On the spectrum in max algebra Müller, Vladimir; Peperko, Aljoša
Linear algebra and its applications,
11/2015, Volume:
485
Journal Article
Peer reviewed
Open access
We give new proofs of several fundamental results of spectral theory in max algebra. This includes the description of the spectrum in max algebra of a given non-negative matrix via local spectral ...radii, the spectral theorem and the spectral mapping theorem in max algebra. The latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra. Our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non-negative matrices.
This research deals with the temporal performance analysis of distributed networked automation systems. It describes a new formal approach that estimates the maximum limit of response time in ...Networked Control Systems (NCS) that use a producer/consumer protocol. Two different tools of discrete event systems have been used: Timed Event Graphs (TEGs) and (Max,+) algebra. First, the contribution is to model all components of NCS with TEGs, and to represent the behaviour of these graphs by (Max,+) linear equations, and then to determine the upper bounds of the response time by an analytical formula. The main novelty of these results is to introduce a modular method capable of providing crucial validation elements regarding temporal performances for an industrial scale NCS that includes inter CPU communication. Moreover, the modular aspect of the presented approach remains in considering the end-to-end delays introduced by the network as model parameters, calculated from other models.
•Modeling of logistic networks characterized by resource sharing.•(Max, +) equations describing the behavior of logistic networks.•Control approach to optimize logistic costs.
The present article ...deals with supply chain management considered as a Discrete Event System (DES). We introduce a new modeling approach based on linear and non-stationary (max, +) equations obtained from a Timed Coloured Petri Nets (TCPN) describing the studied system’s behavior. Our contribution does not lie only on the modeling of the dynamics associated with vehicles and their timetables at different sites (e.g., suppliers, warehouses, customers), but also on the evaluation of loading, unloading, and delivery times of products from suppliers to customers, taking into account their appropriate characteristics (e.g., number, nature, destination). Moreover, a control approach is performed to optimize the number of vehicles to deploy on the network by ensuring product storage times below a given threshold at some specific sites considered strategic (e.g., delivery hubs). The developed models can be used as a decision-making system for logistic companies to result in efficient supply chain management. The developed models are tested and validated on several configurations and scenarios to demonstrate our approach’s applicability.
Several spectral radii formulas for infinite bounded non-negative matrices in max algebra are obtained. We also prove some Perron-Frobenius type results for such matrices. In particular, we obtain ...results on block triangular forms, which are similar to results on Frobenius normal form of
$ n \times n $
n
×
n
matrices. Some continuity results are also established.
We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems ...of rating alternatives from pairwise comparison data. In this framework, we first consider the minimax Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. Recasting this approximation problem as a problem of tropical pseudo-quadratic programming, we then write out a closed-form solution to it. This solution might be either a unique score vector (up to a positive factor) or a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques of maximizing and minimizing the Hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores, and thus are well representative of the entire solution set.
This paper proposes a fractal modification of tropical algebra for noise removal and optimal control. Fractal addition, fractal multiplication, and fractal dot product are defined and explained in a ...fractal space. A fractal tropical polynomials function can model a coastline at any scale, providing a new tool for fractal analysis of any irregular curve. An example is given to solve an optimal control subject to a fractal tropical polynomials function. This paper provides a new window for tropical algebra, combining fractal geometry and algebra.
An analog of Perron–Frobenius theory is proposed for some classes of nonnegative tensors in the max algebra. In the first part some important characterizations of nonnegative matrices can be extended ...to nonnegative tensors over max algebra, especially the Perron–Frobenius theorem for weakly irreducible nonnegative tensors and the Collatz–Wielandt minimax theorem for nonnegative tensors. Then, in the second part, an iterative method is proposed for finding the largest max eigenvalue of a nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for weakly irreducible nonnegative tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.
We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring ...results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of nonnegative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.
This paper presents the new investigations on the disturbance decoupling problem (DDP) for the geometric control of max-plus linear systems. The classical DDP concept in the geometric control theory ...means that the controlled outputs will not be changed by any disturbances. In practical manufacturing systems, solving for the DDP would require further delays on the output parts than the existing delays caused by the system breakdown. The new proposed modified disturbance decoupling problem (MDDP) in this paper ensures that the controlled output signals will not be delayed more than the existing delays caused by the disturbances in order to achieve the just-in-time optimal control. Furthermore, this paper presents the integration of output feedback and open-loop control strategies to solve for the MDDP, as well as for the DDP. If these controls can only solve for the MDDP, but not for the DDP, an evaluation principle is established to compare the distance between two output signals generated by controls solving for the MDDP and DDP, respectively. This distance can be interpreted as the number of tokens or firings that are needed in order for the controls to solve for the DDP. Moreover, another alternative approach is finding a new disturbance mapping in order to guarantee the solvability of the DDP by the same optimal control for the MDDP. The main results of this paper are illustrated by using a timed event graph model of a high throughput screening system in drug discovery.
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