Practical Approach to Tuning MPC
Willy Wojsznis, John Gudaz, Ashish Mehta and Terry Blevins
Emerson Process Management
willy.wojsznis@emersonprocess.com
Keywords
Model Predictive Control, DMC, MPC Tuning, Reference Trajectory
Abstract
This paper presents the results of a heuristic approach for developing MPC tuning rules. The tuning has been applied and
tested in easy-to-use MPC. Process modeling in this MPC uses normalized input/output range. As a result, there is no need
for tuning outputs, a procedure known as adjusting equal concern error. Penalties on moves are set as a function of process
deadtime as a primary factor, with some correction from process gain. The default calculation delivers robust control,
which tolerates up to triple increase in process static gain. If control is too aggressive, further on-line adjustment can be
done by reference trajectory. Test results show that this tuning is robust for process gain change; however, it is much less
efficient in compensating process dead time changes. It was found that dead time mismatch is much better compensated
with the model correction filter. Combining the three handles, i.e. penalties on moves, reference trajectory and model filter
easy and intuitively understandable MPC tuning was achieved. The findings are illustrated by numerous MPC simulated
tests.
Introduction
MPC is a model based control. Therefore, it would seem that after developing process model,
controller design is straightforward. However, MPC controller design -or generation, as it is often
called- involves many parameters which strongly affect controller performance and robustness. The
main issues related to MPC controller generation are the following:


Model equalization – traditional MPC models have inputs/outputs in engineering units. Some
outputs like temperature may have a range of a thousand degrees, others like composition have a
range of a few units. In a multivariable controller, errors from the two outputs should be
equalized. Dynamic Matrix Control (DMC) [7], for example, uses a parameter called equal
concern of error for that purpose. This parameter is manually adjusted in simulation by trial and
error method. An advance in solving this problem is minimizing the condition number of the
system matrix and scaling model accordingly [2].
Penalty on moves – another tuning parameter adjusted by trial and error in a traditional
MPC/DMC procedure.
On-line controller tuning allows further controller performance adjustment by applying range
control, model output filtering and reference trajectory.
Interesting considerations and proposals for MPC controller tuning are provided in [1] by Lee, J. and
Yu, Z. The tuning is based on sensitivity functions in frequency domain. The main tuning parameter
is filter gain. Filter calculations are complemented by deliberate selection of penalty on move factors.
Another controller design is presented in [2]. It features significant complexity of on-line
calculations, which may be more complex than desirable for an MPC embedded in a system
controller. A tuning strategy based on required condition number is discussed in [3].
The current paper presents a tuning approach which features simple and intuitively understandable
calculations of tuning parameters, delivering a well performing and robust MPC controller.
Controller robustness can be easily increased on-line by adjusting reference trajectory time constant.
The procedure was developed for small and medium size controllers and includes the following
steps:
 Reviewing the process model
 Setting the parameters for controller generation
 Testing the response in simulation
 Adjusting MPC once control is in service
The tuning satisfies the requirements for embedded MPC implementation in a scalable control
system.
Model Validation
Model validation shows exactness of process model and provides a good indication about the required
robustness of the controller. If the model demonstrates a significant mismatch with process, the
controller should be more robust. There are several validation techniques available; the simplest one
is to review the model to ensure that it reflects your knowledge of the process. We present a design
which incorporates human inspection. A typical procedure involves the following steps:
1. Inspection of validation errors for all outputs. Simulation software applies real process input
data as process model inputs and displays the root mean square error between the calculated
output and the actual output for a selected data set that represents normal operation. If the average
output error exceeds 1% per scan, associated step responses should be examined in more detail.
Figure 1-A shows model validation for incorrect process gain. This figure shows model validation
where the model gain is approximately two times smaller than true process gain. Notice that this
gain mismatch results in much smaller changes in predicted process output (upper curve). The
average squared error is high (2.35 percent per scan).
Figure 1. A. Model validation with significant gain mismatch. B. Model validation with correct gain.
Figure 1-B shows validation with correct model. This figure shows the same validation plot as
the previous figure but with an acceptable error (.32 percent per scan) and a good model –
process match (model gain is approximately –3 percent off).
2. Inspection of every step response. Inspection verifies if the gain sign is correct and that the gain
value is in the range expected based on the knowledge of the process. It is a significant advantage
when identifier has two or more techniques for model identification, for example ARX and FIR1.
If the model is accurate, step responses developed by those two techniques should match.
Figure 2-A illustrates FIR and ARX step responses for incorrect model. In this figure, FIR and
ARX indicate a different gain sign. One reason for such model inaccuracy might be that the dead
time used for ARX algorithm was defined poorly. Improving dead time identification, the FIR and
ARX matches acceptably (Figure 2-B).
A
B
Figure 2. A. FIR and ARX step responses for incorrect model. B. FIR and ARX responses for correct model.
3. Making minor corrections in process dynamics. Noisy data may introduce error in the process
model. Frequently, this is reflected as irregular shape of the step response. In such situations, it
would be advantageous to use graphical or numerical step response design tools to smooth out the
step response.
1
ARX – AutoRegressive with eXternal inputs, FIR – Finite Impulse Response
4. Correcting step responses that deviate from expected ranges. Noisy data, insufficient
excitation, or short test time for data collection, may produce a process model that is not
satisfactory for control. However, if plant conditions do not allow for a better test, one should
consider correcting the model based on the knowledge of the process, information gathered by
observing trends of the measurement, and the obtained process model. MPC application provides
numerical entry or graphical design option for step response parameters.
5. Inspecting validation after making model changes. If someone has modified the model it would
be a good practice to observe again how well the calculated and actual outputs match in a selected
set of data.
Model scaling assumes 100% range on the process input and 100% range on the process output, with
the same engineering ranges as used for local measurement and control. This simple solution
automatically assures outputs equalization and delivers normalized model for controller generation.
MPC tuning calculations
The MPC controller is generated from the process model and controller design parameters. The MPC
controller minimizes the squared error of a controlled variable over the prediction horizon and the
squared error of controller output over control horizon:
min y[X (k) - R(k)]2 + uU(k)2
(1.1)

U(k)
where
p – prediction horizon
m – control horizon
X(k) - controlled output p-step ahead prediction vector
R(k) - p-step ahead setpoint vector
U(k) - m-step ahead incremental controller output moves vector
y = diag{y1 ,...,yp} - penalty matrix on the output error
u = diag{u1,...,um} - penalty matrix on the control moves
The solution is in the form:
-1
U(k)= (SuTyTySu + uTu) SuTyTyEp(k)
(1.2)
where
Su - process dynamic matrix built from step responses: p x m for SISO model and p*no x m*ni
matrix for MIMO model with ni inputs and no outputs
Ep(k) - error vector over prediction horizon
The performance of the control algorithm can be modified by the adjustable parameters: p, m, u, and
y. From an implementation standpoint, p and m are not convenient to use as tuning parameters. u
and y are usually applied as scalars, that is, control error is multiplied by the scalar y over the
whole prediction horizon, and any controller moves over control horizon are multiplied by the scalar
u.
Figure 3. Controller output change in dependence of penalty on moves.
Process model is a first order with time constant equal 30 MPC scans, p=120
Let us explore in more detail how controller output depends on tuning parameters for a controller with
m=1 for E p (k) =1. After few calculations on (1.2) we have an equation:
p
u(k)= (S   S +   ) S   
uT
yT
y u
uT
u -1 uT
yT
a
i
y
1
 u 
1 ai    y 
 
p
2
(1.3)
2
where ai are the step response coefficients. Figure 3 shows the dependence of controller output
u
. It is a good practice to assume Penalty on Error (PE=  y ) parameter=1, for most cases,
y
and use Penalty on Moves (PM=  u ) for adjusting the controller performance.
change on
Controller Robustness
Sensitivity of control to changes in process dynamics is determined by the controller robustness. The
parameter used in controller generation that most impacts robustness is the Penalty on Moves (PM)
(Figure 3). PM defines how much the MPC controller is penalized for change in the manipulated
output (MV). The Penalty on Moves parameter is defined independently for every MV. High PM
value results in a slow controller with a wide stability margin. With such settings, the control is
relatively insensitive to change in either the process or the model errors.
Low PM values result in a fast controller with a narrow stability margin. When the model used in
generating the control accurately reflects the process gain and dynamics, then changing the PM value
does not affect the controller performance significantly. However, a difference in the controller
performance will occur if the model does not match the real process. Further exploration of controller
robustness has been done experimentally.
Figures 4 and 5 show gain margin and deadtime margin dependence on process deadtimes for
different times to steady state. The obvious conclusion from the plots is, that the gain margin and the
deadtime margin is strongly affected by the deadtime. It follows that deadtime should be accounted
for as a major factor in calculating PM. Experimental formula (1.4) defines PM which provides stable
and responsive MPC operation for model error of up to about 50%.
 6 DTi 3Gi DTi 
PM i  3 1 


p
p 

(1.4)
DTi is the deadtime in MPC scans for each MVi - CVi relation
Gi is gain (no units) for each MVi - CVi relation
MV, CV denote Manipulated Variable and Controlled Variable respectively.
In a particular MPC implementation, PM is calculated with no user involvement. In most cases, the
calculated setting for PM gives good control even if the model error is greater than +/-50%.
Figure 4. Deadtime Margin in dependence of process deadtime
Figure 5. Gain Margin in dependence of process deadtime
A
B
Figure 6. Setpoint Step Response Using Default Settings with good model and with process gain 2.5 times the model gain.
Figure 6 shows an example of a setpoint step response with a good model match and with process
gain equal 2.5 times the model gain. PE equals 1, and PM equals 4 (default calculations according
1.4).
Effect of penalties on moves on controller performance is shown in Figure 7-A, that shows the same
process-model mismatch as in the Figure 6-B except that PM was set to be equal 20 (five times the
default setting).
Control Sensitivity
To meet application requirements, one may consider giving priority to maintaining a specific process
output at the setpoint. Penalty on Error (PE) factor indicates importance placed on a specific
Controlled Variable (CV). The default value of the PE is 1 for all CVs and this value provides a good
control for most applications. Changing the PE from this default value prioritizes control action.
Setting the associated PE to a value greater than 1 realizes more sensitive control for the specific CV,
while a PE value less than 1 relaxes the control action. The primary criterion for adjusting the PE is
acceptable variability on a specific controlled parameter.
A
B
Figure 7. Utilization of PM and Setpoint Filter to compensate for model mismatch.
Testing MPC in simulation
After controller generation, as a rule MPC is tested in simulation. The simulation environment
facilitates the control response for setpoint changes, measured and unmeasured disturbances,
constraints handling, and optimization. Controller performance can be tested by adjusting the
A
B
Figure 8. Utilization of model filter to compensate for dead time mismatch.
maximum MV move and the setpoint trajectory online. An operator interface used in simulation is
shown in Figure 9.
Figure 9. MPC simulation interface
Tuning MPC in operation
Modifying the setpoint reference trajectories is the primary method of on-line MPC tuning. Increasing
the setpoint trajectory filter time constant increases overall controller robustness. It is possible to
select a specific CV which exhibits an oscillatory response and increases the related setpoint filter.
Figure 7-B shows the same process-model mismatch and controller generation settings as in the
Figure 6-B. The controller performance is adjusted online by a setpoint filter time constant that equals
120 (same as Prediction Horizon) for the first step response (up) and 240 (two times the prediction
horizon) for the second step response (down). The default setpoint filter time constant in Figure 6-B
was 0 (zero).
Adjusting the SP filter time constant is recommended in increments of one half of the time to steady
state up to two times the value of time to steady state.
When model mismatch comes from the dead time in a dead time dominant process, SP filter may not
be enough to eliminate chattering of the controlled output. Figure 7-A shows application of SP filter
=450 sec, with MPC prediction horizon =600 sec. A significant chattering on the process output has
been eliminated after application of the model filter with time constant equal to about ten MPC scans.
While adjusting MPC controller, user understanding of process behavior, as well as achievable
control objectives, is important.
Conclusion
A simple and transparent procedure of model validation and automatic calculation of tuning
parameters for MPC controller delivers robust controller performance. Controller operation can be
next adjusted on-line by adjusting reference trajectory time constant and model output filter.
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