Seguimiento de puntos singulares: un método para el análisis de bifurcaciones deslizantes en sistemas no suaves

El principal tópico de esta investigación está relacionado con un nuevo método para mejorar el entendimiento del comportamiento y el análisis de bifurcaciones deslizantes en sistemas no suaves. Este método ha sido llamado seguimiento de puntos singulares (Singular Point Tracking) debido a que está basado en la identificación y seguimiento de algunos puntos con características únicas. Estos puntos han sido poco estudiados motivando el deseo de realizar un más profundo estudio acerca del tema. Como resultado de esta investigación, se ha encontrado que la información de los puntos singulares unida a la información de los puntos regulares puede ser usada para encontrar o detectar bifurcaciones no suaves. Los resultados del método han sido presentados en eventos internaciones, publicados en revistas especializadas y son el principal bloque de este documento. En las publicaciones listadas abajo sólo se ha realizado un pequeño cambio en el formato, en comparación con las versiones publicadas originalmente. En el resto del documento, utilizando un formato de catálogo, se presenta cómo fue desarrollado el método / Abstract: The main objective of this research is to develop a new method for improving the understanding of the behaviour and the analysis of sliding bifurcations in nonsmooth systems. This method is called singular point tracking (SPT) because it is based on the identification and synthesis of certain points with unique characteristics. These points have been not given much attention, thus motivating an in-depth study on the subject. As a result of this research, it has been found that the information about singular points combined with the information on regular ones can be used for finding or detecting nonsmooth bifurcations. The results of the abovementioned method have been presented at international events, published in specialized journals, and constitute the main part of this document. In the publications listed below, only the format and notation has been changed slightly in comparison with the original published versions. In the rest of this document, I discuss how the abovementioned method was developed.Doctorad

Cone-like Invariant Manifolds for Nonsmooth Systems

This thesis deals with rigorous mathematical techniques for higher-dimensional nonsmooth systems and their applications. The dynamical behaviour of these systems is a nonlocal problem due to the lack of smoothness. Motivated by various examples of nonsmooth systems in applications, we propose to explore the concept of invariant surfaces in the phase space which is separated by a discontinuity hypersurface. For such systems the corresponding Poincaré map can be determined; it turns out that under suitable conditions an invariant cone occurs which is characterized by a fixed point of the Poincaré map. The invariant cone seems to serve in a similar way as a generalisation of the classical center manifold for smooth differential systems. Hence, the stability of the whole system can be reduced to investigate the stability on the two-dimensional surface of the cone. Motivated to study the generation of invariant cones out of smooth systems, a numerical procedure to establish invariant cones and their stability is presented. It has been found that the flat degenerate cone in a smooth system develops under nonsmooth perturbations into a cone-like configuration. Also a simple example is used to explain a paradoxical situation concerning stability. Theoretical results concerning the existence of invariant cones and possible mechanisms responsible for the observed behavior for general three dimensional nonsmooth systems are discussed. These investigations reveal that the system possesses a rich dynamic behavior and new phenomena such as, for instance, the existence of multiple invariant cones for such system. Our approach is developed to include the case when sliding motion takes place on the manifold. Sliding dynamical equations are formulated by using Filippov's method. Existence of invariant cones containing a segment of sliding orbits are given as well as stability on these cones. Different sliding bifurcation scenarios are treated by theoretical analysis and simulation. As an application we have investigated the dynamics of an automotive brake system model under the excitation of dry friction force which has served as a motivating example to develop our concepts. This model belongs to the class of nonsmooth systems of Filippov type which is investigated from direct crossing and a sliding motion point of view. Existence of invariant cones and different types of bifurcation phenomena such as sliding periodic doubling and multiple periodic orbits are observed. Finally, extensions to nonlinear perturbations of nonsmooth linear systems have been obtained by using the nonsmooth linear system as basic system. If the basic system possesses an attractive invariant cone without sliding motion, we have shown that locally the Poincaré map contains the necessary information with regard to attractivity of the invariant cone. The existence of a generalized center manifold reduction of nonlinear system has been proven by using Hadamard graph transformation approach. A class of nonlinear systems having a cone-like invariant "manifold" is presented to illustrate the center manifold reduction and associated bifurcation. The scientific contributions of parts of this thesis are presented in [32,39,66]

Finite Elements with Switched Detection for Direct Optimal Control of Nonsmooth Systems with Set-Valued Step Functions

This paper extends the Finite Elements with Switch Detection (FESD) method [Nurkanovi\'c et al., 2022] to optimal control problems with nonsmooth systems involving set-valued step functions. Logical relations and common nonsmooth functions within a dynamical system can be expressed using linear and nonlinear expressions of the components of the step function. A prominent subclass of these systems are Filippov systems. The set-valued step function can be expressed by the solution map of a linear program, and using its KKT conditions allows one to transform the initial system into an equivalent dynamic complementarity system (DCS). Standard Runge-Kutta (RK) methods applied to DCS have only first-order accuracy. The FESD discretization makes the step sizes degrees of freedom and adds further constraints that ensure exact switch detection to recover the high-accuracy properties that RK methods have for smooth ODEs. All methods and examples in this paper are implemented in the open-source software package NOSNOC.Comment: submitted to CDC202

Robust Simulation for Hybrid Systems: Chattering Path Avoidance

The sliding mode approach is recognized as an efficient tool for treating the chattering behavior in hybrid systems. However, the amplitude of chattering, by its nature, is proportional to magnitude of discontinuous control. A possible scenario is that the solution trajectories may successively enter and exit as well as slide on switching mani-folds of different dimensions. Naturally, this arises in dynamical systems and control applications whenever there are multiple discontinuous control variables. The main contribution of this paper is to provide a robust computational framework for the most general way to extend a flow map on the intersection of p intersected (n--1)-dimensional switching manifolds in at least p dimensions. We explore a new formulation to which we can define unique solutions for such particular behavior in hybrid systems and investigate its efficient computation/simulation. We illustrate the concepts with examples throughout the paper.Comment: The 56th Conference on Simulation and Modelling (SIMS 56), Oct 2015, Link\"oping, Sweden. 2015, Link\"oping University Pres

The regularized visible fold revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law

Variational Methods for Control and Design of Bipedal Robot Models

This thesis investigates nonsmooth mechanics using variational methods for the modeling, control, and design of bipedal robots. The theory of Lagrangian mechanics is extended to capture a variety of nonsmooth collision behaviors in rigid body systems. Notably, a variational impact model is presented for the transition of constraints behavior that describes a biped switching stance feet at the conclusion of a step. Next, discretizations of the impact mechanics are developed using the framework of variational discrete mechanics. The resulting variational collision integrators are consistent with the continuous time theory and have an underlying symplectic structure. In addition to their role as integrators, the discrete equations of motion capturing nonsmooth dynamics enable a direct method for trajectory optimization. Upon specifically defining the optimal control problem for nonsmooth systems, examples demonstrate this optimization method in the task of determining periodic gaits for two rigid body biped models. An additional effort is made to optimize bipedal walking motions through modifications in system design. A method for determining optimal designs using a combination of trajectory optimization methods and surrogate function optimization methods is defined. This method is demonstrated in the task of determining knee joint placement in a given biped model.</p

Localización de bifurcaciones deslizantes en un oscilador rotativo de doble leva

En este trabajo se analizan las bifurcaciones no suaves no convencionales también llamadas deslizantes en un sistema con múltiples límites de discontinuidad. El método de seguimiento de puntos singulares (SPT) es probado para localizar las bifurcaciones en un oscilador rotativo de doble leva. Los resultados indican que el método SPT puede ser utilizado para analizar diferentes tipos de sistemas no suaves que presentan dinámicas deslizantes complejas