Semi-global Stability Research Articles

Semi-global stabilization of parabolic PDE–ODE systems with input saturation

This paper addresses the semi-global stabilization problem of parabolic PDE–ODE systems with input saturation. Two low-gain controllers are proposed for two types of systems. It is shown that the proposed controllers can solve the semi-global stabilization problem of the concerned parabolic PDE–ODE systems with input saturation. Distributed-diffusion, counter-convection and Robin boundary condition are simultaneously considered. Moreover, the controllability of PDE–ODE systems with input saturation is discussed. It is shown that semi-global stabilization is impossible if the open-loop ODE system is exponentially unstable, which is another distinctive contribution of this work. Two illustrative examples are given to show the effectiveness of our proposed low-gain controllers.

Semi-Global Stabilization of Discrete-Time Linear Systems Subject to Infinite Distributed Input Delays and Actuator Saturations.

The semi-global stabilization problem of discrete-time systems subject to infinite distributed input delays and actuator saturations is investigated in this article. This article develops two low-gain feedback control laws for two types of systems, respectively. It is shown that the resulting system is semi-globally exponentially stabilized. Our results include those existing results on systems subject to only input saturations and systems subject to bounded delays and input saturations as special cases. Compared with existing results on infinite delays and actuator saturations, this article develops a more accurate scaling utilizing a more general framework. Furthermore, a novel converse Lyapunov theorem for discrete-time linear infinite-delayed systems and a novel stability analysis theorem for perturbed discrete-time linear infinite-delayed systems are developed to handle the nonlinearity induced by saturations. Finally, this article provides two numerical examples to illustrate the effectiveness of the developed theorems.

Stability of KdV equation on a network with bounded and unbounded lengths

In this work, we studied the exponential stability of the nonlinear KdV equation posed on a finite star shaped network with finite number of branches. On each branch of the network we define a KdV equation posed on a finite domain [[EQUATION]] or the half-line [[EQUATION]]. We start by proving well-posedness and some regularity results. Then, we state the exponential stability of the linear KdV equation by acting with a damping term on some branches. The main idea is to prove a suitable observability inequality. In the nonlinear case, we obtain two kinds of results: The first result holds for small amplitude solutions, and is proved using a perturbation argument from the linear case but without acting on all edges. The second result is a semiglobal stability result, and it is obtained by proving an observability inequality directly for the nonlinear system, but we need to act with damping terms on all the branches. In this case, we are able to prove the stabilization in weighted spaces.

Initialization-free Lie-bracket Extremum Seeking

Stability results for extremum seeking control in Rn have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global (i.e., under any initialization) practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all the results via different analytical and/or numerical examples.

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.

A Lyapunov--Krasovskii Method for Semiglobal Stabilization of a Class of Time-Delay Nonlinear Systems With Uncontrollable Linearization by Linear Feedback

A Lyapunov--Krasovskii Method for Semiglobal Stabilization of a Class of Time-Delay Nonlinear Systems With Uncontrollable Linearization by Linear Feedback

Semi-global exponential stabilization of stochastic nonlinear systems with distributed delay under discrete-time state observations and application to circuit systems

This paper is aimed at studying the semi-global exponential stabilization problem for stochastic nonlinear systems with distributed delay under discrete-time state observations. An extended definition of horizontal translation of the segment function along the investigated systems is given, related to the stochastic disturbance and Brownian motion. Basically, the right and upper Dini’s derivative of the constructed functional is presented. Combining this with the global exponential stability of the stochastic nonlinear systems with distributed delay under continuous-time feedback control, the Lyapunov method and Halanay inequality, the maximum observation interval and some stabilization criteria to make the investigated systems under discrete-time state observations reach semi-global exponential stabilization are provided. The results indicate that the semi-global exponential stabilization is dependent upon stochastic disturbance, initial condition range, and maximum observation interval. Finally, an application to the circuit systems is discussed and the involved sufficient conditions for the circuit systems to keep semi-global stabilization are also proposed. Besides, the numerical simulation results confirm the effectiveness of the theoretical results.

Finite-time semi-global stabilisation of polynomial growth nonlinear systems by linear time-varying feedback

The paper studied the finite-time semi-global stabilisation problems for a class of nonlinear systems with a novel polynomial growth condition. By using some key properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions, two time-varying high-gain controllers, where scalar time-varying functions are generated by differential equations, are designed. Both state feedback and observer-based output feedback are considered. It is shown that the proposed methods can achieve finite-time semi-global stabilisation for closed-loop systems and the controls are bound. Two numerical examples verify the effectiveness of the established results.

Stabilization results for delayed fifth-order KdV-type equation in a bounded domain

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to $ 0 $ exponentially as $ t \rightarrow \infty $. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.

A Sampling Control Framework and Its Applications to Robust and Adaptive Control

In this paper, we propose a novel sampling control framework based on the emulation technique where the sampling error is regarded as an auxiliary input to the emulated system. Utilizing the supremum norm of sampling error, the design of periodic sampling and event-triggered control law renders the error dynamics bounded-input-bounded-state (BIBS), and when it is coupled with system dynamics, achieves global or semi-global stabilization. The proposed framework is then extended to tackle the stabilization problem for a system where the state of dynamic uncertainty is not available for feedback and the dynamics is subject to parameter uncertainties. The proposed framework is further extended to solve two classes of event-triggered adaptive control problems where the emulated closed-loop system does not admit an input-to-state stability (ISS) Lyapunov function. For the first class of systems with linear parameterized uncertainties, even-triggered global adaptive stabilization is achieved without the global Lipschitz condition on nonlinearities as often required in the literature. For the second class of systems with uncertainties whose bound is unknown, the event-triggered adaptive (dynamic) gain controller is designed for the first time. Finally, theoretical results are verified by three numerical examples.

Semiglobal stability of PID for uncertain nonaffine systems

Proportional–integral–derivative (PID) controller is the most widely used form of feedback control by far. However, a satisfactory theory is still lacking to explain the rationale of the widespread PID control, and the gap between theory and practice of PID needs to be filled by control scientists. It is worth noting that a recent work (Zhao and Guo, 2022) proved that the classical PID controller has the ability to globally stabilize a basic class of uncertain nonaffine nonlinear systems, while the system nonlinear function is required to have a linear growth rate with respect to the state variables. In this paper, we will consider the more general case where the system nonlinear function may have super-linear growth rates, and show that the classical PID control can always be designed to stabilize such systems in the semiglobal sense, provided that the smallest eigenvalue of the control gain matrix has a positive lower bound. Moreover, a three dimensional parameter set for the stabilizing PID parameters will be constructed. Furthermore, under some additional knowledge about the system uncertain functions, we will propose a simpler PD controller with desired compensation, and show that the proposed controllers can still be designed to achieve semiglobal stability and desired tracking targets.

Research on Variable-Swept Hybrid Aerial Underwater Vehicle Plunge-Diving Control Based on Adaptive Dynamic Surface Control

The variation in aerodynamic parameters during the process of a variable sweepback hybrid aerial underwater vehicle (HAUV) affects flight stability. During the air–water trans-media locomotion, there are medium mutations and solid–liquid gas coupling phenomena, resulting in the complex dynamic process of HAUV. To ensure stable control during the trans-media process of a variable sweepback vehicle, this study proposes a neural-network-based adaptive dynamic surface control method for aircraft flight-path angle. This method aims to establish an effective control model for the entire process of air to media transition, in response to the characteristics of uncertainty and external disturbances in the process of variable backsweeping in the air and media transition. By utilizing the multibody dynamics method, the dynamic equations for variable-swept vehicles are established and transformed into a rigorous feedback system with model uncertainty. The adaptive dynamic surface method in this paper introduces a first-order filter, which overcomes the “differential explosion” problem in traditional backstepping control design through differential filtering; the unknown parameters present in the model are estimated online through adaptive laws, and the uncertain parts of the system are overcome through nonlinear damping items. By analyzing Lyapunov stability, the semi-global stability of the required closed-loop system can be obtained, and adjusting the controller parameters can make the tracking error infinitely small. Numerical simulations are conducted to illustrate the tracking control of flight-path angles for different plunge-diving angle rates and strategy of ingress. The results show that HAUV with variable-swept configuration with different strategy has a great effect on the stability of plunge-diving locomotion; the designed controller can effectively track the target trajectory and has a certain degree of robustness and adaptability.

Global fuzzy adaptive asymptotic tracking control for unknown semi-linear parabolic distributed parameter systems

This paper is mainly devoted to investigating the globally asymptotic tracking problem of unknown semi-linear parabolic distributed parameter systems (SLPDPSs) with uncertain external disturbance. Firstly, an analysis of the well-posedness of the system solution is given based on the operator semigroup theory. Then, from the Lyapunov direct method, a distributed adaptive fuzzy controller (AFC) is designed so that the unknown SLPDPSs can asymptotically track target systems. More importantly, the globally asymptotic tracking condition can be obtained, which overcomes the semi-global stability results in the existing literatures. Finally, several simulation cases are introduced to verify the availability and practicability of the proposed method.

Feedback semiglobal stabilization to trajectories for the Kuramoto–Sivashinsky equation

AbstractIt is shown that an oblique projection-based feedback control is able to stabilize the state of the Kuramoto–Sivashinsky equation, evolving in rectangular domains, to a given time-dependent trajectory. The actuators consist of a finite number of indicator functions supported in small subdomains. Simulations are presented, in the one-dimensional case under periodic boundary conditions and in the two-dimensional case under Neumann boundary conditions, showing the stabilizing performance of the feedback control.

Novel Frequency-Based Approach to Analyze the Stability of Polynomial Fractional Differential Equations

This paper proposes a novel approach for analyzing the stability of polynomial fractional-order systems using the frequency-distributed fractional integrator model. There are two types of frequency and temporal stabilization methods for fractional-order systems that global and semi-global stability conditions attain using the sum-of-squares (SOS) method. Substantiation conditions of global and asymptotical stability are complicated for fractional polynomial systems. According to recent studies on nonlinear fractional-order systems, this paper concentrates on polynomial fractional-order systems with any degree of nonlinearity. Global stability conditions are obtained for polynomial fractional-order systems (PFD) via the sum-of-squares approach and the frequency technique employed. This method can be effective in nonlinear systems where the linear matrix inequality (LMI) approach is incapable of response. This paper proposes to solve non-convex SOS-designed equations and design framework key ideas to avoid conservative problems. A Lyapunov polynomial function is determined to address the problem of PFD stabilization conditions and stability established using sufficiently expressed conditions. The main goal of this article is to present an analytical method based on the optimization method for fractional order models in the form of frequency response. This method can convert it into an optimization problem, and by changing the solution of the optimization problem, the stability of the fractional-order system can be improved.

Fixed-Time Neural Control of Robot Manipulator With Global Stability and Guaranteed Transient Performance

Nowadays, due to many limitations in reality, the optimization of tracking precision and convergence time has attracted the attention of researchers in robotics community. In this article, a fixed-time adaptive neural network (NN) controller is proposed for unknown robot manipulators. A switching mechanism is integrated into the control design such that the semiglobal stability of the conventional NN control systems can be extended to global stability. The time-varying barrier Lyapunov function and the fixed-time control technique are incorporated into the controller design to guarantee the prescribed motion constraints and the fixed-time convergence simultaneously. Compared with some existing fixed-time NN control algorithms, the assumption that NN weight should be upper bounded can be relaxed in our work. Finally, the simulation and experiment studies are respectively carried out based on an unknown 2-degree of freedom robot and a Baxter robot to demonstrate the effectiveness and superiority of the proposed control scheme.

Dynamic event-triggered practical stabilization of random suspension system based on immersion and invariance

This article investigates the practical stabilization problem of random quarter-car active suspension systems. An adaptive dynamic event-trigger strategy is proposed to stabilize the states of vehicle suspension in response to system uncertainty and controller area network resource constraints. Moreover, the model of random active suspension systems is extended to the general random robot systems; the controller is developed with the aid of a double dynamic surface filter, immersion and invariance (I&amp;I) techniques, and event-triggered mechanisms. The results show that the semi-global stability of error systems is achieved, and there are some improvements in triggering times and adaptive estimation performance under the control framework. Finally, simulation comparison results are provided to prove the advantages of the proposed scheme.

Event-Triggered and Self-Triggered Gain Scheduled Control of Linear Systems With Input Constraints

This article proposes static/dynamic event-triggered and self-triggered discrete gain scheduled control with a designable parametric minimal interevent time (MIET) to achieve semiglobal stabilization of linear systems with input constraints. First, a novel static event-triggered discrete gain scheduled control, which can improve the control performance and simultaneously save the communication resources, is proposed by utilizing the properties of the parametric Lyapunov equation (PLE). Moreover, the static self-triggered mechanism, in which the next control law updates based on the previous triggered states, is also designed to avoid the monitoring of all states. In order to further increase the interevent times (IETs), the corresponding dynamic event-triggered and self-triggered discrete gain scheduled control are designed, respectively. All the proposed algorithms can not only avoid the Zeno phenomenon but also provide a designable parametric MIET. This allows to easily find a tradeoff between the IETs and the control performance by adjusting the only design parameter. In addition, by exploiting the properties of the PLE, the designed algorithms avoid the complicated relationship between the MIET and the system matrices. In some cases, the MIET can totally avoid the relationship with the system itself and be designed as an arbitrarily large bounded constant. Finally, applications to the spacecraft rendezvous system show the effectiveness of the established algorithms.

Spill-Free Transfer and Stabilization of Viscous Liquid

This article studies the feedback stabilization problem of the motion of a tank that contains an incompressible, Newtonian, viscous liquid. The control input is the force applied on the tank and the overall system consists of two nonlinear partial differential equations and two ordinary differential equations. Moreover, a spill-free condition is required to hold. By applying the control Lyapunov functional methodology, a set of initial conditions (state space) is determined for which spill-free motion of the liquid is possible by applying an appropriate control input. Semi-global stabilization of the liquid and the tank by means of a simple feedback law is achieved, in the sense that for every closed subset of the state space, it is possible to find appropriate controller gains, so that every solution of the closed-loop system initiated from the given closed subset satisfies specific stability estimates. The closed-loop system exhibits an exponential convergence rate to the desired equilibrium point. The proposed stabilizing feedback law does not require measurement of the liquid level and velocity profiles inside the tank and simply requires measurements of 1) the tank position error and tank velocity, 2) the total momentum of the liquid, and 3) the liquid levels at the tank walls. The obtained results allow an algorithmic solution of the problem of the spill-free movement and slosh-free settlement of a liquid in a vessel of limited height (such as water in a glass) by a robot to a prespecified position, no matter how full the vessel is.

Composite Nonlinear Path-Following Control for Unmanned Ground Vehicles With Anti-Windup ESO

In this article, a composite nonlinear feedback (CNF) controller with anti-windup extended state observer (ESO) is proposed for path-following control of unmanned ground vehicles (UGVs). To describe steering dynamics accurately, a saturation model with bounded disturbances is developed subject to path-following, lateral dynamic, and steering capability. The anti-windup ESO is investigated for real-time disturbance estimation under saturated input. The CNF controller is designed for path-following. In terms of asymptotically null controllability, a practical semiglobal stabilization result is derived for the closed-loop system with saturated input and bounded disturbances. Experimental results show that the proposed strategy is effective for UGVs.