Right invertible analytic Toeplitz operators and optimal 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 ) = I m , 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 ) = I m Given: G ( z ) stable rational m × p matrix function, m ≤ p G ( z ) = D + zC ( I n − zA ) − 1 B , 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 ρ 1 ( z ) 0 · · · 0 . . ... . . G ( z ) = U ( z ) V ( z ) . . ρ m ( z ) 0 · · · 0
Smith McMillan form: ρ 1 ( z ) 0 · · · 0 . . ... . . G ( z ) = U ( z ) V ( z ) . . ρ m ( z ) 0 · · · 0 ρ 1 ( z ) − 1 ... ρ m ( z ) − 1 X ( z ) = V ( z ) − 1 U ( z ) − 1 0 · · · 0 . . . . . . 0 · · · 0 Then X is a stable rational matrix solution of G ( z ) X ( z ) = I m . [Smith-McMillan forms are not numerically viable.]
Right invertible analytic Toeplitz operators G ( z ) stable rational m × p matrix function, m ≤ p T G the Toeplitz operator defined by G : G 0 G 1 G 0 : ℓ 2 + ( C p ) → ℓ 2 + ( C m ) . T G = G 2 G 1 G 0 . . . ... . . . . . . G ( z ) X ( z ) = I m ⇒ T G T X = T GX = I ℓ 2 + ( C m ) ⇒ T G right invertible
Right invertible analytic Toeplitz operators – continued Conversely, if T G is right invertible, then T G T ∗ G is invertible and X defined by I m 0 G ) − 1 [ F C p Fourier transform] X = F C p T ∗ G ( T G T ∗ 0 . . . is an H 2 solution. Indeed, I m I m 0 0 G ) − 1 G ) − 1 ( GX )( · ) = G F C p T ∗ G ( T G T ∗ = F C m T G T ∗ G ( T G T ∗ = I m . 0 0 . . . . . .
I m 0 G ) − 1 X = F C p T ∗ G ( T G T ∗ 0 . . . This H 2 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 � 2 π � 2 π 1 X ( e it ) ∗ X ( e it ) dt ≤ 1 V ( e it ) ∗ V ( e it ) dt . 2 π 2 π 0 0 G ) − 1 is the Moore-Penrose right inverse Reason for (1) : T ∗ G ( T G T ∗ of T G . The fact that X is rational is less trivial. Questions: How to compute X ? State space formula for X ?
Inverting T G T ∗ G z − 1 ) ∗ , and put Set R ( z ) = G ( z ) G ∗ ( z ), where G ∗ ( z ) = G (¯ R 0 R − 1 R − 2 · · · G 1 G 2 G 3 · · · R 1 R 0 R − 1 · · · G 2 G 3 G 4 · · · T R = , H G = . R 2 R 1 R 0 · · · G 3 G 4 G 5 · · · . . . . . . ... ... . . . . . . . . . . . . T R = T G T ∗ G + H G H ∗ G PROP. The operator T G T ∗ G is invertible if and only if G T − 1 (i) T R is invertible and (ii) I − H ∗ R H G is invertible. In that case G ) − 1 = T − 1 + T − 1 G T − 1 R H G ) − 1 H ∗ G T − 1 ( T G T ∗ R H G ( I − H ∗ R . R
First step: inverting T R [a classical topic] R ( z ) = G ( z ) G ∗ ( z ), where G ( z ) = D + zC ( I n − zA ) − 1 B . Thus R 0 = DD ∗ + CPC ∗ , − j = CA j − 1 Γ for j = 1 , 2 , 3 , . . . R j = R ∗ P − APA ∗ = BB ∗ and Γ = BD ∗ + APC ∗ . THM. The Toeplitz operator T R is invertible if and only if the discrete algebraic Riccati equation Q = A ∗ QA + ( C − Γ ∗ QA ) ∗ ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) (ARE) has a ( unique ) stabilizing solution Q, that is, R 0 − Γ ∗ Q Γ is positive definite, Q is an n × n matrix satisfying ( ∗ ) and the matrix A − Γ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) is stable
First step: inverting T R – continued THM. The Toeplitz operator T R is invertible if and only if Q = A ∗ QA + ( C − Γ ∗ QA ) ∗ ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) (ARE) has a ( unique ) stabilizing solution Q. In that case � � T − 1 I m − zC 0 ( I n − zA 0 ) − 1 Γ ∆ − 1 , where = T Ψ T ∗ Ψ , Ψ( z ) = R C 0 = ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) , A 0 = A − Γ C 0 , ∆ = ( R 0 − Γ ∗ Q Γ) 1 / 2 . C CA obs T − 1 N.B.: Q = W ∗ R W obs , where W obs = . CA 2 . . .
Second step: computing T − 1 G T − 1 G T − 1 R H G ( I − H ∗ R H G ) − 1 H ∗ R THM. Let Q be the stabilizing solution of the Riccati equation Q = A ∗ QA + ( C − Γ ∗ QA ) ∗ ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) . (ARE) G T − 1 Then I − H ∗ R H G is invertible if and only if I n − PQ is non-singular, where P − APA ∗ = B. In that case T − 1 G T − 1 G T − 1 R H G ) − 1 H ∗ = K ( I n − PQ ) − 1 PK ∗ , R H G ( I − H ∗ R where C 0 C 0 A 0 : C n → ℓ 2 + ( C m ) . K = C 0 A 2 0 . . . G ) − 1 = T Ψ T ∗ Ψ + K ( I n − PQ ) − 1 PK ∗ . Conclusion: ( T G T ∗
First main theorem – [Frazho-K-Ran, 2010] THM 1. Equation G ( z ) X ( z ) = I m has a stable rational matrix solution if and only if the corresponding Riccati equation (ARE) has a stabilizing solution Q, and I n − PQ is non-singular, where P − APA ∗ = B. In that case the least squares solution is the rational matrix function X is given by � � I p − zC 1 ( I n − zA 0 ) − 1 ( I n − PQ ) − 1 B X ( z ) = D 1 , where A 0 = A − Γ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) , C 1 = D ∗ C 0 + B ∗ QA 0 , with C 0 = ( R 0 − Γ ∗ Q Γ) − 1 ( C − Γ ∗ QA ) , D 1 = ( D ∗ − B ∗ Q Γ)( R 0 − Γ ∗ Q Γ) − 1 + C 1 ( I n − PQ ) − 1 PC ∗ 0 . In particular, X is rational, and the McMillan degree of X ≤ the McMillan degree of G.
The null space of T G Assume T G is surjective. General H ∞ theory tells us: ◮ Beurling-Lax: Ker T G = T Θ ℓ 2 + ( C k ), Θ inner, p × k . ◮ k = p − m . ◮ Θ(0) one to one. ◮ T Θ T ∗ G ) − 1 T G Θ = I ℓ 2 + ( C p ) − T ∗ G ( T G T ∗ Hence I p I p 0 0 Θ( · )Θ(0) ∗ = F C p T Θ T ∗ G ) − 1 T G = I p − F C p T ∗ G ( T G T ∗ 0 0 Θ . . . . . . Θ(0) ∗ � � − 1 � � Θ(0) Θ(0) ∗ Θ(0) = I k
Second main theorem – [Frazho-K-Ran, 2010] THM 2. [Frazho-K-Ran, 2010] Assume T G is surjective. Then Ker T G = T Θ ℓ 2 + ( C p − m ) , where Θ is given by � � I p − zC 1 ( I n − zA 0 ) − 1 ( I n − PQ ) − 1 B ˆ Θ( z ) = D . Here A 0 and C 1 are as in THM 1 , and ˆ D is any one-to-one p × ( p − m ) matrix such that D ∗ = I p − ( D ∗ − B ∗ Q Γ)( R 0 − Γ ∗ Q Γ) − 1 ( D − Γ ∗ QB )+ D ˆ ˆ − B ∗ QB − C 1 ( I n − PQ ) − 1 PC ∗ 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 � � 1 � � � � G ( z ) = 1 + z − z . Obviously, 1 + z − z = 1 . 1 A minimal realization of G is given by � � � � A = 0 , B = 1 − 1 C = 1 , D = 1 0 , . Solution Stein equation: P = 2. Since G ( z ) G ∗ ( z ) = 3 + z + z − 1 , the corresponding Riccati equation is 1 Q = 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: √ � 1 − Q � Q 1 − 2 Q (1 + zQ ) − 1 , where Q = 1 X ( z ) = 2 (3 − 5) . Q 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 √ � � z � Q (1 + zQ ) − 1 , where Q = 1 Θ( z ) = 2 (3 − 5) . 1 + z
Final remarks Ongoing work: Rational matrix solutions, optimal or suboptimal, in the H ∞ norm. Put γ ∞ = inf {� X � ∞ | G ( z ) X ( z ) = I m , 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 ) ∗ ( R 0 − γ − 2 I m − Γ ∗ Q Γ) − 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 = Q γ such that r spec ( PQ γ ) < 1 . N.B. The proof uses the formula for the central solution in the commutant lifting theorem.
Recommend
More recommend
