step response comparison of linear and non linear model

NON-LINEAR MODEL EQUATIONS:

xdot1,xdot2,xdot = states derivatives; x1,x2,x3=states; u1,u2,u3=inputs; y1,y2,y3=outputs

xdot1= (-0.0018*u2*x1^(9/8))+(0.9*u1) -(0.15*u3);

xdot2=((((0.73*u2)-0.16)*x1^(9/8)-x2))/10;

xdot3= ((141*u3)-((1.1*u2)-0.19)*x1)/85;

y1= x1;

y2= x2;

y3= 0.05*((0.13073*x3) + (100*((1-(0.001538*x3))*((0.8*x1)- 25.6))/(x3*(1.0394-0.00123404*x1))) +(((((0.854*u2) -0.147)*x1) + (45.59*u1) -(2.514*u3) -2.096)/9) -67.975);

LINEAR STATE SPACE MODEL FROM THE NON LINEAR MODEL:

a = [-0.0025087 0 0

0.069424 -0.1 0

-0.0066941 0 0];

b= [ 0.9 -0.34904 -0.15

0 14.155 0

0 -1.3976 1.6588];

c=[1 0 0

0 1 0

0.0063465 0 0.004705];

d=[0 0 0

0 0 0

0.25328 0.5124 -0.013967];

I have attached the step response for linear model. i want the step response for non linear model and compare with the linear model for the initial conditions (x10=108;x20=66.65;x30=428)

Answer 1

[Sam Chak]

Hi @kruthika u

To compare the results, you need to solve the linear system and generate the linear outputs, just as you would for the nonlinear system in this link: Step Response of Nonlinear Model. Once you have both the linear and nonlinear results, you can plot them for comparison using the uipanel() and tiledlayout() commands.

% ------ The Beginning of Script ------
%% Task 1: Call ode45 to solve both linear & nonlinear systems
tspan    = [0 1];
x0       = [108; 66.65; 428];     % <-- non-zero initial values
[tL, xL] = ode45(@kruthikaLNS, tspan, x0);
[tN, xN] = ode45(@kruthikaLNS, tspan, x0);

%% Task 2: Generate both linear & nonlinear output vectors
%  Outputs from Linear system
for j = 1:numel(tL)
    [~, yL(j,:)] = kruthikaLNS(tL(j), xL(j,:).');
end

%  Outputs from Nonlinear system
for k = 1:numel(tN)
    [~, yN(k,:)] = kruthikaNLS(tN(k), xN(k,:).');
end

%% Task 3: Plot Output responses
%  Setup UI panels for Tiled Layouts
f                  = figure;
p1                 = uipanel('parent', f);
p2                 = uipanel('parent', f);
p1.BackgroundColor = [1, 1, 1]; 
p2.BackgroundColor = [1, 1, 1]; 
% p.Position       = [X_Start, Y_start, Length, Height];
p1.Position        = [0.0, 0.0, 0.5, 1.0]; 
p2.Position        = [0.5, 0.0, 0.5, 1.0];
T1                 = tiledlayout(p1, 3, 1);
T2                 = tiledlayout(p2, 3, 1);

%  Plot the desired graphs in the tiles
nexttile(T1);
plot(tL, yL(:,1)), grid on, ylabel('y_{1}') % Linear Output y1
nexttile(T1);
plot(tL, yL(:,2)), grid on, ylabel('y_{2}') % Linear Output y2
nexttile(T1);
plot(tL, yL(:,3)), grid on, ylabel('y_{3}') % Linear Output y3
nexttile(T2);
plot(tN, yN(:,1)), grid on, ylabel('y_{1}') % Nonlinear Output y1
nexttile(T2);
plot(tN, yN(:,2)), grid on, ylabel('y_{2}') % Nonlinear Output y2
nexttile(T2);
plot(tN, yN(:,3)), grid on, ylabel('y_{3}') % Nonlinear Output y3

%  Label the axes of shared tiles
title( T1, 'Linear System')
xlabel(T1, 'Time')
ylabel(T1, 'Output states')
title( T2, 'Nonlinear System')
xlabel(T2, 'Time')
ylabel(T2, 'Output states')

% ------ The End of Script ------

% Put local function files at the end of the executable script

%% Linear System
function [xdot, y] = kruthikaLNS(t, x)  
    % Initialzations
    xdot    = zeros(3, 1);
    y       = zeros(3, 1);
    u       = zeros(3, 1);
      
    % Input signals: step functions
    u(1)    = (t >= 0)*1;   
    u(2)    = u(1);
    u(3)    = u(2);
    
    % State matrix
    A       = [-0.0025087 0         0;
                0.069424 -0.1       0;
               -0.0066941 0         0];
    % Input matrix
    B       = [ 0.9      -0.34904  -0.15;
                0        14.155     0;
                0        -1.3976    1.6588];
    % Output matrix
    C       = [ 1         0         0;
                0         1         0;
                0.0063465 0         0.004705];
    % Direct matrix
    D       = [ 0         0         0;
                0         0         0;
                0.25328   0.5124   -0.013967];
    
    % Linear State-Space
    xdot    = A*x + B*u;
    
    % Linear Output equation
    y       = C*x + D*u;
end

%% Nonlinear System
function [xdot, y] = kruthikaNLS(t, x)  
    % Initialzations
    xdot    = zeros(3, 1);
    y       = zeros(3, 1);
    
    % Definitions
    x1      = x(1);
    x2      = x(2);
    x3      = x(3);
    
    % Input signals: step functions
    u1      = (t >= 0)*1;   
    u2      = u1;
    u3      = u2;
    
    % ODEs
    xdot(1) = (-0.0018*u2*x1^(9/8)) + (0.9*u1) - (0.15*u3);
    xdot(2) = ((((0.73*u2)-0.16)*x1^(9/8)-x2))/10;
    xdot(3) = ((141*u3)-((1.1*u2)-0.19)*x1)/85;
    
    % Outputs
    y(1)    = x1;
    y(2)    = x2;
    y(3)    = 0.05*((0.13073*x3) + (100*((1-(0.001538*x3))*((0.8*x1)-25.6))/(x3*(1.0394-0.00123404*x1))) +(((((0.854*u2) -0.147)*x1) + (45.59*u1) -(2.514*u3) -2.096)/9) -67.975);
end