\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {mathtools}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
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\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand {\mC }{\mathbb C}\)
\(\newcommand {\mR }{\mathbb R}\)
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\(\newcommand {\Ipd }[2]{\left \langle #1 , #2 \right \rangle }\)
\(\newcommand {\sbm }[1]{\left [\begin {smallmatrix}#1\end {smallmatrix}\right ]}\)
\(\newcommand {\bbm }[1]{\begin {bmatrix}#1\end {bmatrix}}\)
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\(\)
Chapter I Problem Sheet 9 (Lectures 18–19)
-
1. Consider the first order scalar differential equation
\[ \dot {x}=x+u, \]
and the performance output
\[ y=\bbm {x\\u}, \]
i.e.
\[ A=2,\qquad B=1,\qquad C=\bbm {1\\0},\qquad D=\bbm {0\\1}. \]
-
(a) Determine whether or not \(D\) is injective;
-
(b) Determine whether or not \((A,B)\) is stabilizable ;
-
(c) Determine whether or not the Rosenbrock injectivity condition holds;
-
(d) Determine all symmetric positive semidefinite solutions of the algebraic Riccati equation (18.2), the corresponding state feedback matrices \(F\) and the corresponding
closed-loop matrices \(A+BF\).
-
2. Consider the undamped second order scalar differential equation
\[ \ddot {q}(t)+q(t)=u(t), \]
with the state \(x=\sbm {q\\\dot {q}}\) and the performance output
\[ y=\bbm {q\\\varepsilon u}, \]
where \(\varepsilon >0\), i.e.
\[ A=\bbm {0&1\\-1&0},\quad B=\bbm {0\\1},\quad C=\bbm {1&0\\0&0},\qquad D=\bbm {0\\\varepsilon }. \]
-
(a) Determine whether or not \(D\) is injective;
-
(b) Determine whether or not \((A,B)\) is stabilizable ;
-
(c) Determine whether or not the Rosenbrock injectivity condition holds;
-
(d) Determine all symmetric positive semidefinite solutions of the algebraic Riccati equation (18.2) and the corresponding state feedback matrices;
-
(e) Write down the closed-loop system in second order form.