Control of Multiple-Input, Multiple-Output Processes

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. 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. 18.1. • 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. 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. 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. 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. Control-loop interactions • 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. 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. Relative gain array • 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. 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. Scaling Properties: • 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. Recall, for 2X2 systems... EXAMPLE: Chapter 18 Recommended pairing is Y1 and U1, Y2 and U2. EXAMPLE: Recommended pairing is Y1 with U1, Y2 with U2.

22. 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. EXAMPLE: Ill-conditionedGain Matrix y = Ku 2 x 2 process y1 = 5 u1 + 8 u2 y2 = 10 u1 + 15.8 u2 Chapter 18 specify operating point y, solve for u effect of det K→ 0 ?

24. RGA for Higher-Order Systems: For and n x n system, Chapter 18 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. 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. 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: Selection of Inputs and Outputs Chapter 18

29. 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. 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. 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. 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. 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 Previous chapter Next chapter