INTRODUCTION
All industries tend to increase their production efficiency and reduce as much
cost as possible (Seborg et al., 2004). This situation
leads to more complexity in processing plants, in which more variables are involves.
Besides that, most industries are facing massive challenges such as tighter
product quality specifications, increasing productivity demands, the existence
of new environmental regulations and the fast changes in the economical market.
The conventional ProportionalIntegralDerivative (PID) controllers cannot really
capture the complexity of this type of process, as well as these challenges.
So advanced process control is a promising solution for this problem. One class
of advanced process controller is Model Predictive Control (MPC). MPC has been
used because it is able to respond in an effective way to these challenges.
MPC has been developed and extensively used in the process industry for controlling
key unit operations in chemical plants over the last two decades (Badwe
et al., 2008). This is not a surprising fact since MPC is the most
general way of posing the process control problem in the time domain. Difficult
multivariable control problem with the presence of inequality constrains on
input and outputs also can be handled by this type of modelbased control strategy.
MPC uses a process model and the current state to predict future values of the
output. However, poor modeling of the model based multiinputmultioutput (MIMO)
controller in a single channel can affect multiple outputs.
It is important to detect a modelplant mismatch (MPM) to assist in reidentification.
About 60% of all industrial controllers have some performance problem (Harris
et al., 1999). Reidentification of the process model with large
number of inputs and outputs is costly due to potential production losses and
high manpower effort. For the purpose of modelplant mismatch detection, it
is necessary to identify the effect of the mismatch in the process model.
An MPC controller is similar to the structure of Internal Model Control (IMC)
a s presented by Seborg et al. (2004). For some
cases, the MPC control system is using IMC structure because there is no explicit
model in the lineartime invariant (LTI) framework capable of describing an
MPC (Carlsson, 2010). IMC approach has advantage that
it allows model uncertainty as well as considered the tradeoff between performance
and robustness in systematic manner. Dynamic matrix control (DMC) is one type
of MPC that used this IMC structure (Wang and Wang, 2010).
The feedback structure in DMC algorithm makes this controller robust and suitable
for use in controlling complex and highly interacting processes such as distillation
columns and chemical reactors (AlvarezRamirez et al.,
2004).
Figure 1 shows a block diagram of a closedloop system.
G is the model representing nxm MIMO plant and G is the plant. The modelplant
mismatch, Ĝ is described in Eq. 1.
Plant output:
where, u (k) and v (k) is the vector of manipulated variables (MVs) and Gaussian disturbances acting on the process, respectively.
Model output:
Model residuals:
e (k) = y (k) ŷ (k) =Δu (k)+v
(k) 
(4) 
The objective of this study is to investigate the effect of modelplant mismatch on the performance of MPC controller. The effect of modelplant mismatch was demonstrated on Wood and Berry distillation column for four types of mismatch: (1) gain mismatch, (2) reverse gain mismatch, (3) time constant mismatch and (4) time delay mismatch.
METHODOLOGY
The process started with the process modeling and MPC controller design. MPC controller designed is done by specifying tuning parameters and a discrete process model to represent the actual plant. Then mismatch is added to the process model with several cases. Setpoint change is done to the inputs before the simulation is done. After that data from the process are collected and the response is plotted. All the responses are compared each other. The process is repeated for other different cases.
CASE STUDIES
In this study, the effect of modelplant mismatch on the performance of MPC
controller are illustrated through a case study of Wood and Berry distillation
column. MPC controller performance has been tested by introduced mismatch in
the model plant to portray the real situation in the plant. Several scenarios
have been tested independently: (1) Gain mismatch, (2) Reverse gain mismatch,
(3) Time constant mismatch and (4) Time delay mismatch. In this study, the controller
performance in setpoint tracking was evaluated by making a step change in an
input. In all cases, the disturbance was set to zero. Wood and Berry distillation
column model is a distillation column model based on the study conducted by
Wood and Berry (1973) which later will be called Wood
and Berry distillation column.
The model is shown as follow:
This is MIMO system with two controlled variables and two manipulated variables. In this study, unmeasured disturbance part is eliminated to simplify the process. The controlled variables and manipulated variables are listed below:
• 
Cvs 


X_{D}: 
The distillate composition (output 1) 

X_{B}: 
The bottom composition (output 2) 
• 
Mvs 


R: 
Reflux flow rate (input 1) 

S: 
Steam flow rate (input 2) 
Since the distillation column process is interacting, the controlled variables and manipulated variables are interrelated each other. The relationship between controlled variables and manipulated variables is shown in Table 1.
For all the four scenarios, MATLAB simulation was performed by introducing
a step change at 5th min with magnitude of 1 in reflux flow rate.
Table 1: 
Relationship between controlled variables and manipulated
variables for each channel 

Simulink model was linked with MATLAB script to provide controller tuning parameters
to MPC controller. The responses of each mismatch variation are then observed
and the data is collected. Distillate composition and bottom composition responses
with the mismatch variation are plotted.
Gain mismatch (Variation in Magnitude): Underestimated gain mismatch with 10 to 70% variation of G_{11 }gain magnitude was added to the process model. For underestimated gain mismatch, the mismatch was added so that the new gain magnitude is smaller than its original value. The original value of G_{11} gain is 12.8. The mismatch variation in magnitude as well as the corresponding values of new gain for underestimated gain mismatch is listed in Table 2. In Table 2, it can be seen that as mismatch variation increases, the value of G_{11} new gain become smaller.
From Fig. 2, it can be seen that as there is no mismatch
in process model, the process was able to achieve the new desired setpoint
of 1 mole % as shown by red line. With the existence of gain mismatch in the
process model, the MPC controller cannot bring the process to the desired setpoint.
This is because process gain has significant effect to the process. Decreasing
process gain can reduce the process output and vice versa.
Table 2: 
Mismatch variation (magnitude) and the values of G_{11}
new gain for underestimated gain mismatch 


Fig. 2: 
Distillate composition responses in variation of underestimated
gain mismatch (magnitude) in G_{11} channel 
For underestimated gain mismatch, as the mismatch variation in the process
model is higher, the values of G_{11} new gain become smaller thus make
the process farther from the setpoint. For comparison, the response for 30%
mismatch (yellow line) is farther from the setpoint compared to the 10% mismatch
(green line). From the graph, it shows that with 10% underestimated gain mismatch
of G_{11}, the desired setpoint of distillate composition cannot be
achieved by 0.2 mole %. It is also found that the distillate composition decreases
as the gain mismatch is exceeding 50%. For example, 70% gain mismatch of G_{11}
reduces the composition of distillate by 0.4 mole % from the steady state value.
In Fig. 3, when step change is done to the reflux flow rate,
the process can go back to its initial steady state at 0 whether gain mismatch
exist or not. But there is a difference between the response with and without
mismatch in the process model. With no mismatch (light green line) in the process
model, the bottom composition is increases a bit before going back to its initial
steady state. For the process with gain mismatch in the process model, the response
is increasing and achieved the highest peak at 20 min and then decreasing before
it is going back to the initial steady state. This is shows that MPC controller
can bring back the bottom composition to its initial setpoint although there
is a mismatch in the process model. It can be seen that the responses are close
among each other but as the mismatch variation is higher, the response achieved
its steady state in longer time. The bottom composition was also affected even
though the step input is done to the reflux flow rate. This is due to the additive
effect on each outputs (distillate and bottom composition) when the step changes
was done to the reflux flow rate.

Fig. 3: 
Bottom composition responses in variation of underestimated
gain mismatch (magnitude) in G_{11} channel 
Table 3: 
Mismatch variation (magnitude and direction) and the values
of new gain for underestimated gain mismatch 


Fig. 4: 
Distillate composition responses in variation of underestimated
gain mismatch (magnitude and direction) in G_{11} channel 

Fig. 5: 
Bottom composition responses in variation of underestimated
gain mismatch (magnitude and direction) in G_{11} channel 
All responses come back to the initial steady state after about 150 min.
Gain mismatch (variation in magnitude and direction): For this case,
underestimated gain mismatch with variation of G_{11} gain magnitude
from 10 to 70 % as well as reverse gain sign was added to the process model.
The mismatch variation (magnitude and direction) and the values of new gain
for underestimated gain mismatch are listed in Table 3.
Changing the sign of the G_{11} gain was significantly affected the process. When the gain sign is reverse from its actual sign, the process is away and unable to achieve the desired setpoint. In this case, the actual gain is 12.8. With 0% magnitude gain mismatch and the sign is reverse from the actual sign, the new gain value is 12.8. With this new gain, the process tends to reverse from its actual direction. This is because the relationship between input and output determine by process gain. For example, as the reflux flow rate increases, the distillate composition will also increase when process gain is positive sign. But when the gain sign is reverse (negative), as the reflux flow rate increases, the distillate composition will be decrease. The response also depends on the magnitude of mismatch, as the magnitude increases, the process farther from the set point. As shown in Fig. 4, increasing underestimated gain mismatch as well as reversing the gain sign makes the reduction in distillate composition become smaller. For comparison, 0% mismatch (blue line) reduces the distillate composition by 3 mole % while 50% mismatch (orange line) reduce only 2 mole% of distillate composition. From the graph, it indicates that MPC controller cannot perform well (unable to bring the distillate composition to its desired setpoint) in the presence of gain mismatch with variation in magnitude and reverse gain direction.
From Fig. 5, it can be seen that the process can go back to its initial steady state at 0 whether gain mismatch exist or not. This is shows that MPC controller can bring back the bottom composition to its initial setpoint although there is a mismatch in the process model. It can be seen as the mismatch magnitude smaller, the response achieved its steady state in shorter time. The bottom composition was also affected even though the step input is done to the reflux flow rate. As mention earlier, this is due to the additive effect on each outputs (distillate and bottom composition) when the step changes was done to the reflux flow rate. All responses come back to the initial steady state after about 200 min.
Time constant mismatch: In this case, mismatch is added in the process model with G_{11} time constant varies by 10, 30, 50 and 70%. For underestimated time constant mismatch, the mismatch was added so that the value of new time constant is smaller than its original value. The mismatch variation and the values of new time constant for underestimated time constant mismatch are listed in Table 4.
Table 4: 
Mismatch variation and the values of new time constant for
underestimated time constant mismatch 


Fig. 6: 
Distillate composition responses in variation of underestimated
time constant mismatch in G_{11 }channel 

Fig. 7: 
Bottom composition responses in variation of underestimated
time constant mismatch in G_{11} channel 
From Fig. 6, it can be seen as the mismatch variation higher,
the response is initially faster but the responses became slower in achieving
the desired setpoint.
Figure 7 shows bottom composition responses in variation
of underestimated time constant mismatch in G_{11} channel. It can be
seen that the process can go back to its initial steady state at 0 whether time
constant mismatch exist or not. This is shows that MPC controller can bring
back the bottom composition to its initial setalthough there is a mismatch
in the process model.point
Table 5: 
Mismatch variation and the values of new time delay 


Fig. 8: 
Distillate composition responses in variation of time delay
in G_{11} channel 
It can be seen as the mismatch magnitude greater, the response of the process
is slower and achieved its initial steady state in longer time. The bottom composition
was also affected even though the step input is done to the reflux flow rate.
As mention earlier, this is due to the additive effect on each outputs (distillate
and bottom composition) when the step changes was done to the reflux flow rate.
All responses come back to the initial steady state after about 100 minutes.
In this case, it shows that MPC controller is perform well, where it can bring
the bottom composition to the initial steady state even with the presence of
time constant mismatch in the process model. It can be conclude that the time
constant mismatch does not affect the direction of the process; it only can
make the process slower toward reaching the initial steady state.
Time delay mismatch: For this case, mismatch is added in the process model with G_{11} time delay varies by addition and deduction of sample time. The original time delay for G_{11} is 1. The mismatch variation and the values of new time delay are listed in Table 5.
From Fig. 8, it can be seen that as time delay decreases,
the response is faster and vice versa. For comparison, the distillate composition
response of time delay reduction by 1 sample time (light green line) is faster
than the distillate composition response of time delay addition by 1 sample
time (purple line).

Fig. 9: 
Bottom composition responses in variation of time delay in
G_{11} channel 
This is means that when the time delay decreases, the process can react faster
towards the desired setpoint.
From Fig. 9, it can be seen that reduction and addition of time delay does not much affect the process. But as time delay greater, the response is slower in achieving the initial steady state (blue line). From this study, it can be concluded that time delay mismatch in the process does not have significant effect to the process, it only make the process slower towards achieving the setpoint.
The previous study by Badwe et al. (2010) proposed
a noninvasive methodology for quantifying the impact of MPM on control performance.
The proposed methodology was aid for diagnosing poor quality of control and
also for isolating the role of modelplant mismatch in poor control. In this
work, setpoint direction effect on modelplant mismatch was discussed and it
was shown that the ‘worst’ and the ‘best’ setpoint directions
can be determined using closedloop data. The technique proposed was able to
successfully isolate the causes of degradation in quality of control. In the
recent work, the effect of modelplant mismatch has been investigated. The methodology
is based on the analysis of closed loop data. This works focused on the contribution
of gain mismatch, time constant and time delay mismatch on the MPC controller
performance. From this work, it shows that gain mismatch has greatest effect
on MPC controller performance compared to time constant and time delay.
CONCLUSION
In MPC application, the accuracy of the process model plays a crucial role
to the controller performance. Mismatch between the process model and plant
may have significant effect to the process especially in gain mismatch.

Fig. 10: 
Integral Absolute Value of the error (IAE) plot for all cases 
Table 6: 
Summary of all cases for study on the effect of modelplant
mismatch to the MPC controller performance 

The gain mismatch can make the process away from the setpoint and change the
process direction if the magnitudes of gain mismatch higher than 50% from the
original value. The direction of the process also will be reversing than the
actual direction if the gain sign reverse from the actual sign. Hence, it is
suggested that if the output of MPC process model is away from the setpoint,
the possible cause is gain mismatch. Time constant and delay mismatch also can
affect the process but the effect is insignificant. In the presence of time
constant and time delay mismatch, MPC controller can perform well because it
can bring the process to the setpoint. For time constant mismatch, when the
magnitude of new time constant smaller than the actual time constant, the response
is faster and vice versa. For time delay, increasing delay makes the process
slower and vice versa. From this study, it shows that MPC controller performance
is not good for setpoint tracking in the presence of gain mismatch. The summary
of all cases for study on the effect of modelplant mismatch to the MPC controller
performance are shown in Table 6. The performance of MPC controller
is shown by the integral absolute value of error (IAE) in Fig.
10. The IAE values are calculated based on 50% underestimate mismatch for
all cases. As the IAE higher, the performance of MPC controller is worse. From
Fig. 10, A2 shows highest value of IAE. This is followed
by A1. IAE values for A3 and A4 are very small compared to the first two mismatch
types. This is indicates that MPC controller performance is very bad in the
presence of gain mismatch with variation in magnitude as well as for both magnitude
and direction. Future study can be done to identify the performance of MPC controller
in terms of disturbance rejection.
ACKNOWLEDGMENT
The authors gratefully acknowledge the support from Universiti Teknologi PETRONAS in carrying out this research.