Technische Universität Clausthal
Institut für Elektrische Informationstechnik
Rapid Control Prototyping Using
an STM32 Microcontroller
BACHELORTHESIS
CÁNDIDO OTERO MOREIRA
Matrikelnummer: 461108
Betreuer:
Erstgutachter:
Zweitgutachter:
Dipl. -Ing. Pablo Ballesteros
Prof. Dr.-Ing. Christian Bohn
Prof. Dr.-Ing. Christian Rembe
Erklärung
Eidesstatliche Erklärung
Ich erkläre hiermit eidesstattlich, daß ich die vorliegende Arbeit selbständig
und ohne fremde Hilfe angefertigt und alle Abschnitte, die wörtlich oder
annähernd wörtlich aus einer Veröffentlichung entnommen sind, als solche
kenntlich gemacht habe. Ferner, daß die Arbeit noch nicht veröffentlich und auch
keiner anderen Prüfungsbehörde vorgelegt worden ist.
Clausthal-Zellerfeld, den 28. August 2015
CONTENTS
1 Introduction ............................................................................................................................1
1.1 Introduction to Code Generation .....................................................................................2
1.2 Introduction to Model-Following Control .......................................................................3
1.2.1 The Plant ...................................................................................................................4
1.2.2 State-Space Representation .......................................................................................4
1.2.3 The Controller: State-feedback .................................................................................5
1.2.4 The Model-Following Perspective ............................................................................6
2 Hardware ................................................................................................................................7
2.1 Motor ...............................................................................................................................7
2.1.1 Motor diagram ...........................................................................................................7
2.1.2 Ports ...........................................................................................................................8
2.1.3 Sensors Specifications ...............................................................................................9
2.1.4 Technical Motor Specifications ................................................................................9
2.2 Microcontroller ..............................................................................................................10
2.2.1 Getting Started.........................................................................................................10
2.2.2 Features overview ...................................................................................................11
2.2.3 ADC and DAC peripherals .....................................................................................11
2.2.4 USARTs and UARTs peripherals ...........................................................................13
3 Code Generation ..................................................................................................................14
3.1 Requirements for Rapid Prototyping .............................................................................14
3.2 Toolboxes Overview......................................................................................................14
3.2.1 Embedded Coder Support Package for STMicroelectronics
STM32F4-Discovery Board................................................................................... 14
3.2.2 Target Support Package – STM32F4 Adapter ........................................................15
3.2.3 Waijung Blockset ....................................................................................................17
3.3 Basic Operations ............................................................................................................18
3.3.1 Model Configuration for Code Generation .............................................................19
3.3.2 Analog-to-Digital Converter Configuration ............................................................19
3.3.3 Digital-to-Analog Converter Configuration ............................................................20
3.3.4 USART Communication Configuration ..................................................................21
3.3.5 PC Serial Communication Configuration ...............................................................23
3.3.6 SIL Simulation ........................................................................................................24
3.3.7 PIL Simulation ........................................................................................................26
4 Plant Modelling ....................................................................................................................27
4.1 Theoretical Method: Analysis of a DC Motor ...............................................................27
4.1.1 Velocity Control ......................................................................................................28
4.1.2 Position Control.......................................................................................................29
4.1.3 Results .....................................................................................................................29
4.2 Empirical Method: Grey-Box Model Estimation with MATLAB ................................30
4.2.1 Introduction to Experimental Estimation ................................................................30
4.2.2 Getting the Output ...................................................................................................31
4.2.3 Parameters Estimation .............................................................................................34
4.2.4 Results .....................................................................................................................35
A
5 Signal Adaptation.................................................................................................................38
5.1 Analog Output ...............................................................................................................38
5.1.1 PWM .......................................................................................................................38
5.1.2 DAC ........................................................................................................................39
5.2 Inverting Rotation Sense ...............................................................................................39
5.2.1 H Bridge ..................................................................................................................39
5.2.2 Custom Circuit ........................................................................................................39
5.3 Custom Circuit analysis .................................................................................................40
5.4 Amplifier Calibration ....................................................................................................43
5.4.1 Output Amplifier Calibration ..................................................................................43
5.4.1 Input Amplifier Calibration .....................................................................................44
6 Classic State Feedback .........................................................................................................45
6.1 Controllability................................................................................................................45
6.2 Pole Placement ..............................................................................................................46
6.3 State Observers ..............................................................................................................47
6.3.1 Observer design .......................................................................................................47
6.3.2 Observability ...........................................................................................................48
6.4 The Linear Quadratic Regulator Problem .....................................................................49
7 Model-following Controller .................................................................................................50
7.1 Gains calculation ...........................................................................................................50
7.1.1 Controller Gains ......................................................................................................51
7.1.2 Feed-forward Gain ..................................................................................................52
7.2 Time-discrete Controller................................................................................................54
8 Velocity Control...................................................................................................................55
8.1 MIL Simulation .............................................................................................................55
8.2 SIL Simulation ...............................................................................................................56
8.3 PIL Simulation ...............................................................................................................58
8.4 HIL Simulation ..............................................................................................................60
9 Position Control ...................................................................................................................64
9.1 MIL Simulation .............................................................................................................64
9.2 SIL Simulation ...............................................................................................................65
9.3 PIL Simulation ...............................................................................................................66
9.4 HIL Simulation ..............................................................................................................66
10 Conclusions ........................................................................................................................69
B
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 1: INTRODUCTION
1 INTRODUCTION
The main goal of this project is to dig into rapid control prototyping tools that can save
time, money and effort in control design. The object of study is the microcontroller
STM32F4.
The secondary aim is to study model-following control in dynamical systems. To
achieve this goal, two applications will be performed on a DC motor: velocity and position
control.
To achieve these aims, the present work approaches progressively to the final goal. In
the next sections of this chapter a general vision of the automatic code generation process is
given, establishing the main differences between the MIL, SIL, PIL and HIL simulations.
The model-following controller is introduced as well, starting from the basic control
principles for readers who are not very familiar with the subject.
In Chapter 2 the main hardware is reviewed to clarify several decisions made in further
sections. The motor used is a QET DC Motor Control Trainer (DCMCT). It will be analysed
and this will give a better perspective to understand the equations related to its performance
and some design considerations such as signal adaptation or plant modelling. The STM32F4
microcontroller will also be analysed to understand its capabilities to communicate and
process signals.
The computer setup to perform rapid control prototyping is explained in Chapter 3.
Three Simulink toolboxes can be used for this purpose and each of them is analysed
highlighting its strengths and weak spots. Finally, some of the main generic operations
accomplished in this project are described in detail, such as CAD and DAC conversion and
communication with a computer.
In Chapter 4 a mathematical model of the motor is achieved by using a grey-box
estimation with MATLAB. First, the physical principles of a generic DC motor are analysed
to obtain a theoretical model of the DC motor according to the values provided by the
manufacturer. Then, the grey-box estimation is performed to obtain a more accurate
experimental model for velocity control. Two experiments will be necessary in order to
obtain a model for the motor without the belt and with the belt linked to the potentiometer.
The full model for position control is finally deducted from the theoretical analysis and the
motor specifications.
The problem of adapting the microcontroller analog inputs and outputs to the analog
motor signals is tackled in Section 5. First, some possible options are analysed to solve this
problem. The adopted solution will be to use a custom signal amplification circuit. This
circuit will be analysed and tested. It will be necessary to calibrate it and the followed steps
will be shown.
In Chapter 6 the theoretical control principles of state feedback are analysed. First, the
requirements of a plant to be controllable are investigated. Then, the classic pole placement
technique is introduced, which is the basis of the model-following controller. An
explanation to the need of using an observer can also be found in this chapter with a brief
study of its principles. Finally, an approach to optimal design is given by introducing the
linear-quadratic-regulator problem.
The model-following controller is finally obtained in Chapter 7 by describing its desired
behaviour and deducting its equations. A diagram of the controller is given and the steps to
calculate all the gains are explained. The controller is finally written in its state-space form
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 1: INTRODUCTION
to store it in MATLAB as a state-space model class. The controller is finally discretized in
order to implement it in the microcontroller.
The next two chapters show the model-following controller performance with real
applications. In Chapter 8 the controller will be tuned along the different simulation stages
(MIL, SIL, PIL and HIL) until obtaining a satisfactory velocity response. Communication
computer-microcontroller is also demonstrated by using a computer as a host that gives the
reference to the microcontroller. The ultimate goal is the real plant to track the model output
accurately. The script to obtain a model-following controller for a generic-order system is
used in this chapter. The script can be found in the annexed document. In this chapter a firstorder model will be followed by the motor.
In Chapter 9 the controller will be tested again to perform position control over the DC
motor. The same steps will be followed as for velocity control. In this chapter the motor will
follow a second-order system.
Finally, the results observed along the development of this project are reflected in
Chapter 10. This section is a breakdown of the achievements and points to improve in other
possible works derived from this one.
In this project, MATLAB R2014a version is used. It is recommended for the reader to
use the same version in order to avoid compatibility code problems with the scripts
provided. All activities have been performed under Windows 7-64 bits.
The attached CD contains the Simulink models and scripts to perform all activities from
Chapters 4, 8 and 9. The files should be copied into the user’s computer before executing it.
This is to keep MATLAB from trying to build additional files in the CD, which would lead
to error.
Now, a brief introduction to rapid prototyping is made to have a more clear vision of the
activities intended to do in this project. In section 1.2 the model-following controller is
presented to the reader and a few ideas of control theory are given.
1.1 Introduction to Code Generation
Rapid prototyping allows the user to quickly test a design, being able to make fast
adjustments until the results are satisfactory [1]. It is usual to start using a tool like Simulink
to simulate control algorithms for modelled systems. With rapid prototyping tools, the
design can be quickly tested on the real control hardware, this is, the STM32F4.
During the development of a controller there are several stages [2]. First, the plant is
modelled and the controller is tested in simulation using the model; this is called model-inthe-loop (MIL) simulation. The next step would be testing the code for the controller before
loading it onto a real embedded system, such as the STM32F4. This can be done in Simulink
by adding a new block with the code (whether automatically generated or not) containing the
algorithm and performing a new simulation; this is called software-in-the-loop (SIL)
simulation. The differences between these two simulations are shown in Figure 1.1.
After that, the code should be loaded onto the real embedded system and tested again;
this would be the processor-in-the-loop (PIL) simulation. Finally, the designer should go one
step further and try the embedded system with a physical simulator or with the real plant
instead of the virtual modelled plant; this is the hardware-in-the-loop (HIL) simulation.
Figure 1.2 illustrates the transition from PIL to HIL simulation.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 1: INTRODUCTION
REFERENCE
REFERENCE
CONTROLLER
(BLOCKS)
CONTROLLER
(CODE)
MODELLED
PLANT
MODELLED
PLANT
READINGS
READINGS
MIL simulation
SIL simulation
Figure 1.1: Differences between MIL and SIL simulation
REFERENCE
REFERENCE
STM32
CONTROLLER
STM32
CONTROLLER
REAL
PLANT
MODELLED
PLANT
READINGS
READINGS
PIL simulation
HIL simulation
Figure 1.2: Differences between PIL and HIL simulation
The tools for code generation allow for the focus on design and testing instead of
programming. In fact, some tools allow directly generating C/C++ code and flashing it onto
the board, skipping the SIL and PIL simulation steps.
Nevertheless, it is recommendable, when feasible, to follow all the steps in order to
detect possible errors.
1.2 Introduction to Model-Following Control
This sub-chapter has the intention to give some control notions to the readers who have
not become very familiar with the control subject yet. It is also a good reading to get used to
the notation and ease understanding in next chapters.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 1: INTRODUCTION
1.2.1 The Plant
In control theory the term ‘plant’ is often used to name the system to command. It can
be modelled as a ‘box’ that receives an input and gives an output. In Figure 1.3, it is shown a
DC motor modelled by a first order system to obtain the velocity.
Input
voltage
Input
voltage
M
Shaft
velocity
𝐾
𝑇𝑠 + 1
Shaft
velocity
Figure 1.3: Modelled DC motor by a first-order system
In order to be able to study the control of the plant the motor must be represented with a
mathematical function. This can be achieved by following two methods:
- Studying the physical laws that intervene in the output. Depending on the
complexity of the system, it can lead to very long and complicated equations, even
making simplifications.
- Observing the behaviour of the system. If a known input is given to the plant and the
output is registered with a sensor, there are several techniques to obtain an
approximation of the plant.
1.2.2 State-Space Representation
The state of a dynamical system is the smallest combination of variables which allow
fully know its response to a given input at any further instant [3]. In the case of a physical
system, those state variables are physical parameters, such as voltage, speed, temperature,
etc.
An example of state-space representation for a first-order system will now be analysed.
The results will be used in chapters ahead.
Given a generic first-order transfer function
𝑌(𝑠)
𝐾
(1-1)
𝐺(𝑠) =
=
,
𝑈(𝑠) 𝑇𝑠 + 1
it can be expressed as
𝑇𝑦̇ + 𝑦 = 𝐾𝑢 .
(1-2)
𝑦=𝑥.
(1-3)
Let
The following system can be written:
1
𝐾
𝑇𝑥̇ = −𝑥 + 𝐾𝑢
𝑥̇ = − 𝑥 + 𝑢
{
{
𝑇
𝑇 .
𝑦=𝑥
𝑦=𝑥
4
(1-4)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 1: INTRODUCTION
According to the general state-space equations for a single-output system
𝒙̇ (𝑡) = 𝑨𝒙(𝑡) + 𝑩𝑢(𝑡)
,
𝑦(𝑡) = 𝑪𝒙(𝑡) + 𝐷𝑢(𝑡)
(1-5)
the following equalities can be written:
1
𝐾
(1-6)
; 𝐵 = ; 𝐶 = 1; 𝐷 = 0 .
𝑇
𝑇
Observe that for first-order systems A, B and C are scalar values. For higher orders, they
would be matrices. Also note that for single input systems the control signal is a scalar. If
u(t) was a vector, then D would be a matrix as well.
For the general case A is the state matrix, B is the input matrix, C is the output matrix
and D is the direct transmission matrix.
The previous result will be used in section 4.2.3 when programming the architecture of a
first-order system. From these matrices, a general system with a single input can be
represented in a block diagram as illustrated in Figure 1.4.
𝐴=−
D
𝑢(𝑡)
B
+
𝒙̇
∫dt
+
𝒙
C
+
+
𝑦(𝑡)
A
Figure 1.4: State-space representation in blocks of a continuous time system with single input
1.2.3 The Controller: State-feedback
Plants do not usually behave the desired way. The mission of the controller is to move
the poles of the system so that it accomplishes the given specifications. This is done by
using a state matrix gain that weights each state and modifies the control input as shown in
Figure 1.5.
REFERENCE
+
𝑢(𝑡)
PLANT
-K1
-K2
...
-Kn
Figure 1.5: State-space feedback
5
𝑦(𝑡)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 1: INTRODUCTION
1.2.4 The Model-Following Perspective
The classic method used in state feedback control in order to adjust the output of the
plant is to calculate the zeros and poles the controller needs to modify the whole system so
that it accomplishes the response requirements, such as overshooting, gain, settling time,
stability, etc.
The new focus of the model-following controller is that given an explicit mathematical
model, the real plant must ‘follow’ the model and both responses should be the same.
Figure 1.6 shows that the model-following controller is compensating the actuation by
using the model and state feedback.
MODEL-FOLLOWING CONTROLLER
𝑟(𝑡)
MODEL
𝒙m
𝑢(𝑡)
-Km
+
𝑦m (𝑡)
-K
𝑦(𝑡)
PLANT
𝒙
Figure 1.6: Basic idea of the model-following controller
The theoretical basis is the same in both points of view, so in chapter 6 the classical
state feedback control is analysed and in chapter 7 the analog steps are shown to get a
model-following controller.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE
2 HARDWARE
2.1 Motor
The rapid control prototyping will be done over a DC servomotor; in this case, the QET
DC Motor Control Trainer will be used. To give a more tangible idea, the real system is
illustrated in Figure 2.1.
Figure 2.1: Photograph of the DCMCT system (extracted from the manual)
This device is designed for teaching about control fundamentals and basic controllers
design [4]. The board includes an embedded PIC microcontroller, a servo drive, an encoder,
a potentiometer, several outputs to read the status of each element and one input to drive the
motor. The motor also has its own software, the QICii, to perform different controls via
USB in conjunction with the Quanser QIC Processor Core. The supplied guide gives
information about the technical characteristics of the motor and shows step by step different
control fundamentals.
For this project only the motor, the servo drive and its sensors will be used. In
conclusion, the board will be used in open loop configuration using the STM32F4 to read
and write signals.
2.1.1 Motor diagram
In order to illustrate how the servo works and understand the analysis made in the next
chapter of the theoretical model of the plant, Figure 2.2 is provided. The voltages of interest
are summarized in Table 2.1.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE
Symbol
Vc
Vt
Vp
Vm
Description
COMMAND port input voltage
TACH port output voltage
POT port output voltage
Motor voltage feed
Range
±5 V
±5 V
±5 V
±15 V
Table 2.1: Voltages of interest in the analysis of the DCMCT
Note that to switch from QIC to HIL control, the J6 jumper must be configured. The
right setup for the purpose of this project is the HIL configuration. This is also shown in the
diagram.
Linear amplifier
𝑉m
M
POWER
COMMAND
𝑉c
ENCODER
ENCODER
CURRENT
Adapter
±5 V
TACH
HIL
J6
POT
𝑉t
𝑉p
QIC
D/A
POTENTIOMETER
DCMCT
SERIAL
Figure 2.2: Motor diagram with servo drive, potentiometer and encoder
The previous figure is useful to understand the difference, between the theoretical model
given by the documentation and the empirical model obtained in section 4.2.
2.1.2 Ports
The external connections are summarized in Table 2.2, which is the same as Table A.4
from the motor documentation.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE
Connector Label
D/A
POT
TACH
CURRENT
ENCODER
COMMAND
POWER
Connector Type
RCA
RCA
RCA
RCA
5-Pin DIN
RCA
6-mm Jack
Signal
QIC D/A Output
Potentiometer Output
Tachometer Output
Current Measurement Output
Encoder Output
Command To Linear Amplifier
AC Power Input to DCMCT
Range
±5 V
±5 V
±5 V
±5 V
A, B, Index
±5 V
15VAC, 2.4A
Table 2.2: DCMCT external connections
The connections of interest for closed-loop control will be COMMAND, to drive the
motor; TACH, to measure the velocity and POT, to read the position. The optical encoder is
directly mounted to the rear of the motor, while the potentiometer must be linked with a belt
to the motor shaft.
2.1.3 Sensors Specifications
The sensor parameter specifications are shown in Table 2.3 and have been taken from
the Table A.3 of the documentation.
Description
Potentiometer
Potentiometer Calibration at POT RCA JACK
Potentiometer Calibration at QIC A/D Input
Potentiometer Resistance
Potentiometer Bias Voltage
Potentiometer Electrical Range
Tachometer
Tachometer Calibration at TACH RCA JACK
Tachometer Calibration at QIC A/D Input
Value
Unit
39
78
10
±4.7
350
º/V
º/V
kΩ
V
º
667
1333
RPM/V
RPM/V
Table 2.3: Sensor parameter specifications
2.1.4 Technical Motor Specifications
The most relevant specifications of the DCMCT are given in Table 2.4. The full model
parameters can be consulted in Table A.2 of the documentation. The nomenclature has been
adapted to match with the notation used in this project.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE
Symbol
Ra
La
Km
Kt
Jeq
Description
Motor armature resistance
Motor armature inductance
Electro-motive-force constant
Motor torque constant
Moment of inertia of rotor + load
Linear Amplifier Maximum Output voltage
Linear Amplifier Gain
Value
10.6
0.82
Not specified
0.0502
2.21 · 10−5
15
3
Unit
Ω
mH
V·s/rad
N·m/A
kg·m2
V
V/V
Table 2.4: System parameters
2.2 Microcontroller
The STM32F4DISCOVERY board is a low-cost and easy-to-use development kit to
quickly evaluate and start a development with an STM32F4 high-performance
microcontroller. It is based on an STM32F407VGT6 microcontroller and includes an STLINK/V2 embedded debug tool interface, ST MEMS digital accelerometer, ST MEMS
digital microphone, audio DAC with integrated class D speaker driver, LEDs, pushbuttons
and a USB OTG micro-AB connector. In this text, the reader can find a brief overview of
the STM32F4 features, with special focus on the peripherals used in this project. To consult
the full specifications, refer to the STM32F4DISCOVERY and STM32F407VGT6
datasheets [5] [6]. A picture of this board can be found in Figure 2.3.
Figure 2.3: Photograph of the actual STM32F4 microcontroller used in this project (extracted from the
STM32F4DISCOVERY datasheet)
2.2.1 Getting Started
In order to power the board, the JP1 jumper must be set. To program the STM32F4Discovery board, the two CN3 jumpers must be plugged as shown in Figure 2.4. The power
10
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE
is supplied by the USB connection with the PC using a USB cable ‘type A to mini-B’. It can
also be powered for external application with a 5 V source.
Figure 2.4: JP1 jumper on and CN3 jumper configuration to allow ST-LINK/V2 programming and
debugging (extracted from the STM32F4DISCOVERY datasheet)
2.2.2 Features overview
The STM32F407VGT6 is based on the high-performance ARM® Cortex®-M4 32-bit
RISC core operating at a frequency of up to 168 MHz. The Cortex-M4 core features a
floating point unit (FPU) single precision which supports all ARM single precision dataprocessing instructions and data types. It also implements a full set of DSP instructions and
a memory protection unit (MPU) which enhances application security. The
STM32F407VGT6 incorporates high-speed embedded memories (1 Mbyte of Flash
memory, 192 Kbytes of SRAM), and an extensive range of enhanced I/Os and peripherals
connected to two APB buses, three AHB buses and a 32-bit multi-AHB bus matrix.
It offers three 12-bit ADCs, two DACs, a low-power RTC, twelve general-purpose 16bit timers including two PWM timers for motor control, two general-purpose 32-bit timers
and a true random number generator (RNG). They also feature standard and advanced
communication interfaces:







Three I2Cs.
Three SPIs, two I2Ss full duplex. To achieve audio class accuracy, the I2S
peripherals can be clocked via a dedicated internal audio PLL or via an external
clock to allow synchronization.
Four USARTs plus two UARTs.
An USB OTG full-speed and a USB OTG high-speed with full-speed capability
(with the ULPI).
Two CANs.
An SDIO/MMC interface.
Ethernet and the camera interface.
The STM32F407VGT6 operates in the –40 to +105 °C temperature range from a 1.8 to
3.6 V power supply.
2.2.3 ADC and DAC peripherals
The STM32F4 has two 12-bit buffered DAC channels that can be used to convert two
digital signals into two analog voltage signal outputs. Each channel has its own DAC with
the following features:
 8-bit or 12-bit mode.
 Left or right data alignment in 12-bit mode.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE







Synchronized update capability.
Noise-wave generation.
Triangular-wave generation.
Dual DAC channel independent or simultaneous conversions.
DMA capability for each channel.
External triggers for conversion.
Input voltage reference VREF+.
Also three 12-bit analog-to-digital converters are embedded and each ADC shares up to
16 external channels (check datasheet for ports information), performing conversions in the
single-shot or scan mode. In scan mode, automatic conversion is performed on a selected
group of analog inputs. Additional logic functions embedded in the ADC interface allow:
 Simultaneous sample and hold.
 Interleaved sample and hold.
The ADCs share the input voltage reference with the DACs: VREF+. According to the
electrical schematics provided by the datasheet, the VREF+ port is connected to VDD as shown
in Figure 2.5.
Figure 2.5: Reference voltage ports circuit
VDD is the power voltage, which is given by the 3 V reference as shown in Figure 2.6.
The inductor L1 filters the continuous value of the VDD signal. The capacitors C21, C22,
C34 and C25 are decoupling capacitors, which have infinite impedance to continuous
voltage. According to this analysis and assuming that the VREF+ port consumes no current,
the voltage in VREF+ is given by VDD.
Figure 2.6 explains the need to connect the JP1 to power the board. Alternatively, the
SB17 connection can be permanently closed by solder. The 3 V reference is achieved with a
voltage regulator powered by the 5 V feed of the USB connection. The full scheme can be
consulted in the datasheet for more details.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 2: HARDWARE
Figure 2.6: Circuit showing that the board is powered when the JP1 jumper is on or SB17 is bridged
While testing the DAC and CAD peripherals, it has been detected that the VREF+ voltage
is not exactly 3 volts, for this reason, it is recommended to measure the exact voltage
reference before using the ADC or DAC peripherals.
Since the VREF+ is not available for direct measurement, it must be approximated by
measuring the VDD pin on the board.
2.2.4 USARTs and UARTs peripherals
The STM32F407VGT6 embed four universal synchronous/asynchronous receiver
transmitters (USART1, USART2, USART3 and USART6) and two universal asynchronous
receiver transmitters (UART4 and UART5).
These six interfaces provide asynchronous communication, IrDA SIR ENDEC support,
multiprocessor communication mode, single-wire half-duplex communication mode and
have LIN Master/Slave capability. The USART1 and USART6 interfaces are able to
communicate at speeds of up to 10.5 Mbit/s. The other available interfaces communicate at
up to 5.25 Mbit/s.
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CHAPTER 3: CODE GENERATION
3 CODE GENERATION
3.1 Requirements for Rapid Prototyping
In this project the capability of Simulink to automatically generate C/C++ code will be
used for rapid prototyping.
The general idea to program the microcontroller consists of three parts: first, a toolbox is
used to interact with the ports and peripherals of the board; secondly, a coder generates the
C/C++ code of the Simulink model and finally, a toolchain compiles the previous code and
loads it onto the microcontroller. This last operation usually can also be done in two
different stages: first, generating the binary code and finally, flashing it.
Each available toolbox has its own needs, but the common requirements are:
 MATLAB and MATLAB Coder
 Simulink and Simulink Coder
 Embedded Coder
Furthermore, the STM32F4 ST-LINK must be installed in order to provide all the
drivers for the STM32F4.
In this project MATLAB R2014a is used. For compatibility with previous versions,
check the requirements for each toolbox.
3.2 Toolboxes Overview
3.2.1 Embedded Coder Support Package for STMicroelectronics
STM32F4-Discovery Board
This toolbox is provided by MathWorks [7]. The package can be installed from
MATLAB going to Add-Ons>> Get Hardware Support Packages. After choosing the
installation method, select the package STMicroelectronics STM32F4-Discovery and
continue with the installation. Some additional packages may be automatically installed or
updated, such as the GNU Toolchain. At some point of the process, the installer will ask the
user to provide the path of the Cortex Microcontroller Software Interface Standard (CMSIS)
folder. The latest CMSIS is available for download at the official webpage [8]. The CMSIS
is a third party hardware abstraction layer (HAL) for Cortex-M processors; this layer
provides a standard interface for the software regardless the used hardware.
After finishing these steps, the toolbox will be ready to use. Start a new Simulink model
and configure its code generation parameters to use the ‘ert.tlc’ file as system target. After
that, choose the STM32F4-Discovery as the target hardware. This configuration is shown in
Figure 3.1.
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CHAPTER 3: CODE GENERATION
Figure 3.1: Code generation parameters for the Embedded Coder Support Package for
STMicroelectronics STM32F4-Discovery Board toolbox
The available blocks are shown in Figure 3.2. This toolbox is suitable for SIL and PIL
simulation with different toolchains, but the lack of blocks and configuration parameters
does not allow the user to take advantage of all the features available in the STM32F4.
For instance, it is not possible to have full access neither to the DAC peripherals nor to
the USART modules. The DACs are limited to be used for stereo audio output and the
USART communication can only be used to send information to the computer during PIL
simulation.
Figure 3.2: Embedded Coder Support Package for STMicroelectronics STM32F4-Discovery Board
blocks
For PIL simulation a TTL-RS232 adapter is needed. More information and tutorials can
be found by clicking the ‘[Examples]’ block.
Due to the mentioned limitations, the use of this toolbox for this project will be
discarded, but the STM32F4-Discovery package will be installed in order to perform SIL
and PIL simulations.
3.2.2 Target Support Package – STM32F4 Adapter
This toolbox is provided by STMicroelectronics. The target can be installed by
executing the STM32-MAT/TARGET installer that can be found at the main page [9].
Another software is needed to obtain a preset pin configuration of the board and to
generate the HAL; the STM32CubeMX. This program can also be downloaded from the
main page. Once installed and executed, it is necessary to download the firmware package
for the desired family board to be able to properly configure the ports.
Finally, a toolchain is needed to compile the code. The toolchain will be integrated in
the STM32CubeMX, and can be chosen among the following: EWARM, MDK-ARM,
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TrueSTUDIO and SW4STM32F4. The MDK-ARM Lite version is available for download
at the main page. The SW4STM32F4 is an integrated toolchain of the open source IDE
Eclipse, but it is not yet fully supported by STM32CubeMX and the projects generated for
this toolchain must be manually added to Eclipse. The MDK-ARM Lite version would be
suitable for the purpose of this project.
After installing all the software, the toolbox can be used. First, start a new Simulink
model and configure its code generation parameters to use the ‘STM32F4.tlc’ file as system
target.
After that, go to Code Generation>> STM32F4 Options and check the options
Download Application and STM32CubeMx Path update. Apply changes and close the
window. Ignore the error message that shows up when applying changes, it is an application
bug.
The available block groups appear in Figure 3.3.
Figure 3.3: Block groups in the Target Support Package – STM32F4 Adapter toolbox
Each block group has its own blocks. The first block that must be added belongs to the
MCU CONFIG group: an STM32_Config block. This block contains the port configuration
of the target board. To configure it, once it has been added to the model, double click on it
and go to New ioc file>> Start STM32CubeMx configuration tool. This will open a new
STM32CubeMX instance. Open a new project for the desired board. A graphic pin planner
will appear in the next window. Now, every pin must be configured for its purpose by
clicking on it and choosing the right function, as shown in Figure 3.4. This is a very intuitive
configuration method. The program even alerts the user if a pin is already in use or could
cause trouble with a peripheral. More information can be found in the program
documentation.
Once all the necessary pins have been set, go to Project>> Generate Code>> Project
and choose a valid name and path to save the project. Choose the desired toolchain and click
OK. After generating the code, close the notification window and go back to the
STM32_Config block parameters and click Select STM32F4 configuration file. Go to the
folder where the project template has been generated and select the ‘.ioc’ file. Now the
Simulink model knows what ports and peripherals are available.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 3: CODE GENERATION
Figure 3.4: Example of pin assignment. PA8 is configured as a digital input while PB7 is a serial
reception port
The next step is assembling the diagram and building it. After building it, a new instance
of the STM32CubeMX will show up. Again, generate the code for the project and click the
Open Project button of the notification window. The project will be opened in an Integrated
Development Environment (IDE). According to the chosen toolchain, the user will have to
follow different steps to compile the C code and flash it onto the microcontroller.
This toolchain has the advantage of allowing the user to take almost full control of the
STM32F4 configuration. The blocks permit access to most peripherals and the
STM32CubeMX enables easy configuration of pins, the clock tree, peripherals and
middleware.
The disadvantages are the numerous steps and software required to build and load an
algorithm. Furthermore, since there are many third party applications implied in obtaining
the final code, it has been noticed that some updates in the software make the
STM32CubeMX project not fully supported by the toolchains and some initializations of the
code must be manually made in order to get a successful compilation.
For these reasons, this option will also be discarded.
3.2.3 Waijung Blockset
This toolbox is provided by the STMicroelectronics’ third party member Aimagin Co.,
Ltd [10]. The blockset is released for evaluation purpose only, so to get technical support as
well as other features such as permission for commercial use, a special license must be
purchased.
The toolbox can be directly downloaded from the main page. The file must be
uncompressed and the resulting folder must be saved in a fixed path of the computer. To
install the package, execute the install_waijung MATLAB script file that is located inside
the folder. Make sure the STM32F4 ST-LINK is properly installed. Anyway, a warning
message will be shown saying that the STM32F4 ST-LINK could not be found. Just
continue with the installation process, that message is an application bug.
The package includes the GNU toolchain from ARM Cortex-M & Cortex-R processors,
but the user can install its own compiler. In this project the default GNU toolchain will be
used.
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CHAPTER 3: CODE GENERATION
Once the script has been executed, the toolchain will be ready to use. There is a vast
amount of available blocks, but the mainly used will be the On-chip Peripherals of the
STM32F4 Target, which are shown in Figure 3.5.
Figure 3.5: Main group blocks of the Waijung Blockset toolbox
This toolbox has even more available peripherals than the previous one, such as CAN
and I2C communication. The blocks in this toolbox can be directly configured for almost
any possible need as if programming it manually.
To start the algorithm design, open a new Simulink model. Configuring the target is now
automatically done just by dragging a Target Setup block into the model. This block is
located inside the Device Configuration group block of the STM32F4 Target. Its function is
to configure settings such as the clock tree or the compiler. It can be checked that the system
target file has been automatically set to ‘stm32f4.tlc’, otherwise it should be set manually.
Now the algorithm can be created using the necessary blocks. Once the design is done, it
can be coded, compiled and loaded just by clicking the
Build Model button.
The advantage of this toolbox is the high level of configuration directly from Simulink.
This allows the designer to save a lot of time.
The simplicity and the capability of the blockset have been the reasons to choose this
toolbox for generating the code and flashing the microcontroller of this project.
3.3 Basic Operations
Next, some of the common setups used in the algorithms developed in this project are
described.
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CHAPTER 3: CODE GENERATION
3.3.1 Model Configuration for Code Generation
The basic arrangement for a model that is planned to be executed on the STM32F4
consists of configuring the target and the Simulink solver.
First, open a blank Simulink model and drop a Target Setup block from STM32F4
target. The parameters can be modified by double clicking on it. In this project the GNU
ARM compiler will be used. Also make sure to select the correct microcontroller unit
(MCU) and clock configuration. For this microcontroller, the configuration should look like
Figure 3.6.
Figure 3.6: Target Setup block configuration
The rest of the parameters can remain as default.
Finally, go to Model Configuration Parameters>> Solver>> Solver Options. Configure
the solver type to ‘Fixed-step’, the Solver to ‘discrete (no continuous states)’ and set the
Fixed-step size to Ts. The step size is a variable that must be declared in the console. In this
project the sample time, Ts, will be set to 0.0001 seconds. From the command window,
make Ts equal to 0.0001.
One last thing to take into account is that the model must be saved before building and
both the model’s path and the current workspace directory must be the same.
3.3.2 Analog-to-Digital Converter Configuration
As seen in section 2.2.3, the reference voltage for ADCs and DACs peripherals is not
exactly the nominal value of 3 volts, so it should be directly measured from the VDD pin for
better accuracy. Keep in mind that in this section and the next one, the nominal value of 3
volts is used.
The typical configuration used in this project for ADC reading is shown in Figure 3.7.
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CHAPTER 3: CODE GENERATION
Figure 3.7: ADC configuration to read a single ADC PORT.
ADC port output is an integer value between 0 and 4095 regardless the selected data
output. This range is given by the fact that the ADCs implemented in the STM32F4 have 12bits resolution. In this case, the output has been set to single, but it can be set to a fitter range
data type as long as the consistency of the signals is kept along the algorithm.
The ADC port output is translated into volts by multiplying it by a gain according to the
equation
𝑁port 𝑉ref
𝑉𝑝𝑜𝑟𝑡 =
,
(3-1)
4095
where Nport is the raw reading from 0 to 4095, Vref is the reference voltage and Vport is the
read voltage in volts.
Apart from the data type, the ADC prescaler can be configured. It is not a critical
parameter, but setting it to the lowest value will minimize the conversion time.
Finally, the sample time must be set to Ts.
Here, a single ADC1 port is read, but more pins can be selected from the same ADCx
block. To use another ADC, say ADC2, an additional Regular ADC block must be inserted.
3.3.3 Digital-to-Analog Converter Configuration
As for the ADCs, the real Vref value should be read before executing the conversion. In
this the conversion can be performed either by the DAC1 (PA4) or by the DAC2 (PA5). The
configuration is simple and is shown in Figure 3.8. The desired DAC is selected by checking
the corresponding checkbox. The user must be careful not to let the Input Vref field as
default, which is a value of 3.3 volts and would lead to wrong writings. Furthermore, it has
been detected that the value entered at the Input Vref field must be a number. It is not
possible to use a generic variable and change its value from the command window, since it
leads to compilation error.
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CHAPTER 3: CODE GENERATION
Figure 3.8: Typical configuration for analog value writing
3.3.4 USART Communication Configuration
The STM32F4 has 4 USARTs and 2 UARTs. In this project the communication with the
microcontroller is always made through the USART1, which uses pins PB6 and PB7 for
transmitting and receiving, respectively.
The first step is configuring the USART1 as shown in Figure 3.9. Drag a USART Setup
block into a configured model for code generation and select the module number 1.
Simulink will ask to close the window and reopen it to go on with the configuration. The
baud rate has been set to 460800 bps, which is an acceptable speed for the sample time that
is being managed. The USART1 can work at up to 7.5Mbps, nevertheless, it has been
observed that high rates cause the simulation to work not so fluently. The maximum speed,
besides of the microcontroller, is also a matter of the other used hardware. The baud rate
460800 bps is giving good results and is fast enough for this project.
The rest of the parameters are configured as shown in the picture. By default two buffers
of 512 bytes each are assigned for receiving and transmitting data.
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CHAPTER 3: CODE GENERATION
Figure 3.9: USART Setup block configuration for using USART1 (Tx/Rx: B6/B7 )
The reception of the data will be accomplished by using the UART Rx block. It can be
configured as ‘Non-Blocking’ if it is important to execute the algorithm when all the data
has been received. In that case, the arrangement would look like Figure 3.10.
Figure 3.10: Setup for non-blocking data reception
If the ‘Non-Blocking’ feature is not important, the Enabled Subsystem block will be
omitted, since there is no READY signal. The desired behaviour is that the microcontroller
waits for available data to be received. For example, when the STM32F4 is waiting for a
serial reference to drive the plant, and has an integral control incorporated, the algorithm
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CHAPTER 3: CODE GENERATION
would make the control input to constantly increase if the plant is not connected. In order to
avoid this initial error due to the integral action, a blocking UART Rx block should be used.
The procedure would be: first flash the controller onto the STM32F4, connect it to the plant
and finally give the reference value through the serial channel.
Each quantity of data type must be specified to be received as a binary package. Other
transmission formats are available. The sample time can be different from the rest of the
program, but it must be a multiple of Ts. This is useful in case Ts is very small and the serial
communication does not require such amount of samples. The sample time for
communication between the PC and the STM32F4 has been declared as Tc and has a value
of 0.001 seconds.
The data transmission is analogous to the reception configuration, as shown in Figure
3.11.
Figure 3.11: Setup for blocking data transmission
The data is sent when it is required by the program and the receiver is responsible to
read the information. The rest of the settings are similar to the reception setup.
3.3.5 PC Serial Communication Configuration
The communication with the PC is done through a TTL-RS232 USB adapter. For this
project, the USB to TTL adapter (PL2303 XA/HXA) by D-sun has been used.
The connection is made as shown in Figure 3.12
Tx
PB6
Rx
PB7
USB
MINIUSB
STM32
TTL-RS232
GND
GND
Figure 3.12: Hardware connection for serial communication between PC and microcontroller
The software configuration for PC communication is very similar to USART
communication. The main difference is that this algorithm will be executed on the PC, so
another Simulink model will be needed. This model will not be compiled onto the board, so
a default blank Simulink model will be used.
The used blocks are found under the block group Waijung Blockset>>
Communication>> Host Serial Port. They appear listed in Figure 3.13.
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CHAPTER 3: CODE GENERATION
Figure 3.13: Blocks for PC serial communication
The setup for these three blocks is analogous to the USART blocks. The only additional
action is that the Host Serial Setup block must be configured to work with the
corresponding port. To identify the port number of the serial adapter connected to the PC,
open the device manager and look for the name of the device under PORTS (COM & LPT).
The port number will appear in brackets at the end of the device as shown Figure 3.14.
Figure 3.14: Device manager showing that the USB-TTL adapter has been assigned the COM4
3.3.6 SIL Simulation
The differences between MIL, SIL, PIL and HIL simulations have been explained in
section 1.1. SIL and PIL simulations are the intermediate steps that separate the model
design from the real hardware implementation. In these two sections the configuration to
perform both simulations will be described.
The idea is that one part of the model, which in this case is a controller, must be built
into C/C++ code and tested before real implementation. These simulations are accomplished
by the Embedded Coder included in Simulink. The STMicroelectronics STM32F4Discovery package from section 3.2.1 must be installed in order to run the code on the
STM32F4.
To perform a SIL simulation, open the desired Simulink model. The model must be
configured before being able to generate a C/C++ code block. Open the Configuration
Parameters window and go to Code Generation. Choose ‘ert.tlc’ as the desired System
Target File and the ARM Cortex-M3 (QEMU) as the Target Hardware. Select the Toolchain
that will be used for the final code generation with the Waijung toolbox. In this project the
GNU ARM toolchain will be used. The window must look as Figure 3.15.
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CHAPTER 3: CODE GENERATION
Figure 3.15: Target and toolchain configuration for SIL simulation
Go to Code Generation>> Verification and make sure that the creation block is set to
PIL as shown in Figure 3.16.
Figure 3.16: Block type selection for SIL and PIL simulations
The SIL simulation could also be done using a specific SIL block, but in this case, the
difference between performing a SIL or a PIL simulation will be given by the selected
target. The target previously selected was the QEMU, which is an emulator that executes the
PIL block from the computer without the need of having an STM32F4 connected. Close the
configuration window. Before going on with the explanation, it is recommended at this point
to set the current workspace to a specific location for this generated code.
The controller can be generated from an LTI system, a subsystem or another model. In
this project the controller is built form LTI blocks. The next step is to build the block. Go to
the model, right click on the LTI system that contains the discretized controller and click
C/C++ Code>> Build this Subsystem, as shown in Figure 3.17.
Figure 3.17: Building the PIL block for a LTI System block named ‘Controller_disc’
A new window will appear, click the Build button. After the process is completed, a new
model will be opened containing a PIL block. Copy it and paste it into the previous Simulink
model replacing with it the discretized controller block. This block will be the simulated
controller and its behaviour can be compared with the real model.
Examples of this simulation are given in sections 8.2 and 9.2.
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CHAPTER 3: CODE GENERATION
3.3.7 PIL Simulation
PIL simulation is the next step after SIL simulation. The microcontroller must be
connected via USB and the use of the USB to TTL adapter is highly recommended. Without
the serial adapter, the PIL data will be collected by the ST-LINK interface, which is much
slower than the serial communication and makes it impractical.
The initial configuration is identical to the SIL setup, the only difference is that now the
selected target will be the ‘STM32F4-Discovery’. An additional step must be followed: in
the Configuration Parameters window go to Coder Target>> PIL. Choose the serial
interface and select the COM port as explained in section 3.3.5. The peripheral used for PIL
is the USART2, so make sure that the transmitter terminal is connected to the PA2 pin and
the receiver to the PA3 pin.
Follow the same procedure as for the SIL simulation: set a specific workspace, build the
PIL block, include it the Simulink project and run the simulation in normal mode.
Examples of this simulation are given in sections 8.3 and 9.3.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 4: PLANT MODELLING
4 PLANT MODELLING
Having an accurate model of the plant is very important to achieve a good controller
design. Furthermore, in some cases the plant is not available or it is not possible to measure
all the states of the system. That is why an observer based on the model might be needed to
get the feedback, as it will be seen in section 6.3.
In this section the transfer function that describes the plant will be obtained by two
different methods. First, physical behaviour of the DC motor is described by establishing the
physical laws that command the system. That is the theoretical method. Secondly,
experimental techniques will be used measuring the output of the system for a known input.
That is the empirical method.
The experimental result will be used to design the controller. The theoretical method
will justify the chosen order of the model system and will provide a comparison with the
empirical result.
4.1 Theoretical Method: Analysis of a DC Motor
A reliable approximation of a DC motor is given by the equivalent circuit in Figure 4.1
[11]. Some simplifications will be made regarding the described system:
 La is the armature inductance and it can be neglected. This is the case for most cases,
in which La<<Ra.
 There is no friction opposing the motor shaft, only the load and rotor inertia. So the
total friction will be ignored.
La
Ra
+
Jeq
+
E
Vm
-
ia
M
-
θ, ω
Figure 4.1: Equivalent circuit of a DC motor. The excitation circuit has been omitted
Some of the symbols appearing in Figure 4.1 are shown in Table 2.1.
Symbol
E
ia
θ
ω
Description
Motor back electromotive force
Motor armature current
Angular position
Angular speed
Table 4.1: DC Modelling Nomenclature
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CHAPTER 4: PLANT MODELLING
The rest of the symbols used in the previous figure and in the rest of this chapter can be
consulted in Table 2.1 and Table 2.4.
4.1.1 Velocity Control
The motor back electromotive force can be expressed as
𝐸 = 𝐾m 𝜔 .
According to Newton’s second law for rotations:
𝑑𝜔
𝜏=𝐽
.
𝑑𝑡
The torque is also given by its Kt:
𝜏 = 𝐾t 𝑖a .
(4-1)
(4-2)
(4-3)
Finally:
(4-4)
𝑉m = 𝐸 + 𝑅a 𝑖a .
Using the four equations above, it must be solved for Vm(t) as a function of θ for
position and for ω for angular velocity. It will be solved as a function of ω and then derived:
from (4-1) and (4-4):
𝑉m 𝐾m
(4-5)
𝑖a =
−
𝜔.
𝑅a 𝑅a
From (4-2), (4-3) and (4-5):
𝐾t 𝑉m 𝐾m 𝐾t
𝑑𝜔
(4-6)
𝜏 = 𝐾t 𝑖a =
−
𝜔=𝐽
.
𝑅a
𝑅a
𝑑𝑡
Taking into account that when working with SI units both Kt and Km have the same
value (it just changes its physical meaning), (4-6) can be rewritten as
2
𝑑𝜔 𝐾m
𝐾m 𝑉m
(4-7)
𝐽
+
𝜔=
.
𝑑𝑡
𝑅a
𝑅a
Taking Laplace:
2
𝐾m
𝐾m 𝑉m (𝑠)
(4-8)
𝐽Ω(𝑠)𝑠 +
Ω=
,
𝑅a
𝑅a
and expressing it as a transfer function:
1
Ω(𝑠)
𝐾m
𝐺1 [𝜔, 𝑉m ](𝑠) =
=
.
𝐽𝑅
𝑉(𝑠)
a
𝑠
+
1
2
𝐾m
(4-9)
Writing the state-variable model is easy for first-order systems. Let
𝑥1 = 𝜔 .
(4-10)
Then, rewriting (4-7):
𝑥̇ 1 = −
2
𝐾m
𝐾m
𝑥1 +
𝑢(𝑠) ,
𝐽𝑅a
𝐽𝑅a
(4-11)
being Vm(s) the input.
That is the analytic representation of the angular velocity against the input voltage.
Now, these results will be used to get the transfer function and state-variable model of the
full system considering the angular position.
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CHAPTER 4: PLANT MODELLING
4.1.2 Position Control
Taking into account that
𝜃̇ = 𝜃𝑠 = 𝜔 ,
(4-12)
(4-9) could be written as
𝐺1 [𝜔, 𝑉m ](𝑠) =
𝛩(𝑠)𝑠
,
𝑉m (𝑠)
(4-13)
so the new full transfer function is
1
𝛩(𝑠) 1
1
𝐾m
𝐺2 [𝜃, 𝑉m ] =
= 𝐺1 =
.
(4-14)
𝐽𝑅
(𝑠)
𝑉m
𝑠
𝑠 a𝑠+1
2
𝐾m
As it can be noticed, only the addition of an integrator to the previous model is needed
to get the rotational position. The state-variable model for the full system will now be
written. Let
𝑥1 = 𝜃
(4-15)
𝑥2 = 𝜃̇ .
Now, from (4-7) and (4-12)
2
𝐾m
𝐾m
(4-16)
𝑥̇ 2 = −
𝑥2 +
𝑉 (𝑠) .
𝐽𝑅a
𝐽𝑅a m
Thus, the state equations are
0
1
0
𝑥1
𝑥̇ 1
2
𝐾
𝐾
(4-17)
[ ]=[
m ] [ ] + [ m ] 𝑢(𝑠),
𝑥2
𝑥̇ 2
0 −
𝐽𝑅a
𝐽𝑅a
being Vm(s) the input.
4.1.3 Results
In Figure 4.2 there is a simplified full model of a DC motor with an amplifier working in
open loop operation.
𝑉c
𝑉m
AMPLIFIER
1
𝐾m
𝜔
𝐽𝑅a
𝑠+1
𝐾m2
1
𝑠
𝜃
MOTOR
Figure 4.2: Representation of the mathematical model with the states of interest for controlling
The goal of the demonstration above is to justify that a first-order model can be used if
only velocity control is being performed, but a two-order model will be needed for position
control. According to these considerations, a DC motor can be modelled as a second-order
system, but if the armature inductance had been taken into account, the equations would
have led to a third-order system.
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4.2 Empirical Method: Grey-Box Model Estimation with MATLAB
Now, the real system will be given a known input and the output will be measured. This
procedure is divided into two main parts: measuring the output (from Simulink) and running
the script to obtain the model.
4.2.1 Introduction to Experimental Estimation
The step response of a first-order system is a well-known asymptotic exponential curve
[3]. The expression of the step response can be obtained for a generic first-order transfer
function such as
𝐾
(4-18)
𝐻(𝑠) =
.
𝑇𝑠 + 1
The output will be the system, H(s), multiplied by the input. The Laplace transform of a
unit step in time is an integrator, so
𝐾
1
1 𝐾
1 𝑇
(4-19)
𝑌(𝑠) = 𝐻(𝑠) =
=
.
𝑠
𝑠 𝑇𝑠 + 1 𝑠 𝑠 + 1
𝑇
By calculating the residues and applying the inverse Laplace transform the final result
turns out to be:
𝑌(𝑠) =
𝐴
𝐵
+
𝑠 𝑠+1
𝑇
(4-20)
𝐴 = 𝑌(𝑠)𝑠|𝑠=0 = 𝐾
1
𝐵 = 𝑌(𝑠) (𝑠 + )| 1 = −𝐾
𝑇 𝑠=−
(4-21)
𝑇
𝑡
𝑦(𝑡) = ℒ −1 (𝑌(𝑠)) = 𝐾 − 𝐾𝑒 −𝑇
(4-22)
This last equation describes the evolution in time of the system output. It can be seen
that the first term is independent of time and that the second term does depends of time,
making itself smaller and smaller over time. This proves that in steady state the final value is
K.
It has been demonstrated that the gain value of the system, K, can be approximated just
by observing the final value a cautious time after giving a unit step to it. Now, the time
constant, T, which is responsible for the speed of the system, must be obtained. Making t =
T:
𝑇
1
(4-23)
𝑦(𝑡) = 𝐾 − 𝐾𝑒 −𝑇 = 𝐾 (1 − ) ≈ 𝐾 · 0.6321 .
𝑒
It means that when y(t) is equal to approximately 0.6321 times the system gain, the
elapsed time since t=0, at the start of the step, will be equal to the time constant.
An example of this is shown in Figure 4.3. The final value of the response clearly
reaches amplitude 6. Applying Equation (4-23) it turns out that the time constant is reached
when amplitude is 3.79, thus, the time constant must be around 2 seconds. The conclusion is
that the simulated system in this example must have been
6
(4-24)
𝐻(𝑠) =
2𝑠 + 1
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Step Response
6
5
Amplitude
4
System: H
Time (seconds): 2
Amplitude: 3.79
3
2
1
0
0
2
4
6
8
10
12
14
16
18
Time (seconds)
Figure 4.3: Response example of a first-order system to a step input
Instead of identifying the system manually according to the previous measurements, the
input, the output, and the parametrized system will be given to the ‘greyest’ function. A
grey-box estimation returns the unknown parameters of the modelled plant.
4.2.2 Getting the Output
The input command will be directly given from the microcontroller and one of its ADC
ports will be used to measure and save the output. The required resources for this operation
are:




1 ADC port
1 DAC port
TTL-RS232 adapter
DC motor
The components are connected as shown in Figure 4.4. It is strongly recommended to
read the chapters 3.3.2 and 3.3.3 about CAD and DAC configuration very carefully because
the default value of the Vref parameter could lead to wrong system identification. The setup
explained in section 3.3.4 for receiving serial data in the microcontroller will be used in the
next procedures. Nevertheless, the reader can check most of the ports and settings used just
by looking at the diagrams along the explanation.
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+15 V
MINI-USB
M
COMMAND
±5 V
TACH
±5 V
PA4
PB7
Tx
PA0
PB6
Rx
USB
DRIVER
STM32
TTL-RS232
GND
GND
GND
Figure 4.4: Connections between components for model identification
The Simulink setup for model identification will now be explained. First, a new blank
Simulink model window must be opened and prepared for code generation as seen in section
3.3.1. Now, insert a few blocks and connect them to get the diagram shown in Figure 4.5.
Figure 4.5: Algorithm executed by the STM32F4 for model identification
What the upper part of the program does is to wait for the serial data to be received
through the Rx pin and send the value to an analog output via a Regular DAC block. With
this setup a value can be sent to the microcontroller in execution time and it will send it to
the COMMAND port of the DC motor.
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The lower blocks are read the value of the ADC port, convert it to voltage and then send
it trough the Tx pin. This allows reading the voltage in the TACH port of the DC mottor,
which is the output, and save it for further use.
At this point the values for the sample time, Ts should be entered in the MATLAB
console. It is recommended to click in Display>> Sample Time>> All, now all the blocks
should be coloured the same colour. It means that they are discrete blocks and that they have
the same sample time. This is a good habit to detect possible errors during compilation. Now
the Build Model button can be clicked to generate the code and load it. Wait untill the
program has been completely flashed into the memory of the microcontroller.
Note that the nominal reference voltage value has been replaced by its real measured
value.
A single serial port is being used, so there is a master who is giving the input and
reading the output. That is the next step: to programme the master.
First, a new blank Simulink model window must be opened, but it will not be configured
like the previous one. This diagram is not for generating code for the microcontroller, but for
running in the computer. Go to Model Configuration Parameters>> Solver. In Simulation
time set Stop time to ‘inf’. In Solver options set Type to ‘Fixed-step’, Solver to ‘discrete’ and
the Fixed-step size to ‘Ts’. The block diagram must be built as in Figure 4.6.
Figure 4.6: Simulink diagram executed in PC for model identification
In this diagram the TACH reading sent by the microcontroller is read by the computer
and the COMMAND value is sent to the STM32F4. The step is accomplished by manually
clicking the switch. The step value and the output value are both multiplexed and sent to the
Scope. The Scope is going to be configured to save the ploted data into arrays, so it will
create three arrays: one with time measurements, a second one with the output data and a
last one with the step data. Double click in the Scope >> Parameters>> History. Make sure
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CHAPTER 4: PLANT MODELLING
the checkbox for limiting data is unmarked. Check Save data to workspace and write a
variable name for it.
Once everything is connected as in Figure 4.4, make sure the switch points at the zero
value and click the Start button. A progress bar should now appear at the right bottom of the
window, which indicates that it is running without problems. Now switch to the step value,
the motor should move and reach a steady state. Waiting around 3 seconds for it to settle is
enough. Now press the Stop button. The motor can be let running or manually disconnected.
The values have been saved in a matrix in the workspace with the name ‘Scope_Step’.
4.2.3 Parameters Estimation
The theory previously seen is the basis for transfer function reconstruction from a given
output. Even though those concepts could be directly applied for the data that has just been
obtained, a MATLAB tool called Linear grey-box model estimation will be used [12]. This
function is called by the keyword ‘greyest’.
A grey-box estimation is based on the fact that some information can be given about the
system, but still there are some unknown parameters to approximate. Looking at the input
and the output of the grey box, the algorithm should return a model that fits into its
behaviour but also that looks as the desired system. In other words, the function will be told
that the grey-box is a first-order system and will be given three arrays with the time base, the
input and the output.
The first step is defining the system architecture, its ‘shape’. A new function is created
in MATLAB called ‘myfunc.m’. In section 1.2.2 the generic state-equations for a first-order
system can be found. According to it, the matrices A, B, C and D can be obtained. The
function should look as shown in Script A from the annexed document.
In this script, K is the gain, T the time constant and Ts the sample time. The function
must be saved in the directory of the current workspace dedicated for plant identification.
Now all the means are ready to be used. A new script must be started to execute the
algorithm. The identification motor script is shown in Script B from the annexed document.
The script has been commented to ease its understanding. First, the values saved in
‘Scope_Step’ are used to store the time-based data into an iddata object. After that, the
architecture, the unknown parameters, and the function type are specified. With all this
information a grey-box is created using the ‘idgrey’ function. The model is estimated using
the ‘greyest’ command. Finally the results are shown. More information about each function
can be found in MATLAB documentation. Five measurements for a 2.8 volts step have been
performed. The results are summarised in Table 4.2.
K [Vt/Vc]
0.776442
0.776276
0.777392
0.777333
0.777149
T [s]
0. 089989
0.078369
0.090998
0.087328
0.086350
FIT [%]
98.810256
97.943966
98.951664
98.493917
98.681353
Table 4.2: Model identification measurements for the motor (without belt)
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4.2.4 Results
It is important to make clear that the considered plants in the applications put into
practise in this project include both input and output amplifiers. In the case of velocity
control, the plant is all the system between the COMMAND input and the TACH output.
For position control, the plant is the system between the COMMAND input and the POT
output. This is represented in Figure 4.7.
𝑉p
POSITION CONTROL PLANT
𝑉c
𝑉m
𝐾
𝑇𝑠 + 1
𝜔
VELOCITY CONTROL PLANT
1
𝑠
𝜃
𝑉t
Figure 4.7: DCMCT plant representation for velocity and position control
The average parameters of the motor turned out to be K=0.776918 and T = 0.084761.
The models fit nearly a 98% to the real plant, which is a very acceptable estimation for the
purpose of this project. A step value of 2.8 volts has been chosen because it is close to the
highest value the DAC can reach and it has been checked that the system shows better
response for medium-high inputs. In next chapters an amplifier will be installed, so a second
system identification can be made with a higher step and take a look at the results, but this
estimation is already good enough.
Equation (4-9) gives the velocity in rad/s for a given input in volts (measured at the
motor terminals). In model identification, [Vt]/[Vc] transfer functions have been used, while
in section 4.1 the theorical expresions obtained were [Ω]/[Vm] and [Θ]/[Vm] transfer
functions. Equation (4-9) can be transformed into a [Vt]/[Vc] expresión to compare the
results:
rad
1 rev min · 𝑉t 60 s 3 𝑉m
60 · 3 𝑉t
[ s ][
][
][
][
]=
[ ].
𝑉m 2𝜋 rad 667 rev min 1 𝑉c
2𝜋 · 667 𝑉c
(4-25)
Therefore,
1
60 · 3
𝐾𝑚 · 2𝜋 · 667
𝐺1 [𝑉t , 𝑉c ](𝑠) =
.
𝐽𝑅𝑎
2 𝑠+1
𝐾𝑚
(4-26)
Thus,
𝐾theo =
1
60 · 3
1
60 · 3
𝑉t
·
=
·
= 0.8556 [ ] ,
𝐾m 2𝜋 · 667 0.502 2𝜋 · 667
𝑉c
35
(4-27)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 4: PLANT MODELLING
𝑇theo =
𝐽𝑅a 0.0000221 · 10.6
=
= 0.0929 [s] .
2
𝐾m
0.05022
(4-28)
The values for the parameters in (4-27) and (4-28) have been taken from the table A.2
DCMCT Model Parameter Specifications from the DCMCT motor documentation. As it can
be noticed above, both theoretical and experimental results are of the same order. The
theoretical gain is an 11% bigger than the experimental gain, whereas the theoretical time
constant is nearly a 10% bigger. The simplifications made are not so insignificant, so it is
much more accurate to work with the experimental model.
The transfer function that will be used to design the velocity controller is
𝐺v [𝑉t , 𝑉c ](𝑠) =
0.776918
.
0.084761s + 1
(4-29)
A graphical response comparison of the last measure of Table 4.2, the final model and
the theoretical model is given in Figure 4.8. Both real and model responses are very similar,
while the theoretical response has greater amplitude. This is logical, due to the friction and
other simplifications made.
2
Real output
Model output
Theoretical Output
1.5
1
0.5
0
1
1.1
1.2
1.3
1.4
1.5
1.6
Figure 4.8: Visual responses comparison
For position control, the plant will change its dynamics because an external
potentiometer must be connected with a belt and it adds more load. Thus, the experiment has
been repeated for the motor with the potentiometer. The results appear in Table 4.3.
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CHAPTER 4: PLANT MODELLING
K [Vt/Vc]
0.750745
0.750147
0.753007
0.751658
0.752822
T [s]
0.093776
0.085073
0.093018
0.091294
0.092688
FIT [%]
99.137308
98.689054
99.122953
98.982729
98.761368
Table 4.3: Model identification measurements for the motor (with belt)
The average values have been K=0.7516758 and T=0.0911698 with a high fitting as in
the previous estimation without the belt.
The transfer function for position control is obtained by adding an integrator to the firstorder system as seen in Equation (4-12). With the previous results, undoing the TACH port
amplification gain, as in Equation (4-25), and including the POT port amplification gain, the
modelled plant that will be used for position control is:
𝐺p [𝑉p , 𝑉c ] =
0.7516758
2𝜋667 180
77.133501
·
·
≈
𝑠(0.0911698𝑠 + 1)
60
𝜋39 𝑠(0.0911698𝑠 + 1)
All voltage equivalences are taken from Table 2.3 and Table 2.4.
37
(4-30)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 5: SIGNAL ADAPTATION
5 SIGNAL ADAPTATION
As seen in section 2.2.3 the STM32F4 has two DACs and three ADCs with a common
VREF+ which is connected to VDD. It has been checked that the measured VDD pin voltage is
around 3 volts; this restricts the STM32F4 to directly read or provide analog signals up to
this voltage. Nevertheless, it is desirable that it can reach the full scale of ±5 V of the servo
signals COMMAND, TACH and POT. This requires two operations: amplifying the signals,
so that the full scale of the microcontroller matches with the full scale of the motor and
inverting the signal, so that the motor can run both clockwise and counterclockwise.
Next, some of the possible solutions to solve this problem will be examined.
5.1 Analog Output
It must be decided how the microcontroller will generate the analog output according to
the desired velocity regardless the rotation sense.
5.1.1 PWM
This technique consists in varying the duty cycle of a periodic square wave whose
amplitude alternates between zero and the maximum drive voltage of the motor as shown in
Figure 5.1. The optimum frequency must be selected according to the application and the
load, but the motor roll-off frequency is a good reference. The microcontroller only needs a
digital output to control a switch that commutes allowing a source of the maximum voltage
close the circuit.
ON
FULL CYCLE
V
15
0
t
OFF
Figure 5.1: Two PWM signal cycles. The average value is proportional to the duty cycle (Ton/T)
This is a very commonly used method because it is simple to do in digital control and
working always with the maximum voltage allows the shaft to rotate with a constant torque
near the maximum even at low speed rate.
Even though the motor, due to its inductance, sees the input voltage as an average
continuous value, the fact that its torque is greater at low speed than with variable voltage
control, makes the motor follow a transfer function different from the calculated in section
4.2.3 that transforms a given duty cycle into velocity and position.
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CHAPTER 5: SIGNAL ADAPTATION
5.1.2 DAC
The other option is to directly use one of the two digital-to-analog converter of the
STM32F4. An amplifier is still needed from the range 0-3 volts to 0-5 volts. Since the servo
used in this project has been designed to be controlled with a variable analog voltage, this
will be the selected solution.
Furthermore, generating an analog output is a more general purpose than DC motor
control and the department has shown interest in this matter.
5.2 Inverting Rotation Sense
5.2.1 H Bridge
Using the H bridge arrangement needs just a control bit for rotation sense selection, the
connection is shown in Figure 5.2. The bit makes commutate two mutual transistors that
close the circuit polarizing the motor positively or negatively.
+V
M
CONTROL
CONTROL
CONTROL
CONTROL
Figure 5.2: H bridge connection. The bit CONTROL or its opposite value closes the circuit allowing one
polarization or another.
5.2.2 Custom Circuit
This is a customized solution that solves both the problem of amplifying and inverting
the rotation sense. The idea is to get a lineal amplifier that transforms a COMMAND input
of 0 – 3 V to ±5 V. With an ideal behaviour it would follow the equation
10
(5-1)
𝑉 −5,
3 in1
is the value at the COMMAND port and Vin1 is the output from the
𝑉out1 =
where here Vout1
microcontroller.
For reading values from the servo, another amplifier is needed to do the opposite
operation:
39
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 5: SIGNAL ADAPTATION
3
(5-2)
.
10
For this one, Vout2 is the analog value read at the microcontroller and Vin2 the real
measure at the servo port.
In Figure 5.3 all external connections are shown. The ports have been named and
coloured following the circuit diagram attached in the annexed document for easier
understanding.
This will be the chosen solution.
𝑉out2 = (𝑉in2 + 5)
+15 V
SIGNAL ADAPTER
± 15 V
OUTPUT AMPLIFIER
COMMAND
±5 V
M
Vout1
X3
X1
Vin1
X5
Vout2
+3.3 V
PA4
INPUT AMPLIFIER
TACH
±5 V
Vin2 X7
POT
±5 V
PA0
STM32
GNDs (X2 & X8)
GND
GND
Figure 5.3: Connections of the signal adapter.
The full circuits can be found in the annexed document. The theoretical equations of
each amplifier will be now analysed according to its schematic.
5.3 Custom Circuit analysis
The input and output voltages of the signal adapter are into the ranges shown in Table
5.1.
Symbol
Vin1
Vout1
Vin2
Vout2
Description
Output amplifier in
Output amplifier out
Input amplifier in
Input amplifier out
Range
0-3 V
±5 V
±5 V
0-3 V
Table 5.1: Voltages of the signal adapter
The output amplifier, corresponding to Equation (5-1), is shown in Figure 5.4.
40
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 5: SIGNAL ADAPTATION
i4
33.2K
Vin1
1K
i1
OA1
+
V1
i2
OA2
+
10K
Follower
V2
1K
Amplifier
i3
OA3
+
Vout1
Adder
1K
+5V
Figure 5.4: Output amplifier. Vin1 (0÷3 V) is the STM32F4 output. Vout1 (±5 V) is the servo COMMAND
The first operational amplifier, OA1, is just a voltage follower to uncouple impedances;
its negative and positive terminals must be at the same potential (a small virtual ground is
written at its negative terminal), so
𝑉1 = 𝑉in1 .
(5-3)
The topology of the OA2 is an amplifier with a negative gain. This behaviour is
described by the next equations:
𝑖1 =
𝑉2 = 0 − 𝑖1 · 33.2K

𝑉1
,
10K
33.2K
V2 = −𝑉1
= −𝑉1 · 3.32
10K
(5-4)
Finally, AO3 works as a voltage adder: it adds the voltages coming to the negative
terminal, but it also changes its sign. The output can be deducted as follows:
𝑖2 =
𝑉2
;
1K
𝑖3 =
5
;
1K
𝑖4 = 𝑖2 + 𝑖3 ;
𝑉out1 = 0 − 𝑖4 · 1K = −(𝑉2 + 5) = 𝑉1 · 3.32 − 5
(5-5)
Equation (5-5) is the theoretical equation of the output amplifier, which is very close to
(5-1), which is the ideal equation. The exact gain could not be accomplished due to the
standard resistance values.
For the input amplifier, the same procedure is followed as before. Its schematic is shown
in Figure 5.5.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 5: SIGNAL ADAPTATION
i7
1K
Vin2
i5
OA4
+
V3
Follower
1K
i6
OA5
+
Adder
V4
3.5K
i8
1K
+5V
3.5K
V5
i9
20K
3K
20K
OA6
+
Inverter
V6
OA7
+
Follower
Vout2
Figure 5.5: Input amplifier. Vin2 (±5 V) is a servo output. Vout2 (0÷3 V) is an STM32F4 analog input
A voltage follower is placed after Vin2 and another one before Vout2. According to the
analysis of the output amplifier,
𝑉3 = 𝑉in2
(5-6)
𝑉6 = 𝑉out2 .
(5-7)
and
The OA5 works as an adder, so as seen in (5-5)
𝑉4 = −(𝑉in2 + 5).
(5-8)
The OA6 works as an inverter. It can be proved as follows:
𝑉5
,
20K
𝑉6 = 0 − 𝑖9 · 20K = −𝑉5 .
𝑖9 =
(5-9)
So, according to (5-9) and (5-7), V5 can be expressed as
𝑉5 = −𝑉out2 .
(5-10)
The problem reduces to solve the voltage divider between V4 and V5:
𝑉5 =
𝑉4 · 3𝐾
−𝑉in2 − 5
=
= −𝑉out2 .
7K + (3K ∥ 20K)
3.2029
(5-11)
Thus, the theoretical equation for the input amplifier is
𝑉in + 5
.
3.2029
Again, it is an approximation to the ideal equation, (5-2), for this amplifier.
𝑉out2 =
42
(5-12)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 5: SIGNAL ADAPTATION
5.4 Amplifier Calibration
The importance of having a good equation that accurately calculates the output of each
amplifier is that this conversion must be undone in Simulink, so the mathematical
expression should be as exact as possible.
After testing the signal adapter, it has been observed that the output amplifier is showing
good results, whereas the input amplifier has important errors and does not perform as its
theoretical equation. In order to obtain the best results, both amplifiers will be tested to find
its real equation.
The basis of the calibration will be that the amplifiers are operating linearly, as is has
been theoretically deducted previously, but the mismatch between the nominal value and the
real value of the resistors is making them amplify with different gains. For this reason, the
models will still be linear equations and they will be calculated experimentally. The input in
this calibration will be given by a variable source voltage and both input and output will be
measured with a multimeter.
5.4.1 Output Amplifier Calibration
The transfer function of this amplifier should look like the graph shown in Figure 5.6,
where the ideal maximum should be at (3, 5) and the minimum at (0, -5). The points of
interest for the calibration are A, B and P. A and B are two arbitrary points taken near the
maximum and minimum but within the operation range. The more separate the points, the
better the gain will be measured. P will estimate the offset and it corresponds with the value
of the input that returns zero output.
Vin1
Max
B
P
A
Min
Figure 5.6: Operation range of the output amplifier
During the calibration, the results have been:
𝐴 = (0.1522, −4.58) ,
𝐵 = (2.948, 4.66) ,
𝑃x = 1.539 .
The gain is the slope of the function
43
Vout1
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 5: SIGNAL ADAPTATION
𝑚=
4.66 + 4.58
≅ 3.3050 .
2.948 − 0.1522
(5-13)
And the offset is
0 = 3.305 · 1.539 + 𝑛

𝑛 = −5.0863 .
Hence, the real equation of the output amplifier will be taken as
(5-14)
𝑉out1 = 3.305𝑉in1 − 5.0863 .
(5-15)
It does not show big difference with Equation (5-5), what confirms the good behaviour
previously observed.
5.4.1 Input Amplifier Calibration
This amplifier does need an adjustment. Its transfer function is shown in Figure 5.7,
where the ideal maximum should be at (5, 3) and the minimum at (-5, 0).
Vout2
Max
B
n
A
Min
Vin2
Figure 5.7: Operation range of the input amplifier
The gain is estimated analogously to the previous case. Now, instead of P, the offset can
be directly read by giving a zero input to the amplifier. The measures for this calibration
have been:
𝐴 = (−4.83, 0.0753)
𝐵 = (4.83, 2.701)
𝑛 = 1.388
Thus,
𝑚=
2.701 − 0.0753
≅ 0.2718
4.83 + 4.83
(5-16)
Therefore, the real equation of the input amplifier will be taken as
𝑈out2 = 0.2718 · 𝑈in2 + 1.388 =
which quite differs from (5-12).
44
𝑈in2 + 5.1067
,
3.6792
(5-17)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 6: CLASSIC STATE FEEDBACK
6 CLASSIC STATE FEEDBACK
The design technique presented in this chapter is commonly known as ‘pole placement’.
First, all the system states will be supposed controllable and available for measurement and
feedback, and then system observers will be explained.
6.1 Controllability
A system is controllable in time t=t0 if there exists a control signal u(t) capable to move
the system from the initial state x(t0) to any final state x(tf) in a finite time tf. If all the states
are controllable, then it is said that the system is fully controllable [3].
The conditions for a system to be fully controllable will now be analysed. Let the singleinput system:
𝒙̇ = 𝑨𝒙 + 𝑩𝑢 .
The solution for the equation above is
(6-1)
𝑡
𝑨𝑡
𝑥(𝑡) = 𝑒 𝑥(0) + ∫ 𝑒 𝑨(𝑡−𝜏) 𝑩𝑢(𝜏)𝑑𝜏 .
(6-2)
0
Now, without loss of generality, suppose the final state is the state-space origin and
make t0=0:
𝑡f
𝑥(𝑡f ) = 0 = 𝑒 𝑨𝑡f 𝑥(0) + ∫ 𝑒 𝑨(𝑡f −𝜏) 𝑩𝑢(𝜏)𝑑𝜏 .
(6-3)
0
Solving for x(0):
𝑥(0) = − ∫
−𝑡f
𝑒 𝑨𝜏 𝑩𝑢(𝜏)𝑑𝜏 .
(6-4)
0
According to Sylvester’s formula, 𝑒 𝑨𝜏 can be written as follows:
𝑛−1
𝑒
𝑨𝜏
= ∑ 𝛼𝑘 (𝜏)𝑨𝑘 .
(6-5)
𝑘=0
Replacing equation (6-5) in equation (6-4):
𝑛−1
𝑘
𝑥(0) = − ∑ 𝑨 𝑩 ∫
𝑘=0
−𝑡f
0
𝛼𝑘 (𝜏)𝑢(𝜏)𝑑𝜏 .
(6-6)
Calling
∫
−𝑡f
0
𝛼𝑘 (𝜏)𝑢(𝜏)𝑑𝜏 = 𝛽𝑘 ,
(6-7)
Equation (6-6) can be written as
𝑛−1
𝑥(0) = − ∑ 𝑨𝑘 𝑩𝛽𝑘 =
𝑘=0
= −[𝑩
𝛽0
𝛽1
𝑨𝑩 … 𝑨𝑛−1 𝑩] [ ⋮ ] .
𝛽𝑛−1
45
(6-8)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 6: CLASSIC STATE FEEDBACK
Thus, if the system is fully controllable, for whatever initial state x(t0) Equation (6-8)
must be satisfied. This means that the matrix
(6-9)
𝓒 = [𝑩 𝑨𝑩 … 𝑨𝑛−1 𝑩]
must have full rank. The system described by equation (6-1) is fully controllable if, and only
if, 𝓒nxn has rank n. This matrix is called controllability matrix.
6.2 Pole Placement
Pole placement is a technique that consists in moving the closed-loop poles of a system
to the desired location in order to modify its response characteristics. To explain it, a singleinput system for which all the states are available and measurable will be again used. The
idea is that the control signal depends of the state vector at every moment [3].
Let the system to control be
𝒙̇ (𝑡) = 𝑨𝒙(𝑡) + 𝑩𝑢(𝑡)
y(𝑡) = 𝑪𝒙(𝑡) .
The desired input control is
(6-10)
(6-11)
𝑢(𝑡) = −𝑲𝒙(𝑡) .
The equation (6-11) is known as the state-feedback control law. The matrix K1xn is
called state feedback gain matrix, and it weighs every state at every instant to calculate the
input control. The new regulator system is shown in Figure 6.1. In this case there is no
reference, r(t), to follow, so the regulator has to keep the output to zero in case of
perturbations. For the tracker case, where the system should follow the reference, the input
control will be calculated as the difference between the reference and –Kx.
u(t)
B
+
+
𝒙̇
∫dt
𝒙
B
𝑦(t)
A
-K
Figure 6.1: Regulator based on full-state feedback
Replacing (6-11) into (6-10) it results that:
𝒙̇ (𝑡) = (𝑨 − 𝑩𝑲)𝒙(𝑡) .
The solution for this equation is
(6-12)
(6-13)
𝒙(𝑡) = 𝑒 (𝑨−𝑩𝑲)𝑡 𝒙(0) ,
where x(0) is the initial state. The stability of the system and the dynamic characteristics are
given by the matrix A-BK. The matrix K must be chosen such a way that A-BK becomes
asymptotically stable, this means that whatever the value for x(0) is, x(t) tends to zero over
time. For this to happen, the eigenvalues of A-BK must be negative. This is logical because
the eigenvalues of the matrix A-BK are the closed-loop poles of the system. This can be
expressed as
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 6: CLASSIC STATE FEEDBACK
(6-14)
𝑝cl = |𝑠𝑰 − 𝑨 + 𝑩𝑲| = (𝑠 − 𝜇1 )(𝑠 − 𝜇2 ) … (𝑠 − 𝜇𝑛 ) ,
where 𝜇𝑛 are the eigenvalues of A-BK.
There are several ways to calculate the feedback gain matrix. Trying to directly solve
Equation (6-13) is not always easy, but applying Ackermann’s formula gives a direct result:
𝑲 = [𝟎
𝟎 …
𝟎 𝟏][𝑩
−1
𝑨𝑩 … 𝑨𝒏−𝟏 𝑩] 𝑝cl .
(6-15)
It can be proved that arbitrary pole assignment is possible if and only if the system is
fully controllable.
When the reference is not zero, the control signal should be
𝑢(𝑡) = 𝑟(𝑡) − 𝑲𝒙(𝑡) ,
so, (6-12) would now look like
(6-16)
(6-17)
𝒙̇ (𝑡) = (𝑨 − 𝑩𝑲)𝒙(𝑡) − 𝑩𝑟(𝑡) .
Thus, in steady state, the first term should be zero, according to the previous reasoning:
(6-18)
𝒙̇ (∞) = (𝑨 − 𝑩𝑲)𝒙(∞) − 𝑩𝑟(∞) = −𝑩𝑟(∞) .
This matches with the desired behaviour of a regulator: the poles only affect to the
transient state and the dynamical characteristics while the steady state only depends on the
reference.
6.3 State Observers
Until now it has been assumed that all the states of the system were available for
measure, but in real systems it is not always that easy. One or more variables, such as
voltage or speed, may not be measurable or may require a high cost. For those cases, an
observer is used. An observer estimates the estate vector according to the measures of the
output and the control signal. When the observer can estimate all the states it is called a fullorder observer, otherwise it is a reduced-order observer [3].
6.3.1 Observer design
The basic idea of an observer matches with a clone of the plant that receives the same
control signal and gives an estimate output. But a duplicate of the modelled system is not
enough, because the imprecisions in matrices A and B would lead to bigger and bigger errors
over time. For this reason, the difference between the real and the estimated output must be
taken into account. According to this description, the mathematical model of an observer for
the system described by (6-10) is defined as
̂̇ = 𝑨𝒙
̂ + 𝑩𝑢 + 𝑳(𝑦 − 𝑦̂)
𝒙
̂ + 𝑩𝑢 + 𝑳(𝑦 − 𝑪𝒙
̂)
= 𝑨𝒙
̂ + 𝑩𝑢 + 𝑳𝑦,
= (𝑨 − 𝑳𝑪)𝒙
(6-19)
̂ is the estimated estate vector and 𝑦̂ is the estimated output. L is the observer matrix
where 𝒙
gain; it weights the error of the estimated output at every instant and continuously improves
the observer. This is represented in Figure 6.2.
47
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 6: CLASSIC STATE FEEDBACK
B
+
𝒙̇
+
∫dt
𝒙
𝑦(𝑡)
C
A
𝑢(𝑡)
L
B
+
+
+
OBSERVER
̂̇
𝒙
∫dt
A
C
𝑒
𝑦̂(𝑡)
+ -
̂
𝒙
Figure 6.2: Diagram block of a generic system and its full-order state observer
To obtain the equation that gives the error, (6-19) is subtracted from (6-10):
̂̇ = 𝑨𝒙 − 𝑨𝒙
̂ − 𝑳(𝑪𝒙 − 𝑪𝒙
̂)
𝒙̇ − 𝒙
̂).
= (𝑨 − 𝑳𝑪)(𝒙 − 𝒙
̂ is the error vector, e:
The difference between 𝒙 and 𝒙
̂.
𝒆=𝒙−𝒙
According to this, (6-20) can be rewritten as
(6-20)
(6-21)
(6-22)
𝒆̇ = (𝑨 − 𝑳𝑪)𝒆 .
Following the same reasoning as with (6-12), the error of the estimation would tend to
zero as long as A-LC is asymptotically stable. The poles of the observer must be faster than
the closed-loop system poles so that the error converges to zero faster than the system
evolves. Many sources suggest placing the observer poles around 5 or 10 times further to the
left. This is a design parameter that may require some iterations to get a good response.
Determining the value of L can be done by using the same techniques seen for K, but
Ackermann cannot be directly applied because L is a column vector while K is a row vector.
According to the fact that a matrix, M, has the same eigenvalues as his transposed, M*, it can
be written as follows:
(6-23)
det((𝑨 − 𝑳𝑪)∗ − 𝑠𝑰) = det((𝑨∗ − 𝑪∗ 𝑳∗ ) − 𝑠𝑰) .
*
Therefore, to calculate L the same methods seen for pole placement to determine K can
be applied.
6.3.2 Observability
A system is fully observable if the current state vector x(t0) can be determined in a finite
time tf just by observing the output.
The observability matrix is
𝑪
𝑪𝑨
(6-24)
𝓞=[
]
⋮
𝑪𝑨n−𝟏
48
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 6: CLASSIC STATE FEEDBACK
and the system is observable if and only if this matrix has full rank. The demonstration is
similar to the controllability analysis and can it be found in the references.
6.4 The Linear Quadratic Regulator Problem
One big issue in control theory is choosing the optimal gains for state feedback or state
observation [3] [13]. The technique seen so far consisted in placing the poles in arbitrary
locations using Ackerman’s formula. In this section, an approach to optimal control is
introduced. Consider the linear system
𝒙̇ (𝑡) = 𝑨𝒙(𝑡) + 𝑩𝒖(𝑡)
y(𝑡) = 𝑪𝒙(𝑡)
and the quadratic objective function (or cost function):
1 𝑇
𝐽 = ∫ (𝒙∗ 𝑸𝒙 + 𝒖∗ 𝑹𝒖)d𝑡 .
2 0
For the control input
𝒖 = −𝑲𝒙 .
(6-25)
(6-26)
(6-27)
Equation (6-26) becomes
1 𝑇 ∗
(6-28)
𝐽 = ∫ (𝒙 𝑸𝒙 + 𝑲∗ 𝒙∗ 𝑹𝑲𝒙)d𝑡 .
2 0
In this equation, J represents the weighted sum of energy of the state and control. The
problem is to minimize J with respect to the control input, u. This is known as the linear
quadratic regulator (LQR) problem. The design parameters are Q and R, and they weight the
state energy and the control energy, respectively. In general, several design iterations are
needed to get a good response. The LQR method gives an optimal K that minimizes the cost
function. This minimum may not be global or may not give the expected results, that is why
some iterations are generally necessary before achieving good results.
J is minimized when the Riccati equation
𝑨∗ 𝑷 + 𝑷𝑨 − 𝑷𝑩𝑹−𝟏 𝑩∗ 𝑷 + 𝑸 = 𝟎 ,
(6-29)
which is associated with Equation (6-26), is satisfied. Note that Q must be symmetric
positive semidefinite and R symmetric positive definite.
49
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 7: MODEL-FOLLOWING CONTROLLER
7 MODEL-FOLLOWING CONTROLLER
Consider the idea of having a mathematical model with the desirable behaviour for the
plant desired to control. The prescribed model accomplishes the speed, robustness,
overshoot and other response requirements.
In explicit model-following design, the model appears as a feed-forward compensator as
it was described in Figure 1.6. In this structure the model is exhibited and the gains can be
calculated methodologically as it will be shown in the next section [13] [14] [15] [16]. This
is a very comfortable way of control design where only the desired model must be provided.
7.1 Gains calculation
Given the system
𝒙̇ = 𝑨𝒙 + 𝑩𝑢
𝑦 = 𝑪𝒙
(7-1)
it is desired that it behaves as the model
𝒙̇ m = 𝑨m 𝐱 m + 𝑩m 𝑟
𝑦m = 𝑪m 𝒙m .
(7-2)
The augmented system can be written as follows:
𝒙
𝑨 0
0
𝒙̇
𝑩
[ ]=[
][ ] + [ ]𝑢 + [ ]𝑟 .
0 𝑨m 𝒙m
𝑩
𝒙̇ m
0
m
(7-3)
This is a tracker that follows the r signal. Without a reference signal, the system would
become a regulator and its objective would be to keep the output equal to zero. For the
regulator case, the problem would be figured out just by solving it as a regular LQR. Since
there exists a reference signal, directly applying the LQR will not give an exact track
control.
In order to address this problem, the tracker can be analysed separately as a regulator
and an integral filter can be added to perform a better tracking. Many sources handle this
problem by using the command generator tracker (CGT) approach that basically gives the
same solution.
Using both techniques requires adding the output error as an extra state, which leads to
using an integral filter in either case.
The regulator could be directly used as a tracker, in a similar way as it was done in
section 6.2 with the pole placement technique. Nevertheless, in order to improve the tracking
performance, a feed-forward gain and an integral filter will be added.
The mathematical demonstration in next section will show that the integral action is
necessary in order to achieve good tracking. In fact, this will reduce the value of the feedforward gain almost to zero, making it dispensable for the studied cases in this project. It has
been decided to keep this gain with the aim of building a more general controller to be
implemented in other systems that may require it.
50
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 7: MODEL-FOLLOWING CONTROLLER
7.1.1 Controller Gains
In order to reduce the tracking control to a LQR problem, the CGT is applied. First,
Equation (7-3) can be written as
̅𝑟 ,
̅ 𝒙′ + 𝑩
̅𝑢 + 𝑮
(7-4)
̅̇ = 𝑨
𝒙
where the overlined symbols stand for the augmented matrices. Deriving the previous
expression and assuming that 𝑟̇ = 0, the result is
̅𝝃 + 𝑩
̅𝜇
𝝃̇ = 𝑨
(7-5)
The aim of the tracker is to make the real output value equal to the model output. In
other words, defining the error as
𝒙
̅𝒙
(7-6)
̅,
𝑒 = 𝑦m − 𝑦 = 𝑪m 𝒙m − 𝑪𝒙 = [−𝑪 𝑪m ] [𝒙 ] = 𝑪
m
the tracker must accomplish that 𝑒 = 0. Deriving the error:
̅𝒙
̅𝝃 .
(7-7)
̅̇ = 𝑪
𝑒̇ = 𝑪
Collecting all the dynamics from (7-5) and (7-7):
𝑒̇
̅ 𝑒
0
[ ̇ ] = [0 𝑪] [𝝃] + [ ̅ ] 𝜇 .
̅
𝝃
𝑩
0 𝑨
Defining the derivative of the error as the output:
𝑒
̅ 𝝃 = 𝑒̇ .
̅ ] [𝝃] = 𝑪
𝑦 = [0 𝑪
The final augmented system would look as follows:
̿ 𝝃̅ + 𝑩
̿𝜇
𝝃̅̇ = 𝑨
̅ 𝝃̅ .
𝑦=𝑪
Expressed in its full form:
0
𝑒̈
[ 𝒙̈ ] = [0
𝒙̈ m
0
(7-8)
(7-9)
(7-10)
−𝑪 𝑪m 𝑒̇
0
𝑨
0 ] [ 𝒙̇ ] + [𝑩] 𝑢̇
0 𝑨m 𝒙̇ m
0
(7-11)
𝑒̇
𝑪m ] [ 𝒙̇ ] .
𝒙̇ m
The error has been forced to appear as an additional state in order to obtain an extra gain
for the filter. The objective will be to minimize the error and its integral value. Since the
previous reasoning is made using the derivative of the final system, the cost function of
interest is
𝑦 = [0 −𝑪
1 𝑇 ∗
𝐽 = ∫ (𝑒 𝑄𝑒 + 𝑒̇ ∗ 𝑄𝑒̇ + 𝜇 ∗ 𝑅𝜇)d𝑡 ,
2 0
(7-12)
where Q weights the error and its derivative and R weights the control input. To achieve the
previous equation, QLQR must be defined as
𝑄
0
(7-13)
𝑸LQR = [
].
∗
̅
0 𝐶 𝑄𝐶̅
To see this more clearly, consider the cost function
51
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 7: MODEL-FOLLOWING CONTROLLER
1 𝑇 ∗
𝐽 = ∫ (𝝃̅ 𝑸LQR 𝝃̅ + 𝜇 ∗ 𝑅𝜇)d𝑡 .
2 0
(7-14)
This is the objective function for the system described by Equation (7-8), which after
substituting Equation (7-13) and multiplying gives the desired cost function shown in
Equation (7-12). The resulting gain vector is partitioned as follows:
𝑲 𝑲m ] .
𝑲t = [𝐾i
(7-15)
To sum up this process, the MATLAB function that gives the corresponding LQR
algorithm is now shown:
̿, 𝑩
̿ , 𝑸LQR , 𝑅) .
𝑲t = 𝑙𝑞𝑟(𝑨
(7-16)
The control input in Equation (7-8) is
𝑒
𝒙̇
𝑲 𝑲m ] [ ]
𝒙̇ m
= −𝑲i 𝒆 − 𝑲𝒙̇ − 𝑲m 𝒙̇ m .
𝜇 = −𝑲t 𝝃̅ = −[𝐾i
(7-17)
Since 𝜇 is the derivative of the control input in Equation (7-4), 𝑢 can be written as
𝑇
𝑇
𝑢 = ∫ 𝜇 𝑑𝑡 = − 𝐾i ∫ 𝑒 𝑑𝑡 − 𝑲𝒙 − 𝑲m 𝒙m ,
0
(7-18)
0
which, as it has been anticipated, incorporates an integral filter.
7.1.2 Feed-forward Gain
Even though including an integral filter already corrects the steady error, a feed-forward
gain can be added with the same aim. According to Figure 7.1, the control law is:
𝑢 = −𝑲𝒙 − 𝑲m 𝒙m − 𝑲i 𝑒 + 𝑢ff .
Let 𝜌̇ = 𝑒. Including the error in the state representation:
𝜌
𝜌̇
0
−𝑪
𝑪m
0
0
𝒙
−𝑩𝑲
𝑨
−
𝑩𝑲
−𝑩𝑲
𝑩
[ 𝒙̇ ] = [
] [𝒙 ] + [ 0 ] 𝑟 .
i
m
𝑩m
0
0
𝑨m
0 𝑢m
𝒙̇ m
ff
The desired behaviour is that the error is zero:
𝑒 = 𝑦m∞ − 𝑦∞ = 𝑪m 𝒙m∞ − 𝑪𝒙∞ = 0 .
When time tends to infinite:
0
0
[0] = [−𝑩𝑲i
0
0
−𝑪
𝑨 − 𝑩𝑲
0
𝑪m
−𝑩𝑲m
𝑨m
𝜌∞
0
0
𝒙∞
𝑩] [𝒙 ] + [ 0 ] 𝑟∞ .
𝑩m
0 𝑢m∞
ff∞
Rewriting Equation (7-22):
52
(7-19)
(7-20)
(7-21)
(7-22)
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 7: MODEL-FOLLOWING CONTROLLER
0
[−𝑩𝑲i
0
−𝑪
𝑨 − 𝑩𝑲
0
𝑪m
−𝑩𝑲m
𝑨m
𝜌∞
0
0
𝒙∞
𝑩] [𝒙 ] = [ 0 ] 𝑟∞ .
−𝑩m
0 𝑢m∞
(7-23)
ff∞
It can be solved for the state vector multiplying by the pseudoinverse:
𝜌∞
0
𝒙∞
−𝑩𝑲
[𝒙 ] = [
i
m∞
0
𝑢ff∞
−𝑪
𝑨 − 𝑩𝑲
0
𝑪m
−𝑩𝑲m
𝑨m
0 −1 0
𝑩] [ 0 ] 𝑟∞ .
−𝑩m
0
(7-24)
The feed-forward gain can be obtained from this last equation according to the next
equality:
𝐾ff =
𝑢ff∞
.
𝑟∞
(7-25)
PLANT
𝑟
Kff
𝑢ff
+
𝑢
B
+
𝒙
∫dt
𝒙̇
C
𝑦
A
-K
-Ki
𝜌
∫dt
𝑒=𝜌̇ +
-
-Km
Bm
+
𝒙m
∫dt
𝒙̇ m
Cm
𝑦m
Am
MODEL
Figure 7.1: Model-following controller with feed-forward gain and integral control
53
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 7: MODEL-FOLLOWING CONTROLLER
7.2 Time-discrete Controller
Until now, all the control design has been developed for continuous time simulation.
The diagram described in Figure 7.1 can be directly be translated into a continuous Simulink
model, but before implementing it on the STM32F4, it should be discretized.
In this chapter the state-space equations of the controller described in the previous
section will be deducted and then the controller will be directly discretized it using
MATLAB.
The controller is the whole group of elements corresponding to the observer and the
model. Its inputs are the reference, r, and the system output, y. The output of the controller is
the control signal of the system, u. This is shown in Figure 7.2.
CONTROLLER
𝑢
OBSERVER
z
𝑟
MODEL
𝑦
PLANT
Figure 7.2: Simplified representation of the system
Replacing the real plant from Figure 7.1 with the observer from Figure 6.2 the modelfollowing controller can be described as
𝜌̇ = 𝑪m 𝒙m − 𝑦
̂̇ = 𝑨𝒙
̂ + 𝑳(𝑦 − 𝑪𝒙
̂) + 𝑩(𝑟𝐾ff − 𝒙m 𝑲m − 𝑲i 𝜌 − 𝒙
̂𝑲)
𝒙
𝒙̇ m = 𝑩m 𝑟 + 𝑨m 𝒙m .
(7-26)
In state-space form:
𝜌̇
0
[ 𝐱̂̇ ] = [−𝑩𝑲i
0
𝐱̇ m
0
𝑨 − 𝑳𝑪 − 𝑩𝑲
0
𝜌
𝑪m
0
−𝑩𝑲m ] [ 𝐱̂ ] + [𝑩𝐾ff
𝐱m
𝑩m
𝐀m
−1 𝑟
𝑳 ] [𝑦] .
0
(7-27)
Establishing the control signal as the output:
𝑢 = [−𝐾i
𝜌
−𝑲 −𝑲m ] [ 𝐱̂ ] + [𝐾ff
𝐱m
𝑟
0] [𝑦] .
(7-28)
The controller can be saved as a state-space model class by declaring the previous A, B,
C and D matrices of its state-pace form. The resulting system can be directly discretized
with MATLAB using the ‘c2d’ function.
54
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
8 VELOCITY CONTROL
After having analysed all the means used in this project, velocity control will be now
performed over the DCMCT in open loop configuration. The TACH port will be used to
measure the velocity of the motor.
First, using the modelled plant, the controller will be obtained and discretized with
MATLAB to store it into a state-space model. Then, the MIL, SIL, PIL and HIL simulations
will be performed and the controller will be modified until achieving a satisfactory result.
The MATLAB script to obtain the model-following controller for velocity is Script C
from the annexed document.
8.1 MIL Simulation
The MIL simulation is very important because it is the first time the designed controller
is running. This gives a first impression of the global response in terms of stability and
fitting to the desired model. It is useful because tuning parameters and performing a new
simulation takes no time.
The chosen model for velocity control will be the first-order system
1
(8-1)
.
𝑠+1
The plant is controllable and observable, so the system should be reachable.
At this stage, the simulated control input for a given reference can be checked. This is
also an important design matter, because the control input must be within the actuation range
of the system. In this case, the control input, which is the COMMAND port voltage (Vc),
must be within ±5 volts. In fact, the signal adaptation circuit cannot reach the COMMAND
range limits, so the control input range should be reduced to about ±4.5 volts to avoid not
reachable references.
The MIL simulation has been performed using the following Simulink model given in
Figure 8.1.
𝐺m =
Figure 8.1: Block diagram for MIL simulation in velocity control
Some gains must be added in order to adjust the equivalence between volts and speed
according to Table 2.3. The response is shown in Figure 8.2.
55
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
The model and the controller response are virtually equal and would not be
distinguishable if the simulated output had not been formatted to a dotted line.
The simulated control input for this simulation is within the reachable range. A
simulated control input will be shown in PIL simulation to compare it with the real values in
HIL. As seen in the next section, the controller is going to need some tuning, so the control
input will change.
2200
2000
1800
1600
Velocity (rpm)
1400
1200
Simulated output
Model output
Step reference
1000
800
600
400
200
0
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 8.2: Results for MIL simulation in velocity control
8.2 SIL Simulation
Configuring the Simulink model as explained in section 3.3.6, and adding the gains as it
was done for the MIL simulation in Figure 8.1, the first results for this first SIL simulation
are shown in Figure 8.3. The tracking is not as good as in the MIL simulation, but it still
follows the model output.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
2200
2000
1800
1600
Velocity (rpm)
1400
1200
Simulated output
Model output
Step reference
1000
800
600
400
200
0
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 8.3: First results of the SIL simulation in velocity control
The Q and R weights of the LQR and the position of the observer poles should now be
tuned in order to improve tracking. In this case the new values will be 𝑄 = 1010 , 𝑅 = 1 and
the observer poles will remain five times faster than the closed-loop system poles.
The new response is shown in Figure 8.4. Now the tracker is performing again with
almost no error. In the first simulation, the error had been weighted too heavily and the poles
were too fast.
57
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
2200
2000
1800
1600
Velocity (rpm)
1400
1200
1000
Simulated output
Model output
Step reference
800
600
400
200
0
0
1
2
3
4
5
Time (s)
6
7
8
9
Figure 8.4: Second results of the SIL simulation in velocity control
8.3 PIL Simulation
Now the PIL block will be running on the STM32F4 as explained in section 3.3.7. The
results are shown in Figure 8.5. The visual response is very similar for both SIL and PIL
simulation. This means that the emulator reproduces the behaviour of the real
microcontroller very accurately.
The next step will be HIL simulation. The simulated control input has also been plotted
in Figure 8.6 in order to compare it with the real values. Even though a simulated control
input shows that the reference can be reached, it may not occur with the real system, because
there are mismatches between the modelled plant and the real one. For this reason, a higher
control input value might be needed to correct the error.
For this case, the simulated control input maximum value is about one volt below the
actual range limit, so it should not be a problem for the real system to follow the same step.
58
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
2200
2000
1800
1600
Velocity (rpm)
1400
1200
1000
Simulated output
Model output
Step reference
800
600
400
200
0
0
1
2
3
4
5
Time (s)
6
7
8
9
9
10
Figure 8.5: Results of the PIL simulation in velocity control
4
3.5
Control input (V)
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
Time (s)
6
7
8
Figure 8.6: Control input for PIL simulation in velocity control for a unitary step
59
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
8.4 HIL Simulation
The STM32F4 will be running with the real DCMCT board in HIL configuration. The
connections are shown in Figure 8.7.
± 15 V
+15 V
MINI-USB
M
COMMAND
±5 V
X3
TACH
±5 V
X7
X1
PA4
PB7
Tx
USB
X5
PA0
PB6
Rx
DRIVER
SIGNAL ADAPTER
STM32
TTL-RS232
GND
GNDs (X2 & X8)
GND
GND
Figure 8.7: Physical connections for HIL simulation in velocity control
The Simulink model to load the program onto the STM32F4 board is shown in Figure
8.8. The control input and the reading of the system output have been converted to read the
real values provided by the signal adapter. Equation (5-15) must be undone at the output of
the ‘controller_disc’ block so that the Vout1 pin gives the desired control input. The same
happens with the read value at the A0 pin of the ADC port, where Equation (5-17) gives the
right reading to the controller block.
After the code has been loaded onto the STM32F4, the reference will be given by the
host PC, following the steps explained in section 3.3.5. The executed model in the PC is
shown in Figure 8.9. The gains here convert the reference value into volts as in the previous
simulations.
60
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
Figure 8.8: Simulink block model to load the algorithm onto the STM32F4 for HIL simulation
Figure 8.9: Algorithm executed continuously on the host PC to send a reference value and plot the
results. The reference here is zero.
61
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
The response to a 2000 rpm step is shown in the Figure 8.10.
2000
1800
1600
Position (º)
1400
1200
1000
800
Real output
Model output
Step reference
600
400
200
0
1
2
3
4
Time (s)
5
6
7
Figure 8.10: Comparison between the model and the real plant response to a 2000 rpm step input
The final results are very satisfactory. The model output is over the real output and can
barely be distinguished. The only problem with the response is that there is much of noise in
the signal. This is mainly caused by the DAC and ADC ports of the STM32F4.
The real control input can also be registered sending it with the output value. For the
response shown in the previous figure, the real control input calculated by the controller is
shown in Figure 8.11. Comparing this control input with the previously simulated in Figure
8.6, both are very similar. This means that the plant identification is a good approximation to
the real hardware.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 8: VELOCITY CONTROL
3.5
Control input(V)
3
2.5
2
1.5
1
0.5
0
1
2
3
4
Time (s)
5
Figure 8.11: Real control input in the HIL simulation for a 2000 rpm step input
63
6
7
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 9: POSITION CONTROL
9 POSITION CONTROL
In this section position control of the motor shaft will be demonstrated. The plant will be
given now by Equation (4-30). The desired model to follow will be the underdamped
second-order system:
𝐺m =
1
𝑠2
𝜔𝑜2
+ 𝜁𝜔𝑜 𝑠 + 1
,
𝜔𝑜 = 3,
𝜁 = 0.3.
(9-1)
The MATLAB script to obtain the model-following controller for position is Script D
from the annexed document. The code is basically the same. The controller has been written
in a generic format, so the script is valid for systems of any order as long as they have a
single output. Only the plant, the model and the iteration parameters (Q, R and the observer
poles) have been modified.
The same steps as in velocity control will be followed next.
9.1 MIL Simulation
The gains from Figure 8.1 must be replaced with the equivalence between volts and
degrees as it has been done in Figure 9.1.
Figure 9.1: Simulink diagram for MIL simulation in position control
The electrical range of the potentiometer is of 350º. This means that the maximum
overshoot peak must be within the ±175º range. Otherwise, the controller would enter in the
nonlinear working area.
The first parameters of the simulation process will be 𝑄 = 106 , 𝑅 = 1 and the observer
poles will be two times faster than the closed-loop system. A step of 100 degrees has been
simulated. The results are shown in Figure 9.2.
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 9: POSITION CONTROL
150
Position (º)
100
50
0
Simulated output
Model output
Step reference
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Figure 9.2: Results for MIL simulation in position control
The model is perfectly tracked by the motor and the overshoot is in the reachable range.
9.2 SIL Simulation
For the SIL Simulation the results can be found in Figure 9.3
150
Position (º)
100
50
Simulated output
Model output
Step reference
0
0
0.5
1
1.5
2
Time (s)
Figure 9.3: Results for SIL simulation in position control
65
2.5
3
3.5
4
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 9: POSITION CONTROL
The response is virtually the same.
9.3 PIL Simulation
For the PIL simulation the results are shown in Figure 9.4.
150
Position (º)
100
Simulated output
Model output
Step reference
50
0
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Figure 9.4: Results for PIL simulation in position control
This result is identical to the previous simulations. The HIL simulation may not be as
good as the first attempt because the real plant is not as the estimated one and the states are
given by an observer.
9.4 HIL Simulation
Now connect the Vin2 of the signal adapter to the POT port and link the potentiometer to
the motor using the belt. The rest of the procedure is the same as for velocity control.
The first results are not as good as with the previous simulations where the plant was a
software simulation. This response comparison is shown in Figure 9.5. The system tries to
follow the model but is unable to correct the error.
After several iterations, the parameters have been tuned to 𝑄 = 104 , 𝑅 = 1 and the
observer poles have been set ten times faster than the closed-loop system. The new results
are shown in Figure 9.6.
Now the system accurately follows the shape of the model and corrects the error in
steady state. The problem with the first simulation was that the observer was too slow to
66
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 9: POSITION CONTROL
follow the model. The error has been weighted a bit less heavily, because the actuation was
too big even when the error in steady state was practically zero.
150
Position (º)
100
50
Real output
Model output
Step reference
0
4.5
5
5.5
6
Time (s)
Figure 9.5: First HIL response comparison to a 100º step input
67
6.5
7
7.5
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 9: POSITION CONTROL
140
120
Position (º)
100
80
60
Real output
Model output
Step reference
40
20
0
2
2.5
3
3.5
Time (s)
4
4.5
Figure 9.6: HIL response comparison to a 100º step input after parameter tuning
68
5
5.5
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 10: CONCLUSIONS
10 CONCLUSIONS
This section is a summary of the results obtained along the development of the project.
The initial objectives have been achieved, nevertheless the way is open to improve some
points as it will be analysed next.
The first task has been to become familiar with the STM32F4 and the development of
simple algorithms using Simulink so that it can interact with a computer. It has been
possible to use the microcontroller as a data acquisition card using a TTL-RS232 adapter to
communicate with the computer through a serial data channel and read the STM32F4 pins in
execution time. It is also possible to modify the value of its analog and digital ports from the
computer.
Three toolboxes by different firms have been analysed to achieve the previous aim and
have access to other peripherals without programming the microcontroller by hand.
The toolbox provided by MATLAB is easy to use, but it has very limited features. In
this project the target package associated to this toolbox has been used to perform SIL and
PIL simulations with the GNU toolchain.
STM provides another Simulink toolbox much more complete. The trouble is that its use
is not that simple and several intermediate steps are necessary to code, compile and flash a
Simulink model onto the microcontroller. Furthermore, it has been observed that new
updates of the software involved in this rapid prototyping method are still being released
frequently and the STM32CubeMX seems not to fully support the compilers such as Keil
and some code initializations must be added manually. This might be a better option in
future releases.
The Waijung toolbox has been the chosen software due to its extreme integration with
Simulink and the ease to directly build and load the program onto the STM32 from a block
model. Besides, this is the toolbox that allows more access to the peripherals configuration.
The STM32F4 has a fast processor that can work with 32-bit single precision registers.
Even though in the Simulink models of this project the algorithms use double-precision
values, a data conversion is automatically performed during encoding.
The high-performance of this MCU allows working with small sampling rates for
discrete controllers. Here the controller has been designed in continuous time and then
discretized using MATLAB. The focus of this project is the rapid prototyping, so the sample
time has been chosen in a way that the results are satisfactory. Nevertheless, some criteria
can be applied to determine a proper sampling. One criterion when applying discretization is
to allow the margin phase to deteriorate 5 to 15 degrees compared to the continuous design
[17].
Whereas the performance of the STM32F4 is very high, the built-in DACs and CADs
peripherals resulted very disappointing for three reasons.
The main reason is the noise in the measurements. The voltage reference for conversion
is around 3 volts and the peripherals work with 12-bit data, so the resolution should be
smaller than a millivolt. Nevertheless, the noise makes its accuracy much worse.
The second reason is the voltage reference itself. It is not as steady as desired because it
has been observed that its value may vary some millivolts with temperature. Not much
information is provided about the DAC and CAD processes in the datasheets.
The last negative point is that during the tests it has been observed that the DACs are
unable to provide a voltage lower than approximately 50 millivolts.
69
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
CHAPTER 10: CONCLUSIONS
In spite of these facts, both DAC and CAD peripherals can be used taking into account
the results in these tests if the application does not have requirements too demanding. For
example, rapid control prototyping demonstration has been successfully accomplished in
this project using these peripherals.
The controller used in these demonstrations is an explicit model-following controller
with feed-forward gain and integral action. It has been observed that, for the actual
implemented controllers, when using an integral control, the feed-forward gain is virtually
zero because the error is already corrected.
The implementation and debugging processes of the controller has been guided through
the MIL, SIL, PIL and finally HIL simulations. In fact, when some acceptable parameters
are achieved after SIL and PIL simulations, new parameters can be directly tested on the real
plant or using a simulator.
According to the tuning tests performed in this project, using an observer with too fast
poles introduces noise into the control input even when the steady error is practically zero.
An observer too slow makes the motor not to track the model.
The main gains of the controller have been calculated by solving a linear quadratic
regulator problem that minimizes the tracking error and its integral value. Penalizing the
error too slightly causes the motor to be unable to track the model. On the other hand,
penalizing it too heavily causes undesired effects as well.
Several iterations are necessary to get good response results since there are various
elements involved in the system behaviour.
An introduction to optimal control was made using the LQR to calculate the main gains.
The next step could be the study of optimal observers instead of placing its poles manually
according to the closed-loop system poles.
Identifying the real plant (in this case, the motor) is also an important matter to get a
good controller design. To estimate a mathematical model of the motor, several
simplifications have been made.
The velocity response to a voltage input has been approximated as a first-order system
response after analysing the physical laws of its behaviour. According to this, a grey-box
estimation has been performed using MATLAB and the STM32F4 as a data acquisition
card. The result has been a transfer function of the desired order that fits in a high percentage
to the real measurements.
The motor does not behave like the estimated model for very low voltages. This is
because for small input values the braking torque is bigger than the driving torque and the
motor has trouble to spin. If this was a problem it could be minimized by using PWM. PWM
is more suitable for applications were the DC motor rotates at low frequencies.
Both velocity control and position control could be much more accurate reading the
digital outputs of the DCMCT board. The encoder has 12-bit resolution and that means that
the motor can detect rotations smaller than 0.1 degrees without uncertainty.
The final conclusion of this project is that rapid control prototyping using the
STM32F4-Discovery board and the studied tools can save time, money and effort by
automatizing coding, compilation and flashing.
70
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
REFERENCES
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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http://es.mathworks.com/solutions/rapid-prototyping/embedded-control-systems.html.
[Accessed 2015].
A. S. Basem S. El-Haik, Software Design for Six Sigma: A Roadmap for Excellence, John
Wiley & Sons, 2011.
K. Ogata, Modern Control Engineering, Prentice Hall, 2010.
H. L. Karl Johan Åström & Jacob Akparian, “USB QICii Laboratory Workbook.
Instructor Workbook,” Revision: 01.
STMicroelectronics, “UM1472 User Manual. Discovery kit for STM32F407/417 lines,”
2014.
STMicroelectronics, “STM32F405xx STM32F407xx,” 2015.
MathWorks, “Embedded Coder Support Package for STMicroelectronics STM32F4Discovery Board,” [Online]. Available:
http://www.mathworks.com/matlabcentral/fileexchange/43093-embedded-coder-supportpackage-for-stmicroelectronics-stm32f4-discovery-board.
ARM, “CMSIS,” [Online]. Available: http://www.arm.com/products/processors/cortexm/cortex-microcontroller-software-interface-standard.php.
STMicroelectronics, “Code Generation for STM32 MCUs using MATLAB® and
Simulink®,” [Online]. Available: http://www.st.com/st-webui/static/active/en/resource/sales_and_marketing/presentation/product_presentation/stm32matlab.pdf. [Accessed 2015].
Aimagin, “Waijung Blockset,” [Online]. Available: http://waijung.aimagin.com/.
[Accessed 2015].
H. T. N. Charles L. Phillips, Digital control system analysis and design, Prentice-Hall,
1984.
MathWorks, “Estimating Linear Grey-Box Models,” [Online]. Available:
http://es.mathworks.com/help/ident/ug/estimating-linear-grey-box-models.html. [Accessed
2015].
J. J. S. Tyler, “The Characteristics of Model-Following Systems as Synthesized by
Optimal Control,” Cornell Aeronautical Laboratory, Inc., Buffalo, N. Y., 1964.
V. L. S. Frank L. Lewis, Optimal Control, John Wiley & Sons, 1995.
G.-R. Yu, “Explicit Model-Following Design of Propulsion Control Aircraft,” in Systems,
Man, and Cybernetics, Taipei, 2006.
R. F. Stengel, Optimal Control and Estimation, Dover Publications, Inc., 1994.
B. W. Karl J Åström, Computer-Controlled Systems: Theory and Design, Third Edition,
Courier Corporation, 2013.
71
ANNEXES
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
CONTENTS
1
Signal Adapter: Electrical Circuit .....................................................................................1
2
Signal Adapter: Printed Circuit .........................................................................................2
3
Script A: Architecture of the Model to Estimate ..............................................................3
4
Script B: Model Estimation Using a Grey Box ................................................................4
5
Script C: Velocity Controller ............................................................................................5
6
Script D: Position Controller ............................................................................................7
A
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
1 SIGNAL ADAPTER: ELECTRICAL CIRCUIT
1
Annexes
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
2 SIGNAL ADAPTER: PRINTED CIRCUIT
2
Annexes
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
3 SCRIPT A: ARCHITECTURE OF THE MODEL TO ESTIMATE
function [A,B,C,D] = myfunc( K, T, Ts )
A
B
C
D
=
=
=
=
-1/T;
K/T;
eye(1);
zeros(1);
End
3
Annexes
RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
4 SCRIPT B: MODEL ESTIMATION USING A GREY BOX
s=tf('s'); %frecuency domain s
%Reminder: previous declaration of Ts is assumed
y=Scope_Step(:,2); %Output vector
u=Scope_Step(:,3); %Input vector
data= iddata(y, u, Ts); %Time-domain data
odefun = 'myfunc';
%system architecture
%--Creation of an array for parameters to estimate-K=1; %Initial values must be set for K and T
T=1;
parameters = {'gain', K; 'speed', T};
fcn_type = 'c'; %continuous function
%Creation of the grey-box model
sys = idgrey(odefun,parameters,fcn_type);
%Parameter estimation according to data
m_est = greyest(data,sys);
%Show results
Param=m_est.Structure.Parameters;
fprintf('%s: %f\n',Param(1,1).Name,Param(1,1).Value);
fprintf('%s: %f\n',Param(1,2).Name,Param(1,2).Value);
fprintf('fit: %f percent \n',m_est.Report.Fit.FitPercent);
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
5 SCRIPT C: VELOCITY CONTROLLER
s=tf('s');
Ts=0.0001; %sample time
Tc=0.001; %communication time (for HIL simulation)
%Modelled plant
K=0.776918;
T=0.084761;
G=K/(T*s+1);
%-------------G=ss(G); %to state-space form
A=G.a; B=G.b; C=G.c; D=G.d;
Gm=1/(1*s+1);
%model to follow
p_lc=roots(Gm.den{1}); %closed-loop system poles
Gm=ss(Gm);
%to state-space form
Am=Gm.a; Bm=Gm.b; Cm=Gm.c; Dm=Gm.d;
L=acker(A',C',p_lc*5); %observer poles x5 faster than p_lc
L=L';
Q=1e15; %the error is penalized heavily
R=1;
%in comparison with the control input
%Augmented system including the model and the error----Anew=[0,-C,Cm;
zeros(size(A,1),1),A,zeros(size(A,1),size(Am,2));
zeros(size(Am,1),1),zeros(size(Am,1),size(A,2)),Am];
Bnew=[0;B;zeros(size(Am,1),1)];
Cnew=[0,-C,Cm];
%------------------------------------------------------%Qlqr must minimize the error and its accumulated value-----Qlqr=[-C,Cm]'*Q*[-C,Cm];
Qlqr=[Q,zeros(1,size(Qlqr,2));zeros(size(Qlqr,1),1),Qlqr,];
%----------------------------------------------------------Kt= lqr(Anew, Bnew,Qlqr,R); %gains calculation
Ki=Kt(1); %integral gain
K=Kt(2:size(A,2)+1); %observer gain
Km=Kt(size(A,2)+2:size(Kt,2)); %model gain
%(Continues)==>
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
%Feed-forward gain calculation-------------------------------M=[0,-C,Cm,0;
-B*Ki,A-B*K, -B*Km,B;
zeros(size(Am,1),size(B,2)),zeros(size(Am,1),size(A,2)),...
Am, zeros(size(Am,1),1)];
M_inv=pinv(M);
B_aux=[0;zeros(size(A,2),1);Bm];
Kff=-M_inv(size(M_inv,1),:)*B_aux;
%------------------------------------------------------------%Controller in continuous time to state-space form------------------Controller_cont= ss([zeros(1,size(B,2)),zeros(1,size(A,2)),Cm;
-B*Ki,A-L*C-B*K,-B*Km;
zeros(size(Am,1),size(B,2)),zeros(size(Am,2),size(A,2)),Am],...
[zeros(1,size(B,2)),-1;B*Kff,L;
Bm,zeros(size(Bm,1),1)],[-Ki,-K,-Km],[Kff 0]);
%------------------------------------------------------------------Controller_disc= c2d(Controller_cont,Ts); %to discrete time
Model_disc=c2d(Gm,Ts); %to discrete time (for MIL, SIL and PIL simulations)
Plant_disc=c2d(G,Ts); %to discrete time (for MIL, SIL and PIL simulations)
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
6 SCRIPT D: POSITION CONTROLLER
s=tf('s');
Ts=0.0001; %sample time
Tc=0.001; %communication time (for HIL simulation)
%Modelled plant. Now the estimation is a second-order system
K=77.133501;
T=0.0911698;
G=K/(T*s+1);
G=G/s;
%-----------------------------------------------------------G=ss(G); %to state-space form
A=G.a; B=G.b; C=G.c; D=G.d;
Gm=1/(s^2/3^2+0.3*3*s+1);
%second-order model to follow
p_lc=roots(Gm.den{1}); %closed-loop system poles
Gm=ss(Gm);
%to state-space form
Am=Gm.a; Bm=Gm.b; Cm=Gm.c; Dm=Gm.d;
L=acker(A',C',p_lc*2); %observer poles x2 faster than p_lc
L=L';
Q=1e6;
R=1;
%the error is penalized heavily
%in comparison with the control input
%Augmented system including the model and the error----Anew=[0,-C,Cm;
zeros(size(A,1),1),A,zeros(size(A,1),size(Am,2));
zeros(size(Am,1),1),zeros(size(Am,1),size(A,2)),Am];
Bnew=[0;B;zeros(size(Am,1),1)];
Cnew=[0,-C,Cm];
%------------------------------------------------------%Qlqr must minimize the error and its accumulated value-----Qlqr=[-C,Cm]'*Q*[-C,Cm];
Qlqr=[Q,zeros(1,size(Qlqr,2));zeros(size(Qlqr,1),1),Qlqr,];
%----------------------------------------------------------Kt= lqr(Anew, Bnew,Qlqr,R); %gains calculation
Ki=Kt(1); %integral gain
K=Kt(2:size(A,2)+1); %observer gain
Km=Kt(size(A,2)+2:size(Kt,2)); %model gain
%(Continues)==>
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RAPID CONTROL PROTOTYPING USING AN STM32 MICROCONTROLLER
ANNEXES
%Feed-forward gain calculation-------------------------------M=[0,-C,Cm,0;
-B*Ki,A-B*K, -B*Km,B;
zeros(size(Am,1),size(B,2)),zeros(size(Am,1),size(A,2)),...
Am, zeros(size(Am,1),1)];
M_inv=pinv(M);
B_aux=[0;zeros(size(A,2),1);Bm];
Kff=-M_inv(size(M_inv,1),:)*B_aux;
%------------------------------------------------------------%Controller in continuous time to state-space form------------------Controller_cont= ss([zeros(1,size(B,2)),zeros(1,size(A,2)),Cm;
-B*Ki,A-L*C-B*K,-B*Km;
zeros(size(Am,1),size(B,2)),zeros(size(Am,2),size(A,2)),Am],...
[zeros(1,size(B,2)),-1;B*Kff,L;
Bm,zeros(size(Bm,1),1)],[-Ki,-K,-Km],[Kff 0]);
%------------------------------------------------------------------Controller_disc= c2d(Controller_cont,Ts); %to discrete time
Model_disc=c2d(Gm,Ts); %to discrete time (for MIL, SIL and PIL simulations)
Plant_disc=c2d(G,Ts); %to discrete time (for MIL, SIL and PIL simulations)
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