CHE 185 – PROCESS
CONTROL AND DYNAMICS
PID CONTROL APPLIED TO
MIMO PROCESSES
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• COMPARISON BETWEEN CENTRALIZED AND
DECENTRALIZED CONTROL
• DECENTRALIZED (MULTILOOP) CONTROL HAS
SEVERAL SINGLE LOOP CONTROLS IN PARALLEL
FOR A SINGLE PROCESS. REACTION SYSTEM
EXAMPLE:
AC
TC
LC
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• THE ALTERNATE TO THIS APPROACH IS
CENTRALIZED (COORDINATED) CONTROL AND FOR
THE SAME SYSTEM IT WOULD LOOK LIKE:
A
CONTROLLER
T L
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• THERE ARE SPECIFIC ADVANTAGES TO WORKING
WITH SINGLE INPUT SINGLE OUTPUT (SISO) LOOPS
BECAUSE THEY CAN BE TUNED AND OPTIMIZED
INDEPENDENTLY
• THE INTERACTIONS (COUPLING) CAN TAKE PLACE
THROUGH THE PROCESS
– FOR THE EXAMPLE SYSTEM , THE COMPOSITION MIGHT
BE AFFECTED BY TEMPERATURE CONTROL BECAUSE
THE TEMPERATURE AFFECTS THE REACTION RATE
– SIMILARLY, TEMPERATURE MIGHT AFFECT THE MEDIA
DENSITY, WHICH COULD HAVE SOME IMPACT ON THE
RESIDENCE TIME IN THE UNIT AND THE LEVEL CONTROL
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• THERE ARE SPECIFIC ADVANTAGES TO WORKING
WITH SINGLE INPUT SINGLE OUTPUT (SISO) LOOPS
BECAUSE THEY CAN BE TUNED AND OPTIMIZED
INDEPENDENTLY
• THE INTERACTIONS (COUPLING) CAN TAKE PLACE
THROUGH THE PROCESS
– FOR THE EXAMPLE SYSTEM , THE COMPOSITION MIGHT
BE AFFECTED BY TEMPERATURE CONTROL BECAUSE
THE TEMPERATURE AFFECTS THE REACTION RATE
– SIMILARLY, TEMPERATURE MIGHT AFFECT THE MEDIA
DENSITY, WHICH COULD HAVE SOME IMPACT ON THE
RESIDENCE TIME IN THE UNIT AND THE LEVEL CONTROL
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• MIMO REFERS TO
THE GENERAL
CONFIGURATION OF
THE CONTROL
SYSTEM WITH
MULTIPLE INPUTS
AND OUTPUTS
• CONSIDER THE
SYSTEM SHOWN AS
FIGURE 15.1.1
EXAMPLE OF A 2×2
MIMO PROCESS
• TWO INPUTS:
SETPOINTS FOR
FLOW
CONTROLLER ON
STEAM AND
REFLUX.
• TWO OUTPUTS:
COMPOSITION OF
PRODUCTS B AND
D
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• THE STEADY STATE TRANSFER FUNCTION FOR
THIS COMBINED SYSTEM CAN BE EXPRESS AS:
Y1( s )  G11( s ) G12 ( s )  C1( s ) 
Y ( s )  G ( s) G ( s) C ( s )
 2   21
 2 
22
• STEADY STATE COUPLING
– THE RELATIVE GAIN ARRAY (RGA) IS A MATRIX OF THE
GAINS AND PROVIDES A MEASURE OF THE STEADYSTATE EFFECTS OF COUPLING
– FOR THE 2X2 SYSTEM ABOVE:
 

RGA  



 21
22 
11
12
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• THE Λ TERMS HAVE THE FOLLOWING VALUES AS
PARTIAL FUNCTIONS
 Y1 
 Y1 
11 
21 


 C1  C2
 Y1 



C
 1  Y2
 Y2 


 C1  C2
 Y2 


 C1  Y1
12 
22 


 C2  C1
 Y1 



C
 2  Y2
 Y2 


 C2  C1
 Y2 


 C2  Y1
• THE RELATIVE GAIN ARRAY (RGA) REPRESENTS
THE PROCESS GAIN WITHOUT COUPLING
RELATIVE TO THE PROCESS GAIN WITH
COUPLING
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• EVALUATION OF EACH OF THE TERMS IN THE
RGA HELPS DETERMINE THE DEGREE OF
COUPLING
• AS THE VALUE FOR THE TERM → 0, THERE IS
LESS INTERACTION, COUPLING INCREASES AS
THE VALUE INCREASES.
• EXAMPLES OF INTERACTION ARE SHOWN IN THE
TABLE ON THE NEXT SLIDE
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• THE STEADY STATE GAIN MATRIX FOR THE 2X2
SYSTEM IS:
K K
 11 12 
K

K
K
 21 22 
• THE TWO CONTROLLERS MUST BE TUNED
SIMULTANEOUSLY TO ACHIEVE STABILITY AND
PERFORMANCE
• .THE VALUES OF THE MANIPULATED VARIABLES
THAT MEET CONTROL VARIABLE TARGETS MUST
BE DETERMINED SIMULTANEOUSLY
• .AS THE INTERACTION INCREASES, THE SYSTEM
BECOMES MORE MULTIVARIANT AND LESS
SINGLE VARIABLE
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• OTHER RGA CHARACTERISTICS
• VALUES ARE SCALE INDEPENDENT - SO WILL BE
CORRECT AS LONG AS UNITS ARE CONSISTENT
FOR ALL TERMS
• AS SHOWN IN EQUATION 13.2, THE VALUES CAN
BE DETERMINED FROM THE OPEN LOOP DATA.
• OTHER TERMS CAN BE EVALUATED BY TAKING
ADVANTAGE OF THE FACT THAT THE RGA ROWS
AND COLUMNS MUST SUM TO ZERO.
• RELATIVE GAIN TERMS CAN BE VERY SENSITIVE
TO ERRORS IN THE GAIN CALCULATION
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• DYNAMIC FACTORS IN CONFIGURATION
SELECTION
• THE RGA VALUES DISCUSSED ABOVE ARE
FOR SYSTEMS WITH THE SAME DYNAMIC
BEHAVIOR
• WHEN ONE OF THE LOOPS HAS FASTER
OR SLOWER DYNAMICS THAN THE
OTHERS, THE SELECTION OF PAIRINGS
CAN BE CHANGED
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• A FAST LOOP COUPLED WITH A SLOW
LOOP CAN BE TUNED AS THOUGH IT IS A
SINGLE LOOP CONTROLLER WITHOUT
INTERACTION
• FOR A SLOW LOOP, THE TUNING CAN BE
ADJUSTED BY MULTIPLYING THE SINGLE
LOOP CONTROLLER GAIN BY THE
APPROPRIATE λ TERM IN THE RGA - THUS
PROVIDING A BIAS FOR THE GAIN
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• WHEN LOOPS HAVE SIMILAR DYNAMICS,
BOTH LOOPS NEED TO BE DETUNED
FROM THEIR SINGLE LOOP SETTINGS
• MANUAL TUNING IS RECOMMENDED
• WORKSHOP #14 IN THE CONTROL
STATION PACKAGE IS AN EXAMPLE OF
THIS SITUATION.
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• SENSITIVITY TO DISTURBANCES
• PROCESS VARIABLES HAVE DIFFERENT
RESPONSES TO VARIOUS TYPES OF
DISTURBANCES
– EXAMPLE 15.3 IN BOOK LOOKS AT
DISTILLATION COLUMN DYNAMICS FOR
VARIOUS CONFIGURATIONS
– THE RESPONSES MAY NOT BE THE SAME FOR
ALL TYPES OF DISTURBANCES, COMPOSITION
VS. FLOW FOR DISTILLATION COLUMNS
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• DECOUPLERS
• DECOUPLING IS APPLIED WHEN THERE ARE
TWO SIGNIFICANT OUTPUTS THAT HAVE
EQUAL IMPORTANCE AND WHICH ARE
NEGATIVELY AFFECTED BY COUPLING
• METHODS TO DECOUPLE THE VARIABLES
INCLUDE
– ALTERING THE MANIPULATED VARIABLES
– ALTERING THE CONTROLLED VARIABLES
– ALTERING THE FEEDBACK CONTROL
CALCULATION
PID FOR MULTIPLE INPUT/MULTIPLE
OUTPUT (MIMO) SYTEMS
• DECOUPLERS
• A DECOUPLER ACTS LIKE A FEED FORWARD
CONTROL AND CAN BE INSERTED IN THE
LOOP TO CHANGE THE FEEDBACK SIGNAL