Process Noise Variance Research Articles

Steady-State Performance of Kalman Filter for DPLL

For certain system models, the structure of the Kalman filter is equivalent to a second-order variable gain digital phase-locked loop (DPLL). To apply the knowledge of DPLLs to the design of Kalman filters, this paper studies the steady-state performance of Kalman filters for these system models. The results show that the steady-state Kalman gain has the same form as the DPLL gain. An approximate simple form for the steady-state Kalman gain is used to derive an expression for the equivalent loop bandwidth of the Kalman filter as a function of the process and observation noise variances. These results can be used to analyze the steady-state performance of a Kalman filter with DPLL theory or to design a Kalman filter model with the same steady-state performance as a given DPLL.

Filter Design for Steady-State Tracking of Maneuvering Targets with LFM Waveforms

Modern radars employ linear frequency modulated (LFM) waveforms for target tracking as the range-Doppler coupling enables better track accuracy in range. This paper derives expressions for the sensor-noise only (SNO) covariance matrix and the position and velocity lags during maneuvers for an alphabeta-filter processing measurements from LFM waveforms. These expressions were confirmed through computer simulations. An analysis of the maximum root-mean-square error (RMSE) in position and velocity estimates is performed, and the gains that minimize the maximum RMSE in either position or velocity are plotted versus the tracking index. For deterministic maneuvers, the paper introduces the concept of the deterministic tracking index and shows its relation to the typical tracking index generally used for random maneuvers. Using these relations, the paper also proposes a method to calculate the optimal process noise variance to be used in the tracking filter for a deterministic tracking index.

Riccati Equation and EM Algorithm Convergence for Inertial Navigation Alignment

This correspondence investigates the convergence of a Kalman filter-based expectation-maximization (EM) algorithm for estimating variances. It is shown that if the variance estimates and the error covariances are initialized appropriately, the underlying Riccati equation solution and the sequence of iterations will be monotonically nonincreasing. Further, the process noise variance estimates converge to the actual values when the measurement noise becomes negligibly small. Conversely, when the process noise variance becomes negligible, the measurement noise variance estimates asymptotically approach the true values. An inertial navigation application is discussed in which performance depends on accurately estimating the process variances.

Performance Evaluation of Interacting Multiple Model Kalman Filter

Conventional Kalman filter having constant velocity model (KFcv), conventional Kalman filter having constant acceleration model (KFca) and Interacting Multiple Model Kalman Alter (IMMKF) having constant velocity & constant acceleration models were implemented and their performance was studied. Overall performance of IMMKF was better than the other two. The performance of IMMKF with different process noise variance, measurement noise variance and model transition matrix was explored. It was concluded that Q1 and Q2 higher than their true would give better performance. The measurement noise variance R lesser than the true value gives improved results. The diagonal elements of model transition matrix derived from sojourn time shows superior performance. Results showed that, IMMKF exhibits overall better performance and was more suitable for tracking maneuvering targets than conventional Kalman filters.

Performance Evaluation of Interacting Multiple Model Kalman Filter

Conventional Kalman filter having constant velocity model (KFcv), conventional Kalman filter having constant acceleration model (KFca) and Interacting Multiple Model Kalman Alter (IMMKF) having constant velocity & constant acceleration models were implemented and their performance was studied. Overall performance of IMMKF was better than the other two. The performance of IMMKF with different process noise variance, measurement noise variance and model transition matrix was explored. It was concluded that Q1 and Q2 higher than their true would give better performance. The measurement noise variance R lesser than the true value gives improved results. The diagonal elements of model transition matrix derived from sojourn time shows superior performance. Results showed that, IMMKF exhibits overall better performance and was more suitable for tracking maneuvering targets than conventional Kalman filters.

A Fix-Up for the EKF Parameter Estimator

We have reduced recursive parameter estimation to Kalman filtering, with a few added fixes. By incorporating projections in the parameter gain updates and parameter variance estimates, the recursive maximum likelihood method asymptotically becomes a reformulation and fix-up of the extended Kalman filter used as a parameter estimator (EKFPE), except that an additional n x n linear symmetric matrix must also be updated for each parameter estimate. Estimates for both the process and measurement noise variances, as well as for structural parameters, have been proven globally convergent to a local maximum of the likelihood function. This obviates the usual guesswork in finding noise variances when fitting data using the EKFPE, and assures the existence of the innovations representation for the recursive maximum likelihood method. Slightly non-linear and also slightly unstable linear, as well as drastically time-varying stable linear, system parameters can be estimated even in severe noise environments On average, the rate of convergence of parameter estimates appears to be faster than other methods if no projection limit is hit.

Intelligent tracking algorithm for manoeuvering target using Kalman filter with fuzzy gain

An intelligent filter with fuzzy gain is proposed, which calculates a manoeuver input into the time-varying variance of the overall process noise and compensates an estimation error by using a fuzzy gain. The proposed technique not necessary to predesign the acceleration levels according to the target manoeuver properities. The manoeuver input is esitmated using the difference between the non-manoeuvering filter and the manoeuvering one. To compensate the estimation error, the fuzzy gain is used to describe the relationship between the filter residual and its variation. A fuzzy system is utilised for an universal approximator. It is optimised by using a genetic algorithm. A computer simulations show that the effectiveness of the proposed method is compared with multiple filters.

퍼지 뉴럴 네트워크 기반 다중모델 기법 추적 시스템

본 논문에서는 기동표적의 추적에 대한 새로운 퍼지 뉴럴 네트워크 기반의 다중모델 기법을 소개한다. 표적의 가속도를 효과적으로 다루기 위하여, 이 논문에서는 표적의 가속도를 시변 변수인 표적의 추가적인 잡음으로 두고 각각의 가속도 간격의 정도에 따라 얻어지는 모든 잡음에 대한 변수에 의해 각각의 하부 모델들을 특성화시켰다. 모르는 가속도에 따른 시변 변수를 적응적으로 어립잡기는 어렵기 때문에 정밀한 계산을 위하여 퍼지 뉴럴 네트워크가 이용되었다. 퍼지 뉴럴 네트워크의 동정을 위해서는 오차 역전파 학습법을 사용하였다. 그리고 제안된 알고리즘의 수행 가능성을 보여주기 위하여 몇 가지 예를 제시하였다. This paper presents a new fuzzy-neural-network based interacting multiple model (FNNBIMM) algorithm for tracking a maneuvering target. To effectively handle the unknown target acceleration, this paper regards it as additional noise, time-varying variance to target model. Each sub model characterized by the variance of the overall process noise, which is obtained on the basis of each acceleration interval. Since it is hard to approximate this time-varying variance adaptively owing to the unknown acceleration, the FNN is utilized to precisely approximate this time-varying variance. The error back-propagation method is utilized to optimize each FNN. To show the feasibility of the proposed algorithm, a numerical example is provided.

ESTIMATING DENSITY DEPENDENCE, PROCESS NOISE, AND OBSERVATION ERROR

We describe a discrete-time, stochastic population model with density dependence, environmental-type process noise, and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state-space model (Kalman filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete-time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. We describe two methods for estimating parameters in the Gompertz state-space model, and we compare the statistical qualities of the methods with computer simulations. The methods are maximum likelihood based on observations and restricted maximum likelihood based on first differences. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated-measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. We use the model to analyze a data set from the Breeding Bird Survey. The fitted model suggests that over 70% of the noise in the population's growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accommodate the phenomenon of multimodal likelihood functions to produce statistically valid results.

Adapt the steady-state Kalman gain using the normalized autocorrelation of innovations

A discrete linear time-varying stochastic system with scalar measurement data is considered for which neither measurement noise variance nor power of process noise is known. A novel adaptive Kalman filter that gradually approaches the optimum steady-state gain is proposed in this letter. Different from previous adaptation schemes, our algorithm adjusts the Kalman gain depending on the normalized autocorrelation of the prediction error sequence of the suboptimal filter. We also present how to choose appropriate latency time and sample window length in the adaptation process. In our experiments, the filter shows advantages over several other methods under the same conditions.

Fuzzy-logic-based IMM algorithm for tracking a manoeuvring target

The authors propose a new fuzzy-logic-based interacting multiple model (FIMM) algorithm for tracking a manoeuvring target. The unknown target acceleration is regarded as an additional process noise to the target model, and each sub-model is characterised by the variance of the overall process noise, which is obtained on the basis of each acceleration interval. Since it is hard to approximate this time-varying variance adaptively owing to the unknown acceleration, a fuzzy system is applied as the universal approximator to compute it. To optimise each fuzzy system, a genetic algorithm is utilised. The proposed FIMM algorithm does not require prior information on the statistical properties of the target manoeuvre. An example is included for visualising the effectiveness of the proposed algorithm.

Intelligent Kalman filter for tracking a manoeuvring target

The Kalman filter (KF) has been widely used in the state estimation of a target, but in the presence of a manoeuvre, its performance may be seriously degraded because the manoeuvre appears as extensive noise on the target model and the process noise variance cannot cover it. To solve this problem, a new intelligent KF (IKF) is proposed for tracking a manoeuvring target. The unknown target acceleration is regarded as additive process noise, and the time-varying variance of the overall process noise is computed in an intelligent manner using a fuzzy system as universal approximator. To optimise the fuzzy system, a genetic algorithm (GA) or DNA coding method can be utilised, and the filter is then termed a GA-based IKF or DNA coding-based IKF according to the optimisation tool used. The proposed IKF can effectively treat a target manoeuvre with only one filter and can relax the additional requirements of conventional manoeuvring target tracking methods. The performance of the proposed IKFs is compared with that of multiple model methods through computer simulations.

Image sequence stabilisation using fuzzy adaptive Kalman filtering

Image sequence stabilisation based on fuzzy adaptive Kalman filtering of absolute frame displacements is proposed. Two Kalman filters are operated in parallel, one of which is used as a reference filter and is run with a constant high process noise variance to enable smooth and close tracking of global camera displacements. The process noise variance of the stabilisation filter is adaptively varied according to the residual between the stabilisation filter and the reference filter output.

Real-Time Digital Image Stabilization Using Kalman Filters

This paper presents a novel, real-time stabilization system that uses Kalman filters to remove short-term image fluctuations with retained smooth gross movements. The global camera motion is defined in terms of constant acceleration motion and constant velocity motion models, and Kalman filtering is employed to facilitate smooth operation. It is shown that the process noise variance has a direct effect on stabilization performance, and that it is possible to implement an efficient and robust stabilization system by adaptively changing the process noise variance value according to long-term camera motion dynamics.

Tracking maneuvering targets with multiple sensors: does more data always mean better estimates?

In many multisensor systems the number and type of sensors supporting a particular target track can vary with time due to the mobility, type, and resource limitations of the individual sensors. This variability in the configuration of the sensor system poses a significant problem when tracking maneuvering targets because of the uncertainty in the target motion model. A Kalman filter is often employed to filter the position measurements for estimating the position, velocity, and acceleration of a target. When designing the Kalman filter, the process noise (acceleration) variance Q/sub k/ is selected such that the 65 to 95% probability region contains the maximum acceleration level of the target. However, when targets maneuver, the acceleration changes in a deterministic manner. Thus, the white noise assumption associated with the process noise is violated and the filter develops a bias in the state estimates during maneuvers. The problem of tracking maneuvering targets with multiple sensors is illustrated through an example involving target motion in a single coordinate in which it is shown that with two sensors one can have (under certain conditions that include perfect alignment of the sensors) worse track performance than a single sensor. The Interacting Multiple Model (IMM) algorithm is applied to the illustrative example to demonstrate a potential solution to this problem of track filter performance.

適応カルマン追尾フィルタとそのテザー衛星系への応用

An Kalman filter has been widely used in target tracking. In most circumstances, the exact values of model parameters, however, are unknown and adaptive filtering is necessary. In this paper, a novel target tracking approach based on adaptive Kalman filter is proposed, in which unknown process noise variance is estimated by using the measured data of an actual state. The suggested approach has certain merits over various existing adaptive algorithms and it is simple, efficient and suitable for real-time applications. As an example, the tracking of a tethered subsatellite is analyzed, and the performance of this approach is evaluated by a simulation study on a realistic system.

Tracking targets using adaptive Kalman filtering

A simple algorithm for estimating the unknown process noise variance of an otherwise known linear plant, using a Kalman filter is suggested. The process noise variance estimator is essentially dead beat, using the difference between the expected prediction error variance, computed in the Kalman filter, and the measured prediction error variance. The estimate is used to adapt the Kalman filter. The use of the adaptive filter is demonstrated in a simulated example in which a wildly maneuvering target is tracked. >

Recursive estimation of the observation and process noise covariances in online Kalman filtering

A comparison of several methods for estimating the observation and process noise variances in the Kalman filtering approach to statistical forecasting is undertaken by means of simulated time series analysis. Particular attention is given to the requirement of an online, automatic, forecasting system for recursive estimation procedures.