Detailed principle of Kalman filter

Kalman filter is an algorithm for optimal estimation of system state by using linear system state equation and system input and output observation data. Because the observation data contains the influence of noise and interference in the system, the optimal estimation can also be regarded as a filtering process.

Stanley Schmidt first realized the Kalman filter. When visiting the Ames Research Center of NASA, Kalman found his method very useful for solving the orbit prediction of Apollo program. Later, the navigation computer of Apollo spacecraft used this filter. Swerling (1958), Kalman (1960) and Kalman and Bucy (196 1) have published papers on this kind of filter.

Data filtering is a data processing technology to remove noise and restore real data. Kalman filter can estimate the state of dynamic system from a series of data with measurement noise when the measurement variance is known. Because it is convenient for computer programming and can update and process the data collected in real time, Kalman filter is the most widely used filtering method at present, and it has been well applied in many fields such as communication, navigation, guidance and control.

express

X(k)=A X(k- 1)+B U(k)+W(k)

background

For the first time, Stanley Schmidt

The Kalman filter is now available. When visiting the Ames Research Center of NASA, Kalman found his method very useful for solving the orbit prediction of Apollo program. Later, the navigation computer of Apollo spacecraft used this filter. Swerling (1958), Kalman (1960) and Kalman and Bucy (196 1) have published papers on this kind of filter.

definition

The traditional filtering method can only be realized when the useful signal and noise have different frequency bands. In the 1940s, N. Wiener and A.H. Kolmogorov introduced the statistical properties of signal and noise into the filtering theory, and under the assumption that the signal and noise are stationary processes, the true value of the signal is estimated by the optimization method to achieve the purpose of filtering, which is conceptually linked with the traditional filtering method and is called Wiener filtering. This method requires that both signal and noise must be based on stationary process. In the early 1960s, R.E.Kalman and R.S. Bucy published an important paper "New Achievements of Linear Filtering and Prediction Theory", and put forward a new linear filtering and prediction theory, called Kalman filtering. It is characterized by processing the noisy input and observation signals on the basis of linear state space representation to obtain the system state or real signal.

This theory is expressed in time domain, and the basic concept is: based on the state space representation of linear system, the optimal estimation of system state is obtained from the output and input observation data. The system state mentioned here is a set of minimum parameters, which summarizes the influence of all past inputs and disturbances on the system. Knowing the system state can determine the overall behavior of the system and the future input and interference.

Kalman filtering does not require that both signal and noise are stationary processes. For the system disturbance and observation error (i.e. noise) at each moment, as long as some appropriate assumptions are made on their statistical properties, the estimated value of the real signal with the smallest error can be obtained by processing the observation signal containing noise. Therefore, since the advent of Kalman filter theory, it has been applied in many departments such as communication system, power system, aerospace, environmental pollution control, industrial control, radar signal processing and so on, and has achieved many successful results. For example, in image processing, Kalman filter is applied to restore blurred images caused by some noises. After assuming some statistical properties of noise, we can use Kalman algorithm to recursively get the real image with the smallest mean square error from the blurred image, so as to restore the blurred image.

nature

① Kalman filter is an algorithm, which is suitable for linear, discrete and finite dimensional systems. Every autoregressive moving average system (ARMAX) with external variables or a system that can be represented by rational transfer function can be transformed into a system represented by state space, so it can be calculated by Kalman filter.

② No set of observation data can help to eliminate the certainty of x(t). The gain K(t) is also independent of the observed data.

(3) When the observed data and states obey Gaussian distribution, the conditional mean and conditional variance of Gaussian random variables are calculated by the Kalman recursive formula, so the Kalman filtering formula gives the linear minimum variance estimation of the updating process of the conditional probability density of the calculated states, that is, the minimum variance estimation.

form

Kalman filter has been implemented in many different ways, and the form originally proposed by Kalman is generally called simple Kalman filter. In addition, there are many variants of Schmidt spread filter, information filter and square root filter developed by beermann and Thornton. The most common Kalman filter is phase-locked loop, which is widely used in radios, computers and almost any video or communication equipment.

example

A typical example of Kalman filter is to predict the coordinate position and velocity of an object from a set of limited observation sequences containing noise. It can be found in many engineering applications (radar, computer vision). At the same time, Kalman filter is also an important topic in control theory and control system engineering.

App application

For example, radar, people are interested in tracking the target, but the measured values of the target's position, speed and acceleration are often noisy at any time. Kalman filter uses the dynamic information of the target to try to remove the influence of noise and get a good estimate of the target position. The estimation can be the estimation of current target position (filtering), the estimation of future position (prediction) and the estimation of past position (interpolation or smoothing).

Extended Kalman filter (EKF)

It is a dynamic system with time nonlinearity considered by Kalman filter, which is often used in target tracking system.

State estimation

State estimation is an important part of Kalman filter. Generally speaking, the quantitative inference of random quantity based on observation data is an estimation problem, especially the state estimation of dynamic behavior, which can realize the function of real-time operation state estimation and prediction. Such as aircraft state estimation. State estimation is of great significance for understanding and controlling the system, and the method adopted belongs to the estimation theory in statistics. The most commonly used are least square estimation, linear minimum variance estimation, minimum variance estimation and recursive least square estimation. Other methods, such as risk standard of Bayesian estimation, maximum likelihood estimation and stochastic approximation, are also applicable.

quantity of state

The state quantity disturbed by noise is a random quantity, and the exact value cannot be measured, but a series of observations can be made and estimated according to a group of observations and some statistical viewpoints. Make the estimated value as accurate as possible close to the true value, which is the optimal estimation. The difference between the true value and the estimated value is called the estimation error. If the mathematical expectation of the estimated value is equal to the true value, this estimation is called unbiased estimation. The recursive optimal estimation theory proposed by Kalman adopts state space description method and the algorithm adopts recursive form. Kalman filter can deal with multi-dimensional and non-stationary stochastic processes.

theory

Kalman filter theory overcomes the limitations of Wiener filter theory and has been widely used in engineering, especially in modern engineering such as control, guidance, navigation and communication.