An experimental validation of model based control techniques for
interacting nonlinear systems
KANTHALAKSHMI.S1, WINCY PON ANNAL.A.S2
1Department of Electrical and Electronics Engineering,
PSG College of Technology, Coimbatore, Tamilnadu, INDIA
2Department of Electronics and Instrumentation Engineering,
Government College of Technology, Coimbatore, Tamilnadu, INDIA
Abstract: - Model based controllers are those controllers that has gained significant attention in the arena of
nonlinear process control. Conical tank is a nonlinear process whose nonlinearity increases when it interacts
with another conical tank. Maintaining the level of an interacting nonlinear process operating with constraints is
the control objective of this paper. Model Predictive Control (MPC) has the capability of handling constraints
and exerts a control action with optimization. MPC is employed for this process and the experimental results
obtained are subjected to time domain analysis and the performances are compared with the performance of
Proportional-Integral-Derivative (PID) and Internal Model controller based PID (IMC-PID) controller.
Key-Words: - Non linear system, proportional integral derivative control, internal model control, model
predictive control
Received: July 17, 2022. Revised: October 29, 2023. Accepted: November 27, 2023. Published: December 31, 2023.
1 Introduction
The chemical process control is one area where
automation has a significant impact. In fact, almost
all of the processes are nonlinear. Nonlinear systems
are chaotic, uncertain, or counterintuitive.
Nonlinear systems are usually approximated by
linear equations. This is effective with some
precision and range for the input values, but some
interesting phenomena like chaos and singularities
are obscured by linearization.
To achieve perfection in operation of such a system,
the choice of an appropriate control technique that
will guarantee smooth running of the process at the
desired level of performance is required. The factors
for the selection of such a controller are that, it
should be able to work with constraints, should be
reliable and should have a robust design. When the
process is nonlinear, robust control is a challenging
task [1].
In industries, the PID controller is the most used
controller for decades because of its feasibility and
easy implementation. Its performance is not based
on a process model and so it cannot compensate for
process dynamics such as dead time and
nonlinearity. The limitations of PID controller were
reported in [2,3]..Paper [4] stated various tuning
methods and modifications to be incorporated to
design PID which may work with robustness and
easily adopted in industries. Even if a good PID
design is made, the PID controller cannot be robust
compared to robust controllers when the system
encounters multiple challenges from the operating
environment of the system.
The development of robust control systems, like
model-based control techniques, including Process
Model Based Control, (PMBC) [5], Non Linear
Model Predictive Control (NLMPC) [6] has resulted
from the advancement of technology in control.
Among the control techniques based on model,
Internal Model Control (IMC) has a considerable
importance to be adopted for a process due to its
effective design philosophy. [7]. The ideology
behind IMC is the Internal Model Principle
according to which, the only approach to acquire
better control is to understand the process model.
IMC is able to forecast output constraint violations
and take remedial action using this model. Another
advantage of IMC is that, only the controller and the
nominal plant are responsible for its stability.
The IMC controller possesses the ability to cancel
out any variations in the process variable from the
requirement, hence, attaining prefect control. The
following characters are known to be present in
IMC: dual stability, perfect control, zero offset.
Paper [8] made analysis of the various tuning
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techniques for PID controller. IMC based PID gives
better performance in case of delays and results in
good and robust settings
The major drawback of IMC is its constraint
handling capacity. This problem can be overcome
by MPC. It reduces the operating cost while
satisfying the constraints. It can be used for
processes whose manipulated and controlled
variables are large in number. It enables the
application of constraints to manipulated variables
as well as control variables. It has the capability to
operate nearer to constraints. It allows time delays,
inverse response and inherent nonlinearities[9]. The
paper [10] states that, MPC provides robust
feasibility with trackable real time computation,
with optimal closed loop dynamics. Paper [11]
reported results on stability and computations of
NMPC.
Level control of two conical tanks which are
interacting to each other is the process considered
here. The process dynamics is highly nonlinear
with a significant dead time because of the
interaction. Both the control algorithms are
implemented and the results obtained are
experimentally verified.
2 Plant Modeling
Two conical tanks interacting with each other, and
whose level has to be maintained constant is the
plant chosen. These types of processes are most
commonly used in pharmaceutical industries where
proper drainage of the fermented products is very
crucial. The importance of conical tank in
fermentation process is reported in [12]. A single
conical tank is nonlinear in nature and when it
undergoes interaction with another conical tank it
exhibits high nonlinearity in its character and it
becomes difficult to control such a process.
The plant (Fig. 1) consists of three conical tanks of
which, the two tanks, tank 1 and tank 2 which are
interacting to each other are taken to implement the
control algorithms. The inlets to both the tanks are
from the sump along with the inlet from the other
tank through the pipeline which causes interaction.
The inlet flows of liquid from the sump to the tanks
are controlled by the pneumatic control valves
attached to their pipelines.
Design of effective controllers depends on how well
the process dynamics is known for which modelling
the process becomes essential. Modelling a process
requires the knowledge of all the basic principles of
operations. The significance of order of the system
during modelling is analysed in detail in [13].
Fig. 1 Experimental setup of two interacting
conical tanks
2.1 Experimental approach
Experimental modeling gives a better knowledge on
how the process reacts to various set point changes
and to disturbances. It also gives information
regarding the dead time of the process, which is a
crucial factor for the design of controller.
From the open loop response obtained using
experimental modeling, a transfer function is
obtained which is of the form,
() 1
d
ts
K
H s e
s
(1)
which will have information on the dead time (td) of
the process. Here, K is the gain of the process
around the operating point, τ is the time constant of
the process.
Approximation of dead time is done using first order
Pade’s approximation
12
12
d
d
ts
d
ts
ets
(2)
Therefore,
12
() 112
d
d
ts
K
Hs ts
s
(3)
For the two interacting conical tanks used, the
transfer function around the operating region of (16-
30 cms) was obtained as,
40
2.688
() 115 1
s
H s e
s
(4)
whose dead time when approximated becomes,
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40
1
2.688 2
() 40
115 1 12
s
Hs s
s
(5)
3. Controller Design
3.1 IMC controller
The method used to account for model uncertainty
and disturbance is IMC. The major advantages of
IMC controller in contrast with traditional feedback
controllers is that, it is easy to tune and it shows
clearly how time delay and right hand plane zeros
influence the process's built in controllability. It is a
compensation between closed loop behavior and
robustness to model miscalculations using a single
tuning parameter, λ. The restriction of this method is
that, the system must be stable.
The process model was experimentally developed
and its transfer function was found to be as in
equation 4. This process model was factorized as,
40
() s
H s e
(6)
which was the non-invertible portion and
2.688
() 115 1
Hs s
(7)
which became the invertible component by the
usage of all pass factorization.
The process model’s invertible section was inverted
and cascaded with a first order filter with filter
coefficient λ that makes the controller proper.
115 1
1
() 2.688 1
s
qs s
(8)
λ was adjusted to vary the speed of response and
robustness.
3.2 IMC based PID controller
It makes advantage of dead time approximation for
analysis. The major difference between IMC and
IMC based PID is that, IMC based PID allows the
controller to be improper so as to locate a controller
that is comparable to a PID controller. Also it
permits good set-point tracing for the process with a
small delay time to time constant ratio. Fruehauf in
1990 reported improvement in IMC based PID
controller performance.
The Pade's approximation of dead time was
considered and the process transfer function was
found to be as in equation 5 and the process model
was factorized as,
40
( ) 1 2
s
Hs
(9)
which was the non invertible portions and,
2.688 1
() 40
115 1 12
Hs s
s
(10)
which was the invertible portion.
The process model’s invertible section was inverted
and cascaded with a filter with filter coefficient λ
and the controller was designed.
40
115 1 1
12
() 2.688 1
s
s
qs s
(11)
On expansion,
2
3300 135 1
1
() 2.688 1
ss
qs s
(12)
The filter coefficient λ cannot be made small
randomly. If so, then there will be a restriction on
the performance of the IMC based PID strategy.
Rivera et al. in 1986 recommended that λ>0.8td due
to the model uncertainty caused by Pade’s
approximation. Morari and Zafiriou in 1989
recommended it to be that λ>0.25td for the PID plus
lag formulation.
The PID parameter values were calculated
comparing q(s) with the standard PID equation and
the values obtained were Kc= 0.9658, τI=2.26 mins,
τD= 0.2839 mins. The value of λ was tuned on line
as a tradeoff between performance and robustness.
3.3 Model Predictive Controller (MPC)
MPC controller is analogous to IMC controller as
the model performs laterally with the process and
the outcomes serve as feedback. Yet the collocation
of the control and target computation is an exclusive
character of MPC. Additionally MPC has had larger
brunt on industrial applications than IMC as it is apt
for constraint MIMO problems. The formulation of
multivariable systems with time delays is also made
easy by this method.
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It is a procedure that constructs controllers that can
alter the control action in advance to the actual
change in output target. This anticipating capacity,
when mixed with conventional feedback operation,
facilitates a controller to make variations that are
mild and nearer to the optimal control action. The
targets are computed from an optimization
depending on the steady state model of the process.
The targets are computed each time the control
computations are performed.
The common optimizations comprises of profit
function, reducing the cost function to a minimum,
and increasing the production rate to a maximum.
Here the optimization problem dealt with is,
minimizing the cost function obtained by varying
the M control moves, taking into consideration the
modelling equations and the limitations on inputs
and outputs. The three major steps involved in the
working of a standard MPC are, estimating the
system states, calculating the effective input which
would minimize the required cost function over the
prediction horizon, and implementing the first part
of the optimal input until the next sampling instant.
The general schematic of a process when implanting
MPC is shown in Fig. 2
Fig. 2 General schematic of Model predictive
controller
3.3.1 Choice of MPC parameters
State space model is more appropriate to be used as
it requires less number of model parameters when
compared to step response model to describe
process behavior. The continuous state space model
obtained for this process is,
0.0587 0.0278 0.25
0.0156 0 0
x x u
(13)
0.0935 0.2992yx
(14)
Control interval k is chosen initially, and then is
held constant to tune other controller parameters. As
the value of k reduces, the anonymous disturbance
rejection increases greatly. But as k becomes very
small, the complexity in computation increases
greatly. As a result, a compromise between
performance and effort in computation is the best
option.
When optimizing the manipulated variables at the
given control interval k, the prediction horizon P is
the number of future control intervals the MPC must
calculate through prediction. The value of P is
chosen so that the controller maintains the internal
stability, predicts constraint violations quickly to
permit remedial measures. To make the control
system less responsive to model errors, the
prediction horizon is chosen to be larger than
control horizon. P is increased until any further
increases have a negligible effect on performance.
But at the same time, as P increases, the need for
controller memory and the time to solve the
quadratic problem increases.
The number of manipulated variable moves to be
adjusted, so as to give the best result possible at a
selected control interval k is the control horizon M.
It must be generally greater than 1 and lesser than P.
As M decreases, only fewer variables are to be
computed in the quadratic problem solved at each
time interval, which ultimately leads to a faster
computation. As there is delay in a plant, M is
chosen much lesser than P. When M is smaller it
provides more chance for internal stability of the
controller.
Least square formulation objective function is used
because it penalizes large errors compared to
smaller errors. It is of the form,
1
22
10
ˆ
pM
k i k i k i
ii
r y w u
(15)
where, P Prediction horizon, M – Control Horizon,
r - Set point, ∆u – change in manipulated input,
w – Weights for changes in manipulated input, -
Predicted model output, k – control interval
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The necessary condition for minimum Φ is obtained
from
0
u
, from which the optimal solution for the
control parameter ∆u could be obtained.
1 .... 1 T
i i i i
u k u k u k u k M
(16)
where ki is the sampling instant,
Only the first element of is used for
implementing the control. The future state variables
1
ii
x k k
to
ii
x k P k
are calculated
sequentially using the set of future control
parameters starting from
1
i i i i
x k k Ax k B u k
(17)
to
12P P P
i i i i
x k P k A x k A B u k A B
1 ...... 1
PM
ii
u k A B u k M
(18)
From the predicted state variables the predicted
output variables
1
ii
y k k
to
ii
y k P k
are
calculated.
i i i i
y k P k CAx k CB u k
(19)
It continues till
12P P P
i i i i
y k P k CA x k CA B u k CA B
1 ...... 1
PM
ii
u k CA B u k M
(20)
The predicted output is
1 2 ... 1 T
i i i i i i
Y y k k y k k y k M k
(21)
Based on the predicted output, the current output,
the control action is taken and the process is
repeated until the system meets the requirement.
As the prediction horizon is substantially greater
than the control horizon, control weighting is set to
zero. The prediction horizon and control horizon are
chosen as 30 and 7 respectively. The constraint
considered is the inlet valve stem lift which has to
vary only between 25% and 75 %.
4. Results
4.1.Open loop response
Keeping the manual outlet valve of tank 1 and tank
2, to drain tank, to open by 50 %, and allowing the
solenoid valve fixed in the pipeline of interacting
between the two tanks, tank 1 and tank 2, to open,
letting 100% interaction between the tanks, the level
of liquid in the tank is maintained by varying the
pneumatic actuator's stem lift.
The open loop response of the system was obtained
by initially maintaining the actuator stem lift at
30%, allowing the system to attain steady state, after
which, additional pneumatic signal was given,
which made the stem lift to 60%. The level of the
tank whose output has to be controlled was noted
until steady state had occurred.
From the response curve, it is noted that the initial
steady state occurred at 16 cms of level in the tank
1, and the final steady state occurred at 30 cms.
From the input fed to the tank 1, the output level
obtained and the graph (Fig 3) which shows the
dynamics of the plant, the system is represented by
the transfer function given in equation4.
Even if the system transfer function seems to be first
order, in practical case, the measuring device may at
least be of first order, and the control valve which
implements the controllers action will again be at
least of first order, which ultimately says that a
system in which a controller is implemented will be
of at least third order, which can be seen from the
time constant of the open loop response. Using
Pade's approximation, it becomes as in equation5.
From the open loop response curve (Fig.3) and the
transfer function obtained, it is evident that the
system has a delay, and a larger time constant.
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4.2 PID controller
The PID controller parameters were found by Cohen
Coon's method of tuning, and the controller gains
calculated are, proportional gain KP is 1.15909,
integral gain KI is 0.0115, derivative gain KD is
6.8401.
From the graph (Fig.4), it is noted that, on
implementing PID controller, the system response
speed has increased. It is due to the fact that the time
constant of the system has reduced by a factor of
(1+KP K). So as the proportional gain is increased,
the speed of response increases. The rise time is
about 500 seconds. Also the offset of the system
decreases on increasing the proportional gain and is
found to be 6.5%. Because integral controller takes
action as long as an error persist, the system takes a
longer time of around 800 seconds to settle. The
controller response is stabilized by derivative action,
which permits the usage of higher gains and lesser
integral time constants, but it creates a noisy
environment.
4.3. Internal Model Control based PID
From the graph (Fig. 5), it is seen that, compared to
PID controller, the IMC based PID controller has a
better performance. Irrespective of time delay
approximation, the controller provides good set
point tracking. The offset has reduced to 4 % from
6.5%. The ratio of time delay to time constant
influences the performance of the controller. It
provides flexibility in controller design, to attain the
required performance. As the filter coefficient
decreases, the performance improves. The speed of
response has improved and the rise time has
decreased to 400 seconds which is smaller
compared to PID controller. The system has also
settled quickly at almost 550 seconds which is a
much better performance compared to PID
controller. The performance indices based on error
has also greatly improved (Table 1), which says that
IMC-PID is much better controller than a PID
controller.
4.4. Model Predictive Control
Without constraints control problems would not
exist. So while designing a controller always the
constraints are to be kept in mind. They lead to
additional control objective. The main drawback of
PID and IMC PID are that, they do not have the
capability to handle constraints, which can be very
well handled by MPC. The constraint considered is
the inlet valve stem lift which has to vary only
between 25% and 75 %.
Even if a good initial model for a controlled process
is known, it may not be sufficient for effective
control during long operations because of process
non linearity which may cause the change in process
characteristics based on the operating point. So the
controller is designed such that it is less sensitive to
model errors, for which, the prediction horizon is
made much greater than control horizon. So
prediction horizon is chosen as 30. Increasing
prediction horizon has also lead to decrease in mean
square error.
If control horizon is small, the peak time increases.
Also it provides internal stability of the controller. If
control horizon increases, the control moves get a
tendency to become more violent. Larger weights
are required to reduce the aggressiveness of control
moves. So a control horizon of 7 which is much less
when compared to prediction horizon is chosen.
From the graph (Fig. 6), it is clear that the system
has a rise time of only 200 seconds, which says that
the system response speed has improved.
The process delay which is not compensated by
PID, is well compensated by MPC. Also the weight
on control is made zero as the prediction horizon is
much larger than the control horizon.
From the graph (Fig 6), it is evident that, the MPC
controller shows a much better performance
compared to PID controller and IMC based PID
controller because of its predicting capability. It has
almost no offset and system has settled very
quickly.
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0200 400 600 800 1000 1200
-5
0
5
10
15
20
25
30
35
Open loop response of the system
Time (seconds)
Level of liquid in the tank (cm)
Fig. 3. Open loop response of the system
0200 400 600 800 1000 1200
0
5
10
15
20
25
Response of the system using PID controller
Time(sec)
Level of liquid in the Tank (cm)
Level
Setpoint
Fig. 4. Response of the system using PID controller
0200 400 600 800 1000 1200 1400
-5
0
5
10
15
20
25
Response of the system using IMC-PID
Time (seconds)
Level of liquid in the tank (cm)
Level
Setpoint
Fig. 5. Response of the system using IMC-PID
controller
0200 400 600 800 1000 1200
0
5
10
15
20
25
Response of the system using MPC
Time (seconds)
Level of liquid in the tank (cm)
Level
setpoint
Fig.6. Response of the system using MPC controller
4.5. Comparison of results
Table 1shows the comparison of results of all the
three controllers. From the table, it is clear that, the
offset has reduced much in MPC compared to IMC-
PID and PID. This is because the target is computed
each time the control calculation is done. This has
lead to effective control and ultimately better
tracking of set point. The rise time has drastically
reduced when using MPC because of the use of a
larger control horizon. But the aggressiveness of the
control action is minimized by the choice of a still
large prediction horizon, which has also supported
to reduce errors caused by model mismatches. When
using PID and IMC-PID controllers the ISE
produced is large which paved way to use it as the
optimization function for MPC, as a result of which
the large errors were penalized and a very small ISE
compared to PID and IMC-PID was achieved. The
reduction of the performance parameters IAE and
ITAE by MPC controller also shows that any small
errors caused and the errors that persist for a long
time were also suppressed by MPC. Ultimately,
from the table it is clear that MPC outperforms PID
and IMC based PID controllers
Table 4.1 Comparison of performance of the system subjected to different controllers
Set point
Level
(cm)
Steady state
Level
(cm)
Offset error
Level
(cm)
Rise time
(Sec)
Settling time
(sec)
IAE
ISE
ITAE
PID
20
18.7
1.3
500
805
4.562
42.26015
0.001922
IMC-PID
20
19.2
0.8
400
550
3.729
32.87
0.0014
MPC
20
19.9
0.1
200
220
1.261
11.213
0.0002
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5. Conclusion
An experimental comparison of MPC controller
with PID and IMC-PID controller was made for a
highly non linear process. It was noted that MPC
outperforms the other two controllers. However, the
optimization of MPC parameters could be obtained
by using soft computing techniques. Also the model
mismatches which are taken into consideration in
MPC design could be better handled if online
estimation of parameters were done.
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Conflict of Interest
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that are relevant to the content of this article.
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