Control of Multiple-Input, Multiple- Chapter 18 Output (MIMO)

Control of Multiple-Input, Multiple- Chapter 18 Output (MIMO) Processes 18.1 Process Interactions and Control Loop Interactions 18.2 Pairing of Controlled and Manipulated Variables 18.3 Singular Value Analysis 18.4 Tuning of Multiloop PID Control Systems 18.5 Decoupling and Multivariable Control Strategies 18.6 Strategies for Reducing Control Loop Interactions 1

Control of Multivariable Processes Chapter 18 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 throughput must usually be controlled. Several simple physical examples are shown in Fig. 18.1. Note the "process interactions" between controlled and manipulated variables. 2

Chapter 18 SEE FIGURE 18.1 in text. 3

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Controlled Variables: xD , xB , P, hD , and hB Manipulated Variables: D, B, R, QD , and QB Chapter 18 Note: Possible multiloop control strategies 5! 120 5

In this chapter we will be concerned with characterizing process interactions and selecting an appropriate multiloop control configuration. Chapter 18 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 6

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

Transfer Function Model (2 x 2 system) Chapter 18 Two controlled variables and two manipulated variables (4 transfer functions required) Y1( s ) Y1( s ) GP11( s ), GP12 ( s ) U1 ( s ) U 2( s ) Y2 ( s ) Y2 ( s ) GP 21( s ), GP 22 ( s ) U1 ( s ) U 2( s ) 18 1 Thus, the input-output relations for the process can be written as: Y1( s ) GP11( s )U1( s ) GP12 ( s )U 2 ( s ) Y2 ( s ) GP 21( s )U1( s ) GP 22 ( s )U 2 ( s ) 18 2 18 3 8

Or in vector-matrix notation as, Chapter 18 Y s G p s U s 18 4 where Y(s) and U(s) are vectors, Y1( s ) Y ( s ) Y ( s ) 2 U1( s ) U ( s ) U ( s ) 2 18 5 And Gp(s) is the transfer function matrix for the process GP11( s ) GP12( s ) G p ( s ) G ( s ) G ( s ) P 21 P 22 18 6 9

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Control-loop Interactions Chapter 18 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. 11

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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 Y1( s ) GP11( s ), U1( s ) (second loop open) (18-7) GP12GP 21GC 2 Y1( s ) GP11 (second loop closed) U1( s ) 1 GC 2GP 22 Note that the last expression contains GC2. (18-11) 14

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Relative Gain Array Chapter 18 Provides two types of useful information: 1. Measure of process interactions 2. Recommendation about best pairing of controlled and manipulated variables. Requires knowledge of steady-state gains but not process dynamics. 19

Example of RGA Analysis: 2 x 2 system Chapter 18 Steady-state process model, y1 K11u1 K12u2 y2 K 21u1 K 22u2 The RGA, , is defined as: 11 12 21 22 where the relative gain, ij, relates the ith controlled variable and the jth manipulated variable yi / u j u open-loop gain ij yi / u j y closed-loop gain 18 24 20

Chapter 18 Scaling Properties: i. ij is dimensionless ii. ij i ij 1.0 j For a 2 x 2 system, 1 11 , 12 1 11 21 K12 K 21 1 K11K 22 (18-34) Recommended Controller Pairing It corresponds to the ij which have the largest positive values that are closest to one. 21

Chapter 18 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 Matrix, K : Relative Gain Array, : 1 0 0 1 K12 0 0 1 1 0 K11 0 K12 K 22 1 0 0 1 K11 K 21 0 K 22 1 0 0 1 K11 0 0 K 22 0 K 21 22

For 2 x 2 systems: y1 K11u1 K12u2 Chapter 18 y2 K 21u1 K 22u2 11 1 1 , K12 K 21 12 1 11 21 K11K 22 Example 1: K11 K K 21 K12 2 1.5 K 22 1.5 2 2.29 1.29 Λ 1 . 29 2 . 29 Recommended pairing is Y1 and U1, Y2 and U2. Example 2: 2 1.5 K 1.5 2 0.64 0.36 Λ 0 . 36 0 . 64 Recommended pairing is Y1 with U1 and Y2 with U2. 23

Chapter 18 EXAMPLE: Thermal Mixing System The RGA can be expressed in two equivalent forms: Wh T Tc W Th Tc K Th T T Th Tc Wc Th T Th Tc T Tc Th Tc and Wh Wc W Wc Wh Λ Wh T Wc Wh Wc Wh Wc Wh Wc Wc Wh Note that each relative gain is between 0 and 1. The recommended controller pairing depends on nominal values of T, Th, and Tc. 24

RGA for Higher-Order Systems Chapter 18 For and n x n system, u1 y1 11 y2 21 Λ yn n1 u2 12 22 n1 un 1n 2 n nn 18 25 Each ij can be calculated from the relation, ij Kij H ij 18 37 where Kij is the (i,j) -element of the steady-state gain K matrix, y Ku -1 T . Hij is the (i,j) -element of the H K Note : Λ KH

Example: Hydrocracker Chapter 18 The RGA for a hydrocracker has been reported as, u1 u2 u3 u4 y1 0.931 0.150 0.080 0.164 y2 0.011 0.429 0.286 1.154 Λ y3 0.135 3.314 0.270 1.910 y4 0.215 2.030 0.900 1.919 Recommended controller pairing? 26

Singular Value Analysis Any real m x n matrix can be factored as, Chapter 18 T K W V Matrix is a diagonal matrix of singular values: diag ( 1, 2, , r) The singular values are the positive square roots of the T T eigenvalues of K K ( r the rank of K K). The columns of matrices W and V are orthonormal. Thus, T T WW I and VV I Can calculate , W, and V using MATLAB command, svd. Condition number (CN) is defined to be the ratio of the largest to the smallest singular value, CN 1 r A large value of CN indicates that K is ill-conditioned. 27

Chapter 18 Condition Number CN is a measure of sensitivity of the matrix properties to changes in individual elements. Consider the RGA for a 2x2 process, 1 0 K 10 1 Λ I If K12 changes from 0 to 0.1, then K becomes a singular matrix, which corresponds to a process that is difficult to control. RGA and SVA used together can indicate whether a process is easy (or difficult) to control. 10.1 0 (K ) 0 0.1 CN 101 K is poorly conditioned when CN is a large number (e.g., 10). Thus small changes in the model for this process can make it very difficult to control. 28

Selection of Inputs and Outputs Chapter 18 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: 0.48 K 0.52 0.90 0.90 0.95 0.95 0.006 0.008 0.020 29

Chapter Chapter18 18 0.5714 0.3766 0.7292 W 0.6035 0.4093 0.6843 0.5561 0.8311 0.0066 0 0 1.618 0 1.143 0 0 0 0.0097 0.0151 0.0541 0.9984 V 0.9985 0.0540 0.0068 0.0060 0.0154 0.9999 CN 166.5 (σ1/σ3) The RGA is: 2.4376 3.0241 0.4135 Λ 1.2211 0.7617 0.5407 2.2165 1.2623 0.0458 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). 30

Chapter 18 Question: How sensitive are these results to the scaling of inputs and outputs? 31

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., a steady-state model or transfer function model) 32

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Decoupler Design Equations We want cross-controller, T12, to cancel the effect of U2 on Y1. Thus, we would like, Chapter 18 T12GP11U 22 GP12U 22 0 18 79 Because U22 0 in general, then GP12 T12 GP11 18 80 Similarly, we want T12 to cancel the effect of U1 on Y2. Thus, we require that, T21GP 22U11 GP 21U11 0 GP 21 T21 GP 22 18 76 18 78 Compare with the design equations for feedforward control based on block diagram analysis 34

Variations on a Theme 1. Partial Decoupling: Chapter 18 Use only one “cross-controller.” 2. Static Decoupling: Design to eliminate SS interactions Ideal decouplers are merely gains: K P12 T12 K P11 18 85 K P 21 T21 K P 22 18 86 3. Nonlinear Decoupling Appropriate for nonlinear processes. 35

Chapter 18 Wood-Berry Distillation Column Model (methanol-water separation) CT Feed F Reflux R Distillate D, composition (wt. %) XD Steam S CT Bottoms B, composition (wt. %) XB 36 36

Chapter 18 Wood-Berry Distillation Column Model y1 ( s ) y ( s) 2 12.8e s 16.7 s 1 7s 6.6e 10.9s 1 18.9e 3s 21s 1 3s 19.4e 14.4s 1 u1 ( s ) u ( s) 2 (18 12) where: y1 xD distillate composition, %MeOH y2 xB bottoms composition, %MeOH u1 R reflux flow rate, lb/min u1 S reflux flow rate, lb/min 37 37

Chapter 18