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RSS feed for this sectionECE 515, homework 4
\(\def\Real{\mathbb{R}} \)
This is a somewhat computationheavy 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).…
Math 487, midterm 2
\(\def\Comp{\mathbb{C}}\def\Real{\mathbb{R}}\)
Midterm 2.

 [15] Let \(u_1, u_2, u_3\in\Real^n\) are three unit vectorcolumns 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
 [15] Let \(u_1, u_2, u_3\in\Real^n\) are three unit vectorcolumns such that the angle between any two of them is \(120^\circ\). Find the spectrum of the \(n\times n\) matrix
Math 487, Nov. 9
 Elements of Fourier transform.
 Parseval theorem.
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.
 Example:
 Value iteration algorithm, computational complexity and the curse of dimensionality.
 Dynamic
ECE 515, Nov. 6
 Tracking.
 Internal model principle.
Broad course overview thus far.…
Math 487, Nov. 5
Based on DiAngelo’s text, ch. 4.
 Laurent series.
 Domains of convergence.
 Integrals via residues.
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.
ECE 515, Nov. 1
Tracking. Based on Trentelman et al, Control theory for linear systems, ch. 9.…
Math 487, Oct. 31
 Analytic functions.
 CauchyRiemann equations.
 Harmonic functions and CR equations.
 Integrals along a curve.