CDS 212, Homework 8, Fall 2010: Difference between revisions
From Murray Wiki
Jump to navigationJump to search
No edit summary |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{CDS homework | {{CDS homework | ||
| instructor = J. Doyle | | instructor = J. Doyle | ||
| Line 6: | Line 5: | ||
| title = Problem Set #8 | | title = Problem Set #8 | ||
| issued = 12 Nov 2010 | | issued = 12 Nov 2010 | ||
| due = | | due = 30 Nov 2010 | ||
}} | }} | ||
| Line 19: | Line 18: | ||
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 | ||
<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
- 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.
- 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.
- 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.
- 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).
