FEEDBACK CONTROL OF BORDER COLLISION BIFURCATIONS IN PIECEWISE SMOOTH SYSTEMS
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Feedback control of border collision bifurcations in continuous piecewise smooth discrete-time systems is considered. These bifurcations occur when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides with the border between two regions of smooth operation as a system parameter is quasistatically varied. The goal of the control effort in this work is to modify the bifurcation so that the bifurcated steady state is locally attracting and locally unique. In this way, the system's local behavior is ensured to remain stable and close to the original operating condition. Linear and piecewise linear feedbacks are used since the system linearization on the two sides of the border generically determines the type and stability properties of any border collision bifurcation.
A complete classification of possible border collision bifurcations is only available for one-dimensional maps. These classifications are used in the design of stabilizing feedback controls. For two dimensional piecewise smooth maps, sufficient conditions for nonbifurcation with persistent stability are proved. The derived sufficient conditions are then used as a basis for the design of feedback controls to eliminate border collision bifurcations.
For higher dimensional piecewise smooth maps, only very general results on existence of certain types of border collision bifurcations are currently known. To address these problems Lyapunov techniques are used to find conditions for nonbifurcation with persistent local stability in general finite dimensional piecewise smooth discrete time systems depending on a parameter. A sufficient condition for nonbifurcation with persistent stability in PWS maps of any finite dimension is given in terms of linear matrix inequalities. The Lyapunov-based methodology is used to consider the design of washout filter based controllers. These are dynamic feedback laws that do not alter a system's fixed points, even in the presence of model uncertainty. These ideas are extended to allow nonmonotonically decreasing Lyapunov functions.
Finally, a two-dimensional example of using feedback to quench cardiac arrhythmia is considered. The cardiac model consists of a nonlinear discrete-time piecewise smooth system, and was previously used to show a link between cardiac alternans and period doubling bifurcation.