Reduced order state estimators for discrete-time stochastic systems

A reduced order, least squares, state estimator is developed for linear discrete-time systems having both input disturbance noise and output measurement noise with no output being free of measurement noise. The order reduction is achieved by using a Luenberger observer in connection with some of the system outputs and a Kalman filter to estimate the state of the Luenberger observer. The order of the resulting state estimator is reduced from the order of the usual Kalman filter system state estimator by the number of system outputs selected for use as inputs to the Luenberger Observer. The manner in which the noise associated with the selected system outputs affects the state estimation error covariance provides considerable insight into the compromise being attempted.     

Self-tuning weighted measurement fusion white noise deconvolution estimator and its convergence analysis

White noise deconvolution or input white noise estimation has a wide range of applications including oil seismic exploration, communication, signal processing, and state estimation. For the multisensor linear discrete time-invariant stochastic systems with correlated measurement noises, and with unknown ARMA model parameters and noise statistics, the on-line AR model parameter estimator based on the Recursive Instrumental Variable (RIV) algorithm, the on-line MA model parameter estimator based on Gevers–Wouters algorithm and the on-line noise statistic estimator by using the correlation method are presented. Using the Kalman filtering method, a self-tuning weighted measurement fusion white noise deconvolution estimator is presented based on the self-tuning Riccati equation. It is proved that the self-tuning fusion white noise deconvolution estimator converges to the optimal fusion steady-state white noise deconvolution estimator in a realization by using the dynamic error system analysis (DESA) method, so that it has the asymptotic global optimality. The simulation example for a 3-sensor system with the Bernoulli–Gaussian input white noise shows its effectiveness.     

Combined parameter and output estimation of dual-rate systems using an auxiliary model

For a dual-rate sampled-data system, an auxiliary model based identification algorithm for combined parameter and output estimation is proposed. The basic idea is to use an auxiliary model to estimate the unknown noise-free output (true output) of the system, and directly to identify the parameters of the underlying fast single-rate model from the dual-rate input-output data. It is shown that the parameter estimation error consistently converges to zero under generalized or weak persistent excitation conditions and unbounded noise variance, and that the output estimates uniformly converge to the true outputs. An example is included.     

A nonlinear set‐membership filter for on‐line applications

A guaranteed estimator for a general class of nonlinear systems and on‐line usage is developed and analysed. This filter bounds the linearization error, then applies a linear set‐membership filter such that stability guarantees hold for nonlinear systems. A tight bound on the linearization error is found using interval analysis. This filter recursively estimates an ellipsoidal set in which the true state lies. General assumptions include the use of bounded noises and twice continuously differentiable dynamics. When the system is uniformly observable, it is proven that the nonlinear set‐membership filter is stable. In addition, if no noise is present and the initial error is small, the error between the centre of the estimated set and the true value converges to zero. The result is an estimator which is computationally attractive and can be implemented robustly in real‐time. The proposed method is applied to a two‐state example to demonstrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Observer‐based Fault Estimators Using Iterative Learning Scheme for Linear Time‐delay Systems with Intermittent Faults

This paper deals with the fault estimation problem for a class of linear time‐delay systems with intermittent fault and measurement noise. Different from existing observer‐based fault estimation schemes, in the proposed design, an iterative learning observer is constructed by using the integrated errors composed of state predictive error and tracking error in the previous iteration. First of all, Lyapunov function including the information of time delay is proposed to guarantee the convergence of system output. Subsequently, a novel fault estimation law based on iterative learning scheme is presented to estimate the size and shape of various fault signals. Upon system output convergence analysis, we proposed an optimal function to select appropriate learning gain matrixes such that tracking error converges to zero, simultaneously to ensure the robustness of the proposed iterative learning observer which is influenced by measurement noise. Note that, an improved sufficient condition for the existence of such an estimator is established in terms of the linear matrix inequality (LMI) by the Schur complements and Young relation. In addition, the results are both suit for the systems with time‐varying delay and the systems with constant delay. Finally, three numerical examples are given to illustrate the effectiveness of the proposed methods and two comparability examples are provided to prove the superiority of the algorithm.     

自适应SVD–UKF算法及在穿刺状态估计中的应用

柔性针在实际穿刺过程中会产生不规则形变, 导致柔性针模型存在参数不确定性问题, 影响穿刺精度. 本文针对柔性针穿刺过程存在的不确定性问题以及超声成像等设备存在的量测噪声统计特征不准确性问题, 提出了一种带有噪声估计器的自适应奇异值分解无迹卡尔曼滤波算法. 该算法采用自适应因子实时修正动力学模型误差, 通过奇异值分解抑制系统状态协方差矩阵的负定性, 利用Sage-Husa估计器在线估计噪声的统计特性, 减小了系统状态估计误差. 将新算法应用于带有曲率不定性的柔性针穿刺模型进行计算仿真, 仿真结果显示, 新的算法较现有的UKF算法相比, 估计误差减小了0.28 mm(82.7%), 与AUKF算法相比, 估计误差减小0.06 mm(52%). 因此, 新算法可有效改善滤波性能, 提高穿刺状态的估计精度.     

Unknown disturbance inputs estimation based on a state functional observer design

We consider the problem of function of state plus unknown input estimation of a linear time-invariant system using only the measured outputs. Two reduced-order input estimators built upon a state functional observer are proposed. The first is an extension of a state/input estimator, while the second is based on adaptive observer design technique. The proposed estimator can be designed under less restrictive conditions than those of the previous work, and unlike some of the past studies the proposed observer can be designed for certain nonminimum phase systems.     

Moving data window gradient‐based iterative algorithm of combined parameter and state estimation for bilinear systems

The combined iterative parameter and state estimation problem is considered for bilinear state‐space systems with moving average noise in this paper. There are the product terms of state variables and control variables in bilinear systems, which makes it difficult for the parameter and state estimation. By designing a bilinear state estimator based on the Kalman filtering, the states are estimated using the input‐output data. Furthermore, a moving data window (MDW) is introduced, which can update the dynamical data by removing the oldest data and adding the newest measurement data. A state estimator‐based MDW gradient‐based iterative (MDW‐GI) algorithm is proposed to estimate the unknown states and parameters jointly. Moreover, given the extended gradient‐based iterative (EGI) algorithm as a comparison, the MDW‐GI algorithm can reduce the impact of noise to parameter estimation and improve the parameter estimation accuracy. The numerical simulation examples validate the effectiveness of the proposed algorithm.     

On asymptotic behaviors of a sensitivity penalization based robust state estimator

Asymptotic properties are investigated in this paper for the robust state estimator derived by Zhou (2008) [11]. A new formula is derived for the update of the pseudo-covariance matrix of estimation errors. In the case where plant nominal parameters are time invariant, it is shown that, in order to guarantee that this pseudo-covariance matrix converges to a constant positive definite matrix, it is necessary and sufficient that some stabilizability and detectability conditions are satisfied. It is also proved that when these conditions are satisfied, the robust estimator converges to a stable time-invariant system. Moreover, when the system is exponentially stable, this estimate is asymptotically unbiased and its estimation errors are upper bounded.     

A study of uncertain state estimation

In this paper, we present results of uncertain state estimation of systems that are monitored with limited accuracy. For these systems, the representation of state uncertainty as confidence intervals offers significant advantages over the more traditional approaches with probabilistic representation of noise. While the filtered-white-Gaussian noise model can be defined on grounds of mathematical convenience, its use is necessarily coupled with a hope that an estimator with good properties in idealised noise will still perform well in real noise. In this study we propose a more realistic approach of matching the noise representation to the extent of prior knowledge. Both interval and ellipsoidal representation of noise illustrate the principle of keeping the noise model simple while allowing for iterative refinement of the noise as we proceed. We evaluate one nonlinear and three linear state estimation technique both in terms of computational efficiency and the cardinality of the state uncertainty sets. The techniques are illustrated on a synthetic and a real-life system.     

Two approaches to partial-nodes-based state estimation for delayed complex networks with intermittent measurement transmissions

This paper is concerned with the state estimation problem for delayed complex dynamic networks with non-identical local dynamical systems. The state estimation is conducted based on constrained information of the measurement outputs. Specifically, the network outputs are available only from a portion of network nodes, and such outputs are transmitted from the network nodes to the estimator in an intermittent way. By utilizing the Halanay inequality method as well as the average dwell-time approach, two sets of sufficient conditions are established that ensure the error dynamics of the state estimation to converge to zero exponentially, and explicit expressions of the estimator gains are further characterized. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed approaches.     

PI velocity control of brushless DC motors using the extended dq transformation

This paper is concerned with velocity control in brushless direct current (BLDC) motors. Our control scheme is composed of a classical proportional–integral velocity controller and a proportional electric current controller computed in the extended dq coordinate system. We employ a finite time estimator that has been previously proposed in the literature intended to estimate the unknown back electromotive forces. This estimator has been proven previously to converge locally to the true values. This control strategy is proposed to eliminate torque ripple produced by the fact that back electromotive forces in BLDC motors are nonsinusoidal. We present, for the first time, a formal stability result for this control scheme: (a) the state has an ultimate bound provided that the estimate of the back electromotive forces is close to the true values, and (b) this ultimate bound can be rendered arbitrarily small when the estimate of the back electromotive forces converges to the true values. This result stands when starting from any initial condition such that the finite time estimator is ensured to converge to the true values. We verify our findings through simulations.     

On extended state based Kalman flter for nonlinear time‑varying uncertain systems with measurement bias

This paper addresses the state estimation for a class of nonlinear time-varying stochastic systems with both uncertain dynamics and unknown measurement bias. A novel extended state based Kalman flter (ESKF) algorithm is developed to estimate the original state, the uncertain dynamics and the measurement bias. It is shown that the estimation error of the proposed algorithm is bounded in the mean square sense. Also, the estimation of the measurement bias asymptotically converges to its true value, such that the infuence of measurement bias is eliminated. Furthermore, the asymptotic optimality of the estimation result is proved while the uncertain dynamics approaches to a constant vector. Finally, a simulation study for harmonic oscillator system model is provided to illustrate the efectiveness of proposed method.     

A New Approach To Optimal and Self‐Tuning Filtering,Smoothing and Prediction For Discrete‐Time Systems

A new approach to optimal and self‐tuning state estimation of linear discrete time‐invariant systems is presented, using projection theory and innovation analysis method in time domain. The optimal estimators are calculated by means of spectral factorization. The filter, predictor, and smoother are given in a unified form. Comparisons are made to the previously known techniques such as the Kalman filtering and the polynomial method initiated by Kucera. When the noise covariance matrices are not available, self‐tuning estimators are obtained through the identification of an ARMA innovation model. The self‐tuning estimator asymptotically converges to the optimal estimator.     

White noise estimators for networked systems with packet dropouts

This paper studies the optimal and suboptimal deconvolution problems over a network subject to random packet losses, which are modeled by an independent identically distributed Bernoulli process. By the projection formula, an optimal input white noise estimator is first presented with a stochastic Kalman filter. We show that this obtained deconvolution estimator is time-varying, stochastic, and it does not converge to a steady value. Then an alternative suboptimal input white-noise estimator with deterministic gains is developed under a new criterion. The estimator gain and its respective error covariance-matrix information are derived based on a new suboptimal state estimator. It can be shown that the suboptimal input white-noise estimator converges to a steady-state one under appropriate assumptions.     

Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes

A moving-horizon state estimation problem is addressed for a class of nonlinear discrete-time systems with bounded noises acting on the system and measurement equations. As the statistics of such disturbances and of the initial state are assumed to be unknown, we use a generalized least-squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding-window strategy. For the resulting estimator, the existence of bounding sequences on the estimation error is proved. In the absence of noises, exponential convergence to zero is obtained. Moreover, suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value. The approximate solution can be determined either on-line by directly minimizing the cost function or off-line by using a nonlinear parameterized function. Simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended Kalman filter.     

Closed-form unbiased frequency estimation of a noisy sinusoid using notch filters

In this note, the problem of the frequency estimation of a sinusoid embedded in white noise is considered. The approach used herein is the minimization of the sample variance of the output of constrained notch filters fed by the noisy sinusoid. In particular, this note focuses on closed-form expressions of the frequency estimate, which can be obtained using notch filters having an all-zeros finite-impulse response (FIR) structure. The results presented in this note are as follows: 1) it is shown that the FIR notch filters obtained from standard second-order infinite-impulse response (IIR) filters are inadequate; 2) a new second-order IIR notch filter is proposed, which provides an unbiased estimate of the frequency; 3) the FIR filter obtained from the new IIR filter provides a closed-form unbiased frequency estimate; and 4) the closed-form frequency estimate obtained using the new FIR notch filter asymptotically converges toward the Pisarenko harmonic decomposition estimator and the Yule-Walker estimator.     

An alternative approach to state estimation in linear time-invariant systems

An alternative approach to state estimation problem in linear, time-invariant dynamic systems is presented in this paper. The approach developed first identifies the initial state of the system by using a proportional plus integral parameter identification method. The Lyapunov design technique is used to guarantee the asymptotic convergence of the initial state identifier. A state estimator is then constructed to operate in series with the initial state identifier. The estimator generates an estimate of the unobserved part of the system state. Simulation studies have shown that satisfactory state estimation can be achieved in the presence of measurement or disturbance noise. An example problem is considered to demonstrate the response characteristics of the estimator-identifier combination.     

State estimation for complex systems with randomly occurring nonlinearities and randomly missing measurements

This paper is concerned with the state estimation problem for the complex networked systems with randomly occurring nonlinearities and randomly missing measurements. The nonlinearities are included to describe the phenomena of nonlinear disturbances which exist in the network and may occur in a probabilistic way. Considering the fact that probabilistic data missing may occur in the process of information transmission, we introduce the randomly data missing into the sensor measurements. The aim of this paper is to design a state estimator to estimate the true states of the considered complex network through the available output measurements. By using a Lyapunov functional and some stochastic analysis techniques, sufficient criteria are obtained in the form of linear matrix inequalities under which the estimation error dynamics is globally asymptotically stable in the mean square. Furthermore, the state estimator gain is also obtained. Finally, a numerical example is employed to illustrate the effectiveness of the proposed state estimation conditions.     

Improving the state estimation for optimal control of stochastic processes subject to multiplicative noise

Computational models for the neural control of movement must take into account the properties of sensorimotor systems, including the signal-dependent intensity of the noise and the transmission delay affecting the signal conduction. For this purpose, this paper presents an algorithm for model-based control and estimation of a class of linear stochastic systems subject to multiplicative noise affecting the control and feedback signals. The state estimator based on Kalman filtering is allowed to take into account the current feedback to compute the current state estimate. The optimal feedback control process is adapted accordingly. The resulting estimation error is smaller than the estimation error obtained when the current state must be predicted based on the last feedback signal, which reduces variability of the simulated trajectories. In particular, the performance of the present algorithm is good in a range of feedback delay that is compatible with the delay induced by the neural transmission of the sensory inflow.