We offer a technique, motivated by feedback control and specifically sliding mode control, for the simulation of differential-algebraic equations (DAEs) that describe common engineering systems such as constrained multibody mechanical structures and electric networks. Our algorithm exploits the basic results from sliding mode control theory to establish a simulation environment that then requires only the most primitive of numerical solvers. We circumvent the most important requisite for the conventional simulation of DAEs: the calculation of a set of consistent initial conditions. Our algorithm, which relies on the enforcement and occurrence of sliding mode, will ensure that the algebraic equation is satisfied by the dynamic system even for inconsistent initial conditions and for all time thereafter.

1.
Simeon
,
B.
, 2001, “
Numerical Analysis of Flexible Multibody Systems
,”
Multibody Syst. Dyn.
1384-5640,
6
(
4
), pp.
305
325
.
2.
Simeon
,
B.
, 2006, “
On Lagrange Multipliers in Flexible Multibody Dynamics
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
195
, pp.
6993
7005
.
3.
Brenan
,
K. E.
,
Campbell
,
S. L.
, and
Petzold
,
L. R.
, 1989,
Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations
,
North-Holland
,
New York
.
4.
Gear
,
C. W.
, 2006, “
Towards Explicit Methods for Differential Algebraic Equations
,”
BIT Numerical Mathematics
,
46
(
3
), pp.
505
514
.
5.
Utkin
,
V.
, 1992,
Sliding Modes in Control and Optimization
,
Springer-Verlag
,
New York
.
6.
Utkin
,
V. I.
,
Guldner
,
J. G.
, and
Shi
,
J.
, 1999,
Sliding Mode Control in Electromechanical Systems
,
Taylor & Francis
,
London
.
7.
Leimkuhler
,
B.
,
Petzold
,
L. R.
, and
Gear
,
C. W.
, 1991, “
Approximation Methods for the Consistent Initialization of Differential-Algebraic Equations
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429,
28
(
1
), pp.
205
226
.
8.
Shampine
,
L. F.
,
Reichelt
,
M. W.
, and
Kierzenka
,
J. A.
, 1999, “
Solving Index-1 DAEs in MATLAB and Simulink
,”
SIAM Rev.
0036-1445,
41
(
3
), pp.
538
552
.
9.
Brown
,
P. N.
,
Hindmarsh
,
A. C.
, and
Petzold
,
L. R.
, 1998, “
Consistent Initial Condition Calculation for Differential-Algebraic Systems
,”
SIAM J. Sci. Comput. (USA)
1064-8275,
19
(
5
), pp.
1495
1512
.
10.
Arnold
,
M.
,
Fuchs
,
A.
, and
Fuehrer
,
C.
, 2006, “
Efficient Corrector Iteration for DAE Time Integration in Multibody Dynamics
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
195
, pp.
6958
6973
.
11.
Utkin
,
V. I.
, and
Lee
,
H.
, 2006, “
The Chattering Analysis
,”
Twelfth International Power Electronics and Motion Control Conference 2006
, pp.
2014
2019
, Paper No. EPE-PEMC 2006.
12.
Rao
,
S.
, and
Utkin
,
V.
, 2003, “
Development of a Unified Simulation Methodology for Electric Networks Using Sliding Modes
,”
2003 IEEE International Conference on Industrial Technology
, Vol.
1
, pp.
439
444
.
13.
Schuster
,
M.
, and
Unbehauen
,
R.
, 2006, “
Analysis of Nonlinear Electric Networks by Means of Differential Algebraic Equations Solvers
,”
Electr. Eng.
0948-7921,
88
(
3
), pp.
229
239
.
14.
Tischendorf
,
C.
, and
Schwarz
,
D. E.
, 2001, “
Mathematical Problems in Circuit Simulation
,”
Math. Comput. Model. Dyn. Syst.
,
7
(
2
), pp.
215
223
.
15.
Zienkiewicz
,
O. C.
,
Taylor
,
R. L.
, and
Zhu
,
D. J.
, 2005,
The Finite Element Method: Its Basis and Fundamentals
,
6th ed.
,
Butterworth-Heinemann
.
16.
Meirovitch
,
L.
, 1997,
Principles and Techniques of Vibrations
,
Prentice Hall
.
You do not currently have access to this content.