How to solve matrix in characteristic equation?

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Tianyi Chai
Tianyi Chai 2022-5-9
编辑： Torsten 2022-5-9
Given the system matrix A=[0 1 0 0;3 0 0 2; 0 0 0 1; 0 -2 0 0] and B=[0 0;1 0;0 0;0 1], From the characteristic equation det(A-BF) the eigenvalues{-1,-3,-5,-8} are found. How do I reverse the process to find the gain F?
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Tianyi Chai
Tianyi Chai 2022-5-9
Sorry for the confusion guys here is the complete question, which I misst the system is single-input single-output
Torsten
Torsten 2022-5-9
编辑：Torsten 2022-5-9
If it is satisfactory for you, you should then accept Sam's answer.

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Sam Chak
Sam Chak 2022-5-9
Hi @Tianyi Chai
This is actually very easy if you know algebra and solving simultaneous equations on the desired characteristic equation (from the eigenvalues) and the actual characteristic equation found from . The fancy name for this method is called Pole Placement:
A = [0 1 0 0; 3 0 0 2; 0 0 0 1; 0 -2 0 0] % state matrix
B = [0 0; 1 0; 0 0; 0 1] % input matrix
p = [-1 -3 -5 -8] % desired poles
F = place(A, B, p) % Pole placement design to calculate the control gain matrix F
% check the result
eig(A-B*F)
For more info, please check:
https://www.mathworks.com/help/control/ref/place.html

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