To realize repetitive motion planning (RMP) for redundant robot manipulators in the presence of initial shifts, this paper introduces an initial-rectifying-constraint optimization scheme that reformulates the problem as a constrained time varying quadratic programming task. Using the initial-rectifying zeroing neural network (IRZNN) proposed in this article, the initial-rectifying-constraint optimization problem can be solved, and a repeatable solution with prescribed-time convergence can be obtained. The polynomial rectifying functions are developed to maintain smooth operation of manipulators during rectification. Convergence of the IRZNN model in solving the initial-rectifying-constraint optimization scheme is analyzed. The proposed scheme takes the initial shift problem into account, realizes RMP for redundant manipulators, and achieves prescribed-time convergence of the end-effector position error. The simulation and experiment results validate the effectiveness and practicality of the proposed approach. Note to Practitioners—The motivation of this article is the joint-angular-drift phenomenon found in closed trajectory tracking of redundant manipulators. This phenomenon, where a closed path of the end-effector may result in non-closed joint motion, is regarded as a potential menace in the operation of redundant manipulators. Existing solutions assume the initial state of joints aligns with the desired one, which is too idealistic and usually requires additional adjustment to meet the assumption. This article suggests a novel RMP scheme to alleviate the joint-angular-drift phenomenon in the presence of initial joint shifts. The proposed scheme enables the specification of the settling time for the end-effector position error in accordance with task requirements. To be more practical, polynomial rectifying functions are formed to facilitate smooth operation of robotic manipulators.

In this paper, the robust adaptive learning bipartite consensus problem for heterogeneous nonlinear multiagent systems with unknown control gains, external disturbances, and actuator constraints under signed directed graphs is investigated. A novel neural distributed protocol is proposed, whose key techniques lie in the introduction of deadzone-modified Lyapunov functions and integral Lyapunov functions into the command filtered backstepping design. The former enhances the robustness of the undertaken system by attenuating the impact of complex uncertainties and transforms the robustness problem into a convergence one by introducing the deadzones, while the latter efficiently tackles both the state-dependent control gain functions and the input saturation nonlinearities, simplifying the consensus design significantly. In addition, the requirement of the command filtered design for the control gain functions is relaxed by the appropriate system transformation. Furthermore, the incremental adaptive algorithm takes the place of the integral adaptation for parameter learning, avoiding numerical integration in implementation. The theoretical results of the performance analysis are presented in detail, in which the boundedness of all closed-loop variables is examined, and the asymptotic consensus is achieved, in the sense that the bipartite synchronization error converges to a pre-specified region asymptotically. Numerical results verify the feasibility of the presented scheme. Note to Practitioners—This paper is devoted to the problem of robust adaptive learning bipartite consensus for heterogeneous nonlinear multiagent systems (MASs). It is important to investigate heterogeneous MASs in practical engineering applications, and a typical example is the air-ground cooperative combat system consisting of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs). In addition, the bipartite consensus allows MASs to accomplish more diverse tasks. However, uncertain nonlinearities, external disturbances, and actuator constraints are widespread in system dynamics. Moreover, the unknown control gains make consensus design challenging. These issues are well addressed by adopting the key techniques including deadzone-modified strategy, integral Lyapunov synthesis and incremental adaptive learning mechanism. Furthermore, the proposed scheme guarantees the asymptotic convergence of the bipartite synchronization errors with a pre-specified accuracy, and enhances the system robustness under complex heterogeneous nonlinearities. Therefore, the presented method contributes to practical applications.

In this article, the fuzzy adaptive learning bipartite consensus problem is addressed for strict-feedback multiagent systems subject to state constraints under a structurally unbalanced signed graph. An agent hierarchical categorization strategy is suggested, with the assistance of which the requirements on the network topology can be relaxed, and the bipartition of all agents is easily achieved, even if the communication graph is structurally unbalanced. In addition, taking advantage of the treatment with symmetric fractional barrier Lyapunov functions, which transforms asymmetric constrained scenarios into symmetric cases and subsequently into equivalent unconstrained ones, it facilitates the realization of the bipartite consensus under state constraints and the performance analysis is greatly simplified. Furthermore, the fuzzy logic systems are employed to approximate the uncertainties involved in the system. It is shown that the boundedness of all variables of the closed-loop system undertaken and the convergence of consensus errors are established, even for the structurally unbalanced topology graph. Numerical results demonstrate feasibility of the presented consensus scheme.

Learning control is applicable to systems that operate periodically or over finite time intervals. Currently, there is a lack of research results about learning control approaches to infinite-duration tracking, without requiring periodicity or repeatability. This article addresses the problem of adaptive learning control (ALC) for systems performing infinite-duration tasks. Instead of using integral adaptation, incremental adaptive mechanisms are exploited, by which the numerical integration for implementation can be avoided. The comparison with the conventional integral adaptive mechanisms indicates that the suggested methodology can be an alternative to the adaptive system designs. Using an error-tracking approach, the approximation-based backstepping design is carried out for systems in the strict-feedback form, where a novel integral Lyapunov function is shown to be efficient in the treatment of state-dependent control gain. Theoretical results for the performance analysis are presented in detail. In particular, the robust convergence of the tracking error is established, while the boundedness of the variables of the closed-loop system is characterized, with the aid of a key technical lemma. It is shown that the proposed control method can provide satisfactory tracking performance and simplify the controller designs. Numerical results are presented to demonstrate effectiveness of the learning control schemes.

This article addresses the problem of adaptive learning control (ALC) for nonlinear systems in nonstrict-feedback form. For parameter learning, an incremental adaptive mechanism is proposed and used as an alternative to integral adaptation, with which numerical integration in implementation can be avoided. Taking advantage of the error-tracking approach, a novel integral Lyapunov function, developed specifically for tackling state-dependent control gains, is incorporated into the approximation-based backstepping design. In addition, the technical challenges associated with nonstrict-feedback structures are successfully overcome, by employing the key property of neural networks in the ALC design. It is shown that with the aid of the technique lemma for robustness analysis, the proposed ALC control strategy guarantees robust convergence of the tracking error, despite the complex uncertainties involved. The design method guarantees the tracking performance and facilitates the implementation of the suggested algorithm. Illustrative examples are provided which verify the effectiveness of the presented ALC control scheme.

This article concerns with terminal recurrent neural network (RNN) models for time-variant computing, featuring finite-valued activation functions (AFs), and finite-time convergence of error variables. Terminal RNNs stand for specific models that admit terminal attractors, and the dynamics of each neuron retains finite-time convergence. The might-existing imperfection in solving time-variant problems, through theoretically examining the asymptotically convergent RNNs, is pointed out for which the finite-time-convergent models are most desirable. The existing AFs are summarized, and it is found that there is a lack of the AFs that take only finite values. A finitely valued terminal RNN, among others, is taken into account, which involves only basic algebraic operations and taking roots. The proposed terminal RNN model is used to solve the time-variant problems undertaken, including the time-variant quadratic programming and motion planning of redundant manipulators. The numerical results are presented to demonstrate effectiveness of the proposed neural network, by which the convergence rate is comparable with that of the existing power-rate RNN.

This article presents a repetitive learning control method, which deals with both parametric and nonparametric uncertainties involved in the control-affine systems with time delays. The desired control is taken as a time-varying parameter, and a learning algorithm is applied for estimation. The time-delay nonlinearity and the state-dependent input gain are handled as the nonparametric uncertainties, for which the adaptive bounding technique is adopted such that the requirement for the knowledge about the system undertaken can be reduced, and the troublesome time delays are effectively treated. The presented control method only needs the lower bound of the input gain is to be known as a priori. Furthermore, it is shown applicable to systems with multiple time delays. The fully saturated learning algorithms are utilized to solve the positive accumulation of estimates so that the possible divergence phenomenon can be avoided. The convergence of the tracking error is established, while the boundedness of the variables in the closed-loop system is assured. The numerical results are presented to verify the effectiveness of the proposed learning control method.

This article concerns studies on finite-time systems (FTSs), which are expected for high-level system synthesis. An FTS usually obeys an attracting law (AL), under which the dynamics of the FTS is governed. This article probes into the analysis of finite-duration convergence of the power-rate ALs, and especially on the estimation for the bound of their settling time functions, without selecting the transition state to be one. It is shown that the settling time is a function of the initial state, and the value of the function varies with the initial state. The existence of the upper bound of the function assures that the duration over which the settling time varies is finite with respect to the initial state. The estimate depends on the transition state, revealing that the bound on the settling time cannot be determined by a certain value, as different transition states are chosen. The optimal transition state in certain sense is acquired in this article, and the comparison is made between the existing estimation and the optimal one. Furthermore, we concern ourselves with the two-phase AL (TPAL) also without fixing the transition state, for which we are able to give an accurate bound of its settling time function. To illustrate the usefulness of the TPAL, we present a finite-duration learning control approach which utilizes the merits of the exiting control methods in coping with both time-varying parametric and nonparametric uncertainties.

In this article, an echo state network (ESN)-based backstepping adaptive iterative learning control scheme is proposed for nonlinear strict-feedback systems performing the same operation repeatedly over a finite-time interval. Different from most of the output tracking approaches, an error-tracking approach is presented using the backstepping technique, such that the tracking error can follow a prespecified error trajectory without any requirement on the initial value of system states. Then, a novel Lyapunov function is constructed to deal with the unknown state-dependent gain function of the controller design. The uncertain nonlinearities are approximated by employing ESNs with simple feedback structures, and the weight update laws are developed by combining the parameter adaptation in the time domain and iteration domain. Moreover, the proposed control scheme is further extended to handle the strict-feedback systems with input saturations. Through the Lyapunov-like synthesis, the closed-loop stability and error convergence of the proposed error-tracking control scheme are analyzed in the presence of the approximation errors. Numerical simulations are provided to verify the effectiveness of the proposed scheme.

A specific class of nonlinear systems, admitting finite-duration attractors (FDAs), is introduced in this paper, and used for consensus of multiagent systems (MASs). The advantage is that one can obtain a closed-form solution for the settling time function of the presented nonlinear systems. Such a system exhibits a two-phase convergence process, and its settling time function takes values over a finite duration. However, the duration width changes with respect to the initial condition. The FDA-based consensus protocol is shown to make the MAS undertaken achieve finite-duration convergence. With illustrative examples, the perfect consensus property is verified for MASs with or without a leader agent.

This paper concerns with the adaptive iterative learning control using neural networks for systems performing repetitive tasks over a finite time interval. Two standing issues of such iterative learning control processes are addressed: one is the initial condition problem and the other is that related to the approximation error. Instead of the state tracking, an error tracking approach is proposed to tackle the problem arising from arbitrary initial shifts. The desired error trajectory is prespecified at the design stage, suitable to different tracking tasks. The initial value of the desired error trajectory for each cycle is required to be the same as that of the actual error trajectory. It is just a requirement for the initial value of the desired error trajectory, but does not pose any requirement for the initial value of the actual error trajectory. It is shown that the actual error trajectory is adjustable and is able to converge to a prespecified neighborhood of the origin, while all variables of the closed-loop system are of uniform boundedness. The robustness improvement in case of nonzero approximation error is made possible due to the use of a deadzone modified Lyapunov functional. The resultant estimation for the bound of the approximation error avoids deterioration in tracking performance. The effectiveness of the designed learning controller is validated through an illustrative example.

An adaptive iterative learning control method is presented in this paper, for SISO time-varying discrete-time systems. In order to estimate the time-varying unknowns, two iterative learning algorithms, fully-saturated iterative learning projection algorithm and fully-saturated iterative learning least square algorithm, are given, respectively. A one-step ahead controller is developed on the basis of the certainty equivalence principle. The stability and convergence of the closed-loop system are established with the aid of the iteration-domain key technical lemma, which is a variant of the existing one, tailored for the analysis purpose in the iterative domain. The complete tracking is achieved over the pre-specified time interval excluding initial instants, as iteration goes to infinity, while all the signals in the closed-loop remain bounded.