zero/pole interpolation problem for the class bold0mu mumu ...andreym/zerpole.pdf · 1. classical...

OutlineClassical zero/pole interpolation

Overdetermined 2D systems and their transfer functionsDefinition of the class of functionsI

Zero/pole interpolation problemHermitian realization theorem

Zero/pole interpolation problem for the class I oftransfer functions of overdetermined 2D systems,

invariant in one direction.

Andrey Melnikov

Department of MathematicsBen Gurion University

Notes available at http://www.math.bgu.ac.il/∼andreym

William and Marry college, Willilamsburg

21 februar 2009

Andrey Melnikov Zero/pole interpolation problem in class I




Here is the plan of my lecture:

1. Classical Zero/Pole interpolation problem (for rational matrixfunctions)

2. Overdetermined 2D systems and their transfer functions

3. Definition of a class of functions I4. Zero/pole interpolation problem in the class I5. Hermitian realization theorem








3. Definition of a class of functions I

4. Zero/pole interpolation problem in the class I5. Hermitian realization theorem








3. Definition of a class of functions I4. Zero/pole interpolation problem in the class I

5. Hermitian realization theorem








3. Definition of a class of functions I4. Zero/pole interpolation problem in the class I5. Hermitian realization theorem





(Right) zero and pole functionCanonical right null functionNull and pole chainsZero/pole interpolation problem

Classical zero/pole interpolation(Right) zero and right pole functions

Suppose that S(λ) is an n × n matrix function, whose determinantdoes not vanish identically at points of analyticity.

1. If S(λ) is meromorphic at a point λ0 ∈ C, we say thatanalytic φ(λ) is a right null function for S(λ) at λ0 of orderk > 0 if φ(λ0) 6= 0 and if S(λ)φ(λ) is analytic at λ0 with azero of order k at λ0.

2. We say that analytic ψ(λ) is a right pole function for S(λ) atλ0 of order k > 0 if φ(λ0) 6= 0 and if there is an analyticvector function ξ(λ) such that ξ(λ0) 6= 0 andS(λ)ψ(λ) = (λ− λ0)

−kξ(λ).

Notice that ψ(λ) is a right pole function for S(λ) at λ0 of order kiff ψ(λ) is a right null function for S−1(λ) at λ0 of order k.






Classical zero/pole interpolationCanonical right null function

In order to learn S(λ), it is important to characterize all right nullfunctions. Using local Smith-McMillian form of a matrix (forinvertible E (λ),F (λ), k1 ≥ . . . ≥ kn)

S(λ) = E (λ) diag[(λ− λ0)k1 , . . . , (λ− λ0)

kn ]F (λ)

and here one can read the canonical right null functions:

{F (λ)−1e1, . . . ,F (λ)−1ei}

of orders k1, . . . , ki > 0. And the canonical right pole functions are

{F (λ)−1en−r+1, . . . ,F (λ)−1en}

of orders −kn−r+1, . . . ,−kn.








S(λ) = E (λ) diag[(λ− λ0)k1 , . . . , (λ− λ0)

kn ]F (λ)


{F (λ)−1e1, . . . ,F (λ)−1ei}

of orders k1, . . . , ki > 0.

And the canonical right pole functions are

{F (λ)−1en−r+1, . . . ,F (λ)−1en}

of orders −kn−r+1, . . . ,−kn.








S(λ) = E (λ) diag[(λ− λ0)k1 , . . . , (λ− λ0)

kn ]F (λ)


{F (λ)−1e1, . . . ,F (λ)−1ei}

of orders k1, . . . , ki > 0. And the canonical right pole functions are

{F (λ)−1en−r+1, . . . ,F (λ)−1en}

of orders −kn−r+1, . . . ,−kn.Andrey Melnikov Zero/pole interpolation problem in class I





Classical zero/pole interpolationNull and pole chains

It is possible to construct X0 = [X1, . . . ,Xp] and Jordan blockJ0 = J1 ⊕ · · · ⊕ Jp which characterize Ker S(λi ) for each value λi .The pair (X0, J0) is called right null pair of S(λ). Any other pair(Z ,T ) satisfying (for invertible S)

Z = X0S ,T = S−1J0S

is also called a right null pair.

In the same manner it is definedright pole pair of S(λ) as left zero pair of S−1(λ).






Classical zero/pole interpolationNull and pole chains

It is possible to construct X0 = [X1, . . . ,Xp] and Jordan blockJ0 = J1 ⊕ · · · ⊕ Jp which characterize Ker S(λi ) for each value λi .The pair (X0, J0) is called right null pair of S(λ). Any other pair(Z ,T ) satisfying (for invertible S)

Z = X0S ,T = S−1J0S

is also called a right null pair. In the same manner it is definedright pole pair of S(λ) as left zero pair of S−1(λ).






Classical zero/pole interpolationZero/pole interpolation problem

Given a right zero pair (C ,Aπ) and the right pole pair (Aξ,B), theproblem of reconstruction of the matrix S(λ) is called zero/poleinterpolation problem [BGR]. Necessary and sufficient condition forsolving this problem is existence of a solution X of Sylvesterequation

XAπ − AξX = BC . (1)

X is called the null-pole coupling matrix for S(λ). In this case

S(λ) = D + CX−1(λI − Aπ)−1B

The collection < (C ,Aπ), (Aξ,B),X > is called zero/pole data ofS(λ).





2D systems, invariant in one directionOverdeterminednesst1-invariant vesselFrequency domain analysis

Overdetermined 2D systems and their transfer functions2D systems, invariant in one direction

t1-invariant 2D system is a linear input-state-output (i/s/o) systemof the form

Σ :

∂x∂t1

(t1, t2) = A1(t2)x(t1, t2) + B1(t2)u(t1, t2)

∂x∂t2

(t1, t2) = A2(t2)x(t1, t2) + B2(t2)u(t1, t2)

y(t1, t2) = C (t2)x(t1, t2) + D(t2)u(t1, t2)

(2)

where for some Hilbert spaces H, E , E∗

A1(t2),A2(t2) : H → H, B1(t2),B2(t2) : E → H,C (t2) : H → E∗, D(t2) : E → E∗

are linear operators, satisfying certain regularity assumptions.






Overdetermined 2D systems and their transfer functions2D systems, invariant in one direction

t1-invariant 2D system is a linear input-state-output (i/s/o) systemof the form

Σ :

∂x∂t1

(t1, t2) = A1(t2)x(t1, t2) + B1(t2)u(t1, t2)

∂x∂t2

(t1, t2) = A2(t2)x(t1, t2) + B2(t2)u(t1, t2)

y(t1, t2) = C (t2)x(t1, t2) + D(t2)u(t1, t2)

(2)

where for some Hilbert spaces H, E , E∗

A1(t2),A2(t2) : H → H, B1(t2),B2(t2) : E → H,C (t2) : H → E∗, D(t2) : E → E∗

are linear operators, satisfying certain regularity assumptions.Andrey Melnikov Zero/pole interpolation problem in class I





Σ :

∂x∂t1

(t1, t2) = A1(t2)x(t1, t2) + B1(t2)u(t1, t2)

∂x∂t2

(t1, t2) = A2(t2)x(t1, t2) + B2(t2)u(t1, t2)

y(t1, t2) = C (t2)x(t1, t2) + D(t2)u(t1, t2)

u(t1, t2) ∈ E is called the input, y(t1, t2) ∈ E∗ is called the output,x(t1, t2) ∈ H is called the state.






Overdetermined 2D systems and their transfer functionsOverdeterminedness

If we consider u(t1, t2) continuously differentiable, x(t1, t2) has tosatisfy:

∂

∂t1

( ∂

∂t2x(t1, t2)

)=

∂

∂t2

( ∂

∂t1x(t1, t2)

). (3)

Substituting the expressions for derivatives of x and demandingcompatibility conditions for free evolution (u = 0), we shall obtain:

d

dt2A1(t2) = A2(t2)A1(t2)− A1(t2)A2(t2) (4)

and






B2∂

∂t1u − B1

∂

∂t2u + (A2B1 − A1B2 −

d

dt2(B1))u = 0. (5)

It is convenient to suppose that we have the factorizations:

B2 = Bσ2(t2), B1 = Bσ1(t2), (A2B1 − A1B2 −d

dt2(B1)) = Bγ(t2)

(6)for some operators

B : E → H, σ1(t2), σ2(t2), γ(t2) : E → E .

Then it follows that it is sufficient for the input to satisfy the inputcompatibility condition

σ2∂

∂t1u − σ1

∂

∂t2u + γu = 0. (7)






For the output signal we shall look for the output compatibilityconditions of the same type:

σ2∗∂

∂t1y − σ1∗

∂

∂t2y + γ∗y = 0 (8)

for some operators

σ1∗(t2), σ2∗(t2), γ∗(t2) : E∗ → E∗.

We recall that y = Cx + Du and we insert it into the last equation.Using the system equations (2) and the compatibility condition forthe input, we are naturally led to require






σ2∗CA1 − σ1∗d

dt2C − σ1∗A2 + γ∗C = 0 (9)

andDσ1 = σ1∗D, Dσ2 = σ2∗D,

Dγ = σ2∗CBσ1 − σ1∗CBσ2 − σ1∗d

dt2D + γ∗D

(10)

where D∗ : E∗ → E is the feed through operator for the naturallydefined adjoint system with input state E∗ and the output space E .






Overdetermined 2D systems and their transfer functionst1-invariant vessel

We define a t1-invariant vessel as a collection of operators andspaces:

V = (A1,A2, B,C ,D, D;σ1, σ2, γ, σ1∗, σ2∗, γ∗;H, E , E∗, E , E∗),

which are all function of t2 and satisfy the following axioms:

ddt2

A1 = A2A1 − A1A2d

dt2

(Bσ1

)− A2Bσ1 + A1Bσ2 + Bγ = 0

ddt2

(σ1∗C

)+ σ1∗CA2 − σ2∗CA1 − (γ∗ + d

dt2σ1∗)C = 0

Dσ1 = σ1∗D, Dσ2 = σ2∗D,

Dγ = σ2∗CBσ1 − σ1∗CBσ2 − σ1∗d

dt2D + γ∗D.






The vessel is associated with the system

Σ :

∂

∂t1x(t1, t2) = A1(t2)x(t1, t2) + B(t2) σ1(t2) u(t1, t2)

∂∂t2

x(t1, t2) = A2(t2)x(t1, t2) + B(t2) σ2(t2) u(t1, t2)

y(t1, t2) = C (t2)x(t1, t2) + D(t2)u(t1, t2)

and compatibility conditions for the input/ output signals:

σ2(t2)∂

∂t1u(t1, t2)− σ1(t2)

∂∂t2

u(t1, t2) + γ(t2)u(t1, t2) = 0

σ2∗(t2)∂

∂t1y(t1, t2)− σ1∗(t2)

∂∂t2

y(t1, t2) + γ∗(t2)y(t1, t2) = 0






Remarks: 1. The first equation is the Lax equation, which playsan important role in completely integrable non-linear PDE’s.

Defining the fundamental solution

d

dt2F (t2, t

02 ) = A2(t2)F (t2, t

02 ), F (t0

2 , t02 ) = I ,

we obtain from the Lax equation that the spectrum of A1(t2) isindependent of t2:

A1(t2) = F (t2, t02 )A1(t

02 )F (t2, t

02 )−1. (11)

3. This object is interesting, because it is time variying on the onehand, but has all advantages of time-invariant case on the otherhand: transfer function, functional model.4. We shall always assume that σ1(t2) and σ1∗(t2) are invertiblefor all t2.






Remarks: 1. The first equation is the Lax equation, which playsan important role in completely integrable non-linear PDE’s.Defining the fundamental solution

d

dt2F (t2, t

02 ) = A2(t2)F (t2, t

02 ), F (t0

2 , t02 ) = I ,


A1(t2) = F (t2, t02 )A1(t

02 )F (t2, t

02 )−1. (11)








d

dt2F (t2, t

02 ) = A2(t2)F (t2, t

02 ), F (t0

2 , t02 ) = I ,


A1(t2) = F (t2, t02 )A1(t

02 )F (t2, t

02 )−1. (11)

3. This object is interesting, because it is time variying on the onehand, but has all advantages of time-invariant case on the otherhand: transfer function, functional model.

4. We shall always assume that σ1(t2) and σ1∗(t2) are invertiblefor all t2.







d

dt2F (t2, t

02 ) = A2(t2)F (t2, t

02 ), F (t0

2 , t02 ) = I ,


A1(t2) = F (t2, t02 )A1(t

02 )F (t2, t

02 )−1. (11)







Overdetermined 2D systems and their transfer functionsFrequency domain analysis

Performing a partial separation of variables for the system (2),

u(t1, t2) = uλ(t2)eλt1 ,

x(t1, t2) = xλ(t2)eλt1 ,

y(t1, t2) = yλ(t2)eλt1 ,

we arrive at the notion of a transfer function.

Note that u(t1, t2), y(t1, t2) satisfy PDEs, but uλ(t2), yλ(t2) aresolutions of ODEs with a spectral parameter λ,

λσ2(t2)uλ(t2)− σ1(t2)∂

∂t2uλ(t2) + γ(t2)uλ(t2) = 0,

λσ2(t2)yλ(t2)− σ1(t2)∂

∂t2yλ(t2) + γ∗(t2)yλ(t2) = 0.






Overdetermined 2D systems and their transfer functionsFrequency domain analysis

Performing a partial separation of variables for the system (2),

u(t1, t2) = uλ(t2)eλt1 ,

x(t1, t2) = xλ(t2)eλt1 ,

y(t1, t2) = yλ(t2)eλt1 ,

we arrive at the notion of a transfer function.Note that u(t1, t2), y(t1, t2) satisfy PDEs, but uλ(t2), yλ(t2) aresolutions of ODEs with a spectral parameter λ,

λσ2(t2)uλ(t2)− σ1(t2)∂

∂t2uλ(t2) + γ(t2)uλ(t2) = 0,

λσ2(t2)yλ(t2)− σ1(t2)∂

∂t2yλ(t2) + γ∗(t2)yλ(t2) = 0.






The corresponding i/s/o system becomes{λxλ(t2) = A1(t2)xλ(t2) + B(t2)σ1(t2)uλ(t2)d

dt2xλ(t2) = A2(t2)xλ(t2) + B(t2)σ2(t2)uλ(t2).

The output yλ(t2) = D(t2)uλ(t2) + C (t2)xλ(t2) may be foundfrom the first i/s/o equation:

yλ(t2) = S(λ, t2)uλ(t2),

using the transfer function

S(λ, t2) = D(t2) + C (t2)(λI − A1(t2))−1B(t2)σ1(t2). (12)

Here λ is outside the spectrum of A1(t2), which is independent oft2 by (11).






Remarks: 1. S(λ, t2) is a rational matrix function of λ for each t2.

2. Multiplication by S(λ, t2) maps solutions of the input ODE withthe spectral parameter λ to solutions of the output ODE with thesame spectral parameter. This can be written by means offundamental matrices Φ(λ, τ2, t2) and Φ∗(λ, τ2, t2) of the inputand of the output ODE’s. Namely:

S(λ, t2)Φ(λ, t2, t02 ) = Φ∗(λ, t2, t

02 )S(λ, t0

2 ) (13)






Remarks: 1. S(λ, t2) is a rational matrix function of λ for each t2.2. Multiplication by S(λ, t2) maps solutions of the input ODE withthe spectral parameter λ to solutions of the output ODE with thesame spectral parameter. This can be written by means offundamental matrices Φ(λ, τ2, t2) and Φ∗(λ, τ2, t2) of the inputand of the output ODE’s. Namely:

S(λ, t2)Φ(λ, t2, t02 ) = Φ∗(λ, t2, t

02 )S(λ, t0

2 ) (13)





Definition of the class of functions I

We consider a class I of rational in λ matrix valued functionsS(λ, t2)

that map solutions of one ODE with a spectral parameterλ

λσ2u − σ1∂

∂t2u + γu = 0, (14)

to solutions of another ODE with the spectral parameter λ:

λσ2∗y − σ1∗∂

∂t2y + γ∗y = 0. (15)

Here σ∗1 = σ1, σ∗2 = σ2, σ

∗1∗ = σ1∗, σ

∗2∗ = σ2∗.





Definition of the class of functions I

We consider a class I of rational in λ matrix valued functionsS(λ, t2) that map solutions of one ODE with a spectral parameterλ

λσ2u − σ1∂

∂t2u + γu = 0, (14)

to solutions of another ODE with the spectral parameter λ:

λσ2∗y − σ1∗∂

∂t2y + γ∗y = 0. (15)

Here σ∗1 = σ1, σ∗2 = σ2, σ

∗1∗ = σ1∗, σ

∗2∗ = σ2∗.





Zero/pole interpolation problem

From the formula (13)

S(λ, t2) = Φ∗(λ, t2, t02 )S(λ, t0

2 )Φ−1(λ, t2, t02 )

the spectral data is in t2 independent matrix S(λ, t02 ).

Consequently, zero (C (t2),Aπ) and pole (Aξ,B(t2)σ1) data is forsome constant matrices Api ,Axi . In order to have (for each t2) asufficient zero/pole data for reconstructing S(λ, t2) up tosimilarity, there must exist a solution X (t2) of Sylvester equation

X (t2)Aπ − AξX (t2) = B(t2)σ1C (t2). (16)

Recall that the matrix X (t2) is called null-pole coupling matrix forS(λ, t2).





Remarks: If one starts from a matrix S(λ, t2) with the usualproperties, then one can explicitly evaluate zero-pole data(C (t2),Aπ), (Aξ,B(t2)σ1∗),X (t2) by definition. Thendifferentiating Sylvester equation (16), one obtains:

(X ′ − B(t2)σ2∗C (t2))Aπ − Aξ(X′ − B(t2)σ2∗C (t2)) = 0. (17)

1. Notice that if additionally, the spectrum of Aπ disjoint fromthe spectrum of Aξ, then from the uniqueness of solution of(17) we obtain that

X ′ = B(t2)σ2∗C (t2). (18)

In other words, X (t2) will additionally satisfy the differentialequation (18).





2. Suppose that X (t2) satisfies the differential equation (18) andfor t0

2 the algebraic Sylvester equation (16) is satisfied:

X (t02 )Aπ − AξX (t0

2 ) = B(t02 )σ1C (t0

2 ),

then Solving the differential equation (18) with this initialcondition X (t0

2 ), we obtain that X (t2) satisfies the algebraicdifferential equation (16) for each t2.

3. Notice that invertibility of the matrix X (t2) is not globallypromised. If det X (t0

2 ) 6= 0, then there could be values of t2for them det X (t2) = 0.





TheoremSuppose that (C (t2),Aπ), (Aξ,B(t2)),X (t2) is the zero-pole data,with invertible coupling matrix X (t2) on the interval I. Supposealso that additionally to (16), X (t2) satisfies (18). Then thereexists unique matrix function S(λ, t2), which maps solutions of(14) with spectral parameter λ to solutions of (15) with the samespectral parameter and which is identity at infinity.

Sketch of Proof: We may suppose that the zero/pole datasatisfies ODE’s with the corresponding spectral matrix parameters.Since we consider S(∞, t2) = I , we actually consider the caseσ1∗ = σ1, σ2∗ = σ2. Then one has to check that the constructedmatrix

S(λ, t2) = I + C (t2)(λI − Aπ)−1B(t2)σ1

satisfies the differential equation





Hermitian realization theoremAs a special case of a realization theorem for such matrix functionsone can consider the Hermitian case

TheoremSuppose that S(λ, t2) is as above and additionally satisfies

1. S(∞, t2) = ∞.

2. S(λ, t2)σ−11 S∗(λ, t2) ≥ σ−1

1 for <λ > 0.

3. S(λ, t2)σ−11 S∗(−λ, t2) = σ−1

1 for all t2.

then there exists a conservative vessel with S(λ, t2) as the transferfunction. In this case

S(λ, t2) = I + C (t2)(λI + A1)−1C ∗(t2)σ1,

where C (t2) satisfies the output differential equation with thematrix parameter A1.





Proof: One realize first S(λ, t2)

S(λ, t2) = I + C (t2)(λI + A1)−1X−1(t2)C (t2)

∗σ1

by means of the invertible Hermitian matrix X (t2), satisfying

X (t2)A1 + A∗1X (t2) = −C ∗(t2)σ1C (t2) (19)

Then redefining the inner product on the inner space H, on whichA1 acts by

〈u, u〉 = 〈X (t2)u, u〉and using kinematic equivalence of the vessels, defined byY (t2) =

√X (t2):

C (t2) = C (t2)Y−1(t2)

A1(t2) = Y−1(t2)A1Y (t2)(20)

one obtains that

S(λ, t2) = I + C (t2)(λI + A1(t2))−1C ∗(t2)σ1.





What next?

1. Interpolation theorems of various types (done).

2. Potapov’s theorem (almost done).

3. Schur algorithm for conservative Vessels and Nevanlinna Pickinterpolation (in progress).

4. Scattering theory for special cases (Sturm-Liouville ODE).

5. Generalizations to n-th order time varying ODEs.

6. Liouville extensions (Differential algebra).





J. A. Ball, I. Gohberg, L. Rodman, Interpolation of RationalMatrix Functions, Operator theory: advances and applications,Birkhauser, 1990.


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