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Methods to stabilize discrete-time linear control systems subject to variable sampling rates, i.e., using state feedback controllers, are well known in the literature. Several recent works address the use of the Tikhonov regularization method, originally designed to attenuate the noise effects on ill-posed problems, with the aim of improving performance and stabilizing approximately controllable dynamical systems. Inspired by these works, we propose the use of a feedback controller designed using the Tikhonov method to regularize discrete-time linear systems subject to varying sampling rates. The goal is to minimize an error function, thus improving the performance of the closed loop system and reducing the possibility of instability. Illustrative examples show the effectiveness of the proposed method.

Iterative methods to solve linear large-scale discrete problems are well known in the literature. When the linear system is ill-posed and contaminated by noise, some kind of regularization must be applied in order to achieve a feasible solution. In the first part of this paper, we revisit briefly some known methods to solve large-scale ill-posed discrete linear problems which are easy to implement and have low computational cost, formulating them in a unified manner and also proposing simple modifications in order to improve their performances. Matrix forms of iterative algorithms can be formulated depending on certain conditions on the blurring process, and have the advantage of avoiding the formation and storage in memory of the matrix that represents the blurring process, which is generally of very large dimension. As an original contribution, in the final part of this paper we present the matrix forms of the iterative algorithms revisited and test them in the problem of restoration of an image degraded by blurring and noise.

Numerical problems, including linear algebraic equations, are frequently ill-posed, which means that small perturbations in the observation data can have large effects on the computed solutions. In these cases some kind of regularization method must be used to find a feasible solution. Regularization implies the modification of the original ill-posed problem, so that it becomes well-posed, and the Tikhonov regularization method is the most often used. This method requires the choice of a regularization parameter, and this choice is a crucial issue. Recently, several papers were presented proposing adaptive Tikhonov regularization parameters, instead of constant ones. This paper revisits all these adaptive choices and shows that they can be interpreted in a unified and systematic manner using the Liapunov Optimizing Control (LOC) method, which, in addition, opens up the possibility of new choices of the regularization parameter. Examples of matrices from real applications show that one of the methods designed by the LOC method, called the OVM method by its discoverer, can even outperform the CG method for some matrices, although the CG method performs better for a larger class of matrices.

Iterative gradient descent algorithms to solve algebraic linear equations are well known in the literature. These algorithms can be interpreted as dynamical closed loop control systems, where the step sizes are the control variables that can be optimally calculated by the Liapunov Control Functions (CLF) and Liapunov Optimizing Control (LOC) methods. In this paper matrix versions of gradient descent algorithms are deduced, including an unpublished matrix form of the Barzilai-Borwein algorithm. The step sizes (control variables) are also calculated by CLF/LOC. The main utility of these matrix algorithms is in image deblurring in the cases where the matrix that represents the blurring process is too large to be stored in the memory of the majority of conventional computational systems.

This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding of ill-posed problems, showing that they can be profitably viewed as homotopy methods, in which the homotopy parameter is interpreted as a learning parameter. Moreover, it is shown that the choice of this learning parameter can be made in a natural manner from a control Liapunov function approach CLF. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner. Adaptive regularization parameters for solving linear algebraic ill-posed problems were also proposed. This paper also shows that the discretizations of the ODEs to solve the zero finding problem, as well as the different adaptive choices of the regularization parameter, yield iterative methods for linear systems, which are also derived using the Liapunov optimizing control LOC method.

International audience; This paper extends the idea of Brezinski's hybrid acceleration procedure, for the solution of a system of linear equations with a symmetric coefficient matrix of dimension n, to a new context called cooperative computation , involving m agents (m n), each one concurrently computing the solution of the whole system, using an iterative method. Cooperation occurs between the agents through the communication, periodic or probabilistic, of the estimate of each agent to one randomly chosen agent, thus characterizing the computation as concurrent and asynchronous. Every time one agent receives solution estimates from the others, it carries out a least squares computation , involving a small linear system of dimension m, in order to replace its current solution estimate by an affine combination of the received estimates, and the algorithm continues until a stopping criterion is met. In addition , an autocooperative algorithm, in which estimates are updated using affine combinations of current and past estimates, is also proposed. The proposed algorithms are shown to be efficient for certain matrices, specifically in relation to the popular Barzilai–Borwein algorithm, through numerical experiments.

This paper presents a unified way to design neural networks characterized as second order ordinary differential equations (ODEs) that aim to find the global minimum of nonconvex scalar functions. These neural networks, alternatively referred to as continuous time algorithms, are interpreted as dynamical closed loop control systems. The design is based on the control Liapunov function (CLF) method. For nonconvex scalar functions, the goal of these algorithms is to produce trajectories, starting from an arbitrarily chosen initial guess, that do not get stuck in local minima, thereby increasing the chances of converging to the global minimum.

This article carries out a comparative study of zero-finding neural networks for nonlinear functions. By taking the view that artificial recurrent neural nets are dynamical systems described by ordinary differential equations (ODEs), new neural nets were recently derived in the literature using a unified control Liapunov function (CLF) approach, after interpreting the zero finding problem as a regulation problem for a closed-loop continuous-time dynamical system. The resulting neural net or continuous-time ODE is discretized by Euler’s method and the discretization step size interpreted as a control which is chosen so as to optimize the decrement in the chosen CLF, along system trajectories. Given the viewpoint adopted in this article, the words dynamical system, ODE and neural net are used interchangeably. For standard test functions of two variables, the basins of attraction are found by numerical simulation, starting from a uniformly distributed grid of initial points. For the chosen test functions, analysis of the basins shows a correlation between regularity of the basin boundaries and the predictability of convergence to a zero. In addition, this analysis suggests how to construct a team algorithm with favorable convergence properties.

Resumen La tendinopatia es una lesion frecuente que se produce durante la practica deportiva. El reparto desigual de la carga de trabajo a lo largo del tendon produce roturas heterogeneas en extension y distribucion. Estas roturas ponen en marcha procesos de reparacion defectuosos que producen un tendon degenerado con alteracion estructural y de la respuesta funcional al ejercicio. En este trabajo se estudian y analizan los distintos factores predisponentes, mecanismos de accion de los agentes quimicos y celulares implicados en la fisiopatologia de la tendinopatias. Por otra parte, se analizan los componentes basicos (soporte, celulas y sustancias quimicas) que se usan para la ingenieria tisular. Las posibilidades actuales de uso de los componentes basicos y sus interrelaciones, y el nivel actual de desarrollo.

This paper presents two neural networks to find the optimal point in convex optimization problems and variational inequality problems, respectively. The domain of the functions that define the problems is a convex set, which is determined by convex inequality constraints and affine equality constraints. The neural networks are based on gradient descent and exact penalization and the convergence analysis is based on a control Liapunov function analysis, since the dynamical system corresponding to each neural network may be viewed as a so-called variable structure closed loop control system.

Published methods to determine the Klinkenberg permeability, Klinkenberg slip factor and Forchheimer turbulence factor of core plugs can exhibit considerable error. Jones presented a technique based on gas pressure decay measurements during the transient state, where, with a single run, an algorithm can calculate the parameters with precision. However, this paper shows that Jones' method is based on a linear regression to find a fixed point of a nonlinear error function, and is presented without theoretical justification or convergence conditions. This paper proposes a simple algorithm, based on nonlinear regression, to calculate the unknown parameters, and has the advantage of theoretical justification as well as weaker requirements for convergence. In addition, a strategy to calculate the unknown physical parameters when the measurements are noisy is presented.

This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding, showing that this class of homotopy methods can be designed in a natural manner from the control Liapunov function (CLF) approach proposed earlier by the authors. Moreover, the CLF approach clarifies the interplay between the homotopy parameter, which can be interpreted as a learning parameter and the choice of descent direction, which is the control vector and guides the choice of both. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner.

This paper proposes a generalization of the conjugate gradient (CG) method used to solve the equation $Ax=b$ for a symmetric positive definite matrix $A$ of large size $n$. The generalization consists of permitting the scalar control parameters (= stepsizes in gradient and conjugate gradient directions) to be replaced by matrices, so that multiple descent and conjugate directions are updated simultaneously. Implementation involves the use of multiple agents or threads and is referred to as cooperative CG (cCG), in which the cooperation between agents resides in the fact that the calculation of each entry of the control parameter matrix now involves information that comes from the other agents. For a sufficiently large dimension $n$, the use of an optimal number of cores gives the result that the multithread implementation has worst case complexity $O(n^{2+1/3})$ in exact arithmetic. Numerical experiments, that illustrate the interest of theoretical results, are carried out on a multicore computer., Comment: Expanded version of manuscript submitted to the IEEE-CDC 2012 (Conference on Decision and Control)

On the 21st of March, 2015, experts met at Clínica CEMTRO in Madrid, Spain, under the patronage of The Spanish Society for Sports Traumatology (SETRADE), The Spanish Federation of Sports Medicine (FEMEDE), The Spanish Association of Medical Services for Football Clubs (AEMEF), and The Spanish Association of Medical Services for Basketball Clubs (AEMB) with the aim of establishing a round table that would allow specialists to consider the most appropriate current general actions to be taken when treating muscle tears in sport, based on proven scientific data described in the medical literature. Each expert received a questionnaire prior to the aforementioned meeting comprising a set of questions concerning therapeutic indications generally applied in the different stages present during muscle repair. The present Consensus Document is the result of the answers to the questionnaire and resulting discussion and consensus over which are the best current indications in the treatment of muscle tears in sport. Avoiding immobilization, not taking nonsteroidal anti-inflammatory drugs (NSAIDs) randomly, fostering early mobilization, increasing vascularization of injured, site and regulating inflammatory mechanisms—without inhibiting these from the early stages of the recovery period—all stood out as main points of the Consensus Document. Additionally, there is controversy concerning cell stimulation techniques and the use of growth factors or cell inhibitors. The decision concerning discharge was unanimous, as was the criteria considered when it came to performing sport techniques without pain.

Neural nets that minimize quasi-convex scalar functions are designed as dynamical systems (ordinary differential equations) which correspond to various well known discrete time algorithms, such as steepest descent, Newton, Levenberg-Marquardt, etc. The main contribution is a generalization of the Levenberg-Marquardt algorithm, including an adaptive version, that combines good features of the Newton and Levenberg-Marquardt algorithms and leads to trajectories that converge faster to the minimum of a quasiconvex objective function that is assumed to have known gradient and Hessian.1

International audience; This paper introduces a new paradigm, called cooperative computation, for the solution of systems of linear equations with symmetric coefficient matrices. The simplest version of the algorithm consists of two agents, each one computing the solution of the whole system, using an iterative method. Infrequent unidirectional communication occurs from one agent to the other, either periodically, or probabilistically, thus characterizing the computation as parallel and asynchronous. Every time one agent communicates its current approximation of the solution to the other, the receiving agent carries out a least squares computation to replace its current value by an affine combination of the current approximations, and the algorithm continues until a stopping criterion is met. Deterministic and probabilistic variants of this algorithm are introduced and shown to be efficient, specifically in relation to the popular Barzilai-Borwein algorithm, particularly for ill-conditioned matrices.

This paper presents a unified way to design neural networks characterized as second-order ordinary differential equations (ODE), the trajectories of which can converge to the global minimum of nonconvex scalar functions. These neural networks, sometimes also called continuous-time algorithms, are interpreted as closed-loop control systems and the state feedback design is based on control Liapunov functions. The focus is on a new family of continuous-time versions of conjugate gradient method, named controlled conjugate gradient (CCG) nets, that generalize heavy ball with friction (HBF) nets. For nonconvex functions, the goal of these nets is to produce trajectories that start from an arbitrary initial point and can escape from local minima, thereby increasing the chances of converging to the global minimum. Several numerical examples are given, using benchmark problems, showing that this escape and subsequent convergence to the global minimum occurs.

International audience; Iterative methods for linear systems with a symmetric positive definite coefficient matrix are designed from a control-theoretic viewpoint. In particular, it is shown that a control-theoretic approach loosely based on m-step dead beat control of the error or residual system, with a suitable definition of error norm can be utilized to design new iterative methods that are competitive with the popular Barzilai-Borwein method, that is well known to be an efficient method with low computational cost. Numerical experiments are reported on to confirm the claimed results.

This paper presents a unified approach to the design of neural networks that aim to minimize scalar nonconvex functions that have continuous first- and second-order derivatives and a unique global minimum. The approach is based on interpreting the function as a controlled object, namely one that has an output (the function value) that has to be driven to its smallest value by suitable manipulation of its inputs: this is achieved by the use of the control Liapunov function (CLF) technique, well known in systems and control theory. This approach leads naturally to the design of second-order differential equations which are the mathematical models of the corresponding implementations as neural networks. Preliminary numerical simulations indicate that, on a small suite of benchmark test problems, a continuous version of the well known conjugate gradient algorithm, designed by the proposed CLF method, has better performance than its competitors, such as the heavy ball with friction method or the more recent dynamic inertial Newton-like method.

Considera-se o problema de rastreamento de trajetórias para robôs manipuladores com parâmetros físicos incertos e sob a ação de distúrbios de amplitude limitada. Descreve-se um algoritmo de controle de modo dual adaptativo/robusto que garante desempenho transitório exponencial com relação a um domínio residual arbitrariamente pequeno e com atenuação arbitrária de distúrbios. Na ausência destes, o controlador mostra-se capaz de rastrear assintoticamente a trajetória desejada. We consider the trajectory tracking problem for robot manipulators with unknown physical parameters and affected by bounded external disturbances. We present a non linear adaptive tracking controller by dual mode which guarantees exponential transient performance with respect to an arbitrary small residual set, resulting in arbitrary disturbance attenuation. If the disturbances vanish the controller achieves asymptotic tracking of the desired trajectory.