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
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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.

Chapter 18
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• 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.
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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
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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
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Chapter 18
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• 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
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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 Note that the last
expression contains GC2 .
Chapter 18
(second loop open)
(second loop closed)
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Chapter 18
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Chapter 18
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Chapter 18
Figure 18.6 Stability region for Example 18.2
with 1-1/2-2 controller pairing
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Chapter 18
Figure 18.7 Stability region for Example 18.2
with 1-2/2-1 controller pairing
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• 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
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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
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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
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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.
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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
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EXAMPLE Ill-conditioned Gain 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 ?
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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,
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EXAMPLE Hydrocracker
The RGA for a hydrocracker has been reported as,
Chapter 18
Recommended controller pairing?
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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
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• 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., gt 10). Hence small changes in the model
  for this process can make it very difficult to
  control.

Chapter 18
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Selection of Inputs and Outputs

• Arrange the singular values in order of largest
  to smallest and look for any si/si-1 gt 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

Chapter 18
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• CN 166.5 (s1/s3)
• 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
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Chapter 18
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Chapter 18
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• 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
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Chapter 18
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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
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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
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• 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
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Chapter 18
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Chapter 18
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