24.4: Continuous Time Analysis
- Page ID
- 43184
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As with reachability, the discussion of observability for continuous-time systems is algebraically very similar to the discussion for discrete-time systems. We begin with a theorem.
24.4.1: The Observability Gramian
Define
\[\mathcal{Q}_{t}=\int_{0}^{t}\left(e^{A \tau}\right)^{T} C^{T} C e^{A \tau} d \tau\nonumber\]
The system is then observable if and only if rank(Qt) � n� 8 t � 0. If A is stable, then we may de�ne the observability Gramian as
Once again, Q satis�es a Lyapunov equation: