First-order Nonlinear Systems Research Articles

Design of consensus and cluster consensus controller for first-order nonlinear multi-agent systems based on subgroup structure

It is well known that nonlinear multi-agent systems (MASs) with a directed topology can reach a consensus if the global coupling coefficient is strong enough. In this paper, we propose a novel distributed protocol based on the subgroup structure. Our results demonstrate that the MAS design can follow a part-to-whole approach, starting with subgroups and gradually building up to a more complex system. We firstly consider two cases for the MASs: one with a single leader and one with a leader group, where agents within the leader group can communicate with each other, but cannot receive the information of other group. We show that followers can track the leader or group leaders if intra-coupling within the subgroup and information strength received by the root node are above a certain threshold. Next, we investigate cluster consensus and practical cluster consensus for the Laplacian matrix with unequal row sum, equal row sum, and mixing of partitioned matrices. Our results reveal that row sum and community structure of the partitioned matrix are crucial in determining the final state of each agent. Our proposed control protocol is highly flexible and relies only on the cluster structure and information received by the root node, rather than global information. Moreover, our method is not limited to tracking problems, cooperative control, or multi-objective control, and can accommodate new group agents joining the system, making it highly adaptable. We provide illustrative examples to demonstrate the effectiveness of our approach.

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

Pinning Group Consensus of Multi-agent Systems Under DoS Attacks

In this paper, group consensus is investigated for a class of nonlinear multi-agent systems suffered from the DoS attacks. Firstly, a first-order nonlinear multi-agent system is constructed, which is divided into M subsystems and each subsystem has an unique leader. Then a protocol is proposed and a Lyapunov function candidate is chosen. By means of the stability theory, a sufficient criterion, which involves the duration of DoS attacks, coupling strength and control gain, is obtained for achieving group consensus in first-order system. That is, the nodes in each subsystem can track the leader of that group. Furthermore, the result is extended to nonlinear second-order multi-agent systems and the controller is also improved to obtain sufficient conditions for group consensus. Additionally, the lower bounds of the coupling strength and average interval of DoS attacks can be determined from the obtained sufficient conditions. Finally, several numerical simulations are presented to explain the effectiveness of the proposed controllers and the derived theoretical results.

SPREADING PATTERN OF INFECTIOUS DISEASES: SUSCEPTIBLE INFECTED RECOVERED MODEL WITH VACCINATION AND DRUG-RESISTANT CASES (APPLICATION ON TB DATA IN INDONESIA)

Mycobacterium tuberculosis is the causative agent of the infectious illness tuberculosis (TB). Indonesia is the world's third-highest TB burden country. TB transmission is prevented by the BCG vaccination. A directly observed treatment, short-course (DOTS) treatment approach can cure TB illness. Recurrent TB may occur due to either relapse or reinfection with drug-resistant bacteria. The goals of this article are formulating the SVITR model with relapse and drug-resistant cases, applying the model to the TB data in Indonesia, determining the model accuracy, determining the spreading pattern and interpreting the result, and simulating the parameters. Literature study and application methods are used in this research. The SVITR model with relapse and drug-resistant cases is a first-order nonlinear differential equation system. The model is applied to TB in Indonesia based on annual data from Indonesian Health Profile, World Bank, and WHO. The model is solved by the fourth-order Runge-Kutta method. The model is accurate enough to explain the spread of TB in Indonesia with a MAPE value of 15,5%. The spreading pattern of tuberculosis infection is upward from 2010 to 2050. In 2050, there are still 8.115.976 TB cases in Indonesia. Hence in 2050, Indonesia's free of TB target has yet to be achieved. Simulation is conducted by increasing BCG vaccination to 95%, reducing contact with TB patients to 5%, increasing treatment to 95%, and lowering relapses and drug-resistant cases to 0.00005%, so the Indonesia free of TB target in 2050 can be achieved from 2042.

Prescribed-Time Network-Based Deployment of Nonlinear Multiagent Systems: A Discrete-Space PDE Method.

This article attempts to design the prescribed-time time-varying deployment schemes for first-order and second-order nonlinear multiagent systems (MASs). We assume that all agents can obtain the information of their current and final relative positions with their neighbors, and the final absolute velocities (as well as their current and final relative velocities, the final absolute accelerations for the second-order MASs) through a communication network, whereas two boundary agents are able to obtain their current and final absolute positions (as well as their current and final absolute velocities for the second-order MASs). The neighbor relationship of all agents is described by a spatial variable and two static-feedback controllers are introduced, which can be expressed as a second-order space difference of the spatial variable. Then, the deployment of MASs can be transformed into the stabilization of discrete-space partial differential equation (PDE) systems. Three virtual agents are introduced to constitute the Dirchlet and Neumann boundary conditions. Several algebraic inequality criteria are derived to guarantee that the prescribed-time time-varying deployment can be achieved within a prescribed time under the Dirchlet and mixed boundary conditions. Unlike the published results, our results are derived based on the discrete-space PDE systems instead of continuous-space PDE systems, which is consistent with the discrete spatial distribution of agents. Finally, two numerical examples are given to illustrate the effectiveness of our results.

Fixed-time output consensus for nonlinear heterogeneous multi-agent systems with disturbances

The fixed-time output consensus is considered in this article for heterogeneous nonlinear first-order and second-order multi-agent systems with disturbances. Firstly, we design a distributed observer by the fixed-time consensus theory, realizing distributed control of the system under directed graph. Secondly, designing a fixed-time controller for the first-order nonlinear system with disturbances by using constant exponential and time-varying exponential coefficient methods. For the second-order nonlinear system with both matched and mismatched disturbances, designing a fixed-time tracking controller by using the backstepping method and robust sliding mode control. The designed controller effectively suppresses the mismatched disturbance of system and makes the multi-agent system achieve output consensus in a fixed time. Finally, we demonstrate the effectiveness of the protocol through numerical examples.

Leader-following cluster-delay consensus control for first-order nonlinear multi-agent systems based on event-triggered mechanism

In this paper, the event-triggered-based cluster-delay consensus control problem is investigated for leader-following nonlinear multi-agent systems (MASs). In control design, both state triggering in the agents’ network and constant time-delay in the leaders’ communication network are considered. Under the framework of Lyapunov function stability theory, the Lipschitz condition is used to overcome the influence caused by time-delay. In order to further effectively utilize data transmission resources, and reduce the communication load of the topology network between each agent, an event-triggered mechanism is established. Subsequently, a robust cluster consensus control strategy is proposed based on event-triggered mechanism, which can ensure all signals of the controlled system are bounded, and the tracking errors converge to zero. In addition, it can also effectively avoid the Zeno behavior. Finally, the effectiveness of the presented control method and theory is verified by a simulation example.

Control of transport constraints for shuttle vehicles based on barrier Lyapunov functions

With the rapid development of the warehousing and logistics industry, a stable, accurate and efficient shuttle operation system has become a hot topic of research. In this study, a constraint controller based on the Lyapunov function is designed to improve the constraint control and disturbance problems in the operation of shuttle vehicles. The performance test results show that the difference between the system output and the reference trajectory becomes progressively smaller in the first-order non-linear system, with the difference reaching zero at 2.5 s, after which the two are almost identical. The control output moves sinusoidally at [-14, 14]. In the second order non-linear system the difference between the system output and the reference trajectory is zero after 7 s. The control output at [-18, 18] moves with decreasing amplitude and is zero after 17 s. Application test results show that the velocity constraint is almost identical to the reference velocity, the trajectory stopping accuracy is above 98.0% and the immunity to interference meets the limit and fluctuates above and below 37. It shows that the method used in the study can control the speed, trajectory and external disturbance of the shuttle car with reasonable constraints and improve the operation efficiency and safety of the shuttle car. This study can provide theoretical support for improving the efficiency and safety of shuttle vehicles, and promote the development of the warehousing and logistics industry.

A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

In this paper, we show how to construct a positive semi-discrete Lagrangian–Eulerian scheme on triangular grids that asymptotically satisfies multidimensional scalar hyperbolic problems and first-order nonlinear systems of conservation laws. The construction of such numerical scheme is based on the improved concept of space–time no-flow curves. For scalar equations, through a sequence of computational grid functions that tend to satisfy the underlying hyperbolic conservation law(s) in a weak sense in the space variable and in a strong sense in the time variable, we prove that the proposed scheme converges to the unique weak solution satisfying a type of Kruzhkov entropy condition. For nonlinear systems of first-order hyperbolic conservation laws, we show that the proposed scheme also satisfies a (weak) positivity principle. This provides an effective numerical analysis tool (based on weak asymptotic methods) to construct a family of continuous functions that asymptotically satisfies scalar conservation laws with discontinuous nonlinearity and systems that have irregular solutions. In order to assess the robustness of the proposed approach, we solve a series of numerical tests on several hyperbolic problems (scalar and systems) using the positive semi-discrete Lagrangian–Eulerian scheme on triangular grids. The results confirm that the proposed method is efficient (since it is Riemann-solver-free) and provides good resolution for scalar models and for the solution of systems of conservation laws.

The fixed-time consensus of leader-following nonlinear multi-agent systems under event-triggered impulsive control

This paper studies the leader-following fixed-time consensus problem for a class of first-order nonlinear multi-agent systems via event-triggered impulsive control where the communication topologies are general undirected. For each agent, the controller is updated only when some state-dependent errors exceed a tolerable bound, and the impulsive control time sequence is produced by the event triggered mechanism. This original control protocol can substantially improve the convergence speed, obtain the lower energy consumption and reduce the update times of the controller. Also, it is shown that continuous communication of neighboring agents can be avoided and there is no Zeno-behavior. The results are verified by a numerical simulation.

NARMA-L2–based online computed torque control for robotic manipulators

Computed torque is an effective method for control of robotic manipulators. In this paper, a novel approach is proposed for computed torque control method, and also, it is integrated with online least squares support vector regression algorithm. The motive is to improve control performance and to build an adaptive architecture where no prior information on system model is required; the model is obtained continuously using a machine learning technique. There are two main contributions of the proposed work: the first one is the formulation of the computed torque control law using the nonlinear autoregressive moving average-L2 control methodology and the second contribution is the implementation of online least squares support vector regression method in estimating the dynamical model of the system and calculating the control law. The performance of the proposed method has been evaluated by simulations on a two-link robotic arm. The robustness of the method has also been evaluated for cases with disturbance and model uncertainty. Comparison with an existing computed torque control approach in the literature demonstrates the superiority of the proposed approach. Thus, it can be concluded that the method can be effectively used for first-order and second-order nonlinear systems without inherent instability.

Practical torque sensorless super-twisting control of manipulators based on a novel integral non-singular fast terminal sliding mode with fixed-time convergence

ABSTRACT In this paper, a robust controller has been proposed for first-order uncertain non-linear systems by introducing a novel hybrid sliding surface. Using this method, contrary to previous proposed sliding surfaces, the state trajectory begins moving on the correct sliding mode and the reaching phase can be reduced to the minimum time. Thus, the transient response speed of the system and consequently the efficiency and robustness of the system increase. Moreover, the convergence of the system state in bounded time, independent of the initial conditions of the system, is guaranteed without singularity problems. Then, for torque sensorless control of permanent-magnet DC motors, as one of the most prevalent joint actuators in robotics, an adaptive super-twisting controller, based on the proposed hybrid sliding surface, has been presented. Using this method, continuous finite-time control of the DC motor’s torque without a torque sensor is realized. The DC motor load-side torque is estimated based on an adaptive wavelet neural network and it is compensated in the control loop. In addition to the stability proof and fixed-time convergence of control schemes, a set of simulation and experimental studies have been provided to evaluate the efficiency of control algorithms and investigate their feasibility.

Data-Driven Learning Extended State Observers for Nonlinear Systems: Design, Analysis and Hardware-in-Loop Simulations

This paper considers the disturbance/uncertainty estimation of first-order nonlinear system subject to fully unknown internal dynamic, external disturbance, and unknown control input gain. Compared with existing extended state observer (ESO) where priori knowledge of model parameter such as nominal input gain should be known as a priori, reducedand full-order data-driven learning ESOs are developed for estimating the lumped disturbance and control input gain. A salient feature of the proposed data-driven learning ESOs is that the unknown input gain and lumped disturbance can be estimated synchronously, in the meantime, the estimation convergence is guaranteed benefiting from the data-driven approach. Hardware-in-loop simulation is carried out to substantiate the performance of the proposed data-driven learning ESO for surge speed tracking of a robotic marine vehicle without knowing the model parameter in advance.

A novel space/time integration technology via altogether different space and time stepping methods for nonlinear first-order systems

PurposeThe purpose of this paper is to describe a novel implementation of a multispatial method, multitime-scheme subdomain differential algebraic equation (DAE) framework allowing a mix of different space discretization methods and different time schemes by a robust generalized single step single solve (GS4) family of linear multistep (LMS) algorithms on a single body analysis for the first-order nonlinear transient systems.Design/methodology/approachThis proposed method allows the coupling of different numerical methods, such as the finite element method and particle methods, and different implicit and/or explicit algorithms in each subdomain into a single analysis with the GS4 framework. The DAE, which constrains both space and time in multi-subdomain analysis, combined with the GS4 framework ensures the second-order time accuracy in all primary variables and Lagrange multiplier. With the appropriate GS4 parameters, the algorithmic temperature rate variable shift can be matched for all time steps using the DAE. The proposed method is used to solve various combinations of spatial methods and time schemes between subdomains in a single analysis of nonlinear first-order system problems.FindingsThe proposed method is capable of coupling different spatial methods for multiple subdomains and different implicit/explicit time integration schemes in the GS4 framework while sustaining second-order time accuracy.Originality/valueTraditional approaches do not permit such robust and flexible coupling features. The proposed framework encompasses most of the LMS methods that are second-order time accurate and unconditionally stable.

Predicting greenhouse daily light integral using photosynthetic photon flux density and finite difference method

In the state-of-the-art agricultural technology using greenhouses, it is useful to estimate a dynamic model of internal environmental changes according to external environmental changes. In this article, we propose a method to predict the change in the daily light integral inside the greenhouse according to the change in the daily light integral outside the greenhouse. Since daily light integral corresponds to the integral value of photosynthetic photon flux density, it is necessary to obtain a model of change in photosynthetic photon flux density inside the greenhouse according to the change in photosynthetic photon flux density outside the greenhouse to improve the prediction accuracy. The greenhouse was considered a nonlinear first-order system with a time lag for the photosynthetic photon flux density, and the system parameters were identified using the recursive least-squares method. Here, the photosynthetic photon flux density data measured outside the greenhouse is used as the input to the system, and the photosynthetic photon flux density data measured inside the greenhouse is used as the output of the system. Next, we obtain the daily light integral prediction model. The daily light integral prediction model is estimated from the photosynthetic photon flux density model using the finite difference method. The performance of the proposed method has been evaluated through extensive experimental studies. In Korea, since the weather changes in summer and winter are great, it must compare the daily light integral prediction performance in summer and winter to more accurately verify the proposed method. In this study, the experimental results are presented, respectively, for summer in August 2021 and winter in January 2022. The error performance of the human empirical prediction method and the proposed data-driven prediction method are compared for two seasons, and the robustness of the rapid weather changes of the proposed method is discussed.

An Improved Distributed Nonlinear Observers for Leader-Following Consensus via Differential Geometry Approach

This article is concerned with the leader-following output consensus problem in the framework of distributed nonlinear observers. Instead of certain hypotheses on the leader system, a group of geometric conditions is put forward to develop a novel distributed observers strategy, thereby definitely improving the applicability of the existing results. To be more specific, the improved distributed observers can precisely handle consensus problems for some nonlinear leader systems, which are invalid for the traditional strategies with a certain assumption, such as elastic shaft single linkage manipulator (ESSLM) systems and most of the first-order nonlinear systems. We prove the sufficient conditions for the exponential stability of our distributed observers’ error dynamic by proposing two pioneered lemmas to show the relationship between the maximum eigenvalues of two matrices appearing in Lyapunov type matrices. Then, a partial feedback linearization method with zero dynamic proposed in differential geometry is employed to design the purely decentralized control law for the affine nonlinear multiagent system. With this advancement, the existing results can be regarded as a specific case owing to that the followers can be chosen as an arbitrary minimum phase affine smooth nonlinear system. At last, the novel distributed observers and the improved purely decentralized control law are applied in the distributed control framework to construct a closed-loop system. We also prove the stability of the closed-loop system to achieve leader-following consensus. Our method is illustrated by the ESSLM system and Van der Pol system as leaders.

Towards a Taylor-Carleman bilinearization approach for the design of nonlinear state-feedback controllers

The Carleman bilinearization is an approach that performs an exact conversion of a finite-dimensional nonlinear system into an infinite-dimensional bilinear system. A finite-dimensional system is later obtained through a truncation for analysis and control purposes. This paper investigates the linear matrix inequality (LMI)-based design of a switched state-feedback control law for the model obtained via Carleman bilinearization of a first-order nonlinear system. It is shown that in order to obtain feasible design conditions, the performance requirements must be relaxed in a neighborhood of the zero equilibrium point, so that problems arising from the uncontrollability of the linear part of the model can be avoided. The effectiveness of the proposed approach is shown using a numerical example and experimental results using a multi-input tank system.

Adaptive backstepping for distributed optimization

This paper presents an adaptive backstepping approach to distributed optimization for a class of nonlinear multi-agent systems with each agent represented by the parametric strict-feedback form. In particular, this paper does not assume known gradient functions of the local objective functions, and uses the measured gradient values depending on the agents’ real-time outputs instead. A stepwise method is presented to derive novel distributed adaptive optimization algorithms that steer the outputs of all the agents to the optimal solution of the total objective function. First, a novel distributed adaptive optimization algorithm is developed for first-order nonlinear uncertain multi-agent systems, supported by stability analysis and convergence proofs using Lyapunov arguments. Second, by means of Lyapunov arguments in the spirit of backstepping, a distributed adaptive optimization algorithm is presented for high-order strict-feedback systems with parametric uncertainty. Interesting extensions of the main result to practically important classes of systems with unknown virtual control coefficients, output feedback, and relative-measurement feedback are also discussed.

Distributed Fast Finite-Time Tracking Consensus of Multi-Agent Systems With a Dynamic Leader

Distributed fast finite-time tracking consensus (FTTC) is investigated for first-order nonlinear multi-agent systems (MASs), where the leader has an external input unknown to all followers. Based on global finite-time stability and Lyapunov stability, two distributed tracking protocols are proposed to guarantee fast FTTC. First, a basic distributed non-smooth protocol for solving this problem is proposed and analyzed. A limitation of the designed protocol is that the control gains should depend on some global information like the spectrum of Laplacian matrix. To remove this limitation, a fully distributed protocol is further designed and analyzed. Theoretical studies imply that the states of the considered MASs will also achieve fast FTTC. Finally, two simulation examples are performed on Chua’s circuit system to verify the theoretical results.