Index Theory
and Structural Analysis
for multi-mode DAE Systems
Albert Benveniste (Inria, Rennes) Benoît Caillaud (Inria, Rennes)
Marc Pouzet (ENS, Paris)
Hilding Elmqvist (Mogram, Lund) Martin Otter (DRL, Munich)
INRIA and ENS, France
May 27, 2016
1 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
2 / 34
Compositionality and reuse: Simulink → Modelica
Simulink has become a central tool in systems design: Block Diagram
3 / 34
Compositionality and reuse: Simulink → Modelica
From Block Diagram to Component Diagram
3 / 34
Compositionality and reuse: ODE → DAE












0

x = f (x , u )




y = g (x , u )
from Simulink (ODE):
HS in state space form


the state space form 



depends on the context 









reuse is difficult
−→

to Modelica (DAE):




HS as physical balance equations








0 = f (x 0 , x , u )




0 = g (x , u )



Ohm & Kirchhoff laws, bond graphs,




multi-body mechanical systems;




DAE are context-free





reuse is much easier
4 / 34
Compositionality and reuse: ODE → DAE
I Modeling tools supporting DAE
I
Some modeling tools provide a library of predefined models
ready for assembly (e.g., LMS/AmeSim) ;
I
Spice is a popular language dedicated to EDA
I
Mathworks/Simscape reuses Matlab as a programming language, in
which “=” has been replaced by “==”, indicating a constraint
I
Modelica comes with a full programming language that is a public
standard https://www.modelica.org/
5 / 34
A sketch of Modelica and its semantics [Fritzson]
Summary
model SimpleDrive
..Rotational.Inertia
Inertia1 (J=0.002);
..Rotational.IdealGear IdealGear1(ratio=100)
..Basic.Resistor
Resistor1 (R=0.2)
...
equation
connect(Inertia1.flange_b, IdealGear1.flange_a);
connect(Resistor1.n, Inductor1.p);
...
end SimpleDrive;
model Resistor
package SIunits = Modelica.SIunits; type Voltage =
Real(quantity="Voltage",
parameter SIunits.Resistance R = 1;
unit
="V");
SIunits.Voltage v;
..Interfaces.PositivePin p;
..Interfaces.NegativePin n;
connector PositivePin
equation
package SIunits = Modelica.SIunits;
0 = p.i + n.i;
SIunits.Voltage
v;
v = p.v - n.v;
flow SIunits.Current i;
v = R*p.i;
end PositivePin;
end Resistor;
Sunday, October 12,
2003
Multi-domain Modeling and Simulation with Modelica
35
6 / 34
A sketch of Modelica and its semantics [Fritzson]
I Modelica Reference v3.3:
“The semantics of the Modelica language is specified by means of
a set of rules for translating any class described in the Modelica
language to a flat Modelica structure”
I the good:
I
Semantics of continuous-time 1-mode Modelica models: Cauchy
problem on the DAE resulting from the inlining of all components
I
Modelica supports multi-mode systems
x*x + y*y = 1;
der(x) + x + y = 0;
when x <= 0 do reinit(x,1); end;
when y <= 0 do reinit(y,x); end;
I the bad: What about the semantics of multi-mode systems?
I and . . . : Questionable simulations [Tim Bourke and Marc Pouzet]
6 / 34
Examples of multi-mode systems
Cup-and-Ball game
(a two-mode
extension of
the pendulum)
A Clutch
A Circuit Breaker
7 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
8 / 34
A key notion in DAE/dAE Systems: the index
I Differential Algebraic Equations systems (DAE, continuous time) may involve
more constraints than specified:
x
0
0
=
=
f (x , u )
g (x )
differentiating
=⇒
substituting
=⇒
 0
 x
0

0
 0
 x
0

0
= f (x , u )
= g (x )
= gx0 (x )x 0
=
=
=
f (x , u )
g (x )
gx0 (x )f (x , u )
(e 1 )
(e2 )
(e3 )
I What is the consequence for execution schemes?
9 / 34
A key notion in DAE/dAE Systems: the index
I difference Algebraic Equations systems (dAE, discrete time) may involve
more constraints than specified:
x
0
•
= f (x , u )
= g (x )
shifting
=⇒
substituting
=⇒
 •
 x
0

0
 •
 x
0

0
= f (x , u )
= g (x )
= g (x • )
= f (x , u )
= g (x )
= g (f (x , u ))
(e1 )
(e2 )
(e3 )
9 / 34
A key notion in DAE/dAE Systems: the index
I difference Algebraic Equations systems (dAE, discrete time) may involve
more constraints than specified:
x
0
•
= f (x , u )
= g (x )
shifting
=⇒
substituting
=⇒
 •
 x
0

0
 •
 x
0

0
= f (x , u )
= g (x )
= g (x • )
= f (x , u )
= g (x )
= g (f (x , u ))
(e1 )
(e2 )
(e3 )
I Execution scheme:
1. Given x such that g (x ) = 0
2. Use (e3 ) to evaluate u (constraint solver needed)
3. Use (e1 ) to evaluate x • , which satisfies g (x • ) = 0, and repeat
Adding (e3 ) essential in deriving the execution scheme of each step
9 / 34
A key notion in DAE/dAE Systems: the index
I difference Algebraic Equations systems (dAE, discrete time) may involve
more constraints than specified:
x
0
•
=
=
f (x , u )
g (x )
shifting
=⇒
shifting twice
=⇒
 •
 x
0

0
 •
x



0
0



0
=
=
=
f (x , u )
g (x )
g (x • )
=
=
=
=
f (x , u )
g (x )
g (x • )
g (x •2 )
(e1 )
(e2 )
(e3 )
(e4 )
I Shifting twice is useless since the second shifting introduces
1. eqn (e4 ) but also
2. the fresh variable x •2
Thus, adding (e4 ) does not help getting the value of x •
9 / 34
A key notion in DAE/dAE Systems: the index
I difference Algebraic Equations systems (dAE, discrete time) may involve
more constraints than specified:
x
0
•2
=
=
f (x , u )
g (x )
shifting
=⇒
shifting twice
=⇒
 •2
 x
0

0
 •2
x



0
0



0
= f (x , u )
= g (x )
= g (x • )
=
=
=
=
f (x , u )
g (x )
g (x • )
g ( x •2 )
(e1 )
(e2 )
(e3 )
(e4 )
I This time, shifting twice is useful since the second shifting introduces
1. eqn (e4 ) and
2. the variable x •2 , which is not fresh but already there
9 / 34
A key notion in DAE/dAE Systems: the index
I Differential Algebraic Equations systems (DAE, continuous time) may involve
more constraints than specified:
x
0
0
= f (x , u )
= g (x )
differentiating
=⇒
substituting
=⇒
I
 0
 x
0

0
 0
 x
0

0
= f (x , u )
= g (x )
= gx0 (x )x 0
= f (x , u )
(e1 )
= g (x )
(e2 )
= gx0 (x )f (x , u ) (e3 )
∼ ODE if we have a constraint solver providing x → u by using (e3 )
9 / 34
A key notion in DAE/dAE Systems: the index
I
differentiations
shiftings
make
DAE
dAE
becoming
ODE
TS
-like
(TS: transition system)
I Define the differentiation / difference index as being the minimal number of
differentiations
shiftings
needed until
no further differentiation
no further shifting
can reveal additional latent constraints.
I The differentiation index emerged in the late 1980’s in the applied math
community; other notions of index were proposed [Campbell & Gear 1995].
I Yet, no notion of index was formally defined for multi-mode DAE systems.
9 / 34
Differentiation/Difference Index: (1) exact
F (x , x 0 ) = 0
F (x , x • ) = 0
Fk (x ,v ,w )










z
F (x , x 0 )
d
F (x , x 0 )
dt
..
.x , x 0 )
k
d
F (x , x 0 )
dt k

F (x , x • )
F • (x , x • )
.. •
., x )
F •k ( x , x • )














=def
=def




|
[Campbell & Gear 1995]:
F (0) (x , x 0 , w )
F (1) (x , x 0 , w )
..
.x , x 0 , w )
(k )
F (x , x 0 , w )
}|
{
F (0) (x , x • , w )
F (1) (x , x • , w )
.. •
., x , w )
F (k ) (x , x • , w )

{z
}
Fk (x ,v ,w )




,
v =def x 0
w =def (x (2) , . . . , x (k +1) )

v =def x •

 ,
w =def (x •2 , . . . , x •k +1 )

The index is the smallest k such that
solving Fk (x , v , w ) = 0 while eliminating w
uniquely determines v as a partial function of x
10 / 34
Differentiation/Difference Index: (1) exact
0
0
=
=
x 0 − f (x , u )
g (x )
set
S (x , v ) =def
x 0 − f (x , u )
g (x )
where x is the state and v =def (x 0 , u ); consider the Jacobian
J =def dS /dv
=
1
0
−fu0 (x , u )
is structurally singular
0
v cannot be determined and S has index > 0. Set w = (x 00 , u 0 ) and consider
 0

x 0 − f (x , u )
x − f (x , u )

 g (x )
 ⇒  g (x )

=

 g 0 (x )x 0
g 0 (x )x 0
00
d
x − dt f (x , u )


S 1 (x , v , w )
=def
S (x , v )
d
S (x , v , w )
dt

J1
=def
1
dS1 /dv =  0
1
−fu0 (x , u )
0
0

⇒
1
1
−fu0 (x , u )
0
invertible
MMMhhhhh, only structurally invertible — it may happen that fu0 (x , u ) = 0
11 / 34
Differentiation/Difference Index: (1) exact
0
0
=
=
x 0 − f (x , u )
g (x )
set
S (x , v ) =def
x 0 − f (x , u )
g (x )
where x is the state and v =def (x 0 , u ); consider the Jacobian
J =def dS /dv
=
1
0
−fu0 (x , u )
is structurally singular
0
v cannot be determined and S has index > 0. Set w = (x 00 , u 0 ) and consider
 0

x 0 − f (x , u )
x − f (x , u )

 g (x )
 ⇒  g (x )

=

 g 0 (x )x 0
g 0 (x )x 0
00
d
x − dt f (x , u )


S 1 (x , v , w )
=def
S (x , v )
d
S (x , v , w )
dt

J1
=def
1
dS1 /dv =  0
1
−fu0 (x , u )
0
0

⇒
1
1
−fu0 (x , u )
0
invertible
MMMhhhhh, only structurally invertible — it may happen that fu0 (x , u ) = 0
11 / 34
Differentiation/Difference Index: (2) structural

E


 0
E1
E


 2
E3
:
:
:
:
x0
0
0
0
=def
=
=
=
1
•
− x)
x − f (x , u )
g (x )
g 0 (x )x 0
∂ (x
0
E1
x
E1
x
E2
u
E2
u
E3
x0
E3
x0
structural pivoting
(Pantelides algorithm)
⇔
searching for a consistent orientation
of the incidence graph
12 / 34
Differentiation/Difference Index: (2) structural

E


 0
E1
E


 2
E3
:
:
:
:
x0
0
0
0
=def
=
=
=
1
•
− x)
x − f (x , u )
g (x )
g (x • )
∂ (x
0
E1
x
E1
x
E2
u
E2
u
E3
x•
E3
x•
structural pivoting
(Pantelides algorithm)
⇔
searching for a consistent orientation
of the incidence graph
12 / 34
Differentiation/Difference Index: (2) structural
I Exact differentiation/difference index:
I
adding differentiated/shifted versions of subsets of equations
I
until leading variables can be determined from other variables
I Structural differentiation/difference index:
I
adding differentiated/shifted versions of subsets of equations
I
until leading variables can be determined from other variables,
almost everywhere
I Theorem: the graph-based structural analysis finds the structural index.
12 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
13 / 34
Recall: structural Differentiation/Difference Index

E


 0
E1
E


 2
E3
:
:
:
:
x0
0
0
0
=def
=
=
=
1
•
− x)
x − f (x , u )
g (x )
g 0 (x )x 0
∂ (x
0
E1
x
E1
x
E2
u
E2
u
E3
x0
E3
x0
structural pivoting
(Pantelides algorithm)
⇔
searching for a consistent orientation
of the incidence graph
14 / 34
Recall: structural Differentiation/Difference Index

E


 0
E1
E


 2
E3
:
:
:
:
x0
0
0
0
=def
=
=
=
1
•
− x)
x − f (x , u )
g (x )
g (x • )
∂ (x
0
E1
x
E1
x
E2
u
E2
u
E3
x•
E3
x•
structural pivoting
(Pantelides algorithm)
⇔
searching for a consistent orientation
of the incidence graph
14 / 34
Recall: structural Differentiation/Difference Index
For the previous example,
I the DAE system and
I its 1st -order Euler expansion (a dAE system)
possess identical structural analyses.
I Theorem: This result is true in general
∂ infinitesimal in the sense of
non-standard analysis, yields an exact representation of the DAE system,
not an approximation of it
I Taking, in the Euler approximation,
14 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
15 / 34
Structural analysis of single-mode dAE systems
I Classify the system variables into
I
leading variables (of max differentiation/shifting degree)
I
other variables
I Execution principle:
I
at initialization: evaluate all variables
I
otherwise: evaluate leading variables
16 / 34
Structural analysis of single-mode dAE systems
Pantelides (outside initialization):
I In order to find latent equations of a system S involving
n equations
n variables
1. Fill a bag BS with all equations of S and their shifted versions;
2. Let BS be the collection
I
of all subsets of n or more equations from BS
I
ensuring the evaluation of all the leading variables of S
3. Select an element of BS such that the max shift used is minimal.
16 / 34
Structural analysis of multi-mode DAE systems
non-standard interpretation
(removing logico-numerical fixpoints)
mDAE
mdAE
Applying Pantelides
(adding mode-dependent
latent equations)
standardizing
(within modes and at mode changes)

 standard domain
non-standard domain
Approach :

translating s ↔ ns
17 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
18 / 34
A simple unilateral constraint in discrete time
x
•
0
=
≤
f (x , u )
g (x )
expands as
x • = f (x , u )
γ = [ 0≥g (x )]

when γ do g (x )=0


(e1 )
(γ )
(e3 )
We apply our approach:
1. Check if we can find a successful execution scheme
2. Find countermeasures in case of failure
19 / 34
A simple unilateral constraint in discrete time
x
•
0
=
≤
f (x , u )
g (x )
T
F
⊥
?
=
=
=
=
expands as
evaluated or true
false
dead code
not evaluated yet
x • = f (x , u )
γ = [ 0≥g (x )]

when γ do g (x )=0









in
(e1 )
(γ )
(e3 )
e1 (x • , u , x )
γ(x )

when γ do e3 (x )


19 / 34
A simple unilateral constraint in discrete time
x
•
0
=
≤
f (x , u )
g (x )
=
F
=
⊥ =
? =
T
expands as
evaluated or true
false
dead code
not evaluated yet
x • = f (x , u )
γ = [ 0≥g (x )]

when γ do g (x )=0






in



x , γ known from previous step
stuck at e3 (.) with no variable left
:
(e1 )
(γ )
(e3 )
e1 (x • , u , x )
γ(x )

when γ do e3 (x )


e1 (x • , u , x )
γ(x )

when γ do e3 (x )


I fixpoint involving a guard; we must break this fixpoint
I principle: have guards acting on leading variables
I hence we replace e3 (x ) by e3 (x • )
19 / 34
A simple unilateral constraint in discrete time
x
•
0
=
≤
f (x , u )
g (x )
=
F
=
⊥ =
? =
T
expands as
evaluated or true
false
dead code
not evaluated yet
x • = f (x , u )
γ = [ 0≥g (x )]

when γ do g (x )=0









in
(e1 )
(γ )
(e3 )
e 1 (x • , u , x )
γ(x )

when γ do e3 (x • )


e 1 (x • , u , x )
γ(x )

when γ do e3 (x • )


success in case γ = T
failure in case γ = F
by having 1 equation and 2 variables
e 1 (x • , u , x )
γ(x )

when γ do e3 (x • )





e 1 (x • , u , x )
γ(x )
when γ do ⊥
19 / 34
A simple unilateral constraint in discrete time
x
•
=
≤
0
T
F
⊥
?
=
=
=
=
f (x , u )
g (x )
expands as
evaluated or true
false
dead code
not evaluated yet




x • = f (x , u )
γ = [ 0≥g (x )]

when γ do g (x )=0


in



added stub (state feedback)
success in case γ = F
success in case γ = T
:
(e1 )
(γ )
(e3 )
e 1 (x • , u , x )
γ(x )
•
when
γ
do
e

3 (x )


when not γ do e4 (u , x )




when not γ do h(x , u ) = 0




e1 (x • , u , x )
γ(x )
when γ
do ⊥



when not γ do e4 (u , x )

e1 (x • , u , x )



γ(x )
•
when
γ
do
e

3 (x )


when not γ do ⊥
19 / 34
A simple unilateral constraint in discrete time
x
•
0
=
≤
f (x , u )
g (x )
expands as
complementarity
condition in the
tuple (u , x • ),
with the stub
x • = f (x , u )
γ = [ 0≥g (x )]

when γ do g (x )=0


(e1 )
(γ )
(e3 )
 •
x −f ( x , u ) = 0




 0≤g (x • ) ⊥ |h(x , u )| ≥0
{z
}
|
:

•


 g (x ) ≥ 0



|h(x , u )| ≥ 0
 expands as: 

•
g (x )×|h(x , u )| = 0
I Alternative approach: using complementarity conditions as atomic actions
for handling by non-smooth systems solvers [Acary & Brogliato]
I For the structural analysis:
complementarity conditions ≡ regular systems of smooth equations
I Our structural analysis regards as blocks both:
I
I
regular systems of smooth equations
and complementarity conditions
19 / 34
Cup-and-ball in continuous time
Two-mode pendulum
z 00 =














 when
when
00
γ
not γ
do
do
1
∂ 2 (z
•2
− 2z • + z )
x = fx
y 00 = fy − g
γ = [ x 2 + y 2 ≥ L2 ]
x 2 + y 2 = L2
f =0
(e0 )
(e1 )
(e2 )
(γ )
(e4 )
(e5 )
20 / 34
Cup-and-ball in continuous time
Recalling our approach:
non-standard interpretation
(removing logico-numerical fixpoints)
mDAE
mdAE
Applying Pantelides
(adding mode-dependent
latent equations)
standardizing
(within modes and at mode changes)
20 / 34
Cup-and-ball in continuous time
Two-mode pendulum in non-standard analysis
z 00 =














 when
when
I
1
∂ 2 (z
•2
− 2z • + z )
00
γ
not γ
do
do
x = fx
y 00 = fy − g
γ = [ x 2 + y 2 ≥ L2 ]
x 2 + y 2 = L2
f =0
(e0 )
(e1 )
(e2 )
(γ )
(e4 )
(e5 )
(e0 ) is a generic expansion that applies to each second derivative
I Stuck by a causality fixpoint for the pair
(γ, e4 )
I Countermeasure: let the equations (e4 , e5 ) act on leading variables
I
I
(e5 ): OK
(e4 ): shift (x , y ) two times forward
20 / 34
Cup-and-ball in continuous time
Performing this yields:
z 00 =














 when
when
00
γ
not γ
do
do
1
∂ 2 (z
•2
− 2z • + z )
x = fx
y 00 = fy − g
γ = [ x 2+ y 2 ≥ L2 ]
x 2 + y 2 •2 = L2
f =0
(e0 )
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
20 / 34
Cup-and-ball in continuous time
Performing this yields:
z 00 =








1
∂ 2 (z
•2
− 2z • + z )
(e0 )
00






 when
when
γ
not γ
do
do
x = fx
y 00 = fy − g
γ = [ x 2+ y 2 ≥ L2 ]
x 2 + y 2 •2 = L2
f =0
tick
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
x = x0 = y = y0 = T
γ=
γ=
T
(e1 , e2 , e4•2 ) = x 00 = y 00 = f =
T
F
e5 = f = T
tick
e1 = x
00
=
T
e2 = y 00 = T
The evaluation scheme succeeds
20 / 34
Cup-and-ball in continuous time
Performing this yields:
z 00 =














 when
when
1
∂ 2 (z
•2
− 2z • + z )
00
γ
not γ
do
do
x = fx
y 00 = fy − g
γ = [ x 2+ y 2 ≥ L2 ]
x 2 + y 2 •2 = L2
f =0
(e0 )
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
We proceed to standardization:
(. . . )•2 in (e4•2 )
since forward shifts have no standard counterpart
I Within continuous modes we must get rid of the
I At an event t of mode change, map the forward shift to super-dense time
•
•
•
•
t ≡ (t , 0) → (t , 1) → (t , 2) → . . . → (t , nt )
But we must get rid of second derivatives in (e1 , e2 ) since they involve ∂
20 / 34
Cup-and-ball in continuous time
Standardization within continuous modes: Taylor expansion of (. . . )•2 in (e4•2 )
x2 + y2
0=
expands as
0=
+2∂
+2∂ 2
• 2
− L2
(x 2 + y 2 − L2 )
.
(xx 0 + yy 0 )
2
2
(xx 00 + yy 00 + x 0 + y 0 )
The second equation must hold structurally w.r.t. ∂ , hence the standardization is:



















when
γ
when
not γ
do
and
and
do
x 00 = fx
y 00 = fy − g
γ = [ x 2 + y 2 ≥ L2 ]
x 2 + y 2 = L2
xx 0 + yy 0 = 0
2
2
xx 00 + yy 00 + x 0 + y 0 = 0
f =0
(e1 )
(e2 )
(γ )
(e4 )
(e40 )
(e400 )
(e5 )
which coincides with the equations of the pendulum
20 / 34
Cup-and-ball in continuous time
Standardization at events: non-standard expansion of second derivatives in
(e1 , e2 ) in terms of forward shifts and positions x , y
x •2 − 2x • + x = ∂ 2 fx
y •2 − 2y • + y = ∂ 2 (fy − g )
γ = [ x 2 + y 2 ≥ L2 ]









when


when
γ
not γ
do x 2 + y 2
do f = 0
• 2
= L2
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
(γ, e4•2 , e5 ) are fine but we must get rid of ∂ 2 in (e1 , e2 ):
I at the event
γ:
T
→ F: we eliminate f by substituting f = 0 in (e1 , e2 );
x •2 − 2x • + x = 0
y •2 − 2y • + y = 0
which we solve for the leading variables x •2 , y •2
20 / 34
Cup-and-ball in continuous time
Standardization at events: non-standard expansion of second derivatives in
(e1 , e2 ) in terms of forward shifts and positions x , y
x •2 − 2x • + x = ∂ 2 fx
y •2 − 2y • + y = ∂ 2 (fy − g )
γ = [ x 2 + y 2 ≥ L2 ]









when


when
γ
not γ
do x 2 + y 2
do f = 0
• 2
= L2
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
(γ, e4•2 , e5 ) are fine but we must get rid of ∂ 2 in (e1 , e2 ):
I at the event
γ:
F
→ T: we eliminate f from (e1 , e2 ):
(x •2 − 2x • + x )y − (y •2 − 2y • + y )x = 0
• 2
x2 + y2
− L2 = 0
(e12 )
(e4•2 )
which we solve for the leading variables x •2 , y •2 (f is impulsive)
So far we got the positions at (t , 2), if t = (t , 0) is the instant of mode change.
In addition we need the velocities.
20 / 34
Cup-and-ball in continuous time
Standardization at events: non-standard expansion of second derivatives in
(e1 , e2 ) in terms of forward shifts and velocities x 0 , y 0








when



when
γ
not γ
do
do
x 0• − x 0 = ∂ fx
y 0• − y 0 = ∂(fy − g )
γ = [ x 2 + y 2 ≥ L2 ]
•
(xx 0 + yy 0 ) = 0
f =0
(e1 )
(e2 )
(γ )
•
(e40 )
(e5 )
•
(γ, e40 , e5 ) are fine but we must get rid of ∂ 2 in (e1 , e2 ):
I at the event
γ:
T
→ F: we eliminate f by substituting f = 0 in (e1 , e2 );
x 0• − x 0 = 0
y 0• − y 0 = 0
which we solve for the leading variables x 0• , y 0•
20 / 34
Cup-and-ball in continuous time
Standardization at events: non-standard expansion of second derivatives in
(e1 , e2 ) in terms of forward shifts and velocities x 0 , y 0








when



when
γ
not γ
do
do
x 0• − x 0 = ∂ fx
y 0• − y 0 = ∂(fy − g )
γ = [ x 2 + y 2 ≥ L2 ]
•
(xx 0 + yy 0 ) = 0
f =0
(e1 )
(e2 )
(γ )
•
(e40 )
(e5 )
•
(γ, e40 , e5 ) are fine but we must get rid of ∂ 2 in (e1 , e2 ):
I at the event
γ:
F
→ T: we eliminate f from (e1 , e2 ):
(x 0• − x 0 )y − (y 0• − y 0 )x = 0
•
(xx 0 + yy 0 ) = 0
which we solve for the leading variables x 0• , y 0• (f is impulsive)
Applying this at (t , 1) yields the velocities at (t , 2). We have eveything we need for
the initialization of the new continuous mode.
20 / 34
Cup-and-ball in continuous time
Problem with the above approach: we used elimination based on computer algebra. Not always possible. Here is a first approach to cope with this.
S
:
x •2 − 2x • + x = ∂ 2 fx
y •2 − 2y • + y = ∂ 2 (fy − g )
γ = [ x 2 + y 2 ≥ L2 ]









when


when
do x 2 + y 2
do f = 0
γ
not γ
• 2
= L2
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
Suppose that all expressions involved in the system are polynomial in the variables
(it is the case here). Expand the tension f as a polynomial in ∂ −1 :
f = f0 + f1 ∂ −1 + f2 ∂ −2
and we are left with the following three systems when (e4•2 ) is active:
f0 = f1 = 0
,
 •2
•
 x − 2x + x = f2 x
•2
y − 2y • + y = f2 y
•2

x2 + y2
= L2
20 / 34
Cup-and-ball in continuous time
Problem with the above approach: we used elimination based on computer algebra. Not always possible. Here is a second approach to cope with this.
We can always instead compute the following two steps with ∂ > 0 a small
(standard) step size and solve for the leading variables:
at t
at t + ∂
:
:
x •2 − 2x • + x = ∂ 2 f x
y •2 − 2y • + y = ∂ 2 (f y − g )
γ = [ x 2 + y 2 ≥ L2 ]









when
γ


when not γ






do x 2 + y 2
do f = 0


when
γ



when not γ
do
do
• 2
= L2
x 0• − x 0 = ∂ f x
y 0• − y 0 = ∂(f y − g )
γ = [ x 2 + y 2 ≥ L2 ]
•
(xx 0 + y y 0 ) = 0
f =0
This way we obtain at t + 2∂ a consistent value for the positions and velocities and
can restart the continuous time mode.
20 / 34
Cup-and-ball in continuous time
Summary:
z 00 =








1
∂ 2 (z
•2
− 2z • + z )
00






 when
when
γ
not γ
do
do
x = fx
y 00 = fy − g
γ = [ x 2+ y 2 ≥ L2 ]
x 2 + y 2 •2 = L2
f =0
entering tight rope
continuous time
mode
with γ = F
(t , 1)
mode change
and reset
discrete time
(t , 2)
(t , 1)
(e0 )
(e1 )
(e2 )
(γ )
(e4•2 )
(e5 )
(t , 2)
continuous time
mode
with γ = T
exiting tight rope
20 / 34
Cup-and-ball in continuous time
Using complementarity conditions:


z 00 = ∂12 (z •2 − 2z • + z )


 00
x = fx

y 00 = fy − g



0≤ (x •2 )2 +(y •2 )2 −L2
(e0 )
⊥
f ≥0
(e1 )
(e2 )
(e3 )
Again, we standardize by eliminating the shifts using Taylor expansions to get the
code for continuous time modes:
 00
x = fx




 y 00 =
 fy 2− g 2

x + y − L2



0≤  xx 0 + yy 0



00
00
02
02
xx + yy + x + y
(e1 )
(e2 )
⊥
f ≥0
(e3 )
For the first vector, we equip R3 with the lexicographic order: (u , v , w ) > 0 iff
either [u > 0] or [u = 0 and v > 0] or [u = v = 0 and w > 0]
20 / 34
A simple clutch (continuous time)










when


when




when



when
I ui /ci
I
γ
γ
not γ
not γ
do
do
do
do
z 0 = ∂1 (z • − z )
u10 = f1 (u1 , c1 )
u20 = f2 (u2 , c2 )
u1 = u2
c1 + c2 = 0
c1 = 0
c2 = 0
(e0 )
(e1 )
(e2 )
(e3 )
(e4 )
(e5 )
(e6 )
areare the two speeds/torques
γ = F/T is an input representing the pressed/released mode
21 / 34
A simple clutch (continuous time)










when


when




when



when
γ
γ
not γ
not γ
do
do
do
do
z 0 = ∂1 (z • − z )
u10 = f1 (u1 , c1 )
u20 = f2 (u2 , c2 )
u1 = u2
c1 + c2 = 0
c1 = 0
c2 = 0
(e0 )
(e1 )
(e2 )
(e3 )
(e4 )
(e5 )
(e6 )
I Fails by not having guards acting on leading variables
I Counter-measure: shifting forward u1 , u2
21 / 34
A simple clutch (continuous time)










when


when




when



when
tick
γ
γ
not γ
not γ
γ =
z 0 = ∂1 (z • − z )
u10 = f1 (u1 , c1 )
u20 = f2 (u2 , c2 )
u1• = u2•
c1 + c2 = 0
c1 = 0
c2 = 0
do
do
do
do
γ =
T
γ = T, u1 = u2 =
T
c1 = c2 = T, u1• = u2• =
tick
F
γ = F, u1 = u2 =
T
e1 = e2 = e3• = e4 =
(e0 )
(e1 )
(e2 )
(e3• )
(e4 )
(e5 )
(e6 )
e5 = c1 =
T
e6 = c2 =
T
T
T
e1 = e2 = T, u1• = u2• =
T
The execution scheme succeeds
21 / 34
A simple clutch (continuous time)










when


when




when



when
γ
γ
not γ
not γ
do
do
do
do
z 0 = ∂1 (z • − z )
u10 = f1 (u1 , c1 )
u20 = f2 (u2 , c2 )
u1• = u2•
c1 + c2 = 0
c1 = 0
c2 = 0
(e0 )
(e1 )
(e2 )
(e3• )
(e4 )
(e5 )
(e6 )
Standardization is performed as for the previous example:
super-dense
time at reset
events
get rid of ∂
in (e1 , e2 )
•
•
u1 = u2
u10 = f1 (u1 , c1 )
u20 = f2 (u2 , c2 )
u1 = u2
u10 = u20
↔
↔
Taylor
expansion in
continuous
modes
21 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
22 / 34
Approach inherited from Synchronous Languages
I The structural analysis consists in searching for
I
a mode-dependent assignment variable ←− equation
I
such that the variable can be evaluated using its assigned equation;
I
blocks of {variables, equations} may be needed
I This turns out to be an extension of the constructive semantics
[Berry1996, Benveniste et al.2003], which served as a mathematical
basis for the code generation from synchronous languages
23 / 34
A multi-mode dAE language and its abstraction
var
prop
num
pred
eqn
term
def
system
::= X | var •
::= T | F | var | not prop | prop and prop
::= num
::= pred “predicate on var ”
::=
::=
::=
::=
when prop do “equation on var ”
num | pred | eqn
var : term
def | def ; system
Similar to the intermediate format in Modelica compilers
Questions regarding the structural analysis:
1. How to schedule (blocks of) equations depending on the mode?
2. How to deal with mode changes & varying dimension?
24 / 34
A multi-mode dAE language and its abstraction
var
vars
prop
num
pred
inv
eqn
term
def
system
X | var •
var | var ,vars
T | F | var | not prop | prop and prop
::=
::=
::=
::=
::=
::=
::=
::=
::=
::=
num
pred (vars)
inv prop
when prop do (vars)
num | pred | inv | eqn
var : term
def | def ; system
I Structural abstraction of the hybrid DAE language
I “something(vars )”:
I “inv ”
captures causality constraints
: invariants as predicate abstraction
24 / 34
Scott domain of values for the structural analysis
Scott domain: D = { T, F, ⊥, ? } with ? < ⊥, T,
I F
: false
I T
: evaluated/true (for prop only)
(for prop only)
I
⊥ : dead (like in “dead code”: not involved in the current mode)
I
? : undefined (“not evaluated yet”)
I consistency:
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I execution scheme: moving tuple X
I
I
I
I
F
: vars
from (?, . . . , ?)
to ξ ∈ { ⊥, T, F }X
by following an increasing path in DX
while meeting consistency
25 / 34
Scott domain of values for the structural analysis
Scott domain: D = { T, F, ⊥, ? } with ? < ⊥, T,
I F
: false
I T
: evaluated/true (for prop only)
(for prop only)
I
⊥ : dead (like in “dead code”: not involved in the current mode)
I
? : undefined (“not evaluated yet”)
I consistency:
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I execution scheme: moving tuple X
I
I
I
I
F
: vars
from (?, . . . , ?)
to ξ ∈ { ⊥, T, F }X
by following an increasing path in DX
while meeting consistency
25 / 34
Scott domain of values for the structural analysis
Scott domain: D = { T, F, ⊥, ? } with ? < ⊥, T,
I F
: false
I T
: evaluated/true (for prop only)
(for prop only)
I
⊥ : dead (like in “dead code”: not involved in the current mode)
I
? : undefined (“not evaluated yet”)
I consistency:
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I execution scheme: moving tuple X
I
I
I
I
F
: vars
from (?, . . . , ?)
to ξ ∈ { ⊥, T, F }X
by following an increasing path in DX
while meeting consistency
25 / 34
Scott domain of values for the structural analysis
Scott domain: D = { T, F, ⊥, ? } with ? < ⊥, T,
I F
: false
I T
: evaluated/true (for prop only)
(for prop only)
I
⊥ : dead (like in “dead code”: not involved in the current mode)
I
? : undefined (“not evaluated yet”)
I consistency:
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I execution scheme: moving tuple X
I
I
I
I
F
: vars
from (?, . . . , ?)
to ξ ∈ { ⊥, T, F }X
by following an increasing path in DX
while meeting consistency
25 / 34
Scott domain of values for the structural analysis
Scott domain: D = { T, F, ⊥, ? } with ? < ⊥, T,
I F
: false
I T
: evaluated/true (for prop only)
(for prop only)
I
⊥ : dead (like in “dead code”: not involved in the current mode)
I
? : undefined (“not evaluated yet”)
I consistency:
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I execution scheme: moving tuple X
I
I
I
I
F
: vars
from (?, . . . , ?)
to ξ ∈ { ⊥, T, F }X
by following an increasing path in DX
while meeting consistency
25 / 34
Scott domain of values for the structural analysis
Scott domain: D = { T, F, ⊥, ? } with ? < ⊥, T,
I F
: false
I T
: evaluated/true (for prop only)
(for prop only)
I
⊥ : dead (like in “dead code”: not involved in the current mode)
I
? : undefined (“not evaluated yet”)
I consistency:
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I execution scheme: moving tuple X
I
I
I
I
F
: vars
from (?, . . . , ?)
to ξ ∈ { ⊥, T, F }X
by following an increasing path in DX
while meeting consistency
25 / 34
Structural Analysis: non-det. transition system
I hot/cold initialization of the step assigns a value from
{ ⊥, T,
F}
to every
non-leading num-variable and assigns ? to other vars
I transitions:
I
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I
x = ⊥ in y : pred /eqn/inv (x , . . . ) ⇒ set y := ⊥
I
X = T in y : pred (X )
⇒ set y := T / y :=
I
p = ⊥ or p = F in when p do e
⇒ set e := ⊥
I
(consistency )
F
(non-det.)


pi = T
xi = ?
in yi : when pi do ei (x1 , . . . , xn , Zi ) ⇒ set xi , yi := T

Zi = T
where i = 1, . . . , n
(solving enabled blocks of n eqns/vars)
I termination by making a tick
I
I
requires success, i.e., all leading variables have been evaluated;
completes the current step and sets the initialization for the next step
26 / 34
Structural Analysis: non-det. transition system
I hot/cold initialization of the step assigns a value from
{ ⊥, T,
F}
to every
non-leading num-variable and assigns ? to other vars
I transitions:
I
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I
x = ⊥ in y : pred /eqn/inv (x , . . . ) ⇒ set y := ⊥
I
X = T in y : pred (X )
⇒ set y := T / y :=
I
p = ⊥ or p = F in when p do e
⇒ set e := ⊥
I
(consistency )
F
(non-det.)


pi = T
xi = ?
in yi : when pi do ei (x1 , . . . , xn , Zi ) ⇒ set xi , yi := T

Zi = T
where i = 1, . . . , n
(solving enabled blocks of n eqns/vars)
I termination by making a tick
I
I
requires success, i.e., all leading variables have been evaluated;
completes the current step and sets the initialization for the next step
26 / 34
Structural Analysis: non-det. transition system
I hot/cold initialization of the step assigns a value from
{ ⊥, T,
F}
to every
non-leading num-variable and assigns ? to other vars
I transitions:
I
x = ? in y : pred /eqn/inv (x , . . . ) ⇒ y = ?
I
x = ⊥ in y : pred /eqn/inv (x , . . . ) ⇒ set y := ⊥
I
X = T in y : pred (X )
⇒ set y := T / y :=
I
p = ⊥ or p = F in when p do e
⇒ set e := ⊥
I
(consistency )
F
(non-det.)


pi = T
xi = ?
in yi : when pi do ei (x1 , . . . , xn , Zi ) ⇒ set xi , yi := T

Zi = T
where i = 1, . . . , n
(solving enabled blocks of n eqns/vars)
I termination by making a tick
I
I
requires success, i.e., all leading variables have been evaluated;
completes the current step and sets the initialization for the next step
26 / 34
Structural Analysis: theoretical results
I The transition system is non-deterministic
(i) evaluation of predicates yields T/F
(ii) selection of one among all enabled moves
Lemma 1: Given the value of predicates, the transition system is confluent
I Partial order reduction:
I
I
Priority: tick > inv > block > pred
(to avoid enumerating modes)
Priorize minimal blocks with respect to Pryce order on variables
I Perform backward pruning of paths reaching a configuration violating an inv
I Lemma 2: There exists reachable deadlocks if and only if
I
I
either the system is over-/under-constrained at some reachable mode
or the initial configuration is inconsistent
27 / 34
Structural Analysis: theoretical results
I The transition system is non-deterministic
(i) evaluation of predicates yields T/F
(ii) selection of one among all enabled moves
Lemma 1: Given the value of predicates, the transition system is confluent
I Partial order reduction:
I
I
Priority: tick > inv > block > pred
(to avoid enumerating modes)
Priorize minimal blocks with respect to Pryce order on variables
I Perform backward pruning of paths reaching a configuration violating an inv
I Lemma 2: There exists reachable deadlocks if and only if
I
I
either the system is over-/under-constrained at some reachable mode
or the initial configuration is inconsistent
27 / 34
Structural Analysis: theoretical results
I The transition system is non-deterministic
(i) evaluation of predicates yields T/F
(ii) selection of one among all enabled moves
Lemma 1: Given the value of predicates, the transition system is confluent
I Partial order reduction:
I
I
Priority: tick > inv > block > pred
(to avoid enumerating modes)
Priorize minimal blocks with respect to Pryce order on variables
I Perform backward pruning of paths reaching a configuration violating an inv
I Lemma 2: There exists reachable deadlocks if and only if
I
I
either the system is over-/under-constrained at some reachable mode
or the initial configuration is inconsistent
27 / 34
Structural Analysis: theoretical results
I The transition system is non-deterministic
(i) evaluation of predicates yields T/F
(ii) selection of one among all enabled moves
Lemma 1: Given the value of predicates, the transition system is confluent
I Partial order reduction:
I
I
Priority: tick > inv > block > pred
(to avoid enumerating modes)
Priorize minimal blocks with respect to Pryce order on variables
I Perform backward pruning of paths reaching a configuration violating an inv
I Lemma 2: There exists reachable deadlocks if and only if
I
I
either the system is over-/under-constrained at some reachable mode
or the initial configuration is inconsistent
27 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
28 / 34
Multi-mode DAE and Pantelides
∆
var
vars
prop
num
pred
inv
eqn
term
def
system
::=
::=
::=
::=
::=
::=
::=
::=
::=
::=
::=
o | +
(zero|positive duration)
X | var + | var 0
var | var ,vars
T | F | var | not prop | prop and prop
num
pred (vars)
inv prop
when (prop, ∆) do (vars)
num | pred | inv | eqn
var : term
def | def ; system
I x + : right-limit (used for resets); x + is mapped to x •
I x 0 : derivative; x 0 is mapped to 1 (x •
∂
I
− x)
∆: duration type (zero | positive)
I This way every mDAE system is mapped to an mdAE system,
which is then submitted to the constructive semantics
29 / 34
Multi-mode DAE and Pantelides
I Until the constructive semantics succeeds, do
1. chose non-deterministically a guarded equation
“when γ do e(.)”
starving from dependent variables and replace it by
“when γ do e• (.)”
(unlocking action)
2. chose non-deterministically
I
I
•`
•`
a tuple (x1 1 , . . . , xk k ) of leading variables that is starving from
equations for its evaluation and
a numerical equation e(.) that depends on a subset of the set
•`0i
{xi
| i = 1, . . . , k ; `0i < `i } of variables,
and shift e(.) to e• (.)
(Pantelides action)
I This algorithm may not terminate, in which case the compilation fails
I If this algorithm terminates, we can move to the standardization phase
29 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
30 / 34
Structural Analysis: standardization
I At events:
I
map discrete non-standard time to Ed Lee’s superdense-time
I
map x(•t ,k ) to x(t ,k +1)
I
get rid of the infinitesimal ∂ arising in the non-standard expansion
of derivatives and compute a consistent reset state
for the next continuous mode
I Within continuous time modes:
I
use Taylor expansion to replace a constraint
[f (x )]• = 0
by the pair
f (x ) = 0
f 0 (x ).x 0 = 0
31 / 34
Structural Analysis: standardization
I At events:
I
map discrete non-standard time to Ed Lee’s superdense-time
I
map x(•t ,k ) to x(t ,k +1)
I
get rid of the infinitesimal ∂ arising in the non-standard expansion
of derivatives and compute a consistent reset state
for the next continuous mode
I Within continuous time modes:
I
use Taylor expansion to replace a constraint
[f (x )]• = 0
by the pair
f (x ) = 0
f 0 (x ).x 0 = 0
31 / 34
Structural Analysis: standardization
I At events:
I
map discrete non-standard time to Ed Lee’s superdense-time
I
map x(•t ,k ) to x(t ,k +1)
I
get rid of the infinitesimal ∂ arising in the non-standard expansion
of derivatives and compute a consistent reset state
for the next continuous mode
I Within continuous time modes:
I
use Taylor expansion to replace a constraint
[f (x )]• = 0
by the pair
f (x ) = 0
f 0 (x ).x 0 = 0
31 / 34
Motivation: physical system modeling
Background: Exact and Structural DAE index
Using NonStandard Analysis, DAE become dAE and index is preserved
Structural analysis of multi-mode DAE systems: approach
Examples
Discrete time multi-mode dAE
Continuous time multi-mode DAE
Formalization: Structural Analysis of multi-mode dAE systems
Formalization: Structural Analysis of multi-mode DAE and Pantelides
Standardization
Conclusions
32 / 34
Conclusions
I Modelica is much more powerful than classical (Simulink-like) modeling:
I
I
models for simulation by assembling sub-models from libraries
DAEs, multi-mode
I The compilation of Modelica with its multi-mode extension is difficult
I
I
problems in Modelica tools
we proposed a systematic approach
I Other uses of Modelica
I
Requirements: expressing abstract properties of systems as an early
phase of system design. Requires supporting under-determined
multi-mode DAE systems (less equations than variables)
I
Fault detection and diagnosis: generating parity models
F (X and derivatives, Y , U ) where some of the Y ’s and U’s are
observed; check if F = 0 holds when feeding with measurements.
Requires extensions of Modelica compilation techniques.
33 / 34
Conclusions
I Modelica is much more powerful than classical (Simulink-like) modeling:
I
I
models for simulation by assembling sub-models from libraries
DAEs, multi-mode
I The compilation of Modelica with its multi-mode extension is difficult
I
I
problems in Modelica tools
we proposed a systematic approach
I Other uses of Modelica
I
Requirements: expressing abstract properties of systems as an early
phase of system design. Requires supporting under-determined
multi-mode DAE systems (less equations than variables)
I
Fault detection and diagnosis: generating parity models
F (X and derivatives, Y , U ) where some of the Y ’s and U’s are
observed; check if F = 0 holds when feeding with measurements.
Requires extensions of Modelica compilation techniques.
33 / 34
Conclusions
I Modelica is much more powerful than classical (Simulink-like) modeling:
I
I
models for simulation by assembling sub-models from libraries
DAEs, multi-mode
I The compilation of Modelica with its multi-mode extension is difficult
I
I
problems in Modelica tools
we proposed a systematic approach
I Other uses of Modelica
I
Requirements: expressing abstract properties of systems as an early
phase of system design. Requires supporting under-determined
multi-mode DAE systems (less equations than variables)
I
Fault detection and diagnosis: generating parity models
F (X and derivatives, Y , U ) where some of the Y ’s and U’s are
observed; check if F = 0 holds when feeding with measurements.
Requires extensions of Modelica compilation techniques.
33 / 34
Conclusions
I Modelica is much more powerful than classical (Simulink-like) modeling:
I
I
models for simulation by assembling sub-models from libraries
DAEs, multi-mode
I The compilation of Modelica with its multi-mode extension is difficult
I
I
problems in Modelica tools
we proposed a systematic approach
I Other uses of Modelica
I
Requirements: expressing abstract properties of systems as an early
phase of system design. Requires supporting under-determined
multi-mode DAE systems (less equations than variables)
I
Fault detection and diagnosis: generating parity models
F (X and derivatives, Y , U ) where some of the Y ’s and U’s are
observed; check if F = 0 holds when feeding with measurements.
Requires extensions of Modelica compilation techniques.
33 / 34
Conclusions
I Modelica is much more powerful than classical (Simulink-like) modeling:
I
I
models for simulation by assembling sub-models from libraries
DAEs, multi-mode
I The compilation of Modelica with its multi-mode extension is difficult
I
I
problems in Modelica tools
we proposed a systematic approach
I Other uses of Modelica
I
Requirements: expressing abstract properties of systems as an early
phase of system design. Requires supporting under-determined
multi-mode DAE systems (less equations than variables)
I
Fault detection and diagnosis: generating parity models
F (X and derivatives, Y , U ) where some of the Y ’s and U’s are
observed; check if F = 0 holds when feeding with measurements.
Requires extensions of Modelica compilation techniques.
33 / 34
Conclusions
I Formal definition of the structural analysis
I Ideas from synchronous languages and their constructive semantics
I Can be used to compute at compile time: (i) the “index” and (ii) a
mode-dependent static scheduling of blocks of equations
I Formal definition of when and why the analysis may fail
I Implementation: S UNDAE determines index, computes static scheduling
I Future: import {Modelica intermediate format}
→ S UNDAE
I Open issues and extensions:
1. Efficient compilation algorithms
2. Generating efficient code
3. Handling under/over-constrained models for
I
I
partial designs
parity models in diagnosis
34 / 34
Conclusions
I Formal definition of the structural analysis
I Ideas from synchronous languages and their constructive semantics
I Can be used to compute at compile time: (i) the “index” and (ii) a
mode-dependent static scheduling of blocks of equations
I Formal definition of when and why the analysis may fail
I Implementation: S UNDAE determines index, computes static scheduling
I Future: import {Modelica intermediate format}
→ S UNDAE
I Open issues and extensions:
1. Efficient compilation algorithms
2. Generating efficient code
3. Handling under/over-constrained models for
I
I
partial designs
parity models in diagnosis
34 / 34
Conclusions
I Formal definition of the structural analysis
I Ideas from synchronous languages and their constructive semantics
I Can be used to compute at compile time: (i) the “index” and (ii) a
mode-dependent static scheduling of blocks of equations
I Formal definition of when and why the analysis may fail
I Implementation: S UNDAE determines index, computes static scheduling
I Future: import {Modelica intermediate format}
→ S UNDAE
I Open issues and extensions:
1. Efficient compilation algorithms
2. Generating efficient code
3. Handling under/over-constrained models for
I
I
partial designs
parity models in diagnosis
34 / 34