {1&0\\1&1},\qquad C=\bbm {1&0}. \]

    • (a) Use the Kalman observability matrix to show that this system is not observable.

    • (b) Use the Hautus observability matrix to show that this system is not observable.

    • (c) Show using only the definition of observability that this system is not observable, i.e. show that there exists a nonzero \(x^0\in \mR ^2\) so that with the initial condition \(x(0)=x^0\) the unique solution satisfies \(y(t)=0\) for all \(t\geq 0\).

  • 4. Consider the second order scalar differential equation

    \[ \ddot {q}+4\dot {q}+3q=0,\qquad y=q. \]

    • (a) Write this in the standard form \(\dot {x}=Ax\), \(y=Cx\) with \(x=\sbm {q\\\dot {q}}\).

    • (b) Determine the infinite-time observability Gramian \(S\) by solving the observation Lyapunov equation and use this to show that the system is observable.

    • (c) Use the infinite-time observability Gramian to determine \(\int _0^\infty |h(t)|^2\,dt\) where \(h\) is the impulse response of

      \[ \ddot {q}(t)+4\dot {q}(t)+3q(t)=u(t),\qquad y(t)=q(t). \]