Stability of Nonlinear Differential-Algebraic Systems Via Additive Identity

The stability analysis for nonlinear differentialalgebraic systems is addressed using tools from classical control theory. Sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a smallgain-like condition. The study of stability properties for constrained mechanical systems, for a class of Lipschitz differential-algebraic systems and for an academic example is used to illustrate the theory.

Class of numerical methods for differential-algebraic systems with discontinuous right-hand sides

Numerical methods for Differential-Algebraic systems with discontinuous right-hand sides is discussed. A class of continuous Rosenbrock methods are constructed, and numerical experiments show that the continuous Rosenbrock methods are effective. Applying the methods, a fast and high-precision numerical algorithm is given to deal with typical discontinuous parts, which occur frequently in differential-algebraic systems(DAS).

Delay and Its Time-Derivative Dependent Stable Criterion for Differential-Algebraic Systems

In this paper, the stable problem for differential-algebraic systems is investigated by a convex op-timization approach. Based on the Lyapunov functional method and the delay partitioning approach, some delay and its time-derivative dependent stable criteria are obtained and formulated in the form of simple linear matrix inequalities (LMIs). The obtained criteria are dependent on the sizes of delay and its time-derivative and are less conservative than those produced by previous approaches.

Solution of Multi-Delay Dynamic Systems by Using Hybrid Functions

In this paper, we present a method for finding the solution of the linear multi-delay systems (MDS) by using the hybrid of the Block-Pulse functions and the Bernoulli polynomials. In this approach, the MDS is reduced to a system of linear algebraic equations by expanding various time functions for the hybrid functions and using operational matrices. To demonstrate the validity and the applicability of the technique, some examples are presented.

A multiscale differential-algebraic neural network-based method for learning dynamical systems

The objective of dynamical system learning tasks is to forecast the future behavior of a system by leveraging observed data.However,such systems can sometimes exhibit rigidity due to significant variations in component parameters or the presence of slow and fast variables,leading to challenges in learning.To overcome this limitation,we propose a multiscale differential-algebraic neural network(MDANN)method that utilizes Lagrangian mechanics and incorporates multiscale information for dynamical system learning.The MDANN method consists of two main components:the Lagrangian mechanics module and the multiscale module.The Lagrangian mechanics module embeds the system in Cartesian coordinates,adopts a differential-algebraic equation format,and uses Lagrange multipliers to impose constraints explicitly,simplifying the learning problem.The multiscale module converts high-frequency components into low-frequency components using radial scaling to learn subprocesses with large differences in velocity.Experimental results demonstrate that the proposed MDANN method effectively improves the learning of dynamical systems under rigid conditions.

STABILITY OF N-DIMENSIONAL LINEAR SYSTEMS WITH MULTIPLE DELAYS AND APPLICATION TO SYNCHRONIZATION

This paper further investigates the stability of the n-dimensional linear systems with multiple delays. Using Laplace transform, we introduce a definition of characteristic equation for the n-dimensional linear systems with multiple delays. Moreover, one sufficient condition is attained for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. In particular, our result shows that some uncommensurate linear delays systems have the similar stability criterion as that of the commensurate linear delays systems. This result also generalizes that of Chen and Moore （2002）. Finally, this theorem is applied to chaos synchronization of the multi-delay coupled Chua＇s systems.

Variable structure control for three-variable autocatalytic reaction

In this paper, two irreversible exothermic autocatalytic reactions which carry out in continuous stirred tank reactor （CSTR） are considered. A differential-algebraic system is applied to model these chemical reactions. The stability and the dynamic behavior are studied for the differential-algebraic system. The Hopf bifurcation appears when the parameter exceeds a critical value. In order to eliminate this complex behavior, the differential-algebraic system is described by a single-input and single-output system with parameter varying within definite intervals, and then variable structure control with sliding mode based on a special power reaching law is designed to stabilize this chemical system. Numerical simulations are given to illustrate the effectiveness of the method.