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Relationship between the covariance of continuous and discrete noise processes.

08/11/2011

Relationship between the covariance of continuous and discrete noise processes.

Covariance of continuous and discrete systems

But you should know what you have now is for the continuous or the discrete system.

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Discretize a continuous LTI system: zero-order holding

08/11/2011

The continuous LTI system is

\dot{x}=Ax+Bu

The corresponding discrete system is

x_k=Fx_{k-1}+Gu_{k}

What are F and G? Usually we have two methods: use zero-order hold method to discretize it; or use approximation to a rigorous discrete system. Today I will show they are the same to the first order !!! So in the future, we are confident to directly use zero-order holding method to discretize a system as long as you want the first order accuracy!!!

1) Zero-order holding method:

x_k=x_{k-1}+(Ax_{k-1}+Bu_k)\Delta t

where \Delta t=t_{k}-t_{k-1}. Then

x_k=(I+\Delta t A)x_{k-1}+\Delta t Bu_k

So

>>>F=I+\Delta t A

>>>G=\Delta t B

2) Rigorous discrete system

From

x(t)=e^{A(t-t_0)}x_0+\int_{t_0}^te^{A(t-\tau)} Bu(\tau) \mbox{d}\tau

it is easy to see

>>>F=e^{A\Delta t}=I+\Delta t A + \frac{1}{2}A^2\Delta t^2\cdots

>>>G=\left(\int_0^{\Delta t}e^{A\alpha}d\alpha\right)B=\left(\Delta t+\frac{1}{2}A\Delta t^2+\cdots\right)B

Conclusion: Obviously, method 1 and 2 are equivalent to the first order!!!

Error model of the navigation equation

08/11/2011

Linearized error model of the 6-DOF kinematic model is

error model of 6-DOF

Note: it is based on perturbation, only valid to first order.

Memo: skew-symmetric matrix associating a vector

01/11/2011

Let a be a vector, then a\times b=[a]_{\times}b, what is [a]_{\times}?

If a=[a1,a2,a3]^T, then

[a]_{\times}=

[0 -a3 a2]
[a3 0 -a1]
[-a2 a1 0]

You can check it using the definition of cross product, see http://en.wikipedia.org/wiki/Cross_product.

Point transformation and Vector transformation between two frames

21/10/2011

You think you are familiar with homography? No, There are many things you don’t know yet.

See here about point transformation and vector transformation with application in Homography. Note it is easy to be confused when a vector starting from original.

Homography in different frames

Minimum variance estimation & Maximum Likelihood Estimation

17/10/2011

Are all MLE MVE? At least, MLE of means of Gaussian distribution is MLE. Here is a good note!!! MLE vs MVE

Multiple classes with the same covariance

15/10/2011

If there are multiple classes, each class has many samples. All classes have the same covariance matrix but different mean. So can we use all samples across classes to estimate the covariance matrix? Click here MLE mutiple class, same covariance