Show that (C, A) is an observable pair if and only if the pair (A^H ·C^H) is controllable.

Step 1: Define observability and controllability. - A pair ((C, A)) is called observable if the

Show that (C, A) is an observable pair if and only if the...

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Key Concepts

Observability

Observability refers to the ability to determine the complete state of a system from its output measurements over time. It is typically characterized by an observability matrix whose full rank indicates that the internal state can be uniquely reconstructed from the outputs.

Controllability

Controllability is the property that allows a system to be driven from any initial state to any desired final state within a finite time interval through an appropriate choice of control inputs. It is analyzed using a controllability matrix, and full rank of this matrix confirms that all states are accessible through the control inputs.

Duality Principle

The duality principle in control theory establishes a deep connection between observability and controllability. Specifically, it shows that the observability of a system described by the pair (C, A) is equivalent to the controllability of the dual system formed by (A^H, C^H). This equivalence uses the transposition (or conjugate transposition in complex spaces) of the matrices and underpins many fundamental results in linear system analysis.

Show that $(C, A)$ is an observable pair if and only if the pair $\left(A^H \cdot C^H\right)$ is controllable.

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