Multistep variable methods for exact integration of perturbed stiff linear systems

A family of real and analytical functions with values within the ring of M(m, R) is introduced. The solution for linear systems of differential equations is expressed as a series of Φ-functions. This new multistep method is defined for variable-step and variable-order, maintains the good properties of the Φ-function series method. It incorporates to compute the coefficients of the algorithm a recurrent algebraic procedure, based in the existing relation between the divided differences and the elemental and complete symmetrical functions. In addition, under certain hypotheses, the new multistep method calculates the exact solution of the perturbed problem. The new method is implemented in a computational algorithm which enables us to resolve in a general manner some physics and engineering IVP’s modeled by means systems of differential equations. The good behaviour and precision of the method is evidenced by contrasting the results with other-reputed algorithms and even with methods based on Scheifele’s G-functions.This work has been supported by GRE09-13 project of the University of Alicante and the project of the Generalitat Valenciana GV/2011/032

ANALYSIS OF DIFFERENT STRATEGIES FOR CIRCUIT OPTIMIZATION

AbstractThe process of analog circuit optimization is defined mathematically as a controllable dynamical system. In this context, we can formulate the problem of minimizing the CPU time as the minimization problem of a transitional process of a dynamical system. To analyse the properties of such a system, we propose to use the concept of the Lyapunov function of a dynamical system. This function allows us to analyse the stability of the optimization trajectories and to predict the CPU time for circuit optimization by analysing the characteristics of the initial part of the process.Keywords: Circuit optimization, control theory, Lyapunov function, minimal-time system design, time-optimal strategy.ANÁLISIS DE DIFERENTES ESTRATEGIAS PARA LA OPTIMIZACIÓN DE CIRCUITOSResumenEl proceso de la optimización del circuito analógico es definido matemáticamente como un sistema dinámico controlable. En este contexto, podemos formular el problema de minimizar el tiempo de la CPU como el problema de minimización de un proceso de transición de un sistema dinámico.  Para analizar las propiedades de tal sistema, proponemos de usar el concepto de la función de Lyapunov de un sistema dinámico. Esta función permite analizar la estabilidad de las trayectorias de optimización y predecir el tiempo de la CPU para la optimización del circuito analizando las características de la parte inicial del proceso.Palabras Claves: Diseño del sistema en el tiempo mínimo, estrategia óptima en el tiempo, función de Lyapunov, optimización del circuito, teoría de control

Accurate integration of forced and damped oscillators

The new methods accurately integrate forced and damped oscillators. A family of analytical functions is introduced known as T-functions which are dependent on three parameters. The solution is expressed as a series of T-functions calculating their coefficients by means of recurrences which involve the perturbation function. In the T-functions series method the perturbation parameter is the factor in the local truncation error. Furthermore, this method is zero-stable and convergent. An application of this method is exposed to resolve a physic IVP, modeled by means of forced and damped oscillators. The good behavior and precision of the methods, is evidenced by contrasting the results with other-reputed algorithms implemented in MAPLE

Neograničeni regulatori s promijenjivim pojačanjem za upravljanje robotskim manipulatorima s direktnim pogonom

This paper addresses the position-control problem with variable gains for robot manipulators. We present a new regulator based on a hyperbolic-sine structure with tuning rules for control gains. It is demonstrated that the equilibrium point of the closed-loop system is globally, asymptotically stable according to Lyapunov theory. By using a similar methodology, this concept can be extended to other unbounded controllers such as PD and PID. In order to show the usefulness of the proposed scheme and with the purpose of validating its asymptotical stability property, an experimental comparison involving constant gains controllers, for example: simple PD, PID and hyperbolic-tangent schemes vs variable-gains hyperbolic-sine and PD control schemes, was performed by using a three degree-of-freedom, direct-drive robot manipulator.Ovaj rad se bavi problemom kontrole pozicije s promjenjivim pojačanjem robotskog manipulatora. U radu je predstavljen novi regulator baziran na hiperbolično-sinusnoj stukturi s pravilima ugađanja upravljačkih pojačanja. Pokazano je da je točka ravnoteže sustava u zatvorenoj petlji globalno i asimptotski stabilna prema Lzapunovljevoj teoriji stabilnosti. Korištenjem slilčne metodologije, predstavljeni koncept se može primijeniti na ostale neograničene kontrolere, npr. PD i PID. Kako bi pokazali korisnost predložene sheme i s ciljem provjere asimptotske stabilnosti, provedena je eksperimentalna usporedba između kontolera s konstantnim pojačanjem (npr. jednostavni PD, PID i hiperbolični-tangencijalna shema) i hiperbolično-sinusnih i PD upravljačkih shema s promjenjivim pojačanjem korištenjem robotskog manipulatora s direktnim pogonom i tri stupnja slobode

Output-feedback proportional-integral-derivative-type control with multiple saturating structure for the global stabilization of robot manipulators with bounded inputs

"An output-feedback proportional integral derivative-type control scheme for the global regulation of robot manipulators with constrained inputs is proposed. It guarantees the global stabilization objective-avoiding input saturation-releasing the feedback not only from the exact knowledge of the system structure and parameter values but also from velocity measurements. With respect to previous approaches of the kind, the proposed controller is expressed in a generalized form whence multiple saturating structures may be adopted, thus enlarging the degree of design flexibility. Furthermore, experimental tests on a two-degree-of-freedom direct-drive manipulator corroborate the efficiency of the proposed scheme.

Global Saturated Regulator with Variable Gains for Robot Manipulators

In this paper, we propose a set of saturated controllers with variable gains to solve the regulation problem for robot manipulators in joint space. These control schemes deliver torques inside the prescribed limits of servomotors. The gamma of variable gains is formed by continuous, smooth, and differentiable functions of the joint position error and velocity of the manipulator. A strict Lyapunov function is proposed to demonstrate globally asymptotic stability of the closed-loop equilibrium point. Finally, the functionality and performance of the proposal are illustrated via simulation results and comparative analysis against Proportional-Derivative (PD) control scheme on a two-degrees-freedom direct-drive robot manipulator

A generalized scheme for the global adaptive regulation of robot manipulators with bounded inputs

"In this work, a generalized adaptive control scheme for the global position stabilization of robot manipulators with bounded inputs is proposed. It gives rise to various families of bounded controllers with adaptive gravity compensation. Compared with the adaptive approaches previously developed in a bounded-input context, the proposed scheme guarantees the adaptive regulation objective: globally, avoiding discontinuities in the control expression as well as in the adaptation auxiliary dynamics, preventing the inputs to reach their natural saturation bounds, and imposing no saturation-avoidance restriction on the control gains. Experimental results corroborate the efficiency of the proposed adaptive scheme.

Output-feedback adaptive control for the global regulation of robot manipulators with bounded inputs

"In this paper, an output-feedback adaptive scheme for the global position stabilization of robot manipulators with bounded inputs is proposed. Compared to the previous output-feedback adaptive approaches developed in a bounded-input context, the proposed free-of-velocity feedback controller guarantees the adaptive regulation objective: globally, avoiding discontinuities throughout the scheme, preventing the inputs to reach their natural saturation bounds, and imposing no saturation-avoidance restriction on the control gains. Moreover, the developed scheme is not restricted to the use of a specific saturation function to achieve the required boundedness, but may involve any one within a set of smooth and non-smooth (Lipschitz-continuous) bounded passive functions that include the hyperbolic tangent and the conventional saturation as particular cases. Experimental results corroborate the efficiency of the proposed scheme.

Output-feedback adaptive SP-SD-Type control with an extended continuous adaptation algorithm for the global regulation of robot manipulators with bounded inputs

"In this work, an output-feedback adaptive SP-SD-type control scheme for the global position stabilization of robot manipulators with bounded inputs is proposed. Compared with the output-feedback adaptive approaches previously developed in a bounded-input context, the proposed velocity-free feedback controller guarantees the adaptive regulation objective globally (i.e. for any initial condition), avoiding discontinuities throughout the scheme, preventing the inputs from reaching their natural saturation bounds and imposing no saturation-avoidance restrictions on the choice of the P and D control gains. Moreover, through its extended structure, the adaptation algorithm may be configured to evolve either in parallel (independently) or interconnected to the velocity estimation (motion dissipation) auxiliary dynamics, giving an additional degree of design flexibility. Furthermore, the proposed scheme is not restricted to the use of a specific saturation function to achieve the required boundedness, but may involve any one within a set of smooth and non-smooth (Lipschitz-continuous) bounded passive functions that include the hyperbolic tangent and the conventional saturation as particular cases. Experimental results on a 3-degree-of-freedom manipulator corroborate the efficiency of the proposed scheme

Seismic model analysis by means of a series method

La respuesta desde la Mecánica Estructural a los fenómenos sísmicos, hace necesario mejorar el cálculo de las estructuras así como su análisis. Para ello los métodos especialmente basados en el análisis estático no lineal necesitan tener una mayor precisión. El análisis no lineal se puede abordar mediante modelos discretos o continuos. Los modelos discretos representan la estructura a través de un número finito de grados de libertad; en este caso las ecuaciones de movimiento son ecuaciones diferenciales ordinarias que se resuelven por métodos numéricos. En este trabajo se muestra una aplicación del método de series ɸ-funciones para calcular la respuesta ante un terremoto de las estructuras modeladas mediante sistemas SDOF (Single Degree Of Freedom system) y 2DOF (Two Degree Of Freedom systems). Además, en el caso de SDOF, el método se ha aplicado tomando como la frecuencia forzada la frecuencia natural de vibración. La solución de los modelos sísmicos se ha obtenido mediante la generación de un algoritmo numérico y su implementación computacional. El método de series ɸ-funciones integra osciladores forzados y es una adaptación de los métodos de Scheifele, con la ventaja de integrar, sin error de truncamiento, el problema perturbado con sólo las dos primeras ɸ-funciones. El cálculo de coeficientes de la serie se efectúa por recurrencias algebraicas sencillas en las que se implica la función de perturbación. El buen comportamiento y precisión del método de series ɸ-funciones se ilustra cuando se contrasta con otros métodos de integración ya conocidos e implementados en MAPLE, comparándose también con los métodos clásicos de Ingeniería de Estructuras.The seismic events have attracted interest and the need to improve the structures and their analysis to sustain this type of oscillation. To do this, new methods especially those based on static non-linear analysis need to have increased accuracy. The non-linear analysis can be approached by means of discrete or continuous models. The discrete models represent the structure through a finite number of degrees of freedom; in this case the equations of motion are ordinary differential equations which are solved by numerical methods. This paper shows an application of the ɸ-functions series method to calculate the response of structures, modeled as both SDOF(Single Degree Of Freedom system) and 2DOF (Two Degree Of Freedom systems) systems, to an earthquake. Furthermore, in the case of SDOF, the method has been applied taking as the forcing frequency the natural frequency of vibration. The solution of the seismic models has been obtained by the generation of the numerical algorithm and its computational implementation. The ɸ-functions series method integrates forced oscillators and it is an adaptation of Scheifele's methods, with the advantage of integrating, without truncation error, the perturbed problem with just the first two ɸ-functions. The calculation of series coefficients is effected by simple algebraic recurrences in which the perturbation function is takes part. The good precision of ɸ-functions series method is illustrated when contrasted with other methods of integration already known and implemented in MAPLE and even with classic methods of Structural Engineering