Process Control and Optimization

Advance Process Control I

Carlos VelázquezPharmaceutical Engineering Research Laboratory

ERC on Structured Organic Particulate SystemsDepartment of Chemical Engineering

University of Puerto Rico at Mayaguez

Process Control and Optimization

• Control and optimization are terms that are many times erroneously interchanged.

• Control has to do with adjusting flow rates to maintain the controlled variables of the process at specified setpoints.

• Optimization chooses the values for key setpoints such that the process operates at the “best” economic conditions.

Review of Basic Control

Conservation Equations:Mass, Moles, or Energy Balances

Systemthewithinaction

byGenerationofRate

Systemthe

LeavingRate

Systemthe

EnteringRate

onAccumulati

ofRate

Re

Laplace Transforms and Transfer Functions

• Provide valuable insight into process dynamics and the dynamics of feedback systems.

• Provide a major portion of the terminology of the process control profession.

• Are NOT generally directly used in the practice of process control.

Method for Solving Linear ODE’s using Laplace Transforms

Laplace Domain

Time Domain

dy/dt = f(t,y)

sY(s) - y(0) =F(s,Y)

Y(s) = H(s)

y(t) = h(t)

Transfer Functions

• Defined as G(s) = Y(s)/U(s)

• Represents a normalized model of a process, i.e., can be used with any input.

• Y(s) and U(s) are both written in deviation variable form.

• The form of the transfer function indicates the dynamic behavior of the process.

Example of Derivation of a Transfer Function

)()()()()(

212211 tTFFtTFtTFdt

tdTM • Dynamic model of

CST thermal mixer• Apply equation to the

steady state• Substitute deviation

variables• Equation in terms of

deviation variables.

0220110 TTTTTTTTT

)()(

)()()(

21

2211

tTFF

tTFtTFdt

tTdM

)0()()0()0()0(

212211 TFFTFTFdt

dTM

Derivation of a Transfer Function

21

1

1 )(

)()(

FFsM

F

sT

sTsG

• Apply Laplace transform to each term considering that only inlet and outlet temperatures change.

• Determine the transfer function for the effect of inlet temperature changes on the outlet temperature.

• Note that the response is first order.

21

2211 )()()(

FFsM

sTFsTFsT

What if the Process Model is Nonlinear

• Before transforming to the deviation variables, linearize the nonlinear equation.

• Transform to the deviation variables.

• Apply Laplace transform to each term in the equation.

• Collect terms and form the desired transfer functions.

Use Taylor Series Expansion to Linearize a Nonlinear Equation

...)()()(0

00

xxdx

dyxxxyxy

• This expression provides a linear approximation of y(x) about x=x0.

• The closer x is to x0, the more accurate this equation will be.

• The more nonlinear that the original equation is, the less accurate this approximation will be.

General form of a Transfer Function

1

0

1

1)(

mp

k

sm

nLp

ss

esKsG

Dynamic Modeling Approach for Process Control Systems

Actuator Process Sensorc y y su

Dynamic Model for Sensors• These equations

assume that the sensors behave as a first order process.

• The dynamic behavior of the sensor is described by the time constant since the gain is unity

• T and L are the actual temperature and level.

sTs

s TTdt

dT

1

sLs

s LLdt

dL

1

Dynamic Model for Sensors• The units of KT

depend on the input to the sensor and the type of signal of the sensor.

• For instance, if input is pressure and the signal from the sensor is in mA, then KT = psig/mA

1)(

s

KsH

T

T

• input : concentration,

level, pressure, temperature, force, velocity

• Signals : mA, mV, %TO

Actuator System

• Control Valve– Valve body– Valve actuator

• I/P converter

• Instrument air system

Dynamic Model for Actuators

FFdt

dFspec

v

1

QQdt

dQspec

H

1

• These equations assume that the actuator behaves as a first order process.

• The dynamic behavior of the actuator is described by the time constant since the gain is unity

Dynamic Response of an Actuator (First Order Process)

0 2 4 6 8 10Time (seconds)

Fspec

F

Parameter Estimation

Estimation of Transfer Functions

• Factors involved– Sampling time– Signal-to-noise ratio– Input type

1) Sampling time (T)

• If proper sampling is not used, the data could be corrupted by too much noise (fast sampling) or could lack of enough dynamic data (too slow sampling)

Figure 3.6 Effect of a double pulse and T= 10.0 in the SSE surface of a FO model

K

p

p

Figure 3.7 Effect of a double pulse and T= 100.0 in the SSE surface of a FO model

Kp

p

2) Signal-to-Noise ratio

• Measures how big is the signal of the measured variable compared to the signal of the noise

Figure 3.14 Effect of a double pulse and a S/N = 10:1 in the SSE surface of a FO model, T = 1.0

Kp

p

Figure 3.15 Effect of a double pulse and a S/N = 1:1 in the SSE surface of a FO model, T =1.0

Kp

p

3) Input wave form

• Virtually any type of input can be used. The amplitude of the signal is the key factor for an input to be proper.

0

1

step pulse

0

1

0

1

doublet

-1

0

1

sinusoidal

-1

0

1

PRBS

-1

Figure 3.1 Effect of a step change in the SSE surface of a FO model

Kp

p

Figure 3.2 Effect of a pulse in the SSE surface of a FO model

Kp

p

Figure 3.3 Effect of a double pulse in the SSE surface of a FO model

Kp

p

Figure 3.4 Effect of sinusoid in the SSE surface of a FO model

Kp

p

Figure 3.5 Effect of PRBS in the SSE surface of a FO model

Kp

p

Feedback Controllers

• Direct acting

• Reverse acting

t

DI

c dt

teddtteteKctc

00

)()(

1)()(

t

DI

c dt

teddtteteKctc

00

)()(

1)()(

Position Form of the PID Algorithm

Definition of Terms

• e(t)- the error from setpoint [e(t)=ysp-ys].

• Kc- the controller gain is a tuning parameter and largely determines the controller aggressiveness.

• I- the reset time is a tuning parameter and determines the amount of integral action.

• D- the derivative time is a tuning parameter and determines the amount of derivative action.

Properties of Proportional Action

11

1

)(

)(

)()( 0

sKK

KK

KK

sY

sY

teKctc

pc

p

pc

pc

sp

c

• Closed loop transfer function base on P-only control applied to a first order process.

• Properties of P control– Does not change order of process

– Closed loop time constant is smaller than open loop p

– Does not eliminate offset.

Offset Resulting from P-only Control

Time

Setpoint1.0

1

2

3

0

Offset

Properties of Integral Action

• Based on first order process

• Properties of I control– Offset is eliminated

– Increases the order by 1

– As integral action is increased, the process becomes faster, but at the expense of more sustained oscillations

pcp

I

pc

pII

pc

I

pc

pIsp

t

I

c

KK

KK

sKK

sKK

sY

sY

dtteK

ctc

2

1

1

1

)(

)(

)()(

2

00

Integral Action for the Response of a PI Controller

Time

ys

ysp

Controller Tuning

• Ziegler-Nichols

• Integral Criteria

• Internal Model Control

• Frequency Techniques

Stability

• Substitution

• Root Locus

• Frequency techniques

Stability of the Control Loop

• For a feedback control loop to be stable, all the roots of its characteristic equation must be either negative real numbers or complex numbers with negative real parts.

Stability of a Controlled System

• Method 1: Direct Substitution– Select a P-Only controller.

– Write the characteristic equation in a polynomial form of s.

– Substitute s=wui and Kc=Kcu

– Solve for Kcu and wu

01)()()()( sGsGsGsG cTpa

Example

skg

C

ssGP /130

50)(

C

TO

ssGT

%

110

1)(

CO

skg

ssGv %

/

13

016.0)(

TO

CKsG cc %

%)(

Exchanger Sensor/Transmitter

Valve P-Only controller

Solution

0)()()()(1 sGsGsGsG cTPv

013

016.0

130

50

110

11

cK

sss

08.0143420900 23 cKsss

08.0143420900 ,23 ucuuu Kiwiwiw

Characteristicequation

SubstitutingcorrespondingTFs

Rearranging

Substituting s=iwu, and Kc=Kcu

Solution

;%

%8.23/2186.0

;%

%25.10

043900

080.01420

;004390080.01420

3

2

32

TO

COKsradwFor

TO

COKwFor

K

iiK

cuu

cuu

uu

cuu

uucuu

ww

w

www

The real and imaginary parts have to be each one equal to zero to make the equation equal to zero.

Solution

• To determine the correct range, first we need to determine the sign of the multiplication of the gain of the process, sensor, and actuator.

• If KpKTKV is (+) then Kc must be +.

• If KpKTKV is (-) then Kc must be -.

• In this case, Kc must be positive, hence Kc = 23.8 is the only physical sound solution

Feedback advance strategies

Cascade

• Main purpose: Reject disturbances that affect an intermediate controlled variable before it hit the main controlled variable.

• Results: Improve performance in rejecting some process disturbances.

Cascade control scheme

Cascade diagram

Gv

GL2

GM1

GM2

Gc2Gc1Km Gp

GL1

+

-

+

-

+

+

+

+

C1C2PE2

R2E1R1Ysp

L2 L1

Cascade design considerations

)()()()()()()()(1

)()(

)(

)(

11222

2

2

1

sGsGsGsGsGsGsGsG

sGsG

sL

sC

MpccvMcv

Lp

)()()()()()()()(1

)()()(1)(

)(

)(

11222

221

1

1

sGsGsGsGsGsGsGsG

sGsGsGsG

sL

sC

MpccvMcv

McvL

)()()()()()()()(1

)()()()(

)(

)(

11222

1121

sGsGsGsGsGsGsGsG

KsGsGsGsG

sY

sC

MpccvMcv

Mpccv

sp

Comparison for load change

Feedforward Controller Design Based on Dynamic Models

• Objective: Measure important load variables and take corrective action before they upset the process.

Feed-forward

• Main purpose: Reject disturbances that would affect directly the main controlled variable before it hit the process.

• Results: Improved performance in rejecting disturbances compared to simple feedback.

Feedforward Controller Design Based on Dynamic Models

Transfer Function

• Relates the process variable to the disturbance (load)

• Permits the design of the feedforward controller.• Stability of the closed loop system.

2517,1)(

)(

mpvc

pvftL

GGGG

GGGGG

sL

sC

Ideal Feedforward Transfer Function

• If the set point is constant, C(s) should be equal to 0 despite the change in L(s).

• The ideal transfer function could not be physically realizable.

pvt

Lf

pvftL

GGG

GG

GGGGG

0

Time-Delay Compensation

• Time delay occurs because of the presence of distance velocity lags, recycle loops, and dead time from composition analysis.

• Limits performance of a conventional feedback control system by adding phase lag.

• Controller gain must be reduced below the value that could be used if dead time were not present, hence sluggish response would be obtained compared to that of no dead time

Block Diagram of Smith Predictor

Block Diagram of Smith Predictor

Set Point TrackingC

ontr

olle

r ou

tput

Con

trol

ler

outp

ut 70

60

50

40

70

60

50

40

Pro

cess

var

iabl

e/se

t poi

ntP

roce

ss v

aria

ble/

set p

oint

60

55

50

60

55

50

Smith Predictor - PI Comparison

Relationships from the block diagram

)~

(~~

')1918( 211 CCCRCEE

11~~

')2018( CRCEE

Assuming ideal model is perfect and disturbance is zero

)1(1)2118( '

sc

cc eGG

G

E

PG

For the inner loop,

For the closed loop servo transfer function,

GG

eGG

R

C

c

sc

1)2218(

For the closed loop servo transfer function,conventional feedback

sc

sc

eGG

eGG

R

C

1)2318(

Control of Multiple-Input, Multiple-Output Processes

Examples

• One input affects two or more outputs, and one output is affected by two or more inputs.

Compact Representation of MIMO Processes

)()()()()(

)()()()()(

2221212

2121111

sMsGpsMsGpsC

sMsGpsMsGpsC

)(

)()(,)(,

)(

)()(

)()()(

2

1

22

12

21

11

2

1

sM

sMs

G

G

G

Gs

sC

sCs

where

sss

p

p

p

pMGpC

MGpC

Block Diagram

Block Diagram

19.2 Pairing of controlled and manipulated variables

• To determine the best pair of variables to be used within the control strategy.

• Method: Bristol’s Relative gain Array– A measure of process interactions– A recommendation concerning the most

effective pairing of controlled and manipulated variable.

Concept of relative gain

• M means all the M are kept constant except Mj.

• C means all the C are kept constant except Ci.

gainloopclosed

gainloopopen

M

C

M

C

Cj

i

Mj

i

ji

Example

)()()(

)()()(

2221212

2121111

sMKsMKsC

sMKsMKsC

?

2

2

1

1

111

1

C

M

M

C

KM

C

Example

)()(

)()(

0)(

122

2112111

122

212

2

sMK

KKKsC

sMK

KsM

sC

21122211

221111

22

21122211

1

1

2

KKKK

KK

K

KKKK

M

C

C

Example

?

1

1

2

1

122

1

C

M

M

C

KM

C

)()(

)()(

0)(

221

2211121

221

221

2

sMK

KKKsC

sMK

KsM

sC

Example

21122211

211221

2111 1

KKKK

KK

21122211

211212

21

22112112

2

1

2

KKKK

KK

K

KKKK

M

C

C

Example

21122211

2211

21122211

2112

21122211

2112

21122211

2211

KKKK

KK

KKKK

KK

KKKK

KK

KKKK

KK

Summary• Ci should be paired with Mj

• Ci should be paired with Mk=/j

• Indicates interaction between loops, worst interaction =.5

• Too much interaction, closed loop gain is reduced by closing the other control loops

• Gains have different sign, pairing these two variables causes severe interaction

0

1

10

0

1

ij

ij

ij

ij

ij

Example 19.3

110

21

5.1

1

5.1110

2

)(

s

s

s

ssGp

K11

K21

K12

K22

21122211

2211

21122211

2112

21122211

2112

21122211

2211

KKKK

KK

KKKK

KK

KKKK

KK

KKKK

KK

Example 19.3

5.15.122

225.15.122

5.15.1

5.15.122

5.15.15.15.122

22

28.2

28.1

28.1

28.2

19.4 Decoupling Control Systems

• Benefits– Interactions are eliminated and the stability of

the closed-loop system is determined by the stability characteristics of the individual control loops.

– A set point change for one controlled variable has no effect on the other controlled variable.

Decoupling diagram

Ideal Controller

0)()()()( 2221121 sMsGsMsG pp

)()(0)(

)()()(

21222

21222

sMsMsM

but

sMsMsM

0)()()()()(

)()()(

1121221121

112121

sMsDsGsMsG

sMsDsM

Since

pp

)(

)()(

22

2121 sG

sGsD

p

p

Ideal ControllerIn an analogous fashion

• Careful with unrealizable controllers, specially with transfer functions containing delay times

)(

)()(

11

1212 sG

sGsD

p

p

Controller Design by Direct Synthesis


• Assume the process + measurement device + final control element can be represented by

• Assume also that the closed loop behavior follows an ideal

y(s) ggc

1 ggc

yd (s)

y(s) q(s)yd (s)


• where

• The choice of the reference trajectory q(s) depends on the desired closed loop response and the open loop response of the process.

• Next table presents some typical responses

q(s) ggc

1 ggc



• The key question is what if the form of gc(s) that produces the desired response q(s) given the g(s) of the process.

• This can be achieving manipulating the expression of q(s) to obtain

gc (s) 1

g(s)

q(s)

1 q(s)


• Assume a first order model for g(s) and an underdamped second order for the reference trajectory.

• Therefore, gc(s) is

gc (s) s1

k

1

r2s2 2 r rs1

1 1

r2s2 2 r rs1


• After some algebra

gc (s) s1

k

1

r2s2 2 r rs

Model Predictive Control

MPC best suited for

• MIMO processes with significant interaction between SISO loops

• Either equal or unequal number of inputs

• Complex and usually problematic dynamics (longtime delays, and inverse response, or unusually large time constant)

• Constraints in inputs and/or outputs

Basic Elements of MPC

1. Trajectory

2. Output Prediction

3. Control Action Sequence

4. Error Prediction

Ogunaike

MPC Architecture

PredictionControl

Calculations

Set-point Calculations

Process

Model +-Model

Output

Residuals

Inputs

InputsPredicted

Output

Set points

Prediction models• Finite convolution models

– Impulse-Response

– Step-Response

where

y(k) g(i)u(k i)i0

k

y(k) (i)u(k i)i0

k

g(i) (i) (1)

and (i) g( j)j1

i

Prediction models• Discrete State-Space

• Discrete Transfer function

where A(z-1) and B(z-1) are polynomials in z

y(k) a(i)y(k i)i0

k

b(i)u(k i m)i0

k

y(k) z mB(z 1)

A(z 1)u(z)

Dynamic Matrix Control

Unconstrained Problem Solution

Unconstrained Problem Solution

Unconstrained Problem solution



• In DMC, it is assumed that there exist disturbances unaccounted for by the model.

• It is further assumed that the best estimate of the values of these disturbances in the future of the elements– w(k+i), i=1,2, …, p of the vector w(k+1) is

estimated as

Model Algorithmic Control

• Similar to DMC with difference in the coefficients of the model and the reference trajectory– Process representation

– u is the actual input

– Output prediction

ˆ y (k 1) ˆ y 0(k)Gu(k) w(k 1)

ˆ y *(k 1) ˆ y 0(k)Gu(k) w(k 1)

Model Algorithmic Control

• In MAC, ŷ*(k+1) is defined typically as a reference trajectory based on the current measured value, y(k), and the current set point ysetpoint(k) as follows

with subsequent values determined according to

y*(k 1) y(k) (1 )ysetpoint(k); 0 < < 1

y*(k i) y(k i 1) (1 )ysetpoint (k); i > 1

Application of MPC to batchfluid bed drying

Modeling and Optimal Control of a FBDProcess Understanding: Drying theory

114

Drying CurveDrying Curve

Preliminaries Results

Dr. Velazquez, PhERL

4.5

5.5

6.5

7.5

8.5

9.5

10.5

1 2 3 4 5 6 7 8 9 10 11Sample number

% M

C

AF 40 T60

AF 50 T60

AF 60 T60

AF*60 T60

AF 70 T60

AF*80 T60

115

Moisture Content Curve Modeling

)(11.0

45.0)(

11.0

03.0)( sT

ssQ

ssWp

60m³/hr ; 60°C

4

5

6

7

8

9

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

t (hr)

Wp

Exp

Modelo en Laplace

AF1 Ave

Modelo en tiempo

116

In-Line NIR Measurements & Optimization

• Experimental design: 18 experimental combinations.

• Factors studied: bulk mass, air flow, and probe axial position.

• Response variable: residual (static – in-line NIR spectra).

• Region: drying equilibrium point (5 - 6 % H2O).

• Constant air’s inlet temperature = 70 °C

(same as in the calibration) 117

Axial Position Comparison

Analysis of Variance

118

Results

119

120

Results

121

Results

Rewriting of Transfer Function to fit MPC algorithm

• Can be transformed to describe the drying curve parting from any given point of the curve knowing the starting time and the current time of the process. Equation below describes the step wise function

Rewriting of Transfer Function to fit MPC algorithm

• The low order transfer equation proposed can be turned into the matrix form described by

Rewriting of Transfer

Function to fit MPC

algorithm

Closed loop of FBD

NIR

DCS-MPC

HXBlower

AirFlow

AirTemperature

MoistureMeasurement

125

FBD performance under MPC

126


127

128


Robustness test for error in modeling

Dr. Velazquez, PhERL 129

Comparison of performance for base and comparison cases

Comparison of performance for base and comparison cases

Conclusions of example• The drying curve can be described accurately with a

first order transfer function.

• The position of the sensor probe is critical for meaningful monitoring data.

• The MPC algorithm demonstrated to be capable of handling the model mismatch.

• The MPC application would minimize the energy consumption when compare to the open loop case. In the comparison case, the minimization decreased compare to the base case.

Process Control and Optimization

Documents