Difference between revisions of "CDS 212, Homework 2, Fall 2010"

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{{CDS 212 draft HW}}
 
 
{{CDS homework
 
{{CDS homework
 
  | instructor = J. Doyle
 
  | instructor = J. Doyle
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=== Reading ===
 
=== Reading ===
* {{DFT}}, Chapters 3
+
* {{DFT}}, Chapter 3
 
*(FBS 9.1-9.3, 11.1-11.2)  
 
*(FBS 9.1-9.3, 11.1-11.2)  
  

Latest revision as of 20:48, 5 October 2010

J. Doyle Issued: 5 Oct 2010
CDS 212, Fall 2010 Due: 14 Oct 2010

Reading

  • DFT, Chapter 3
  • (FBS 9.1-9.3, 11.1-11.2)

Problems

  1. [DFT 2.2, page 28]
    Consider the Venn diagram shown below, which relates the finiteness of norms (as described in DFT).
    Hw2-venn.png

    Show that the functions defined below are contained in the locations shown in the diagram. All functions are zero for <amsmath>t < 0</amsmath>.

    1. <amsmath>\textstyle u_2 =\begin{cases} \frac{1}{t^{1/4}} \quad \text{if} \ t\leq 1 \\ 0 \quad \text{if} \ t> 1 \end{cases} </amsmath>
    2. <amsmath>\textstyle u_4 = 1/(1+t) </amsmath>
    3. <amsmath>\textstyle u_5 = u_2 + u_4 </amsmath>
    4. <amsmath>\textstyle u_9 = \begin{cases} 1 \quad t \in [2^{2k}, 2^{2k+1}],\, k = 0,1,2,\dots \\ 0 \quad \text{elsewhere} \end{cases}</amsmath>
  2. Consider a second order mechanical system with transfer function
    <amsmath>
     \widehat G(s) = \frac{1}{s^2 + 2 \omega_n \zeta s + \omega_n^2}
    
    </amsmath>

    (<amsmath>\omega_n</amsmath> is the natural frequence of the system and <amsmath>\zeta</amsmath> is the damping ratio). Setting <amsmath>\omega_n = 1</amsmath>, write a short MATLAB program to generate a plot of the 2-norm as a function of the damping ratio <amsmath>\zeta > 0</amsmath>.

  3. [DFT 3.1, page 44]
    Show that for a unity feedback system it suffices to check only two transfer functions to determine internal stability.
  4. [DFT 3.2, page 44]
    Let
    <amsmath>
     \widehat P(s) = \frac{1}{10s + 1} \quad
     \widehat C(s) = k \quad
     \widehat F(s) = 1.
    
    </amsmath>

    Find the least positive gain <amsmath>k</amsmath> such that the following are all true:

    1. The feedback system is internally stable
    2. <amsmath>|e(\infty)| \leq 0.1</amsmath> when <amsmath>r(t)</amsmath> is the unit step and <amsmath>n = d = 0</amsmath>.
    3. <amsmath>\|y\|_\infty \leq 0.1</amsmath> for all <amsmath>d(t)</amsmath> such that <amsmath>\|d\|_2 \leq 1</amsmath> when <amsmath>r = n = 0</amsmath>.
  5. [DFT 3.3, page 44]
    Consider a unity gain feedback system with <amsmath>r = n = 0</amsmath> and <amsmath>d(t) = \sin(\omega(t)) 1(t)</amsmath>. Prove that if the feedback system is internally stable then <amsmath>y(t) \to 0</amsmath> as <amsmath>t \to \infty</amsmath> if and only if <amsmath>\widehat P</amsmath> has a zero at <amsmath>s = j \omega</amsmath> or <amsmath>\widehat C</amsmath> has a pole at <amsmath>s = j\omega</amsmath>.
  6. [DFT 3.4, page 44]
    Consider a feedback system with plant <amsmath>\widehat P</amsmath> and sensor <amsmath>\widehat F</amsmath>. Assume that <amsmath>\widehat P</amsmath> is strictly proper and <amsmath>\widehat F</amsmath> is proper. Find conditions on <amsmath>\widehat P</amsmath> and <amsmath>\widehat F</amsmath> for the existence of a proper controller such that
    1. The feedback system is internally stable.
    2. <amsmath>(y(t) - r(t)) \to 0</amsmath> when <amsmath>r</amsmath> is a unit step.
    3. <amsmath>y(t) \to 0</amsmath> when <amsmath>d = A \sin (100 t)</amsmath>.