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A connection between time domain model order reduction and moment matching Manuela Hund joint with Jens Saak September 2, 2016 Partners: Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1. A connection between time domain model order reduction and moment matching Manuela Hund joint with Jens Saak September 2, 2016 Partners:

  2. Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-invariant (LTI) system x ( t ) = ˙ x ( t ) + B u ( t ) E A y ( t ) = x ( t ) C M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 2/24

  3. Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-invariant (LTI) system x ( t ) = ˙ x ( t ) + B u ( t ) E A y ( t ) = x ( t ) C E r = V T EV ∈ R m × m A r = V T AV ∈ R m × m B r = V T B ∈ R m × p C r = CV ∈ R q × m M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 2/24

  4. Introduction Model order reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-invariant (LTI) system x ( t ) = ˙ x ( t ) + B u ( t ) E A y ( t ) = x ( t ) C E r = V T EV ∈ R m × m A r = V T AV ∈ R m × m B r = V T B ∈ R m × p C r = CV ∈ R q × m Reduced LTI system: E r x r ( t ) = A r x r ( t )+ B r u ( t ) ˙ y r ( t ) = C r x r ( t ) M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 2/24

  5. Time domain MOR based on orthogonal polynomials Basic idea [ JIANG/CHEN 2012 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Input Single-Output (SISO) system ( p = q = 1): E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) . M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 3/24

  6. Time domain MOR based on orthogonal polynomials Basic idea [ JIANG/CHEN 2012 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Input Single-Output (SISO) system ( p = q = 1): E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) . Approximation of state and input: ( ∀ i : v i ∈ R n , w i ∈ R , g i : [ t 0 , t f ] → R ) m − 1 m − 1 � � x ( t ) ≈ x m ( t ) = v i g i ( t ) , u ( t ) ≈ u m ( t ) = w i ˙ g i ( t ) . i =0 i =1 Artificial initial condition: m − 1 � x 0 = x ( t 0 ) ≈ x m ( t 0 ) = v i g i ( t 0 ) . i =0 M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 3/24

  7. Time domain MOR based on orthogonal polynomials Restriction [ HUND 2015 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Input Single-Output (SISO) system ( p = q = 1): E ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) . Approximation of state and input: ( ∀ i : v i ∈ R n , w i ∈ R , g i : [ t 0 , t f ] → R ) m m � � x ( t ) ≈ x m +1 ( t ) = v i g i ( t ) , u ( t ) ≈ u m +1 ( t ) = w i ˙ g i ( t ) . i =1 i =1 Fixed initial condition:   0 .  ∈ R n . . x 0 = x ( t 0 ) =   .  0 M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 3/24

  8. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  9. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. application of differential recurrence formula in x m +1 ( t ) ⇒ application to state equation leads to expressions depending on g i ( t ) ˙ M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  10. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. application of differential recurrence formula in x m +1 ( t ) ⇒ application to state equation leads to expressions depending on g i ( t ) ˙ coefficient comparison leads to a linear system of equations Hv = f , where H ∈ R mn × mn , v ∈ R mn × 1 , f ∈ R mn × 1 M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  11. Time domain MOR based on orthogonal polynomials Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem 1 (Differential recurrence formula) [ HUND 2015 ] For three sequenced orthogonal polynomials g i ( t ), where i ∈ N 0 , it holds: g n ( t ) = α n ˙ g n +1 ( t ) + β n ˙ g n ( t ) + γ n ˙ g n − 1 ( t ) , n = 1 , 2 , . . . , where α n , β n , γ n are diffential recurrence coefficients. application of differential recurrence formula in x m +1 ( t ) ⇒ application to state equation leads to expressions depending on g i ( t ) ˙ coefficient comparison leads to a linear system of equations Hv = f , where H ∈ R mn × mn , v ∈ R mn × 1 , f ∈ R mn × 1 determine projection matrix V by orthogonalization of span { v 1 , . . . , v m } M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 4/24

  12. Time domain MOR based on orthogonal polynomials Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   E − β 1 A − γ 2 A 0 0 · · · · · · · · · . ...  .  − α 1 A E − β 2 A − γ 3 A .     . ...   .  0 − α 2 A E − β 3 A − γ 4 A .    . .  ... ... ... ... ...  . . H = ,   . .     . ... ... ... ...  .  . 0     . ... ... ...   .  . − γ m A  0 0 − α m − 1 A E − β m A · · · · · · · · ·   v 1 .   . v =  , .  v m   Bw 1 .   . f =  . .  Bw m M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 5/24

  13. Time domain MOR based on orthogonal polynomials Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   E − β 1 A − γ 2 A 0 0 · · · · · · · · · . ...  .  − α 1 A E − β 2 A − γ 3 A .     . ...   .  0 − α 2 A E − β 3 A − γ 4 A .    . .  ... ... ... ... ...  . . H = ,   . .     . ... ... ... ...  .  . 0     . ... ... ...   .  . − γ m A  0 0 − α m − 1 A E − β m A · · · · · · · · ·   v 1 .   . v =  , .  v m   Bw 1 .   . f =  . .  Bw m M. Hund, hund@mpi-magdeburg.mpg.de A connection between time domain MOR and moment matching 5/24

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