Control and Optimization in Applied Mathematics
Volume:4 Issue: 2, Summer-Autumn 2019

Guaranteed cost control (GCC) is an impressive method of controlling nonlinear systems‎, ‎incredibly uncertain switched systems‎. ‎Most of the recent studies of GCC on uncertain switched linear systems have been concerned with asymptotic stability analysis‎. ‎In this paper‎, ‎a new robust switching law for time-delay uncertain switched linear systems is designed‎. ‎First‎, ‎the switching law is designed‎, ‎and second‎, ‎a state-feedback controller based on Lyapunov-Krasovskii Functional (LKF) is designed‎. ‎Also‎, ‎using Linear Matrix Inequality (LMI) particular condition for the existence of a solution of obtained switching law and controller is achieved‎. ‎Consequently‎, ‎in the presented theorems‎, ‎the exponential stability of the overall system under switching law and controller is analyzed‎. ‎Finally‎, ‎theoretical results are verified via presenting an example.

In the present study‎, ‎a novel methodology is developed to enlarge the Region of Attraction (ROA) at the point of equilibrium of an input-affine nonlinear control system‎. ‎Enlarging the ROA for non-polynomial dynamical systems is developed by designing a nonlinear state feedback controller through the State-Dependent Riccati Equation (SDRE)‎. ‎Consequently‎, ‎its process is defined in the form of Sum-of-Squares (SOS) optimization problem with control and non-control constraints‎. ‎Of note‎, ‎the proposed technique is effective in estimating the ROA for a nonlinear system functioning on polynomial or non-polynomial dynamics‎. ‎In the present study‎, ‎the application of the proposed scheme are shown by numerical simulations‎.

Product reviews in E-commerce websites such as restaurants‎, ‎movies‎, ‎E-commerce products‎, ‎etc.‎, ‎are essential resources for consumers to make purchasing decisions on various items‎. ‎In this paper‎, ‎we model discovering groups with maximum descriptively from E-commerce website of the form $<i,u,s>$‎, ‎where $iin mathcal{I}$ (the set of items or products)‎, ‎$uin mathcal{U}$ (the set of users) and $s$ is the integer rating that user $u$ has assigned to the item $i$‎. ‎Labeled groups from user's attributes are found by solving an optimization problem‎. ‎The performance of the approach is examined by some experiments on real data-sets‎.

Experiences of teaching Integral have indicated that the vast majorities of Iranian university students commit numerous errors while solving integral problems and have weak skills in this field; we might even say that they hide away from integral and consider it the nightmare of mathematics‎. ‎On the other hand, Integral is the base of pure and applied mathematics for all students of science‎, ‎especially engineering‎, ‎which some of their lessons are dependent on it directly or indirectly‎, ‎so it is important to pay attention to it‎. ‎Through descriptive method-exposed factor, ‎an exam has been conducted in the form of three questions‎, ‎the first of which is consisted of 4 sections on fifty students from different fields, and then interviews were conducted with a few of those students about their answers in order to study the students' behaviors when solving integral problems and to determine the type of their errors‎. ‎By analyzing the performance of students in this test‎, ‎we can see that students often struggle with integral and mostly have a feeble performance in solving trigonometric integrals‎. ‎They want to learn computational integral instead of how to conceptualize integral in their minds correctly‎. ‎The error most committed by university students was procedural errors‎, ‎which arise from using derivative instead of integral‎. ‎Besides most of the mistakes happen in solving definite integrals‎, ‎and calculating finite areas between two curves‎. ‎This is due to a lack of understanding of integrals and a lack of information in other areas of mathematics.

In this study‎, ‎$R$ and $M$ are assumed to be a commutative ring with non-zero identity $M$ and an $R$-module‎, ‎respectively‎. ‎Scalar Product Graph of $M$‎, ‎denoted by $G_R(M)$‎, ‎is a graph with the vertex-set $M$ and two different vertices $a$ and $b$ in $M$ are connected if and only if there exists $r$ belong to $R$ such that $a=rb$ or $b=ra$‎. ‎This paper studies some properties of such weakly perfect graphs‎.

Conventional model predictive control (MPC) methods are usually implemented to systems with discrete-time dynamics laying on smooth vector space $ mathbf{R}^n$‎. ‎In contrast‎, ‎the configuration space of the majority of mechanical systems is not expressed as Euclidean space‎. ‎Therefore‎, ‎the MPC method in this paper has developed on a smooth manifold as the configuration space of the attitude control of a 3D pendulum‎. ‎The Lie Group Variational Integrator (LGVI) equations of motion of the 3D pendulum have been considered as the discrete-time update equations since the LGVI equations preserve the group structure and conserve quantities of motion‎. ‎The MPC algorithm is applied to the linearized dynamics of the 3D pendulum according to its LGVI equations around the equilibrium using diffeomorphism‎. ‎Also‎, ‎as in standard MPC algorithms‎, ‎convex optimization is solved at each iteration to compute the control law‎. ‎In this paper‎, ‎the linear matrix inequality (LMI) is used to solve the convex optimization problem under constraints‎. ‎A numerical example illustrates the design procedure‎.