Dynamics Of Original System Research Articles

Robust Cooperative Control for Heterogeneous Uncertain Nonlinear High-Order Fully Actuated Multiagent Systems.

This research is intended to address a robust cooperative control problem of heterogeneous uncertain nonlinear high-order fully actuated multiagent systems (HUN-HOFAMASs). A nonlinear HOFA system model is used to describe the multiagent systems (MASs) with heterogeneous uncertain nonlinear dynamics, which is called the HUN-HOFAMASs. A predictive terminal sliding-mode control-based robust cooperative control scheme is presented to address this problem. In this scheme, heterogeneous nonlinear dynamics of original system are offset to establish a linear constant HOFA system with the help of full actuation feature. Then, a terminal sliding-mode variable for enhancing the system robustness is introduced to handle the uncertainties. Furthermore, a linear incremental prediction model is developed in a HOFA form by means of a Diophantine equation. According to this model, the multistep terminal sliding-mode predictions are yielded to optimize the robust cooperative control performance and compensate for the network-induced communication constraints in the feedback and forward channels. Based on a linear matrix inequality (LMI) method, a necessary and sufficient criterion is derived to discuss the simultaneous consensus and stability of closed-loop HUN-HOFAMASs. The simulation and comparison results of cooperative flying around of multiple spacecraft system are shown to illustrate the capability and advantage of the presented predictive terminal sliding-mode control for robust cooperative control.

Assessing Credibility in Bayesian Networks Structure Learning.

Learning Bayesian networks from data aims to create a Directed Acyclic Graph that encodes significant statistical relationships between variables and their joint probability distributions. However, when using real-world data with limited knowledge of the original dynamical system, it is challenging to determine if the learned DAG accurately reflects the underlying relationships, especially when the data come from multiple independent sources. This paper describes a methodology capable of assessing the credible interval for the existence and direction of each edge within Bayesian networks learned from data, without previous knowledge of the underlying dynamical system. It offers several advantages over classical methods, such as data fusion from multiple sources, identification of latent variables, and extraction of the most prominent edges with their respective credible interval. The method is evaluated using simulated datasets of various sizes and a real use case. Our approach was verified to achieve results comparable to the most recent studies in the field, while providing more information on the model's credibility.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

Dynamical analysis of a parameter-aware reservoir computer.

Reservoir computing is a useful framework for predicting critical transitions of a dynamical system if the bifurcation parameter is also provided as an input. This work shows how the dynamical system theory provides the underlying mechanism behind the prediction. Using numerical methods, by considering dynamical systems which show Hopf bifurcation, we demonstrate that the map produced by the reservoir after a successful training undergoes a Neimark-Sacker bifurcation such that the critical point of the map is in immediate proximity to that of the original dynamical system. Also, we compare and analyze how the framework learns to distinguish between different structures in the phase space. Our findings provide insight into the functioning of machine learning algorithms for predicting critical transitions.

Improved State-Space Approach Based on Lumped Mass Matrix for Transient Analysis of Large-Scale Locally Nonlinear Structures

Due to the assumption of acceleration variation in traditional step-by-step integration methods such as Newmark, sufficiently small time steps are required to ensure numerical stability and accuracy in dynamic systems. In contrast, the state-space approach, based on piecewise interpolation of discrete load functions, does not rely on predetermined acceleration assumptions and has demonstrated high efficiency in terms of stability and accuracy. The original state-space method requires the calculation of the inverse of the structural mass in the transition matrix. However, when a lumped mass matrix is used, this computation renders the entire mass matrix singular, resulting in an invalid solution expression. To address this issue, this study proposes an improved state-space approach for the transient analysis of large-scale structural systems with local nonlinearities. In this approach, a nonlinear force corrector is introduced as an external force term applied to the linear elastic system to account for the nonlinear behavior of locally yielding components. Consequently, the original nonlinear dynamic system can be transformed into an equivalent linear elastic transient system. Furthermore, based on the lumped mass matrix, a first-order ordinary differential state-space equation for such an equivalent linear elastic transient system is derived. Simulation results from three transient system examples show that the state-space approach outperforms the Newmark method in terms of accuracy and stability for dynamic systems characterized by high frequency and low damping. The prediction results show that the state-space approach appears to be insignificantly affected by the choice of the consistent or lumped mass matrix. The numerical results show that the root-mean-square errors between the consistent and lumped matrices in the top displacement time histories of a 15-storey plane frame under various seismic intensities are all less than 1%, and in the base reaction time histories responses the discrepancies are only about 0.5%, indicating that the use of lumped mass matrices is quite reliable. When many nodes or degrees of freedom have no assigned mass, the dimensionality of the state-space equation can be significantly reduced using the lumped mass approach. Therefore, the simulation of large-scale systems can be simplified by employing the improved state-space approach with lumped mass matrices, yielding results nearly identical to those obtained using traditional methods. In conclusion, the improved state-space approach has great potential for the simulation of transient behavior in large-scale systems with local nonlinearities.

Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system.

Method orderable cascade decomposition for control system

In this paper, we propose a method to determine the state function and control function of the dynamical system represented by a system of equations with first-order partial derivatives under boundary and the first-order partial derivatives conditions of state function where are real matrices with corresponding sizes. The basis of the theory is a method to prove the orderable cascade decomposition to transform the original system into the equivalent system in the type . In the final step, we obtain the state function satisfying the conditions and substituting this in the previous step. Continuing this process, we can find out the state function and controllable function of the original dynamical system.

Data-Driven Stochastic Averaging

Abstract Stochastic averaging, as an effective technique for dimension reduction, is of great significance in stochastic dynamics and control. However, its practical applications in industrial and engineering fields are severely hindered by its dependence on governing equations and the complexity of mathematical operations. Herein, a data-driven method, named data-driven stochastic averaging, is developed to automatically discover the low-dimensional stochastic differential equations using only the random state data captured from the original high-dimensional dynamical systems. This method includes two successive steps, that is, extracting all slowly varying processes hidden in fast-varying state data and identifying drift and diffusion coefficients by their mathematical definitions. It automates dimension reduction and is especially suitable for cases with unavailable governing equations and excitation data. Its application, efficacy, and comparison with theory-based stochastic averaging are illustrated through several examples, numerical or experimental, with pure Gaussian white noise excitation or combined excitations.

Adaptive dynamic programming-based hierarchical decision-making of non-affine systems

In this paper, the problem of multiplayer hierarchical decision-making problem for non-affine systems is solved by adaptive dynamic programming. Firstly, the control dynamics are obtained according to the theory of dynamic feedback and combined with the original system dynamics to construct the affine augmented system. Thus, the non-affine multiplayer system is transformed into a general affine form. Then, the hierarchical decision problem is modeled as a Stackelberg game. In the Stackelberg game, the leader makes a decision based on the information of all followers, whereas the followers do not know each other’s information and only obtain their optimal control strategy based on the leader’s decision. Then, the augmented system is reconstructed by a neural network (NN) using input–output data. Moreover, a single critic NN is used to approximate the value function to obtain the optimal control strategy for each player. An extra term added to the weight update law makes the initial admissible control law no longer needed. According to the Lyapunov theory, the state of the system and the error of the weights of the NN are both uniformly ultimately bounded. Finally, the feasibility and validity of the algorithm are confirmed by simulation.

Vector Control Lyapunov Function Based Stabilization of Nonlinear Systems in Predefined Time

Predefined-time stability is the stability of dynamical systems whose solutions approach the equilibrium point within a pre-decided time duration. In this technical note, we develop general results of predefined-time stability of nonlinear systems using vector Lyapunov functions. A vector comparison system, which is predefined time convergent, is constructed, and after that the stability of the original dynamical system is proved using differential inequalities and comparison principles. Moreover, we design predefined-time controllers for large-scale systems using vector control Lyapunov functions (VCLFs). Sliding mode control is introduced in the design approach to mitigate matched bounded disturbances/uncertainties. Also, we aggregate comparison systems to reduce their dimensionality in order to effectively apply the derived results on practical systems. The theoretical results are implemented on a 2 DOF Helicopter model.

Projective embedding of dynamical systems: Uniform mean field equations

We study embeddings of continuous dynamical systems in larger dimensions via projector operators. We call this technique PEDS, projective embedding of dynamical systems, as the stable fixed point of the original system dynamics are recovered via projection from the higher dimensional space. In this paper we provide a general definition and prove that for a particular type of rank-1 projector operator, the uniform mean field projector, the equations of motion become a mean field approximation of the dynamical system. While in general the embedding depends on a specified variable ordering, the same is not true for the uniform mean field projector. We prove a variety of results on the relationship between the spectrum of the Jacobian for fixed points in the original and in the embedded system. Direct applications of PEDS can be non-convex optimization and machine learning.

THUẬT TOÁN ĐIỀU KHIỂN THEO ĐẦU RA HỆ THỐNG TUYẾN TÍNH KHÔNG DỪNG, SỬ DỤNG CÁC PHƯƠNG PHÁP NHẬN DẠNG THAM SỐ

The problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. For synthesis of the observer of state variables, a new approach is proposed, providing the possibility of obtaining monotonous estimates of convergence with the regulation of transition time. The main idea of the synthesis of observers is based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. Then the DREM algorithm is used to estimate the state variables. Some simulations on Matlab/Simulink prove the correctness of the theory built, ensuring the good quality of the response processes.

Renormalization in one-dimensional dynamics

The study of the dynamical and topological properties of interval exchange transformations and their natural generalizations is an important problem, which lies at the intersection of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory, and geometric group theory. The purpose of the survey is to make a systematic presentation of the existing results on the ergodic and geometric characteristics of the one-dimensional maps under consideration, as well as on the measured foliations on surfaces and two-dimensional complexes that one can associate with these maps. These results are based on the research of the ergodic properties of the renormalization process - an algorithm that takes an original dynamical system and builds a sequence of equivalent dynamical systems with a smaller support set. For all dynamical systems considered in the paper these renormalization algorithms can be viewed as multidimensional fraction algorithms. Bibiliography: 74 titles.

CONTROL VARIATE POLYNOMIAL CHAOS: OPTIMAL FUSION OF SAMPLING AND SURROGATES FOR MULTIFIDELITY UNCERTAINTY QUANTIFICATION

We present a multifidelity uncertainty quantification numerical method that leverages the benefits of both sampling and surrogate modeling, while mitigating their downsides, for enabling rapid computation in complex dynamical systems such as automotive propulsion systems. In particular, the proposed method utilizes intrusive generalized polynomial chaos to quickly generate additional information that is highly correlated with the original nonlinear dynamical system. We then leverage a Monte Carlo-based control variate to correct the bias caused by the surrogate approximation. In contrast to related works merging adaptive surrogate approximation and sampling in a multifidelity setting, the intrusive generalized polynomial chaos (gPC) surrogate is selected because it avoids statistical errors by design by providing analytical estimates of output statistics. Moreover, it enables theoretical contributions that provide an estimator design strategy that optimally balances the computational efforts allocated to sampling and to gPC construction. We deploy our approach to multiple numerical examples including simulations of hybrid-electric propulsion systems, where the proposed estimator is shown to achieve orders-of-magnitude reduction in mean squared error of statistics estimation under comparable costs of purely sampling or purely surrogate approaches.

Accelerating metabolic models evaluation with statistical metamodels: application to Salmonella infection models

Mathematical and numerical models are increasingly used in microbial ecology to model the fate of microbial communities in their ecosystem. These models allow to connect in a mechanistic framework species-level informations, such as the microbial genomes, with macro-scale features, such as species spatial distributions or metabolite gradients. Numerous models are built upon species-level metabolic models that predict the metabolic behaviour of a microbe by solving an optimization problem knowing its genome and its nutritional environment. However, screening the community dynamics with these metabolic models implies to solve such an optimization problem by species at each time step, leading to a significant computational load further increased by several orders of magnitude when spatial dimensions are added. In this paper, we propose a statistical framework based on Reproducing Kernel Hilbert Space (RKHS) metamodels that are used to provide fast approximations of the original metabolic model. The metamodel can replace the optimization step in the system dynamics, providing comparable outputs at a much lower computational cost. We will first build a system dynamics model of a simplified gut microbiota composed of a unique commensal bacterial strain in interaction with the host and challenged by a Salmonella infection. Then, the machine learning method will be introduced, and particularly the ANOVA-RKHS that will be exploited to achieve variable selection and model parsimony. A training dataset will be constructed with the original system dynamics model and hyper-parameters will be carefully chosen to provide fast and accurate approximations of the original model. Finally, the accuracy of the trained metamodels will be assessed, in particular by comparing the system dynamics outputs when the original model is replaced by its metamodel. The metamodel allows an overall relative error of 4.71% but reducing the computational load by a speed-up factor higher than 45, while correctly reproducing the complex behaviour occurring during Salmonella infection. These results provide a proof-of-concept of the potentiality of machine learning methods to give fast approximations of metabolic model outputs and pave the way towards PDE-based spatio-temporal models of microbial communities including microbial metabolism and host-microbiota-pathogen interactions.

Parallel Algorithm for Parametric Identification of Dynamical Systems with Interval Parameters

The paper presents a parallel algorithm for the parametric identification of dynamical systems with interval parameters. The algorithm is based on the previously developed, substantiated and tested adaptive interpolation algorithm, which makes it possible to explicitly obtain the dependence of the states of a dynamic system on interval parameters. The solution of the problem of parametric identification is reduced to the problem of minimizing a certain objective function in the space of boundaries of interval parameter estimates. Due to the use of the adaptive interpolation algorithm when calculating the gradient of the objective function, there is no need for additional analysis and modeling of the original dynamic system, so it is convenient to use first-order methods for optimization. However, the task of calculating the objective function and the gradient includes a set of conditional minimization problems for explicit functions that can be solved independently of each other. The article discusses the main aspects and features of parallelization and implementation of the parametric identification algorithm and tests it on several representative examples. The acceleration and efficiency of parallelization are analyzed.

Robust Passivity-Based Dynamical Systems for Compliant Motion Adaptation

Motivated by human compliant behaviors during interacting with unknown environments and how motions and impedance to are adapted skilfully complete a task, this article develops a motion planning scheme that is capable of generating a compliant trajectory online such that tracking desired contacting forces under a predefined motion task. First, an improved dynamical system (DS) is designed to generate an adaptive compliant scanning trajectory online from the original DS in terms of the contact forces and the desired scanning forces. Inspired by passivity analysis for the robot control system, a robust term is formulated to guarantee stability by considering the balance between environmental and robotic energy. Furthermore, we develop a state-constrained controller based on barrier Lyapunov function to track the compliant DS motion and to ensure safety during scanning for the patient. Finally, comparative simulations are conducted to validate the general compliant capability of the proposed framework. We also instantiate our methodology through a use case of liver ultrasound scanning to demonstrate the stable and dynamic force tracking performance.

Backcasting COVID-19: a physics-informed estimate for early case incidence.

It is widely accepted that the number of reported cases during the first stages of the COVID-19 pandemic severely underestimates the number of actual cases. We leverage delay embedding theorems of Whitney and Takens and use Gaussian process regression to estimate the number of cases during the first 2020 wave based on the second wave of the epidemic in several European countries, South Korea and Brazil. We assume that the second wave was more accurately monitored, even though we acknowledge that behavioural changes occurred during the pandemic and region- (or country-) specific monitoring protocols evolved. We then construct a manifold diffeomorphic to that of the implied original dynamical system, using fatalities or hospitalizations only. Finally, we restrict the diffeomorphism to the reported cases coordinate of the dynamical system. Our main finding is that in the European countries studied, the actual cases are under-reported by as much as 50%. On the other hand, in South Korea-which had a proactive mitigation approach-a far smaller discrepancy between the actual and reported cases is predicted, with an approximately 18% predicted underestimation. We believe that our backcasting framework is applicable to other epidemic outbreaks where (due to limited or poor quality data) there is uncertainty around the actual cases.

Research on fault-tolerant consensus of heterogeneous multi-agent systems based on adaptive robust sliding mode observer

Abstract The fault-tolerant consensus control for heterogeneous multi-agent systems with actuator faults and external disturbances is studied in this paper. Firstly, considering the unknown actuator fault information and the influence of external disturbance on the observer, a robust sliding mode observer with adaptive laws modification was proposed to realize the actuator fault reconstruction. Secondly, a fault-tolerant controller is designed based on the reconstructed values obtained from the adaptive robust sliding mode observer, and the original heterogeneous complex dynamic high-dimensional system is converted into two low-dimensional closed-loop error systems by decoupling method. Furthermore, Lyapunov and Ricatti theories are used to obtain the controller gain solution method. Finally, a simulation experiment of fault-tolerant consensus control for heterogeneous multi-agent systems is carried out with one leading agent and four following agents, and the effectiveness and feasibility of the proposed method are verified.

A wavelet-based dynamic mode decomposition for modeling mechanical systems from partial observations

Dynamic mode decomposition (DMD) has emerged as a popular data-driven modeling approach to identifying spatio-temporal coherent structures in dynamical systems, owing to its strong relation with the Koopman operator. For dynamical systems with external forcing, the identified model should not only be suitable for a specific forcing function but should generally approximate the input–output behavior of the underlying dynamics. A novel methodology for modeling those classes of dynamical systems is proposed in the present work, using wavelets in conjunction with the input–output dynamic mode decomposition (ioDMD). The wavelet-based dynamic mode decomposition (WDMD) builds on the ioDMD framework without the restrictive assumption of full state measurements. Our non-intrusive approach constructs numerical models directly from trajectories of the full model’s inputs and outputs, without requiring the full-model operators. These trajectories are generated by running a simulation of the full model or observing the original dynamical systems’ response to inputs in an experimental framework. Hence, the present methodology is applicable for dynamical systems whose internal state vector measurements are not available. Instead, data from only a few output locations are only accessible, as often the case in practice. The present methodology’s applicability is explained by modeling the input–output response of an Euler–Bernoulli finite element beam model. The WDMD provides a linear state-space representation of the dynamical system using the response measurements and the corresponding input forcing functions. The developed state-space model can then be used to simulate the beam’s response towards different types of forcing functions. The method is further validated on a real (experimental) data set using modal analysis on a simple free–free beam, demonstrating the efficacy of the proposed methodology as an appropriate candidate for modeling practical dynamical systems despite having no access to internal state measurements and treating the full model as a black-box.