Abstracts — Fall 2009

Wednesday September 2: Anders Rantzer

3-day course: Distributed and Event-based Control, Lecture 1
Time: 09:15-12:00. Place: Visionen.

How can control synthesis for a complex distributed system be done in a systematic way? Can a global objective for the total system be split into individual objectives for subsystems? What information transfer is most important for control performance? These are some of the main issues in this course, which combines ideas for decentralized optimization and control from the 1960-70s with recent work on networked control systems.

Thursday September 3: Anders Rantzer

3-day course: Distributed and Event-based Control, Lecture 2
Time: 09:15-12:00. Place: Visionen.


Friday September 4: Karl-Henrik Johansson

3-day course: Distributed and Event-based Control, Lecture 3
Time: 09:15-12:00. Place: Visionen.


Tuesday September 8: Jan van Schuppen and Jana Nemcova

System identification of biochemical reaction systems & Realization theory for rational systems
Time: 10:15-11:00. Place: Glashuset.

System identification of biochemical reaction systems

A model for the protein network GATA1 leads to a problem of system identification for a biochemical reaction system. The problem is motivated by the wish for understanding of the development of red blood cells in humans. A key network is that of the Gata1 protein and its related proteins. The cell develops in about ten days via several stages and separates out into six different types. The currently formulated model is likely to be partly incorrect but there is no theory for how to modify it. The measurement data are quite limited.

System identification of biochemical reaction systems is motivated by systems biology. The main models are for metabolic, signaling, and genetic networks. The systems are rational positive systems. The many open problems and possible approaches are described.

The paper is based on joint research with Dirk De Vos, Jana Nemcova, and Mihaly Petreczky.

Realization theory for rational systems

Rational systems are dynamical systems whose dynamics and output functions are determined by rational functions. Rational systems are widely used as models of phenomena in life sciences, economy and engineering. In this talk we present realization theory for the class of rational systems. The results can be applied in system identification, model reduction, and control and observer synthesis.

Realization theory for rational systems deals with the problem of the existence of rational systems whose input-output behavior is given by a considered input-output or response map. The realization problem concerns specifying the conditions for the existence of rational realizations which are controllable, observable, and/or minimal. We provide necessary and sufficient conditions for the existence of such rational realizations. The algorithms for constructing rational realizations follow from the proofs of the corresponding theorems. Further, we relate minimality, controllability, and observability of rational realizations, and provide the characterization of canonical and minimal realizations up to a birational equivalence.

The obtained results are applied in system identification to derive the characterization of identifiability properties of the parametrizations of rational systems with parameters. Namely, we determine when the parameter values of parametrized rational systems can be determined uniquely.

Thursday September 17: Jean Thomas

Model Predictive Control for Hybrid Systems
Time: 13:15-14:00. Place: Glashuset.

Hybrid systems become more and more attractive to researchers, as they cover several domains of applications. "Hybrid dynamical systems" indicates categories of dynamical systems including explicitly and simultaneously continuous and discrete type dynamical phenomena. The major challenge of Hybrid systems remains the computation time of the optimization problem.

This work investigates two major directions. The first aims at developing new techniques of application of the predictive strategy for hybrid systems under the MLD form. The objective of these approaches is to reduce the computing time for real time applications purposes. Three solutions are proposed to simplfy the optimisation problem. Also the dual problem of the previous one, the state estimation of hybrid systems modelled under the MLD form, and in particular the application to the sensors fault detection is examined. The techniques developed are validated on a steam generator benchmark.

The second direction of research investigates the robust tracking and regulation control problems for discrete-time, piecewise affine systems affected by parameter variations and bounded disturbances, where reachability technique based on polyhedral approach is developed and the robust control problems are investigated. Checking attainability and calculating the state space regions for which a robust control is assured despite the possible disturbance and the parameter variations is performed using a geometrical approach. A model predictive control law derived from a quadratic cost function minimization is further examined as an alternative sub-optimal approach for decreasing the computational load. The proposed technique is applied in simulation to a two-tank benchmark.

Thursday September 24: Michael Felsberg

DIPLECS - Dynamic Interactive Perception-action LEarning in Cognitive Systems
Time: 13:15-14:00. Place: Glashuset.

To make progress toward the goal of reducing fatalities and accidents on our roads, designers of safety systems have focused on the prevention or mitigation of accidents. Since humans (drivers) play an important role in the pre-crash phase, systems that protect occupants and pedestrians must be seen as distributed, cognitive systems rather than as purely technical systems.

The DIPLECS project aims to design an Artificial Cognitive System capable of learning and adapting to respond in the everyday situations humans take for granted. The primary demonstration of its capability, will be the driving of a car. The system will learn by watching humans, how they act and react while driving, building models of their behaviour and predicting what a driver would do when presented with a specific driving scenario. The end goal of which is to provide a flexible cognitive system architecture demonstrated within the domain of a driver assistance system, thus potentially increasing future road safety.

In order to achieve these goals, the DIPLECS architecture must allow for learning and adaptation in dynamic, real-time and real-world scenarios. Starting from a basic, rudimentary capability, it must constantly refine and improve its capability by observing a human driver, the car data and the surrounding environment.

The architecture applies a hierarchical design principle, where adjacent levels are connected by feedback-loops. Learning occurs in two ways, either by analysing human-car-environment interaction or by (cognitive) bootstrapping of its own capabilities. We extend the established system architecture from the COSPAL project in order to establishes the fundament for implementing the three lower levels of the well-known ECOM model: tracking, regulating, and monitoring.

We designed a suitable mobile platform for the demonstrator, by modifying suspension and gearbox of a standard RC car. For offline experiments, we recorded more than 20 hours of data in an instrumented vehicle.

The DIPLECS system applies learning techniques at all levels and number of examples are presented. At lowest level, we apply learnable detectors for generic objects based on the WaldBoost technique. In this particular case, the front and rear views of cars are detected. The detector is trained from manually labelled images and is based on a cascade of weak classifiers.

These detections are input to a learnable object tracker, which is based on channel representations. This tracker is trained from detection results without known correspondence, but produces consisting tracking results for individual objects. Tracking establishes a Neisser perceptual cycle at the lowest level of the ECOM model.

A later goal is to learn perception-action cycles, which control the system directly from optical flow. In order to achieve this goal, automatic labelling of training-data is required. A method for labelling real-world sequences based on successful 2D tracking using certain constraints has been implemented.

Progress has also been made in learning more general perception-action mappings at different levels. Applying techniques from data mining, low-level behavioral patterns have been learned, for example the dependency between visual stimuli and basic driving atoms. Using these learned mappings prediction of action like turn control becomes possible.

Thursday October 8: Roland Toth

Identification of linear parameter-varying systems: challenges and solutions
Time: 13:15-14:00. Place: Glashuset.

In the past 15 years, the rapid evolution of the Linear Parameter-Varying (LPV) framework has been driven by the industrial need for high performance control of nonlinear/time-varying processes. A particular strength of the LPV systems is that they represent an intermediate step between the well-understood class of Linear Time-Invariant (LTI) systems and the vast universe of nonlinear time-varying dynamics. Therefore, the LPV framework especially offers a useful alternative to model mildly nonlinear systems or position dependent processes while at the same time it is supported by powerful, practically applicable control synthesis approaches.

Despite the advances of LPV control, identification of such systems is still in its infancy as there are numerous fundamental difficulties connected to this model class. Many problems are related to the gaps in LPV system theory, which were uninvestigated in the past, as most theoretical developments were based on LPV state-space representations exclusively. To develop novel approaches of LPV system identification, where each step of the identification cycle from experiment design to validation can be treated appropriately, these gaps in the system theory must be filled. To achieve this objective, an LPV behavioral approach is introduced as the extension of the LTI behavioral theory. By this new LPV system theoretical framework, which is compatible with existing results of LPV identification and control, concepts of representations, minimality, equivalence transformations, and realizability can all be expressed in a unified manner. This gives the common ground for the comparison of existing LPV model structures; solving the problem of converting input-output models to state-space descriptions and vice-versa.

Based on the developed behavioral approach, it can be also shown that LPV systems, similarly to LTI systems, have a series expansion representation in terms of Orthonormal Basis Functions (OBFs). This gives not only the possibility to introduce identification methods based on truncated expansion structures but also to derive impulse response representation of these systems. The latter provides the ingredient to establish a prediction error framework for the analysis of different LPV model structures provided predictors and also to understand consistency, bias and variance properties of the obtained model estimates.

In the LTI case, truncated OBF expansions based model structures provide efficient alternatives for system identification with many attractive properties. Using the established tools of the LPV behavioral system theory and the introduced predication error framework, OBFs based LPV model structures are introduced with linear regression based identification methods with the aim to overcome the disadvantages of other type of LPV model structures and to provide an efficient approximate identification approach for LPV control. The used model structures are a linear combination of globally fixed (scheduling-independent) OBFs with scheduling-dependent weights. We distinguish a local and a global identification approach, both based on a fixed set of OBFs. For the selection of the model structure, i.e. the OBFs that guarantee the least worst-case modeling error in an asymptotic sense (fastest convergence rate of the series expansion), a basis selection tool, the so called Fuzzy Kolmogorov c-Max approach is introduced.

Thursday October 15: Jonas Stålhand

Mathematical Modelling of Arteries - Can Biomechanics Predict Arterial Diseases?
Time: 13:15-14:00. Place: Glashuset.

The initiation and development of many cardiovascular diseases are related to mechanics. For instance, the proteins responsible for the structural integrity of the arterial wall undergo micro structural changes with age making the artery stiffer. The increased stiffness results in an elevated blood pressure which is an important risk factor for contracting other cardiovascular diseases.

Clinical measures for the arterial stiffness treat arterial wall as a single material even though it comprises of several constituents with different ageing processes. As a consequence, it is of great interest develop mechanical models which has a better resolution in terms of the material description.

This presentation focuses on how fundamental postulates of mechanics can be used to develop a state-of-the-art model which resolves different constituents in the arterial wall and is suitable for clinical measured signals. In addition, results from a recent study where the model was applied to 30 humans will also be shown.

Thursday October 22: Henrik Mosskull

Experiences from Gröna Tåget and future trends in the railway industry
Time: 13:15-14:00. Place: Glashuset.

In this talk I will present new trends in the railway industry, exemplified by the Bombardier concept ECO4; Energy, Efficiency, Economy and Ecology. Focus will be on experiences from Gröna Tåget, a test train equipped with permanent magnet synchronous motors. A high-light is the new Nordic train speed record of 303 km/h, set in Västergötland on regular track last summer.

Thursday October 29: Matthijs Spaan

Decision-theoretic Planning under Uncertainty for Active Cooperative Perception
Time: 13:15-14:00. Place: Glashuset.

As robots leave research labs to operate more often in human-inhabited, larger environments, cooperation between sensor networks and mobile robots becomes crucial. For example, in urban scenarios, employing mobile robots is a need to augment the limited sensor coverage and improve detection and tracking accuracy. The fusion of sensory information between fixed surveillance cameras and each robot, with the goal of maximizing the amount and quality of perceptual information available to the system can be called cooperative perception. A promising decision-theoretic planning framework for cooperative active perception is that of Partially Observable Markov Decision Processes (POMDPs). The suitability of POMDPs for the previously depicted scenario arises from their ability to inherently trade off task completion, which could be react to a potential event that has been detected, and information gathering in a efficient way, that is decide to send a robot to improve situational awareness. In this talk we will discuss how planning under uncertainty can be applied to active cooperative perception problems.

Friday October 30: Per Skoglar

Planning Methods for Aerial Exploration and Ground Target Tracking
Time: 10:15-12:00. Place: Visionen.

This thesis considers unmanned airborne surveillance systems equipped with electrooptical vision sensors. The aim is to increase the level of autonomy and improve the system performance by the use of planning methods for aerial exploration and target tracking. The general problem is very complex due to the “curse-of-dimensionality” and suboptimal approaches are necessary in order to handle advanced surveillance missions. A general planning framework is proposed and the planner contains a high-level scheduler and a number of planning modes. Each mode consists of planning modules that solve smaller sub-tasks and in this thesis a number of these modules are developed. In particular, two major approaches are treated; information based planning, and Bayesian target search. In addition, the on-road target tracking problem is treated in detail and an algorithm based on the Particle filter is presented. In information based planning, different information measures are used to solve the optimal trajectory planning problem for bearings-only estimation. Thus, the problem is how to maneuver an unmanned aerial vehicle (UAV) to achieve the best possible estimate of a target location while observing it with a vision sensor. Approaches based on the Information filter and the differential entropy are proposed. The Information filter approach is also used to develop an exploration framework where the UAV flight trajectory and the sensor pointing direction are considered concurrently. In Bayesian target search, the aim is to find a target as quickly as possible given some prior knowledge of where it might be. Methods based on both gradient search and combinatorial optimization routines are proposed for the search problem where a UAV is equipped with a controllable vision sensor with limited field-of-view.

Thursday November 5: Adrian Bishop

An Analysis of the Optimal Geometries for Localization and Some Problems in Formation Control
Time: 13:15-14:00. Place: Glashuset.

During this presentation, I will discuss the optimality of certain sensor-target localization geometries for a number of localization problems. It is known that the localization performance can be explicitly characterized by certain measures, for example, by the Cramer-Rao bound. It is also known that the relative sensor-target geometry can significantly affect the performance of any particular localization algorithm. The aim of this presentation is to identify those relative sensor-target geometries which result in a measure of the localization performance being optimized. The optimal geometries for an arbitrary number of sensors are identified and it is shown that an optimal sensor-target configuration is not, in general, unique. The importance of understanding the influence of the sensor-target geometry on the potential localization performance is highlighted via formal analytical results and a number of illustrative examples.

Thursday November 12: Erik G. Larsson

Optimal scheduling in OFDMA under a control signaling cost constraint
Time: 13:15-14:00. Place: Glashuset.

I will discuss the problem of user scheduling in orthogonal frequency-division multiple access (OFDMA) wireless communication systems. In particular I will present a new algorithm for scheduling that maximizes the throughput, taking into account the amount of signaling needed to transmit scheduling maps to the users. A combinatorial problem is formulated and solved via a dynamic programming approach reminiscent of the Viterbi algorithm. The total computational complexity of the algorithm is upper bounded by O(K^4N) where K is the number of users that are being considered for scheduling in a frame and N is the number of resource blocks per frame.

Tuesday November 17: Janne Harju Johansson

High Voltage Direct Current, HVDC
Time: 10:15-11:00. Place: Glashuset.

In the talk fundamentals of power transmission links using high voltage direct current, HVDC, will be presented. The focus will be on voltage source converters, VSC, and the core control algorithms for such converters. Furthermore, the different aspects of implementation and verification of a control scheme into a commercial power link are discussed.

Thursday November 19: Jonas Sjöberg

Modeling and simulation projects at the Mechatronic group at Chalmers University
Time: 13:15-14:00. Place: Glashuset.

This talk gives an overview of current and past research activities during the last couple of years at the Mechatronic research group at Chalmers. Most research activities are in close cooperation with the automotive industry, either concerning hybrid electric vehicles (HEV) or active traffic safety. Modeling, simulation, estimation, optimization, and control are important issues in all projects.

For the design of hybrid vehicles, to decide upon the sizing of components, optimal control techniques are used. We work on automatic model simplifications so that this can be done with reasonable computational effort. New certification methods are needed for HEV, and especially for heavy vehicles where almost no vehicles are identical, these need to be model based due to cost reasons. We are working on hardware-in-the-loop project to investigate the possibilities for this.

In automotive active safety functions, it is extremely important to avoid unnecessary interventions. Such intervention could cause an accident in a situation where the driver was capable of maneuvering the vehicle without any risk. Hence, active safety functions are often based on a threat assessment algorithm which calculates if it is possible for the driver to avoid an accident. The algorithm needs models of driver, vehicle and traffic, including the road to for the estimation, and the models need to be simple so that the estimation can be done in real time. Also active safety system needs to be certified, and methods for this are missing today. It is very challenging since traffic situations can be so very different. We are working on all these issues.

Friday November 20: Christian Lundquist

Automotive Sensor Fusion for Situation Awareness
Time: 10:15-12:00. Place: Visionen.

The use of radar and camera for situation awareness is gaining popularity in automotive safety applications. In this thesis situation awareness consists of accurate estimates of the ego vehicle’s motion, the position of the other vehicles and the road geometry. By fusing information from different types of sensors, such as radar, camera and inertial sensor, the accuracy and robustness of those estimates can be increased.

Sensor fusion is the process of using information from several different sensors to compute an estimate of the state of a dynamic system, that in some sense is better than it would be if the sensors were used individually. Furthermore, the resulting estimate is in some cases only obtainable through the use of data from different types of sensors. A systematic approach to handle sensor fusion problems is provided by model based state estimation theory. The systems discussed in this thesis are primarily dynamic and they are modeled using state space models. A measurement model is used to describe the relation between the state variables and the measurements from the different sensors. Within the state estimation framework a process model is used to describe how the state variables propagate in time. These two models are of major importance for the resulting state estimate and are therefore given much attention in this thesis. One example of a process model is the single track vehicle model, which is used to model the ego vehicle’s motion. In this thesis it is shown how the estimate of the road geometry obtained directly from the camera information can be improved by fusing it with the estimates of the other vehicles’ positions on the road and the estimate of the radius of the ego vehicle’s currently driven path.

The positions of stationary objects, such as guardrails, lampposts and delineators are measured by the radar. These measurements can be used to estimate the border of the road. Three conceptually different methods to represent and derive the road borders are presented in this thesis. Occupancy grid mapping discretizes the map surrounding the ego vehicle and the probability of occupancy is estimated for each grid cell. The second method applies a constrained quadratic program in order to estimate the road borders, which are represented by two polynomials. The third method associates the radar measurements to extended stationary objects and tracks them as extended targets.

The approaches presented in this thesis have all been evaluated on real data from both freeways and rural roads in Sweden.

Thursday November 26: Rickard Karlsson

Sensor fusion for monitoring of tire pressures
Time: 13:15-14:00. Place: Glashuset.

TPI from NIRA Dynamics is a second generation indirect Tire Pressure Monitoring System (TPMS), which is able to detected puncture for 1-3 wheels, and diffusion up to four wheels. The system is used in production vehicles both in the US and in Europe. Its signal processing algorithm uses internal vehicle sensor data information together with data from the Anti-locking Brake System (ABS). Particularly the ABS wheel speed data is very important for system performance. The TPI system fuses sensor data and analyzes relative changes in wheel radius as well as spectral properties from the tire vibration in order to detect pressure deflations.

The main algorithm design challenge is to maintain fast detection without causing nuisance warnings, under a large range of driving conditions, road surface changes, load changes etc. The final software is typically integrated in the vehicle's ESP unit, hence both CPU load and memory requirements are extremely important, forcing a highly optimized integer implementation of the signal processing algorithms.

TPI is compliant with current US TPMS regulation FMVSS 138, and implemented for the upcoming EU-legislation ECE R-64. In the presentation system aspects, legislation, test procedures, and implementation are addressed. Current research projects with alternative detection algorithms based on wheel speed correlation and analysis methods to classify tires in order to reduce testing will also be discussed briefly.

Thursday December 3: Lars Ulander

Signal processing algorithms for ultra-wide band synthetic aperture radar
Time: 13:15-14:00. Place: Glashuset.

Lars Ulander is Director of Research at FOI and Adjunct Professor at Chalmers University of Technology. His research profile includes synthetic-aperture radar (SAR), signal processing algorithms and electromagnetic scattering models. His focus is on developing systems and signal/data processing algorithms for low-frequency SAR in the VHF- and UHF-bands. He will talk about time-domain algorithms for solving general SAR imaging problems and for detecting changes in SAR images.

Tuesday December 8: Christian Lyzell

Initialization Methods for System Identification
Time: 10:15-12:00. Place: Visionen.

In the system identification community a popular framework for the problem of estimating a parametrized model structure given a sequence of input and output pairs is given by the prediction-error method. This method tries to find the parameters which maximize the prediction capability of the corresponding model via the minimization of some chosen cost function that depends on the prediction error. This optimization problem is often quite complex with several local minima and is commonly solved using a local search algorithm. Thus, it is important to find a good initial estimate for the local search algorithm. This is the main topic of this thesis.

The first problem considered is the regressor selection problem for estimating the order of dynamical systems. The general problem formulation is difficult to solve and the worst case complexity equals the complexity of the exhaustive search of all possible combinations of regressors. To circumvent this complexity, we propose a relaxation of the general formulation as an extension of the nonnegative garrote regularization method. The proposed method provides means to order the regressors via their time lag and a novel algorithmic approach for the ARX and LPV-ARX case is given.

Thereafter, the initialization of linear time-invariant polynomial models is considered. Usually, this problem is solved via some multi-step instrumental variables method. For the estimation of state-space models, which are closely related to the polynomial models via canonical forms, the state of the art estimation method is given by the subspace identification method. It turns out that this method can be easily extended to handle the estimation of polynomial models. The modifications are minor and only involve some intermediate calculations where already available tools can be used. Furthermore, with the proposed method other a priori information about the structure can be readily handled, including a certain class of linear gray-box structures. The proposed extension is not restricted to the discrete-time case and can be used to estimate continuous-time models.

The final topic in this thesis is the initialization of discrete-time systems containing polynomial nonlinearities. In the continuous-time case, the tools of differential algebra, especially Ritt's algorithm, have been used to prove that such a model structure is globally identifiable if and only if it can be written as a linear regression model. In particular, this implies that once Ritt's algorithm has been used to rewrite the nonlinear model structure into a linear regression model, the parameter estimation problem becomes trivial. Motivated by the above and the fact that most system identification problems involve sampled data, a version of Ritt's algorithm for the discrete-time case is provided. This algorithm is closely related to the continuous-time version and enables the handling of noise signals without differentiations.

Thursday December 10: Ulf Jönsson

Traffic Control: A Case for Scalable Robust Control
Time: 13:15-14:00. Place: Algoritmen.

Traffic control refers to the problem of achieving a fair, stable and robust congestion control mechanism for the Internet. In this talk we survey some tools from optimization and control theory that can be used to study fluid flow models of Internet traffic. In particular, various types of generalized Nyquist criteria that can be used to establish stability of this class of systems will be discussed.

Friday December 11: Henrik Tidefelt

Differential-algebraic equations and matrix-valued singular perturbation
Time: 13:15-15:00. Place: C3.

With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero.

The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions.

Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations.

While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly.

One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution.

One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

Thursday December 17: Vaclav Smidl

On Combining Distributional Approximations in Bayesian Filtering
Time: 13:15-14:00. Place: Glashuset.

The task of Bayesian filtering is solvable only for a few special cases of system models, such as linear models with Gaussian disturbances, which is equivalent to the Kalman filter. For more complex models, the solution of Bayesian filtering is is typically analytically intractable and the some forms of distributional approximations must be used to obtain practically applicable solutions. Different type of algorithms arise under different choices of approximations. For example, the famous particle filter arise as a consequence of approximation of the posterior density by empirical distributions. The well known Extended Kalman filter arise as a projection of the posterior density to the Gaussian distribution. Each of these approximations has some advantages and disadvantages.
In this talk, we will discuss possible ways of combination of various distributional approximations in Bayesian filtering. This may be advantageous especially for high dimensional systems, where each part of the space may be approximated by different density. Preliminary results in this area will be illustrated in simulation and in the application of early warning system for radiation protection.