How to solve matrix in characteristic equation?

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Tianyi Chai 2022-5-9

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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 2022-5-9

Sorry for the confusion guys here is the complete question, which I misst the system is single-input single-output

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 2022-5-9

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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)