Global Stability Of System Research Articles

Global output feedback stabilization of stochastic uncertain nonlinear systems with large constant delays

This paper studies the problem of output feedback stabilization for a class of stochastic uncertain nonlinear systems with large constant delays in state and input. For stochastic nonlinear systems, the maximal open sector Δ of output function is given. As long as output function belongs to any closed sector included in Δ, by constructing a reduced-order observer and C 2 Lyapunov–Krasovskii functional and using the homogeneous domination method, an output feedback controller can be developed to guarantee the closed-loop system globally asymptotically stable in probability.

Path planning and tracking control of orchard wheel mower based on BL-ACO and GO-SMC

This research proposes an improved ant colony algorithm (BL-ACO) path planning algorithm and a tracking controller based on global optimal sliding mode variable structure control (GO-SMC) for the problem of path planning and tracking control of lawn mowers in quadrilateral orchard environments. The novelty of this research lies in two aspects. On one hand, we analyze the operating scenarios of lawn mowers in standardized orchards, then transform the path planning problem into a traveling salesman problem, and mathematically model the U-shaped and T-shaped turning strategies based on the characteristics of the wheeled lawn mower. In order to make the ant colony algorithm suitable for orchard operation path optimization problems, we modified its pheromone update rules, heuristic functions, state transition probabilities, and other equations. In order to accelerate the convergence speed of the ant colony algorithm, we use the bilayer ant colony algorithm optimization strategy. On the other hand, we establish a kinematic model with the wheeled lawn mower as the control object, and design a control law using a hyperbolic tangent function to ensure the global stability of the trajectory tracking control system. Furthermore, we demonstrate through Lyapunov stability analysis that the GO-SMC controller can ensure the mower tracks the reference path accurately. The simulation experiments of path planning and tracking control show that BL-ACO and GO-SMC perform the best compared to similar algorithms. Field experiments shows that BL-ACO & GO-SMC, with a time reduction rate of 47.58 % and a fuel consumption rate reduction of 47.59 % compared to line by line & SMC.

Global exponential stability of nonlinear time-delay system under impulsive control with distributed delay

This paper investigates the global exponential stability of nonlinear time-delay system (NTDS) under impulsive control with distributed delay. By establishing a connection between the distributed delay in impulsive control and the number of impulsive times, a novel Razumikhin-type impulsive inequality is formulated. Using this inequality and the concept of the average impulsive interval, some delay-independent sufficient conditions of the global exponential stability of NTDS are derived. This indicates that the upper bounds of the impulsive interval and state time delay are no longer constrained, and the mutual constraint between the distributed delay in impulse and state time delay is removed, which greatly improves the results of recent studies. Finally, two numerical examples are presented to showcase the validity and practicality of the results.

Global Stability of Fractional Nonlinear Differential Systems With State-Dependent Delayed Impulses.

The fractional-order (FO) nonlinear differential system with state-dependent (SD) delayed impulses (DI) is considered in this brief. The considered impulses are related to the delayed state of the system and the delays are SD. A novel lemma for the monotonicity of the solution of Caputo's FO derivative equation is given. By means of linear matrix inequality (LMI) and several comparative arguments, criteria of uniform stability, uniform asymptotical stability, and Mittag-Leffler stability are obtained. Compared with other works on integer-order (IO) impulsive delayed systems with SD delays or fixed delays, how to impose constraints on parameters and impulses is explored, without imposing the boundedness on the state delays. Two examples are implemented to examine the practicality and sharpness of our theoretical analysis.

Adaptive control of nonlinear time-varying systems with unknown parameters and model uncertainties

This paper investigates the adaptive control problem for nonlinear time-varying systems with unknown parameters and model uncertainties. A novel class of switching functions is designed, and its construction method is detailed, along with a proof of the continuity of its n−1 order derivatives. Two simple examples are provided to illustrate how the proposed congelation of variables method handles unknown high-frequency time-varying parameters in both the feedback and input paths. A new neural network control scheme is then developed, integrating an adaptive neural network controller with a robust controller. The smooth transition between these two controllers is ensured by the novel switching function, which guarantees global system stability. Furthermore, by combining the congelation of variables method with adaptive backstepping, a new adaptive tracking control scheme is proposed. This scheme effectively handles unknown high-frequency time-varying parameters while achieving asymptotic tracking of arbitrary reference signals. Simulation results show that the proposed novel adaptive control method delivers superior control accuracy while reducing energy consumption: it achieves an order of magnitude improvement over the traditional adaptive robust control method and two orders of magnitude improvement over the conventional sliding mode control method.

Adaptive robust integrated guidance and control for thrust-vector-controlled aircraft by solving LQR online

In order to enhance the matching relationship between guidance subsystem and control subsystem of the thrust-vector-controlled aircraft, a kind of sufficient modeling and adaptive robust design facing to integrated guidance and control (IGC) is proposed. With respect to the researched aircraft, a linear state-dependent mathematical model facing to IGC design is first established. Based on the as-built model, a new kind of adaptive robust IGC law is proposed and it is composed by an adaptive optimal IGC law and a robustness-improved IGC law. In order to guarantee the global stability of time-varying control system, the adaptive optimal IGC law is designed by solving Riccati matrix equation on line and using matrix Sign function method. Furthermore, in order to enhance the robust ability against the unmatched system uncertainties, the robustness-improved IGC law is designed by using dynamic surface control approach and extended state observing strategy. Simulation results present that, the proposed IGC scheme presents more performance advantages compared with traditional IGC schemes, including the improvement of guidance precision and attitude stabilization. Furthermore, in the conditions of 256 simulation combinations, the minimum relative distances between aircraft position and target position are distributed from 0.103m to 5.333m, and the average value is 2.208m, which means the proposed IGC scheme possesses strong robustness against different and time-varying model uncertainties.

Improving extensions for the global uniform asymptotic stability of continuous-time nonlinear systems

This paper considers the classical problems of global asymptotic stability (GAS) of nonlinear autonomous (time-invariant) systems and global uniform asymptotic stability (GUAS) of nonlinear non-autonomous (time-varying) systems. By imposing new conditions on the Lyapunov function and its derivative, the state convergence to the origin from any initial condition with bounded norm in the classical setting is extended to the initial conditions with unbounded norms in the present work. The proposed new notions of stability are called extended global asymptotic stability (EGAS) and extended global uniform asymptotic stability (EGUAS) in this paper. The proposed Lyapunov-based criteria yield the global asymptotic stability of the origin and fixed-time/predefined-time attractiveness of any selected neighbourhood of the origin. The result is that the dynamical behaviour of the state trajectories from the convergence time point of view is improved considerably. The relationship between the smoothness and input-to-state stability (ISS) properties is also studied from a quantitative point of view. Under some new sufficient conditions, it is shown that dynamical systems with smaller Lipschitz constants represent smaller ultimate bounds, and as a result, are more robust. The desirable dynamical behaviour and robustness property of the proposed stability notion makes it comparable with the finite/fixed/predefined/prescribed (exact) stable systems that are naturally non-Lipschitz at the origin. Numerical results illustrate the effectiveness of the proposed theoretical results.

Controllability and Optimal Fast Operation of Nonlinear Systems

A new method for solving the problem of controllability and optimal transient behavior of nonlinear systems subject to boundary conditions and constraints on control values was proposed. Unlike existing methods, this new approach is based on constructing a general solution of the integral equation for a linear controlled system, followed by transforming the original problem into a special initial optimal control problem. We propose a new method for studying the global asymptotic stability of dynamical systems with a cylindrical phase space with a countable equilibrium position based on a non-singular transformation of the equation of motion and estimation of improper integrals along the solution of the system. Conditions for global asymptotic stability were obtained without involving any periodic Lyapunov function, as well as the frequency theorem. The effectiveness of the proposed method is shown with an example.

A reliable Bayesian regularization neural network approach to solve the global stability of infectious disease model

The purpose of this study is to perform the numerical results of the global stability of infectious disease mathematical model by using the stochastic computing scheme. The design of proposed solver is presented by one of the efficient and reliable schemes named as Bayesian regularization neural network (BRNN). The global stability of infectious disease mathematical nonlinear model is categorized into susceptible, infected, recovered and vaccinated. The construction of dataset is performed through the Runge-Kutta scheme in order to lessen the mean square error (MSE) by dividing the statics as training 74 %, while 13 % for both testing and endorsement. The proposed stochastic process contains log-sigmoid merit function, twenty neurons and optimization through RBNN for the numerical solutions of the global stability of infectious disease mathematical system. The best training values for each model's case are performed around 10–11. The scheme's correctness is performed by the matching of the results and the minor calculated absolute error performances. Moreover, the regression, state transmission, error histogram and MSE indicate the trustworthiness of the designed solver.

Simultaneous compensation of input delay and state/input quantization for linear systems via switched predictor feedback

We develop a switched predictor-feedback law, which achieves global asymptotic stabilization of linear systems with input delay and with the plant and actuator states available only in (almost) quantized form. The control design relies on a quantized version of the nominal predictor-feedback law for linear systems, in which quantized measurements of the plant and actuator states enter the predictor state formula. A switching strategy is constructed to dynamically adjust the tunable parameter of the quantizer (in a piecewise constant manner), in order to initially increase the range and subsequently decrease the error of the quantizers. The key element in the proof of global asymptotic stability in the supremum norm of the actuator state is derivation of solutions’ estimates combining a backstepping transformation with small-gain and input-to-state stability arguments, for addressing the error due to quantization. We extend this result to the input quantization case and illustrate our theory with a numerical example.

Self-Triggered Impulsive Control for Lyapunov Stability of Nonlinear Systems in Discrete Time.

This article studies Lyapunov stability for nonlinear systems based on discrete-time self-triggered impulsive control (STIC). With the help of comparison approach, some Lyapunov-based sufficient conditions guaranteeing nonZeno behavior and global asymptotic stability of nonlinear systems under STIC are derived. Different from the existing self-triggered mechanisms using implicit expressions to compute the next impulse instant, which may result in computational burden, we propose a novel discrete-time self-triggered mechanism (STM) by which the next impulse instant can be predicted directly by the information of the comparison system. Moreover, the resulting algorithm is easy to be implemented. It is shown that the designed STIC strategy can not only achieve a tradeoff between computational complexity, communication resource usage and control performance, but also has strong robustness and flexibility. Finally, an illustrative simulation is provided showing the effectiveness of the results.

Adaptive Multi-Switching-Based Global Tracking Control for Switched Nonlinear Systems With Prescribed Performance

This paper is concerned with the tracking control for a class of switched nonlinear systems subject to prescribed performance. Firstly, a barrier function and a normalized function are introduced to achieve prescribed performance control, which ensures that the tracking error evolves within prescribed boundary. Then, multi-dimensional Taylor network provides a new approach for how to estimate the nonlinearity arising from the backstepping control process. Significantly, the proposed multi-switching-based adaptive controller realizes that all signals in the closed-loop system are globally uniformly ultimately bounded while ensuring asymptotic tracking. Different from most existing network-approximation-based control strategies, the developed method in this paper is not only independent of the initial state, but also can achieve global stability. Finally, it is easy to verify the effectiveness of the proposed control method through two simulations. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This research is motivated by the fact that the control ideas of many practical engineering systems can be provided by making a profound study on switched nonlinear systems. However, most of the existing results focused on semi-global stability of systems. Therefore, this study devotes to develop a novel adaptive prescribed performance control strategy, which can not only ensure global stability of closed-loop system but also realize the asymptotic tracking of the switched nonlinear systems. It is worth noting that the proposed control approach has great significance for many practical systems, such as circuit systems and single-link inverted pendulum systems.

Event‐triggered stabilization for systems with an unknown control direction

AbstractIn this article, we present an event‐triggered control law for systems with an unknown control direction, where the adaptation dynamics of the Nussbaum gain are sampled. Specifically, we employ the emulation method and introduce a specific set of sampling errors. This approach enables the design of an event‐triggered law without encountering Zeno behavior. Additionally, we introduce a lemma to address the non‐integrability issue of the Nussbaum gain dynamics during the event‐triggered law design. We apply this design philosophy to co‐design both the continuous‐time controller and the event‐triggered law, aiming to achieve the global stabilization of systems with an unknown control direction. Finally, we validate the theoretical results through a numerical example.

An improved algebraic approach to proving global stability of autonomous polynomial differential systems with applications to epidemic models

Global stability of autonomous polynomial differential systems is an important issue in applications. By means of the Lyapunov direct method, an improved algebraic approach to proving the global stability of such systems in the positive cone is presented in this paper. This approach is based on an important finding that the derivative of a Volterra-type Lyapunov function along a polynomial differential system can be expressed as a linear combination of some Volterra-type functions. Applications to some epidemic models demonstrate the effectiveness and universality of the approach. Moreover, according to the property of the Volterra-type function and the relation between the endemic equilibrium and the disease-free equilibrium, the applicable Lyapunov function for the disease-free equilibrium can be derived by reformulating the Lyapunov function for the endemic equilibrium. The approach proposed here greatly simplifies the discussion on global stability of the feasible equilibria of some models described by polynomial differential systems.

Adaptive control for nonlinear systems with unknown actuator dynamics based on a novel extended Nussbaum function

AbstractIn this article, we introduced a novel definition for a class of extended Nussbaum functions that have looser requirements than the traditional Nussbaum function. The extended Nussbaum function achieves comparable performance to the traditional Nussbaum function while addressing the challenges of unknown control directions and time‐varying actuator faults. We then provide a specific example function based on the extended Nussbaum function definition. Unlike the traditional Nussbaum function, which oscillates with fixed frequency and constantly increasing amplitude, this example function constrains the amplitude and oscillates with decreasing frequency. Consequently, its amplitude and derivative remain bounded. The oscillation of the system caused by the too large amplitude of the traditional Nussbaum function can be avoided. Furthermore, this example function is constructed using portions of elliptic curve. As a result, it is continuous and differentiable, in contrast to the normal decreasing‐frequency Nussbaum function using trigonometric function, which is only continuous. The overall closed‐loop system's global stability and asymptotic convergence of system states are successfully achieved. Simulation results demonstrate the effectiveness of the proposed function.

Anti-unwinding adaptive robust backstepping attitude maneuver control for rigid spacecraft

In this paper, an anti-unwinding adaptive robust backstepping control law is proposed to solve the attitude control problem of rigid spacecraft with external disturbances and inertia uncertainties. A new variable based on hyperbolic sine functions is designed to prevent the unwinding phenomenon, and then a novel robust backstepping controller is developed. Initially, a nonlinear tracking law is designed to obtain the desired angular velocity and to ensure that the attitude maneuvering system could have two equilibrium points. Subsequently, an adaptive robust control law is devised, which incorporates the adaptive estimation method to update the inertial parameters in real-time. Moreover, the bounded external disturbances are taken into account in the design of the control law, and thus the robustness of the attitude control system is improved. Furthermore, the global asymptotic stability of the attitude control system is verified by the Lyapunov-Krasovskii theorem. Finally, numerical simulations are carried out to demonstrate the effectiveness and robustness of the proposed control law, as well as the anti-unwinding characteristics are perfectly validated during the attitude maneuver of rigid spacecraft.

Global Stabilization of Incommensurate Real Order Time-Varying Nonlinear Uncertain Systems

Global Stabilization of Incommensurate Real Order Time-Varying Nonlinear Uncertain Systems

Global stability of multi-agent systems with heterogeneous transmission and perception functions

In this brief, we consider an extended opinion dynamic framework by including both transmitting and perceiving behaviors in the agents’ interactions: the first represents how the agent transmits his own opinion to the neighbors, while the latter models how personal features of the agent affect the final perception of external opinions. The agents’ interactions are modeled by general piecewise linear functions that can be heterogeneous and not necessarily monotone, thus generalizing the analytical framework usually considered in the literature, in particular, the well-known interval consensus (Fontanet al., 2020). In the considered novel multi-agent scenario, we formulate sufficient operative LMI conditions to assess global network asymptotic stability for either a consensus or a cluster equilibrium, without necessarily requiring strong network connectivity. The proposed approach is validated through illustrative examples.

A novel adaptive synchronization algorithm for a general class of fractional-order complex-valued systems with unknown parameters, and applications to circuit realization and color image encryption

This article proposes an adaptive synchronization (AS) algorithm to synchronize a general class of fractional-order complex-valued systems with completely unknown parameters, which may appear in physical and engineering problems. The analytical and theoretical concepts of the algorithm rely on the mathematical framework of the Mittag-Leffler global stability of fractional-order systems. A specific control system is established analytically based on the fractional-order adaptive laws of parameters, and the corresponding numerical results are executed to verify the accuracy of the AS algorithm. The proposed synchronization method is evaluated using the fractional-order complex Rabinovich system as an attractive example. The electronic circuits of the new system with different fractional orders are designed. By utilizing the Multisim electronic workbench software, various chaotic/hyperchaotic behaviors have been observed, and a good agreement is found between the numerical results and experimental simulation. In addition, the approximation of the transfer function for different fractional-order are presented. And the corresponding resistor and capacitor values in the chain ship model (CSM) are estimated, which can be utilized in designing electronic circuits for other fractional-order systems. Furthermore, two strategies for encrypting color images are proposed using the AS algorithm and fractional-order adaptive laws of parameters. In the first strategy, the color image is treated as a single package and divided into two vectors. The first vector is embedded into transmitter parameters, while the second vector is injected into the transmitter state signals. In the second strategy, the primary RGB channel components of the original color image are extracted and separated into two vectors, and the same process is followed as in the first strategy. These strategies complicate the decryption task for intruders. Different scales of white Gaussian noise are added to color images to examine the robustness of the proposed color images encryption strategies.

Observer-based dynamic event-triggered control for distributed parameter systems over mobile sensor-plus-actuator networks

Abstract The objective of this paper is to develop a policy of observer-based dynamic event-triggered state feedback control for distributed parameter systems over a mobile sensor-plus-actuator network. It is assumed that the mobile sensing devices that provide spatially averaged state measurements can be used to improve state estimation in the network. For the purpose of decreasing the update frequency of controller and unnecessary sampled data transmission, an efficient dynamic event-triggered control policy is constructed. In an event-triggered system, when an error signal exceeding a specified time-varying threshold it indicates the occurrence of a typical event. The global asymptotic stability of the event-triggered closed-loop system and the boundedness of the minimum inter-event time can be guaranteed. Based on the linear quadratic optimal regulator, the actuator selects the optimal displacement only when an event occurs. A simulation example is finally used to verify the effectiveness of such a control strategy can enhance the system performance.