Intro LinSys H2Opt DataDriven Conclusions - Part 1 An Introduction to System-theoretic Methods for Model Reduction - Part II - Interpolatory Methods Serkan Gugercin Department of Mathematics, Virginia Tech Division of Computational Modeling and Data Analytics, Virginia Tech ICERM Semester Program - Spring 2020 Model and dimension reduction in uncertain and dynamic systems January 31, 2020, Providence, RI Thanks to: NSF , NIOSH, The Simons Foundation, and ICERM Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Outline Linear dynamical systems: E ˙ x ( t ) = A x ( t ) + B u ( t ) , y ( t ) = C x ( t ) Rational interpolation problem Projection-based rational interpolation Optimal rational interpolation Optimality in the H 2 norm Iterative Rational Krylov Algorithm Data-driven (frequency-domain) rational interpolation Loewner framework Time-domain Loewner: See Peherstorfer’s talk this afternoon. If time allows: E ˙ x ( t ) = A x ( t ) + N x u ( t ) + H ( x ⊗ x ) + B u ( t ) , y ( t ) = C x ( t ) Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Outline Linear dynamical systems: E ˙ x ( t ) = A x ( t ) + B u ( t ) , y ( t ) = C x ( t ) Rational interpolation problem Projection-based rational interpolation Optimal rational interpolation Optimality in the H 2 norm Iterative Rational Krylov Algorithm Data-driven (frequency-domain) rational interpolation Loewner framework Time-domain Loewner: See Peherstorfer’s talk this afternoon. If time allows: E ˙ x ( t ) = A x ( t ) + N x u ( t ) + H ( x ⊗ x ) + B u ( t ) , y ( t ) = C x ( t ) Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Outline Linear dynamical systems: E ˙ x ( t ) = A x ( t ) + B u ( t ) , y ( t ) = C x ( t ) Rational interpolation problem Projection-based rational interpolation Optimal rational interpolation Optimality in the H 2 norm Iterative Rational Krylov Algorithm Data-driven (frequency-domain) rational interpolation Loewner framework Time-domain Loewner: See Peherstorfer’s talk this afternoon. If time allows: E ˙ x ( t ) = A x ( t ) + N x u ( t ) + H ( x ⊗ x ) + B u ( t ) , y ( t ) = C x ( t ) Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Outline Linear dynamical systems: E ˙ x ( t ) = A x ( t ) + B u ( t ) , y ( t ) = C x ( t ) Rational interpolation problem Projection-based rational interpolation Optimal rational interpolation Optimality in the H 2 norm Iterative Rational Krylov Algorithm Data-driven (frequency-domain) rational interpolation Loewner framework Time-domain Loewner: See Peherstorfer’s talk this afternoon. If time allows: E ˙ x ( t ) = A x ( t ) + N x u ( t ) + H ( x ⊗ x ) + B u ( t ) , y ( t ) = C x ( t ) Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Indoor-air environment in a conference room light vent light inlet Z inlet table window inlet window inlet X Y Figure: Geometry for our Indoor-air Simulation: Example from [Borggaard/Cliff/G., 2011] , research under EEBHUB Four inlets, one return vent Thermal loads: two windows, two overhead lights and occupants A FE model for thermal energy transfer with frozen velocity field v : Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 ∂ T 1 ∂ t + v · ∇ T = RePr∆ T + Bu , E ˙ = ⇒ x ( t ) = A x ( t ) + B u ( t ) , y ( t ) = C x ( t ) and E , A ∈ R n × n with n = 202140 , x ∈ R n , u ∈ R m with m = 2 inputs (forcing) B ∈ R n × m and the temperature of the inflow air at all four vents, and 1 a disturbance caused by occupancy around the conference table, 2 y ∈ R q with q = 2 outputs (measurements) C ∈ R q × n and the temperature at a sensor location on the max x wall, 1 the average temperature in an occupied volume around the table, 2 Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Linear Dynamical Systems E ˙ x ( t ) = A x ( t ) + B u ( t ) u ( t ) − → y ( t ) → − S : y ( t ) = C x ( t ) + D u ( t ) A , E ∈ R n × n , B ∈ R n × m , C ∈ R q × n and D ∈ R q × m x ( t ) ∈ R n : states, u ( t ) ∈ R m : Input, y ( t ) ∈ R q : Output State-space dimension, n , is quite large What is important is the mapping “ u �→ y ”, NOT the complete state information x ( t ) = Remove the unimportant states. ⇒ Parametrized linear dynamical systems (see Beattie’s talk on Feb 4) x ( t ; p ) = A ( p ) x ( t ; p ) + B ( p ) u ( t ) , y ( t ; p ) = C ( p ) x ( t ; p ) , p ∈ C ν E ( p ) ˙ Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Linear Dynamical Systems E ˙ x ( t ) = A x ( t ) + B u ( t ) u ( t ) − → y ( t ) → − S : y ( t ) = C x ( t ) + D u ( t ) A , E ∈ R n × n , B ∈ R n × m , C ∈ R q × n and D ∈ R q × m x ( t ) ∈ R n : states, u ( t ) ∈ R m : Input, y ( t ) ∈ R q : Output State-space dimension, n , is quite large What is important is the mapping “ u �→ y ”, NOT the complete state information x ( t ) = Remove the unimportant states. ⇒ Parametrized linear dynamical systems (see Beattie’s talk on Feb 4) x ( t ; p ) = A ( p ) x ( t ; p ) + B ( p ) u ( t ) , y ( t ; p ) = C ( p ) x ( t ; p ) , p ∈ C ν E ( p ) ˙ Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Project the dynamics onto r -dimenisonal dimensional subspaces E r ˙ x r ( t ) = A r x r ( t ) + B r u ( t ) u ( t ) − → y r ( t ) ≈ y ( t ) S r : → − y r ( t ) = C r x r ( t ) + D r u ( t ) with A r , E r ∈ R r × r , B r ∈ R r × m , C r ∈ R q × r , and D r ∈ R q × m such that � y − y r � is small in an appropriate norm Important structural properties of S are preserved The procedure is computationally efficient . For simplicity of notation, assume m = q = 1 : B → b ∈ R n , C → c T ∈ R n , and, D → d ∈ R = ⇒ u ( t ) , y ( t ) ∈ R For the MIMO case details, see [Antoulas/Beattie/G.,20] . Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Project the dynamics onto r -dimenisonal dimensional subspaces E r ˙ x r ( t ) = A r x r ( t ) + B r u ( t ) u ( t ) − → y r ( t ) ≈ y ( t ) S r : → − y r ( t ) = C r x r ( t ) + D r u ( t ) with A r , E r ∈ R r × r , B r ∈ R r × m , C r ∈ R q × r , and D r ∈ R q × m such that � y − y r � is small in an appropriate norm Important structural properties of S are preserved The procedure is computationally efficient . For simplicity of notation, assume m = q = 1 : B → b ∈ R n , C → c T ∈ R n , and, D → d ∈ R = ⇒ u ( t ) , y ( t ) ∈ R For the MIMO case details, see [Antoulas/Beattie/G.,20] . Gugercin Interpolatory methods for model reduction
c T b T c r Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt n r n E , A r E r , A r b r Model Reduction = ⇒ Figure: Projection-based Model Reduction Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Model Reduction via Projection Choose V r = Range ( V r ) : the r -dimensional right modeling subspace (the trial subspace) where V r ∈ R n × r and W r = Range ( W r ) , the r -dimensional left modeling subspace (test subspace) where W r ∈ R n × r Approximate x ( t ) ≈ V r x r ( t ) by forcing x r ( t ) to satisfy ���� ���� ���� n × r n × 1 r × 1 W rT ( EV r ˙ x r − AV r x r − b u ) = 0 (Petrov-Galerkin) Leads to a reduced order model: E r = W rT EV r A r = W rT AV r b r = W rT b , , , c r = V r c , d r = d ���� � �� � � �� � � �� � ���� 1 × 1 r × r r × r r × 1 q × r Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Impulse Response and Transfer Functions � t S : u ( t ) �→ y ( t ) = ( S u )( t ) = h ( t − τ ) u ( τ ) d τ. −∞ h ( t ) = c T e A t b ( impulse response ) Let E = I and d = 0 : � ∞ h ( τ ) e − s τ d τ = c T ( s E − A ) − 1 b + d . H ( s ) = 0 = Transfer function � − 3 � � 1 � � 0 � − 2 Take E = I 2 , A = , b = , c = , d = 0 . 1 0 0 1 s + 1 + − 1 1 1 h ( t ) = e − t − e − 2 t ⇐ ⇒ H ( s ) = s 2 + 3 s + 2 = s + 2 Gugercin Interpolatory methods for model reduction
Intro LinSys H2Opt DataDriven Conclusions - Part 1 Settings Proj Meas Intrplt Impulse Response and Transfer Functions � t S : u ( t ) �→ y ( t ) = ( S u )( t ) = h ( t − τ ) u ( τ ) d τ. −∞ h ( t ) = c T e A t b ( impulse response ) Let E = I and d = 0 : � ∞ h ( τ ) e − s τ d τ = c T ( s E − A ) − 1 b + d . H ( s ) = 0 = Transfer function � − 3 � � 1 � � 0 � − 2 Take E = I 2 , A = , b = , c = , d = 0 . 1 0 0 1 s + 1 + − 1 1 1 h ( t ) = e − t − e − 2 t ⇐ ⇒ H ( s ) = s 2 + 3 s + 2 = s + 2 Gugercin Interpolatory methods for model reduction
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