Sums of finite products of Chebyshev polynomials of two different types

<abstract><p>In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.</p></abstract>

Inductive linearization for binary quadratic programs with linear constraints

A linearization technique for binary quadratic programs (BQPs) that comprise linear constraints is presented. The technique, called “inductive linearization”, extends concepts for BQPs with particular equation constraints, that have been referred to as “compact linearization” before, to the general case. Quadratic terms may occur in the objective function, in the set of constraints, or in both. For several relevant applications, the linear programming relaxations obtained from applying the technique are proven to be at least as strong as the one obtained with a well-known classical linearization. It is also shown how to obtain an inductive linearization automatically. This might be used, e.g., by general-purpose mixed-integer programming solvers.

Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials

In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of finite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefficients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 .

Evaluating Fractional PID Control in a Nonlinear MIMO Model of a Hydroelectric Power Station

In this paper a Fractional PID Control is presented. This control was designed for a hydropower plant with six generation units working in an alternation scheme. The parameters and other features of such a set of hydrogeneration units have been used to perform the respective tuning up. In order to assess the behavior of this controlled system, a model of such nonlinear plant is regulated through a classical PID by classical linearization of its set points, and then a pseudo-derivative part is substituted into a Fractional PID. Both groups of signals contain variations of voltage suggesting some abrupt changes in the supply of electricity fed to the network. Both sets of resulting signals are compared; the simulations show that the Fractional PID has a faster response with respect to those plots obtained from the classical PID used.