MATLAB &Simulink Sample Model Description...MATLAB®&Simulink® Sample Model Description ―...

MATLAB®&Simulink® Sample Model Description ― Simulation of Cascade PID Control and Model Predictive Control for a gas

separation plant by membrane separation ―

UpperController

PID

PID+－

+－

InnerLoop

Outer Loop

InnerLoop

InnerLoop

Simulation of Cascade PID Control and Model Predictive Control for a gas separation plant by membrane separation

1. Introduction 3

2. About Plant 5

2.1. Overall Structure of Plant and Control System 5

2.2. Gas Separator 7

2.3. About Membrane Separator 8

3. Introducing Mathematical Model 8

3.1. Introduction of Mathematical Model of Permeable Juxtamembrane 9

3.2. Introduction of Mathematical Model for Gas Separator 11

3.3. Mathematical-Based Simulink Model 14

4. System Identification 14

4.1. Introduction 14

4.2. Identification of Inner Loop 15

4.3. Identification of Outer Loop 18

5. Control System Design 28


5.2. Inner Control System Design ~ PID Control 30

5.3. Upper Control System Design ~ Cascade PID Control 33

5.4. Outer Control System Design ~ Model Predictive Control 35

5.5. Computing Unit 37

1

6. Multivariable Model Predictive Control in MATLAB[3] 38

6.1. The Model Predictive Control (MPC) Toolbox and the MPC Blockset 38

6.2. Control Structure of Model Predictive Control 39

6.3. Operation of Model Predictive Control 41

6.4. Linear Model of Plant 41

6.5. Optimization Problem 42

6.6. Cost Function of Model Predictive Control 44

7. Evaluation of Control System 46


7.2. Simulation Model of Cascade PID Control 46

7.3. Simulation Model of Model Predictive Control 46

7.4. Comparison of Cascade PID Control and Model Predictive Control 47

8. Conclusion 51

9. References 52

10. Exemption from Responsibility 52

11. Author 52

2

1. Introduction This example demonstrates simulation of membrane separation control using

MATLAB®&Simulink® where a particular gas is extracted from mixed gas consisting of more than

2 components. The process assumes oxygen enrichment, air as mixed gas, and oxygen as an

extracted gas. The controlled object (plant) is a gas separator. The gas separator has functionality

which extracts only a particular gas through permeable membrane. This process produces specific

gas as a product. It controls pressure of gas separator so that concentration level of the particular gas

tracks given set points. “Plant” indicates simulation model rather than actual unit in this document.

Figure 1 shows flow of the control system described in this example. This figure is a typical flow of

control system design. Plant models are modeled mathematically in a plant modeling. The main

reason why plant model is created is that parametric model is identified from plant model, and

control system is designed based on parametric model. Control system design requires either

parametric or non-parametric models. Parametric model is identified from transient response data in

system identification. In control system design, PID control is designed for inner control system, and

cascade PID control system and Model Predictive Control are designed for outer control system

based on the parametric model. Performance evaluation of the designed control system is performed

through simulations in the evaluation of the control system.

Plant Modeling

System Identification

Control Design PID Control , Cascade PID Control

Model Predictive Control

START

END

Evaluation

Figure 1. Flowchart of Control System Design

A schematic diagram of gas separator is shown in figure 2. Gas separator separates oxygen from air

by utilizing pressure differences. Here, gas separator is assumed to be connected at tri-level. Gas

3

extraction method is permeable membrane method utilizing pressure differences. Gas separator has

permeable membrane in the middle. Permeable membrane is assumed to be more permeable with

oxygen than other gas. The upper portion is where gas is to be flown into, and the lower portion is

where gas is permeated through permeable membrane which separates them. By creating pressure

differences between the two, oxygen concentration level becomes higher in the lower portion. Gas in

the lower portion flows into the next gas separator. By repeating this operation, oxygen

concentration becomes higher after going through each gas separator. Gas extracted after permeation

from the third gas separator is the final product. Positive pressure differences can be created by

controlling gas inlet flow and outlet flow, which results in variance of pressures in the upper and

lower portions.

Permeation Emission gas

Material Emissiongas

MaterialPart

Agitator

PermeationPart

Low Pressure

MaterialPart

Agitator

Flow gas

Low Pressure

MaterialPart

Agitator

High Pressure

PermeationPart

High Pressure

High Pressure

(A)

(A)

(A)

(A)

(A)

(A)

(B)

(B)

(B)

(C)

(D)

(C)

(C)

(C)

(C)

(C)

(D)

(D)

(D)

(D)

(D)

NotePressuresensor

InhalationControlValve

Pump Manipulatedvariable switcher (refer 5.2.)

(A) Densitysensor

(B)

(C) Emission gas ControlValve

(D)

Material Inhalation gas

PermeationInhalation gas

Low Pressure

Selectivelypermeable membrane



PermeationPart

Figure 2. Schematic Diagram of Gas Separator

Figure 3 shows structure of the control system. This plant controls concentration and pressure. The

upper control system (outer loop) deals with concentration control, while the lower control system

(inner loop) deals with pressure control. The outer and inner controls work in combination to control.

In concentration control, pressure differences are computed for targets for the material and

permeation parts so that emission gas concentration of permeation part of each gas separator tracks

4

the given set point. Pressure differences are allocated appropriately to set point of pressure in

material and permeation parts by Computing Unit (Refer 5.5.). In pressure control, emission and

inhalation quantities are controlled so that pressure of permeation and material parts tracks set point

given to concentration control. Emission and inhalation quantities vary depending on opening of

each control valve. Computing Unit has functionality to switch between emission and inhalation

whether manipulated value of PID controller that controls whether pressure is positive or negative.

In order to perform this sample, MATLAB Version 7.0.1, Simulink 6.1 and Model Predictive

Control Toolbox Version 2.1 are required.

OuterControllertype1:Cascade PIDtype2:MPC

PIDcontroller

ComputingUnit

SV: Seｔpoint - PressureMV: Manipulated Variable - Emission or InhalationVolume of airCV: Controlled Variable - Pressure

SV1

MV

Pressure sensorControl Valve

CV

PIDcontroller

SV2 MV

CV

Control Valve

+ －

+－

Gas Separator2

Gas Separator3

Same as above

Same as above

MV2

MV3

Inner Loop

Pressuresensor

Density Sensor

Inner LoopOuter LoopSV: Setpoint - Oxygen DensityMV: Manipulated Variable - Pressure DifferenceCV: Controlled Variable - Oxygen Density

SV

MV1

PV

Material Part

PermeationPart

Gas Separator1

Outer Loop

Figure 3. Control System Structure ~Inner Loop and Outer Loop

2. About Plant 2.1. Overall Structure of Plant and Control System

Figure 4 shows overall structure of plant and control system. The plant consists of three-level gas

separators. The control system consists of the outer and inner control systems. The outer control

system and Computing Unit assume a program in calculator module of control system that monitors

and controls this plant. PID controller assumes one loop controller or PID control module. The

Computing Unit assumes sequence logic within PID control module. PID controller of the inner

control system has three PID controllers attached to material part of each gas separator unit, and

three controllers for permeation part. The Computing Unit receives pressure difference as set point

from the outer control system, and allocates them to pressure set point of each PID controller.

5

YAQ1OxygenDensity

Inlet Flow GasVolume F1Density XAF1

GasSeparator1

Outer Controllertype1:Cascade PIDtype2:MPC

ComputingUnit

Set point of Oxygen density

Note

Sensor

ControlValve

Pump

U

L

Control valve of inner control

L

***_SV U

Manipulatedvariable switcher (Refer 5.2.)

PID

MaterialPart

＋

－

＋

－

PtW1_SV

PID

PtW1Pressure CV

PtQ1Pressure CV

U

L

L

L

L

L

L

YAQ2Oxygen

Density

＋

－ PID

L

L

LL

L

YAQ3 OxygenDensity

＋

－ PID

L

L

L

L

L

L

PID＋

－L

PID＋

－

L

PtW2_SV

U

PtW1_SV

UU

U

U

GasSeparator2

GasSeparator3

PermeationPart

Controlled variable of inner control (Pressure)

Controlled variable of outer control (Density)

Manipulated variable of outer control=Set point of inner control

PtQ3Pressure CV

PtQ2Pressure CV PtW3

Pressure CV

PtW2Pressure CV

Figure 4. Overall Structure of Plant and Control System

6

2.2. Gas Separator

Gas separator assumes as follows (Refer to figure 5).

Gas separator shape is rectangular. Gas assumed to be air that flows into inlet line of gas separator.

Permeable membrane is placed in the middle of gas separator to separate gases. Oxygen in the gas

can penetrate permeable membrane, but nitrogen can not. By crating pressure differences between

material and permeation parts, membrane permeation flow of oxygen in the gas occurs from material

to permeation part. Nitrogen is assumed to be significantly less permeable than oxygen. This makes

mol fraction of oxygen of permeation part higher. Agitator makes gas concentration uniform by

agitating inside. There are pressure sensors to material and permeation parts. Control valves are

attached to each inhalation and emission line to control pressures of material and permeation parts.

Flow of inhalation and emission are controlled by control valve opening. When inhalation and

emission flows change, pressures at both material and permeation parts change. To make it simple,

assume that inhalation and emission capabilities are constant. There is a concentration sensor

attached to permeation part to measure concentration of emission gas.

area

z

Selectivelypermeablemembrane

PressureSensor

h

Inlet Flow Gas

Inlet Line

area

Density Sensor

Agitator

Control Valve

Pressure SensorEmission

Inhalation

Pump

Material Part

Permeation Part

Inhalation

Emission

Figure 5. Gas Separator

7

Numerical specification of gas separator is described in table 1.

Table 1. Numerical Specification of Gas Separator

Specification Value Unit Symbol

Cross-section area 5 m2 Area

Permeable membrane cross-section area 5 m2 Area

Cubic volume of material and

permeation parts

25 m3 volume ( = h*area)

Height of material and permeation parts 5 m h

Mass transfer coefficient (oxygen) 0.00001 mol・m-2・s-1・㎩-1 －

Mass transfer coefficient (nitrogen) 0.000001 mol・m-2・s-1・㎩-1 －

2.3. About Membrane Separator

Permeable membrane is barrier diffusion that selectively permeates according to material type.

Concentration difference, pressure difference and difference in potential are used to drive mass

transfer (energy). Practical examples included oxygen enrichment, water purification and salt

manufacturing. [1] Among such examples, gas separation is described in table 2.

Table 2. Description of gas separation [1]

Permeable material Separation method

Driving force Residue material

Separation mechanism

Practical example

Easy-to-permeate gaseous molecule

Gas separation

Concentration difference (partial pressure difference)

Difficult-to-permeate gaseous molecule

Solution-diffusion mechanism, Knudsen diffusion

Oxygen enrichment, Hydrogen separation, Helium separation, Uranium concentration (gas diffusion method)

3. Introducing Mathematical Model This chapter describes physical environment controlling gas separator, permeable juxtamembrane,

and entire gas separator. Although the actual unit is considered to be quite complex, for the purpose

of modeling, the following assumptions are made:

<Hypothesis>

(1) Gas is ideal gas.

(2) Gas is air. Air consists of two components. Components are component A (oxygen) and

component B (nitrogen).

(3) Mol fraction of component A in the air is 0.2, and that of component B is 0.8.

(4) Ignore friction with gas and container wall and piping, and pressure loss.

(5) Gas is at normal temperature and constant (20 degrees, Celsius).

(6) Gas component distribution within gas separator is uniform.

8

(7) Mass transfer coefficients of component A and B are constant and do not vary depending on

conditions such as temperature and pressure.

(8) No chemical change of gas.

(9) Ignore diffusion term of fluid.

(10) Dynamics of gas separator is governed by mass balance equation and ideal gas equation.

(11) There is no back flow of gas between gas separator 1, 2 and 3.

(12) Ignore dynamics within piping.

(13) Gas emission and inhalation solely depend on valve opening, and pump does not have any

impact on them.

(14) Pressure in initial state at time 0 within gas separator equals to air pressure.

(15) Mol fraction in initial state at time 0 within gas separator equals to that of air.

3.1. Introduction of Mathematical Model of Permeable Juxtamembrane

The three governing rules of permeable juxtamembrane are as follows. To make it generalized, they

are formulated as component A and B.

<Governing Rules>

(1) Permeable membrane flow rate of component within gas is proportionate to partial pressure

difference of components inside and outside of membrane.

(2) Positive permeable membrane flow rate occurs from higher partial pressure side to lower side.

(3) Component A is easy to penetrate the permeable membrane, whereas component B is not.

Where, permeable membrane flow rates NA of gas component A, and NB of gas component B are

described in the following formula[1] .

)()( yPxPKPPKN ,, AitQAitWmAiAQiAWmAA −=−= (3-1)

)()( yPxPKPPKN ,, BitQBitWmAiBQiBWmBB −=−= (3-2)

（KmA >> KmB）

9

δ[m]

(W)

(Q)

PAW:Partial pressure of gas component A

PAQ,i

PAW,i

PAQ:Partial pressure of gas component A

NA:Permeablemembrane flow rate of component A

NB:Permeable membrane flowrate of component B

PｔW:Total pressure of gas

PｔQ:Total pressure of gas

Material Part

PermeationPart

Selectivelypermeablemembrane

Figure 6. Diagram of permeable juxtamembrane

[Symbol description]

NA: Permeable membrane flow rate of component A [mol・m-2・s-1]

NB: Permeable membrane flow rate of component B [mol・m-2・s-1]

KmA: Mass transfer coefficient of partial pressure standard within membrane of component A [mol・

m-2・s-1・㎩-1]

KmB: Mass transfer coefficient of partial pressure standard within membrane of component B [mol・

m-2・s-1・㎩-1]

PAW,i: Partial pressure of gas component A [㎩] in the material part. Subscript i means surface of

separation, and this applies the following.

PBW,i: Partial pressure of gas component B [㎩] in the material part

PAQ,i: Partial pressure of gas component A [㎩] in the permeation part

PBQ,i: Partial pressure of gas component B [㎩] in the permeation part

PtW: Total pressure of gas in the material part [㎩]

PtQ: Total pressure of gas in the permeation part [㎩]

xAi: Mol fraction of component A in the material part

yAi: Mol fraction of component A in the permeation part

PmA: Permeable membrane coefficient [mol・m-1・s-1・Pa-1]

10

Note: Although modeling is not provided using this model, mass transfer coefficient is inversely

proportional to film thickness, the following equation holds true using permeable membrane

coefficient.

δ/mAmA PK = (3-3)

3.2. Introduction of Mathematical Model for Gas Separator

Dynamics of gas is assumed to be controlled by material balance equation and ideal gas equation.

Material balance equation determines number of mol of total gas, component A (oxygen) and

component B (nitrogen) within gas separator plant. Ideal gas equation determines total pressure, and

partial pressures of component A and B within gas separator plant.

Material balance equation, pressure and mol fraction are described as follows:

F[mol・s-1]

XAF:Mol fraction of component A in F[-]

Selectivelypermeablemembrane（Area[m3]）

XAW: mol fraction [-]

NB:Permeablemembrane flowrate of

component A[mol・m-2・s-1]

Agitator

W:Emission gas quantity from the material partper unit time [mol・s-1]

YAQ: mol fraction [-]

Q:Emission gas quantityfrom the permeation partper unit time [mol・s-1]

Material Part

PermeationPart

Emission

Inhalation

Emission

Inhalation

NA:Permeablemembrane flowrate of

component A[mol・m-2・s-1]

Figure 7. Diagram of gas separator

[Symbol description]

N: Number of mol in gas

F: material gas quantity flown into the material part per unit time [mol・s-1]

xAF: Mol fraction of component A in F [-]

W: Emission gas quantity from the material part per unit time [mol・s-1]

x: mol fraction [-]

y: mol fraction [-]

11

Q: Emission gas quantity from the permeation part per unit time [mol・s-1]

Area: membrane area [m2]

t: time [s]

P: pressure [Pa]

R: Gas constant 8.314[J・mol-1・K-1]

V: Volume in the material part (permeation part) [m3]

[Meaning of subscript]

|t=0: Value at time t=0

tW: All gases in the material part

tQ: All gases in the permeation part

AW: Gas component A in the material part

AQ: Gas component A in the permeation part

(Material balance equation of all gases in the material part)

Changes in number of mol of all gases in the material part per unit time can be expressed by the

following equation.

AreaNAreaNWFdt

dNBA

tW ** −−−= (3-4)

Considering integral from time 0 to t, it can be expressed by the following equation. This is the

number of mol in the material part at time t.

00)**( == +−−−= ∫ ttW

t

BAtttW NdtAreaNAreaNWFN (3-5)

(Material balance equation of gas component A in the material part)

Changes in number of mol of gas component A in the material part per unit time can be expressed by

the following equation.

AreaNXWXFdt

dNAAWAF

AW ⋅−⋅−⋅= (3-6)


number of mol of gas component A in the material part at time t.

00)( == +⋅−⋅−⋅= ∫ tAW

t

AAWAFttAW NdtAreaNXWXFN (3-7)

(Pressure of all gases in the material part)

It is assumed that the ideal gas equation is approved. P=nRT/V, instead of PV=nRT

VTRNP tWtW /⋅⋅= (3-8)

12

(Mol fraction in the material part)

WAWAW NNX /= (3-9)

(Material balance equation of all gases in the permeation part)

Change of mol number of all gases in the permeation part per unit time can be expressed by the

following equation.

QAreaNAreaNdt

dNBA

tQ −+= ** (3-10)


number of mol in the permeation part at time t.

00)**( == +−+= ∫ ttQ

t

BAtttQ NdtQAreaNAreaNN (3-11)

(Material balance equation of gas component A in the permeation part)

This can be expressed by the following equation in the same way:

AQAAQ YQAreaN

dtdN

⋅−⋅= (3-12)

00)( == +⋅−⋅= ∫ tAQ

t

AQAttAQ NdtYQAreaNN (3-13)

(Pressure of all gases in the permeation part)

It is assumed that the ideal gas equation is approved. P=nRT/V, instead of PV=nRT

VTRNP tQtQ /⋅⋅= (3-14)

(Mol fraction in the permeation part)

WAQAQ NNX /= (3-15)

13

3.3. Mathematical-Based Simulink Model

The Simulink model in figure 8 can be obtained by expressing equation (3-1) to (3-15) in block

diagram in Simulink.

Figure 8. Gas separator model in Simulink

4. System Identification 4.1. Introduction

In designing control system, parametric model or non-parametric model of controlled object (plant)

is necessary. Then, parametric model is acquired by employing system identification method to

Simulink model. As control system design requires both outer and inner loops, system identification

is performed for outer and inner loops. Some of the purposes of model acquisition by system

identification of each loop are described in table 3 (refer to figure 3 for outer and inner loops).

Table 3. Purposes of model acquisition by system identification of the outer and inner loops

Purpose

Inner loop To construct inner loop which can track rapidly change of pressure set points that

the outer loop outputs.

Outer loop To improve precision of model used by outer loop in order to enhance control

performance of the outer loop.

14

4.2. Identification of Inner Loop

In this example, model is acquired by linearization and model order reduction for the inner loop

system identification.

PID parameter tuning is performed for the acquired model in the “Control System Design” of the

next chapter.

Figure 9 summarizes purposes of each operation, procedures and tools used in this example for inner

loop system identification (including PID parameter tuning).

Simulink+SimulinkControl Design

Closed-loop structure of the PID control is cut, to make it open loop structure, and linearization.

Linearization of non linear plant model

SimulinkDynamic simulation by SimulinkEvaluation control performanceby above PID parameter calculated by Ziegler-Nichols Method

Control System Toolbox

（１） pole-zero cancellation of transfer function（２） Evaluation by pole-zero map（３） Evaluation by step response in time domain

Model order reduction of high order plantto apply PID parameter tuning method (Ziegler-Nichols Method)

ToolsProcedurePurpose

Linearization of plant

Tuning of PID Parameters

Evaluation

START

END

Model order reduction of plant

Simulation

OK

Not OK

Figure 9. Purposes of each operation, procedures and tools used in this example

15

Figure 10 shows model used for system identification of the inner loop and PID control parameter

tuning.

Signal Builder Block

Gas Separator1 Gas Separator2 Gas Separator3

Permeation partPID controller1

Material PartPID controller1





Data storing part Data display part

Figure 10. Model used in system identification of the inner loop and tuning of PID control parameter

The Simulink Control Design is used for linearization in Simulink. It is possible to specify input and

output points and linearize input and output points in the Simulink Control Design. Although the

inner loop is closed loop structure in figure 10, linearization as a closed loop is possible without

disconnecting closed loop structure (refer to figure 11). GUI of the Simulink Control Design is

shown in figure 11. Linearization results are displayed on GUI such as step response, bode diagram,

impulse response, Nichols diagram, Nyquist diagram, pole-zero map and singular value map and so

on. Figure 11 shows step response in parallel before and after linearization.

16

PlantPIDController

+

－

×Set Value

Input point

Simplify

×：It means to cut closed loop.

GUI of the Simulink Control Design

Step response of a model which is applied model order reduction by pole-zerocancellation with linearizedtransfer function(Green)

Step response of linearized transfer function(Blue)

Example.Permeation part of Gas separator2

Output point

Input point Output point

Figure 11. Simulink Control Design

Based on the linearization results from the Simulink Control Design, it is assumed that model

response between input and output points that are order reduced in all loops can be linear

approximation to the some hundredth seconds order, it is considered to be integrated system for

simplicity. Linearized transfer functions of all inner loops are summarized in table 4.

Table 4. Transfer function with input and output points linearized

Input and output points of the inner loop Transfer Function

Gas separator 1 material part pressure(y)－material part valve opening (u) -0.0008/s

Gas separator 1 permeation part pressure (y)－permeation part valve opening

(u)

-0.0009/s

Gas separator 2 material part pressure (y)－material part valve opening (u) -0.0009/s

Gas separator 2 permeation part pressure (y)－permeation part valve opening

(u)

-0.0008/s

Gas separator 3 material part pressure(y)－material part valve opening (u) -0.0008/s

Gas separator 3 permeation part pressure(y)－permeation part valve opening

(u)

-0.0008/s

17

4.3. Identification of Outer Loop

System identification in the outer loop models impact that pressure differences between the material

and permeation parts of each gas separator unit 1, 2 and 3 have on oxygen concentration in the

permeation part of gas separator 1, 2 and 3. Open loop response between control quantity 1, 2 and 3,

and manipulated variable 1, 2 and 3 in figure 12 are identified. Although set point of pressure

tracking control of material and permeation parts of each plant is performed for the inner loop, the

inner loop shows very high speed set point tracking compared to outer loop dynamics, therefore it is

assumed to be included in low speed dynamics of the outer loop.

Plant (including inner loop)

Outer controlsystem

Gas separator 2Set point of pressuredifference between material part and permeation part

Gas separator 3Oxygen concentrationof permeation part



MV1

MV2

MV3

CV1

CV2

DV1

DV2Inlet flow gasdensity

CV3

MV: Manipulated VariableCV: Controlled VariableDV: Disturbance



Inlet flow gas quantity

Figure 12. Manipulated variable, controlled variable, and disturbance of outer loop

Simulate the model in figure 13, and log impact of changes of each manipulated variable on

responses of each controlled variable as time series data. Extract necessary parts from time series

data, and identify extracted data.

18

Signal Builder Block

Gas Separator1 Gas Separator2 Gas Separator3







Data storing part

Data display part

Figure 13. Simulink model for outer system identification

Major parts of this model have no significant differences from the model in figure 10, except for the

contents of the Signal Builder Block which create signal for system identification. The settings of the

Signal Builder Block in figure 13 are shown in figure 14.

19

Gas separator2Set point of pressure in material partGas separator2Set point of pressure in permeation partGas separator3Set point of pressurein material partGas separator3Set point of pressure in permeation part

Inlet flow gasquantity

Inlet flow gasdensity

Gas separator1 Set point of pressurein permeation part

Gas separator1Set point of pressure in material part

MV2

MV3

DV1

DV2

Signal Name

MV1MV1

MV2

MV3

DV1

DV2

Figure 14. Setup Signal for the Signal Builder Block

Table 5 describes what symbols used in Simulink model of this example mean, including figure 14.

20

Table 5. Meaning of symbols in Simulink model

Name Contents Symbol

Pressure set point in the material part of plant 1 PTW1_SV Manipulated

variable 1 (outer) Pressure set point in the permeation part of plant 1 PTQ1_SV





Gas flow quantity of plant 1 F1 Disturbance

Oxygen concentration mol fraction of gas flow quantity

of plant 1 XAF1

Controlled

variable 1 (outer)

Oxygen concentration mol fraction in the permeation

part of plant 1 YAQ1

Controlled

variable 2 (outer)



Controlled

variable 3 (outer)




part of plant 1 YAQ1_SV



Set point (outer)



Figure 15 show each manipulated variable and open loop response of each controlled variable

obtained from simulating the model in figure 13.

21

MV1

MV2

MV3

DV1

DV2

CV3

CV2

Oxygendensity

Pressuredifference

CV1Impact of DV1Impact of MV1 Impact of MV3 Impact of DV2Impact of MV2

Figure 15. Open loop response to each manipulated variable

Figure 16 shows system identification flow of the outer control system using this simulation data.

Items in the flow are described below.

(1) Sampling of simulation data

In this example, system identification is performed from data acquired in simulation not real plant,

the right path is relevant. When acquiring real plant data, the left path is relevant.

(2) Extraction of data

Each manipulated value is considered as step input signal. Extract data should cover proximity of

steady state so that plant transient response is adequately included.

(3) Trimming of abnormal data

Remove essentially impossible data due to noise and so on, so that model should not be impacted by

abnormal data. Abnormal data does not exist such as noise impact in this simulation.

(4) Removal of bias (direct-current component) from data

Remove low frequency disturbance such as bias (direct-current component) which is undesirable for

identification.

22

Removalof bias

4. Removal of bias(direct-current component)

from data

5. System Identification

START

END

Ex. Transient response by step input

Steady state

3. Trimming of abnormal data

1.Acquiring real plat data

1.Sampling of simulation data

2. Extraction of data

Figure 16. System identification flow of the outer control system

(5) System Identification

System Identification Toolbox and Simulink Parameter Estimation are MATLAB&Simulink tools

used for system identification. It is considered that model is approximated by transfer function of

first order delay + dead time system from step input response characteristics in figure 15.

)exp(1

)( LsTs

KsG −+

= (4-1)

K: gain, T: time constant, L: dead time

or

Transfer function of second order delay system

22

2

2)(

nn

n

ssK

sGωζω

ω++

= (4-2)

K: gain, ζ: damping ratio, ωn: eigen frequency

Model identification is possible using either tool, method using the Simulink Parameter Estimation is

introduced in this guide. Simulink Parameter Estimation estimates parameter within Simulink model

based on optimization. It is necessary to provide appropriate initial value to estimation value in the

23

same way as general optimization in order to derive highly precise estimate with less number of

estimates.

In this simulation, non-linearity of model, that has time constant varying a few hundred times in

vertical direction of step input, can be observed from figure 15, and although

Upward time constant of step input < Downward time constant of step input

applies, derive upward model of step input.

For example, from simulation results, relationship between manipulated value 2 and control value 2

is shown in figure 17. From figure 17, transfer function is considered to be approximated as first

order delay + dead time system. Initial values of each parameter are gain=0.25(=magnitude of

manipulated value 2(0.1)/magnitude of manipulated value 2(0.4)), time constant=250, and dead time

L=30.

Figure 17. Step response of manipulated variable 2 and controlled variable 2

(1) and (2) in figure 18 are Simulink models used for parameter estimation. (3) in figure 19 is GUI

for the Simulink Parameter Estimation. (4) is graph of evaluation function with GUI (square sum

error). X-axis denotes number of estimation, and y-axis denotes value of evaluation function. (5) is a

graph indicating calculated values from models of estimation data and estimation parameter with

GUI. (4) and (5) are updated dynamically every time estimation is performed. Estimation finishes

when conditions set as optimization option is satisfied. Conditions include tolerance of evaluation

function, maximum estimation number, etc.

24

(1)：first order delay+dead time system

(2)：Second order delaysystem

Figure 18. Models used for first order delay + dead time system, parameter estimation of second

order delay

25

Cost Function

Measureddata (gray)

Simulated data (blue)

Measured and Simulated

GUI of Simulink Parameter Estimation

Iterations

(3)

(4)

(5)

Figure 19. Simulink Parameter Estimation

If 0<ζ<1 for Second order delay system, overview of step response include vibrations as in figure 20,

and the relationship between parameter of second order system ζ, and parameter: time constant T of

first order system ωn is assumed to be T= 1/ ςωn. ζ=0.6 is assumed in figure 20.

Display simulation data in graph, derive initial value of ζ、ωn by rough estimation. Relatively good

initial values can be provided by determining initial value referencing figure 21.

26

nωπ2

11+Ts

n

Tξω

1=

Second order delay system

First order delay system

Figure 20. Relationship between first order delay system and second order delay system (0＜ζ＜1) (Figure in [4] is partially modified.）

1.02 22

2

=++

n

nn

n

ssω

ωξωω

Figure 21. Graph of second order delay system (0＜ζ≦2)

27

From the above, transfer function matrix of the derived model is shown in (4-3). Step response of the

model is shown in figure 22.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

−

)(3)(2)(1

1 + 234.2s0.2884

10*1.23 + s 0.0071 + s10*3.641

10*3.4489 + 0.0032ss10*9.2372

01 + 205.5s

0.2462 1 + 736.7s

0.2348

001 + s 313.2

0.182

)(3)(2)(1

)41(5-2

6-

6-2

7-

)8.43(

)6.3(

sUsUsU

e

e

e

sYsYsY

s

s

s

(4-3)

CV1

MV2 MV3MV1

CV2

CV3

Figure 22. Step Response

5. Control System Design 5.1. Introduction

This example outer and inner control systems work together. As this is multi-inputs and

multi-outputs, control system is multi-variable control. Outer and inner control systems are designed

separately in control system design. Cascade PID control and multi-variable Model Predictive

28

Control is designed and compared as multi-variable control.

Figure 23 shows why multi-variables Model Predictive Control is adopted. Model Predictive Control

is control rules that are theoretically valid for plant with long time constant and dead time. PID

control design for multi-input multi-output system generally is becoming more complex. Model

Predictive Control is a control rules that optimized control is possible where stress put on plant is

treated as constraints.

The gas separators in the back and forth influence each other because it connects.

incidenceInterference

It is not an optimizing control of the entire plant. The stress given to the plant is not evaluated enough.

There is no norm of control.

Optimality

Cascade PID Control

Description

Structure of control

A control system is closed in a gas separator.It only controls Set Value tracking.

EpexegesisItem

Improvement of tracking Set Value

Entire optimization of multi variable control system

Improvement item by MPC installation

Expecting effect by MPC installation

・Running cost reduction・Stabilization of control・Reduction of stress to plant

Figure 23. Why multivariable Model Predictive Control is adopted

29

Table 6 summarizes outer and inner control systems.

Table 6. Outer control system and inner control system

Control method Function Set point, controlled variable, manipulated

variable

Cascade PID control

(position type)

To track set point of oxygen

concentration in the

permeation part of each gas

separator.

Set point: oxygen concentration in emission gas

in the permeation part

Controlled variable: oxygen concentration in

emission gas in the permeation part

Manipulated variable: pressure difference

between the material and permeation parts

Outer control

system

Multivariable Model

Predictive Control

Same as above Same as above

Inner control

system

PID control (position

type)

To control inhalation and

emission control valve

opening and follow set points

of oxygen concentration in

the material and permeation

parts of each gas separator.

Set point: pressures in the material and

permeation parts of gas separator

Controlled variable: pressures in the material and

permeation parts of gas separator

Manipulated variable: inhalation (emission)

5.2. Inner Control System Design ~ PID Control

Inner control system is assumed to be loop control by PID controller. Manipulated variable switcher

switches between emission and inhalation, depending on sign of manipulated variables, W and Q

that PID controllers of loop1 and loop2 output.

The function of manipulated variable switcher is assumed as follows.

In the loop 1, total pressure, PtW of pressure sensor is treated as CV (Controlled Variable) , and W

(emission gas flow in the material part) as MV (Manipulated variable). If W is positive, emission gas

value is positive, and emission value control valve is controlled. Inhalation control valve then is fully

closed. If W is negative, emission gas value is negative, and inhalation value control valve is

controlled. Emission value control valve then is fully closed. To match sign of inhalation value and

valve opening, if W is negative, it is absolute value of W.

In the loop 2, total pressure, PtQ of pressure sensor is treated as CV (Controlled Variable), Q

(emission gas flow in the permeation part) as MV(manipulated variable). If Q is positive, emission

gas value is positive, and emission value control valve is controlled. Inhalation value control valve

then is fully closed. If Q is negative, emission gas value is negative, and inhalation value control

valve is controlled. Emission value control valve then is fully closed. Switching between emission

and inhalation is not modeled in Simulink model.

30

In the loop 1, when W (emission gas flow in the material part) increases, total pressure, PtW

decreases, then proportional gain of PID controller is negative. In the loop 2, when Q (emission gas

flow in the permeation part) increases, total pressure, PtQ decreases, then proportional gain of

controller is negative. In the loop 1 and 2, set points are supplied by the outer control system.

F: Inlet gas quantity [mol・s-1]

W: Emission gas quantity of material part [mol・s-1]

Material part

Permeation part

Gas separator

Inhalationcontrol valve

Emissioncontrolvalve

Total PressurePtQ

PressureSensor

Total PressurePtW

loop1

loop2

PIDcontroller

－Outer ControlSystem

＋

－＋

Set point2

Emissioncontrolvalve

DensitySensor

Sign of MVof PID Controller

0

1


Setpoint

0

1PID

controller

AbsoluteABS

ManipulatedVariableSwitcher

ABS

PressureSensor

Set point1

Inhalationcontrol valve

ManipulatedVariableSwitcherSign of MV

of PID Controller

Q: Emission gas quantity of permeation part [mol・s-1]

Absolute

Figure 24. PID controller attached to gas separator

In this model, structure of PID control adds rate limits, upper and lower limits to outputs

(manipulated variable) of basic position PID controller (see figure 25). Structure of PID controller is

common for both outer loop and inner loop. Upper and lower limits, and rate limits of all PID

controller are shown in table 7.

Table 7. Upper and lower limits and upper and lower rate limits of PID Controller

Parameter Name Value [%]

MV upper limit 20

MV lower limit -20

MV upper rate limit 10

MV lower rate limit -10

31

PID Controller Plant

Inside of subsystem

ProportionalGain

Integral Time

Derivative Time

AppropriateDifferential Operator

Integrator

Upper and lower limit

Ratelimiter

Figure 25. Structure of PID controller

PID control parameter is derived using Ziegler-Nichols Method. An example of PID control

parameter using Ziegler-Nichols Method is displayed in table 8. Additionally, results applying

tuning visually from simulation results are shown in table 9.

Table 8. Example of PID control parameters using Ziegler-Nichols Method [2]

Control law Proportional gain Kp Integral time Ti Derivative time Td

PI Control 0.45*(gain of stability limit) 0.833*(frequency of

stability limit)

－

Table 9. PID Control Parameter of inner loop

Input and output points Transfer

Function

Proportional

gain

Integral time Derivative

time

Gas separator 1 material part

pressure (y)－material part valve

opening (u)

-0.0008/s -20 67 －

Gas Separator 1 permeation part

pressure (y)－ permeation part

valve opening (u)

-0.0009/s -17 67 －

Gas Separator 2 material part

pressure (y)－material part valve

opening (u)

-0.0009/s -20 67 －

Gas Separator 2 pressure in the

permeation part (y)－valve

-0.0008/s -20 67 －

32

opening in the permeation part (u)

Gas Separator 3 material part (y)

－material part valve opening (u)

-0.0008/s -30 67 －

Gas Separator 3 permeation part

pressure (y)－permeation part

valve opening (u)

-0.0008/s -30 67 －

5.3. Upper Control System Design ~ Cascade PID Control

When designing multiple loop control using multi-variable control, it is important to determine

appropriately which manipulated variables control which controlled variables, and its combinations.

In this example, based on a basic theory that one manipulated variable controls one controlled

variable, manipulated variable 1－controlled variable 1, manipulated variable 2－controlled variable

2, manipulated variable 3－controlled variable 3.

For simplicity, where equation (4-3) of the outer system model is equation (5-1), Y2(s) is found to

be interfered by U1(s), and Y3(s) is interfered by U1(s) and U2(s). Decoupling can be used for

cancelling interferences, but in this example, each gas separator controls independently each PID

controller (outer) does not have decoupling functionality to avoid mutual interference.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

)(3)(2)(1

)()()(0)((s)G00)(

)(3)(2)(1

333231

2221

11

sUsUsU

sGsGsGsG

sG

sYsYsY

(5-1)

Figure 26 shows structure of cascade PID control. PID controller (outer) and PID controller (inner)

are double structure. The outer loop behavior is as follows:

PID controller (outer) outputs pressure difference SV (Set point) as manipulated variable so that

emission concentration in the permeation part of each gas separator is controlled to follows SV(Set

point). Computing unit(refer 5.5.) assigns pressure difference SV(Set point) that PID controller

(outer) outputs as SV(Set point) of pressure for the material and permeation parts of gas separator.

The inner loop behavior:

The inner loop controls emission and inhalation value to control to follow pressure SV (Set point) of

pressure provided from Computing Unit.

33

Outer loop

plant

PID controller(inner)+

－

1/2+

1.3

+

+

1.0+ +

－

+－

1/2+

1.3

+

+1.0

++

－

+

－

1/2

+

1.3

+

+

1.0

Base pressure [atm]

++

－

SV of pressure difference between material part and permeation partIn gas separator3

SV of pressurein permeation part of gas separator3

Computing unit

Inner loop

Y1

Y2

Y3

U1

U2

U3

Manipulatedvariable switcher

PID controller(inner)





PID controller(outer)+

－

PID controller(outer)

－

PID controller(outer)

－SV of densityin gas separator3






Base pressure[atm]

Base pressure [atm]

Base pressure[atm]

Base pressure [atm]

Base pressure [atm]

SV of densityin gas separator2

SV of densityin gas separator1

SV: Set pointCV: Controlled Variable

CV of densityin permeation part of gas separator3





SV of pressurein material part of gas separator1





Figure 26. Structure of Cascade PID Control

PID parameters of cascade PID controller are derived from CHR method.

If plant can be expressed as transfer function of first order delay + dead time system, an example of

CHR method parameters are shown in table10.

)exp(1

)( LsTs

KsG −+

=

K: gain, T： time constant, L： dead time

Table 10. Parameter examples of CHR method [2]

External input Overshoot Control law Proportional

gain Kp

Integral

time Ti

Derivative

time Td

Set point change None PI Control 0.35T/KL 1.17T －

34

The results obtained from applying CHR method to transfer function of the outer system are shown

in table 11.

Table 11. PID parameter of cascade PID control (host)

PID Parameter

Input and output points Transfer Function Proportional

gain

Integral

time

Derivative

time

Gas Separator

manipulated value 1 (y)

－control value 1 (u)

)6.3exp(12.313

182.0 ss

−+

167.3 366.4 －

Gas Separator



)8.43exp(15.205

2462.0 ss

−+

6.67 240.4 －

Gas Separator



)41exp(12.234

2884.0 ss

−+

6.93 274.0 －

Upper and lower limits, and rate limits of all cascade PID controller (outer) are shown in table 12.

Table 12. Upper and lower limits, and upper and lower rate limits

Value

MV upper limit 1

MV lower limit 0

MV upper rate limit 0.05

MV lower rate limit -0.05

5.4. Outer Control System Design ~ Model Predictive Control

Detailed structure of Model Predictive Control system is shown in figure 27. What differs from

cascade PID control is that cascade PID control is single-input single-output, and Model Predictive

Control system is multi-inputs and multi-outputs.

35

plant

+－

1/2+

1.3

+

+

1.0+ +

－

+－

1/2+

1.3

+

+1.0

++

－

+

－

1/2

+

1.3

+

+

1.0+

+－

Y1

Y2

Y3

U1

U2

U3

SV of densityin each gas separators







Manipulatedvariableswitcher

Manipulatedvariableswitcher











Model PredictiveController

Base pressure [atm]

Base pressure[atm]

Base pressure [atm]

Base pressure[atm]

Base pressure [atm]

Base pressure [atm]

Computing unit

Outer loop


Inner loop






SV: Set pointCV: Controlled Variable

Figure 27. Control structure of Model Predictive Control (Outer system) and PID control (Inner

system)

Design parameters for Model Predictive Control system and its design basis are shown in table 13.

See chapter 6 for details of Model Predictive Control in MATLAB.

Table 13. Design parameters for Model Predictive Control system and its design basis

Time related parameter Value Design basis

Control interval 60 [seconds] As time constant of plant ranges from approximately a

few hundred to kilo seconds order, prediction horizon

is set at 1200 seconds to include generally plant’s

dynamics. Control interval is set at 60 seconds as this

does not require high speed control.

Prediction horizon 20 [samples]

Control horizon 10 [samples]

For calculation speed-up, control horizon is set at half

of prediction horizon.

36

Variable

type

Variable Weight Rate

Weight

Design basis

U12 1 1

U34 1 1

Manipul

ated

variables U56 1 1

YAQ1 10

YAQ2 1000

Output

variables (Controlled

Variables)

YAQ3 10000

To make significantly bigger weight factor of

controlled variables than that of manipulated

variables. This evaluates highly set point tracking

of controlled variables within the evaluation

function. Manipulated variable can move as long

as it satisfies set point tracking. This calculates Δu

vector that places more importance on set point

tracking than manipulated variable stability.

Furthermore, place weight in set value tracking so

that set point of Y3 (gas concentration of final

product). Apply more loose constraints for

intermediate product (Y2) compared with Y3.

Intermediate product (Y1) is not required to track

setpoint.

Variable type Variable Constraints Design basis

U12 0 <= U12 <= 1

-0.05 <= U12/rate <= 0.05

U34 0 <= U34 <= 1

-0.05 <= U34/rate <= 0.05

U56 0 <= U56 <= 1

Manipulated

variables

-0.05 <= U56/rate <= 0.05

U1, U2 and U3 are operated within the

upper and lower limits.

U1, U2 and U3 are 1[atm] at

maximum, 0[atm] at minimum. This

means to define the upper and lower

limits of stress put on the plant. Rate

limit is to be strict in order to mitigate

stress put on the plant.

YAQ1 No constraints

YAQ2 No constraints

Output

variables

YAQ3 0 <= YAQ3 <= 1

Upper and lower limits are placed only

for the final product, YAQ3, and no

limits are placed for intermediate

product, YAQ1 and YAQ2.

5.5. Computing Unit

Functionalities of Computing Unit are described in figure 28. The Computing Unit allocates half of

the output from the outer control system to both the material and permeation parts. Half of the

37

manipulated variable is added to the base pressure in the material part, and half of the manipulated

variable is subtracted from the base pressure in the permeation part. The base pressure in the

material part is 1.3[atm], and 1.0[atm] in the permeation part. This is because it assumes the case

where outer control system is stopped and control is performed only by the inner control system. For

example, they are transient state such as launching and stopping plant, and malfunction of the outer

control system and maintenance.

Functionalities of Computing Unit are the same for cascade PID control and Model Predictive

Control.

Model PredictiveController output(Manipulated Value)

U

Gas separator

Permeationpart

Material part

Base pressure1.3[atm] +

+

Base pressure1.0[atm]

+ －

½×U

Figure 28. Functions of Computing Unit

6. Multivariable Model Predictive Control in MATLAB[3]

6.1. The Model Predictive Control (MPC) Toolbox and the MPC Blockset

The Model Predictive Control (MPC) Toolbox is a MATLAB product which is used for design and

simulation of Model Predictive Control system. In order to simulate the designed Model Predictive

Control system use the MPC Controller Block provide in this tool. The MPC toolbox provides

function library for command line and mpctool for dedicated GUI. Model Predictive Control

designed in mpctool is MPC object form, and can be treated on MATLAB Workspace. Simulink and

the MPC Toolbox can be seamlessly used by specifying MPC object in the MPC Controller Block.

Examples of Functionalities of mpctool are described in figure 29.

(1) in figure 29 sets control structure of Model Predictive Control system to be designed. (2) sets

38

inequality constraints of Model Predictive Control system to be designed. There are upper and lower

limits and rate limit constraints of manipulated variables, and upper and lower limits of controlled

valriables in inequality constraints. (3) sets control interval of Model Predictive Control system to be

designed, prediction horizon, control horizon and so on. (4) sets weight factor within the cost

function of Model Predictive Control system to be designed.

Model Predictive Control system described in chapter 6 refers to the Model Predictive Control

(MPC) Toolbox. MPC Structure Overview

Controlled Variables

ManipulatedVariables

Model and Horizons

Constraints

Weight Tuning

(1)(2)

(3) (4)

Figure 29. Example of various GUIs for designing mpctool

6.2. Control Structure of Model Predictive Control

Control structure of Model Predictive Control is shown in Fig30. The structure is the same in SISO

(single-input single-output) and MIMO (multi-input multi output). Model Predictive Control is a

kind of feedback control.

39

Figure 30. Block Diagram of a SISO Model Predictive Control Toolbox Application [3]

Description of symbols in Figure 30 is shown in Table14.

Table 14. Description of Model Predictive Control Toolbox Signals [3]

Symbol Description

d Unmeasured disturbance. Unknown but for its effect on the plant output. The controller

provides feedback compensation for such disturbances.

r Setpoint (or reference). The target value for the output.

u Manipulated variable (or actuator). The signal the controller adjusts in order to achieve

its objectives.

v Measured disturbance (optional). The controller provides feedforward compensation for

such disturbances as they occur to minimize their impact on the output. y Output (or controlled variable). The signal to be held at the setpoint. This is the “true”

value, uncorrupted by measurement noise.

y Measured output. Used to estimate the true value, y .

z Measurement noise. Represents electrical noise, sampling errors, drifting calibration,

and other effects that impair measurement precision and accuracy.

40

6.3. Operation of Model Predictive Control

Dynamic optimization is performed in Model Predictive Control. (1) and (2) as follows are repeated

in dynamic optimization procedure.

(1) Prediction future controlled variable of the plant as many as p pieces of prediction horizon, p for

every control interval. (2) Solving manipulated variable rate vector [ ])k1(),( +−ΔΔ mkukku K that minimizes quadratic

cost function described by manipulated variable rate vector [ ])k1(),( +−ΔΔ pkukku K ,

manipulated variable vector [ )k1(),( +−pkukku K ] and controlled variable vector

[ )k1(),( +−pkykky K ] within the constraints. The procedures for dynamic optimization is shown

to the left in figure 31. To the right, prediction horizon and control horizon are described.

present(k)

,M

Prediction future controlled variable of the plant as many as p pieces of prediction horizon, p for every control interval.

Solving manipulated variable rate vector that minimizes quadratic cost functiondescribed by manipulated variable rate vector,

manipulated variable vector and controlled variable vector within the constraints.

Set u(k) to plant

k=k+1（Shift sample time )

Procedure of MPC

repeat

Dynamic optimization

Figure 31. Behavior overview of Model Predictive Control [3]

6.4. Linear Model of Plant

The linear model used in Model Predictive Control Toolbox for prediction and optimization is

depicted in Figure 32.Inputs of the model are manipulated variables and observable and

non-observable disturbances.

41

Figure 32. Linear Model of a Plant used by MPC Toolbox [3]

The model of the plant is a linear time-invariant system described by the equations (6-1).

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) )16(

1

−+++=++=

+++=+

kuDkdDkvDkxCkykdDkvDkxCky

kdBkvBkuBkAxkx

uuduvuuu

dmvmmm

dvu

where x(k) is the nx-dimensional state vector of the plant, u(k) is the nu-dimensional vector of

manipulated variables (MV), i.e., the command inputs, v(k) is the nv-dimensional vector of measured

disturbances (MD), d(k) is the nd-dimensional vector of unmeasured disturbances (UD) entering the

plant, ym(k) is the vector of measured outputs (MO), and yu(k) is the vector of unmeasured outputs

(UO). The overall output vector y(k) collects ym(k) and yu(k).

6.5. Optimization Problem

Assume that the estimates of x(k), xd(k) are available at time k. The Model Predictive Control

action at time k is obtained by solving the optimization problem.

where the subscript “( )j” denotes the j-th component of a vector, “(k+i|k)” denotes the value

predicted for time k+i based on the information available at time k; r(k) is the current sample of the

output reference, subject to with respect to the sequence of input increments

{Δu(k|k),…,Δu(m-1+k|k)} and to the slack variable ε, and by setting u(k)=u(k-1)+Δu(k|k)*, where

Δu(k|k)* is the first element of the optimal sequence.

42

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )⎪⎭

⎪⎬⎫

+⎟⎟⎠

⎞+−+++Δ+

⎪⎩

⎪⎨⎧

⎜⎜⎝

⎛++−+++−ΔΔ

∑∑

∑ ∑

==

Δ

−

= =+

2

1

2arg,

1

2,

1

0 1

2,1 11,

min1,...,

ερωω

ωε

ε

uu

y

n

jetjtj

uji

n

jj

uji

p

i

n

jjj

yji

ikukikukiku

ikrkikykkmukku (6-2)

where the subscript “( )j” denotes the j-th component of a vector, “(k+i|k)”

denotes the value predicted for time k+i based on the information available at

time k; r(k) is the current sample of the output reference, subject to

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

1,...,001,...,,0

1 maxmaxminmin

maxmaxminmin

maxmaxminmin

−=≥−==+Δ

+≤++≤−+Δ≤+Δ≤−Δ

+≤+≤−ΔΔ

pipmhkhku

iViykikyiViyiViukikuiViu

iViukikuiViu

j

yjjj

yjj

ujjj

ujj

ujjj

ujj

ε

εεεε

εε

(6-3)

with respect to the sequence of input increments {Δu(k|k),…,Δu(m-1+k|k)} and to the slack variable

ε, and by setting u(k)=u(k-1)+Δu(k|k)*, where Δu(k|k)* is the first element of the optimal sequence.

When the reference r is not known in advance, the current reference r(k) is used over the whole

prediction horizon, namely r(k+i+1)=r(k) in Equation (6-3).

ji,y

ji,u

ji,u w, w,wΔ are nonnegative weights for the corresponding variable. The smaller w, the less

important is the behavior of the corresponding variable to the overall performance index.

maxj,minj,maxj,minj,maxj,minj, y ,y ,u ,u ,u ,u ΔΔ are lower/upper bounds on the corresponding

variables. In Equation (6-3), the constraints on u, Δu, and y are relaxed by introducing the slack

variable 0≥ε . The weight ερ on the slack variable ε penalizes the violation of the constraints.

The larger ερ with respect to input and output weights, the more the constraint violation is

penalized. The Equal Concern for the Relaxation (ECR) vectors

have nonnegative entries which represent the

concern for relaxing the corresponding constraint; the larger V, the softer the constraint. maxminmax

umin

umax

umin

u V ,V ,V ,V ,V ,V yyΔΔ

V=0 means that the constraint is a hard one that cannot be violated. By default, all input constraints

are hard ( )0V V VV maxu

minu

maxu

minu ==== ΔΔ and all output constraints are soft

( )1VV maxy

miny == . As hard output constraints may cause infeasibility of the optimization

problem (for instance, because of unpredicted disturbances, model mismatch, or just because of

numerical round off).

43

6.6. Cost Function of Model Predictive Control

Cost Function (6-2) is shown as follows.

22

2

arg

arg2

arg

arg

)(

)1(

)(

)1(

)(

)1(

)(

)1(

)1(

)0(

)1(

)0(

)1(

)0(

)1(

)0(

)1(

)0(

)1(

)0(),(

εσ

ε

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−Δ

Δ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−Δ

Δ+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

Δ

pr

r

py

yW

pr

r

py

y

pu

uW

pu

u

pu

u

pu

uW

pu

u

pu

uzJ

y

T

u

T

ett

ett

u

T

ett

ett

LLLL

LL

LLLL

(6-4)

Constraint (6-3) is shown as follows.

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−Δ

−Δ−−−

−−

−

≤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−Δ

Δ−

≤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−Δ

−Δ−−−

−−

−

Δ

Δ

Δ

Δ

)1()1(

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

)1()1()()(

)1()1(

)1(

)0()1(

)0()(

)1(

)1()1(

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

)1()1()()(

)1()1(

maxmax

maxmax

maxmax

maxmax

maxmax

maxmax

minmin

minmin

minmin

minmin

minmin

minmin

pVpu

VupVpu

VupVpy

Vy

pu

upu

upy

y

pVpu

VupVpu

VupVpy

Vy

u

u

u

u

y

y

u

u

u

u

y

y

ε

εε

εε

ε

ε

εε

εε

ε

L

L

L

L

L

L

L

L

L

(6-5)

Cost Function (6-4) is shown as follows.

2222),( εσε ++ΔΔ+= Δ YWYUWUUWUzJ yT

uT

uT (6-6)

Constraint (6-6) is shown as follows.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ≤

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ≤

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Δ max

max

max

min

min

min

UUY

UUY

UUY

(6-7)

Δu that minimizes J is calculated sequentially-iterated by the quadratic programming with

constraints. Symbols that used in equation (6-6) and (6-7) are shown as follows.

44

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−Δ

Δ=Δ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

)(

)1(

)(

)1(

)1(

)0(

)1(

)0(

)1(

)0(

arg

arg

pr

r

py

yY

pu

uU

pu

u

pu

uU

ett

ett

LL

L

LL

(6-8)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−Δ

−Δ=Δ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−Δ

−Δ=Δ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

Δ

Δ

Δ

Δ

)1()1(

)1()1(

)1()1(

)1()1(

)()(

)1()1(

)1()1(

)1()1(

)1()1(

)1()1(

)()(

)1()1(

maxmax

maxmax

min

maxmax

maxmax

max

maxmax

maxmax

max

minmin

minmin

min

minmin

minmin

min

minmin

minmin

min

pVpu

VuU

pVpu

VuU

pVpy

VyY

pVpu

VuU

pVpu

VuU

pVpy

VyY

u

u

u

u

y

y

u

u

u

u

y

y

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

L

L

L

L

L

L

(6-9)

45

7. Evaluation of Control System 7.1. Introduction

This chapter evaluates control performance of cascade PID control and Model Predictive Control of

the outer control system designed in chapter 5.

7.2. Simulation Model of Cascade PID Control

Figure 33 shows simulation model of cascade PID control.

After 0 to 25000 seconds, it is controlled in combination of cascade PID control and the inner

control system.

CascadePID

ControllerComputing Unit

Signal Builder Block Gas Separator1 Gas Separator2 Gas Separator3







Data storing part

Data display part

CascadePID

Controller

CascadePID

Controller

Figure 33. Simulation Model

7.3. Simulation Model of Model Predictive Control

Model that simulates Model Predictive Control is shown in figure 34. Model Predictive Control is

performed from time 2000 seconds. So, Model Predictive Controller (MPC Controller Block) is

located within Enabled Subsystem. It is controlled in the inner control system only up to 2000

seconds, and in combination of the Model Predictive Control and the inner control system after 2000

to 25000 seconds.

46

Computing Unit

Signal Builder Block Gas Separator1 Gas Separator2 Gas Separator3







Data storing part

Data display part

Model PredictiveController

Figure 34. Simulation Model

7.4. Comparison of Cascade PID Control and Model Predictive Control

Perform simulations for cascade PID control and Model Predictive Control. The results obtained are

shown in figure 35 to 38. (Figure 35: Set points and controlled variables, Figure 36: superimpose of

Figure 35, Figure 37: manipulated variables, Figure 38: disturbance)

In (1) in figure 35 (left side of figure 36), set point tracking control apparently stops and windup that

requires time for a while to restart set point tracking is confirmed in cascade PID control. Windup

occurs because: integrator is affected if output signal (MV) is reached to upper or lower limit in PID

calculation when error that is difference between set point and controlled variable occurs. An

appropriate integrated value is stored in integrator and normal calculation is performed if MV does

not go over the limits. If MV goes over the limits, excessive integrated values are stored in integrator.

It requires substantial time for integrated value to become 0 when this state is controlled to be

normal range. Until reaching this time, MV output does not return to normal range, and the state of

going over the limits continues. The functionality to initialize integrated value in a short time is

called anti-reset windup. This PID controller does not have anti-reset windup.

(1) of figure 35 (left side of figure 36) indicate the state which can not track set point. Each gas

separator constructs own closed cascade PID control system, and in the gas separator 3, it confirms

47

that manipulated variable of cascade PID controller is at the upper limit in (1) of figure 37. The

cascade PID control shows that the ability to make oxygen of more than 95% concentration set here

can not be obtained.

Manipulated variable of cascade PID control always stick to the upper limit within of the upper and

lower limits of manipulated variable, 0~1.5[atm].Manipulated variable of the upper and lower limits

of Model Predictive Control is 0~1[atm], and manipulated variable of Model Predictive Control

fluctuate within the range, stress put on the plant is lower in Model Predictive Control.

In Model Predictive Control, set point tracking control is achieved within the upper and lower limits

of the manipulated variables.

Weight factor of set point tracking of the gas separator 3 is weighted and set point tracking of the gas

separators 2 and 3 are lowered for Model Predictive Control in the design of cost function. Figure 35

shows that set point tracking is the best in the gas separator 3, followed by the gas separator 2 and 1,

respectively, which confirms behavior in accordance with the design concept. It also shows that the

ability to make oxygen with more than 95% concentration can be achieved in the current Model

Predictive Control performance. Weight factors for the manipulated variables 1, 2 and 3 are

minimized for large behavior range in the cost function. Each manipulated variables indicates larger

behavior range for control value set point tracking. As impact of disturbance (figure 38) is not so

significant for both cascade PID control and Model Predictive Control, it can be considered as good

for disturbance absorbability.

In conclusion, Model Predictive Control is considered to have the better control performance in this

example. However, this cascade PID control is simple and there are some possibilities for improved

control performance using feedforward and decoupling and so on.

48

Cascade PID control Model Predictive Control

(3)(2)

Windup

Figure 35. Setpoints and manipulated variables

Windup

Figure 36. Superimpose of figure 35

49

Cascade PID Control Model Predictive ControlManipulated Variables

(1)

Figure 37. Manipulated variables

50

Disturbance

Figure 38. Disturbance

8. Conclusion This example shows simulation for gas separator process control using permeable membrane in

Simulink.

The control object (plant) is gas separator. Plant model is created on a formula basis using Simulink,

and as control system for the plant, cascade PID control and Model Predictive Control are compared.

Control system for the gas separator process control consists of combination of the outer and inner

control systems (cascade control). Outputs of the outer control system is manipulated variables that

is set points to track controlled variables of inner control system.

The procedures for simulation are as follows:

(1) Plant modeling

Governing rules of the plant are assumed and modeled on a formula basis. The governing rules are

derived from mass balance equation and ideal gas equation.

(2) System identification

Identify parametric model from plant input and output response in time domain.

(3) Control system design (PID control, Model Predictive Control)

Design of PID control for the inner control system, and cascade PID control and Model Predictive

51

Control for the outer control system. The control object is the above parametric model.

(4) Evaluation of Control System

Performance of designed control system is evaluated in some diagram. Validity of Model Predictive

Control is confirmed in this example for multi-inputs multi-outputs system.

Plant model and control system in this example can be created in MATLAB&Simulink and

simulated. Logics of MATLAB&Simulink can be changed and extended as a more detailed model,

and improved gradually, a larger system can be constructed. It is also possible to embed

experimental equation or table format if processes are difficult to equate theoretically. Efficient and

intuitive modeling is possible in Simulink’s graphical environment.

9. References [1] Kenji Hashimoto and Fumimaru Ogino, Today’s Chemical Engineering, Sangyo-Tosho (2001)

[2] Iori Hashimoto, Shinji Hasebe and Manabu Kano, Process Control Engineering, Asakura

Publishing Co., Ltd. (2002)

[3] Model Predictive Control Toolbox User’s Guide Version 2 The Mathworks, 2004

[4] Shuichi Adachi, Control Engineering using MATLAB, Tokyo Denki University Press (1999)

10. Exemption from Responsibility Under no circumstances will CYBERNET SYSTEMS CO.,LTD be liable in any way for in this

content, or for any loss or damage of any kind incurred as a result of the use of this content.

11. Author Hiroumi Mita

[email protected]

Senior Application Engineer

CDA Engineering Group, Field Application Engineering Department,

Applied Systems First Division

CYBERNET SYSTEMS CO.,LTD.

52

mailto:[email protected]

Created in February 2005

This model is created using MATLAB Version 7.0.1 and Simulink : Version 6.1.

MATLAB&Simulink Sample Model Description ― Simulation of Cascade PID Control and Model Predictive Control for a gas separation plant by

membrane separation ―

MATLAB and Simulink are registered trademarks of the MathWorks, Inc. All other names such as products are trademarks or registered trademarks of

respective owners. Reprint, copy or reproduction of the whole or any part of this material is prohibited. The contents of this material may be altered

without prior notice.

http://www.cybernet.co.jp/MATLAB Applied Systems First Division, Sales, E-mail: [email protected]

Applied Systems First Division, Engineering, E-mail: [email protected]

53

mailto:[email protected]

MATLAB &Simulink Sample Model Description...MATLAB®&Simulink® Sample Model Description ―...

Documents

Transcript of MATLAB &Simulink Sample Model Description...MATLAB®&Simulink® Sample Model Description ―...