How can I obtain the State space from the Mass,stiffness,damping matrices

I've modeled a system using Matlab and I want to obtain the State space from Mass,stiffness,damping matrices..is there a function on matlab doing that ?

[Michael]

Can you given an example of your mass, spring, damping matricies?

[aiman]

Mass

[m1, 0 ]

[ 0, m2 ]

Stiffness

[ k2, -k2 ]

[ -k2 k2 ]

Damping

[c1, -c2]

[ -c2, c2]

Answer 1

[Sam Chak]

Hi @aiman

If the matrices are known, then you can apply these formulas:

A = [zeros(n) eye(n); M\K M\C]; % to get the State matrix

B = [zeros(n); M\eye(n)]; % to get the Input matrix

Example:

tspan  = [0 30];
x0     = [1 0.5 0.25 0 0 0];        % initial values
[t, x] = ode45(@odefcn, tspan, x0);
plot(t, x), grid on, xlabel({'$t$'}, 'interpreter', 'latex')

function xdot = odefcn(t, x)
    xdot = zeros(6, 1);	
    M = diag([2 3 5]);                  % Mass matrix
    C = diag(randi([-20, -10], 1, 3));  % Damping matrix
    K = diag(randi([-10, -1], 1, 3));   % Stiffness matrix
    A = [zeros(3) eye(3); M\K M\C];     % State matrix
    % B = [zeros(3); M\eye(3)];         % Input matrix
    xdot = A*x;                         % if u = 0
    % xdot = A*x + B*u;
end

[aiman]

@Sam Chak Thanks alot for your comment,It helped to get A&B but I struggling to find the output C and D to be able to use sys = ss(A,B,C,D)

[Sam Chak]

What are your desired C and D? They don't come from the mass spring damper equations of motion.

If y is your defined output that can be constructed from state vector x, then you can construct the output matrix C.

If all states x are measurable individual outputs, then C is obviously an Identity matrix.

There is no indication of the direct feedforward from the input u. So, D must be an array of zeros.