PID control

(1)

PID control - Simple tuning methods

(2)

Introduction

Lab processes Control System

Dynamical System

Step response model Self-oscillation model

PID control

PID structure

Step response method (Ziegler-Nichols) Self-oscillation method (Ziegler-Nichols)

Experiment

Level control in a tank

(3)

PID control

Introduction

Lab processes

Control System

Dynamical System

PID control

PID structure

Experiment

(4)

Tank process

• Tank with level control

• Two connected tanks

• Pump for in-flow of water

• Level measurements

(5)

PID control

Introduction

Lab processes

Control System Dynamical System

PID control

PID structure

Experiment

(6)

Open-loop system

u _Actuator Control object Sensor y

System

• u control signal (pump voltage)

• Actuator (tubes to pump, power amplifier , in-flow)

• Control object (level in Tank with in- and out-flow)

• Sensors (pressure sensors, tubes, elektronics)

(7)

Closed-loop system

r + e Controller u System y

−

Give reference (set-point of water level in tank)r to controller in

(8)

PID control

Introduction

Dynamical System

Step response model

Self-oscillation model

PID control

PID structure

Experiment

(9)

Time constant and stationary gain

System u y u(t) t Step t y(t) Step response 1 Kp T Example–Stove plate • u(t)power to stove plate

• y(t)temperature on plate

• Time constantT

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Time constant and stationary gain

System u y u(t) t Step t y(t) Step response ∆u Kp∆u T

• Step response size proportional to step size

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Dead-time (time delay)

System u y u(t) t Step t y(t) Step response 1 Kp L T

Example– roll transport time

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Step response model

System u y u(t) t Step t y(t) Step response A 1 Kp L T

Step response model from unit step:

• T Time constant

• KpStationary gain

• LDead-time

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Step response model

System u y u(t) t Step t y(t) Step response A∆u ∆u Kp∆u L T

Step response model from experiment:

• T Time constant

• KpStationary gain

• LDead-time

(14)

PID control

Introduction

Dynamical System

Step response model

Self-oscillation model PID control

PID structure

Experiment

(15)

Self-oscillation model

r + Ku e System u _y − _T u Experiment

• P-control (closed-loop system!)

• Crank up gain to self-oscillation

• Reference-step starts oscillation

Self-oscillation model

• Ultimate gainKu

(16)

PID control

Introduction

Dynamical System

PID control

PID structure

Experiment

(17)

PID structure

r + e Controller u System y

−

• r reference, set-point (SP)

• y output, measured signal to be regulated

• e=r−y control error PID-controller: e u P I D +

(18)

P-controller

K e u e(t) t e0 t u(t) Ke0

(19)

P-controller with offset

K e + u u0 e(t) t e0 t u(t) Ke0 u0Offset

Control signal Proportional to error plus ’offset’

Example: speed control of car

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I-part (integrator)

1 Ti R e e I-del e(t) t t I-del e0 e0 Ti I-part

• ∝surface undere(t)so far

• becomes as big as constant errore0in timeTi

(21)

D-part (derivator)

Tdde_dt e D-del e(t) t t D-del D-part • ∝slope one(t)

(22)

PID-controller structure

Control signal = P + I + D u(t) =K[e(t) + 1 Ti Z t e(s)ds+T_dde(t) dt ]

(23)

PID control

Introduction

Dynamical System

PID control

PID structure

Step response method (Ziegler-Nichols)

Self-oscillation method (Ziegler-Nichols)

Experiment

(24)

Ziegler-Nichols step response method

• MeasureAandLfrom step response

• Tp expected time constant for closed-loop system

Controller K T_i T_d Tp

P 1/A 4L

PI 0.9/A 3L 5.7L

(25)

PID control

Introduction

Dynamical System

PID control

PID structure

Step response method (Ziegler-Nichols)

Self-oscillation method (Ziegler-Nichols) Experiment

(26)

Ziegler-Nichols self-oscillation method

• Measure ultimate gainKuand periodTu from experiment

• Tp expected time constant for closed-loop system

Controller K Ti Td Tp

P 0.5Ku Tu

PI 0.4Ku 0.8Tu 1.4Tu

(27)

PID control

Introduction

Dynamical System

PID control

PID structure

Experiment

(28)

Step response

Step response model for the left tank

0 100 200 300 2 3 4 5 6 7 8 9 10 20 30 40 50 60 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4

Estimated from figure:

• A∆u=0.1,∆u=0.5

⇒A=1/5

(29)

P-control of the left tank

0 20 40 60 80 100 120 140 160 180 2.5 3 r, y 0 20 40 60 80 100 120 140 160 180 0 2 4 u Valve opens/closes Ziegler-Nichols P-control • K =1/A=5

• Stationary error after valve

(30)

PI-control of the left tank

0 50 100 150 200 2.5 3 r, y 0 50 100 150 200 0 2 4 u Valve opens/closes Ziegler-Nichols PI-control • K =0.9/A=4.5 • Ti=3L=9

• No stationary error after valve

(31)

PID-control of the left tank

0 50 100 150 200 2.4 2.6 2.8 3 3.2 r, y 0 50 100 150 200 5 u

Valve opens/closes _{Ziegler-Nichols PID-control} • K =1.2/A=6

• Ti=2L=6 • Td =L/2=1.5

• No stationary error after valve

disturbance

• Fast and damped

(32)

Self-oscillation experiment

Self-oscillation model for the left tank

0 20 40 60 80 100 0

1 2 3

r (bˆ¶rvˆ⁄rde) och y (ˆ⁄rvˆ⁄rde)

0 20 40 60 80 100 120 0

5 10

u (styrsignal)

Ultimate gain and period

• Ku=15 (from tuning) • Tu =10 (from figure) Z-N PID-tuning: K =0.6Ku=8 Ti =0.5Tu =5 Td =0.125Tu =1.25

(33)

Ziegler-Nichols PID from self-oscillation model

0 50 100 150 200 250 0 5 t, y 0 50 100 150 200 250 0 5 10 u

(34)

PID control

Introduction

Dynamical System

PID control

PID structure

Experiment

(35)

Step response

Step response model for the right tank

0 100 200 300 400 2 3 4 5 6 7 8 30 40 50 60 70 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Roughly estimated (bad precision!):

• A∆u≈0.2,∆u=0.5

⇒A=2/5

• L≈10

From larger figure:

(36)

PI-control of the right tank

0 100 200 300 2 3 r, y 0 100 200 300 0 1 2 3 u Valve opens/closes Ziegler-Nichols PI-control • K =0.9/A=2.25 • Ti=3L=30 Disturbance eliminated in 60s (CompareTp=5.7L=57s)

(37)

PID-control of the right tank (step response tuning)

0 100 200 300 3 4 5 6 r, y 0 100 200 300 0 5 10 u Valve opens/closes Ziegler-Nichols PID-control • K =1.2/A=3 • T_i=2L=20 • Td =L/2=5 Disturbance eliminated i 40s (CompareTp=3.4L=34s)

(38)

Self-oscillation experiment

Self-oscillation model for the right tank

0 20 40 60 80 100 4.7 4.8 4.9 y 0 20 40 60 80 100 0 1 2 3 u

Ultimate gain and period

• Ku=10 (from tuning) • Tu =25 (from figure) Z-N PID-tuning: K =0.6Ku =6 Ti =0.5Tu≈13 Td =0.125Tu≈3

(39)

PID-control of the right tank (self-oscillation tuning)

0 50 100 150 200 250 3 4 5 r, y 0 50 100 150 200 250 0 5 10 u Valve opens/closes