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Theory of differential-algebraicequations
Seminar 18-2-2004Arie Verhoeven
Technische Universiteit Eindhoven
Theory of differential-algebraic equations – p.1
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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
Theory of differential-algebraic equations – p.2
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Survey
• Introduction
• Example
• Theory of linear DAEs
• Transferable DAEs
• Index
• Summary
Theory of differential-algebraic equations – p.3
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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).
Theory of differential-algebraic equations – p.4
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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)
Theory of differential-algebraic equations – p.5
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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
Theory of differential-algebraic equations – p.6
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Varactor circuit
0
1
2
v=V(t)
i=i_V
Model equations:
CV1 = iVV1 = −V (t)
iV = iLL d
dtiL = −V2
Theory of differential-algebraic equations – p.7
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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!
Theory of differential-algebraic equations – p.8
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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)
Theory of differential-algebraic equations – p.9
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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).
Theory of differential-algebraic equations – p.10
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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.
Theory of differential-algebraic equations – p.11
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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)
Theory of differential-algebraic equations – p.12
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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
.
Theory of differential-algebraic equations – p.13
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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.
Theory of differential-algebraic equations – p.14
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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.
Theory of differential-algebraic equations – p.15
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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.
Theory of differential-algebraic equations – p.16
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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
Theory of differential-algebraic equations – p.17
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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.
Theory of differential-algebraic equations – p.18
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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.
Theory of differential-algebraic equations – p.19
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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.
Theory of differential-algebraic equations – p.20
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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.
Theory of differential-algebraic equations – p.21
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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
Theory of differential-algebraic equations – p.22
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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.
Theory of differential-algebraic equations – p.23
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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.
Theory of differential-algebraic equations – p.24
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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
Theory of differential-algebraic equations – p.25