Fishwick P.A. (editor) Handbook of Dynamic System Modeling

Подождите немного. Документ загружается.

36-18 Handbook of Dynamic System Modeling

FIGURE 36.10 Automatically constructed parameter dialog.

Furthermore, other people and organizations are developing free and commercial Modelica libraries. For

more information and especially for downloading the free libraries, see http://www.modelica.org/library.

Version 2.2.1 of the Modelica standard library from March 2006 contains about 650 model classes and

500 functions in the following sublibraries:

Sublibrary Content

.Blocks Continuous, discrete, logical, table input/output blocks

.Electrical.Analog Analog electrical components (e.g., resistor, diode, and MOS transistors)

.Electrical.Digital Digital electrical components based on the VHDL standard for 9-valued logic with

conversions to 2-, 3-, 4- valued logic

.Electrical.Machines Uncontrolled asynchronous, synchronous, DC-machines

.Electrical.MultiPhase Analog electrical components for 2, 3, and more phases

.Math Mathematical functions (e.g., sin, cos) and functions operating on matrices (e.g., norm,

solve, eigenValues, and singularValues)

.Mechanics.MultiBody Three-dimensional mechanical components (bodies, joints, sensors, forces, etc.)

.Mechanics.Rotational One-dimensional rotational mechanical components. Includes convenient to use

speed- and torque-dependent Coulomb friction elements

.Mechanics.Translational One-dimensional translational mechanical components

.Media Large library of ﬂuids (1240 gases and mixtures between the gases, IF97 detailed

water medium, simple liquid water medium, dry and moist air, table-based media)

.SIunits 450 type deﬁnitions based on ISO 31-1992 (e.g., Angle and Length)

.StateGraph Hierarchical state machines with Modelica as action language

.Thermal.FluidHeatFlow Simple components for one-dimensional incompressible thermo-ﬂuid

ﬂow (e.g., to model cooling of machines with air or water).

.Thermal.HeatTransfer One-dimensional heat transfer with lumped elements

.Utilities Utility functions especially for scripting (e.g., operating on ﬁles, streams, strings,

and system)

Multidomain Modeling with Modelica 36-19

Transmission

Chassis

Driveline Brakes

Acc...

driver Enviro...

Road AtmosphereWorld

ControlBus

Engine

FIGURE 36.11 A VehiclesInterfaces architecture model consisting of accessory, engine, transmission, drive line,

chassis, brakes, driver environment, road, and atmosphere components with standardized interfaces.

Component libraries in package Modelica are realized by specialists in the respective area. There is also

a growing number of commercial Modelica libraries available, e.g.:

Library name Content

Air Conditioning Detailed models of vehicle air conditioning systems (from Modelon)

AlternativeVehicles Alternative vehicle architectures, such as hybrid cars (from arsenal research and DLR)

FlexibleBodies Modeling large motions of beams and of ﬂexible bodies exported from ﬁnite element programs (DLR)

HyLib Hydraulic components (from Modelon)

PneuLib Pneumatic components (from Modelon)

PowerTrain Vehicle power trains (from DLR)

SmartElectricDrives Detailed models of controlled machines (from arsenal research)

Transmission Detailed models of vehicle transmissions (from Ricardo)

VehicleDynamics Three-dimensional mechanical models of complete vehicles (from Modelon)

The automotive related libraries are based on the free VehicleInterfaces package that provides standard

component interfaces and vehicle architectures for complete vehicle system models, in order that automo-

tive libraries can work seamlessly together (Dempsey et al., 2006). This library is based on ideas of Tiller

et al. (2003) and of the PowerTrain library (PowerTrain, 2002). A screenshot of one of theVehicleInterfaces

architecture models is shown in Figure 36.11.

36.4 Symbolic Processing of Modelica Models

The Modelica Language Speciﬁcation (Modelica, 2005) deﬁnes how a Modelica model shall be mapped

into a mathematical description as a mixed system of differential-algebraic equations (DAE) and discrete

equations with Real, Integer, and Boolean variables as unknowns. There are no general-purpose solvers

for such problems. There are numerical DAE solvers, which could be used to solve the continuous part.

36-20 Handbook of Dynamic System Modeling

However, if a DAE solver is used directly to solve the original model equations, the simulation will be

very slow and initialization might be not possible for higher index systems (see below). It is therefore

assumed that Modelica models are ﬁrst symbolically transformed into a form that is better suited for

numerical solvers. In this section, the transformation techniques are sketched that have been initially

designed for the Dymola modeling language (Elmqvist, 1978), further developed in Omsim for the

Omola language (Mattsson and Söderlind, 1993) and in the commercial Modelica simulation environment

Dymola (Mattsson et al., 2000; Dynasim, 2006):

Dymola converts the differential-algebraic system of equations symbolically to ordinary differential

equations in state-space form, i.e., solves for the derivatives. Efﬁcient graph-theoretical algorithms are

used to determine which variables to solve for in each equation and to ﬁnd minimal systems of equations

to be solved simultaneously (algebraic loops). The equations are then, if possible, solved symbolically

or code for efﬁcient numeric solution is generated. Discontinuous equations are properly handled by

translation to state or time events as required by numerical integration routines.

36.4.1 Sorting and Algebraic Loops

The behavior of a Modelica model is deﬁned in terms of genuine equations and a Modelica translator

must assign an equation for each variable as part of the sorting procedure, which also identiﬁes algebraic

loops. To be able to process problems with hundred thousand unknowns, the idea is to focus on the

structural properties, i.e., which variables that appear in each equation rather than how they appear. This

information can be represented by a “structure” Jacobian, where for a system of equations, h(x) =0,each

element i, j,iszeroifx

does not appear in the expression h

, otherwise it is one. The sorting procedure is

to order unknowns and equations to make the structure Jacobian become Block Lower Triangular (BLT).

A BLT partitioning reveals the structure of a problem. It decomposes a problem into subproblems, which

can be solved in sequence. There are efﬁcient algorithms (see, e.g., Duff et al., 1986), for constructing BLT

partitions with diagonal blocks of minimum size (with respect to permutation of equations and variables).

Each nonscalar block on the diagonal constitutes an algebraic loop. This sorting procedure identiﬁes all

algebraic loops in their minimal form that is unique. The sorting procedure is done in two steps. The ﬁrst

step is to assign each variable, x

, to a unique equation, h

= 0 such that x

appears in this equation. It can be

viewed as permuting the equations to make all diagonal elements of the structure Jacobian nonzero. If it is

impossible to pair variables and equations in this way then the problem is structurally singular. The second

step of the BLT partition procedure is to ﬁnd the loops in a directed graph that has the variable/equation

pairs of the ﬁrst step as nodes. The basic algorithm was given by Tarjan (1972).

36.4.2 Reduction of Size and Complexity

A Modelica model has typically many simple equations, v

or v

=−v

being the result of connections.

These are easy to exploit for elimination.

From the BLT partition it is rather straightforward to ﬁnd unknowns that actually are constant and can

be calculated and substituted at translation. This may have considerable impact on the complexity of the

problem that has to be solved numerically. For example, a multibody component is developed for free

motion in a three-dimensional space. When using it we connect it to other components and set parameters

implying restrictions on its motion. For example, it may be restricted to move in a plane. It means that

coefﬁcients in the equations become zero and terms disappear. This in turn may make algebraic loops to

decompose into smaller loops or even disappear.

A linear small algebraic loop is solved symbolically. Otherwise code for efﬁcient numeric solution is

generated. To obtain efﬁcient simulation, it is very important to reduce the size of the problem sent to a

numerical solver. The work to solve a system of equations increases rapidly with the number of unknowns,

because the number of operations is proportional to the cube of n, i.e., O(n

), where n is the number of

unknowns. One approach to reduce size is called tearing (Elmqvist and Otter, 1994). Let z represent the

Multidomain Modeling with Modelica 36-21

unknowns to be solved from the system of equations. Let z be partitioned as z

and z

such that

L ·z

= f

)

0 = f

, z

)

where L is the lower triangular with nonzero diagonal elements. A numerical solver needs then only

consider z

as unknown. A numerical solver provides guesses for z

and would like to have the f

residuals

calculated for these guesses. When having a value for z

, it is simple to calculate z

from the ﬁrst set of

equations. Note, that it is very important to avoid divisions by zero. The assumption that the diagonal

elements are nonzero guarantees this. It is then straightforward to calculate the f

residuals. The z

variables

are in fact hidden from the numerical solver. The general idea of tearing is to decompose a problem into

two sets, where it is easy to solve for the ﬁrst set when the solution to the second set is known and to iterate

over the second set. The aim is of course to make the number of components of z

as small as possible.

It is a hard (NP-complete) problem to ﬁnd the minimum. However, there are fast heuristic approaches to

ﬁnd good partitions of z. If the equations are linear, they can be written as

= Az

0 = Bz

+Cz

and it is possible to eliminate z

to get Jz

=b,where

J = C + BL

−1

b = b

+BL

−1

This may be interpreted as Gauss elimination of z

. The procedure may be iterated. Note, since L isalower

triangular matrix, the determination of J and b is at most O(n

When solving a linear equation system, a major effort is to calculate an LU or QR factorization of the

Jacobian, J. Back substitutions are much less computationally demanding. In some cases, the elements of

the Jacobian does not vary continuously with time. The Jacobian may, for example, only change at events

and it is then only necessary to calculate and factorize it during event iterations and not during continuous

simulation. In other cases, it may depend only on parameters and constants and then it needs only to be

calculated once, at the start of a simulation.

When using Newton methods for nonlinear equation systems, it is necessary to calculate the Jacobian. If

this is made numerically from residuals, then n residual calculations are needed. Dymola provides analytic

Jacobians. These are more accurate and much less computationally demanding, because there are many

common subexpressions to exploit. Modelica provides facilities to provide derivatives also for external

functions.

36.4.3 Index Reduction

When solving an ordinary differential equation (ODE) the problem is to integrate, i.e., to calculate the states

when the derivatives are given. Solving a DAE may also include differentiation, i.e., to calculate the deriva-

tives of given variables. Such a DAE is said to have high index. It means that the number of states needed for

a model is less than the number of variables appearing differentiated. The number of states is equal to the

number of independent initial conditions that can be imposed. Higher index DAEs are typically obtained

because of constraints between models. To support reuse, model components are developed to be“general.”

Their behavior is restricted when they are used to build a model and connected to other components. Take

as a very simple example two rotating bodies with inertia J

and J

connected rigidly to each other. The

angles and the velocities of the two bodies should be equal. Not all four differentiated variables can be state

variables with their own independent start values. The connection equation for the angles, ϕ

=ϕ

, must

be differentiated twice to get a relation for the accelerations to allow calculation of the reaction torque.

The reliability of a direct numerical solution is related to the number of differentiations needed to

transform the system algebraically into ODE form. Modern numerical integration algorithms for DAEs,

36-22 Handbook of Dynamic System Modeling

such as used by most simulators, can handle systems where equations needed to be at most differentiated

once. However, reliable direct numerical solutions for nonlinear systems are not known if two or more

differentiations are required. Furthermore, if mixed continuous and discrete systems are solved, the hybrid

DAE must be initialized at every event instant. In this case, it is, in general, not sufﬁcient to just fulﬁll the

original DAE. Instead, also some differentiated equations have to be fulﬁlled, in order that the initialization

is consistent. Direct numerical methods have problems at events to determine consistent restart conditions

of higher index systems.

Higher index DAEs can be avoided by restricting how components may be connected together and/or

include manually differentiated equations in the components for the most common connection structures.

The drawback is (1) physically meaningful component connections may no longer be allowed in the model

or (2) unnecessary“stiff” elements have to be introduced in order that a connection becomes possible. For

example, if a stiff spring is introduced between the two rotating bodies discussed above, the problem has

no longer a higher index.

Since most Modelica libraries are designed in a truly object-oriented way, i.e., every meaningful physical

connection can also be performed with the corresponding Modelica components, this leads often to

higher index systems, especially in the mechanical and thermo-ﬂuid ﬁeld. Also modern controllers based

on nonlinear inverse plant models lead to higher index DAEs (Looye et al., 2005) and can be conveniently

treated with Dymola.

Dymola transforms higher index problems by differentiating equations analytically. The standard algo-

rithm by Pantelides (1988) is used to determine how many times each equation has to be differentiated.

The algorithm by Pantelides is based on the structure of the equations. It means that there are examples

where it does not give the optimal result (Reissig et al., 2000). However, the practical experience is very

good. Moreover, for large problems a structural analysis is the only feasible approach. Selection of which

variables to use as state variables is done statically during translation or in more complicated cases during

simulation with the dummy derivative method (Mattsson and Söderlind, 1993; Mattsson et al., 2000). Let

us make the example above a bit more realistic and put a gearbox with ﬁxed gear ratio n between the two

bodies. Dymola differentiates the position constraint twice to calculate the reaction torque in the coupling,

and it is sufﬁcient to select the angle and velocity of either body as state variables. The constraint leads

to a linear system of simultaneous equations involving angular accelerations and torques. The symbolic

solution contains a determinant of the form “J

”. Dymola thus automatically deduces how inertia

is transformed through a gearbox.

36.4.4 Example

To illustrate how Dymola’s symbolic processing reduces the size and complexity, we will show the structure

Jacobian at different stages when translating a mechanical model with a kinematic loop.

Figure 36.12 shows the structure Jacobian of the original model. There are about 1200 unknown variables

and equations. Each row corresponds to an equation and each column corresponds to a variable. A blue

marker indicates that the variable appears in the equation. There are 3995 markers. The upper half of the

matrix has a banded structure. These equations are the equations appearing in the component models

and such equations refer typically only to the local variables of the component. The equations in the lower

part are equations deduced from the connections, which includes references to variables of two or more

components.

Figure 36.13 shows the structure of the problem after exploitation of simple equations to eliminate alias

variables and utilizing zero constants. The number of unknowns is reduced from about 1200 to 330.

Then equations are differentiated to reduce the DAE index and states are selected. After some further

simpliﬁcations the number of unknowns is reduced to 250. A BLT partitioning reveals that there are three

algebraic loops as indicated by Figure 36.14.

Figure 36.15 shows the structure after tearing. The ﬁrst algebraic loop is a nonlinear loop with 12

unknowns. This loop includes the positional constraints of the kinematics loop. The tearing procedure

reduces the number of iteration variables to 2. This is illustrated by turning the eliminated part from grey

Multidomain Modeling with Modelica 36-23

0 200 400 600 800

1000 1200

200

400

600

800

1000

1200

nz  3995

FIGURE 36.12 The structure Jacobian of the original model.

0 50 100 150 200

250 300

100

150

200

250

300

nz  1017

FIGURE 36.13 The structure Jacobian after elimination of alias variables.

36-24 Handbook of Dynamic System Modeling

0 50 100 150 200

250

100

150

200

250

nz  895

FIGURE 36.14 The BLT partitioning.

0 50 100 150 200

250

100

150

200

250

nz  916

FIGURE 36.15 The structure after tearing.

Multidomain Modeling with Modelica 36-25

to green. The second loop includes the velocity constraints due to the kinematic loop. It means that this

loop includes the equations of the positional constraints differentiated. This loop has also 11 unknowns,

but it is linear. The remaining two by two system can be solved symbolically or numerically. The third loop

includes acceleration and force/torque as unknown variables. The loop is linear and has 62 unknowns and

the tearing procedure eliminates 57 so a linear 5 by 5 system remains to be solved numerically.

36.5 Outlook

The Modelica Association will continue to further improve the Modelica language and new versions of the

Modelica standard library will continue to occur every 6–12 months.

In June 2006, Dassault Systemes, the world leader in Computer Aided Design (CAD) and Product

Lifecycle Management (PLM) announced its new product line CATIA Systems (see http://www.3ds.com/

news-events/press-room). A central part will be behavioral modeling and simulation with Modelica and

Dymola. This will add a new dimension to Modelica because system simulation with close integration of

CAD and product databases will become possible. This means that the time consuming and error prone

process to parameterize system models will be considerably improved. We expect that Dassault Systems’

strategic decision will further accelerate the growth of the Modelica user community.

Acknowledgments

Up to 2006, about 50 people, mostly from the Modelica Association, have contributed to the Modelica

language and the Modelica Standard Library. The contributors are listed at the end of the Model-

ica language speciﬁcation (Modelica, 2005) and in the Modelica.UsersGuide of the Modelica standard

library (http://www.modelica.org/library/Modelica/Modelica_2_2_1.zip). This chapter is based on their

contributions.

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