An improved adaptive hybrid firefly differential evolution algorithm for passive target localization

Abstract

This paper considers a passive target localization problem based on the noisy time of arrival measurements obtained from multiple receivers and a single transmitter. The maximum likelihood (ML) estimator for this localization problem is formulated as a highly nonlinear and non-convex optimization problem, where conventional optimization methods are not suitable for solving such a problem. Consequently, this paper proposes an improved adaptive hybrid firefly differential evolution (AHFADE) algorithm, based on hybridization of firefly algorithm (FA) and differential evolution (DE) algorithm to estimate the unknown position of the target. The proposed AHFADE algorithm dynamically adjusts the control parameters, thus maintaining high population diversity during the evolution process. This paper aims to improve the accuracy of the global optimal solution by incorporating evolutionary operators of the DE in different searching stages of the FA. In this regard, an adaptive parameter is employed to select an appropriate mutation operator for achieving a proper balance between global exploration and local exploitation. In order to efficiently solve the ML estimation problem, this paper proposes the well-known semidefinite programming (SDP) method to convert the non-convex problem into a convex one. The simulation results obtained from the proposed AHFADE algorithm and well-known algorithms, such as SDP, DE and FA, are compared against Cramér–Rao lower bound (CRLB). The statistical analysis has been performed to compare the performance of the proposed AHFADE algorithm with several well-known algorithms on CEC2014 benchmark problems. The obtained simulation results show that the proposed AHFADE algorithm is more robust in high-noise environments compared to existing algorithms.

A Derivation of CRLB

From Eq. (8), the first partial derivative of the log-likelihood function with respect to $\mathbf {x}$ is given as:

$$\begin{aligned}&\frac{{\partial \ln \left( {f\left( {{\tilde{\mathbf{d}}}\left| \mathbf {x} \right. } \right) } \right) }}{{\partial \mathbf {x}}}\nonumber \\&= - \frac{1}{2}\frac{\partial }{{\partial {\mathbf {x}}}}\left( {{{\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) }^T}{{\mathbf {C}}^{ - 1}}\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) } \right) ,\nonumber \\ \end{aligned}$$

(64)

where

$$\begin{aligned}&\frac{\partial }{{\partial {\mathbf {x}}}}\left( {{{\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) }^T}{{\mathbf {C}}^{ - 1}}\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) } \right) \nonumber \\&\quad = - 2\frac{{\partial {\mathbf {d}}{{\left( {\mathbf {x}} \right) }^T}}}{{\partial {\mathbf {x}}}}{{\mathbf {C}}^{ - 1}}\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) . \end{aligned}$$

(65)

Then, Eq. (64) can be expressed as:

$$\begin{aligned} \frac{{\partial \ln \left( {f\left( {{{\tilde{\mathbf{d}}}}\left| {\mathbf {x}} \right. } \right) } \right) }}{{\partial {\mathbf {x}}}} = \frac{{\partial {\mathbf {d}}{{\left( {\mathbf {x}} \right) }^T}}}{{\partial {\mathbf {x}}}}{{\mathbf {C}}^{ - 1}}\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) . \end{aligned}$$

(66)

Hence, taking the partial derivative with respect to x follows:

$$\begin{aligned} \frac{{\partial \ln \left( {f\left( {{{\tilde{\mathbf{d}}}}\left| {\mathbf {x}} \right. } \right) } \right) }}{{\partial {{x}}}} = \frac{{\partial {\mathbf {d}}{{\left( {\mathbf {x}} \right) }^T}}}{{\partial {{x}}}}{{\mathbf {C}}^{ - 1}}\left( {{{\tilde{\mathbf{d}}}} - {\mathbf {d}}\left( {\mathbf {x}} \right) } \right) . \end{aligned}$$

(67)

Then, the first diagonal element of the FIM can be derived as follows:

$$\begin{aligned} {I_{xx}}&= E\left[ {\left( {\frac{{\partial \ln \left( {f\left( {\left. {\tilde{\mathbf{d}}} \right| \mathbf {x}} \right) } \right) }}{{\partial {x}}}} \right) {{\left( {\frac{{\partial \ln \left( {f\left( {\left. {\tilde{\mathbf{d}}} \right| \mathbf {x}} \right) } \right) }}{{\partial {x}}}} \right) }^T}} \right] \nonumber \\&= E\left[ {{\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) }^T}{\mathbf {C}^{ - 1}}\left( {{\tilde{\mathbf{d}}} - \mathbf {d}\left( \mathbf {x} \right) } \right) {{\left( {{\tilde{\mathbf{d}}} - \mathbf {d}\left( \mathbf {x} \right) } \right) }^T}\right. \nonumber \\&\quad \left. {{\left( {{\mathbf {C}^{ - 1}}} \right) }^T} \left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) \right] \nonumber \\&={\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) ^T}{\mathbf {C}^{ - 1}}E\left[ {\mathbf {n}{\mathbf {n}^T}} \right] {\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) \nonumber \\&={\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) ^T}{\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) . \end{aligned}$$

(68)

The remaining elements of the FIM can be derived in the same way as above:

$$\begin{aligned} {I_{xy}}&= {I_{yx}} = E\left[ {\left( {\frac{{\partial \ln \left( {f\left( {\left. {\tilde{\mathbf{d}}} \right| \mathbf {x}} \right) } \right) }}{{\partial {x}}}} \right) {{\left( {\frac{{\partial \ln \left( {f\left( {\left. {\tilde{\mathbf{d}}} \right| \mathbf {x}} \right) } \right) }}{{\partial {y}}}} \right) }^T}} \right] \nonumber \\&={\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) ^T}{\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial y}}} \right) , \end{aligned}$$

(69)

$$\begin{aligned} {I_{yy}}&= E\left[ {\left( {\frac{{\partial \ln \left( {f\left( {\left. {\tilde{\mathbf{d}}} \right| \mathbf {x}} \right) } \right) }}{{\partial {y}}}} \right) {{\left( {\frac{{\partial \ln \left( {f\left( {\left. {\tilde{\mathbf{d}}} \right| \mathbf {x}} \right) } \right) }}{{\partial {y}}}} \right) }^T}} \right] \nonumber \\&={\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial y}}} \right) ^T}{\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial y}}} \right) . \end{aligned}$$

(70)

According to Eq. (4), the partial derivative of $\mathbf {d}\left( \mathbf {x} \right)$ with respect to the components of $\mathbf {x}$ can be expressed as:

$$\begin{aligned} \frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial \mathbf {x}}} = \begin{bmatrix} {{\frac{{x - x_1^r}}{{{{\left\| {{\mathbf {x}} - \mathbf {x}_1^r} \right\| }_{2}}}} + \frac{x}{{{{\left\| {\mathbf {x}} \right\| }_{{2}}}}}}} &{} {{\frac{{y - y_1^r}}{{{{\left\| {{\mathbf {x}} - \mathbf {x}_1^r} \right\| }_{2}}}} + \frac{y}{{{{\left\| {\mathbf {x}} \right\| }_{{2}}}}}}}\\ {{\frac{{x - x_2^r}}{{{{\left\| {{\mathbf {x}} - \mathbf {x}_2^r} \right\| }_{2}}}} + \frac{x}{{{{\left\| {\mathbf {x}} \right\| }_{{2}}}}}}}&{} {{\frac{{y - y_2^r}}{{{{\left\| {{\mathbf {x}} - \mathbf {x}_2^r} \right\| }_{2}}}} + \frac{y}{{{{\left\| {\mathbf {x}} \right\| }_{{2}}}}}}}\\ \vdots &{} \vdots \\ {{\frac{{x - x_N^r}}{{{{\left\| {{\mathbf {x}} - \mathbf {x}_N^r} \right\| }_{2}}}} + \frac{x}{{{{\left\| {\mathbf {x}} \right\| }_{{2}}}}}}} &{} {{\frac{{y - y_N^r}}{{{{\left\| {{\mathbf {x}} - \mathbf {x}_N^r} \right\| }_{2}}}} + \frac{y}{{{{\left\| {\mathbf {x}} \right\| }_{{2}}}}}}}\\ \end{bmatrix}, \end{aligned}$$

(71)

where $\partial \mathbf {d}\left( \mathbf {x} \right) /\partial \mathbf {x}$ is the $N \times 2$ Jacobian matrix.

Performing the matrix multiplication, the following expressions can be easily obtained

$$\begin{aligned} {I_{xx}}&= {\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) ^T}{\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) \nonumber \\&= \sum \limits _{i = 1}^N {\frac{1}{{\sigma _{ni}^2}}{{\left( {\frac{{x - x_i^r}}{{{{\left\| {\mathbf {x} - \mathbf {x}_i^r} \right\| }_{{2}}}}} + \frac{x}{{{{\left\| \mathbf {x} \right\| }_{{2}}}}}} \right) }^2}}. \end{aligned}$$

(72)

Similarly, the expressions for the remaining elements of FIM can be obtained as:

$$\begin{aligned} {I_{xy}}&= {I_{yx}} = {\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial x}}} \right) ^T}{\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial y}}} \right) \nonumber \\&= \sum \limits _{i = 1}^N {\frac{1}{{\sigma _{ni}^2}}{{\left( {\frac{{x - x_i^r}}{{{{\left\| {\mathbf {x} - \mathbf {x}_i^r} \right\| }_{{2}}}}} + \frac{x}{{{{\left\| \mathbf {x} \right\| }_{{2}}}}}} \right) }}{{\left( {\frac{{y - y_i^r}}{{{{\left\| {\mathbf {x} - \mathbf {x}_i^r} \right\| }_{{2}}}}} + \frac{y}{{{{\left\| \mathbf {x} \right\| }_{{2}}}}}} \right) }}}, \end{aligned}$$

(73)

$$\begin{aligned} {I_{yy}}&= {\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial y}}} \right) ^T}{\left( {{\mathbf {C}^{ - 1}}} \right) ^T}\left( {\frac{{\partial \mathbf {d}\left( \mathbf {x} \right) }}{{\partial y}}} \right) \nonumber \\&= \sum \limits _{i = 1}^N {\frac{1}{{\sigma _{ni}^2}}{{\left( {\frac{{y - y_i^r}}{{{{\left\| {\mathbf {x} - \mathbf {x}_i^r} \right\| }_{{2}}}}} + \frac{y}{{{{\left\| \mathbf {x} \right\| }_{{2}}}}}} \right) }^2}}. \end{aligned}$$

(74)

Finally, Eqs. (55)-(57) of the corresponding elements of FIM are obtained.