Global Exponential Stability Research Articles

A GES joint position trajectory tracking smooth controller of torque‐driven robot manipulators affected by disturbances

SummaryThis paper presents a controller for joint position tracking of torque‐driven robot manipulators affected by torque disturbances. In particular, the proposed approach allows concluding global exponential stability (GES) of the state‐space origin of the closed‐loop system when constant disturbances affect the robot dynamics. Besides, when time‐varying disturbances are presented, the trajectories of the closed‐loop system are proven to be bounded as result of the input‐to‐state stability. The main contribution is the design of the control law and a nonlinear observer based on an alternative energy shaping approach. The nonlinear observer is designed to compute and compensate for unknown disturbances, whether they are constant or time‐varying. Furthermore, a detailed stability analysis of the closed‐loop system based on Lyapunov theory and input‐to‐state stability is presented. As far as the authors know, this is the first energy‐shaping controller for trajectory tracking control of robot manipulators affected by constant disturbances that achieves global exponential stability. Real‐time experiments on a manipulator arm of two degrees‐of‐freedom illustrate the performance of the proposed controller.

An Exponential Stabilization Analysis for Switched Interval Type-2 Fuzzy Sampled-Data Control Systems

This work is focused on designing the admissible edge-dependent average dwell time (AE-DADT) based fuzzy sampled-data control problem for a class of switched nonlinear systems. The upper and lower grade of membership functions is used to characterize uncertain parameter-dependent nonlinear systems in the interval type-2 fuzzy (IT2F) model. Unlike from the existing literature, a new fuzzy dependent refined looped-Lyapunov functional (LF) is introduced which involves the new exponent terms and state information such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{e^{2\gamma _{v}(\varrho -\varrho _{\mathscr {k}})}-1}{2\gamma _{v}}, \ \frac{e^{2\gamma _{v}(\varrho _{\mathscr {k}+1}-\varrho)}-1}{2\gamma _{v}}, \ \mathscr {z}(\varrho _{\mathscr {k}})$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {z}(\varrho _{\mathscr {k}+1})$</tex-math></inline-formula> with tuning parameters and few slack variables. The AE-DADT switching property has also been taken into account for nonlinear systems, and it contributes to getting more information about previous modes. Based on the proposed new refined looped-LFs, new sufficient conditions are derived in terms of linear matrix inequalities (LMI), which ensure the global uniform exponential stability (GUES) of the nominated switched IT2F systems. The feasibility of the obtained results is verified by the numerical example, and the tunnel diode circuit system model and Lorenz chaotic system are established to illustrate the feasibility and efficacy of the theoretical findings.

On Contraction Analysis of Switched Systems With Mixed Contracting-Noncontracting Modes via Mode-Dependent Average Dwell Time

This paper studies contraction analysis of switched systems that are composed of a mixture of contracting and noncontracting modes. The first result pertains to the equivalence of the contraction of a switched system and the uniform global exponential stability of its variational system. Based on this equivalence property, sufficient conditions for a mode-dependent average dwell/leave-time based switching law to be contractive are established. Correspondingly, LMI conditions are derived that allow for numerical validation of contraction property of nonlinear switched systems, which include those with all non-contracting modes.

Global classical solutions to equatorial shallow-water equations

In this article, we study the equatorial shallow-water equations with slip boundary condition in bounded domain. By exploring the dissipative structures of the system, we obtaining a priori estimates of the solution for small initial data. Then the existence of classical global solutions and exponential stability results are given.&#x0D; For more inofrmation see https://ejde.math.txstate.edu/Volumes/2023/62/abstr.html

Global exponential stability for delayed Clifford-valued coupled neural networks with impulsive effects

In the paper, the global exponential stability for impulsive delayed Clifford-valued coupled neural networks is studied. Firstly, a delayed Clifford-valued coupled neural network model is established. So as to avoid the non-commutativity issue of Clifford numbers multiplication, the original n-dimensional Clifford-valued models are decomposed into 2mn-dimensional real-valued models. Then, by using Lyapunov-Krasovskii (L-K) functional approach and the technique about linear matrix inequality, some new sufficient conditions are presented for the global exponential stability of the considered neural network models. Finally, the effectiveness of the results are verified by two numerical simulations.

Stability analysis of optimal control problems with time-dependent costs

We present stability conditions for deterministic time-varying nonlinear discrete-time systems whose inputs aim to minimize an infinite-horizon time-dependent cost. Global asymptotic and exponential stability properties for general attractors are established. This work covers and generalizes the related results on discounted optimal control problems to more general systems and cost functions.

Lyapunov conditions for exponential stability of nonlinear delay systems via impulsive control involving stabilizing delays

The problem of global exponential stability (GES) for nonlinear delay impulsive systems is investigated in this paper. By extending the traditional comparison principle, delay effects on continuous and discrete dynamics of the system are estimated, based on which, the internal relationship between delays, parameters of impulsive control, and continuous dynamics of the system is revealed. Then some sufficient criteria are obtained for GES, which quantitatively shows the beneficial influences of delays in impulses on the system performance. Finally, two numerical examples are given to illustrate the effectiveness of the proposed result.

Random periodic oscillations and global mean-square exponential stability of discrete-space and discrete-time stochastic competitive neural networks with Dirichlet boundary condition

The current article explores the affects of space-time discrete stochastic competitive neural networks. In line with a discrete-space and discrete-time constant variation formula, boundedness and stability are addressed to the space-time discrete stochastic competitive neural networks. Notably, the best convergence speed can be computed by a non-linear optimization problem. In the end, random periodic sequences with respect to time variable of the discrete-space and discrete-time stochastic competitive neural networks are discussed. The results indicate that spatial diffusion with non-negative density factors has no effect on the global mean square boundedness and stability and random periodicity of the network model. The current article is precursory in consideration of space-time discrete competitive neural networks.

Asymptotically almost periodic synchronization in fuzzy competitive neural networks with Caputo-Fabrizio operator

This paper researches a class of Caputo-Fabrizio fractional-order fuzzy competitive neural networks (CF-FCNNs) with different time scales and time delays. Firstly, according to Banach contracting mapping principle and some appropriate assumptions, it derives CF-FCNNs possess a unique weak solution. Then, exponential Euler difference scheme of discrete-time CF-FCNNs (DCF-FCNNs) is presented by the constant variation methods and the difference format is implicit Euler difference and is solved by fractional PECE algorithm. Next, the existence of a unique bounded asymptotically almost periodic sequence solution, global exponential stability and synchronization of DCF-FCNNs are considered. In the end, numerical examples are given to describe the effectiveness of the acquired conclusions.

Event-Triggered Control for a Second Order ODE-Heat System Coupling at Intermediate Point

Motivated by the thermoelastic coupling effect arising in microelectromechanical systems (MEMS), event-triggered control of second-order ODE-heat systems coupled at intermediate point is investigated. The event-triggered control includes two parts, one is the feedback control signal, and the other is the event-triggered mechanism that decides when to update the control input. First, we design an event-triggered controller by using Zero-Order Hold to a continuous-time controller. Then, a dynamic triggering condition is established. Next, we derive that the existence of a minimal dwell-time, which avoids the occurrence of Zeno behavior. By utilizing Lyapunov-Krasovskii functional method, the global exponential stability of the closed-loop system is demonstrated. Finally, the simulation data based on the thermoelastic coupling system is displayed to demonstrate the availability of the theoretical results.

On the Existence of Invariantly Differentiable Functionals for Globally Exponentially Stable Time-Delay Systems

This note argues over the existence of invariantly differentiable functionals for globally exponentially stable time-delay systems. We show that the existence of invariantly differentiable, Lipschitz on bounded sets Lyapunov-Krasovskii functional is necessary and sufficient condition for the global exponential stability of time-invariant nonlinear systems described by retarded functional differential equations with discrete type delay. The presented converse results are in terms of the invariant derivative definition. To prove converse results, both point-wise and history-wise dissipation conditions are provided. The validity of the results is illustrated by an example.

Growth conditions for global exponential stability and exp-ISS of time-delay systems under point-wise dissipation

Growth conditions for global exponential stability and exp-ISS of time-delay systems under point-wise dissipation

Fuzzy-aided finite-time frequency controller of renewable-integrated power systems with hydrogen energy storage

Reliable frequency regulation of power systems integrated with low-inertia variable renewable energy generations (REGs) is crucial in islanded mode operation. It demands a resilient control framework to uphold grid stability/security against undeniable physical disturbances/cyber-attacks. This work maidenly develops and applies a fuzzy disturbance observer (FDO)-based resilient frequency controller to advance the dynamic stability of REGs-integrated large-scale power plants. Further, to enhance the damping of frequency/power oscillations, hydrogen energy storage (HES) has been incorporated, and its effectiveness is analyzed. The estimated disturbance using FDO is augmented into secondary control law derived from a non-singular terminal sliding mode strategy to increase active disturbance rejection proficiency of the overall closed-loop system. The global exponential finite-time stability of the testbed under the developed control framework has been rigorously reviewed using the Lyapunov theory. The qualitative assessment of simulation outcomes has confirmed the proficiency and resiliency of the developed control framework regarding superior frequency regulation against unknown disturbance inputs. The practicability of the developed control framework has been further quantified on IEEE 68-bus large-scale testbed, and a comparative study is presented following unknown/uncertain disturbances. Lastly, the cyber-tolerance of the applied resilient controller has been established against detectable DoS cyber-attack.

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).

Weighted Pseudo-θ-Almost Periodic Sequence and Finite-Time Guaranteed Cost Control for Discrete-Space and Discrete-Time Stochastic Genetic Regulatory Networks with Time Delays

This paper considers the dual hybrid effects of discrete-time stochastic genetic regulatory networks and discrete-space stochastic genetic regulatory networks in difference formats of exponential Euler difference and second-order central finite difference. The existence of a unique-weight pseudo-θ-almost periodic sequence solution for discrete-time and discrete-space stochastic genetic regulatory networks on the basis of discrete constant variation formulation is discussed, as well as the theory of semi-flow and metric dynamical systems. Furthermore, a finite-time guaranteed cost controller is constructed to reach global exponential stability of these discrete networks via establishing a framework of drive, response, and error networks. The results indicate that spatial diffusions of non-negative dense coefficients have no influence on the global existence of the unique weighted pseudo-θ-almost periodic sequence solution of the networks. The present study is a basic work in the consideration of discrete spatial diffusion in stochastic genetic regulatory networks and serves as a foundation for further study.

Global mean-square exponential stability and random periodicity of discrete-time stochastic inertial neural networks with discrete spatial diffusions and Dirichlet boundary condition

Global mean-square exponential stability and random periodicity of discrete-time stochastic inertial neural networks with discrete spatial diffusions and Dirichlet boundary condition

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

Earthquake Control: An Emerging Application for Robust Control. Theory and Experimental Tests

This article addresses the possibility of using robust control theory for preventing earthquakes through fluid injections in the Earth’s crust. The designed robust controllers drive aseismically a fault system to a new equilibrium point of lower energy by tracking a slow reference signal. The control design is based on a reduced-order nonlinear model able to reproduce earthquake-like instabilities. Uncertainties related to the frictional and mechanical properties of the underlying physical process and external perturbations are considered. Two types of controllers are derived. The first one is based on the sliding-mode theory and leads to local finite-time convergence of the tracking error and rejection of Lipschitz w.r.t. time perturbations. The second controller is based on linear quadratic regulator (LQR) control and presents global exponential stability of the tracking error and rejection of Lipschitz w.r.t. state perturbations. Both the controllers generate a continuous control signal, attenuating the chattering effect in the case of the sliding-mode algorithms. The developed controllers are tested extensively and compared on the basis of numerical simulations and experiments in the laboratory. The present work opens new perspectives for the application of the robust nonlinear control theory to complex geosystems, earthquakes, and sustainable energy production.

Predictor‐based switching control for networked systems

SummaryThis paper concerns a networked predictive control (NPC) for linear systems with data time‐varying delays, packet losses and disordering. By utilizing the received time‐delayed state measurement and control input, a predictive algorithm is firstly proposed for approximating the future state prediction. Afterwards, a probability dependent switching control law is proposed. Although the large data transmission delays emerge in the forward and backward channels, the packet disordering in the backward channel can be naturally excluded thus the actuator always receives valid control command during each sampling period with correct time sequence. The proposed NPC not only guarantees the global uniform exponential stability of the overall networked system but also brings merits in tolerating locally unstable sub‐systems. Numerical examples are provided to demonstrate the effectiveness and improvements of the proposed method.

Weighted pseudo [formula omitted]-almost periodic sequence solution and guaranteed cost control for discrete-time and discrete-space stochastic inertial neural networks

This article considers the dual hybrid influences of both discrete spatial diffusions and discrete time in a stochastic inertial neural networks via the methods of exponential Euler difference and second order central finite difference. Based on a non-decomposed constant variation formula, and the theories of semi-flow and metric dynamical systems, the existence of a unique weighted pseudo θ-almost periodic sequence solution is addressed to the discrete space and time stochastic inertial neural networks. Further, a guaranteed cost controller is designed to complete a global exponential stabilization for this discrete networks by establishing a framework of drive, response and error networks. Meanwhile, global exponential stability in the sense of mean square is achieved as well. This discussion is pioneering in considering discrete spatial diffusions in inertial neural networks and offers the studying bases for future researches.