Intuitive Understanding of Kalman Filtering with MATLAB®

Armando Barreto, Malek Adjouadi, Francisco R. Ortega, Nonnarit O-larnnithipong

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9780429575457
• Edition: 1
• Topic: Informatik
• Subtopic: Informatik Allgemein

The emergence of affordable micro sensors, such as MEMS Inertial Measurement Systems, which are being applied in embedded systems and Internet-of-Things devices, has brought techniques such as Kalman Filtering, capable of combining information from multiple sensors or sources, to the interest of students and hobbyists. This will book will develop just the necessary background concepts, helping a much wider audience of readers develop an understanding and intuition that will enable them to follow the explanation for the Kalman Filtering algorithm

I
Background

THE OBJECTIVE OF THE three chapters in this part of the book is to provide the reader with the background concepts that will be essential to understand the elements involved in the kind of estimation challenge that the Kalman Filter addresses. This background is also necessary to follow the reasoning steps that will lead us to the Kalman Filter algorithm, in Part II.

We expect that the reader of this book will be primarily interested in understanding and applying the Kalman Filter. As such, we have tried to keep the background chapters in this part of the book very concise and “to the point.” Clearly, any of the concepts outlined in these chapters could be developed much more extensively, and put into a wider, more formal context. Instead we have tried to communicate only the concepts that will be used in our future reasoning (Part II) and we have emphasized descriptions of these concepts in familiar terms, whenever possible.

We strongly suggest that the reader reviews these short background chapters. We hope these chapters will “fill any gaps” in the knowledge of some of our readers, so that they will be well equipped to follow and assimilate our reasoning through Part II. Even if you believe you are fairly familiar with all these background items, it will likely be worthwhile reviewing these chapters to make sure you identify our specific view of these concepts, which should facilitate your reading of Part II.

CHAPTER 1

System Models and Random Variables

This chapter presents the reader with the concept of a model for a system and justifies the need to address some variables as random (and not deterministic) variables. The remainder of this chapter is devoted to providing an intuitive understanding of the typical mechanisms that are used to characterize random variables, emphasizing the understanding of how the characterization applies to the digitized values of a signal. The latter portion of this chapter focuses on the Gaussian (Normal) probability distribution, its notation, the effect of processing samples that have a Gaussian distribution through a transformation represented by a straight line, and the characteristics exhibited by a random variable that is the result of multiplying two variables that have Gaussian probability distributions.

1.1 DETERMINISTIC AND RANDOM MODELS AND VARIABLES

Engineers very often aim at providing real-life solutions to real-life problems. However, engineers very seldom attack the problem directly on the physical reality or circumstance where the “problem” lies. Instead, engineers abstract the critical functional aspects of a real-life problem in an operational model. The model frequently is an intangible representation of the real situation that can, nonetheless, be manipulated to predict what would be the outcomes of different changes in the inputs or the structure of the real-life situation.

Engineers use knowledge of the physical world to emulate real-life constraints in the manipulations of their abstract models, so that results obtained from the models are good predictors of what will happen if the same manipulations applied to the model are implemented in the real world. Those real-life constraints and the manipulations performed in the model are commonly expressed and achieved in mathematical terms. For example, Kirchhoff’s Voltage Law states that the net sum of voltages around a loop is zero (Rizzoni 2004):

$${\displaystyle \sum _i{V}_i}=0$$

(1.1)

Using this and the model of the relationship between the current i_R through a resistor R and the...