metodika nastavovania pid-imc2

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D.E. Rivera, ChE 461/598: Process Dynamics and Control; Introduction toInternal Model Control with Application to PID Controller Tuning

Copyright 1999 by D.E. Rivera, All Rights Reserved

1

Introduction to Internal Model Control

with Application to PID Tuning

Daniel E. Rivera, Ph.D.

Department of Chemical, Bio and Materials Engineeringand

Control Systems Engineering LaboratoryComputer-Integrated Manufacturing Systems Research Center

Arizona State University, Tempe AZ [email protected]

Session Focus

u(t) = u +

Proportional

Kc e(t) +

Integral KcI

t0 e(t

)dt +

Derivative KcD

de

dte(t) = r(t) y(t)

r yu

d

e+ ++

PC

We will emphasize the use of the Internal Model Controldesign procedure as a means for tuning PID controllers:

PID Controller Forms

c(s) = Kc 1 +1Is

+ Ds

Ideal Form

c(s) = Kc 1 +1

I s

Ds+1

Ds + 1

Interactive "Electronic" Form

Ideal Form with Filter

c(s) = Kc 1 +1

I s+ Ds

1Fs + 1

Low-Order Difference Equation Form

uk=Kc'

ek - I'

ek-1 + D'

ek-2 + F'

uk-1

PID ue

-y

r

ue

-y

r

PD

I

ue

-y

r

D

PI

Equation A

Equation B

Equation C

Honeywell PID Eqn Forms


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2

r yu

d

e+ ++

PC

r yu

d

+

+

++

P

P~

Q

C = Q(I-PQ)-1 Q = C(I+PC)-1

Internal Model Control Structure

r yu

d

+

+

++

P

P~

Qe = r - y

IMC-Closed Loop Transfer Functions

In the absence of plant/model mismatch (p = p):

y = pq r + (1 pq) d

= r + d

u = q r q d

= p1 r p1 d

e = (1 pq) r (1 pq) d

= r d

= pc(1 + pc)1 = (1 + pc)1

Requirements for Physical Realizability ofthe IMC Controller q(s)

Internal Stability: Bounded inputs to the controlsystem must lead result in bounded outputs elsewherein the control system.

"Properness": The controller must not differentiatestep changes (e.g., avoid subjecting step inputssubject to pure derivative action).

Causality: The controller must not require prediction,i.e., it must rely on past and current measurements of

the process.

Step 1 (Nominal Performance). Obtain an optimal Q

must specify input type (e.g., step, ramp)and objective function

closed-form solutions available.

Step 2 (Robust Stability and Performance). Augment Q witha filter F(s)

F(s) contains adjustable parameters which can be tuned forRobust Stability and Performance

Qfinal(s) = Q(s) F(s)

Internal Model Control Design Procedure


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3

Performance and Robustness Measures

Integral Square Error (ISE):

J = I SE =0 (y r)

2dt

Integral Absolute Error (IAE):

J2 = I A E =0 |y r|dt

Maximum Peak of the ComplementarySensitivity Function:

M = sup

Step 1: Factor out all deadtime and Right-Half Plane Zerosin the model

which leads to a controller that is stable and causal

P(s) = P+(s) P-(s)

P+(s) =e-s -is + 1i

n

Re(i)>0

P+(s) =e-s-is + 1

is + 1

i

n

Re(i)>0

Q(s) = P--1(s)

ISE-Optima

IAE-Optimal

Internal Model Control Design Procedure (Cont)

(for step inputs)

(for step inputs)

Step 2: Augment the optimal controller with a filter F(s)

which insures that the final control system is proper:

Qfinal

(s) = Q(s) F(s)

Example:F(s) =

1

( s + 1)nf

To note:

Closed-loop transfer function

Filter time constants reflect closed-loop speed-of-response

Filter time constants can be optimized for robust performance(Zafiriou and Morari, 1986)

P C 1+P C -1= P+(s) F(s)

Internal Model Control Design Procedure (Cont) Some IMC-PID Tuning RulesSelected rules for plants without integrator, > 0, no offset for step

setpoint/disturbance changes, is an adjustable parameter.

Plant KKc I D F Controller

K(-s + 1)

s + 1

+ 0 0 PI

::

::

::

::

::

::

K(-s + 1)

2s2 + 2 s + 1

2

+ 2

20 PID

::

::

::

::

::

::

K(-s + 1)

2s2 + 2 s + 1

2

2 + 2

2

2 +

PID w/filter

c(s) = Kc 1 +1

I s+ Ds

1Fs + 1


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4

A PI tuning rule arises from applying IMC to the first-order model:

p =K

s + 1 > 0 (1)

under the condition that d and r are step input changes.

Step 1: Factor and invert p; since p+ = 1, we obtain:

q=s + 1

K

Step 2: Augment with a first-order filter

f =1

(s + 1)

The final form for q is

q=s + 1

K(s + 1)(2)

Example 1: PI Control

We can now solve for the classical feedback controller equivalent c(s) to

obtain

c =q

1 pq=

K(1 +

1

s) (3)

which leads to the tuning rule for a PI controller

Kc =

K(4)

I = (5)

The corresponding nominalclosed-loop transfer functions for this con-

trol system are

=1

s + 1p1 =

s + 1

k(s + 1) =

s

s + 1(6)

Example 1: PI Control (Continued)

Example 1b: PI ControlConsider now the first-order model with Right Half Plane (RHP) zero:

p(s) =K(s + 1)

(s + 1) , > 0

again under the assumption that the inputs to r and d are steps.

Step 1: Use the IAE-optimal factorization for step inputs:

p+ = (s + 1) p =K

(s + 1)q =

(s + 1)

K

Step 2: Use a first-order filter

f =1

(s + 1)q =

(s + 1)

K(s + 1)

Solving for the classical feedback controller leads to another tuning rule

for a PI controller:

c(s) = Kc(1 +1

Is)

Kc =

K(+ )I =

Example 1c: PI with Filter Control

Consider now the first-order model with Left Half-Plane (LHP) zero:

p(s) =K(s + 1)

(s + 1) > 0 > 0


Step 1: No nonminimum phase behavior in p; since p+ = 1, we obtain:

p =K(s + 1)

(s + 1)q =

(s + 1)

K(s + 1)

Step 2: Use a first-order filter (q is now strictly proper).

f =1

(s + 1)q =

(s + 1)

K(s + 1)(s + 1)


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Example 1c: PI with Filter Control

(cont.)

Solving for the classical feedback controller c = q1pq leads to a tuning rule

for an PI with filtercontroller:

c(s) = Kc

1 + 1

Is

1

(Fs + 1)

Kc =

KI =

F =

In IMC design, the presence of a Left-Half Plane zero in the model

leads a low-pass filter element in the classical feedback controller!

Example 2: PID Control

Consider now the second-order model with RHP zero:

p(s) =K(s + 1)

(1s + 1)(2s + 1), 1, 2 > 0 (7)


Step 1: Use the IAE-optimal factorization for step inputs:

p+ = (s + 1) p =K

(1s + 1)(2s + 1)(8)

q =(1s + 1)(2s + 1)

K(9)

Step 2: Use a first-order filter (even though this means that q will still be

improper).

f =1

(s + 1)q =

(1s + 1)(2s + 1)

K(s + 1)(10)

Example 2: PID Control (Continued)

Solving for the classical feedback controller c = q1pq leads to a tuning rule

for an ideal PID controller:

c(s) = Kc(1 +1

Is+ Ds) (11)

Kc =1 + 2

K(+ )(12)

I = 1 + 2 (13)

D =12

1 + 2 (14)

This PID tuning rule can be rearranged to conform to the inter-

active form

c(s) = Kc(1 +1

Is)(1 + Ds) (15)

Example 3: PID with Filter Control

Consider a second-order model with RHP zero

p(s) =K(s + 1)

(1s + 1)(2s + 1), 1, 2 > 0 (16)

and subject to step inputs to the closed-loop system. Applying the IMCdesign procedure gives:

Step 1: Use the ISE-optimal factorization

p+ = s + 1s + 1 p = K(s + 1)(1s + 1)(2s + 1)

(17)

Step 2: A first-order filter leads to q which is semiproper:

q =(1s + 1)(2s + 1)

K(s + 1)(s + 1)f =

1

s + 1(18)


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Example 3: PID with Filter Control(Continued)

Solving for c(s) as before results in a filtered ideal PID controller

c = Kc(1 +1

Is+ Ds)

1

(Fs + 1)

with the associated tuning rule

Kc =(1 + 2)

K(2+ )(19)

I = 1 + 2 (20)

D =

121 + 2 (21)

F =

2+ (22)

Note the insight given by IMC design procedure regarding on-lineadjustment (by changing the value for the IMC filter parameter ).

Example 4: Deadtime Compensation (PI + Smith Predictor)Consider the first-order with delay plant

p(s) =Kes

s + 1(23)

and step setpoint/output disturbance changes to the closed-loop system.

Step 1: The optimal factorization (IAE, ISE, or otherwise) is p+ = es,

resulting in:

q = p1 =s + 1

K

Step 2: A first-order filter makes q semiproper;

q =s + 1

K(s + 1) =

es

(s + 1)(24)

The corresponding feedback controller is

c(s) =s + 1

K(s + 1 es)(25)

which can be expressed as a PI controller using the Smith Predictor struc-

ture (see Figure 17.4, page 605 in Ogunnaike and Ray).

Plants With Integrator

For plants with integrator, the practical problem will most likely demand

no offset for ramp output disturbances (d = As2

). This requires a Type-2filter meeting the requirement

d

ds(p+f) =

d

ds()|s=0 = 0

Plant = pq= p+f Controller c(s) No Offset ConditionsK(s+1)

ss+1

s+1 P Steps onlyK(s+1)

s

(s+1)(s+1)(s+1)

P with filter Steps OnlyK(s+1)

s(s+1)[(+2)s+1]

(s+1)2PI Steps and Ramps

K(s+1)s

(s+1)s+1)

[2(+)s+1](s+1)2

PI with filter Steps and RampsK(s+1)

s(s+1)(s+1)

s+1)[2(+)s+1]

(s+1)2PID with filter Steps and Ramps

Note the progression in controller sophistication as closed-loop per-

formance requirements increase!

One such filter transfer function which meets the condition dds

()|s=0 = 0

is

f(s) =(2 p+(0))s + 1

(s + 1)2(26)

Specific forms for p+(0) for various simple factorizations of nonminimum

phase elements are shown below:

d

ds(es)|s=0 = (27)

d

ds(s + 1)|s=0 = (28)

d

ds(s + 1

s + 1)|s=0 = 2 (29)

Equation (26) for simple plants with integrator leads to IMC-PID rules,

as summarized in the previous table.

Plants with Integrator (Cont.)


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IMC-PID Tuning Rules for First-Order With DeadtimeProcesses

Controller KKc I D F

Recommended/

(>0.2 always )

PI 2+2

+2

0 0 >1.7

PID 2+2+

+2

2+0 >0.8

PID w/filter

2+2(+)

+2

2+

2(+)>0.25

Rivera, Morari, Skogestad, "Internal Model Control. 4. PID Controller Design," Ind.Eng. Chem. Proc. Des. Dev., Vol 25, No. 1, 1986, pgs. 252-265.

p(s) = K e-s

s + 1, is an adjustable parameter

Ideal Form ParametersWe can obtain a PID tuning rule for plants with deadtime by using a

first-order Pade approximation in lieu of the time delay.

p =Kes

s + 1

K(2s + 1)

(2s + 1)(s + 1)

(26)

The Pade-approximated plant is a se cond-order plant with RHP zero;

using the analysis from Example 2: PID Control leads to a PID

tuning rule:

Kc =2+

K(2 + )(27)

I = +

2(28)

D =

2+ (29)

As shown in Rivera et al. the ISE objective function for this control

system can be plotted as a function of

, independent of.

IMC-PID Tuning - FODT Plant

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

IMC-PID Tuning - Effect of Filter

Parameter on Closed-loop Response

Controlled Variable Response

Solid: / = 0.8

Dashed: / = 0.4

Dotted: / = 2.5

IMC-PID Tuning Performance Chart

J = ISE for a unit step input

Jopt = ISE for "optimal" controller (perfect Smith Predictor)

M = Maximum peak, sensitivity function


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IMC-PID Tuning - Comparison With Other Rules

J = ISE for a unit step input

Jopt = ISE for "optimal" controller

(perfect Smith Predictor)

O.L. Z-N: Open-Loop Ziegler-Nichols; C.L. Z-N: Closed-loop Ziegler-Nichols,

C-C: Cohen-Coon

"Original" IMC-PI tuning

The Improved PI rule arises by incorporating the delay in the timeconstant of the internal model p

p =Kes

s + 1

K

( + 2)s + 1(30)

resulting, as shown in Example 1: PI control in the tuning rule:

Kc =

2 +

2K (31)

I = +

2(32)

For the Improved PI rules, closed-loop performance varies as a function of/.

"Improved" IMC-PI TuningImproved IMC-PI Tuning - Worst-Case

Analysis (over all Theta/Tau)


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"Improved" IMC-PI Tuning Comparison(lambda/theta = 1.7)

Solid: J/Jopt Dashed: M

C.L. Z-N: Closed-loop Ziegler-Nichols, C-C: Cohen-Coon

IMC-PID with Filter

The tuning rule for a PID with filter controller can be obtained from the

Pade-approxim ated plant and t he analysi s of Example 3: PID with

Filter Control,

Kc =2+

2K( + )(37)

I = +

2(38)

D =

2+ (39)

F = 2( + )

(40)

The IMC-PID with filter tuning parameters lead to higher ISE than the

IMC-PID for the same value of/; however, the PID with filter settings

display much smoother closed-loop responses. In industrial practice, the

smoothness of the response may well be worth the loss of performance in

terms of ISE.

IMC-PID with Filter PerformanceIMC-PID with filter (solid) versus IMC-PID (dashed) response

(lambda/theta = 0.45 vs lambda/theta = 0.8)

metodika nastavovania pid-imc2

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