Include a new condition to solve a system using ODE45

Hello,

I'm programming a SMC control algorithm for a DC motor where it has to track a given trajectory. I'm using ODE45 to solve the equation system, but I have an issue, I dont know how can I include a step perturbation in the system for example in the second 4 of the loop. Here is my code:

Is it possible to do so?

clear variables;
close all;
clc;

%% Control of a DC motor using SMC control

% kirchoff
% 𝑅𝑖 + L ( d𝑖 / 𝑑𝑡) = 𝑣 − 𝐾𝑒m

%newton second law
% 𝐽𝜔̇ + 𝑏𝜔 = 𝐾𝑡𝑖

%expressed as space state
% dx1 = -b/j * x1 + Kt/j * x2
% dx2 = -Ke/L * x1 - R/L *x2 + 1/L *u
% y = x1

%% Init variables

% space state definition
A11 = -5.5; 
A21 = -0.01852;
A12 = 37.5;
A22 = -0.1481;

A= [A11 A21; A21 A22];

B11 = 0;
B21 = 0.3704;

B = [B11; B21];

C = [1 0];

D = 0;

%convert to transfer function
[a,b]=ss2tf(A,B,C,D);

%% Controller design

% 𝜔̈ + 5.648𝜔̇ + 1.509𝜔 = 13.89𝑣

% x1 = w
% x2 = dw
% u = v

% dx1 = x2
% dx2 = -5.648𝜔̇ - 1.509𝜔 + 13.89u

%sliding variable
%s = ce + de

%where e = wd - w

%% CONTROL

% solving the ode
t0=0;
tfinal=8;
tspan  = linspace(t0, tfinal, 8e4);
x0     = [0 0 0];
[t, x] = ode45(@(t,x) SMCf(t,x), tspan, x0);

s = zeros(numel(t),1);
u = s;
wd = s;
dwd = s;

for i = 1:numel(t)
  [~, s(i),u(i),wd(i),dwd(i)] = SMCf(t(i), x(i,:));
end

%compute tracking error
y = x(:,1);
e = wd - y;

%% PID

sys=ss(A,B,C,D);
[gains,info] = pidtune(sys,'PID',10);

closedLoop = feedback(sys*gains,1);

%% FUNCTION

% describing the ode
function [dot,s,u,wd,dwd,ddwd] = SMCf(t, x)

    %variables
    c=5;
    r=10;
    % perturbation
    f = step function 

    % yc, desired track
    wd      = 2*cos(t);
    % compute desired velocity and acceleration 
    %     dyc     = gradient(yc,t);
    dwd =  -2*sin(t);
    ddwd =  -2*cos(t);

    %sliding variable
    s= dwd+c*wd-c*x(1)-x(2);

    %control law U to satisfy ss'<0
    u       = r*tanh(s/0.01);

    x1dot = x(2);
    x2dot = -5.648*x(2)-1.609*x(1)+13.89*u;
    
    % somehow include this step in the 4th second so: 
    x2dot = (-5.648*x(2)-1.609*x(1)+13.89*u) + f;
    x3dot = x(1);

    dot = [x1dot x2dot x3dot]';
    
end

Answer 1

[Sam Chak]

Hi @Mikel

You can try the two approaches in the code.

However, the step value should not exceed fs = 130, or else your designed SMC cannot track well (unless you increase r).

% solving the ode
t0     = 0;
tfinal = 20;
tspan  = linspace(t0, tfinal, 20001);
x0     = [0 0 0];
[t, x] = ode45(@(t, x) SMCf(t, x), tspan, x0);

% plotting the output
plot(t, 2*cos(t), t, x(:,1)), grid on
xlabel('t')
legend('\omega_{d}(t)', 'y(t)')

% describing the ode
function [dot,s,u,wd,dwd,ddwd] = SMCf(t, x)

    %variables
    c  = 5;
    r  = 10;
    % perturbation
    ts = 4;                     % step time
    fs = 139/1.08;              % step output value (shouldn't exceed 130 or it cannot track)
    f  = fs*(t > ts);           % step function (not a math expression, but a program syntax)
    % f  = fs*heaviside(t - ts);  % Heaviside function (an acceptable math function for step)

    % yc, desired track
    wd    =   2*cos(t);
    % compute desired velocity and acceleration 
    %     dyc     = gradient(yc,t);
    dwd   = - 2*sin(t);
    ddwd  = - 2*cos(t);

    %sliding variable
    s     = dwd + c*wd - c*x(1) - x(2);

    %control law U to satisfy ss'<0
    u     = r*tanh(s/0.01);

    x1dot = x(2);
    x2dot = - 5.648*x(2) - 1.609*x(1) + 13.89*u + f;
    x3dot = x(1);

    dot = [x1dot x2dot x3dot]';
    
end