Convergence Of Trajectories Research Articles

Collocated Adaptive Control of Underactuated Mechanical Systems

Collocated adaptive control of underactuated mechanical systems is still a concern for the control community. The main difficulty comes from the nonlinearity of the collocated inverse dynamics with respect to the base parameters, which forbids the direct application of classical adaptive control schemes. This paper extends and encompasses the Slotine's adaptive control, which was developed for fully actuated mechanical systems, to stabilize the collocated state space of an underactuated mechanical system. The key point is to define the sliding variable as the difference between the system's velocity and an exogenous state whose dynamics is considered as control input. We first revisit the Slotine's result in view of this definition and then show how to extend it to the underactuated case. Stability and convergence of time-varying reference trajectories for the collocated dynamics are shown to be in the sense of Lyapunov. Global well-posedness of the control laws is achieved by means of a new algebraic property of the mass matrix. Simulations, comparisons to existing control strategies, and experimental results on a two-link manipulator verify the soundness of the proposed approach.

PMU-Based Model-Free Approach for Real-Time Rotor Angle Monitoring

In this letter, we present an algorithm for rotor angle stability monitoring of power system in real-time. The proposed algorithm is model-free and can make use of high resolution phasor measurement units (PMUs) data to provide reliable, timely information about the system's stability. The theoretical basis behind the proposed algorithm is adopted from dynamical systems theory. In particular, the algorithm approximately computes the system's Lyapunov exponent (LE), thereby measuring the exponential convergence or divergence rate of the rotor angle trajectories. The LE serves as a certificate of stability where the positive (negative) value of the LE implies exponential divergence (convergence) of nearby system trajectories, hence, unstable (stable) rotor angle dynamics. We also show the proposed model-free algorithm can be used for the identification of the coherent sets of generators. The simulation results are presented to verify the developed results in the paper on the modified IEEE 162-bus system.

Model of a Dynamical System of the “Fire–Water” Conflict Type

UDC 517.9 We propose a model of dynamical system generated by the conflict interaction between the alternative elements of the “fire–water” type with systematic external replenishment of the resources and internal mixing. The behavior of the trajectories is investigated. We establish the convergence of trajectories of the system to ! -limit cyclic orbits.

Optimal control with connected initial and terminal conditions

An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of L2. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, dual trajectories, and terminal variables are proved.

Stabilization of continuous-time linear systems subject to input quantization

This paper deals with the stabilization of continuous-time linear time-invariant systems subject to uniform input quantization. Specifically, the right-hand side of the closed-loop system is rewritten as a linear system subject to a discontinuous perturbation due to the quantization error. Then, the controller design is performed to achieve finite-time convergence of the closed-loop trajectories toward a compact invariant set surrounding the origin. Furthermore, a computationally tractable design procedure for the proposed controller based on linear matrix inequalities, and some insights on the simulation of the closed-loop system are presented. In addition, the effectiveness of the proposed control design procedure is shown in a numerical example.

Convergence and synchronization in heterogeneous networks of smooth and piecewise smooth systems

This paper presents a framework for the study of convergence in networks where the nodes’ dynamics may be both piecewise smooth and/or nonidentical. Sufficient conditions are derived for global convergence of all node trajectories towards the same bounded region in the synchronization error space. The analysis is based on the use of set-valued Lyapunov functions and bounds are derived on the minimum coupling strength required to make all nodes in the network converge towards each other. We also provide an estimate of the asymptotic bound on the mismatch between the node state trajectories. The analysis is performed both for linear and nonlinear coupling protocols. The theoretical analysis is extensively illustrated and validated via its application to a set of representative numerical examples.

Horn growth patterns in Alpine chamois

The analysis of horn growth may provide important information about the allocation of metabolic resources to secondary sexual traits. Depending on the selective advantages offered by horn size during intra- and inter-specific interactions, ungulates may show different investment in horn development, and growth variations within species may be influenced by several parameters, such as sex, age, or resource availability. We investigated the horn growth patterns in two hunted populations of Alpine chamois (Rupicapra r. rupicapra) in the Central Italian Alps. We tested the role of individual heterogeneity on the growth pattern and explored the variation in annulus length as a function of different factors (sex, age, hunting location, cohort). We then investigated the mechanisms underlying horn growth trajectories to test for the occurrence of compensatory or recovery growth and their potential differences between sexes and populations. Annulus length varied as a function of sex, age of individuals and, marginally, hunting location; no effect of cohort or individual heterogeneity was detected. Male and female chamois showed compensatory horn growth within the first 5½ years of life, though the partial convergence of horn trajectories in chamois suggests that this mechanisms would best be described as ‘recovery growth’. Compensation rates were greater in males than in females, while only compensatory growth rates up to 2½ years of age were different in the two populations. Besides confirming the sex- and age-dependent pattern of horn development, our study suggests that the mechanism of recovery growth supports the hypothesis of horn size as a weakly selected sexual trait in male and female chamois. Furthermore, the greater compensation rates in horn growth shown by male chamois possibly suggest selective effects of hunting on age at first reproduction, while different compensation rates between populations may suggest the occurrence of some plasticity in resource allocation to sexual traits in relation to different environments.

Stability Verification of a Control System using the Reachability Analysis

Model checking is a formal method technique used for the verification of safety critical and/or complex software and hardware systems. It provides accurate results by exhaustively exploring the abstract mathematical model of the system. This paper discusses the framework proposed for the verification of an underactuated inverted pendulum based two parallel wheeled vehicle. The physical model of the perceived system is designed for 2 DOF that is the pitch along x-axis and roll along y-axis. The mathematical model of the system is developed using the state-space approach and the system is analyzed using Matlab. Stability is introduced in to the system using a Linear Quadratic Regulator (LQR) with an Observer. This controlled system is analyzed and formally verified using the SpaceEx model checker. The stability verification of this system is performed using the reachability analysis. The model checker results shows the stability verification of the system over the convergence of infinite trajectories for a range of input values provided, showing advantage over simulation based techniques. The results also provide the phase portraits of the output variables in 2-D and 3-D planes.

Integrated guidance and control law for cooperative attack of multiple missiles

An integrated guidance and control law is developed, using the dynamic surface control theory and disturbance observation technology, for multiple missiles attacking targets cooperatively. The modeling error, aerodynamic parameters perturbation and external disturbance are regarded as system uncertainties, and then integrated guidance and control model of missile is built. Considering multiple missiles' cooperation of flight position and impact time, the constraint of maximum target look angle and the requirement of trajectory convergence, a cooperative strategy expressed by desired target look angle command is proposed. To analyze the effect of nonlinear saturation of missile control input on integrated guidance and control system, a supplementary subsystem is introduced. Three disturbance observers are designed to estimate the system uncertainties. With the signals generated by supplementary subsystem and uncertainty estimate from disturbance observers, a robust controller is designed using dynamic surface control theory to track the desired target look angle command. Based on Lyapunov stability theory and comparison principle, the stability of system under the condition of unknown uncertainties and control input saturation is proved, and the transient tracking error is also discussed. Simulation results demonstrate the effectiveness and robustness of the integrated guidance and control law.

Learning in Games via Reinforcement and Regularization

We investigate a class of reinforcement learning dynamics where players adjust their strategies based on their actions’ cumulative payoffs over time—specifically, by playing mixed strategies that maximize their expected cumulative payoff minus a regularization term. A widely studied example is exponential reinforcement learning, a process induced by an entropic regularization term which leads mixed strategies to evolve according to the replicator dynamics. However, in contrast to the class of regularization functions used to define smooth best responses in models of stochastic fictitious play, the functions used in this paper need not be infinitely steep at the boundary of the simplex; in fact, dropping this requirement gives rise to an important dichotomy between steep and nonsteep cases. In this general framework, we extend several properties of exponential learning, including the elimination of dominated strategies, the asymptotic stability of strict Nash equilibria, and the convergence of time-averaged trajectories in zero-sum games with an interior Nash equilibrium.

Transparent Higher Order Sliding Mode Control for Nonlinear Master-Slave Systems without Velocity Measurement

Transparency has been a major objective in bilateral teleoperation systems, even in the absence of time delay induced by the communication channel, since a high degree of transparency would allow humans to drive the remote teleoperator as if he or she were directly interacting with the remote environment, with the remote teleoperator as a physical and sensorial extension of the operator. When fast convergence of position and force tracking errors are ensured by the control system, then complete transparency is obtained, which would ideally guarantee humans to be tightly kinaesthetically coupled. In this paper a model-free Cartesian second order sliding mode (SOSM) PD control scheme for nonlinear master-slave systems is presented. The proposed scheme does not rely on velocity measurements and attains very fast convergence of position trajectories, with bounded tracking of force trajectories, rendering a high degree of transparency with lesser knowledge of the system. The degree of transparency can easily be improved by tuning a feedback gain in the force loop. A unique energy storage function is introduced; such that a similar Cartesian-based controller is implemented in the master and slave sides. The resulting properties of the Cartesian control structure allows the human operator to input directly Cartesian variables, which makes clearer the kinaesthetic coupling, thus the proposed controller becomes a suitable candidate for practical implementation. The performance of the proposed scheme is evaluated in a semi-experimental setup.

Existence and approximation for vibro-impact problems with a time-dependent set of constraints

We consider a discrete mechanical system subjected to perfect time-dependent unilateral constraints, which dynamics is described by a second order measure differential inclusion. The transmission of the velocity at impacts is given by a minimization property of the kinetic energy with respect to the set of kinematically admissible post-impact velocities. We construct a sequence of feasible approximate positions by using a time-stepping algorithm inspired by a kind of Euler discretization of the differential inclusion. We prove the convergence of the approximate trajectories to a solution of the Cauchy problem and we obtain as a by-product a global existence result.

Mathematical and numerical approaches to a one-dimensional dynamic thermoviscoelastic contact problem

In this paper, we propose numerical schemes for solving a nonlinear system which consists of a coupled partial differential equations and two conditions, called normal compliance contact condition and Barber’s heat exchange condition. The convergence of numerical trajectories is shown by using a time discretization and passing the limit of the time step size. The uniqueness of the weak solution is proved as well. We derive the extensive form of an energy balance which will be a criterion to examine numerical stability. An example is provided to present and discuss numerical results.

Predictive feedback control and synchronization of hyperchaotic systems

The paper deals with the problem of control and synchronization of hyperchaotic systems. The proposed control method is based on the predictive principle which employ an instantaneous control input to guarantee the convergence of the chaotic trajectory towards an unstable equilibrium point. The synchronization is constructed around the drive-response principle. Numerical simulations on some hyperchaotic systems are given to demonstrate the main results.

A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving nonsmooth convex objective functions. Based on the Yosida regularization of the subdifferential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions, a key property for applications in cooperative games, inverse problems, and numerical multiobjective optimization.

Breaking the Glass Ceiling? The Effect of Board Quotas on Female Labor Market Outcomes in Norway.

In late 2003, Norway passed a law mandating 40 percent representation of each gender on the board of publicly limited liability companies. The primary objective of this reform was to increase the representation of women in top positions in the corporate sector and decrease gender disparity in earnings within that sector. We document that the newly (post-reform) appointed female board members were observably more qualified than their female predecessors, and that the gender gap in earnings within boards fell substantially. While the reform may have improved the representation of female employees at the very top of the earnings distribution (top 5 highest earners) within firms that were mandated to increase female participation on their board, there is no evidence that these gains at the very top trickled-down. Moreover the reform had no obvious impact on highly qualified women whose qualifications mirror those of board members but who were not appointed to boards. We observe no statistically significant change in the gender wage gaps or in female representation in top positions, although standard errors are large enough that we cannot rule economically meaningful gains. Finally, there is little evidence that the reform affected the decisions of women more generally; it was not accompanied by any change in female enrollment in business education programs, or a convergence in earnings trajectories between recent male and female graduates of such programs. While young women preparing for a career in business report being aware of the reform and expect their earnings and promotion chances to benefit from it, the reform did not affect their fertility and marital plans. Overall, in the short run the reform had very little discernible impact on women in business beyond its direct effect on the newly appointed female board members.

Inducing sustained oscillations in feedback-linearizable single-input nonlinear systems

The present paper concerns the induction of stable sustained oscillation in feedback-linearizable single-input affine nonlinear dynamical systems via continuous-time state feedback control. The proposed application-oriented control approach is based on the conception of a state feedback controller that ensures the tracking of a limit cycle characterized in terms of the feedback-linearized system. Boundedness and convergence of the closed-loop trajectories are established following the Lyapunov theoretical framework and applying LaSalle׳s stability principle. The proposed approach is demonstrated with computer-simulated control experiments, showing that it ensures the convergence of the state trajectories of the controlled system to a designed limit cycle and that the methodology can, in principle, be applied to any single input feedback linearizable system.

Dynamic frictionless contact of a nonlinear beam with two stops

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right end and are interpreted as a pair of complementarity conditions. We formulate a time discretization based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.

Caputo standard α-family of maps: fractional difference vs. fractional.

In this paper, the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo standard α-Families of maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter α) and differences in properties of falling factorial- and power-law memory maps are investigated.

Flow-plate interactions: Well-posedness and long-time behavior

We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent resultson existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting setscorresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to aPDE system with delay.