STABILITY OF NON-LINEAR DYNAMICAL SYSTEM

The mainobjective of this research is to study the stability of thenon-lineardynamical system by using the linearization technique of three dimension systems toobtain an approximate linear system and find its stability. We apply this technique to reaches to the stability of the public non linear dynamical systems of dimension. Finally, some proposed examples (example (1) and example (2)) are given to explain this technique and used the corollary.

ON A STOCHASTIC LINEAR DYNAMICAL SYSTEM DRIVEN BY A VOLTERRA PROCESS

The aim of this note is considering a dynamical system expressed by a Langevin equation driven by a Volterra process. An Ornstein - Uhlenbeck process as the solution of this kind of equation is described and a problem of state estimation (filtering) for this dynamical system is investigated as well.

On error estimation for reduced-order modeling of linear non-parametric and parametric systems

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Moreover, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It is shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate the output error of reduced-order modeling for steady linear parametric systems.

Reflections on Termination of Linear Loops

This paper shows how techniques for linear dynamical systems can be used to reason about the behavior of general loops. We present two main results. First, we show that every loop that can be expressed as a transition formula in linear integer arithmetic has a best model as a deterministic affine transition system. Second, we show that for any linear dynamical system f with integer eigenvalues and any integer arithmetic formula G, there is a linear integer arithmetic formula that holds exactly for the states of f for which G is eventually invariant. Combining the two, we develop a monotone conditional termination analysis for general loops.

New formulations of algebraic criteria for controllability and observability of a linear MIMO-system

New formulations of algebraic criteria for controllability and observability of a linear dynamical system with multiple inputs and outputs (MIMO-systems) are given, the corresponding theorems are formulated. The criteria are based on algebraic relations between linear combinations of the control matrix columns and own vectors of the free dynamics matrix. Keywords algebraic criterion; controllability; observability; linear MIMO-system; own value; own vector; Krylov vector and matrix; kernel; cokernel

On the regulator problem for linear systems over rings and algebras

The regulator problem is solvable for a linear dynamical system Σ \Sigma if and only if Σ \Sigma is both pole assignable and state estimable. In this case, Σ \Sigma is a canonical system (i.e., reachable and observable). When the ring R R is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).

Linear dynamical systems of dimension two over the ring of integers modulo pt

In this paper, we investigate the linear dynamical system [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text] ([Formula: see text] is a prime). In order to facilitate the visualization of this system, we associate a graph [Formula: see text] on it, whose nodes are the points of [Formula: see text], and for which there is an arrow from [Formula: see text] to [Formula: see text], when [Formula: see text] for a fixed [Formula: see text] matrix [Formula: see text]. In this paper, the in-degree of each node in [Formula: see text] is obtained, and a complete description of [Formula: see text] is given, when [Formula: see text] is an idempotent matrix, or a nilpotent matrix, or a diagonal matrix. The results in this paper generalize Elspas’ [1959] and Toledo’s [2005].