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• The different-level nonlinear system is first investigated. • The zeroing equivalency between different levels is proposed. • The new algorithm is proposed based on the equivalency and a discretization formula. • The problem is solved in a discrete-time manner with the future-instant solution predicted. • The problem of robotic system control with the end-effector orientation and fault tolerance considered simultaneously is solved. In this paper, we consider a special kind of time-dependent nonlinear system that includes different-level subsystems (i.e., a subsystem with respect to time-dependent solution and a subsystem with respect to its time derivative). To solve this kind of time-dependent nonlinear system, an equivalency between different-level subsystems is proposed and termed as zeroing equivalency based on the zeroing neural network method. This kind of time-dependent nonlinear system can be solved in a discrete-time manner with future-instant solution predicted at current instant based on the zeroing equivalency and a discretization formula. Prediction ability can perfectly meet the requirements of real-time computation, which is crucial in engineering fields. In addition, different-level algorithms are further employed to solve the problem of robotic system control considering additional restrictions. The position tracking control and end-effector orientation control are simultaneously achieved with even fault tolerance by utilizing the proposed algorithms. [ABSTRACT FROM AUTHOR]

Abstract Discrete time-varying linear system (LS) is a fundamental topic in science and engineering. However, conventional methods essentially designed for time-invariant LS generally assume that LS is time-invariant during a small time interval (i.e., sampling gap) for solving time-varying LS. This assumption quite limits their precision because of the existing of lagging errors. Discarding this assumption, Zhang neural dynamics (ZND) method improves the precision for LS solving, which is a great alternative for the solving of discrete time-varying problems. Note that precision solutions to discrete time-varying problems depend on discretization formulas. In this paper, we propose a new ZND model to solve the discrete time-varying LS. The discrete time-varying division is a special case of discrete time-varying LS with the solution being a scalar while it is usually studied alone. Considering the above inner connection, we further propose a special model for solving the discrete time-varying division. Moreover, as an application of discrete time-varying LS, the discrete time-varying quadratic programming (QP) subject to LS is also studied. The convergence and precision of proposed models are guaranteed by theoretical analyses and substantiated by numerous numerical experiments. [ABSTRACT FROM AUTHOR]

Being two famous neural networks, the error back-propagation (BP) algorithm based neural networks (i.e., BP-type neural networks, BPNNs) and Hopfield-type neural networks (HNNs) have been proposed, developed, and investigated extensively for scientific research and engineering applications. They are different from each other in a great deal, in terms of network architecture, physical meaning and training pattern. In this paper of literature-review type, we present in a relatively complete and creative manner the common natures of learning between BP-type and Hopfield-type neural networks for solving various (mathematical) problems. Specifically, comparing the BPNN with the HNN for the same problem-solving task, e.g., matrix inversion as well as function approximation, we show that the BPNN weight-updating formula and the HNN state-transition equation turn out to be essentially the same. Such interesting phenomena promise that, given a neural-network model for a specific problem solving, its potential dual neural-network model can thus be developed. [ABSTRACT FROM AUTHOR]