# Difference between revisions of "Hw 5 ex 2"

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− | Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix <math> \tilde{A}=A-BK</math> must have an eigenvalue with positive real part, to which corresponds a certain eigenvector <math>v</math>. | + | Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix <math> \tilde{A}=A-BK</math> must have an eigenvalue with positive real part, to which corresponds a certain eigenvector <math>v</math>. |

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+ | One can rewrite the Algebraic Riccati equation using <math> \tilde{A}</math>, where you should note the changes of signs: | ||

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+ | <math>P\tilde{A} + \tilde{A}^T P + PBQ_u^{-1}B^T P + Qx=0 </math> | ||

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+ | Pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and <math>v^* Q_v v</math> is zero. Which contradicts the initial assumption. | ||

--[[User:Franco|Elisa]] | --[[User:Franco|Elisa]] | ||

[[Category:CDS 110b FAQ - Homework 5]] | [[Category:CDS 110b FAQ - Homework 5]] |

## Latest revision as of 01:28, 14 February 2007

Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix \( \tilde{A}=A-BK\) must have an eigenvalue with positive real part, to which corresponds a certain eigenvector \(v\).

One can rewrite the Algebraic Riccati equation using \( \tilde{A}\), where you should note the changes of signs\[P\tilde{A} + \tilde{A}^T P + PBQ_u^{-1}B^T P + Qx=0 \]

Pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and \(v^* Q_v v\) is zero. Which contradicts the initial assumption.

--Elisa