Real-time control often requires process models that can be solved adequately fast on prescribed control hardware. One class of applications is model predictive control, which can be considered as one of the most powerful control techniques. However, for real-time control of systems with fast dynamics such as vehicles, appropriate models are necessary. Thus, on the one hand, the design process requires models of adequate complexity to cover all relevant effects, and, on the other hand, real-time control requires models with a limited number of computations. In this contribution, an approach for a generation of generic models of the dynamic of technical systems is presented. In this context generic is twofold. First, the model is dimensionless. In particular, the model is robust with respect to variations of the dimensionless parameters and can be easily projected onto a large variety of geometries. Second, equation-based reduction techniques are used to limit the number of necessary operations for a simulation. Hence, the models can be reduced for given controller hardware. As an application example, a standard vehicle model with nonlinear tire characteristics is used. Here the equation-based reduction techniques lead to a reduction of the required computations by a factor of five. In this case the maximum error compared to the reference model is smaller than 5%.

References

References
1.
Brennan
,
S.
, and
Alleyne
,
A.
, 2005, “
Dimensionless Robust Control With Application to Vehicles
,”
IEEE Trans. Control Syst. Technol.
,
13
(
4
), pp.
624
630
.
2.
Brennan
,
S.
, 2002, “
On Size and Control: The Use of Dimensional Analysis in Controller Design
,” Ph.D. thesis, University of Illinois at Urbana-Champaign, Urbana, IL.
3.
Li
,
Y.
,
Hencey
,
B.
, and
Alleyne
,
A.
, 2008, “
Dimensional Analysis for Robust Control of Planar Vehicle Dynamics
,”
Int. J. Robust Nonlinear Control
,
18
(
6
), pp.
587
616
.
4.
Szirtes
,
T.
, and
Rózsa
,
P.
, 1997,
Applied Dimensional Analysis and Modeling
,
McGraw-Hill
,
New York
.
5.
Langhaar
,
H.
, 1951,
Dimensional Analysis and Theory of Models
,
Wiley
,
New York
.
6.
Lapaponga
,
S.
,
Guptaa
,
V.
,
Callejasb
,
E.
, and
Brennana
,
S.
, 2009, “
Fidelity of Using Scaled Vehicles for Chassis Dynamic Studies
,”
Veh. Syst. Dyn.
,
47
(
11
), pp.
1401
1437
.
7.
Petersheim
,
M.
, and
Brennan
,
S.
, 2009, “
Scaling of Hybrid-Electric Vehicle Powertrain Components for Hardware-In-The-Loop Simulation
,”
Mechatronics
,
19
(
7
), pp.
1078
1090
.
8.
Antoulas
,
A.
,
Sorensen
,
D.
, and
Gugercin
,
S.
, 2001, “
A Survey of Model Reduction Methods for Large-Scale Systems
,”
Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference
,
Boulder, CO
, June 27-July 1,
280
, p.
193
.
9.
Gugercin
,
S.
, and
Antoulas
,
A.
, 2002, “
A Comparative Study of 7 Algorithms for Model Reduction
,”
Proceedings of the 39th IEEE Conference on Decision and Control
,
3
, pp.
2367
2372
.
10.
Ersal
,
T.
,
Fathy
,
H.
,
Rideout
,
D.
,
Louca
,
L.
, and
Stein
,
J.
, 2008, “
A Review of Proper Modeling Techniques
,”
J. Dyn. Syst., Meas., Control
,
130
, p.
061008
.
11.
Gorban
,
A.
,
Kazantzis
,
N.
,
Kevrekidis
,
I.
,
Ottinger
,
H.
, and
Theodoropoulos
,
C.
, eds., 2006,
Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena
,
Springer
,
New York
.
12.
Burgermeister
,
B.
,
Arnold
,
A.
, and
Eichberger
,
A.
, 2009, “
Smooth Velocity Approximation for Constrained Systems in Real-Time Simulation
,”
Proceedings of ECCOMAS Thematic Conference on Multibody Dynamics
.
13.
Ersal
,
T.
,
Fathy
,
H.
, and
Stein
,
J.
, 2009, “
Realization-Preserving Structure and Order Reduction of Nonlinear Energetic System Models Using Energy Trajectory Correlations
,”
J. Dyn. Syst., Meas., Control
,
131
, p.
031004
.
14.
Ersal
,
T.
,
Kittirungsi
,
B.
,
Fathy
,
H.
, and
Stein
,
J.
, 2009, “
Model Reduction in Vehicle Dynamics Using Importance Analysis
,”
Veh. Syst. Dyn.
,
47
(
7
), pp.
851
865
.
15.
Borchers
,
C.
, 1998, “
Symbolic Behavioral Model Generation of Nonlinear Analog Circuits
,”
IEEE Trans. Circuits Syst. II: Analog Digital Signal Process.
,
45
(
10
), pp.
1362
1371
.
16.
Wichmann
,
T.
, 2004, “
Symbolische Reduktionsverfahren f¨ur nichtlineare DAE-Systeme
,”
Berichte aus der Mathematik.
Shaker Verlag, Aachen
,
Germany
.
17.
Mikelsons
,
L.
,
Ji
,
H.
,
Brandt
,
T.
, and
Lenord
,
O.
, 2009, “
Symbolic Model Reduction Applied to Realtime Simulation of a Construction Machine
,”
Proceedings of the 7th Modelica Conference
,
Como, Italy
, Sept.
20
22
.
18.
Buckingham
,
E.
, 1914, “
On Physically Similar Systems: Illustrations of the use of Dimensional Equations
,”
Phys. Rev.
,
4
(
4
), pp.
345
376
.
19.
Isaacson
,
E.
, and
Isaacson
,
M.
, 1975,
Dimensional Methods in Engineering and Physics: Reference Sets and the Possibilities of Their Extension
,
Halsted Press
,
New York
.
20.
Riekert
,
P.
, and
Schunck
,
T.
, 1940, “
Zur Fahrmechanik des gummibereiften Kraftfahrzeugs
,”
Arch. Appl. Mech. (Ingenieur Archiv)
,
11
(
3
), pp.
210
224
.
21.
Mitschke
,
M.
, and
Wallentowitz
,
H.
, 2004,
Dynamik der Kraftfahrzeuge
,
Springer
,
Berlin
.
22.
Pacejka
,
H.
, 2006,
Tyre and Vehicle Dynamics
,
Butterworth-Heinemann Ltd.
,
Oxford, UK
.
23.
Bakker
,
E.
,
Nyborg
,
L.
, and
Pacejka
,
H.
, 1987, “
Tyre Modelling for Use in Vehicle Dynamics Studies
,”
Proceedings of Society of Automotive Engineers International Congress and Expo
,
23
.
24.
Heydinger
,
G.
,
Bixel
,
R.
,
Garrott
,
W.
,
Pyne
,
M.
,
Howe
,
J.
, and
Guenther
,
D.
, 2004, “
Measured Vehicle Inertial Parameters-NHTSA’s Data Through November 1998
,”
Prog. Technol.
,
101
, pp.
531
554
.
25.
Sommer
,
R.
,
Halfmann
,
T.
, and
Broz
,
J.
, 2008, “
Automated Behavioral Modeling and Analytical Model-Order Reduction by Application of Symbolic Circuit Analysis for Multi-Physical Systems
,”
Simul. Modell. Pract. Theory
,
16
(
8
), pp.
1024
1039
.
26.
Wichmann
,
T.
, 2003, “
Transient Ranking Methods for the Simplification of Nonlinear DAE Systems in Analog Circuit Design
,”
Proc. Appl. Math. Mech.
,
2
(
1
), pp.
448
449
.
27.
Maly
,
T.
, and
Petzold
,
L.
, 1996, “
Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems
,”
Appl. Numer. Math.
,
20
(
1
), pp.
57
82
.
28.
Li
,
S.
, and
Petzold
,
L.
, 2000, “
Software and Algorithms for Sensitivity Analysis of Large-Scale Differential Algebraic Systems
,”
J. Comput. Appl. Math.
,
125
(
1–2
), pp.
131
145
.
29.
Cao
,
Y.
,
Li
,
S.
, and
Petzold
,
L.
, 2002, “
Adjoint Sensitivity Analysis for Differential-Algebraic Equations: Algorithms and Software
,”
J. Comput. Appl. Math.
,
149
(
1
), pp.
171
191
.
30.
Cao
,
Y.
,
Li
,
S.
,
Petzold
,
L.
, and
Serban
,
R.
, 2003, “
Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and its Numerical Solution
,”
SIAM J. Sci. Comput.
,
24
(
3
), p.
1076
.
31.
Hindmarsh
,
A.
,
Brown
,
P.
,
Grant
,
K.
,
Lee
,
S.
,
Serban
,
R.
,
Shumaker
,
D.
, and
Woodward
,
C.
, 2005, “
SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers
,”
ACM Trans. Math. Softw.
,
31
(
3
), pp.
363
396
.
32.
Blochwitz
,
T.
, and
Beutlich
,
T.
, 2009, “
Real-Time Simulation of Modelica-Based Models
,”
Proceedings of the 7th Modelica Conference
, Como, Italy, Sept.
20
22
.
33.
Rill
,
G.
, 1994,
Simulation von Kraftfahrzeugen
.
Vieweg
,
Braunschweig
.
34.
Dongarra
,
J.
,
Luszczek
,
P.
, and
Petitet
,
A.
, 2003, “
The LINPACK Benchmark: Past, Present and Future
,”
Concurrency Comput.: Pract. Exper.
,
15
(
9
), pp.
803
820
.
35.
Fritzson
,
P.
,
Aronsson
,
P.
,
Bunus
,
P.
,
Engelson
,
V.
,
Saldamli
,
L.
,
Johansson
,
H.
, and
Karstom
,
A.
, 2002, “
The Open Source Modelica Project
,”
Proceedings of the 2nd International Modelica Conference
, pp.
18
19
.
You do not currently have access to this content.