Help with longitudinal car following model based on spring / damper style interactions

This is my attempt at creating a car following model using a pedal input u to produce an acceleration/deceleration and thus a velocity using euler integration. It's in it's early days so there may be some fundamental issues, but the theory is that a pedal input u is used to scale a max accel/brake at specific vehicle speeds using a lookup table / known values. The cruise control part is ok but I can't stop the cars from driving through eachother (ie. the cars won't follow). I've tried a range of values of gains and I now suspect the theory in my logic is flawed. The script is below and any help is appreciated.

close all
clear all

%% ------------------------------------------------------------------------
%VEHICLE  (acceleration lookup table
Vehicle_Mass = 1675;
Total_Force = [0, 150.3, 293.4, 466.1, 611, 757.1, 884.9, 1024, 1164, 1316, 1481, 1639, 1799, 1974, 2141, 2325, 2502, 2673, 2861, 3030, 3218, 3428, 3633, 3862, 4106, 4363, 4618, 4888, 5199, 5558, 5898, 6296, 6734, 7151, 7610, 8037, 8550, 9121, 9705, 10420, 11170, 11850, 12620, 13380, 14240, 14960, 15740, 16610, 17570, 18460, 19100, 19800, 20750, 21300, 21550, 21550, 21200, 20600, 19850, 18950, 18000, 17150, 16100, 15000, 13000];
Vehicle_Speed = [73.484, 71.9, 70.4, 68.6, 67.1, 65.6, 64.3, 62.9, 61.5, 60, 58.4, 56.9, 55.4, 53.8, 52.3, 50.7, 49.2, 47.8, 46.3, 45, 43.6, 42.1, 40.77, 39.2, 37.7, 36.2, 34.8, 33.4, 31.9, 30.3, 28.9, 27.4, 25.9, 24.6, 23.3, 22.2, 21, 19.8, 18.7, 17.5, 16.4, 15.5, 14.6, 13.8, 13, 12.4, 11.8, 11.2, 10.6, 10.1, 9.7, 9.2, 8.3, 7.4, 6.3, 5.2, 3.8, 2.9, 2.1, 1.5, 1, 0.7, 0.4, 0.2, 0];

Max_Acceleration = Total_Force/Vehicle_Mass;        
Max_Deceleration = -15*ones(size(Vehicle_Speed));

%% ------------------------------------------------------------------------
figure
plot(Vehicle_Speed, Max_Acceleration, 'Color', 'r', 'LineWidth', 2)
hold on
plot(Vehicle_Speed, Max_Deceleration, 'Color', 'r', 'LineWidth', 2); 
ylim([-20 15])
xlim([0 74])
line([0,0], ylim, 'Color', 'k', 'LineWidth', 2);
line(xlim, [0,0], 'Color', 'k', 'LineWidth', 2)
xlabel('Vehicle Speed (m/s)')
ylabel('Max Vehicle Acceleration (m/s^2)')
grid on
grid MINOR

%% ------------------------------------------------------------------------

N = 1000;                                %    length of sim 
h = 0.01;                                %    timestep
cars = 5;                                %    number of cars
in_spac = 10;                            %    initial spacing  
a = zeros(cars, N);                      %    acceleration matrix
v = zeros(cars, N);                      %    velocity matrix
d_euler = zeros(cars, N);                %    distance matrix
d = zeros(cars, N);                      %    distance matrix    
u_rec = zeros(cars, N);                  %    input matrix                              
v_target = randi([22 35],cars,1);        %    Random values of target speed between 50-80mph
k1 = 1;                                  %    gain 1
k2 = 1;                                  %    gain 2


for i = 1:N
    for j = 1:cars
                if v(j,i) < 0
            v(j,i) = 0.1;
                end

        if j == cars || i == 1 
        u = ( -k1 * ( v(j,i) - v_target(j) ) ); %leading car (j = 5) pedal input according to target speed only
        else 
        u = ( -k1 * ( v(j,i) - v_target(j) ) ) + ( -k2 * ( d(j+1,i-1) - d(j,i-1) ) ); %hooks law type interaction for second term
        end
                
        %conditions to keep u between -1 and 1
        if u > 1
        u = 1;
        elseif u < -1
        u = -1;
        end
    
        if u < 0 
        a(j,i) = u*15; %uniform max rate deceleration at all speeds
        else
        a(j,i) = u*interp1(Vehicle_Speed, Max_Acceleration, v(j,i),'spline'); 
        end

        v(j,i+1) = v(j,i) + a(j,i)*h; 
        d_euler(j,i+1) = d_euler(j,i) + v(j,i)*h;
        
        if i == 1
        d(j,i) = j*in_spac; %setting initial spacing
        else
        d(j,i) = d(j,1) + d_euler(j,i);
        end

        u_rec(j,i) = u; %recording the pedal input
             
    end
end

figure
subplot(4,1,1)
plot(a')
ylabel('Acceleration')
grid on 
grid minor
subplot(4,1,2)
plot(v')
ylabel('Velocity')
grid on 
grid minor
subplot(4,1,3)
plot(d')
ylabel('Distance')
grid on 
grid minor
subplot(4,1,4)
plot(u_rec')
ylabel('u')
grid on 
grid minor