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

Chapter 18
Control of Multiple-Input,
Multiple-Output Processes
Multiloop controllers
• Modeling the interactions
• Relative Gain Array (RGA)
• Singular Value Analysis (SVA)
• Decoupling strategies
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 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
SEE FIGURE 18.1
in text.
Chapter 18
• Controlled Variables: xD , xB , P, hD , and hB
Chapter 18
• Manipulated Variables: D, B, R, QD , and QB
 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
Multiloop Control Strategy
Chapter 18
•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
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.
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)
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)
Or in vector-matrix notation as,
Chapter 18
Y ( s )  GP ( s )U ( s )
where Y(s) and U(s) are vectors,
Y1 ( s) 
U1 ( s) 
Y ( s)  
, U ( s)  


Y
(
s
)
U
(
s
)
 2 
 2 
And Gp(s) is the transfer function matrix for the process
G P11 (s) G P12 (s) 
G P (s)  

G
(
s
)
G
(
s
)
P 22
 P 21

Chapter 18
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
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)
(second loop open)
 GP11 ( s),
U1 ( s )
GP12GP 21GC 2 (second loop closed)
Y1 ( s)
 GP11 
U1 ( s )
1  GC 2GP 22
Note that the last expression contains GC2 .
Chapter 18
Chapter 18
Chapter 18
Figure 18.6 Stability region for Example 18.2 with 1-1/2-2 controller pairing
Chapter 18
Figure 18.7 Stability region for Example 18.2 with 1-2/2-1 controller pairing
Relative gain array
Chapter 18
• 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.
Example of RGA Analysis: 2 x 2 system
• Steady-state process model,
Chapter 18
Y1  K11U1  K12U 2
Y2  K 21U1  K 22U 2
The RGA is defined as:
 11 12 
RGA  



22 
 21
where the relative gain, ij, relates the ith controlled
variable and the jth manipulated variable
open- loop gain
 ij 
closed- loop gain
Chapter 18
Scaling Properties:
i) ij is dimensionless
ii)  ij   ij  1.0
i
j
For 2 x 2 system,
1
11 
,
K K
1  12 21
K11K 22
12  1  11   21
Recommended Controller Pairing
Corresponds to the ij which has the largest positive
value.
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,  :
 K 11
 0

0 
K 22 

1 0 
0 1 


 0
K
 21
K 12 
0 

0 1 
1 0 


 K 11
 0

K 12 
K 22 

 K 11
K
 21
0 
K 22 

1 0 
0 1 


1 0 
0 1 


Recall, for 2X2 systems...
Y1  K11U1  K12U 2
Chapter 18
Y2  K 21U1  K 22U 2
EXAMPLE:
K12   2 1.5
K
K   11



K 21 K 22  1.5 2 
 2.29  1.29
  

 1.29 2.29 
11 
1
,
K K
1  12 21
K11K 22
12  1  11   21
Recommended pairing is Y1
and U1, Y2 and U2.
EXAMPLE:
 2 1.5
0.64 0.36
K



1
.
5
2
0
.
36
0
.
64




Recommended pairing is Y1 with U1, Y2 with U2.
Chapter 18
EXAMPLE: Thermal Mixing System
The RGA can be expressed in terms of the manipulated variables:
Wh
W  Wc
W  W
h
  c
 Wh
T W  W
h
 c
Wc
Wh 
Wc  Wh 

Wc 
Wc  Wh 
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
EXAMPLE: Ill-conditioned Gain Matrix
y = Ku
Chapter 18
2 x 2 process
y1 = 5 u1 + 8 u2
y2 = 10 u1 + 15.8 u2
specify operating point y, solve for u
Adj K
u=K y =
y
det K
-1
RGA : 11
K11K 22
K11K 22


K11K 22  K12K21
det K
effect of det K → 0 ?
RGA for Higher-Order Systems:
For and n x n system,
Chapter 18
U1 U 2
Y1  11 12
Y2 21 22



Yn n1 n1
Un
1n 
2 n 


nn 
Each ij can be calculated from the relation
ij  KijHij
Where Kij is the (i,j) -element in the steady-state gain matrix, K :
Y  KU
 
And Hij is the (i,j) -element of the H  K
Note that,
  KH
1 T
.
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 

Y3  0.135 3.314  0.270  1.910


Y4  0.215  2.030 0.900 1.919 
Recommended controller pairing?
Singular Value Analysis
K = W S VT
Chapter 18
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
CN is a measure of sensitivity of the matrix properties to changes in a
specific element.
Consider
 1 0
K 

10 1
 (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.
10.1 0 
 (K ) = 

 0 0.1
CN = 101
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.
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 
0.7292 
 0.5714 0.3766
W   0.6035 0.4093 0.6843 
 0.5561 0.8311 0.0066 
Chapter
18
Chapter18
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 as follows:
 2.4376 3.0241 0.4135 
 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).
Chapter 18
Chapter 18
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
Design Equations:
We want cross-controller, GC12, to cancel out the effect of U2 on Y1.
Thus, we would like,
Chapter 18
T12GP11U2  GP12U2  0
Since U2  0 (in general), then
GP12
T12  
GP11
Similarly, we want G21 to cancel the effect of M1 on C2. Thus, we
require that...
T21GP 22U1  GP 21U1  0
T21  
GP 21
GP 22
cf. with design equations for FF control based on block diagram
analysis
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:
GP12 ( s )
T12 ( s )  
GP11 ( s )
GP 21 ( s )
T21 ( s )  
GP 22 ( s )
Variations on a Theme:
Chapter 18
•Partial Decoupling:
Use only one “cross-controller.”
•Static Decoupling:
Design to eliminate SS interactions
Ideal decouplers are merely gains:
K P12
T12  
K P11
K P 21
T21  
K P 22
•Nonlinear Decoupling
Appropriate for nonlinear processes.
Chapter 18
Chapter 18
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