# CDS 212, Homework 6, Fall 2010

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 J. Doyle Issued: 2 Nov 2010 CDS 212, Fall 2010 Due: 11 Nov 2010

### Reading

KYP paper: (On the Kalman—Yakubovich—Popov lemma, Anders Rantzer, 1996)

### Problems

1. Proof of Lemma 5 in the KYP paper:
1. Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).
2. Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.

(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)

2. Consider the following state space equation:
<amsmath>

\dot{x}=ax+bu, \quad y=cx

</amsmath>

where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:

1. Frequency analysis.
2. KYP Lemma.
3. Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students); each team needs to hand in only a single write-up.