Nth-order System Research Articles

Symmetries and Invariant Solutions of Higher-Order Evolution Systems

In this paper, we investigate evolution systems in two components, characterized by higher-order spatial derivatives and the presence of two arbitrary functions. Our study begins with an analysis of a fourth-order system. We perform a detailed group classification and identify specific forms of the constitutive functions that allow the system to exhibit additional symmetries in addition to spatial and temporal translations. We extend these results to nth-order systems. Moreover, we derive invariant solutions for these systems. Finally, for each order n, we are able to find non-negative solutions.

Delayed feedback implementation of decentralized derivative-dependent control of large-scale systems with input delays and disturbed measurements

We study decentralized derivative-dependent control of large-scale nth-order systems with input delays via delayed feedback implementation. The unavailable derivatives can be approximated by finite differences giving rise to a time-delayed feedback. In the centralized case, an efficient simple linear matrix inequalities (LMIs)-based method for designing of such static output-feedback and its sampled-data implementation was recently suggested. In the present paper, we extend this design to large-scale systems in the presence of input delays and disturbed measurements. Under the assumption of the stabilizability of the system with small enough input delays and small enough interactions by a state-feedback that depends on the output and its derivatives, a delayed static output-feedback that stabilizes the system is presented by using the current and past disturbed measurements. To compensate the errors due to the input delays, we add the appropriate terms to the corresponding Lyapunov–Krasovskii functional that lead to LMIs conditions. The efficient bounds on the delays preserving that the resulting system is input-to-state stable (ISS) are found by verifying the LMIs. In addition, we employ the vector Lyapunov functional method that may allow larger couplings compared with the existing method. Finally, the effectiveness of the proposed methods is illustrated by numerical examples.

Solution of Time Independent System of First Order Ordinary Differential Equations Using Embedding Method in Fuzzy Environment

In this study, time independent system of first order fuzzy ordinary differential equations is solved using embedding method. Results for the system of FODEs are obtained and a case of non-homogeneous system of FODEs is established and used to find the Eigen values in an n x n matrix with negative real numbers R- consisting of n fuzzy numbers. Examples are constructed using the principles of existence and uniqueness. It is recommended that further study should look at the possibilities of generalizing this work to nth order system of first order ordinary differential equations using embedding method in fuzzy environment.

High-order leader following consensus for multi-agent systems via intermittent sampled position data control

This paper considers the leader-following consensus problem of multi-agent systems, where agents are described by high-order dynamics under a directed graph. An algorithm depending on a distributed control protocol with time delay and intermittent sampled position information only is proposed. The protocol makes use of past position data. We show that such protocol cannot work if not enough past position information is taken into account. For nth-order system, we take n past position data. To achieve consensus, a necessary and sufficient condition depending on communication width, time delay, and sampling period is derived. Furthermore, the control protocol can be extended to leaderless consensus problem. Numerical simulations illustrate that appropriately chosen control gains can verify the theoretical analysis.

Nonsingular Chattering-Free Barrier Function Finite Time Tracker for Perturbednth-Order Nonlinear Systems and its Application to Chaotic Color Image Scrambling

Nonsingular Chattering-Free Barrier Function Finite Time Tracker for Perturbed<i>n</i>th-Order Nonlinear Systems and its Application to Chaotic Color Image Scrambling

Stabilizing design for discrete-time reversible switched linear control systems: A deadbeat control approach

In this work, we address the feedback stabilization problem for discrete-time reversible switched linear control systems. We focus on the question “how many state feedback gain matrices are needed for an nth order system with m subsystems?” We prove that n+m is an upper bound for the question. For completely controllable systems, a constructive design scheme is developed to steer any initial state to the origin within n(n+1)2 steps. When the controllability is incomplete, we prove that the system is stabilizable iff the uncontrollable sub-mode is stabilizable, and propose a mixed time-driven and state-driven switching law to stabilize both the controllable sub-mode and the uncontrollable sub-mode simultaneously.

Robust PI-Based Non-Singular Terminal Synergetic Control for Nonlinear Systems via Hybrid Nonlinear Disturbance Observer

This paper proposes a novel robust proportional integral non-singular terminal synergetic control (PI-NTSC) for a class of affine nonlinear systems. To counteract the effects of the disturbances, a hybrid nonlinear disturbance observer (HNDOB) scheme is proposed to estimate the matched and mismatched disturbances. A novel proportional integral non-singular terminal synergetic control (PI-NTSC) is proposed for a class of affine nth-order systems. The stability of the PI-NTSC approach is proven using the Lyapunov stability theory. Its effectiveness and applicability is evaluated through simulations of a four-bar linkage mechanism and a servo-hydraulic system. The proposed HNDOB-based PI-NTSC is characterized by a finite time convergence rate, simple implementation, better disturbance rejection ability to both matched and mismatched uncertainties, and chattering free dynamics compared to conventional sliding mode control.

Oscillation and Asymptotic Behavior of Solution of nth Order System Nonlinear of Neutral Differential Equation

In this paper, authors obtained sufficient conditions to ensure that every bounded solution of nth order nonlinear neutral differential system oscillates or nonoscillatory converges to zero as t→∞, some examples were given to explain the results obtained.

Modified GESO based Control for Systems with Sinusoidal Disturbances

This work addresses the problem of handling sinusoidal disturbances by proposing a modified Generalized Extended State Observer (GESO) designated as Higher order GESO (HGESO). The HGESO simultaneously estimates the mismatched disturbances along with their derivatives and matched disturbance. This work basically analyzes and extends the work in Li et al. (2012) for the generalized nth order system where a GESO based Control (GESOBC) is proposed for handling a constant type of mismatched disturbance only. The practical stability of the overall system in HGESO is proved and the results are verified through simulation of an illustrative example.

Finite-time terminal synergetic control of a class of nonlinear systems with unmatched uncertainties

Abstract In this paper, the problem of finite-time tracking for nth-order uncertain nonlinear systems with unmatched uncertainties is addressed. Using a terminal synergetic manifold, a controller is provided to force the tracking error to the origin in finite time in the presence of unmatched uncertainties. With this method, chattering problem is completely removed without defining a new function. Lyapunov theory is used to prove the stability of the proposed method. The proposed controller provides the convergence of finite-time tracking error to zero by suitable performance that is theoretically analysed and proved through simulation.

An enhanced state convergence architecture incorporating disturbance observer for bilateral teleoperation systems

To bilaterally control an nth-order teleoperation system modeled on state space, state convergence methodology provides an elegant way to design control gains through a solution of 3 n + 1 equations. These design conditions are obtained by allowing the master–slave error to evolve as an autonomous system and then assigning the desired dynamic behavior to the slave and error systems. The controller, thus obtained, ensures the motion synchronization of master and slave systems with adjustable force reflection to the operator. Although simple to design and easy to implement, state convergence method suffers from its dependence on model parameters, and thus the performance of the controller may degrade in the presence of parametric uncertainties. To address this limitation, we propose to integrate an extended state observer in the existing state convergence architecture which will not only compensate the modeling inaccuracies by treating them as a disturbance but will also provide the estimates of the master and slave states. These estimated states are then used to construct the bilateral controller which is designed by following the method of state convergence. In this case, 2 n + 2 additional design equations are required to be solved to fix the observer gains. To validate the proposed enhancement in the state convergence architecture, simulations and semi-real-time experiments are performed in MATLAB/Simulink environment on a single degree-of-freedom teleoperation system.

Integral Non-Singular Terminal Sliding Mode Controller for nth-Order Nonlinear Systems

In this study, a new integral non-singular terminal sliding mode control method for nonlinear systems is introduced. The proposed controller is designed by defining a new sliding surface with an additional integral part. This new manifold is first introduced into the second-order system and then expanded to nth-order systems. The stability of the control system is demonstrated for both second-order and nth-order systems by using the Lyapunov stability theory. The proposed controller is applied to a robotic manipulator as a case study for second-order systems, and a servo-hydraulic system as a case study for third-order systems. The results are presented and discussed.

Model Order Reduction Based on Agglomerative Hierarchical Clustering.

This paper presents an improved method for reducing high-order dynamical system models via clustering. Agglomerative hierarchical clustering based on performance evaluation (HC-PE) is introduced for model order reduction. This method computes the reduced order denominator of the transfer function model by clustering system poles in a hierarchical dendrogram. The base layer represents an nth order system, which is used to calculate each successive layer to reduce the model order until finally reaching a second-order system. HC-PE uses a mean-squared error (MSE) in every reduced order, which modifies the pole placement process. The coefficients for the numerator of the reduced model are calculated by using the Padé approximation (PA) or alternatively a genetic algorithm (GA). Several numerical examples of reducing techniques are taken from the literature to compare with HC-PE. Two classes of results are shown in this paper. The first sets are single-input single-output models that range from simple models to 48th order systems. The second sets of experiments are with a multi-input multioutput model. We demonstrate the best performance for HC-PE through minimum MSEs compared with other methods. Furthermore, the robustness of HC-PE combined with PA or GA is confirmed by evaluating the third-order reduced model for the triple-link inverted pendulum model by adding a disturbance impulse signal and by changing model parameters. The relevant stability proofs are provided in Appendixes A and B in the supplementary material. HC-PE with PA slightly outperforms its performance with GA, but both approaches are attractive alternatives to other published methods.

Time-varying leader-following consensus of high order multi-agent systems

In this paper, a time-varying leader-following consensus of multi-agent systems under fixed, connected and undirected communication topology is presented. In the proposed method, the dynamics of each agent including the followers and their corresponding leader is a linear nth order system. Moreover, the communication topology between the leader and its neighbours depends upon bounded and time-varying functions, assumed to remain connected as time passes. To tackle this problem, a set of time-varying distributed control laws for each follower agent is designed, based on algebraic graph theory, algebraic Riccati equation and Lyapunov direct method. Simulation results indicate the effectiveness of the proposed method and display convergence to consensus is achieved in a finite time via time-varying distributed control laws.

Multidimensional output stabilization of a certain class of uncertain systems

Consideration was given to the indeterminate nth order system with l observed coordinates and l controls l < n. With the use of a backstepping-based construction of the observer and quadratic Lyapunov function, designed were continuous or pulse controls under which the system becomes globally asymptotically stable.

Deadbeat controller with two additional steps

An nth-order closed-loop hybrid system will exhibit a deadbeat response if the system output approaches the reference input in n steps. This paper studies the problem of obtaining a deadbeat response for an nth-order single-input single-output plant (process) in n + 2 steps with a minimum deadbeat controller output deviation from the steady state value. We identify the requirements for the design of a deadbeat controller with two additional steps and present worked examples for the problem. A deadbeat controller is then used to control a piezoelectric micro-robot model. The second order continuous transfer function is used to simulate the robot’s piezoelectric actuator. This research presents the advantages of a deadbeat controller with two additional steps when applied to a robot control problem.DOI: http://dx.doi.org/10.5755/j01.eie.23.5.19115

Analysis of Multidimensional Time Delay Systems Using Lambert W Function

In this note, the analysis of time delay systems (TDSs) using Lambert W function approach is reassessed. A common canonical (CC) form of time delay systems is defined. We extended the recent results of Cepeda–Gomez and Michiels (2015, “Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using the Matrix Lambert W Function,” Automatica, 53, pp. 339–345) for second-order into nth order system. The eigenvalues of a time delay system are either real or complex conjugate pairs and therefore, the whole eigenspectrum can be associated with only two real branches of the Lambert W function. A new class of time delay systems is characterized to extend the applicability of the above-said method. Moreover, this approach has been exploited to design a controller which places a subset of eigenvalues at desired locations. Stability is guaranteed by using a new algorithm developed in this paper, which is based on the Nyquist plot. The approach is validated through numerical examples.

Time-varying leader-following consensus of high order multi-agent systems

In this paper, a time-varying leader-following consensus of multi-agent systems under fixed, connected and undirected communication topology is presented. In the proposed method, the dynamics of each agent including the followers and their corresponding leader is a linear nth order system. Moreover, the communication topology between the leader and its neighbours depends upon bounded and time-varying functions, assumed to remain connected as time passes. To tackle this problem, a set of time-varying distributed control laws for each follower agent is designed, based on algebraic graph theory, algebraic Riccati equation and Lyapunov direct method. Simulation results indicate the effectiveness of the proposed method and display convergence to consensus is achieved in a finite time via time-varying distributed control laws.

Adaptive Sliding Mode Control of a 6-DOF RLED Robot Manipulator with Uncertain Parameters and External Disturbances

This paper addresses the end-effector position and orientation tracking problem of a 6-degrees of freedom (DOF) rigid link electrically driven (RLED) revolute joint serial robot manipulator with uncertain parameters and external disturbances. System uncertainties and external disturbances are bounded but their upper limits are unknown. The input matrix, which is always invertible and not necessarily a diagonal matrix, is also assumed to be uncertain with a certain bound. An adaptive sliding mode control (ASMC) is designed to solve the tracking problem given these uncertainties. Note that the designed control method can be applied to any RLED serial robot manipulator. No regression matrices are required and the uncertainty upper limits are not necessarily known. Furthermore, the controller computation time can be reduced by considering some parts of the system dynamics and the nth time-derivative reference signals of nth order system, in this case desired joint jerk signals, as unknown bounded uncertainties. The desired trajectory is defined in operational space while the designed ASMC works in joint space. The methods to convert the end-effector pose, velocity, acceleration, and jerk from operational space to joint space are also explained. The Lyapunov stability criterion is applied to prove the convergence of the adaptive gain of ASMC, the sliding surface, and system stability. Finally, simulation results verified ASMC performance.

MLP technique based reinforcement learning control of discrete pure-feedback systems

The reinforcement learning control with neural networks (NNs) is investigated for a class of pure-feedback systems in discrete time using minimal-learning-parameter (MLP) technique. To make the dynamics feasible for controller design, the nth order system is transformed into the prediction model. By selecting the “strategic” utility function including the future performance, the critic NN is designed. The action NN is employed to minimize both the strategic utility function and the tracking error. A radial basis function (RBF) NN is employed to approximate the unknown control with the MLP technique which greatly reduces the number of the online adaptive parameters. The uniformly ultimate boundedness (UUB) of the closed-loop tracking error is guaranteed. The feasibility of the proposed controller is verified by a simulation example.