Theory of differential-algebraic equations · "Differential-algebraic equations and their numerical treatment" by Griepentrog/März "Solving ordinary differential equations II" by

ScientificComputingGroup

Theory of differential-algebraicequations

Seminar 18-2-2004Arie Verhoeven

[email protected]

Technische Universiteit Eindhoven

Theory of differential-algebraic equations – p.1


Seminar: DAEs

• 18-2: "Theory of DAEs" by AV

• 10-3: "Methods for DAEs" by SandraAllaart-Bruin

• 17-3: "DAEs in electrical circuits" by PieterHeres

• 31-3: "DAEs in multibody systems" byMichael Sizov



Survey

• Introduction

• Example

• Theory of linear DAEs

• Transferable DAEs

• Index

• Summary



Introduction to DAEs

• Solution x : R → Rm satisfies

f(t, x, x) = 0,

where f : R × Rm × R

m → Rm is a given

function.

• If ∂f∂x is invertible, x is also determined by

ODE:x = g(t, x).



Applications

• Mathematical models of physicalphenomenons, such as electrical circuits ormechanical multibody-systems

d

dtq(t, x) + j(t, x) = 0.

• Singular perturbations of ODEs{

x1 = f1(t, x1, x2, ε)

εx2 = f2(t, x1, x2, ε)



Applications

• Constrained variational problems, e.g.optimalcontrol problems

J [x, u] =∫ t1

t0g(x, u, s)ds

x = f(x, u, t)

Euler-Lagrange equations:

x = f(x, u, t)

λ = − ∂g∂x(x, u, t) −

(

∂f∂x

)T

λ

0 = ∂g∂u(x, u, t) +

(

∂f∂u

)T

λ



Varactor circuit

0

1

2

v=V(t)

i=i_V

Model equations:

CV1 = iVV1 = −V (t)

iV = iLL d

dtiL = −V2



Varactor circuit

Parameters:

C 10−4 F

L 10−3 H

T 10−3 s

f 105 Vs2

V (t) ft2 V

Exact solution:

V1(t) = −V (t) = −ft2

iL(t) = iV (t) = −CV (t) = −2Cft

V2(t) = −LCV (t) = −2LCf

So, V2 is constant!



Numerical solution

• High accuracy(Numform 8)

• 1.3 million stepsizes

• No convergence

0.020.0u

40.0u60.0u

80.0u100.0u

120.0u140.0u

160.0u180.0u

-200.0m

-175.0m

-150.0m

-125.0m

-100.0m

-75.0m

-50.0m

-25.0m

0.0

25.0m

Index 3 varactor circuit

T

- y1-axis -

VN(2)



Classification

• Linear implicit or quasilinear DAE’s:

A(x, t)x + g(x, t) = 0.

• Semi-explicit DAE’s:

x1 = f1(t, x1, x2)

0 = f2(t, x1, x2)

• Linear DAE’s:

A(t)x + B(t)x = q(t).



Linear timeindependent form

Ax + Bx = q(t).

¬∀λλA+B singular ⇔ λA+B regular pencil ⇔ DAE solvable

Theorem 1 Suppose that λA + B is a regular pencil. Thenthere exist nonsingular matrices P and Q such that

PAQ =

I 0

0 N

PBQ =

C 0

0 I

Here, N is a nilpotent matrix with index k.



Solution

New variables: x = Qy, r = Pq

y1 + Cy1 = r1(t)

N y2 + y2 = r2(t)

y2 = (Nd

dt+ I)−1r2 =

k−1∑

i=0

(−1)iN ir(i)2

Consistent initial solution if k ≥ 2:

y2(0) = r2(0) + . . . + (−1)k−1Nk−1r(k−1)2 (0)

Steady-state solution at t = 0: y2(0) = r2(0)



Linear example 1

x1 + x3 = q1

x2 + x1 = q2

x2 = q3

The DAE can also be written in the next form:

Ax + Bx = q,

where A =

1 0 0

0 1 0

0 0 0

, B =

0 0 1

1 0 0

0 1 0

.



Linear example 2

P = I, Q =

0 1 0

0 0 1

1 0 0

Decomposition:

PAQ =

0 1 0

0 0 1

0 0 0

, PBQ = I.

Index k = 3.



General DAEs

f(x, x, t) = 0.

It is assumed that

• ∂f∂x has constant rank on G

• ker( ∂f∂x) = N(t) depends only on t

• N(t) is smooth

Define a projector Q(t) onto the nullspace of ∂f∂x , such that

∀x∈Rm

∂f∂x

Q(t)x = 0.



Transferability

Definition 1 The general DAE f(x, x, t) = 0 iscalled transferable if

∂f∂x

+∂f∂x

Q(t)

has a bounded inverse.Theorem 2 Assume that the DAE is transferableand | ∂f

∂x| is bounded on G. Then, its IVP isuniquely solvable.



Index

• The index of a DAE is a measure of thedifficulty.

• The local index of a DAE is the nilpotency

index of the matrix pencil λ ∂f∂x + ∂f

∂x at t.

• Geometrical index

• Differential index

• Tractability index

• Perturbation index



Differential index

f(x, x, t) = 0ddt

f(x, x, t) = 0...

dµ

dtµf(x, x, t) = 0

The differential index is equal to µ if x dependsuniquely on x and t.

If f is twice differentiable, transferable DAEs have

differential index one.



Hessenberg form

The DAE is in Hessenberg form of size µ if

x1 = f1(x1, x2, . . . , xµ, t)

x2 = f2(x1, x2, . . . , xµ−1, t)...

xi = fi(xi−1, xi, . . . , xµ−1, t), 3 ≤ i ≤ µ − 1...

0 = fµ(xµ−1, t)

∂fµ∂xµ−1

∂fµ−1

∂xµ−2

· · ·∂f1∂xµ

invertible ⇒ differential index = µ.



Nonlinear example

x1 = ex3 + q1

x2 = ex1 + q2

0 = ex2 + q3

Because

∂f3∂x2

∂f2∂x1

∂f1∂x3

= ex2ex1ex3 6= 0,

the differential index is equal to 3.



Perturbation index

The DAE has the perturbation index k along asolution x if k is the smallest integer such that,for all functions x(t) having the defect

f(xδ, xδ, t) = δ(t),

there exists an estimate

‖x(t) − xδ(t)‖ ≤ C (‖x(t0) − xδ(t0)‖ + maxt ‖δ(t)‖

+ maxt ‖δ′(t)‖ + . . . + maxt ‖δ

(k−1)(t)‖)

for a constant C > 0, if δ is sufficiently small.



Overview of indices

1. Local index λn

2. Geometrical index γ

3. Differential index δ

4. Perturbation index π

5. Tractability index τ

Linear timeindependent DAEs: λn = δ = τAlways: δ ≤ π ≤ δ + 1First integral: δ = π

Solvable and δ ≤ 1: λn = δ



Index reduction

• Reducing the index by means ofdifferentiating the algebraic equations.

• Index reduction must only be applied toresponsible components.

• It needs an enormous amount of symboliccomputations to get the underlying ODE.Furthermore, high smoothness of the inputsources is necessary.

• The underlying ODE is not unique.

• The asymptotical and stability behaviour isnot well reflected.



Summary

• DAEs are more general than ODEs and havemore applications. However, they are alsoessentially more complex than ODEs.

• For linear time-independent DAEs, there is anice theory.

• Transferable DAEs have also nice properties.

• There are many different index concepts,which are not equivalent.

• Index reduction can be attractive for highindices, but has also disadvantages.



References

• "Numerical solution of intitial-value problemsin differential-algebraic equations" byBrenan/Campbell/Petzold

• "Differential-algebraic equations and theirnumerical treatment" by Griepentrog/März

• "Solving ordinary differential equations II" byHairer/Wanner

• "Ordinary differential equations in theory andpractice" by Mattheij/Molenaar

• "Numerical analysis of differential-algebraicequations" by C.Tischendorf


Documents

Theory of differential-algebraic equations · "Differential-algebraic equations and their numerical treatment" by Griepentrog/März "Solving ordinary differential equations II" by