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Implementation and Experimental Validation of Classic MPC on
Programmable Logic Controllers
B. Huyck, L. Callebaut, F. Logist, H. J. Ferreau, M. Diehl, J. De Brabanter, J. Van Impe and B. De Moor
Abstract— Over the last years, a number of publications were
written about Model Predictive Control (MPC) on industrial
Programmable Logic Controllers (PLC). They focussed on
explicit MPC strategies to provide a fast solution. When
sufficient time is available to solve a classic MPC problem, an
online solution to the corresponding Quadratic Problem (QP)
can be provided. This paper investigates the use of an online
quadratic programming solver to exploit MPC on a PLC. This
will be illustrated with the classic Hildreth QP algorithm and
qpOASES, a recently developed online active set strategy. These
algorithms will be investigated on a MISO system.
I. INT RODU CTION
Model Predictive Control has become a mature control
technique. It is widely spread for complex and large instal-
lations, particularly in the process industry. In the recent
years, the MPC community evolved towards high speed
MPC and embedded systems. The hardware used are typical
FPGA devices and microcontrollers. A PLC is a widely used
microcontroller adapted to meet industry standard in- and
outputs and communication protocols. These devices can
be easily programmed and expanded to meet the needs of
an industrial control or automation problem. The common
PLC programming languages are standardized and differ
from the typical PC programming languages. In order to
use algorithms from literature, a manual time-consuming
translation of the algorithm is needed. This is one of the
reasons why trials to run MPC on PLCs employ explicit MPC
[1], [2], [3] as less PLC code is needed. This approach uses
a precomputed solution stored in look-up tables. For MIMO
systems with a large control horizon, i.e. more than 10, these
solutions are sub-optimal as the time to search the look-
up tree becomes prohibitive. These methods therefore seem
perfectly suited for small SISO (Single Input - Single Output)
systems. For large MIMO systems Pannocchia et al. [4]
suggested to use online QP solvers. This paper investigates
the possibility of using standard industry hardware for classic
MPC applications on small MIMO (Multiple Input - Multiple
Output) systems. Classic MPC solves the corresponding
quadratic problem online. This article examines two online
QP algorithms to solve the MPC problem online. First, the
Bart Huyck, Lennert Callebaut and Jos De Brabanter are with the
Department of Industrial Engineering, KAHO St-Lieven, B-9000 Gent,
Belgium prename.name@kahosl.be
Bart Huyck, Hans Joachim Ferreau, Jos De Brabanter, Moritz
Diehl and Bart De Moor are with the Department of Electrical
Engineering, (ESAT - SCD), KU Leuven, B-3001 Leuven, Belgium
prename.name@esat.kuleuven.be
Bart Huyck, Filip Logist and Jan Van Impe are with the Department
of Chemical Engineering (CIT - BioTeC), KU Leuven, B-3001 Leuven,
Belgium prename.name@cit.kuleuven.be
Hildreth QP algorithm [5], a classic but easy to implement
algorithm, will be used. It will be compared to qpOASES,
a state-of-the-art online active set algorithm [6]. Parts of
the code which are not necessary for this application have
been removed. The application consists of a heating device
where fan speed and resistor power can be manipulated
independently. Although small scale, it is an interesting
test application [1]. Both algorithms will be compared with
regard to their applicability and the time needed to solve the
online MPC problem for this application. This paper is struc-
tured as follows: Section II deals with the MPC formulation
used in this paper. Section III describes the experimental set-
up and the applied model. Section IV contains the practical
implementation of the MPC algorithm on a Siemens PLC.
Results are discussed in Section V and finally, the main
conclusions will be summarized in Section VI .
II. MOD EL P RE D IC TIV E CO NTRO L FOR MU L ATION
Linear Model Predictive Control is well known in litera-
ture [7], [8], [9] and the reader is invited to read these works
for a detailed description. The basic formulation needed for
this paper is given below.
A linear discrete-time system is described by:
x(k+ 1) = Ax(k) + Bu(k)(1)
y(k) = Cx(k),
where A∈Rnxn,B∈Rnxmand C∈Rpxnwith m,n
and prespectively the number of inputs, states and outputs.
The objective of the controller is finding the optimal input for
this system by means of minimizing a cost function, which
for the purpose of this paper, is assumed to be the following:
J=
Hp
X
i=1
ky(k+i|k)−r(k+i|k)k2
Wy
+
Hc
X
j=1
ku(k+j|k)k2
Wu.(2)
Hpand Hcare respectively the prediction and control
horizon of the controller. Hc≤Hpis assumed. Wy∈Rpxp
and Wu∈Rmxmare positive definite weighting matrices
and ris the output reference. The formulation y(k+i|k)
represents the vector yon time k+iat calculation time k.
The power of MPC lies in the possibility of adding
constraints. For this paper only input constraints are taken
into account. Output and state constraints are omitted. Given
Preprints of the 2012 20th Mediterranean Conference on
Control & Automation (MED), Barcelona, Spain, July 3-6,
2012
ThA3.5
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the constraints for the input: uMin ≤u≤uM ax , the
optimization problem can be formulated as:
Minimize J=
Hp
X
i=1
ky(k+i|k)−r(k+i|k)k2
Wy
+
Hc
X
j=1
ku(k+j|k)k2
Wu(3)
Subject to x(k+ 1) = Ax(k) + Bu(k)(4)
y(k) = Cx(k)
u(j)≤uMax
u(j)≥uMin
In order to solve this problem, the optimization problem
is reformulated by elimination of the states in the form of a
Quadratic Problem:
Minimize J=θTHθ +gTθ(5)
Subject to P θ ≤α(6)
To this end, following steps are taken. First the state space
model is rewritten in terms of a new input u′= [∆uy]Tto
add integration. The prediction along the prediction horizon
is written in matrix formulation and is be formulated as:
Y=Fx(k) + HU′,(7)
with Yand U′column matrices of respectively outputs
and inputs. E.g. U′∈R(p+m)Hcx1 composed of u′(k|k)to
u′(k+Hc|k). The matrices Fand Hcan be found in many
works on MPC [7], [9]. The matrix His postprocessed by
including the weight matrices as follows:
H′=HTWybd H+Wubd (8)
with Wybd and Wubd blockdiagonal matrices of respec-
tively Wyand Wu. As for the purpose of this work, the aim
is to minimize the online calculation work, matrices that can
be computed in advance and are calculated offline. Matrix
H′is one of the precomputed matrices that does not change
during runtime. Vector gfrom (5) has to be calculated online
and contains two parts depending on the current state and the
reference of the in- and outputs. So, ghas to be calculated
according to:
g=G1x−G2Yref (9)
where G1and G2are gradient matrices. Yref is an
RpHcx1 matrix of the references r(k|k)to r(k+Hc|k). The
gradient matrices are constant and are computed offline:
G1 = H′TWT
ybd F;(10)
G2 = H′TWT
ybd ;(11)
The constraints, i.e. the minimum and maximum admis-
sible value for U′, are calculated online. For this paper, the
Pt100
sensor
Pt100
sensor
Flow
direction
Fig. 1. Schematic overview of the temperature control set-up.
constraints are constant during the experiment. Finally, the
QP problem to be solved is:
min
U′
1
2U′TH′U′+gU′(12)
Subject to U′
Min ≤U′≤U′
Max (13)
g=G1x−G2Yref (14)
Both the Hildreth [5] and the qpOASES [6] algorithm will
be used to solve this problem.
III. EXP ER IME NTAL T EMP ER ATU RE C ONT ROL SE T-UP
A. Hardware and software
The temperature control set-up consists of a resistor and
a fan which can be controlled separately. The fan is driven
by a 24V DC motor and the resistor has a maximum power
of 1400 W. The heating power delivered by the resistor can
be adapted by a solid state relay with analog control (Gefran
GTT 25A 480VAC - analog control voltage 0-10 V). The
fan is controlled by a custom made DC Drive based on a
TEXAS INSTRUMENTS DRV102T chip and adapted for
an analog control voltage of 0-10 V. Temperature sensors
measure the environmental temperature and the temperature
of the heated air as indicated in Fig. 1. Both sensors are
of the PT100 class B type. The controller is a Siemens
CPU319-3DP/PN equipped with an 8Mb memory card and
an additional SM334 analog card to connect to the solid
state relay and DC drive. The Siemens CPU is programmed
using Step 7 Professional 2010. To code the problem, the
Structured Control Language (S7-SCL) is used.
B. Model Identification
As MPC requires a model, this temperature control set-
up is modeled. Based on a multisine input signal, a two-
input (fan speed and resistor power) and single output
(temperature) model is created. Both the construction of the
input signal and the identification are performed using the
Matlab System Identification Toolbox [10]. The resulting
model consists of first order transfer functions:
Tn=−0.98
1+5.17se−1.34s0.83
1+7.17se−1.53suF an,n
uP ower,n
(15)
In (15) the index nindicates a normalized variable. uF an,n
and uP ower,n are respectively the normalized and detrended
input signal for the motor drive and the normalized and de-
trended voltage applied to the solid state relay of the resistor.
After detrending, a zero output of the model corresponds to
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42◦C. For the inputs, the zero input corresponds to 5 V.
More information about this set-up and this identification
procedure can be found in [11].
C. Controller Design
Model (15) is converted to a discrete state space model of
order 4. The sampling rate is 1 Hz. It has been verified that
the model is controllable, observable and stable. The control
horizon Hcof the controller is set to 7 and the prediction
horizon Hpis 22. These horizons have been chosen similar to
those in [11] to compare the different controllers and control
algorithms for this temperature control set-up.
IV. IMP L EM ENTATI ON O N T H E HAR DWARE
The current evolution towards code generation and MPC
toolkits that produce ready-to-run C/C++-code, makes it
possible to quickly implement MPC algorithms on micro-
controllers and computers. This type of devices are often
lab equipment or development tools. A lot of standard
industrial hardware is not able to run this generated code as
a different programming standard is employed or compilers
are unavailable. Closed software or the fact that the real-time
behavior of the device cannot be guaranteed are often reasons
for this situation. Siemens S7-300 PLC CPUs do not have
a port to C/C++. Therefore, to run MPC on these devices,
the applied QP solvers have to be translated in a suitable
supported language. In this paper S7-SCL is used.
A. Hildreth algorithm
This algorithm has been chosen due to its easy implemen-
tation which does not require the use of matrix operations
like, e.g. transpose. The employed PLC has a limited compu-
tational power, so constant parts of the algorithms which be
calculated offline are calculated offline. The data is stored in
matrices to keep the link with the theory, but the computation
of the optimal input is executed element-wise.
The Hildreth algorithm calculates the solution in two
steps [9]. First, the unconstrained solution is calculated and
if no constraints are violated, this solution is presented at the
outputs of the PLC. If a constraint is violated, a QP will be
solved. The solution of the QP is then passed to the inputs of
the heating device. For more information about the solution
routing, see [5], [9], [12]. Important for the online use of
this algorithm is that there is no guaranteed solution that
fulfills the constraints if it is stopped early. So, if the time
needed to calculate the solution exceeds the available time,
the algorithm will be stopped, but it cannot be ensured that
the constraints are satisfied. In such a case, the solution of
the previous QP is used.
B. qpOASES algorithm
qpOASES is an open-source C++ implementation of the
recently proposed online active set strategy [6]. It builds on
the idea that the optimal sets of active constraints do not
differ much from one QP to the next. At each sampling
instant, it starts from the optimal solution of the previous
QP and follows a homotopy path towards the solution of the
current QP. Along this path, constraints may become active
or inactive as in any active set QP solver and the internal
matrix factorizations are adapted accordingly. While moving
along the homotopy path, the online active set strategy de-
livers sub-optimal solutions in a transparent way. Therefore,
such sub-optimal feedback can be reasonably passed to the
process in case the maximum number of iterations is reached.
As no port to C/C++ exists, a simplified version of
qpOASES has been translated to S7-SCL and was used
for controlling the heating device. Note that our simplified
implementation does not allow for hotstarting the QP so-
lution and is not fully optimized for speed. Moreover, it
only handles bounds on the control inputs but no general
constraints.
C. Steps to compute a solution
To compute the inputs, the following sequence of actions
presented in Algorithm 1 are programmed. In advance, con-
stant matrices are precomputed and the reference trajectory
for the output temperature is defined. The input reference is
set to zero for the algorithm, which corresponds to 5 V for
both inputs.
1Offline: Calculate H’, G1, G2 and G3;
2Online: Start PLC;
3Store all precomputed matrices in working memory,
together with the reference for inputs and output;
4while PLC is running do
5if 1 second passed since last call then
6Calculate current state with current information
of the system in- and outputs;
7Calculate g,U′
min and U′
max;
8while maximum numbers of iterations is not
reached and solution not found do
9Solve one iteration of the QP;
10 Check if solution is found or maximum
numbers of iterations is reached;
11 end
12 switch max. numbers of iterations is reached do
13 case Hildreth
14 Use solution of the former QP;
15 endsw
16 case qpOASES
17 Use last sub-optimal solution;
18 endsw
19 endsw
20 Apply the inputs to the system;
21 end
22 end
Algorithm 1: Steps to compute the inputs of the system.
D. Programming the PLC
To calculate the appropriate inputs of the system and
solve the QP, following build-in function blocks (FB) and
organization blocks (OB) are programmed.
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0 100 200 300 400 500 600 700
20
25
30
35
40
45
50
55
Time (s)
Temperature (°C)
Meas. temperature (qp)
Meas. temperature (H)
Reference
Fig. 2. Measured temperature for both the Hildreth (H) and qpOASES (qp)
algorithm. The weight matrix for the inputs of the systems is the identity
matrix (W11
u).
1) OB 100 - Cold Start: This block is called once when
the controller is started. It calls function FB 1, which is used
to initialize the precomputed matrices which are stored in
FB 2. This procedure is followed to reduce the look-up time
of the variables during the calculation of the solution and to
overcome the limitation that an array of constants cannot be
larger than 256 elements at compilation time. In the case a
matrix needs to contain more than 256 elements, more arrays
are combined in this block at runtime into a combined array.
For these experiments, only matrix G2is initialized in this
function.
2) OB 1 - main loop: This loop is started as soon as OB
100 is finished. When this function finishes, it restarts again.
This loop is used to program standard tasks of the PLC. In
this experiment, it is not used. It will run during the idle
time of the CPU between two OB35 calls.
3) OB 35 - timed loop: This organization block is called
every second. It contains the necessary code to read the
current controlled and environmental temperature. This in-
formation is scaled and employed to calculate the current
state (FB 3). Together with the reference for the in- and
outputs, the state is used to update matrix g(FB 2). Now
the QP is solved and the scaled solution will be passed to
the outputs of the PLC. The absolute maximum execution
time of an OB is six seconds, otherwise the system will
raise a system failure. This is a hardware setting that cannot
be overruled by the user. For the current experiments, the
maximum execution time for OB 35 is set to 980 ms.
V. R ESU LTS
Two cases with a different weight matrix Wuin the cost
function are presented. In the first case, the weight matrix is
the identity matrix Wu=I2x2. We will use W11
uto refer to
this case. W15
uwill be used for the second case where the
weight matrix is:
Wu=1 0
0 5.(16)
0 100 200 300 400 500 600 700
0
2
4
6
8
Heating (V)
0 100 200 300 400 500 600 700
2
3
4
5
6
7
8
Fan speed (V)
Time (s)
Fig. 3. The applied inputs to the system for both the Hildreth (H) and
qpOASES (qp) algorithm. The weight matrix is the identity matrix (W11
u).
Fig. 2 depicts the measured temperature and its reference
temperature for both the Hildreth and qpOASES algorithms
in case W11
u. This reference temperature is a constant tem-
perature of 40◦C during 200 s, followed by a jump towards
45◦C during 120 s, again a temperature of 40◦C during
120 s and a set-point of 45◦C for 60 s. From here, the
temperature ramps up to 50◦C at a rate of one degree per
second. This ramp is followed by a sine wave for one and a
half period with a period length of 50 s. After a large jump
towards 47◦C, the temperature is kept constant until the end
of the experiment. The corresponding inputs of the system
are displayed in Fig. 3. Fig. 4 depicts the same reference
trajectory as Fig 2 for W15
u. In this case, the deviation from
the input corresponding to the heating is punished five times
more than the deviation from the reference of the fan speed.
The inputs of the corresponding experiment are plotted in
Fig. 5. The logging of the experiments is done on a computer.
Every experiment starts when the PLC is switched on. This
means a connection has to be established between PLC and
PC at the start of each experiment. Hence, it is unknown in
advance when the logging starts. To deal with this problem,
every experiment begins with a fixed fan speed and switched-
off heating for 30 s. This shortens the 200 s starting set-point
at 40◦C to 170 s. All plots only show the last 5 s of these
startup period. This start-up phase also gives the estimator
the time to calculate the best possible current state of the
system.
A. Hildreth Algorithm
For W11
u, the Hildreth algorithm follows the desired
reference temperature correctly with some small disturbances
caused by the environmental conditions. A large overshoot is
observed at the start caused by the inaccurate state estimation
of the model. After all, the model is a linearized model
around 42◦C. The environmental temperature is almost three
times the standard deviation away form its nominal value
resulting in a bad state estimation. Every further step is taken
smoothly and over- of undershoots are part of the properties
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0 100 200 300 400 500 600 700
20
25
30
35
40
45
50
55
Time (s)
Temperature (°C)
Meas. temperature (qp)
Meas. temperature (H)
Reference
Fig. 4. Measured temperature for both the Hildreth (H) and qpOASES
(qp) algorithm. The weight matrix for the inputs of the systems W15
u.
of the algorithm. The ramp and sine wave are followed close
by both experiments, but a small delay appears. For W15
u,
the first part of the experiment until 450 s, is very similar
to the W11
u. The error for the ramp and sine wave is clearly
larger, certainly in the lower temperature range. The sum of
squared error is nearly doubled as printed in Table I.
TABLE I
THE S UM O F S QU AR ED E RR OR S F OR T HE R A MP A N D SI N E WAVE (FRO M
421 S UN TI L 651 S)F OR T HE EX P ER IM EN TS DEP I CT ED I N FI G . 2 AND 4
Hildreth qpOASES
W11
u1397 762
W15
u2406 1406
For W11
u, the two inputs remain closely to their reference
(5 V, zero for the algorithm). The constraints are hit four
times by the fan speed. Fig. 6 depicts the required number
of iterations of both algorithms and Fig. 7 plots the cor-
responding calculation time. For W15
u, the constraints are
also hit four times, but for a longer time as deviation of the
reference for the heating input is punished five times more
than in the first case. For both W11
uand W15
u, the Hildreth
algorithm needs between 10 and 25 iterations if a QP has to
be solved. Each iteration takes approximately 1 ms resulting
in a maximum runtime of 25 ms for these experiments. It
has to be noted that in the described experiment only one
constraint is active.
B. qpOASES Algorithm
For both W11
uand W15
u, the reference is followed in a
similar way for both the Hildreth and qpOASES algorithm.
The differences can only be explained by differences in the
solution of the QP and different environmental conditions.
From both Fig 2 and 4, it can be noticed that the Hildreth
algorithm reacts a little bit slower. It is typically one second
behind the qpOASES solution, resulting in e.g. a larger
overshoot at the start, although both algorithms start at the
same moment in similar circumstances.
0 100 200 300 400 500 600 700
0
2
4
6
8
Heating (V)
0 100 200 300 400 500 600 700
2
3
4
5
6
7
8
Time (s)
Fan speed (V)
Fig. 5. The applied inputs to the system for both the Hildreth (H) and
qpOASES (qp) algorithm. The weight matrix is W15
u.
The first step on the homotopy path for qpOASES takes
approximately 10 ms. If no constraint is active, the optimal
solution is found immediately. If a constraint is active,
maximum 7 iterations were needed for these experiments by
qpOASES performing cold-starts. Again, it has to be noted
that only one constraint was active during the experiment.
For the additional iterations, qpOASES needs approximately
15 ms resulting in a maximum runtime to get a solution
of 120 ms. For W15
u(Fig. 3), the inputs of the system are
similar for both algorithms. qpOASES hits the constraints at
the same four moments, but one second earlier than Hildreth.
The constraints are also left one second earlier. For W11
u
(Fig. 5) the same evolutions are seen. The clearly different
path of the fan speed during the first 400 s in Fig. 5 is caused
by different environmental conditions.
C. Comparison of both algorithms
Both algorithms solve the QP online. Compared with
respect to computing time, the Hildreth algorithm is faster. If
no constraints are active, this algorithm calculates a solution
10 times faster. If constraints are hit, the number of iterations
is lower for qpOASES. Due to the advanced mathematics
in the algorithm, it still takes more time to get a solution
than with the Hildreth algorithm. Table I lists the sum of
squared errors (SSE) for the ramp and sine wave. The SSE
values are only calculated for a part of the experiment,
but are representative for the whole experiment. When both
algorithms are compared, the SSE of qpOASES is clearly
lower than the one for the Hildreth algorithm. For W15
u
the SSE is nearly doubled for the Hildreth algorithm, for
W15
uthe SSE is 70% higher than for qpOASES. It is
clear that qpOASES has a higher accuracy and is preferred
over the Hildreth algorithm. Although not described in this
paper, identical experiments are set up with different start-
up conditions. During the first 30 s of the experiment,
the inputs passed to the estimator were the inputs of the
former QP solution, not a manually fixed set as depicted in
the plots. This resulted in an estimated state corresponding
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0 100 200 300 400 500 600 700
0
5
10
15
20
25
# iterations
qpOASES Wu
11
Hildreth Wu
11
0 100 200 300 400 500 600 700
0
5
10
15
20
Time (s)
# iterations
qpOASES Wu
15
Hildreth Wu
15
Fig. 6. Number of iterations needed to solve the QP problem. Above: W11
u
- both inputs are punished equally. Below W15
u- the input controlling the
heating is punished five times more than the input controlling the fan.
to approximately 50◦C and inputs touching both heating
and fan constraints. The Hildreth algorithm needed more
than 60 iterations at that point, the maximum allowed, and
was stopped early. The former solution was passed to the
estimator (but not to the process). qpOASES, on the other
hand, calculated the solution in maximum 14 iterations and
provided this to the estimator. Maximum 20 iterations were
allowed. This behavior demonstrates that for the Hildreth
algorithm appropriate measures need to be taken to handle
these situations. For this application, the current output is
not changed.
VI. CO NCL US I ON
This paper illustrates that it is possible to run an online
model predictive controller on a Programmable Logic Con-
troller. Two online QP algorithms are tested to solve a Model
Predictive Control (MPC) problem. Both algorithms success-
fully control a temperature along a reference trajectory. The
Hildreth algorithm is able to present a solution in a shorter
time than the qpOASES algorithm. However, the latter has a
much higher performance. Hence, it depends on the specific
application which QP solver is preferred.
ACK NOWL EDG EME NT S
Research supported by Research Council KUL: OT/10/035,
GOA/11/05 Ambiorics, GOA/10/09 MaNet , PFV/10/002 (OPTEC),
IOF-SCORES4CHEM, several PhD/postdoc & fellow grants; Flem-
ish Government: FWO: PhD/postdoc grants, projects: G0226.06
(cooperative systems and optimization), G0321.06 (Tensors),
G.0302.07 (SVM/Kernel), G.0320.08 (convex MPC), G.0558.08
(Robust MHE), G.0557.08 (Glycemia2), G.0588.09 (Brain-
machine) research communities (WOG: ICCoS, ANMMM,
MLDM); G.0377.09 (Mechatronics MPC) IWT: PhD Grants,
Eureka-Flite+, SBO LeCoPro, SBO Climaqs, SBO POM, O&O-
Dsquare Belgian Federal Science Policy Office: IUAP P6/04
(DYSCO, Dynamical systems, control and optimization, 2007-
2011) ; IBBT EU: ERNSI; FP7-HD-MPC (INFSO-ICT-223854),
COST intelliCIS, FP7-EMBOCON (ICT-248940), FP7-SADCO
0 100 200 300 400 500 600 700
0
20
40
60
80
100
120
140
Computation time (ms)
qpOASES Wu
11
Hildreth Wu
11
0 100 200 300 400 500 600 700
0
20
40
60
80
100
120
140
Time (s)
Computation time (ms)
qpOASES Wu
15
Hildreth Wu
15
Fig. 7. Time needed to solve the QP problem. Above: W11
u- both inputs are
punished equal. Below W15
u- the input controlling the heating is punished
five times more than the input controlling the Fan.
(MC ITN-264735), ERC HIGHWIND (259 166), Interreg
IVa-7-022-BE i-MOCCA; Contract Research: AMINAL Other:
Helmholtz: viCERP ACCM The scientific responsibility is assumed
by its authors. BDM is a full professor at the KU Leuven, Belgium.
JVI holds the chair Safety Engineering sponsored by the Belgian
chemistry and life sciences federation essenscia. The fourth author
holds a PhD fellowship of the Research Foundation – Flanders
(FWO). We thank Andreas Potschka for providing a simplified
Matlab implementation of qpOASES.
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