CDS 212, Homework 8, Fall 2010

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J. Doyle Issued: 12 Nov 2010
CDS 212, Fall 2010 Due: 30 Nov 2010

Reading

Problems

  1. Suppose <amsmath>M</amsmath> is a real matrix. Consider 3 cases where there is just one type of uncertainty, either real repeated scalar, complex repeated scalar, or complex full block. The exact answer for the minimum norm <amsmath>\Delta</amsmath> that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using standard linear algebra. Compare this with the LMI upper bound and show that they are equal.
  2. Suppose <amsmath>M</amsmath> is a complex matrix that is rank one, so that <amsmath>M=xy^T</amsmath> where <amsmath>x</amsmath> and <amsmath>y</amsmath> are vectors. Assume there is one block of each type of uncertainty. Again compute the analytic answer and compare with the LMI solution.
  3. Suppose <amsmath>M</amsmath> is a full complex matrix. Use the robust control toolbox to write a short program to set up and compute <amsmath>\mu(M)</amsmath> for the block uncertainty in this handout. Compute upper and lower bounds for some random matrices of moderate size.
  4. Suppose <amsmath>M</amsmath> is a real matrix and there is no real repeated scalar, just the complex repeated scalar and full block. Suppose the complex repeated scalar is treated as if it were a z transform variable for a discrete time system. Compare the LMI conditions for <amsmath>det(I-M\Delta)=0</amsmath> with LMIs that would arise in computing whether the discrete time <amsmath>H_\infty</amsmath> norm is less than 1 (discrete version of KYP).