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midterm redo

posted on compass. Reply here if any questions arise.…

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ECE 515, homework 4

\(\def\Real{\mathbb{R}} \)

 

This is a somewhat computation-heavy homework; feel free to use your preferred software.

Consider the linear controlled system
$$
\dot{x}=Ax+Bu, y=Cx, x\in\Real^3, u\in\Real^2, y\in \Real,
$$
with
$$
A=\left(
\begin{array}{ccc}1&1&0\\0&1&0\\1&0&1\\\end{array}
\right);
B=\left(
\begin{array}{cc}1&0\\1&0\\0&1\\\end{array}
\right);
C=(1\quad 0\quad 1).…

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Math 487, midterm 2

\(\def\Comp{\mathbb{C}}\def\Real{\mathbb{R}}\)

Midterm 2.

    1. [15] Let \(u_1, u_2, u_3\in\Real^n\) are three unit vector-columns such that the angle between any two of them is \(120^\circ\). Find the spectrum of the \(n\times n\) matrix
      $$
      A=u_1\cdot u_1^*+u_2\cdot u_2^*+u_3\cdot u_3^*
      $$
      (here \(u*\) is
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Math 487, Nov. 9

  • Elements of Fourier transform.
  • Parseval theorem.
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ECE 515, Nov. 8

  • Overview of optimal control.
  • Basic concepts: cost, value function, principle of optimality, finite and infinite horizon problems.
  • Discrete time optimal control.
    • Example:
      $$
      V(x)=\sum_t \left[f(x_t)+q(x_{t+1}-x_t)\right].
      $$
      Dualities, Hamilton formalism.
  • Value iteration algorithm, computational complexity and the curse of dimensionality.
  • Dynamic
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ECE 515, Nov. 6

  • Tracking.
  • Internal model principle.

 

Broad course overview thus far.…

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Math 487, Nov. 5

Based on DiAngelo’s text, ch. 4.

  • Laurent series.
  • Domains of convergence.
  • Integrals via residues.
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Math 487, Nov. 2

Based on DiAngelo’s text, ch. 4

  • Cauchy formula and corollaries:
  • Liouville’s theorem (complex analytic functions bounded on complex plane are constants);
  • Each polynomial factors over \({\mathbb{C}}\).
  • Expansions into Laurent series in annuli.
    • Same function can have many expansions.
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ECE 515, Nov. 1

Tracking. Based on Trentelman et al, Control theory for linear systems, ch. 9.…

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Math 487, Oct. 31

  • Analytic functions.
  • Cauchy-Riemann equations.
  • Harmonic functions and CR equations.
  • Integrals along a curve.
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