\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
Due by noon 12.18.
Numbers in brackets indicate the points for each (sub)problem.
Questions? Misprints suspected? Post as a comment here!
- Consider the vector field \(\bv\) given by
\[
\dot{x}=-xy^2-x^5;\\
\dot{y}=-x^2y-y^5.
\]- [3] Find a Lyapunov function and prove thus that the origin is Lyapunov stable. Is it asymptotically stable? Is it exponentially stable?
- [1] Show that one can find an arbitrarily small linear perturbation of the vector field \(\bv\) (that is add arbitrarily small linear terms to \(\bv\) so that the origin becomes an unstable equilibrium.
- [3] What is the maximal degree of perturbation of the vector field that can make the origin unstable?
- Consider the system given by
\[
\begin{array}{ccl}
\dot{x}_1&=&a_1(t)x_2,\\
\dot{x}_2&=&u,\\
\dot{x}_3&=&a_3(t)x_2,
\end{array}
\]
where
\[
a_1=\left\{\begin{array}{ccc}
0 &\mathrm{for}& 0\leq t<1;\\
1 &\mathrm{for}& 1\leq t\leq 2;
\end{array}\right. \mathrm{ and }\quad
a_3=\left\{\begin{array}{ccc}
1 &\mathrm{for}& 0\leq t<1;\\
0 &\mathrm{for}& 1\leq t\leq 2;
\end{array}\right.
\]- [3] Find the controllability Grammian \(W(0,t), 0\leq t\leq 2\).
- [1] Is this system controllable on \([0,1]\)? on \([1,2]\)?
- [2] Is this system controllable on \([0,2]\)?
- [5] Find the rank of the minimal realization of the following transfer function (in this \(101\times 101\)-matrix, the entries on the $k$-th anti-diagonal are \(1/(s+k-1)\)):
\[
P(s)=\left(
\begin{array}{ccccc}
\frac{1}{s}&\frac{1}{s+1}&\frac{1}{s+2}&\cdots&\frac{1}{s+100}\\
\frac{1}{s+1}&\frac{1}{s+2}&\frac{1}{s+3}&\cdots&\frac{1}{s+101}\\
\frac{1}{s+2}&\frac{1}{s+3}&\frac{1}{s+4}&\cdots&\frac{1}{s+102}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
\frac{1}{s+100}&\frac{1}{s+101}&\frac{1}{s+102}&\cdots&\frac{1}{s+200}
\end{array}\right)
\] - Consider the optimal control problem
\[
\dot{x}_1=x_2;\\
\dot{x}_2=x_1+u;\\
|u|\leq 1; L\equiv 1; (x_1(0),x_2(0))=(x_o,v_o).
\]- [4] For \((x_o,v_o)=(1,-1)\), find the trajectory stopping at \((0,0)\) at the shortest time. (This is the shortest time problem, so the terminal time is not fixed, hence the Hamiltonian is \(\equiv 0\).)
- [3]Find for which \((x_o,v_o)\) it is possible at all to stop at \((0,0)\).
- [5] Consider the infinite horizon LQR
\[
\begin{array}{ccl}
\dot{x}_1&=&x_2+b_1u;\\
\dot{x}_2&=&-x_1+b_2u;
\end{array}\\
\int_0^\infty u^2+x_1^2 dt\to\min.
\]
Assume that the control vector \(b\) has norm one, i.e.
\[
b_1^2+b_2^2=1.
\]
Among such vectors, find one providing the speediest convergence of the close-loop feedback LQR regulator, that is minimizing the (negative) maximal real part of \(\dot{x}=(A-bk)x\).
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