Abstract
Kalman filtering with random packet drops has been studied extensively since the work of Sinopoli (IEEE Trans. Autom. Control 49(9), 1453–1464, 2004, [1]), which showed that for i.i.d. Bernoulli packet drops, there exists a critical threshold such that if the packet arrival rate exceeds this threshold, then the expected error covariance remains bounded but diverges otherwise.
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Notes
- 1.
We say that a matrix \(X>0\) if X is positive definite, and \(X\ge 0\) if X is positive semi-definite.
- 2.
In practice this can be determined using simple error detecting codes.
- 3.
In practice, this can be achieved by periodically sending pilot signals either from the sensor to the remote estimator to allow the remote estimator to estimate the channel, or from the remote estimator to the sensor under channel reciprocity.
- 4.
In wireless communications, online computation of powers at the base station, which is then fed back to the mobile transmitters, is commonly done in practice [12], at time scales on the order of milliseconds.
- 5.
We measure energy on a per channel use basis and we will refer to energy and power interchangeably in this chapter.
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Appendix
Appendix
2.1.1 Proof of Lemma 2.2
Proof
The proof uses similar ideas to the proof of Proposition 3.1 in [36]. The decreasing property follows from the relation
and the assumption that f(.) is an increasing function.
For the proof of convexity, let \(u^1\) and \(u^2\) be two average transmit powers, where \(u^1 \ne u^2\), with \(\mathfrak {p}^*(u^1)\) and \(\mathfrak {p}^*(u^2)\) the corresponding traces of the expected error covariances. We want to show that
Let \(\{u_k^1 (P_k,g_k)\}\) be the optimal power allocation policy that achieves \(\mathfrak {p}^*(u^1)\), and \(\{u_k^2(P_k,g_k)\}\) be the optimal power allocation policy that achieves \(\mathfrak {p}^*(u^2)\). Define a new policy \(\{u^\lambda _k(P_k,g_k)\}\) such that
We will first show that for a given \(P_k\), we have:
where \(P_{k+1}^j\) is the value of \(P_{k+1}\) that follows from using policy \(\{u_k^j(.)\}\), for \(j=1,2,\lambda \), respectively. For (1), this clearly follows from the definition of \(u_k^\lambda \). For (2), we have
where the inequality comes from the strict concavity of f(.).
From (1) and (2), we have
and
By the definition of \(\mathfrak {p}^*(u)\) being the minimum expected error covariance such that the average transmit power is less than or equal to u, we then have \(\mathfrak {p}^*(\lambda u^1 + (1-\lambda )u^2) \le \frac{1}{K} \sum _{k=1}^K \mathbb {E}[\text {tr}(P_{k+1}^\lambda )] < \lambda \mathfrak {p}^*(u^1) + (1-\lambda ) \mathfrak {p}^*(u^2)\).
2.1.2 Proof of Theorem 2.3
We first establish the inequality
by verifying conditions (W) and (B) of [48], that guarantee the existence of solutions to (2.26) for MDPs with general state space. Denote the state space by \(\mathscr {S}\) and action space by \(\mathscr {A}\), i.e. \((P_k,g_k,H_k,B_k) \in \mathscr {S}\) and \(u_k \in \mathscr {A}\). Condition (W) of [48] in our notation says that:
(1) The state space \(\mathscr {S}\) is locally compact.
(2) Let \(U(\cdot )\) be the mapping that assigns to each \((P_k,g_k,H_k,B_k)\) the nonempty set of available actions. Then \(U(P_k,g_k,H_k,B_k)\) lies in a compact subset of \(\mathscr {A}\) and \(U(\cdot )\) is upper semicontinuous.
(3) The transition probabilities are weakly continuous.
(4) \(\mathbb {E} \big [\mathrm {tr}\big (\mathscr {L}(P,\gamma )\big )\big |P,g,u\big ]\) is lower semicontinuous.
By our assumption that \(u_k \le B_k \le B_{\text {max}}\), (0) and (1) of (W) can be easily verified. The conditions (2) and (3) follow from the definition (2.23).
Define \(w_{\delta }(P_0,g_0,H_0,B_0) = v_{\delta } (P_0,g_0,H_0,B_0) - m_{\delta }\), where
and \(m_{\delta } = \inf _{(P_0,g_0,H_0,B_0)} v_\delta (P_0,g_0,H_0,B_0)\). Condition (B) of [48] in our notation says that
Following Sect. 4 of [48], define the stopping time
for some \(\varsigma \ge 0\). Given \(\varsigma > 0\) and an arbitrary \((P_0,g_0,H_0,B_0)\), consider a suboptimal power allocation policy where the sensor transmits based on the same policy as the one that achieves \(m_\delta \) (with a different initial condition) until \(v_{\delta } (P_N,g_N,H_N,B_N) \le m_{\delta } + \varsigma \) is satisfied at some time N. By the exponential forgetting property of initial conditions for Kalman filtering, we have \(N < \infty \) with probability 1 and \(\mathbb {E}[N] < \infty \). Since \(\tau \le N\), we have \(\mathbb {E}[\tau ] < \infty \). Then by Lemma 4.1 of [48],
where the second inequality uses Wald’s equation, with Z being an upper bound to the expected error covariance, which exists by Theorem 2.2. Hence, condition (B) of [48] is satisfied and a solution to the inequality (2.26) exists.
To show equality in (2.26), we will require a further equicontinuity property of the optimal cost for the related discounted cost MDP to be satisfied. This can be shown by a similar argument as in the proof of Proposition 3.2 of [49]. The assumptions in Sects. 5.4 and 5.5 of [33] may then be verified to conclude the existence of a solution to the average cost optimality equation (2.24).
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Leong, A.S., Quevedo, D.E., Dey, S. (2018). Optimal Power Allocation for Kalman Filtering over Fading Channels. In: Optimal Control of Energy Resources for State Estimation Over Wireless Channels. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-65614-4_2
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