How to express a piecewise function with time

Question

ZC Song 2019-4-29

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https://ww2.mathworks.cn/matlabcentral/answers/459203-how-to-express-a-piecewise-function-with-time

编辑： ZC Song 2019-4-29

Hello,

I would like to express a piecewise function with time. Specifically, I have a time periods of rainfall as well as Intensity for each duration. For example, in the duration of [0,0.1], the intensity will be 0.00365 m/h and in the duration of [0.24 0.45], the intensity is 0.0996 m/h

i_q = -(1/3600)*[0.0365 0.0697 0.0996 0.0493 0.0153];       %FOR STEP LOADING: vector of rainfall intensity steps [m/s]
T_q = 2*3600*[0.1 0.24 0.45 0.78 1];           %FOR STEP LOADING: vector of time intervals of rainfall intensities (2 hrs)

I know for a given piecewise, in this case is 5, the intensity can be expressed by heaciside function and the result is below:

 q = @(t) (i_q(1).*(heaviside(t)-heaviside(t-T_q(1)))+ i_q(2).*(heaviside(t-T_q(1))-heaviside(t-T_q(2))) + i_q(3).*(heaviside(t-T_q(2))-heaviside(t-T_q(3))) + i_q(4).*(heaviside(t-T_q(3))-heaviside(t-T_q(4)))...
               + i_q(5).*(heaviside(t-T_q(4))-heaviside(t-T_q(5))));

but now my problem is for arbitary piecewise n? Imaging I have n pairs of Intensity/Duration,how can I compute the q versus time (t)? I tried using other summation funcion but it doesn't work.

             i_q = -(1/3600)*I';                 %I' contains arbitray numbers which can be import by excel  
             T_q = time_final*T';                %T' contains arbitray numbers which can be import by excel  
             n = length(I');
             syms n k
             q_aux = @(t)(i_q(k).*(heaviside(t-T_q(k-1))-heaviside(t-T_q(k))));            
             q = symsum(q_aux,k,2,n)+ @(t) (i_q(1).*(heaviside(t)-heaviside(t-T_q(1))));

Does anyone knows how to deal with this problem?