Steady-state Kalman Filter Research Articles

Estimating High Dimensional Multivariate Stochastic Volatility Models

This paper proposes tree main results that enable the estimation of high dimensional multivariate stochastic volatility models. The first result is the closed-form steady-state Kalman filter for the multivariate AR(1) plus noise model. The second result is an accelerated EM algorithm for parameters estimation. The third result is an estimator of the correlation of two elliptical random variables with time-varying variances that is consistent and asymptotically normal regardless of the variances evolution. Speed and precision of our methodology are evaluated in a simulation experiment. Finally, we implement our method and compare its performance with other approaches in a minimum variance portfolio composed by the constituents of the CAC40 and S&P100 indexes.

On the Identification of Noise Covariances and Adaptive Kalman Filtering: A New Look at a 50 Year-Old Problem.

The Kalman filter requires knowledge of the noise statistics; however, the noise covariances are generally unknown. Although this problem has a long history, reliable algorithms for their estimation are scant, and necessary and sufficient conditions for identifiability of the covariances are in dispute. We address both of these issues in this paper. We first present the necessary and sufficient condition for unknown noise covariance estimation; these conditions are related to the rank of a matrix involving the auto and cross-covariances of a weighted sum of innovations, where the weights are the coefficients of the minimal polynomial of the closed-loop system transition matrix of a stable, but not necessarily optimal, Kalman filter. We present an optimization criterion and a novel six-step approach based on a successive approximation, coupled with a gradient algorithm with adaptive step sizes, to estimate the steady-state Kalman filter gain, the unknown noise covariance matrices, as well as the state prediction (and updated) error covariance matrix. Our approach enforces the structural assumptions on unknown noise covariances and ensures symmetry and positive definiteness of the estimated covariance matrices. We provide several approaches to estimate the unknown measurement noise covariance R via post-fit residuals, an approach not yet exploited in the literature. The validation of the proposed method on five different test cases from the literature demonstrates that the proposed method significantly outperforms previous state-of-the-art methods. It also offers a number of novel machine learning motivated approaches, such as sequential (one sample at a time) and mini-batch-based methods, to speed up the computations.

A Novel Repetitive Controller Assisted Phase-Locked Loop with Self-Learning Disturbance Rejection Capability for Three-Phase Grids

The synchronization between the power grid and distributed power sources is a crucial issue in the concept of smart grids. For tracking the real-time frequency and phase of three-phase grids, phase-locked loop (PLL) technology is commonly used. Many existing PLLs with enhanced disturbance/harmonic rejection capabilities, either fail to maintain fast response or are not adaptive to grid frequency variations or have high computational complexity. This article, therefore, proposes a low computational burden repetitive controller (RC) assisted PLL (RCA-PLL) that is not only effective on harmonic rejection but also has remarkable steady-state performance while maintaining fast dynamic. Moreover, the proposed PLL is adaptive to variable frequency conditions and can self-learn the harmonics to be canceled. The disturbance/harmonic rejection capabilities together with dynamic and steady-state performances of the RCA-PLL have been highlighted in this article. The proposed approach is also experimentally compared to the synchronous rotation frame PLL (SRF-PLL) and the steady-state linear Kalman filter PLL (SSLKF-PLL), considering the effect of harmonics from the grid-connected converters, unbalances, sensor scaling errors, dc offsets, grid frequency variations, and phase jumps. The computational burden of the RCA-PLL is also minimized, achieving an experimental execution time of only $12~\mu \text{s}$ .

Integrated system identification and modal state estimation for control of flexible space structures

A novel approach of integrated system identification and modal state estimation is proposed for control of linear dynamical systems including flexible space structures. There are four steps involved in this approach. First, the relation between a stochastic state space model of a dynamical system and the coefficients of its autoregressive model with exogenous input is derived. Second, an adaptive least-squares transversal predictor is used to estimate the coefficients of the model. Third, a state space model and a steady state Kalman filter gain of the dynamical system are then identified from the coefficients of the model by using the eigensystem realization algorithm. Fourth, a modal state estimator is constructed using the modal parameters of the identified model. On-line implementation of this algorithm can continually improve the modal parameters and the filter gain. It can also gradually update the system model when the system characteristics are slowly changing. A numerical example is used to illustrate the feasibility of the new approach.

Frequency stability analysis of pulsar‐aided clocks

This paper discusses the use of a pulsar signal in clock calibration. The analysis method is presented from the perspective of filtering theory, where Hadamard variance is used to analyze the frequency stability of both the stand-alone and the pulsar-aided clock systems. The aided clock is constructed using a steady state Kalman filter and accounts for the fractional power spectral density of the pulsar timing noise. The results of this simplified analysis indicate that the frequency stability of the aided clock has noticeable improvement at large averaging time due to the suppression of the random walk frequency noise in the stand-alone clock.

Speed Sensor Fault Tolerant PMSM Machines: From Position-Sensorless to Sensorless Control

New sensorless observers (i.e., from stator currents/voltages measurements only), to be included into a simple observer-based sensorless control for the tracking of non-definitely zero speed references in nonsalient-pole surface permanent magnet synchronous machines (PMSMs), are proposed. They are obtained through a “minimum distance” modification from recently presented position-sensorless observers (i.e., from rotor speed and stator currents/voltages measurements only). The rotor speed estimate is here directly provided by the phase locked loop (PLL)-based third-order steady-state linear kalman filter (SSLKF) that has been previously used to mitigate the distortions on the estimated position. Experimental results illustrate the effectiveness of the proposed approach in a speed sensor fault-tolerant scenario.

A Novel Iterative Learning Control Approach Based on Steady-State Kalman Filtering

This paper presents a novel off-line iterative learning control algorithm for multiple-input–multiple-output time-varying discrete stochastic systems. Using the steady-state Kalman filtering method, we provide a novel framework for the selection of optimal/sub-optimal fixed learning gain matrices in real applications, which is convenient for engineers. Meanwhile, this framework considerably decreases the calculation about the operations of inverting matrix by introducing a matrix Riccati equation at every iteration. It is strictly proved that the input error covariance converges to its steady-state value asymptotically in the mean square sense, and accordingly, the tracking error covariance also converges. The numerical simulations verify the theoretical results.

Steady-State Linear Kalman Filter-Based PLLs for Power Applications: A Second Look

In three-phase power and energy applications, the synchronous reference frame phase-locked loop (SRF-PLL) is a popular tool for synchronization purposes. The SRF-PLL can be easily and effectively customized for different scenarios by changing its loop filter. Recently, some supposedly different PLLs using the steady-state linear Kalman filter have been developed. The main aim of this letter is to analyze these PLLs. It is demonstrated that they are actually equivalent to some well-known SRF-PLL structures and, therefore, provide no advantage compared to them.

Numerical and Experimental Analyses of a Smart Fin Using Piezoelectric Actuator

This paper introduces the comprehensive design of a thin piezoelectric-driven fin. Because of its hinge and flexible spar component having a low torsional stiffness with a thin composite actuator, the divergence speed is also predicted to be low, which is ascertained by a wind-tunnel test. As the divergence is related to the first pitch mode, the control input magnitude may be significantly reduced at the divergence speed. An improved design of the fin is also suggested to ensure its operability up to Mach 0.6. Its equivalent aeroelastic analysis is conducted by rational function approximation, and a linear quadratic regulator control law is designed using a steady-state Kalman filter with disturbance and an integral state for the tip deflection state augmentation. Both open- and closed-loop simulations are conducted while considering the input voltage saturation and other nonlinear losses in the actuator. In the relevant simulation, it is shown that a sufficient amount of pitch rotation can be achieved without any overshoots while consuming half of the zero-load voltage. Additionally, an external disturbance such as a gust can be sufficiently suppressed by increasing the closed-loop bandwidth.

Rapid calibration of an intracortical brain–computer interface for people with tetraplegia

Objective. Brain–computer interfaces (BCIs) can enable individuals with tetraplegia to communicate and control external devices. Though much progress has been made in improving the speed and robustness of neural control provided by intracortical BCIs, little research has been devoted to minimizing the amount of time spent on decoder calibration. Approach. We investigated the amount of time users needed to calibrate decoders and achieve performance saturation using two markedly different decoding algorithms: the steady-state Kalman filter, and a novel technique using Gaussian process regression (GP-DKF). Main results. Three people with tetraplegia gained rapid closed-loop neural cursor control and peak, plateaued decoder performance within 3 min of initializing calibration. We also show that a BCI-naïve user (T5) was able to rapidly attain closed-loop neural cursor control with the GP-DKF using self-selected movement imagery on his first-ever day of closed-loop BCI use, acquiring a target 37 s after initiating calibration. Significance. These results demonstrate the potential for an intracortical BCI to be used immediately after deployment by people with paralysis, without the need for user learning or extensive system calibration.

State estimation problems in PRF-shift magnetic resonance thermometry

PurposeThe purpose of this paper is to apply the Steady State Kalman Filter for temperature measurements of tissues via magnetic resonance thermometry. Instead of using classical direct inversion, a methodology is proposed that couples the magnetic resonance thermometry with the bioheat transfer problem and the local temperatures can be identified through the solution of a state estimation problem.Design/methodology/approachHeat transfer in the tissues is given by Pennes’ bioheat transfer model, while the Proton Resonance Frequency (PRF)-Shift technique is used for the magnetic resonance thermometry. The problem of measuring the transient temperature field of tissues is recast as a state estimation problem and is solved through the Steady-State Kalman filter. Noisy synthetic measurements are used for testing the proposed methodology.FindingsThe proposed approach is more accurate for recovering the local transient temperatures from the noisy PRF-Shift measurements than the direct data inversion. The methodology used here can be applied in real time due to the reduced computational cost. Idealized test cases are examined that include the actual geometry of a forearm.Research limitations/implicationsThe solution of the state estimation problem recovers the temperature variations in the region more accurately than the direct inversion. Besides that, the estimation of the temperature field in the region was possible with the solution of the state estimation problem via the Steady-State Kalman filter, but not with the direct inversion.Practical implicationsThe recursive equations of the Steady-State Kalman filter can be calculated in computational times smaller than the supposed physical times, thus demonstrating that the present approach can be used for real-time applications, such as in control of the heating source in the hyperthermia treatment of cancer.Originality/valueThe original and novel contributions of the manuscript include: formulation of the PRF-Shift thermometry as a state estimation problem, which results in reduced uncertainties of the temperature variation as compared to the classical direct inversion; estimation of the actual temperature in the region with the solution of the state estimation problem, which is not possible with the direct inversion that is limited to the identification of the temperature variation; solution of the state estimation problem with the Steady-State Kalman filter, which allows for fast computations and real-time calculations.

A robust predictor for dead-time systems based on the Kalman filter

Abstract Dead time processes are common in the industry and represent a challenge for control. One way to avoid (or attenuate) the problem is to use a predictor structure alongside the controller. In this paper, an observer-predictor structure based on a steady-state Kalman filter is explored. It is shown that this structure is equivalent to the Filtered Smith Predictor (FSP) for linear systems, hence, all the tools used to analyze the closed-loop properties of the FSP can also be used in the proposed predictor, specially the frequency domain ones. With the results of this analysis, tuning guidelines for the predictor are given to achieve different closed-loop characteristics, e.g., better disturbance rejection or better robustness. Furthermore, the proposed predictor can easily be used with all kind of systems, including unstable and non-minimum phase systems.

Real-time implementation of Kalman filter for unsteady flow measurement in a pipe

A Kalman filter which estimates unsteady laminar flow in a pipe is implemented on a real-time computing system. The plant model is the optimised finite element model of pipeline dynamics considering unsteady laminar friction. A steady-state Kalman filter is built based on the model of pipeline dynamics. Pressure signals at both ends of a target section of a pipe are input to the model of pipeline dynamics, and as an output of the model an estimated pressure signal at a mid-point of the pipe is obtained. Difference between measured and estimated pressure signals at the mid-point is fed back to the model of pipeline dynamics to modify state variables of the model. According to the Kalman filter principle, the state variables of the model are adjusted so that they converge to real values. It is demonstrated that real-time implementation of the Kalman filter is possible with the sampling time of 0.1 ms.

Real-time implementation of Kalman filter for unsteady flow measurement in a pipe

A Kalman filter which estimates unsteady laminar flow in a pipe is implemented on a real-time computing system. The plant model is the optimised finite element model of pipeline dynamics considering unsteady laminar friction. A steady-state Kalman filter is built based on the model of pipeline dynamics. Pressure signals at both ends of a target section of a pipe are input to the model of pipeline dynamics, and as an output of the model an estimated pressure signal at a mid-point of the pipe is obtained. Difference between measured and estimated pressure signals at the mid-point is fed back to the model of pipeline dynamics to modify state variables of the model. According to the Kalman filter principle, the state variables of the model are adjusted so that they converge to real values. It is demonstrated that real-time implementation of the Kalman filter is possible with the sampling time of 0.1 ms.

Best Achievable Performance of Cooperative Kalman Filters for Decoupled Multi-agent Systems under Environmental Disturbances

Abstract This paper deals with a distributed state estimation problem for a class of multi-agent systems. The agents are assumed to be decoupled and identical, but affected by a common environmental disturbance in addition to individual process noises. In the presence of such a disturbance, the resulting optimal filter consists of decoupled steady-state Kalman filters with information exchange. The changes in the performance caused by the existence of the environmental noise and the required information exchange are evaluated based on block decomposition of the stabilizing solution of the associated algebraic Riccati equation. Moreover, an optimal filter for the system with an additional global measurement output is derived. The performance improvement due to the additional measurement is also evaluated.

Toeplitz matrices for LTI systems, an illustration of their application to Wiener filters and estimators

ABSTRACTThe Wiener–Kolmogorov theory of filtering has been with us since the first half of the twentieth century. A later matrix-based approach which was more general was derived with the steady-state Kalman filter. This approach uses a novel method of representing causal and uncausal systems in the form of convolution matrices and leads to a Wiener solution which is much easier to calculate than either the Kalman or Wiener approaches. For coloured additive noise, it avoids the use of Diophantine equations. The key idea missing in previous work is the close link between polynomials and Toeplitz matrices which are lower triangular in form. There is already a reasonably sized literature in the mathematics field on such matrices and so the area is ripe for exploration. Although the method does not offer a different or better solution, it shows a completely new way of defining linear time-invariant (LTI) systems which is neither transfer-function nor state-space-based. This is achieved by exploiting the connection between polynomials and Toeplitz matrices. The application here is the Wiener filter but there could well be many more as this is a generic approach.

A Comparison of Intention Estimation Methods for Decoder Calibration in Intracortical Brain-Computer Interfaces.

Recent reports indicate that making better assumptions about the user's intended movement can improve the accuracy of decoder calibration for intracortical brain-computer interfaces. Several methods now exist for estimating user intent, including an optimal feedback control model, a piecewise-linear feedback control model, ReFIT, and other heuristics. Which of these methods yields the best decoding performance? Using data from the BrainGate2 pilot clinical trial, we measured how a steady-state velocity Kalman filter decoder was affected by the choice of intention estimation method. We examined three separate components of the Kalman filter: dimensionality reduction, temporal smoothing, and output gain (speed scaling). The decoder's dimensionality reduction properties were largely unaffected by the intention estimation method. Decoded velocity vectors differed by <5% in terms of angular error and speed vs. target distance curves across methods. In contrast, the smoothing and gain properties of the decoder were greatly affected (> 50% difference in average values). Since the optimal gain and smoothing properties are task-specific (e.g. lower gains are better for smaller targets but worse for larger targets), no one method was better for all tasks. Our results show that, when gain and smoothing differences are accounted for, current intention estimation methods yield nearly equivalent decoders and that simple models of user intent, such as a position error vector (target position minus cursor position), perform comparably to more elaborate models. Our results also highlight that simple differences in gain and smoothing properties have a large effect on online performance and can confound decoder comparisons.

State-Space Model and Kalman Filter Gain Identification by a Kalman Filter of a Kalman Filter

This paper describes an algorithm that identifies a state-space model and an associated steady-state Kalman filter gain from noise-corrupted input–output data. The model structure involves two Kalman filters where a second Kalman filter accounts for the error in the estimated residual of the first Kalman filter. Both Kalman filter gains and the system state-space model are identified simultaneously. Knowledge of the noise covariances is not required.

Event‐triggered state estimator for stochastic systems with unknown inputs

This article studies the problem of state estimation for stochastic systems with unknown inputs. To reduce the communication cost from the sensor to the remote processor, an event-triggered communication mechanism is proposed in terms of an event generator function for the innovation vectors. The event-triggered estimator is developed by introducing an input term in the steady-state Kalman filter for the corresponding nominal system. The input gain matrix is determined by treating the nominal estimator error dynamics as the desired performance. It is shown that the estimation error is bounded in mean square under certain conditions. A numerical example is provided to verify the effectiveness of the proposed estimator.

Real-time identification of a high-magnitude boundary heat flux on a plate

Identification of high-magnitude heat flux in real time is a challenging problem, since most of the currently available algorithms require large computation time in comparison with the time scale of the real physical problem. This paper presents a methodology that allows for quantifying the unsteady heat flux in real time by using the steady-state Kalman filter. Two different cases have been used to verify this algorithm and the estimates are in excellent agreement with the reference values.