A Recurrent Neural Network With Explicitly Definable Convergence Time for Solving Time-Variant Linear Matrix Equations

Time-variant linear matrix equations (TVLMEs) are ubiquitous in engineering. To solve TVLMEs, various zeroing neural network (ZNN) models have been developed. These ZNNs globally converge to the solution of TVLMEs either in infinity long time or in finite time. However, even the convergence time of a finite-time convergent ZNN is implicit and closely dependent on the initial condition of a problem. This may reduce its applicability to time-critical applications in practice. To overcome this problem, this paper for the first time accelerates a ZNN to fixed-time convergence using a novel activation function. The convergence time of the proposed ZNN can be antecedently defined as an explicit parameter. Theoretically, its fixed-time convergence and robustness properties are rigorously proved. Comparative numerical results substantiate the superior convergence and robustness performance of the fixed-time convergent ZNN for TVLMEs solving. Additionally, the fixed-time convergent ZNN is applied to motion planning of a redundant robotic arm.

I. Introduction

Linear matrix equations play a significant role in science and engineering fields such as control systems design, stability analysis, image processing, and robot inverse kinematics [1], [2]. For solving linear matrix equations, there exist two typical classes of approaches: numerical algorithms and neural-dynamic methods [3], [4]. Many numerical algorithms are intrinsically developed for solving time-invariant linear matrix equations and they require minimal arithmetic operations that are usually proportional to the cube of the matrix size [5], [6]. When applied these numerical algorithms to solving time-variant linear matrix equations (TVLMEs), numerical algorithms must be executed at each sampled time step and calculations must be completed within every time step. A numerical algorithm fails if the sampling is too frequent to allow the algorithm to perform the computations within a time step, not to mention a more challenging case to continuously solve TVLMEs in real time. Due to the parallel distributed nature and convenience of hardware implementation [4], [7], [8], neural-dynamic methods have been maturely investigated as powerful alternatives for computation in real time. One of the popular neural-dynamic solutions is gradient-based neural networks (GNNs, or termed Hopfield neural networks) with scalar-valued energy functions and explicit dynamics, which have been successfully applied to a broad range of problems solving [4], [9], [10]. For instance, a new energy function was formulated for solving nonlinear algebraic equations in [9], where the modified GNN could provide fast convergence and extremely accurate solutions for all solvable nonlinear algebraic equations. In [10], a GNN was applied to optimal control of an automotive vehicle system by solving an algebraic Riccati matrix equation. However, like the aforementioned numerical algorithms, GNNs are also inherently proposed for time-invariant problems solving and there always exist lagging-behind errors between the GNN solutions and the theoretical solutions for TVLMEs solving [4]. Due to this fact, a more powerful approach that can eliminate these lagging-behind errors is desirable and preferable.