Algorithms for noise removal are either complex or ineffective, and the optimal control with inequality constrains makes the algorithm even more complex. Now the condition is changed completely, and the tropical algebra is an extremely simple tool for this purpose. The tropical algebra-based filter has obvious advantages over traditional ones, and an inequality constrain can be converted to a tropical polynomial, making the tropical algebra much attractive in engineering applications. In this paper, idempotents on multiplicative semigroups of tropical matrices are studied. First, the concepts of tropical algebra and the semigroup of tropical matrices under multiplication are introduced. Second, the structure of idempotents on the semigroup of [Formula: see text] tropical matrices under multiplication is given. Finally, as examples, the tropical addition is used as a filter for noise removal, and a simple optimization is given where the constraint is given by the tropical algebra. The paper has opened the path for a new way to noise removal and optimization.
This paper addresses the problem of adaptive tracking control for a class of stochastic nonlinear systems with time-varying input delays, and the nonlinear functions of the systems are with not only the unknown parameters but also the unknown state time-varying delays, which is different from the previous work. In this paper, through a state transformation, the system can be easily transformed into a system without the time-varying input delay; the appropriate Lyapunov–Krasovskii functionals are used to compensate the unknown time-varying delay terms, and the quadratic functions instead of the quartic functions often utilized in the existing results are used as Lyapunov functions to analyze the stability of systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms. Fuzzy logic systems (FLSs) in Mamdani type are used to approximate the unknown nonlinear functions. Then, based on the backstepping technique, the adaptive fuzzy controller is designed. The three main advantages of the developed scheme are that (i) unlike the existing results which deal with the nonlinearly parameterized functions by using the separation principle, the nonlinearly parameterized functions are lumped into the continuous functions which can be approximated by using the FLS; (ii) the number of the adjusted parameters only depend on the order of the investigated systems, which can reduce the computational burden greatly; and (iii) the existence of the time-varying input delay such that the controller design becomes much more difficult, and in this paper, it can be dealt with by using an appropriate state transformation. It is proven that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded in probability, whereas the tracking error converges to a small neighborhood of the origin. Finally, simulation results are provided to show the effectiveness of the proposed approach.
The aim of this paper is to characterize the ( * , ∼)-good congruences on regular ortho-lc-monoids by making use of the compatible congruence systems on the semi-spined product components of regular ortho-lc-monoids.
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