PID控制器设计
1、the original system step response diagram, The root locus diagram and Bode diagram.
The transfer function of controlled object is:
G s=
15
(s+2)(s2+3s+5)
Simulation with matlab: clear
s1=tf([15],[1,5,15,10])
s2=feedback(s1,1)
figure(1),step(s2),grid
figure(2),rlocus(s1),grid
figure(3),bode(s1),grid
Figure 1 Step Response Diagram We can find:
×100%=16.7% Overshoot:σ=0.7−0.6
0.6
Adjustment time:t s=4.5s
Figure 2 Root Locus Plot Diagram
Figure 3 Bode diagram
2、PID parameter tuning
It can be seen from Figure 2 that the root trajectory of the system has two branches that are routed through the imaginary axis. According to the characteristics of the system stability, it can be known that the system is unstable when the trajectories are distributed in the right half of the complex plane The system is stable if distributed over the left half plane. The point at which the root trajectory intersects the imaginary axis is the critical stability of the system, and the value of K is the critical ratio of the amplitude of the amplitude. From the root trajectory curve of the system, we can see that the open-loop gain of the intersection of the root locus and the imaginary axis is K =4. At this time, the integral time constant and the differential time constant are obtained. The amplitude curve is plotted by matlab. The program and result are as follows:
s3=tf([20],[1,5,7,4])
s4=feedback(s3,1)
figure(4),step(s4),grid
Figure 4 Step Response Diagram
Calculate the PID parameters and simulate them Using empirical formula ：
k d k i k T T T T 125.0,
5.0,
7.1===δδ
The Simulink model of PID:
Figure 5 Step Response Diagram
The PID parameter ：
s T s T K d i p 3.0,
2.1,
4.1===
The transfer function of the PID regulator is:
s s s s s s G p 833
.04.13.03.02.114.1)(2++=
++=
After adding the PID regulator, the open-loop transfer function of the system is:
)43)(1(495
.12215.4)(22'
+++++=
s s s s s s s G
Using Matlab to simulate and analyze the effect of PID control, procedures and results are as follows:
Figure 6 Step Response Diagram
Secondary tuning of PID parameters: The PID parameter ：
s T s T K d i p 4.0,
1,
2.1===
The output response curve is:
Figure 6 Step Response Diagram
As we can see,
σ=0;t s=2.3s;e ss=0
For the original control system,the overshoot is very large compare with the step signals,just as the settling time is too long.While the overshoot of the system with step response has been reduced to 6% and it also eliminate steady-state error after adding the PID controller.What’s more, system settling time change into 2.3s.Therefore,it will be the small overshoot and short settling time with the PID controller.。