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final

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)
Due by noon 12.18.

Numbers in brackets indicate the points for each (sub)problem.

Questions? Misprints suspected? Post as a comment here!

  1. Consider the vector field \(\bv\) given by
    \[
    \dot{x}=-xy^2-x^5;\\
    \dot{y}=-x^2y-y^5.
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parabola construction

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)

Consider the optimal control problem
\[
\dot{x}=u; x(0)=x_o; \int_0^T u^2/2dt+M(x(T))\to\min.
\]
The task is to find the cost function \(S(x_o)\) (the optimal achievable value).

It can be solved as follows. Assume for …

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ece515, homework 5

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of december 4.

Use MATLAB or octave if needed.

  1. Consider the optimal control problem
    \[{x}”’=u; \int_0^\infty u^2+x^2 dt\to\min.
    \]
    • Write this system as a LQR;
    • Solve the ARE;
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notes for lectures 10-14

here

Many thanks to James!…

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solutions to midterm

here

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ece515, homework 4

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of november 8.

  1. Consider the system \(\dot{x}=Ax+bu\),
    \[A=\left(
    \begin{array}{ccc}
    1&1&0\\
    1&1&1\\
    0&1&1\\
    \end{array}\right);
    b=\left(
    \begin{array}{c}
    b_1\\
    b_2\\
    b_3\\
    \end{array}\right).
    \]
    • Derive conditions on \(b\) making the system controllable.
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ece515, midterm

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Midterm.
Open book. Points for each (sub)problem shown. Full credit given for 25 points.

  1. Consider the smallest linear space \(V\) of functions on the real line containing the functions
    \(te^t, te^{-t}\) and
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ece515, homework 3

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by noon of october 14.

  1. Consider the system
    \[
    \begin{array}{ccl}
    \dot{x}_1&=&x_2\\
    \dot{x}_2&=&-ax_2-a^2x_1-x_1^3\\
    \end{array}
    \]
    For which values of \(a\) the system is
    unstable? Lyapunov stable? (Locally) asymptotically stable?
  2. Let
    \[
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october 2

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

Quadratic forms as Lyapunov functions

If the operator defines an asymptotically stable system, it is Hurwitz. For a Hurwitz operator, a quadratic form exists which is a (strict) Lyapunov function. A strict Lyapunov function implies …

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solutions

for homework 1 and homework 2.

Thanks to Cheng!…

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