INTRODUCTION

Being nonlinear and non Gaussian most of the time, the general form of a discrete-time nonlinear dynamic system is modelled in Eq. 1 and 2, where: X_k is the unobserved state at step k. Let P(X₀) be its initial distribution, its evolution is related to a Markovian process having P(X_k|X_k-1) as a density function. The noted observations Z_k are independent and related to X_k, they are generated according to the density function P(Z_k|X_k).

(1)

(2)

Where:

Nonlinear systems are often sullied with noises related at the same time on the process of state and the collected observations. Kalman filter (Kalman, 1960) is applied to Gaussian linear systems with discrete or continuous states; its success was in short following the state models evolution, being nonlinear non-Gaussian in majority.

PF is a method based on sampling has the advantage of being applied to any type of system, however it has some form of degeneracy due to the increase in the variance of samples (Doucet et al., 2001).

To remedy this problem, other researches has been conducted and resulted in algorithms such as Sigma Point Particle Filter (SPPF) also known as Unscented Particle Filter (UPF) (Van der Merwe et al., 2000), GMSPF (Gaussian Mixture Sum Particle Filter) (Van der Merwe et al., 2003) and so on. These approaches have helped to correct the particles, thus, the problem of computing time was inevitable. The second part of this study is devoted to the main methods for filtering nonlinear systems, given next Particle Swarm Optimization (PSO) (Kennedy and Eberhart, 1995; Clerc, 2004) will be briefly defined following a comparison of PF and PSO. Both of PF and PSO use random sampling at initialization stage and the correction of the particles for searching the best solution, Table 1 shows similarities between these two approaches.

Table 1:	Comparison of PF and PSO: Comparison

The PF is a Markovian process, so each particle at step k remembers only its previous state k-1, in fact, the particles in PSO are designed to save on memory all previous states.

FILTERING NONLINEAR SYSTEMS

Unscented Kalman Filter (UKF): Extended Kalman filter EKF (Kalman, 1960) was the first filter applied to nonlinear dynamic systems, supposing state process distribution, observation and all noises Gaussian. The principle of EKF is founded on linearization of functions f and h by adopting a first order Taylor development, then, after obtained a linear system, Kalman filter will be applied. This approach does not take into account the uncertainty of states around the mean; for this reason EKF diverge in most time.

Julier and Uhlmann (1997) and Wan and Van der Merwe (2001) introduced a new approach called Unscented Kalman filter or Sigma Point Kalman filter, assuming that the fundamental task in filtering and estimation is to calculate statistics of the random variable representing state. They propose to approximate mean and covariance through nonlinear transformation called unscented transformation. Let x be a random variable with mean and covariance P_x, a set of points can be generated from the rows or the colons of matrix where l is the size of the vector x. These points have 0 mean and covariance P_x. By adding to each point, we obtain a set of 2l symmetric points which have the same statistics, we have then 2l+l Sigma vectors forming a new matrix, weights W_i are associated to these vectors here k ε R, is the ith row or colon related to the square root of (l+k)P_x (Eq. 3). To calculate the statistics of the observation variable Z_i, Sigma points are propagated through the nonlinear equation Z_i = h(X_i), i = 0,….,2l:

(3)

UKF filtering is applied to the augmented variable obtained by the concatenation of the original state vector and noises vectors. We observe that particles are generated in a deterministic way because they are extracted from the covariance matrix.

Particle Filter (PF): PF, also known as sequential Monte Carlo method (Doucet et al., 2001) is a sophisticated estimation technique based on simulation; its usually used to estimate Bayesian models. The PF expresses a prior distribution by using a set of a weighted particles. Providing the prior density P(X₀), it’ll be easy to construct the posterior probability P(Z_k|X_k). Assuming that the importance density function is π(X_k|Z_k) and all particles (i = 1..N) are drawn from this distribution, that is to say:

(4)

Based on PF,

(5)

Where:

(6)

Therefore, as the samples were drawn from an importance density, the weights in Eq. 6 are defined to be:

(7)

Normalized weights of particles are:

(8)

When we obtain a measurement at time k, the particles and their weights can be computed recursively by using these formulas. After several iterations, only one particle will probably have a larger weight, the others will be equivalent to zero. This degeneracy can be expressed by the variance of the importance weights, this is remedied in the so called re-sampling stage: eliminate samples with low importance weight and multiply those having high weights.

Although, the re-sampling step reduces the effects of the degeneracy problem, it introduces other inconvenience. The particles that have high weights are statistically selected many times, this leads to a loss of diversity among the particles as the resultant sample will contain many repeated points, this problem, is known as sample impoverishment, complete description of PF is shown in Fig. 1.

Fig. 1:	Algorithm PF

Sigma-point Particle Filter (SPPF): PF has the advantage of being applied to nonlinear and non-Gaussian systems. The major problem of PF is the degeneration; an idea suggested (Van der Merwe et al., 2000) was to move the particles towards a space with high probability. Instead of generating them in an indeterminist way, a Gaussian distribution will be used. Van der Merwe et al. (2000) supposed that since, the Unscented transformation makes it possible to capture a random variable statistics as well as possible former, the alternative is to use UKF for all particles which offers a better approximation of state. This new filter named SPPF (Sigma-Point Particle Filter) is viewed as hybridization between PF and UKF.

Particle swarm optimization (PSO): PSO have been suggested by Kennedy and Eberhart (1995). They estimate that finding a source of food is similar to finding a solution in a common field of research. The PSO is initialized randomly with a population of stochastic particles; each particle is characterized by a position and a speed. Positions and speeds are adjusted as the particle moves. At each iteration, all particles keep track of their coordinates which are associated with the best solution their have achieved so far (pbest) and the coordinates which are associated with the best solution achieved by any particle in the neighborhood (gbest).

Each particle updates its position and its speed according to the following system Eq. 9-10, where, v is the speed of the particle, X is the current particle (solution), rand is a random number between 0 and 1, c₁ and c₂ are constants representing social confidence coefficients as follow:

(9)

(10)

Several variants and parameters were suggested for this algorithm and in particular the values of c₁ and c₂; we will consider the values of c₁ and c₂ in our tests as follows:

c₁ = c₂ = 2, this assignment is the most used to ensure equity between the local experience and the neighborhood one of the particle. Some versions of PSO were proposed and adapted to different problems, we can find an overview of this approach (Clerc, 2004).

PFPSO

In this approach called PFPSO, we propose to use PSO to reduce the degeneracy of PF: let consider the best particle at step k over its previous steps having the higher weight which represents the best hypothesis and the best particle in the neighborhood of the particle. Unlike Zhang et al. (2008), we suggest replacing re-sampling step of PF by one iteration of PSO equations, thus, for each particle are applied the tow equations below:

(11)

(12)

This will allow us to retain more diversity in the samples.

The process is Markovian, in fact, the system does not keep in memory all previous states, so, will be defined only in relation to X_k and X_k-1. The problem does not arise for gbest that will be assessed against all the particles in step k.

Fig. 2:	Algorithm 2 PFPS

Resulting samples will be used for the calculation of the new solution X_k as:

where, N represents the number PF particles.

The main advantage of our approach is that it ensures convergence towards the best solution; the variance is not likely to increase rapidly with the number of iterations as in PF because no particles will be eliminated regardless of their weight and the risk of degeneracy will be reduced, it will not be entirely avoided but will occurs more later, the cost in computing time will not be affected, this is due to the simplicity of the 2 equations of PSO. The PFPSO is more defined in Fig. 2.