Controller Designing Of MIMO System

(Phase 1)

Project Report submitted in partial fulfillment of the requirement for the

degree of B.Tech.

In

Instrumentation and Control Engineering

By

Sulagna Sarkar (IC-513)

Madhumita Mantri (IC-505)

Pratik Nath (IC-530)

Debayan Sen (IC-526)

Under the Supervision of

Mr. Parikshit Kr. Paul

Calcutta Institute of Engineering and Management

24/1A Chandi Ghosh Road Kolkata-40

Year- 2013-14

1

DECLARATION

We declare that this written submission represents my ideas in my own words and where others' ideas or

words have been included, I have adequately cited and referenced the original sources. I also declare that I

have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or

falsified any idea/data/fact/source in my submission. I understand that any violation of the above will be

cause for disciplinary action by the Institute and can also evoke penal action from the sources which have

thus not been properly cited or from whom proper permission has not been taken when needed.

NAME :

SULAGNA SARKAR(IC-513)

MADHUMITA MANTRI(IC-505)

PRATIK NATH(IC-530)

DEBAYAN SEN(IC-526)

SIGNATURE:

DATE: 25/11/2013

2

RECOMMENDATION /CERTIFICATE OF APPROVAL

This Project Report entitled Controller Designing Of MIMO System

by

Name: University Roll No:

Sulagna Sarkar 16504010018

Madhumita Mantri 16504010015

Pratik Nath 16504010007

Debayan Sen 16504010037

Is approved for the partial fulfillment of the requirement for the degree of B.Tech in

Instrumentation and control Engineering .

Teacher in charge Supervisor

________________________ ________________________

Date: 25/11/2013

Place: Kolkata

3

4

ACKNOWLEDGEMENT

We are very thankful to Mr. Parikshit Kr Paul for his guidance and support which he has showed to us for the completion of this project. We also convey our special thanks to the entire ICE faculty members and teacher in charge Mr. Somnath Gorai for their valuable suggestions. Without their support and guidance it wouldn’t have been possible for us to complete this work …

Signature of the students

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Table of Contents

1-ABSTRACT.....................................................................................................................................................7

2 - INTRODUCTION...........................................................................................................................................8

2.1ADVANTAGES OF CONTROL SYSTEM ..................................................................................................9

2.2 OPEN LOOP CONTROLSYSTEM.........................................................................................................10

2.3 CLOSED LOOP CONTROL..................................................................................................................10

3 – AUTOMATIC CONTROL SYSTEM...............................................................................................................11

3.1 FUNCTIONS OF AUTOMATIC CONTROL............................................................................................11

3.2ELEMENTS OF AUTOMATIC CONTROL..............................................................................................11

3.3CONTROL SYSTEM CLASSIFICATION………………………………………………...12

4 – SINGLE INPUT SINGLE OUTPUT(SISO).................................................................................13

4.1TRANSFER FUNCTION OF SISO………………………………………………………..13

5-MULTIPLE INPUT MULTIPLE OUTPUT(MIMO)……………………………………............15

5.1TRANSFER FUNCTION MODEL OF (2X2 MULTI LOOP MODEL)………………......175.2BLOCK DIAGRAM OF 2X2 MULTIPLE LOOP SYSTEM AND ANALYSIS………….18

5.2.1HIDDEN FEEDBACK LOOP…………………………………………………....185.3PROCESS INTERACTIONS IN MIMO SYSTEM(2X2 SYSTEM)………………………19

6-RELATIVE GAIN ARRAY(RGA)………………………………………………………………..20

6.1BRISTOL’S RGA METHOD……………………………………………………………...206.2CALCULATION OF RGA………………………………………………………………...20

7-STRATEGIES FOR DEALING WITH UNWANTED CONTROL LOOP INTERACTIONS……………………………………………………………………………………22

7.1DECOUPLING METHOD…………………………………………………………..........227.1.1BLOCK DIAGRAM OF DECOUPLING SYSTEM……………………………237.1.2DECOUPLER DESIGN EQUATIONS…………………………………………237.1.3DIFFERENT TYPES OF DECOUPLING………………………………………24

8-CONCLUSION……………………………………………………………………………………26

9-REFERANCE………………………………………………………………………………..........27

10-SCOPE OF PROJECT…………………………………………………………………….........28

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1. ABSRACT

In this paper, some information about the MIMO system, its transfer functions and block diagram is given. In addition, problems related to MIMO systems such as process interactions are presented. Also different strategies such as decoupling method, to reduce these control interactions are presented. Finally, relative gain array (RGA) is applied here for pairing of manipulated and controlled variable to get a desired output.

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2.INTRODUCTION:

A control system consists of subsystems and processes (or plants) assembled for the purpose of

obtaining a desired output with desired performance, given a specified input. Figure 1 shows a control

system in its simplest form, where the input represents a desired output.

Input

.

Fig1. Schematic diagram of control system

For example, consider an elevator. When the fourth-floor button is pressed on the first floor, the elevator

rises to the fourth floor with a speed and floor-leveling accuracy designed for passenger comfort. The

push of the fourth-floor button is an input that represents our desired output, shown as a step function in

Fig 2. The performance of the elevator can be seen from the elevator response curve in the figure.

Two major measures of performance are apparent: (1) the transient response and (2) the steady-state

error. In our example, passenger comfort and passenger patience are dependent upon the transient

response. If this response is too fast, passenger comfort is sacrificed; if too slow, passenger patience is

sacrificed. The steady-state error is another important performance specification since passenger safety

and convenience would be sacrificed if the elevator did not properly level.

Fig .2 elevator response curve

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Control system

2.1 ADVANTAGES OF CONTROL SYSTEM:

With control systems we can move large equipment with precision that would otherwise be impossible.

We can point huge antennas toward the farthest reaches of the universe to pick up faint radio signals,

controlling these antennas by hand would be impossible. Because of control systems, elevators carry us

quickly to our destination, automatically stopping at the right floor (Figure 1.3). We alone could not

provide the power required for the load and the speed; motors provide the power, and control systems

regulate the position and speed.

We build control systems for four primary reasons:

1. Power amplification

2. Remote control

3. Convenience of input form

4. Compensation for disturbances

Robots designed by control system principles can compensate for human disabilities. Control systems

are also useful in remote or dangerous locations. For example, a remote-controlled robot arm can be

used to pick up material in a radioactive environment.

Control systems can also be used to provide convenience by changing the form of the input. For

example, in a temperature control system, the input is a position on a thermostat. The output is heat.

Thus, a convenient position input yields a desired thermal output.

Another advantage of a control system is the ability to compensate for disturbances. Typically, we

control such variables as temperature in thermal systems, position and velocity in mechanical systems,

and voltage, current, or frequency in electrical systems. The system must be able to yield the correct

output even with a disturbance. For example, consider an antenna system that point in a commanded

direction. If wind forces the antenna from its commanded position, or if noise enters internally, the

system must be able to detect the disturbance and correct the antenna's position. Obviously, the system's

input will not change to make the correction. Consequently, the system itself must measure the amount

that the disturbance has repositioned the antenna and then return the antenna to the position commanded

by the input.

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2.2 OPEN LOOP SYSTEM:

Those systems on which output has no effect on the control action are called open loop control system.

In other words, in open loop control system the output is neither measured nor fed back for comparison

with input .One practical example is washing machine. Soaking, washing and rinsing in the washer

operate on a time basis .The machine does not measure the output signal, that is, the cleanliness of the

clothes.

Fig.3 open loop control system

2.3 CLOSED LOOP SYSTEM:

The disadvantages of open-loop systems, namely sensitivity to disturbances and inability to correct for

these disturbances, may be overcome in closed-loop systems .Feedback control systems are often

referred to as closed loop control systems .In closed loop control system the actuating error

signal ,which is the difference between the input signal and the feedback signal(which may be the

output signal itself or the functions of the output signal and its derivatives and/or integrals),is fed to the

controller so as to reduce the error and to bring the output of the signal to the desired value.

Fig.4 Closed loop control system

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3.AUTOMATIC CONTROL SYSTEM

A machine (or system) work by machine-self, not by manual operation. Automatic control is the

application of control theory for regulation of processes without direct human intervention. Automatic

control can self-regulate a technical plant (such as a machine or an industrial process) operating

condition or parameters by the controller with minimal human intervention. A regulator such as

a thermostat is an example of a device studied in automatic control.

3.1Functions   of   Automatic   Control

In any automatic control system, the four basic functions that occur are

Measurement

Comparison

Computation

Correction

In the water tank level control system in the example above, the level transmitter measures the level

within the tank.  The level transmitter sends a signal representing the tank level to the control  device,

where  it  is  compared  to  a  desired  tank  level. The level control device then computes how far to

open the supply valve to correct any difference between actual and desired tank levels.

3.2 Elements   of   Automatic   Control

The three functional elements needed to perform the functions of an automatic control system are

A measurement element

An error detection element

A final control element

Example , WATER LEVEL CONTROL SYSTEM

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M

Water pool

val ve

fl oat

ampl i fi er

motor

Gearassembl y

+

-

Fig .5 water level automatic control system

3.3CONTROL SYSTEM CLASSIFICATION:

12

amplifierMotor GearingValve

Actuator

WatercontainerProcesscontroller

Float

measurement (Sensor)

Error

Feedback signal

resistance comparatorActual

water level

Output

Fig. 1.8

Fig .6 types of control system

4. SISO SYSTEM (SINGLE INPUT SINGLE OUTPUT):-

In control engineering, a single-input and single-output (SISO) system is a simple single

variable control system with one input and one output. In radio it is the use of only one antenna both in

the transmitter and receiver.

Fig. 7 block diagram of SISO control system

Frequency domain techniques for analysis and controller design dominate SISO control system

theory. Bode plot, Nyquist stability criterion , Nichols plot , and root locus are the usual tools for SISO

system analysis. Controllers can be designed through the polynomial design, root locus design methods

to name just two of the more popular. Often SISO controllers will be PI, PID, or lead-lag.

4.1The transfer function for a SISO linear system:

The first thing we do is look at our linear systems formalism and see how it appears in the Laplace

transform scheme .We suppose we are given a SISO linear system _ = (A, b, ct,D), and we fiddle with

Laplace transforms a bit for such systems. Note that one takes the Laplace transform of a vector

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CONTROL SYSTM

SISO TITO MIMO

function of time by taking the Laplace transform of each component. Thus we can take the left causal

Laplace transform of the linear system

If the system is LTI and Lumped, we can take the Laplace Transform of the state-space equations, as

follows:

L [ X ' ( t ) ]=L [ AX (t )+L [ BU ( t ) ] ] …(1)

L [Y (t)]=L [CX ( t )+L [DU (t )] ] …(2)

This gives us the result:

sX (s )−X (0 )=AX ( s )+BU (s) ...(3)

Y (s )=CX (s )+DU (s ) ...(4)

Where X (0) are the initial conditions of the system state vector in the time domain. If the system is

relaxed, we can ignore this term, but for completeness we will continue the derivation with it. We can

separate out the variables in the state equation as follows:

sX (s )−AX ( s )=X (0 )+B U (s) ...(5)

Then factor out an X(s):

X ( s) [ sI−A ]=X (0 )+BU (s) ...(6)

And then we can multiply both sides by the inverse of [sI - A] to give us our state equation:

X ( s)=[ sI−A ]−1X(0)+[ sI −A ]−1BU(s) ...(7)

Now, if we plug in this value for X(s) into our output equation, above, we get a more complicated

equation:

Y (s )=C ( [ sI −A ]−1X (0 )+ [ sI −A ]−1

BU (s ) )+DU (s) ...(8)

and we can distribute the matrix C to give us our answer:

Y (S )=C [ sI−A ]−1X (0 )+C [ sI−A ]−1

BU (s )+ DU (s) ... (9)

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Now, if the system is relaxed, and therefore X(0) is 0, the first term of this equation becomes 0. In this

case, we can factor out a U(s) from the remaining two terms:

Y (s )=(C [ sI−A ]−1B+D )U (s ) … (10)

We can make the following substitution to obtain the Transfer Function Matrix, or more simply,

the Transfer Matrix, H(s):

Transfer Matrix

C [ sI−A ]−1B+ D=H (s) …(11)

Rewrite our output equation in terms of the transfer matrix as follows:

Y (s )=H ( s ) U (s) ...(12)

If Y(s) and X(s) are 1 × 1 vectors (SISO system), then we have our external description:

Y (s )=H ( s ) X (s) …(13)

Now, since X(s) = X(s), and Y(s) = Y(s), then H(s) must be equal to H(s). These are simply two

different ways to describe the same exact equation, the same exact system.

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5.MULTIPLE INPUT MULTIPLE OUTPUT (MIMO) SYSTEM:

MIMO control system is a Manipulated Input Manipulated Output system. In this type of system more

than one controlled variables are applied into the process and get more than one output. The main

difference between a scalar (SISO) system and a MIMO system is the presence of directions.

Directions are relevant for vectors and matrices, but not for scalars. In many practical control problems

typically a number of process variables which must be controlled and number of variables which can

be manipulated, these problems are referred to as MIMO System. The application of MIMO system is

inverted pendulum, aircraft control, and four tank controls.

Fig 8. Block diagram of MIMO control system

In above diagram, multiple inputs are applied into the process and get the multiple outputs. Two Input

Two Output system, i.e. TITO system also a part of MMIO system.

In TITO system, two inputs are applied into the process and get the two outputs.

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Fig 9. Block diagram of TITO control system

Examples are described where two controlled variable and two manipulated variables are shown in the

fig 10.

Fig.10 line blending system

In above example, illustrate a characteristic feature of MIMO system that is process interaction.

Process interaction means each manipulated variable can affect both controlled variable. The first

examples, the in-line blending system two controlled variable A and B are applied to the system to

produce product stream with mass flow rate w and composition x, the mass fraction of A. Adjusting

manipulated flow ratew A ¿wB affected both w and x. MIMO control system is more complex than SISO

control because of the process interactions occur between the controlled variable and manipulated

variable

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5.1 TRANSFER FUNCTION MODEL OF(2x2) MULTI-LOOP MODEL:

Transfer function of a MIMO system is very important to determine the effect of the manipulated

variables on the controlled variables. There are two controlled variable two manipulated variable so

that four transfer functions are required to completely characterize the process dynamics.

Y 1(S )U 1(S )

=GP11 (S ) Y 1(S )U 2(S )

=GP12(S )

Y 2(S )U 1(S )

=GP21(S ) Y 2(S )U 2(S )

=GP22(S ) ……(14)

The above expression is used to determine the effect of a change in either U 1 or U 2 on Y 1 or Y 2 from

the principle of Superposition Theorem. Input –Output relation for the process.

Y 1(S)=GP 11(S)U 1(S)+GP12(S)U 2(S) ...(15)

Y 2 (S )=GP21 (S ) U 1 ( S )+GP22 ( S )U 2 (S ) ...(16)

These input-output relationships can also be expressed in vector matrix form

Y(S) =GP ( S )U (S) ...(17)

Where Y(S) = output vector and U(S) = input vector, written as:

Y(S) =(Y 1(S)Y 2(S)) U (S)=(U 1(S)

U 2(S)) …(18)

GP =process transfer function matrix, written as:

GP(S )=(GP11 (S) GP 12(S)GP21(S ) GP 22(S)) ...(19)

The steady state process transfer matrix (S=0) is called the process gain matrix. It is defined by

K.

GP(S )=(GP11 (0) GP12(0)GP21(0) GP22(0)) ...(20)

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5.2 BLOCK DIAGRAM OF 2x2 MULTILOOP SYSTEM AND ANALYSIS:

Fig 11.(a) 1-1/2-2 controller pairing.(b)1-2/2-1 controller pairing

The two possible configurations are shown in the above diagram.the fig11.(a) shows that, theY 1is

controlled by adjustingU 1 whileY 2 is controlled by adjusting U 2. Hence the control scheme is known

as 1-1/2-2 .similarly in the fig 11.(b) Y 1 is controlled by U 2 andY 2 is controlled by U 1, so the

configuration scheme is 1-2/2-1.

5.2.1 Hidden feedback loop

In 1-1/2-2 scheme inititla changes in U 1 has two effects on Y 1 – a) direct effect b) indirect effect

through control loop interactions. The control loop interaction is generated due to the presence of

third feedback loop contains two controllers and four process transfer function. This third

feedback loop is also known as Hidden feedback loop. For 1-1/2-2 configuration scheme the

hidden feedback loop contains GC 1 ,GC 2 ,GP12, GP21.The 1-1/2-2 pairing the dark line shows the the

third feedback loop.

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Fig 12.hidden feedback loop

The presence of hidden feedback loop causes two problems:

It usually destabilizes the whole system.

It makes the controller tuning much more difficult.

5.3 Process interactions in Multiple Input and Multiple Output system( 2x2 system)

For the multiloop control configuration 1-1/2-2, the transfer function between a controlled and a

manipulated variable depends on whether the other feedback control loops are open or closed. If

the controller of the second loop GC 2is out of service or in manual mode, then(U ¿¿2=0)¿. For

this situation the transfer function between Y 1∧U 1 is:

Y 1

U 1

=GP11 ...(21)

If both loops are closed then Y 1 is:

Y 1=GP11 U1+GP12U 2 …(22)

If the second feedback controller is in the automatic model Y SP 2=0, then

Y 2=GP21 U 1

1+GC 2GP22

...(23)

The signal to the first loop from the second loop is:

GP12U 2=−GP12GC 2Y 2 …(24)

The overall transfer function is:

Y 1

U 1

=GP11−GP12GP21GC 2

1+GC 12GP 22

…(25)

The transfer function between Y 1 ,U 1 depends on the second loop controller GC 2 so that the two

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controllers should not be tuned independently.

6. RELATIVE GAIN ARRAY METHOD

A systematic approach for determining the best pair of controlled and manipulated variable is

Relative Gain Array or RGA method.

6.1BRISTOL’S RELATIVE GAIN ARRAY METHOD:

Bristol(1966) developed a systematic approach for the analysis of multivariable process control

problems. It requires only steady state information (the process gain matrix K) but not process

dynamics. It provides two useful information:

1. Measure of process interactions

2. Recommendation about best pairing of controlled and manipulated variables.

Consider a process with ‘n’ controlled variable and ‘n’ manipulated variables. The relative gain λ ij

, relates the ith controlled variable and the jth manipulated variable

λ ij≜¿¿

λ ij≜open loop gain

closed loop gain

... (26)

For i=1, 2, 3 …n; j=1, 2, 3 ….n

¿= partial derivative evaluated with all of the manipulated variables except u j held constant.

¿= partial derivative evaluated with all of the controlled variables except Y i held constant.

6.2CALCULATION OF RGA:

The relative gain array can be calculated by the Steady-state process of (2x2) model.

y1=K11 u1+K12u2 ... (27)

y1=K11 u1+K12u2 … (28)

Where K ij denotes the steady state gain between y i and u j.this model can be expressed in matrix

notation is

K=YU ... (29)

Now from above equation,

( ∂ y1

∂ u1)

u

=K11 ... (30)

21

To calculate (∂ y1/∂ u1 )y 2 , eliminate u2.this done by the solving u2 and consider the y2 is its

nominal value, i.e. y2=0

u2=−K21

K22

u1 ... (31)

Then,

y1=K11 (1− K12 K21

K11 K22) ... (32)

It follows that,

( ∂ y1

∂ y2)

y 2

=K11(1−K12 K21

K11 K22) … (33)

So that, the relative gain λ11 can be expressed as

λ11=1

1−K12 K21

K11 K22

… (34)

Each row and columns of RGA ( Λ) sums to one. so other relative gain are easily calculated from

the λ11 for the (2x2) process

λ12= λ21=1− λ11 ... (35)

And,

λ22= λ11 ... (36)

RGA can be described for 2x2 model,

Λ=( λ 1−λ1−λ λ ) ... (37)

For higher order matrix, the RGA can be calculated as:

Λ=K⨂H ... (38)

⨂Denotes element by element multiplication

λ ij=K ij H ij ... (39)

K ij= The (i , j) element of K (Steady state gain matrix)

H ij=The (i , j ) element of H=( K−1 )

7. Strategies for Dealing with unwanted control loop interactions

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Process interactions between manipulated and controlled variables can result in undesirable

control loop interactions. When control loop interactions are a problem, a number of alternate

strategies are available:

Select different manipulate or controlled variables.

Retune one or more PID controllers, taking process interactions into account.

Use a decoupling control scheme to reduce process interactions. e.g., nonlinear

functions of original variables.

Consider a more general multivariable control method such as model predictive

control.

Decoupling Control Systems

Basic Idea: Use additional controllers (Decoupler) to compensate for process interactions and thus

reduce control. Ideally, decoupling control allows set point changes to affect only the desired

controlled variables. Typically, decoupling controllers are designed using a simple process model

(e.g., a steady-state model or transfer function model)

7.1.DECOUPLING METHOD

One of the early approaches to multivariable control is decoupling control. It is used to reduce

control loop interactions. By adding additional controllers called decouplers to a conventional

multiloop configuration, control loop interactions can be reduced. Decouplers are designed using a

simple process model that can be either a steady-state or a dynamic model.

The benefits of decoupling control are:

1. Control loop interactions are limited.

2. A set-point change of one controlled variables has no effect on other controlled variables.

7.1.1Block Diagram of Decoupling system

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Fig: 13. Decoupling control system for a2x2 process and a 1-1/2-2 configuration

7.1.2Decoupler Design Equations

 We want cross-controller, T12, to cancel the effect of U2 on Y1.

Thus, we would like GP11T12U12+G12U22= 0 …40

or, GP11U12+G12U22= 0 …41

 Because U22 is not equal to 0 in general, then

T12=- GP12

GP11 …42

Similarly, we want T12 to cancel the effect of U1 on Y2. Thus, we require that,

  GP22T21U11+GP21U11= 0 …43

T21=- GP21

GP22 …44

Compare with the design equations for feed forward control based on block diagram analysis

7.1.3. Different types of Decoupling

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Partial Decoupling:

Only one of the two decouplers (T12 or T21) is used and the other one is set to zero. This is called

partial or one-way decoupling. It is advantageous for high interacting processes and also provides

better control than complete decoupling.

Static Decoupling(Steady-state Decoupling):

In this type of decoupling, Decoupler is designed in such a way that only steady-state interactions

between control loops are eliminated. The design equations for ideal static Decoupler is obtained

from the equations:

T12=- GP12

GP11 …45

T21=- GP21

GP22 …46

By setting s=0.

The process transfer functions is replaced with the corresponding steady-state gains.

T12 = -KP12

KP11 …47

T21 = -KP21

KP22 …48

Because static decouplers are merely gains, they are always physically realizable and easily

implemented.

One advantage is that less process information is required and a disadvantage is that control loop

interactions still exists during transient conditions. However, if the dynamics of two loops are

similar static decoupling produces excellent transient responses.The performance of decoupling

controllers depends strongly on the accuracy of the process model how well decouplers are tuned.

Decoupling control strategies can be extended to general nxn control problems (seldom used for

n>3).

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8. CONCLUSION

In this project we have considered control problems with multiple inputs (manipulated variable) and multiple outputs (controlled variable) using a set of single-loop controllers (multiloop control). Such MIMO control problems are more difficult than SISO control problems due to the presence of process interactions. They produce undesirable control loop interactions for multiloop control. If these interactions are unacceptable, then different model-based multivariable control strategies are taken. One such is the decoupling method which is used to reduce the control loop interactions.

In MIMO system another problem is that on changing the input variable (manipulated variable) output variable (controlled variable) also changes. So relative gain array (RGA) is used for pairing the manipulated and controlled variable to get a desired output.

In this phase we have studied upto this and in our next phase we want to design a controller on MIMO and implement this on MATLAB simulink.

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9. REFERENCES

[1] http://www.google.com

[2] http://www.daenotes.com/electronics/industrial-electronics/process-control

[3] Benjamin C.Kuo, “Automatic Control Systems”, 7th Edition

[4] Dale E. Seborg, Thomas F. Edgar, Duncan A. Mellichamp, “Process Dynamics and

Control”,2nd Edition

[5] N. Jensen, D.G. Fisher, S.L. Shah, Interaction analysis in multivariable control system,

AIChE J. 32 (6) (1986) 959–970.

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10. SCOPE OF PROJECT

We want to design controller for MIMO system and to implement that in MATLAB in our phase 2.

28