Kalman Filter Principle

The following is a more detailed information. http://jpkc.nwpu.edu.cn/jp2005/40/ebook/kcsj/chp1/Kalman.htm

1. Kalman Filter Algorithm
In this section, we will describe the Kalman Filter derived from Dr Kalman. The following description involves some basic conceptual knowledge, including probability, random variable, Gaussian or normal distribution, and state-space model. However, the detailed proof of the Kalman filter cannot be described here.
First, we need to introduce a discrete control process system. The system can be described by a linear random differential equation (Linear Stochastic difference equation:

X (K) = a x (k-1) + B u (k) + W (k)

Plus the system measurement value:

Z (K) = H x (k) + V (k)

In the previous two formulas, x (k) is the system state at K moment, and U (k) is the control value for the system at K moment. A and B are system parameters. For multi-model systems, they are matrices. Z (k) is the measured value at K time, and H is the parameter of the measurement system. H is the matrix of multiple measurement systems. W (K) and V (k) represent the process and measurement noise respectively. They are assumed to be white Gaussian noise and Their covariance is Q and R (Here we assume they do not change with the system status ).

Kalman filter is the optimal information processor to meet the above conditions (process and measurement of linear random differential systems are Gaussian white noise. Next we will use them together with their covariances to estimate the optimal output of the system (similar to the temperature example in the previous section ).

First, we need to use the system process model to predict the next state system. Assume that the current system status is K. Based on the system model, you can predict the status based on the previous status of the system:

X (k | k-1) = a x (K-1 | k-1) + B u (k )........... (1)

In formula (1), x (k | k-1) is the result of using the previous state prediction, x (K-1 | k-1) is the result of the optimum of the previous State, u (k) the current State control. If there is no control, it can be 0.

Our system results have been updated so far, but the covariance for X (k | k-1) has not been updated yet. We use P to represent covariance:

P (k | k-1) = a p (K-1 | k-1) a' + q ......... (2)

In formula (2), P (k | k-1) is the covariance of X (k | k-1), P (K-1 | k-1) is the covariance of X (K-1 | k-1, a' indicates the transpose matrix of A, and Q is the covariance of the system process. Formula 1 and 2 are the first two of the five formulas of Kalman filter, that is, the prediction of the system.

Now we have the prediction result of the current state, and then we collect the measurement value of the current state. Combined with the predicted values and measured values, we can obtain the optimal estimation value of the current State (k) x (k | K ):

X (k | K) = x (k | k-1) + KG (k) (Z (k)-H x (k | k-1 ))......... (3)

Where kg is Kalman gain (Kalman gain ):

Kg (K) = P (k | k-1) H '/(h p (k | k-1) H' + r )......... (4)

So far, we have obtained the optimal estimate value x (k | K) in K State ). But to keep the other Kalman filter running until the system process ends, we need to update the covariance of X (k | K) in K State:

P (k | K) = (I-kg (k) H) P (k | k-1 )......... (5)

Where I is a matrix of 1, for a single measurement of a single model, I = 1. When the system enters the k + 1 State, P (k | K) is the formula (2) P (K-1 | k-1 ). In this way, the algorithm can be used to perform self-regression operations. The principle of Kalman filter is basically described. Formula 1, 2, 3, 4, and 5 are his five basic formulas. According to these five formulas, it is easy to implement computer programs.

Next, I will use a program to give an example of actual operation.
2. Simple Example

Here we use section 3 of section 2 to give a very simple example to illustrate the operation process of Kalman filter. The example above is a further example of section 2, which is also matched with a program simulation result.

According to the description in section 2, the room is regarded as a system, and then the system is modeled. Of course, the model we see does not need to be very accurate. The temperature of the room we know is the same as that of the previous time, so a = 1. There is no control, so u (K) = 0. Therefore, it is concluded that:

X (k | k-1) = x (K-1 | k-1 )........... (6)

The sub-Statement (2) can be changed:

P (k | k-1) = P (K-1 | k-1) + q ......... (7)

Because the measured value is a thermometer and corresponds directly to the temperature, H = 1. Format 3, 4, and 5 can be changed to the following:

X (k | K) = x (k | k-1) + KG (k) (Z (k)-X (k | k-1 ))......... (8)
Kg (K) = P (k | k-1)/(P (k | k-1) + r )......... (9)
P (k | K) = (1-kg (k) P (k | k-1 )......... (10)

Http://blog.csdn.net/luckydongbin/archive/2007/03/01/1518594.aspx

Kalman filter is an optimal recursive processing.Algorithm

Summary

In the process of target detection and tracking, to improve the tracking speed, estimate the possible position of the target at the next time when the target is determined, then, we take the estimated location as the center and search for the target within a certain range. This will narrow down the search range of the target and increase the tracking speed.

 

 

 

 

1) The optimal (optimal) depends on the performance evaluation criterion. The kalman filter makes full use of the following information to estimate the current value of the variable of Interest:. dynamic Characteristics of systems and measuring devices; B. statistical description of system noise, measurement errors, and uncertainty of dynamic models; C. information about the initial conditions of variables of interest. (2) recursion means that Kalman does not need to store the previous data and does not need to process the original data when performing a new measurement.

(3) filter (DPA) is actually a data processing algorithm, but a program processed in computing. Therefore, it can process Discrete Time Measurement samples instead of continuous time input.

• Basic Assumptions

It is reasonable to use a linear model. This is a typical engineering model that is linear at some major points or tracks. linear models are simpler than non-linear models. Therefore, linear models are used for approximation.

White Noise means that the noise value is irrelevant to the time. White Noise refers to the noise with the same frequency characteristics throughout the frequency. In practical applications, the frequency is set to a constant value. The noise with a bandwidth that greatly exceeds the system bandwidth is called white noise. It can be simulated with Gaussian white noise to greatly simplify the model.

It is feasible to use Gaussian density functions in practice. It is easy to use Gaussian Functions in mathematics. In the absence of Higher-Order Statistics, except for the Gaussian density, there is no better function form that can be expressed. Gaussian white noise can be fully described using first-order and second-order statistics.