2.1. Extended Nonlinear Kalman FilterFor a nonlinear state-space model, the extended Kalman filter is a frequently used method to estimate the system state. The key point of this algorithm is to use first-order Taylor expansion to approximate the structural functions of the model. More precisely, a general nonlinear discrete-time state-space model is given as follows:
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where the system state and the noise are r-dimensional, whiled the observation value and the noise are n-dimensional. Moreover, the following assumptions are satisfied:
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If is a twice continuously differentiable function and is a continuously differentiable function, then we use Taylor expansion of and to approximate these function. In this case, the nonlinear state-space model (1) can be described approximately as a linear model. Let denote the information flow, namely, the observed data obtained through date be summarized. Suppose that the initial value of the state vector of the state-space model (1) is normally distributed and the distur-
bances and are normal. Then and obey normal distributions as well. Since h has
continuous first-order partial derivatives and has continuous second-order partial derivative functions, it is not difficult to linearize the nonlinear state-space model (1) with the help of Taylor expansion. Now define
, , Discrete space model is approxi-
mated by linear model can be used as follows:
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Proposition 2.1 If the discrete time linear system (3) satisfies (2), and the disturbances and are independent and obey normal distributions, then we have
(1) state estimates
,
and
(2) mean square error matrix of system.
.
The proof of Proposition 2.1 can be found in [8] . Although the EKF algorithm is simple and effective for some nonlinear state-space models, it is likely to lead to sub-optimal state estimate due to removing higher-order terms of structural equations in Taylor expansion. In what follows, two modified nonlinear Kalman filter algorithms will be given for the purpose of reducing errors of the state estimation.
2.2. Second-Order Nonlinear Kalman Filter AlgorithmsIn the extended nonlinear Kalman filter algorithm, an application of first-order Taylor expansion on results in a linear state equation. This linearization approximation will cause some errors and instability of algorithm in some sense. It is natural to use second-order Taylor expansion to approximate.
Proposition 2.2 Under the same conditions as Proposition 2.1, if is twice continuously differentiable and h is continuously differentiable, then the nonlinear state-space model (1) has the following nonlinear Kalman filter algorithm, namely,
(1) state estimates are given by
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and
(2) mean square error matrix of system
.
Proof: To guarantee normality of, we take the linear part of Taylor expansion of. Thus we have
.
Note that is normal, so is. By reference [8] ,
,
.
An application of Taylor expansion to gives that
,
Taking conditional expectations on both sides of (5), the state estimates are given by
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where we have applied the following fact that, if has mean and variance, then
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On the other hand, the residual is expressed as
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Thus the mean square error matrix of system is
.
It should be pointed out that the above nonlinear Kalman filter algorithm needs to calculate Hessian matrix of. Such an approximation, on one hand, increases running time, on the other hand, cannot guarantee normality of, so it will influence on multi-step predictions of the state. In next proposition, we will apply linearization approximation to the quadratic in (5) and deduce corresponding recursive formula of nonlinear Kalman filter algorithm.
Proposition 2.3 Under the same conditions as Proposition 2.2 we have the following nonlinear Kalman filter algorithm for the nonlinear system (1):
(1) state estimates are
.
and
(2) mean square error matrix of system state is
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Proof: Firstly, a second-order Taylor expansion of gives that
,
Applying first-order Taylor expansion to, we have
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This also implies that
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It is not hard to get that
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Moreover, the mean square error matrix of system is given by
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Note that in this nonlinear filter algorithm, we have applied first-order Taylor expansion to deal with the qua-
dratic, thus still obeys normal distribution in this case. This also implies
that this algorithm can give multi-step predictions of the system state.
2.3. Extended Particle Filter AlgorithmThe extended particle filter (EKPF) algorithm was initially given by Freitas in [9] , under the framework of EKPF algorithm. This algorithm applies recursive formula of EKF to update each particle and will get an approximate posterior probability density of. By the posterior probability distribution, new samples from the distribution are represented by a set of particles. Each particle has a weight assigned to it that represents the probability of that particle being sampled from the probability density function. In the resampling step, the particles with negligible weights are replaced by new particles in the proximity of the particles with higher weights. EKPF, SEKPF and SLEKPF algorithm are as follows [10] :
(1) Set initial value;
(2) Draw N particles from prior distribution (important distribution);
(3) Apply recursive formulas of EKF, SEKF, and SLEKF to update sampling particles;
(4) Update weights;
(5) Normalize weights;
(6) Resample,
(7),
(8) If the number of particles is enough, the procedure stops.