The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.

/pdf/kalman-filtering-with-state-constraints-a-survey-of-linear-z9bwow7t97.pdf

Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

Kalman filters are commonly used to estimate the states of a dynamic system. However, in the application of Kalman filters there is often known model or signal information that is either ignored or dealt with heuristically. For instance, constraints on state values (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. A rigorous analytic method of incorporating state equality constraints in the Kalman filter is developed. The constraints may be time varying. At each time step the unconstrained Kalman filter solution is projected onto the state constraint surface. This significantly improves the prediction accuracy of the filter. The use of this algorithm is demonstrated on a simple nonlinear vehicle tracking problem.

/pdf/kalman-filtering-with-state-equality-constraints-45ynbm3xls.pdf

Kalman filtering with state equality constraints

Polylactide (PLA), a biodegradable aliphatic polyester, has been studied extensively for its wide applications in various fields from food packaging to interior materials for automobiles. One of PLA's advantages is that the raw material, lactic acid can be derived from renewable resources, which makes PLA very attractive for the packaging and considered as green packaging. Although, the cost of PLA is relatively higher compared to petroleum based packaging materials and it is predicted that the price will go down after commercial success of the process. This paper presents a review on polylactide with focus on its chemistry, synthesis, properties, and applications. Plasticization effect on PLA, sorption isotherm and mechanical properties of PLA based packaging materials have been discussed. PLA/layered organo nanocomposites exhibit significant improvement in mechanical, thermal, optical, and physicochemical properties over pure PLA. This review includes brief overview of PLA based nanocomposite, their mec...

/pdf/polylactides-chemistry-properties-and-green-packaging-23wnjezmvg.pdf

Polylactides—Chemistry, Properties and Green Packaging Technology: A Review

The state space description of some physical systems possess nonlinear equality constraints between some state variables. In this paper, we consider the problem of applying a Kalman filter-type estimator in the presence of such constraints. We categorize previous approaches into pseudo-observation and projection methods and identify two types of constraints-those that act on the entire distribution and those that act on the mean of the distribution. We argue that the pseudo-observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only. We propose a new method that utilizes the projection method twice-once to constrain the entire distribution and once to constrain the statistics of the distribution. We illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation

/pdf/on-kalman-filtering-with-nonlinear-equality-constraints-2g9152vdgp.pdf

On Kalman Filtering With Nonlinear Equality Constraints

Brief paper: State estimation for linear systems with state equality constraints

The authors describe an algorithm for implementing a multisensor system in a model-based environment with consideration of the constraints. Based on an environment model, geometric features and constraints are generated from a CAD model database. Sensor models are used to predict sensor response to certain features and to interpret raw sensor data. A constrained MMS (minimum mean squared) estimator is used to recursively predict, match, and update feature location. The effects of applying various constraints in estimation were shown by simulation system mounted on a robot arm for localization of known object features. >

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