1. Control Loop Interaction
Lecture 11
Control Loop Interaction

2. MIMO systems
Thus far in the course, we have focused on SISO control systems. However, most process unit operations have a number of manipulated inputs and controlled outputs.
The MIMO systems can be controlled by associating one manipulated variable to one controlled variable, hence forming several SISO control loops.
Unfortunately, there is often interaction between these loops and this may cause the response of one loop to deteriorate once the other loops are closed.

3. MIMO systems
The proper choice of input output pairs can minimize the interaction.
This pairing problem is known as control structure selection.
After studying this chapter, we should be able to use the relative gain array (RGA) to determine the proper input-output pairings.

4. Understanding the Interaction
Let us consider mixing two streams: alcohol with mass flow rate (u1) and water with mass flow rate (u2). Assume that both the concentration (y1) of Alcohol in the outlet stream and the total flow rate (y2 ) are to be controlled.
With simple intuition, we know that changes in both u1 and u2 will affect y1 and y2.

5. Understanding the Interaction
Let us assume that we make a closed-loop to control the composition (y1) using alcohol mass flow rate (u1).
The other loop is open.
The response of the system due to a step set point change in outlet concentration y1.
The interaction is obvious as y2 changes as well.

6. Understanding the Interaction
Let us assume that we make a closed-loop to control the total flow rate (y2) using water mass flow rate (u2).
The other loop is open.
The response of the system due to a step set point change in outlet total flow rate y2.
The interaction obvious as y1 changes as well.

7. Understanding the Interaction
Since we desire to control both (y1) and (y2) at the same time, we need to have both loops closed simultaneously. Consider the system shown below with pairing (y1) with (u1) and (y2) with (u2).
A set-point change in y1 will cause a change in u1. The change in u1 will also affect (y2, total flow). Once y2 is disturbed, u2 will change. Input u2 will disturb the concentration y1, causing the concentration controller to change u1, and we are back to where we started.
The effect of one control loop on the other is referred to as control loop interaction.
Although it is natural to think of these interactions as occurring in a sequential fashion, they actually happen simultaneously.

8. Understanding the Interaction
For the system in the previous slide, the response to a set-point change in the outlet concentration with both loops closed is very oscillatory as shown.
It is interesting to note that, in this example, if we pair (y1) with (u2) and (y2) with (u1), the system will perform better.
Hence, how should our outputs and inputs be paired?

9. The Pairing Problem for 2x2 Processes
For simplicity, we will first treat 2x2 systems, for which there is only two pairing choices as shown below.
The process to be controlled

10. The Pairing Problem for 2x2 Processes
For a 2x2 system, the open-loop input-output relationships are
Which can be rewritten in matrix form as

11. The Pairing Problem for 2x2 Processes
Let us consider the relationship between y1 and u1 under two conditions:
loop 2 is open (b) loop 2 is closed.
What about this case?

12. The Pairing Problem for 2x2 Processes
If the second loop (y2-u2) is closed, there is an additional effect of u1 on y1 through the second controller. This effect can be positive or negative.
This relationship can be derived as (assuming no set-point change in loop 2, r2 = 0),
g11,eff(s) is the effective input-output relationship between u1 and y1, with loop 2 closed. This means that g11,eff(s) should be used as the process transfer function to design controller 1 if loop 2 is closed.

13. Steady-State Effective Gain
Naturally, g11,eff(s) is a function of the controller gc2. What we would really like is a method to determine g11,eff(s) without knowing gc2(s).
We can easily determine this relationship for a limiting case, the steady state. For the steady state, we simply let s → 0. Assuming integral action is used in the controller, we find that as s → 0, gc2(s) → ∞.
Where k11, k21, k12, and k22 are the steady state gains.
We now have a steady-state effective gain relationship between u1 and y1 with the loop between u2 and y2 closed.

14. The Relative Gain Array
Bristol developed a heuristic technique to predict possible interactive effects between control loops when multiple SISO loops are used. This simple measure of interaction is known as the relative gain array (RGA).

15. Definition of the Relative Gain
The relative gain (λij) between input (j) and output (i) is defined as
Gain between input j and output i with all other loops open
Gain between input j and output i with all other loops closed

16. Relative gain between u1 and y1 for 2x2 System
Based on the results we just obtained about the relationship between y1 and u1 when the other loop is open or closed, we can deduce that

17. The RGA
The RGA is simply the matrix whose elements are the individual relative gains.
For a 2 x 2 system, the RGA is

18. The RGA Below is a general relative gain array for n x n system.
Different columns represent the different manipulated (input) variables

19. The RGA
Different rows represent the different controlled (output) variables
Note that in the RGA, the elements of any row or column sum to 1.0.

20. How to calculate the RGA for n x n process?
Given the steady-state gain matrix Kss = G(0), the relative gain array can be calculated using the following relationship:
where:
“ ” denotes element-wise multiplication (not matrix multiplication!!).
“T ” denotes the transpose of a matrix.

21. Example
Given the following steady state gain matrix
The RGA is then

22. Significance of the RGA
Think of it this way: if we want to use a u1-y1 pairing for a SISO controller, we do not want it to matter whether the other loops in the system are closed or not. This tells us that we desire a relative gain close to 1.0.
A value of 1 indicates that specific input variable is the only influence on that output variable.
A value of 0 indicates no relationship between u1 and y1.
Negative values indicate an unstable relationship. Why?

23. Negative relative gain
If λ11 is negative, this means that the gain between u1 and y1 with loop 2 closed (k11,eff) has the opposite sign compared with the gain between them (k11) with loop 2 open.
For open-loop stable processes, the controller gain must have the same sign as the process gain. Hence, a controller for loop 1 designed based on g11 (loop 2 open) will be destabilized when loop 2 is closed and vice versa.
This is unacceptable in practice. Hence, avoid input-output pairings based on a negative relative gain.

24. Input-output Pairing Guidelines
Try to pair on relative gains close to 1.
Do not pair on a negative or zero relative gain
The higher the value of λij, the greater the interaction. Therefore try not to pair yi with uj if the value of λij is large.

25. Example 2 Consider a 2x2 system having the following RGA.
The recommended pairings are y1-u1 & y2-u2
However, for the case
The recommended pairings are y1-u2 & y2-u1

26. Example 3
Consider the following RGA for a system with three inputs and three outputs: How would you choose input-output pairings for this process? Solution 1- First of all, we know that we do not want to pair on a negative relative gain. 2- Second, we do not want to pair with a relative gain of 0 because that means that the particular input does not have an effect on the particular output, when all of the other loops are open. Applying these guidelines, we get the following pairings y1-u1, y2-u3, y3-u2.

27. Example 4
Given the following RGA for a 4x4 distillation column, suggest the input-output pairing.
Looking at column 3, we see only one relative gain that is not negative, therefore, we must pair y2 with u3.
Looking at row 4, we notice that there is only one relative gain that is not negative, therefore, we must pair y4 with u1.
Looking at column 2, there are two nonnegative gains, but we have already used output 2, so we must pair y3 with u2.
This leaves y1 and u4, which, fortunately, has a favorable relative gain (0.8546). If the y1-u4 relative gain had not been favorable, we would have been forced to drop output 1 and input 4 and simply have three control loops for our system.

28. RGA & Controller Tuning
When tuning a set of SISO controllers in a MIMO system, the RGA gives a valuable insight on how to tune the controller to avoid any failure in the control loops.
When a control loop fails, it becomes open. This may happen in a number of ways:
The most obvious way is for an operator to put loop 2 on manual control (i.e., open loop).
Another way is if the second manipulated variable hits a constraint, say if the valve goes all the way open or closed—this means that the second input can no longer affect the second output, which is equivalent to being open loop.

29. Relative Gains Between 0 and 1
If a relative gain is less than 1, this means that
k11,eff (other loops closed) > k11 (other loops open).
If a controller for loop 1 is based on k11 (implicitly assuming that loop 2 is open), then when loop 2 is closed, the control system will respond too rapidly and there will be a chance of instability.
For safety purposes, if the relative gain is less than 1, then loop 1 should be tuned based on loop 2 closed.

30. Relative gains greater than 1
If a relative gain is greater than 1, this means that
k11,eff < k11.
If a controller for loop 1 is based on k11,eff (implicitly assuming that loop 2 is closed), then if loop 2 is opened, the control system will respond too rapidly and there will be a chance of instability.
For safety purposes, if the relative gain is greater than 1, then loop 1 should be tuned based on loop 2 open.

31. Dynamic Considerations in pairing I/O
Note that the RGA gives recommendations to pair I/O variables taking into account the steady-state effects only. If dynamic considerations are taken into account, the recommended pairing may differ.
For example, consider transfer function matrix
Pairings based on RGA: (u1-y1), (u2-y2)
Pairings based on time constants: (u1-y2), (u2-y1).

32. Reference
Bequette, B. W. Process control Modeling, Design, and Simulation. Prentice Hall PTR, 2002, Chapter 13.