CDS 212, Homework 8, Fall 2010 - Revision history

Sojoudi at 23:52, 25 November 2010

2010-11-25T23:52:41Z

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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
	}}		}}

Sojoudi at 14:44, 17 November 2010

2010-11-17T14:44:00Z

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	~~{{CDS 212 draft HW}}~~
	{{CDS homework		{{CDS homework
	\| instructor = J. Doyle		\| instructor = J. Doyle

Sojoudi: /* Problems */

2010-11-12T22:50:25Z

Problems

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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.

Sojoudi: /* Problems */

2010-11-12T22:48:50Z

Problems

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	\| due = 25 Nov 2010		\| due = 25 Nov 2010
	}}		}}

			=== Reading ===
			*[http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Notes 1]
			*[http://www.cds.caltech.edu/~sojoudi/RobustnessDetCond.pdf Notes 2]

	=== Problems ===		=== Problems ===

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #8 | issued = 12 Nov 2010 | due = 25 Nov 2010 }}…'

2010-11-12T22:45:33Z

Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #8 | issued = 12 Nov 2010 | due = 25 Nov 2010 }}…'

New page

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #8
| issued = 12 Nov 2010
| due = 25 Nov 2010
}}

=== Problems ===
<ol>
<li>
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
delta 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.
</li>

<li>
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.
</li>

<li>
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.
</li>

<li>
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).
</li>

</li>