Traslación controlada de actuador con colchón de aire ante fenómenos tipo Jerk.

Tesis (Maestría en Control y Sistemas Dinámicos)"Los procesos de manufactura pueden requieren movimientos rápidos y precisos; por ejemplo, para el transporte de materiales, paquetes, ensamble, etc. Los motores lineales de magnetos permanentes son una alternativa para garantizar precisión en movimientos lineales a altas velocidades. Sin embargo, los movimientos a altas velocidades ocasionan problemas que reducen el desempeño del sistema. Estos sistemas presentan fenómenos de sacudida al tener un lazo cerrado de control. Es por esto que, queremos un m´ınimo de sacudidas como requerimiento de diseño. El sistema en lazo cerrado puede ser descrito por una ecuación diferencial tipo Jerk al considerar la derivada temporal de las fuerzas que actúan en el sistema (por ejemplo fricción, reluctancia, inducida por la ley de control, entre otras ) obteniendo una ecuación diferencial de tercer orden. En el presente trabajo se busca la disminución de fenómenos de cambios temporales en la aceleración (sacudidas o Jerk) mediante una ley de control en un actuador lineal magnético para lograr un perfil de movimiento suave, el cual nos permite reducir el error durante el movimiento, menor desgaste del actuador y nos ayuda a evitar resonancia mecánica en la estructura. Sin embargo, ya que hay incertidumbre en la expresión exacta de estas fuerzas, si el sistema es observable, se puede construir un observador que estime los estados internos no medibles, relacionados a las fuerzas desconocidas, y con ellos diseñar una ley de control que aproxime la linealización del modelo y sea capaz de lograr el seguimiento de una referencia que garantice un mínimo de sacudidas. La aproximación es construida por un control dinámico no lineal que converge a uno geométrico linealizante.""Manufacturing processes might required fast and accurate movements, for example to transfer of materials, packages, assembly, etc. The permanent magnet linear motors are an alternative to ensure accuracy in linear movements at high speeds. Nevertheless, high speed movements bring inertia troubles reducing the system performance. This thesis seeks the compensation phenomena of temporal changes in acceleration, (known as Jerk), via a control law on a magnetic linear actuator and smooth motion, which allows us to reduce the movement error, less deterioration on the actuator and this helps us to avoid mechanical resonance in the structure. These systems present shaken phenomena as a result of having a closed loop control, this is why when closing the control loop, low shaken is required from design. This system can be described by a differential equation type Jerk considering the time derivative of the forces acting on the system (friction, reluctance, induced by the control law among others) obtaining a third-order differential equation, however, because we don’t know the exact expression of these forces, and knowing the system is observable, an observer is built to estimate the unmeasured internal states related to the unknown forces and from them designing a control law that approximates to a linear type control capable to achieve the tracking of a reference that guarantees bounded Jerk.

Una aproximación determinista de orden fraccionario al movimiento Browniano

"A partir de la ecuación de Langevin, un modelo determinista para la generación de movimiento Browniano es propuesto. Reemplazando el término estocástico por una variable de estado adicional da un grado de libertad más a la ecuación de Langevin y la transforma en un sistema de tres ecuaciones diferenciales lineales, además derivadas fraccionarias son consideradas; las cuales nos permiten obtener mejores propiedades estadísticas propias del movimiento Browniano real. Como parte de la aceleración fluctuante se establecen superficies de conmutación en el modelo. El sistema final no contiene términos estocásticos, esto es, el movimiento obtenido es completamente determinista. Además, del análisis de series de tiempo, encontramos que el comportamiento del sistema presenta las propiedades características de movimiento Browniano, tales como: crecimiento lineal en tiempo para el desplazamiento cuadrado promedio, distribución de probabilidad Gaussiana para el desplazamiento promedio. Adicionalmente, usamos el análisis de fluctuación sin tendencia para probar el carácter Browniano de las series obtenidas.""From the Langevin equation, a deterministic model for Brownian motion generation is proposed. Replacing the stochastic term with an additional state variable gives a degree of freedom to the Langevin equation and transforms it into a system of three linear differential equations, also fractional derivatives are considered; which allow us to obtain better statistics properties of the real Brownian movement. As a part of the fluctuating acceleration, switching surfaces are established in the model. The final system does not contain a stochastic terms, that is, the obtained motion is completely deterministic. In addition, from the time series analysis, we found that the system behavior exhibits statistics properties of Brownian motion, such as, a linear growth in time of the mean square displacement, Gaussian probability distribution for the average displacement. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion."Conacyt beca 26224

¿Es el movimiento Browniano un proceso estocástico o determinista?

"Con el afán de comprender el comportamiento del universo se han desarrollado distintas técnicas y herramientas para la generación de conocimiento estructurado que han permitido entender diferen­ tes fenómenos en la naturaleza, buscando formular leyes que describan en muchas ocasiones compor­ tamiento complejo. En lo que respecta a las cien­ cias exactas, se han desarrollado modelos matemá­ ticos que nos ayudan a describir la evolución tem­ poral del universo, ya sea de manera determinista o estocástica. No obstante fue hasta el siglo XX que se empezó a considerar la teoría probabilística.

Bistable behavior via switching dissipative systems with unstable dynamics and its electronic design

"In this work we present a design of a bistable system and its electronic circuit which is generated by a switching system. The switching system is comprised by dissipative subsystems with unstable dynamics based on the jerk equation. For this system with unstable dynamics, it is necessary to use a switching control law in order to change the equilibrium point of the linear part and get bounded trajectories. Also the dynamics of the piecewise linear (PWL) system is illustrated by numerical simulations to depict the bistable states. We present an easy electronic design of the proposed system by employing resistors, capacitors and comparators, to exhibit the capability to generate bistable behavior.

A class of Chua-like systems with only two saddle-foci of different type

"Since the reported Chua’s system, several generalizations of this system have been presented, these approaches include new equilibria in order to obtain three or more scrolls in the attractor. One of these generalizations requires at least the same number of saddle-foci with local two-dimensional unstable manifolds as the desired number of scrolls. In this work, we present the generation of a double-scroll chaotic attractor called Chua-like system. Once that an equilibrium point has been removed from the Chua’s system and there are only two saddle-foci of different class, i.e. the dimension of one of the local unstable manifolds is one while the other is of dimension two. The new class is constructed based on the existence of a heteroclinic loop by linear affine systems with two saddle-focus equilibrium points of different type. Furthermore, the chaotic behavior of the proposed system is tested by the maximum Lyapunov exponent and the 0 — 1 chaos test.

Multistability in piecewise linear systems versus eigenspectra variation and round function

"A multistable system generated by Piecewise Linear (PWL) subsystems based on the jerk equation is presented. The system’s behavior is characterized by means of the Nearest Integer or the round(x) function to control the switching events and to locate the corresponding equilibria on each of the commutation surfaces. These surfaces are generated through the switching function dividing the space into regions equally distributed along one axis. The trajectory of the system is governed by the eigenspectrum of the coefficient matrix, which can be adjusted by a bifurcation parameter. The behavior of the system can change from multiscroll oscillations in a mono-stable state into the coexistence of several single-scroll attractors in multistable states. The dynamics and bifurcation analysis are illustrated by numerical simulations to depict the multistable states.