Systems Engineering/Process control L9

The PID controller

◮ The algorithm

◮ Frequency analysis

◮ Practical modifications

◮ Tuning methods

Reading: Systems Engineering and Process Control: 9.1–9.6

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The PID controller

“Based on a survey of over eleven thousand controllers in

the refining, chemicals and pulp and paper industries, 97%of regulatory controllers utilize PID feedback.”

[Desborough and Miller, 2002]

“School-book form”:

u(t) = K(

e(t) + 1Ti

∫ t

0

e(τ )dτ + Tdde(t)dt

)

Transfer function:

Gc(s) = K(

1+ 1

sTi+ sTd

)

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The P part

◮ P controller:

u = K (r − y) + u0 = K e+ u0

umax

umin

u

e

0e– e

0

u0

Proportionalband

◮ u0 can be chosen to eliminate stationary error at setpoint

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Example: P control of Gp(s) = (s+ 1)−3

0 10 200

1

Ou

tpu

t

0 10 20−2

0

2

4

6

Inp

ut

Time

K = 5

K = 2K = 1

K = 5

K = 2K = 1

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The I part

◮ Introduce automatic/online/dynamic selection of u0:

replacements

K e

u0

u∑

1

1+ sTi

U (s) = K E(s) + 1

1+ sTiU (s)

U (s) = K(

1+ 1

sTi

)

E(s)

◮ Assume stationarity: How does u and u0 relate? What is e?

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Example: PI control of Gp(s) = (s+ 1)−3 (K = 1)

0 10 200

1

Ou

tpu

t

0 10 200

2

Inp

ut

Time

Ti = ∞

Ti = 5

Ti = 2Ti = 1

Ti = ∞Ti = 5

Ti = 2Ti = 1

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The D part

◮ A P controller gives the same control in both these cases:

B

Areplacements

Control error

Time

e

ep

ep

t t+ Td◮ Predicted error:

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

◮ PD controller:

u(t) = K(

e(t) + Tdde(t)dt

)

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Example: PD control of Gp(s) = (s+ 1)−3 (K = 5)

0 10 200

1

Ou

tpu

t

0 10 20

−2

0

2

4

6

Inp

ut

Time

Td = 0.1Td = 0.5

Td = 2

Td = 0.1Td = 0.5

Td = 2

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Parallel and serial form

◮ PID controller on standard form (parallel form):

Gc(s) = K +K

sTi+ sKTd

e u

K

KTis

KTds

Σ

◮ PID controller on serial form (common in industry):

G′c(s) = K ′(1+ 1

sT ′i

)(1+ sT ′d)

e uK ′ 1+ 1

T ′is 1+ T ′ds

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Parallel and serial form

Transformation parallel form Q serial form:

K = K ′ T′i+T ′d

T ′i

K ′ = K2

(

1+√

1− 4TdTi

)

Ti = T ′i + T ′d T ′i = Ti2

(

1+√

1− 4TdTi

)

Td = T ′iT′d

T ′i+T ′d

T ′d = Ti2

(

1−√

1− 4TdTi

)

◮ Identical parameters for PI and PD controller

◮ Parallel → serial only possible if Ti ≥ 4Td◮ Parallel form more general

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Frequency analysis of PID controller

Frequency function for PID controller on serial form:

G′c(iω ) =K ′

iωT ′i(1+ iωT ′i )(1+ iωT ′d)

◮ For low frequencies (small ω ):

pG′c(iω )p (K ′

ωT ′iargG′c(iω ) ( −90○

◮ Zero at s = −1/T ′i bends amplitude curve up and increases

phase with 90○ around ω = 1/T ′i◮ The same holds for the zero at s = −1/T ′d

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Frequency analysis of PID controller

10-2

10-1

100

101

100

101

10-2

10-1

100

101

-100

-50

0

50

100

1/T ′i 1/T ′d

K ′

Frequency [rad/s]

Ga

inP

ha

se

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Repetition: Amplitude and phase margin

10-1

100

10-2

10-1

100

101

10-1

100

-250

-200

-150

-100

-50

1/Am

ϕm

ω c

ω o

Frekvens [rad/s]

Fö

rstä

rkn

ing

Fa

s

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Frequency analysis of PID controller

The P part:

◮ Affects gain at all frequencies

◮ Higher gain [ faster system but worse margins

The I part:

◮ Increases gain and reduces phase for low frequencies

◮ Eliminates low frequency (constant) control errors but gives

worse phase margin

The D part:

◮ Increases gain and phase at high frequencies

◮ Gives better phase margin (to a limit) but amplifies noise

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Practical modifications of PID controllers

School-book form:

e(t) = r(t) − y(t)

u(t) = K e(t)︸ ︷︷ ︸

P(t)

+ KTi

∫ t

0

e(τ )dτ︸ ︷︷ ︸

I(t)

+ KTdde(t)dt

︸ ︷︷ ︸

D(t)

Modifications:

◮ The P part: reference weighting

◮ The I part: anti-windup

◮ The D part: reference weighting and limited gain

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Modification of P part

◮ Introduce reference weighting β :

P(t) = K(β r(t) − y(t)

), 0 ≤ β ≤ 1

◮ Can be used to limit overshoot after reference changes

(moves a zero in closed-loop system)

◮ Note! Works only if also I part used

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Example: Reference weighting with PI control

(reference change at t = 0, load disturbance at t = 25):

0 500

1

0 500

2

β = 0β = 0.5

β = 1

β = 0β = 0.5

β = 1

Time

Input

Output

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Modification of I part

Input is always limited in practice (umin ≤ u ≤ umax)

◮ Let v be the input the controller wants to use

◮ Let u be the input the controller can use

umax

umin

u

v

Integrator windup: I part keeps growing when signal saturated

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Example: PI control with integrator windup

Gp(s) = 1/s, K = Ti = 1, −0.3 ≤ u ≤ 0.3:

0 10 20

Out

put

0

1

2

Time0 10 20

Inpu

t

0

1

2v

u

I

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Anti-windup

I(t) =∫ t

0

(K

Tie(τ ) + 1

Tt

(u(τ ) − v(τ )

))

dτ

PI controller with anti-windup:

K

K

Ti

1

s

1

Tt

∑ ∑

∑e v u

I − +

Actuator(or model)

Rule of thumb for constant Tt:

◮ PI controller: Tt = 0.5Ti◮ PID controller: Tt =

√TiTd

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Example: PI control with anti-windup

Same example as before, but with anti-windup (Tt = 0.5):

0 10 20

Out

put

0

1

2

Time0 10 20

Inpu

t

0

1v

u

I

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Modification of D part

◮ Reference weighting: derivate only measurement, not

reference

D(t) = −KTddy(t)dt

◮ Limit gain with low-pass filter (extra pole):

D(s) = − sKTd

1+ sTd/NY(s)

(“fuskderivata”)

Maximal derivative gain N typically chosen in interval 5–20

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Example: Limited derivative gain

y(t) = sin t+ 0.01 sin 100t, Td = 1, N = 5

0 5 10 15

-2

2Brusfri signal

0 5 10 15

-2

2Brusig signal y

0 5 10 15

-2

2Brusig signal y

0 5 10 15

-2

2Brusfri derivata

0 5 10 15

-2

2Derivatan av y

0 5 10 15

-2

2Fuskderivatan av y

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Summary: Practical modifications

−1sKTd

1+ sTd/N

β K

K

Ti

1

s

1

Tt

∑

∑

∑

∑

∑

y

r

v u

− +

Actuator(or model)

(More to think about: bumpless transfer between manual/automatic

control, bumpless parameter changes, sampling filters, sampling,

. . . )

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Tuning methods for PID controllers

◮ Manual tuning (lab 1)

◮ Ziegler–Nichols methods

◮ The Lambda method

◮ Arresttidstrimning (project)

◮ Model-based tuning (lab 2)

◮ Relay methods

◮ Optimization-based methods

◮ . . .

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Ziegler–Nichols step response method

Experiment on open-loop system, read a and b in step response:

a

b

y

t

Controller K Ti TdP 1/aPI 0.9/a 3b

PID 1.2/a 2b 0.5b

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Ziegler–Nichols frequency method

(Ziegler–Nichols’ ultimate-sensitivity method)

Experiment on closed-loop system

1. Disconnect I and D parts in PID controller

2. Increase K until oscillations with constant amplitude. This

K = K0.3. Measure period time T0 for oscillations.

Controller K Ti TdP 0.5K0PI 0.45K0 T0/1.2PID 0.6K0 T0/2 T0/8

(Note that T0 = 2π /ω 0, where ω 0 is frequency that gives

−180○ phase shift)

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Ziegler–Nichols methods – warning

◮ Ziegler–Nichols’ methods give aggressive control with bad damping

◮ Recommendation: K lowered with 30–50 % for better robustness

◮ Example: PID control of Gp(s) = 1/(s+ 1)4:

0 25 50

Out

put

0

1

Step response methodUltimate sensitivity methodUltimate sensitivity, 40% lower gain

Time0 25 50

Inpu

t

0

2

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Lambda method

1. Read deadtime L, time constant T and static gain Kp = ∆y∆u

:

∆y

∆u

Process output

Control signal

63%

L T

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Lambda-method

2. Choose λ = desired time constant for closed-loop system

◮ λ = T common choice◮ λ = 2T a bit slower for more robustness

3. PI controller:

K = 1

Kp

T

L + λ, Ti = T

PID controller (in serial form):

K ′ = 1

Kp

T

L/2+ λ, T ′i = T , T ′d =

L

2

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Model based tuning (Lab 2)

1. Find process transfer function Gp(s)2. Choose controller type Gc(s)3. Compute closed-loop system transfer function:

G(s) = Gp(s)Gc(s)1+ Gp(s)Gc(s)

4. Choose controller parameters to place poles for G(s) to

achieve desired behavior (pole placement)

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