\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
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\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
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\(\let \delimiter \mathchar \)
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\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
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\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\)
Chapter G Problem Sheet 7 (Lectures 14–15)
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1. Consider the undamped second order scalar differential equation
\[ \ddot {q}+q=0,\qquad y=q. \]
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(a) Write this in the standard form \(\dot {x}=Ax\), \(y=Cx\) with \(x=\sbm {q\\\dot {q}}\).
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(b) Use the Kalman observability matrix to show that this system is observable.
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(c) Use the Hautus observability matrix to show that this system is observable.
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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.
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3. Consider the state-output system \(\dot {x}=Ax\), \(y=Cx\) with
\[ A=\bbm {1&0\\1&1},\qquad C=\bbm {1&0}. \]
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(a) Use the Kalman observability matrix to show that this system is not observable.
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(b) Use the Hautus observability matrix to show that this system is not observable.
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(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 \(t\geq 0\).
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4. Consider the second order scalar differential equation
\[ \ddot {q}+4\dot {q}+3q=0,\qquad y=q. \]
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(a) Write this in the standard form \(\dot {x}=Ax\), \(y=Cx\) with \(x=\sbm {q\\\dot {q}}\).
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(b) Determine the infinite-time observability Gramian \(S\) by solving the observation Lyapunov equation and use this to show that the system is observable.
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(c) Use the infinite-time observability Gramian to determine \(\int _0^\infty |h(t)|^2\,dt\) where \(h\) is the impulse response of
\[ \ddot {q}(t)+4\dot {q}(t)+3q(t)=u(t),\qquad y(t)=q(t). \]