differential-algebraic equations. what they are and why we ... · simulation and control of systems...

Differential-algebraic equations.What they are and why we want to

solve them.

Volker MehrmannInstitut für Mathematik

Technische Universität Berlin

DFG Research Center MATHEONMathematics for key technologies

24.04.14

Outline

1 Introduction2 Applications3 Research Projects4 Setting for this lecture5 Linear constant coefficient DAEs6 Linear variable coefficient DAEs7 Derivative arrays8 Nonlinear DAEs

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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 ,p),

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

or (linearization along stationary solutions)

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

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

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

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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


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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 (MODELICA) 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


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Automatic transmissionModeling, 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 fuel consumption, improve smoothness ofswitching

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

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A current half-toroid model

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Hydraulic system

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Variator

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Modeling components

. multi-body system,

. elastic contact, oil film,

. hydraulic system,

. friction,

. model switching.

Large hybrid control system of nonlinear partial differentialequations, ordinary differential equations and algebraicequations, (PDAE).

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Verification measurement/simulation

control voltage contact pressure

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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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The brake project

Current project with Audi and Opel and several SMEs (2012-14)Joint with U. von Wagner, TU Berlin, Mechanics andN. Hoffmann, TU Hamburg-Harburg, MechanicsGoals:. Model and simulate brake behavior near squeaking frequency.. Actively control oscillations of brake to avoid squeaking, by

feedback when brake gets into resonance region.. Develop industrial controller.

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Disk brake model

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Measurement of brake vibrations

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A new turbineNew project collaborative research center, SFB1029, TurbInSignificant increase of efficiency via the interactive use ofinstationary effects of combustion and flow in gas turbines.

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Optimal fuel injection

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Our tasks

. Generate simple input/output based simplified models.

. Model reduction.

. Discretization of Input/Output maps.

. Controller design for fuel injection.

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Outline


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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)ξ = 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.36 / 78

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


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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

,

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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[

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

N i f (i)2 (t)

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

i=0 N i 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


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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.48 / 78

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 = range 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.52 / 78

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).


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 .55 / 78

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). 57 / 78

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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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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Outline


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Derivative arraysFor numerical methods and for the design of controllers, we usederivative arrays (Campbell 1989). We assume that derivativesof original functions are available or can be obtained viacomputer algebra or automatic differentiation.Linear case: We put E(t)x = A(t)x + f (t) and its derivatives upto order µ into a large DAE

Mk (t)zk = Nk (t)zk + gk (t), k ∈ N0

for zk = [xT , xT , . . . , x (k)T].

M2 =

E 0 0E − A E 0−2A + E 2E − A E

, N2 =

A 0 0A 0 0A 0 0

, z2 =

xxx

.63 / 78

Remodeled system


Under some constant rank assumptions, for every linear DAEthere exist integers µ, a, d and v such that:1. corank Mµ+1(t)− corank Mµ(t) = v .2. rank Mµ(t) = (µ + 1)m − a− v on I, and there exists a smooth

matrix function Z2,3 (left nullspace of Mµ) with Z T2,3Mµ(t) = 0.

3. The projection Z2,3 can be partitioned into two parts: Z2 (leftnullspace of (Mµ,Nµ)) so that the first block column A2 ofZ ∗2 Nµ(t) has full rank a and Z ∗3 Nµ(t) = 0. Let T2 be a smoothmatrix function such that A2T2 = 0, (right nullspace of A2).

4. rank E(t)T2 = d = `− a− v and there exists a smooth matrixfunction Z1 of size (n,d) with rank E1 = d, where E1 = Z T

1 E.

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Reduced problem

. Z T2,3 operates on the derivative array

Mµ(t)zµ = Nµ(t)zµ + gµ(t),

and picks out the algebraic and the redundant part.. It extracts all constraint equations including redundancies (Z T

3 )and all explicit and implicit (hidden) constraints (Z T

2 .. Z T

1 operates on the original system

E(t)x = A(t)x + f (t),

and picks out the dynamic part.

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Reduced problem, remodeled DAE

We obtain a numerically computable strangeness-freecondensed form

E1(t)x = A1(t)x + f1(t), dµ equations0 = A2(t)x + f2(t), aµ equations0 = f3(t), vµ equations

where A1 = Z T1 A, f1 = Z T

1 f , and f2 = Z T2 gµ, f3 = Z T

3 gµ.The partitioning is the same as in the canonical form

x1(t) = A13(t)x3(t) + f1(t), d equations0 = x2(t) + f2(t), a equations0 = f3(t), v equations.

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Circuit model

Circuit model as linear DAE with constant coefficientsEx = Ax + f . 0 C C

0 0 00 0 0

x1

x2

x3

=

−1/R 1/R 01 0 −10 0 1

x1

x2

x3

− 0

U0

Derivative array

0 C C 0 0 00 0 0 0 0 00 0 0 0 0 0

R−1 −R−1 0 0 C C1 0 −1 0 0 00 0 1 0 0 0

x1x2x3x1x2x3

=

−R−1 R−1 0 0 0 0

1 0 −1 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

x1x2x3x1x2x3

−

0U0000

.

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Remodeling

Z T2,3 =

[0 1 0 0 0 00 0 1 0 0 0

], Z T

2,3N1I3 =

[1 0 −10 0 1

]has full row rank, hence Z T

2,3 = Z T2 and Z3 is void.

A2 =

[1 0 −10 0 1

], f2 =

[−U0

], T2 =

010

.

ET2 =

C00

, Z T1 =

[1 0 0

]E1 =

[0 C C

], A1 =

[−R−1 R−1 0

], f1 = 0.

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Circuit model

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

x3 = 0.

DAE 0 C C0 0 00 0 0

x1

x2

x3

=

−1/R 1/R 01 0 −10 0 1

x1

x2

x3

− 0

U0

Remodeled strangeness-free DAE 0 C C

0 0 00 0 0

x1

x2

x3

=

−1/R 1/R 01 0 −10 0 1

x1

x2

x3

− 0

U0

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Semi-discretized Stokes equation

Derivative array.I 0 0 00 0 0 0A B I 0

BT 0 0 0

ddt

uh

ph

uh

ph

=

A B 0 0

BT 0 0 00 0 0 00 0 0 0

uh

ph

uh

ph

plus boundary and initial conditions.

Z T2 =

[0 I 0 0−BT 0 0 I

], A2 =

[BT 0−BT A −BT B

].

Choose T2,Z T1 =

[UT

1 UT2

]such that A2T2 = 0, Z T

1 ET2 = Id .

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Remodeling

Semi-discretized Stokes.

uh = Auh + Bph, BT uh = 0,

Remodeled strangeness-free system UT1 0

0 00 0

[ uh

ph

]=

UT1 A + UT

2 BT UT1 BT

BT 0−BT A −BT B

[ uh

ph

]

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Outline


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Derivative arrays, nonlinear problems

Analogous approach for F(t , x , x) = 0 yields derivative array:

0 = Fk (t , x , x , . . . , x (k+1)) =

F (t , x , x)

ddt F (t , x , x)

. . .dk

dtk F (t , x , x)

.We set

Mk (t , x , x , . . . , x (k+1)) = Fk ;x ,...,x (k+1)(t , x , x , . . . , x (k+1)),

Nk (t , x , x , . . . , x (k+1)) = −(Fk ;x (t , x , x , . . . , x (k+1)),0, . . . ,0),

zk = (t , x , x , . . . , x (k+1)).

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Nonlinear Hypothesis

Hypothesis: There exist integers µ, r , a, d , and v such thatL = F−1

µ ({0}) 6= ∅.We have rank Fµ;t ,x ,x ,...,x (µ+1) = rank Fµ;x ,x ,...,x (µ+1) = r , in aneighborhood of L such that there exists an equivalent systemF (zµ) = 0 with a Jacobian of full row rank r . On L we have1. corank Fµ;x ,x ,...,x (µ+1) − corank Fµ−1;x ,x ,...,x (µ+1) = v .2. corank Fx ,x ,...,x (µ+1) = a and there exist smooth matrix functionsZ2 (left nullspace of Mµ) and T2 (right nullspace of A2 = Fx ) withZ T

2 Fx ,x ,...,x (µ+1) = 0 and Z T2 A2T2 = 0.

3. rank FxT2 = d , d = `− a− v , and there exists a smoothmatrix function Z1 with rank Z T

1 Fx = d .

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The solution set L forms a (smooth) manifold of dimension(µ + 2)n + 1− r .The DAE can locally be transformed (by application of theimplicit function theorem) to a reduced DAE of the form

x1 = G1(t , x1, x3), (d differential equations),x2 = G2(t , x1, x3), (a algebraic equations),0 = 0 (v redundant equations).

The variables x3 represent undetermined components (controls).

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The chemical reactor

1 0 00 1 00 0 0

cTR

=

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

R − k3 exp(−k4T )c

,M0 = Fx =

1 0 00 1 00 0 0

, N0 = Fx =

k1 0 1k1 k3 k2

w1 w2 1

with appropriate w1,w2.

Z T2 =

[0 0 1

], A2 =

[w1 w2 1

], T2 =

1 00 1−w1 −w2

ET2 = I2, Z T

1 =

[1 0 00 1 0

].

The system is strangeness-free.76 / 78

Black-box modeling with DAEs

Modeling becomes extremely easy, 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!

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What can we do about it?

. Keep modeling easy.

. Take the model and make it better for numerics and control.

. Balance modeling, simulation and control errors.

. Make this as automatic as possible.

. Use equation based setting→ PDAEs

. Use modern modeling framework→ modelica, simscape.

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differential-algebraic equations. what they are and why we ... · simulation and control of systems...

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