Min-max System Research Articles

Disjointed Cycle Time Assignment for Min–Max Systems Using an Output Feedback

In min-max systems, the cycle time plays an important role on analyzing the performance metric. In this work, we mainly investigate the performances of the min-max systems with cycle time assignment, which corresponding to the feedback problem in linear systems. It is achieved that the cycle time can be assigned disjointedly by an output feedback in linear systems, if and only if the system is reachable and at least has an output node which has only one observable condensed node. In this way, for the min-max system with max-plus inputs and outputs, the sufficient condition for the output feedback cycle time assignability is can be obtained.

Upper critical dimension of the Kardar-Parisi-Zhang equation

Numerical results for the directed polymer model in 1+4 dimensions in various types of disorder are presented. The results are obtained for a system size that is considerably larger than considered previously. For the extreme "strong" disorder case (min-max system), associated with the directed percolation model, the expected value of the meandering exponent, ζ=0.5, is clearly revealed, with very weak finite size effects. For the "weak disorder" case, associated with the Kardar-Parisi-Zhang equation, finite size effects are stronger, but the value of ζ is clearly seen in the vicinity of 0.57. In systems with strong disorder it is expected that the system will cross over sharply from min-max behavior at short chains to weak disorder behavior at long chains. Our numerical results agree with that expectation. To complete the picture we obtain the energy fluctuation exponent ω for weak disorder, and we find that the value of ω is in the vicinity of 0.14. Thus, the meandering exponent and the energy fluctuation exponent obey the strong coupling scaling relation 2ξ-ω=1. Our results indicate that 1+4 is not the upper critical dimension in the weak disorder case, and thus 4+1 does not seem to be the upper critical dimension for the Kardar-Parisi-Zhang equation.

Independent cycle time assignment for min-max systems

A variety of problems in digital circuits, computer networks, automated manufacturing plants, etc., can be modeled as min-max systems. The cycle time is an important performance metric of such systems. In this paper, we focus on the cycle time assignment of min-max systems which corresponds to the pole assignment problem in traditional linear control systems. For the minmax system with max-plus inputs and outputs, we show that the cycle time can be assigned disjointedly by a state feedback, if and only if the system is reachable. Furthermore, a necessary and sufficient condition for the cycle time to be assigned independently by a state feedback is given. The methods are constructive, and some numerical examples are given to illustrate how the methods work in practice.

The balance problem of min–max systems is co-NP hard

Min–max systems are timed discrete event dynamic systems containing min, max and plus operations. A min–max system is balanced if it admits a global cycle time (generalized eigenvalue) for every possible assignment of parameters. This paper proves that the testing of the balance property for min–max systems is co-NP hard by showing that the test of inseparability is co-NP hard. The latter result is established on a new complexity result, namely, that the testing of existence of non-trivial fixed points for monotone Boolean functions is NP-complete.

Cycle time assignment of min-max systems

A variety of problems in computer networks, digital circuits, communication networks, manufacturing plants, etc., can be modelled as discrete event systems with maximum and minimum constraints. Systems with mixed constraints are non-linear and are called min-max systems. The cycle time vector of such a system arises as a performance measure for discrete event systems and provides the appropriate non-linear generalization of the spectral radius. This paper gives a complete account of the cycle time assignment by the state feedback for min-max systems. We describe some new definitions and results about such assignment which generalize the initial earlier works and shed new light on aspects of linear control theory. For an arbitrary min-max system, by introducing the concept of colouring graph and constructing the total condensation and its matrix representation, we give the canonical structure form. In order to design the state feedback system in which the internal structure property is unchanged and the cycle time can be assigned, we introduce and characterize the assignability, uniform state feedback and unmerged assignment for min-max systems. We present an algorithm for the unmerged assignment and illustrate our algorithm by means of an example.

A measure-theoretic axiomatization of fuzzy sets

Using measurement theory, this paper examines three empirical structures that underlie the representation of fuzzy sets: the fuzzy membership structure, the fuzzy component structure, and the fuzzy system structure. These qualitative structures justify the use of the standard min-max system to represent fuzzy sets. The results of this study facilitate the development of a sound measurement-theoretic axiomatization of fuzzy systems.