Control Parameter Space Research Articles

Interpretable machine learning for inferring the phase boundaries in a nonequilibrium system.

Still under debate is the question of whether machine learning is capable of going beyond black-box modeling for complex physical systems. We investigate the generalizing and interpretability properties of learning algorithms. To this end, we use supervised and unsupervised learning to infer the phase boundaries of the active Ising model, starting from an ensemble of configurations of the system. We illustrate that unsupervised learning techniques are powerful at identifying the phase boundaries in the control parameter space, even in situations of phase coexistence. It is demonstrated that supervised learning with neural networks is capable of learning the characteristics of the phase diagram, such that the knowledge obtained at a limited set of control variables can be used to determine the phase boundaries across the phase diagram. In this way, we show that properly designed supervised learning provides predictive power to regions in the phase diagram that are not included in the training phase of the algorithm. We stress the importance of introducing interpretability methods in order to perform a physically relevant classification of the phases with deep learning.

Extreme Events in Lasers with Modulation of the Field Polarization

We develop a theoretical model for a unidirectional ring laser consisting of an isotropic active medium inside a cavity containing a birefringent Kerr cell. We analyze the dynamical behavior of the system as we modulate the voltage applied to the Kerr cell. We discuss the bifurcation diagram and we study the regions of control parameter space where it becomes possible to observe and predict extreme events.

Non-Gibbs states on a Bose-Hubbard lattice

We study the equilibrium properties of the repulsive quantum Bose-Hubbard model at high temperatures in arbitrary dimensions, with and without disorder. In its microcanonical setting the model conserves energy and particle number. The microcanonical dynamics is characterized by a pair of two densities: energy density $\varepsilon$ and particle number density $n$. The macrocanonical Gibbs distribution also depends on two parameters: the inverse nonnegative temperature $\beta$ and the chemical potential $\mu$. We prove the existence of non-Gibbs states, that is, pairs $(\varepsilon,n)$ which cannot be mapped onto $(\beta,\mu)$. The separation line in the density control parameter space between Gibbs and non-Gibbs states $\varepsilon \sim n^2$ corresponds to infinite temperature $\beta=0$. The non-Gibbs phase cannot be cured into a Gibbs one within the standard Gibbs formalism using negative temperatures.

Сложная динамика неавтономного осциллятора с управляемой фазой внешнего воздействия

AbstractThe dynamics of a non-autonomous oscillator in which the phase of external force linearly depends on a dynamic variable has been investigated. This control of the external-drive phase leads to the formation of a hierarchy of various periodic and chaotic oscillations. The structure of the space of control parameters has been studied. It is established that dynamics of the system exhibits oscillatory regimes analogous to those of a non-autonomous oscillator with potential in the form of a periodic function.

Complex Dynamics of a Non-Autonomous Oscillator with a Controlled Phase of an External Force

The dynamics of a non-autonomous oscillator in which the phase of external force linearly depends on a dynamic variable has been investigated. This control of the external-drive phase leads to the formation of a hierarchy of various periodic and chaotic oscillations. The structure of the space of control parameters has been studied. It is established that dynamics of the system exhibits oscillatory regimes analogous to those of a non-autonomous oscillator with potential in the form of a periodic function.

Analysis of Second Order Cluster Consensus for Directed Networks with Time Delay

Abstract In this paper, we consider the cluster consensus problem in delayed directed networks where agents are described by second-order discrete-time dynamics. Unlike first order systems, delay is shown to lead to instability in second order systems if the controller parameters are not chosen properly. Furthermore, the numerical results reveal that as the amount of delay increases, the stability region in the controller parameter space shrinks.

Dynamics Feature and Synchronization of a Robust Fractional-Order Chaotic System

Exploring the dynamics feature of robust chaotic system is an attractive yet recent topic of interest. In this paper, we introduce a three-dimensional fractional-order chaotic system. The important finding by analysis is that the position of signalx3descends at the speed of 1/cas the parameterbincreases, and the signal amplitude ofx1,x2can be controlled by the parametermin terms of the power function with the index −1/2. What is more, the dynamics remains constant with the variation of parametersbandm. Consequently, this system can provide rich encoding keys for chaotic communication. By considering the properties of amplitude and position modulation, the partial projective synchronization and partial phase synchronization are realized with linear control scheme. The distribution map of optimal synchronization region in the control-parameter space is charted by defining the power consumption of controller. Numerical simulations are executed to confirm the theoretical analysis.

Prediction of exotic ion-crystal structures in a Paul trap

Trapped Coulomb crystals are of prime importance for the fields of one-component plasmas, quantum computing, quantum simulations, artificial atoms, nonlinear spectroscopy, and structural phase transitions. In all these applications it is essential to be able to accurately predict the structure of the crystals given the trapping parameters. For the Paul trap, one of the most promising platforms for quantum computing, this is nontrivial, since the confining radio-frequency fields are time-dependent. The pseudopotential approach eliminates this time-dependence. However, the standard pseudopotential, commonly used in atomic physics and quantum computing applications, is not even powerful enough to predict all stable two-ion crystal configurations. In this paper, we develop an improved pseudopotential, applicable to few-ion Coulomb crystals. Our potential is vastly more accurate than the standard version, and is powerful enough to predict analytically the existence and structural phase boundaries of new three- and four-ion configurations that are completely missed by the standard pseudopotential. In particular, we make quantitative predictions of the border lines between different crystal configurations in the Paul trap’s (q, a) stability diagram, which can be used to accurately switch between configurations. In addition, our improved pseudopotential accurately predicts the tilt angles of two-ion crystals. We also delineate the regions in (q, a) control parameter space where no two-, three-, and four-ion crystals exist. While this region is known for two-ion crystals, the regions for three- and four-ion crystals are new.

Three-Dimensional Modeling of a Fin-Actuated Robotic Fish With Multimodal Swimming

Dynamical model is always an important factor in controller design for robots. Existing models of robotic fish typically incorporate only planar motion, rarely considering spatial motion. This paper formulates a complete three-dimensional (3-D) dynamic model for the robotic fish actuated by pectoral and caudal fins, in which the fluid forces mainly contain quasi-steady lift and drag, gravity and buoyancy, and waterjet strike force. The critical lift and drag of flapping fins are derived with an explicit 3-D angle of attack. Taking a bioinspired central pattern generator as the system actuation, our model can produce multimodal maneuvers, including forward/backward swimming, turning, and ascending/descending, as well as complicated motions, such as rolling and spiraling. Motions simulated in a 3-D environment are experimentally validated with a free-swimming robotic fish. Furthermore, systematic comparisons between simulations and experiments are conducted over a wide range of the control parameter space for beating frequency, amplitude, and offset. The overall results demonstrate the effectiveness and the versatility of the developed 3-D dynamic model in the prediction of the robot trajectory, velocity, and attitude.

Additional food supplements as a tool for biological conservation of predator-prey systems involving type III functional response: A qualitative and quantitative investigation

Provision of additional food supplements for the purpose of biological conservation in ecosystems has of late been intensely researched by agriculturalists, biologists and mathematicians. The study of these ecosystems is usually done using the predator-prey systems. In these ecological studies it has been observed that the quality and quantity of additional food supplements provided play a crucial role in the growth of the predators and thereby influence the eventual state of the ecosystem. Also, in some of the ecological experiments it has been observed that predators exhibit non-optimal foraging behaviour in the presence of additional food. Findings also show that the predators exhibit a Holling type II response towards a target prey with predation highest at low prey densities. The results suggest that predation by predators is unlikely to stabilize low density prey populations. This can be attributed to the prey detectability independent nature of the type II response. In nature, sigmoidal functional responses such as the Holling type III response, have been documented in organisms from various taxa. In this kind of type III response the predators exhibit low detectability nature at low prey densities. Due to this the ecosystem tends to get stabilized at low prey densities avoiding the oscillations encountered in type II response. Motivated by these studies, in this paper, we consider a predator-prey system provided with additional food where the predator is assumed to exhibit Holling type III functional response towards the available food and the additional food supplements provided are assumed to be of constant density. We also assume that the predators are not optimal foragers. The model is analyzed in the control parameter space using the control parameters, quality and quantity of additional food. It is observed that the system exhibits apparent competition only when the predators are provided with high quality additional food supplements. Further, it has been shown that the ecosystem tends to get stabilized at low prey densities and the system can be steered to a desired state by a suitable choice of additional food supplements. Provision of low quality additional food supplements can result in completely opposite results to the expected ones.

PID Controller Synthesis Using a <inline-formula> <tex-math notation="LaTeX">$\sigma$ </tex-math> </inline-formula>-Hurwitz Stability Criterion

This letter proposes a new approach to design Proportional-Integral (PI) and Proportional-Integral-Derivative (PID) controllers. The main result is a constructive determination of the set $\mathcal {S}$ ( $\sigma $ ) of stabilizing PI and PID controllers with closed loop poles having real parts less than $-\sigma $ . The signature method is applied to obtain the set $\mathcal {S}$ ( $\sigma $ ) in the controller parameter space. Illustrative examples are included to describe the procedure of the proposed method and to find the maximum achievable $\sigma $ for a given plant.

Computation of PI Controllers Ensuring Desired Gain and Phase Margins for Two-Area Load Frequency Control System with Communication Time Delays

This article deals with computation of all proportional-integral (PI) controllers which achieve the desired frequency-domain specifications namely, gain and phase margins (GPMs) of a traditional two-area load frequency control (LFC) system with constant communication delays. An efficient graphical method based on extracting the boundaries of stability regions in PI controller parameter space having user defined GPMs is proposed to obtain GPMs based stability regions. Regions are determined for a wide range of GPMs and time delay. The theoretical region boundaries are validated using a powerful numerical method known as the quasi-polynomial mapping-based root finder (QPmR) and time-domain simulations. Results indicate that the proposed scheme gives faster disturbance rejection and desired frequency response as compared to the recently developed scheme for LFC systems with communication delays.

One-to-three resonant Hopf bifurcations of a maglev system

This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.

Design and Evaluation of Robust Cooperative Adaptive Cruise Control Systems in Parameter Space

This paper is on the design of cooperative adaptive cruise control systems for automated driving of platoons of vehicles in the longitudinal direction. Longitudinal models of vehicles with simple dynamics, an uncertain first order time constant and vehicle to vehicle communication with a communication delay are used in the vehicle modeling. A robust parameter space approach is developed and applied to the design of the cooperative adaptive cruise control system. D-stability is chosen as the robust performance goal and the feedback PD controller is designed in controller parameter space to achieve this D-stability goal for a range of possible longitudinal dynamics time constants and different values of time gap. Preceding vehicle acceleration is sent to the ego vehicle using vehicle to vehicle communication and a feedforward controller is used in this inter-vehicle loop to improve performance. Simulation results of an eight vehicle platoon of heterogeneous vehicles are presented and evaluated to demonstrate the efficiency of the proposed design method. Also, the proposed method is compared with a benchmark controller and the feedback only controller. Time gap regulation and string stability are used to assess performance and the effect of the vehicle to vehicle communication frequency on control system performance is also investigated.

Phenomenology of coupled nonlinear oscillators.

A recently introduced model of coupled nonlinear oscillators in a ring is revisited in terms of its information processing capabilities. The use of Lempel-Ziv based entropic measures allows to study thoroughly the complex patterns appearing in the system for different values of the control parameters. Such behaviors, resembling cellular automata, have been characterized both spatially and temporally. Information distance is used to study the stability of the system to perturbations in the initial conditions and in the control parameters. The latter is not an issue in cellular automata theory, where the rules form a numerable set, contrary to the continuous nature of the parameter space in the system studied in this contribution. The variation in the density of the digits, as a function of time is also studied. Local transitions in the control parameter space are also discussed.

Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models

This paper derives new formulations for designing dominant pole placement based proportional-integral-derivative (PID) controllers to handle second order processes with time delays (SOPTD). Previously, similar attempts have been made for pole placement in delay-free systems. The presence of the time delay term manifests itself as a higher order system with variable number of interlaced poles and zeros upon Pade approximation, which makes it difficult to achieve precise pole placement control. We here report the analytical expressions to constrain the closed loop dominant and non-dominant poles at the desired locations in the complex s-plane, using a third order Pade approximation for the delay term. However, invariance of the closed loop performance with different time delay approximation has also been verified using increasing order of Pade, representing a closed to reality higher order delay dynamics. The choice of the nature of non-dominant poles e.g. all being complex, real or a combination of them modifies the characteristic equation and influences the achievable stability regions. The effect of different types of non-dominant poles and the corresponding stability regions are obtained for nine test-bench processes indicating different levels of open-loop damping and lag to delay ratio. Next, we investigate which expression yields a wider stability region in the design parameter space by using Monte Carlo simulations while uniformly sampling a chosen design parameter space. The accepted data-points from the stabilizing region in the design parameter space can then be mapped on to the PID controller parameter space, relating these two sets of parameters. The widest stability region is then used to find out the most robust solution which are investigated using an unsupervised data clustering algorithm yielding the optimal centroid location of the arbitrary shaped stability regions. Various time and frequency domain control performance parameters are investigated next, as well as their deviations with uncertain process parameters, using thousands of Monte Carlo simulations, around the robust stable solution for each of the nine test-bench processes. We also report, PID controller tuning rules for the robust stable solutions using the test-bench processes while also providing computational complexity analysis of the algorithm and carry out hypothesis testing for the distribution of sampled data-points for different classes of process dynamics and non-dominant pole types.

Handling and Stability Analysis of Vehicle Plane Motion

The work analyzes the properties of handling and bicycle vehicle model motion stationary states manifold stability taking into account drift force nonlinear characteristics. Determining single two-axle vehicle nonlinear model stationary states and analyzing their stability were based on a graphical method (Y. M. Pevzner, H. Pacejka). It has its disadvantages: the absence of evident analytical stability criteria for the entire wheeled vehicle circular stationary states manifold. And also the absence of global stability threshold characteristics in the controlled parameter space. The task part suggests developing methods for building bifurcation manifold or critical parameters manifold (longitudinal velocity and wheel turning angle) with which the divergent loss of stability occurs. Known H. Troger, K. Zeman and Fabio Della Rossaa, GiampieroMastinub, Carlo Piccardia results are based on parameter continuation numerical methods which makes the quality analysis of drift force nonlinear characteristics impact on the entire stationary states manifold stability conditions more difficult. A compelling grapho-analytic approach towards bifurcation manifold building and getting circular stationary states analytical stability conditions based on moving from nonlinear drift forces on axles dependencies to their inverse dependence is developed in the suggested work. This methodology allows defining dangerous/safe stability threshold conditions in the control parameters space.

Функциональный синтез управления в однофазных электроприводах малой мощности

This paper dwells upon the problem of using the spectral method to formalize the description of non-linear system motion in the space of controllable parameters, with evidence from a series-wound commutator motor power from a gate converter in the discontinuous-current region; it also covers the functional relationships of the controlled motion coordinates. The paper presents data on how the motor speed affects the change in its dynamic parameters. It gives approximated dependencies that enable generating the corrective signal by means of typical linear links. Results of the study might find use in designing motor drive systems as an electrified tool, or traction drives for optimal speed and minimum losses in dynamic modes.

Leader-following consensus of discrete-time fractional-order multi-agent systems

Leader-following consensus of fractional order multi-agent systems is investigated. The agents are considered as discrete-time fractional order integrators or fractional order double-integrators. Moreover, the interaction between the agents is described with an undirected communication graph with a fixed topology. It is shown that the leader-following consensus problem for the considered agents could be converted to the asymptotic stability analysis of a discrete-time fractional order system. Based on this idea, sufficient conditions to reach the leader-following consensus in terms of the controller parameters are extracted. This leads to an appropriate region in the controller parameters space. Numerical simulations are provided to show the performance of the proposed leader-following consensus approach.

Excess Entropy Production in Quantum System: Quantum Master Equation Approach

For open systems described by the quantum master equation (QME), we investigate the excess entropy production under quasistatic operations between nonequilibrium steady states. The average entropy production is composed of the time integral of the instantaneous steady entropy production rate and the excess entropy production. We propose to define average entropy production rate using the average energy and particle currents, which are calculated by using the full counting statistics with QME. The excess entropy production is given by a line integral in the control parameter space and its integrand is called the Berry-Sinitsyn-Nemenman (BSN) vector. In the weakly nonequilibrium regime, we show that BSN vector is described by $\ln \breve{\rho}_0$ and $\rho_0$ where $\rho_0$ is the instantaneous steady state of the QME and $\breve{\rho}_0$ is that of the QME which is given by reversing the sign of the Lamb shift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropy production approximately reduces to the difference between the von Neumann entropies of the system. Additionally, we point out that the expression of the entropy production obtained in the classical Markov jump process is different from our result and show that these are approximately equivalent only in the weakly nonequilibrium regime.