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(Solved): Problem 6 (Observability/Controllability and internal stability, 25 Points). Consider the LTI syst ...
![b) (5 points) Under what conditions on \( A \) is the system:
\[
\dot{X}(t)=A^{\top} X(t)+X(t) A
\]
asymptotically stable?
Hi](https://media.cheggcdn.com/media/9bd/9bdad7e4-f591-43dc-8e25-539377e3bb7c/phpUPiZMC)
Problem 6 (Observability/Controllability and internal stability, 25 Points). Consider the LTI system: \[ \begin{aligned} \dot{x}(t) &=A x(t)+B u(t) \\ y(t) &=C x(t) \end{aligned} \] We will explore the relation between the Lyapunov equation and the Observability/Controllability Gramian in linear time-invariant systems.
b) (5 points) Under what conditions on \( A \) is the system: \[ \dot{X}(t)=A^{\top} X(t)+X(t) A \] asymptotically stable? Hint: What are the eigenvalues of \( \mathcal{L}(X)=A^{\top} X+X A \) ?
Expert Answer
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the obse