CDS 212, Homework 4, Fall 2010

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 J. Doyle Issued: 19 Oct 2010 CDS 212, Fall 2010 Due: 28 Oct 2010

Problems

1. [DFT 6.1]
Show that any stable transfer function can be uniquely factored as the product of an all pass function and a minimum phase function, up to a choice of sign.
2. [DFT 6.4]
Let
<amsmath>

P(s) = 4\frac{s-2}{(s+1)^2}

</amsmath>

and suppose that <amsmath>C</amsmath> is an internally stabilizing controller such that <amsmath>\| S \|_\infty = 1.5.</amsmath> Give a positive lower bound for

<amsmath>
 \max_{0 \leq \omega \leq 0.1} |S(j\omega)|.

</amsmath>
3. [DFT 6.5]
Define <amsmath>\epsilon = \|W_1 S\|_\infty</amsmath> and <amsmath>\delta = \| C S \|_\infty</amsmath>, so that <amsmath>\epsilon</amsmath> is a measure of tracking performance and <amsmath>\delta</amsmath> measures control effort. Show that for every point <amsmath>s_0</amsmath> with Re <amsmath>s_0 \geq 0</amsmath>,
<amsmath>
 |W_1(s_0)| \leq \epsilon + |W_1 (s_0) P(s_0)|\, \delta.

</amsmath>

Hence <amsmath>\epsilon</amsmath> and <amsmath>\delta</amsmath> cannot both be very small and so we cannot get good tracking without exerting some control effort.

4. [DFT 6.6]
Let <amsmath>\omega</amsmath> be a frequency such that <amsmath>j \omega</amsmath> is not a pole of <amsmath>P</amsmath>. Suppose that
<amsmath>
 \epsilon := |S(j\omega)| < 1.

</amsmath>

Derive a lower bound for <amsmath>C(j\omega)</amsmath> that blows up as <amsmath>\epsilon \to 0</amsmath>. Hence good tracking at a particular frequency requires large controller gain at this frequency.

5. [DFT 6.7]
Consider a plant with transfer function
<amsmath>
 P(s) = \frac{1}{s^2 - s + 4}

</amsmath>

and suppose we want to design an internally stabilizing controller such that

1. <amsmath>|S(j\omega| \leq \epsilon</amsmath> for <amsmath>0 \leq \omega \leq 0.1</amsmath>
2. <amsmath>|S(j\omega| \leq 2</amsmath> for <amsmath>0.1 \leq \omega \leq 5</amsmath>
3. <amsmath>|S(j\omega| \leq 1</amsmath> for <amsmath>5 \leq \omega \leq \infty</amsmath>

Find a (positive) lower bound on the achievable <amsmath>\epsilon</amsmath>.