Matlab solve a first-order transfer functin using differential equation and step responce simulation

Most of the time, we use transfer function to simulate the signal relating to time by using tf() and lsim(). In this paper, I try to use differential equation to solve it.

Main

• Transfer funtion:
  $$0.5\dot{y}+y=F(t)$$
• Input :
  $$\begin{gathered} F(t)=100(t>0) \\ F(t)=75(t\le 0) \end{gathered}$$
• matlab code

syms y(t) inp;
ode = 0.5*diff(y,t)+y== inp;
cond = y(0) == 75;
ySol(t,inp) = dsolve(ode, cond);
x=[0:0.1:10];
result=ySol(x,100);
input=x*0+100;
input(1)=75;
plot(x, result);
hold on;
plot(x, input);
legend({'Output','Input'},'Location','southwest');
xlabel('Time')
ylabel('Amplitude')
hold off;

• result 1

ySol(t, inp) =
inp - exp(-2*t)*(inp - 75)

• result 2

Note

1. This method is not suitable for second order system practically. I guess primarily because of its complexcity of the differential equation.
2. This method is useful for periodic input.

最后

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