In a distributed system with *l* sensors, let *x*_{k}∈*R*^{n} is a deterministic parameter vector to be estimated. Suppose \(\hat {x}_{k}^{(i)},i=1,\ldots, l\) are sensor estimators of *x*_{k} at time *k*, and we have the prior information that the state *x*_{k} 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 *a*_{i} 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 *X*_{1}>0, then *X*≥0 if and only if \(X_{3}-X_{2}'X_{1}^{-1}X_{2}\geq 0\).

###
**Lemma 2**

Let *ε*,*γ*_{i} be non-negative scalars, *z* and *B*_{i} 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 *ζ*_{k} 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*,*γ*_{i},*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 *P*_{k}, so we can only require the *P*_{k} 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 *W*_{k}, 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 *W*_{k} 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 *W*_{k}=*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 *γ*_{l}. 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. □