9 chapter 9 PID controller

11 PID CONTROLLER Ziegler–Nichols 2 – Example Consider the control system shown in Figure in which a PID controller is used to control the system... PID CONTROLLER Ziegler–Nichols – Ex

SYSTEM DYNAMICS & CONTROL

CHAPTER 9 PID CONTROLLER

Dr Vo Tuong Quan

HCMUT - 2011

PID CONTROLLER

Ziegler–Nichols 1

Unit step response of a plant

PID CONTROLLER

Ziegler–Nichols 2

the output first exhibits sustained oscillations (If the output does not exhibit

experimentally determined Ziegler and Nichols suggested that we set the

7

PID CONTROLLER

Ziegler–Nichols 2

𝑃𝑐𝑟

Type of Controller

11

PID CONTROLLER

Ziegler–Nichols 2 – Example

Consider the control system shown in Figure in which a PID controller is used

to control the system Design a PID controller

PID CONTROLLER

Ziegler–Nichols 2 – Example

Let use the second method of Ziegler–Nichols tuning rules By setting

PID CONTROLLER

Ziegler–Nichols – Example

Design a PID controller to control the angle position of a DC motor, Providing that by experiment the critical gain of the system is 20, the critical is 𝑇 = 1 𝑠𝑒𝑐

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PID CONTROLLER

Ziegler–Nichols – Example

Applying Zeigler-Nichols method 2:

PID CONTROLLER

Design PID controller – Frequency Approach

Consider the system shown in Figure Using a frequency-response approach, design a PID controller such that the static velocity error constant is 4 sec−1, phase margin is 50° or more, and gain margin is 10 dB or more Obtain the unit-step and unit-ramp response curves of the PID controlled system with MATLAB

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PID CONTROLLER

Design PID controller – Frequency Approach

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PID CONTROLLER

Design PID controller – Frequency Approach

The phase margin of at least 50° and gain margin of 10 dB or more From the Bode diagram, The gain crossover frequency is approximately

𝜔 = 1.8 𝑟𝑎𝑑/𝑠𝑒𝑐 Assume the gain crossover frequency of the compensated system to be somewhere between 𝜔 = 1, 𝜔 = 10𝑟𝑎𝑑/𝑠𝑒𝑐

PID CONTROLLER

Design PID controller – Frequency Approach

-20 0 20 40 60

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PID CONTROLLER

Design PID controller – Frequency Approach

PID CONTROLLER

Design PID controller – Frequency Approach

-40 -20 0 20 40 60

-135 -90 -45 0 45

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PID CONTROLLER

Design PID controller – Frequency Approach

PID CONTROLLER

Design PID controller – Frequency Approach

0 0.2 0.4 0.6 0.8 1 1.2

PID CONTROLLER

Design PID controller – Frequency Approach

0 2 4 6 8 10 12 14 16 18

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PID CONTROLLER

Design PID controller – Analytical Method

Design PID controller so that the control system satisfies the following

requirements:

is state controllable if each state variable can be influenced by the input

initial states 𝑥 𝑡0 Qualitatively, the system is state observable if all state variable 𝑥 𝑡 influences the output 𝑦(𝑡)

system and the input-output data

Evaluate the observability of the system

−6

3

−8det 𝜕 = 10 → 𝑟𝑎𝑛𝑘 𝜕 = 2

→ 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑖𝑠 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒

PID CONTROLLER

Control system design in state space - Pole Placement Method

State Feedback Control

Consider a system described by the state-space equation:

𝑦 𝑡 = 𝐶𝑥(𝑡)The state feedback controller: 𝑢 𝑡 = 𝑟 𝑡 − 𝐾𝑥(𝑡)

The characteristic equation of the closed-loop system:

𝑥 𝑡 = [𝐴 − 𝐵𝐾]𝑥 𝑡 + 𝐵𝑟(𝑡)

𝑦 𝑡 = 𝐶𝑥(𝑡)

If the system is controllable then it is possible to determine the feedback gain 𝐾

so that the closed-loop system has the poles at any location

Step 1: the characteristic equation of the closed-loop system:

det 𝑠𝐼 − 𝐴 + 𝐵𝐾 = 0 (1)

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PID CONTROLLER

Control system design in state space - Pole Placement Method

State Feedback Control

Step 3: balance the coefficient of the equation (1), (2), we can find the state feedback gain 𝐾

PID CONTROLLER

Control system design in state space - Pole Placement Method

State Feedback Control

Given a system described by the state-state equation:

−3

, 𝐵 =

0

3 1

, 𝐶 = [0 0 1]

Determine the state feedback controller 𝑢 𝑡 = 𝑟 𝑡 − 𝐾𝑥(𝑡) so that the

at -20

+

0

31

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PID CONTROLLER

Design PID controller with computational optimization approach

Consider the PID-controlled system shown in Figure below The PID controller

PID CONTROLLER

Design PID controller with computational optimization approach

Assume that the region to search for 𝐾 and 𝑎 is: 2 ≤ 𝐾 ≤ 3, 0.5 ≤ 𝑎 ≤ 1.5

K = [2.0 2.2 2.4 2.6 2.8 3.0]; a = [0.5 0.7 0.9 1.1 1.3 1.5];

t = 0:0.01:5;

g = tf([1.2],[0.36 1.86 2.5 1]); k = 0;

for i = 1:6; for j = 1:6;

gc = tf(K(i)*[1 2*a(j) a(j)^2], [1 0]); % controller

G = gc*g/(1 + gc*g); % closed-loop transfer function

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PID CONTROLLER

Design PID controller with computational optimization approach

Consider the PID-controlled system shown in Figure below The PID controller

PID CONTROLLER

Design PID controller with computational optimization approach

Assume that the region to search for 𝐾 and 𝑎 is: 3 ≤ 𝐾 ≤ 5, 0.1 ≤ 𝑎 ≤ 3.0

t = 0:0.01:8;k = 0;

for K = 3:0.2:5;

for a = 0.1:0.1:3;

num = [4*K 8*K*a 4*K*a^2];

den = [1 6 8+4*K 4+8*K*a 4*K*a^2]; y = step(num,den,t);

s = 801;while y(s)>0.98 & y(s)<1.02; s = s-1;end;

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PID CONTROLLER

Design PID controller with computational optimization approach

K = sortsolution(1,1), a = sortsolution(1,2)

num = [4*K 8*K*a 4*K*a^2];

den = [1 6 8+4*K 4+8*K*a 4*K*a^2];