Control and Optimization in Applied Mathematics
Volume:7 Issue: 2, Summer-Autumn 2022

Switched linear systems are noted as a major category of control systems‎. ‎Fault detection of these systems is affected by switching phenomena and therefore their integrated fault detection and robust control (IFDRC) are the central issues of recent studies‎. ‎Existing studies on IFDRC do not consider the effects of all of the parameter uncertainties‎, ‎input disturbance‎, ‎and mode-dependent time-varying state delay in the presence of mode-dependent average dwell time (MDADT) switching together in these systems. To address the issue based on output feedback‎, ‎in this paper‎, ‎the IFDRC design problem is formulated as a multi-objective or mixed H∞/H- optimization problem‎. ‎H∞ performance indicator guarantees the robustness of residual to disturbance‎, ‎and H- performance represents the sensitivity index of residual to the fault‎. ‎A piecewise Lyapunov-Krasovskii function is employed together with the MDADT scheme and therefore‎, ‎sufficient conditions are derived in terms of linear matrix inequalities (LMIs) in order to deal with the problem‎. ‎Then to clarify the design procedure‎, ‎we also present an algorithm in the light of the proposed approach‎. ‎Eventually‎, ‎to illustrate the efficiency of the suggested approach‎, ‎the designed IFDRC framework is simulated for a case study of an Electrical Circuit system.

This paper presents a capable recurrent neural network, the so-called µRNN for solving a class of non-convex quadratic programming problems‎. ‎Based on the optimality conditions we construct a new recurrent neural network (µRNN)‎, ‎which has a simple structure and its capability is preserved‎. ‎The proposed neural network model is stable in the sense of Lyapunov and converges to the exact optimal solution of the original problem‎. ‎In a particular case‎, ‎the optimality conditions of the problem become necessary and sufficient‎. ‎Numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network.

In a water distribution network‎, ‎in order to analyze and determine its parameters such as head and flow rate‎, ‎we have to solve nonlinear hydraulic equations in each component of the network‎. ‎Contrary to most of the water distribution network simulation software‎, ‎solving these equations by using the gradient method‎, ‎we propose a trust-region method to solve them‎, as the trust-region method is newer than the gradient method and has well worked in mathematical problems. ‎To prove the effectiveness of our method‎, we made a comparison between our proposed method and the well-known gradient method‎. ‎The results show that‎ ‎the trust-region method is convergent in all instances‎, but the gradient method diverges when the dimension of nonlinear hydraulic equations of water distribution networks increases‎. ‎In addition‎, ‎our results convince the solution obtained from the trust-region method is more accurate compared to the gradient method‎. Thus, ‎using the trust-region method in solving the network equations can lead to a better hydraulic analysis of the network.

In this paper‎, ‎we present a numerical method for solving the fractional optimal control problems in which fractional integral operational matrices of basic B-spline functions are used‎. ‎In the proposed method‎, ‎we use the Riemann-Liouville fractional integral‎. ‎With the help of the operational matrix of the fractional integral and the collocation method‎, ‎we transform the fractional optimal control problem into a nonlinear programming problem‎ ‎and then solve it with an appropriate optimization algorithm‎. ‎Compared to similar numerical techniques‎, ‎our method has better accuracy and efficiency, and also it is easy to use‎. ‎To provide a clear view of the applicability and efficiency of our numerical method‎, several illustrative examples are presented.

Social‎, ‎economic‎, ‎and environmental issues such as population growth‎, ‎reduction of natural resources‎, ‎climate change‎, ‎market fluctuations‎, ‎and changing consumer behavior have attracted the attention of politicians to the supply chain of agricultural products‎. ‎Designing an effective supply chain for each product can lead to optimal management of the agricultural sector and create coordination and links between activities‎. ‎In this article‎, ‎the design of a two-echelon supply chain network of shrimp in Golestan province is investigated‎. ‎The objective is to minimize the total cost associated with fixed opening and operating costs of shrimp farming companies and to determine the target market for these producers‎. ‎Also‎, ‎this study involves deciding on the amount of inputs purchased by each company and determining the best mode of transport‎. ‎To characterize and solve this problem‎, ‎we developed a mixed-integer programming (MIP) model that solves with GAMS software‎. ‎The results show that with the implementation of the MIP model‎, ‎the total costs of the chain are reduced by nearly 20 percent compared to the current situation‎. ‎In addition‎, ‎without increasing production‎, ‎it is possible to supply 0.053 percent of global market demand‎, ‎which is 76 percent more than before.

Solving fuzzy linear systems has been widely studied during the last decades‎. ‎However‎, ‎there are still many challenges to solving fuzzy linear equations‎, ‎as most of the studies have used the principle of extension‎, ‎which suffers from shortcomings such as the lack of solution‎, ‎achieving solutions under very strong conditions‎, ‎large support of the obtained solutions‎, ‎inaccurate or even incorrect solutions due to not utilizing all the available information‎, complicated process and high computational load‎. ‎These problems motivated us to present a fuzzy Cramer method for solving fuzzy linear equations‎, ‎which uses arithmetic operations based on the Transmission Average (TA)‎. ‎In this study‎, ‎fully fuzzy linear systems in the form of $ \tilde{A}\tilde{X}=\tilde{B} $‎, ‎and dual fuzzy linear systems in the form of $ \tilde{A}\tilde{X}+\tilde{B}=\tilde{C}\tilde{X}+\tilde{D} $ are solved using the proposed fuzzy Cramer method‎, ‎and numerical examples are provided to confirm the effectiveness and applicability of the proposed method.