\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
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 \(\dot{x}=(A-bk)x\).
On question 2(a),
When you say from (0,t), do you mean from (0,2) or do you mean to some undecided t?
If this is for some undecided t, can we assume a1(t > 2) = 1 and a3(t > 2) = 0?
Only for t between 0 and 2.
For the question 5, is it okay to just get a numerical value of b1 and b2? It seems a little bit complex to calculate in analytical way.
Sure.
I don’t know if it is ok to ask but when you say a linear perturbation to the vector field v, do you mean: \dot(v) = f(v) + \epsilon * v, where epsilon is some small positive number ?
Arbitrarily small (in absolute value) number.
For question 5, I’m not quite sure what you mean when you say ‘ minimizing the (negative) maximal real part of x_dot = (A-bk)x’
Do you mean, the largest negative value and take it to zero?
or take the largest negative value and take it further away from zero?
Further away from the origin (as stable as possible).
In Problem 3, you asked for “the rank of the minimal realization”. Does it mean “the degree of the minimal realization” or “the rank of matrix A of the minimal realization” ?
Latter (i.e. dimension of the state space of the minimal realization).
Q4. (b) Do want to reach the origin only using the optimal control or do you want to just get there by ANY control (irrespective of whether or not it is optimal, as long as we get there) ?
Reach the origin in minimal time – that is optimally for L=1.
Does L mean the Lagrangian of the system?
Yes.
Is it possible for there be a hint for Q4a? I can’t seem to solve it using the method from Page 222-224 of the class notes… Thank you!
Form long Hamiltonian, then the short one (maximizing in the control). Write down the differential equations for the covectors. Figure out how many switches can there be.
Will solutions be posted?
Yeah, hopefully next week.