CDS 212, Homework 3, Fall 2010

J. Doyle	Issued: 12 Oct 2010
CDS 212, Fall 2010	Due: 21 Oct 2010

Reading

• DFT, Chapter 4

Problems

1. [DFT 4.1, page 62]
   Consider a unity feedback system. True or false: If a controller internally stabilizes two plants, they have the same number of poles in Re <amsmath> s \geq 0</amsmath>.
2. [DFT 4.2, page 62]
   Unity-feedback problem: Let <amsmath>P_{\alpha}(s)</amsmath> be a plant depending on a real parameter <amsmath>\alpha</amsmath>. Suppose that the poles of <amsmath>P_{\alpha}</amsmath> move continuously as <amsmath>\alpha</amsmath> varies over the interval <amsmath>[0, \ 1]</amsmath>. True or false: If a controller internally stabilizes <amsmath>P_{\alpha}</amsmath> for every <amsmath>\alpha</amsmath> in <amsmath>[0, \ 1]</amsmath>, then <amsmath>P_{\alpha}</amsmath> has the same number of poles in Re <amsmath>s \geq 0</amsmath> for every <amsmath>\alpha</amsmath> in <amsmath>[0, \ 1]</amsmath>.
3. [DFT 4.6, page 63]
   Consider the unity feedback system with <amsmath>C(s) = 10 </amsmath> and plant <amsmath>

    P(s) = \frac{1}{s-a},
   

   </amsmath>

   where <amsmath>a</amsmath> is real.

   1. Find the range of <amsmath>a</amsmath> for the system to be internally stable.
   2. For <amsmath>a = 0</amsmath> the plant is <amsmath>P(s) = 1/s</amsmath>. Regarding <amsmath>a</amsmath> as a perturbations, we can write the plant as <amsmath>

      \widetilde P = \frac{P}{1 + \Delta W_2 P}

      </amsmath> with <amsmath>W_2(s) = -a</amsmath>. Then <amsmath>\widetilde P</amsmath> equals the true plant when <amsmath>\Delta(s) = 1</amsmath>. Apply robust stability theory to see when the feedback system <amsmath>\widetilde{P}</amsmath> is internally stable for all <amsmath>\| \Delta\|_\infty \leq 1.</amsmath> Compare this to your result for part (a).
4. [DFT 4.10, page 64]
   Suppose that the plant transfer function is <amsmath>

   \tilde{P}(s) = [1 + \Delta(s)W_2(s)]P(s),

   </amsmath>

   where

   <amsmath>

   W_2(s) = \frac{2}{s + 10}, \ P(s) = \frac{1}{s-1},

   </amsmath>

   and the stable perturbation <amsmath>\Delta</amsmath> satisfies <amsmath>\| \Delta \|_{\infty} \leq 2</amsmath>. Suppose that the controller is the pure gain <amsmath>C(s) = k</amsmath>. We want the feedback system to be internally stable for all such perturbations. Determine over what range of <amsmath>k</amsmath> this is true.

5. Consider the feedback system in the figure below. Uncertainty in the plant is described using two stable weighting functions <amsmath>W_1</amsmath> and <amsmath>W_2</amsmath>, and stable <amsmath>\Delta_1</amsmath> and <amsmath>\Delta_2</amsmath> satisfying <amsmath>\| \Delta_1 \|_{\infty} \leq 1</amsmath> and <amsmath>\| \Delta_2 \|_{\infty} \leq 1</amsmath>. We will assume all plants in the uncertainty set have the same unstable poles.
   
   1. Write down an expression for <amsmath>\tilde{P}</amsmath> (dotted box). If <amsmath>L = PC</amsmath> and <amsmath>\tilde{L} = \tilde{P}C</amsmath>, write an expression for <amsmath>1 + \tilde{L}</amsmath> with <amsmath>1 + L</amsmath> as a factor.
   2. State and prove a necessary and sufficient condition for robust stability of the closed loop system as a function of <amsmath>P</amsmath>, <amsmath>C</amsmath> and the sensitivity functions.
   3. Apply the condition to the case <amsmath>P=1/(s-1)</amsmath>, <amsmath>C=0.5</amsmath> and <amsmath>W_1 = W_2 = 0.1</amsmath>.