Definition (Observability Gramian)
Consider the system setup in the Observability Problem with Transition Matrix Φ A \Phi_{A} Φ A Observability Gramian as: M ( t 0 , t 1 ) = ∫ t 0 t 1 Φ A ⊤ ( τ , t 0 ) C ⊤ ( τ ) C ( τ ) Φ A ( τ , t 0 ) d τ M(t_{0},t_{1})=\int\limits _{t_{0}}^{t_{1}}\Phi_{A}^{\top}(\tau,t_{0})C^{\top}(\tau)C(\tau)\Phi_{A}(\tau,t_{0}) \, d\tau M ( t 0 , t 1 ) = t 0 ∫ t 1 Φ A ⊤ ( τ , t 0 ) C ⊤ ( τ ) C ( τ ) Φ A ( τ , t 0 ) d τ
Lemma (Kernel Equivalence Lemma)
Consider the Observability Problem . We then have that Ker ( L ^ ) = Ker ( M ( t 0 , t 1 ) ) \text{Ker}(\hat{L})=\text{Ker}(M(t_{0},t_{1})) Ker ( L ^ ) = Ker ( M ( t 0 , t 1 )) M ( t 0 , t 1 ) = ∫ t 0 t 1 Φ ⊤ ( τ , t 0 ) C ⊤ ( τ ) C ( τ ) Φ ( τ , t 0 ) d τ M(t_{0},t_{1})=\int\limits _{t_{0}}^{t_{1}}\Phi^{\top}(\tau,t_{0})C^{\top}(\tau)C(\tau)\Phi(\tau,t_{0}) \, d\tau M ( t 0 , t 1 ) = t 0 ∫ t 1 Φ ⊤ ( τ , t 0 ) C ⊤ ( τ ) C ( τ ) Φ ( τ , t 0 ) d τ Observability Gramian .
Proposition (Properties of the Observability Gramian)
Let M ( t 0 , t 1 ) M(t_{0},t_{1}) M ( t 0 , t 1 ) Observability Gramian for an LTVC System on the pair ( t 0 , t 1 ) (t_{0},t_{1}) ( t 0 , t 1 )
M ( t 0 , t 1 ) M(t_{0},t_{1}) M ( t 0 , t 1 ) Positive Semidefinite for t 1 > t 0 t_{1}>t_{0} t 1 > t 0 M M M { d d t M ( t , t 1 ) = A ( t ) M ( t , t 1 ) + M ( t , t 1 ) A ⊤ ( t ) − C ( t ) C ⊤ ( t ) M ( t 1 , t 1 ) = 0 \begin{cases}
\frac{d}{dt}M(t,t_{1})=A(t)M(t,t_{1})+M(t,t_{1})A^{\top}(t)-C(t)C^{\top}(t) \\ \\
M(t_{1},t_{1})=0
\end{cases} ⎩ ⎨ ⎧ d t d M ( t , t 1 ) = A ( t ) M ( t , t 1 ) + M ( t , t 1 ) A ⊤ ( t ) − C ( t ) C ⊤ ( t ) M ( t 1 , t 1 ) = 0 M M M M ( t 0 , t 1 ) = M ( t 0 , t ) + Φ A ⊤ ( t , t 0 ) M ( t , t 1 ) Φ A ( t , t 0 ) M(t_{0},t_{1})=M(t_{0},t)+\Phi_{A}^{\top}(t,t_{0})M(t,t_{1})\Phi_{A}(t,t_{0}) M ( t 0 , t 1 ) = M ( t 0 , t ) + Φ A ⊤ ( t , t 0 ) M ( t , t 1 ) Φ A ( t , t 0 )