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

==