Piecewise affine (PWA) systems belong to a subclass of switched systems and provide good flexibility and traceability for modeling a variety of nonlinear systems. In this paper, application of the PWA system framework to the modeling and control of an automotive all-wheel drive (AWD) clutch system is presented. The open-loop system is first modeled as a PWA system, followed by the design of a piecewise linear (i.e., switched) feedback controller. The stability of the closed-loop system, including model uncertainty and time delays, is examined using linear matrix inequalities based on Lyapunov theory. Finally, the responses of the closed-loop system under step and sine reference signals and temperature disturbance signals are simulated to illustrate the effectiveness of the design.

Issue Section:
Research Papers
Keywords:
piecewise affine systems, time delay systems, automotive clutch systems, Lyapunov stability
Topics:
Closed loop systems, Delays, Design, Stability, Temperature, Torque, Uncertainty, Control equipment, Control modeling, Wheels, Feedback, Feedforward control

References

1.
Sontag
,
E. D.
,
1996
, “
Interconnected Automata and Linear Systems: A Theoretical Framework in Discrete-Time
,”
Hybrid Systems III—Verification and Control
,
Springer-Verlag
,
New York
, pp.
436
–
448
.
2.
Schutter
,
B. D.
, and
Moor
,
B. D.
,
1999
, “
The Extended Linear Complementarity Problem and the Modeling and Analysis of Hybrid Systems
,”
Hybrid Systems Verification and Control: Lecture Notes in Computer Science
, Vol.
1567
,
Springer-Verlag
,
New York
, pp.
70
–
85
.
3.
Kulkarni
,
V.
,
2003
, “
Optimal Mode Changes for Highway Transportation Safety
,”
IEEE International Conference on Systems
,
Man and Cybernetics
, Vol.
2
, pp.
1235
–
1240
.
4.
Dai
,
D.
,
Hu
,
T.
,
Teel
,
A. R.
, and
Zaccarian
,
L.
,
2009
, “
Piecewise-Quadratic Lyapunov Functions for Systems With Dead Zones or Saturations
,”
Syst. Control Lett.
,
58
, pp.
365
–
371
.10.1016/j.sysconle.2009.01.003
Google Scholar
Crossref
Search ADS
 
5.
Johansson
,
M.
,
2003
,
Piecewise Linear Control Systems
,
Springer-Verlag
,
New York
.
6.
Moezzi
,
K.
,
Rodrigues
,
L.
, and
Aghdam
,
A.
,
2009
, “
Stability of Uncertain Piecewise Affine Systems With Time Delay: Delay Dependent Lyapunov Approach
,”
Int. J. Control
,
82
(
8
), pp.
1423
–
1434
.10.1080/00207170802443699
Google Scholar
Crossref
Search ADS
 
7.
Bemporad
,
A.
,
Ferrari-Trecate
,
G.
, and
Morari
,
M.
,
2000
, “
Observability and Controllability of Piecewise Affine and Hybrid Systems
,”
IEEE Trans. Autom. Control
,
45
, pp.
1864
–
1876
.10.1109/TAC.2000.880987
Google Scholar
Crossref
Search ADS
 
8.
Rodrigues
,
L.
, and
How
,
J. P.
,
2003
, “
Observer-Based Control of Piecewise Affine System
,”
Int. J. Control
,
76
, pp.
459
–
477
.10.1080/0020717031000091432
Google Scholar
Crossref
Search ADS
 
9.
Habets
,
L. C. G. J. M.
,
Collins
,
P. J.
,
van Schuppen
,
J. H.
,
2006
, “
Reachability and Control Synthesis for Piecewise-Affine Hybrid Systems on Simplices
,”
IEEE Trans. Autom. Control
,
51
(
6
), pp.
938
–
948
.10.1109/TAC.2006.876952
Google Scholar
Crossref
Search ADS
 
10.
Branicky
,
M. S.
,
1998
, “
Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems
,”
IEEE Trans. Autom. Control
,
43
(
4
), pp.
475
–
482
.10.1109/9.664150
Google Scholar
Crossref
Search ADS
 
11.
Rodrigues
,
L.
,
2005
, “
State Feedback Control of Piecewise-Affine Systems With Norm Bounded Noise
,”
Proceedings of American Control Conference
, pp.
1793
–
1798
.
12.
Hallowell
,
S. J.
,
2005
, “
Torque Distribution Systems and Methods for Wheeled Behicles
,” U.S. Patent No. 6,909,959.
13.
Kulkarni
,
V.
,
Jun
,
M.
, and
Hespanha
,
J.
,
2004
, “
Piecewise Quadratic Lyapunov Functions for Piecewise Affine Time-Delay Systems
,”
Proceedings of the 2004 American Control Conference
, Vol.
5
, No.
30
, pp.
3885
–
3889
.
14.
Xu
,
S.
, and
Lam
,
J.
,
2005
, “
Improved Delay-Dependent Stability Criteria for Time-Delay Aystems
,”
IEEE Trans. Autom. Control
,
50
, pp.
384
–
387
.10.1109/TAC.2005.843873
Google Scholar
Crossref
Search ADS
 
15.
Duan
,
S.
,
Ni
,
J.
, and
Ulsoy
,
A. G.
,
2012
, “
An Improved LMI-Based Approach for Stability of Piecewise Affine Time-Delay Systems With Uncertainty
,”
Int. J. Control
,
85
, pp.
1218
–
1234
.10.1080/00207179.2012.680915
Google Scholar
Crossref
Search ADS
 
16.
Osborn
,
R. P.
, and
Shim
,
T.
,
2007
, “
Independent Control of All-Wheel-Drive Torque Distribution
,”
Veh. Syst. Dyn.
,
44
(
7
), pp.
529
–
546
.10.1080/00423110500485731
Google Scholar
Crossref
Search ADS
 
17.
Mäki
,
R.
,
2005
, “
Wet Clutch Tribology-Friction Characteristics in Limited Slip Differentials
,” Ph.D. thesis, Luleå University of Technology, Luleå, Sweden.
18.
Kapila
,
V.
, and
Grigoriadis
K. M.
,
2002
,
Actuator Saturation Control
,
M. Dekker
,
New York
.
19.
Yi
,
S.
,
Nelson
,
P.
, and
Ulsoy
,
A. G.
,
2010
,
Time-Delay Systems-Analysis and Control Using the Lambert W Function
,
World Scientific
,
Hackensack, NJ
.
20.
Yi
,
S.
,
Duan
,
S.
,
Nelson
,
P.
, and
Ulsoy
,
A. G.
,
2012
, “
The Lambert W Function Approach to Time Delay Systems and the LambertW_DDE Toolbox
,” IFAC Workshop on Time Delay Systems, Boston, MA.
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