Differential-algebraic equations. Control and Numerics Ivm).pdf · DAEs provide a unified framework for the analysis, simulation and control of (coupled) dynamical systems (continuous

Differential-algebraic equations.Control and Numerics I

Volker MehrmannInstitut für Mathematik

Technische Universität Berlin

P. Kunkel

DFG Research Center MATHEONMathematics for key technologies

8.3.10 Elgersburg

Outline

1 Introduction2 Applications3 Research projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs

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Theses. Modern key technologies need modeling, simulation, control

and optimization of complex dynamical systems.. Simulation and control of systems form the third pillar of

scientific development besides theory and experiment.. Most complex systems in key technologies are multi-physics

systems.. We need mathematical techniques to analyze the dynamics of

complex systems.. Modeling, analysis, numerical methods and

control/optimization techniques should go hand in hand.. New levels of interdisciplinary cooperation and a new

modeling paradigm is needed.. Differential-Algebraic Equations (DAEs) equations provide the

ideal framework for such a paradigm.

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What are DAEs/descriptor systems ?Differential-algebraic equations (DAEs), descriptor systems,singular differential eqns, algebro-differential eqns, . . .are implicit systems of differential equations of the form

0 = F(t , ξ,u, ξ, p, ω),

y1 = G1(t , ξ,u,p, ω),

y2 = G2(t , ξ,u,p, ω),

with F ∈ C0(R× Dξ × Du × Dξ × Dp × Dω,R`),Gi ∈ C0(R× Dξ × Du × Dp × Dω,Rpi ), i = 1,2.. t ∈ I ⊂ R is the time,. ξ denotes the state (finite or infinite dimensional), ξ = d

dt ξ,. u denotes control inputs, ω denotes

uncertainties/disturbances,. y1 denotes controlled, y2 measured outputs,. p denotes parameters.

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Linear DAEs/descriptor systems

In the linear case (linearization along non-stationary solutions)we get

E(t ,p)ξ = A(t ,p)ξ + B1(t ,p)u + B2(t ,p)ω + φ(t ,p),

y1 = C1(t ,p)ξ + D11(t ,p)u + D12(t ,p)ω + ψ1(t),

y2 = C2(t ,p)ξ + D21(t ,p)u + D22(t ,p)ω + ψ2(t).

or (linearization along stationary solutions)

E(p)ξ = A(p)ξ + B1(p)u + B2(p)ω,

y1 = C1(p)ξ + D11(p)u + D12(p)ω,

y2 = C2(p)ξ + D21(p)u + D22(p)ω.

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Why DAEs?

DAEs provide a unified framework for the analysis,simulation and control of (coupled) dynamical systems(continuous and discrete time).

. Automatic modeling leads to DAEs. (Constraints at interfaces).

. Conservation laws lead to DAEs. (Conservation of mass,energy, momentum).

. Coupling of solvers leads to DAEs (discrete time).

. Control problems are DAEs (behavior).

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Outline

1 Introduction2 Applications3 Research projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs

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Applications

Classical applications of DAE modeling.. Electronic circuit simulation (Kirchhoff’s laws).. Simulation and control of mechanical multibody systems

(position or velocity constraints).. Flow simulation and flow control (mass conservation).. Metabolic networks (balance equations).. Simulation and control of systems from chemical engineering

(mass balances).. Simulation and control of traffic systems (mass conservation).. . . .

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A simple circuit

x1

x2

x3

e

e

R

CU

Figure: A simple electrical network

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DAE modeling. Charging a capacitor via a resistor (ideal electronic units).. Associate a potential xi , i = 1,2,3, with each node of the

circuit, zero potential x3 = 0.. The voltage source increases the potential x3 to x1 by U, i. e.,

x1 − x3 − U = 0.. By Kirchhoff’s first law the sum of the currents vanishes in

each node.. For the second node we obtain that

C(x3 − x2) + (x1 − x2)/R = 0, R is resistance of the resistorand C is the capacity of the capacitor.

. We get the DAE:

C(x3 − x2) + (x1 − x2)/R = 0,x1 − x3 − U = 0,

x3 = 0.

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A physical pendulum

z-

6

,,,

,,,

?

m

mg

l

x

y

Figure: A mechanical multibody system

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DAE modeling. Mass point with mass m in Cartesian coordinates (x , y) moves

under influence of gravity in a distance l around the origin.. Kinetic energy T = 1

2m(x2 + y2)

. potential energy U = mgy , where g is the gravity constant,

. Constraint equation x2 + y2 − l2 = 0,

. Lagrange function L = 12m(x2 + y2)−mgy − λ(x2 + y2 − l2)

. Equations of motion

ddt

(∂L∂q

)− ∂L∂q

= 0

for the variables q = x , y , λ, i. e.,. DAE model:

mx + 2xλ = 0,my + 2yλ + mg = 0,

x2 + y2 − l2 = 0.

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A chemical reactorChemical reactor in which a first order isomerization reactiontakes place and which is externally cooled.

. c0 the given feed reactant concentration,

. T0 the initial temperature,

. c(t) and T (t) the concentration and temperature at time t ,and

. R the reaction rate per unit volume,

. DAE model 1 0 00 1 00 0 0

cTR

=

k1(c0 − c)− Rk1(T0 − T ) + k2R − k3(T − TC)

R − k3 exp(−k4T )c

,. TC is the cooling temperature (control input),. k1, k2, k3, k4 are constants.

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Semi-discretized PDEs. The non-stationary Stokes equation is a linear model for the

laminar flow of a Newtonian fluid

ut = ∆u +∇p, ∇ · u = 0,

together with initial and boundary conditions.. u describes the velocity and p the pressure of the fluid.. Discretizing first the space variables with finite element or

finite difference methods gives the DAE

uh = Auh + Bph, BT uh = 0,

where uh and ph are semi-discrete approximations for u and p.. The non-uniqueness of a free constant in the pressure must

be fixed by the discretization method.

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Multi-physics systemsDAE modeling is standard in multi-physics systems.

Packages like MATLAB (SIMULINK, DYMOLA (MODELLICA) andSPICE like circuit simulators proceed as follows:. Modularized modeling of uni-physics components.. Network based connection of components.. Identification of input and output parameters.. Numerical simulation and control on full model.

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Outline

1 Introduction2 Applications3 Research projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs

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Automatic gearboxesModeling, simulation and control of automatic gearboxes.Project with Daimler AG (Peter Hamann)→ film.

Technological Application. Modeling of multi-model: multibody-system, including

elasticity, hydraulics.. Development of control methods for coupled system.. Real time control of gearbox.Goal: Decrease full consumption, improve smoothness ofswitching

Space discretization leads to a large hybrid control systemof nonlinear DAEs.

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Drop size distributionswith M. Kraume (Chemical Eng., TU Berlin), M. Schäfer (Mech. Eng.TU Darmstadt)

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Technological Application, Tasks

Chemical industry: pearl polymerization and extractionprocesses

. Modeling of coalescence and breakage in turbulent flow.

. Numerical methods for simulation of coupled system ofpopulation balance equations/fluid flow equations. → film.

. Development of optimal control methods for large scalecoupled systems

. Model reduction and observer design.

. Feedback control of real configurations via stirrer speed.

Goal: Achieve specified average drop diameter and smallstandard deviation for distribution by real time-control ofstirrer-speed.

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Mathematical system components

. Navier Stokes equation (flow field)

. Population balance equation (drop size distribution).

. One or two way coupling.

. Initial and boundary conditions.

Space discretization leads to an extremely large control systemof nonlinear DAEs.

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Active flow control

Project in Sfb 557 Control of complex shear flows, with F.Tröltzsch, M. Schmidt

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Technological Application, Tasks

Control of detached turbulent flow on airline wing

. Test case (backward step to compare experiment/numerics.)

. modeling of turbulent flow.

. Development of control methods for large scale coupledsystems.

. Model reduction and observer design.

. Optimal feedback control of real configurations via blowingand sucking of air in wing.

Ultimate goal: Force detached flow back to wing.

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Controlled flowMovement of recirculation bubble following reference curve.

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Black-box modeling with DAEs

Modeling becomes extremely convenient, but:. Numerical simulation does not always work, instabilities and

convergence problems occur (e.g. SIMULINK) !. Consistent initialization is difficult.. The discretized system may be unsolvable even if the DAE is

solvable and vice versa.. Numerical drift-off phenomenon.. Model reduction is difficult.. Classical control is difficult (non-proper transfer functions).Black-box DAE modeling pushes all difficulties into thenumerics. In general the methods cannot handle this!Today several packages (e.g. Dymola) use computer algebra toturn back to ODE, this is bad.

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Outline

1 Introduction2 Applications3 Research projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs

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Setting for this lecture

. We assume that space discretization has been done, i.e., weonly discuss differential-algebraic systems (DAEs).

. This is justified for the analysis.

. However, for the numerical solution methods typically spaceand time discretization have to be considered together.

. We ignore the dependence on parameters and disturbancesand consider only one type of outputs.

. We will discuss first, constant coefficient, then variablecoefficient and then nonlinear systems.

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DAE theory in behavior frameworkWe first take a mathematical (behavioral) point of view: In

0 = F (t , ξ,u, ξ), t ∈ Iy = G(t , ξ,u),

with F ∈ C0(R× Dξ × Du × Dξ,R`), or

E(t)x = A(t)ξ + B(t)u + φ(t), t ∈ Iy = C(t)ξ + D(t)u,

we introduce x = [yT , ξT ,uT ]T and obtain an over- andunder-determined DAE system

0 = F(t , x , x), t ∈ IE(t)x = A(t)x + f (t), t ∈ I.

In practice and computation, we keep variables separate.27 / 60

Classical solutionsDefinitionConsider an initial value problem for general DAEs

F (x , x , t) = 0, x(t0) = x0.

. A function x is called (classical) solution of the DAE if x is onetimes continuously differentiable and x satisfies the equationpointwise.

. It is called solution of the initial value problem if it is a solutionand satisfies the initial condition.

. An initial condition is called consistent if the correspondinginitial value problem is solvable, i.e. has at least one solution.

Other solvability concepts, weak or distributional solutions.

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Outline

1 Introduction2 Applications3 Research projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs

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Linear DAEs with constant coefficientsWeierstraß/Kronecker 1890-1896 Consider

Ex = Ax + f (t), x(t0) = x0,

where E ,A ∈ C`,n and f ∈ C(I,C`).Scaling from the left and changes of basis with nonsingularmatrices.

PEQ ˙x = PAQx + Pf (t), x(t0) = x0.

DefinitionTwo pairs of matrices (Ei ,Ai), i = 1,2, are called (strongly)equivalent if there exist invertible matrices P ∈ C`,`,Q ∈ Cn,n withE2 = PE1Q, A2 = PA1Q. Write as:

[E2,A2] = P[E1,A1]

[Q 00 Q

].

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Kronecker canonical form (KCF)Theorem (Kronecker 1896)For every pair E ,A ∈ C`,n there exist nonsingularP ∈ C`,`,Q ∈ Cn,n such that P(λE − A)Q =Diag (Lε1 , . . . ,Lεp ,Mη1 , . . . ,Mηq , Jρ1 , . . . , Jρv ,Nσ1 , . . . ,Nσw ),

Lεj = λ

0 1

. . .. . .0 1

−

1 0

. . .. . .1 0

,

Jρj = λ

1

. . .1

−

λj 1

. . .

. . . 1λj

, Mηj = λ

1

0. . .

. . . 10

−

0

1. . .

. . . 01

,

Nσj = λ

0 1

. . .. . .

. . . 10

−

1

. . .1

, Lεj = MTεj

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Regularity and index

DefinitionA matrix pencil λE − A, E ,A ∈ C`,n, is called regular if ` = n andif

P(λ) = det(λE − A)

is not identically 0, otherwise singular. The size ν of the largestN-block in the KCF is the (differentiation) d-index of λE − A.

Control systems in behavior form have singular pencils.

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Weierstraß canonical form (WCF)

Theorem (Weierstraß 1867)

For every regular pair E ,A ∈ Cn,n there exist nonsingularP,Q ∈ Cn,n such that

P(λE − A)Q = λ

[I 00 N

]−[

J 00 I

],

where J = diag(Jρ1 , . . . , Jρv ) and N = diag(Nσ1 , . . . ,Nσw ),

Jρj = λ

1

. . .1

−

λj 1

. . .

. . . 1λj

, Nσj = λ

0 1

. . .. . .

. . . 10

−

1

. . .1

,

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Solution formula in WCF

LemmaConsider a regular constant coefficient DAE in WCF

λ

[I 00 N

] [x1

x2

]=

[J 00 I

] [x1

x2

]+

[f1f2

],

[x1(t0)x2(t0)

]=

[x1,0

x2,0

]with d-index ν. The solution is

x1(t) = eJ(t−t0)x1,0 +

∫ t

t0eJ(t−s)f1(s)ds

x2(t) = −ν−1∑i=0

f (i)2 (t)

Consistent initial values have to satisfy x2(t0) = −∑ν−1

i=0 f (i)(t0).

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Solvability

Theorem (Campbell 1982)

Consider a linear constant coefficient system with regularλE − A and let f ∈ Cν(I,Cn).Then the system is solvable and every consistent initial conditionfixes a unique solution.

Note that this is not an -if and only if- result.[10

]x =

[01

]x +

[0−1

]is not regular but has unique solution x = 1.

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The Drazin inverseLet

TET−1 =

[J1

J0

]be the Jordan form of E with J0 nilpotent of nilpotency index ν,then

ED = T−1[

J−11

0

]T

is the unique Drazin inverse satisfying

EDEED = ED, EDE = EED, EDEν+1 = Eν

Lemma (Campbell 1982)

Let (E ,A) be a regular pair. Then for all λ ∈ R such that(λE − A)−1 exist, E = (λE − A)−1E and A = (λE − A)−1Acommute.

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Solution formula

Theorem (Campbell 1982, Kunkel/M. 2006)

Consider the regular DAE Ex = Ax + f , x(t0) = x0, let ν be thed-be index of (E ,A) and let λ ∈ R such that (λE − A)−1 exist. Iff = (λE − A)−1f is sufficiently smooth, then the solution is

x(t) = eEDA(t−t0)EDEv +∫ t

t0eEDA(t−s)ED f (s) ds −

− (I − EDE)∑ν−1

i=0 (EAD)iAD f (i)(t).

. The formula is independent of the choice of λ.

. An initial condition is consistent iff there is a vector v such that

x0 = EDEv − (I − EDE)∑ν−1

i=0 (EAD)iAD f (i)(t0).

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Outline

1 Introduction2 Applications3 Research projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs

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Linear systems with variable coeff.

E(t)x(t) = A(t)x(t) + f (t), x(t0) = x0.

Scaling from left and basis changes

PEQ ˙x = (PAQ − PEQ)x + Pf , x(t0) = x0.

DefinitionTwo pairs of matrix functions (Ei(t),Ai(t)) in C`,n are calledglobally equivalent if there exist P ∈ C(I,C`,`) andQ ∈ C1(I,Cn,n), P(t),Q(t) nonsingular for all t ∈ I such that

[E2(t),A2(t)] = P(t)[E1(t),A1(t)]

[Q(t) −Q(t)

0 Q(t)

].

Regularity and d-index at time t are not invariant.39 / 60

Example 1

The system[−t t2

−1 t

]x(t) =

[−1 00 −1

]x(t), t ∈ R

is uniformly regular and of uniform d-index ν = 2 but

x(t) = c(t)

[t1

]is a solution for all c ∈ C1(R,C).

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Example 2

The system[0 01 −t

]x(t) =

[−1 t0 0

]x(t) +

[f1(t)f2(t)

],

is uniformly singular, because the pencil is singular for all t .But the system has the unique solution[

f1 + tf2 − tf1f2 − f1

]independent of any initial condition.

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Local version of global equivalence

DefinitionTwo pairs of matrices

(Ei ,Ai), Ei ,Ai ∈ R`,n, i = 1,2

are called locally equivalent if there exist matrices P ∈ C`,`,Q,R ∈ Cn,n with P,Q nonsingular such that

[E2,A2] = P[E1,A1]

[Q −R0 Q

].

By Hermite interpolation there always exists a function Q(t) suchthat at any point t one has Q(t) = Q and Q(t) = R.

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InvariantsTheorem (Kunkel/M. 1994)Let E ,A ∈ C`,n and

(a) T basis of kernel E(b) Z basis of Co-range E = kernel E∗

(c) T ′ basis of Co-kernel E = kernel E∗

(d) V basis of Co-range (Z ∗AT ).

Then, the quantities (convention rank ∅ = 0)

(a) r = rank E (rank)(b) a = rank (Z ∗AT ) (algebraic part)(c) s = rank (V ∗Z ∗AT ′) (strangeness)(d) d = r − s (differential part)(e) v = `− r − a− s (redundant part)

are invariant under the local equivalence transformation.43 / 60

Local canonical form

Theorem (Kunkel/M. 1994)

(E ,A) is locally equivalent to the canonical form:

sdasv

Is 0 0 00 Id 0 00 0 0 00 0 0 00 0 0 0

,

0 0 0 00 0 0 00 0 Ia 0Is 0 0 00 0 0 0

.

Eigenvalues are not invariants of this canonical form.

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Global canonical form

Applying the local canonical form for all t we get integerfunctions r(t),a(t), s(t).

Theorem (Kunkel/M. 1994)

Let E ,A be sufficiently smooth and let r ,a, s be constant in I.Then (E(t),A(t)) is globally equivalent to a pair of matrixfunctions of the form

sdasv

Is 0 0 00 Id 0 00 0 0 00 0 0 00 0 0 0

,

0 A12(t) 0 A14(t)0 0 0 A24(t)0 0 Ia 0Is 0 0 00 0 0 0

.

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More equivalence transformations

(a) x1 = A12(t)x2 + A14(t)x4 + g1(t)(b) x2 = A24(t)x4 + g2(t)(c) 0 = x3 + g3(t)(d) 0 = x1 + g4(t)(e) 0 = g5(t).

Insert the derivative of (d) in (a), which becomes an algebraicequation. This gives

sdasv

0 0 0 00 Id 0 00 0 0 00 0 0 00 0 0 0

,

0 A12(t) 0 A14(t)0 0 0 A24(t)0 0 Ia 0Is 0 0 00 0 0 0

,

for which we can again compute characteristic values r ,a, s,d , v .46 / 60

Inductive procedure

Proceeding inductively we get a sequence of pairs of matrixfunctions (Ei(t),Ai(t)) and integers ri ,ai , si ,di , vi , i ∈ N0, whichwe assume to be constant in I.We start with (E0(t),A0(t)) = (E(t),A(t)), and then(Ei+1(t),Ai+1(t)) is derived from (Ei(t),Ai(t)) by bringing it intocanonical form and inserting the derivative of (d) into (a). Theprocedure stops after finitely many steps.

DefinitionThe number µ of steps is called the strangeness-index or s-indexµ. If µ = 0, then the system is called strangeness-free.

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Existence, uniqueness, consistencyTheorem (Kunkel/M. 1994)Let the s-index µ be well–defined for (E(t),A(t)) and let f ∈ Cµ(I,Cl).Then the system is equivalent to a remodeled DAE in normal form

x1(t) = A13(t)x3(t) + f1(t), dµ equations,0 = x2(t) + f2(t), aµ equations,0 = f3(t), vµ equations,

where the inhomogeneity is determined by f (0), . . . , f (µ).

. The problem is solvable if and only if f3(t) ≡ 0.

. An initial condition is consistent if and only if in additionx2(t0) = −f2(t0) holds.

. The problem is uniquely solvable if again in addition we haveuµ = n − dµ − aµ = 0.

. Otherwise, we can choose x3 ∈ C(I,Cuµ) arbitrarily (control). 48 / 60

Example 1 cont.We get the canonical form([

−t t2

−1 t

],

[−1 00 −1

])∼([

0 00 0

],

[1 00 0

])We have µ = 1 with

r0 = 1,a0 = 0, s0 = 1,d0 = 0,u0 = 0,

r1 = 0,a1 = 1, s1 = 0,d1 = 0,u1 = 1.

The problem is solvable, since f (t) = 0, but not uniquelysolvable, since uµ 6= 0. The general solution is given by

x(t) =

[1 t0 1

] [0

x2(t)

]= x2(t)

[t1

].

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Example 2 cont.

([0 01 −t

],

[−1 t0 0

], f)∼(

0,[−1 00 −1

],

[f1

f2 − f1

])We have µ = 1 with

r0 = 1,a0 = 0, s0 = 1,d0 = 0,u0 = 0,

r1 = 0,a1 = 2, s1 = 0,d1 = 0,u1 = 0.

The problem is uniquely solvable for every consistent initialcondition with

x(t) =

[f1(t) + tf2(t)− t f1(t)

f2(t)− f1(t)

].

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Circuit model

C(x3 − x2) + (x1 − x2)/R = 0,x1 − x3 − U = 0,

x3 = 0.

This system has d-index 1, s-index 0.

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Semi-discretized Stokes equation

uh = Auh + Bph, BT uh = 0,

where uh and ph are semi-discrete approximations for u and p.If the non-uniqueness in p is not fixed then the d-index is notdefined and the s-index is 1. If it is fixed then the d-index is 2.

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Evaluation of the algebraic approach

. The algebraic approach is essential for the theoreticalunderstanding of DAEs.

. It can be used to study controllable, observable, autonomousbehavior in the sense of Willems, see Ilchmann/M. 2005.

. The approach allows to do bifurcation analysis, the pointswhere ranks change are a superset of the set of critical points.

. But, it cannot be used for the nonlinear case, for numericalmethods or the design of controllers, since one would needderivatives of computed transformation matrices.

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References

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