The Beauty of Control (Volume 2) - Kalman Filter

dirty

The Beauty of Control (Volume 2) - Kalman Filter [Copy link]

 

This article talks about studying the last chapter of this book - Kalman filter, understanding its concepts and mathematical methods, and doing simulations with cases.

1. Concept and Idea of Kalman Filter

Kalman filter is an optimization, recursive, digital processing algorithm that has the characteristics of both filter and observer.

The Kalman filter effectively solves the above-mentioned uncertainty problems by fusing data from multiple sensors or measuring devices and using probability theory and linear system theory to estimate and predict the state. It can recursively update the state estimate based on prior knowledge and measurement information and provide the best prediction of the future state.

2. Mathematical methods included in the Kalman filter

1. Recursive Algorithms and Fusion

The Kalman filter combines the idea of recursive algorithm and data fusion. The algorithm that infers the current estimated value from the previous estimated value is called a recursive algorithm. The measured value is fused with the previous estimated value by adjusting (coefficient). This embodies the idea of data fusion.

2. Introduction to probability theory, data fusion and covariance matrix

Kalman filter is a probability-based computational method used to deal with uncertainty in the system. This section involves the integration of probability theory with expectation, variance, normal distribution, covariance, etc. in mathematical statistics.

3. Derivation of Linear Kalman Filter and Description of Its Algorithm

Here we describe the derivation of five important formulas of the Kalman filter: prior state estimation, Kalman gain, a posteriori and prior state estimation error, optimal estimated Kalman gain, and the covariance matrix of the a posteriori state estimation error. The formula derivation focuses on the introduction and calculation of mathematical foundations. The Kalman filter algorithm can be divided into two parts: time update and measurement update.

3. Kalman filter case analysis and simulation

Taking drone altitude observation as an example, this paper introduces the use of Kalman filter. In this example, the state equation of the control unit for drone altitude control is given. A and B are matrix constants, and the three-dimensional measurement matrix λ is given. The initial altitude and speed are given, and the simulation is performed after adding noise. After a series of processing, the real value and the measured value are simulated, and intuitive results can be seen.

It can be seen that the true value of the height is well matched with the measured value.

The combination of Kalman filter and MPC controller and extended Kalman filter are introduced later in this chapter. Here is a diagram of the combination of Kalman filter and MPC controller.

The extended Kalman filter is mainly used when the system has a nonlinear dynamic model or observation equation, such as a simple pendulum system.

At this point, I have reviewed and studied the entire book "The Beauty of Control (Volume 2)", and have a more systematic understanding and study of control theories and methods. This has improved my knowledge and has guiding significance for future engineering control, and I have benefited a lot.

Jacktang

I feel that the Kalman filter is difficult in theory.

yangfang0916

How long do I have to chew it slowly? It's too difficult.

kevin-----

Thanks a lot for sharing

ew123

I have seen the term Kalman filter many times, but I have not studied it in depth. I read this post here and it seems to be quite useful and worth studying in depth.

13620203064

Good! I learned! Good! I learned! Good! I learned! Good! I learned! Good! I learned! Good! I learned!

dirty

Jacktang posted on 2024-2-29 07:45 I feel that the Kalman filter is very difficult in theory,,

It is necessary to have a solid foundation in mathematical theory and deduce a lot of formulas.

dirty

yangfang0916 posted on 2024-2-29 08:21 Take your time, how long will it take? It's too difficult

Haha, read often and learn new things, review the old and learn new things, that's it

dirty

13620203064 Published on 2024-2-29 09:49 Not bad! I learned! Not bad! I learned! Not bad! I learned! Not bad! I learned! Not bad! I learned!

It is good to expand your knowledge and ability

Gen_X

It is suitable for use in situations where the discreteness is large but the accuracy requirements are relatively high.

For example, the GPS positioning information received by the missile has somewhat discrete data, but it must be credible, so only a similar method can be used.

freebsder

Thanks for sharing, Kalman is so difficult