# Archive | ece 515

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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. ## 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 …

## 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;

## notes for lectures 10-14

here

Many thanks to James!…

here

## 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.

## 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

## 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
\[

## 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 …

## solutions

for homework 1 and homework 2.

Thanks to Cheng!…