Semi-global Practical Stability Research Articles

Adaptive finite-time fuzzy control for nonlinear systems with input quantization and unknown time delays

We investigate the finite-time adaptive control issue for a class of nonlinear systems with quantized input signals and unknown time delays in this paper. In the control design process, the nonlinear terms are approximated by the fuzzy logic systems. A hysteretic quantizer is introduced to avoid chattering in the quantized signals, and the fuzzy state observer is constructed to estimate the unmeasurable states of the system. In order to propose the finite-time control strategy, firstly, a semi-global practical finite-time stability criterion is given. On this basis, a fuzzy adaptive controller is designed by combining the dynamic surface control technology with backstepping method. The controller can not only ensure that the observer and tracking error converge to a small neighborhood of the origin in a finite time, but also keep all the signals in the closed-loop system bounded. Finally, the effectiveness and feasibility of the control method are verified by two simulation examples.

On Practical Stability Preservation Under Fast Sampling and Accurate Quantization of Feedbacks for Nonlinear Time-Delay Systems

In this article, we deal with the problem of stabilizing a nonlinear system with state-delays by means of quantized sampled-data state feedback control laws. Quantization in the state measurements and in the input signal are simultaneously considered. Fully nonlinear (i.e., possibly nonaffine in the control) time-delay systems are studied. Sufficient conditions are provided such that suitably fast sampling and accurate quantization of the state feedback at hand yield semiglobal practical stability, with arbitrarily small final target ball of the origin. Nonlinear delay-free systems are addressed as a special case: it is shown that the above sufficient conditions, ensuring the semiglobal practical stability, are satisfied if the continuous-time static state feedback controller is a global stabilizer. The theory of stabilization in the sample-and-hold sense is used. The theoretical results are validated through an example.

Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality

In this paper, we propose a novel state-feedback backstepping control design approach for a single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunov functions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adopted in the stability proof. Semiglobal practical finite-time stability, or global asymptotically stability, is guaranteed by a continuous control law using a commonly used recursive backstepping-like approach. Unlike the inductive design of typical finite-time backstepping controllers, the proposed method has the advantage of reduced design complexity. The virtual control laws are designed by directly canceling the nonlinear terms in the derivative of the specific Lyapunov functions. The terms with exponents are transformed into linear forms as their bases. The stability proof is simplified by applying several inequalities in the final proof, instead of in each step. Furthermore, the singularity problem no longer exists. The weakness of the concept of practical finite-time stability is discussed. The method can be applied to smoothly extend numerous design methodologies with asymptotic stability with a higher convergence rate near the equilibrium. Two numerical case studies are provided to present the performance of the proposed control.

Acceleration-Actuated Source Seeking Without Position and Velocity Sensing

We study the problem of source seeking for an acceleration-controlled unicycle. The objective is to asymptotically stabilize the unicycle around states where a smooth position-dependent signal (or cost) function attains a minimum value. An implementation of the proposed control strategy only requires measurements of the cost function value at the current position. We do not assume that the unicycle can measure its current positions or its current forward velocity. For this purpose, we extend a recently introduced approach to extremum seeking control, which is based on the approximation of symmetric products of vector fields. An additional high-gain observer is used to estimate the derivative of the sensed cost signal. The estimates provided by the observer compensate for the missing velocity measurements and allow a reduction of the kinetic energy without velocity-dependent damping. Under suitable assumptions on the cost function, the control law leads to semi-global practical asymptotic stability.

Optimizing the strategy of activities using numerical methods for determining equilibrium

The paper considers issues on the theoretical substantiation of options for choosing an optimal strategy to integrate an agricultural enterprise into the wholesale market by using methodological tools of the non-cooperative game theory. We have proposed modeling the behavior of an agrarian enterprise in the market by achieving a Nash equilibrium under various scenarios of competitors’ activities and volumes of information on market conditions. The methodology has been substantiated to apply the iterative algorithms to calculate equilibria in a general class of non-quadratic convex polyhedra in order to form the methodologies and construct algorithms for a behavior of agricultural enterprises in market activity. It was determined that decision-making occurs in parallel to the real conditions of activity of an agricultural enterprise in the wholesale market. The comprehensive application of numerical methods based on solving the optimization problems provides a smooth approach to the Nash equilibrium. A game can have multiple isolated Nash equilibria if players have non-quadratic payment functions when solving such problems. Based on the above, the results were determined of local convergence, since global results have strong constraints in non-quadratic problems. However, there is a connection with semi-global practical asymptotic stability if players have quadratic payoff functions. It has been shown that there is a shift in the convergence in proportion to the amplitudes of disturbance signals and the third derivative of payoff functions for non-quadratic payoff functions. This shift in the convergence corresponds to the shift in a numerical example. It has been determined that the learning strategy developed in accordance with the main provisions of the theory of games remains attractive if one has partial information on the state of the market. Application of the indicated action strategy provides a company with a possibility to improve its initial position by measuring its own payoff values only and not using estimates of potentially uncertain parameters. It has been proposed to use applied tools from the game theory to determine an optimal action strategy for an agricultural enterprise for its integration into the wholesale market of vegetable products

Practical Finite-Time Fuzzy Control for Hamiltonian Systems via Adaptive Event-Triggered Approach

This paper is to present our new results on practical finite-time fuzzy control scheme for a class of Hamiltonian systems by an adaptive event-triggered approach. An unknown function of the system is approximated by an adaptive fuzzy system. According to a semi-global practical finite-time stability criterion, a novel adaptive fuzzy finite-time controller is presented. In the interest of saving resources, an event-triggered approach is developed where the controller is updated under the respective triggering conditions. Based on the special structure of Hamiltonian systems and Lyapunov theory, a sufficient condition is obtained so that the states of original systems converge to a small neighborhood containing the origin in a finite time. Meanwhile, with the proposed controller, there exists a positive lower bound for the interexecution time, and the Zeno phenomenon is avoided. The design processes of the proposed controller, adaptive law and event-triggered conditions are also proved by a circuit system simulation example.

Finite-Time Adaptive Fuzzy Command Filtered Backstepping Control for a Class of Nonlinear Systems

In this article, the problem of adaptive fuzzy finite-time command filtered control is considered for a class of nonstrict-feedback nonlinear systems. The explosion of complexity problem is dealt with by employing the command filter approach. In order to design a finite-time control scheme, a finite-time semi-global practical stability criterion is first presented. Based on this criterion, by applying the command filter technology and backstepping technique, adaptive fuzzy finite-time tracking control scheme is developed with the help of fuzzy logic systems approximation. Under the presented adaptive control scheme, all the closed-loop variables are semi-global practical finite-time stable and the tracking error goes into an adjustable neighborhood around the origin in a finite time. Finally, simulations are utilized to verify the effectiveness of designed control scheme.

Extremum Seeking Control With Input Dead-Zone

This article develops a novel extremum seeking control (ESC) scheme to address an input dead-zone. The new ESC scheme includes two parts: one is the learning mechanism that can rapidly compensate the effect of the dead-zone; the other is the classic ESC scheme, which can regulate the output of the system to an arbitrarily small neighborhood of its extremum. The proposed scheme has a two-time-scale structure, and the boundary layer system is uniformly, practically, and asymptotically stable. By applying results that exist in the literature, semi-global practical asymptotic stability of the new ESC scheme is established. Finally, numerical examples are presented to illustrate the effectiveness of the proposed scheme.

Finite-time Adaptive Integral Backstepping Fast Terminal Sliding Mode Control Application on Quadrotor UAV

This paper presents a reliable and novel quadrotor flight control system designed to enhance trajectory tracking performance, robustness and adaptiveness against the uncertain parameters and the external wind disturbance. By combining a recursive control methodology with a robust control algorithm, a finite-time adaptive integral backstepping fast terminal sliding mode control is designed for major control loops related to position tracking and attitude stabilization. To estimate quadrotor mass and inertia moments, only four adaptation laws are developed. To compensate the unknown upper bound on the disturbances, a robust and adaptive switching gain is designed. The designed controller guarantees that all the closed signals are semi-global practical finite-time stability while the tracking error converges to a small neighborhood of the origin. The obtained numerical results and comparison studies show the effectiveness, robustness, adaptiveness and energy efficiency of the proposed flight control system.

Observed-based adaptive finite-time tracking control for a class of nonstrict-feedback nonlinear systems with input saturation

This paper concentrates upon the problem of adaptive neural finite-time tracking control for uncertain nonstrict-feedback nonlinear systems with input saturation. The design difficulty of non-smooth input saturation nonlinearity is solved by applying a smooth non-affine function to approximate the saturation signal. Neural networks, as a kind of specialized function estimators, are used to estimate the uncertain function. Meanwhile, a neural network-based observer is constructed to observe the unavailable states, and thus an observer-based adaptive finite-time tracking control strategy is developed by combining dynamic surface control (DSC) technique and backstepping approach. Furthermore, the stability of the considered system is analyzed via semi-global practical finite-time stability theory. Under the proposed control method, all the signals in the closed-loop system are bounded, and the system output can almost surely track the desired trajectory within a specified bounded error in a finite time. In the end, two examples are adopted to illustrate the validity of our results.

Constrained extremum seeking of a MIMO dynamic system

Constraints on outputs or measured states arise in many practical engineering problems, yet are not readily handled by conventional extremum seeking techniques that require the output or states to be mapped to a corresponding input region. In this work, an alternative approach utilising non-linear programming techniques along with measurements of the constraint functions is proposed to enable a constrained extremum to be sought directly. Under reasonable assumptions and tuning, the proposed approach is shown to provide semi-global practical asymptotic stability guarantees for a class of MIMO plants. Simulation results are provided to demonstrate the efficacy of the proposed approach.

Sampled-data control of underwater gliders: digital redesign approach

This paper discusses a sampled-data control problem for a nonlinear underwater glider based on the digital redesign (DR) technique. An analog controller is pre-designed based on the fully feedback-linearised model. The condition for the DR problem is proposed in terms of linear matrix inequality, using the delta-operator discrete-time models. We show that the DRed sampled-data controller guarantees the semiglobal practical asymptotic stability of the nonlinear steady gliding motion in the sampling period. The numerical simulations demonstrate the advantage of the developed methodology in terms of the larger sampling period and the better preservation of the original analog control performance than the conventional emulation technique.

Fixed-time adaptive neural tracking control for a class of uncertain nonstrict nonlinear systems

This paper addresses the fixed-time adaptive neural control of nonstrict feedback nonlinear system. With the help of neural networks and the backstepping technical, a fixed-time adaptive neural control scheme is presented. To guarantee closed-loop stability, a new semiglobal practical fixed-time stability (SPFTS) criterion is set up. Based on the established SPFTS criterion, we can show that both the tracking performance and the closed-loop stability can be preserved in a fixed time via the presented approach. Compared with the existing finite-time control, the convergence time of the propose fixed-time control scheme does not rely on the initial states. Finally, the proposed technique is demonstrated with simulation results.

Fuzzy Adaptive Finite Time Fault-tolerant Control for Multi-input and Multi-output Nonlinear Systems with Actuator Faults

This paper investigates the problem of fuzzy adaptive finite-time fault tolerant control (FTC) for a class of multi-input and multi-output (MIMO) nonlinear systems with actuator failure. In control design, the fuzzy logic systems (FLSs) are adopted to identify the unknown nonlinear functions and a state observer is constructed to estimate the unmeasurable states. By combining dynamic surface control (DSC) technique with backstepping design, a novel finite-time fuzzy adaptive FTC strategy is proposed based on fault-tolerant control technique to overcomes the “explosion of complexity” problem. The presented control method demonstrates that all signals of the closed-loop system are semi-global practical finite-time stability (SGPFS), and the tracking errors converge to a small neighborhood of zero in a finite time. Finally, a numerical example is provided to illustrate the effectiveness of the presented control method.

Fuzzy Adaptive Practical Finite-Time Control for Time Delays Nonlinear Systems

This paper focuses on fuzzy adaptive practical finite-time output feedback control problem for a class of single-input and single-output nonlinear system with time-varying delays in nonstrict feedback form. Fuzzy logic systems are adopted to approximate the unknown nonlinear functions, and state observer is constructed to estimate the unmeasured states. By combining practical finite-time Lyapunov stability theory with the backstepping design, an observer-based fuzzy adaptive practical finite-time control strategy is proposed. Meanwhile, the stability of the closed-loop system is proved, which means that the output can follow the given reference signal in a finite time, and the closed-loop system is semi-global practical finite-time stability. Finally, two simulation examples are provided to elaborate the effectiveness of the presented control strategy.

Adaptive amplitude fast proportional integral phasor extremum seeking control for a class of nonlinear system

In this paper, we present a modification of the fast phasor extremum-seeking control for the fast optimization of a class of Wiener-Hammerstein nonlinear dynamical systems. The proposed technique provides a significant improvement of the closed-loop system's performance. This study introduces a new adaptive amplitude technique that is used to adaptively adjust the perturbation amplitude to a small predetermined value in a neighbourhood of the system's unknown optimal equilibrium. An analysis of the system demonstrates that semi-global practical stability analysis of the overall system to the unknown optimum is achieved. The effectiveness of the proposed approach is illustrated using numerical examples. The approach is also implemented for the optimal operation of a lean burn combustion system.

Adaptive Scan for Atomic Force Microscopy Based on Online Optimization: Theory and Experiment

A major challenge in atomic force microscopy is to reduce the scan duration while retaining the image quality. Conventionally, the scan rate is restricted to a sufficiently small value in order to ensure a desirable image quality as well as a safe tip–sample contact force. This usually results in a conservative scan rate for samples that have a large variation in aspect ratio and/or for scan patterns that have a varying linear velocity. In this paper, an adaptive scan scheme is proposed to alleviate this problem. A scan line-based performance metric balancing both imaging speed and accuracy is proposed, and the scan rate is adapted such that the metric is optimized online in the presence of aspect ratio and/or linear velocity variations. The online optimization is achieved using an extremum-seeking approach, and a semiglobal practical asymptotic stability result is shown for the overall system. Finally, the proposed scheme is demonstrated via both simulation and experiment.

Efficient PID Tracking Control of Robotic Manipulators Driven by Compliant Actuators

This brief aims to show that a linear proportional–integral–derivative (PID) controller is theoretically valid for tracking control of robotic manipulators driven by compliant actuators. The control problem is formulated into a three-time-scale singular perturbation formula, including a slow time scale at the rigid robot dynamics, one actual fast time scale at the actuator dynamics, and another virtual fast time scale at the controller dynamics. A PID-type controller is derived to guarantee semiglobal practical exponential stability of the rigid robot dynamics, and a derivative-type controller is applied to establish global exponential stability of the actuator dynamics. Based on a state transformation to the closed-loop rigid robot dynamics and the extended Tikhonov’s theorem, it is proven that the entire system has semiglobal practical exponential stability under a proper choice of control parameters. The proposed controller is not only structurally simple and model-free resulting in low implementation cost, but also robust against external disturbances and parameter variations. The current design is only valid while the spring stiffness is relatively large compared with other parameters of the robot dynamics. Experimental results based on a single-link compliant robotic manipulator have verified effectiveness of the proposed approach.

Finite-Time Adaptive Fuzzy Output Feedback Dynamic Surface Control for MIMO Nonstrict Feedback Systems

This paper investigates the finite-time adaptive fuzzy control problem for a class of multi-input and multi-output (MIMO) nonlinear nonstrict feedback systems. During the control design process, fuzzy logic systems (FLSs) are utilized to approximate the unknown nonlinear functions, and fuzzy state observer is constructed to estimate the unmeasured states. By combining adaptive backstepping with the dynamic surface control (DSC) technique, a finite-time fuzzy adaptive control scheme is presented to overcome the “explosion of complexity” problem. The stability of the close-loop systems can be proved based on the finite-time Lyapunov stability theory. The presented control scheme demonstrates that the closed-loop systems are semiglobal practical finite-time stability, and tracking errors converge to a small neighborhood of the origin in a finite time. Finally, two simulation examples are provided to show the effectiveness of the presented control method.

Adaptive Finite-Time NN Control for 3-DOF Active Suspension Systems With Displacement Constraints

In this paper, the displacement constraints and finite-time control problems are handled for 3 degrees of freedom (DOF) active suspension systems (ASSs). In order to ensure that seat and active suspension's displacements do not violate their limits, the Barrier Lyapunov functions (BLFs) are employed to handle displacement constraints problem, and the ride comfort and safety are improved at the same time. The unknown functions in the ASSs are approximated by using the neural networks (NNs). The traditional finite-time controller often contains the sign functions or absolute value functions, and they cause the problem about buffeting of the controller. By using NNs, finite-time theory, and BLFs, an adaptive finite-time control approach is constructed to avoid this problem. Then, the boundedness of all signals of 3-DOF ASSs is testified by the semi-global practical finite-time stability analysis and a zero dynamics analysis. In addition, the results of two numerical examples demonstrate the effectiveness of the established adaptive finite-time NN control scheme.