Transcript Model Predictive Control

Chapter 16
Model Predictive Control
Single Loop Controllers
y1,sp
+
-
GC1
u1
G11(s)
+
+
C on trol Loop 1
y1
G21(s)
G12(s)
-
GC2
u2
G22(s)
C on trol Loop 2
++
+
y2,sp
y2
MPC Controller
d1
G'd1(s)
y1,sp
y2,sp
G'11(s)
+ +
+
u1
y1
G'21(s)
MPC
Controller
G'12(s)
G'22(s)
+ +
+
u2
G'd2(s)
Process
d2
y2
Model Predictive Control
• Most popular form of multivariable control.
• Effectively handles complex sets of
constraints.
• Has an LP on top of it so that it controls
against the most profitable set of constraints.
• Several types of industrial MPC but DMC is
the most widely used form.
DMC is based on Step Response
Models
• Allow the development of empirical
input/output process models.
• The coefficients correspond to the step
response behavior of the process
yˆ (ti )
ai 
u (t0 )
Example of a SRM
•
• t
i
u
y(t)
ai
•
•
•
•
•
•
•
•
•
0
1
2
3
4
5
6
7
8
1
0
0
0
0
0
0
0
0
0
0
0.63
0.87
0.95
0.98
0.99
1.00
1.00
0
0
0.63
0.87
0.95
0.98
0.99
1.00
1.00
0
1
2
3
4
5
6
7
8
Open-Loop Step Test for Thermal
Mixer
Temperature (ºC)
51
50
49
48
0
20
40
60
Time (s)
80
100
SRM for the Thermal Mixer (u=0.05)
i
0
1
2
3
4
5
6
7
8
9
ti
5
10
15
20
25
30
35
40
45
50
yi
50.00
49.77
49.35
49.08
48.94
48.87
48.84
48.83
48.82
48.82
yˆ i
0.0
-0.23
-0.65
-0.92
-1.06
-1.13
-1.16
-1.17
-1.18
-1.18
ai
0.0
-4.68
-13.0
-18.4
-21.2
-22.6
-23.2
-23.5
-23.7
-23.7
Example of SRM Applied to a
Process with Complex Dynamics
2.5
2.3
a6
2.1
a5
a1 a2
y
• •
1.9
•
1.7
•
•
•
a8
•
•
a7
a4
a3
1.5
0
100
200
Time
300
400
• SRM: [0 0 -0.3 -0.1 0.05 0.1 0.2 0.3 . . .]
Complex Dynamics
• An integrating process (e.g., a level) is
modeled the same except that the Tss is
based on attaining a constant slope (i.e., a
constant ramp rate).
• An integrating variable is referred to as a
“ramp variable”.
SRM for Ramp Function
•
y4
y3
y2
y0
•
y1
•
•
•
 Ts
Tss
 u=1
Time
Using SRM to Calculate y(t)
• Given: SRM: [0 0.4 0.8 0.9 1.0]
– y0=5
u=5
Ts=20 sec
• Solve for y(t):
–
–
–
–
–
–
–
y = y0 + u SRM
y1=5 + 5×0 = 5;
y2 = 5 + 5×0.4 = 7;
y3 = 5 +5×0.8 = 9;
y4 = 5 + 5×.9 = 9.5
y5 = 5 + 5×1 = 10
or in vector form
y = [5 7 9 9.5 10 10 10 …]
y(20 s)=5; y(40 s)=7; y(60 s)=9; y(80 s)=9.5
y(100 s)=10; y(120 s)=10; y(140 s)=10 ...
Class Exercise
• Given: SRM: [0 0.4 0.8 0.9 1.0]
– y0=5
u=-2
• Solve for y(t):
Ts=20 sec
Solution to Class Exercise
• Given: SRM: [0 0.4 0.8 0.9 1.0]
– y0=5
u=-2
Ts=20 sec
• Solve for y(t):
–
–
–
–
–
y = y0 + u SRM
y1=5 + -2×0 = 5;
y2 = 5 + -2×0.4 = 4.2;
y3 = 5 +-2×0.8 = 3.4;
y4 = 5 + -2×.9 =3.2
y5 = 5 + -2×1 = 3
or in vector form
y = [5 4.2 3.4 3.2 3 3 3 …]
Calculating SRM from Step
Response Data
• Remember previous equation:
– y = y0 + u SRM
• Solving for SRM yields:
– SRM = (y-y0)/u
Example of the Calculation of
SRM from Step Response Data
Time (s)
0
10
20
30
40
50
u(t)
1000 lb/h
0
0
0
0
0
yi
351 F
352 F
355 F
356 F
357 F
357 F
ai
0
0.001 F-h/lb
0.004 F-h/lb
0.005 F-h/lb
0.006 F-h/lb
0.006 F-h/lb
Class Exercise for the
Calculation of the SRM from
Step Response Data
Time (s)
0
10
20
30
40
50
u(t)
500 lb/h
0
0
0
0
0
yi
55 psig
53 psig
48 psig
46 psig
45 psig
45 psig
ai
Class Exercise for the
Calculation of the SRM from
Step Response Data
Time (s)
0
10
20
30
40
50
u(t)
500 lb/h
0
0
0
0
0
yi
55 psig
53 psig
48 psig
46 psig
45 psig
45 psig
ai
0
-.004 psig-h/lb
-.014 psig-h/lb
-.018 psig-h/lb
-0.02 psig-h/lb
-0.02 psig-h/lb
Simulation Demonstration
• A thermal mixing process mixes two streams
with different temperatures (25ºC and 75ºC)
to produce a product (about 50ºC).
• Consider a 10% increase (0.05 kg/s) increase
in the flow rate of colder stream.
• Develop SRM from step response.
Step Test Result and SRM
Time (s)
5
12
19
26
33
40
47
54
61
68
75
Outlet Temperature (ºC)
50
49.77
49.35
49.08
48.94
48.87
48.84
48.82
48.816
48.813
48.811
SRM Coefficients (ºC-s/kg)
-4.68
-13.0
-18.4
-21.2
-22.6
-23.2
-23.5
-23.7
-23.7
-23.8
Comparison for the Same u
(+10%)
CV
Controlled Variable
50.2
50
49.8
49.6
49.4
49.2
49
48.8
48.6
T Out
MPC OpenLoop Model
Output
0
200
400
Time (sec)
600
Comparison for the Same u
(-10%)
Controlled Variable
51.5
CV
51
T Out
50.5
MPC Open-Loop
Model Output
50
49.5
0
200
400
Time (sec)
600
Demonstration Conclusions
• MPC model agrees closely for the same step
input change, but shows significant
mismatch for a different size of input
change due to process nonlinearity.
Nonlinear Effects
• This SRM will not accurately represent
negative changes in the MV or different
magnitude MV changes due to process
nonlinearity.
• Therefore, a better way to develop a SRM is
to make a number of MV changes with
different magnitude and directions and
average the results to get the SRM.
SRM’s for Different Size u’s
u
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
+10%
0.05
-4.68
-13.0
-18.4
-21.2
-22.6
-23.2
-23.5
-23.7
-23.7
-23.8
+5%
0.025
-4.70
-13.2
-18.7
-21.6
-23.0
-23.7
-24.1
-24.2
-24.3
-24.4
-5%
-0.025
-4.73
-13.4
-19.2
-22.4
-24.0
-24.8
-25.2
-25.4
-25.5
-25.6
-10%
-0.05
-4.75
-13.5
-19.5
-22.8
-24.5
-25.4
-25.8
-26.1
-26.2
-26.3
Average
-4.72
-13.3
-19.0
-22.0
-23.3
-24.3
-24.7
-24.9
-24.9
-25.0
Another Approach
• Use the average ai’s
Using Average ai’s for u=+10%
F1
Time
0
1
2
3
4
5
6
7
8
9
0.5
0.5
0.5
0.5
0.5
0.5
0.514376
0.524619
0.531917
0.537116
T Out
50
50
50
50
50
50
49.99804
49.98688
49.96313
49.92698
T1
MPC Open-Loop
50
50
50
50
50
50
50
49.96629
49.93257
49.89886
Output
F1 S.P.
Model
0.5
25
0.5
25
0.5
25
0.5
25
0.5
25
0.5
25
0.55
25
0.55
25
0.55
25
0.55
25
Using Average ai’s for u=-10%
Time
F1
T Out
0
0.5
50
1
0.5
50
2
0.5
50
3
0.5
50
4
0.5
50
5
0.5
50
6 0.485624 50.00196
7 0.475381 50.01314
MPC Open-Loop
T1
50
50
50
50
50
50
50
50.03371
Model
F1 S.P.
Output
25
0.5
25
0.5
25
0.5
25
0.5
25
0.5
25
0.5
25
0.45
25
0.45
Using Average ai’s for u=+5%
CV
Controlled Variable
50.2
50
49.8
T Out
49.6
49.4
49.2
0
200
400
Time (sec)
600
MPC OpenLoop Model
Output
Relation to MPC Model
Identification
• Developing SRM’s that represent an
“average” response is the basis of good
MPC models.
Class Exercise
• Using the composition mixer simulator
(CMIXER), use a +5% step input change
(0.025 kg/min) to generate a SRM for this
process
• First, implement the step input change
• Next, determine the model interval
• Then, calculate the SRM coefficients
• Finally, compare simulator result to MPC
model prediction for u (+10% to -10%)
SRM Summary
• SRM’s represent the dynamic behavior of a
dependent variable (output variable) for a
process for a unit step change in an input.
• SRM’s are empirical models that are
flexible enough to model complex dynamics
• SRM’s are linear approximations of process
behavior and can be estimated using step
responses for the process.
SISO MPC Controller
Application of SRM for a Series of
Input Changes
A Series of Input Changes
• Definition of the SRM is based on a single
step input change.
• But feedback control requires a large
number of changes to the input variables.
• To use a SRM to predict the output behavior
for a series of input changes, linear
superposition is assumed, i.e., the process
response is equal to the sum of the
individual step responses.
Superposition of Two Step Input
Changes
Combined
Response
u1
u2
u2
u
u1
Time
Superposition of Two Step Input
Changes
u1
Combined
Response
u2
u1
u2
u
Time
Equation Form for Two
Sequential Input Changes
• Consider the the response of a process to two step
input changes separately
u0 is applied at t  0 u1 is applied at t  Ts
y1  y0  u0 a1
y2  y0  u0 a2
y2  y0  u1a1
y3  y0  u0 a3
y3  y0  u1a2
y4  y0  u0 a4
y4  y0  u1a3
.
.
.
.
Combined Response for Two
Input Changes
Using superposition to calculate
the combined response
y1  y0  u0 a1
y2  y0  u0 a2  u1a1
y3  y0  u0 a3  u1a2
y4  y0  u0 a4  u1a3
.
.
Combined Response for a Series
of 4 Input Changes
Using superposition to calculate
the combined response
y1  y0  u0 a1
y2  y0  u0 a2  u1a1
y3  y0  u0 a3  u1a2  u2 a1
y4  y0  u0 a4  u1a3  u2 a2  u3a1
y5  y0  u0 a5  u1a4  u2 a3  u3 a2  u4 a1
Matrix Form for Combined Response
for a Series of 4 Input Changes
 y1  y0   a1
 y  y  a
0
 2
 2
 y3  y0   a3

 
 y4  y0    a4
 y  y  a
5
0
4

 
 y6  y0   a4
 y  y  a
 7
0
  4
0 0 0 

a1 0 0 
 u0 
a2 a1 0  

  u1 
a3 a2 a1 
 u2 
a4 a3 a2  

  u3 
a4 a4 a3 
a4 a4 a4 
The Dynamic Matrix
 a1
a
 2
 a3

A   a4
a
4

 a4
a
 4
0 0 0 

a1 0 0 
a2 a1 0 

a3 a2 a1 
a4 a3 a2 

a4 a4 a3 

a4 a4 a4 
• The Dynamic Matrix
determines the dynamic
response of the process
to a series of input
changes, nc, for a
prediction horizon of np
steps into the future.
• The dynamic Matrix is
constructed from the
coefficients of the SRM
• A is np×nc; SRM is m
(this case has 4 inputs
and 7 output predictions)
Zero’s in Dynamic Matrix
• Where do they come from?
• Consider the first row: y1 is cannot be
affected by u1, u2, and u3 because y1 is
measured before u1, u2, and u3 are
applied.
• Likewise, y2 is cannot be affected by u2
and u3
The Matrix Form of the
Generalized Model Equation
 y1  y0 
y  y 
0
 2
y  = Au where y    . 


 . 
 . 


Dynamic Matrix Example
SRM: (0.1 0.5 0.9 1.0) nc  m; n p  1.5m
0.1 0 0 0 
0.5 0.1 0 0 


0.9 0.5 0.1 0 
A

1.0 0.9 0.5 0.1 
1.0 1.0 0.9 0.5 


1.0 1.0 1.0 0.9 
Class Exercise
SRM: (0.3 1.0)
A is a 3 by 3 matrix (i.e., n p =1.5m
and nc  m)
Solution to Class Exercise
SRM: (0.3 1.0)
0.3 0 


A  1.0 0.3 
1.0 1.0 
SISO MPC Controller
Prediction Vector
Previous Inputs Affect Future
Response of the Process
u
Future Inputs
Past Inputs
-10
-5
0
Time
5
10
Previous Inputs Affect Future
Response of the Process
Past Behavior
y
Setpoint
Future Prediction
-10
-5
0
Time
5
10
Prediction Vector
• The prediction vector, yP, contains the
combined effect of the previous input
changes on the future values of the output
variable.
• The prediction vector is calculated from the
product of the previous input changes and
the prediction matrix, which is constructed
using the coefficients of the SRM.
Process Model
• The future values of the output variable are
equal to the sum of the contribution from
previous inputs (prediction vector) and the
contribution from future change in the input
variable:
y = y + Au
where u is the vector of input changes
P
Process Model
• In order to correct for the mismatch
between the predicted value of y at the
current time and the measured value, a
correction vector, e, is added to the process
model equation.
y = y + Au + e
P
where
e  y (t0 )  y (t0 )
P
Correction for Mismatch in
Ramp Variable
• The error between the predicted value of
y(t0) and the measured value is applied as
before: e= y(t0) - yP(t0)
• An unmeasured disturbance can affect the
slope of the modeled SRM
• Therefore, an additional tuning factor for
ramp variables is the “rotation factor (RF)”,
which is the fraction of the mismatch e that
is used to change the slope of the SRM of
the ramp variable. Slopenew=Slopeold+RF×e
SISO MPC Controller
DMC Controller
Development of DMC Control
Law
Minimize the error from setpoint:

2
Min    ( ysp  yi ) 
i


Substituting for yi
   ( ysp  y  Aui  e )
P
i
i
2
DMC Control Law
Define Ei  ysp  y  e
P
i
   ( Ei  Aui )
2
i
DMC Control Law: u = (A A) A E
T
-1
T
Analysis of DMC Controller
• (ATA)-1AT is a constant matrix that is
determined explicitly from the coefficients
of the SRM.
• If you double the number of coefficients in
the SRM, you will increase the number of
terms in the dynamic matrix by a factor of 4
• E takes into account changes in the setpoint,
the influence of previous inputs, and the
correction for model mismatch.
• A full set of future moves are determine by
this control law at each control interval.
Moving Horizon Controller
Past Behavior
y
Setpoint
Future Prediction
-10
-5
0
Time
5
10
Moving Horizon Controller
• Even though the DMC controller calculates
a series of future moves, only the first of the
calculated moves is actually implemented.
• The next time the controller is called (Ts
later), a new series of moves is calculated,
but only the first is applied.
• If the SRM were perfect, the subsequent set
of first moves would be equal to the first
series of moves calculated by the DMC
controller.
SISO MPC Controller
Implementation Details
Implementation Details
• Choose Ts based on when new information
is available on the CV. E.g., analyzer update.
For temperature sensor, use Ts=10-15 s.
• Model horizon: m= Tss / Ts (normally set
m=60-90 coefficients. Always set m>30.
• Control horizon: nc=½m
• Prediction horizon: np=m+nc
Controller Implementation
Example
• Consider a SISO DMC temperature control
loop that has an open loop time to steadystate equal to 20 minutes.
• Applying the previous equations:
Ts  10s
Tss
20  60 s
m

 120
Ts
10s
n  1.5m  180
Class Example
• Consider a SISO DMC controller applied to
the overhead product of a C3 splitter. The
overhead composition analyzer update
every 10 min and the time to steady-state
for the overhead composition for a reflux
flow rate change is 7 hours. Determine m,
np and nc for this controller.
Solution
Ts  10 min
7  60 min
m
 42
10 min
n  1.5m  63
SISO MPC Controller
Tuning
DMC Controller Tuning
• Previous DMC control law results in
aggressive control because it is based on
minimizing the error from setpoint without
regard to changes in the MV.
• (ATA)-1 is usually ill-conditioned due to
normal levels of process/model mismatch.
• These problems can be overcome by adding
a diagonal matrix, Q, to the Dynamic
Matrix, A.
DMC Controller with Move
Suppression
 q 0 0 ... 0 
0 q 0 ... 0 


Q = 0 0 q ... 0  q is the move suppression factor


...

0 0 0 ... q 
Q is a diagonal matrix (nc  nc )
The Dynamic Matrix is augmented with Q,
A
A 
Q 
The Dynamic Matrix with the
Move Suppression Factor Added
 a1
a
 2
 a3

a3

Example Dynamic Matrix: A 
a
 3
q
0

0
0 0 
a1 0 
a2 a1 

a3 a2 
a3 a3 

0 0
q 0

0 q 
Move Suppression

A



  a1

 a
 Minimize   2

  a3
 Error
 a

 3

  a3


 Minimize   q

 0
 Move

 Size
 0


0 0
a1 0
a2 a1
a3 a2
a3 a3
0
0
q
0
0
q
Δu 
E
 e1 
e 

 2

 e3 

 

e4 

  u1   
   u    e5 
  2   e6 
  u3   
e7

 

0 

 

0 
 0 
DMC Control Law
DMC control law remains the same as before
except the A now contains Q.
DMC Control Law: u = (A A) A E
T
-1
T
Effect of Move Suppression
Factor
• The larger q, the greater the penalty for
MV moves. The smaller q, the greater the
penalty for errors from setpoint.
Tmixer DMC Tuning Example
Q=300
CV
Controlled Variable
56
55
54
53
52
51
50
49
T Out
T Out S.P.
0
50
100
Time (sec)
150
200
Tmixer DMC Tuning Example
Q=100
CV
Controlled Variable
56
55
54
53
52
51
50
49
T Out
T Out S.P.
0
50
100
Time (sec)
150
200
Tmixer DMC Tuning Example
Q=60
CV
Controlled Variable
56
55
54
53
52
51
50
49
T Out
T Out S.P.
0
50
100
Time (sec)
150
200
Tmixer DMC Tuning Example
Q=30
Controlled Variable
58
CV
56
54
T Out
52
T Out S.P.
50
48
0
50
100
Time (sec)
150
200
Tmixer DMC Tuning Example
Q=20
Controlled Variable
58
CV
56
54
T Out
52
T Out S.P.
50
48
0
50
100
Time (sec)
150
200
Tmixer DMC Tuning Example
Q=10
CV
Controlled Variable
60
58
56
54
52
50
48
T Out
T Out S.P.
0
50
100
Time (sec)
150
200
Controller Tuning
• Reliability versus performance.
• Meeting the overall process objectives.
Checking the Controller Tuning
for a Setpoint Changes in the
Opposite Direction Q=30
CV
Controlled Variable
51
50
49
48
47
46
45
44
T Out
T Out S.P.
0
50
100
Time (sec)
150
200
Class Exercise
SISO DMC Tuning
• Tune the DMC controller for the CMixer
SISO MPC Controller
Testing
Controller Testing
• Use disturbance upsets to test controller
performance
Controller Disturbance Rejection
Q=60
Controlled Variable
52
CV
51.5
51
T Out
50.5
T Out S.P.
50
49.5
0
50
100
Time (sec)
150
200
Controller Disturbance Rejection
Q=30
Controlled Variable
52
CV
51.5
51
T Out
50.5
T Out S.P.
50
49.5
0
50
100
Time (sec)
150
200
Controller Disturbance Rejection
Q=10
Controlled Variable
52
CV
51.5
51
T Out
50.5
T Out S.P.
50
49.5
0
50
100
Time (sec)
150
200
Class Exercise
Testing the Controller Tuning
• Using disturbance upset tests, test the
performance of your tuned DMC controller.
Model Identification:
The Least Squares Solution
N
    ys (ti )  y p (ti ) 
2
i 1
ys (ti ) is the measured value of the CV at time ti
y p (ti ) is the predicted value of the CV from the
SRM at time ti
Model ID: adjust the coefficients of the SRM
until  minimized (results in averaged
SRM coeficients if enough tests are used)
Controller Disturbance Rejection
Ts=7, Q=30
Controlled Variable
52
CV
51.5
51
T Out
50.5
T Out S.P.
50
49.5
0
50
100
Time (sec)
150
200
What if the SRM become worse?
Ts=11; Q=30
Controlled Variable
52
CV
51.5
51
T Out
50.5
T Out S.P.
50
49.5
0
50
100
Time (sec)
150
200
What if the SRM become worse?
Ts=15; Q=30
Controlled Variable
52
CV
51.5
51
T Out
50.5
T Out S.P.
50
49.5
0
50
100
Time (sec)
150
200
SRM Model Mismatch
• The controller still works in this case, but
the performance is penalized.
MIMO DMC Control
MIMO DMC Control
– Extension of SISO to MIMO
– Constraint control
– Economic LP
– Large-scale applications
MIMO DMC Control
Extension of DMC to MIMO
processes
Dynamic Matrix for a MIMO
Process
 A11 A12 ... A1j 


A
A
...
A
21
22
2j 

 .


A
 .

 .



 A k1 A k2 ... A kj 
A lm is the SISO
Dynamic Matrix
th
for the l CV
th
and the m MV
with the move
suppression factor
added
Recall
The SISO Dynamic Matrix
 a1
a
 2
 a3

a3

Example Dynamic Matrix: A 
a
 3
q
0

0
0 0 
a1 0 
a2 a1 

a3 a2 
a3 a3 

0 0
q 0

0 q 
Example of a Partitioned
Dynamic Matrix
1 2 
A11  

3 4 
9 10 
A 21  

11
12


 A11
A
 A 21
5 6 
A12  

 7 8
13 14 
A 22  

15
16


1

A12  3



A 22  9

11
2 5 6

4 7 8
10 13 14 

12 15 16 
Dynamic Matrix for a MIMO
Process
 A11 A12 ... A1j 


A
A
...
A
21
22
2j 

 .


A
 .

 .



 A k1 A k2 ... A kj 
A lm is the SISO
Dynamic Matrix
th
for the l CV
th
and the m MV
A Partitioned Control Vector
 u 1 
 u 
 2
. 
u  

. 
. 


 u j 
An Example of a Partitioned
Control Vector
1 
u1   
2
then
3 
u 2   
 4
1 
 2
u   
3 
 
 4
Control Law for MIMO DMC
Controller
1
u  [ A W A ] A W E
T
2
T
2
W is a partitioned diagonal matrix, which
contains matrices, Wi . Wi is a diagonal
matrix, which contains wi on its diagonal.
wi is the relative weighting factor for
the i -th CV.
Relative Weighting Factor
• The relative weighting factor, wi , allows the
application control engineer to quantitatively
rank the importance of each CV and constraint.
• For each of the constraints and CV’s, the
control engineer must determine how large a
violation of the constraint or deviation from
setpoint requires drastic action to maintain
reliable operation (Equal Concern Errors).
1
wi 
(Equal Concern Error)i
Equal Concern Errors
• ECE’s are based on process experience.
• Consider a distillation column
– 0.5 psi above the differential pressure limit of 2
psi will cause drastic action to prevent flooding.
– 1% impurity levels above the specified impurity
levels in the products will require rerunning the
product.
– Reboiler temperatures greater than 10ºF above
the reboiler temperature constraint will result in
excessive fouling of the reboiler.
Feedforward Variables
• Measured disturbances should be modeled
as inputs for the MPC controller to reduce
the effect of those disturbances on the CV.
• FF variables are treated as inputs that are
not manipulated.
MIMO DMC Control
Constraint Control
Constraint Control
• With DMC, constraints are treated as CV’s.
• As different combinations of constraints become
active, different weighting factors, wi, are used
which are based on the equal concern errors.
Therefore, as different combinations of active
constraints are encountered, the wi’s automatically
determine the relative importance of each.
• The DMC controller determines the optimal control
action based on considering that the wi’s change
along the entire path to the setpoints. Therefore, the
entire control law must be solved at each control
interval.
MIMO DMC Control Law
1
u  [ A W A ] A W E
T
2
T
2
W is a partitioned diagonal matrix, which
contains matrices, Wi . Wi is a diagonal
matrix, which contains wi on its diagonal.
wi is the relative weighting factor for
the i -th CV.
MIMO DMC Control
MPC Model Identification
MPC Model ID
N
    ys (ti )  y p (ti ) 
2
i 1
ys (ti ) is the measured value of the CV at time ti
y p (ti ) is the predicted value of the CV from the
SRM's at time ti
Model ID: adjust the coefficients of the SRM's
until  minimized
MIMO Model Identification
• For DMC, impulse models are used for
identification so that bad data can be
“sliced” out of the training data.
• Sources of bad data: analyzer failure, data
not recorded, process not operated in
normal fashion (by-pass opened, atypical
feed to the process, saturated regulatory
controls, etc.), and large unmeasured
disturbance to the process.
Model Identification
• SRM’s have the flexibility to model a full
range of process dynamics (e.g., recycle
systems).
• MPC controllers that use preset functional
forms for models (e.g., state-space models
or a set of preset transfer function models)
can be inferior to SRM based models in
certain cases.
MIMO DMC Control
Economic LP
Economic LP
• The LP combines process optimization with
the DMC controller.
• The LP is based on economic parameters
(e.g., product values, energy costs, etc.) and
the steady-state gains of the process.
• The last term in each SRM is used to
provide the necessary gain information;
therefore, the LP is consistent with the MPC
controller.
Example of an LP
C on strai n ts
Fe asi bl e
Re gi on
O pti m u m
Ve rte x
Economic LP and PID Control
Performance
C on strai n ts
Operator
PID
Performance
O pti m u m
Ve rte x
Economic LP and MPC Control
Performance
C on strai n ts
C on trol
Re gi on
for MPC
O pti m u m
Ve rte x
MPC Control Performance
(i.e., size of control circle)
• Depends on accuracy of models (e.g.,
process nonlinearity, type and size of
unmeasured disturbances, changes in the
process and operating conditions).
• Depends on the performance of regulatory
controls and sensors.
• Can also depend on the control MPC
technology used.
Economic LP
• Consider the case with 5 MV’s and 10 control
objectives (i.e., upper and lower limits on CV’s,
upper and lower limits on MV’s, and upper rate of
change limit on MV’s): because there are 5 degreesof-freedom (i.e., one for each MV) the LP will
determine which 5 constraints to simultaneously
operate against.
• The 5 constraints, in this case, become the setpoints
for the MPC controller.
• The LP is applied each control interval along with
the MPC controller.
The LP and the MPC Controller
• Both use the steady-state gains from the SRM.
• The LP determines the setpoints for the controller
from the economic parameters and the process
gains.
• The controller drives the process to the most
profitable set of constraints, i.e., keeps the process
making the most profit even though the most
profitable set of process constraints changes with
time parameters.
• Balanced ramp variables are constraints for the LP.
MIMO DMC Control
Large-Scale Applications
Size of a Typical Large MPC
Application
•
•
•
•
MV’s: 40
CV’s: 60
Constraints: 75-100
There are some companies that have even
larger applications.
MPC Project Organization
MPC Project Organization
•
•
•
•
•
•
•
•
•
•
Understand the process
Set the scope of the MPC controller
Choose the control configuration
Design the Plant Test
Conduct the Pretest
Conduct the Plant Test and collect the data
Analyze data and determine MPC model
Tune the controller
Commission the controller
Post audit
MPC Project Organization
Understand the process
Understand the Process
• Process understanding is the single most
important issue for a successful MPC
application.
• Stated another way, if you do not fully
understand the process and its preferred
operation, it is highly unlikely that you will
be able to develop a successful MPC
application.
How to Develop Process
Understanding
•
•
•
•
•
Study PFD’s
Study P&ID’s
Talk to the operators (how it really works!)
Talk to plant engineers (how they want it to work)
Interview plant economic planners (how much its
worth).
• If available, run steady-state process simulator
• Read the plant operating manuals
• Spend time at the process on graveyards.
Be able to answer the following
questions
•
•
•
•
What is the purpose of the plant?
Where does the feed come from?
Where do the product go?
How much flexibility is there for feed
supply and product demand?
• How do the seasons affect the operation?
Also, feed supply and product demand?
Be able to answer the following
questions
• What are the 3 or 4 most important
constraints that operators worry about?
• Where is energy used and how expensive is
it?
• What are the product specifications?
• Are there environmental or tax issues that
affect plant operations?
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Set the scope of the MPC controller
Set the Scope of the Project
(Very Important Step)
• What part of the plant should be included in
the controller? That is, how much of the
plant should be included in the MPC
controller to meet the project objectives?
• To answer this question, you must discuss
the objectives of the project with the
operations personnel, the technical staff for
that portion of the plant, and the scheduling
people. Only then can you be sure that you
are solving the “correct” problem.
MPC Project Organization
Select the control configuration
MPC Configuration Selection
• Which PID loops do you open (i.e., turn
over to MPC controller) and which do you
leave closed. That is, if you leave a PID
loop closed, the MPC controller sets the
PID loop setpoint (e.g., flow controller).
• What are the MV’s for the distillation
columns in the process? Should the MPC
controller be responsible for accumulator or
reboiler level control?
MPC Configuration Selection
• From an overall point of view, what are the
MV’s and the CV’s for the MPC controller?
• A challenging problem that requires process
knowledge and control experience while
keeping focused on the overall process
objectives.
Reasons for Leaving a PID Loop
Closed
• The PID loop may be more effective in
eliminating certain disturbance due to the
higher frequency of application. For
example, flow control loops. In addition,
certain pressure, level, and temperature PID
loops should be left closed.
Situations for which a PID loop
should be opened
• Loops with very long dynamic and/or large
deadtime
• Process lines with two control valves in
series.
• Certain levels for which allowing the MPC
controller control the level adds important
flexibility to better meet the primary
objectives of the process. For example, it
can provide more effective decoupling.
Use Good Control Engineering
• Ensure that the MV’s have a direct and
immediate affect on the CV’s.
• Use computed MV’s to reject certain
disturbances, e.g., use internal reflux control to
reduce the effect of rain storms or computed heat
input for a pumparound.
• Use inferential measurements to reduce the
effect of deadtime (e.g., inferential temp control)
• Use CV transformations that linearize the overall
response of the process.
Variable Transformations
• For a control valve, P=K×Flow2
• For high purity distillation columns, use log
transformed compositions: x’=log(x)
• Use different linear models depending on
the operating range.
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Design the Plant Test
Plant Test Guidelines
• Make from 5 to 15 step tests for each MV
• The MV changes should be as random as
possible to prevent correlated data.
• MV moves should be as large as possible
but not so large that it upsets the process
(e.g., 1-10%).
• For each MV, make a change after 1tp, 2tp,
3tp, 4tp, and 5tp, where tp is Tss/4.
• Above all, remember the product specs and
the process constraints
Develop the Testing Plan
• Identify all the MV’s you plan to move.
Tabulate the following data for each MV:
– Tag number of the MV and physical description
– Nominal value
– Range of move sizes
• Make a proposed testing sequence and
indicate when each MV is moved and by
how much.
• Discuss the MV move sizes with operations.
Remember the spec and constraints.
Choose Sampling Period
• Small sample period: high time resolution but
increased size of data set. In general, sample
as new information become available.
• For fast responding processes, sample every
few seconds.
• For slow processes, sample every 5 min
• Look at the smallest expected Tss.
– Estimate sample period as Tss/50
– Compare with sensor dynamics
Make MV Changes one MV at a Time
to Generate Uncorrelated Results
Develop a Roughed-out Gain
Matrix
• Using your process knowledge, determine
whether the steady-state process gain is
positive, negative, or zero for each input
(MV & DV)/ output (CV) pair.
• Using + for positive, - for negative, or 0 for
zero, construct the roughed-out gain matrix.
Inputs are listed vertically while outputs are
listed horizontally.
Roughed-out Gain Matrix Ex.
PT
LC
L
z
AT
AT
y
S
LC
MV's: L an d S
DV: z
C V's: y an d x
x
AT
Ex. Roughed-out Gain Matrix
CV's
L
Inputs
S
z
y
x
+
+
+
-
MPC Project Organization
Conduct the Pretest
Pretest
• Check all the sensors, control valves and regulatory
control loops for proper operation.
• Ensure that the process equipment is in proper repair.
Check material and energy balances on the unit.
Make sure that all the major pieces of equipment are
in operation. Check the feed to the unit.
• Then, apply step input changes for each MV and
compare to your roughed out gain matrix.
• Reconcile any deviations from what you expected
(e.g., roughed out gain matrix, dynamic response,
etc.). This will enhance your process understanding.
MPC Project Organization
Conduct the Plant Test and collect the
data
Testing in Optimal Operating
Mode
• Because you will want your plant to operate
effectively at its optimum operating conditions,
you should, if possible, test you plant near the
optimum operating conditions to reduce the
effects of process/model mismatch.
• Therefore, you will need to determine the
optimum operating conditions for your plant,
e.g., discuss with an experienced plant engineer
or use a nonlinear process optimizer.
Plant Test Overview
• The plant test is the most important step
in an MPC project because if the MPC
models do not agree with the process,
controller reliability and performance will
be poor.
• If the process changes (e.g., different
equipment configuration, different feed,
different regulatory controller tuning, etc.),
mismatch between the MPC model and the
process will result, affecting controller
performance.
Plant Test Check List
• Meet (5-10 min) with the operators at the
start of each shift and explain what you are
doing.
• Make sure that each MV move is made on
time, in the proper direction, and by the
correct amount.
• Make sure that a technical person is
observing entire testing period (24-7).
Plant Test Check List
• Make real-time plots of the data and explain
all the behavior that you see.
• Identify abnormal periods and make notes:
significant unmeasured disturbances,
instrument failures, utility outages. The
MPC model should not be trained using
data collected during these periods.
• Talk to the operators about the process.
• Buy pizza, donuts, ice cream and barbeque
for the operators
Using Roughed-out Gain Matrix
• During the testing phase, make sure that the
plant results agree with your roughed-out
gain matrix.
• If not, you will need to explain the
difference. That is, your process knowledge
needs to be improved a bit or something is
not working properly (e.g., an analyzer is
off-line)
Unmeasured Disturbances
• Training an MPC controller on data
collected during periods with unmeasured
disturbances will reduce the accuracy of
your model and reduce the effectiveness of
the overall project.
• Some unmeasured disturbances are
inevitable, but you must strive to keep them
to an absolute minimum.
Examples of Unmeasured
Disturbances to be Avoided
•
•
•
•
Back flushing a condensor
Changes in by-pass streams
Product grade changes
Changes in the controller settings for the
regulatory controllers.
• Any change in the process operating
conditions not modeled in the controller
or process equipment changes
Identified Periods with
Unmeasured Disturbances
• Slice out the data from those periods so that
this data is not used to train the MPC
model; therefore, the identified model will
not be corrupted.
MPC Project Organization
Analyze the data and develop the
MPC model
Obtain the MPC Process Models
• Use the accepted plant test data with the
MPC model identification software to
develop each of the input/output SRM
models.
Evaluate Process Models
• For each input/output pair, plot the SRM on
a small graph.
• Arrange the SRM models into a matrix
similar to the roughed-out gain matrix.
• Review the results to ensure that you have
models that make sense.
• Use the statistical tools from the MPC
model ID software to evaluate each SRM
Evaluate Process Models
• Examine the difference between the plant
test data and your models (residual).
• Plot the residual for positive and negative
MV changes. This will help identify the
degree of process nonlinearity. Also, it will
help evaluate CV transformations that
behave more linearly.
Plant Economics
• The LP will require incremental costs (feed
costs, utility costs) and incremental
revenues (product values)
• For previous distillation example, you will
need the incremental cost of steam ($/lb),
the incremental cost of feed ($/lb), and the
incremental value of both products ($/lb).
MPC Project Organization
Tune the controller
General Approach to Tuning
• Use the SRM’s of the process (MPC model)
off-line as the process to develop the
preliminary settings for the controller.
• Three areas that you need to get right:
– Economics
– Constraints
– Dynamics
Testing Economics
• Run plant/controller simulations (off-line)
with different combinations of constraints to
ensure that the controller pushes the plant in
the correct direction.
• Run test cases to see which LP constraints
are operative.
Constraints
• Check the equal concern errors for each
constraint to ensure that the proper relative
weighting factors for each combination of
active constraints is used. That is, when
constraint 1 and 5 are active, what is the
relative importance of the CV’s and
constraints.
• Also, use the plant/controller simulation
(off-line) to test the relative weighting.
Dynamics
• Select the move suppression factors for
each MV using off-line simulations of the
plant. Run the simulations a number of
times for setpoint changes and unmeasured
disturbances to get the MSF’s right.
Operators are a good source for letting you
know how much the MV’s should move in
certain situations and how fast the process
should move to CV’s, but remember that
they are most familiar with how the plant
used to operate.
Operator Training
• In some states, it is a legal requirement.
• Typically operators receive one to two hours
of training with MPC.
• Nevertheless, better operator training
increases your chance for success.
Watchdog Timer
• The controller should run every minute 24-7
• Otherwise, a DCS alarm should sound.
• The most common approach is to use a
watchdog timer (a small program running
on the DCS) to communicate with the MPC
controller. If the watchdog fails to receive a
healthy response from the controller after an
appropriate period of time, the watchdog
forces each MV into a fallback (shed)
position and writes an alarm to the operator.
MPC Project Organization
Commission the controller
General Approach
• Now it is time to install the controller in the
closed loop in the process.
• Since the controller will directly affect the
plant operation, it is essential to proceed
methodically and with caution.
• Use a checklist and be observant
Commissioning Checklist
• Make sure that the operators have been
trained
• Install control software, database, and other
real-time components.
• Check controller database
– Control CV’s, DV’s, and MV’s match raw DCS
inputs exactly
– Controller models, tuning parameters, etc.
match off-line versions exactly
Commissioning Checklist (cont.)
• Turning controller on the prediction mode:
–
–
–
–
–
Each MV configured correctly in the DCS?
Correct shed mode for each MV?
MV’s out of cascade (controller cannot change)?
Controller turned on the prediction mode?
The size of the calculated MV moves make
sense?
– The CV predictions make sense?
Commissioning Checklist (cont.)
• First-time MV check
– Use conservative controller settings
– Reduce difference between upper and lower
limits on MV (or rate limits)
– Make sure controller is running
– Put MV’s into cascade one at a time.
– Ensure that the calculated MV change is
applied to DCS setpoint.
Commissioning Checklist (cont.)
• Close the loop
– Ensure that the controller is running
– Turn ON the controller master switch
– With MV’s clamped, put one or two MV’s into
cascade
– Relax the MV limits somewhat and check for
good controller behavior
– Add more MV’s gradually, ensuring good
control performance.
Final Checks
• Ensure that the standard deviations of the
CV’s decrease and that the MV’s are not
moving around too much.
• Ensure that the LP is driving the process in
the correct direction.
MPC Project Organization
Post audit
Post audit
• Collect post commissioning data for the key
CV’s and compare to pre-project data.
• Ensure that the standard deviations of the
key CV’s have decreased.
• Evaluate whether the upper and lower limits
on the MV’s are still set correctly.
• Ensure that the controller is running the
process at the economically best set of
constraints most of the time.
Advantages of MPC
• Combines multivariable constraint control
with process optimization.
• A generic approach that can be applied to a
wide range of processes.
• Allows for more systematic controller
maintenance.
• Depends on the particular process
MPC Offers the Most Significant
Advantages
• For high volume processes, such as refineries and
high volume chemical intermediate plants.
• For processes with unusual process dynamics.
• For processes with significant economic benefit to
operated closer to optimal constraints.
• For processes that have different active constraints
depending on product grade, changes in product
values, summer/winter operation, or day/night
operation.
• For processes that it is important to have smooth
transitions to new operating targets.
Application Example
• C3 splitter in an ethylene plant (reboiler duty
is free; therefore, run at maximum reboiler
duty and apply single-ended control.)
• DCS 70% of cost
35% of inc. benefit
• AdPID 10% of cost
60% of inc. benefit
• MPC 10% of cost
3% of inc. benefit
• RTO 10% of cost
2% of inc. benefit
Application Example
• Reformer. Control is easy since it only
involve inlet temperature controllers.
• DCS 70% of cost
35% of inc. benefit
• AdPID 10% of cost
30% of inc. benefit
• MPC 10% of cost
5% of inc. benefit
• RTO 10% of cost
30% of inc. benefit
Application Examples
• FCC Unit. Optimal constraints change with
economics, operating conditions, etc.
Finding the optimal riser temperature
requires RTO (yield model and nonlinear
problem).
• DCS 70% of cost
35% of inc. benefit
• AdPID 10% of cost
10% of inc. benefit
• MPC 10% of cost
35% of inc. benefit
• RTO 10% of cost
30% of inc. benefit
Limitations of MPC
• It is a linear method; therefore, it is not
recommended for highly nonlinear
processes (e.g., pH control).
• It does not adapt to process changes. If the
process changes significantly, the MPC
model must be re-identified.
• For certain cases, the economic LP may not
be accurate enough. For these cases, it may
be necessary to add nonlinear optimization
on top of the LP.
Different Forms of MPC
Primary Commercial MPC
Software
•
•
•
•
•
•
ABB
Aspentech DMCplus
Fisher-Rosemount
Foxboro
Honeywell RMPCT
Yokogawa
• MPC software based on models with
specified dynamic characteristics can be
inferior to SRM based approaches.
Conclusions
Conclusions
• The challenges of this course:
– A very difficult area with considerable
mathematics and industrial practice.
– Only two days
– Wide variety of backgrounds of the students.
Conclusions
• You should now be familiar with MPC
terminology and technology and the major
issues associated with the commercial
application of MPC.
• Are you an MPC expert? Not yet! But with
more study and a lot more experience, you
can become one.
Conclusions
•
•
•
•
“This is not the end.
It is not even the beginning of the end.
But it is, perhaps, the end of the beginning”
(Sir Winston Churchill, Nov. 10, 1942)