04 PID Controllers

8/11/2019 04 PID Controllers

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Proportional-Integral-Derivative

PID Controllers

Cheng-Liang ChenPSELABORATORY

Department of Chemical EngineeringNational TAIWAN University


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Chen CL 1

Outline

Proportional-Integral-Derivative Controller: PID Control based on CurrentFuture Error withConstantReset Bias

PID Control based on CurrentFuture Error withConstantReset Bias







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Steady Offset ofPID when using Constant bias

Series PID Parallel PID

Response ofPID Controllers to Typical Inputs

Operational Aspects of PID Controllers

problems ofD action: sensitivity to noise

problems of I action: moving PB and reset windup

manual control requirement bump-less transfer


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Chen CL 4

PID Controller: A Survey

More Than 95% of Controllers Are of PID Type

Ender (1993)

30%of installed process controllers operate in manual

20%of loops use default parameters (factory tuning) 30%of loops function poorly because of equipment

problems (valves, sensors )

C C


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Why PID Controllers Are So Popular ?

Structure: simple Easy for understanding

A few adjustable parameters, easy for tuning

Performance: good or acceptable level

Robustness: strong to uncertain operating conditions

Applicability: wide to different processes/industries

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A Heat Exchanger with Feedback ControlValve: AFC (ATO); Controller: Reverse Action

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Simplest Controller: On-Off Control

u(t) =

ub+ u if y(t)< ysp d

ub u if y(t)> ysp+d

Problem: oscillatory response !

On-off cares about control direction only;

On-off givessamecontrol action fordifferenterror magnitudes

Solution: control action considering error magnitude

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Current-Error-Based P Controlwith Constant Bias (SS value)

current control corrective action, u(t)

current error magnitude, e(t) ysp y(t)

u(t) e(t) ysp y(t)

u(t) = Kc e(t) p(t) (Kc: adjustable proportionality)

u(t) = Kc e(t)

u(t)+ ub

SS valuefore(t) = 0

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Problem of Current-Error-Based P Control

u(t) =Kc e(t) u(t)

+ ubSS valuefore(t) = 0

Too late to correct current error

It takes time to know effect of control on output

conservative action to guarantee stability

limited achievable control performance

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Solution: Future-Error-Based P ControlBack to the Future

Control action based onfuture condition (error)

take control action in advance better control performance

u(t) = Kc e(t+Td)

u(t)

+ ub

SS value

fore(t) = 0

u(t) = Kc e(t+Td)

u(t)

+ ub

SS value

fore(t) = 0

Chen CL 11


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Chen CL 11

Future-Error-Based P ControlGives Better Performance

HE: pulse decrease in input temperature

Att1: e(t1+Td)> e(t1)P control based on e(t1+Td) has larger action

more steam smaller undershoot in T

( Pre-Act P)

Kc e(t1+Td) +ub > Kc e(t1) + ub

( Normal P)

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Future-Error-Based P ControlGives Better Performance

HE: pulse decrease in input temperature

Att2: e(t2+Td)< e(t2)P control based on e(t2+Td) has smaller action

less steam smaller overshoot in T

( Pre-Act P)

Kc e(t2+Td) +ub < Kc e(t2) + ub

( Normal P)

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Implement Future-Error-Based P Control

As Current-Error-Based PD Control

Original P: u(t) = Kc e(t) + ub

PreAct P: u(t) = Kc e(t+Td) + ub

Kc e(t) +Tdde(t)dt e(t)e(t+Td)

+ ub

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u(t) = Kc e(t+Td) + ub

Kc e(t) +Tdde(t)dt e(t)e(t+Td)

+ ub

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Sub-SummaryP control based oncurrent error with constant bias

e(t) e(t+Td)

P control based onfuture error with constant bias

u(t) = Kce(t+Td) + ub

Kc

e(t) + Td

de(t)

dt

e(t)

+ub

PDcontrol based oncurrent error with constant bias

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Steady Offset for P (PD) Control

Non-zero Steady State Offset (I): Setpoint Change

Initial steady state: (all variables [0, 100%])

y=ysp=y e= 0 u=ub=u =

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Case I: Setpoint change ysp:y y+ New steady state: y=?, u=?, e=?

(change in y) = Kp {change in process input}= Kp {(change in CO)+(change in load)}

y y = Kp

u u

change in CO+

change in load=0

= Kp

Kc

e

(y+ )

new spy

+ub

uu+

=0

y = y+ KcKp1 +KcKp

new steady state

= y+ desired sp

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Non-zero Steady State Offset (II): Load Change

Initial steady state: (all variables [0, 100%])

y=ysp=y e= 0 u=ub=u =

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Case II: Load change : +New steady state: y=?, u=?, e=?

(change in y) = Kp {change in process input}= Kp {(change in CO)+(change in load)}

y y = Kp

u u

change in CO+

change in load=

= Kp

Kc

e

(y)

spy

+ub

uu+

y = y+ Kp

1 +KcKp

new steady state

= ydesired sp

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Current-Error-Based P Controlwith Reset Bias

Question:

usingP or PD control with constant bias,

how to guarantee zero steady-state offset ?

(at steady state: e= 0, y=ysp)

Answer: u=ub at SS

desired: e = 0 (y = ysp) at SS

u = ub at SS

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Reset Bias for Zero SS OffsetSimplest Method

Simplest method to guarantee u=ub at SS:

Let ub(t) = u(t) for all t

then ub = u at steady state

e = 0 y = ysp

P control based on current error withreset biasvia a unity-gain dynamics (not workable !)

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Reset Bias for Zero SS OffsetFeasible Method

Feasible method to guarantee u=ub at SS:

desired: ub = u at SS

simplest method: ub(t) equals u(t) t

feasible method: ub(t) tracks u(t) t

ub(t) tracks u(t) with first-order dynamics:

(use Ti to adjust tracking velocity)

Tidub(t)

dt +ub(t) =u(t)

equal tracking

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P control based on current error withreset biasvia first-order dynamics

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Reset Bias Integral Action

(1) u(t) = Kce(t) +ub(t) (P with reset bias)

(2) u(t) = Tidub(t)

dt +ub(t) (ub(t) tracks u(t))

Tidub(t)

dt

= Kce(t) = p(t)

ub(t) ub = 1

Ti

t0

p(t)dt

change in bias=

KcTi

t0

e(t)dt

=ub(t)

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(1) u(t) = Kce(t) p(t)+

ub(t) 1

Ti t

0

p(t)(t)dt

ub(t)+ub (PI)

= Kce(t) +KcTi

t0

e(t)dt+ub

= Kc e(t) + 1Ti t

0

e(t)dt +ub Reset bias: interpreted as integral action

current-error-based P control with reset bias

(reseting bias according to integral of error)= current-error-basedPI control with constant bias

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Reset Bias Integral Action


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Future-Error-Based P Control with Reset Bias

P control based on future error PD withreset bias PI Interpreted As PD +PI (series PID ) Controller

e(t+Td) e(t) +Tdde(t)

dt e(t)

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ub(t) = 1

Ti

Kce

(t)dt+ub

u(t) = Kce(t)

p(t)

+

ub

(t) 1

Ti

Kce

(t) p(t)

dt +ub

ub(t)

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Kpc = Kc 1 + TdTi

Kc = Kpc 0.5 +

0.25

Tpd

Tpi

Tpi = Ti

1 + TdTi

Ti = Tpi

0.5 +

0.25 T

pdTpi

Tpd = Td/

1 + TdTi

Td = T

pd /

0.5 +

0.25

Tpd

Tpi

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p(t) = Kc e(t) +Tdde(t)dt u(t) = p(t) +ub(t)

Tidub(t)dt + ub(t) = u(t)

P(t) = Kc

E(t) +Td

dE(t)dt

U(t) = P(t) +Ub(t)

TidUb(t)dt + Ub(t) = U(t)

P(s) = Kc (E(s) +TdsE(s)) = Kc(1 +Tds)E(s)

U(s) = P(s) +Ub(s)

TisUb(s) + Ub(s) = U(s) Ub(s) = 1Tis+1

U(s)

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U(s) = P(s) +Ub(s) =P(s) + 1Tis+1

U(s)

U(s) = Tis+1Tis P(s)

= Tis+1Tis Kc(1 +Tds)E(s) (series PID)

= KcTis

TiTds2 + (Ti+Td)s+ 1

E(s)

= KcTi+TdTi

1 + 1(Ti+Td)s+ TiTdTi+Td

s

E(s)

= Kc

1 + TdTi

1 + 1

Ti

1+

TdTi

s

+Td

1 + TdTi

s

E(s)

= Kpc

1 + 1

Tpi s

+Tpd s

E(s) (parallelPID)

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P Control Based on {Current, Future} Errorwith {Constant, Reset} Bias

const. bias reset biascurrent error P PI

future error PD PID

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P Action to Step Error

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PD Action to Step Error

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PI (P-Reset) Action to Step Error

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S


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PID Action to Step Error

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PD A i R E


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PD Action to Ramp Error


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f ( )


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Problem of D Action (I):Sensitive to High-Frequency Noise

Derivative action without filter:

Problem: sinusoidal input output mag. frequency

e(t) =A sin(t) D(t) =Tdde(t)

dt =TdAcos (t)

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S l i (fi d ) l fil


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Solution: (first-order) low-pass filter

Td

N

dD(t)

dt +D(t) =Td

de(t)

dt D action

equal

follow

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P bl ( i )


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Problem: (again)

derivative + filtering large inner signal (ed)

Solution: filtering derivative

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P bl f D A i (II)


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Problem of D Action (II):Bump Response to Step Input

Solution 1: D action on measurement Derivative on measurement:

y(t+Td) y(t) +Tddy(t)

dte(t+Td) = ysp y(t+Td)

= ysp y(t) +Tddy(t)dt e(t) Td

dy(t)

dt For noise: filtering + derivative action

Td

N

dyf(t)

dt +yf(t) = y(t) (yf(t) follows y(t))

D(t) = KcTddyf(t)

dt (derivative on yf(t))

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S l ti 2 filt i t i t i l


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Solution 2: filtering on setpoint signal

(use r(t) as the practical setpoint)

Solution 3: setpoint weighting (skip)

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P bl f R (I)


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Problem of Reset (I):Moving Proportional Band

(Reverse) P with reset bias: ([umin, umax] = [0%, 100%])

u(t) = Kc

ysp y(t)

+ub(t) (Kc> 0)

y(t) = ymax = ? u(t) = Kc

ysp ymax

+ub(t) =umin= 0%

y(t) = ymin = ? u(t) = Kc ysp ymin +ub(t) =umax= 100%y(t) > ymax u(t) < umin (saturation !)

y(t) < ymin u(t) > umax (saturation !)

y(t) {ymin, ymax} u(t) {umax, umin} (normal operation)

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P ti l B d (f l ti )


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Proportional Band: (for normal operation)

PB = [ymin, ymax] = ysp+

ub(t)uminKc

, ysp+ub(t)umax

Kc =

ysp+

ub(t)0Kc

, ysp+ub(t)100

Kc

If y(t) PB, Then controller output is normal

width of PB: affected by Kc

ymax ymin=umax umin

Kc=

100

Kc%

position of PB: affected by Kc, ysp, ub(t)

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Example:


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Example:

ysp= 50%, Kc = 2, ub(t) = 50%t,

umin= 0%, umax= 100%

PB =

50 +

ub(t) 100

Kc% , 50 +

ub(t) 0

Kc%

= [ 25% , 75% ]

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Mo i PB d e to Resetti Bias E a le


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Moving PB due to Resetting Bias: Example

Process: G(s) = 0.0163s+1 5030s+1 110s+1 ZN-PI tuning: Kc= 3.40, Ti= 37 ( PB =

100%Kc%/%

= 29.4%)


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Process: G(s) = 0.0163s+1 5030s+1 110s+1 Slower reset: Kc= 3.40, Ti= 74 ( PB =

100%Kc%/%

= 29.4%)

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Problem ofReset (II)

Reset Windup

HE Example

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Anti Reset Windup:


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Anti Reset-Windup:Use A Saturation Modelto Limit Control Signal

u(t) =

100% for v(t) 100%

v(t) for 0% v(t) 100%

0% for v(t) 100%

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Anti Reset Windup for Parallel PID


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Anti Reset-Windup for Parallel PID

If v > u = umax

Then es = u v < 0

1Tt

esdt < 0 v until v = umax

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Modification of PID (I):


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Modification of PID (I):High-Frequency Noise

Problem of D Action: sensitive to high frequency noise

Solution: low-pass filter on D (P & D)

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Modification of PID (II):


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Modification of PID (II):Anti Reset-Windup

Problem of Reset Action: reset windup (saturation)

Solution: limiter

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Modification of PID (III):


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Modification of PID (III):Manual Control

Manual control: with an integrator

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Modification of PID (IV):


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Modification of PID (IV):Bumpless A/M Transfer

When in automatic control: m(t) (manual) tracks u(t) (auto)

m(t): always ready to control

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Modification of PID (V):


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Modification of PID (V):Bumpless M/A Transfer

Method 1: resetting bias during M/A (keep original ysp)

let ub(t) = m(t) Kce(t) (when M/A transfer)

thus u(t) = ub(t) +Kce(t) (auto control action)

= m(t) (final manual action)


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A Complete PID Controller (I)


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A Complete PID Controller (I) Automatic control Anti-reset windup Bias tracking: bias tracks final outputu(t) Manual tracking: m(t) tracks final output u(t) Manual setpoint adjustment

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A Complete PID Controller (II)


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A Complete PID Controller (II) Manual control Anti-reset windup Bias tracking: bias tracks final outputu(t) Manual tracking: m(t) tracks final output u(t) (Manual setpoint adjustment)

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A Complete PID Controller (III)


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A Complete PID Controller (III) External setpoint (used in Cascade Control) Anti-reset windup Bias tracking: bias tracks final outputu(t) Manual tracking: m(t) tracks final output u(t) Setpoint tracking:

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A Complete PID Controller (IV)


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A Complete PID Controller (IV) Automatic control (used in Override Control) Anti-reset windup Bias tracking: bias tracks external signalz(t) Manual tracking: m(t) tracks final output u(t) Setpoint tracking:

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Thank You for Your Attention

04 PID Controllers

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