Modifica di System logic (sezione)

==== Mathematical formalism in 'Systems Theory'====
The "systems theory" studies oriented systems, in which it becomes possible to classify the quantities of interest into two categories:

*quantities that vary over time independently from the others ('''inputs''')
*quantities whose evolution over time is to be studied, depending on the inputs, called '''outputs'''.

A real system can have multiple inputs and multiple outputs. In particular, we indicate with:

* <math>u(t)= (u_1(t),..., u_r(t))</math>the vector of the inputs at time <math>{t}</math>
*<math>y(t)= (y_1(t),..., u_m(t))</math>the vector of the output at time <math>{t}</math>

It is also generally defined as the state vector of the system in a generic instant <math>\tilde{t}</math> the information instantly <math>\tilde{t}</math> necessary to uniquely determine the output <math>y(t)</math> for each <math>t\geq{\displaystyle {\tilde {t}}}</math> once the entrance has been assigned <math>u(t), </math><math>t\geq{\displaystyle {\tilde {t}}}</math>.

We denote the state vector, whose components are defined as state variables, with the notation <math>x(t)= (x_1(t),..., x_n(t))</math> .

The inputs act on the state of the system and modify its characteristics at a given moment in time; these changes are recorded by the state variables. The values of the system outputs, usually the only measurable variables, in turn depend on the system state variables and the inputs.

The input, status and output quantities are functions of the time variable.

This takes values in an ordered subset <math>T \subseteq \R</math>, which can be continuous or discrete. In the following discussion we will consider a discrete subset of times:<math>T = \{t_0,..., t_s\}</math>

Therefore, given a set of times <math>T = \{t_0,..., t_s\}</math>, we can formally define a system as the pair of equations

<math>x(t_{k+1})=f\bigl(x(t_k), u(t_k), t_k\bigr)
</math>

<math>y(t_k)=g\bigl(x(t_k), u(t_k), t_k\bigr)
</math>

with <math>x(t_0)=x_0
</math>, where <math>f
</math> is called generating function e <math>g
</math> is called the output transformation.

In the field of biosignals, the (<math>g</math>) models are used to analyze EEG and vibration systems in vehicles, human hearing systems and vascular systems, and so on. While much is still unknown about the physiological mechanism or pattern of internal changes in the tested system, the output transfer or transformation function <math>g</math> in our context allows us to reconstruct a wave function by interpolating the points detected by the instrument which has its own particular sampling frequency. This <math>g</math> function, for our purposes, is a reconstruction of a wave function on which to search for latencies, amplitudes and integral areas and make the necessary conclusions,<ref>{{cita libro 
 | autore = Haebeom L
 | autore2 = Hyunho K
 | autore3 = Jungkuk K
 | autore4 = Hwan-Sup O
 | autore5 = Young-Jae P
 | autore6 = Young-Bae P
 | titolo = Feasibility study of transfer function model on electrocardiogram change caused by acupuncture
 | url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5299648/pdf/12906_2017_Article_1615.pdf
 | volume = 
 | opera = BMC Complement Altern Med
 | anno = 2017
 | editore = 
 | città = 
 | ISBN = 
 | DOI = 10.1186/s12906-017-1615-5
 | PMID = 28178964
 | PMCID = PMC5299648
 | oaf = yes<!-- qualsiasi valore -->
 | LCCN = 
 | OCLC = 
 }}</ref><ref>{{cita libro 
 | autore = Smith RJ
 | autore2 = Kamali G
 | autore3 = Hays M
 | autore4 = Coogan C
 | autore5 = Crone NE
 | autore6 = Kang JY
 | autore7 = Sarma SV
 | titolo = Transfer Function Models for the Localization of Seizure Onset Zone From Cortico-Cortical Evoked Potentials
 | url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7758451/pdf/fneur-11-579961.pdf
 | volume = 
 | opera = Front Neurol
 | anno = 2020
 | editore = Frontiers in Neurology
 | città = 
 | ISBN = 
 | DOI = 10.3389/fneur.2020.579961
 | PMID = 33362689
 | PMCID = PMC7758451
 | oaf = y<!-- qualsiasi valore -->
 | LCCN = 
 | OCLC = 
 }}</ref> and, obviously, by retesting the system in subsequent epochs, the integrity of the system itself can be compared. 

In the engineering field, various mathematical modeling of a system are possible, depending on whether or not they explicitly consider the state variables.
[[File:Finite Elements - electric field within the intracranial brain tissue - FEM.jpg|thumb|center|'''Figure 5:''' A. Positioning of the electrodes for the delivery of the electrical stimulus. B. Representation of the electric field within the brain structure. C. Localization of the induced electric field at the level of the trigeminal roots ]]