In a distributed system with l sensors, let xk∈Rn is a deterministic parameter vector to be estimated. Suppose \(\hat {x}_{k}^{(i)},i=1,\ldots, l\) are sensor estimators of xk at time k, and we have the prior information that the state xk lies in the non-empty intersection of l ellipsoids \(S_{k}^{(i)}\)defined by
$${}S_{k}^{(i)}\,=\,\left\{\!x_{k}:\left(\!x_{k}-\hat{x}_{k}^{(i)}\!\right)^{T} \!\bar{P}_{k}^{(i)}\left(x_{k}-\hat{x}_{k}^{(i)}\right)\leq a_{i},i=1,\ldots,l\right\},$$
where \(P_{k}^{(i)}\) are some known positive semi-definite matrices, and ai is a positive scalar. It is easy to standardize the ellipsoids as
$$S_{k}^{(i)}=\{x_{k}:(x_{k}-\hat{x}_{k}^{(i)})^{T} P_{k}^{(i)}(x_{k}-\hat{x}_{k}^{(i)})\leq 1,i=1,\ldots,l\}.$$
When the fusion central receives l sensor estimators and the ellipsoids \(S_{k}^{(i)},i=1,\ldots, l\), they should be fused to obtain optimal fusion estimation. Therefore, we first maximize the estimation error over the intersection of \(S_{k}^{(i)}\), then choose the fusion estimate \(\hat {x}_{k}\) to minimize the estimation error. Finally, the fusion estimation is designed to solve the following minimax problem:
$$\begin{array}{@{}rcl@{}} &&\min_{\hat{x}_{k}}\max_{x_{k}\in \bigcap_{i=1}^{l}S_{k}^{(i)}}\|x_{k}-\hat{x}_{k}\|_{2}^{2}. \end{array} $$
(1)
Generally speaking, problem (1) is an NP-hard problem when l≥2. In the sequel, we seek for approximation solution of problem (1) by designing a convex relaxation strategy.
The following lemmas are necessary for further derivation.
Lemma 1
(see [15]) Let X be an symmetric matrix partitioned as
$$X=\left(\begin{array}{cc} X_{1} & X_{2} \\ X_{2}' & X_{3} \\ \end{array} \right). $$
Then X>0 if and only if \(X_{3}-X_{2}'X_{1}^{-1}X_{2}>0\). Furthermore, if X1>0, then X≥0 if and only if \(X_{3}-X_{2}'X_{1}^{-1}X_{2}\geq 0\).
Lemma 2
Let ε,γi be non-negative scalars, z and Bi be any compatible vector and Hermitian matrices, i=1,…,l. Then the following inequality holds for any vector y and scalar t
$$\begin{array}{@{}rcl@{}} \epsilon t^{2} + \sum_{i = 1}^{l} \gamma_{i} y^{T} B_{i} y - (y^{T} y - 2 t y^{T} z) \geq 0, \end{array} $$
(2)
if and only if
$$\begin{array}{@{}rcl@{}} \left(\begin{array}{cc} \epsilon & z^{T} \\ z & \sum\limits_{i = 1}^{l}\gamma_{i} B_{i} - I\\ \end{array} \right)\geq 0. \end{array} $$
(3)
Furthermore, if (2) or (3) holds, then
$$ {\begin{aligned} \max_{y, t} &\left\{y^{T} y - 2t y^{T} z : y^{T} B_{i} y \leq 1, i = 1, \ldots, l, t^{2} \leq 1\right\} \\ &\quad\leq \epsilon + \sum_{i = 1}^{l}\gamma_{i}. \end{aligned}} $$
(4)
Proof
Note that (3) is equivalent to
$$\left(\begin{array}{c} t \\ y \\ \end{array} \right)^{T} \left(\begin{array}{cc} \epsilon & z^{T} \\ z & \sum\limits_{i = 1}^{l} \gamma_{i} B_{i} - I \\ \end{array} \right) \left(\begin{array}{c} t \\ y \\ \end{array} \right) \geq 0 $$
holds for any vector y and scalar t, which implies the equivalence between (2) and (3). At the same time, from (2),
$$\begin{array}{@{}rcl@{}} y^{T} y - 2t y^{T} z \leq \epsilon t^{2} + \sum_{i = 1}^{l} \gamma_{i} y^{T} B_{i} y. \end{array} $$
Obviously,
$${{} \begin{aligned} \max_{y, t} &\left\{\epsilon t^{2} + \sum_{i = 1}^{l} \gamma_{i} y^{T} B_{i} y: y^{T} B_{i} y \leq 1, i = 1, \ldots, l, t^{2} \leq 1\right\} \\&\quad\leq \epsilon + \sum_{i = 1}^{l} \gamma_{i}. \end{aligned}} $$
Therefore, (4) follows from (2).
A positive semi-definite relaxation of problem (1) is presented in the following theorem, which can be solved by some efficient semidefinite programming solvers such as SeDuMi (see [16]). □
Theorem 2
The positive semi-definite relaxation of the optimal problem (1) is
$$\begin{array}{@{}rcl@{}}{} &&\min_{\epsilon \geq 0,\gamma_{i}\geq 0, \tau\geq 0, \hat{x}_{k},\alpha_{k}^{(i)}} \tau\\ && {}s.t.\left(\begin{array}{ccc} \epsilon & 0 & \left(\sum_{i=1}^{l}\alpha_{k}^{(i)}\hat{x}_{k}^{(i)}-\hat{x}_{k}\right)^{T} \\ 0 & \sum_{i=1}^{l}\gamma_{i} Q_{k}^{(i)} & \left(\sum_{i=1}^{l}\alpha_{k}^{(i)}A_{k}^{(i)}\right)^{T} \\ \sum_{i=1}^{l}\alpha_{k}^{(i)}\hat{x}_{k}^{(i)}-\hat{x}_{k} & \sum_{i=1}^{l}\alpha_{k}^{(i)}A_{k}^{(i)} & I \\ \end{array} \right)\!\geq\! 0~~ \\ &&{\text{and}}~~~~~~~~ \epsilon+\sum_{i=1}^{l}\gamma_{i}\leq \tau,\sum_{i=1}^{l}\alpha_{k}^{(i)}=1, \end{array} $$
(5)
where \(A_{k}^{(i)}=(0,\ldots,I\ldots,0)_{n\times nl}, Q_{k}^{(i)}=\left (A_{k}^{(i)}\right)^{T} P_{k}^{(i)}A_{k}^{(i)}, i=1,\ldots,l\). The optimal fusion estimation \(\hat {x}_{k}\) is given by the solution of the SDP problem (5).
Proof
Note that problem (1) is the same as
$$\begin{array}{@{}rcl@{}} &&\min_{\hat{x}_{k},\tau>0}\tau \\ &&s.t. ~\|x_{k}-\hat{x}_{k}\|_{2}^{2}\leq \tau ~~ {\text{and}} \\ &&~~\left(x_{k}-\hat{x}_{k}^{(i)}\right)^{T} P_{k}^{(i)}\left(x_{k}-\hat{x}_{k}^{(i)}\right)\leq 1,i=1,\ldots,l. \end{array} $$
(6)
Denoted by
$$\begin{array}{*{20}l} {}\eta_{k}^{(i)}&=x_{k}-\hat{x}_{k}^{(i)},\eta_{k}=\left(\begin{array}{c} \eta_{k}^{(1)} \\ \ldots \\ \eta_{k}^{(l)} \\ \end{array} \right),\\ A_{k}^{(i)}&=(0,\ldots,I\ldots,0)_{n\times nl},i=1,\ldots,l, \end{array} $$
then \(\eta _{k}^{(i)}=A_{k}^{(i)}\eta _{k},i=1,\ldots,l\). For any real scalars \(\alpha _{k}^{(i)},i=1,\ldots, l\) satisfy \(\sum \limits _{i=1}^{l}\alpha _{k}^{(i)}=1\), let
$$\zeta_{k}=\sum_{i=1}^{l} \alpha_{k}^{(i)}\hat{x}_{k}^{(i)}.$$
Note that the ζk is not the linear combination of local estimator \(\hat {x}_{k}^{(i)}\), unless the \(\alpha _{k}^{(i)}\) is independent of local estimators \(\hat {x}_{k}^{(i)}\). In fact, the \(\alpha _{k}^{(i)}\) is given by the optimal problem (1), thus it is not independent of \(\hat {x}_{k}^{(i)}\). Therefore, problem (6) is the same as
$$\begin{array}{@{}rcl@{}} &&\min_{\hat{x}_{k},\tau>0}\tau \\ &&s.t. ~\|x_{k}-\zeta_{k}+\zeta_{k}-\hat{x}_{k}\|_{2}^{2}\leq \tau ~~ {\text{and}} \\ &&~~\eta_{k}^{T} \left(A_{k}^{(i)}\right)^{T}P_{k}^{(i)}A_{k}^{(i)}\eta_{k}\leq 1,i=1,\ldots,l. \end{array} $$
(7)
Let \(\beta _{k}=\zeta _{k}-\hat {x}_{k}, Q_{k}^{(i)}=(A_{k}^{(i)})^{T} P_{k}^{(i)}A_{k}^{(i)}\), so \(x_{k}-\zeta _{k}=\sum _{i=1}^{l}\alpha _{k}^{(i)}A_{k}^{(i)}\eta _{k}\). Denote \(A_{k}=\sum _{i=1}^{l}\alpha _{k}^{(i)}A_{k}^{(i)}\), then problem (7) is equivalent to
$$\begin{array}{@{}rcl@{}} &&\min_{\hat{x}_{k},\tau>0}\tau \\ &&s.t. ~\|A_{k}\eta_{k}+\beta_{k}\|_{2}^{2}\leq \tau ~~ {\text{and}} \\ &&~~\eta_{k}^{T}Q_{k}^{(i)}\eta_{k}\leq 1,i=1,\ldots,l. \end{array} $$
(8)
Considering the constrains of problem (8), it is the same as
$$\max_{\eta_{k}}\{\|A_{k}\eta_{k}+\beta_{k}\|_{2}^{2}:\eta_{k}^{T}Q_{k}^{(i)}\eta_{k}\leq 1,i=1,\ldots,l\}\leq \tau.$$
Equivalently,
$${}\begin{aligned} \max_{\eta_{k},t}\left\{\|A_{k}t\eta_{k}+\beta_{k}\|_{2}^{2}:\eta_{k}^{T}Q_{k}^{(i)}\eta_{k}\leq 1,i=1,\ldots,l,t^{2}\leq 1\right\}\leq \tau, \end{aligned} $$
which can be reformulated to
$$\begin{array}{*{20}l} &\max_{\eta_{k},t}\left\{(A_{k}\eta_{k})^{T} A_{k}\eta_{k}+2t(A_{k}\eta_{k})^{T}\beta_{k}:\eta_{k}^{T}Q_{k}^{(i)}\eta_{k}\right.\\ &\qquad\left.\leq 1,i=1,\ldots,l,t^{2}\leq 1{\vphantom{Q_{k}^{(i)}}}\right\}\\ &\qquad\leq \tau-\beta_{k}^{T}\beta_{k}. \end{array} $$
(9)
From Lemma 2, for non-negative scalars γ,γi,i=1,…,l, the constrains
$$\begin{array}{@{}rcl@{}} \left(\begin{array}{cc} \gamma & -\beta_{k}^{T}A_{k} \\ -(\beta_{k}^{T}A_{k})^{T} & \sum_{i=1}^{l}\gamma_{i} Q_{k}^{(i)}-A_{k}^{T}A_{k} \\ \end{array} \right)\geq 0 \end{array} $$
(10)
and
$$\begin{array}{@{}rcl@{}} && \gamma +\sum_{i=1}^{l}\gamma_{i}\leq \tau-\beta_{k}^{T}\beta_{k} \end{array} $$
are sufficient for (9) holding. Let \(\epsilon =\gamma +\beta _{k}^{T}\beta _{k}\), then (10) can be rewritten as
$$\begin{array}{@{}rcl@{}} \left(\begin{array}{cc} \epsilon & 0 \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} \\ \end{array} \right)- \left(\begin{array}{cc} \beta_{k}^{T}\beta_{k} & \beta_{k}^{T}A_{k} \\ (\beta_{k}^{T}A_{k})^{T} & A_{k}^{T}A_{k} \\ \end{array} \right) \geq 0. \end{array} $$
From Lemma 1, which is equivalent to
$$\begin{array}{@{}rcl@{}} &&\left(\begin{array}{ccc} \epsilon & 0 & \beta_{k}^{T} \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} & A_{k}^{T} \\ \beta_{k} & A_{k} & I \\ \end{array} \right)\geq 0. \end{array} $$
Therefore, we can relax the problem (8) to be the following SDP problem:
$$\begin{array}{@{}rcl@{}} &&\min_{\epsilon \geq 0,\gamma_{i}\geq 0, \tau\geq 0, \beta_{k},\alpha_{k}^{(i)}} \tau\\ && s.t.\left(\begin{array}{ccc} \epsilon & 0 & \beta_{k}^{T} \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} & A_{k}^{T} \\ \beta_{k} & A_{k} & I \\ \end{array} \right)\geq 0~~ {\text{and}}\\ &&~~~~~~~~\epsilon+\sum_{i=1}^{l}\gamma_{i}\leq \tau,\sum_{i=1}^{l}\alpha_{k}^{(i)}=1 \end{array} $$
which is equivalent to
$$\begin{array}{@{}rcl@{}} &&\min_{\epsilon \geq 0,\gamma_{i}\geq 0, \tau\geq 0, \hat{x}_{k},\alpha_{k}^{(i)}} \tau\\ && s.t.\left(\begin{array}{ccc} \epsilon & 0 & (\sum_{i=1}^{l}\alpha_{k}^{(i)}\hat{x}_{k}^{(i)}-\hat{x}_{k})^{T} \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} & (\sum_{i=1}^{l}\alpha_{k}^{(i)}A_{k}^{(i)})^{T} \\ \sum_{i=1}^{l}\alpha_{k}^{(i)}\hat{x}_{k}^{(i)}-\hat{x}_{k} & \sum_{i=1}^{l}\alpha_{k}^{(i)}A_{k}^{(i)} & I \\ \end{array} \right)\geq 0\\ &&{\text{and}}\ \epsilon+\sum_{i=1}^{l}\gamma_{i}\leq \tau,\sum_{i=1}^{l}\alpha_{k}^{(i)}=1. \end{array} $$
Then, we finish the proof from the equivalence between the problem (8) and problem (1). □
Remark 1
Theorem 2 gives a approximate solution of problem (1), and the approximation is exactly when l=1 (see [3]). Furthermore, the proof of the Theorem 2 does not require the positive semi-definite of Pk, so we can only require the Pk to be Hermitian if problem (5) is of sense.
Just like the idea of weighted least squares estimation, we can generalize the result by introducing a positive semi-definite weight-valued matrix Wk, and replace problem (1) by
$$\begin{array}{@{}rcl@{}} &&\min_{\hat{x}_{k}}\max_{x_{k}\in \bigcap_{i=1}^{l}S_{k}^{(i)}}\|x_{k}-\hat{x}_{k}\|_{W_{k}}, \end{array} $$
(11)
where \(\|x_{k}-\hat {x}_{k}\|_{W_{k}}=(x_{k}-\hat {x}_{k})^{T}W_{k}(x_{k}-\hat {x}_{k})\).
Obviously, it is of sense to introduce weight matrix Wk if one wants to give different punishments to the estimation error in different dimension. Then, problem (1) is a special of problem (11) by taking Wk=I. Similar to the proof of Theorem 2, we can derived the positive semi-definite relaxation of problem (11).
Colloary 1
The positive semi-definite relaxation of optimal problem (11) is
$$ {{} \begin{aligned} &{}\min_{\epsilon \geq 0,\gamma_{i}\geq 0, \tau\geq 0, \hat{x}_{k},\alpha_{k}^{(i)}} \tau\\ &{} s.t.\left(\!\! \begin{array}{ccc} \epsilon & \!0 & \!\!\!\left(\sum_{i=1}^{l}\alpha_{k}^{(i)}\hat{x}_{k}^{(i)}-\hat{x}_{k}\right)^{T}W_{k}^{\frac{T}{2}} \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} & \!\left(\sum_{i=1}^{l}\alpha_{k}^{(i)}A_{k}^{(i)}\right)^{T}W_{k}^{\frac{T}{2}} \\ W_{k}^{\frac{1}{2}}\left(\sum_{i=1}^{l}\alpha_{k}^{(i)}\hat{x}_{k}^{(i)}-\hat{x}_{k}\right) & W_{k}^{\frac{1}{2}}\left(\sum_{i=1}^{l}\alpha_{k}^{(i)}A_{k}^{(i)}\right) & I \\ \end{array} \! \!\right)\\ &{}\geq 0\ and~~~~~~~~ \epsilon+\sum_{i=1}^{l}\gamma_{i}\leq \tau,\sum_{i=1}^{l}\alpha_{k}^{(i)}=1, \end{aligned}} $$
(12)
where \(A_{k}^{(i)}, Q_{k}^{(i)}\) is defined as in Theorem 2. The optimal fusion estimation \(\hat {x}_{k}\) is given by the solution of the SDP (12).
Theorem 2 provides a way to fusion sensor estimates when the prior knowledge \(x_{k}\in \bigcap _{i=1}^{l}S_{k}^{(i)}\) is available. With appropriate assumption, the feasible and unique of the solution to problem (1) can be derived.
Proposition 1
Suppose the set \(\bigcap _{i=1}^{l}S_{k}^{(i)}\) has non-empty inner point, then the optimal fusion estimation given by problem (1) is feasible and unique.
Proof
Note that problem (1) can be rewritten as
$$\begin{array}{@{}rcl@{}} &&\min_{\hat{x}_{k}}\left\{\|\hat{x}_{k}\|_{2}^{2}+\max_{x\in \bigcap_{i=1}^{l} S_{k}^{(i)}}\left(-2x_{k}^{T}\hat{x}_{k}+\|x_{k}\|_{2}^{2}\right)\right\}, \end{array} $$
thus \(\max _{x_{k}\in \bigcap _{i=1}^{l} S_{k}^{(i)}}\left (-2x_{k}^{T}\hat {x}_{k}+\|x_{k}\|_{2}^{2}\right)\) is convex in \(\hat {x}_{k}\). Therefore, \( \|\hat {x}_{k}\|_{2}^{2}+\max _{x_{k}\in \bigcap _{i=1}^{l}S_{k}^{(i)}}\left (-2x_{k}^{T}\hat {x}_{k}+\|x_{k}\|_{2}^{2}\right)\) is strictly convex in \(\hat {x}_{k}\), which implies the solution of problem (1) is feasible and unique.
In the sequel, some properties of RFE are considered. □
Theorem 3
Suppose that there is one \(P_{k}^{(i)}>0\), then the feasible solution set of problem (5) is always non-empty, thus the solution of problem (5) always exists.
Proof
Without loss of generality, suppose \(P_{k}^{(l)}>0\), note that problem (5) is the same as
$$\begin{array}{@{}rcl@{}} &&{}\min_{\epsilon \geq 0,\gamma_{i}\geq 0, \tau\geq 0, \hat{x}_{k},\alpha_{k}^{(i)}} \tau\\ &&{}\left(\! \begin{array}{ccc} \epsilon & 0 & (\sum_{i=1}^{l-1}\alpha_{k}^{(i)}(\hat{x}_{k}^{(i)}-\hat{x}_{k}^{(l)})+\hat{x}_{k}^{(l)}-\hat{x}_{k})^{T} \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} & (\sum_{i=1}^{l-1}\alpha_{k}^{(i)}(A_{k}^{(i)}-A_{k}^{(l)})+A_{k}^{(l)})^{T} \\ \sum_{i=1}^{l-1}\alpha_{k}^{(i)}(\hat{x}_{k}^{(i)}-\hat{x}_{k}^{(l)})+\hat{x}_{k}^{(l)}-\hat{x}_{k} & \sum_{i=1}^{l-1}\alpha_{k}^{(i)}(A_{k}^{(i)}-A_{k}^{(l)})+A_{k}^{(l)} & I \\ \end{array}\!\! \right)\!\geq\! 0\\ &&~~~~~~~~ and~~\epsilon+\sum_{i=1}^{l}\gamma_{i}\leq \tau. \end{array} $$
Let \(\alpha _{k}^{(i)}=0, i=1,\ldots,l-1\), and \(\hat {x}_{k}=\hat {x}_{k}^{(l)}\), then the matrix constrain is
$$\begin{array}{@{}rcl@{}} &&\left(\begin{array}{ccc} \epsilon & 0 & 0 \\ 0 & \sum_{i=1}^{l}\gamma_{i}Q_{k}^{(i)} & (A_{k}^{(l)})^{T} \\ 0 & A_{k}^{(l)} & I \\ \end{array} \right)\geq 0, \end{array} $$
equivalently,
$$\begin{array}{@{}rcl@{}} &&\left(\begin{array}{ccccc} \varepsilon & 0 & \ldots & 0 &0 \\ 0 & \gamma_{1}P_{k}^{(1)} & \ldots &0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & \gamma_{l}P_{k}^{(l)} & I \\ 0 & 0 & \ldots & I & I \\ \end{array} \right)\geq 0 \end{array} $$
(13)
Note that \(P_{k}^{(i)}\!\geq 0,i=1,\ldots,l-1\), and \(P_{k}^{(l)}>0\), then (13) always holds by taking large enough γl. In other words, the feasible solution set of problem (5) is always non-empty, thus the positive semi-definite relaxation solution exists. □
Proposition 2
Suppose \(P_{k}^{(i)}>0, i=1,\cdots,l\), then each local estimate \(\hat {x}_{k}^{(i)}\) is a feasible fusion estimations of problem (5). In other words, the fusion estimation \(\hat {x}_{k}\) always has better tracking performance than that of local estimates \(\hat {x}_{k}^{(i)}\).
Proof
Suppose \(P_{k}^{(l)}>0\), from the proof of Theorem 3, there exist appropriate ε, \(\alpha _{k}^{(i)}, \gamma _{i},i=1,\cdots,l\) and τ, such that \((\epsilon,\gamma _{i},\tau,\hat {x}_{k}^{(l)}, \alpha _{k}^{(i)})\) is a feasible solution of problem (5). Therefore, the local estimate \(\hat {x}_{k}^{(l)}\) is a feasible fusion estimation. The proofs of local estimates \(\hat {x}_{k}^{(i)}, i=1, \cdots, l-1\) are similar. □