1. Control of Multiple-Input, Multiple-Output Processes
Multiloop controllers
Modeling the interactions
Relative Gain Array (RGA)
Singular Value Analysis (SVA)
Decoupling strategies
Chapter 18

2. Chapter 18 Control of multivariable processes
In practical control problems there typically are a number of process variables which must be controlled and a number which can be manipulated
Example: product quality and through put
must usually be controlled.
Several simple physical examples are shown in Fig
Note "process interactions" between controlled and manipulated variables.
Chapter 18

3. Chapter 18
SEE FIGURE 18.1
in text.

4. Chapter 18

5. Controlled Variables:
Manipulated Variables:
Chapter 18

6. In this chapter we will be concerned with characterizing process
interactions and selecting an appropriate multiloop control
configuration.
If process interactions are significant, even the best multiloop
control system may not provide satisfactory control.
In these situations there are incentives for considering
multivariable control strategies
Definitions:
Multiloop control: Each manipulated variable depends on
only a single controlled variable, i.e., a set of conventional
feedback controllers.
Multivariable Control: Each manipulated variable can depend
on two or more of the controlled variables.
Examples: decoupling control, model predictive control
Chapter 18

7. Chapter 18 Multiloop Control Strategy Typical industrial approach
Consists of using n standard FB controllers (e.g. PID), one for
each controlled variable.
Control system design
1. Select controlled and manipulated variables.
2. Select pairing of controlled and manipulated variables.
3. Specify types of FB controllers.
Example: 2 x 2 system
Chapter 18
Two possible controller pairings:
U1 with Y1, U2 with Y2 …or
U1 with Y2, U2 with Y1
Note: For n x n system, n! possible pairing configurations.

8. Chapter 18 Transfer Function Model (2 x 2 system)
Two controlled variables and two manipulated variables
(4 transfer functions required)
Chapter 18
Thus, the input-output relations for the process can be written as:

9. Chapter 18 Or in vector-matrix notation as,
where Y(s) and U(s) are vectors,
Chapter 18
And Gp(s) is the transfer function matrix for the process

10. Chapter 18

11. Chapter 18 Control-loop interactions Example: 2 x 2 system
Process interactions may induce undesirable interactions between two or more control loops.
Example: 2 x 2 system
Control loop interactions are due to the presence of a third feedback loop.
Problems arising from control loop interactions
i) Closed -loop system may become destabilized.
ii) Controller tuning becomes more difficult
Chapter 18

12. Chapter 18 Block Diagram Analysis
For the multiloop control configuration the transfer function between a controlled and a manipulated variable depends on whether the other feedback control loops are open or closed.
Example: 2 x 2 system, 1-1/2 -2 pairing
From block diagram algebra we can show
Note that the last expression contains GC2 .
Chapter 18
(second loop open)
(second loop closed)

13. Chapter 18

14. Chapter 18

15. Chapter 18
Figure 18.6 Stability region for Example 18.2 with 1-1/2-2 controller pairing

16. Chapter 18
Figure 18.7 Stability region for Example 18.2 with 1-2/2-1 controller pairing

17. Chapter 18 Relative gain array 1) Measure of process interactions
Provides two useful types of information:
1) Measure of process interactions
2) Recommendation about best pairing of controlled and manipulated variables.
Requires knowledge of s.s. gains but not process dynamics.
Chapter 18

18. Chapter 18 Example of RGA Analysis: 2 x 2 system
Steady-state process model,
The RGA is defined as:
where the relative gain, ij, relates the ith controlled variable and the jth manipulated variable
Chapter 18

19. Chapter 18 Scaling Properties: Recommended Controller Pairing
i) ij is dimensionless
ii)
For 2 x 2 system,
Recommended Controller Pairing
Corresponds to the ij which has the largest positive value.
Chapter 18

20. In general:
1. Pairings which correspond to negative pairings should
not be selected.
2. Otherwise, choose the pairing which has ij closest
to one.
Examples:
Process Gain Relative Gain
Matrix, : Array, :
Chapter 18

21. Chapter 18 Recall, for 2X2 systems... EXAMPLE:
Recommended pairing is Y1
and U1, Y2 and U2.
EXAMPLE:
Recommended pairing is Y1 with U1, Y2 with U2.

22. Chapter 18 EXAMPLE: Thermal Mixing System
The RGA can be expressed in terms of the manipulated variables:
Chapter 18
Note that each relative gain is between 0 and 1. Recommended
controller pairing depends on nominal values of W,T, Th, and Tc.
See Exercise 18.16

23. Chapter 18 EXAMPLE: Ill-conditioned Gain Matrix y = Ku 2 x 2 process
y1 = 5 u1 + 8 u2
y2 = 10 u u2
Chapter 18
specify operating point y, solve for u
effect of det K → 0 ?

24. Chapter 18 RGA for Higher-Order Systems: For and n x n system,
Each ij can be calculated from the relation
Where Kij is the (i,j) -element in the steady-state gain matrix,
And Hij is the (i,j) -element of the
Note that,

25. Chapter 18 EXAMPLE: Hydrocracker
The RGA for a hydrocracker has been reported as,
Chapter 18
Recommended controller pairing?

26. Singular Value Analysis
K = W S VT
S is diagonal matrix of singular values
(s1, s2, …, sr)
The singular values are the positive square roots of the eigenvalues of
KTK (r = rank of KTK)
W,V are input and output singular vectors Columns of W and V are orthonormal. Also
WWT = I
VVT = I
Calculate S, W, V using MATLAB (svd = singular value decomposition)
Condition number (CN) is the ratio of the largest to the smallest singular value and indicates if K is ill-conditioned.
Chapter 18

27. CN is a measure of sensitivity of the matrix properties to changes in a specific element.
Consider
 (RGA) = 1.0
If K12 changes from 0 to 0.1, then K becomes a singular matrix, which corresponds to a process that is hard to control.
RGA and SVA used together can indicate whether a process is easy (or hard) to control.
K is poorly conditioned when CN is a large number (e.g., > 10). Hence small changes in the model for this process can make it very difficult to control.
Chapter 18

28. Selection of Inputs and Outputs
Arrange the singular values in order of largest to smallest and look for any σi/σi-1 > 10; then one or more inputs (or outputs) can be deleted.
Delete one row and one column of K at a time and evaluate the properties of the reduced gain matrix.
Example:
Chapter 18

29. Chapter 18 Chapter 18 CN = 166.5 (σ1/σ3) The RGA is as follows:
Preliminary pairing: y1-u2, y2-u3,y3-u1.
CN suggests only two output variables can be controlled. Eliminate one input and one output (3x3→2x2).
Chapter 18
Chapter 18

30. Chapter 18

31. Chapter 18

32. Chapter 18 Alternative Strategies for Dealing with Undesirable
Control Loop Interactions
1. "Detune" one or more FB controllers.
2. Select different manipulated or controlled variables.
e.g., nonlinear functions of original variables
3. Use a decoupling control scheme.
4. Use some other type of multivariable control scheme.
Decoupling Control Systems
Basic Idea: Use additional controllers to compensate for process
interactions and thus reduce control loop interactions
Ideally, decoupling control allows setpoint changes to affect only
the desired controlled variables.
Typically, decoupling controllers are designed using a simple
process model (e.g. steady state model or transfer function model)
Chapter 18

33. Chapter 18

34. Chapter 18 Design Equations:
We want cross-controller, GC12, to cancel out the effect of U2 on Y1.
Thus, we would like,
Since U2  0 (in general), then
Chapter 18
Similarly, we want G21 to cancel the effect of M1 on C2. Thus, we
require that...
cf. with design equations for FF control based on block diagram
analysis

35. Chapter 18 Alternatives to Complete Decoupling
Static Decoupling (use SS gains)
Partial Decoupling (either GC12 or GC21 is set equal to zero)
Process Interaction
Corrective Action (via “cross-controller” or “decoupler”).
Ideal Decouplers:
Chapter 18

36. Chapter 18 Variations on a Theme: Partial Decoupling:
Use only one “cross-controller.”
Static Decoupling:
Design to eliminate SS interactions
Ideal decouplers are merely gains:
Nonlinear Decoupling
Appropriate for nonlinear processes.
Chapter 18

37. Chapter 18

38. Chapter 18
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