Lazaros Moysis

Closed form solution for the equations of motion for constrained linear mechanical systems and generalizations: An algebraic approach

Abstract In this paper, a mathematical methodology is presented for the determination of the solution of motion for linear constrained mechanical systems applicable also to systems with singular coefficients. For mathematical completeness and also to incorporate some other interesting cases, the methodology is formulated for a general class of higher order matrix differential equations. Thus, describing the system in an autoregressive moving average (ARMA) form, the closed form solution is derived in terms of the finite and infinite Jordan pairs of the system׳s polynomial matrix. The notion of inconsistent initial conditions is considered and an explicit formula for the homogeneous system is given. In this respect, the methodology discussed in the present note provides an alternative view to the problem of computation of the response of complex multi-body systems. Two interesting examples are provided and applications of the equation to such systems are illustrated.

Introduction to Control Systems Design Using Matlab

Control systems theory is a wide area covering a range of artificial and physical phenomena. Control systems are systems that are designed to operate under strict specifications, to satisfy certain aims, like safety regulations in the industry, optimal production of goods, disturbance rejection in vehicles, smooth movement and placement of objects in warehousing, regulation of drug administration in medical operations, level control in chemical processes and many more. The present work provides an introduction to the fundamental principles of control systems analysis and design through the programming environment of Matlab and Simulink. Analysis of transfer function models is carried out though multiple examples in Matlab and Simulink, analyzing the dynamics of 1st and 2nd order systems, the role of the poles and zeros in the systems dynamic response, the effects of delay and the possibility to approximate higher order systems by lower order ones. In addition, examples are given from the fields of mechanical systems, medically induced anesthesia, neuroprosthetics and water level control, showcasing the use of controllers that satisfy certain design specifications.

New Discrete Time 2D Chaotic Maps

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, meteorology, economics, electrical circuits, engineering, computer science and more. Of these innumerable chaotic systems, perhaps the most interesting are those that exhibit attracting behavior. By that, the authors refer to systems whose trajectories converge with time to a set of values, called an attractor. This can be a single point, a curve or a manifold. The attractor is called strange if it is a set with fractal structure. Such systems can be both continuous and discrete. This paper reports on some new chaotic discrete time two dimensional maps that are derived from simple modifications to the well-known HA©non, Lozi, Sine-sine and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes.

Analysis of a Dynamical Model for HIV Infection with One or Two Inputs

A dynamical model that describes the interaction of the HIV virus and the immune system is presented. The effect of introducing antiretroviral therapy on the model, consisting of RTI and PI drugs is investigated, along with the result of undesired treatment interruption. Furthermore, the effect of both drugs can be combined into a single parameter that further simplifies the model into a single input system. Drug administration can be adjusted by feedback control, through monthly blood tests that measure the viral load. Furthermore, the system is linearized around the equilibrium, leading to a system of linear differential equations of first order that can be integrated into courses of control systems engineering, linear and nonlinear systems in higher education.

Observability of linear discrete-time systems of algebraic and difference equations

ABSTRACT The notion of observability for higher order discrete-time systems of algebraic and difference equations is studied. Such systems are also known as polynomial matrix descriptions . Attention is first given to a special form of descriptor systems with a state lead in the output. This system is transformed into its causal and noncausal subsystems and observability criteria are given in terms of the subsystems matrices, and the fundamental matrix sequence of the matrix pencil (σE − A). Afterwards, the higher order system is studied. By transforming it into a first-order descriptor system of the above form, an observability criterion is provided for the higher order system in terms of the Laurent expansion at infinity of the systems polynomial matrix. In addition, observability is connected with the coprimeness of the polynomial matrices of the higher order system and the coprimeness of the matrix pencils of the descriptor system.

Existence of Reachable and Observable Triples of Linear Discrete-Time Descriptor Systems

This work studies the reachability and observability of discrete-time descriptor systems and considers the following problem: Given a matrix pair (E,xa0A), find a matrix B (C) such that the corresponding descriptor system is not reachable (observable). The computation of such a matrix can give us a set of conditions that can then be taken into account when constructing the matrix B (C) to make the system reachable (observable). The above problem is solved by working on the equivalent causal and noncausal subsystems that are obtained through the Weierstrass decomposition of discrete-time descriptor systems. Positive descriptor systems are also considered. The developed theory is illustrated through physical and numerical examples.

Construction of algebraic and difference equations with a prescribed solution space

Abstract This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ)β(k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ). This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ)β(k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

Reachability of discrete time causal ARMA representations

The reachability subspace of discrete time causal ARMA representations is examined. First, the conditions under which a system is causal are studied and a formulafor the solution of a causal system is presented. Then, an important result regarding the set of admissible initial conditions is derived and the reachable subspace is provided.

Modeling of discrete time auto-regressive systems with given forward and backward behavior

We study the behavior of discrete time AR-representations. A theorem is provided connecting the backward behavior of a system, due to its infinite elementary divisors, with the forward behavior of its dual system. We first use this result to construct a system satisfying a certain backward behavior. In addition to this, we propose a way to combine this result with previous ones to create an algorithm for computing a system satisfying a given forward and backward behavior.

On the modeling of discrete time Auto-Regressive representations

It is well known [2], [6], that given the discrete-time AutoRegressive representation A(σ)β(k) = 0; where σ denotes the shift forward operator and A(σ) a polynomial matrix, we can always construct the forward-backward behavior of this system, by using the finite and infinite elementary divisor structure of A(σ). The main theme of this work is to study the inverse problem: given a specific forward-backward behavior, find a family of polynomial matrices A(σ), such that the system A(σ)β(k) = 0 has exactly the prescribed behavior. As we shall see, the problem can be reduced either to a linear system equation problem or to an interpolation problem.