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Algebraic Properties of Riccati equations

Ruth Curtain University of Groningen, The Netherlands

Special Recognition

Peter Falb Jan Willems Tony Pritchard Hans Zwart

Riccati equation

˙ P(t) + A(t)∗P(t) + P(t)A(t) = −P(t)B(t)B(t)∗P(t) + C(t)∗C(t), P(T) = Q. W.T. Reid: Riccati Differential Equations, Academic Press, 1972.

Algebraic Riccati equation

Let A, B, C be complex matrices. If (A, B) is stabilizable and (A, C) is detectable, then A∗P + PA − PBB∗P + C∗C = 0 has a unique stabilizing solution P. Stabilizing solution: P = P∗ ≥ 0 and A − BB∗P is stable. (A, B) is stabilizable if ∃ F s.t. A + BF is stable. (A, C) is detectable if ∃ L s.t. A + LC is stable.

Algebraic question

Let A, B, C ∈ A, a Banach algebra with involution †. When will A†P + PA − PBB†P + C†C = 0 have a unique stabilizing solution P ∈ An×n? Stabilizing solution: P = P† and A − BB†P is exponentially stable, i.e., the spectrum of A is contained in the open left half-plane.

Hilbert space Riccati eqns

Suppose that H is a Hilbert space and A, B, C ∈ L(H). If (A, B) is exponentially stabilizable and (A, C) is exponentially detectable, then ∃ a unique exponentially stabilizing solution P ∈ L(H), P = P∗ ≥ 0 of A∗P + PA − PBB∗P + C∗C = 0. A Positive result: L(H) is a Banach algebra with the involution the adjoint operation.

Gelfand Representation Theorem for commutative Banach algebras

Every commutative semi-simple Banach algebra is isomorphic to an algebra of continuous functions on its maximal ideal space M(A) (a compact Hausdorff space, equipped with the weak * topology). Denote the Gelfand transform of a ∈ A by ˆ

• a. Then

ˆ a(ϕ) = ϕ(a), ∀ϕ ∈ M(A), ∀a ∈ A.

Example (Even-Weighted Wiener algebras on unit circle T)

Let α = (αn)n∈Z be any sequence of positive real numbers satisfying αn+m ≤ αnαm, α−n = αn (even). An even-weighted Wiener algebra is Wα(T) :=

• f : f(φ) =
• n∈Z

fnφn (φ ∈ T) and

• n∈Z

αn|fn| < +∞

• ,

with norm fWα(T) :=

n∈Z αn|fn|.

This is a commutative semi-simple Banach algebra and the maximal ideal space is isomorphic to the annulus around T: A(ρ) = {φ ∈ C : 1/ρ ≤ |φ| ≤ ρ}, ρ = inf

n>0

n

√αn = lim

n→∞

n

√αn. Gelfand transform: ˆ f(ϕ) = f(ϕ) for ϕ ∈ A(ρ) .

Chris Byrnes, CDC 1980

Let A be a commutative, unital, complex, semisimple Banach algebra, with an involution ·†. Let A ∈ An×n, B ∈ An×m, C ∈ Ap×n be such that for all ϕ ∈ M(A) (the maximal ideal space) ( A(ϕ), B(ϕ)) is controllable and ( A(ϕ), C(ϕ)) is

• bservable. Then there exists a solution P ∈ An×n

such that PA + A†P − PBB†P + C†C = 0, and A − BB†P is stable.

Example (Counterexample to Chris Byrnes’ claim)

Let ˆ a(ϕ) = 2 + ϕ, ˆ b(ϕ) = 1,ˆ c(ϕ) = 1 ∈ Wα(T) for arbitrary even weights αn = α−n. Now (ˆ a(ϕ), ˆ b(ϕ)) is controllable and and (ˆ a(ϕ),ˆ c(ϕ)) is observable for all ϕ ∈ C. According to Byrnes we should have a unique solution to the Riccati equation P ∈ Wα(T) for for arbitrary even weights αk. Now P(ϕ) = ˆ P(ϕ) = 1 2

• 4 + ϕ + 1/ϕ +
• (4 + ϕ + 1/ϕ)2 + 4
• .

Note: Singularity at ϕ = −1 ± √

5−1 2

+ i

• −1 ±

√

5+1 2

• .

But elements of Wα(T) are analytic in the interior of A(ρ), where ρ = limn→∞

n

√αn. Thus P / ∈ Wα(T) for for arbitrary symmetric weights αk.

Correct result with Amol Sasane, SIAM 2011 Let A be a commutative, unital, complex, semisimple Banach algebra. For all ϕ ∈ M(A) let A ∈ An×n, B ∈ An×m, C ∈ Ap×n satisfy:

• (A†)(ϕ) = (

A(ϕ))∗

• (BB†)(ϕ) =

B(ϕ)( B(ϕ))∗

• (C†C)(ϕ) = (

C(ϕ))∗ C(ϕ) ( A(ϕ), B(ϕ)) is stabilizable ( A(ϕ), C(ϕ)) is detectable. Then there exists a P ∈ An×n such that PBB†P − PA − A†P − C†C = 0, A − BB†P is exponentially stable, and P† = P.

Essential, but restrictive condition on the involution: Gelfand transform of involution must match complex conjugation:

• (A†)(ϕ) = (

A(ϕ))∗. (1) Symmetric Banach algebras always satisfy (1). Two possible involutions for Wα(T) are f †(φ) := f(φ), f ∼(φ) := f(1/φ)∗. In our counter example neither satisfies (1):

• (2 + z)† = 2+z,
• (A + z)

∗

= 2 + z ,

• (2 + z)∼ = 1+1

z.

Example ( Symmetric even-weighted Wiener algebras Wα(T))

The Gelfand-Raikov-Shilov condition: ρ = 1 = ⇒ A(ρ) = T . Involution ∼ reduces to f ∼(φ) := f(φ)∗ (φ ∈ T), f ∈ Wα(T). With this involution Wα(T) is a symmetric Banach algebra and (1) is always satisfied. Exponential weights: ρ > 1. Subexponential weights: ρ = 1, where αn = eα|n|β, α > 0, 0 ≤ β < 1.

Corollary, Curtain & Sasane, SIAM 2011 Let A be a commutative, unital, complex, symmetric Banach algebra. For all ϕ ∈ M(A) let A ∈ An×n, B ∈ An×m, C ∈ Ap×n satisfy: ( A(ϕ), B(ϕ)) is stabilizable ( A(ϕ), C(ϕ)) is detectable. Then there exists a P ∈ An×n such that PBB†P − PA − A†P − C†C = 0, A − BB†P is exponentially stable, and P† = P. Example : A, B, C are matrices with entries from Wα(T) where αn are subexponential weights.

Platoons of vehicles

Spatially Invariant Systems (Bamieh 2002) Σ(A, B, C) with A, B, C matrices whose entries are bounded convolution operators on the Hilbert space ℓ2(Z; Cn) = {x = (xr)r∈Z :

• r=Z

|xr|2 < ∞}.

Example

˙ zr(t) = zr(t) + ur(t) + 2ur−1(t), yr(t) = zr(t) − zr−1(t), −∞ ≤ r ≤ ∞.

L(ℓ2(Z; C)) is a Banach algebra with the adjoint

• peration ·∗ as involution.

Convolution operators (Tx)l =

r∈Z Tr−lxr form a subalgebra:

Under the Fourier transform F : ℓ2(Z) → L2(T) and spatially invariant systems Σ(A, B, C) are isometrically isomorphic to systems on L2(T; Cn): Σ(FAF−1, FBF−1, FCF−1) := Σ(ˆ A, ˆ B, ˆ C). Since ˆ A, ˆ B, ˆ C are multiplication operators on L∞(T; Cn) they are much easier to handle mathematically.

Example (Fourier transform of example)

˙ zr(t) = zr(t) + ur(t) + 2ur−1(t) , yr(t) = zr(t) − zr−1(t), −∞ ≤ r ≤ ∞. ˙ ˆ z(φ) = (10 − φ − 1/φ)ˆ u(φ), ˆ y(φ) = ˆ z(φ), φ ∈ T. The Riccati equation can be solved pointwise: ˆ A(φ)∗ˆ P(φ) + ˆ P(φ)ˆ A(φ) + ˆ C(φ)∗ˆ C(φ) = ˆ P(φ)ˆ B(φ)ˆ B(φ)∗ˆ P(φ), φ ∈ T. ˆ P(φ) =

1 10−φ−1/φ = 1 4 √ 6

• k∈Z δ−|k|φk ,

δ = 5 + √ 24.

Control of Platoon-type spatially invariant systems Spatially invariant operators are isomorphic to L(L2(T; Cn)) = L∞(T; Cn×n). Existence of Riccati solutions If (ˆ A(φ), ˆ B(φ), ˆ C(φ)) is stabilizable and detectable for all φ ∈ T the Riccati equation has a unique stabilizing solution P ∈ L∞(T; Cn×n). Even for simple examples ˆ P(φ) =

r∈Z prφr.

But for an implementable control law you need to truncate and the truncation should approximate ˆ P, i.e.,

r∈T αrpr < ∞ for some (αr), i.e.,

ˆ P must be in a Wiener algebra Wα(T) ⊂ L∞(T).

Spatially distributed systems Introduced in Bamieh et al IEEE AC Trans 2002 and further studied in Motee & Jadbabaie in IEEE AC

• Trans. 2008. Unlike spatially invariant systems,

spatially distributed systems allow for the interaction

• f an array of infinitely many distinct linear systems.

The requirement of implementable control laws corresponds to the algebraic properties of the Riccati equation for noncommutative Banach algebras. Claim in Motee & Jadbabaie in IEEE AC Trans. 2008 was false. Bunce 1985 had proven a positive result for C∗-algebras, but this does not cover the spatially distributed case.

Algebraic properties for noncommutative Banach algebras (Curtain 2011) Suppose A is a unital symmetric Banach algebra and A is a Banach *-subalgebra of L(Z), where Z is a Hilbert space; A has the inverse-closed property: D ∈ A, D−1 ∈ L(Z) = ⇒ D−1 ∈ A; AA ≥ MAL(Z). If A, B, C ∈ A and (A, B, C) is exponentially stabilizable and detectable wrt Z, then P ∈ A, where P ∈ L(Z) is the unique nonnegative solution to the control Riccati equation: A∗P + PA − PBB∗P + C∗C = 0.

Applications Using results by Gröchenig & Leinert (2006) it can be shown that a subclass of the spatially distributed systems studied in Motee & Jadbabaie in 2008 satisfies the conditions of the previous theorem which leads the way to designing implementable control laws. Again, only subexponential weights are allowed and not exponential ones.

Conclusions For commutative Banach algebras we have an elegant result for the LQR Riccati equation; in particular, for symmetric algebras. Application to platoon-type spatially invariant systems: design of implementable control laws. There are generalizations to other Riccati equations, including H∞ type equations. Algebraic properties of the LQR Riccati equation for inverse-closed noncommutative algebras have direct applications to spatially distributed systems. Engineers provide motivation for interesting mathematical problems.

Open: spatially invariant systems with unbounded A Linear p.d.e.’s on infinite domains are examples of spatially invariant systems with unbounded A. Under the Fourier transform these corresponds to systems with multiplicative operators on L∞(iR). Semigroup properties have been well studied by the Nagel school. An algebraic property of the Riccati equation is shown in Curtain 2011. System theoretic properties need closer investigation.