When I do my research, I found that there are two definitions of uniformly observable, I can't help thinking are they equivalent?

These two definitions are listed as follows.

For a linear stochastic system with the description $$x_{k+1} = F_{k}x_{k}+w_{k}$$ and $$z_k = H_k x_k+v_k$$ where $F_{k}$ and $H_k$ are system matrix and measurement matrix, respectively, $x_k$ is the system state, $z_k$ is the output, while $w_k$ and $v_k$ are Gaussian while noise sequences, $R_k$ is the covariance of $v_k$.

The first definition of uniformly observable is coming form

J. C. Spall and K. D. Wall, “Asymptotic distribution theory for the Kalman filter state estimator,” Communications in Statistics - Theory and Methods, vol. 13, no. 16, pp. 1981–2003, Jan. 1984.

it says

The system is said to be uniformly completely observable if there exists an integer $1 \leq m< \infty$ and constants $0<\beta_1 \leq \beta_2 <\infty$ such that \begin{align} \beta_1 I \leq \mathcal{O}(l,k) \triangleq \sum_{l=k-m}^{k} \phi^T(l,k) H_l^T R_l^{-1} H_l \phi(l,k) \leq \beta_2 I \end{align} for all $k\geq m$, where $\phi(k,k) =I$ and \begin{align} \phi(l,k) = (F_l)^{-1} (F_{l+1})^{-1} \cdots (F_{k-1})^{-1}. \end{align}

The second one is found in

K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 714–728, Apr. 1999.

it says

Consider time-varying matrices $F_k$, $H_k$, $k \geq 0$, and let the observability gramian be given by \begin{align} M_{k+m,k}=\sum_{l=k}^{k+m} \phi^T(l,k) H_l^T H_l \phi(l,k) \end{align} for some integer $m\geq 0$ with $\phi(k,k)=I$ for \begin{align} \phi(l,k) = (F_{l-1}) (F_{l-2}) \cdots (F_{k}), \end{align} for $i>n$. The matrices $F_k$, $H_k$, $k \geq 0$ are said to satisfy the uniform observability condition, if there are real numbers $\alpha_1$, $\alpha_2$ and an integer $m>0$, such that the following inequality holds: \begin{align} \alpha_1 I \leq M_{k+m,k} \leq \alpha_2 I. \end{align}

My question is that are those two definition equivalent?

  • 1
    $\begingroup$ You have not defined $R_l$. If $c_1 I<R_l<c_2I $ with $c_2>c_1>0$ for all $l$ then they are obviously equivalent. $\endgroup$ – RTJ Dec 4 '15 at 15:10
  • $\begingroup$ @CTNT I have defined $R_k$ and it's bounded. But I cannot see they are equivalent, can you write an answer to me? $\endgroup$ – wayne Dec 4 '15 at 15:59
  • 3
    $\begingroup$ I did not have the time to write the complete solution but you can prove $$\min_{k-m\leq l\leq k}\{\lambda_{min}(R_l)\}\sigma_{min}^2(F_{k-1}\cdots F_{k-m})\mathcal{O}(k-m,k)\leq M_{k,k-m}\leq \max_{k-m\leq l\leq k}\{\lambda_{max}(R_l)\}\sigma_{max}^2(F_{k-1}\cdots F_{k-m})\mathcal{O}(k-m,k)$$ where $\sigma(\cdot)$ denotes a singular value and $\mathcal{O}(k-m,k)$ the function you have defined above with $l$ ($l$ is a dummy variable for the sum so $\mathcal{O}(l,k)$ is not properly defined). $\endgroup$ – RTJ Dec 9 '15 at 18:49
  • $\begingroup$ Yes, I got it, thanks for your helps. $\endgroup$ – wayne Dec 11 '15 at 3:19
  • $\begingroup$ You are welcome! $\endgroup$ – RTJ Dec 11 '15 at 8:00

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