How to solve matrix in characteristic equation?

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?

[Torsten]

???

What is F in the forward process (from which you found the eigenvalues) ?

[Sam Chak]

Hi @Torsten

The problem is to find the matrix F such that eig(A - B*F) gives the desired eigenvalues {-1,-3,-5,-8}.

I tried not to use the special function from the toolbox. Mathematically, the solution can be found by comparing terms. But there are more variables than available equations. I must be missing something. Perhaps you may offer some insights on solving this:

syms s k11 k12 k13 k14 k21 k22 k23 k24
A = sym([0 1 0 0; 3 0 0 2; 0 0 0 1; 0 -2 0 0]);
B = sym([0 0; 1 0; 0 0; 0 1]);
F = sym([k11 k12 k13 k14; k21 k22 k23 k24]);
Cpoly = charpoly(A-B*F, s)		% characteristic polynomial
Dpoly = (s+1)*(s+3)*(s+5)*(s+8)	
Dpoly = expand(Dpoly)		% desired characteristic polynomial

[Ccoef, Cterms] = coeffs(Cpoly, s)
[Dcoef, Dterms] = coeffs(Dpoly, s)

[Torsten]

You know how you make F unique ? Seems you have more unknowns than equations, or am I mistaken ?

[Tianyi Chai]

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

[Torsten]

If it is satisfactory for you, you should then accept Sam's answer.

Answer 1

[Sam Chak]

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