山东科技控制理论与控制工程专业英语作业

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.