Lecture 17 PID: Tuning and Practial Issues Process Control Prof. - - PowerPoint PPT Presentation

Lecture 17 PID: Tuning and Practial Issues

Process Control

• Prof. Kannan M. Moudgalya

IIT Bombay Monday, 2 September 2013

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Outline

• 1. Recalling PI controller
• 2. Derivative mode
• 3. PID controller

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• 1. Recalling PI controller

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I order system and proportional controller

ysp

Gc u G −

e

y

◮ Take proportional controller, i.e., Gc = Kc ◮ Take first order system, i.e., G =

K τs + 1

◮ Steady state error =

1 1 + KKc

◮ SS error can be made small by making Kc large ◮ Because of unmodelled dynamics, G may

actually be a second order system!

◮ Recall SBHS!

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Shortcomings of indefinite increase in Kc

× ×

◮ Increase Kc to bring the closed loop poles

inside desired region

◮ Indefinite increase of Kc will take root locus

• utside desired region

◮ Large overshoot and lots of oscillations in this

case!

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Trade off between offset & control action

Recall the closed loop system:

∆Ysp

Kc ∆U G −

E

∆Y

◮ Zero error E ⇒ ∆U = 0 ◮ This implies zero control action with respect to

nominal value of U

◮ Servo/tracking control (set point changes)

cannot be implemented

◮ Can we have both E = 0 and ∆U = 0 with

finite Kc?

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Introducing Integral Mode

u(t) = u + 1 τi t e(w)dw

◮ Even a 0 value of e(t) can give rise to nonzero

values of u(t)!

◮ τi is reset time or integral time ◮ Recall u(t) ↔ U(s) ◮

∆U(s) = 1 τisE(s)

◮ Normally, we use a proportional-integral (PI)

controller

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PI controller

PI controller: u(t) = u + Kc

• e(t) + 1

τi t e(w)dw

• r

∆U(s) = Kc

• 1 + 1

τis

• E(s)

◮ What is the steady state offset now?

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MCQ on PI controller terminology

Consider the PI controller: ∆U(s) = Kc

• 1 + 1

τis

• E(s)

Integral action/effort increases implies

• 1. τi increases
• 2. τi decreases
• 3. This has nothing to do with τi

Answer: 2

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Integral mode generalises proportional mode

PI controller: ∆U(s) = Kc

• 1 + 1

τis

• E(s)
• r

∆u(t) = Kc

• e(t) + 1

τi t e(w)dw

• = Kce(t) + Kc

1 τi t e(w)dw With e constant, after every t = τi, Kce gets added!

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MCQ on PI Controller

A PI controller is put in a closed loop with a first

• rder system. A step input is given in the set point.
• 1. As the integral mode generalises the

proportional mode, increasing the integral action makes the error very small, but not zero

• 2. As the integral mode generalises the

proportional model, increasing the integral efforts results in larger oscillations in response and also the offset - the offset does not reach a steady state

• 3. The offset goes to zero

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Why not use integral mode ALONE?

Covered in the last class: integral windup

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• 2. Derivative mode

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Derivative Mode

y(t) tr tp ts

Mp ◮ Often need to reduce the oscillations ◮ Need to use the slope information

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Derivative Mode - ctd

◮ Derivative mode is given by

u(t) = u + τd de(t) dt τd is the derivative time

◮ Intuitive explanation in B. C. Kuo ◮ Problem: noise; so used with prop. cont. ◮ For an open loop stable system, increase in

derivative mode generally results in

◮ decreased oscillations. ◮ Will it have any effect at steady state?

◮ Remember this while tuning the derivative

mode

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PD Controller

◮ Derivative mode is not used alone ◮ PD controller is usually used ◮ Reason: noise ◮ ∆U(s) = Kc (1 + τds) E(s) ◮ What is the effect of this controller on

• verdamped second order system?

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PD control to II order overdamped system

◮ G(s) =

1 (s + 1)(s + 2) = 1 s2 + 3s + 2

◮ What is the effect of PD controller:

Gc = Kc (1 + τds)

◮ Use root locus ◮ But root locus works only with Kc ◮ Answer: Combine the zero with G ◮ Study root locus of G1 =

(1 + τds) (s + 1)(s + 2)

◮ Effect of derivative mode can be studied by

addition of zeros!

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Derivative mode on an oscillating system

× ×

◮ Try the PD controllers for

G(s) = 1 (s + 1)(s + 2)

• 1. Kc(1 + 0.5s)
• 2. Kc(1 + 1.5s)
• 3. Kc(1 + 3s)

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Zero to the left of poles

asymptotic directions

• pen loop poles
• pen loop zeroes
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 2.0

• 6
• 5
• 4
• 3
• 2
• 1

Evans root locus Real axis Imaginary axis 19/30 Process Control PID: Tuning and Practical Issues

Zero to the left of poles

asymptotic directions

• pen loop poles
• pen loop zeroes
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 2.0

• 6
• 5
• 4
• 3
• 2
• 1

Evans root locus Real axis Imaginary axis

◮ Only Kc: poles would have gone to ∞ ◮ The zero brings the locus back to real axis ◮ Can make it as damped as required ◮ What happens to other locations?

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Effect of zero on an overdamped II order system

◮ G(s) =

1 (s + 1)(s + 2) = 1 s2 + 3s + 2

◮ Gc = Kc (1 + τds) ◮ Zero < −2 results in a root locus plot as given

in the previous page

◮ 0 > Zero > −2 results in the root locus lying

entirely on real axis, making the closed loop system overdamped and stable

◮ An example of the derivative mode reducing

• scillations

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Effect of derivative mode on oscillating system

A system oscillates a lot and reaches a steady state with offset. Increase in derivative action is expected to

• 1. Increase the oscillations
• 2. Decrease the oscillations and offset
• 3. Decrease the oscillations only
• 4. Increase both oscillations and offset

Ans: 3

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• 3. PID controller

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PID controller

PID controller has proportional, integral and derivative modes: u(t) = u + Kc

• e(t) + 1

τi t e(w)dw + τd de(t) dt

• ◮ e is error, u is control effort and u is the steady

state value of u

◮ It has three tuning parameters, Kc, τi, τd ◮ Recall the tuning guidelines to change these

parameters

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How does one design the PID controller?

◮ By trial and error, using tuning methods ◮ By direct synthesis ◮ Using advanced control techniques and

implementing them as PID

◮ Will study the most popular tuning method

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Reaction Curve - Ziegler Nichols Tuning

◮ Applicable only to stable systems ◮ Give a unit step input and get R = K/τ L τ K

• 1. the time lag after which the system starts

responding (L),

• 2. the steady state gain (K) and
• 3. the time the output takes to reach the steady

state, after it starts responding (τ)

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Reaction Curve - Ziegler Nichols Tuning

R = K/τ L τ K

Calculate slope, R = K/τ

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Reaction Curve Method Ctd.

R = K/τ L τ K

PID settings are given below: Kc τi τd P 1/RL PI 0.9/RL 3L PID 1.2/RL 2L 0.5L Consistent units should be used

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What we learnt today

◮ P, I, D modes ◮ Guidelines for tuning ◮ Ziegler-Nichols methods

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Thank you

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