Aim

PID control algorithm is used in most advanced motion control application for increased accuracy and efficiency. There are three control loops that can be controlled using a PID controller which are torque, velocity and position respectively. In vehicles which do not have the electronic controls do not have the linear motion though same power is transferred independently to the wheels. The non-linear motion of the vehicle is caused mainly by the mechanical vibrations and the difference in the surfaces in which the vehicle operates.

Hence, for minimizing deviation of vehicle from straight direction a closed loop PID controller that changes the power input the motors of the vehicle based on the change in their rotation [1]. Basically, the PID controller minimizes the gap between the measured power and the desired power to the wheels to maintain linear motion by changing the control inputs of the system. In this assignment, the vehicle is a brick escalator which moves in an incline plane from base to about 10 meters of height in full loaded condition [1].

The angle of inclination of the plane is about 45 degrees and the cart is moved through the rails installed in the inclined plane. The PID controller is installed in an Arduino board and the controller settings as specified below is written in MATLAB environment as given in the later sections [2]. The Arduino board is installed in the brick cart and the total weight of the cart is 200 kgs.

Aim:

The aim of the assignment is to move the loaded brick cart from base through the rails in the inclined plane to the top of the plane which is 10 meters above the ground. PID control system must be employed to control the movement of the brick cart.

Method:

The PID controller is designed in the Arduino board installed in the brick cart that provides signals and supply voltage which drives the motors of the engine coupled with the cart wheels. Hence, at first the hardware design of the PID controller components are is installed in the Arduino board followed by electric motor with supply installations in the cart [2]. Then the MATLAB programming is done in some capable machine and the motion of the cart will be controlled remotely via some signal transmitter installed in the machine and signal sensors installed the cart and work with the hardware system in synchronism.

Method

The inclined track followed by the brick cart is given by following schematic diagram.

The programming of the PID controller is done in the trainsim.m MATLAB file with the given environmental conditions and the initial conditions of the brick cart.

1. The mass of the system in full loaded condition (with bricks) is m=200 kg and moved in full loaded condition.
2. Initially the cart has no motion and stationary at the base i.e. at height h = 0 m.
3. Destination height is h = 10 m.
4. Inclination angle is θ = 45 degrees.
5. No brakes are installed in the cart or the escalator, braking is done with the help of gravity and the controller installed in the engine.
6. After reaching the destination the cart should not go beyond 0.1 m of the ground.
7. The acceleration of the cart should be below 0.5 ms^(-2) and the velocity of the cart must not exceed 2 m/s second to avoid damage to the bricks.
8. The maximum driving force that can be delivered by the engine cart is 10 kN.
9. The dynamic friction coefficient of the rail is 1000 N-s/m

Hence, the force balancing equation of the brick is

F_net(t) = F_d(t) – F_f(t) – F_g(t)

Here, F_d(t) = driving force of the cart engine = K_pe(t) + K_I e_I(t) + K_D e_D(t)

Where, e(t) = error in driving force in proportional control part at time t.

e_I(t) = error in driving force in integral control part at time t.

e_D(t) = error in driving force in derivative control part at time t.

K_p, K_I and K_D are the proportional, integral and derivative constants which are needed to be determined to move the cart at the top of inclined plane satisfying the conditions given above.

F_f(t) = the dynamic friction force acting on the cart opposite of its motion = bv(t) = 1000v(t).

v(t) =  velocity of the cart at time t.

F_g(t) = force due to gravity acting opposite of the carts moving direction = m∗g∗sin(θ)

MATLAB code:

function [error,F_drive] = trainsim(Kp,Ki,Kd

%% PID controller coefficients

% Kp = proportional

% Ki = integral

% Kd = derivative

% 350 50 10 (chosen PID values)             

%% physical parameters

m = 200;      % mass of train in kgs

b = 1000;    % coefficient of dynamic friction in N-s/m

d = 10;      % distance between point A and B

alpha = 45;  % incline angle (degrees)

g = 9.8;    % acceleration due to gravity

F_reaction= m*g*sind(alpha);

%% simulation time

T = 50;     % simulation time

dt = 0.01;  % simulation resolution

N = T/dt +1;% number of samples

%% simulation variables

t=0:dt:T;        % time vectors

v=zeros(1,N);    % velocity

a=zeros(1,N);    % acceleration

x=zeros(1,N);    % distance traveled

F_net=zeros(1,N);% net force

e = d*ones(1,N); % error

eI=0;

for(n=2:N)

  %% Process

  a(n) = F_net(n-1)/m;                 % train acceleration

  v(n) = v(n-1) + 0.5*(a(n-1)+a(n))*dt;% train velocity

  x(n) = x(n-1) + 0.5*(v(n-1)+v(n))*dt;% distance traveled

  %% Feedback error

  e(n) = d-x(n);  % end value indicates the amount by which the cart overshoots final height

  %% Control

  format short

  eI = eI + 0.5*(e(n)+e(n-1))*dt;              % integral

  eD = (e(n)-e(n-1))/dt;                       % derivative

  F_drive = Kp*e(n) + Ki*eI + Kd*eD;           % driving force in Newton

  F_friction = b*v(n-1);                       % friction force in Newton

  F_net(n) = F_drive – F_friction – F_reaction;% net force in Newton

end

error = e(end);

figure

plot(t,a,’b-‘,t,v,’k-‘,t,x,’r-‘)

xlim([0,15])

ylabel(‘dynamic position of distance,velocity and acceleration’);

xlabel(‘time (s)’);

legend(‘acceleration in m/s^2′,’velocity in m/s’,’distance in m’)

title(‘Dynamic characteristics of brick cart at time t’)

grid on

The chosen values of the coefficients K_P, K_D and K_I are calculated by trial error method such that it satisfies all the environmental and boundary conditions of carts velocity, position, acceleration.

Result and Discussion:

The modified trainsim.m file is executed in the MATLAB command window and the results are given below.

>> [error,F_drive] = trainsim(350,35,10)

error =

   -0.0011

F_drive =

   1.3857e+03

Hence, it is clearly visible that the cart attains its destination at about 15 seconds after the launch from the base point. During its runtime its velocity is always under 2 m/sec, acceleration goes below 0.2 m/s^2 in less than 0.4 seconds and the driving force to the cart in the first 10 seconds is approximately 1385.7 N. The cart do not overshoot the destination height of 10 m, whereas it undershoots by approximately 1.1 mm, the amount which can be neglected.

Conclusion:

Hence, the MATLAB programming of the PID controller settings is performed appropriately as most of the environmental and brick cart’s dynamic conditions are met and it is proved that the cart can be successfully delivered in its destination with the calculated coefficients of PID controller. Although, the acceleration exceeds its limitation of 0.2 m/s^2 for the first 0.4 sec (approx.), it can be assured that the time interval is much smaller to execute significant disturbance in the dynamics that can partially unload the brick cart. The initial acceleration for sufficiently small time interval is necessary to provide the driving force that will drive the cart in its later course. Therefore, it can be concluded that PID controllers can be used to control the motion of dynamic bodies for better performance and outcomes.

References:

[1]        “Drive with PID Control – MATLAB & Simulink Example – MathWorks India”, In.mathworks.com, 2018

[2]        P. Pal and R. Dey, “Optimal PID Controller Design for Speed Control of a Separately Excited DC Motor: A Firefly Based Optimization Approach”, The International Journal of Soft Computing, Mathematics and Control, vol. 4, no. 4, pp.