Plotting the aircraft pitch angle response to elevator

The image attached below are the equations needed and the simulink model I have so far, using the following code. The Step 1 is: Step time = 1, Initial value = 0 and final value = 5. Step 2 is: Step time = 4, Initial value = 5, and final value = 0. The bottom two are just the opposite of these two. However the pitch rate and pitch graphs do not look right, but I can't seem to see whats gone wrong.

% short period 

% constant variables
rho = 0.905;
S = 64.8;
c = 2.51;
a = 5.3;
h_n = 0.63;
h_fwd = 0.18;
h_ac = 0.27;
I_yy = 136182.4308;
g = 9.81;
m = 18000;
C_D_0 = 0.011;
k = 0.041263048;
V_bar_t = 0.72;
a_1 = 4.33;
a_2 = 2.16;
a_3 = 0.47;
DWG = 0.4;
d_T = 11.2259;
V_cruise = 138.283;

% damping ratio
zeta = sqrt((rho*S*c)/(8*(h_n-h_fwd)*a*I_yy))*(V_bar_t*a_1*d_T);

% omegas for cruise velocity
omega_n_cruise = V_cruise*sqrt(((rho*S*c*a*(h_n-h_fwd))/(2*I_yy)));

omega_d_cruise = omega_n_cruise*sqrt(1-(zeta^2));

k_eta = (-1*V_bar_t*a_2)/((h_n - h_fwd)*a);

eta_coefficient = k_eta * (omega_n_cruise)^2;
theta = (omega_n_cruise)^2;
theta_dot = 2*zeta*omega_n_cruise;

Answer 1

[Sam Chak]

Hi @Daniel Jackson,

I just replied to your other question, and then I noticed this new but similar question. Anyway, the second-order differential equation of the pitch dynamics you posted is linear. Therefore, you can use a single state-space block to represent that system. The values for the parameters , ζ, and can be calculated by hand or computed in MATLAB.

https://www.mathworks.com/help/simulink/slref/statespace.html

Please write out the state-space model. This topic should be covered in the Aircraft course. By the way, I don't see any errors in the block configuration. But of course, I don't know what values you entered in the blocks.

[Daniel Jackson]

Hi @Sam Chak

Thank you for your help, I have both calculated those values by hand and in MATLAB, I just have a few more questions if thats okay.

Where would I impliment the single state-space block? Also which parameters go with the variables .

[Sam Chak]

No worries. This chunk of blocks can be replaced by a single State-space block. You just to enter the values for matrices A, B, C, D, as well as the initial values of the pitch θ and pitch rate .

Can you rewite the 2nd-order differential equation in the state-space form? The state-space contains two equations: one is the Matrix 1st-order differential equation , and the other is the Matrix output equation , where the state vector and the input u is the elevator signal, η.