Abstract
For hybrid positioning systems (HPSs), the estimator design is a crucial and important problem. In this paper, a finite-element-method- (FEM-) based state estimation approach is proposed to HPS. As the weak solution of hybrid stochastic differential model is denoted by the Kolmogorov's forward equation, this paper constructs its interpolating point through the classical fourth-order Runge-Kutta method. Then, it approaches the solution with biquadratic interpolation function to obtain a prior probability density function of the state. A posterior probability density function is gained through Bayesian formula finally. In theory, the proposed scheme has more advantages in the performance of complexity and convergence for low-dimensional systems. By taking an illustrative example, numerical experiment results show that the new state estimator is feasible and has good performance than PF and UKF.
Highlights
In recent years, a new concept called hybrid positioning systems (HPSs) is more and more popular in navigation and location-based services research community [1]
Compared with Particle Filters, assuming that the number of particles is defined as N in a two-dimensional state space and the recursive call was adopted in the process of particle resampling, so the computational complexity is calculated as O(N!)
As the real point in these experiments is generated by the random number, every simulation result will be different, ; the estimation precision will be measured by using the root mean square error (RMSE) and standard deviation of state, which is defined as follows: R = ∑Si=T1 √(∑tk=1 (Sk − Yk)2) /t, ST (43)
Summary
A new concept called hybrid positioning systems (HPSs) is more and more popular in navigation and location-based services research community [1]. With the development of computing technology, PF is in a golden age as it could deal with nonlinear, non-Gaussian, non-steady-state recursive estimation problem It still has some disadvantages in practice, such as the fast growing computational complexity and the sample impoverishment. Speaking, a typical hybrid positioning system should be described by a partial differential equation (Kolmogorov’s forward equation) and a difference equation separately, while the former reflects how the conditional density of a dynamic system evolves, and the latter means how it is works by the new measuring information [22]. FEM has more advantages than PF or other classic filters like EKF and UKF It outperforms others in case of model mismatches, large state variations, and arbitrary initial conditions.
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