Difference between revisions of "CDS 212, Homework 8, Fall 2010"

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(Problems)
 
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{{CDS 212 draft HW}}
 
 
{{CDS homework
 
{{CDS homework
 
  | instructor = J. Doyle
 
  | instructor = J. Doyle
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  | title = Problem Set #8
 
  | title = Problem Set #8
 
  | issued = 12 Nov 2010
 
  | issued = 12 Nov 2010
  | due = 25 Nov 2010
+
  | due = 30 Nov 2010
 
}}
 
}}
  
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type of uncertainty, either real repeated scalar, complex repeated
 
type of uncertainty, either real repeated scalar, complex repeated
 
scalar, or complex full block.  The exact answer for the minimum norm
 
scalar, or complex full block.  The exact answer for the minimum norm
delta that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using
+
<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
 
standard linear algebra.  Compare this with the LMI upper bound and show
 
that they are equal.
 
that they are equal.

Latest revision as of 23:52, 25 November 2010

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).