Transfer function of first-order delay system

　・ In Japanese
＜premise knowledge＞
　・RL Circuit , Transfer function , Scilab , Complex number

Understand the behavior of an object with a first-order delay transfer function.

■Transfer function of first-order delay system

The differential equation of the RL circuit and the transfer function G (s) of V (t) and i (t) are as follows.



The above transfer function is the general formula of the transfer function of the first-order lag system below, where K is the gain and T is the time constant. The output when a unit step signal is input is as follows. Generally, the first-order delay system is called a low-pass filter.


This is equal to the following equation. (Click here for details)

■Confirm the operation of the first-order delay system with Scilab

Compare equation (1) with the state of the transfer function. The parameters are as follows.

　V = 1(V) , R = 2(Ω) , L = 0.5(H)



The simulation results are as follows. You can see that the behaviors of both are the same.

■Stability of first-order delay system

Since the stability of the system can be known from Eq. (2), it can also be determined from the transfer function obtained by converting Eq. (2).

y (t) depends on the value of e^{(-t / T)}. In particular, it varies greatly depending on whether T is greater than 0 or less than 0. When T> 0, e^{(-t / T)} converges to 0, that is, y (t) converges to K, so it is stable. On the contrary, when T <0, e^{(-t / T)} diverges, that is, y (t) also diverges, so it is unstable.

■Characteristic equation of first-order delay system

I explained that the stability of an object is determined by the value of T. In the case of a transfer function, the following equation with a denominator of 0 is called a characteristic equation, and the stability can be determined by the solution of the characteristic equation.



Here, -1 / T in the above formula is called a pole.

■Position of the pole of the first-order delay system

Since s is a complex number, we represent the poles on the complex plane. As shown below, if the pole is a positive real number, it will be unstable, and if it is a negative real number, it will be stable.

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