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 \( \tilde{A}=A-BK\) 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.