7 Minimal realization and coprime fraction

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Cornelius Spencer Ellis
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1 7 Minimal realization and coprime fraction 7 Introduction If a tranfer function i realizable, what i the mallet poible dimenion? Realization with the mallet poible dimenion are called minimal-dimenional or minimal realization

2 7 implication and Coprimene Conider ĝ() Conider ŷ() () () Let a peudo tate () + α + + α ()û() α + α vˆ() ()û()(or ()vˆ() û())

3 The realization i It controllability matrix can be computed a It determinant i for any α i Hence the realization i called a controllable canonical form u x bu Ax x + α α α α + & [ ]x cx y α α α α α α α + α α α α C

4 Theorem 7 The controllable canonical form i obervable if and only if () and () are coprime If the controllable canonical form i a realization of ĝ(), then we have, by definition, ĝ() c(i A) b Taking it tranpoe yield the tate equation (a different realization) x& A x + c u α α α α x + u y b x [ ]x

5 It i called an obervable canonical form The equivalent tranformation with will get the different controllable and obervable canonical form x Px P

6 7 Minimal realization Let R() be a greatet common divior (gcd) of () and () Then, the tranfer function can be reduce to (coprime fraction) ĝ () () / () where (S) ()R() and () ()R() We call () a characteritic polynominal of ĝ() It degree i defined a the degree of ĝ()

7 Theorem 7 A tate equation (A, b, c, d) i a minimal realization of a proper reational function ĝ() if and only if (A, b) i controllable and (A, c) i obervable or if and only if dim(a) deg(ĝ()) The Theorem provide a alternative way of checking controllability and obervability Theorem 7 All minimal realization of ĝ() are equivalent

8 If a tate equation i controllable and obervable, then every eigenvalue of A i a pole of ĝ() and every pole of ĝ() i an eigenvalue of A Thu we conclude that if (A, b, c, d) i controllable and obervable, then we have Aymptotic tability BIBO tability

9 7 Computing coprime fraction Let write () () () () which implie ()( ()) + ()() Let () + + S + + () () + () +

10 Sylverter reultant (Homogeneou linear algebraic equation) () and () are coprime if and only if the Sylverter reultant i noningular S :

11 Theorem 7 eg ĝ() number of linearly independent -column : µ and the coefficient of a coprime fraction [ ] µ µ equal the monic null vector of the matrix that conit of the primary dependent - column and all it LHS linearly independent column of S

12 7 QR ecompoition Conider an n m matrix M Then there exit an n n orthogonal matrix uch that Q M R where R i an upper triangular matrix Becaue i orthogonal, we have Q Q Q : Q and M QR Q

13 7 Balanced realization The diagonal and modal form, which are leat enitive to parameter variation, are good candidate for practical implementation A different minimal realization, called a balanced realization Conider a table ytem x & Ax + y cx bu

14 Then the controllability Gramian W c and the obervability W o are poitive definite if the ytem i controllable and obervable AW c + W c A -bb A W o + W o A -c c ifferent minimal realization of the ame tranfer function have different controllability and obervability

15 Theorem 75 Let (A, b, c) and (A, b, c) be minimal and equivalent Then W c W o and W c W o are imilar and their eigenvalue are all real and poitive Theorem 76 A balanced realization For any minimal tate equation (A, b, c) an equivalent tranformation x Px uch that the equivalent controllability and obervability have the property Σ W c Wo

16 75 Realization from Markov parameter Conider the trictly proper rational function ĝ() n + α n + n + α n + + n n + + α + n Expend it into an infinite power erie a ĝ() h() + h() + h() + ( h() for trictly proper) The coefficient h(m) are called Markov parameter n + α n

17 Let g(t) be the invere Laplace tranform of ĝ() Then, we have m d h (m) g(t) m t dt Hankel matrix (finding Markov parameter) h() h() T( α, ) h() h( α) h() h() h() h( α + ) h() h() h(5) h( α + ) h() h( + ) h( + ) h( α + ) h() ; h() -α h() + ; h() -α h() - α h() + ; h(n) -α h(n-)-α h(n-)- -α n- h()+ n

18 Theorem 77 A trictly proper rational function ĝ() ha degree n if and only if ρt(n, n) ρt(n+k, n+l) n where ρ denote the rank

19 76 egree of tranfer matrice Given a proper rational matrix Ĝ(), aume that every entry of Ĝ() i a coprime fraction efinition 7 The characteritic polynomial of Ĝ() i defined a the leat common denominator of all minor of Ĝ() It degree i defined a the degree of Ĝ()

20 77 Minimal realization-matrix cae Theorem 7M A tate equation (A, B, C, ) i a minimal realization of a proper rational matrix Ĝ() if and only if (A, B) i controllable and (A, C) i obervable or if and only if dim A deg Ĝ() Theorem 7M All minimal realization of are equivalent Ĝ ()

21 78 Matrix polynomial fraction The degree of the calar tranfer function ĝ() () () () () ()() i defined a the degree of () if () and () are coprime fraction Every q p proper rational matrix can be expreed a (right fraction polynomial) Ĝ() () ()

22 The expreion (left polynomial fraction) Ĝ() ()() The right fraction i not unique (The ame hold for left fraction) Ĝ() [()R()][()R()] () efinition 7 A quare polynomial matrix M() i called a unimodular matrix if it determinant i nonzero and independent of ()

23 efinition 7 A quare polynomial matrix R() i a greatet common right divior (gcrd) of () and () if (i) R() i a common right divior of () () (ii) R() i a left multiple of every common right divior of () and () If a gcrd i a unimodular matrix, then () and () are aid to be right coprime

24 efinition 7 Conider Ĝ() () ()() () (right coprime) (left coprime) Then, it characteritic polynomial i defined a det () or det () and it degree i defined a deg Ĝ() deg det() deg det ()

25 78 Column and row reducedne efine δ ci M() degree of ith column of M() δ ri M() degree of ith row of M() For example: M() δ c, δ c, δ c, δ r, and δ r efinition 75 A noningular matrix M() i column reduced if deg detm() um of all column degree

26 It i row reduced if deg det M() um of all row degree Let δ ci M() k ci and define H c () diag( kc, kc, ) Then the polynomial matrix M() can be expreed a M() M hc H c () + M lc () M hc : The column-degree coefficient matrix M lc (): The remaining term and it column ha degree le than k ci M() i column reduced M hc i noningular

27 Row form of M() M() H r ()M hr + M lr () H r () diag( kr, kr, ) M hr : the row-degree coefficient matrix M() i row reduced M hr i noningular Theorem 78 Let () i column reduced, Then () - () i proper (trictly proper) if and only if δ ci () δ ci () [δ ci ()<δ ci ()]

28 78 Computing matrix coprime fraction Conider Ĝ() expreed a Ĝ() Imply Auming ()() ()() () ()() () () + () + () + () +

29 A generalized reultant (the matrix verion) Theorem 7M Let µ i, be the number of linear independent Then and a right coprime fraction obtained by computing monic null vector SM : p Ĝ() deg µ µ µ

30 79 Realization from matrix coprime fraction efine (for µ and µ ) and, H() : µ µ µ µ L() :

31 Let and define Then, we have Let define û() () Ĝ()û() ŷ() ()û() vˆ() ()vˆ() ŷ() and û(), ()vˆ() µ µ () x () x () x () x () x () x : () vˆ () vˆ () vˆ () vˆ () vˆ () vˆ () vˆ () vˆ L()vˆ() xˆ() 6 5

32 Expre () a (S) hc H() + lc L() Then we have H()vˆ() and ŷ() ()vˆ() hc lc xˆ() + hc û() xˆ() L()vˆ()

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