Discrete-form time-dependent quadratic programming (DF-TDQP) problem, as a type of time-dependent problems, is prevalent in science and engineering. Currently, there are many studies on the discretization of continuous time-dependent problems despite the fact that researchers have made different breakthroughs using recurrent neural networks (RNNs) in dealing with discrete-form time-dependent problems, relatively limited research has been devoted to the direct discretization approach. To solve the DF-TDQP problem, a novel direct discretization approach is introduced. Based on this approach, the corresponding discrete-form recurrent neurodynamics (DFRN) model is developed. Furthermore, on this basis, a novel enhanced discrete-form recurrent neurodynamics (E-DFRN) model is further established to obtain high-accuracy optimal solutions for such DF-TDQP problems. The accuracy and convergence of both the DFRN model and E-DFRN model are explained through theoretical analysis, and their effectiveness and superiority are validated based on numerical experiments. Finally, the applicability of these models to the tracking tasks of planar manipulators is verified.

Recurrent neural network (RNN) is a neurodynamic method designed to tackle time-varying problems in various technical domains, which are widely derived from scientific research and practical applications. It should be noted that traditional models often lack an effective capability to suppress nonlinear time-varying noise during the design process, and thus may encounter many difficulties in practical applications. This article presents a novel RNN model for solving the continuous time-varying matrix pseudoinverse, which has a significant characteristic of double integral-reinforcing (DIR) term and is termed DIR continuous-time RNN (DIR-CT-RNN) model. Correspondingly, using the discretization formula, a DIR discrete-time RNN (DIR-DT-RNN) is presented for solving the discrete time-varying matrix pseudoinverse. The theoretical results present that the DIR-DT-RNN model converges toward the theoretical solution under the discrete time-unvarying constant (DTU-C) noise or discrete time-varying linear (DTV-L) noise interference. Under the discrete time-varying quadratic (DTV-Q) noise interference, the proposed model converges to a constant that relates to the design parameters. In addition, simulation results, including an application for trajectory tracking of three-link robotic manipulator, which come from practical engineering background, verify the effectiveness and superiority of DIR-DT-RNN model for solving the time-varying matrix pseudoinverse under various types of noise interference.

Discrete time-variant equation systems represent a typical and complex problem across various disciplines. With the increasing complexity of systems in various fields, traditional methods have been unable to effectively deal with the current discrete time-variant equation systems, especially in the dynamic engineering problem. Generally speaking, traditional methods are typically limited to considering discrete time-variant equation systems in ideal state, and there is a lack of deep research about more intricate disturbance states. This paper introduces a new recurrent neural network (RNN) algorithm, termed the discrete-time double-integration-enhanced RNN (DT-DIE-RNN) algorithm, for handling discrete time-variant equation systems (including discrete time-variant linear and nonlinear equation system) under discrete square-time-variant disturbance. Firstly, the continuous-time double-integration-enhanced RNN (CT-DIE-RNN) algorithm is presented for solving discrete time-variant linear and nonlinear equation systems by using double-integral-type error function. Secondly, the corresponding discrete-time RNN algorithm is presented, and the convergence and precision of such an algorithm are theoretically analyzed. Finally, the effectiveness and superiority of the proposed DT-DIE-RNN algorithm for solving discrete time-variant linear and nonlinear equation systems are supported by comparative numerical experiments, and these theoretical results are further verified by robot manipulator applications.Note to Practitioners—Generally speaking, previous algorithms are unable to guarantee the operation of robot manipulator stably with discrete square-time-variant disturbance. In this study, we design a new RNN algorithm which possesses stronger anti-disturbance capability for solving discrete time-variant equation systems, and such the algorithm is applied to the tracking control of robot manipulator. In the discrete-time environment, in order to solve the discrete time-variant problem smoothly and stably, it is necessary to ensure that the error in each time period is within an acceptable range. The basic guidelines of this paper are as follows. First of all, a double-integration RNN algorithm is constructed, i.e., discrete-time double-integration-enhanced RNN (DT-DIE-RNN) algorithm. Then the effectiveness and superiority of the DT-DIE-RNN algorithm are verified by numerical experiment and engineering simulation, respectively. In summary, the proposed RNN algorithm can be seen as a new breakthrough in the research field of discrete-time RNN algorithm.

In this article, the discrete-form time-variant multi-augmented Sylvester matrix problems, including discrete-form time-variant multi-augmented Sylvester matrix equation (MASME) and discrete-form time-variant multi-augmented Sylvester matrix inequality (MASMI), are formulated first. In order to solve the above-mentioned problems, in continuous time-variant environment, aided with the Kronecker product and vectorization techniques, the multi-augmented Sylvester matrix problems are transformed into simple linear matrix problems, which can be solved by using the proposed discrete-time recurrent neural network (RNN) models. Second, the theoretical analyses and comparisons on the computational performance of the recently developed discretization formulas are presented. Based on these theoretical results, a five-instant discretization formula with superior property is leveraged to establish the corresponding discrete-time RNN (DTRNN) models for solving the discrete-form time-variant MASME and discrete-form time-variant MASMI, respectively. Note that these DTRNN models are zero stable, consistent, and convergent with satisfied precision. Furthermore, illustrative numerical experiments are given to substantiate the excellent performance of the proposed DTRNN models for solving discrete-form time-variant multi-augmented Sylvester matrix problems. In addition, an application of robot manipulator further extends the theoretical research and physical realizability of RNN methods.