CDS 212, Homework 7, Fall 2010

J. Doyle	Issued: 9 Nov 2010
CDS 212, Fall 2010	Due: 18 Nov 2010

Problems

1. Show that <amsmath>E(s) = D+C(sI-A)^{-1}B</amsmath> has <amsmath>H_\infty</amsmath> norm <amsmath>< \gamma</amsmath> if the following LMI is satisfied: <amsmath>

   \left[\begin{array}{ccccccc} A^TP+PA& PB& C^T\\ B^TP& -\gamma^2 I& D^T\\ C& D& -I\end{array}\right]\leq  0,
   

   </amsmath>

   for some <amsmath>P>0.</amsmath>

2. Formulate the model fitting problem <amsmath> min ||(G-\hat{G})||_{H_\infty}</amsmath> where <amsmath>\hat{G}=\hat{D} + \hat{C} (sI - \hat{A})^{-1}\hat{B}</amsmath> with <amsmath>\hat{A}</amsmath> and <amsmath>\hat{B}</amsmath> given and <amsmath>\hat{C}</amsmath> and <amsmath>\hat{D}</amsmath> to be optimized as an LMI. Write a MATLAB/cvx code for this problem.
3. Consider the system <amsmath>G(s) =\frac{P(s)}{(s+0.1)}</amsmath> where <amsmath>P(s)</amsmath> is a 10th order Pade approximation to a 1 second delay. Calculate the Hankel singular values for this system (using balancmr). Output the truncated balanced truncations of orders 1:10. (note that balancmr can produce a set of output ss systems). and compare the norm of the error with the upper and lower bounds.
4. For <amsmath>G(s)</amsmath> as above calculate the optimal Hankel norm approximations. Note the Hankel singular values of the error system and comment. Note that the better error bound on the <amsmath>H_\infty</amsmath> norm requires a non-zero <amsmath>D</amsmath>-term but the hankelmr function does not output this. By examining the Nyquist plot of the error in an example demonstrate that there exists such a <amsmath>D</amsmath>-term. Note how the poles positions vary with the order of the approximation.
5. Use cvx to examine improvements to the above <amsmath>H_\infty</amsmath> norm errors that can be achieved by optimizing the <amsmath>C</amsmath> and <amsmath>D</amsmath> terms with the <amsmath>A</amsmath> and <amsmath>B</amsmath> terms from the balanced and Hankel-norm approximants.