Defined domain integral(convoulution) in simulink

Hello,

I trying to use simulink to somve the following system:

Where A is a NxN matrix, B is a Nx1 column vector and x(t) and u(s) are known functions.

I thought that handiling the domain of the integration can be done by using a transport delay block and substracting the delayed signal from the non delayed one. But I'm still puzzeld on how to compute the integral. I have treid to use the "Conv" which works for scalar A,B and x but does not work when these are matrices.

Does anyone have some suggestion on the matter?

Thank you in advance.

[Paul]

Should the x(t) on the right hand side be x(t-h)? Actually, I guess it doesn't matter beause x(t) is known function.

Is h constant?

Is there an intial condition on z(t), perhaps z(h)?

[Jpk]

Pardon the ambiguity I wrote the post in a rush. I'll update it soon.

• x(t) is the output of a filter (which yes is a t-h backwards in time filter - but it should not matter as stated-).
• h is constant in a simple setting, however it should me made variable in time.
• z(t) does indeed have an intial condition, and I could try to formulate (if my inference is correct on your final goal) the solution to the problem as the solution of linear system. I'm trying, however, to avoid this and would like to expicitly compute the solution the integral as this will be needed in future steps

Thanks for you time!

Answer 1

[Swetha Polemoni]

Hi

It is my understanding that you want to do convolution of two matrices. You may find this documentation "2-D Convolution" useful.

[Jpk]

Hi Swetha,

I have given it a quick try but maybe do not undestand the procedure well.

When I use the 2d convolution block I do not get a Nx1 (i.e. row) vector that I'm expecting but I get a matrix. How is this possible? I would expect that if I feef the 2-d convolution block expm(A*t) and B*u(t) this yiields a vector of Nx1 size.

Thanks in advance for your patience.

[Jpk]

Alternatively I could solve, which is obtanied from a change of variables:

Is there any to solve this in simulink?

Answer 2

[Paul]

Assuming h is constant and h >= 0 ....

It seems like the model can be expressed as follows:

wdot(t) = A*w(t) + B*u(t)
z(t) = expm(A*h)*x(t) + w(t) - w(t-h)  % w(t) - w(t-h) is the value or the integral

These equations can be implemented in Simulink assuming you have an initial condition w(0) and assuming that w(t-h) is known (probably should be w(0)) for t - h < 0.