Algebraic Properties of Riccati equations Ruth Curtain University - PowerPoint PPT Presentation

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 ∈ A n × 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 = ( α n ) n ∈ Z be any sequence of positive real numbers satisfying α n + m ≤ α n α m , α − n = α n ( even ) . An even-weighted Wiener algebra is � � � � f n φ n ( φ ∈ T ) and W α ( T ) := f : f ( φ ) = α n | f n | < + ∞ , n ∈ Z n ∈ Z with norm � f � W α ( T ) := � n ∈ Z α n | f n | . This is a commutative semi-simple Banach algebra and the maximal ideal space is isomorphic to the annulus around T : √ α n = lim √ α n . A ( ρ ) = { φ ∈ C : 1 /ρ ≤ | φ | ≤ ρ } , ρ = inf n n n > 0 n →∞ ˆ for ϕ ∈ A ( ρ ) . f ( ϕ ) = f ( ϕ ) Gelfand transform :

Chris Byrnes, CDC 1980 Let A be a commutative , unital, complex, semisimple Banach algebra, with an involution · † . Let A ∈ A n × n , B ∈ A n × m , C ∈ A p × n be such that for all ϕ ∈ M ( A ) (the maximal ideal space) ( � B ( ϕ )) is controllable and ( � A ( ϕ ) , � A ( ϕ ) , � C ( ϕ )) is observable. Then there exists a solution P ∈ A n × n such that PA + A † P − PBB † P + C † C = 0 , and A − BB † P is stable.

Example ( Counterexample to Chris Byrnes’ claim) a ( ϕ ) = 2 + ϕ, ˆ c ( ϕ ) = 1 ∈ W α ( T ) for arbitrary Let ˆ b ( ϕ ) = 1 , ˆ a ( ϕ ) , ˆ even weights α n = α − n . Now (ˆ b ( ϕ )) is controllable and c ( ϕ )) is observable for all ϕ ∈ C . According to and (ˆ a ( ϕ ) , ˆ Byrnes we should have a unique solution to the Riccati equation P ∈ W α ( T ) for for arbitrary even weights α k . Now � � � P ( ϕ ) = 1 ( 4 + ϕ + 1 /ϕ ) 2 + 4 P ( ϕ ) = ˆ 4 + ϕ + 1 /ϕ + . 2 � � � √ � √ 5 + 1 5 − 1 Note: Singularity at ϕ = − 1 ± + i − 1 ± . 2 2 But elements of W α ( T ) are analytic in the interior of A ( ρ ) , √ α n . where ρ = lim n →∞ n ∈ W α ( T ) for for arbitrary symmetric weights α k . Thus P /

Correct result with Amol Sasane, SIAM 2011 Let A be a commutative, unital, complex, semisimple Banach algebra. For all ϕ ∈ M ( A ) let A ∈ A n × n , B ∈ A n × m , C ∈ A p × 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 ∈ A n × 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 W α ( T ) are f † ( φ ) := f ( φ ) , f ∼ ( φ ) := f ( 1 /φ ) ∗ . In our counter example neither satisfies (1): ∗ ( 2 + z ) ∼ = 1 + 1 � � � ( 2 + z ) † = 2 + z , ( A + z ) = 2 + z , z .

Example ( Symmetric even-weighted Wiener algebras W α ( T ) ) The Gelfand-Raikov-Shilov condition : ρ = 1 ⇒ A ( ρ ) = T . = Involution ∼ reduces to f ∼ ( φ ) := f ( φ ) ∗ ( φ ∈ T ) , f ∈ W α ( T ) . With this involution W α ( T ) is a symmetric Banach algebra and (1) is always satisfied. Exponential weights: ρ > 1. Subexponential weights: ρ = 1, where α n = e α | 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 ∈ A n × n , B ∈ A n × m , C ∈ A p × n satisfy: ( � A ( ϕ ) , � B ( ϕ )) is stabilizable ( � A ( ϕ ) , � C ( ϕ )) is detectable. Then there exists a P ∈ A n × 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 W α ( T ) where α n are subexponential weights.

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Spatially Invariant Systems (Bamieh 2002) Σ( A , B , C ) with A , B , C matrices whose entries are bounded convolution operators on the Hilbert space � | x r | 2 < ∞} . ℓ 2 ( Z ; C n ) = { x = ( x r ) r ∈ Z : r = Z Example ˙ z r ( t ) = z r ( t ) + u r ( t ) + 2 u r − 1 ( t ) , y r ( t ) = z r ( t ) − z r − 1 ( t ) , −∞ ≤ r ≤ ∞ .

L ( ℓ 2 ( Z ; C )) is a Banach algebra with the adjoint operation · ∗ as involution. Convolution operators ( Tx ) l = � r ∈ Z T r − l x r form a subalgebra: Under the Fourier transform F : ℓ 2 ( Z ) → L 2 ( T ) and spatially invariant systems Σ( A , B , C ) are isometrically isomorphic to systems on L 2 ( T ; C n ) : Σ( F A F − 1 , F B F − 1 , F C F − 1 ) := Σ(ˆ B , ˆ A , ˆ C ) . Since ˆ B , ˆ A , ˆ C are multiplication operators on L ∞ ( T ; C n ) they are much easier to handle mathematically.

Example ( Fourier transform of example) ˙ z r ( t ) = z r ( t ) + u r ( t ) + 2 u r − 1 ( t ) , y r ( t ) = z r ( t ) − z r − 1 ( t ) , −∞ ≤ r ≤ ∞ . ˙ ˆ z ( φ ) = ( 10 − φ − 1 /φ )ˆ u ( φ ) , φ ∈ T . ˆ y ( φ ) = ˆ z ( φ ) , 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 φ ∈ T . P ( φ ) , � k ∈ Z δ −| k | φ k , ˆ 1 1 P ( φ ) = 10 − φ − 1 /φ = √ 4 6 √ δ = 5 + 24.

Control of Platoon-type spatially invariant systems Spatially invariant operators are isomorphic to L ( L 2 ( T ; C n )) = L ∞ ( T ; C n × 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 ; C n × n ) . P ( φ ) = � Even for simple examples ˆ r ∈ Z p r φ r . But for an implementable control law you need to truncate and the truncation should approximate ˆ P , i.e., � r ∈ T α r p r < ∞ for some ( α r ) , i.e., ˆ P must be in a Wiener algebra W α ( 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 of 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 − 1 ∈ L ( Z ) = ⇒ D − 1 ∈ A ; D ∈ A , � A � A ≥ M � A � L ( 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.