Semi-global Practical Stability Research Articles

Adaptive Finite-Time-Based Neural Optimal Control of Time-Delayed Wheeled Mobile Robotics Systems.

For nonlinear systems with uncertain state time delays, an adaptive neural optimal tracking control method based on finite time is designed. With the help of the appropriate LKFs, the time-delay problem is handled. A novel nonquadratic Hamilton-Jacobi-Bellman (HJB) function is defined, where finite time is selected as the upper limit of integration. This function contains information on the state time delay, while also maintaining the basic information. To meet specific requirements, the integral reinforcement learning method is employed to solve the ideal HJB function. Then, a tracking controller is designed to ensure finite-time convergence and optimization of the controlled system. This involves the evaluation and execution of gradient descent updates of neural network weights based on a reinforcement learning architecture. The semi-global practical finite-time stability of the controlled system and the finite-time convergence of the tracking error are guaranteed.

On robustification of digital event-based controllers for control-affine nonlinear systems

In this paper, the robust digital stabilization problem of nonlinear systems is investigated. In particular, a methodology for the design of robust quantized sampled-data stabilizers updated via an event-triggered mechanism is provided for time-varying control-affine nonlinear systems affected by actuation disturbances and measurement noises. The notion of time-varying steepest descent feedback (TSDF), continuous or not, and the Input-to-State Stability (ISS) redesign methodology are used for the development of the proposed robust event-based digital controller. Under the assumption that the actuation disturbances and measurement noises are bounded with a-priori known bounds and that the amplitude of the measurement noises satisfies a certain condition related to the new added robustification term, the following result is proved: there exist a suitably fast sampling and an accurate quantization of the input/output channels such that the digital implementation of robustified TSDF controllers, updated through a proposed event-triggered mechanism, ensures semi-global practical stability of the related closed-loop system regardless of the above uncertainties. In the methodology here proposed, time-varying sampling periods and the non-uniform quantization of the input/output channels are allowed. Moreover, the theory here developed includes the analysis of the intersampling system behaviour. Possible discontinuities in the function describing the TSDF at hand are also managed. The provided results are validated through a numerical example.

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 finite-time neural control of nonstrict-feedback nonlinear systems with input dead-zone and output hysteresis

This paper explores the adaptive finite-time neural control issue for nonlinear systems with input dead zone and output hysteresis in nonstrict-feedback form. The unknown functions are estimated by employing the radial basis function neural networks (RBFNN) approach. A systematic adaptive finite-time control method is introduced using the backstepping technique and neural network approximation properties. The stability of the system is also examined by using semi-global practical finite-time stability theory. The established control approach guarantees the boundedness of all signals within the closed-loop system, enabling the system output to accurately follow the desired signal within a finite time framework while maintaining a small and bounded tracking error. Finally, simulation results are shown to demonstrate the efficacy of the suggested strategy.

Distributed finite-time adaptive consensus output-feedback control of multi-agent systems under switching topologies

The paper investigates the finite-time adaptive consensus control of nonlinear multi-agent systems (MASs) by output-feedback. During the process of control design, the fuzzy logic system (FLS) is employed to approximate nonlinear function, as well as constructing a fuzzy state observer to estimate unmeasured states. Combining the dynamic surface control (DSC) with adaptive backstepping method conquers the complexity explosion issue. Then, semi-global practical finite-time stability (SGPFS) for closed-loop system is obtained under arbitrary switching communication topologies. All the signals of the overall system are uniformly ultimately bounded (UUB). Finally, the availability of developed control approach will be demonstrated through a simulation.

On the robustification of digital event-based stabilizers for nonlinear time-delay systems

In this paper, the robust stabilization problem by means of quantized sampled-data event-based (QSE) controllers is investigated for nonlinear systems affected by state delays and unknown disturbances. In particular, a methodology for the design of robust QSE stabilizers is provided for control-affine nonlinear systems affected by unknown actuation disturbances and unknown measurement errors. Firstly, the notion of Steepest Descent Feedback (SDF), continuous or not, is suitably revised in order to deal with the robustification of event-based controllers. Then, Input-to-State Stability (ISS) redesign methodologies are used to provide the robustification term which is added to the SDF at hand in order to arbitrarily attenuate the effects of unknown external disturbances affecting the considered control scheme. A spline approximation approach is used in order to cope with the problem of the possible non-availability in the buffer of suitable past values of the system state required for the correct application of the proposed robust QSE controller. It is proved that there exist a suitably fast sampling and an accurate quantization of the input/output channels such that: the robust QSE implementation of SDFs, continuous or not, ensures the semi-global practical stability of the related closed-loop system, regardless of the above disturbances, provided that the observation errors affects marginally the new added control term. The stabilization in the sample-and-hold sense theory is used as a tool to prove the results. The provided results include the case of non-uniform quantization of the input/output channels and the case of aperiodic sampling. Applications are presented in order to validate the results.

Semi-Global Practical Stability of Random Systems and Its Application

Semi-Global Practical Stability of Random Systems and Its Application

Robust optimal triple event-triggered intermittent control for uncertain input-constrained nonlinear systems

This paper investigates the concept of finite time (FT) optimal event-triggered intermittent (ETI) control for nonlinear systems with input constraints and dynamic uncertainty. Firstly, we address a key challenge faced by existing controllers, wherein the unbounded nature of the signal function renders it impractical to handle the input-constrained FT control problem. To overcome this limitation, we derive a novel lemma of semiglobal practical finite-time stability (SGPFTS) specifically for intermittent control with the additional constant terms. By leveraging the lemma and the Hamilton–Jacobi–Bellman (HJB) equation, we obtain the optimal FT guaranteed cost intermittent control law and establish FT stability criteria for uncertain nonlinear systems. Recognizing that the general HJB theory is ill-equipped to address input-constrained optimal control problems, We introduce the model predictive control (MPC) algorithm, which has been proven to be an effective online optimization method in handling robust optimal control and constrained problems. Moreover, conventional ETI optimal control methods struggle to address intermittent control for nonlinear systems with input constraints and dynamic uncertainty. To address this limitation, we propose a triple ETI Lyapunov-based MPC strategy, which not only enhances intermittent control performance but also resolves input constraints and dynamic uncertainty issues. Finally, we present two straightforward examples of uncertain nonlinear systems to demonstrate the effectiveness of the developed strategy in practice.

A distributed strategy for games in Euler–Lagrange systems with actuator dead zone

This paper explores a distributed Nash equilibrium (NE) seeking problem for games in which the involved players are of Euler–Lagrange (EL) dynamics with actuator dead zone. To find NE of such games, a novel distributed algorithm with a dynamical dead-zone compensator is proposed. The compensating module in this new strategy is regarded as a fast dynamics that rapidly addresses the effect caused by actuator dead-zone characteristic. The other module of this algorithm is a slow optimizer that enables the actions of EL players to approach the NE. Since the designed strategy is a two-time-scale model, semi-global practical asymptotic stability of this strategy can be obtained by utilizing the generalized singular perturbation method. Finally, an electricity market game with six turbine-generator dynamics is utilized to validate the theoretical result of the proposed method.

Adaptive Control for Complex Systems with Dynamics and Time-Varying Powers

This work focuses on presenting a control algorithm to investigate nonlinear systems, which contain time-varying powers, inverse dynamics, and uncertainties. First, some appropriate transformations are introduced to obtain a new system. Then, a Lyapunov function, which covers quadratic and high-order components, is recursively constructed for control design. Subsequently, by introducing the neural networks, the uncertain functions encountered during the design are approximated. Based on the inequality techniques, the nonlinear terms are skillfully estimated. By defining the bounds of some unknown parameters and using the adaptive technique, some virtual controllers are selected in each step to dominate the nonlinear functions and guarantee that the derivative of the Lyapunov function satisfies the required form. Finally, a new adaptive controller is constructed and semiglobal practical finite time stability (SGPFS) is guaranteed. The proposed approach is verified with a numerical example.

Underactuated Source Seeking by Surge Force Tuning: Theory and Boat Experiments

We extend source seeking algorithms, in the absence of position and velocity measurements, and with the tuning of the surge input, from velocity-actuated (unicycle) kinematic models to force-actuated generic Euler–Lagrange dynamic underactuated models. In the design and analysis, we employ a symmetric product approximation, averaging, passivity, and partial-state stability theory. The proposed control law requires only real-time measurement of the source signal at the current position of the vehicle and ensures semi-global practical uniform asymptotic stability (SPUAS) with respect to the linear motion coordinates for the closed-loop system. The performance of our source seeker with surge force tuning is illustrated with both numerical simulations and experiments of an underactuated boat.

On Sontag’s formula for the sampled-data observer-based stabilization of nonlinear time-delay systems

In this paper, a new methodology for the design of sampled-data dynamic output feedback (DOF) stabilizers for control-affine nonlinear time-delay systems is proposed. In particular, the methodology here developed is based on the Artstein’s theory of control Lyapunov functions and related Sontag’s formula, here extended, for the first time in the literature, to the case of DOF controllers. Firstly, the new notion of Steepest Descent Error Dynamics (SDED) is introduced and used to suitably re-formulate the well-known Sontag’s universal formula for the case of DOF controllers. Then, it is shown that, under suitably fast sampling, a proposed sampled-data DOF controller based on the new provided version of Sontag’s universal formula ensures the semiglobal practical stability of the related sampled-data closed-loop system. Time-varying sampling intervals, as well as the intersampling system behavior, are taken into account. In the theory here developed, the case of delay-free systems is addressed as a special case. Applications are presented for the validation of the proposed theoretical results.

Fixed-Time Nash Equilibrium Seeking in Time-Varying Networks

In this paper we introduce first-order and zeroth-order Nash equilibrium seeking dynamics with fixed-time convergence certificates for non-cooperative games having finitely many players. The first-order algorithms achieve exact convergence to the Nash equilibrium of the game in a finite time that can be additionally upper bounded by a constant that is independent of the initial conditions of the actions of the players. Moreover, these fixed-time bounds can be prescribed a priori by the system designer under an appropriate tuning of the parameters of the algorithms. When players have access only to measurements of their cost functions, we consider a class of distributed zeroth-order model-free adaptive dynamics that achieve semi-global practical fixed-time stability, qualitatively preserving the fixed-time bounds of the first-order dynamics. Moreover, by leveraging the property of fixed-time input-to-state stability, further results are obtained for mixed games where some of the players implement different seeking dynamics. Fast and slow switching communication graphs are also incorporated using tools from hybrid dynamical systems theory. We consider potential games as well as gener

Adaptive fuzzy fixed-time dynamic surface control for stochastic nonstrict nonlinear systems with unknown dead-zones

This paper focuses on an adaptive fuzzy fixed-time control problem for stochastic nonstrict nonlinear systems with unknown dead-zones by using dynamic surface control (DSC) technology. Fuzzy logic systems (FLSs) and DSC technology are used to approximate nonlinear functions and reduce the computational complexity, respectively. At the same time, the influence of the dead-zone disturbance is offset by transforming the dead-zone model into the nonlinear model that can be approximated by the FLSs. Then, based on the fixed-time stability theory, an adaptive fuzzy fixed-time tracking control strategy is proposed, which can ensure semi-global practical fixed-time stability of the system and the tracking error converging to a small neighborhood near the origin. Finally, two simulation examples are given to prove the effectiveness of the proposed control strategy.

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.

Finite-Time Fuzzy Adaptive PPC for Nonstrict-Feedback Nonlinear MIMO Systems.

This article addresses the issue of the fuzzy adaptive prescribed performance control (PPC) design for nonstrict feedback multiple input multiple output (MIMO) nonlinear systems in finite time. Unknown nonlinear functions are handled via fuzzy-logic systems (FLSs). By combining the adaptive backstepping control algorithm and the nonlinear filters, a novel dynamic surface control (DSC) method is proposed, which can not only avoid the computational complexity issue but also improve the control performance in contrast to the traditional DSC control methods. Furthermore, to make the tracking errors have the prescribed performance in finite time, a new Lyapunov function is constructed by considering the transform error constraint. Based on the designed Lyapunov functions, it is proved that all the signals of the controlled systems are semiglobal practical finite-time stability (SGPFS). Finally, a simulation example is provided to illustrate the feasibility and validity of the put forward control scheme.

Adaptive Fuzzy Finite-Time Fault-Tolerant Consensus Tracking Control for High-Order Multiagent Systems With Directed Graphs.

This article investigates the distributed adaptive fuzzy finite-time fault-tolerant consensus tracking control for a class of unknown nonlinear high-order multiagent systems (MASs) with actuator faults and high powers (ratio of positive odd rational numbers). The fault models include both loss of effectiveness and bias fault. Compared with existing similar results, the MASs considered here are more general and complex, which include the special case when the powers are equal to 1. Besides, the functions in this article are completely unknown and do not need to satisfy any growth conditions. In the backstepping framework, an adaptive fuzzy fault-tolerant consensus tracking controller is designed via adding one power integrator technique and directed graph theory so that the controlled systems are semiglobal practical finite-time stability (SGPFTS). Finally, numerical simulation results further verify the effectiveness of the developed control scheme.

Digital Event-Based Stabilization of Nonlinear Time-Delay Systems

In this paper, the stabilization problem of nonlinear time-delay systems via quantized sampled-data event-based controllers is investigated. Fully nonlinear (i.e., possibly non-affine in the control) systems affected by state delays are studied. Sufficient conditions are provided for the existence of a suitably fast sampling and of an accurate quantization of the input/output channels such that the digital implementation of the continuous-time controller at hand, updated through a proposed event-triggered mechanism, ensures the semi-global practical stability property, with arbitrarily small final target ball of the origin, of the related closed-loop system. A spline approximation methodology is used in order to cope with the problem of the possible non-availability in the buffer of suitable past values of the system variables needed for the digital implementation of the controller. The stabilization in the sample-and-hold sense theory is used as a tool to prove the results. In the theory here developed, the case of non-uniform quantization of the input/output channels and the case of aperiodic sampling are both included. The proposed theoretical results are validated through an application concerning the plasma glucose regulation problem in type-2 diabetic patients.

Robust tracking control of a robot arm actuated by permanent magnet synchronous motors

AbstractIn this article, we show the use of the recently developed enhanced extended observer design technique in tracking time‐dependent references by a PMSM‐actuated robot arm. It is proven that this robust control framework guarantees achievement of semiglobal practical stability and makes trajectories of the system arbitrarily close to those that would had been obtained by the non‐robust nominal feedback‐linearizing control if there were no plant uncertainties.

Prescribed Performance Fuzzy Adaptive Output Feedback Control for Nonlinear MIMO Systems in a Finite Time

This article studies the fuzzy adaptive output feedback control design problem for nonstrict feedback multi-input–multi-output nonlinear systems with full-states prescribed performance in finite time. Fuzzy logic systems are introduced to solve the problem of unknown nonlinear dynamics. And on this basis, a fuzzy-based state observer is designed to observe the unmeasurable state. Further, by combining the adaptive back-stepping control algorithm and the nonlinear filters, a novel dynamic surface control (DSC) method is proposed, which not only solves the computational complexity explosion problem inherent in the back-stepping control algorithm, but also improves the control performance, in contrast to the traditional DSC methods with linear filters. Besides, to further improve the tracking performance under the structure of full-states prescribed performance, a new Lyapunov function is designed considering the transform error constraint. Based on the Lyapunov theory, the stability of the closed-loop system is analyzed to ensure that all signals of the closed-loop system are semiglobal practical finite-time stability. Finally, to elaborate on the feasibility and effectiveness of the proposed control method, a simulation example is provided.