Matrices do not have the same number of columns

I am simulating a half-car model using a Mass & Stiffness Matrices and have run into an error nested with the "ss" function which states: "Error using ss (line 355); The "A" and "C" matrices must have the same number of columns."

In the workspace, I can observe that the A matrix is a 10x10 matrix, where b,c,d are 5x5. The first 5x5 within the A matrix is simply the identity matrix but this should not be included. I am using "A = [ zeros(5,5) eye(5); -inv(M)*K -inv(M)*C ]" in place of A = [ 0 i ; -inv(M)*k -inv(M)*C]

The code is as follows:

% Parameters
m1 = 125; %kg %unsprung mass 
m2 = 100; %kg %unsprung mass 
m3 = 1425; %kg %sprung mass 
m4 = 75; %kg %driver mass
I = 8500; %kg-m^2 %moment of Inertia 
kt = 350000; %N/m %tire stiffness 
k1 = 100000; %N/m %front suspension stiffness
k2 = 100000; %N/m %rear suspension stiffness 
k3 = 30000; %N/m %seat stiffness
C = 0.002; %N-s/m %damping constant

% Mass Matrix 
M = [m1 0 0 0 0 
    0 m2 0 0 0 
    0 0 m3 0 0
    0 0 0 m4 0 
    0 0 0 0 I];
% Stiffness Matrix 
K = [(kt +k1) 0 -k1 0 k1
    0 (kt+k2) -k2 0 k2
    -k1 -k2 (k3+k2+k1) -k3 (-k3+k2-k1)
    0 0 -k3 k3 k3 
    -k1 -k2 (k3+k2+k1) -k3 (-k3+k2-k1)]; 
% Identity Matrix 
i = [1 0 0 0 0 
    0 1 0 0 0 
    0 0 1 0 0 
    0 0 0 1 0 
    0 0 0 0 1];
% A Matrix 
A = [ zeros(5,5) eye(5); -inv(M)*K -inv(M)*C ];
% b Matrix 
b = [0 0 0 0 0 500/m1 0 0 0 0]';
%c state variable 
c = [C * i];
% d state variable 
d=[0];
% Part 1 
sys1=ss(A,b,c,d);

Error using ss (line 370)
The "A" and "C" matrices must have the same number of columns.

% Bode plot 
bode(sys1,{1,100});
% pole-zero plot 
pzmap(sys1);axis([-80 80 -80 80]);
% impulse 
impulse(sys1,30);
%RMS output for 30 sec
rms(impulse(sys1,30));
%undamped natural frequencies 
sort(sqrt(abs(eig(-inv(M)*K)))) ; 

End of code

any help is much appreciated!

Answer 1

[Walter Roberson]

A = [ zeros(5,5) eye(5); -inv(M)*K -inv(M)*C ];

That has 10 columns -- 5 of zeros and 5 of eye

C = 0.002; %N-s/m %damping constant

A constant.

i = [1 0 0 0 0 
    0 1 0 0 0 
    0 0 1 0 0 
    0 0 0 1 0 
    0 0 0 0 1];

That has 5 columns. It could have been written as i = eye(5);

c = [C * i];

Constant times matrix with 5 columns gives you 5 columns of output.

sys1=ss(A,b,c,d);

A has 10 columns, c has 5 columns; the combination is not permitted.

https://www.mathworks.com/help/control/ref/ss.html#d126e233984

The state-space matrices are:

• A is an Nx-by-Nx real- or complex-valued matrix.
• B is an Nx-by-Nu real- or complex-valued matrix.
• C is an Ny-by-Nx real- or complex-valued matrix.
• D is an Ny-by-Nu real- or complex-valued matrix.

If A is 10 x 10, then B must be 10 x something; we see by examination that indeed B is 10 x 1.

If B is 10 x 1 then Nu must be 1.

If C is something by Nx then C must be something by 10. Suppose it were 5 x 10.

Then D must be the same number of rows as C is, and must have Nu columns, where we have already found the Nu is 1. So D would have to be 5 x 1.

Or, working backwards,

If D is 1 x 1 then Ny = 1 and Nu = 1.

Then C would have to be 1 x Nx, value of Nx to be determined.

Then B would have to be Nx x 1, value of Nx to be determined.

With A being 10 x 10, that gives us our Nx, and it follows that:

A is 10 x 10

B is 10 x 1

C is 1 x 10

D is 1 x 1

Or if we assume the C is the correct size, 5 x 5 then

A would have to be 5 x 5

B would have to be 5 x Nu

C would be 5 x 5 by assumption

D would be 5 x Nu

[Paul]

ss automagically expands D = 0 to the appropriate size

sys = ss(rand(4),rand(4),rand(4),0); % D should be 4 x 4
sys.D

ans = 4×4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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