The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

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Unscented filtering and nonlinear estimation

The Kalman Filter (KF) is one of the most widely used methods for tracking and estimation due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the Extended Kalman Filter (EKF) which simply linearizes all nonlinear models so that the traditional linear Kalman filter can be applied. Although the EKF (in its many forms) is a widely used filtering strategy, over thirty years of experience with it has led to a general consensus within the tracking and control community that it is difficult to implement, difficult to tune, and only reliable for systems which are almost linear on the time scale of the update intervals. In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely sampled points can be used to parameterize mean and covariance, the estimator yields performance equivalent to the KF for linear systems yet generalizes elegantly to nonlinear systems without the linearization steps required by the EKF. We show analytically that the expected performance of the new approach is superior to that of the EKF and, in fact, is directly comparable to that of the second order Gauss filter. The method is not restricted to assuming that the distributions of noise sources are Gaussian. We argue that the ease of implementation and more accurate estimation features of the new filter recommend its use over the EKF in virtually all applications.

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New extension of the Kalman filter to nonlinear systems

The Interacting Multiple Model (IMM) estimator is a suboptimal hybrid filter that has been shown to be one of the most cost-effective hybrid state estimation schemes. The main feature of this algorithm is its ability to estimate the state of a dynamic system with several behavior modes which can "switch" from one to another. In particular, the IMM estimator can be a self-adjusting variable-bandwidth filter, which makes it natural for tracking maneuvering targets. The importance of this approach is that it is the best compromise available currently-between complexity and performance: its computational requirements are nearly linear in the size of the problem (number of models) while its performance is almost the same as that of an algorithm with quadratic complexity. The objective of this work is to survey and put in perspective the existing IMM methods for target tracking problems. Special attention is given to the assumptions underlying each algorithm and its applicability to various situations.

Interacting multiple model methods in target tracking: a survey

Interacting Multiple Model Methods in Target Tracking: A Survey

Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book's practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include MATLAB computations, and the numerous end-of-chapter exercises include computational assignments. MATLAB/GNU Octave source code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.

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Bayesian Filtering and Smoothing

In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for discrete-time pole assignment by output feedback for linear time-invariant systems The resultant output feedback gain is a periodic function of time with a period equal to the sampling interval It is also shown that periodic discrete-output feedback gains can be used in stabilizing a linear time-invariant system if and only if its unstable modes are controllable and observable

On the design of linear time invariant systems by periodic output feedback Part I. Discrete-time pole assignment†

Differential equations are developed for the smoothing density function and for the smoothed expectation of an arbitrary function of the state. The exact equations developed herein are difficult to solve except in trivially simple cases. Approximations to these equations are developed for the smoothed expectation of the state and the smoothing covariance matrix. For linear systems these equations are shown to reduce to previously derived results. An iterative technique is suggested for even greater accuracy in approximations for severely nonlinear systems.

Nonlinear Smoothing Theory

The problem of finite time controllability of linear time invariant systems by piecewise constant output feedback is considered It is shown that complete controllability and complete observability are both necessary and sufficient conditions for output feedback controllability by piecewise constant gain Dead-beat output feedback controllers that drives the system to the origin in at most (μd × vd) steps, are also presented, where μd and vg are the indices of controllability and observability

On the finite time control of linear systems by piecewise constant output feedback

This is the first part of a two-part paper which summarizes work pursued by the author in 1966 [1]. The paper describes the application of minimum-variance estimation techniques for in-flight alignment and calibration of an inertial measurement unit (IMU) relative to another IMU and/or some other reference. The first part formulates the problem, and the second part [2] reports numerical results and analyses. The approach taken is to cast the problem into the framework of Kalman-Bucy estimation theory, where velocity and position differences between the two IMU's are used as observations and the IMU parameters of interest become part of the state vector. Instrument quantization and computer roundoff errors are considered as measurement noise, and environmental induced random accelerations are considered as state noise. Typical applications of the technique presented might include the alignment and calibration of IMU's on aircraft carriers, the initialization of rockets or rocket airplanes which are launched from the wing of a mother ship, the alignment and calibration of IMU's which are only used in the latter phases of rocket flight, and for the initialization/updating of SST guidance systems.

In-Flight Alignment and Calibration of Inertial Measurement Units - Part I: General Formulation

In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for the complete controllability of linear time-invariant systems by discrete output feedback. In the case where the desired final state is the origin, the resultant, output feedback gain is periodic and the transfer is accomplished in at most vd periods, where vd is the discrete-time index of observability. In the more general case where the desired final state is non-zero, the transfer is accomplished using at most vd output measurements.

Cornelius T. Leondes

Papers

On the design of linear time invariant systems by periodic output feedback Part I. Discrete-time pole assignment†

Nonlinear Smoothing Theory

On the finite time control of linear systems by piecewise constant output feedback

In-Flight Alignment and Calibration of Inertial Measurement Units - Part I: General Formulation

On the design of linear time-invariant systems by-periodic output feedback Part II. Output feedback controllability†