1. Coupling Analysis of Multivariable Systems ( 多变量系统的关联分析 ) Lei XIE Zhejiang University, Hangzhou, P. R. China

2. Control Problem Discussion For the two-input-two-input controlled system, design your control schemes. Suppose that Initial states:

3. Examples of Multivariable Control Systems: Blending Tank For the two-input- two-input controlled system, the simplest control schemes? #1: F 1  F, F 2  C; #2: F 1  C, F 2  F. Which one is better ?

4. Multi-Loop Control Scheme #1 for a Blending Process

5. Multi-Loop Control Scheme #2 for a Blending Process

6. Simulation Results of Multiloop Scheme (u 1 -y 1, u 2 -y 2 ) Analyze response differences

7. Simulation Results of Multiloop Scheme (u 1 -y 2, u 2 -y 1 ) Analyze response differences

8. Simulation Result Analysis Different Control Paths for PID2

9. Analysis on the Control Path for Controller PID2 Case 1: PID1 is in “M” model Case 2: PID1 is in “A” model

10. Analysis on the Control Path for Controller PID2 (cont.) Case 1: PID1 is in “M” model Case 2: PID1 is in “A” model

11. Concept of Relative Gain Definition of Relative Gain Calculation of Relative Gain Meaning of Relative Gain Matrix Calculation of Relative Gain Matrix Relative gain for control path u 2  y 2 ?

12. Concept of Relative Gain for a 2*2 Multivariable Process Open-loop Gain K 22 Closed-loop Gain K’ 22 Relative Gain Kinds of Process Gain Relative gain for other paths ?

13. Example of Relative Gain Calculation Steady-state model in incremental mode: Other relative gains ?

14. Computation of Relative Gain for 2*2 Control System Steady-state equation: μ 11 ?

15. Definition of Relative Gain Matrix The relative gain describes the “effect” on this loop of the other loops.

16. Calculation Example of Relative Gain Matrix Steady-state model in incremental mode: Properties of Relative Gain Matrix ?

17. Property of Relative Gain Matrix Summation of all the terms in each row and in each column must equal 1. 2×2 systems: 3×3 systems:

18. Computation of Relative Gain for n×n Systems Note: “ ● ” means the multiplication of matrix elements

19. Application of Relative Gain Meaning of Relative Gain CVs and MVs Pairing Application Examples Example 1: Example 2: Problem: which are the best pairs and why?

20. Significance of Relative Gain : no interaction between the particular loop and all other loops, or possible offsetting interaction. : the open-loop gain is very small, or the closed-loop gain is very large. : the open-loop gain is very large, or the closed-loop gain is very small. : the signs of the closed-loop gain and the open-loop gain are different

21. Pairing Rule of Multiloop Systems Bristol (1966) : To minimize the interaction between loops, always pair on RGM elements that are closest to 1.0. Avoid negative pairings. Example 1: Example 2: Problem: which are the best pairs and why?

22. Pairing of a Blending Process Steady-state model: Problem: it is a nonlinear model, how can you analyze the coupling between two loop ? Suppose C 1 > C 2

23. Pairing of a Blending Process (cont.) 1. Obtaining steady-state process gain:

24. Pairing of a Blending Process (cont.) 2. Obtaining relative gain matrix:

25. Pairing of a Blending Process (cont.) 3. Pairing CVs and MVs using RGM If F 1 >F 2, the correct pairing is F － F 1, C － F 2 ; If F 2 >F 1, the correct pairing is F － F 2, C － F 1. If F 2 =F 1, which is the correct pairing ?

26. Exercise 8.1 It is assumed that the open-loop gain matrix of a 3×3 multivariable controlled process is Please calculate the relative gain matrix of the process, select the best pairing of CVs and MVs, and analyze if a decoupler is needed.

27. Next Topic: Decoupling Control of Multivariable Systems When Necessary to Design Decoupler Linear Decoupler Design from Block Diagrams Nonlinear Decoupler Design from Basic Principles Application Examples

28. Problem Discussion for Next Topic For the controlled system, design your decoupling control systems and simulate your solution with SimuLink. If Initial states: