1 Solution Theory 1.1 Motivation: Modeling of electrical circuits L - PDF document

SAMM DAEs, Elgersburg 2014: Lecture 1 Version: 22. September 2014 If you have any questions concerning this material (in particular, specific pointers to literature), please don’t hesitate to contact me via email: trenn@mathematik.uni-kl.de 1 Solution Theory 1.1 Motivation: Modeling of electrical circuits L u R C Basic components: • Resistors: v R ( t ) = Ri R ( t ) • Capacitor: C d dt v C ( t ) = i C ( t ) • Coil: L d dt i L ( t ) = v L ( t ) • Voltage source: v S ( t ) = u ( t ) All components have the same form: E,A ∈ R ℓ × n , B ∈ R ℓ × m E ˙ x = Ax + Bu � v R � • Resistor: x = , E = [0 , 0], A = [ − 1 ,R ], B = [] i R � v C � • Capacitor: x = , E = [ C, 0], A = [0 , 1], B = [] i C � v C � • Inductor: x = , E = [0 ,L ], A = [1 , 0], B = [] i C � v C � • Voltage source x = , E = [0 , 0], A = [ − 1 , 0], B = [1] i C i RC v RC R C Connecting components: Component equations remain unchanged! + Kirchhoffs laws: v RC = v R , i RC = i R + i C , v R + v C = 0 Stephan Trenn, TU Kaiserslautern 1/5

SAMM DAEs, Elgersburg 2014: Lecture 1 Version: 22. September 2014 Results again in E ˙ x = Ax + Bu with x = ( v R , i R , v C , i C , v RC , i RC ) and     0 0 − 1 R 0 0 1 C         E = 0 , A = 1 − 1         0 − 1 − 1 1     0 1 1 Altogether: x = ( v R , i R , v C , i C , v L , i L , v S , i S )       0 0 − 1 0 R C 0 0 1 0             0 1 0 0 L             0 0 1 0 1       E = , A = , B =       0 1 1 − 1 0             0 − 1 1 0             0 − 1 1 0       0 1 1 1 0 1.2 DAEs: What is different to ODEs Example:       0 1 0 1 0 0 f 1  ˙  x + 0 0 0 x = 0 1 0 f 2     0 0 0 0 0 0 f 3 x 1 = − f 1 − ˙ x 2 = x 1 + f 1 ˙ f 2 0 = x 2 + f 2 x 2 = − f 2 0 = f 3 no restriction on x 3 Observations: • For fixed inhomogeneity, initial values cannot be chosen arbitrarily ( x 1 (0) = − f 1 (0) − ˙ f 2 (0), x 2 (0) = f 2 (0)) • For fixed inhomogeneity, solution not uniquely determined by initial value ( x 3 free) • Inhomogeneity not arbitrary - structural restrictions ( f 3 = 0) - differentiability restrictions ( ˙ f 2 must be well defined) 1.3 Special DAE-cases a) ODEs: x = Ax + f ˙ • Initial values: arbitrary • Solution uniquely determined by f and x (0) • Inhomogeneity constraints - no structural constraints - no differentiability constraints Stephan Trenn, TU Kaiserslautern 2/5

SAMM DAEs, Elgersburg 2014: Lecture 1 Version: 22. September 2014 b) nilpotent DAEs:   0 ...   1   x = x + f ˙   ... ...     1 0 ⇔ 0 = x 1 + f 1 − → x 1 = − f 1 x 2 = − f 2 − ˙ x 1 = x 2 + f 2 ˙ − → f 1 . . . . . . . . . n � f ( n − i ) x n − 1 = x n + f n ˙ − → x n = − i i =1 In general: with N nilpotent, i.e. N n = 0 N ˙ x = x + f N d dt x + N ˙ f = x + f + N ˙ ⇒ N 2 ¨ x = N ˙ f N d N d n − 1 � ⇒ 0 = N n x ( n ) = x + dt dt N i f ( i ) ⇒ · · · i =0 n − 1 � N i f ( i ) ⇒ x = − i =0 is unique solution of N ˙ x = x + f • Initial values: fixed by inhomogeneity • Solution uniquely determined by f • Inhomogeneity constraints: - no structural constraints - differentiability constraints: ( N i f ) ( i ) needs to be well defined c) underdetermined DAEs n     1 0 0 1 ... ... ... ...      ˙ x =  x + f n − 1   1 0 0 1     0 1 0     x 1 ˙ x 1 ... ... .    .  .   . .  .   .    ⇔  =  +  + f   . . ...       0 1    ˙ x n − 1 x n − 1 x n 0 ⇔ ODE with additional “input” x n • Initial values: arbitrary • Solution not uniquely determined by x (0) and f • Inhomogeneity constraints: none Stephan Trenn, TU Kaiserslautern 3/5

SAMM DAEs, Elgersburg 2014: Lecture 1 Version: 22. September 2014 d) overdetermined DAEs n     0 1 ... ...     1 0         ... ... ... ...     x = ˙ x + f n + 1         ... ...     0 1     1 0   0   f 1 ...   1 .    .  ⇔ x = x + ˙  ∧ ˙ x n = f n +1   . ... ...      f n 1 0 � �� � N n − 1 n � � N i f ( i ) ∧ ˙ ! f n − i +1 ⇔ x = − x n = − = f n +1 i i =0 i =1 � �� � n +1 � f ( n +1 − i ) ⇔ = 0 i i =1 • Initial valus: fixed by inhomogeneity • Solution uniquely determined by f • Inhomogeneity constraints - structural constraint: � n +1 i =1 f ( n +1 − i ) = 0 i - differentiability constraint: f n +1 − i needs to be well defined i We will see: There are no other cases ! 1.4 Solution behavior, equivalence and normal forms Solution behavior of E ˙ x = Ax + f � � � � x ∈ C 1 ( R → R n ) , f : R → R m , E ˙ B [ E,A,I ] := ( x,f ) x = Ax + f Fact 1: For any invertible matrix S ∈ R m × m : ( x,f ) ∈ B [ E,A,I ] ⇔ ( x,Sf ) ∈ B [ SE,SA,I ] Fact 2: For coordinate transformation x = Tz , T ∈ R n × n invertible: ( x,f ) ∈ B [ E,A,I ] ⇔ ( T − 1 x,f ) ∈ B [ ET,AT,I ] Together: ( x,f ) ∈ B [ E,A,I ] ⇔ ( T − 1 ,Sf ) ∈ B [ SET,SAT,I ] Definition 1. ( E 1 ,A 1 ), ( E 2 ,A 2 ) are called equivalent : ⇔ ( E 2 ,A 2 ) = ( SE 1 T, SA 1 T ) short: S,T ( E 1 ,A 1 ) ∼ ∼ = ( E 2 ,A 2 ) or ( E 1 ,A 1 ) = ( E 2 ,A 2 ) Stephan Trenn, TU Kaiserslautern 4/5

SAMM DAEs, Elgersburg 2014: Lecture 1 Version: 22. September 2014 Theorem 1 (Quasi-Kronecker Form) . For any E,A ∈ R ℓ × m , ∃ invertible S ∈ R ℓ × ℓ and invertible T ∈ R n × n :       E U A U         I   J         S,T       ∼ N I ( E,A ) , =                         E O A O       where ( E U ,A U ) consists of underdetermined blocks on the diagonal, N is nilpotent, and ( E O ,A O ) consists of overdetermined diagonal bolcks Example:             0 1 0 1 0 0 0 1 1 0  ∼  ,  , 0 0 0 0 1 0 0 0 0 1 =          0 0 0 0 0 0 | | Corollary 1. E ˙ x = Ax + f has solution x for any sufficiently smooth f and each solution x is uniquely determined by x (0) and f ⇔ �� I � � J �� 0 0 ( E,A ) ∼ = , , N nilpotent 0 0 N I ( E,A ) is then called regular. Stephan Trenn, TU Kaiserslautern 5/5