In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potentialWe analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on t is given.

In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented.

This paper considers the robust stabilization problem for interval plants with parametric uncertainty and uncertain time-delay based on the value set characterization of closed-loop control systems and the zero exclusion principle. Using Kharitonov’s polynomials, it is possible to establish a sufficient condition to guarantee the robust stability property. This condition allows us to solve the control synthesis problem using conditions similar to those established in the loopshaping technique and to parameterize the controllers using stable polynomials constructed from classical orthogonal polynomials.

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The presence of faults in dynamic systems causes the potential loss of some of the control objectives. For that reason, a fault-tolerant controller is required to ensure a proper operation, as well as to reduce the risk of accidents. The present work proposes a passive fault-tolerant controller that is based on robust techniques, which are utilized to adjust a proportional-derivative scheme through a linear matrix inequality. In addition, a nonlinear term is included to improve the accuracy of the control task. The proposed methodology is implemented in the control of a two degrees of a freedom robotic helicopter in a simulation environment, where abrupt faults in the actuators are considered. Finally, the proposed scheme is also tested experimentally in the Quanser® 2-DOF Helicopter, highlighting the effectiveness of the proposed controller.