Multiloop and Multivariable Control

Multiloop and Multivariable Control

Multiloop and Multivariable Control • Process Interactions and Control Loop Interactions • Pairing of Controlled and Manipulated Variables • Singular Value Analysis • Tuning of Multiloop PID Control Systems • Decoupling and Multivariable Control Strategies

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Control of Multivariable Processes • Control systems that have only one controlled variable

and one manipulated variable. Single-input, single-output (SISO) control system Single-loop control system • In practical control problems there typically are a

number of process variables which must be controlled and a number of variables which can be manipulated. Multi-input, multi-output (MIMO) control system Example: product quality and throughput must usually be controlled. Note the "process interactions" between controlled and manipulated variables. 2


Several simple physical examples

Process interactions : Each manipulated variable can affect both controlled variables

3


MIMO

SISO

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• 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 5


Multiloop Control Strategy • •

Typical industrial approach Consists of using several 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

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.

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Process Interactions Transfer Function Model (2 x 2 system) • Two controlled variables and two manipulated variables (4 transfer functions required) Y1( s ) Y1( s ) = G p11( s ), = G p12 ( s ) U1 ( s ) U2( s ) Y2 ( s ) Y2 ( s ) = G p 21( s ), = G p 22 ( s ) U1 ( s ) U2( 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) 7


In vector-matrix notation as

Y ( s ) = G p ( s )U ( s )

(18 − 4)

where Y(s) and U(s) are vectors

 Y1 (s )  U1 (s )  Y (s ) =  U (s ) =    Y ( s ) U ( s )  2   2 

(18 − 5)

And Gp(s) is the transfer function matrix for the process  G p11( s ) G p12 ( s )   G p( s ) =  G p 21( s ) G p 22 ( s )  

(18 − 6 )

The steady-state process transfer matrix (s=0) is called the process gain matrix K  G p11( 0 ) G p12 ( 0 )   K  =  11 K = G p 21( 0 ) G p 22 ( 0 )  K 21  

K12  K 22  8


Block Diagram for 2x2 Multiloop Control

1-1/2-2 control scheme

1-2/2-1 control scheme

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Control-loop Interactions • Process interactions may induce undesirable

interactions between two or more control loops. Example: 2 x 2 system Change in U1 has two effects on Y1 (1) direct effect : U1 Gp11 Y1 (2) indirect effect : U1 Gp21 Y2 Gc2 U2 Gp12 Y1

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• Control loop interactions are due to the presence of a

third feedback loop. Example: 1-1/2-2 pairing

The hidden feedback control loop (in dark lines)

• Problems arising from control loop interactions i. Closed-loop system may become destabilized. ii. Controller tuning becomes more difficult.

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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 ) = G p11( s ) U1( s )

(second loop open)

G p12G p 21Gc 2 Y1( s ) = G p11 − U1( s ) 1 + Gc 2G p 22

(second loop closed)

Note that the last expression contains GC2. The two controllers should not be tuned independently 12


Example: Empirical model of a distillation column  12.8e − s  X D ( s )  16.7 s + 1  X (s)  =  −7 s  B   6.6e 10.9s + 1

−18.9e −3s  21s + 1   R( s )    −19.4e −3s   S ( s )  14.4s + 1 

xD set-point response

Single-loop ITAE tuning Pairing

Kc

τI

xD - R

0.604

16.37

xB - S

-0.127

14.46

xB set-point response

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Closed-Loop Stability • Relation between controlled variables and set-points

• Closed-loop transfer functions

where

• Characteristic equation 14


Example: Two P controllers are used to control the process 1.5   2 10s + 1 s + 1  G p (s) =   2   1.5  s + 1 10 s + 1 

Stable region for Kc1 and Kc2 1-1/2-2 pairing

1-2/2-1 pairing

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Pairing of Controlled and Manipulated Variables • Control of distillation column • Controlled variables: xD , xB , P, hD , hB • Manipulated variables: D, B, R, QD , QB

Possible multiloop control strategies

= 5! = 120

Column pressure Condenser heat duty

Reflux drum liquid level Top flow rate Reboiler heat duty Base liquid level

Reflux flow rate

Bottom flow rate Bottom composition

Top composition

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• One of the practical pairing R → xD QB → xB QD → P

PC

D → hD B → hB AC1

LC1

AC2 LC2

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Relative Gain Array (RGA) (Bristol, 1966) • 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.

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Example of RGA Analysis: 2 x 2 system • Steady-state process model y1 = K11u1 + K12u2 y2 = K 21u1 + K 22u2

or

y = Ku

The RGA, Λ, is defined as:

 λ11 λ12  Λ =   λ λ  21 22  where the relative gain, λij , relates the ith controlled variable and the jth manipulated variable

( ∂yi / ∂u j )u open-loop gain λij ≜ = ( ∂yi / ∂u j ) y closed-loop gain

( ∂yi / ∂u j )u : partial derivative evaluated with all of the manipulated variables except uj held constant (Kij)

( ∂yi / ∂u j ) y : partial derivative evaluated with all of the controlled variables except yi held constant

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Scaling Properties: λij is dimensionless

i.

ii.

∑λ = ∑λ ij

i

ij

=1

j

For a 2 x 2 system, λ11 =

1 , K12 K 21 1− K11K 22

 λ 1− λ  Λ =  λ  1 − λ

λ12 = 1 − λ11 = λ21

( λ = λ11 )

Recommended Controller Pairing It corresponds to the λij which have the largest positive values that are closest to one. 20


In general: 1. Pairings which correspond to negative pairings should not be selected. 2. Otherwise, choose the pairing which has λij closest to one. Examples: Process Gain Matrix, K :

Relative Gain Array, Λ :

⇒

1 0 0 1   

K12  0 

⇒

0 1  1 0  

 K11  0 

K12  K 22 

⇒

 K11 K  21

0  K 22 

⇒

 K11  0 

0  K 22 

 0 K  21

1 0 0 1    1 0 0 1   

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For 2 x 2 systems: y1 = K11u1 + K12u2

λ11 =

y2 = K 21u1 + K 22u2

1−

1 , K12 K 21

λ12 = 1 − λ11 = λ21

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. 22


EXAMPLE: Blending System Controlled variables: w and x Manipulated variables: wA and wB

Steady-state gain matrix:

Steady-state model: w = wA + wB xw = wA

⇒ x=

The RGA is:

wA wA + wB wA

 1 K = 1 − x   w

1  − x  w

wB

1− x Λ=  x 1 − x x  w x

Note that each relative gain is between 0 and 1. The recommended controller pairing depends on the desired product composition x. For x = 0.4, w-wB / x-wA (large interactions) For x = 0.9, w-wA / x-wB (small interactions)

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RGA for Higher-Order Systems For a n x n system,

u1 u2 ⋯ y1  λ11 λ12 ⋯ y2 λ21 λ22 ⋯ Λ = ⋮  ⋮ ⋮ ⋱  yn  λn1 λn1 ⋯

un

λ1n  λ2 n 

(18 − 25)

⋮   λnn 

Each λij can be calculated from the relation,

λij = Kij H ij

(18 − 37 )

where Kij is the (i,j) -element of the steady-state gain K matrix,

( ).

Hij is the (i,j) -element of the Η = K In matrix form,

Λ= K ⊗H

-1 T

⊗ : Schur product

(element by element multiplication)

Note :

Λ ≠ KH 24


Example: Hydrocracker The RGA for a hydrocracker has been reported as,

Λ =

u1 u2 u3 u4 y1  0.931 0.150 0.080 −0.164  y2  −0.011 −0.429 0.286 1.154    y3  −0.135 3.314 −0.270 −1.910    y4  0.215 −2.030 0.900 1.919 

Recommended controller pairing?

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Dynamic Consideration An important disadvantage of RGA approach is that it ignores process dynamics

Example:

 −2e − s 1.5e − s  10s + 1  s 1 +  G p (s) =  −s 2  1.5e −s  e  s + 1 10s + 1 

λ11 = 0.64 Recommended controller pairing?

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Singular Value Analysis • Any real m x n matrix can be factored as, T K=WΣV • Matrix Σ is a diagonal matrix of singular values: Σ = diag (σ1, σ2, …, σr) • 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 Σ, 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 ≜

σ1 σr

• A large value of CN indicates that K is ill-conditioned. 27


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

∑ (K ) =   0

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 •

•

•

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 .4 8 K =  0 .5 2  0 .9 0

0 .9 0 0 .9 5 − 0 .9 5

− 0 .0 0 6  0 .0 0 8  0 .0 2 0 

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Multiloop and Multivariable Chapter 18 Control

 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

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Matrix Notation for Multiloop Control Systems Gp

Multi-loop

Single loop CLTF

Y=

G p Gc 1 + G p Gc

Ysp

(

Y = Ι + G pGc

)

-1

G pGc Ysp

Y : (n x 1) vector of control variables Ysp : (n x 1) vector of set-points Gp : (n x n) matrix of process transfer functions Gc : (n x n) diagonal matrix of controller transfer functions

Characteristic equation

1 + G p Gc = 0

(

)

det Ι + G pGc = 0 32


Tuning of Multiloop PID Control Systems • Detuning method – Each controller is first designed, ignoring process interactions – Then interactions are taken into account by detuning each controller • More conservative controller settings (decrease controller gain, increase integral time) – Tyreus-Luyben (TL) tuning

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Biggest log-modulus tuning (BLT) method (Luyben, 1986) • Log-modulus : a robustness measure of control systems – Single loop Lc = 20log

G p Gc

= 20log

1 + G p Gc

G 1+ G

 G  Lc max = max Lc = max 20log  ω ω  1+ G 

A specification of Lc max = 2 dB has been suggested. – Multi-loop Define W = −1 + det Ι + G pGc

(

)

W Lc = 20log 1+W max = max Lc = 2n Luyben suggest that Lc ω where n is the dimension of the multivariable system.

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Tuning Procedure of BLT method 1. Calculate Z-N PI controller settings for each control loop

K c,ZN = 0.45 K cu , τ I ,ZN = Pu 1.2 2. Assume a factor F; typical values between 2 and 5 3. Calculate new values of controller parameters by K ci ,ZN K ci = , τ Ii = F τ Ii ,ZN ; i = 1, 2,⋯, n F 4. Compute W = −1 + det Ι + G G for 0 ≤ ω < ∞

(

p c

(detuning)

)

for example, 2x2 system

(

)

(

det Ι + G pGc = 1 + Gc1G p11 + Gc 2G p 22 + Gc1Gc 2 G p11G p 22 − G p12G p 21

5. Determine

)

 W  Lc max = max 20log  ω  1+W 

6. If Lc max ≠ 2n , select a new value of F and return to step 2 until Lc max = 2n 35


Multiloop IMC Controller • Design IMC controller based in diagonal process transfer functions G p11 ⋯ G p1n    Gp =  ⋮ ⋱ ⋮  G pn1 ⋯ G pnn   

• The IMC controller is designed as Gc = diag [Gc1 Gc 2 ⋮ Gcn ]

with

Gci = G −pii1 − fi

i = 1, 2,⋯, n

• Since the off-diagonal terms of Gp have been dropped, modeling error are always present. 36


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 (decoupler) 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)

37


A Decoupling Control System

decoupler 38

38


Decoupler Design Equations We want cross-controller, T12, to cancel the effect of U2 on Y1. Thus, we would like G U + G U = 0 p11

or

12

p12

22

G p11T12U 22 + G p12U 22 = 0

Because U22 ≠ 0 in general, then

T12 = −

G p12 G p11

Similarly, we want T12 to cancel the effect of U1 on Y2. Thus, we require that,

G p 22T21U11 + G p 21U11 = 0

∴ T21 = −

G p 21 G p 22

Compare with the design equations for feedforward control based on block diagram analysis 39


Variations on a Theme 1. Partial Decoupling: Use only one “cross-controller.”

2. Static Decoupling: Design to eliminate Steady-State interactions Ideal decouplers are merely gains:

T12 = −

K p12

T21 = −

K p 21

K p11 K p 22

3. Nonlinear Decoupling Appropriate for nonlinear processes. 40


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

41 41


Wood-Berry Distillation Column Model  12.8e− s  16.7 s + 1  y1 ( s )    y ( s)  =   2   −7 s  6.6e 10.9s + 1

−18.9e−3s   21s + 1   −3s  −19.4e  14.4 s + 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

4242

43
