Second-order Dynamics Research Articles

A Katz-centrality-based protocol design for leader-following formation of discrete-time multi-agent systems with communication delays

This paper aims at constructing an improved structure for the consensus protocol in discrete-time multi-agent systems. A consensus protocol is proposed by using the centrality measure for agent which is determined by the information flow through multi-agent systems. In analyzing multi-agent systems, a graph representation is used. The concept of centrality measure which was introduced in the field of social network analysis is attempted to find the most central node within the graph. This work is intended to observe the Katz-centrality that is one of centrality measures. Here, the advantage of Katz-centrality is that a node’s centrality depends not only on how many others the node is connected to (its degree), but also on their centrality. The alternative consensus protocol will be applied to the leader-following formation problems for discrete-time multi-agent systems with second-order dynamics and time-varying communication delay. To achieve this, sufficient conditions for the aforementioned problems will be established in terms of linear matrix inequality by utilizing the Lyapunov method and some mathematical techniques. Finally, the discrete-time multi-agent systems modeled with a point-mass dynamics of motion for aircraft and its simulation results are given to illustrate the advantages of the alternative consensus protocol in point of the robustness on time delay and H∞ performance.

Relativistic Dissipative Fluid Dynamics from the Non-Equilibrium Statistical Operator

We present a new derivation of second-order relativistic dissipative fluid dynamics for quantum systems using Zubarev’s formalism for the non-equilibrium statistical operator. In particular, we discuss the shear-stress tensor to second order in gradients and argue that the relaxation terms for the dissipative quantities arise from memory effects contained in the statistical operator. We also identify new transport coefficients which describe the relaxation of dissipative processes to second order and express them in terms of equilibrium correlation functions, thus establishing Kubo-type formulae for the second-order transport coefficients.

Distributed-observer-based output synchronization for heterogeneous double-integral networks

This paper considers the output synchronization for nonidentical second-order integral dynamics with external disturbance. The leader’s signal can only be received by parts of the followers, and all the followers have different dynamics. An adaptive distributed compensator is utilized to estimate the leader’s signal and system matrix. Since the states’ information of the agents can not be measured by the others, a novel distributed reduced-dimension observer is designed to estimate the unmeasurable states. Based on the adaptive distributed compensator and reduced-dimension observer, a dynamic output feedback control law is designed to solve the output synchronization problem. Finally, an example is given to illustrate the validity of the theoretical results.

( 3+1 )-dimensional anisotropic fluid dynamics with a lattice QCD equation of state

Anisotropic hydrodynamics improves upon standard dissipative fluid dynamics by treating certain large dissipative corrections nonperturbatively. Relativistic heavy-ion collisions feature two such large dissipative effects: (i) Strongly anisotropic expansion generates a large shear stress component which manifests itself in very different longitudinal and transverse pressures, especially at early times. (ii) Critical fluctuations near the quark-hadron phase transition lead to a large bulk viscous pressure on the conversion surface between hydrodynamics and a microscopic hadronic cascade description of the final collision stage. We present a new dissipative hydrodynamic formulation for nonconformal fluids where both of these effects are treated nonperturbatively. The evolution equations are derived from the Boltzmann equation in the 14-moment approximation, using an expansion around an anisotropic leading-order distribution function with two momentum-space deformation parameters, accounting for the longitudinal and transverse pressures. To obtain their evolution we impose generalized Landau matching conditions for the longitudinal and transverse pressures. We describe an approximate anisotropic equation of state that relates the anisotropy parameters with the macroscopic pressures. Residual shear stresses are smaller and are treated perturbatively, as in standard second-order dissipative fluid dynamics. The resulting optimized viscous anisotropic hydrodynamic evolution equations are derived in $3+1$ dimensions and tested in a ($0+1$)-dimensional Bjorken expansion, using a state-of-the-art lattice equation of state. Comparisons with other viscous hydrodynamical frameworks are presented.

Deterministic robust control design for an iron core linear drive including second-order electrical dynamics

This work proposes a deterministic robust controller to improve tracking performance for a linear motor, taking into account the electrical dynamics imposed by a commercial current controller. The design is split in two parts by means of the backstepping technique, in which the first part corresponds to a typical deterministic robust controller, neglecting the electrical dynamics. In the second part, a second-order electrical dynamics is considered using a particular state transformation. There, the proposed control law is composed of a term to compensate the known part of the model and a robust control term to impose a bound on the effect of uncertainties on tracking error. Stability and boundedness results for the complete controller are given. To this effect, a general result on boundedness and stability of nonlinear systems with conditionally bounded state variables is derived first. Finally, experimental results for the complete controller show an improvement on tracking error of up to 31.7% when compared with the results from the typical controller that neglects the electrical dynamics.

A multiplicative eigenvalues perturbation and its application to natural frequency assignment in undamped second-order systems

In this article, a novel theorem for eigenvalue perturbation inspired by the results of Brauer is introduced. The new result is valid for single and multiple eigenvalue perturbations, with preservation—no spillover of the remaining spectrum. An application of the proposed results is given since the model updating and the partial eigenvalue assignment problems for second-order linear dynamics have gained crescent attention in the last decades. Furthermore, its effectiveness is illustrated with numerical examples.

Adaptive distributed convex optimization for multi-agent and its application in flocking behavior

This paper studies adaptive optimization problem of continuous-time multi-agent systems. Multi-agents with second-order dynamics are considered. Each agent is equipped with a time-varying cost function which is known only to an individual agent. The objective is to make multi-agents velocities minimize the sum of local functions by local interaction. First, a distributed adaptive algorithm is presented, in which each agent depends only on its own velocity and neighbors velocities. It is indicated that all agents can track the optimal velocity. Then we apply the distributed adaptive algorithm to flocking of multi-agents. It is proved that all agents can track the optimal trajectory. The agents will maintain connectivity and avoid the inter-agent collision. Finally, two simulations are included to illustrate the results.

Interpolation-based contour error estimation and component-based contouring control for five-axis CNC machine tools

High accuracy contour error estimation and direct contour error control are two major approaches to reduce the contour error. However, two key factors make them complex for five-axis machine tools: the nonlinear kinematics and the coupling between the tool position and orientation. In this study, by finding the reference point nearest to the current actual position, and interpolating the point with two neighboring reference points and using the distance ratio, a new contour error estimation method for five-axis machine tools is proposed, which guarantees high accuracy while depending on only the reference points. By adding a weighted contour error on the tracking error in the workpiece coordinate system, and specifying a desired second-order error dynamics based on the error variable, an effective contouring control method is proposed, which can alleviate the problem: when the contour error components are introduced into the controller, the contour errors increase instead in some regions of the tracking trajectory. A series of experiments are performed on a tilting-rotary-table (TRT) type five-axis machine tool. The results reveal that the proposed estimation method has high accuracy, and compared with the case without contour error control, the proposed control approach can reduce the contour error along the whole trajectory.

Cooperative global robust group output regulation for multi-agent systems with heterogeneous uncertain second-order nonlinear dynamics

In this paper, we study the global robust group output regulation problem for multi-agent systems with heterogeneous uncertain second-order nonlinear dynamics. The intelligence factor is introduced to construct a designed graph matrix which overcomes the communication influence between multiple coupled subnetworks with different exosystems. In terms of internal model principle, the global robust group output regulation problem is converted into the global robust dynamic stability problem of the so-called augmented multi-agent system. Further, the augmented multi-agent system is globally stabilized via a distributed state feedback control law. A special case of the main result of this paper leads to the solution of the global robust multi-tracking control problem for nonlinear multi-agent networks. A simulation is performed to illustrate the effectiveness of the theoretical results.

Safety Verification and Control for Collision Avoidance at Road Intersections

This paper presents the design of a supervisory algorithm that monitors safety at road intersections and overrides drivers with a safe input when necessary. The design of the supervisor consists of two parts: safety verification and control design. Safety verification is the problem to determine if vehicles will be able to cross the intersection without colliding with current drivers' inputs. We translate this safety verification problem into a jobshop scheduling problem, which minimizes the maximum lateness and evaluates if the optimal cost is zero. The zero optimal cost corresponds to the case in which all vehicles can cross each conflict area without collisions. Computing the optimal cost requires solving a Mixed Integer Nonlinear Programming (MINLP) problem due to the nonlinear second-order dynamics of the vehicles. We therefore estimate this optimal cost by formulating two related Mixed Integer Linear Programming (MILP) problems that assume simpler vehicle dynamics. We prove that these two MILP problems yield lower and upper bounds of the optimal cost. We also quantify the worst case approximation errors of these MILP problems. We design the supervisor to override the vehicles with a safe control input if the MILP problem that computes the upper bound yields a positive optimal cost. We theoretically demonstrate that the supervisor keeps the intersection safe and is non-blocking. Computer simulations further validate that the algorithms can run in real time for problems of realistic size.

Clearance-driven motion planning for mobile robots with differential constraints

SUMMARYThis paper presents an approach that integrates the geometric notion of clearance (distance to the closest obstacle) into sampling-based motion planning to enable a robot to safely navigate in challenging environments. To reach the goal destination, the robot must obey geometric and differential constraints that arise from the underlying motion dynamics and the characteristics of the environment. To produce safe paths, the proposed approach expands a motion tree of collision-free and dynamically feasible motions while maintaining locally maximal clearance. In distinction from related work, rather than explicitly constructing the medial axis, the proposed approach imposes a grid or a triangular tessellation over the free space and uses the clearance information to construct a weighted graph where edges that connect regions with low clearance have high cost. Minimum-cost paths over this graph produce high-clearance routes that tend to follow the medial axis without requiring its explicit construction. A key aspect of the proposed approach is a route-following component which efficiently expands the motion tree to closely follow such high-clearance routes. When expansion along the current route becomes difficult, edges in the tessellation are penalized in order to promote motion-tree expansions along alternative high-clearance routes to the goal. Experiments using vehicle models with second-order dynamics demonstrate that the robot is able to successfully navigate in complex environments. Comparisons to the state-of-the-art show computational speedups of one or more orders of magnitude.

Stochastic heavy ball

This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f. The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f. We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms.

Distributed Constraint Optimization with Flocking Behavior

This paper studies distributed optimization having flocking behavior and local constraint set. Multiagent systems with continuous‐time and second‐order dynamics are studied. Each agent has a local constraint set and a local objective function, which are known to only one agent. The objective is for multiple agents to optimize a sum of the local functions with local interaction and information. First, a bounded potential function to construct the controller is given and a distributed optimization algorithm that makes a group of agents avoid collisions during the evolution is presented. Then, it is proved that all agents track the optimal velocity while avoiding collisions. The proof of the main result is divided into three steps: global set convergence, consensus analysis, and optimal set convergence. Finally, a simulation is included to illustrate the results.

Bipartite Consensus on Coopetition Networks With Time-Varying Delays

In this paper, the bipartite consensus problem is investigated for multiagent systems with second-order dynamics and antagonistic interactions. We consider bipartite consensus under two kinds of protocols. One is an absolute damping protocol, and the other is a relative damping protocol. When there are no time delays, necessary and sufficient conditions are derived for the system to asymptotically achieve the bipartite consensus under the given protocols. When there are communication delays, sufficient conditions are obtained for the system to asymptotically achieve the bipartite consensus. In case $\mathcal {G} (\mathcal {A})$ is structurally unbalanced, consensus can also be asymptotically achieved, while this case is trivial.

No spillover partial eigenvalue assignment in second-order linear systems using the Brauer’s theorem and dense influence matrices

The control of vibrations for systems with second-order linear dynamics is a cornerstone for several engineering application, as structural vibration control in civil and aerospace structures. In such systems, sometimes only a few eigenvalues must be mitigated in situations as resonances, flutter, and another dangerous vibratory phenomenon. The use of active vibration control is an attractive alternative due to its flexibility and the availability of general design approaches. In the case of partial eigenvalue assignment, no spillover can be achieved, that is, only a small spectrum or eigenstructure partition is reallocated, with the remaining being unperturbed. In this paper, the no spillover property of Brauer’s spectrum perturbation theorem is applied to compute a dense influence matrix structure of actuation, reassigning then the dangerous target eigenvalues and kept the remaining of the spectrum unchanged. A Sylvester equation must be solved using only the part of the spectrum to be reassigned. The results are tested in a benchmark taken from the specialized literature, and thus confirming its effectiveness.

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence.

The hierarchical graphs and Sierpiński graphs are constructed iteratively, which have the same number of vertices and edges at any iteration, but exhibit quite different structural properties: the hierarchical graphs are nonfractal and small-world, while the Sierpiński graphs are fractal and "large-world." Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpiński graphs, focusing on three important quantities of consensus problems, that is, convergence speed, delay robustness, and coherence for first-order (and second-order) dynamics, which are, respectively, determined by algebraic connectivity, maximum eigenvalue, and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of Laplacian matrix. For both graphs, based on the explicit recursive relation of eigenvalues at two successive iterations, we evaluate the second smallest eigenvalue, as well as the largest eigenvalue, and obtain the closed-form solutions to the sum of reciprocals (and square of reciprocals) of all nonzero eigenvalues. We also compare our obtained results for consensus problems on both graphs and show that they differ in all quantities concerned, which is due to the marked difference of their topological structures.

Distributed finite‐time tracking of second‐order multi‐agent systems: An edge‐based approach

Distributed finite-time tracking problem of a multi-agent system with second-order non-linear dynamics is studied in this article. Based on measurements of edges, a finite-time tracking protocol is proposed without relying the complete position and velocity measurements. It means that the proposed protocol only requires a binary information between the neighbouring agent. By designing a carefully constructed Lyapunov function, the distributed finite-time tracking problem is solved if the sub-topology among the followers is connected and undirected and the topology of whole systems has a spanning tree with a leader been a root node. Furthermore, it has successfully estimated the finite settling time in theory with the proposed control protocol. The authors provide a numerical example to illustrate the effectiveness of the analytical results.

Containment control for heterogeneous multi-agent systems with asynchronous updates

This paper is concerned with the asynchronous containment control problem for heterogeneous multi-agent systems with time-varying delays. Here, asynchrony means that each agent updates the state information by its own clock that is independent of the other agents’ update times, and the update intervals of each agent are not necessarily equispaced. It is assumed that the leaders are stationary, and the followers can be classified into two generic categories: the followers with the first-order dynamics and the followers with second-order dynamics. For different kinds of followers, two distributed containment control protocols are presented to guarantee that all the followers can asymptotically converge into the convex hull formed by the leaders. The properties of the product of infinite nonnegative matrices are explored to arrive at that the asynchronous containment control can be achieved under the proposed protocols if and only if the communication topology among the agents contains a directed spanning forest rooted at the leaders. At last, simulation results are given to demonstrate the validity of the theoretical findings.

A Non-Linear Control Method to Compensate for Muscle Fatigue during Neuromuscular Electrical Stimulation

Neuromuscular electrical stimulation (NMES) is a promising technique to artificially activate muscles as a means to potentially restore the capability to perform functional tasks in persons with neurological disorders. A pervasive problem with NMES is that overstimulation of the muscle (among other factors) leads to rapid muscle fatigue, which limits the use of clinical and commercial NMES systems. The objective of this article is to develop an NMES controller that incorporates the effects of muscle fatigue during NMES-induced non-isometric contraction of the human quadriceps femoris muscle. Our previous work that used the RISE class of non-linear controllers cannot accommodate fatigue and muscle activation dynamics. A totally new control design approach and associated stability proof is required to derive a new class of NMES control design that accounts for muscle fatigue dynamics and a first-order activation dynamics, in addition to the second-order musculoskeletal dynamics. Motivated from a control method for robotic systems in a strict-feedback form, a backstepping based-non-linear NMES controller was designed to accommodate for the additional muscle activation dynamics. Further, experimentally identified estimates of the fatigue and activation dynamics were incorporated in the control design. The developed controller uses a neural network-based estimate of the musculoskeletal dynamics and error due to fatigue estimation. A globally uniformly ultimately bounded stability is proven the new controller that accounts for an uncertain non-linear muscle model and bounded non-linear disturbances (e.g., spasticity and changing load dynamics). The developed controller was validated through experiments on the left and right legs of 3 able-bodied subjects and was compared with a proportional-derivative (PD) controller and a PD augmented with a neural network. The statistical analysis showed improved control performance compared with the PD controller.

A Multiplicative Eigenvalues Perturbation and Its Application to Natural Frequency Assignment in Undamped Second-Order Systems

In this note, a novel theorem for eigenvalue perturbation inspired by the results of Brauer is introduced. The new result is shown to be valid for single and multiple eigenvalue perturbations, with preservation - no spillover of the remaining spectrum. Since in the last decades, the model update and the partial eigenvalue assignment problems for second-order linear dynamics have gained crescent attention, an application of the proposed results is given, and its effectiveness is illustrated in numerical examples.