Convergence Of Trajectories Research Articles

Tubularity and Asymptotic Convergence of Penalty Trajectories in Convex Programming

In this paper, we give a sufficient condition for the asymptotic convergence of penalty trajectories in convex programming with multiple solutions. We show that, for a wide class of penalty methods, the associated optimal trajectory converges to a particular solution of the original problem, characterized through a minimization selection principle. Our main assumption for this convergence result is that all the functions involved in the convex program are tubular. This new notion of regularity, weaker than that of quasianalyticity, is defined and studied in detail.

Context in temporal sequence processing: a self-organizing approach and its application to robotics

A self-organizing neural net for learning and recall of complex temporal sequences is developed and applied to robot trajectory planning. We consider trajectories with both repeated and shared states. Both cases give rise to ambiguities during reproduction of stored trajectories which are resolved via temporal context information. Feedforward weights encode spatial features of the input trajectories, while the temporal order is learned by lateral weights through delayed Hebbian learning. After training, the net model operates in an anticipative fashion by always recalling the successor of the current input state. Redundancy in sequence representation improves noise and fault robustness. The net uses memory resources efficiently by reusing neurons that have previously stored repeated/shared states. Simulations have been carried out to evaluate the performance of the network in terms of trajectory reproduction, convergence time and memory usage, tolerance to fault and noise, and sensitivity to trajectory sampling rate. The results show that the model is fast, accurate, and robust. Its performance is discussed in comparison with other neural-networks models.

A Turnpike Theorem in the Closed Dynamic Leontief Model with a Singular Matrix of Capital Coefficients

The paper refers to the well-known Tsukui turnpike theorem on convergence of optimal growth trajectories in the closed dynamic Leontief model to the maximum balanced growth trajectory, called turnpike. In the original proof of this theorem, the assumption that the matrix B of capital coefficients is non-singular plays an essential role. For many reasons this assumption, very convenient for theoretical analysis, is not always satisfied in input-output systems built for empirical purposes. This paper fills the gap between theory and empirical studies, presenting a proof that convergence of optimal trajectories towards the turnpike is also a characteristic feature of the closed Leontief model in the case when matrix B is singular. The general idea of the proof is based on the approximation of a singular matrix B by an infinite sequence of non-singular matrices.

Triviality of Hierarchical Ising Model¶in Four Dimensions

Existence of critical renormalization group trajectory for a hierarchical Ising model in 4 dimensions is shown. After 70 iterations of renormalization group transformations, the critical Ising model is mapped into a vicinity of the Gaussian fixed point. Convergence of the subsequent trajectory to the Gaussian fixed point is shown by power decay of the effective coupling constant. The analysis in the strong coupling regime is computer-aided and Newman's inequalities on truncated correlations are used to give mathematical rigor to the numerical bounds. In order to obtain a criterion for convergence to the Gaussian fixed point, characteristic functions and Newman's inequalities are systematically used.

Averaging of three time scale singularly perturbed control systems

We investigate the asymptotic properties of singularly perturbed control systems with three time scales. We apply the averaging method to construct a limiting system for the slowest motion in the form of a differential inclusion. Sufficient conditions for the uniform convergence of the slowest trajectories are given.

Global stabilization for nonlinear uncertain systems with unmodeled actuator dynamics

Second-order sliding-mode control (2-SMC) algorithms are analyzed to assess their global convergence properties. While standard first-order sliding-mode control (1-SMC) algorithms derive their effectiveness from the global solution of the well known reaching condition ss/spl dot//spl les/-k/sup 2/|s| (s=0 being the actual sliding manifold), 2-SMC is based on more complex differential inequalities, for which a global solution could not exist. The approach presented introduces a suitable commutation logic (based on an online simple predictor) that prevents an uncontrollable growth of the uncertainties. Due to this new commutation logic, the global convergence of the state trajectory to the designed sliding manifold is ensured.

Analysis of Nonlinear Discrete-Time Systems with Higher-Order Iterative Learning Control

In this paper, an iterative learning control method is proposed for a class of nonlinear discrete-time systems with well-defined relative degree, which uses the output data from several previous operation cycles to enhance tracking performance. A new analysis approach is developed, by which the iterative learning control is shown to guarantee the convergence of the output trajectory to the desired one within bound and the bound is proportional to the bound on resetting errors. It is further proved effective to overcome initial shifts and the resultant output trajectory can be assessed as iteration increases. Numerical simulation is carried out to verify the theoretical results and exhibits that the proposed updating law possesses good transient behavior of learning process so that the convergence speed is improved.

More on the applications of the contraction mapping method in robotics

The objective of this paper is two-fold. Firstly, we extend the applications of the control scheme, proposed by Ailon (1996, Automatica, 32(10), 1455–1461), which is based on the contraction mapping theorem, to a full model of a robot manipulator (Tomei, 1991, IEEE Transactions on Automatic Control, 36 (10), 1208–1213). Next, we present relaxed sufficient conditions on the gains involved in the control scheme that ensure convergence of the system trajectory to the desired neighborhood of the equilibrium point. Simulation results demonstrate the contribution of this note.

Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system

In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter boundary control system governed by a scalar viscous Burgers' equation. Namely, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectories of the open-loop zero dynamics as the gain parameters are increased to infinity. We also prove a robust version of this result, valid for perturbations by an unknown disturbance with arbitrary L2 norm. For the controlled Burgers' equation forced by a disturbance we prove that, for all initial data in L2(0, 1), the closed-loop trajectories converge in L2(0, 1), uniformly in t∈[0, T] and in H1(0, 1), uniformly in t∈[t0, T] for any t0>0, to the trajectories of the corresponding perturbed zero dynamics. We have also extended these results to include the case when additional boundary controls are included in the closed-loop system. This provides a proof of convergence of trajectories in case the zero dynamics is replaced by a non-homogeneous Dirichlet boundary controlled Burgers' equation. As an application of our convergence of trajectories results, we demonstrate that our boundary feedback control scheme provides a semiglobal exponential stabilizing feedback law in L2, H1 and L∞ for the open-loop system consisting of Burgers' equation with Neumann boundary conditions and zero forcing term. We also show that this result is robust in the sense that if the open-loop system is perturbed by a sufficiently small non-zero disturbance then the resulting closed-loop system is ‘practically semiglobally stabilizable’ in L2-norm. Copyright © 1999 John Wiley & Sons, Ltd.

Neural Network Based, Discrete Adaptive Sliding Mode Control for Idle Speed Regulation in IC Engines

An adaptive sliding mode control design method is proposed for discrete nonlinear systems where explicit knowledge of the system dynamics is not available. Three-layer feedforward neural networks are used as function approximators for the unknown dynamics. The control law is designed based on the outputs of the approximators, and the sliding surface is defined in terms of a stable polynomial of the system outputs. Convergence of the state trajectories into a small sliding sector is proved. The method is applied to the internal combustion (IC) engine idle speed control problem. Simulation and experimental results are provided. [S0022-0434(00)01702-0]

Visuomotor processing as reflected in the directional discharge of premotor and primary motor cortex neurons.

Premotor and primary motor cortical neuronal firing was studied in two monkeys during an instructed delay, pursuit tracking task. The task included a premovement "cue period," during which the target was presented at the periphery of the workspace and moved to the center of the workspace along one of eight directions at one of four constant speeds. The "track period" consisted of a visually guided, error-constrained arm movement during which the animal tracked the target as it moved from the central start box along a line to the opposite periphery of the workspace. Behaviorally, the animals tracked the required directions and speeds with highly constrained trajectories. The eye movements consisted of saccades to the target at the onset of the cue period, followed by smooth pursuit intermingled with saccades throughout the cue and track periods. Initially, an analysis of variance (ANOVA) was used to test for direction and period effects in the firing. Subsequently, a linear regression analysis was used to fit the average firing from the cue and track periods to a cosine model. Directional tuning as determined by a significant fit to the cosine model was a prominent feature of the discharge during both the cue and track periods. However, the directional tuning of the firing of a single cell was not always constant across the cue and track periods. Approximately one-half of the neurons had differences in their preferred directions (PDs) of >45 degrees between cue and track periods. The PD in the cue or track period was not dependent on the target speed. A second linear regression analysis based on calculation of the preferred direction in 20-ms bins (i.e., the PD trajectory) was used to examine on a finer time scale the temporal evolution of this change in directional tuning. The PD trajectories in the cue period were not straight but instead rotated over the workspace to align with the track period PD. Both clockwise and counterclockwise rotations occurred. The PD trajectories were relatively straight during most of the track period. The rotation and eventual convergence of the PD trajectories in the cue period to the preferred direction of the track period may reflect the transformation of visual information into motor commands. The widely dispersed PD trajectories in the cue period would allow targets to be detected over a wide spatial aperture. The convergence of the PD trajectories occurring at the cue-track transition may serve as a "Go" signal to move that was not explicitly supplied by the paradigm. Furthermore, the rotation and convergence of the PD trajectories may provide a mechanism for nonstandard mapping. Standard mapping refers to a sensorimotor transformation in which the stimulus is the object of the reach. Nonstandard mapping is the mapping of an arbitrary stimulus into an arbitrary movement. The shifts in the PD may allow relevant visual information from any direction to be transformed into an appropriate movement direction, providing a neural substrate for nonstandard stimulus-response mappings.

Gamma-electron dynamics and radio frequency capacitive sheath diagnostics

Time-resolved measurements of gamma-electrons energy spectra in low-pressure capacitively coupled RF discharge are presented. Time dependence of the sheath voltage is calculated from experimental results and compared with theoretical predictions. Good quantitative agreement is observed for collisional sheath. In low pressure discharge sheath voltage time dependence is close to [1+cos(/spl omega//spl tau/)], which qualitatively corresponds to the theoretical results. Asymmetry of electron energy spectra for increasing and decreasing sheath voltage is observed and explained by divergence and convergence of electron trajectories in plasma (klystron-effect).

Continuous Dependence with Respect to the Input of Trajectories of Control-Affine Systems

We study the continuous dependence on the input of trajectories of control-affine systems belonging to the class C0 (m) of all systems $\Sigma$ of the form $$\Sigma: \ \ \ \ \ \ \dot{x}=f_0(x)+\sum_{i=1}^m u_i(t)f_i(x),$$ where f0 ,. . . fm are continuous vector fields on some open subset of $\mathbb{R}^n$ and the control functions belong to $L^1([0,T],\mathbb{R}^m)$. We give a simple necessary and sufficient condition for a control sequence $\{u^j\}_{j=1}^{\infty}$ to "${\cal T}^0$-converge" to a control $u^\infty$, i.e., to be such that, for every system $\Sigma$ in C0 , the trajectories generated by the uj converge as $j\rightarrow\infty$ to the trajectories generated by $u^\infty$. We also characterize ${\cal T}^k$-convergence (the concept of control convergence that arises when we use, instead of C0 (m), the class Ck (m) of systems $\Sigma$ where the fi are of class Ck) for $k\ge 1$ in the scalar input case, and we explain how the analogous characterization for the multi-input case fails to be true, unless one restricts oneself to the class $C^k_{comm}(m)$ of systems for which the vector fields f0 ,. . . fm commute. As a preliminary, we define a "topology of trajectory convergence" (or "T-convergence") on the set of all time-varying vector fields $\Omega\times I\ni (x,t)\mapsto f(x,t)\in\mathbb{R}^n$, where $\Omega$ is an open subset of $\mathbb{R}^n$ and I is an interval, and we study some of its properties. This enables us to make the definition of ${\cal T}^k$-convergence precise for sequences and, more generally, for nets, by saying that a net $\{u^\alpha\}_{\alpha\in A}$ in $L^1([0,T],\mathbb{R}^m)$ ${\cal T}^k$-converges to a limit $u^\infty$ if for every system $\Sigma$ in C0 (m) the time-varying vector fields $(x,t)\mapsto f_0(x)+\sum_{i=1}^m u_i^\alpha(t)f_i(x)$ ${\cal T}^k$-converge to $(x,t)\mapsto f_0(x)+\sum_{i=1}^m u_i^\infty(t)f_i(x)$.

A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex Minimization Problems

We study the existence and asymptotic convergence when t→+∞ for the trajectories generated by $$ \nabla^2 f(u(t),\epsilon(t))\dot u(t)+\dot\epsilon(t)\frac{\partial^2 f}{\partial\epsilon\,\partial x}(u(t),\epsilon(t))+\nabla f(u(t),\epsilon(t))=0, $$ where $\{f(\cdot,\epsilon)\}_{\epsilon>0}$ is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t)→ 0 when t→+∞ . This method is obtained from the following variational characterization of Newton's method: $$u(t)&\in&\mathop{\rm Argmin} \{f(x,\epsilon(t))-e^{-t}\langle\nabla f(u_0,\epsilon_0),x\rangle : x \in H\} ,\leqno{$(P^\epsilon_t)$}$$ where H is a real Hilbert space. We find conditions on the approximating family $f(\cdot,\epsilon)$ and the parametrization $\epsilon(t)$ to ensure the norm convergence of the solution trajectories u(t) toward a particular minimizer of f . The asymptotic estimates obtained allow us to study the rate of convergence as well. The results are illustrated through some applications to barrier and penalty methods for linear programming, and to viscosity methods for an abstract noncoercive variational problem. Comparisons with the steepest descent method are also provided.

An approach to information propagation in 1-D cellular neural networks-Part I: Local diffusion

This is the first of two companion papers devoted to a deep analysis of the dynamics of information propagation in the simplest nontrivial Cellular Neural Network (CNN), which is one-dimensional and has connections between nearest neighbors only. We will show that two behaviors are possible: local diffusion of information between neighboring cells and global propagation through the entire array. This paper deals with local diffusion, of which we will first give an accurate definition, before computing the template parameters for which the CNN has this behavior. Next we will compute the number of stable equilibria, before examining the convergence of any trajectory toward them, for three different kinds of boundary conditions: fixed Dirichlet, reflective, and periodic.

Output tracking control of uncertain nonlinear second-order systems

The solution of a tracking problem for a secondorder nonlinear system with uncertain dynamics and incomplete state measurement is obtained by means of a procedure directly inspired by the solution of the classical minimum-time optimal control problem. Two different types of uncertainty are considered in the paper: in the first case a constant bound on the uncertain dynamics is assumed to be known; in the second case, the bound is a function of both the measurable and the unmeasurable state variable of the system. In both cases, the possibility of applying the proposed control algorithms is proved to be determined by a proper choice of the control signal features. The resulting system is characterized by a suitable feedback switching logic and the convergence of the system trajectory to the desired one (or to a δ-vicinity of this latter) is proved also in the uncertain case.

On the local structure of $\omega$-limit sets of maps

Let X be a Banach space and let \(F:X\to X\) be \(C^1$, $F(0)=0\). It is proved that, under certain conditions, the \(\omega\)-limit set of a trajectory contains a point of the unstable manifold of 0 different from 0 as soon as it contains 0. The conditions on F involve the spectrum of \(F'(0)\) (implying the existence of stable, unstable, center-unstable and center manifolds of 0) and the dynamics of F on the center manifold of 0. In addition, it is assumed that either the center-unstable space of 0 is finite dimensional or the trajectory is relatively compact.¶In a number of particular cases this result allows to prove convergence of trajectories.

On the local structure of

Let X be a Banach space and let F : X → X be C1, F (0) = 0. It is proved that, under certain conditions, the ω-limit set of a trajectory contains a point of the unstable manifold of 0 different from 0 as soon as it contains 0. The conditions on F involve the spectrum of F ′(0) (implying the existence of stable, unstable, center-unstable and center manifolds of 0) and the dynamics of F on the center manifold of 0. In addition, it is assumed that either the center-unstable space of 0 is finite dimensional or the trajectory is relatively compact. In a number of particular cases this result allows to prove convergence of trajectories. Mathematics Subject Classification (1991). 34D05, 34B40, 34C40.

Sub-Optimal Sliding Mode Control of Uncertain Second Order Dynamical Systems

The solution of a tracking problem for a second order non linear system with uncertain dynamics and non complete available state is obtained by means of a procedure inspired by the solution to the classical minimum time optimal control problem. Two type of uncertain dynamics are considered with a constant bound, with a bound linearly dependent on the state-amplitude. In both cases, the applicability of the proposed control algorithm is proved provided that a proper control signal is chosen. The resulting system is characterized by a feedback switching logic causing the convergence of the system trajectory to the desired one (or to its δ-vicinity).

Numerical Scheme and Program for the Solution and Stability Analysis of a Steady Periodic Vibration Problem

A steady-state periodic solution is essential in the study of vibration theory. A new computer-aided scheme to determine a periodic solution and evaluate its stability is discussed. It is a remarkably precise, speedy and powerful method following after some classical schemes with the same aim, e.g. the well-known harmonic balance method. Since the algorithm consists of a purely numerical procedure, the scheme is very effective for cases in which it is difficult to find steady solutions via conventional analytical methods. The successive improvement scheme for determining particular initial conditions that generate the periodic solution exclusively, is based on a certain strategy which involves the Newton-Raphson method and numerical integration, such as the RKG scheme, for relevant differential equations. Illustrative examples are provided which demonstrate the convergence of successive trajectories in a phase plane. A full listing of the program is given with its instruction manual.