Trying to write a function using step uni function and ramp function (elementary functions)

As seen, I’m trying to express this graph in elementary functions, I found the corner points of every line, and I used if statement, to find out if it’s a ramp function with a slop or if it’s a unit function (just a vertical line), or if it’s a horizontal line due to a ramp function of the opposite sign, however I got the equations for the first 3 lines (a1 & a2 & a3) correct but the rest aren’t and matlab is miscalculating the slope I don’t know why, please help, here’s my code

clc 
clear 
syms t r u f t0 t1 

x   = [-1; 
        0; 
        0;
        1;
        2;
        2.5;
        2.5;
        4;
        4]; 
y   = [0, 0.5, 1, 1, 2, 1, 0.5, 0.5, 0];   

figure 
plot(x,y)    

p1=(y(2)-y(1))/(x(2)-x(1));       
p2=(y(3)-y(2))/1/x(2)-x(2);          
p3=(y(4)-y(3))/1/x(4)-x(3);             
p4=(y(5)-y(4))/x(5)-x(4);                
p5=(y(6)-y(5))/x(6)-x(5);                   
p6=(y(7)-y(6))/1/x(7)-x(6);                      
p7=(y(8)-y(7))/1/x(8)-x(7);                         
p8=(y(9)-y(8))/1/x(9)-x(8);                                                  
p = [p1,p2,p3,p4,p5,p6,p7,p8];                 
n1= (y(2)-y(1));                 
n2= (y(3)-y(2));                  
n3= (y(4)-y(3));                  
n4 =(y(5)-y(4));                  
n5=(y(6)-y(5));                  
n6=(y(7)-y(6));                  
n7= (y(8)-y(7));                  
n8=(y(8)-y(7));    
n=[n1,n2,n3,n4,n5,n6,n7,n8];              

if p(1)==inf & p(1)==-1*inf               
    t1=x(1); 
    a1= n(1)*u*(t-t1);     
else  p(1)>0 & p(1)<0;                  
    t0=x(1);         
    a1=p(1)*r*(t-t0);     
end          

if p(2)==inf & p(2)==-1*inf               
    t1=x(2); 
    a2= n(2)*u*(t-t1);     
else 
    if p(2)>0 & p(2)<0                  
        t0=x(2);         
        a2=p(2)*r*(t-t0);                  
    else                           
        a2= -1*p(1)*r*(t-t0);         
    end              
end                           

if p(3)==inf & p(3)==-1*inf             
    t1=x(3); 
    a3= n(3)*u*(t-t1);       
elseif p(3)>0 & p(3)<0                  
    t0=x(3);         
    a3=p(3)*r*(t-t0);               
elseif p(2)~= 0 && p(2)~= -1*inf && p(2)~=inf                                                        
    a3= -1*p(2)*r*(t-t0);                             
else                                  
    a3= -1*p(1)*r*(t-t0);     
end                           

if p(4)==inf & p(4)==-1*inf             
    t1=x(4); 
    a4= n(4)*u*(t-t1);       
elseif p(4)>0 & p(4)<0                  
    t0=x(4);         
    a4=p(4)*r*(t-t0);               
elseif p(3)~= 0 && p(3)~= -1*inf && p(3)~=inf                                                        
    a4= -1*p(3)*r*(t-t0);                                              
elseif p(2)~= 0 && p(2)~= -1*inf && p(2)~=inf                                                        
    a4= -1*p(2)*r*(t-t0);                     
else                              
    a4= -1*p(1)*r*(t-t0);     
end                                   

if p(5)==inf & p(5)==-1*inf             
    t1=x(5); 
    a5= n(5)*u*(t-t1);       
elseif p(5)>0 & p(5)<0                  
    t0=x(5);         
    a5=p(5)*r*(t-t0);               
elseif p(4)~= 0 && p(4)~= -1*inf && p(4)~=inf                                                        
    a5= -1*p(4)*r*(t-t0);                                              
elseif p(3)~= 0 && p(3)~= -1*inf && p(3)~=inf                                                        
    a5= -1*p(3)*r*(t-t0);                 
elseif p(2)~= 0 && p(2)~= -1*inf && p(2)~=inf                                                        
    a5= -1*p(2)*r*(t-t0);                    
else                              
    a5= -1*p(1)*r*(t-t0);     
end                                                

if p(6)==inf & p(6)==-1*inf             
    t1=x(6); 
    a6= n(6)*u*(t-t1);       
elseif p(6)>0 & p(6)<0                  
    t0=x(6);         
    a6=p(6)*r*(t-t0);               
elseif p(5)~= 0 && p(5)~= -1*inf && p(5)~=inf                                                        
    a6= -1*p(5)*r*(t-t0);                                              
elseif p(4)~= 0 && p(4)~= -1*inf && p(4)~=inf                                                        
    a6= -1*p(4)*r*(t-t0);                 
elseif p(3)~= 0 && p(3)~= -1*inf && p(3)~=inf                                                        
    a6= -1*p(3)*r*(t-t0);               
elseif p(2)~= 0 && p(2)~= -1*inf && p(2)~=inf                                                        
    a6= -1*p(2)*r*(t-t0);     
else                              
    a6= -1*p(1)*r*(t-t0);     
end                                             

if p(7)==inf & p(7)==-1*inf             
    t1=x(7); 
    a7= n(7)*u*(t-t1);       
elseif p(7)>0 & p(7)<0                  
    t0=x(7);         
    a7=p(7)*r*(t-t0);               
elseif p(6)~= 0 && p(6)~= -1*inf && p(6)~=inf                                                        
    a7= -1*p(6)*r*(t-t0);                                              
elseif p(5)~= 0 && p(5)~= -1*inf && p(5)~=inf                                                        
    a7= -1*p(5)*r*(t-t0);                 
elseif p(4)~= 0 && p(4)~= -1*inf && p(4)~=inf                                                        
    a7= -1*p(4)*r*(t-t0);               
elseif p(3)~= 0 && p(3)~= -1*inf && p(3)~=inf                                                        
    a7= -1*p(3)*r*(t-t0);             
elseif p(2)~= 0 && p(2)~= -1*inf && p(2)~=inf                                                        
    a7= -1*p(2)*r*(t-t0);     
else                              
    a7= -1*p(1)*r*(t-t0);     
end                                                                        

if p(8)==inf & p(8)==-1*inf             
    t1=x(8); 
    a8= n(8)*u*(t-t1);       
elseif p(8)>0 & p(8)<0                  
    t0=x(8);         
    a8=p(8)*r*(t-t0);               
elseif p(7)~= 0 && p(7)~= -1*inf && p(7)~=inf                                                        
    a8= -1*p(7)*r*(t-t0);                                              
elseif p(6)~= 0 && p(6)~= -1*inf && p(6)~=inf                                                        
    a8= -1*p(6)*r*(t-t0);                 
elseif p(5)~= 0 && p(5)~= -1*inf && p(5)~=inf                                                        
    a8= -1*p(5)*r*(t-t0);               
elseif p(4)~= 0 && p(4)~= -1*inf && p(4)~=inf                                                        
    a8= -1*p(4)*r*(t-t0);             
elseif p(3)~= 0 && p(3)~= -1*inf && p(3)~=inf                                                        
    a8= -1*p(3)*r*(t-t0);                         
elseif p(2)~= 0 && p(2)~= -1*inf && p(2)~=inf                                                        
    a8= -1*p(2)*r*(t-t0);                   
else                              
    a8= -1*p(1)*r*(t-t0);     
end                         

f=a1+a2+a3+a4+a5+a6+a7+a8;

Answer 1

[Aashray]

Hello @Fatima Noneed,

From the code, I understand that you are trying to build a piecewise function using ramp and unit step functions (r(t) and u(t)), based on the slopes between corner points. You're getting correct equations for the first few segments, but incorrect slopes later.

The trick here lies in the operator precedence in MATLAB. The division operator takes precedence over subtraction. So, while calculating p2, the following line:

p2 = (y(3)-y(2))/1/x(2)-x(2);

is interpreted by MATLAB as:

temp = (y(3)-y(2))/1;
temp2 = temp / x(2);
p2 = temp2 - x(2);

which is not the slope between two points.

To compute slope between point (x(i), y(i)) and (x(i+1), y(i+1)), you should write it as:

p2 = (y(3)-y(2)) / (x(3)-x(2));

So, just fixing all the slope assignments should work.

I have also attached the output plot that I got after fixing the slope assignments for reference.

Documentation link for operator precedence: https://www.mathworks.com/help/matlab/matlab_prog/operator-precedence.html