# Discrete stochastic processes and optimal filtering pdf

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- Discrete Stochastic Processes and Optimal Filtering
- The Kalman-Bucy method of optimal filtering and its generalizations
- Optimal Filtering of Discrete-Time Hybrid Systems

*Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using Matlab. Chapter 1.*

This paper is concerned with the optimal Kalman filtering problem for a class of discrete stochastic systems with multiplicative noises and random two-step sensor delays. Three Bernoulli distributed random variables with known conditional probabilities are introduced to characterize the phenomena of the random two-step sensor delays which may happen during the data transmission. By using the state augmentation approach and innovation analysis technique, an optimal Kalman filter is constructed for the augmented system in the sense of the minimum mean square error MMSE.

## Discrete Stochastic Processes and Optimal Filtering

In statistics and control theory , Kalman filtering , also known as linear quadratic estimation LQE , is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe.

The filter is named after Rudolf E. The Kalman filter has numerous applications in technology. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and dynamically positioned ships.

Kalman filters also are one of the main topics in the field of robotic motion planning and control and can be used in trajectory optimization.

Due to the time delay between issuing motor commands and receiving sensory feedback , use of the Kalman filter supports a realistic model for making estimates of the current state of the motor system and issuing updated commands. The algorithm works in a two-step process. In the prediction step, the Kalman filter produces estimates of the current state variables , along with their uncertainties. Once the outcome of the next measurement necessarily corrupted with some amount of error, including random noise is observed, these estimates are updated using a weighted average , with more weight being given to estimates with higher certainty.

The algorithm is recursive. It can run in real time , using only the present input measurements and the previously calculated state and its uncertainty matrix; no additional past information is required. Optimality of the Kalman filter assumes that the errors are Gaussian. In the words of Rudolf E. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear.

Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The underlying model is a hidden Markov model where the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Also, Kalman filter has been successfully used in multi-sensor fusion , [6] and distributed sensor networks to develop distributed or consensus Kalman filter.

Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, leading to it sometimes being called the Kalman—Bucy filter. Stanley F. Schmidt is generally credited with developing the first implementation of a Kalman filter.

He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. This Kalman filter was first described and partially developed in technical papers by Swerling , Kalman and Kalman and Bucy The Apollo computer used 2k of magnetic core RAM and 36k wire rope [ Clock speed was under kHz [ The fact that the MIT engineers were able to pack such good software one of the very first applications of the Kalman filter into such a tiny computer is truly remarkable.

Kalman filters have been vital in the implementation of the navigation systems of U. Navy nuclear ballistic missile submarines , and in the guidance and navigation systems of cruise missiles such as the U.

Navy's Tomahawk missile and the U. They are also used in the guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station. This digital filter is sometimes called the Stratonovich—Kalman—Bucy filter because it is a special case of a more general, nonlinear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich.

The Kalman filter uses a system's dynamic model e. As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for all place limits on how well it is possible to determine the system's state.

The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a weighted average. The purpose of the weights is that values with better i.

The weights are calculated from the covariance , a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursively and requires only the last "best guess", rather than the entire history, of a system's state to calculate a new state.

The relative certainty of the measurements and current state estimate is an important consideration, and it is common to discuss the response of the filter in terms of the Kalman filter's gain. The Kalman gain is the relative weight given to the measurements and current state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on the most recent measurements, and thus follows them more responsively. With a low gain, the filter follows the model predictions more closely.

At the extremes, a high gain close to one will result in a more jumpy estimated trajectory, while a low gain close to zero will smooth out noise but decrease the responsiveness. When performing the actual calculations for the filter as discussed below , the state estimate and covariances are coded into matrices to handle the multiple dimensions involved in a single set of calculations.

This allows for a representation of linear relationships between different state variables such as position, velocity, and acceleration in any of the transition models or covariances. As an example application, consider the problem of determining the precise location of a truck.

The truck can be equipped with a GPS unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and the angle of the steering wheel.

This is a technique known as dead reckoning. Typically, the dead reckoning will provide a very smooth estimate of the truck's position, but it will drift over time as small errors accumulate.

In this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to the physical laws of motion the dynamic or "state transition" model.

Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds.

Next, in the update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull the position estimate back towards the real position but not disturb it to the point of becoming noisy and rapidly jumping.

The Kalman filter is an efficient recursive filter that estimates the internal state of a linear dynamic system from a series of noisy measurements.

It is used in a wide range of engineering and econometric applications from radar and computer vision to estimation of structural macroeconomic models, [17] [18] and is an important topic in control theory and control systems engineering.

The Kalman filter, the linear-quadratic regulator, and the linear—quadratic—Gaussian controller are solutions to what arguably are the most fundamental problems in control theory. In most applications, the internal state is much larger more degrees of freedom than the few "observable" parameters which are measured.

However, by combining a series of measurements, the Kalman filter can estimate the entire internal state. In the Dempster—Shafer theory , each state equation or observation is considered a special case of a linear belief function and the Kalman filter is a special case of combining linear belief functions on a join-tree or Markov tree.

Additional approaches include belief filters which use Bayes or evidential updates to the state equations. A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the "simple" Kalman filter, the Kalman—Bucy filter , Schmidt's "extended" filter, the information filter , and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps the most commonly used type of very simple Kalman filter is the phase-locked loop , which is now ubiquitous in radios, especially frequency modulation FM radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment.

Kalman filters are based on linear dynamical systems discretized in the time domain. They are modeled on a Markov chain built on linear operators perturbed by errors that may include Gaussian noise.

The state of the system is represented as a vector of real numbers. At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the observed outputs from the true "hidden" state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space as opposed to a discrete state space as in the hidden Markov model.

There is a strong analogy between the equations of the Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani , [19] and Hamilton , Chapter In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations one must model the process in accordance with the following framework.

This means specifying the following matrices:. At time k an observation or measurement z k of the true state x k is made according to. Many real dynamical systems do not exactly fit this model. In fact, unmodeled dynamics can seriously degrade the filter performance, even when it was supposed to work with unknown stochastic signals as inputs.

The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability it diverges. On the other hand, independent white noise signals will not make the algorithm diverge.

The problem of distinguishing between measurement noise and unmodeled dynamics is a difficult one and is treated in control theory under the framework of robust control. The Kalman filter is a recursive estimator.

This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. The Kalman filter can be written as a single equation, however, it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep.

This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep. In the update phase, the current a priori prediction is combined with current observation information to refine the state estimate. This improved estimate is termed the a posteriori state estimate.

Typically, the two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction steps performed. Likewise, if multiple independent observations are available at the same time, multiple update steps may be performed typically with different observation matrices H k.

The formula for the updated a posteriori estimate covariance above is valid for the optimal K k gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the derivations section, where the formula valid for any K k is also shown.

In our case:.

## The Kalman-Bucy method of optimal filtering and its generalizations

This course gives thorough knowledge of linear estimation theory. The main theme of the course is optimal linear estimation, Kalman and Weiner filtering, which are systematic methods to solve estimation problems with applications in several technical disciplines, for example in telecommunications, automatic control and signal processing but also in other disciplines, such as econometrics and statistics. The course also provides an introduction to optimal filtering for non-linear systems. The course assumes familiarity with basic concepts from matrix theory, stochastic processes, and linear systems theory. The course is directed towards the students who intend to work with development and research within these fields. The following topics are covered; Basic estimation theory, time discrete and time continuous Wiener filters, time discrete Kalman filters, properties of Wiener and Kalman filters, smoothing, Extended Kalman filters, sigma-point filters and particle filters.

## Optimal Filtering of Discrete-Time Hybrid Systems

In statistics and control theory , Kalman filtering , also known as linear quadratic estimation LQE , is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. The Kalman filter has numerous applications in technology. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and dynamically positioned ships. Kalman filters also are one of the main topics in the field of robotic motion planning and control and can be used in trajectory optimization.

Ogni modulo equivale a 3 crediti ECTS. La descrizione del modulo scarica il pdf riporta le informazioni linguistiche per ogni modulo, suddivise nelle seguenti categorie:. The goal of this module is to introduce the students to the powerful world of statistical digital signal processing. While at the bachelor level digital signal processing is most often taught with deterministic signals, in the real world most interesting signals are stochastic in nature. Hence in more advanced applications, such as prediction or noise removal, the theories presented in this module are essential.

The locally optimal filter is designed for a class of discrete-time systems subject to stochastic nonlinearity functions, finite-step correlated noises, and missing measurements. The multiplicative noises are employed to describe the random disturbances in the system model. The phenomena of missing measurements occur in a random way and the missing probability is characterized by Bernoulli distributed random variables with known conditional probabilities.