Chapter 4

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# Chapter 4 - PowerPoint PPT Presentation

Chapter 4. Multi-Variable Control. Topics. Synthesis of Configurations for Multiple-Input, Multiple-Output Processes Interaction Analysis and Decoupling Methods Optimal Control Approaches. Examples of Multivariable Control - (1) Distillation Column. Available MV ’ s Reflux Flow

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### Chapter 4

Multi-Variable Control

Topics
• Synthesis of Configurations for Multiple-Input, Multiple-Output Processes
• Interaction Analysis and Decoupling Methods
• Optimal Control Approaches

Examples of Multivariable Control

• - (1) Distillation Column
• Available MV’s
• Reflux Flow
• Distillate Flow
• Steam: Flow to Reboiler
• (Heat Duty)
• 4. Bottom: Flow
• 5. Condenser Duty
• Available CV’s
• Level in Accumulator
• Level in Reboiler
• Bottom Composition
• Pressure in the Column

Problem: How to ‘pair’ variables?

Stream A

FT

CT

• Examples of Multivariable Control
• (2) Blending or two Streams

Stream C

Stream B

MV’s: Flow of Stream A;

Flow of Stream B.

CV’s: Flow of Stream C

Composition of Stream C

Hot

Cold

LT

TT

Examples of Multivariable Control:

(3) Control of a Mixing Tank

MV’s: Flow of Hot Stream CV’s: Level in the tank

Flow of Cold Stream Temperature in the tank

2. Issue of Degree of Freedom (DOF)
• Question: How many variables can be measured and controlled?
• Answer: The number of variables to be controlled is related to the degree of freedom of a system.
• F=V-E
• If F>0, then F variables must be either (1) externally defined or (2)throught a control system
• To control n CV’s, we need n MV’s.

Fi

h

F0

h

Equations

Variables: h, Fi, Fo, x

F=4-2=2

• Externally Specified: Fi
• Controlled Variables: 2-1=1
• Hence, we can control one variable: (1) control flow; (2) control level
• The manipulated variable is valve position or we may choose to have no control; externally specified x

vapor

Fv

y i

Feed

P，T

Z f ，Pf，T f，Ff

h

Steam Ts

FL

Xi

Liquid

Equations:

• More Equations: yi=Kixi (Phase Equilibrium)
• More Equations: Σyi=1; Σxi=1
• Total number of Equations=2N+3
• Variables: zi (N), Tf, ρf, Ff, P, T,y(N-1),FL,xL (N-1),Ts: Total 3N+8
• External Specified: zi (N), Tf, ρf, Ff, Total: N+1
• DOF=3N+8-(N+1)-(2N+1)=4
• There are four variables that must be fixed by controllers or other means.
• Typically: Specify or control T, P, h, Ff
• Ff determines the production rate
• Ff may also be externally specified

PC

TC

.

• What are the manipulated variables? Look for possible valve positions:

LC

Paring : 4!=24 possibilities;

If Ff is specified 3!=6 possibilities

Practical Tips
• Choose the manipulated variable (MV) that has a fast directed effect on the controlled variable (CV) (Calculate or estimate steady state gain, check for controllability)
• Avoid long dead times in the loop
• Reduce or avoid interaction (use relative gain array)
3. Interaction Problem
• Consider a process with two inputs and two outputs.
• Two control loops can be established.
• Question: Will actions in one loop affect the other loop?
• Answer: Most often, yes. This is called interaction
• Interaction is usually decremented to control loop performance
Approaches
• Design single loop controllers and detune them such that one loop does not degrade another loop performance
• not always possible
• selection of pairing very important
• Modify controlled or manipulated variables such that interaction is reduced
• Dynamic interaction
• Determine interacting element’s transfer functions and try to compensate for them
• Solve the multivariable control problem (such as model predictive control –MPC)

+

+

+

+

Process

Block Diagram Analysis

Loop 1

-

+

+

+

+

+

+

-

Loop 2

Block Diagram Analysis-Continued (Both Loops Closed)

P11,P12,P21 and P22have a common denominator:

Roots of Q(s) determine stability

process

m1

y1

y2

m2

process

m1

y1

Controller

Set

point

y2

m2

Measuring Interaction

• If λ11 is unity the other loops do not affect loop 1 and hence there is no interaction
• If λ11 is zero then m1 cannot use to control c1
• if λ11 →∞then other loops will make c1 uncontrollable with m1

Hot

Cold

LT

TT

Examples of Multivariable Control:

(3) Control of a Mixing Tank

MV’s: Flow of Hot Stream CV’s: Level in the tank

Flow of Cold Stream Temperature in the tank

Relative Gain Array
• All rows add up to 1. All columns add up to 1.
• λij is dimensionless. Λ can be used to find suitable
• pairings of input and output
• (3) If Λ is diagonal, there is no interaction between loops
• (4) Elements of Λ significantly different from 1 indicate
• problems with interaction
Derivation of Λ

Choose control loop pairs such that λij is

Positive and as close to unity as possible.

XA

F

CT

FT

XA

FA

F

FB

Example: Blending Problem

Example – Blending Problem - Continued

If FB/F≒1, FA/F≒0 then pair FA→xA; FB→F

If FA/F≒1, FB/F≒0 then pair FB→xA; FA→F

• In the Blending Problem:
• Hence to control F, without changing x is possible by changing ratio constant of FA and FB.
• Change ratio of FA and FB while keeping FA + FB constant
• Let m1= FA + FB, m2=FB/ FA, then FA =m1/ (1 + m2)

FB =m1 m2 / (1 + m2)

FA

COMPUTER

FB

F

X

c1(s)

m1(s)

g11

g12

g21

m2(s)

c2(s)

g22

Dynamic Decoupling
• Let C(s)=G(s)m(s)
• Or c1(s)=g11(s)m1(s)+g12(s)m2(s)

c2(2)=g21(s)m1(s)+g22(s)m2(s)

• One way to decouple the system is to define new input variables m(s)=D(s)u(s)
Dynamic Decoupling-Continued
• If D(s) is a matrix, then c(s)=G(s)D(s)u(s)
• If D(s) is chosen such that G(s)D(s)=diagonal, then
• u1(s) affects only c1, u2 only affects c2.
• The system is called decoupled.

y1s

-

+

y1

+

m1

u1

plant

Gc1

+

D2

D1

y2

+

+

Gc2

u2

m2

+

-

y2s

Dynamic Decoupling-Continued
• For simplicity, we use
• Then
• We want g11D1+g12=0;g21+g22D2=0
• Hence,D1=-g12/g11;D2=-g21/g22

m1(s)

c1(s)

g11

g12

g21

m2(s)

c2(s)

g22

Dynamic Decoupling-Continued
• This leads to m1=u1-g12/g11u2; m2=u2-g22/g21u2
• c1=(g11+D2g12)u1;c2=(g21D1+g22)u2
Conclusion
• Multivariable control is essential in the industrial applications
• Steady state decoupling is very useful in case of no dynamic system such as mixing
• Dynamic decoupling is very important for high valued-added system such as distillation systems
Homework #4 Multivariable Systems
• Textbook p698
• 21.3, 21.4, 21.7, 21.8