Adaptive Pole Positioning in MIMO Linear Systems by Periodic Multirate-Input Controllers

The paper is focused on systems with delay terms at the left (and the right) side of differential equations. Analysis and synthesis of delay systems can be conveniently studied through a special ring of RQ-meromorphic functions. The control methodology is based on a solution of Diophantine equations in this ring. Final controllers result in the Smith predictor like structure. Controller parameters are tuned through a pole-placement problem as a desired multiple root of the characteristic closed loop equation. The methodology is illustrated by a stable second order transfer function with a dead-time term. Then the paper brings an autotuning method as a combination of biased-relay feedback estimation and the proposed algebraic control design. The developed approach is illustrated by examples in the Matlab and Simulink environment.

The Undecidability of Exponential Diophantine Equations

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

This work proposes a distributed composite adaptive synchronization algorithm for multiple uncertain EulerLagrange (EL) systems, where parameter convergence is achieved under a relaxed mathematical condition as compared to the state-of-the-art. Classical adaptive controllers require an analytical condition, called persistence of excitation (PE), to ensure parameter convergence, which results in better transient performance and robustness to disturbance. The PE condition is extended to Cooperative-PE (C-PE) condition for distributed adaptive controllers with cooperative estimation strategies. The PE and C-PE conditions are restrictive in nature since these conditions are not satisfied in most practical applications. Recent literature in adaptive control has relaxed the PE condition to Initial Excitation (IE), which is shown to be sufficient for parameter convergence. The IE condition is argued to be significantly milder than PE and can be satisfied in many practical setting. The proposed result further extends the IE condition to Cooperative-IE (C-IE) condition in distributed adaptive control architecture in the context of synchronizing multiple EL systems. It is established that the C-IE condition is milder than PE, IE, and C-PE conditions. Two-tier filter based estimation algorithm with strategic switching ensures parameter convergence under the C-IE condition and thereby provides exponential convergence of tracking and parameter estimation error to zero. Simulation results validate the efficacy of the proposed algorithm as compared to conventional distributed adaptive controllers in terms of superior tracking and estimation performance.

Sign function and ANN based pole placement for computing interval controls

Diophantine equation is an important concept while studying number theory. But there is as such no standard method or procedure to find the solution of the Diophantine equation. In this article, the numerical solution of Diophantine equation has been discussed using the concept of Artificial Neural Network (ANN). A four layer ANN architecture has been constructed for solving Diophantine eqution. Detail ANN procedure with few example problems and their network architectures have been addressed. Convergence plots and/or convergence tables for different example problems have also been included to validate the proposed method.

Since Diophantine equations play an important role in solving important real-world problems such as business investment problems, network flow problems, pole placement problems, and data privacy problems, researchers are increasingly interested in developing new techniques for analyzing the nature and solutions of the various Dio phantine equations. In this study we investigate the Diophantine problem 22x+40y = z2, where x, y, z are non-negative integers, and discover that it does not have a non-negative integer solution.

A new generic optimal controller structure and a related regulator design method is introduced. The optimality is expressed by two simple serial compensators. It is shown how these filters-depending on the reference signal, the output disturbance and the invariant process factors-should be selected by the solutions of Diophantine equations if optimality is important. The optimal sensitivity properties of the new scheme is also analyzed and if is shown that this controller has the minimum variance to unknown process variations.

We show how the solution to certain diophantine equations involving the discriminant of complex quadratic fields leads to the divisibility of the class numbers of the underlying fields. This not only generalizes certain results in the literature such as [2], [4]–[6] but also shows why certain hypotheses made in these results are actually unnecessary since, as our criteria demonstrate, these hypotheses are forced by the solution of the diophantine equations involved. Our methods are based only on the most elementary properties of a principal ideal in a complex quadratic field.

In this paper, we are focusing on solving Diophantine equations with geometry. Integers that are solutions of Diophantine equations are represented in space by a lattice. We will use the fact that equations can be interpreted geometrically as curves or surfaces. To determine the properties of the solutions, we will use tangents (of the surface).

We extend the Kannan contraction principle and obtain a result that holds for both contractive and non-expansive mappings. Such mappings admit multiple fixed-points and the fixed-point sets and domains of these mappings display interesting algebraic, geometric and dynamical features. Since contraction mappings admit only one fixed-point, almost all the existing results on contractive mappings can be generalized in the light of our theorem. As an application of our main theorem, we obtain the integral solutions of a nonlinear Diophantine equation; the solutions are Pythagorean triples, which represent right angled triangles, and each integer of the triple belongs to a Fibonacci type sequence. These results can be generalised to obtain integral solutions of Diophantine equations of the type (n+k)2 – n2 = p2, k > 1, and to check whether the related sequences are Fibonacci sequences.

Neste trabalho estudamos condições necessárias e suficientes para que uma equação diofantina linear sobre um domínio euclidiano tenha solução. Apresentamos uma série de algoritmos (funções) que podem ser implementados em planilhas eletrônicas (por exemplo LibreOffice Calc, Microsoft Excel, etc.), com o intuito de determinar (caso existam) soluções de equações diofantinas sobre Z[i].

A reinforced adaptive fuzzy sliding-mode controller was developed to track the desired state for a certain class of nonlinear dynamic system. Initially, adaptive fuzzy logic was used to derive the update laws for approximating the uncertain nonlinear functions of a dynamic system. Subsequently, a reinforced adaptive mechanism and sliding-mode control were incorporated into the adaptive fuzzy control scheme. In the reinforced adaptive mechanism, both the model and controller parameters were adjusted simultaneously to attenuate the effects caused by the unmodeled dynamics and disturbances. Performance analysis demonstrated the superiority of the proposed reinforced adaptive law over the conventional direct adaptive scheme regarding faster tracking and convergence of parameters. To verify its effectiveness and extend its application, the proposed reinforced adaptive fuzzy sliding-mode controller was applied to balance a two-wheeled robot steering on a bumpy road. A number of simulations and experiments were performed. In addition, a conventional direct adaptive scheme under the same environment was also performed for comparison. These results demonstrated that the balance performance of a nonlinear dynamic two-wheeled robot improved because of embedding the proposed reinforced adaptive fuzzy sliding-mode controller into the microcontroller.

The linear time-varying multivariable system whose controllability indices or observability indices are not constant is called the non-lexicographically-fixed system. Valasek proposed the pole placement design method for such a continuous system by augmenting the system equation so that the augmented system is lexicographically-fixed. This paper concerns the pole placement and the observer design method for linear time-varying discrete non-lexicographically-fixed system. Using the Valasek's idea, the procedure to augment a discrete non-lexicographically-fixed system to a lexicographically-fixed augmented system will be presented. Then, the simple pole placement technique can be applied to the augmented system without transforming the system into any canonical form. Further, using the property of the anti-causal dual system, it will be shown that the pole placement design method can be used for the augmented observer for non-lexicographically-fixed systems. Finally, the stability and the separation principle of the total closed loop system are also shown for the case where both of the augmented pole placement controller and the augmented observer are used.

Combined Cooperative Adaptive Cruise Control using Collective Initial Excitation based Distributed Parameter Estimator