Bifurcations; Linear complementarity problems; Piecewise linear equations; Topological equivalence; Bifurcation; Bifurcations theory; Control engineering; Nonsmooth bifurcations; Piecewise linear; Piecewise linear equation; Piecewise-linear; Control and Systems Engineering
Abstract :
[en] Many systems of interest to control engineering can be modeled by linear complementarity problems. We introduce a new notion of equivalence between linear complementarity problems that sets the basis to translate the powerful tools of smooth bifurcation theory to this class of models. Leveraging this notion of equivalence, we introduce new tools to analyze, classify, and design nonsmooth bifurcations in linear complementarity problems and their interconnection.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Castaños, Fernando; Automatic Control Department, CDMX, Mexico
Miranda-Villatoro, Felix A.; Department of Engineering, University of Cambridge., Cambridge, United Kingdom
Franci, Alessio ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Brain-Inspired Computing ; Department of Mathematics, Universidad Nacional Autónoma de México, CDMX, Mexico
Language :
English
Title :
A notion of equivalence for linear complementarity problems Applications to the design of nonsmooth bifurcations
Acary, V., Bonnefon, O., and Brogliato, B. (2011). Nonsmooth modeling and simulation for switched circuits. Lecture Notes in Electrical Engineering. Springer.
Arnold, V.I., Varchenko, A.N., and Gusein-Zade, S.M. (1985). Singularities of Differentiable Maps, volume 1. Birkhäuser.
Brogliato, B. (1999). Nonsmooth mechanics: models, dynamics and control. Springer-Verlag, London, second edition.
Castaños, F. and Franci, A. (2017). Implementing robust neuromodulation in neuromorphic circuits. Neurocomputing, 233, 3-13.
Castaños, F., Miranda-Villatoro, F., and Franci, A. (2019). A notion of equivalence for linear complementarity problems with application to the design of non-smooth bifurcations. ArXiv:1911.03774.
Clarke, F. (1990). Optimization and nonsmooth analysis. Society for Industrial and Applied Mathematics.
Cottle, R.W., Pang, J.S., and Stone, R.E. (2009). The Linear Complementarity Problem. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics.
Danao, R.A. (1994). Q-matrices and boundedness of solutions to linear complementarity problems. Journal of Optimization Theory and Applications, 83(2), 321-332.
Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Tost, G.O., and Piiroinen, P.T. (2008). Bifurcations in nonsmooth dynamical systems. SIAM review, 50(4), 629-701.
Garcia, C.B., Gould, F.J., and Turnbull, T.R. (1983). Relations between PL maps, complementary cones, and degree in linear complementarity problems. In Homotopy Methods and Global Convergence, 91-144. Plenum Publishing Corporation.
Golubitsky, M. and Schaeffer, D. (1985). Singularities and Groups in Bifurcation Theory, volume I of Applied Mathematical Sciences. Springer, New York.
Leenaerts, D.M. and Bokhoven, W.M.G. (1998). Piecewise linear modeling and analysis. Kluwer Academic. Springer, New York.
Leine, R.I. and Nijmeijer, H. (2004). Dynamics and bifurcations on non-smooth mechanical systems. Springer, Berlin.
Murty, K.G. (1988). Linear complementarity, linear and nonlinear programming. Helderman Verlag, Berlin.
Nagurney, A. (1999). Network economics: A variational inequality approach. Advances in Computational Economics. Springer-Science+Business Media.
Sikorski, R. (1969). Boolean Algebras. Springer-Verlag, Berlin, Germany.
van der Schaft, A.J. and Schumacher, J. (1998). Complementarity modeling of hybrid systems. IEEE Trans. Autom. Control, 43, 483 - 490.
