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Controllability Gramian

Definition (Controllability Gramian)

For a LTVC system with transition matrix ΦA\Phi_{A} we define the controllability Gramian W(t0,t1)W(t_{0},t_{1}) as W(t0,t1)=∫t0t1ΦA(t0,τ)B(τ)B⊤(τ)ΦA⊤(t0,τ) dτW(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

Proposition (Properties of controllability Gramian)

Let WW be the controllability gramian for some LTVC system.

  1. W(t0,t1)W(t_{0},t_{1}) is symmetric and positive semidefinite ∀t1>t0\forall t_{1}>t_{0}
  2. WW satisfies {ddtW(t,t1)=A(t)W(t,t1)+W(t,t1)AT(t)−B(t)BT(t)W(t1,t1)=0\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. WW satisfies W(t0,t1)=W(t0,t)+Φ(t0,t)W(t,t1)ΦT(t0,t)W(t_{0},t_{1})=W(t_{0},t)+\Phi(t_{0},t)W(t,t_{1})\Phi^{T}(t_{0},t)for t∈Jt\in J.

Theorem (Stationary Image & Kernel of Controllability Gramian)

Consider an LTIC system with Controllability Gramian W(t0,t1)W(t_{0},t_{1}). Then. we have that Ker(W(t0,t1))=Ker(WT)\text{Ker}(W(t_{0},t_{1}))=\text{Ker}(W_{T})andImage(W(t0,t1))=Image(WT)\text{Image}(W(t_{0},t_{1}))=\text{Image}(W_{T})where WT=[B,AB,A2B,…,An−1B][B,AB,A2B,…,An−1B]TW_{T}=[B,AB,A^{2}B, \dots,A^{n-1}B][B,AB,A^{2}B, \dots,A^{n-1}B]^{T}In particular, Image(W(t0,t1))\text{Image}(W(t_{0},t_{1})) and Ker(W(t0,t1))\text{Ker}(W(t_{0},t_{1})) are independent of t0t_{0} and t1t_{1}.

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