7.1 introduction • if a transfer function is realizable ...tuky/linear/pdf/ch7.pdf · 7 minimal...
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7 Minimal realization and coprime fraction
• 7.1 Introduction• If a transfer function is realizable, what is
the smallest possible dimension?• Realizations with the smallest possible
dimension are called minimal-dimensional or minimal realizations.
7.2 implications and Coprimeness
• Consider
• Consider
• Let a pseudo state
432
23
14
432
23
1ssss
sss)s(D)s(N)s(g
α+α+α+α+
β+β+β+β==
)s(u)s(D)s(N)s(y 1−=
))s(u)s(v)s(D or)(s(u)s(D)s(v 1 == −
• The realization is
• Its controllability matrix can be computed as
• Its determinant is 1 for any αi. Hence the realization is called a controllable canonical form.
u
0001
x
010000100001
buAxx
4321
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ α−α−α−α−
=+=&
[ ]xcxy 4321 ββββ==
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
α−α−αα−
α−αα+α−α−αα−
=
1000100
1021
C1
2211
321312
211
• Theorem 7.1 The controllable canonical form is observable if and only if D(s) and N(s) are coprime.
• If the controllable canonical form is a realization of ĝ(s), then we have, by definition,
• Taking its transpose yields the state equation (a different realization)
b)AsI(c)s(g 1−−=
ux
000100010001
ucxAx
4
3
2
1
4
3
2
1
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
ββββ
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
α−α−α−α−
=′+′=& [ ]x0001xby =′=
• It is called an observable canonical form.• The equivalent transformation with
• will get the different controllable and observable canonical form.
Pxx =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
0001001001001000
P
• 7.2.1 Minimal realizations• Let R(s) be a greatest common divisor (gcd) of
N(s) and D(s). Then, the transfer function can be reduce to (coprime fraction)
where • We call a characteristic polynominal of ĝ(s).
Its degree is defined as the degree of ĝ(s).
).s(D/)s(N)s(g =
(s)R(s)D D(s)and )s(R)s(N)S(N ==
)s(D
• Theorem 7.2 A state equation (A, b, c, d) is a minimal realization of a proper reationalfunction ĝ(s) if and only if (A, b) is controllable and (A, c) is observable or if and only ifdim(A) = deg(ĝ(s))
• The Theorem provides a alternative way of checking controllability and observability.
• Theorem 7.3 All minimal realizations of ĝ(s) are equivalent.
• If a state equation is controllable and observable, then every eigenvalue of A is a pole of ĝ(s) and every pole of ĝ(s) is an eigenvalue of A.
• Thus we conclude that if (A, b, c, d) is controllable and observable, then we have Asymptotic stability ⇔ BIBO stability.
7.3 Computing coprime fractions
• Let write
which implies • Let
D(s)=D0+D1s+D2S2+D3s3+D4s4
N(s)=N0+N1s+N2s2+N3s3+N4s4
)s(D)s(N
)s(D)s(N
=
0)s(D)s(N))s(N)(s(D =+−
33
2210 sDsDsDD)s(D +++=
33
2210 sNsNsNN)s(N +++=
• Sylverster resultant (Homogeneous linear algebraic equation)
• D(s) and N(s) are coprime if and only if the Sylverster resultant is nonsingular.
0
DN
DN
DN
DN
ND000000NDND0000NDNDND00NDNDNDNDNDNDNDND00NDNDND0000NDND000000ND
:S
3
3
2
2
1
1
0
0
44
3344
223344
11223344
00112233
001122
0011
00
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
−
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
• Theorem 7.4 Deg ĝ(s) = number of linearly independent N-columns =: µ andthe coefficients of a coprime fraction
equals the monic null vector of the matrix that consists of the primary dependent N-column and all its LHS linearly independent columns of S.
[ ]′−−− µµ DN..DNDN 1100
• 7.3.1 QR Decomposition• Consider an n×m matrix M. Then there
exists an n×n orthogonal matrix such that
where R is an upper triangular matrix.• Because is orthogonal, we have
QRMQ =
RQ Mand Q:QQ 1 ==′=−Q
7.4 Balanced realization
• The diagonal and modal forms, which are least sensitive to parameter variations, are good candidates for practical implementation.
• A different minimal realizations, called a balanced realization.
• Consider a stable systembuAxx +=&
cxy =
• Then the controllability Gramian Wc and the observability Wo are positive definite if the system is controllable and observableAWc + WcA’ = -bb’A’Wo + WoA = -c’c
• Different minimal realizations of the same transfer function have different controllability and observability.
• Theorem 7.5 Let (A, b, c) and be minimal and equivalent. Then WcWo and
are similar and their eigenvalues are all real and positive.
• Theorem 7.6 A balanced realizationFor any minimal state equation (A, b, c)an equivalent transformation such that the equivalent controllability and observability have the property
)c,b,A(
ocWW
Pxx =
Σ== oc WW
7.5 Realizations from Markov parameters
• Consider the strictly proper rational function
• Expend it into an infinite power series as
( h(0) = 0 for strictly proper)• The coefficient h(m) are called Markov
parameters.
n1n2n
21n
1n
n1n2n
21n
1s...sss
s...ss)s(gα+α++α+α+
β+β++β+β=
−−−
−−−
...s)2(hs)1(h)0(h)s(g 21 +++= −−
• Let g(t) be the inverse Laplace transform of ĝ(s). Then, we have
• Hankel matrix (finding Markov parameters)
h(1) = β1; h(2) = -α1h(1) + β2;h(3) = -α1h(2) - α2h(1) + β3; …h(n) = -α1h(n-1)-α2h(n-2)- … -αn-1h(1)+βn
0t1m
1m)t(g
dtd)m(h =−
−=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−β+α+α+αα
+β+β
=βα
)1(h.)2(h)1(h)(h.....
)2(h.)5(h)4(h)3(h)1(h.)4(h)3(h)2(h
)1(h.)3(h)2(h)1(h
),(T
• Theorem 7.7 A strictly proper rational function ĝ(s) has degree n if and only ifρT(n, n) = ρT(n+k, n+l) = nwhere ρ denotes the rank.
7.6 Degree of transfer matrices
• Given a proper rational matrix , assume that every entry of is a coprime fraction.
• Definition 7.1 The characteristic polynomial of is defined as the least common denominator of all minors of . Its degree is defined as the degree of .
)s(G
)s(G
)s(G
)s(G
)s(G
7.7 Minimal realizations-Matrix case
• Theorem 7.M2 A state equation (A, B, C, D) is a minimal realization of a proper rational matrix if and only if (A, B) is controllable and (A, C) is observable or if and only ifdim A = deg
• Theorem 7.M3 All minimal realizations of are equivalent.
)s(G
)s(G
)s(G
7.8 Matrix polynomial fractions
• The degree of the scalar transfer function
is defined as the degree of D(s) if N(s) and D(s) are coprime fraction.
• Every q×p proper rational matrix can be expressed as (right fraction polynomial)
)s(N)s(D)s(D)s(N)s(D)s(N)s(g 11 −− ===
)s(D)s(N)s(G 1−=
• The expression (left polynomial fraction)
• The right fraction is not unique (The same holds for left fraction)
• Definition 7.2 A square polynomial matrix M(s) is called a unimodular matrix if its determinant is nonzero and independent of s.
)s(N)s(D)s(G 1−=
)s(D)s(N)]s(R)s(D)][s(R)s(N[)s(G 11 −− ==
• Definition 7.3 A square polynomial matrix R(s) is a greatest common right divisor (gcrd) of D(s) and N(s) if
(i) R(s) is a common right divisor of D(s) N(s)(ii) R(s) is a left multiple of every common
right divisor of D(s) and N(s).If a gcrd is a unimodular matrix, then D(s) and N(s) are said to be right coprime.
• Definition 7.4 Consider
Then, its characteristic polynomial is defined as
and its degree is defined as
coprime)(left )s(N)s(D
coprime)(right )s(D)s(N)s(G1
1
−
−
=
=
(s)Ddet or D(s)det
(s)Ddet degdetD(s) deg (s)G deg ==
• 7.8.1 Column and row reducedness• Define
δciM(s) = degree of ith column of M(s)δriM(s) = degree of ith row of M(s)
• For example:δc1 = 1, δc2 = 3, δc3 = 0, δr1 = 3, and δr2 = 2.
• Definition 7.5 A nonsingular matrix M(s) is column reduced if deg detM(s) = sum of all column degrees
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−+−+=0s1s15s2s1s)s(M 2
3
It is row reduced ifdeg det M(s) = sum of all row degrees
• Let δciM(s) = kci and define Hc(s) = diag(skc1, skc2, …). Then the polynomial matrix M(s) can be expressed asM(s) = MhcHc(s) + Mlc(s)Mhc: The column-degree coefficient matrixMlc(s): The remaining term and its column has degree less than kci.
• M(s) is column reduced⇔Mhc is nonsingular.
• Row form of M(s)M(s) = Hr(s)Mhr + Mlr(s)Hr(s) = diag(skr1, skr2, …).Mhr: the row-degree coefficient matrix.
• M(s) is row reduced⇔Mhr is nonsingular.• Theorem 7.8 Let D(s) is column reduced,
Then N(s)D-1(s) is proper (strictly proper) if and only if δciN(s)≤δciD(s) [δciN(s)<δciD(s)]
• 7.8.2 Computing matrix coprime fraction• Consider expressed as
• Imply• Assuming
)s(G
)s(D)s(N)s(N)s(D)s(G 11 −− ==
)s(N)s(D)s(D)s(N =
44
33
2210 sDsDsDsDD)s(D ++++=
33
2210 sDsDsDD)s(D +++=
44
33
2210 sNsNsNsNN)s(N ++++=
33
2210 sNsNsNN)s(N +++=
• A generalized resultant (the matrix version)
• Theorem 7.M4 Let µi, be the number of linear independent. Then and a right coprime fraction obtained by computing monic null vectors.
0
DN
DN
DN
DN
ND000000NDND0000NDNDND00NDNDNDNDNDNDNDND00NDNDND0000NDND000000ND
:SM
3
3
2
2
1
1
0
0
44
3344
223344
11223344
00112233
001122
0011
00
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
−
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
p21 ...)s(Gdeg µ++µ+µ=
7.9 Realization from matrix coprime fraction
• Define (for µ1 = 4 and µ2 = 2)
and ,s00s
s00s:)s(H 2
4
2
1
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎥⎥⎦
⎤
⎢⎢⎣
⎡= µ
µ
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
= −µ
−µ
10s0010s0s0s
10..
s001..0s
:)s(L
2
3
1
1
2
1
• Let
and define• Then, we have
• Let define
)s(uD)s(N)s(u)s(G)s(y 1−==)s(u)s(D)s(v 1−=
(s)vN(s)(s)y and ),s(u)s(v)s(D ==
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=⎥⎦
⎤⎢⎣
⎡
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
== −µ
−µ
)s(x)s(x)s(x)s(x)s(x)s(x
:
)s(v)s(vs)s(v)s(vs)s(vs)s(vs
)s(v)s(v
10..
s001..0s
)s(v)s(L)s(x
6
5
4
3
2
1
2
2
1
1
12
13
2
11
1
2
1
• Express D(s) as D(S) =DhcH(s) + DlcL(s)
• Then we have
and )s(uD)s(xDD)s(v)s(H 1
hclc1
hc−− +−=
)s(x
)s(v)s(L)s(v)s(N)s(y
222221214213212211
122121114113112111
222221214213212211
122121114113112111
⎥⎦
⎤⎢⎣
⎡ββββββββββββ
=
⎥⎦
⎤⎢⎣
⎡ββββββββββββ
==