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Discretize a continuous LTI system: zero-order holding


The continuous LTI system is


The corresponding discrete system is


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


>>>F=I+\Delta t A

>>>G=\Delta t B

2) Rigorous discrete system


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


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


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

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


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

You can check it using the definition of cross product, see

Point transformation and Vector transformation between two frames


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


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


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

Estimate Mean and Covariance Matrix Using MLE


See here MLE mean variance!!!