Model Predictive Control

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CAB4 523 – Multivariable Process Control 114/03/2011

Single variable

Model Predictive Control

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Objectives

End of the chapter, you should be able to

• Explain the concept of

• Model predictive control

• Internal model control

• Smith predictor (dead time

compensator)

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Introduction

• Most modifications to single loop feedback

control presented so far have used additional

measurements to improve control

performance

• An alternative to the PID algorithm is provided.

• The most remarkable feature of PID is the

success of this single algorithm in so many

different applications

• The development of PID lacked a fundamental

structure from which the algorithm could be

derived, limitations could be identified, and

enhancements could be developed

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Introduction

• A general development is presented that gives great

insight into the roles of both the control algorithm

and the process in the behavior of feedback systems

• It also provides a method for tailoring the feedback

control algorithm to each specific application

• Since the model of the process is an integral part of

the control algorithm, the controller equation

structure depends on the process model

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Introduction

• Although the control algorithm is different, the

feedback concept is unchanged, and the selection

criteria for manipulated and controlled variables are

the same

• The algorithms could be used as replacements for

the PID controller in nearly all applications

• PID controller algorithm is considered the standard

algorithm

• Alternative algorithm is selected only when it

provides better performance

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Predictive Control Structure

• The derivation of control algorithms is

based on the predictive control structure

• Although many methods are possible for

deriving practical control algorithms, only

two of these will be dealt with

– Internal Model Control

– Smith Predictor

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Model Predictive Control Structure

• Consider the typical thought process used by a

human operator implementing a feedback control

manually

• The approach used by the operator has three

important characteristics:

– It uses a model of the process to determine the

proper adjustment to the manipulated variable ,

because the future behavior of the controlled

variable can be predicted from the values of the

manipulated variable

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– The important feedback information is the

difference between the predicted model

response and the actual process response. If

this difference is zero, the control would be

perfect, and no further correction would be

needed

– This feedback approach can result in the

controlled variable approaching its set point after

several iterations, even with modest model

errors

• These characteristics provide the basis for the

predictive control structure

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• A continuous version of the approach can be

automated with the general predictive control

structure

)(sGd

)(sGp

)(sGm

)(sGcp

D(s)

Tp(s)SP(s) CV(s)

MV(s)

Em(s)

++

+

+

-

-

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• The variable Em is equal to the effect of the disturbance,

Gd(s)D(s), if the model is perfect (Gm(s)=Gp(s))

• The structure highlights the disturbance for feedback

correction

• However, the model is essentially never exact

• The feedback signal includes the effect of the

disturbance and the model error, or mismatch

• The feedback signal can be considered as a model

correction

• It is used to correct the set point so as to provide a better

target value, Tp(s), to the predictive control algorithm

• The controller calculates the value of the manipulated

variable based on the corrected target

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• The closed-loop transfer functions for the

setpoint and disturbances are:

)()()(1

)()(

)()()()()(1

)()()(

)(SP

)(CV

sGsGsG

sGsG

sGsGsGsGsG

sGsGsG

s

s

mpcp

pcp

mspvcp

pvcp

)()()(1

)()()(1

)()()()()(1

)()()(1

)(

)(CV

sGsGsG

sGsGsG

sGsGsGsGsG

sGsGsG

sD

s

mpcp

dmcp

mspvcp

dmcp

(1)

(2)

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• The controller algorithm, Gcp(s), for the predictive structure

is to be determined to give good dynamic performance

• Let us determine a few properties of the predictive

structure that establish important general features of its

performance and give guidance for designing the

controller

• A very important control performance objective is to

ensure that the controlled variable returns to its set point

in steady state

• This objective can be evaluated from the closed-loop

transfer functions by applying the final value theorem and

determining whether the final value of the controlled

variable, expressed as a deviation variable from the initial

set point, reaches the set point

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• The application of the final value theorem is

performed for the following conditions:

1. The input is steplike, in that it reaches a steady state

after a transient, SP(s) = ΔSP/s and D(s)= ΔD/s

2. The process without control reaches a steady state

after a steplike input, Gp(0) = Kp and Gm(0)=Km

3. The closed-loop system is stable, which can be

achieved via tuning

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• Under these conditions, application of the final

value theorem yields,

if and only if

SP

)0()0()0(1

)0()0(SP)(CV)(CV limlim

0 mpcp

pcp

st GGG

GG

sssst

)0()0( 1mcp GG

(3)

(4)

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0

)0()0()0(1

)0()0()0(1)(CV)(CV limlim

0 mpcp

dmcp

st GGG

GGG

s

Dssst

)0()0( 1mcp GGIf and only if

• The predictive control system will satisfy both of the

foregoing equations, thus providing zero steady-

state offset for a steplike input, if

)0()0( 1mcp GG or Kcp = 1/Km

(5)

(6)

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• The next control performance objective is

perfect control

• Perfect control means the controlled variable

never deviates from the set point

• CV(s)/D(s) = 0 and CV(s)/SP(s) = 1 provide the

basis for the following condition:

• Perfect control can be achieved if the controller

could be set equal to the inverse of the

process dynamic model

)()( 1 sGsG mcp (7)

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)(

)(

1)()(1

)()(

)()()(1

)()(

)(

)(MV

sG

sG

sGsG

sGsG

sGsGsG

sGsG

sD

s

p

d

pcp

cpd

mpcp

cpd

• The perfect control system must invert the true

process in some manner

• Block diagram algebra can be applied to derive

the following condition for the behavior of the

manipulated variable

(8)

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• The following are four reasons why an exact

inverse of the process is not possible:

– Dead time

– Numerator dynamics

– Constraints

– Model mismatch

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Dead time

• In most physical processes, the feedback transfer function includes dead time in the numerator

• Application of equations (7) and (8) to a typical process model with dead time gives, when the model is factored into two terms

• The perfect controller would have to include the ability to use future information in determining the current manipulated variable – not physically realizable

s

mm esgsG )()(

s

mmcp esgsGsG11

)()()(

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Numerator Dynamics

• Some process models have dynamic elements in

the numerators of feedback transfer function

• Application of IMC equation to an example gives

• The controller would not be able to provide perfect

control when 2 < 0

2

1

2

1

1)(

s

sKsGm

1

11)()(

2

2

11

s

s

KsGsG mcp

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Constraints

• The manipulated variable must observe constraints

• There is no guarantee that the controller would

observe constraints

• Thus, in some cases, values of the manipulated

variables that are required to achieve perfect

control performance would not be possible

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Model mismatch

• The model used in the predictive system will almost

certainly be different from the true process

• If the difference is large, the closed-loop system

could become unstable, a situation that precludes

acceptable control performance

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Model Predictive Control

• Since the perfect controller is not possible, a

manner for deriving an approximate inverse of the

model is required

• The approximate inverse is the Gcp(s) that contains

important features for control performance

• Many methods exist for developing approximate

inverse

• Each method would result in a different control

algorithm giving different control performance

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IMC Controller

• Based on Brosilow (1979) and Garcia and Morari

(1983)

• Since an exact inverse is not possible, the IMC

approach segregates and eliminates the aspects of

the model transfer function that make the

calculation of realizable inverse impossible

• The first step is to factor the model into the product

of two factors

)()()( sGsGsG mmm

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IMC Controller

• - The noninvertible part has an inverse that is

not causal or is unstable

• The steady state gain of this term must be 1.0

• - The invertible part has an inverse that is

causal and stable, leading to realizable, stable

controller

• The IMC Controller (idealized)

• This design ensures the controller is realizable and

the system is internally stable

)(sGm

)(sGm

1)()(

~sGsG mcp

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IMC Controller

• Example 1: Apply the IMC procedure to design a

controller for a process described by

3)15(

039.0)(

ssGm

3)15(

039.0)(

ssGm

0.1)(sGm

039.0

)15()()(

31 s

sGsG mcp

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IMC Controller

Drawbacks of the design:

• The controller involves first, second and third order

derivatives of the feedback signal

• These derivatives can not be calculated exactly,

although they can be estimated numerically

• Appearance of higher-order derivatives of a noisy

signal could lead to unacceptable control

• High derivatives can lead to extreme sensitivity to

model errors

• The controller can not be used without modification

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IMC Controller

• All realistic processes are modeled by transfer functions having a denominator order greater than the numerator order

• Thus, the IMC controller, the inverse of the process model, will have a numerator order greater than denominator

• Results in first- or higher-order derivatives in the controller that lead to unacceptable manipulated variable behavior, and, thus, poor performance and poor robustness when model errors occur

• Achieving good control performance requires modification that modulates the manipulated variable behavior and increase the robustness of the system

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IMC Controller

• All realistic processes are modeled by transfer functions having a denominator order greater than the numerator order

• Thus, the IMC controller, the inverse of the process model, will have a numerator order greater than denominator

• Results in first- or higher-order derivatives in the controller that lead to unacceptable manipulated variable behavior, and, thus, poor performance and poor robustness when model errors occur

• Achieving good control performance requires modification that modulates the manipulated variable behavior and increase the robustness of the system

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IMC Controller

• A filter of the feedback signal is used

• The filter is placed before the controller as shown in

fig.

)(sGd

)(sGp

)(sGm

)(sGcp

D(s)

Tp(s)

SP(s) CV(s)

MV(s)

Em(s)

++

+

+

-

-

)(sG f

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IMC Controller

• To make the controller proper or semiproper, add a

filter to make the controller proper

• For tracking setpoint changes,

• Adjust the filter-tuning parameter to vary speed of the

response of the closed-loop system

• : small --- response is fast

large --- the closed loop response is more robust

(insensitive to model error)

)()()(~

)(1

sGsGsGsG fmcpcp

nfs

sG1

1)(

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IMC Controller

1

15.10

039.0

1)()(

s

ssGsG cpf

• The modified IMC Controller for the example

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Example 2

Design an IMC controller using the alternative first-

order-plus-dead-time approximate model for the

process

)15.10(

039.0)(

5.5

s

esG

s

m

)15.10(

039.0)(

ssGm

s

m esG 5.5)(

039.0

)15.10()()(

~ 1 ssGsG mcp

• The controller is proportional-derivative, which

still might be too aggressive but can be modified

to give acceptable performance

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Example 2

• To make the controller semiproper,

• Filter tuning parameter is adjusted to provide the

required performance

1039.0

)15.10()(

s

ssGcp

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Example 3

• Consider the following transfer function:

• This system has a RHP zero and will exhibit

inverse response characteristics

• An all-pass factorization of the model is to be used

13115

)16()(

ss

ssGm

)115)(115(

16)(

ss

ssGm

)16(

16)(

s

ssGm

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Example 3

• An idealized controller would be:

• Add the filter to make the controller semi-proper

)1)(16(

)13)(115()(

ss

sssGcp

)16(

)13)(115()()(

~ 1

s

sssGsG mcp

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IMC-Based PID Controllers

• Although the IMC procedure is clear and easily

implemented, the most common industrial controller

is still the PID controller

• The IMC block diagram can be rearranged to form

the standard feedback control diagram

• The IMC law is equivalent to PID-type feedback

controller for a number of common process transfer

functions


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IMC-Based PID Controllers


)(sGd

)(sGp

)(sGm

)(sGcp

D(s)

SP(s) CV(s)

MV(s)

++

++

+-

The inner loop is given by:

)()(1

)()(

sGsG

sGsG

cpm

cp

c: IMC-based PID

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IMC-Based PID Controller Design

Procedure

1. Find the IMC controller transfer function, Gcp(s),

which includes a filter, Gf(s),

– to make Gcp(s) semiproper, or

– the order of numerator of Gcp(s) is one order greater than

the denominator of Gcp(s) (to give derivative action)

– *Major difference from IMC procedure

2. Find the equivalent standard feedback controller

using the transformation:

Write this in the form of a ratio between two

polynomials

3. Show this in PID form and find


)()(1

)()(

sGsG

sGsG

cpm

cp

c

DIcK and ,

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IMC-Based PID Controller Design for

First-Order Process

• First order process:

• Find the equivalent standard feedback controller:


1)(

s

ksG

p

p

m

1

11)(

1

11)()()(

1

s

s

ksG

sk

ssGsGsG

p

p

cp

p

p

mfcp

sk

s

sk

s

s

k

sk

s

sGsG

sGsG

p

p

p

p

p

p

p

p

cpm

cp

c

1

)1(

1

11

)1(

1

)()(1

)()(

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IMC-Based PID Controller Design


• The IMC-based PID design procedure for a first-

order process has resulted in a PI control law


skskksG

pp

p

pp

p

c

11

1)(

pI

p

p

ck

sK )(

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First-Order Process with Time Delay

• First order process:

• Use a first-order Pade approximation

• The idealized controller is


1)(

s

eksG

p

s

p

m

15.0

15.0

s

se s

15.01

15.0)(

ss

sksG

p

p

m

15.01)(

ss

ksG

p

p

m 15.0)( ssGm

p

p

mcpk

sssGsG

15.01)()(

~ 1

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• Note: The numerator order is one degree higher

than the denominator to realize a PID controller



1

115.01)()()(

1

sk

sssGsGsG

p

p

mfcp

)()(~

)(1

)()(~

)()(1

)()(

sGsGsG

sGsG

sGsG

sGsG

fcpm

fcp

cpm

cp

c

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ssk

s

ss

k

s

ss

k

sGsG

sGsG

sGsGsGsG

sGsGsG

p

p

pp

p

pp

p

p

p

fm

fcp

fmmm

fcp

c

25.0

11

5.0

5.0

5.0

15.05.01

5.0

15.011

)()(1

)()(~

)()()()(1

)()(~

)(

2

1

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The Smith Predictor

• Time delay compensation that deals with occurrence

of significant dead time

)(sGd

)(sGp

)(sGm

)(sGc

D(s)

SP(s) Y(s)

MV(s)

++

+

+

-

-

)(sGm

+

-

)(sGcp

E E

2

~Y1

~Y

P

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The Smith Predictor

• The process model is split into two parts:

– The part without a time delay,

– The time delay term,

• The model of the process without time delay is

used to predict the effect of control actions on the

undelayed output

• The controller uses the predicted undelayed

response to calculate its output signal

• The predicted delayed output is compared with

the actual process output (delayed response)

• This step corrects for modeling errors and for

disturbances entering the process

mGs

m eG

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The Smith Predictor

• If the process model is perfect and there are no disturbances,

• Then

• For this ideal case, the controller responds to the error signal that would occur if no time delay were present

121

~)

~(

~YYYYYEE sp

YY2

~

1

~YYE sp

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The Smith Predictor

• An alternative configuration for the Smith predictor

• Somewhat similar to that in cascade control

)(sGd

)(sGp)(sGc

D(s)

SP(s) Y(s)++

+

-

)1)(( s

m esG

+

-

)(sGcp

E E P

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The Smith Predictor

• Assuming there is no model error, the inner loop has the effective transfer function

• The closed-loop setpoint transfer function is

• The Smith predictor has the theoretical advantage of eliminating the time delay term in the characteristic equation

)1(1 s

mc

c

eGG

G

E

PG

mc

s

pc

sp GG

eGG

Y

Y

1

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The Smith Predictor through Direct

Synthesis approach

• The Direct Synthesis approach can be used to

derive a controller with time-delay compensation

• The controller design is based on a process model

and a desired closed-loop transfer function

• Do not always have PID structure, however it

produce PI or PID controllers for common process

models

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Synthesis approach

• Consider the block diagram of a feedback control

system

• The closed-loop transfer function for setpoint

changes is

mmmpv

c

c

mpvc

pvcm

sp

KGGGGG

GG

GG

GGGG

GGGK

Y

Y

assume and with

11

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Synthesis approach

• Rearranging and solving for Gc

• As Y/Ysp is not known a priori, the above design

cannot be used.

• Also, distinguish between the plant G and the

model

• The practical design equation

sp

sp

cYY

YY

GG

/1

/1

G~

dsp

dsp

cYY

YY

GG

/1

/~1

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Synthesis approach

• The selection of the desired closed-loop transfer

function is the key decision

• Note: the controller transfer function has the

inverse of

• Desired closed-loop transfer functions

G~

1

1

1

s

e

Y

Y

sY

Y

c

s

dsp

cdsp

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Synthesis approach

• By substituting in the controller design equation,

we get respectively,

• Approximating the time delay in the denominator

with a truncated Taylor series expansion

• Both are integral controllers, eliminate offset

s

c

s

c

c

c

es

e

GG

sGG

1~1

1~1

s

e

GG

c

s

c)(

~1

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Conclusions

• You have learnt the principles of

– Model Predictive Controller structure

– IMC Controller

– Smith Predictor

Documents

Model Predictive Control