Question

  

How do I calculate the steady-state pitch angle for the unity step input for the above system in matlab workspace?

Answer 1

consider the system with the dynamic equations in the Laplace domain and the open-loop transfer function of the aircraft pitch as given below :

sA(s) = -0.213 A(s) + 46.7 Q(s) + 0.332 (s)

sQ(s) = -0.0139 A(s)-0.426Q(s) +0.0203 (s)

s(s) = 46.7 Q(s)

P(s) = (s) / (s) = 1.161s + 0.1784 / s³ + 0.738 s² + 0.821s

For a step reference, the design criteria are the following.

• Overshoot less than 10%
• Rise time less than 2 seconds
• Settling time less than 10 seconds
• Steady-state error less than 2%

From the analysis of open loop response we can see that the open loop response is unstable. In order to stabilize the system a feedback controller is needed.

The closed-loop transfer function for the above with the controller C(s) simply set equal to one can be generated using the MATLAB command feedback as shown below.

sys_cl = feedback(P_pitch,1)

sys_cl =
 

           1.161 s + 0.1784
  ----------------------------------
  s^3 + 0.738 s^2 + 2.072 s + 0.1784
 
Continuous-time transfer function.

the corresponding step response can be generated by adding the above and following commands to your m-file. Note that the response is scaled to model the fact that the pitch angle reference is a 0.2 radian (11 degree) step. Running your m-file at the command line will produce the plot which will show the rise time, settling time and final value by clicking 'characteristics'.

         step(0.2*sys_cl);
         ylabel('pitch angle (rad)');
         title('Closed-loop Step Response');

We can transform the output back to the time domain to generate a time function for the system's response to get some insight into how the poles and zeros of the closed-loop transfer function affect the system's response. Assuming the closed-loop transfer function has the form Y(s) / R(s), the output Y(s) in the Laplace domain is calculated as follows where R(s) is a step of magnitude 0.2.

Y(s) = 1.161 s + 0.1784 / s^3 + 0.738 s^2 + 2.072 s + 0.1784 X R(s)

Y(s) = 1.161 s + 0.1784 / s^3 + 0.738 s^2 + 2.072 s + 0.1784 X 1/s  

  Y(s) = 1.161 s + 0.1784 / s^4 + 0.738 s^3 + 2.072 s^2 + 0.1784s

Based on the above, the denominator of our output Y(s) can be factored into a first-order term for the real pole of the transfer function, a second-order term for the complex conjugate poles of the transfer function, and a pole at the origin for the step input. Therefore, it is desired to expand Y(s) as shown below.

Y(s) = A/s + B / s+ 0.08905 + Cs+D / s^2 + 0.640s + 2.023

The specific values of the constants A, B, C, and D can be determined by hand calculation or by using the MATLAB command residue to perform the partial fraction expansion ( [r,p,k] = residue(num,den) ) . so the resulting values of A, B,C&D are 0.2, 0.0891, 0.12 & 0.087 respectively.

entering the following code in the MATLAB command window will generate the desired plot with steady state pitch angle

t = [0:0.1:70];
y = 0.2 - 0.0891*exp(-0.08905*t) - exp(-0.3255*t).*(0.12*cos(1.3816*t)+0.0320*sin(1.3816*t));
plot(t,y)
xlabel('time (sec)');
ylabel('pitch angle (rad)');
title('Closed-loop Step Response');