Conditions For Global Convergence Research Articles

A Guiding Vector-Field Algorithm for Path-Following Control of Nonholonomic Mobile Robots

In this paper, we propose an algorithm for path-following control of the nonholonomic mobile robot based on the idea of the guiding vector field (GVF). The desired path may be an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the robot's kinematic model, we design a GVF, whose integral curves converge to the trajectory. A nonlinear motion controller is then proposed, which steers the robot along such an integral curve, bringing it to the desired path. We establish global convergence conditions for our algorithm and demonstrate its applicability and performance by experiments with wheeled robots.

Stability of epidemic models over directed graphs: A positive systems approach

We study the stability properties of a susceptible–infected–susceptible (SIS) diffusion model, so-called the n-intertwined Markov model, over arbitrary directed network topologies. As in the majority of the work on infection spread dynamics, this model exhibits a threshold phenomenon. When the curing rates in the network are high, the disease-free equilibrium is the unique equilibrium over the network. Otherwise, an endemic equilibrium state emerges, where some infection remains within the network. Using notions from positive systems theory, we provide novel proofs for the global asymptotic stability of the equilibrium points in both cases over strongly connected networks based on the value of the basic reproduction number, a fundamental quantity in the study of epidemics. When the network topology is weakly connected, we provide conditions for the existence, uniqueness, and global asymptotic stability of an endemic state, and study the stability of the disease-free equilibrium. Finally, we demonstrate that the n-intertwined Markov model can be viewed as a best-response dynamical system of a concave game among the nodes. This characterization allows us to cast new infection spread dynamics; additionally, we provide a sufficient condition for global convergence to the disease-free equilibrium, which can be checked in a distributed fashion.

Extinction, periodicity and multistability in a Ricker model of stage-structured populations

We study the dynamics of a second-order difference equation that is derived from a planar Ricker model of two-stage (adult-juvenile) biological populations. We obtain sufficient conditions for global convergence to zero in the non-autonomous case yielding general conditions for extinction in the biological context. We also study the dynamics of an autonomous special case of the equation that generates multistable periodic and non-periodic orbits in the positive quadrant of the plane.

Status devices and Status mobility: How does success cross local and global markets?

This paper analyzes status mobility between markets under conditions of local and global art market convergence, using data of Korean artists and their sales movement from 1998 to 2012. We identify...

A new approach for estimation of electrical parameters and flux observation of permanent magnet synchronous motors

SummaryIn this paper, a new class of estimators for permanent magnet synchronous motors is proposed. Using a novel representation of the permanent magnet synchronous motor dynamics and some suitable filtering, we obtain new solutions to two important problems: (a) estimation of the stator resistance and inductance and (b) flux estimation with known electrical parameters. Neither one of the schemes requires knowledge of the mechanical parameters nor of the magnetic flux constant. Conditions for global convergence are given in terms of persistency of excitation requirements that, for the flux observation problem, is imposed only to the rotor speed. Both stability proofs require the availability of rotor speed. However, for the flux observer, extensive simulations using the speed estimate obtained from the estimator (instead of the actual rotor speed) illustrate the robustness of the proposed scheme vis‐à‐vis this assumption. Simulations also show that the new flux observer outperforms other existing designs. Copyright ©2015 John Wiley & Sons, Ltd.

One-Layer Continuous-and Discrete-Time Projection Neural Networks for Solving Variational Inequalities and Related Optimization Problems

This paper presents one-layer projection neural networks based on projection operators for solving constrained variational inequalities and related optimization problems. Sufficient conditions for global convergence of the proposed neural networks are provided based on Lyapunov stability. Compared with the existing neural networks for variational inequalities and optimization, the proposed neural networks have lower model complexities. In addition, some improved criteria for global convergence are given. Compared with our previous work, a design parameter has been added in the projection neural network models, and it results in some improved performance. The simulation results on numerical examples are discussed to demonstrate the effectiveness and characteristics of the proposed neural networks.

The Global Construction of EU Development Policy

At the turn of the twentieth century, 191 countries agreed to realize the Millennium Development Goals (MDGs). The European Union (EU) has incorporated many of these goals into its development policies. However, the effect of the MDGs on the construction of EU development policy has not been achieved through a homogenous diffusion of global development norms, but through a heterogeneous process: some MDGs have had a greater impact on EU policy formation than others. By reconceptualizing the EU as a receiver of norms, this paper aims to locate the scope conditions of global norm convergence in EU development policy through a comparison of disaster risk reduction and urban development in slum dwellings. Informed through world society theory, the findings point to the importance of norm ‘theorization’ in explaining the scope conditions of norm diffusion.

Contraction Analysis for a Class of NonDifferentiable Systems with Applications to Stability and Network Synchronization

In this paper we extend to a generic class of piecewise smooth dynamical systems a fundamental tool for the analysis of convergence of smooth dynamical systems: contraction theory. We focus on switched nondifferentiable systems satisfying Carathéodory conditions for the existence and uniqueness of a solution. After generalizing the classical definition of contraction to this class of dynamical systems, we give sufficient conditions for global convergence of their trajectories. The theoretical results are then applied to solve a set of representative problems such as proving global asymptotic stability of switched linear systems, giving conditions for incremental stability of piecewise smooth systems, and analyzing the convergence of networked switched systems.

A Matrix-Free Trust-Region SQP Method for Equality Constrained Optimization

We develop and analyze a trust-region sequential quadratic programming (SQP) method for the solution of smooth equality constrained optimization problems, which allows the inexact and hence iterative solution of linear systems. Iterative solution of linear systems is important in large-scale applications, such as optimization problems with partial differential equation constraints, where direct solves are either too expensive or not applicable. Our trust-region SQP algorithm is based on a composite-step approach that decouples the step into a quasi-normal and a tangential step. The algorithm includes critical modifications of substep computations needed to cope with the inexact solution of linear systems. The global convergence of our algorithm is guaranteed under rather general conditions on the substeps. We propose algorithms to compute the substeps and prove that these algorithms satisfy global convergence conditions. All components of the resulting algorithm are specified in such a way that they can b...

Switched LQR/H∞ steering vehicle control to detect critical driving situations

This paper proposes a switching steering vehicle control designed using the linear quadratic regulator (LQR) problem, the Linear Matrix Inequality (LMI) framework and the H∞ norm. The proposed switched control law comprises two levels: the first level is a switched Proportional–Integral-Derivative controller of lateral deviation (PIDy) and the second is a switched Proportional-Derivative controller of yaw angle (PDψ). These two levels are used to ensure an accurate tracking of the vehicle's lateral deviation y and yaw angle ψ. This control strategy makes use of a common Lyapunov function design method used for the stability analysis of switched continuous-time systems. Sufficient conditions for global convergence of the switched control law are presented and proved under arbitrary switching signals. All these conditions are expressed in terms of LMIs. The switched steering control was developed for an application seeking to identify approximately the maximum achievable speed in a bend. This application requires a steering control for simulating a realistic nonlinear four-wheel vehicle model and for performing a speed extrapolation test to evaluate the physical limits of a vehicle in a bend. This study includes the performance tests using experimental data from the Peugeot 307 prototype vehicle developed by IFSTTAR Laboratory.

Convergence of evolutionary algorithms on the n-dimensional continuous space

Evolutionary algorithms (EAs) are random optimization methods inspired by genetics and natural selection, resembling simulated annealing. We develop a method that can be used to find a meaningful tradeoff between the difficulty of the analysis and the algorithms' efficiency. Since the case of a discrete search space has been studied extensively, we develop a new stochastic model for the continuous n-dimensional case. Our model uses renewal processes to find global convergence conditions. A second goal of the paper is the analytical estimation of the computation time of EA with uniform mutation inside the (hyper)-sphere of volume 1, minimizing a quadratic function.

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non-proportional viscoelastic structures is developed. The central idea is the construction of two complex-valued functions of a complex variable, whose fixed points are precisely the eigenvalues. This important property allows the use of these functions in a fixed-point iterative scheme. With help of some results in fixed point theory, necessary conditions for global and local convergence are provided. It is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping. In addition, under certain conditions the recursive method can also be used for the calculation of non-viscous real eigenvalues. In order to validate the mathematical results, two numerical examples are analyzed, one for single degree-of-freedom systems and another for multiple ones.

Global convergence conditions in maximum likelihood estimation

Maximum likelihood estimation has been widely applied in system identification because of consistency, its asymptotic efficiency and sufficiency. However, gradient-based optimisation of the likelihood function might end up in local convergence. In this article we derive various new non-local-minimum conditions in both open and closed-loop system when the noise distribution is a Gaussian process. Here we consider different model structures, in particular ARARMAX, BJ and OE models.

From simulated annealing to stochastic continuation: a new trend in combinatorial optimization

Simulated annealing (SA) is a generic optimization method that is quite popular because of its ease of implementation and its global convergence properties. However, SA is widely reported to converge very slowly, and it is common practice to allow extra freedom in its design at the expense of losing global convergence guarantees. A natural way to increase the flexibility of SA is to allow the objective function and the communication mechanism to be temperature-dependent, the idea being to gradually reveal the complexity of the optimization problem and to increase the mixing rate at low temperatures. We call this general class of annealing processes stochastic continuation (SC). In the first part of this paper, we introduce SC starting from SA, and we derive simple sufficient conditions for the global convergence of SC. Our main result is interesting in two respects: first, the conditions for global convergence are surprisingly weak--in particular, they do not involve the variations of the objective function with temperature--and second, exponential cooling makes it possible to be arbitrarily close to the best possible convergence speed exponent of SA. The second part is devoted to the application of SC to the problem of producing aesthetically pleasing drawings of undirected graphs. We consider the objective function defined by Kamada and Kawai (Inf Process Lett 31(1):7---15, 1989), which measures the quality of a drawing as a weighted sum of squared differences between Euclidean and graph-theoretic inter-vertex distances. Our experiments show that SC outperforms SA with optimal communication setting both in terms of minimizing the objective function and in terms of standard aesthetic criteria.

Stability and convergence in discrete convex monotone dynamical systems

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on $n$ letters, where $n$ is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.

Convergence of iterative tâtonnement without price normalization

This paper presents global convergence conditions for a non-normalized fixed-point iteration in computing the equilibria of exchange economies. The usual conditions for the stability of Walras’ tâtonnement are obtained as a limiting case from these conditions. The iteration leads to an equilibrium under a strengthened form of the weak axiom of revealed preferences introduced in the paper. Results for economies with global gross substitutability and no trade at equilibrium are also presented. Furthermore, it is shown that a rather general class of non-normalized iterations converges under the same conditions as the fixed-point iteration.

Global Output Convergence of a Class of Recurrent Delayed Neural Networks with Discontinuous Neuron Activations

This paper studies the global output convergence of a class of recurrent delayed neural networks with time-varying inputs. We consider non-decreasing activations which may also have jump discontinuities in order to model the ideal situation where the gain of the neuron amplifiers is very high and tends to infinity. In particular, we drop the assumptions of Lipschitz continuity and boundedness on the activation functions, which are usually required in most of the existing works. Due to the possible discontinuities of the activations functions, we introduce a suitable notation of limit to study the convergence of the output of the recurrent delayed neural networks. Under suitable assumptions on the interconnection matrices and the time-varying inputs, we establish a sufficient condition for global output convergence of this class of neural networks. The convergence results are useful in solving some optimization problems and in the design of recurrent delayed neural networks with discontinuous neuron activations.

Numerical method of solving a linear problem on the minimum of resource consumption

A simple algorithm for developing a quasioptimal control over resource consumption is considered. The control is used as an initial approach to an iterative procedure of computing an optimal control. A system of linear algebraic equations is derived which approximately relate increments of initial conditions of an adjoint system to increments of amplitudes of a quasioptimal control with respect to ultimate values. Local convergence of the computing process with a quadratic rate is proved, and the convergence radius is found. A condition for global convergence of the method is specified.

Global asymptotic stability of neural networks with discontinuous activations

Without assuming the boundedness and monotonicity of the neuron activations, we discuss the dynamics of a class of neural networks with discontinuous activation functions. The Leray-Schauder theorem of set-valued maps is successfully employed to derive the existence of an equilibrium point. A Lyapunov-like approach is applied to differential equations with discontinuous right-hand sides modeling the neural network dynamics, which yields conditions for global convergence or convergence in finite time. The obtained results extend previous works on global stability of neural networks with continuous neuron activations or discontinuous neuron activations.

Unknown Input Observers for Switched Nonlinear Discrete Time Descriptor Systems

In this paper, a linear matrix inequality technique for the state estimation of discrete-time, nonlinear switched descriptor systems is developed. The considered systems are composed of linear and nonlinear parts. An observer giving a perfect unknown input decoupled state estimation is proposed. Sufficient conditions of global convergence of observers are proposed. Numerical examples are given to illustrate this method.