Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov-Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov-Poincaré-Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations

Increasing the reactant conversion through induced oscillations in a continuous stirred tank reactor by using PI control

We report a strategy to increase the reactant conversion in a continuous stirred tank reactor (CSTR) to produce propylene glycol through induced oscillations generated by two controllers PI1 and PI2 that manipulate the reactor outlet flow and the coolant flow rate respectively. It is shown that an adequate parameter choice for the PI controllers allows one to derive sustained oscillations in the concentrations and reactor temperature, which in turn allows increasing the propylene glycol production. For a suitable choice of the PI1 and PI2 controller parameters, we use a complete reactor model that provides with physically feasible parameters. The issues of external disturbance rejection, self-oscillations and stability have also been discussed. The analytical calculations are verified by means of full numerical simulations

Steady-state self-oscillations and chaotic behavior of a controlled electromechanical device by using the first Lyapunov value and the Melnikov theory

In this paper regular and chaotic oscillations in a controlled electromechanical transducer are investigated. The nonlinear control laws are defined by an electric tension excitation and an external force applied to the mobile piece of the transducer. The paper shows that an Andronov–Poincaré–Hopf bifurcation appears as long as adequate parameters are chosen for the nonlinear control laws. The stability of the weak focuses associated to such bifurcation is examined according to the sign of the first Lyapunov value, showing that chaotic behavior can arise when the first Lyapunov value is varied harmonically. The appearance of a homoclinic orbit is investigated assuming an approximated model for the device. On the basis of the parametric equations of the homoclinic orbit and the presence of harmonic disturbances on the platform, it is demonstrated that chaotic oscillations can also appear, and they have been examined by means of the Melnikov theory. Chaotic behavior is corroborated by means of the sensitive dependence, Lyapunov exponents and power spectral density, and it is applied to drive the transducer mobile piece to a predefined set point assuming that noise due to the measurement process can appear in the control signals. The steady-state error associated to such random noise is eliminated by adding a PI linear controller to the control force. Numerical simulations are used to corroborate the analytical results.This work was supported by the “Generalitat Valenciana” (Spain) under Project GV/2012/099 and by “Ministerio de Ciencia e Innovación” (Spain) under Project FIS2011-29803-C02-01

Desarrollo de software en Comproductive Control y Matlab para asignaturas de Control en Ingeniería Química

El principal objetivo de este trabajo es implementar programas de software utilizando CC (Comproductive Control) y Matlab que permita a los estudiantes comprender los conceptos químicos de control y que así tengan la capacidad de interpretar los cálculos y desarrollar las matemáticas propias de las asignaturas de control de procesos químicos. El CC es un programa que utiliza unos comandos muy similares a la notación que los estudiantes manejan en las clases de teoría y, por tanto, no requiere un conocimiento previo de programación para poder utilizarlo. Además, a pesar de su sencillez, permite estudiar las técnicas básicas para el control de procesos industriales. Sin embargo, el CC tiene sus limitaciones ya que solamente es capaz de simular el control de los procesos químicos más sencillos. Por este motivo, otro de los programas con los que se pretende que el estudiante se familiarice es el Matlab. Este programa está más orientado al desarrollo de algoritmos con los cuales se pueden simular el control de procesos químicos más complejos. Por tanto, el CC y el Matlab son dos programas que se complementan mutuamente para el aprendizaje del estudiante en la materia

Note on an apparently forgotten theorem about solid rigid dynamics

We re-derive a general procedure to substitute any rigid body by an equivalent system of exactly four masses, located at the vertices of an irregular tetrahedron

Path tracking and stability of a rolling controlled wheel on a horizontal plane by using the nonholonomic constraints

This paper studies the stabilization, tracking of a predefined trajectory and how to reach a desired set point for a wheel which is rolling on a horizontal plane without slipping. For this purpose, the wheel is controlled by small torque generated by internal servomechanisms whose dynamics can be neglected. An efficient procedure to determine the kinetic energy of the wheel is developed by introducing a set of reference systems, which in combination with the Lagrange equations with multipliers allow deriving the mathematical model of the rolling wheel. In this model, the Euler angles, the coordinates of the plane–wheel contact point and a control law of proportional+integral+derivative (PID) type provide an efficient computational procedure to track arbitrary trajectories. It is shown that the nonholonomic constraints are fulfilled with admissible reaction forces, even when the desired trajectory has cusp points. A circumference and a family of astroids are used as trajectories to verify the motion conditions derived from the energy conservation and dynamical equilibrium of the wheel along such trajectories. The results of the analytical calculations are corroborated through numerical simulations

Coarse-Grained Simulations of Release of Drugs Housed in Flexible Nanogels: New Insights into Kinetic Parameters

Funding: This research was funded by “Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía, Programa Operativo FEDER Andalucía 2014-2020”, grant number P20_00138 as well as “Ministerio de Ciencia e Innovación”, grant numbers PID2020-116615RA-I00 and PGC2018-098770-B-I00.The diffusion-controlled release of drugs housed in flexible nanogels has been simulated with the help of a coarse-grained model that explicitly considers polymer chains. In these in silico experiments, the effect of its flexibility is assessed by comparing it with data obtained for a rigid nanogel with the same volume fraction and topology. Our results show that the initial distribution of the drug can exert a great influence on the release kinetics. This work also reveals that certain surface phenomena driven by steric interactions can lead to apparently counterintuitive behaviors. Such phenomena are not usually included in many theoretical treatments used for the analysis of experimental release kinetics. Therefore, one should be very careful in drawing conclusions from these formalisms. In fact, our results suggest that the interpretation of drug release curves in terms of kinetic exponents (obtained from the Ritger–Peppas Equation) is a tricky question. However, such curves can provide a first estimate of the drug diffusion coefficient.Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía, Programa Operativo FEDER Andalucía 2014-2020, grant number P20_00138Ministerio de Ciencia e Innovación, grant numbers PID2020-116615RA-I00 and PGC2018-098770-B-I0

Estimation of the virial coefficients by means of chaotic oscillations of pressure and density: Application to quantum gases with cubic equations of state

This paper analyzes how to determine the virial coefficients B and C of real gases by using a theoretical device whose pressures and densities oscillate in chaotic regime. The device is formed by a valve, a pressure controller, a pressure probe and a gas accumulator, for which the thermodynamic model has been derived from the force-mass-energy balances. This model allows keeping the gas temperature almost constant with chaotic oscillations in the inlet to the accumulator. The chaotic data are used to obtain variability in the pressures and densities, so that they can be used as experimental values from which the virial coefficients are estimated. For this purpose, several cubic and high precision equations of state for polar and non-polar gases and mixtures are used. In particular, the virial coefficient B for dry air is estimated by using high precision state equations, whereas, the virial coefficients B and C are also estimated for quantum gases (He4, He3 H2, D2, Ne) by using several modified cubic equations of state at moderate and high pressures. Furthermore, the values for the virial coefficient B obtained from numerical simulations are used to estimate the intermolecular potential and the radial distribution function. The results are in good agreement with the currently known experimental data for virial coefficients published in the literature

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

In this paper we investigate the stability and the onset of chaotic oscillations around the pointing-up position for a simple inverted pendulum that is driven by a control torque and is harmonically excited in the vertical and horizontal directions. The driven control torque is defined as a proportional plus integral plus derivative (PID) control of the deviation angle with respect to the pointing-down equilibrium position. The parameters of the PID controller are tuned by using the Routh criterion to obtain a stable weak focus around the pointing-up position, whose stability is investigated by using the normal form theory. The normal form theory is also used to deduce a simplified mathematical model that can be resolved analytically and compared with the numerical simulation of the complete mathematical model. From the harmonic prescribed motions for the pendulum base, necessary conditions for chaotic motion are deduced by means of the Melnikov function. When the pendulum is close to the unstable pointing-up position, the PID parameters are changed and the chaotic motion is destroyed, which is achieved by employing very small control signals even in the presence of random noise. The results of the analytical calculations are verified by full numerical simulations

Leadership and Teamwork in Innovation Ecosystems

As experts acknowledge innovation is rarely driven by individuals acting in an isolated capacity, it is generally a social and collaborative element that triggers the concepts of organizational behavior. The question is then how to create environments in projects and in organizations where individual’s creativity and contribution fosters pollination to drive innovation. Studies confirm that the key impacting element in this area is teamwork quality, rather than team composition. Thus, organizations need to create teams with key traits that drive positive collaborations such as communication, coordination, balance of member contributions, mutual support, effort, and cohesion. These traits will allow a social group to deal with the inevitable creative tension needed for innovation ecosystems to flourish. Since human behavior is not mathematical, the only way to do this is creating the conditions for these traits to appear. In this context, leaders as social architects become very important, setting the tone of the organization, clearly defining the mission, identifying and living shared values, setting example, and understanding how organizations and social groups behave. When they are able to build high quality and performing environments, they become innovation brokers generating models that are scalable to be able to impact communities