A CNN-based approach for a class of non-standard hyperbolic partial differential equations modeling distributed parameters (nonlinear) control systems

Abstract

The present paper considers a systematic approach within the framework of Neural Mathematics for constructing a computational procedure. This procedure aims to solve a class of problems arising from the control of the systems with distributed parameters; these systems are modeled by second-order one-dimensional hyperbolic partial differential equations (hPDEs) with non-standard boundary conditions. The procedure reveals an explicit algorithmic parallelism and is mainly based on the combination of two powerful “tools”: a convergent Method of Lines (MoL) and the Cellular Neural Network (CNN) paradigm. The role of the Courant-Isaacson-Rees rule and of the Riemann invariants for a correct application of the MoL is emphasized. The procedure is illustrated on a control engineering application – the overhead crane with flexible cable – within a more general context which includes modeling based on the generalized Hamilton variational principle, synthesis of a stabilizing controller via the Control Lyapunov Functional (CLF), qualitative analysis, numerical solving using the proposed computational procedure, numerical simulations and the evaluation of the performances for the closed loop system. The procedure ensures the convergence of the approximation, preserves the basic properties and the Lyapunov stability of the solution of the initial problem and reduces the systematic errors.