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CoDIT'13
Implementation of a Predictive controller
on the STM32 board
1
Smeti A.,2Chagra W., 3Ksouri M.
Laboratory of Analysis Conception and Control of Systems,
1
: Faculty of Sciences of Tunis,
2
: Preparatory Institute for Engineering Studies El Manar, Tunis
3
: National School of Engineering of Tunis, Tunisia.
[email protected], [email protected], [email protected]
Absract— In the literature, there is few practical
works about the implementation on miniaturized
devices like FPGA (Field Programmable Gate Arrays)
and microcontrollers of the Generalized Predictive
Control (GPC) technique. The reason can be
attributed to its programming complexity. In this
work, we took advantage of the advancement in the
field of embedded systems and we realized an
application based on a GPC implemented on an
STM32 microcontroller to control a basic second
order system. Also, a PID (Proportional-IntegralDerivative) controller has been implemented in the
STM32 to make a performance comparison with the
GPC.
Key words- Predictive control; PID; STM32;
Microcontroller
I.
INTRODUCTION
Implementation of control techniques on
miniaturized devices is often used because it saves
a lot of space thus reducing clutter [1], [5], [6], [7],
[8], [9], [10]. In addition, there exist several
softwares that facilitate the communication and the
programming of the mentioned devices. Another
reason is the appearance of several types of
microcontrollers with low cost that are becoming
more and more efficient, thanks to a decreased time
processing. The STM microcontrollers are an
example of the advancement in the embedded
technology that combines low cost, low
consumption and more performance. In this paper,
the implementation of the GPC (Generalized
Predictive Control) control of a system of second
order on the STM32VLDiscovery board is
performed using the software IAR Embeded
Workbench. The results will encourage us to look
for the implementation of more advanced control
techniques on the mentioned microcontroller
applied to nonlinear and complex systems. The
paper is organized as follows. In section II, we
present the GPC controller. The section III deals the
stm32 board and Iar Embeded Workbench. In the
section IV, we present the experimental results.
978-1-4673-5549-0/13/$31.00 ©2013 IEEE
II.
THEORICAL REVIEW OF GPC
We consider systems that can be modelized by the
following discrete state space model:
x ( k + 1) = F x ( k ) + G Δ u ( k ) (1)
y (k ) = H x(k )
The matrices F, G and H, are given as mentioned in
[2]:
⎛ a1 1 " 0 ⎞
⎛ b0 ⎞
⎜
⎟
⎜
⎟
#
# " #⎟
# ⎟
F= ⎜
, G= ⎜
,
⎜ an−1 0 " 1 ⎟
⎜ bn− 2 ⎟
⎜
⎟
⎜
⎟
⎝ an 0 " 0 ⎠
⎝ bn−1 ⎠
H= (1
0 " 0)
H ∈ R (1,n ) , F ∈ R ( n ,n ) , G ∈ R ( n ,1) .
The state of the system is x(k)∈ Rn, the measured
1
output is y (k ) ∈ R1 , the control increment is
with
Δu(k) and Δ = 1 − z
−1
is an operator that denotes
the integral action which ensures static error
elimination. Using the equation (1) we obtain the
state expression at (k+j):
j −1
x(k + j / k ) = F j x(k ) + ∑ F j−1−iGΔu(k + i)
i =0
(2)
The use of both equations (1) and (2) leads to the
output j-step ahead predictor:
j −1
y(k + j / k ) = HF j x(k ) + HF j −1−iGΔu(k + i) (3)
∑
i=0
The GPC is based on the minimization of a
quadratic cost function wich is given by:
Hp
Hc
j=1
j=1
J =∑(y(k + j / k) −wk
( + j))2 +λ∑Δu(k + j −1)2 (4)
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CoDIT'13
where Hp is the prediction horizon; Hc is the
control horizon, λ is the weighting factor, w(k+j)
denotes the set-point at time k+j and y(k + j / k) and
j∈[1,Hp] is described in equation (3). In this
section, we adopt the output deviation method [2].
Using F, G and H above , we define the following
matrices:
0
⎛ HG
⎜
HG
L= ⎜ HFG
⎜
#
#
⎜⎜ H p −1
H p −2
⎝ HF G HF G
⎞
⎟
#
0
⎟;
⎟
#
#
⎟
H −H
# HF p c G ⎟⎠
#
0
⎡ HF ⎤
⎢
2 ⎥
M= ⎢ HF ⎥
⎢ # ⎥
⎢
Hp ⎥
⎣⎢ HF ⎦⎥
The matrix L has dimension (Hp, Hc) where Hc is
the control horizon and the vector M has Hp
elements.
The output sequence can be written as follows:
(5)
Y = LΔu + Mx
Where:
T
y = ⎡ y(k +1/ k ) y(k + 2 / k ) " y(k + H p / k ⎤
⎣
⎦
Δu = [ Δu(k) Δu(k +1) " Δu(k + Hc −1]
The STM32 BOARD
III.
EMBEDDE WORKBENCH
AND
IAR
A microcontroller is a computer system that
includes a single integrated circuit memory ROM
(Read Only Memory) and RAM (Random Access
Memory), a processor that executes information and
input-output circuits for the interaction with the
outside environment.
A. STM32F100RB microcontroller
The choice of the STM32 value line discovery
is based on the following characteristics: low cost,
low power consumption, as well as a great
performance. Indeed, the mentioned card includes a
microcontroller type STM32F100rb, where it
integrates the cortex M3 32-bit processor. The
cortex M3 is based on the RISC (Reduced
Instruction Set Computer) architecture, which
constitutes the main cause of the success of the
combination between low cost and fast processing
and enables also the STM32vldiscovery to reach a
frequency speed of 74 MHz. The mentioned
microcontroller has also a 128 kb flash memory and
8 kb RAM [3].
Adding to this, the board includes a large number of
input pins / outputs, some of which are connected to
the ADC and DAC via a bus 32 MHz. Power can be
provided either through a USB cable or an external
power supply from 3.3V to 5V.
For debugging, the card has an interface called STlink that enables communication between the PC
and board. Fig.1 shows the stm32 boards.
T
The objective function (4) can be expressed as:
J = (Y − W ) T (Y − W ) + λ Δ u T Δ u
(6)
In which
W = ⎡⎣ w( k + 1) w( k + 2) ... w(k + H p ) ⎤⎦
Minimizing the equation (6), we obtain the optimal
control sequence:
−1
Δ u = ⎡⎣ LT L + λ I H c ⎤⎦ LT [W − Mx ( k ) ]
Where
I Hc ∈ R ( Hc , Hc ) is the identity matrix.
Finally, we have the recursive form of the
command described by the following equation:
u (k ) = u ( k − 1) + Δu (1)
In wich Δu(1) is the first element of Δu.
978-1-4673-5549-0/13/$31.00 ©2013 IEEE
Fig 1. STM32VLDiscovery board
T
W = ⎡⎣ w(k + 1) w(k + 2) " w(k + H p ) ⎤⎦
B. IAR Tool Chain
IAR Embedded Workbench is a development
tool easy to use. It allows debugging embedded
applications using assembler, C and C++.
Programming becomes easier using a standard
library given by the manufacturer which provides a
firmware of initializing [3], the peripherals of the
microcontroller. Table I shows the source code
provided by Stmicroelectronics.
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CoDIT'13
TABLE I.
Source Code Provided By STMicroelectronics
File name
Roles of peripherals
p
stm32f10x_rcc
stm32f10x_adc
stm32f10x_dac
stm32f10x_gpio
stm32f10x_dma
Configurattion of all the clocks
that will bee used
Initialization of Analog /
Digital connverter
Initialization of the Digital /
Analog connverter
Configurattion of input and
output portts
Initialization the DMA (Direct
Memory Access)
A
⎡ 2.586 1 0 ⎤
F = ⎢⎢ −2.2129 0 1 ⎥⎥ ,
⎣⎢ 0.6269 0 0 ⎥⎦
⎡ 0.022 ⎤
G = ⎢⎢0.0188⎥⎥ ,
⎢⎣ 0 ⎥⎦
H = [1 0 0] .
In order to use the GPC coontroller algorithm, the
steps illustrated in the flow
wchart, (Fig.2) must be
followed:
After that, we need routines for matrices
computation. Routines will be sum
mmarized in the
following table:
File name
Zero_matrix
Product_matrix
Sum_matrix
Tr_matrices
Prod_vect_mat
IV.
TABLE II.
GPC firmwaree
Roles of peeripherals
Initializatioon all the matrices
at zeros
Multiplicatiion
of
two
matrices
Addition off two matrices
Transposition of a matrice
Multiplicatiion of a matrix
with a vectoor
EXPERIMENTAL RESUL
LTS
In this work, we are interestedd in the control
results when using the STM32 miccrocontroller, so
we chose a basic second order system
s
an RLC
serie. We configured the pin PC4 in input mode and
connected it to the capacity output and
a the pin PA4
in output mode and connected it too the resistance.
The process has a second order diffeerential equation
that leads to the following transfer function:
fu
H ( p) =
Vs ( p)
1
=
2
p +1
Ve ( p ) LCp + RCp
In our work, we have consideredd the following
values:
R=268; L= 11.48 H and C= 680 μF
w
a sampling
The discretized transfer function with
time Te = 0.02s, is given by the following
expression:
G( z) =
0.02197 z −1 + 0.0188 z −2
1 − 1.586 z −1 + 0.6269 z −2
The matrices F, G and H are given by:
b
978-1-4673-5549-0/13/$31.00 ©2013 IEEE
Fig 2. Flowchart of the GPC method
In order to evaluate the perfformance of the control
technique mentioned above we
w use the well known
PID (Proportional-Integral-D
Derivative) controller.
The expression of the discrrete PID controller has
only the next form:
u(k) = kp (e(k) +
k
T n
e( j) + d (e(k) −e(k −1)))
∑
ki j=0
T
where: Te is the sample timee, u is the control input
and e is the difference between the reference input
and the system output.
The latter was implemeented on the same
microcontroller. The param
meters of the PID are
obtained by the method off Takahashi [4]. The
closed loop simulation resultts are obtained with the
following GPC and PID parrameters: Hp = 4; Hc =
1; λ = 1; kp = 0.22; ki=0.00998; kd=0.0392, where kp
is the proportional gain, ki iss the integral gain, kd is
the derivative gain.
The results are displayed inn Fig. 3, Fig. 4, Fig. 5
and Fig. 6, where we can see
s the behavior of the
RLC system output, the refference output and the
252
CoDIT'13
control input using respectively the GPC and the
PID controller.
Fig 3. RLC system response (vs(k) solid line and ve(k) dashed
line) (GPC controller)
Based on this table, we note that the computing
time of the GPC is greater than that of the PID.
Indeed, the GPC algorithm has several matrix
resolutions, see Table II.
The implementation of the PID and the GPC
controller on STM32 was successful. From the
experimental results we can see that both
controllers have a good tracking of the reference
input. Although the execution time of the PID
algorithm is less than that of the GPC; the latter has
the advantage of anticipating the change in the
reference input and thus acts before it occurs, which
is not the case of the PID controller.
V.
Fig 4. GPC control input
Conclusion
In this work, we have implemented a PID and a
GPC controllers on a STM32 microcontroller. The
IAR embeded workbench was used to load the
firmware which enables the microcontroller to
control a second order system. The presented work
is the first step that enables us to consider in the
future the control of complex and nonlinear systems
with the GPC strategy implemented on the STM32
microcontroller.
REFERENCES
[1] E.Aridhi, M. Abbes, A. Mami. “FPGA Implementation of
Predictive Control”, MELLECON’2012, March 25-28,
Hammamet Tunisia.
Fig 5. RLC system response (vs(k)solid line and ve(k) dashed
line) (PID controller)
Fig 6. PID control input
In the table below, we compare the performance of
execution time between PID and GPC controller.
TABLE III.
Performance comparison between GPC and PID Controller
PID
GPC controller
Average execution time (μ s)
4.613
532.142
978-1-4673-5549-0/13/$31.00 ©2013 IEEE
[2] K. Watanabet, K. Ikeda, T. Fukuda, S.G. Tzafestas.
“Adaptive Generalized Predictive Control using A State-Space
Approach”,IEEE International Workshop on Intelligent Robots
and Systems IROS’91, p. 1609-1614, November 3-5, 1991,
Osaka Japan.
[3] STMicroelectronics. Datasheet STM32F100, 2009.
www.st.com
[4] M. Mathelin and I. Bara. Commande numériques des
systèmes. Cours, Ecole Nationale Supérieure de Physique de
Strasbourg, 2ème année, 2007
[5] J. Ghabi. Synthèse et implémentation sur FPGA de la
commande prédictive robuste des systèmes linéaires
multivariables. PhD thesis, Ecole Nationale d’Ingénieurs de
Monastir, 2010.
[6] A. Kheriji, F. Bouani, and M. Ksouri, “Efficient
implementation of constraind robust model predictive control
using a state space model,” in Proceedings of International
Conference on Informatics in Control, Automation and Robotics
(ICINCO), 2010.
[7] Y. F. Chan, M. Moallem, W. Wang, “Efficient
implementation of PID control algorithm using FPGA
technology”, Proceedings of the 43ed IEEE Conference on
Decision and Control, V5, PP. 4885-4890, Bahamas2004.
[8] U. R. Y. Jayaraman, “Fpga implementation of predictive
control strategy for power factor correction,” in Proceedings of
World Academy of Science, Engineering and Technology, 2008.
[9] G. Palomo, K. Hilton, and J. Rossiter, “Predictive control
implementation in a plc using the iec 1131.3 programming
standard,” in Proceedings of American Control Conference,
2009.
[10] G. Pannochia, “Robust model predictive control with
guaranteed set point tracking,” Journal of Process Control, vol.
14,pp.927–937,
2004
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