Question

I required to design a PID controller that has overshoot less than 10% with minimise rise time, settling time, peak time and steady-state error.

The transfer function of the plant is shown below:

= 3.163 S^3 + 92.59 s^2 + 61.66 s

and the step response of the open loop system by using unit-step is shown below:

Then I have designed my PID controller by referring to the example from Modern Control Engineering 5th Edition by Katsuhiko Ogata page 572 by using Ziegler Nichols 2nd Method.

I get Kcr = 5604.68 and Pcr = 0.8078. My PID controller transfer function is (339.5595*(s+4.9517)^2)/s. But the overshoot of my system exceeds 10%. How can I tune my PID controller so that I can get overshoot less than 10%?

Answer 1

CODE :

clc close all s=tf('s'); G = 3.163/(s^3+92.59*5^2+61.66*5); sisotool (G)]

REsult:

L SYSTEM VIEW ? model806 900 Editor - C:\Users\user\lab 3\model808.m +57 model803.m Xmodel804.m X model805.m clc close all 3 s=tf('s'); 4 G = 3.163/(s^3+92.59*5^2+61.66*5); Store Retrieve compare Export Preferences DESIGNS RESULTS PREFERENCES ave Edit Multimodel Tuning New ssion Architecture Configuration Methods Plot ARCHITECTURE TUNING METHODS ANALYSIS IOTransfer_r2y: step X and Fixed Blocks PID Tuning Compensator Step Response From: r To: y (1 + 5.9s + (2.982) C = 7.4145 x 1.2 S Select Loop to Tune 1 ........ Loop Transfer_c Add new loop ... 0.8 Amplitude 0.6 Specifications Tuning method: Robust response time Controller Type: POL PL PD O PID Design with first order derivative filter Design mode: LC у u 0.4 Time < >> 0.9279 0.2 Slower Response Time (seconds) Faster 0.88 Reset Parameters Aggressive Transient Behavior Robust 0 0 5 Update Compensator Help 10 15 20 25 Time (seconds) Compensator tuned using PID tun g to move Data Browser...

Control System Designer - IOTransfer_r2y: step Х CONTROL SYSTEM VIEW ? model806 C: Users user Editor - C:\Users\user\lab 3\model808.m +57 model803.m X model804.m X model805.m X 1 - clc 2 - close all s=tf('s'); 4 - G = 3.163/(s^3+92.59*5^2+61.66*5); POOT . Store Retrieve Compare نيا Export Preferences DESIGNS RESULTS PREFERENCES Open Save Edit Multimodel Tuning New Session Session Architecture Configuration Methods Plot FILE ARCHITECTURE TUNING METHODS ANALYSIS Data Browser IOTransfer_r2y: step X Controllers and Fixed Blocks PID Tuning Compensator Step Response From: r To: y IF (1 + 5.9s + (2.982) 7.4145 x с 1.2 S G Select Loop to Tune 1 Designs Loop Transfer_C System: 10 Transfer_r2y I/O:r to y Peak amplitude: 1.06 Overshoot (%): 5.8 At time (seconds): 4.02 Add new loop ... 0.8 Amplitude 0.6 Specifications Tuning method: Robust response time Controller Type: OPOL PL PD Design with first order derivative filter Design mode Time Responses Loop Transfer_C IO Transfer_r2y IO Transfer_r2u O PID 0.4 » 0.9279 ميحد حمص دما 0.2 Slower Response Time (seconds) Faster Preview 0.88 Reset Parameters Aggressive Transient Behavior Robust 0 0 5 10 15 20 25 Time (seconds) Compensator tuned using PID tun Update Compensator Help

code:

clc close all s=tf('s'); G = 3.163/(s^3+92.59*5^2+61.66*5); = = G_PID 7.4145+ (1+5.9*5+ (2.9*s)^2)/s; sys feedback (G_PID*G, 1); stepinfo (sys) step (sys)

Х Figure 3 File Edit View Insert Tools Desktop Window Help 7 System: sys Peak amplitude: 1.06 ponse Overshoot (%): 5.98 At time (seconds): 3.84 1.2 1 0.8 Amplitude 0.6 0.4 0.2 0 0 5 20 25 10 15 Time (seconds)

Command Window New to MATLAB? See resources for Getting Started. ans = struct with fields: RiseTime: 0.9186 SettlingTime: 10.1200 SettlingMin: 0.9038 SettlingMax: 1.0598 Overshoot: 5.9789 V Undershoot: 0 Peak: 1.0598 PeakTime: 3.8396