CN110689108A - Nonlinear system state estimation method - Google Patents

Nonlinear system state estimation method Download PDF

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CN110689108A
CN110689108A CN201910885904.XA CN201910885904A CN110689108A CN 110689108 A CN110689108 A CN 110689108A CN 201910885904 A CN201910885904 A CN 201910885904A CN 110689108 A CN110689108 A CN 110689108A
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韩雪
王栋
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Changchun Institute of Optics Fine Mechanics and Physics of CAS
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Abstract

A nonlinear system state estimation method relates to the technical field of signal processing and solves the problem of poor particle diversity in the nonlinear system state estimation process of the existing particle filtering method. The invention adopts the firefly algorithm to optimize the particle distribution, and improves the firefly algorithm in the process of combining the firefly algorithm and the particle filter algorithm. The invention effectively increases the diversity of particles, improves the number of effective particles, reduces errors, improves the estimation precision and is widely suitable for the state estimation problem of a nonlinear system. The invention adopts the firefly algorithm to replace the resampling process in the particle filtering method, maintains the diversity of the particles and avoids the problem of the lack of the diversity of the particles. In the process of combining the firefly algorithm and the particle filter algorithm, the standard firefly algorithm is optimized from four sides, the quality of the particles is effectively improved, the distribution of the particles is closer to the state quantity to be estimated, and the estimation precision is improved.

Description

Nonlinear system state estimation method
Technical Field
The invention relates to the technical field of signal processing, in particular to a firefly algorithm-based particle filter nonlinear system state estimation method.
Background
The nonlinear system state estimation has important values in the field of signal processing, such as voice recognition, fault diagnosis, target tracking, environment monitoring, parameter estimation and the like. The nonlinear state estimation refers to estimation of an unmeasured internal state of a nonlinear system based on input and output observation data of the nonlinear system. The Extended Kalman Filtering (EKF) is a classic method in the field of nonlinear system estimation, and the core idea of the EKF is that local approximate linearization processing is carried out on a nonlinear model, some simple weak nonlinear systems can be processed, but for a strong nonlinear system, the algorithm estimation error is large, and the algorithm convergence is difficult to ensure. Unscented Kalman Filtering (UKF) avoids the drawbacks of EKF, whose core idea is to approximate the probability density function of the system state, but it must satisfy the assumption that the system noise follows gaussian distribution. In recent years, particle filter methods have been rapidly developed in terms of nonlinear system state estimation. The particle filter generates a group of random sample (particle) sets with weights according to the empirical condition distribution of the system state, the size and the particle position of the particle weights are continuously adjusted according to the available observed quantity to enable the particle weights to approach the posterior distribution of the state, and finally the weighted sum of the group of samples is used as the estimated value of the state.
The principle of the particle filter resampling process is that the particles with larger weight are copied for many times, the particles with small weight are eliminated, and after a plurality of iterations, the same particles are more and more, even one particle can be copied for many times, so that the final particle type is simplified, the diversity of the particles is lost, and the problem of particle shortage is generated. The lack of the types of the particles means that a large amount of useful information is lost in the whole particle set, and the information contained in the particle set is not complete enough, which results in a large deviation and poor estimation precision in the estimation of the state.
The Gaussian particle filter algorithm approximates the system state distribution by a single Gaussian distribution under the basic framework of particle filtering, and omits the resampling process. However, when the probability distribution is non-gaussian, the gaussian distribution is often not used to obtain an effective approximation effect, and the final estimation result is not ideal. An Auxiliary Particle Filter (APF) algorithm incorporates the latest observations during the importance sampling process and uses auxiliary variables to complete the resampling process. In practical application, the estimation accuracy is better than that of a particle filter algorithm when the system noise is low. The Regularized Particle Filter (RPF) algorithm resamples a continuous approximate distribution of system states to overcome the lack of particle diversity. But it has the theoretical drawback that the discrete approximation distribution it obtains cannot be guaranteed to approach progressively the true distribution of the system state.
In order to avoid the problem of poor particle diversity caused by the resampling process, the method adopts the firefly algorithm to optimize the particle distribution, effectively increases the particle diversity, and improves the overall quality of the particle set, thereby improving the estimation precision.
Disclosure of Invention
The invention aims to solve the problem of poor particle diversity in the state estimation process of a nonlinear system in the conventional particle filtering method. A nonlinear system state estimation method is provided.
A nonlinear system state estimation method is realized by the following steps:
step one, initializing k to be 0: from the prior distribution p (x)0) Sampling to obtain NpParticles ofAll the particles have the same importance weight which is 1/N, and the initialized particles enter an iteration process;
step two, from the importance probability density distribution function
Figure BDA0002207291040000022
The middle sampling results in particles at time k:
Figure BDA0002207291040000023
step three, according to the observation likelihood functionCalculating the weight of each particle:
Figure BDA0002207291040000025
wherein y iskFor an observed value that is known at the current time,
Figure BDA0002207291040000026
normalizing the weight value as a state transfer function, wherein the weight value of the normalized particle is as follows:
step four, optimizing the particles by adopting an improved firefly algorithm, wherein the specific process is as follows:
step four, sorting all the particle weights calculated in the step three to obtain the particle with the maximum weight, and taking the particle as the optimal particle
Figure BDA0002207291040000028
The relative fluorescence brightness of each particle was calculated:
Figure BDA0002207291040000029
in the formula I0Maximum fluorescence brightness, gamma is the light intensity absorption coefficient, riAre particles
Figure BDA00022072910400000210
With optimal particles
Figure BDA00022072910400000211
The distance of the space between the two plates,
Figure BDA0002207291040000031
step four, calculating the attraction degree of each particle:
wherein beta is0N is the current iteration number for the maximum attraction degree;
step four and step three, updating the position of each particle:
wherein rand is [0.1 ]]Subject to uniformly distributed random factors, KiFor the perturbation factor, it is expressed by the following formula:
Figure BDA0002207291040000034
whereinAnd
Figure BDA0002207291040000036
the average value and the maximum value of the weight values of the particles at the moment k are obtained;
fourthly, calculating the particle weight after the position is updated through formulas (1) and (2), and normalizing the particle weight; the weight of the ith particle after the update is obtained as
Figure BDA0002207291040000037
Step five, obtaining a system state estimation value at the current moment:
Figure BDA0002207291040000038
and step six, judging whether a new observed value exists, if so, returning to the step two, and if not, exiting.
The invention has the beneficial effects that: the method adopts the firefly algorithm to optimize the particle distribution, and improves the firefly algorithm in the process of combining the firefly algorithm with the particle filter algorithm. The method effectively increases the diversity of the particles, improves the number of effective particles, reduces errors, improves the estimation precision, and can be widely applied to the state estimation problem of a nonlinear system.
The method of the invention adopts the firefly algorithm to replace the resampling process in the particle filtering method, thus keeping the diversity of the particles and avoiding the problem of the lack of the diversity of the particles.
In the process of combining the firefly algorithm and the particle filter algorithm, the standard firefly algorithm is optimized from four sides, the quality of the particles is effectively improved, the distribution of the particles is closer to the state quantity to be estimated, and the estimation precision is improved.
Drawings
FIG. 1 is a flow chart of a method for estimating a state of a nonlinear system according to the present invention;
FIG. 2 is a comparison diagram of RPF, APF, and particle Filter method (FAPF) state estimation based on the firefly algorithm;
FIG. 3 is a comparison diagram of state estimation errors of RPF, APF and FAPF;
FIG. 4 is a comparison graph of PF and FAPF particle distributions;
FIG. 5 is a diagram illustrating the relationship between the disturbance degree and the particle weight.
Detailed Description
In the first embodiment, the present embodiment is described with reference to fig. 1 to 5, and a method for estimating a state of a nonlinear system is implemented by the following steps:
step 1, initializing k to 0: from the prior distribution p (x)0) Sampling to obtain NpParticles of
Figure BDA0002207291040000041
At the moment, all the importance weights of the particles are the same and are 1/N, and the initialized particles enter an iteration process;
step 2, from the importance probability density distribution function
Figure BDA0002207291040000042
The middle sampling results in particles at time k:
Figure BDA0002207291040000043
step 3, obtaining the observed value y of the current momentkThen, based on the observed likelihood function
Figure BDA0002207291040000044
Calculating the weight of each particle:
wherein
Figure BDA0002207291040000046
Normalizing the weight value as a state transfer function, wherein the weight value of the normalized particle is as follows:
Figure BDA0002207291040000047
and 4, updating the particle position by adopting an improved firefly algorithm:
(1) selecting the particle with the largest weight as the optimal particle
Figure BDA0002207291040000048
The relative fluorescence brightness of each particle was calculated:
Figure BDA0002207291040000049
wherein I0Maximum fluorescence brightness, gamma is the light intensity absorption coefficient, riAre particles
Figure BDA00022072910400000410
With optimal particles
Figure BDA00022072910400000411
The spatial distance between
Figure BDA0002207291040000051
(2) Calculate the attraction for each particle:
Figure BDA0002207291040000052
wherein beta is0For maximum attraction, k is the current iteration number.
(3) Update the position of each particle:
Figure BDA0002207291040000053
wherein rand is [0.1 ]]Subject to uniformly distributed random factors, KiIs a disturbance factor, defined as
Figure BDA0002207291040000054
Wherein
Figure BDA0002207291040000055
And
Figure BDA0002207291040000056
is the average and maximum of the particle weights at that moment.
(4) Calculating the weight of the particle after the position is updated through the formulas (1) and (2), and normalizing the weight to obtain the weight of the ith particle after the update
Figure BDA0002207291040000057
Step 5, obtaining a system state estimation value:
Figure BDA0002207291040000058
and 6, judging whether a new observed value exists, if so, returning to the step 2, and if not, exiting.
In this embodiment, the FAPF, RPF, and APF methods are applied to state estimation of the following nonlinear systems:
state model xk＝0.5xk-1+25xk-1/(1+xk-1 2)+8cos(1.2(k-1))+uk-1
Observation model
Figure BDA0002207291040000059
Wherein the initial value of the state is set as x00.1, initial distribution is highS distribution p (x)0) N (0,5), process noise u to N (0,8), observation noise v to N (0, 1). Number of particles N p100, time step 60, 100 independent experiments were performed. The estimation conditions of the three methods are shown in fig. 2, and it can be seen that the estimation deviation of the RPF is large and deviates from the true state value seriously, the APF can estimate the true state value approximately, but a large error occurs at the inflection point position, and the FAPF can estimate the true state of the state well at both the whole and the inflection point.
In order to make the estimation accuracy of the three algorithms more clear, the estimation errors of each point are compared, as shown in fig. 3, it is obvious from the figure that the estimation error of the RPF is large, the error fluctuation degree is also severe, and the estimation condition is unstable; the error fluctuation of the APF is smaller, but compared with the FAPF of the method of the invention, the error value is larger; the method of the invention is optimal in terms of both error and estimation stability.
Table 1 shows the error comparison for 100 independent simulation experiments for the three estimation methods. As can be seen from the table, the method of this embodiment is clearly superior to the other two methods in comparison of the overall estimation performance.
TABLE 1
Figure BDA0002207291040000061
To better see how the particle filter method (PF) is improved by the method of this embodiment in terms of both particle diversity and particle quality, we compare the particle distributions of the two methods at a certain time, as shown in fig. 4. It can be clearly seen that the values of the particles in the PF are more identical, indicating that the particle diversity has been lost, and that some particles deviate significantly from the true signal, and that the particle distribution as a whole cannot be well concentrated around the true value. The FAPF has fewer particles with the same value, which shows that the types of the particles are rich and the diversity is improved, and meanwhile, the particles are more concentrated on two sides of the true value, which shows that the proportion of the particles with high weight is increased and the quality of the particles is improved.
In order to further improve the quality of the particles, in the process of combining the firefly algorithm with the particle filter algorithm, the firefly algorithm is optimized in four aspects:
(a) the optimal particles guide the position updating process of each particle, the defect that a standard firefly algorithm is easy to fall into a local optimal solution is overcome, and the global optimization capability of the particles is improved.
(b) When calculating the relative fluorescence brightness of the particles, any two particles in the standard firefly algorithm need to be interacted, and the calculation complexity is
Figure BDA0002207291040000071
NpIs the number of particles. The method adopts the optimal particles to interact with each particle, and the calculation amount of the algorithm is reduced to Np-1；
(c) The attraction of the firefly algorithm determines the particle travel distance. The attraction of the standard firefly algorithm is
Figure BDA0002207291040000072
In the later stage of the algorithm iteration, the particles have gradually moved to the vicinity of the target value, and the distance between the particles is gradually reduced. The method of the invention has an attraction degree of
Figure BDA0002207291040000073
The formula introduces the iteration times, so that the attraction degree is reduced along with the increase of the iteration times, and finally a minimum value is kept, thereby avoiding the problem of particle oscillation in the later period of iteration.
(d) In the particle position updating process, the perturbation factor K is introduced into the formula (5)iAnd the disturbance factor is fully integrated with the weight information of the particle set at the current moment. FIG. 5 depicts the relationship between the particle weight and the corresponding degree of perturbation. The particles with low weight can experience a stronger disturbance process under the guidance of the optimal particles, which is beneficial to improving the convergence speed of the movement of the particles to the optimal particles; the particles with high weight are good particles, and the corresponding disturbance factors are comparedSmall means that these good particles will undergo a weak perturbation process, which is beneficial to preserve their good properties and effectively increase the particle diversity.

Claims (2)

1. A nonlinear system state estimation method is characterized in that: the method is realized by the following steps:
step one, initializing k to be 0: from the prior distribution p (x)0) Sampling to obtain NpParticles ofAll the particles have the same importance weight which is 1/N, and the initialized particles enter an iteration process;
step two, from the importance probability density distribution functionThe middle sampling results in particles at time k:
step three, according to the observation likelihood function
Figure FDA0002207291030000014
Calculating the weight of each particle;
step four, optimizing the particles by adopting an improved firefly algorithm, wherein the specific process is as follows:
step four, sorting all the particle weights calculated in the step three to obtain the particle with the maximum weight, and taking the particle as the optimal particle
Figure FDA0002207291030000015
The relative fluorescence brightness of each particle was calculated:
in the formula I0The maximum fluorescence intensity is the maximum fluorescence intensity,gamma is the light intensity absorption coefficient, riAre particles
Figure FDA00022072910300000115
With optimal particles
Figure FDA0002207291030000017
The distance of the space between the two plates,
Figure FDA0002207291030000018
step four, calculating the attraction degree of each particle:
Figure FDA0002207291030000019
wherein beta is0N is the current iteration number for the maximum attraction degree;
step four and step three, updating the position of each particle:
Figure FDA00022072910300000110
wherein rand is [0.1 ]]Subject to uniformly distributed random factors, KiFor the perturbation factor, it is expressed by the following formula:
Figure FDA00022072910300000111
wherein
Figure FDA00022072910300000112
And
Figure FDA00022072910300000113
the average value and the maximum value of the weight values of the particles at the moment k are obtained;
fourthly, calculating the particle weight after the position is updated through formulas (1) and (2), and normalizing the particle weight; the weight of the ith particle after the update is obtained as
Figure FDA00022072910300000114
Step five, obtaining a system state estimation value at the current moment:
Figure FDA0002207291030000021
and step six, judging whether a new observed value exists, if so, returning to the step two, and if not, exiting.
2. The nonlinear system state estimation method according to claim 1, wherein: the concrete process of the third step is as follows: the weight of the ith particle at time k is expressed by the following formula:
Figure FDA0002207291030000022
in the formula, ykFor an observed value that is known at the current time,
Figure FDA0002207291030000023
normalizing the weight value as a state transfer function, wherein the weight value of the normalized particle is as follows:
Figure FDA0002207291030000024
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112583380A (en) * 2020-12-15 2021-03-30 哈尔滨工程大学 Distributed multi-rate particle filtering algorithm based on convergence optimization
CN112613222A (en) * 2021-01-04 2021-04-06 重庆邮电大学 Improved particle filter-based inclination detection ionosphere MUF short-term prediction method
CN113345420A (en) * 2021-06-07 2021-09-03 河海大学 Countermeasure audio generation method and system based on firefly algorithm and gradient evaluation

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112583380A (en) * 2020-12-15 2021-03-30 哈尔滨工程大学 Distributed multi-rate particle filtering algorithm based on convergence optimization
CN112613222A (en) * 2021-01-04 2021-04-06 重庆邮电大学 Improved particle filter-based inclination detection ionosphere MUF short-term prediction method
CN112613222B (en) * 2021-01-04 2023-09-15 重庆邮电大学 Method for short-term prediction of ionized layer MUF by tilt detection based on improved particle filtering
CN113345420A (en) * 2021-06-07 2021-09-03 河海大学 Countermeasure audio generation method and system based on firefly algorithm and gradient evaluation

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