State-space Estimation Research Articles

Environmental time series analysis and forecasting with the Captain toolbox

The Data-Based Mechanistic (DBM) modelling philosophy emphasises the importance of parametrically efficient, low order, ‘dominant mode’ models, as well as the development of stochastic methods and the associated statistical analysis required for their identification and estimation. Furthermore, it stresses the importance of explicitly acknowledging the basic uncertainty in the process, which is particularly important for the characterisation and forecasting of environmental and other poorly defined systems. The paper focuses on a Matlab ® compatible toolbox that has evolved from this DBM modelling research. Based around a state space and transfer function estimation framework, Captain extends Matlab ® to allow, in the most general case, for the identification and estimation of a wide range of unobserved components models. Uniquely, however, Captain focuses on models with both time variable and state dependent parameters and has recently been implemented with the latest methodological developments in this regard. Here, the main innovations are: the automatic optimisation of the hyper-parameters, which define the statistical properties of the time variable parameters; the provision of smoothed as well as filtered parameter estimates; the robust and statistically efficient identification and estimation of both discrete and continuous time transfer function models; and the availability of various special model structures that have wide application potential in the environmental sciences.

State Space Models for Dynamic Style Analysis of Portfolios

This paper presents a framework and methods for the estimation of linear and non-linear state space (SS) models, occasionally subject to restrictions, to construct and estimate several models for style analysis with time varying exposures. The study is conducted by applying these models to an artificial portfolio and to return series of Brazilian investment funds. The results confirm the belief that dynamic allocations in a portfolio are a more realistic assumption for investment funds management.

State-space harmonic distortion modeling in weakly nonlinear, fully balanced G/sub m/-C filters-a modular approach resulting in closed-form solutions

State space harmonic distortion modeling and estimation are introduced for G/sub m/-C filters with fully balanced, weakly nonlinear tranconductors. The proposed method provides compact closed-form answers expressed explicitly in terms of the values of the circuit elements. It can be easily implemented in MATLAB and applied to G/sub m/-C filters of any order. The filter is viewed as a composition of three operators in signal space representing the input stage, the filter core, and the output stage. Each operator is then decomposed into the superposition of a linear operator and a nonlinear operator. The total distortion at the output is shown to be approximately the sum of the distortion introduced by the nonlinear operator of each stage. The theoretical results are found to be in good agreement with Cadence simulations for the cases of a lossy integrator and a third-order Butterworth low-pass filter.

De-embedding intrinsic parameters of high-Q dielectric resonators from noisy measurements

The low losses of electromagnetically coupled dielectric resonators render the measured scattering parameters very sensitive to background “noise” contributed by the coupling mechanism, package modes, radiation, and so forth. It becomes important to properly calibrate out this parasitic influence in order to accurately measure the unloaded Q. We report a robust software calibration procedure based on a state-space spectral estimation method, which filters out the background noise and facilitates linear extraction of the unloaded Q. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1453–1458, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21730

Dense SLAM: The Unidirectional Information Flow (UIF)

This paper presents the concept of Unidirectional Information Flow (UIF) for state-space estimation. The notion of UIF is relevant to systems where the statevector is divided into two groups: (i) states that both give and receive information to and from the rest of the system and (ii) states that only receive information. This paper presents an application of the UIF concept to the problem of Simultaneous Localisation and Mapping (SLAM). Traditional SLAM maps are sparse and are built up of isolated landmarks observed in the environment. Although a dense representation may sometimes be desirable, the construction of a consistent dense map has previously not been possible for the incremental SLAM paradigm. It is also shown that by using the UIF, it is possible to obtain a more detailed environment representation (Dense SLAM) without increasing the computational burden of the algorithm and without loss of consistency. The same concept is also used to evaluate the error propagation inside the local dense maps. Experimental results are finally presented to validate the algorithms.

Observer-based sliding mode control for nonlinear state-delayed systems

An observer-based sliding mode control problem is studied for state-delayed systems with unmeasurable states and nonlinear uncertainties. The main advantage of the proposed scheme is that it eliminates the need for state variables to be fully accessible for its control. This is possible through the use of a sliding mode controller, which performs its control by employing state estimates obtained from the sliding mode observer. By means of linear matrix inequalities, a sufficient condition is then given to ensure the asymptotic stability of the overall closed-loop state-delayed system composed of the observer dynamics and the estimation error dynamics. Furthermore, it is shown that the proposed control scheme ensures the reachability of the sliding surfaces in both the state estimate space and the estimation error space.

Expectational Prediction of Individual Stocks

This study revisits the predictability of individual stock returns by focusing on the investors' expectations of future stock prices. It hypothesizes that expectations cause events, and thus provide predictive power to forecast stock returns. Based on this premise, a model that relies on the investors' information set at time t and formulates expectational positions on stock prices for the near- and medium-term future periods is designed and estimated. The methodology is structural time series analysis under a statespace (time-varying) estimation process. The estimated model is put further to test to predict the one-, five-, and ten-days step-ahead ex-ante forecasts of a test sample of individual stocks in the Russel-1000 portfolio. The results are encouraging, particularly in terms of the Rasps that are computed.

Disturbance Tracking Control and Blade Load Mitigation for Variable-Speed Wind Turbines

A composite linear state-space controller was developed for a multi-objective problem in the variable-speed operation of wind turbines. Disturbance Tracking Control theory was applied to the design of a torque controller to optimize energy capture under the influence of persistent wind disturbances. A limitation in the theory for common multi-state models is described; this led to the design of a complementary pitch controller. The goal of the independent blade pitch design was to minimize blade root fatigue loads. A SymDyn model of a two-bladed, 600-kW machine was used for the simulation studies. Results indicate a 24% reduction in blade fatigue damage using the proposed controllers, compared to a conventional torque-only design. However, energy capture was not improved as much as expected, partly due to nonlinearity effects degrading the performance of the state-space estimator design. Tower base fatigue damage was shown to decrease significantly using active pitch.

Ensemble Square Root Filters*

Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be performed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman square root filters. The nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods.

Time-series estimation of biological factors in optical diffusion tomography

We apply state space estimation techniques to the time-varying reconstruction problem in optical tomography. We develop a stochastic model for describing the evolution of quasi-sinusoidal medical signals such as the heartbeat, assuming these are represented as a known frequency with randomly varying amplitude and phase. We use the extended Kalman filter in combination with spatial regularization techniques to reconstruct images from highly under-determined time-series data. This system also naturally segments activity belonging to different biological processes. We present reconstructions of simulated data and of real data recorded from the human motor cortex (Franceschini et al 2000 Optics Express 6 49–57). It is argued that the application of these time-series techniques improves both the fidelity and temporal resolution of reconstruction in optical tomography.

Real-time human motion estimation using biomechanical models and non-linear state-space filters.

In the field of sports biomechanics and rehabilitation engineering, the possibility of computing, in real time, the angular displacements and derivatives of human joints, from a video of motion sequences, represents an appealing goal. In particular, applications of biofeedback protocols in rehabilitation can benefit from this capability. The focus of the investigation was concerned with the application of biomechanical models, comprising of a kinematic chain and surface envelopes, and state-space filters, to the computation, in real time and with high accuracy, of the angular data and derivatives. By minimising the distances, measured with TV cameras, between the 2D marker projections and the corresponding back-projected markers located on the mannequin, the configuration of the biomechanical model was automatically updated. The use of state-space estimation allowed the computation of smooth derivatives of the orientation data. Owing to the non-linearity of the functions involved, the derivatives of the observation model were obtained through a multidimensional extension of Stirling's interpolation formula. Proper algorithms were developed to cope with the model calibration, initialisation and data labelling. Extensive experiments on real and simulated motions proved the reliability (maximum angular error less than 1 degree, maximum point reconstruction less than 1 mm) of the developed system, which is robust to false matching caused by marker occlusions. Moreover, orientation artifacts due to skin motion can be reduced by a factor of 50%.

Advances in real-time flood forecasting.

This paper discusses the modelling of rainfall-flow (rainfall-run-off) and flow-routeing processes in river systems within the context of real-time flood forecasting. It is argued that deterministic, reductionist (or 'bottom-up') models are inappropriate for real-time forecasting because of the inherent uncertainty that characterizes river-catchment dynamics and the problems of model over-parametrization. The advantages of alternative, efficiently parametrized data-based mechanistic models, identified and estimated using statistical methods, are discussed. It is shown that such models are in an ideal form for incorporation in a real-time, adaptive forecasting system based on recursive state-space estimation (an adaptive version of the stochastic Kalman filter algorithm). An illustrative example, based on the analysis of a limited set of hourly rainfall-flow data from the River Hodder in northwest England, demonstrates the utility of this methodology in difficult circumstances and illustrates the advantages of incorporating real-time state and parameter adaption.

A framework for state-space estimation with uncertain models

Develops a framework for state-space estimation when the parameters of the underlying linear model are subject to uncertainties. Compared with existing robust filters, the proposed filters perform regularization rather than deregularization. It is shown that, under certain stabilizability and detectability conditions, the steady-state filters are stable and that, for quadratically-stable models, the filters guarantee a bounded error variance. Moreover, the resulting filter structures are similar to various (time- and measurement-update, prediction, and information) forms of the Kalman filter, albeit ones that operate on corrected parameters rather than on the given nominal parameters. Simulation results and comparisons with /spl Hscr//sub /spl infin// guaranteed-cost, and set-valued state estimation filters are provided.

Robot angular link velocity estimation in the presence of high-level mixed uncertainties

The estimation problem of the angular link velocity via a nonlinear high-gain observer is considered. It is solved in the presence of essential parametric uncertainty and external noise perturbations based only on current position measurements. The obtained state space estimation error turns out to be robust with respect to given uncertainties, and its bound can be expressed by a linear combination of the internal (parametric) and external (noise type) uncertainty levels. The simulation results, concerning a robot manipulator with two degrees of mobility and with unknown friction parameter subjected to variations within a given interval, illustrate the effectiveness of the suggested technique.

Robustness property of high-gain observers for closed-loop nonlinear systems: Theoretical study and robotics control application

The robust stability problem for a wide class of closed-loop nonlinear systems is considered. A nonlinear feedback is assumed to be a function of current state estimates obtained by a nonlinear high-gain observer. The problem is solved in the presence of essential parametric uncertainties as well as external perturbation noise (mixed uncertainties). A two Riccati-equation approach is applied. We demonstrate that the highgain observer under consideration provides sufficiently good state-space estimates which are bounded 'on average'. The same property is valid for the trajectories of the closed-loop nonlinear system. The observation error bound is derived and turns out to be a linear combination of a priori given uncertainties levels of external input and output perturbations. If no uncertainties are considered in the given model description, this bound is zero and corresponds to the asymptotic globally stability property for the observation error as well as for the state-space trajectories. The simulations results, concerning a robot manipulator, illustrate the effectiveness of the suggested technique.

State-space estimation of rational bubbles in the Yen/Deutsche Mark exchange rate

State-Space Estimation of Rational Bubbles in the Yen/Deutsche Mark Exchange Rate. — This paper considers a series that uncovered interest parity predicts to be white noise and inspects it for evidence of stochastic rational bubbles. State-space methods are used that specify a bubble component of the series as an unobserved state. The technique’s effectiveness is demonstrated by Monte Carlo experiments. One span of the series is found in which a stochastic rational bubble specification clearly dominates the white noise specification. It coincides with a period of general financial turm-oil in the associated economies, i.e. Japan and Germany during 1989 and early 1990.

Genetic Algorithm Filter for Non-Linear State Space Estimation.

A new method for non-linear state space estimation using genetic algorithm is developed. This method is named the Genetic algorithm filter (GA filter). State vactors are coded with genes and their survival rates are decided by a probability density functions. A crowd of genes represents the characteristic of a current state of a system. GA are general techniques for doing numerical optimization and are based on stochastic simulation. Their strength is that they can be applied to problems where an analytical approach is difficult or impossible. GA filter can handle non-linear system equations and measurement equations as well as non-Gaussian and non-additive noise. Implementing the GA requires less analytical work than implementing the extended Kalman filter, but the GA filters are more computer intensive. We will demonstrate effectiveness of the resulting new filters with simple examples.

Dynamic detection of rhythmic oscillations in heart-rate tracings: a state-space approach based on fourth-order cumulants.

This paper presents a new procedure specifically aimed at providing a dynamical detection of the oscillations occurring in long-term heart-rate (HR) tracings. The procedure is based on a time-variant state-space modelling of the fourth-order cumulants of the HR signal. The state-space estimator was selected because of its demonstrated capability to distinguish between deterministic and stochastic components of the signal, while the fourth-order cumulants of the signal were used as input of the model to further reduce adverse effects of coloured, white and l/f Gaussian noise possibly present in the input data. The procedure was tested by the analysis of simulated signals and its performance was compared with the results obtained by state-space modelling applied directly on the test signals (instead of on the fourth-order cumulants of the signals) and by the more traditional auto-regressive modelling. The comparison has shown a clear superiority of the proposed procedure over the other techniques in discriminating deterministic oscillations from coloured noise. Finally, the applicability of the procedure to biological data was verified by analysing five experimental HR tracings recorded in normal subjects during laboratory and daily life conditions.

An extended inverse QR adaptive filtering algorithm

A so-called inverse QR algorithm was recently introduced for recursive adaptive filtering under the exponentially weighted least-squares criterion. It has some attractive features, including the absence of inversions. The extension to the multi-channel case does require inversion however. We present a new derivation of the inverse QR algorithm, based on the technique of Sayed and Kailath, for reformulating the above adaptive filtering problem as a state-space estimation problem. A well-known square-root covariance algorithm for the latter problem is shown to directly give (a multi-channel version of) the inverse QR algorithm. A new extended square-root covariance algorithm is then applied to get a new inversion-free ‘extended inverse QR’ algorithm, even in the multi-channel case.

Airborne vector gravimetry using precise, position-aided inertial measurement units

Vector gravimetry using a precise inertial navigation system continually updated with external position data, for example using GPS, is studied with respect to two problems. The first concerns the attitude accuracy requirement for horizontal gravity component estimation. With covariance analyses in the space and frequency domains it is argued that with relatively stable uncompensated gyro drift, the short-wavelength gravity vector can be estimated without the aid of external attitude updates. The second problem concerns the state-space estimation of the gravity signal where considerable approximations must be assumed in the gravity model in order to take advantage of the ensemble error estimation afforded by the Kalman filter technique. Gauss-Markov models for the gravity field are specially designed to reflect the attenuation of the signal at a specific altitude and the omission of the long-wavelength components from the estimation. With medium accuracy INS/GPS systems, the horizontal components of gravity with wavelengths shorter than 250 km should be estimable to an accuracy of 4–6 mgal (µg); while high accuracy systems should yield an improvement to 1–2 mgal.