A discontinuous recurrent neural network with predefined time convergence for solution of linear programming

Abstract

The aim of this paper is to introduce a new recurrent neural network to solve linear programming. The main characteristic of the proposed scheme is its design based on the predefined-time stability. The predefined-time stability is a stronger form of finite-time stability which allows the a priori definition of a convergence time that does not depend on the network initial state. The network structure is based on the Karush-Kuhn-Tucker (KKT) conditions and the KKT multipliers are proposed as sliding mode control inputs. This selection yields to an one-layer recurrent neural network in which the only parameter to be tuned is the desired convergence time. With this features, the network can be easily scaled from a small to a higher dimension problem. The simulation of a simple example shows the feasibility of the current approach.

Predefined-time stabilization of high order systems

“…Definition 2.3 ( [15], [16]): Let T be the set of all the bounds of the settling time function for the system (1), i.e.,…”

“…To overcome the above, a class of first-order dynamical systems with the minimum upper bound of the fixed stabilization time equal to their only tuning gain has been studied [15], [16], similarly a class of second-order systems with similar features is presented in [17]. It is said that these systems exhibit the property of predefined-time stability.…”

Predefined-time tracking of a class of mechanical systems

“…Definition 2.3 (Settling-time set and its minimum bound [13], [14]): Let T be the set of all the bounds of the settling time function for the system (1), i.e.,…”

“…Definition 2.4 (Predefined-time stability [13]): For the case of fixed time stability when the time T f defined in (3) can be tuned by a particular selection of the parameters ρ of the system (1), it is said that the origin of the system (1) is predefined-time stable.…”

“…they are much larger than the true fixed stabilization time (see for example [12], where the upper bound estimation is approximately 100 times larger than the actual true fixed stabilization time). To overcome the above, another class of dynamical systems which exhibit the property of predefined-time stability, have been studied [13], [14]. For this systems, the minimum upper bound of the fixed stabilization time appears explicitly in their tuning gains.…”