The Behavioral Approach to Systems Theory Paolo Rapisarda, Un. of - PowerPoint PPT Presentation

The Behavioral Approach to Systems Theory Paolo Rapisarda, Un. of Southampton, U.K. & Jan C. Willems, K.U.Leuven, Belgium MTNS 2006 Kyoto, Japan, July 24–28, 2006

Lecture 4: Bilinear and quadratic differential forms Lecturer: Paolo Rapisarda

Part I: Basics

Outline Motivation and aim Definition Two-variable polynomial matrices The calculus of B/QDFs

Dynamics and functionals in systems and control Instances: Lyapunov theory, performance criteria, etc. ⇒ quadratic and bilinear functionals. Linear case =

Dynamics and functionals in systems and control Instances: Lyapunov theory, performance criteria, etc. ⇒ quadratic and bilinear functionals. Linear case = Usually: state-space equations, constant functionals. However, tearing and zooming = ⇒ state space eq.s

Dynamics and functionals in systems and control Instances: Lyapunov theory, performance criteria, etc. ⇒ quadratic and bilinear functionals. Linear case = Usually: state-space equations, constant functionals. However, tearing and zooming = ⇒ state space eq.s ¡High-order differential equations! ...involving also latent variables ...

Example : a mechanical system d 2 w 1 m 1 + k 1 w 1 − k 1 w 2 = 0 dt 2 d 2 w 2 − k 1 w 1 + m 2 + ( k 1 + k 2 ) w 2 = 0 dt 2

Example : a mechanical system d 2 w 1 m 1 + k 1 w 1 − k 1 w 2 = 0 dt 2 d 2 w 2 − k 1 w 1 + m 2 + ( k 1 + k 2 ) w 2 = 0 dt 2 dt 4 w + ( k 1 m 1 + k 2 m 1 + k 1 m 2 ) d 2 d 4 m 1 m 2 dt 2 w + k 1 k 2 w = 0

Example : a mechanical system d 2 w 1 m 1 + k 1 w 1 − k 1 w 2 = 0 dt 2 d 2 w 2 − k 1 w 1 + m 2 + ( k 1 + k 2 ) w 2 = 0 dt 2 dt 4 w + ( k 1 m 1 + k 2 m 1 + k 1 m 2 ) d 2 d 4 m 1 m 2 dt 2 w + k 1 k 2 w = 0 ¿Stability, stored energy, conservation laws?

Aim An effective algebraic representation of bilinear and quadratic functionals of the system variables and their derivatives: Operations/properties of functionals � algebraic operations/properties of representation ...a calculus of these functionals!

Outline Motivation and aim Definition Two-variable polynomial matrices The calculus of B/QDFs

Bilinear differential forms (BDFs) � Φ k ,ℓ ∈ R w 1 × w 2 � Φ := k ,ℓ = 0 ,..., L L Φ : C ∞ ( R , R w 1 ) × C ∞ ( R , R w 2 ) → C ∞ ( R , R )   Φ 0 , 0 Φ 0 , 1 . . .   Φ 1 , 0 Φ 1 , 1 . . . w 2   . .   � � dw 2 . . ⊤ L Φ ( w 1 , w 2 ) := dw 1 · · · w ⊤ . . . . .     dt 1 dt   . Φ k , 0 Φ k , 1 . . . .   . . . . . . . · · · � � ⊤ � � = � d k d ℓ dt k w 1 Φ k ,ℓ dt ℓ w 2 k ,ℓ

Quadratic differential forms (QDFs) � Φ k ,ℓ ∈ R w × w � k ,ℓ = 0 ,..., L symmetric, i.e. Φ k ,ℓ = Φ ⊤ Φ := ℓ, k Q Φ : C ∞ ( R , R w ) → C ∞ ( R , R )   Φ 0 , 0 Φ 0 , 1 . . .   Φ 1 , 0 Φ 1 , 1 . . . w   . . � �   dw . . ⊤ Q Φ ( w ) := dw w ⊤ . . · · ·     . . . dt dt   . Φ k , 0 Φ k , 1 . . . .   . . . . . · · · . . � � ⊤ � � = � L d k d ℓ dt k w Φ k ,ℓ dt ℓ w k ,ℓ = 0

Example: total energy in mechanical system �� d � d � 2 � � 2 1 dt w 1 + dt w 2 2 + 1 � � k 1 w 2 1 + k 2 w 2 2 2     1 2 k 1 0 0 0 w 1 1 w 2 0 2 k 2 0 0     � � d d w 1 w 2 dt w 1 dt w 2     d 1 dt w 1 0 0 0     2 d 1 dt w 2 0 0 0 2

Outline Motivation and aim Definition Two-variable polynomial matrices The calculus of B/QDFs

Two-variable polynomial matrices for BDFs � Φ k ,ℓ ∈ R w 1 × w 2 � k ,ℓ = 0 ,..., L L ( d k d ℓ � dt k w 1 ) ⊤ Φ k ,ℓ L Φ ( w 1 , w 2 ) = dt ℓ w 2 k ,ℓ = 0 Φ( ζ, η ) = � L k ,ℓ = 0 Φ k ,ℓ ζ k η ℓ

Two-variable polynomial matrices for BDFs � Φ k ,ℓ ∈ R w 1 × w 2 � k ,ℓ = 0 ,..., L L ( d k d ℓ � dt k w 1 ) ⊤ Φ k ,ℓ L Φ ( w 1 , w 2 ) = dt ℓ w 2 k ,ℓ = 0 Φ( ζ, η ) = � L k ,ℓ = 0 Φ k ,ℓ ζ k η ℓ

Two-variable polynomial matrices for BDFs � Φ k ,ℓ ∈ R w 1 × w 2 � k ,ℓ = 0 ,..., L L ( d k d ℓ � dt k w 1 ) ⊤ Φ k ,ℓ L Φ ( w 1 , w 2 ) = dt ℓ w 2 k ,ℓ = 0 Φ( ζ, η ) = � L k ,ℓ = 0 Φ k ,ℓ ζ k η ℓ 2-variable polynomial matrix associated with L Φ

Two-variable polynomial matrices for QDFs � Φ k ,ℓ ∈ R w × w � k ,ℓ = 0 ,..., L symmetric (Φ k ,ℓ = Φ ⊤ ℓ, k ) L ( d k d ℓ � dt k w ) ⊤ Φ k ,ℓ Q Φ ( w ) = dt ℓ w k ,ℓ = 0 Φ( ζ, η ) = � L k ,ℓ = 0 Φ k ,ℓ ζ k η ℓ symmetric : Φ( ζ, η ) = Φ( η, ζ ) ⊤

Example: total energy in mechanical system     1 2 k 1 0 0 0 w 1 1 w 2 0 2 k 2 0 0 � �     d d Q E ( w 1 , w 2 ) = w 1 w 2 dt w 1 dt w 2     d 1 dt w 1 0 0 0     2 d 1 dt w 2 0 0 0 2 � 1 � � 1 � 2 k 1 0 2 ζη 0 E ( ζ, η ) = + 1 1 0 2 k 2 0 2 ζη

Historical intermezzo

Historical intermezzo stability tests (‘60s)

Historical intermezzo path integrals (‘60s) stability tests (‘60s)

Historical intermezzo Lyapunov functionals (‘80s) path integrals (‘60s) stability tests (‘60s)

Historical intermezzo Lyapunov functionals (‘80s) path integrals (‘60s) stability tests (‘60s) QDFs (1998)

Outline Motivation and aim Definition Two-variable polynomial matrices The calculus of B/QDFs

The calculus of B/QDFs Using powers of ζ and η as placeholders, B/QDF � two-variable polynomial matrix

The calculus of B/QDFs Using powers of ζ and η as placeholders, B/QDF � two-variable polynomial matrix Operations algebraic and properties operations/properties � of B/QDF on two-variable matrix

Differentiation • Φ ∈ R w × w [ ζ, η ] . Φ derivative of Q Φ : s Φ : C ∞ ( R , R w ) → C ∞ ( R , R ) Q • Φ ( w ) := d Q • dt ( Q Φ ( w )) • Φ( ζ, η ) = ( ζ + η )Φ( ζ, η ) Two-variable version of Leibniz’s rule

Integration D ( R , R • ) C ∞ -compact-support trajectories L Φ : D ( R , R w 1 ) × D ( R , R w 2 ) → D ( R , R ) � L Φ : D ( R , R w 1 ) × D ( R , R w 2 ) → R � + ∞ � L Φ ( w 1 , w 2 ) := −∞ L Φ ( w 1 , w 2 ) dt Analogous for QDFs

Part II: Applications

Outline Lyapunov theory Dissipativity theory Balancing and model reduction

Nonnegativity and positivity along a behavior B Q Φ ≥ 0 if Q Φ ( w ) ≥ 0 ∀ w ∈ B

Nonnegativity and positivity along a behavior B Q Φ ≥ 0 if Q Φ ( w ) ≥ 0 ∀ w ∈ B B B Q Φ > 0 if Q Φ ≥ 0, and [ Q Φ ( w ) = 0 ] = ⇒ [ w = 0 ]

Nonnegativity and positivity along a behavior B Q Φ ≥ 0 if Q Φ ( w ) ≥ 0 ∀ w ∈ B B B Q Φ > 0 if Q Φ ≥ 0, and [ Q Φ ( w ) = 0 ] = ⇒ [ w = 0 ] B Prop.: Let B = ker R ( d ≥ 0 iff there exist dt ) . Then Q Φ D ∈ R •× w [ ξ ] , X ∈ R •× w [ ζ, η ] such that Φ( ζ, η ) = D ( ζ ) ⊤ D ( η ) + R ( ζ ) ⊤ X ( ζ, η ) + X ( η, ζ ) ⊤ R ( η ) � �� � � �� � ≥ 0 for all w = 0 if evaluated on B

Lyapunov theory B autonomous is asymptotically stable iflim t →∞ w ( t ) = 0 ∀ w ∈ B B = ker R ( d dt ) stable ⇐ ⇒ det ( R ) Hurwitz

Lyapunov theory B autonomous is asymptotically stable iflim t →∞ w ( t ) = 0 ∀ w ∈ B B = ker R ( d dt ) stable ⇐ ⇒ det ( R ) Hurwitz Theorem: B asymptotically stable iff B B exists Q Φ such that Q Φ ≥ 0 and Q • < 0 Φ

Example � � d 2 r ( ξ ) = ξ 2 + 3 ξ + 2 dt 2 + 3 d B = ker dt + 2

Example � � d 2 r ( ξ ) = ξ 2 + 3 ξ + 2 dt 2 + 3 d B = ker dt + 2 B < 0, e.g. Ψ( ζ, η ) = − ζη ; Choose Ψ( ζ, η ) s.t. Q Ψ

Example � � d 2 r ( ξ ) = ξ 2 + 3 ξ + 2 dt 2 + 3 d B = ker dt + 2 B < 0, e.g. Ψ( ζ, η ) = − ζη ; Choose Ψ( ζ, η ) s.t. Q Ψ d Find Φ( ζ, η ) s.t. dt Q Φ ( w ) = Q Ψ ( w ) for all w ∈ B : ( ζ + η )Φ( ζ, η ) = Ψ( ζ, η ) + r ( ζ ) x ( η ) + x ( ζ ) r ( η ) � �� � = 0 on B

Example � � d 2 r ( ξ ) = ξ 2 + 3 ξ + 2 dt 2 + 3 d B = ker dt + 2 B < 0, e.g. Ψ( ζ, η ) = − ζη ; Choose Ψ( ζ, η ) s.t. Q Ψ d Find Φ( ζ, η ) s.t. dt Q Φ ( w ) = Q Ψ ( w ) for all w ∈ B : ( ζ + η )Φ( ζ, η ) = Ψ( ζ, η ) + r ( ζ ) x ( η ) + x ( ζ ) r ( η ) � �� � = 0 on B d dt Q Φ ( w ) = Q Ψ ( w ) for all w ∈ B

Example � � d 2 r ( ξ ) = ξ 2 + 3 ξ + 2 dt 2 + 3 d B = ker dt + 2 B < 0, e.g. Ψ( ζ, η ) = − ζη ; Choose Ψ( ζ, η ) s.t. Q Ψ d Find Φ( ζ, η ) s.t. dt Q Φ ( w ) = Q Ψ ( w ) for all w ∈ B : ( ζ + η )Φ( ζ, η ) = Ψ( ζ, η ) + r ( ζ ) x ( η ) + x ( ζ ) r ( η ) � �� � = 0 on B Equivalent to solving polynomial Lyapunov equation 0 = Ψ( − ξ, ξ ) + r ( − ξ ) x ( ξ ) + x ( − ξ ) r ( ξ ) ξ 2 ξ 2 − 3 ξ + 2 ξ 2 + 3 ξ + 2 ❀ x ( ξ ) = 1 6 ξ