Commonly used algorithms for drones - Kalman filter (10)

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4 Knowledge related to Kalman filtering

4.1 Linear system

Judgement of linear systems. When the mathematical equations describing a dynamic system have linear properties, the corresponding system is called a linear system. The basic characteristic of a linear system is that it satisfies the superposition principle, that is, if the mathematical description of the system is L, then for any two input variables u1 and u2 and any two finite constants c1 and c2, there must be

The following description was obtained from the Internet:

A system in which state variables and output variables satisfy the superposition principle for all possible input variables and initial states. The superposition principle means that if the state and output of a system corresponding to any two input and initial states (u1, x01) and (u2(t), x02) are (x1(t), y1(t)) and (x2(t), y2(t)), respectively, then when the input and initial state are (C1u1(t)+C2u2(t), C1x01＋C2x02), the state and output of the system must be (C1x1(t)＋C2x2(t), C1y1(t)＋C2y2(t)), where x represents the state, y represents the input, u represents the output, and C1 and C2 are arbitrary real numbers. A system composed of linear components must be a linear system. However, the opposite proposition may not hold in some cases. The causal relationship between the state variables (or output variables) and input variables of a linear system can be described by a set of linear differential equations or difference equations, which are called the mathematical model of the system. As a direct result of the superposition property, an important property of a linear system is that the response of the system can be decomposed into two parts: zero input response and zero state response. The former refers to the response caused by a non-zero initial state; the latter refers to the response caused by the input. Both can be calculated separately. This property brings great convenience to the analysis and research of linear systems.

4.2 Taylor series expansion

Suppose function f(x) is expanded at x=a, the expansion formula is:

[font=微软雅黑, For the nonlinear system, the state equation is:

X(k+1)=f(k,X(k))+W(k), where there is no input item.

In order to get the one-step prediction state X (k +1| k), the nonlinear function in the above formula is expanded by Taylor series around X (k | k) (which is already a specific value) and its first-order term is taken. The complete Taylor series expansion is:

Where: n is the dimension of the state vector, ei is the Ith Cartesian basis vector. For example, there are 4 Cartesian basis vectors in the case of four-dimensional vectors:

[font=微软雅黑, And the first-order derivative fX (k) of f(k,X(k)):

is also the Jacobian matrix of the vector f, evaluated at the most recent estimate of the state X (k | k).

Similarly, the Hessian matrix of the ith component of the vector f can be calculated as follows:

It can be seen that the second-order Taylor expansion requires a lot of calculations, so the first-order expansion is generally used. The first-order Taylor expansion linearization equation of the state equation is: X(k+1)= f(k,X(k|k))+ f(k)[X(k)-X(k|k)]+W(k)

Taylor series first-order expansion of a function is to linearize the function. If the second-order expansion is taken, it is just another form of approximation of the original nonlinear function, which is still nonlinear.

And the first-order derivative fX (k) of f(k,X(k)):

is also a vector f The Jacobian matrix of is taken at the nearest estimate X (k | k) of the state.

Similarly, the Hessian matrix of the ith component of the vector f can be obtained as follows:

It can be seen that the second-order Taylor expansion requires a lot of calculations, so the first-order expansion is generally taken. The first-order Taylor expansion linearization equation of the state equation is: X(k+1)= f(k,X(k|k))+ f(k)[X(k)-X(k|k)]+W(k)

Taylor series first-order expansion of a function is to linearize the function. If the second-order expansion is taken, it is just another form of approximation of the original nonlinear function, which is still nonlinear.

And the first-order derivative fX (k) of f(k,X(k)):

is also a vector f The Jacobian matrix of is taken at the nearest estimate X (k | k) of the state.

Similarly, the Hessian matrix of the ith component of the vector f can be obtained as follows:

It can be seen that the second-order Taylor expansion requires a lot of calculations, so the first-order expansion is generally taken. The first-order Taylor expansion linearization equation of the state equation is: X(k+1)= f(k,X(k|k))+ f(k)[X(k)-X(k|k)]+W(k)

Taylor series first-order expansion of a function is to linearize the function. If the second-order expansion is taken, it is just another form of approximation of the original nonlinear function, which is still nonlinear.