Systems Structure and Control

The title of the book System, Structure and Control encompasses broad field of theory and applications of many different control approaches applied on different classes of dynamic systems. Output and state feedback control include among others robust control, optimal control or intelligent control methods such as fuzzy or neural network approach, dynamic systems are e.g. linear or nonlinear with or without time delay, fixed or uncertain, onedimensional or multidimensional. The applications cover all branches of human activities including any kind of industry, economics, biology, social sciences etc

A Max-Plus Model of Asynchronous Cellular Automata

This paper presents a new framework for asynchrony. This has its origins in our attempts to better harness the internal decision making process of cellular automata (CA). Thus, we show that a max-plus algebraic model of asynchrony arises naturally from the CA requirement that a cell receives the state of each neighbour before updating. The significant result is the existence of a bijective mapping between the asynchronous system and the synchronous system classically used to update cellular automata. Consequently, although the CA outputs look qualitatively different, when surveyed on "contours" of real time, the asynchronous CA replicates the synchronous CA. Moreover, this type of asynchrony is simple - it is characterised by the underlying network structure of the cells, and long-term behaviour is deterministic and periodic due to the linearity of max-plus algebra. The findings lead us to proffer max-plus algebra as: (i) a more accurate and efficient underlying timing mechanism for models of patterns seen in nature, and (ii) a foundation for promising extensions and applications.Comment: in Complex Systems (Complex Systems Publications Inc), Volume 23, Issue 4, 201

Cyclic classes and attraction cones in max algebra

In max algebra it is well-known that the sequence A^k, with A an irreducible square matrix, becomes periodic at sufficiently large k. This raises a number of questions on the periodic regime of A^k and A^k x, for a given vector x. Also, this leads to the concept of attraction cones in max algebra, by which we mean sets of vectors whose ultimate orbit period does not exceed a given number. This paper shows that some of these questions can be solved by matrix squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit period in max-min algebra. Hence the computational complexity of such problems is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal similarity scaling A -> X^{-1}AX, called visualization scaling, and to study the role of cyclic classes of the critical graph. For powers of a visualized matrix in the periodic regime, we observe remarkable symmetry described by circulants and their rectangular generalizations. We exploit this symmetry to derive a concise system of equations for attraction cpne, and we present an algorithm which computes the coefficients of the system.Comment: 38 page

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.Comment: 27 pages, 6 figures, revised versio

Minimum Input Selection for Structural Controllability

Given a linear system $\dot{x} = Ax$, where $A$ is an $n \times n$ matrix with $m$ nonzero entries, we consider the problem of finding the smallest set of state variables to affect with an input so that the resulting system is structurally controllable. We further assume we are given a set of "forbidden state variables" $F$ which cannot be affected with an input and which we have to avoid in our selection. Our main result is that this problem can be solved deterministically in $O(n+m \sqrt{n})$ operations