MATLAB: Simulating a 1 degree step input on a state space system (using transfer functions)

Question

HI all,

I've been trying to simulate the aircraft response to different control inputs using the state space method. The code first creates the matrices for the state space, then I let MATLAB create the state space model using the ss command. Then to look at the responses, I use the tf command to make transfer functions of the state space. These transfer functions are used to calculate the response to a a 1 degree step input. However, the command step does not give the correct results. So I tried using the lsim command, however, these results weren't correct either.

So my question is; how do i simulate a 1 degree step input on the state space in MATLAB?

P.S. I'm using college values for the state space and I have figures of how the output should look like.

— Used code: —

% State matrix (aerodynamic stability derivatives, u – w – q – theta)

A = [0.00501 0.00464 -72.9 -31.34;

-.08570 -.54500 309 -7.4;

0.00185 -0.00767 -0.395 0.00132;

0 0 1 0];

% Input matrix (control coefficients, aileron – rudder)

B = [5.63;

-23.8;

-4.51576;

0];

% Output matrix

C = [1 0 0 0;

0 1 0 0;

0 0 1 0;

0 0 0 1;

0 0.00316 0 0;

0 -0.00316 0 1];

% Direct matrix

D = 0;

% Creation of state space & transfer functions (u, w, q, theta)

state_space = ss(A,B,C,D);

transfer_function = tf(state_space);

% Input of elevator

input = [ones(1,10), zeros(1,991)]; % Step of 1 for 1 second (in combination with time vector)

t = 0:0.1:100; % Time vector

% Calculation of response to elevator

[u,Y1] = lsim(transfer_function(1,1),input,t); % Velocity, x-axis

[w,Y2] = lsim(transfer_function(2,1),input,t); % Velocity, z-axis

[q,Y3] = lsim(transfer_function(3,1),input,t); % Angular velocity, pitch rate

[theta,Y4] = lsim(transfer_function(4,1),input,t); % Pitch angle

[alfa,Y4] = lsim(transfer_function(5,1),input,t); % Pitch angle

[gamma,Y4] = lsim(transfer_function(6,1),input,t); % Pitch angle

subplot(6,1,1);

plot(t,u);

title('Linear velocity u [ft/s]');

subplot(6,1,2);

plot(t,w);

title('Linear velocity w [ft/s]');

subplot(6,1,3);

plot(t,q);

title('Angular velocity q [ft/s]');

subplot(6,1,4);

plot(t,theta);

title('Pitch angle Theta [rad]');

subplot(6,1,5);

plot(t,alfa);

title('Angle Alfa [rad]');

subplot(6,1,6);

plot(t,gamma);

title('Angle Gamma [rad]');

sgtitle('Boeing 747 response to 1 degree 1 second on elevator');

Answer 1

Your original lsim command didn't really generate a step resonse. It generate the response to a positivie unit step input at t = 0 followed by a negative step input at t = 1. If you all you really want is the unit step response, then for sure just use the step command.

As for the units, the step command doesn't know anything about units. It just applies a unit step; the units are implicit in your model coefficients. If you developed your model coefficients in the B-matrix in units of rad^-1 (e.g., B(1,1) has units of ft/s^2/rad, then the step elevator input has units of rad, and the outputs of your model need to be intepreted appropriately, e.g. the fourth output would be: rad Theta per rad Elevator. Of course deg Theta per deg elevator would be the same. But if you wanted to deg Theta per rad Elevator you'd mutliply the fourth output of the step response by 180/pi.