Finite-time Convergence Property Research Articles

Zeroing Neural Network With Coefficient Functions and Adjustable Parameters for Solving Time-Variant Sylvester Equation.

To solve the time-variant Sylvester equation, in 2013, Li et al. proposed the zeroing neural network with sign-bi-power function (ZNN-SBPF) model via constructing a nonlinear activation function. In this article, to further improve the convergence rate, the zeroing neural network with coefficient functions and adjustable parameters (ZNN-CFAP) model as a variation in zeroing neural network (ZNN) model is proposed. On the basis of the introduced coefficient functions, an appropriate ZNN-CFAP model can be chosen according to the error function. The high convergence rate of the ZNN-CFAP model can be achieved by choosing appropriate adjustable parameters. Moreover, the finite-time convergence property and convergence time upper bound of the ZNN-CFAP model are proved in theory. Computer simulations and numerical experiments are performed to illustrate the efficacy and validity of the ZNN-CFAP model in time-variant Sylvester equation solving. Comparative experiments among the ZNN-CFAP, ZNN-SBPF, and ZNN with linear function (ZNN-LF) models further substantiate the superiority of the ZNN-CFAP model in view of the convergence rate. Finally, the proposed ZNN-CFAP model is successfully applied to the tracking control of robot manipulator to verify its practicability.

Adaptive fast trajectory tracking control for the airship based on prescribed performance

A fast trajectory tracking controller is designed for the airship, involving dynamic uncertainties of parameters, capable of guaranteeing the prescribed performance of tracking errors, and fast response of the closed-loop system. To achieve the decoupling of velocity control and attitude control of the airship, a new guidance and control integration framework is proposed first. Then, based on the dynamics model established by the spinor method, the lumped disturbance, caused by unknown aerodynamic parameter uncertainties and exogenous disturbances, is estimated online using a high-order finite-time observer. The control performance, namely, overshoot and steady-state performance of tracking errors, is significantly improved by utilizing the prescribed output performance constraints, while the convergence rate is further enhanced by combining the finite-time convergence property of the fast terminal sliding mode control. Simulation results attest to the effectiveness of the strategy in this paper. Compared with the adaptive fast terminal sliding mode control, the strategy effectively utilizes the capability of the actuator within the allowed range and reduces the convergence time by more than half with similar control inputs.

Event-Triggered Adaptive Fuzzy Finite-Time Output Feedback Control for Stochastic Nonlinear Systems With Input and Output Constraints

This paper focuses on the problem of designing an adaptive fuzzy event-triggered finite-time output feedback control for stochastic nonlinear systems with input and output constraints. A fuzzy observer is designed to estimate the unmeasured states. The quartic asymmetric time-varying barrier Lyapunov function is established to ensure constraint satisfaction. By utilizing the stochastic theory, finite-time command filtered backstepping method and event-triggered mechanism, a finite-time event-triggered controller is recursively designed, which can not only guarantee finite-time convergent property, but also reduce communication pressure. Meanwhile, the matter of “explosion of complexity” is removed by introducing the finite-time command filter and the effect of filtered errors is offset by constructing error compensation signals. Moreover, an auxiliary system is introduced to handle the input constraint. Finally, the effectiveness of the theoretical results is demonstrated by the simulation example.

Modulated multi-sliding-surface attitude tracking control for all-electric propulsion satellites

This work presents an attitude tracking control method suitable for satellites only using on–off electric thrusters of constant magnitude. It is practically a globally stable finite-time controller modulated by the Pulse-Width-Pulse-Frequency (PWPF) algorithm. Two sliding surfaces are designed to insure a finite-time convergence property. An automatic generator for producing the time-related key parameter is proposed, leaving only two scalar parameters to fine tune values. Typical satellite control modes are used to validate the effectiveness of the proposed method. Results show that it is feasible not only for attitude tracking problem with various errors taken into account, but also for six degree-of-freedom control. Flexibility of trading-off the control accuracy and number of thruster firings can be achieved by the PWPF parameter tuning, which makes it a promising candidate for future use in real practise.

Adaptive finite-time optimised impedance control for robotic manipulators with state constraints

In this paper, an adaptive finite-time impedance control strategy based on optimised backstepping (OB) technique is proposed for robotic manipulators subject to state constraints. The existing OB methods approximately solve the intractable Hamilton-Jacobi-Bellman equation by reinforcement learning (RL) with Bellman residual error, which has intricate actor-critic updating laws and persistence excitation requirement. To overcome this drawback, we construct the simplified RL updating laws by converting the problem to the solution of a positive-definite function, which is composed of actor-critic network weights. Then, the simplified RL updating laws can significantly reduce the controller complexity and relax the persistence excitation. Based on the barrier Lyapunov function, a barrier-type performance index function is constructed for the optimised controller under state constraints. The finite-time stability theory can guarantee the finite-time convergence property of the closed loop system without violating the prescribed constraints. Finally, we demonstrate the effectiveness of the proposed method in the simulation example with environment-robot interaction.

Adaptive neural network observer‐based filtered backstepping control for nonlinear systems with fuzzy dead‐zone and uncertainty

SummaryThis study proposes a high gain observer‐based filtered backstepping control scheme for nonlinear systems with unknown dynamics and dead‐zone constraint. Majority of the previously papers describe dead‐zone by the conventional certain models, assume that all state variables are available and invoke the backstepping or conventional dynamic surface control (CDSC) approaches for controller design. This work describes the dead‐zone by the fuzzy model, invokes radial basis function neural network (RBFNN) to model the unknown functions and proposes a RBFNN‐based high gain observer to estimate unmeasurable states. To avoid the complexity explosion in the backstepping approach and eliminate the sensitivity to the time‐constant of the first‐order filters in the CDSC, a nonlinear tracking differentiator (NTD) with finite time convergence property is used instead of the first‐order filters in the CDSC to obtain derivative of the virtual inputs. Moreover, the error compensation mechanism and auxiliary signal are proposed to eliminate the influence of the filtering error and input nonlinearity, respectively. Despite the fuzzy dead‐zone, the proposed scheme makes the closed‐loop signals uniformly ultimately bounded. In comparison with the existing results, (i) some assumptions like certain model of dead‐zone and full state measurement are removed, (ii) “explosion of complexity” and sensitivity to the time constant of the first‐order filters are eliminated, (iii) effect of the filtering error and input nonlinearity are compensated, (iv) number of adjustable parameters and online computational burden are decreased effectively. Simulation results on the two well‐known examples verify the effectiveness and applicability of the proposed new design technique.

Full Attitude Control of a Quadrotor Using Fast Nonsingular Terminal Sliding Mode With Angular Velocity Planning

In this article, a fast nonsingular terminal sliding mode combined with angular velocity planning (FNTSM-AVP) controller is developed to achieve high-speed, accurate, and robust attitude tracking performance for a quadrotor. The proposed controller employs a sliding surface function with a continuous arctangent function, which can effectively eliminate chattering behavior on the quadrotor while retaining the speed and accuracy of attitude tracking. Unlike conventional methods, the proposed controller preferentially recovers the roll and pitch angles at desired attitude by planning the shortest arrival path, which is effective to improve the tracking performance. Finally, the finite time convergence and zero tracking error properties of the closed-loop control system under the proposed FNTSM-AVP are analyzed and proved. The aggressive flight experiments demonstrate the advantage of faster convergence and fewer chattering behavior of the proposed method compared to the existing FNTSM controller.

Indirect adaptive fuzzy finite time synergetic control for power systems

Introduction. Budget constraints in a world ravenous for electrical power have led utility companies to operate generating stations with full power and sometimes at the limit of stability. In such drastic conditions the occurrence of any contingency or disturbance may lead to a critical situation starting with poorly damped oscillations followed by loss of synchronism and power system instability. In the past decades, the utilization of supplementary excitation control signals for improving power system stability has received much attention. Power system stabilizers (PSS) are used to generate supplementary control signals for the excitation system in order to damp low-frequency oscillations caused by load disturbances or short-circuit faults. Problem. Adaptive power system stabilizers have been proposed to adequately deal with a wide range of operating conditions, but they suffer from the major drawback of requiring parameter model identification, state observation and on-line feedback gain computation. Power systems are nonlinear systems, with configurations and parameters that fluctuate with time that which require a fully nonlinear model and an adaptive control scheme for a practical operating environment. A new nonlinear adaptive fuzzy approach based on synergetic control theory which has been developed for nonlinear power system stabilizers to overcome above mentioned problems. Aim. Synergetic control theory has been successfully applied in the design of power system stabilizers is a most promising robust control technique relying on the same principle of invariance found in sliding mode control, but without its chattering drawback. In most of its applications, synergetic control law was designed based on an asymptotic stability analysis and the system trajectories evolve to a specified attractor reaching the equilibrium in an infinite time. In this paper an indirect finite time adaptive fuzzy synergetic power system stabilizer for damping local and inter-area modes of oscillations for power systems is presented. Methodology. The proposed controller design is based on an adaptive fuzzy control combining a synergetic control theory with a finite-time attractor and Lyapunov synthesis. Enhancing existing adaptive fuzzy synergetic power system stabilizer, where fuzzy systems are used to approximate unknown system dynamics and robust synergetic control for only providing asymptotic stability of the closed-loop system, the proposed technique procures finite time convergence property in the derivation of the continuous synergetic control law. Analytical proofs for finite time convergence are presented confirming that the proposed adaptive scheme can guarantee that system signals are bounded and finite time stability obtained. Results. The performance of the proposed stabilizer is evaluated for a single machine infinite bus system and for a multi machine power system under different type of disturbances. Simulation results are compared to those obtained with a conventional adaptive fuzzy synergetic controller.

Distributed Finite-Time Safety Consensus Control of Vehicle Platoon With Senor and Actuator Failures

Due to the numerical subsystems and functionalities among the platoon, failures easily occur and lead vehicle safety and stability to deteriorate. With the surge of platoon connectivity, the abnormal states arising in faulty vehicles will have an impact on adjacent healthy vehicles, potentially resulting in severe traffic accidents. To address the aforementioned problems and improve the functional safety of platoons, a systemic anti-fault safety consensus strategy is proposed in this article to deal with sensor faults, LOE actuator faults, and bias actuator faults simultaneously. To tackle the necessary defects, this safety consensus technique can be split into fault detection tasks and fault-tolerant control tasks. Initially, a distributed finite-time observer is employed to detect sensor faults through the index between the reconstructed states and received states. In the presence of actuator faults, a novel adaptive finite-time fault parameter estimation law collaboration with a feasible differential observer is developed to assess the precise fault extent. In addition, to improve the estimation performance of the adaptive fault parameter estimation law, an integral discounted cost function is constructed in this paper to obtain the optimal learning gain. Via theoretical analysis, the solution for the minimal value of the cost function and the finite-time convergence property are demonstrated in this paper. Furthermore, by utilizing this novel adaptive optimal finite-time estimation law, a novel distributed adaptive finite-time fault-tolerant controller is constructed to compensate for failures in platoon. Finally, the effectiveness and feasibility of our safety consensus strategy are verified by real-time simulations through the dSPACE platform.

Admissible edge-dependent finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time

Considering that the property of finite-time convergence is of great significance for practical filters, this article is focused on the finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time. Different from the case of the general mode-dependent switching, the information of the admissible switching edge is fully exploited to derive more preferable filtering scheme. In order to further reduce the design conservatism, the multiple discontinuous Lyapunov function approach, which incorporates the concept of admissible edge-dependent average dwell time, is used to analyze the finite-time stability and ℋ∞ performance of the filtering error system. This approach assigns each subsystem of the error system with a discontinuous function rather than a continuous one, so it is less conservative compared to the conventional multiple Lyapunov function approach. First, the finite-time boundedness and ℋ∞ performance is analyzed for general switched nonlinear systems with admissible edge-dependent average dwell time. Then, a sufficient condition of the finite-time boundedness and ℋ∞ performance is derived for the filtering error system based on the obtained criterion. Subsequently, an admissible edge-dependent finite-time filtering scheme is proposed for the underlying system in terms of linear matrix inequalities. Finally, the proposed filter design scheme is applied to a DC–DC power converter circuit, and its effectiveness and superiority are verified by the simulation results.

Fixed-Time Stable Neurodynamic Flow to Sparse Signal Recovery via Nonconvex L1-β2-Norm.

This letter develops a novel fixed-time stable neurodynamic flow (FTSNF) implemented in a dynamical system for solving the nonconvex, nonsmooth model L1-β2, β∈[0,1] to recover a sparse signal. FTSNF is composed of many neuron-like elements running in parallel. It is very efficient and has provable fixed-time convergence. First, a closed-form solution of the proximal operator to model L1-β2, β∈[0,1] is presented based on the classic soft thresholding of the L1-norm. Next, the proposed FTSNF is proven to have a fixed-time convergence property without additional assumptions on the convexity and strong monotonicity of the objective functions. In addition, we show that FTSNF can be transformed into other proximal neurodynamic flows that have exponential and finite-time convergence properties. The simulation results of sparse signal recovery verify the effectiveness and superiority of the proposed FTSNF.

Sparse signal reconstruction via recurrent neural networks with hyperbolic tangent function

In this paper, several recurrent neural networks (RNNs) for solving the L1-minimization problem are proposed. First, a one-layer RNN based on the hyperbolic tangent function and the projection matrix is designed. In addition, the stability and global convergence of the previously presented RNN are proved by the Lyapunov method. Then, the sliding mode control technique is introduced into the former RNN to design finite-time RNN (FTRNN). Under the condition that the projection matrix satisfies the Restricted Isometry Property (RIP), a suitable Lyapunov function is constructed to prove that the FTRNN is stable in the Lyapunov sense and has the finite-time convergence property. Finally, we make a comparison of the proposed RNN and FTRNN with the existing RNNs. To achieve this, we implement experiments for sparse signal reconstruction and image reconstruction. The results further demonstrate the effectiveness and superior performance of the proposed RNN and FTRNN.

Unified framework for implicit-Euler implementation of second-order sliding mode controllers

Chattering, which mainly occurs by inappropriate discretization, has been one of the significant problems of restricting sliding mode controllers’ practical applications. Toward this problem, this paper presents a unified implicit-Euler discretization framework for the chattering-attenuated implementation of a class of homogeneous second-order sliding mode controllers. The discretization framework calculates a proper value inside the set-valued signum function by considering a particular relation between the current system state and the nonlinear sliding surface. An equivalent relation is employed for mapping an equation containing two cascade set-valued signum functions into an equation including nested project functions. The finite-time convergence and accuracy properties of the proposed framework are analyzed. The results of numerical examples show that the proposed framework significantly attenuates the chattering. It is also shown that the proposed framework provides higher accuracy than the explicit-Euler method.

Model free based finite time fault‐tolerant control of robot manipulators subject to disturbances and input saturation

AbstractThis article proposed a model free based finite time fault‐tolerant control (FTC) method of robot manipulators subject to unknown disturbances, input saturation, and actuator faults. First, an accuracy‐driven integral terminal sliding mode surface (ADITSMS) was constructed, and the finite‐time convergence and the parametric analysis were investigated. Second, based on the proposed ADITSMS, a double‐loop control framework which consists of the position loop and velocity loop was proposed, and the finite time command filtered algorithm was utilized to estimate the derivative of the virtual control law of the position loop, thus a novel finite‐time robust adaptive FTC was proposed. Third, to eliminate the negative effects of the faults and disturbances and implement model free control, the uncertainty and disturbance estimation is designed, and a novel adaptive algorithm is developed to suppress the estimation error whose upper bound or derivative was not required. Meanwhile, an auxiliary system is constructed to deal with the input saturation. The proposed controller possess the superiorities of model free nature and global finite time convergence property. Finally, numerical simulations and comparison studies were conducted to demonstrate the effectiveness of the proposed controller.

Finite-time trajectory tracking control for under-actuated unmanned surface vessels with saturation constraint

This article investigates the robust finite-time trajectory tracking control problem of under-actuated unmanned surface vessels (USVs) subject to model uncertainties, saturation constraints, and external disturbances. Firstly, the kinematic and dynamic models of under-actuated USVs are transformed into an equivalent tracking error dynamic by resorting to a novel output redefinition-based dynamic transformation (ORDT). Secondly, a smooth dead-zone operator-based model (DOBM) is introduced to deal with the control input saturation constraints problem. On basis of these, a sliding mode-based controller (SMC) is developed, which possesses the properties of chattering-free and finite-time convergence. Later, Lyapunov stability proved that the proposed control strategy is capable of guaranteeing the boundedness of all closed-loop signals, in spite of the parametric uncertainties, external disturbance, and input saturation constraints. Finally, numerical simulations illustrate the effectiveness of the proposed control approach.

Fuzzy Approximation-Based Fractional-Order Nonsingular Terminal Sliding Mode Controller for DC–DC Buck Converters

This article presents a simple and systematic approach to synthesize a robust adaptive fuzzy fractional-order nonsingular terminal sliding mode controller (AFFO-NTSMC) to improve the output voltage tracking control performance of the dc–dc buck converters. The hybrid control method of fractional-order (FO) calculus and NTSMC are combined to create a FO-NTSMC, in which a new FO nonsingular terminal sliding mode surface is established. The idea behind this strategy is the increased flexibility achieved by FO calculation, improving robustness to disturbances and parameters variations provided by the traditional sliding mode controllers as well as finite time convergence properties of the output voltage error to the equilibrium point during the output load changes, simultaneously. In addition, a fuzzy logic system with online adaptive learning algorithm is designed to provide smooth chattering in switching control signal. The stability of the closed-loop system is carefully demonstrated by Lyapunov's theorem. Experimental measurements from a laboratory prototype are presented to demonstrate the effectiveness of the proposed AFFO-NSTSMC algorithm.

Global fast terminal sliding mode based radial basis function neural network for accurate fault estimation in nonlinear systems

The problem of quick and accurate fault estimation in nonlinear systems is addressed in this article by combining the technique of radial basis function neural network (RBFNN) and global fast terminal sliding mode control (GFTSMC) concept. A new strategy to update the neural network weights, by using the global fast terminal sliding surface instead of conventional error back propagation method, is introduced to achieve real time, quick and accurate fault estimation which is critical for fault tolerant control system design. The combination of online learning ability of RBFNN, to approximate any nonlinear function, and finite time convergence property of GFTSMC ensures quick detection and accurate estimation of faults in real time. The effectiveness of the proposed strategy is demonstrated through simulations using a nonlinear model of a commercial aircraft and considering a wide range of sensors and actuators faults. The simulation results show that the proposed method is capable of quick and accurate online fault estimation in nonlinear systems and shows improved performance as compared to conventional RBFNN and other techniques existing in literature.

Practical Finite-Time Event-Triggered Trajectory Tracking Control for Underactuated Surface Vessels With Error Constraints

This paper proposes a robust trajectory tracking control method for underactuated unmanned surface vessels (USVs) with input saturation based on a relative threshold event-triggered mechanism (ETM), prescribed performance and finite-time convergence. First, to address the challenges posed to controller design by the nondiagonal inertia matrix, coordinate transformation technology is adapted to change the vessel position. An asymmetric smooth saturation function is employed to address the input saturation problem. Second, an adaptive neural network (NN) is developed to approximate the uncertain nonlinear dynamics and external environmental disturbances. Third, the performance function constraints are integrated into the tan-type barrier Lyapunov function (BLF) to greatly enhance the robustness of the system by combining with the finite-time convergence property. In addition, the dynamic surface control (DSC) technique is employed to address the differential explosion problem. Subsequently, a relative threshold ETM is incorporated into the controller to reduce the communication burden. The rationality and feasibility of the proposed algorithm are confirmed by theoretical analysis and numerical simulations.

Convergence and robustness of bounded recurrent neural networks for solving dynamic Lyapunov equations

Recurrent neural networks have been reported as an effective approach to solve dynamic Lyapunov equations, which widely exist in various application fields. Considering that a bounded activation function should be imposed on recurrent neural networks to solve the dynamic Lyapunov equation in certain situations, a novel bounded recurrent neural network is defined in this paper. Following the definition, several bounded activation functions are proposed, and two of them are used to construct the bounded recurrent neural network for demonstration, where one activation function has a finite-time convergence property and the other achieves robustness against noise. Moreover, theoretical analyses provide rigorous and detailed proof of these superior properties. Finally, extensive simulation results, including comparative numerical simulations and two application examples, are demonstrated to verify the effectiveness and feasibility of the proposed bounded recurrent neural network.

Adaptive incremental sliding mode control for a robot manipulator

An adaptive incremental sliding mode control (AISMC) scheme for a robot manipulator is presented in this paper. Firstly, an incremental backstepping (IBS) controller is designed using time-delay estimation (TDE) to reduce dependence on the mathematical model. After substituting IBS controller into the nonlinear system, a linear system w.r.t. tracking errors is obtained while TDE error is the disturbance. Then, the AISMC scheme, including a nominal controller and an SMC, is developed for the resulted linear system to improve control performance. According to the equivalent control method, the SMC in the AISMC scheme is to handle TDE error. To receive optimal control performance at the sliding manifold, an LQR controller is selected as the nominal controller. The SMC is designed based on positive semi-definite barrier function (PSDBF) since it prevents switching gains from being over/under-estimated, and two practical problems are addressed in this paper: A new PSDBF is designed and conservative (large) setting bounds affecting tracking precision and/or system stability are avoided; An improved PSDBF based SMC is developed where the PSDBF and an adaptive parameter are used simultaneously to regulate switching gains, and the system is still stable when sliding variable occasionally exceeds the predefined vicinity. Moreover, finite-time convergence property of the sliding variable is strictly analyzed. Finally, real-time experiments are conducted to verify the effectiveness of the proposed control method.