\(\newcommand{\footnotename}{footnote}\)
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\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
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\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
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\(\newcommand {\LWRabsorboption }[1][]{}\)
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\(\def \OE {\unicode {x0152}}\)
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\( \newcommand {\multicolumn }[3]{#3}\)
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\(\)
Chapter F Problem Sheet 6 (Lectures 12–13)
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1. Consider the first order scalar differential equation
\[ \dot {x}+2x=4u. \]
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(a) Write this in the standard form \(\dot {x}=Ax+Bu\).
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(b) Determine the Kalman controllability matrix and use this to show that the system is controllable.
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(c) Determine the Hautus controllability matrix and use this to show that the system is controllable.
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(d) Determine the controllability Gramian \(Q_T\) and use this to show that the system is controllable. Moreover, determine a control which steers the system from a given \(x^0\in \mR \) to a given \(x^1\in \mR \) in a given time \(T>0\).
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(e) Determine the infinite-time controllability Gramian \(Q\) by solving the control Lyapunov equation and use this to show that the system is controllable.
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2. Consider the second order scalar differential equation
\[ \ddot {q}+q=u. \]
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3. For the input-state system \(\dot {x}=Ax+Bu\) with
\[ A=\bbm {5&4\\-3&-2},\qquad B=\bbm {1\\-1}, \]
determine the reachable subspace.
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4. Determine whether or not \(\dot {x}=Ax+Bu\) is controllable when
\[ A=\bbm {0&1&-1\\2&1&0\\-3&5&1},\qquad B=\bbm {1\\0\\1}. \]
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5. Determine whether or not \(\dot {x}=Ax+Bu\) is controllable when
\[ A=\bbm {0&1&2\\0&1&5\\0&0&1},\qquad B=\bbm {1&0\\0&1\\0&1}. \]
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6. Solve the control Lyapunov equation for
\[ A=\bbm {-2&0\\0&-3},\qquad B=\bbm {1\\2}. \]
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7. Show using only the definition of controllability that \(\dot {x}=Ax+Bu\) with
\[ A=\bbm {1&0\\0&1},\qquad B=\bbm {1\\2}, \]
is not controllable, i.e. show that for all \(T>0\) there exist \(x^0,x^1\in \mR ^2\) such that there does not exist a control \(u:[0,T]\to \mR \) we have \(x(0)=x^0\) and \(x(T)=x^1\).