Costandin Marius-Simion Asistent universitar

Transcription

Curriculum vitae
INFORMAŢII PERSONALE
Costandin Marius-Simion
48 Fantanele, 510041 Alba-Iulia (România)
0746879877
[email protected]
Data naşterii 09/07/1990 | Naţionalitatea română
LOCUL DE MUNCĂ PENTRU
CARE SE CANDIDEAZĂ
Asistent universitar
EXPERIENŢA PROFESIONALĂ
01/10/2015–Prezent
Asistent universitar plata cu ora
Universitatea Tehnica Cluj-Napoca, Cluj-Napoca (România)
EDUCAŢIE ŞI FORMARE
2005–2009
Diploma de bacalaureat
Colegiul National "Horea Closca si Crisan", Alba-Iulia (România)
limba romana, limba engleza, matematica, fizica, informatica, biologie
2009–2013
Diploma de licenta
Universitatea Tehnica Facultatea de Automatica si Calculatoare, Cluj-Napoca (România)
Analiza Matematica;
Algebra Liniara si Geometrie Analitica;
Matematici Speciale (analiza complexa , transformatele Fourier, Laplace , Z);
Bazele Circuitelor Electronice;
Electrotehnica;
Programarea Calculatoarelor;
Circuite Analogice si Numerice;
Modelarea Proceselor;
Calcul Numeric;
Teoria Sistemelor I,II;
Electronica de Putere in Automatica;
Ingineria Reglarii Automate I II;
Identificarea Proceselor(Sistemelor);
Sisteme cu Evenimente Discrete ;
Ingineria Sistemelor de Programare;
Transmisia Datelor;
Optimizari;
Sisteme de conducere a proceselor continue;
Sisteme de Control Distribuit;
Sisteme de Conducere a Robotilor;
Microsisteme si Achizitii de date;
5/1/16
© Uniunea Europeană, 2002-2015 | http://europass.cedefop.europa.eu
Pagina 1 / 28
Curriculum vitae
2010–2013
Diploma de licenta
Universitatea Babes-Bolyai; Facultatea de Matematica si Informatica, Cluj-Napoca (România)
Pentru o lista cu disciplinele studiate, vezi matematica linia romana din tabelul de la adresa
urmatoare:
http://www.cs.ubbcluj.ro/files/curricula/2011/disc/syllabus.init/clean.php?file=lista.htm
2013–2015
Diploma de disertatie (master)
Universitatea Tehnica; Facultatea de Automatica si Calculatoare, Cluj-Napoca (România)
Sisteme Adaptive;
Matematici Avansate;
Automatizarea Proceselor Dinamice;
Sisteme inglobate;
Control Optimal;
Sisteme Robuste;
Sisteme Neliniare si Stohastice
Conducerea Proceselor Neconventionale
2013–2015
Diploma disertatie master
Universitatea Babes-Bolyai, Cluj-Napoca (România)
Analiza Neliniara Aplicata;
Spatii Sobolev;
Mecanica Fluidelor;
Fenomene de transfer in medii poroase;
Metode Numerice pentru Ecuatii Operatoriale;
Capitole Speciale de Analiza Numerica;
Biomatematica
Modele Stohastice;
Metode Topologice pentru Ecuatii cu derivate partiale;
Aproximarea Proceselor Liniare;
Calcul Variational;
Capitole speciale de Analiza Reala si Complexa;
Relativitate si Cosmogologie;
2015–Prezent
Teza de doctorat
Universitatea Tehnica Cluj-Napoca, Cluj-Napoca (România)
COMPETENŢE PERSONALE
Limba(i) maternă(e)
română
Alte limbi străine cunoscute
engleză
ÎNȚELEGERE
VORBIRE
SCRIERE
Ascultare
Citire
Participare la
conversaţie
Discurs oral
B2
B2
B1
B1
B2
Document elaborat in anul III al studiilor de licenta de la UBB valabil 2 ani
5/1/16
Pagina 2 / 28
Curriculum vitae
Niveluri: A1 și A2: Utilizator elementar - B1 și B2: Utilizator independent - C1 și C2: Utilizator experimentat
Cadrul european comun de referinţă pentru limbi străine
Competenţe de comunicare
Competenţă digitală
-bune abilitati de comunicare dobandite in din diferite situatii din viata printre care, importanta, este
activitatea in biserica din care fac parte.
- cunoasterea pachetului Office oferit de Microsoft, sau Open Office, dobandite pe parcursul anilor de
liceu, cat mai important, in faculate, din necesitatea de a tehnoredacta diferite rapoarte, teme, si chiar
lucrarea de licenta
- relativ buna cunostere a limbajului Matlab ca urmare a multelor ore petrecute folosind acest
program, pentru diferite proiecte pentru faculate, cat si din propriu interes;
- relativ buna cunoastere a limbajului C ca urmare a cursurilor si temelor din facultate;
- cunoastere a limbajului C++ ca urmare a cursurilor din facultate, cat si a catorva ore bune petrecute
in C++ din pura placere, (pentru anumite proiecte) ;
- relativ buna cunoastere a limbajului Java, ca urmare a cursurilor, temelor si proiectelor din faculate,
cat si cateva proiecte individuale;
- in facultate am folosit si Transact SQL pentru baze de date. Chiar mai mult, limbajul SQL a fost folosit
deseori impreuna cu Java, sau C# pentru diferite teme si proiecte in cadrul facultatii;
- cunosterea limbajului C# ca urmare a cursurilor, temelor, proiectelor din facultate;
- notiuni elementare am facut si despre limbajul LISP in cadrul unei discipline studiate la facultate;
- am folosit in cadrul unei discipline si libraria OpenGL in cadrul facultatii, insa desi ma impresioneaza,
nu am avut ocazia/timp sa aprofundez;
- Pascal, studiat in liceu;
- utilizez in timpul liber, fiind aproape pasionat de electronica, programul AVR Studio pentru editare de
firmware, pentru microcontrolerele AVR oferite de Atmel;
- utilizez programul avrdude pentru incarcarea codului .hex pe microcontroler;
- am utilizat ISIS Proteus pentru simulare de circuite si pentru proiectare PCB;
- utilizez mai nou programul Eagle pentru proiectare de PCB, desi mai am multe de invatat;
- utilizez LTSpice si Multisim de la NI pentru proiectare/testare circuite electronice;
Permis de conducere
B1
INFORMAŢII SUPLIMENTARE
Publicaţii
" A mathematical approach of fractional order systems" publicata la forumul studentilor in anul 2012 cu
ocazia conferintei AQTR (Automation, Quality and Testing Robotics) organizata de UTC-N;
" Fractional Order PI controller design method" publicata la forumul studentilor in anul 2014 in cadrul
aceeleiasi conferinte AQTR;
" Limit cycle based controller design method" pulicata la forumul studentilor in anul 2014 in cadrul
conferintei AQTR;
" Asupra unor sheme de probabilitate" prezentare la conferinta Didactica Matematicii in anul 2012
organizata de UBB.
5/1/16
Pagina 3 / 28
Curriculum vitae
"A convergence theorem and its applications concerning the Riemann's Zeta function";
Distincţii
In anul 2011 cu ajutorul lui Dumnezeu am obtinul locul II la faza nationala a concursului studentesc de
matematica "Traian Lalescu"
Proiecte
Proiectele mele se impart in doua categorii:
1. Simulari;
1.1 In anul 2014 am lucrat la simularea/modelarea in Matlab a unui convertor DC-DC coborator.
Proiectul din Matlab consta in integrarea numerica succesiva a doua sisteme dinamice
corespunzatoare situatiilor din realitate. Detalii se pot furniza la cerere;
1.2 Am implementat cu succes o serie de metode numerice de optimizare printre care si metode
bazate pe algoritmi genetici.
1.3 Am simulat cu succes in Matlab masina asincrona/inductie trifazata. In cadrul acestui proiect s-au
implementat transformarile Clarke si Park pentru obtinerea sistemului de doua axe sincron cu campul
invartitor. Se intentioneaza implementarea controlului vectorial pentru modelul masinii asincrone. Mai
multe detalii pot fi furnizate la cerere.
1.4 Am implementat modelul neliniar al motorului de curent continuu.
1.5 Am implementat cu succes un algoritm avansat de control numit GPC (Generalised Predictive
Control) ca munca facultativa cadrul unei discipline de la master;
1.6 Am modelat sistemul neliniar format de un pendul invers pe un carucior, si am implementat pentru
el un sistem de comanda cu reactie de la stare. Acesta este prezentat la sfarsitul unui capitol intr-o
carte de control: Ogata : Modern Control Engineering. Mai multe detalii exista;
1.7 Ca proiect facultativ la o disciplina de la UBB, Astronomie, am simulat aproximativ miscarea
sistemului Pamant-Luna in jurul Soarelui, in Matlab;
1.8 Am conceput in cadrul unui proiect la o disciplina de la UTC-N, Sisteme de Control Distribuit, un
program care simuland o retea Petri, programeaza rute pentru trenuri. Mai multe detalii exista;
1.9 In anul 2012 m-am inscris impreuna cu un coleg de facultate la un concurs organizat la Brasov de
catre Route66. Obiectivul concursului era crearea unui program care sa identifice semnele rutiere.
Datoria mea era sa extrag din imagine acele parti unde exista un semn rutier. In acest scop dupa
multa munca, am implemantat propriile retele neuronale feedforward cu algoritmul de invatare
backpropagation, deoarece am considerat ca in aceasta directie trebuie sa fie viitorul. Programul meu
insa avea ceva probleme deoarece pe langa semnele rutiere extragea si alte bucati de imagine,
colegul nu se descurca nici el prea bine cu clasificarea lor, au venit examene si colocvii la facultate si
nu am mai participat. Momentan consider ca asa numitele "Dynamical Bayesian Networks" sunt ceva
ce merita studiat. Exista si aici mai multe detalii.
5/1/16
Pagina 4 / 28
Curriculum vitae
2. Realizari practice;
2.1 Fiind aproape pasionat de electronica mi-am proiectat si implementat mai multe variante de
placute de dezvoltare pentru microcontrolerele de la Atmel, pe care le-am folosit in proiectele de mai
jos;
2.2 Am lucrat la proiectarea si implementarea unui driver sensorless pentru motoarele BLDC, modulul
cu microcontroler si semnalele de comanda, cat si puntea trifazata. Am reusit insa doar sa-l conduc in
bucla deschisa, nereusind sa-l fac sa se autopiloteze.
2.3 Am proiectat si implementat un variator de tensiune alternativa, cu triac;
2.4 Am construit in IDE-ul Qt folosind libbajul C++, un program care comunica pe portul serial. Apoi
folosind un circuit de level shifting de exemplu max232 am reusit sa comunic cu microcontrolerul
Atmega 48pa. Aceata a permis realizarea unui sistem de identificare. Mai multe detalii pot fi furnizate
la cerere.
ANEXE
▪ lista_lucrari.pdf
▪ Paper fractional_7.pdf
▪ Costandin_paper_2_rewd_.pdf
▪ CostandinMarius (2).pdf
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Pagina 5 / 28
Paşaport european al competenţelor
lista_lucrari.pdf
Lista documente anexate
1. A fractional Order PI Controller Design Method;
2. Limit Cycle Controller Design Method;
3. A convergence theorem and it’s application concerning Riemann’s Zeta function;
5/1/16
Pagina 6 / 28
Paper fractional_7.pdf
Fractional Order PI Controller Design Method
Costandin Marius Simion
Automation
Technical University of Cluj-Napoca
Cluj-Napoca, Romania
[email protected]
This paper describes an original method of tuning a fractional
order PI controller, along with a briefly mathematical introduction
of the needed notions. The resulted controller is tested on a
nonlinear model of a continuous current motor. The mathematical
tools described, are not sophisticated and can be used as well for
tuning an integer order controller.
I.
INTRODUCTION
Fractional calculus has become useful over the last 40 years
due to its applications in applied sciences. Among the
applications are: acoustic wave propagation in inhomogeneous
porous material, diffusive transport, fluid flow, dynamical
processes in self-similar structures, dynamics of earthquakes,
optics, geology, bioengineering, medicine, economics,
probability and statistics, astrophysics, chemical engineering,
physics, fluid mechanics, electromagnetic waves, nonlinear
control, signal processing, control of power electronics,
converters, neural networks, etc. [1]. Some researchers
consider this tool as being useful: see [2, 3] for applications in
physics, [4-8] for applications in electrical engineering , [912] for applications in control systems [9-12], [13], for
robotics etc.
In control engineering a major impact had the work of
Podlubny [11], proposing a generalization of the PID
controller, namely the PIDμ controller, involving an integrator
of order  and a differentiator of order μ. He demonstrated a
better response of this type of controller, in comparison with
the classical PID controller, when used for the control of
fractional order systems. The fractional order controller design
techniques are in general based on extensions of the classical
PID control theory. In [18]. In [19] are presented different
fractional controller design methods.
The classical PID controller can be considered as a
particular form of lead-lag compensation in the frequency
domain. Its transfer function can be expressed as:
K


C s   K P 1  i  K d s 
s


 K

C F( s )  K P 1  i  K d s   , ,   
 s

As can be seen by adding more parameters the number of
variables to tune increase, an so the performance of the
controller because one has the possibility to meet more
constrains.
This controller has five parameters therefore allowing up to
five design specifications, while classical PID has up to three.
In other words, fractional-order PID controllers have two
extra degrees of freedom to better adjust the dynamical
properties of a fractional order control system. It is essential to
study which specifications are more interesting for the case
studied and to add more requirements regarding robustness to
plant uncertainties, load disturbances, and high-frequency
noise.
Based on these references, the authors propose a new
fractional order PI controller design method, with case study
which highlights the advantages of the presented algorithm.
II.
The generalized form of the PID controller involves an
integrator of order λ and a differentiator of order μ where λ and
μ can non integer:
The squared modulus of a sum of let’s say two complex
numbers z1 and z2 can be expressed using only the modulus of
each individual complex number and the angle between them.
So the following formula holds:
z1  z 2
2
if
2
2
 z1  z 2  2  z1  z 2  cos(z1, z 2) (1)
one
considers
each
complex
z1  x1  jy1 , and z 2  x2  jy 2 , then
z1  z 2  ( x1  x2)  j ( y1  y 2)
number
(2)
and therefore
z1  z 2
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METHOD DESCRIPTION
A. Mathematical background
The presented method of tuning a fractional order PI
controller intensively uses the next two observations on
complex numbers:
Indeed,
(1)
(2)
2
2
2
 x1  x 2   y1  y 2
(3)
Pagina 7 / 28
After making the calculations one can obtain:
z1  z 2
2
be used. From the Figure 1 one can express the area of the
parallelogram using the following two formulas:
2
2
2
2
 x1  2 x1x 2  x 2  y1  2 y1y 2  y 2 (4)
While evaluating the squared modulus of each complex
number one can get:
2
2
2
z1  x1  y1
z2
2
(10)
A  z1  sinz1, z 2  z1  z 2
(11)
Where, h is the height of the parallelogram perpendicular
on the vector z1, leaving the tip of the vector z2.
(5)
2
2
 x2  y 2
A  h  z1
(6)
Complex plane and the two dimensional plane 
2
are the
2
same, so a complex number is also a vector in  , so
z1  x1, y1 and z 2  x2, y 2 .
In 
2
exists a scalar product, so for each
v1  x1, y1 ,
2
v2  x2, y 2 from  , v1 v2  x1x2  y1y 2 . It
is well known that
v1 v2  v1  v2  cos(v1, v2) .
So applying for the vectors z1 and z2 one obtains:
z1 z 2  x1x2  y1y 2  z1 z 2 cos(z1, z 2)
(7)
z1  z 2
2
2
 z1  z 2  2 z1 z 2
From the above figure the absolute angle between vector z1
and vector z1 + z2 is angle 2, and the absolute angle between
z2 and z1 is angle 1.
The height of the parallelogram h, can be expressed how
follows next:
h  z 2  sin angle1
Finally, from the above equations (4), (5), (6):
2
Figure 1
(12)
(8)
Finally, from equations (10), (11), (12):
From equation (8) and (7):
z1  z 2
2
2
2
 z1  z 2  2  z1  z 2  cos(z1, z 2)
(9)
So equation (1) is proved.
sin angle 2 
sin(angle 2) 
The second observation on complex numbers is that
information about the argument of the sum of two complex
numbers z1  x1  jy1 , and z 2  x2  jy 2 can be
written only in terms of the modulus of each individual
complex number and the angle between them. For proving
this, one can consider the simpler case in which one of the
numbers is positive real thus, it is on the real positive axis.
In this case, as in the above case, the complex numbers can
h
z1  z 2
z 2  sin(angle1)
z1  z 2
(13)
(14)
So, information about angle 2 is expressed in terms of angle
between and modulus of z1 and z2.
The above reasoning is valuable for this method of tuning a
fractional controller because the controller’s transfer function
is:
C ( s)  Kp  Ki  s
a
(15)
2
be seen as vectors in  , so their sum is another vector.
To obtain the resulting vector, the rule of parallelogram can
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Pagina 8 / 28
It can be seen that the controller is the sum of two complex
numbers: Kp , and Ki  s
a
, because
ln  j   ln    j 
s will be replaced by
j .
 j 
a
 j 
a
In the following lines the modulus and phase of each
controller’s components will be determined:
Kp  Kp
(16)
Kp  0
(18)
Before proceeding, the complex logarithm must be
remembered. For any given nonzero complex numbers,
z  x  jy and, w  w  e
iw
z
e  w  Lnw  z
e
x  jy
 w e
(18)
(19)
(20)
therefore,
x
e  w  x  ln  w  ;
jy
e
jw
 y  w  2k
(21)
(22)
After all, can be obtained:
Lnw  ln  w   j w  2k 
(23)
In equations (22) and (23) k is an integer number. If k = 0 then
it is said that the principal determination of the logarithm is
used, and this is denoted by writing the logarithm with small
letter “l” that is ln( ). In this paper the argument of a complex
number z is denoted by either arg z , or z .

The mathematical background section will end with the
determination of the modulus and the phase of the fractional
order integrator component of the controller.
s
a
e
By replacing
a  ln s
5/1/16
(24)
s with j , it is obtained:
ln  j   ln  j   jj
(26)
2
a  ln    j  a 

2
a ja
  e
2
(27)
(28)
From equation (28):
 j 
a

a
(29)
a

 j   a 
2
(30)
Therefore,
jw
x jy
jw
e e  w e
e
e

(25)
a
a
Ki   j   Ki  
(31)
a
a

 Ki   j    Ki   j 


(32)
a


 Ki   j    a 
2


(33)
B. Controller design
Using the mathematical background from above, the controller
is tuned by imposing conditions on the modulus and phase of
the controller, around a desired working frequency. The aim of
this paper was mainly to show that the modulus and the phase
of a transfer function can be determined also using another set
of equations than the classical ones.
Given the transfer function H s of the process, and the


controller’s transfer function, C s , like in (15) one can
obtain the open loop transfer function:
Hd s   C s   H s 
(34)
And therefore, the closed loop transfer function of the process,
considering a single loop and negative feedback, is given by
the next equation:
H 0 s  
Hd s 
1  Hd s 
(35)
Pagina 9 / 28
Let
k  0.98 and  0 be the desired working frequency, then
C  j0   
a first condition arises :
H 0  j0   k  0.98
(36)
Hd  j 0 
 k  0.98
1  Hd  j 0 
(37)
Hd
1  Hd
k
Hd
2
(39)
In this moment, one can choose Hd  j 0   
Hd
2
j   0 is:
2 
2
 k  1  Hd 


The process is a nonlinear model of the continuous current
motor, see [20]. The model is implemented in Simulink
Matlab. First the process is identified using a step response as
a first order function, therefore from now on the process
transfer function H s is available.

Applying sine function to equation (46) one gets:
 

sin C  j0   sin   H  j0 
2


(40)

2
,
sinC  j0   
 

 H  j 0 
 2

a
(42)
one obtains:
2
sin C  j 0  
2
2

1  k   H  j 0 



The known right hand term of equation (43) shall be noted
with the small Greek letter

2

k


a
 
Ki   0  sin  a  
 2
(50)
(51)

There are few observations on the above equations:
(44)
2
2

1  k   H  j 0 



The angle between the controller’s terms is a 

2
since one of the terms is a positive real number, and
represents the angle1 from Figure 1.
It now remains to impose Hd  j 0   

2
. From

The modulus of the integrator term of the controller
is, according to equation (31), Ki   0
equation (34) it is obtained:
5/1/16
a
 
Ki   0  sin  a  
 2
.
2
Hd  C  H
(49)
Using equation (14) with z1  Kp and z 2  Ki   j 0  ,
(43)
2
(48)
  sin 
Using equation (34) and (42):
k
(47)
The right term of equation (47) is known and for simplicity
shall be noted with the letter  , so equation (47) becomes:
(41)
2 
2
2
Hd  1  k   k


2
C  j 0  
(46)

2 2
 1  Hd  k
therefore the equation (40) written on
 H  j0 
Using a bode plot, for example, the phase and magnitude of
the transfer function in the desired working frequency is
determined, so  can be computed, and H s is now
known.
2 
2

 k  1  Hd  2  Hd  cosHd 


2
2
In this moment information about the controlled plant is
needed, therefore the process shall be presented.
(38)
Using the above mathematical observation:
Hd

a
;
(45)
Pagina 10 / 28

 
D  4    1  
 2
The modulus of the result of the sum of this two
terms of the controller is the modulus of the
controller and is equal, according to equation (43)
and (44),with  .
And one can obtain, if
Because this paper focuses on showing a new way of working
with transfer functions and not very necessary on designing
best controllers, the variable “ a ” ,the power of s will be
chosen as an arbitrary negative number in such a way that
a

2
(57)
  2:
a
  1
Kp   Ki  0  cos a     D (58)
 2 2
So here the design of the controller finishes.
belongs to the third quadrant. The motivation is below:

In equation (50) the output must be negative if
negative; (this is the simulated case).

It is desirable that the cosine function on a 

2

SIMULATIONS AND RESULTS
is
In the following lines numerical results shall be given:
to be
Let
H s  
also negative, for partially avoiding of getting
negative Kp as a solution of a future equation (58);
From equation (51)
Ki 
Ki is obtained:
 
(52)
a
 
 0  sin  a  

2218
s  30.79
(59)
be the identified transfer function of the process. The process
is a DC motor. The input is the armature voltage, and the
output is the rotational speed of the rotor.
The working frequency was chosen to be:
2
0  4.35 [rad/s]
Another observation is that it is not allowed a to approach
even integer numbers, since that would imply Ki to approach
infinity.
It remains to Kp to be determined. Using equation (43) and
(44) one can obtain:
2
2
C  j 0   
(53)
The phase and modulus of the transfer function
(60)
H s  at the
working frequency are  0.14 radians and  70 units which
correspond to  37 decibels. The parameter a was chosen to
3
. In this conditions   0.07 ,   0.99 ,
4
D  0.27 and the controller’s parameters are:
be 
Kp  0.2510;
Using equation (9):
2
a2
a  
2
Kp  Ki   0
 2 KpKi0 cos a     (54)
 2
Ki  0.0754;
(63)
The Simulink model is visible in the next figure:
From equation (51):
 
 
sin  a  
 2
Ki   0
a2
 Ki   0

a
 
 
sin  a  
 2
(55)
2
(56)
Equation (54) is seen as a equation with Kp being the
unknown variable, so after computing the determinant, using
the equations (55) and (56):
5/1/16
Figure 2.Simulink simulation .
Pagina 11 / 28
The controller’s model can be seen in Figure 3:
can be resolved using this kind of approach, like phase margin,
settling time and others.
REFERENCES
[1]
[2]
[3]
[4]
Figure 3.Simulink controller’s model.
[5]
The controller’s fractional transfer function was implemented
as a integer order transfer function using the function
crone1.m from NINTEGER tool.
The overall output of the system can be seen in figure
Figure 4:
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Figure 4.Motor’s speed.
III.
CONCLUSIONS AND FUTURE WORK
[16]
[17]
[18]
This paper has shown a way of tuning a fractional order PI
controller using simpler calculations, but a slightly different
approach.
[19]
G. Anasstasiou, 2011, “Advances on Fractional Inequalities”, Springer,
DOI 10.1007/978-1-4614-0703-4
Parada F. J. V., Tapia J. A. O. and Ramirez J. A., 2007, Effective
medium equations for fractional Fick’s law in porous media, Physica A,
373, 339–353.
Torvik P. J. and Bagley R. L., 1984, On the appearance of the fractional
derivative in the behavior of real materials, Transactions of the ASME,
51, 294–298.
Arena P., Caponetto R., Fortuna L. and Porto D., 2000, Nonlinear
Noninteger Order Circuits and Systems – An Introduction, World
Scientific, Singapore.
Bode H. W., 1949, Network Analysis and Feedback Amplifier Design,
Tung Hwa Book Company, Shanghai.
Carlson G. E. and Halijak C. A., 1964, Approximation of fractional
capacitors (1/s)1/n by a regular Newton process, IEEE Trans. on Circuit
Theory, 11, 210–213.
Nakagava M. and Sorimachi K., 1992, Basic characteristics of a
fractance device, IEICE Trans. fundamentals, E75-A, 1814–1818.
Westerlund S., 2002, Dead Matter Has Memory!, Causal Consulting,
Kalmar, Sweden.
Axtell M. and Bise E. M., 1990, Fractional calculus applications in
control systems, Proc. of the IEEE Nat. Aerospace and Electronics
Conf., New York, 563–566.
Oustaloup A., 1995, La Derivation Non Entiere: Theorie, Synthese et
Applications, Hermes, Paris.
Podlubny I., 1999, Fractional-order systems and PIλ Dμ -controllers,
IEEE Transactions on Automatic Control, 44, 208–213.
Podlubny I. 1999, Fractional Differential Equations, Mathematics in
Science and Engineering, volume 198. San Diego: Academic Press
Marcos da Graca, M., Duarte F. B. M. and Machado J. A. T., 2008,
Fractional dynamics in the trajectory control of redundant manipulators,
Communications in Nonlinear Science and Numerical Simulations, 13,
1836–1844.
Tseng C. C., 2007, Design of FIR and IIR fractional order Simpson
digital integrators, Signal Processing, 87, 1045–1057.
Vinagre B. M., Chen Y. Q. and Petr´aˇs I., 2003, Two direct Tustin
discretization methods for fractional-order differentiator/integrator, J.
Franklin Inst., 340, 349–362.
Oldham K. B. and Spanier J., 1974, The Fractional Calculus, Academic
Press, New York.
Magin R. L., 2006, Fractional Calculus in Bioengineering, Begell
House Publishers, Redding.
C. A. Monje, Yang Quan Chen, Blas M. Vinagre, Dingyü Xue, Vicente
Feliu, 2010, Fractional-order Systems and Controls, DOI 10.1007/9781-84996-335-0, Springer Verlag
Luca Zaccarian; DC motors: dynamic model and control techniques
A future development can be a way of translating classical
performances requirements in equivalent requirements which
5/1/16
Pagina 12 / 28
Costandin_paper_2_rewd_.pdf
Limit cycle based controller design
method
Faculty of Automation and Computer Science
Technical University of Cluj-Napoca
Cluj-Napoca, Romania
[email protected]
Abstract—This paper presents a method for calculating a
discontinuous controller, capable of stabilizing the process, and
capable of following exponential inputs with an error smaller
than the α percent of the input value. In addition sinusoidal
inputs can be followed with the above mentioned error given the
frequency of the input sine is smaller than a certain value. The
work refers to a second order process, but it can be generalized.
I.
INTRODUCTION
The idea of this paper is to replace a stable point with a
stable limit cycle, due to which the process value oscillates, but
with a small enough amplitude, around the desired value.
Therefore as controller, a discontinuous element will be
used to generate oscillations. This will result in an ever
oscillating command to the process.
The oscillating command, carry valuable information about
the needs of the process. If the oscillations are desired to
disappear the adaptive techniques are required, in which the
real process is fed in open loop with the mean value of the
oscillations generated by the proposed discontinuous controller
and a model of the process. The model of the process must be
updated. The adaptive part was not implemented here.
Overall a simple controller is obtain which stays stable,
follows step, sine and even exponential references. By
following exponential references, the process can follow any
order polynomial input.
There are some others methods in literature in which a relay
in used, but just for tuning a continuous controller, which
actually is supposed to control the process thereafter.
The difference between the proposed method and the
existing methods using a relay is that the proposed method
does not replace the relay with a PI or PID controller, but the
relay remains the controller. This paper presents a method for
„tuning‟ the relay, therefore the process model is required.
Other methods of control from literature in which an ever
oscillating command are present, are the methods based on
sliding modes.
5/1/16
II.
METHOD DESCRIPTION
A. Automation science background
First of all some notions about linear systems shall be
presented. The method that will be presented in the following
text will use a second order system, having the Laplace
transform:
𝐻(𝑠) =
𝑘∙𝜔 2
𝑠 2 +2∙𝜁∙𝑠+ 𝜔 2
(1)
In this transfer function by replacing
𝑠 =𝑗∙𝜔
(2)
becomes a complex number which has real and imaginary
part.
𝐻 𝑗∙𝜔 =
𝑘∙𝜔 𝑛 2
(3)
−𝜔 2 +2∙𝜁 ∙𝑗 ∙𝜔 + 𝜔 𝑛 2
𝐻 𝑗 ∙ 𝜔 = 𝑘 ∙ 𝜔𝑛 2 ∙
𝜔 𝑛 2 −𝜔 2 −2∙𝑗 ∙𝜁 ∙𝜔
(4)
(𝜔 𝑛 2 −𝜔 2 )2 +(4∙𝜁∙𝜔 )2
Therefore:
𝐻 𝑗 ∙ 𝜔 = 𝑘 ∙ 𝜔𝑛 2 ∙
2∙𝜁∙𝜔
𝜔 𝑛 2 −𝜔 2
(𝜔 𝑛 2 −𝜔 2 )2 +(4∙𝜁∙𝜔 )2
+ 𝑘 ∙ 𝜔𝑛 2 ∙ 𝑗 ∙
(5)
(𝜔 𝑛 2 −𝜔 2 )2 +(4∙𝜁 ∙𝜔 )2
𝜔 𝑛 2 −𝜔 2
(𝜔 𝑛 2 −𝜔 2 )2 +(4∙𝜁 ∙𝜔 )2
𝐼𝑚 𝐻 𝑗 ∙ 𝜔
= 𝑘 ∙ 𝜔𝑛 2 ∙
𝑅𝑒 𝐻 𝑗 ∙ 𝜔
= 𝑘 ∙ 𝜔𝑛 2 ∙ 𝑗 ∙
2∙𝜁∙𝜔
(𝜔 𝑛 2 −𝜔 2 )2 +(4∙𝜁 ∙𝜔 )2
(6)
(7)
Where ω varies between 0 and ∞.
If one will draw for each ω in the complex plane the values
from (6) and (7) it will obtain a curve where ω is the
parameter.
The resulted curve is known as the Nyquist plot. It shows
for each ω the magnitude and the phase of the complex number
which is 𝐻(𝑗 ∙ 𝜔).
Pagina 13 / 28
In order to draw the Nyquist plot several steps must be
made. These steps can be found in most of the System Theory
books [1].
In the following figure the Nyquist plot for the system from
[1] with
We shall define and obtain the description function for the
hysteresis relay. The nonlinear elements for which the
description function is defined have one input and one output
and the following characteristics:

𝑦 𝑡 = 𝑓(𝑢(𝑡))
where
𝑓
is
piecewise
monotonous and continuous univalent or
polyvalent function. 𝑢(𝑡) is the input of the
nonlinear element and 𝑦(𝑡) is the output of the
nonlinear element.

The nonlinearity is symmetrical with respect to the
origin of the plane 𝑢 𝑦

If 𝑢(𝑡)is periodic then 𝑦(𝑡) is periodic with the
same period.

The linear system which succeeds the nonlinear
element is a low pass filter steep enough around
the cutting frequency.
𝑘 =1,
𝜁 = 0.5
𝜔 = 1 can be seen:
Considering the above hypotheses being met and a sine
input to the system:
𝑢 𝑡 = A ∙ sin(𝜔 ∙ 𝑡)
(8)
Then the output of the nonlinear element, 𝑦(𝑡) can be
expanded in Fourier series :
Figure 1.
𝑦 𝑡 = 𝑐0 +
In the following paragraphs some approximative method to
analyze the nonlinear systems shall be given.
The method consists in harmonic linearization of the
nonlinear element. We shall consider for this paper just the
relay with hysteresis nonlinear element. The nonlinear element
shall be replaced with a linear approximation which can be
used in analysis of the element and the overall process where it
belongs to. The utility of this method consists in the possibility
of deciding the existence and the stability of the limit cycles.
For more details about this method see [2].
∞
𝑘=1(𝑐𝑘
∙ cos 𝑘 ∙ 𝜔 ∙ 𝑡 + 𝑠𝑘 ∙ sin(𝑘 ∙ 𝜔 ∙ 𝑡))
(9)
Due to the fact that the system meets the above
requirements, it is safe to consider that 𝑐0 = 0 and the higher
harmonics from the Fourier expansion are severe attenuated
therefore, the following identity holds:
𝑦 𝑡 ≅ 𝑐1 ∙ cos(𝜔 ∙ 𝑡) + 𝑠1 ∙ sin(𝜔 ∙ 𝑡)
(10)
From (10) one can obtain:
𝑦 𝑡 ≅ Y ∙ sin(𝜔 ∙ 𝑡 + 𝜑)
(11)
Where
𝑐1 2 + 𝑠1 2
𝑌=
𝜑=
(12)
𝑐
tan−1 ( 1 )
𝑠1
(13)
From the above equations one can see that the effects of the
nonlinear element on the input are:
𝑌
𝑐1 2 +𝑠1 2

Amplifies the signal with =

Shifts the phase of the input with 𝜑 = tan−1 ( 1 )
𝐴
𝐴
;
𝑐
𝑠1
It is therefore defined, the describing function of the
nonlinear element:
𝑁 𝐴 = 𝑁(𝐴) ∙ 𝑒 𝑗 ∙𝜑
(14)
From (12) (13) and (14) one can obtain:
Figure 2.
5/1/16
𝑁 𝐴 =
𝑐1 2 +𝑠1 2
𝐴
∙𝑒
𝑐
𝑗 ∙tan −1 ( 1 )
𝑠1
(15)
Pagina 14 / 28
With more calculations can be obtain:
𝑁 𝐴 =
𝑠1 +𝑗 ∙𝑐1
(16)
𝐴
In the next figure the curve corresponding to 𝑁𝑖 (𝐴) can be
seen.
For the relay in Figure 2 the describing function is
𝑁 𝐴 =
4∙𝑒∙𝑏
∙
𝜋∙𝐴2
𝐴 2
𝑒
−1−𝑗
The Loeb criterion states that if starting from the
intersection point and advancing on the Nyquist curve for
increasing ω, the curve corresponding to 𝑁𝑖 (𝐴) for increasing
𝐴 remains at left, then the limit cycle is stable, see [2].
(20)
The following figure shows a negative feedback system.
Figure 3.
If there is a limit cycle in the system presented in Figure 3
and some other conditions are met see [2], then the input can be
expressed like in the following equation:
𝑢 𝑡 ≅ 𝑢0 + 𝑢1 ∙ sin(𝜔 ∙ 𝑡)
Figure 4
The next figure shows a plot containing both, the Nyquist
plot and the curve corresponding to 𝑁𝑖 (𝐴) :
(21)
From equation (21) can be obtain that if the nonlinearity is
symmetrical with respect to the origin of the plane (as in Figure
2) and 𝑤 = 0 then the existence of a limit cycle is equivalent
with
𝐻 𝑗∙𝜔 =
−1
𝑁(𝑢1)
(22)
Which is known as the “equation of harmonic balance”.
We define
𝑁𝑖 𝐴 =
−1
𝑁(𝐴)
(23)
Therefore if the equation (22) takes place then in the system
𝜔
is a limit cycle with the frequency
and having the amplitude
2∙𝜋
𝑢1 .
It is necessary to remember that the information so far
about describing function and limit cycles can be found in [2].
There is a method of determining the stability of the limit
cycle determined by the above system known as the Loeb
method, which will not be proven here.
In order to determine the pulsation at which the limit cycle
occurs ant the amplitude of the oscillations at the input of the
nonlinear element one can draw the Nyquist plot having ω as
parameter and the describing function 𝑁𝑖 (𝐴) having 𝐴 as
parameter. At the point where if the curves intersect there is a
limit cycle with the corresponding frequency of oscillations
and the corresponding amplitude of the input to the nonlinear
element.
5/1/16
Figure 5.
From Figure 5 can be seen that there will always will be a
stable limit cycle in the system. This is a very important
observation for this work. This is explained as follows: if one
starts from the intersection of the curves seen in figure 4, and
advances on the Nyquist plot approaching origin (the sense for
which ω increases) then the curve corresponding to 𝑁𝑖 (𝐴) for
increasing 𝐴 remains at left. The conclusion comes from
Loeb‟s criterion.
B. Controller design
According with the above presented already well known
theory it is easily to see that no matter what the parameters of
Pagina 15 / 28
the system might be as long as the system is a second order
system the feedback loop is stable. This is the first observation
which makes the relay controller robust stable.
If the reference is 𝑤 = 0 then there is a limit cycle
oscillating at amplitude 𝑢1 , this being from (21), (22) and
Figure 5. It can be seen from figure 3 that the input to the
nonlinear element is actually the error. In short it is wanted that
the oscillations to be small.
So the idea is, knowing the process, to set the parameters of
the relay is such a way that the limit cycle creates oscillations
at the input of the nonlinear element (the relay) of a given
small amplitude.
It can be seen from Figure 4 that is 𝐴 = 𝑒 then the real part
is 0. On the other hand, if 𝐴 = ∞ then the real part equals −∞,
therefore must be a value of 𝐴 such that a limit cycle takes
place. The algorithm is as follows:

First decide the desired amplitude of the
oscillations, say 𝐴 = 0.02. This means that after
all the amplitude of the error is 0.02 which means
that the system is in steady state.

Decide the maximum command that the controller
can output, say 𝑏 = 100; This means that the
controller can output 100 or −100.

Find 𝑒 and 𝜔 such that there is a limit cycle.
𝐻 𝑗∙𝜔
= 0.02 ∙
𝜋
(31)
4∙𝑏
From equation (31) one can find 𝜔 using for example the
Bode plot for the magnitude. Knowing ω and the process then
𝐼𝑚 𝐻(𝑗 ∙ 𝜔)
can find 𝑒.
is also known and using equation (25) one
III.
SIMULATIONS RESULTS
This paragraph will show the implementations that were
done in Matlab Simulink. The parameters of the controller are:

𝑏 = 39;

𝑒 = 4 ∙ 10−4 ;
As an observation here: if the relay is to be implemented
digitally then is required that the sampling time to be
smaller then 𝑒.
With the above steps completed one will have a relay with
the parameters 𝑒 and 𝑏 which will produce a limit cycle
causing the error of the feedback loop to oscillate but with an
amplitude of 0.02 so it can be considered as being in steady
state.
Figure 6.
The following equations give an analytic solution for 𝑒 and
𝜔.
From equation (20) and (23) one can obtain:
−1
𝑁(𝐴)
= −
𝜋∙𝑒
4∙𝑏
𝐴 2
∙
𝑒
𝐼𝑚 𝐻(𝑗 ∙ 𝜔) = −
0.02 2
𝑒
(24)
(25)
4∙𝑏
2
𝐼𝑚 𝐻(𝑗 ∙𝜔 )
1
𝑒 = 0.02 ∙
𝑒 = 0.02 ∙
𝜋∙𝑒
𝑅𝑒 𝐻(𝑗 ∙𝜔 )
−1 =
𝑒 = 0.02 ∙
−1+𝑗
𝑅𝑒 𝐻 (𝑗 ∙𝜔 ) 2
+1
𝐼𝑚 𝐻 (𝑗 ∙𝜔 )
𝐼𝑚 𝐻(𝑗 ∙𝜔 ) 2
𝑅𝑒 𝐻(𝑗 ∙𝜔 ) 2 + 𝐼𝑚 𝐻(𝑗 ∙𝜔 ) 2
𝐼𝑚 𝐻(𝑗 ∙𝜔 )
𝐻 𝑗 ∙𝜔
(26)
(27)
(28)
(29)
From equation (25) and (29):
𝑒 = 0.02 ∙
Therefore,
5/1/16
𝜋∙𝑒
4∙𝑏
∙
1
𝐻 𝑗 ∙𝜔
(30)
Figure 7.
IV.
CONCLUSIONS AND FUTURE WORK
In conclusion we can say that the proposed controller is
simple robust stable and reliable, following well the references
even if the theory was developed around the reference 𝑤 = 0;
In the future some ways must be found to avoid the
oscillating command, and the more profound study of limit
cycles, along with the necessary elimination of the oscillating
Pagina 16 / 28
respond to non differentiable references, such as a step
function.
[3]
[4]
REFERENCES
[1]
[2]
5/1/16
Teoria Sistemelor. Realizări de stare, Petru Dobra, Ed. Mediamira, ClujNapoca, 2002.
Sliding Mode Control in Electromechanical Systems, Vadim Utkin
Jurgen Guldner Jingxin Shi
Sisteme Neliniare si Stohastice, Petru Dobra
Modern control engineering (3rd edn) by Katsuhico Ogata, PrenticeHall, Upper Saddle River, NJ, 1997
Pagina 17 / 28
CostandinMarius (2).pdf
A convergence theorem and it’s aplication
concerning Riemann’s Zeta function
Costandin Marius
10-05-2015
Abstract
The present paper presents the Euler-McLaurin integral formula along
with it’s demonstration and a new convergence criterion for a certain type
of complex number series. An asymptotic development for Riemann’s
Zeta function is derived, using the Euler-McLaurin integral formula and
the newly presented convergence criterion.
i
5/1/16
Pagina 18 / 28
1
1.1
Euler-MacLaurin formula
Bernoulli polynomials
Benolulli polynomials Bn (x) for n ∈ {0, 1, 2, ...} are defined reccurently
with B0 (x) = 1 and Bn (x) satisfing
Bn0 (x) = nBn−1 (x);
and
Z
(1)
1
Bn (x)dx = 0;
(2)
0
for all n ∈ {1, 2, 3, ...}.
0
Exemple 1.1.
R 1 For n = 1 one can compute B1 (x) = 1 ⇒ B1 (x) = x + c.
From here 0 (x + c)dx = 21 + c = 0 ⇒ c = − 12 , so B1 (x) = x − 12 .
Lemma 1.2. For n > 1 the Bernoulli polynomials satisfy Bn (1) = Bn (0).
R0
Proof. Let n be a natural number n > 1. Then from (2) one has 1 nBn−1 (x)dx =
R0 0
0 so 1 Bn (x)dx = 0. Therefore Bn (1) − Bn (0) = 0.
Definition 1.3 (Bernoulli number). For all n ∈ {0, 1, 2, ...} the Bernoulli
number Bn is defined as Bn = Bn (1).
Definition 1.4 (Periodic Bernoulli polynomials). For all n ∈ {0, 1, 2, ...}
the periodic Bernoulli polynomial Pn (x) is defined as Pn (x) = Bn (x−[x]),
where [x] denotes the integer part of x not greater then x.
1.2
The Euler-MacLaurin formula
The following theorem is a particular case of Euler-McLaurin integral formula, for functions defined on positive real semiaxis and infinitly derivable.
Theorem 1.5 (Euler-McLaurin integral formula). Let f ∈ C ∞ [0, ∞),
then the following relation in true, for n, p ∈ {1, 2, ...}:
n
X
n
Z
f (k) =
f (x)dx +
0
k=1
p
X
f (n) − f (0)
2
i
Bk h (k−1)
f
(n) − f (k−1) (0)
k!
k=2
Z
(−1)(p+1) n (p)
+
f (x)Pp (x)dx
p!
0
+
(−1)k
Proof. The proof follows somehow [1].Let k ∈ N. Then
Z k+1
Z k+1
f (x)dx =
f (x)P0 (x)dx
k
(3)
(4)
k
and using equation (1) for P1 (x), the following relation is true
Z
Z k+1
1 k+1
f (x)P10 (x)dx
f (x)P0 (x)dx =
1 k
k
(5)
1
5/1/16
Pagina 19 / 28
Integrating by parts one obtains the following
Z k+1
Z k+1
f (x)dx = f (x)P1 (x)|k+1
−
f 0 (x)P1 (x)dx
k
k
(6)
k
k+1
Z
f (x)dx = f (k + 1)P1 (k + 1) − f (k)P1 (k)−
k
k+1
Z
−
f 0 (x)P1 (x)dx
(7)
k
Z
k+1
f (x)dx = f (k + 1)B1 (1) − f (k)B1 (0)−
k
k+1
Z
−
f 0 (x)P1 (x)dx
(8)
k
Using the above derivation, the integral from 0 to n can be expressed in
the following way:
n
Z
f (x)dx =
0
n−1
X Z k+1
=
n−1
X
k=0
n
Z
−
f (x)dx
k
k=0
[f (k + 1)B1 (1) − f (k)B1 (0)] −
f 0 (x)P1 (x)dx;
(9)
0
Therefore
Z
n
f (x)dx =
0
n−1
X
k=0
n
Z
−
[f (k + 1) + f (k)] B1 (1)−
f 0 (x)P1 (x)dx
(10)
0
The expresion (10) can be further modified:
n
Z
f (x)dx = [f (0) + f (n)] B1 (1) + 2
0
n−1
X
f (k)B1 (1)−
k=1
n
Z
−
f 0 (x)P1 (x)dx
(11)
0
n
Z
f (x)dx = [f (0) − f (n)] B1 (1) + 2
0
n
X
f (k)B1 (1)−
k=1
Z
−
n
f 0 (x)P1 (x)dx
(12)
0
2
5/1/16
Pagina 20 / 28
n
X
2B1 (1)
n
Z
f (x)dx + [f (n) − f (0)] B1 (1)+
f (k) =
0
k=1
Z
+
n
f 0 (x)P1 (x)dx
(13)
0
Using the above Bernoulli number definition, Equation (13) is rewritten
as
Z n
n
X
[f (n) − f (0)]
f (k) =
f (x)dx +
+
2
0
k=1
Z n
+
f 0 (x)P1 (x)dx
(14)
0
Let us evaluate
R k+1
k
Z
k+1
f 0 (x)P1 (x)dx :
f 0 (x)P1 (x)dx =
k+1
Z
f 0 (x)
k
k
P20 (x)
dx
2
(15)
because P20 (x) = B20 (x − [x]) = 2B1 (x − [x]) = 2P1 (x). Integrating again
by parts one can obtain:
k+1 Z k+1
Z k+1
P2 (x) P2 (x)
−
dx
(16)
f 00 (x)
f 0 (x)P1 (x)dx = f 0 (x)
2 k
2
k
k
k+1
Z
k
B2 (1)
f 0 (x)P1 (x)dx = f 0 (k + 1) − f 0 (k)
−
2
Z k+1
1
−
f 00 (x)P2 (x)dx
2 k
(17)
Hence again
Z
n
f 0 (x)P1 (x)dx =
0
n−1
X Z k+1
k=0
f 0 (x)P1 (x)dx
k
n−1
B1 (1) X 0
f (k + 1) − f 0 (k) −
2
k=0
Z
1 n 00
−
f (x)P2 (x)dx
2 0
=
n
Z
f 0 (x)P1 (x)dx =
0
1
B2 0
f (n) − f 0 (0) −
2
2
n
Z
f 00 (x)P2 (x)dx
(18)
(19)
0
Replacing Equation (19) in Equation (14) one obtains
n
X
k=1
n
Z
f (n) − f (0)
+
2
Z n
1
B2 0
f (n) − f 0 (0) −
+
f 00 (x)P2 (x)dx
2
2 0
f (k) =
f (x)dx +
0
(20)
3
5/1/16
Pagina 21 / 28
R k+1
Let us now evaluate k f 00 (x)P2 (x)dx:
Z k+1
Z k+1
P 0 (x)
f 00 (x)P2 (x)dx =
f 00 (x) 3
dx
3
k
k
k+1 Z
P3 (x) = f 00 (x)
−
3 k
k+1
f 000 (x)
k
P3 (x)
dx
3
B3 00
f (k + 1) − f 00 (k)] −
3
Z
1 k+1 000
f (x)P3 (x)dx
−
3 k
=
Therefore
Z n
Z
1 n 000
B3 00
f 00 (x)P2 (x)dx =
f (x)P3 (x)dx
f (n) − f 00 (0) −
3
3 0
0
n
X
(21)
(22)
n
Z
f (n) − f (0)
B2 0
+
f (n) − f 0 (0) −
2
2
Z
1 n 000
1 B3 00
f (n) − f 00 (0) −
f (x)P3 (x)dx
(23)
−
2 3
3 0
f (k) =
f (x)dx +
0
k=1
The reader can see now that the process can be repeated, so after p steps
the following relation holds:
Z n
n
X
f (n) − f (0)
f (k) =
f (x)dx +
2
0
k=1
p
X
i
Bk h (k−1)
f
(n) − f (k−1) (0)
k!
k=2
Z
(−1)(p+1) n (p)
+
f (x)Pp (x)dx
p!
0
+
(−1)k
(24)
which ends the demostration.
1
Then
Exemple 1.6. Let f (x) = (1+x)
s with x ∈ R+ and s > 1.
Rn
Qk−1
(n+1)1−s
1
1
f (x)dx =
− 1−s and f (k) (x) = (−1)k p=0
(s + p) (1+x)
k+s .
1−s
0
Applying the Theorem 1.5 one obtains:
n
X
k=1
(n + 1)−s+1
(1 + n)−s − 1
1
1
=
−
+
(1 + k)s
−s + 1
−s + 1
2
k−2
m
X Bk Y
1
−1
−
(s + p)
s+k−1
k! p=0
(n + 1)
k=2
−
1
m!
n m−1
Y
Z
0
(s + p)
p=0
1
Pm (x)dx
(1 + x)s+m
(25)
The last term in Equation(25) is the remainder term, it shall be denoted
Rm,n , where m denotes how many derivatives are considered and n is the
upper limit of the integral or the sum.
4
5/1/16
Pagina 22 / 28
1.3
A convergence criterion
In this subsection a convergence criterion for some series is enounced and
an original proof is given. This convergence criterion resembles an already
known theorem due to Cauchy and MacLaurin, but the reader will notice
differences in demonstration and in formulation of it. First the known
theorem, see [2]
Theorem 1.7 (Cauchy, MacLaurin). If f(x) is positive, continuous, and
tends monototonically to 0, then an Euler constant γf , which is defined
below, exists
!
Z n
i=n
X
γf = lim
f (i) −
f (x)dx
(26)
n→∞
1
i=1
Proof. The theorem and the proof follows closely the presentation
R n from
[2]. The continuity of f guarantees the existence of the integral 1 f (x)dx
for n ∈ 1, 2, .... Since f is decreasing, the maximum and minimum of f
over a closed interval is known:
inf
f (x) = f (k + 1)
(27)
f (x) = f (k)
(28)
x∈[k,k+1]
sup
x∈[k,k+1]
therefore the following inequatity holds:
Z k+1
f (k + 1) ≤
f (x)dx ≤ f (k)
(29)
k
and summing form k = 1 to n − 1, one obtains
n
X
Pn
k=1
f (x)dx ≤
1
k=2
Substracting
n
Z
f (k) ≤
n−1
X
f (k)
(30)
k=1
f (k) from both sides in Equation (30)
n
Z
−f (1) ≤
1
n
X
f (k) ≤ −f (n)
(31)
f (x)dx ≥ f (n) ≥ 0
(32)
f (x)dx −
k=1
and after multiplying with −1
f (1) ≥
n
X
1
k=1
The sequence sn =
Pn
k=1
n
Z
f (k) −
f (k) −
Rn
1
f (x)dx is therefore bounded and
Z
n+1
sn+1 − sn = f (n + 1) −
f (x)dx ≤ 0
(33)
n
monotonically decreasing, so it has a limit.
The almost novel convergence criterion this paper presents is enounced
below:
5
5/1/16
Pagina 23 / 28
Theorem 1.8. Let f : D(D ⊆ R) → R be a function
two times
Pnat least
00
derivable with |f 00 | monotonically decreasing P
with
k=1 |f (k)| convern
0
gent for n → ∞. Then the sequence un =
k=1 f (k) − f (n) is also
convergent.
Proof. Let > 0, we shall prove ∃n ∈ N such that ∀n, m > n one
has |un − um | < meaning that (un ) is a Cauchy sequence. Because
R is a complete space, that will make (un ) convergent. Let us evaluate
|un − um |, presuming n > m:
n
m
X
X
|un − um | = f 0 (k) − f (n) −
f 0 (k) + f (m)
k=1
k=1
n
n
X
X
=
f 0 (k) −
(f (k) − f (k − 1))
(34)
k=m+1
k=m+1
Using the Lagrange’s mean value theorem ∃ck such that f (k) − f (k − 1) =
f 0 (ck ) (k − (k − 1)) = f 0 (ck ), hence
n
X
|un − um | = f 0 (k) − f 0 (ck ) (35)
k=m+1
where ck ∈ (k − 1, k). Using again the Lagrange mean value theorem
∃dk ∈ (ck , k) such that f 0 (k) − f 0 (ck ) = f 00 (dk ) (k − ck ).
|un − um | ≤
n
X
0
f (k) − f 0 (ck )
k=m+1
≤
n
n
X
X
00
00
f (dk ) ≤
f (k − 1)
k=m+1
(36)
k=m+1
since |f 00 | is monotonically decreasing. The above sum is the rest of a
convergent series. This ends the demonstration.
Observation 1.9. The Theorem 1.7 give similar results with Theorem1.8,
if instead of f one considers f 0 .
Observation 1.10. Theorem 1.7 asks for the function to be positive,
monotonically decresing to zero, whereas Theorem
for the second
P 1.8 asks
00
derivative to be monotonically decreasing and n
k=1 |f (k)| to be convergent which implies it’s convergence to zero. Note that it not necesary to
be positive.
The Theorem 1.8 can be genreralised for the following case:
Theorem 1.11. Let f : R → C P
be a double differentiable function with
n
00
|f 00 | monotonically
and
k=1 |f (k)| is a convergent real series.
Pn decreasing
0
Then un =
f
(k)
−
f
(n)
is
a
convergent
sequence.
k=1
Proof. Let > 0, we shall prove ∃n ∈ N such that ∀n, m > n one
has |un − um | < meaning that (un ) is a Cauchy sequence. Because
6
5/1/16
Pagina 24 / 28
C is a complete space, that will make (un ) convergent. Let us evaluate
|un − um |, presuming n > m:
n
m
X
X
0
0
|un − um | = f (k) − f (n) −
f (k) + f (m)
k=1
k=1
n
n
X
X
=
f 0 (k) −
(f (k) − f (k − 1))
k=m+1
k=m+1
!
n
n
X 0
X
≤ <
f (k) −
(f (k) − f (k − 1)) +
k=m+1
k=m+1
!
n
n
X
X
+ =
f 0 (k) −
(f (k) − f (k − 1)) (37)
k=m+1
k=m+1
Using again Lagrange’s mean value theorem for real and imaginary parts,
independently, will result in the existence of rk and ik such that < (f (k) − f (k − 1)) =
< (f 0 (rk )) and = (f (k) − f (k − 1)) = = (f 0 (ik )), therefore
|un − um | ≤
n
X
n
X
|< f 0 (k) − f 0 (rk ) | +
|= f 0 (k) − f 0 (ik ) |
k=m+1
k=m+1
n
n
X
X
<(f 0 )(k) − <(f 0 )(rk ) +
=(f 0 )(k) − =(f 0 )(ik )
≤
k=m+1
k=m+1
(38)
where rk and ik are in the interval (k − 1, k). Using again the mean value
theorem
|un − um | ≤
n
n
X
X
<(f 00 )(ck ) +
=(f 00 )(dk )
k=m+1
k=m+1
n
n
X
X
00
00
f (ck ) +
f (dk )
≤
k=m+1
k=m+1
n
n
X
X
00
00
f (k − 1) +
f (k − 1)
≤
k=m+1
(39)
k=m+1
The above sums are converging to zero, being the rests of convergent
series.
Exemple 1.12. Let us apply Theorem 1.11 for the function of real vari1
able x and complex values, f (x) = 1−s
(1 + x)1−s , for x ∈ R+ and
1
s ∈ C with < (s) > 0. The reader can verify that f 0 (x) = (1+x)
s and
−s
,
therefore
f
satisfies
the
condition
in
Theorem1.11,
f 00 (x) = (1+x)
s+1
P
n
1−s
1
1
hence the sequence Zn (s) =
is converk=1 (1+k)s − 1−s (1 + n)
1
gent. For s ∈ C with < (s) > 1 one has limn→∞ 1−s
(1 + n)1−s = 0, hence
limn→∞ Zn (s) = ζ(s) − 1 punctually. It is important to mention that the
convergence is uniform if s ∈ K with K being a compact subset of the
7
5/1/16
Pagina 25 / 28
right complex semiplane. Indeed from Equation (39)
n
n
X
X
00
00
f (k − 1) +
f (k − 1)
|Zn (s) − Zm (s)| ≤
k=m+1
with f 00 (x) =
−s
(1+x)s+1
(40)
k=m+1
so
|Zn (s) − Zm (s)| ≤ 2 |s|
n
X
1 ks+1 (41)
k=m+1
Because s ∈ Kand K is compact,
∃M ∈ R+ with |s| < M ∀s ∈ K, and
1 1 ≤ s +1 ∀s ∈ K, therefore ∀s ∈ K
∃s0 ∈ K such that s+1
k
k 0
|Zn (s) − Zm (s)| ≤ 2M
n
X
1 ks0 +1 (42)
k=m+1
hence Zn converges uniformly in K. Moreover Zn (s) is an holomorfic
function which converges unifromly on every compact subset of the right
complex plane, so using Weierstrass’s theorem, it’s limit is a holomorfic
function, which has the same values with ζ(s) − 1 on the interval (1, ∞).
Using the holomofic functions zeros theorem one obtains that the two
function are identical for s ∈ C with < (s) > 1. But ∃ limn→∞ Zn (s) =
Z(s) for s ∈ (C) with < (s) > 0 and s 6= 1, so 1 + Z(s) ≡ ζ(s), being it’s
analytical continuation on s ∈ C with < (s) > 0 and s 6= 1. Therefore
!
n
X
1
1
1−s
−
(1
+
n)
= ζ(s) − 1
(43)
lim
n→∞
(k + 1)s
1−s
k=1
n+1
X
lim
n→∞
k=2
lim
n→∞
1
1
−
(1 + n)1−s
(k)s
1−s
n+1
X
k=1
!
1
1
(1 + n)1−s
−
ks
1−s
= ζ(s) − 1
(44)
!
= ζ(s)
(45)
Exemple 1.13. Let
Theorem 1.11 for f (x) P
= ln(x). Because
P us apply
n
1
1
|ln00 (x)| = x12 and n
k=1 k2 is convergent, follows that
k=1 k − ln(n) is
also convergent. It’s limit is γ, the Euler-Mascheroni constant.
2
Asymptotic expansion for ζ(s)
An asymptotic expansion for ζ(s) is given below:
Theorem 2.1. For s > 1 the following relation holds:
ζ(s) =
k−2
m
X
1
Bk Y
1
−
+
(s + p) − Rm,∞ (s)
2
1−s
k! p=0
(46)
k=2
if ∃ Rm,∞ (s) = limn→∞ Rm,n (s)
8
5/1/16
Pagina 26 / 28
Proof. Using Equation (25) one has:
n
X
k=1
(n + 1)−s+1
(1 + n)−s − 1
1
1
=
−
+
(1 + k)s
−s + 1
−s + 1
2
k−2
m
X Bk Y
1
−
(s + p)
−1
s+k−1
k! p=0
(n + 1)
k=2
−
1
m!
n m−1
Y
Z
0
(s + p)
p=0
1
Pm (x)dx
(1 + x)s+m
(47)
From Equation(43) one has:
n
X
lim
n→∞
1
1
(1 + n)1−s
−
(k + 1)s
1−s
k=1
!
= ζ(s) − 1
(48)
hence
n
X
k=1
1
1
= Zn (s) +
(1 + n)1−s
(k + 1)s
−s + 1
Using both equations one can obtain, denoting Rm,n =
p) (1+x)1s+m Pm (x)dx
Zn (s) +
1
m!
(49)
R n Qm−1
0
p=0
(s +
(n + 1)−s+1
(1 + n)−s − 1
1
1
(1 + n)1−s =
−
+
−s + 1
−s + 1
−s + 1
2
k−2
m
X
Bk Y
1
−
−
1
(s + p)
k! p=0
(n + 1)s+k−1
k=2
− Rm,n
(50)
therefore
(1 + n)−s − 1
1
+
−s + 1
2
k−2
m
X
Bk Y
1
−
(s + p)
−1
s+k−1
k! p=0
(n + 1)
Zn (s) = −
k=2
− Rm,n
(51)
Letting n → ∞ in Equation (51), because ∃ limn→∞ Zn (s) = ζ(s) − 1 and
∃ limn→∞ Rm,n (s) = Rm,∞ (s) it follows that ∃
k−2
m
X
1
Bk Y
−1
(s + p)
s+k−1
n→∞
k! p=0
(n + 1)
lim
k=2
k−2
m
X
Bk Y
1
=
−1
(s + p) lim
n→∞ (n + 1)s+k−1
k! p=0
k=2
=−
k−2
m
X
Bk Y
(s + p)
k! p=0
(52)
k=2
9
5/1/16
Pagina 27 / 28
and
ζ(s) − 1 = lim Zn (s)
n→∞
=−
k−2
m
1
1 X Bk Y
− +
(s + p) − Rm,∞ (s)
−s + 1
2
k! p=0
(53)
k=2
therefore
ζ(s) =
k−2
m
X
1
Bk Y
1
−
+
(s + p) − Rm,∞ (s)
2
−s + 1
k! p=0
(54)
k=2
References
[1] Tom M. Apostol An Elementary View of Euler’s Summation Formula
American Mathematical Monthly Vol. 106, No. 5(May 1999), pp.409418
[2] Victor Kac, Kuat Yessenov 18.704 Seminar in Algebra and Number
Theory Fall 2005.
10
5/1/16
Pagina 28 / 28

Costandin Marius-Simion Asistent universitar

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