UNIVERSITATEA „POLITEHNICA” din BUCURE ŞTI FACULTATEA FACULTATEA DE INGINERIE AEROSPAŢIALĂ ECHIPAMENTE ȘI INSTALAȚII DE AVIAȚIE
Proiect de curs la disciplina: Comanda și filtrarea optimală
Aspecte privind utilizarea filtrului optimal op timal Kalman
Profesor:
profesor univ. dr.ing dr.ing Adrian-Mihail Adrian-Mihail Stoica Masterand: Ing. Mihai Adrian
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CUPRINS CAPITOLUL CAPITOLUL I ........................... .......................... .......................... .......................... .......................... ..... 3 Introduce Introducere re în metodolo metodologi giaa filtrulu filtruluii Kalman Kalman ................................ .. .............................. ........................... .......................... ....... 3 1 .Introduce .Introducere re ............................. .......................... ........................... .......................... .......................... 3 1.1 Definirea Definirea filtrului filtrului Kalman ................................ ............................... .......................... ................ 3 1.2 Consideraţii privind algoritmii de filtrare adaptivă ........................... .......................... ................ 4 1.2.1 Algoritmul celor mai mici pătrate generalizate generalizate ............................... .......................... ........... 5 1.2.2 Algoritmul Algoritmul Celor Mai Mici Ptrate........................... .......................... .......................... ........ 5 -locked loop (PLL) ........................................ .......................... . 6 1.3 Sistemul in buclă închisă, phase -locked CAPITOLUL CAPITOLUL II............................... .......................... .......................... .......................... .......................... 6 Sisteme Sisteme dinamice dinamice lineare .............................. .......................... .......................... .......................... ............. 6 2.1 Sisteme dinamice dinamice lineare continue continue ............................ .......................... .......................... ................. 6 2.1.1 Modelul intrare – ieşire al sistemelor lineare dinamice continue .............................................. 6 ................................................... ............ 8 2.1.2 Traiectoria de stare şi matricea de tranziţie a stărilor ................................................... 2.2 Sisteme dinamice dinamice liniare discrete .............................. .......................... .......................... ................. 9 2.2.1 Traiectoria de stare şi matricea de tranziţie a stărilor în cazul discret ....................................... 9 2.3 Controlabilitatea Controlabilitatea şi observabilitate sistemelor dinamice lineare .............................................. .................................................... ...... 10 10 2.3.1 Control Controlabili abilitatea tatea sistemel sistemelor or lineare lineare discrete discrete ........................ .......................... ........................ 10 2.3.2 Controlabili Controlabilitatea tatea sistemelor sistemelor lini liniare are continue continue ............................ .......................... ................... 11 2.3.3 Observabili Observabilitate tateaa sisteme sistemelor lor lini liniare are discrete discrete .................................... .......... .......................... ............................ ........... 11 CAPITOLUL III.................................................................................................................................... 12 Procese aleatoare şi sisteme stohastice ................................................................................................... 12 3.1 Introducere Introducere în procese procese aleatoare........................... ........................... .......................... ................... 12 Definiţia axiomatica a probabilităţii: probabilităţii: .............................................................................................. 13 Definiţia probabilităţii ca frecventa relativa: ............................. .......................... .......................... . 13 Funcţia masei de probabilitate ....................................................................................................... 14 Funcţia de probabilitatea a densităţii densităţii ........................... .......................... .......................... ............... 14 3.2 Proprietăţile statistice ale variabilelor aleatoare si ale proceselor aleatoare ............................... .... 17 Momente....................................................................................................................................... 17 Funcţia variabilelor variabilelor aleatoare....................................................................................................... 18 3.3 Proprietăţi statistice ale proceselor aleatoare ................................................................................ 18 Procese Procese aleatoare aleatoare (PA) ................................................................ ................................... ............................. ............................ ...................... 18 Modele Modele de sisteme sisteme lineare lineare ale proceselor proceselor aleatoare aleatoare ..................................................................... ........................................... .......................... ... 18 3.4 Corelaţia, covarianţa şi independenţa ........................................................................................... 19 CAPITOLUL CAPITOLUL IV ................................................. .......................... .......................... ........................... ... 20 Filtre Filt re Kalman Kalman .................................................. .......................... .......................... .......................... ........ 20 4.1 Metoda Metoda filtrulu filtruluii Kalman Kalman discret............................... discret............................... .......................... .......................... .......... 20 20 Estimarea Estimarea procesului procesului ............................ .......................... .......................... .......................... ........... 21 Calculele de bază ale filt rului ........................................................................................................ 21 Originil Originilee filtrului filtrului ............................. .......................... .......................... .......................... ................ 23 Algoritm Algoritmul ul filtrului filtrului KalmanKalman- discret discret ................................ ............................... .......................... ..... 23 Parametrii filtrului şi ajustarea. ............................... .......................... .......................... ................... 24 4.2 Filtrul Filtrul Kalman extins extins (EKF) ............................ .......................... .......................... ........................ 26
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CAPITOLUL I Introducere în metodologia filtrului Kalman 1 .Introducere
Teoretic, un filtru Kalman este un estimator pentru problema liniar pătratică. Aceasta se defineşte ca fiind o problemă în estimarea unei stări instantanee al unui sistem dinamic liniar perturbat de zgomot alb, folosind măsurători legate liniar liniar de stare, dar perturbate de zgomot alb. Estimatorul rezultant este unul optim din punct de vedere statistic. Practic este una dintre cele mai mari descoperiri în istoria estimărilor statistice. Primele aplicaţii au fost reglarea unui sistem dinamic complex cum sunt procesele continue de fabricaţi e, avioane, vapoare, nave spaţiale. Pentru aceste aplicaţii nu este întotdeauna posibil sau de dorit sa măsuram fiecare variabilă pe care vrem sa o reglăm, iar filtrul Kalman furnizează mijloacele pentru a deduce informaţiile informaţiile lipsa din măsurătorile indirecte si asociate de zgomot. Filtrul Kalman este, de asemenea, folosit pentru a prezice cursul probabil pe viitor al sistemelor dinamice in care sunt şanse puţine ca ele sa fie reglate, cum ar fi cursul râului in timpul unei inundaţii, traiectoria unui corp ceresc, sau preturile articolelor de comerţ. co merţ.
Un filtru Kalman este un algoritm optimal recursiv de procesare a datelor. Sunt multe posibilităţi de a defini cuvântul „ optimal” depinde însă, de criteriul ales pentru a evalua performanţa. Filtru Kalman este optimal (FKO) deoarece conţine toate informaţiile informaţiile care ii sunt su nt furnizate. FKO prelucrează toate măsurătorile disponibile, indiferent de precizia lor şi estimează estimează valoarea curentă a variabilelor de interes cu ajutorul: cunoştinţelor despre sistem şi măsurăt orile dispozitivului dinamic; descrierii statistice a zgomotului din sistem, eroriore de măsurare şi incertitudinile modemului dinamic; informaţiilor disponibile despre condiţiile iniţiale ale variabilelor de interes. Un filtru este de fapt un algoritm de procesare a datelor.
1.1 Definirea filtrului Kalman i) FK- instrument .
Nu rezolva probleme de unul singur. Nu este un instrument fizic ci unul matematic. Este realizat din modele matematice. ii) FK- program pe calculator. Este considerat potrivit pentru implementarea pe calculatoare digitale, în parte pentru ca foloseşte o reprezentare finită a problemei de estimare (un numar finit de variabile).
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sunt de asemenea, folositoare pentru analizele statistice si modelele predictive al sistemelor senzoriale. Filtrare Kalman Cele Mai Mici
Sisteme
Patrate
Stohastice
Generalizate Cele Mai Mici
Teoria
Sisteme
Patrate
Probabilitatii
Dinamice
Fundamente matematice
Fig.1.1 Concepte fundamentale in filtrarea Kalman elor Kalman. Aplicatiile filtrelor În Figura 1.1 se prezintă elementele de bază pentru teoria filtr elor Kalman se referă în special la estimare si analiza performanţelor estimatoarelor. Avantaje relative ale filtrelor Kalman si Wiener: 1. Implementarea filtrului Wiener în circuite electronice analogice poate determina la o ef icientă icientă mult mai mare decât filtrul filtrul digital Kalman. 2. Filtrul Kalman este implementat în forma unui algoritm pentru calculatoarele digitale, care inlocuieşte circuitele analogice pentru estimare şi conducere în vremea în care filtrul Kalman a fost introdus. Aceasta implementare poate fi mai înceata, dar este capabilă de o precizie mult mai mare decât poate fi atinsă de un filtru filtru analogic. 3. Filtrul Wiener nu are nevoie de modele deprocese stohastice finite dimensional pentru semnal si zgomot. 4. Filtrul Kalman este compatibil cu formularea controlerelor optimale în spaţiul stărilor ale sistemelor dinamice, iar Kalman a reuşit să dovedească utilitatea celor două proprietăţi ale estimării şi conducerii pentru pe ntru aceste sisteme. 1.2 Consideraţii privind algoritmii de filtrare adaptivă
Este important de subliniat încă de la început ca o aplicaţie de filtrare adaptivă nu admite o soluţie unică. În fapt, avem la dispoziţie un întreg arsenal de tehnici diferite, fiecare având avantaje si dezavantaje specifice, iar alegerea uneia sau alteia dintre variantele posibile trebuie sa ia în considerare criterii precum viteza de convergenta, volumul de calcul si de memorie necesar, ori efectul apariţiei erorilor specifice implementării algoritmului folosind circuite care oferă precizie limitata.
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contextul utilizării unor filtre liniare, însa cu modificări specifice se regăsesc si în cazul filtrelor neliniare, în particular al reţelelor neurale. 1.2.1 Algoritmul celor mai mici pătrate generalizate
Una dintr e abordările cele mai des utilizate în teoria filtrării adaptive este cea bazata pe formularea unei astfel de aplicaţii sub forma unei probleme de optimizare. În mod concret, se defineşte un criteriu de performanta (denumit de regula funcţie de eroare) care depinde de setul de coeficienţi ai filtrului si de proprietăţile statistice ale semnalelor de la intrarea si ieşirea dorita a acestuia, care asociază fiecărui vector de coeficienţi o valoare scalara. Aspectul geometric al suprafeţei multidimensionale rezultate poate fi foarte complicat, însa în cazul particular al unui filtru discret liniar de tip FIR si al erorii pătratice medii aceasta se prezintă ca o suprafaţa parabolica de forma convexa, având o valoare minima unica. Mai mult, vectorul de coeficienţi -Hopf! În corespunzător acestei valori minime este chiar setul optim definit de ecuaţiile Wiener -Hopf! principiu, determinarea acestui sistem de ecuaţii s-ar putea efectua “dintr-un singur foc” prin calcul algebric, însa în multe situaţii practice apar dificultăţi datorate dimensiunilor mari ale matricelor implicate sau probleme de stabilitate ale metodelor numerice folosite în inversarea matricei de autocorelaţie R. Determinarea valorilor extreme ale unei funcţii de mai multe variabile poate fi asigurata printr o paleta larga de metode, expuse cu acurateţe în textele referitoare la tehnicile de optimizare. Una dintre cele mai des folosite soluţii o reprezintă descreşterea după gradient (gradient descent ), ), care în esenţa se bazează pe modificarea succesiva a variabilelor pe direcţia si în sens invers gradientului funcţiei supuse procesului de optimizare. În cazul particular al unui filtru liniar adaptiv modul de operare al algoritmului. 1.2.2 Algoritmul Celor Mai Mici Ptrate
Problema de filtrare liniara optimala poate fi abordata si dintr-o perspectiva complementara celei specifice specifice filtrului Wiener, Wiener, renunţând la punctul de vedere statistic, statistic, bazat pe considerarea considerarea unui ansamblu de realizări individuale ale unor procese aleatoare cu rol de intrare si ieşire dorite ale filtrului si înlocuindu- l cu o abordare temporala. În mod concret, acţiunea operatorului de mediere statistic E{.} este înlocuita cu simpla mediere aritmetica a valorilor unor realizări particulare unice ale proceselor aleatoare menţionate anterior.
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1.3 Sistemul in buclă închisă, phase-locked phase-locked loop (PLL)
Fi
Detector de
U
Frecventa
Pompa
Filtru
de
Trece Jos
D Oscilator de
Convertor
control al
de iesire
Fo
tensiunii
VCO ÷N
Fig. 1.2 Schema unui sistem cu buclă închisă Un sistem de reglare în buclă deschisă este acela în care acţiune de reglare este independentă de ieşire. Un sistem de reglare în buclă închisă este acela în care acţiunea de reglare este dependentă de ieşirea sistemului. Reglarea în buclă închisă a unui sistem este de cele mai multe ori denumită reglarea pe reacţie sistemului.
PLL sunt construite în general din: detector de fază , filtru trece - jos jos şi un oscilator de reglare a voltajului (VCO) plasat pe reacţia negativă. Acesta poate fi divizat în partea de reacţie şi partea de referinţă. Oscilatorul generează un semnal periodic de ieşire. ieşire.
CAPITOLUL II
Sisteme dinamice lineare
2.1 Sisteme dinamice lineare continue
2.1.1 Modelul intrare – ieşire al sistemelor lineare dinamice continue
Fie cvartetul matricial (A,B,C,D) care satisface ecuaţiile:
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| u (.) : n continuă pe porţiuni unde u Є U u (.) |u po rţiuni,, x (t ) : n , y(t): n , se numeşte sistem linear dinamic. Dacă matricele reprezentării de stare sunt dependente de timp atunci sistemul se scrie:
x A( t) x + B(t)u y C( t ) x + D(t)u
(2.2) şi sistemul se
numeşte liniar variabil.
Intrările, care sunt sub controlul nostru şi cunoscute de noi sau măsurate de noi, pe care le-am notat cu „u” sunt aleatoare dar cunoaştem proprietăţile statistice ale acestora Variabilele de stare, care nu pot fi măsurate direct, dar pot fi măsurate cu ajutorul ecuaţiilor Ieşirile care pot fi verificate prin măsurători pe care le-am notat cu „y”
Pentru intrări: intrări:
x(t )
d dt
x (t ) A(t ) x (t ) B (t )u (t )
(2.3)
unde elementele componente ale matricelor şi vectorilor sunt funcţii dependente de timp:
a11 (t ) a12 (t ) a13 (t ) a1n (t ) a (t ) a (t ) a (t ) a (t ) 21 22 23 2n A(t ) ........................................ an1 (t ) an 2 (t ) an 2 (t ) ann (t )
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Pentru ieşiri: ieşiri: y (t )
d dt
y (t ) C (t ) x (t ) D (t )u (t )
c11 (t ) c12 (t ) c13 (t ) c1n (t ) c (t ) c (t ) c (t ) c (t ) 21 22 23 2n C (t ) ........................................ cl1 (t ) cl 2 (t ) cl 2 (t ) cln (t ) d11 (t ) d12 (t ) d13 (t ) d1r (t ) d (t ) d (t ) d (t ) d (t ) 21 22 23 2r D (t ) ........................................ ( ) ( ) d ( ) d ( ) d t d t t t l l l l r 1 2 2 y(t ) y1 (t )
T
y2 (t ) y3 (t ) yl (t )
Matricea C se numeşte matricea măsurărilor sensibile , () matricea D se numeşte matricea cuplajului intrare – ieşire, iar y(t) reprezintă vectorul mărimilor măsurate le ieşire.
2.1.2 Traiectoria de stare şi matricea de tranziţie a stărilor
Pentru sistemele dinamice lineare, traiectoria de stare se obţine o bţine ca soluţie unică A ( t t 0 )
x(t ) e
t
x (t0 ) e A (t ) Bu ( )d t 0
matrice de tranziţie a stărilor stărilor funcţia:
(2.4) se numeşte
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t
rezultând
A ( t t 0 )
y (t ) Ce
x(t0 ) C e
A ( t )
Bu ( )d
(2.7)
t 0
Un sistem dinamic linear continuu este reprezentat prin mulţimea sistemelor dinamice a căror ecuaţie diferenţială de transfer:
d n y dy du dm u F n , ..., , y( t ); u ( t), , ..., m 0 dt dt dt dt
(2.8)
este o ecuație diferențială lineară ordinară: dn y dy du dm u a n (t (t ) n ... a1 ( t ) a 0 (t (t) y( t ) b0 ( t) u ( t ) b1 ... bm m 0 dt dt dt dt
(2.9)
cu n m Discretizarea semnalelor netede se face cu ajutorul extrapolatorului de ordin zero şi este descrisă de sistemul de ecuaţii:
x ( t 1) A d x ( t ) + bd u(t) cu t şi cu reprezentarea de st are: T y( t ) Cd x(t) + dd u(t) Ad e
Ah
T
cd
cT ,
,
h A bd e d b 0 d d d
2.2 Sisteme dinamice liniare discrete
Cvartetul matriceal (A,B,C,D) care satisface ecuaţiile
(
1) A ( )
B ()
(2.10)
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x(t ) A x (0) t
t-1
A
t- -1
Bu ( )
(2.12)
t t Z
(2.13)
=0
se numeşte matrice de tranziţie a stărilor funcţia:
e At (t ) t A
cu ajutorul acesteia se poate scrie traiectoria sistemului liniar discret x(t ) (t ) x (0)
t 1
(t 1)Bu ( )
(2.14)
0
răspunsul sistemului liniar discret, condiţia ca matricea D să fie nulă, se obţine y(t ) CA x( 0) C t
t 1
A
t 1
Bu ( )
(2.15)
0
2.3 Controlabilitatea şi observabilitate sistemelor dinamice lineare 2.3.1 Controlabilitatea sistemelor lineare discrete
Controlabilitatea prin definiție reprezintă posibilitatea aducerii sistemului, prin comandă, în timp finit cu valori negative, dintr-un punct al spaţiului de stare, considerat drept condiţie iniţială în origine. x (t 1) Ax (t ) Bu (t ),
t Z , x n , A n xn , B nxm se consideră un şir
{u(0),u(1)…u(k-1)} de k paşi de comandă care aduce sistemul din starea x(0) în x(k) -1)} cu x(0) dat se calculează şirul : {x(0),x(1)…x(k -1)} x(1) Ax (0) Bu (0);
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Notăm cu : Rk [ B
AB ... A k-1B] matricea de controlabilitate în k – paşi şi
u (k 1) u (k 2) uk ... u (0)
vectorul cu elementele şirului de k – paşi de comandă
în acest caz se poate scrie : x (k ) Ak x (0) Rku k iar dimensiunea spaţiului de stare Х=n O stare x a sistemului liniar discret este controlabilă în k – paşi, dacă există un sir de k – paşi de comandă care conduce sistemul din starea iniţială, x(0), dată, în starea finală x(k). x(k). Sistemul discret este controlabil controlabil în în k – paşi sau perechea (A,B) este controlabilă în k – paşi, dacă orice stare x este controlabilă în k – paşi. Sistemul discret este controlabil în k – paşi dacă d i m k
n
2.3.2 Controlabilitatea sistemelor liniare continue
Ecuaţia diferenţială de stare a unui sistem liniar liniar continuu cont inuu multivariabil la intrare x ( t
1)
A x (t ) B u ( t ) ,
t Z , x n , A n x n , B n xm
tranziţia sistemului dintr -o stare x(0) în starea x(t) este soluţia so luţia ecuaţiei diferenţiale diferenţiale t
x(t ) e x0 At
e A(t ) Bu ( )d
(2.16)
0
Sistemul liniar continuu este controlabil dacă orice stare x n este controlabil co ntrolabilăă
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x(0) x(1) Ax (0) x(k 1) Ak 1 x (0)
y(0) y(1) Cx (0) y(2) Cx (1) CAx (0) y(k 1) Ak 1Cx (0)
şi
se consideră
y (0 ) y (1) (1 ) y k y ( k 1)
C CA 2 si matricea Q k C A pkx n k 1 CA
Sistemul discret este observabil în k – paşi dacă orice stare x este observabilă în k – paşi Sistemul linear discret este observabil în k – paşi dacă şi numai dacă În cazul sistemelor lineare continue avem sistemul de ecuaţii :
x Ax x(0) x0
C CA
ran gQ
k
n
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-o Se numeşte sistem de evenimente o mulţime de evenimente, care pot apare intr -o anumita experienţa. Definiţie: Evenimentul, care se realizează întotdeauna intr -o experienţa data, se numeşte eveniment sigur sau total si se notează cu S. Definiţie: Evenimentul care consta in nerealizarea unui eveniment A se numeşte complementarul (opusul sau contrarul) lui A si se notează cu Ā. Definitie: Complementul evenimentului sigur se numeşte eveniment imposibil si se notează cu Ø. Definiţie: Definiţie:
Definiţia axiomatica a probabilităţii:
Se numeşte probabilitate pe câmpul de evenimente F, funcţia P(A), pozitiva, definita pentru orice eveniment din F si care are proprietăţile: 1) P(A) este pozitiva P(A)≥0
2) Probabilitatea evenimentului sigur este egala cu 1 P(S)=1 (S este evenimentul sigur) 3) Daca A si B sunt evenimente mutual exclusive, atunci
P(A B)=P(A)+P(B)
daca A B=
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Proprietăţile de mai sus sunt doar câteva. Celelalte se pot deduce din acestea prin calcule simple si pot fi generalizate pentru un număr finit finit sau infinit infinit de eveni mente. Pentru a estima cel mai bine un semnal sau un proces aleator avem nevoie de teoria probabilităţilor. probabilităţilor. Spre exemplu, posibilitatea ca evenimentul A sa se producă pro ducă se calculează după formula următoare: P( A)
Posibilitatea ca evenimentul A sa se produca Totalul tuturor posibilitatilor
(3.2)
Zgomotul alb este definit ca un proces de zgomot necorelat cu putere egală pe toate frecvenţele. Un zgomot care are putere egală pe toate frecvenţele în domeniu trebuie să aibă putere infinită şi este în consecinţă consecinţă doar un concept teoretic. Funcţia masei de probabilitate
Pentru variabilele aleatoare discrete, X reprezintă o sumă de valori discrete dintr -un set de N valori x1 , x 2 , ...x n fiecare x i poate fifi considerat considerat un eveniment eveniment si desemnat o frecvenţă de apariţie. Probabilitatea ca o variabilă aleatoare cu valoare discretă să ia o valoare x i P(X=x i ) se numeşte funcţia de probabilitate în masă. Funcţia de probabilitatea a densităţii
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1) Ai Aj ; i j;
2) A=S; i i=1
3) pi=1; pi=P(Ai) i 1
care ia valoarea x1ЄR când se realizează evenimentul A1 cu probabilitatea p1=P(A1), ia valoarea x2ЄR când se realizează evenimentul A2 cu probabilitatea p2=P(A2), ia valoarea xi Є R c ând se realizează evenimentul Ai cu probabilitatea pi, … Daca pentru un rezultat al unui experiment ξ Є S o operaţie de evaluare furnizează un număr real X(ξ), funcţia reala X, care asociază fiecărui ξ Є S valoarea X(ξ) este o variabila aleatoare atunci când mulţimea {ξ: X(ξ)
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VARIABILA ALEATOARE
DISTRIBUITIA PROBABILITATII
R
SPATIUL DE PROBA
UN MODEL PENTRU CUNOASTEREA
AL TUTUROR
MASURAREA
VARIANTELOR
PROBABILITATII
POSIBILE
[0,1]
PROCESULUI FIZIC
p:A→[0,1]
SPATIUL PROBABILITATII
PROCEDURA DE
EXPERIMENT STATISTIC
MODELARE
Proces fizic necunoscut
Fig. 3.1 Modelul conceptual pentru o variabila aleatoare Variabilele aleatoare f trebuie sa aibă proprietatea ca, pentru orice valoare reala a si b astfel încât a b , ieşirile O precum a
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Pf (x (x)=
d dx
Pf (x (x)
(3.7)
este numita funcţia de densitate de probabili p robabilitate tate a variabilei aleato are f, şi diferenţială: p f ( x ) d x d P f ( x )
(3.8)
reprezintă măsura probabilităţii măsura probabilităţii lui f definită pe o algebră sigma care conţine co nţine intervalele deschise (numita „algebra lui Borel peste R”). 3.2 Proprietăţile statistice ale variabilelor aleatoare si ale proceselor aleatoare Valorile aşteptate ale variabilelor aleatoare. Simbolul aşteptare al variabilelor aleatoare. Este numit numit şi expectanţa
“E” este folosit ca operator de valoare aşteptată şi expresia
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Funcţia variabilelor aleatoare.
O funcţie a variabilelor variabilelor aleatoare y=f(x) unde x şi y reprezintă intrarea şi respectiv ieşirea; proprietăţile statistice statistice alea lui y în funcţie de x sunt descrise de ecuaţiile:
Ey
f ( x ) p( x)dx
unde y este un scalar
Ey
n
)]n p ( x )dx [ f ( x )]
Probabilitatea densităţii lui y se poate obţine o bţine din densitatea lui x.
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unde x(t) reprezintă datele de intrare si h(t, ) reprezintă funcţia pondere a sistemului. Daca
sistemul este invariant in timp, atunci ecuaţia de mai sus devine: y(t)= h( x(t-)d . -
Figura 3.3 Diagrama bloc a unui sistem liniar Acest tip de integrala este numită integrală de convoluţie. co nvoluţie. Calcule asupra ultimei ecuaţii conduc la relaţia de autocorelare auto corelare funcţiilor funcţiilor x(t) si y(t):
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În teoria probabilitătilor şi statisticii, corelaţia, care se mai numeşte si coeficientul de corelare, indică puterea şi direcţia a relaţie re laţie liniare dintre două variabile aleatoare. Corficientul de corelaţie intre două variabile aleatoare X ,Y , cu valorile aşteptate, X şi Y şi deviaţia standard , X şi Y se defineşte : X ,Y
cov( X , Y ) E (( X X )(Y Y )) X Y
X Y
(3.19)
unde E reprezină valoare aşteptată şi cov este covarianţa lor. Dacă X
E ( X ), ), X2 E ( X 2 ) E 2 ( X )
şi analog pentru Y putem scrie: E ( X ,Y ) E ( X ) E (Y )
(3.20)
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Estimarea procesului
Filtrul Kalman abordează problema generală a estimării stării x R a unui process controlat în timp discret care este guvernat de ecuaţii diferenţiale stohastice liniare. xk
Axk 1 Bu k 1 wk 1
(4.1)
Hxk vk
(4.2)
cu măsuratoarea z Rm : zk
Variabilele aleatoare wk şi vk reprezintă procesul şi măsuratoarea zgomotului. Se presupune a fi independente (una de cealaltă), curate şi cu distibuţii cu probabilităţi probabilităţi normale
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Eroarea de estimare a prori a covarianţei este:
Pk
E[ek ek T ]
Eroarea de estimare a posteriori a covarianţei co varianţei este:
(4.5)
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Originile filtrului
Justificarea pentru (4.7) işi are rădăcinile în probabilitatea estimării a priori xˆk
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Ecuaţiile specifice actualizării timpului şi măsurătorii măsurătorii sunt prezentate în tabelele 4.1 şi 4.2. Tabelul 4.1: Ecuaţiile de actualizare a timpului ale filtrului Kalman discret xˆ
Axˆ Bu
(4.11)
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Şi într -un caz şi în altul, chiar dacă nu există o bază raţională pentru alegerea parametrilor, de cele mai multe ori performanţele superioare ale filtrului filtrului (din punct de vedere ea parametrilor Q şi R. Ajustarea este făcută de obicei off statistic) pot fi obţinute prin ajustar ea line, de cele mai multe ori cu ajutorul unui alt filtru Kalman într- un proces care poartă numele de
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4.2 Filtrul Kalman extins (EKF)
4.2.1 Estimarea procesului
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