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1. Consider the first order scalar differential equation \(\dot {x}=w+u\), \(z=x\), with exo-system \(\dot {x}_e=0\), \(w=x_e\), \(x_e(0)=d\in \mR \), i.e.
\[ A=0,~~B_1=1,~~B_2=1,~~C_1=1,~~D_{11}=0,~~A_e=0,~~C_e=1. \]
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(a) Show that the Rosenbrock matrix \(\sbm {sI-A&-B_2\\C_1&0}\) is surjective for \(s=0\) (the only eigenvalue of \(A_e\)).
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(b) Show that \((A,B_2)\) is stabilizable.
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(c) What does (a) allow us to conclude about the solvability of the regulator equations and, in combination with (b), about solvability of the full-information disturbance rejection problem?
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(d) Why can we not use the transfer function \(C_1(sI-A)^{-1}B_2\) to draw the conclusions in (c)?
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(e) Find a control \(u\) which solves the full-information disturbance rejection problem.
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2. Consider the undamped second order scalar differential equation
\[ \ddot {q}+q=u. \]
Let \(x:=\sbm {q\\\dot {q}}\). The exo-system is \(\dot {x}_e=A_ex_e\), \(w=C_ex_e\) with
\[ A_e=\bbm {0&1\\-1&0},\qquad C_e=\bbm {1&0}. \]
Define \(z:=q-w\), so that
\[ A=\bbm {0&1\\-1&0},\quad B_1=\bbm {0\\0},\quad B_2=\bbm {0\\1},\quad C_1=\bbm {1&0},\quad D_{11}=-1. \]
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(a) Show that the Rosenbrock matrix \(\sbm {sI-A&-B_2\\C_1&0}\) is surjective for all \(s\in \mC \).
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(b) Show that \((A,B_2)\) is stabilizable.
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(c) What does (a) allow us to conclude about the solvability of the regulator equations and, in combination with (b), about solvability of the full-information disturbance rejection problem?
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(d) Why can we not use the transfer function \(C_1(sI-A)^{-1}B_2\) to draw the conclusions in (c)?
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(e) Find a control \(u\) which solves the full-information output regulation problem.
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3. Consider the second order scalar differential equation \(\ddot {q}+2\dot {q}+q=w+u\), \(z=\dot {q}\) and \(x:=\sbm {q\\\dot {q}}\). Further consider the exo-system \(\dot {x}_e=0\), \(w=x_e\), \(x_e(0)=d\in \mR \). This means that
\[ A=\bbm {0&1\\-1&-2},\quad B_1=\bbm {0\\1},\quad B_2=\bbm {0\\1},\quad C_1=\bbm {0&1},\quad D_{11}=0,\quad A_e=0,\quad C_e=1. \]
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(a) Show that the Rosenbrock matrix \(\sbm {sI-A&-B_2\\C_1&0}\) is not surjective for \(s=0\).
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(b) Show that the regulator equations have a solution.
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(c) Why does the combination of (a) and (b) not contradict Proposition 10.4?
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(d) Find a control \(u\) which solves the full-information output regulation problem.
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4. Consider the first order scalar differential equation
\[ \dot {x}+2x=3w+u,\qquad z=x,\qquad y=x, \]
with exo-system
\[ \dot {x}_e=0,\qquad x_e(0)=d\in \mR , \qquad w=x_e. \]
Solve the measurement feedback disturbance rejection problem.
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5. Consider the first order scalar differential equation
\[ \dot {x}+x=u,\qquad z=x-w, \]
with exo-system
\[ \dot {x}_e=0,\qquad w=x_e,\qquad x_e(0)=r\in \mR , \]
and measurement
\[ y=x. \]
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(a) Show that \(\left (\sbm {A&B_1C_e\\0&A_e},\bbm {C_2&D_{21}C_e}\right )\) is not detectable.
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(b) By considering the closed-loop system
\[ \bbm {\dot {x}\\\dot {x}_c}=\bbm {A&B_2C_c\\B_cC_2&A_c}\bbm {x\\x_c}+\bbm {B_1\\B_cD_{21}}w,\qquad z=\bbm {C_1&0}\bbm {x\\x_c}+D_{11}w, \]
with \(A\), \(B_1\), \(B_2\), \(C_1\), \(D_{11}\), \(C_2\) and \(D_{21}\) from above and arbitrary \(A_c,B_c,C_c\in \mR \), show that the regulation requirement in the measurement feedback output regulation problem cannot be satisfied, i.e.
show that there exist \(x(0)\), \(x_c(0)\) and \(x_e(0)\) such that \(\lim _{t\to \infty }z(t)=0\) does not hold.