Chapter C Problem Sheet 3 (Lectures 6–7)

  • 1.

    • (a) Determine the frequency response \(F\) of \(\dot {x}=2x+5u\), \(y=x\).

    • (b) Calculate \(|F(\omega )|\) (in such a way that only real numbers are involved).

  • 2.

    • (a) Determine the frequency response \(F\) of \(\ddot {q}+4\dot {q}+3q=6u\), \(y=5q\).

    • (b) Calculate \(|F(\omega )|\) (in such a way that only real numbers are involved).

  • 3.

    • (a) Determine the frequency response \(F\) of \(\ddot {q}+4\dot {q}+3q=6u\), \(y=\bbm {q\\\dot {q}}\).

    • (b) Calculate \(\|F(\omega )\|\) (in such a way that only real numbers are involved); here the norm is the usual Euclidean norm.

  • 4. Consider the frequency response

    \[ F(\omega )=\frac {1}{-\omega ^2+2\zeta i\omega +1}, \]

    of the second order scalar differential equation \(\ddot {q}+2\zeta \dot {q}+q=u\), \(y=q\).

    • (a) Show that

      \[ |F(\omega )|^2=\frac {1}{(1-\omega ^2)^2+4\zeta ^2\omega ^2}. \]

    • (b) Show that when \(\zeta \in [0,\frac {1}{\sqrt {2}})\phantom {]}\), \(|F(\omega )|\) has a maximum at \(\omega _{\rm max}:=\sqrt {1-2\zeta ^2}\) and that when \(\zeta \geq \frac {1}{\sqrt {2}}\), \(|F(\omega )|\) is monotonically decreasing for \(\omega >0\).

      [Hint: it suffices to show these statements for \(|F(\omega )|^2\) rather than for \(|F(\omega )|\) since the squareroot function is strictly increasing. Also note that one can work only with the denominator and subsequently draw conclusions about the fraction.]

  • 5. Consider the second order scalar differential equation

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

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

    • (b) Calculate the impulse response by determining \(\e ^{At}\) and forming \(h(t)=C\e ^{At}B\).

    • (c) Calculate the step response by solving \(\ddot {q}+4\dot {q}+3q=3\), \(q(0)=\dot {q}(0)=0\), \(y=q\) and then differentiate this to obtain the impulse response.

    • (d) Determine the transfer function and find the inverse Laplace transform of this to determine the impulse response.

    • (e) Use the initial value problem from Remark 7.2 to determine the impulse response.

  • 6. For a system with impulse response \(\e ^{-t}\), determine the output for initial condition zero and input \(\e ^{-2t}\).