Basics on Differential-Algebraic Equations (DAEs) Stephan Trenn - PowerPoint PPT Presentation

Basics on Differential-Algebraic Equations (DAEs) Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern ICCAS 2014, Seoul, Korea October 23rd, 2014, Tutorial Session TA06

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Contents Motivation: Modeling of electrical circuits 1 DAEs: Differences to ODEs 2 Special DAE-cases 3 Nilpotent DAEs Underdetermined DAEs Overdetermined DAEs Equivalence and quasi-Kronecker form/quasi-Weierstrass form 4 Wong sequences 5 Inconsistent initial values 6 Motivating example Consistency projector Switched DAEs 7 Definition and solution theory Impulse-freeness Stability Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Modeling of electrical circuits v L i L Basic circuit elements L i R i C Resistor : v R ( t ) = R i R ( t ) i S i C ( t ) = C d Capacitor : d t v C ( t ) v S v C v R u ( t ) C R v L ( t ) = L d Inductor : d t i L ( t ) Voltage source : v S ( t ) = u ( t ) DAEs All components are given by a differential-algebraic equation (DAE) E ˙ x = Ax + Bu Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Hierarchical model building v L i L Overall model ⇒ Again DAE: L i S i C i R v S v C v R E ˙ x = Ax + Bu u ( t ) C R           v R ˙ 0 0 -1 R v R ˙ i R C 0 0 1 i R                     v C ˙ 0 L 1 0 v C                     ˙ 0 0 i C -1 0 i C 1           = + u           0 v L ˙ 1 -1 v L                     ˙ 0 1 1 -1 i L i L                     0 1 1 -1 v S v S ˙           0 ˙ 1 -1 i S i S Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Contents Motivation: Modeling of electrical circuits 1 DAEs: Differences to ODEs 2 Special DAE-cases 3 Nilpotent DAEs Underdetermined DAEs Overdetermined DAEs Equivalence and quasi-Kronecker form/quasi-Weierstrass form 4 Wong sequences 5 Inconsistent initial values 6 Motivating example Consistency projector Switched DAEs 7 Definition and solution theory Impulse-freeness Stability Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Recall ODEs Ordinary differential equations (ODEs): x = Ax + f ˙ Initial values: arbitrary Solution uniquely determined by f and x (0) No inhomogeneity constraints Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Simple DAE example DAE 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 Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Conclusions from example Solution of example: x 1 = − f 1 − ˙ f 2 x 2 = − f 2 x 3 free f 3 = 0 necessary Differences to ODEs 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 ( d d t f 2 must be well defined) Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Contents Motivation: Modeling of electrical circuits 1 DAEs: Differences to ODEs 2 Special DAE-cases 3 Nilpotent DAEs Underdetermined DAEs Overdetermined DAEs Equivalence and quasi-Kronecker form/quasi-Weierstrass form 4 Wong sequences 5 Inconsistent initial values 6 Motivating example Consistency projector Switched DAEs 7 Definition and solution theory Impulse-freeness Stability Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs 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 x 3 = − f 3 − ˙ f 2 − ¨ x 2 = x 3 + f 3 ˙ − → f 1 . . . . . . . . . n � f ( n − i ) x n − 1 = x n + f n ˙ − → x n = − i i =1 Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs General nilpotent DAE with N nilpotent, i.e. N n = 0 In general: N ˙ x = x + f N d ⇒ N 2 ¨ d t x + N ˙ f = x + f + N ˙ x = N ˙ f N d ⇒ N 3 ... x + N 2 ¨ f + N 2 ¨ d t x = N 2 ¨ f = x + f + N ˙ f . . . N d n − 1 n − 1 � � ⇒ N n x ( n ) d t N i f ( i ) N i f ( i ) = x + ⇒ x = − � �� � i =0 i =0 =0 Properties Initial values: fixed by inhomogeneity Solution uniquely determined by f Inhomogeneity constraints: no structural constraints i =0 N i f ( i ) needs to be well defined differentiability constraints: � n − 1 Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Underdetermined DAEs n     1 0 0 1 ... ... ... ...          ˙ x =  x + f n − 1     1 0 0 1   1 0 0 1         x 1 ˙ 0 1 x 1 0 . . . ... ...         . . . . . .         ⇔  =  +  + f         x n − 2 ˙ 0 1 x n − 2 0      x n − 1 ˙ 0 x n − 1 x n ⇔ ODE with additional “input” x n Properties Initial values: arbitrary Solution not uniquely determined by x (0) and f Inhomogeneity constraints: none Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Overdetermined DAEs n     0 1 ... ...     1 0      ...   ...  x = ˙ x + f n + 1     0 1     1 0 0 1     1 0     0 f 1 ... .     . 1 .     ⇔ x = x + ˙  ∧ ˙ x n = f n +1  ...    f n − 1  0   1 0 f n � �� � =: 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 i =1 f ( n +1 − i ) = 0 i Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)

Motivation DAEs vs. ODEs Special DAE-cases QKF/QWF Wong sequences Inconsistent initial values Switched DAEs Overdetermined DAEs properties n − 1 n +1 � � N i f ( i ) ∧ f ( n +1 − i ) x = − = 0 i i =0 i =1 Properties Initial valus: fixed by inhomogeneity Solution uniquely determined by f Inhomogeneity constraints i =1 f ( n +1 − i ) structural constraint: � n +1 = 0 i differentiability constraint: f ( n +1 − i ) needs to be well defined i No other cases All DAEs are combinations of ODEs, nilpotent DAEs, underdetermined DAEs, overdetermined DAEs Stephan Trenn Technomathematics group, Dept. of Mathematics, University of Kaiserslautern Basics on Differential-Algebraic Equations (DAEs)