Right invertible analytic Toeplitz operators and optimal solutions - - PowerPoint PPT Presentation

Right invertible analytic Toeplitz operators and

• ptimal solutions to the rational

Corona-Bezout equation

Rien Kaashoek, VU Amsterdam

Workshop in honor of Bill Helton

Best wishes for Bill and many happy returns

Corona-Bezout equation

(∗) G(z)X(z) = Im, z ∈ D Given: G(z) an m × p matrix-valued H∞ function, m ≤ p. Problem: Find p × m matrix-valued H∞ functions X such that (∗) holds. Condition for solvability: G(z)G(z)∗ ≥ δ > 0 for each |z| < 1. Carleson 1962 – scalar case (m = 1) Fuhrmann 1968 – matrix case (m > 1) See Section 5B in Helton’s CBMS lecture notes for further info and references.

Rational Corona-Bezout equation

(∗) G(z)X(z) = Im Given: G(z) stable rational m × p matrix function, m ≤ p G(z) = D + zC(In − zA)−1B, A stable Problem: Find stable rational matrix functions X such that (∗) holds, preferable in state space form. Condition for solvability: G(z) has full column rank for each |z| ≤ 1 (⇔ Carleson’s condition). Necessity obvious. Sufficiency: use Smith McMillan form G(z) = U(z)    ρ1(z) · · · ... . . . . . . ρm(z) · · ·    V (z)

Smith McMillan form: G(z) = U(z)    ρ1(z) · · · ... . . . . . . ρm(z) · · ·    V (z) X(z) = V (z)−1           ρ1(z)−1 ... ρm(z)−1 · · · . . . . . . · · ·           U(z)−1 Then X is a stable rational matrix solution of G(z)X(z) = Im. [Smith-McMillan forms are not numerically viable.]

Right invertible analytic Toeplitz operators

G(z) stable rational m × p matrix function, m ≤ p TG the Toeplitz operator defined by G: TG =      G0 G1 G0 G2 G1 G0 . . . . . . . . . ...      : ℓ2

+(Cp) → ℓ2 +(Cm).

G(z)X(z) = Im ⇒ TGTX = TGX = Iℓ2

+(Cm) ⇒ TG right invertible

Right invertible analytic Toeplitz operators – continued

Conversely, if TG is right invertible, then TGT ∗

G is invertible and X

defined by X = FCpT ∗

G(TGT ∗ G)−1

     Im . . .      [FCp Fourier transform] is an H2 solution. Indeed, (GX)(·) = GFCpT ∗

G(TGT ∗ G)−1

     Im . . .      = FCmTGT ∗

G(TGT ∗ G)−1

     Im . . .      = Im.

X = FCpT ∗

G(TGT ∗ G)−1

     Im . . .      This H2 solution has two special properties: (1) X is the least squares solution, and (2) X is rational. Property (1) means that for any other solution V we have 1 2π 2π X(eit)∗X(eit) dt ≤ 1 2π 2π V (eit)∗V (eit) dt. Reason for (1) : T ∗

G(TGT ∗ G)−1 is the Moore-Penrose right inverse

• f TG. The fact that X is rational is less trivial.

Questions: How to compute X? State space formula for X?

Inverting TGT ∗

G

Set R(z) = G(z)G ∗(z), where G ∗(z) = G(¯ z−1)∗ , and put TR =      R0 R−1 R−2 · · · R1 R0 R−1 · · · R2 R1 R0 · · · . . . . . . . . . ...      , HG =      G1 G2 G3 · · · G2 G3 G4 · · · G3 G4 G5 · · · . . . . . . . . . ...      . TR = TGT ∗

G + HGH∗ G

• PROP. The operator TGT ∗

G is invertible if and only if

(i) TR is invertible and (ii) I − H∗

GT −1 R HG is invertible.

In that case (TGT ∗

G)−1 = T −1 R

+ T −1

R HG(I − H∗ GT −1 R HG)−1H∗ GT −1 R .

First step: inverting TR [a classical topic]

R(z) = G(z)G ∗(z), where G(z) = D + zC(In − zA)−1B. Thus R0 = DD∗ + CPC ∗, Rj = R∗

−j = CAj−1Γ for j = 1, 2, 3, . . .

P − APA∗ = BB∗ and Γ = BD∗ + APC ∗.

• THM. The Toeplitz operator TR is invertible if and only if the

discrete algebraic Riccati equation (ARE) Q = A∗QA + (C − Γ∗QA)∗(R0 − Γ∗QΓ)−1(C − Γ∗QA) has a (unique) stabilizing solution Q, that is, R0 − Γ∗QΓ is positive definite, Q is an n × n matrix satisfying (∗) and the matrix A − Γ(R0 − Γ∗QΓ)−1(C − Γ∗QA) is stable

First step: inverting TR – continued

• THM. The Toeplitz operator TR is invertible if and only if

(ARE) Q = A∗QA + (C − Γ∗QA)∗(R0 − Γ∗QΓ)−1(C − Γ∗QA) has a (unique) stabilizing solution Q. In that case T −1

R

= TΨT ∗

Ψ, Ψ(z) =

• Im − zC0(In − zA0)−1Γ
• ∆−1, where

C0 = (R0 − Γ∗QΓ)−1(C − Γ∗QA), A0 = A − ΓC0, ∆ = (R0 − Γ∗QΓ)1/2. N.B.: Q = W ∗

• bsT −1

R Wobs,

where Wobs =      C CA CA2 . . .     .

Second step: computing T −1

R HG(I − H∗ GT −1 R HG)−1H∗ GT −1 R

• THM. Let Q be the stabilizing solution of the Riccati equation

(ARE) Q = A∗QA + (C − Γ∗QA)∗(R0 − Γ∗QΓ)−1(C − Γ∗QA). Then I − H∗

GT −1 R HG is invertible if and only if In − PQ is

non-singular, where P − APA∗ = B. In that case T −1

R HG(I − H∗ GT −1 R HG)−1H∗ GT −1 R

= K(In − PQ)−1PK ∗, where K =      C0 C0A0 C0A2 . . .      : Cn → ℓ2

+(Cm).

Conclusion: (TGT ∗

G)−1 = TΨT ∗ Ψ + K(In − PQ)−1PK ∗.

First main theorem – [Frazho-K-Ran, 2010]

THM 1. Equation G(z)X(z) = Im has a stable rational matrix solution if and only if the corresponding Riccati equation (ARE) has a stabilizing solution Q, and In − PQ is non-singular, where P − APA∗ = B. In that case the least squares solution is the rational matrix function X is given by X(z) =

• Ip − zC1(In − zA0)−1(In − PQ)−1B
• D1,

where A0 = A − Γ(R0 − Γ∗QΓ)−1(C − Γ∗QA), C1 = D∗C0 + B∗QA0, with C0 = (R0 − Γ∗QΓ)−1(C − Γ∗QA), D1 = (D∗ − B∗QΓ)(R0 − Γ∗QΓ)−1 + C1(In − PQ)−1PC ∗

0 .

In particular, X is rational, and the McMillan degree of X ≤ the McMillan degree of G.

The null space of TG

Assume TG is surjective. General H∞ theory tells us:

◮ Beurling-Lax: Ker TG = TΘℓ2 +(Ck), Θ inner, p × k. ◮ k = p − m. ◮ Θ(0) one to one. ◮ TΘT ∗ Θ = Iℓ2

+(Cp) − T ∗

G(TGT ∗ G)−1TG

Hence Θ(·)Θ(0)∗ = FCpTΘT ∗

Θ

     Ip . . .      = Ip − FCpT ∗

G(TGT ∗ G)−1TG

     Ip . . .      Θ(0)∗ Θ(0)

• Θ(0)∗Θ(0)

−1 = Ik

Second main theorem – [Frazho-K-Ran, 2010]

THM 2. [Frazho-K-Ran, 2010] Assume TG is surjective. Then Ker TG = TΘℓ2

+(Cp−m), where Θ is given by

Θ(z) =

• Ip − zC1(In − zA0)−1(In − PQ)−1B
• ˆ

D. Here A0 and C1 are as in THM 1, and ˆ D is any one-to-one p × (p − m) matrix such that ˆ D ˆ D∗ = Ip − (D∗ − B∗QΓ)(R0 − Γ∗QΓ)−1(D − Γ∗QB)+ − B∗QB − C1(In − PQ)−1PC ∗

1 .

Furthermore, ˆ D is uniquely determined up to a constant unitary matrix on the right, Θ is inner and rational, and the McMillan degree of Θ is less than or equal to the McMillan degree of G.

Example

G(z) =

• 1 + z

−z

• . Obviously,
• 1 + z

−z 1 1

• = 1.

A minimal realization of G is given by A = 0, B =

• 1

−1

• ,

C = 1, D =

• 1
• .

Solution Stein equation: P = 2. Since G(z)G ∗(z) = 3 + z + z−1, the corresponding Riccati equation is Q = 1 3 − Q , and the stabilizing solution is given by Q = 1

2(3 −

√ 5). We see that 1 − PQ = √ 5 − 2 = 0, as expected.

Example – continued

G(z) =

• 1 + z

−z

• .

Least squares solution of G(z)X(z) = 1: X(z) = Q 1 − 2Q (1 + zQ)−1 1 − Q Q

• , where Q = 1

2(3 −

√ 5). All stable rational 2 × 1 matrix solutions of G(z)V (z) = 1 are given by V (z) = X(z) + Θ(z)ϕ(z), where ϕ is any scalar stable rational function and Θ(z) =

• Q(1 + zQ)−1
• z

1 + z

• , where Q = 1

2(3 −

√ 5).

Final remarks

Ongoing work: Rational matrix solutions, optimal or suboptimal, in the H∞ norm. Put γ∞ = inf{X∞ | G(z)X(z) = Im, X is H∞ function}. Question: Does there exists a stable rational p × m matrix solution X with tolerance γ > γ∞, that is, X∞ < γ? New discrete algebraic Riccati equation: Q = A∗QA + (C − Γ∗QA)∗(R0 − γ−2Im − Γ∗QΓ)−1(C − Γ∗QA).

• THM. [Frazho-ter Horst-K] Let γ > 0 be given. Then γ > γ∞ if

and only if the above Riccati equation has a stabilizing solution Q = Qγ such that rspec(PQγ) < 1. N.B. The proof uses the formula for the central solution in the commutant lifting theorem.

• THM. [continued] In that case a stable rational p × m matrix

solution X such that X∞ < γ is given by X(z) = D1,γD−1

2,γ + z

• γ−2D1,γD−1

2,γC0,γ − C1,γ

• (In − zA×)−1×

× (In − PQγ)−1BD1,γD−1

2,γ.

Here C0,γ = (R0 − γ−2Im − Γ∗QγΓ)−1(C − Γ∗QγA) C1,γ = D∗C0,γ + B∗QγA0,γ A0,γ = A − ΓC0,γ A× = A0,γ + γ−2(In − PQγ)−1BD1,γD−1

2,γC0,γ

D1,γ = (D∗ − B∗QγΓ)(R0 − Γ∗QγΓ)−1 + C1,γ(In − PQγ)−1PC ∗

0,γ

D2,γ = I + γ−2(R0 − Γ∗QγΓ)−1 + γ−2C0,γ(In − PQγ)−1PC ∗

0,γ.

N.B.: X is the unique maximal entropy solution.

Final remarks – continued

◮ Analogous results for the equation G(z)Y (z) = F(z), where

F is a given stable rational matrix function.

◮ A state space version of Tolokonnikov’s lemma. ◮ Continuous analogue [work in progress]. ◮ In his CBMS notes Bill puts the corona problem in the context

• f classical interpolation, H∞ approximation, Wiener-Hopf

factorization and integrable systems. For all these topics state space theory for rational solutions is well developed. The present talk closes the circle.

Thank you!