How to add trim values on to a linear models output?

I am comparing a nonlinear model to a linear model, a short elevator input over 10s is used for the input for both the linear and non linear models. And as linear models are defined by deltas around an operating point I was wondering how I would go about adding trim values to the linear model output.

%% Set aircraft parameters

m = 18000;

S = 64.8;

rho = 0.905;

g = 9.81;

a1 = 4.33;

a2 = 2.16;

a3 = 0.47;

b1 = -0.11;

b2 = -0.64;

DWG = 0.4;

S_T = 10.7;

l_T = 11;

c = 2.51;

h_fwd = 0.18; % 0.1 - 0.3

h_aft = 0.66; % 0.6 - 0.7

psi_T = -5.5 * pi / 180;

I_yy = 136182.4308;

I_xx = 236000;

I_zz = 355000;

I_xz = 0;

% Set Aero parameters

a = 5.3;

h_AC = 0.27;

C_L_Max = 1.2;

C_M_0 = -0.06;

% Set CG

h = h_fwd;

% Calculate intermediate groups

V_T = S_T * l_T / (S * c);

d_T = l_T - (h-h_AC)*c;

h_n = h_AC + V_T * (a1/a)*(1-DWG);

% Calculate speed parameters

V_Stall = sqrt(2*m*g/(rho * S * C_L_Max)) * 1.94;

V_Max = 336;

V_C = 268.8;

V_1 = 210;

V_2 = 315;

C_L_Cruise = 2 * m * g / ((V_C/1.94)^2 * S * rho);

C_L_VMax = 2 * m * g / ((V_Max/1.94)^2 * S * rho);

% Set Speed

V = V_C/1.94;

% Drag parameters

C_D_0 = 0.011;

k = 0.041263048;

% Set thrust derivative

dEpsdT = 300;

%% Run 3DoF model

eta_step = -5;

eta_step_t = 0.5;

A = [-0.00787490219865814,0.0914386059970311,-8.15800578286344,-9.80848204812432;

-0.0705390429309272,-1.20661074126797,138.553434891686,0.172568561106160;

0.0103919494442555,-0.176494149311203,-2.63846804739098,0;

0,0,1,0];

B = [0.016666666666667,0;0,0;0,-16.272914167767752;0,0];

C = [1,0,0,0;

0,1,0,0;

0,0,1,0;

0,0,0,1;

0.998271075633783,0.0587780534943110,0,0;

0.000423493150696744,-0.00719249682384279,0,1];

D_0 = [0,0;0,0;0,0;0,0;0,0;0,0];

The linear model has been constructed by trimming the non linear model and using its matrix outputs. Attached is the simulink model and MATLAB code.

Answer 1

[Nipun]

Hi Daniel,

I understand that you are trying to compare a nonlinear model to a linear model and need to add trim values to the linear model output. Here is how you may do it:

% Define the trim values (assumed example values, update as needed)
trim_values = [trim_x1, trim_x2, trim_x3, trim_x4]; % Replace with actual trim values

% Set up your linear model response
linear_response = lsim(A, B, C, D_0, input_signal, t);

% Add trim values to the linear model output
linear_response_trimmed = linear_response + trim_values;

% Plot to compare
figure;
plot(t, nonlinear_response, 'r', t, linear_response_trimmed, 'b');
legend('Nonlinear Model', 'Linear Model with Trim');
xlabel('Time (s)');
ylabel('Response');
title('Comparison of Nonlinear and Linear Models');

Refer to the following MathWorks documentation for more information for plotting dynamic system reponse: https://www.mathworks.com/help/control/ref/lsim.html

Hope this helps.

Regards,

Nipun