Controllability Gramian

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Controllability Gramian

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For a LTVC system with transition matrix $\Phi_{A}$ we define the controllability Gramian $W(t_{0},t_{1})$ as $W(t_{0},t_{1})=\int\limits _{t_{0}}^{t_{1}}\Phi_{A}(t_{0},\tau)B(\tau)B^{\top}(\tau)\Phi^{\top}_{A}(t_{0},\tau) \, d\tau$

Let $W$ be the controllability gramian for some LTVC system. 1. $W(t_{0},t_{1})$ is symmetric and positive semidefinite $\forall t_{1}>t_{0}$ 2. $W$ satisfies $\begin{cases} \frac{d}{dt}W(t,t_{1})=A(t)W(t,t_{1})+W(t,t_{1})A^{T}(t)-B(t)B^{T}(t)\\ \\ W(t_{1},t_{1})=0 \end{cases}$ 3. $W$ satisfies $W(t_{0},t_{1})=W(t_{0},t)+\Phi(t_{0},t)W(t,t_{1})\Phi^{T}(t_{0},t)$ for $t\in J$ .

Consider an LTIC system with Controllability Gramian $W(t_{0},t_{1})$ . Then. we have that $\text{Ker}(W(t_{0},t_{1}))=\text{Ker}(W_{T})$ and $\text{Image}(W(t_{0},t_{1}))=\text{Image}(W_{T})$ where $W_{T}=[B,AB,A^{2}B, \dots,A^{n-1}B][B,AB,A^{2}B, \dots,A^{n-1}B]^{T}$ In particular, $\text{Image}(W(t_{0},t_{1}))$ and $\text{Ker}(W(t_{0},t_{1}))$ are independent of $t_{0}$ and $t_{1}$ .

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Rank

Fixed endpoint problem

Controllability Gramian

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Linear Time Varying Control System