1. Multi-Variable Control
Chapter 4
Multi-Variable Control

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

3. 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
Overhead Composition
Bottom Composition
Pressure in the Column
Problem: How to ‘pair’ variables?

4. Examples of Multivariable Control (2) Blending or two StreamsFT CT
Stream A
Stream C
Stream B
MV’s: Flow of Stream A;
Flow of Stream B.
CV’s: Flow of Stream C
Composition of Stream C

5. Examples of Multivariable Control: (3) Control of a Mixing Tank
Hot
Cold LT TT
MV’s: Flow of Hot Stream CV’s: Level in the tank
Flow of Cold Stream Temperature in the tank

6. 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.

7. 2. Issue of Degree of Freedom (DOF)-Continued (The Level System)Fi F0 h
Equations
Variables: h, Fi, Fo, x
F=4-2=2

8. 2. Issue of Degree of Freedom (DOF)-Continued (The Level System)
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

9. 2. Issue of Degree of Freedom (DOF)-Continued – The flash drum
Feed
Z f ，Pf，T f，Ff
P，T h
vapor Fv
y i
Liquid FL Xi
Steam Ts
Equations:

10. 2. Issue of Degree of Freedom (DOF)-Continued – The flash drum
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

11. 2. Issue of Degree of Freedom (DOF)-Continued – The flash drum
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

12. 2. Issue of Degree of Freedom (DOF)-Continued – The flash drum
What are the manipulated variables? Look for possible valve positions: PC . TC LC
Paring : 4!=24 possibilities;
If Ff is specified 3!=6 possibilities

13. 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)

14. 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

15. 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
Steady state interaction
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)

16. Block Diagram Analysis+
Process
Loop 1 - + + + + + + -
Loop 2

17. Block Diagram Analysis-Continued (One Loop Closed)
Changing y1sp will affects y2

18. Block Diagram Analysis-Continued (Both Loops Closed)
P11,P12,P21 and P22 have a common denominator:
Roots of Q(s) determine stability

19. Measuring Interaction – Brisol’s Relative Gain
Process
FBC C1 C2 m1 m2

20. Measuring Interaction
process m1 m2 y1 y2
process m1 m2 y1 y2
Controller
Set
point

21. Measuring Interactions – Brisol’s Relative Gain - Continued
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

22. Examples of Multivariable Control: (3) Control of a Mixing Tank
Hot
Cold LT TT
MV’s: Flow of Hot Stream CV’s: Level in the tank
Flow of Cold Stream Temperature in the tank

23. 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

24. Derivation of Λ Choose control loop pairs such that λij is
Positive and as close to unity as possible.

25. Example: Blending ProblemCT FT XA F FA FB XA F

26. Example – Blending Problem - Continued

27. 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

28. Steady State Decoupling
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

29. 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)
g11
g21
g12
g22
m1(s)
m2(s)
c2(s)
c1(s)

30. 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.

31. y1s - + y1 + m1 u1
plant
Gc1 + D2 D1 y2 + +
Gc2 u2 m2 + -
y2s

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

33. Dynamic Decoupling-Continued
This leads to m1=u1-g12/g11u2; m2=u2-g22/g21u2
c1=(g11+D2g12)u1;c2=(g21D1+g22)u2
g11
g21
g12
g22
m1(s)
m2(s)
c2(s)
c1(s)

34. 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

35. Homework #4 Multivariable Systems
Textbook p698
21.3, 21.4, 21.7, 21.8