Filter Gain Matrix Research Articles

Event-based recursive filtering for a class of nonlinear stochastic parameter systems over fading channels

In this paper, the recursive filtering problem is studied for a class of time-varying nonlinear systems with stochastic parameter matrices. The measurement transmission between the sensor and the filter is conducted through a fading channel characterized by the Rice fading model. An event-based transmission mechanism is adopted to decide whether the sensor measurement should be transmitted to the filter. A recursive filter is designed such that, in the simultaneous presence of the stochastic parameter matrices and fading channels, the filtering error covariance is guaranteed to have an upper bound and such an upper bound is then minimized by appropriately choosing filter gain matrix. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed filtering scheme.

The Square Root UKF-SLAM Algorithm Based on the Smallest Proportion of Skewness in Single Line Sampling

In view of the distortion in the filter gain matrix calculation as well as the high computational complexity and the nonlocal effect of symmetric sampling that exists in the UKF-SLAM algorithm, the square root UKF-SLAM algorithm based on the smallest proportion of skewness in single line sampling was proposed. According to the mended algorithm, the square root of covariance matrix is brought into iteration operation instead of covariance matrix, moreover, the smallest proportion of skewness in single line sampling is utilized for the optimization of sampling strategy. The results of simulation show that the algorithm can effectively improve the position accuracy in robot as well as the estimation accuracy of feature map. Furthermore, the computational complexity is reduced and the algorithm stability is improved.

State estimation for stochastic discrete-time systems with multiplicative noises and unknown inputs over fading channels

This paper deals with robust state estimation problem for a class of stochastic discrete-time systems with multiplicative noises and unknown inputs over fading channels. An unbiased unknown input insensitive filter is designed such that the variance of the estimation error is minimized in the sense of the so-called P-estimation. The filter gain matrix is derived through solving a recursive Riccati equation and a generalized Lyapunov equation. A necessary and sufficient condition that guarantees the existence of the filter is given, which establishes a fundamental limit on the mean square capacity of each fading channel. Unknown input estimation and finite horizon stability of the proposed filter are also discussed. To illustrate the effectiveness of the proposed approach, the proposed algorithm is applied to a faulty remote controlled uninterruptible power system, where both the state and the fault are estimated.

Resilient Dissipative Filtering for Uncertain Markov Jump Nonlinear Systems with Time-Varying Delays

The resilient dissipative filtering problem for a class of uncertain time-delay Markov jump nonlinear systems is studied in this paper. The nonlinear functions are assumed to belong to sector sets with arbitrary boundaries. The sector boundaries can have positive and/or negative slopes, and therefore, we cover the most general case in our approach. Using the special structure of the system, and by constructing a new multiple Lyapunov–Krasovskii function, the sufficient conditions regarding the existence of desired resilient dissipative filters are obtained in terms of linear matrix inequalities, which ensure the filtering error system is stochastically stable and strictly dissipative. The designed filter can tolerate additive uncertainties in the filter gain matrix, which results from filter implementations. A numerical example is presented to show the effectiveness of the proposed theoretical results.

Envelope-constrained [formula omitted] filtering with fading measurements and randomly occurring nonlinearities: The finite horizon case

In this paper, the envelope-constrained H∞ filtering problem is investigated for a class of discrete time-varying stochastic systems over a finite horizon. The system under consideration involves fading measurements, randomly occurring nonlinearities (RONs) and mixed (multiplicative and additive) noises. A novel envelope-constrained performance criterion is proposed to better quantify the transient dynamics of the filtering error process over the finite horizon. The purpose of the problem addressed is to design a time-varying filter such that both the H∞ performance and the desired envelope constraints are achieved at each time step. By utilizing the stochastic analysis techniques combined with the ellipsoid description on the estimation errors, sufficient conditions are established in the form of recursive matrix inequalities (RMIs) reflecting both the envelope information and the desired H∞ performance index. The filter gain matrix is characterized by means of the solvability of the deduced RMIs. Finally, a simulation example is provided to show the effectiveness of the proposed filtering design scheme.

On designing an extended Hi/H∞-FDF for a class of nonlinear Systems

This paper investigates the problem of fault detection (FD) for a class of discrete-time nonlinear systems by using Hi/H∞ optimization. A finite horizon index Hi and H∞ norm are introduced to represent the sensitivity of residual to fault and robustness of residual against unknown input, respectively. Then the problem of residual evaluation is formulated as to a nonlinear Hi/H∞ optimization. It is shown that a local optimal Hi/H∞ FDF can be obtained by recursively computing of extended Kalman filter and a post-filter. Moreover, the local optimal solution is not unique and, for an arbitrary feasible filter gain matrix, a dynamic post-filter is also presented. A numerical example is given to demonstrate the effectiveness of the proposed method.

An Improved Strong Tracking UKF Based on Fading Factor

STUKF (Strong tracking UKF) algorithm uses the time-varying fading factor to fade the past data and reduce the impact on current filter value, thus achieves the goal of adjusting the filter gain matrix in real time. But STUKF algorithm needs three UT for each filtering, and compared with the UKF filter, calculating amount of three UT increases seriously, and it is not conducive to application of engineering, therefore this paper presents an improved STUKF algorithm. Compared with the traditional STUKF filter, this new algorithm introduces the formulas of redefined fading factor. By changing the position of the fading factor, it improves the accuracy and robustness of the algorithm and reduces the computational complexity of the algorithm. Finally simulation results show that the new algorithm has higher precision and stronger robustness.

Model-based robust estimation and fault detection for MEMS-INS/GPS integrated navigation systems

In micro-electro-mechanical system based inertial navigation system (MEMS-INS)/global position system (GPS) integrated navigation systems, there exist unknown disturbances and abnormal measurements. In order to obtain high estimation accuracy and enhance detection sensitivity to faults in measurements, this paper deals with the problem of model-based robust estimation (RE) and fault detection (FD). A filter gain matrix and a post-filter are designed to obtain a RE and FD algorithm with current measurements, which is different from most of the existing priori filters using measurements in one-step delay. With the designed filter gain matrix, the H-infinity norm of the transfer function from noise inputs to estimation error outputs is limited within a certain range; with the designed post-filter, the residual signal is robust to disturbances but sensitive to faults. Therefore, the algorithm can guarantee small estimation errors in the presence of disturbances and have high sensitivity to faults. The proposed method is evaluated in an integrated navigation system, and the simulation results show that it is more effective in position estimation and fault signal detection than priori RE and FD algorithms.

An Efficient Nonlinear Filter for Spacecraft Attitude Estimation

Increasing the computational efficiency of attitude estimation is a critical problem related to modern spacecraft, especially for those with limited computing resources. In this paper, a computationally efficient nonlinear attitude estimation strategy based on the vector observations is proposed. The Rodrigues parameter is chosen as the local error attitude parameter, to maintain the normalization constraint for the quaternion in the global estimator. The proposed attitude estimator is performed in four stages. First, the local attitude estimation error system is described by a polytopic linear model. Then the local error attitude estimator is designed with constant coefficients based on the robustH2filtering algorithm. Subsequently, the attitude predictions and the local error attitude estimations are calculated by a gyro based model and the local error attitude estimator. Finally, the attitude estimations are updated by the predicted attitude with the local error attitude estimations. Since the local error attitude estimator is with constant coefficients, it does not need to calculate the matrix inversion for the filter gain matrix or update the Jacobian matrixes online to obtain the local error attitude estimations. As a result, the computational complexity of the proposed attitude estimator reduces significantly. Simulation results demonstrate the efficiency of the proposed attitude estimation strategy.

Improved Minimum Entropy Filtering for Continuous Nonlinear Non-Gaussian Systems Using a Generalized Density Evolution Equation

This paper investigates the filtering problem for multivariate continuous nonlinear non-Gaussian systems based on an improved minimum error entropy (MEE) criterion. The system is described by a set of nonlinear continuous equations with non-Gaussian system noises and measurement noises. The recently developed generalized density evolution equation is utilized to formulate the joint probability density function (PDF) of the estimation errors. Combining the entropy of the estimation error with the mean squared error, a novel performance index is constructed to ensure the estimation error not only has small uncertainty but also approaches to zero. According to the conjugate gradient method, the optimal filter gain matrix is then obtained by minimizing the improved minimum error entropy criterion. In addition, the condition is proposed to guarantee that the estimation error dynamics is exponentially bounded in the mean square sense. Finally, the comparative simulation results are presented to show that the proposed MEE filter is superior to nonlinear unscented Kalman filter (UKF).

In-Flight Alignment Algorithm Based on ADD2 for Airborne POS

The Position and Orientation System (POS) is a special Strapdown Inertial Navigation System (SINS)/Global Positioning System (GPS) integrated system, widely employed in airborne remote sensing. In-Flight Alignment (IFA) is an effective way to improve the accuracy and speed of initial alignment for an airborne POS. IFA is normally accomplished with references from the position and velocity of GPS for SINS, so that unstable GPS measurements will result in poor alignment accuracy. To improve alignment accuracy under unstable GPS conditions, an adaptive filtering algorithm of the Second-order Divided Difference filter (DD2) based on adaptive innovation estimation is proposed, which introduces calculated innovation covariance directly into computation of the filter gain matrix. Then, the adaptive DD2 algorithm is used for the IFA of the POS with a large initial heading error. To validate the proposed algorithm, simulations are undertaken, followed by IFA experiments for the prototype of the airborne POS (TX-F30) under a turning manoeuvre in a car-mounted experiment, and under an “8” manoeuvre in-flight. The simulations and experimental results show that the proposed algorithm can reach better alignment accuracy under unknown statistical characteristic of GPS measurement noises.

Study on Innovation Adaptive EKF for In-Flight Alignment of Airborne POS

The Position and Orientation measurement System (POS) is a dedicated Strapdown Inertial Navigation System (SINS)/Global Positioning System (GPS) integrated system for airborne remote sensing. In-flight alignment (IFA) is an effective way to improve the accuracy and speed of initial alignment for an airborne POS. During IFA, the GPS provides the position and velocity references for the SINS, so the alignment accuracy will be degraded by unstable GPS measurements. To improve the alignment accuracy under unstable GPS measurement, an adaptive filtering algorithm of an extended Kalman filter (EKF) combined with innovation-based adaptive estimation is proposed, which introduces the calculated innovation covariance into the computation of the filter gain matrix directly. Then, this innovation adaptive EKF algorithm is used for the IFA of the POS with a large initial heading error. Moreover, it is optimized by blocked matrix multiplication to reduce the computational burden and improve the real-time performance. To validate the proposed algorithm, the car-mounted IFA experiment is carried out for the prototype of the airborne POS (TX-D10) under a turning maneuver, taking Applanix's POS/AV510 as a reference and changing the GPS measurement artificially. The experiment results demonstrate that the proposed algorithm can reach a better alignment accuracy than the EKF under unknown GPS measurement noises.

A closed-form solution to the discrete-time Kalman filter and its applications

This paper presents a closed-form solution to the discrete-time Kalman filter and its applications. We first represent the Kalman filter in terms of model parameters without using the Riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises. Replacing the initial time with the fixed-lag time to achieve the finite memory with respect to inputs and outputs, and choosing the proper initial covariances on the recent finite horizon, we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm. As another application, a closed-form solution to the difference Riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a Luenberger-type filter as in Ackermann’s formula.

${\cal H}_{\infty }$ Filter for Bilinear Systems Using LPV Approach

The aim of this technical note is to present a linear paramater varying (LPV) approach for the design of a functional filter for bilinear systems with a disturbance attenuation specification. The unbiasedness requirement on the observation error is guaranteed by the solution of Sylvester equations obtained by parametrizing the filter gain matrix. This parametrization leads to a non convex optimization problem which is overcome by introducing a constraint on the gain matrix. To take into account the whole set of the inputs in the filter design, an LPV approach is used to obtain the gain matrix guaranteeing the quadratic stability of the observation error and a given level of disturbance attenuation. This approach is then applied to the high gain observer design in order to consider the level of disturbance attenuation.

Distribution function tracking filter design using hybrid characteristic functions

A new tracking filtering algorithm for a class of multivariate dynamic stochastic systems is presented. The system is expressed by a set of time-varying discrete systems with non-Gaussian stochastic input and nonlinear output. A new concept, such as hybrid characteristic function, is introduced to describe the stochastic nature of the dynamic conditional estimation errors, where the key idea is to ensure the distribution of the conditional estimation error to follow a target distribution. For this purpose, the relationships between the hybrid characteristic functions of the multivariate stochastic input and the outputs, and the properties of the hybrid characteristic function, are established. A new performance index of the tracking filter is then constructed based on the form of the hybrid characteristic function of the conditional estimation error. An analytical solution, which guarantees the filter gain matrix to be an optimal one, is then obtained. A simulation case study is included to show the effectiveness of the proposed algorithm, and encouraging results have been obtained.

Optimal filtering for linear systems with state and multiple observation delays

This article solves the optimal filtering problem for linear systems with state and multiple observation delays. The optimal estimate equation similar to the traditional Kalman–Bucy one is derived, and the system of equations for determining the filter gain matrix consists of an infinite set of equations. It is then demonstrated that a finite set of the filtering equations can be obtained in case of commensurable delays. In the example, the designed optimal filter is compared to the traditional Kalman–Bucy filter.

Design of Critical Control Systems for Non-Minimum Phase Plants via LTR Technique

An application of the loop transfer recovery (LTR) technique to critical control systems design is proposed for non-minimum phase plants. The controller structure is chosen as the Davison type integral controller with the Kalman filter. First, a critical control system is designed on the assumption that the state of the minimum phase part of the plant can be used for the feedback. A quadratic performance index with tuning parameters is used for determining the partial state feedback gain matrix. Second, the Kalman filter gain matrix is determined such that the output feedback controller performs as in the partial state feedback controller. The formal partial loop recovery procedure using the Riccati equation is adopted for this purpose. The proposed design method requires much simpler numerical search than the conventional one-step approach. An illustrative design example is presented.

Optimal filtering for linear systems with state and observation delays

AbstractIn this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman–Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman–Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman–Bucy filter available for linear systems without delays and two versions of the extended Kalman–Bucy filter for time delay systems. Copyright © 2005 John Wiley & Sons, Ltd.

OPTIMAL LINEAR FILTERING WITH STATE AND OBSERVATION DELAYS

In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman-Bucy one is derived; however, the resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is then demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman-Bucy filter available for linear systems without delays.

On regional nonlinear H∞-filtering

The structure of the nonlinear H ∞-filter in the neighborhood of the estimated trajectory is investigated and a bound on the size of the neighborhood that allows this structure is determined, both for finite and infinite horizons. Riccati inequalities that depend on the estimated trajectory are derived for finding the filter gain matrix and an algorithm for calculating the bound on the size of the above neighborhood is presented. Explicit formulas are obtained in the infinite horizon case for the minimum achievable disturbance attenuation level, the size of the neighborhood, and the corresponding filter gain.