Control of the Grushin equation: non-rectangular control region and minimal time

This paper is devoted to the study of the internal null-controllability of the Grushin equation. We determine the minimal time of controllability for a large class of non-rectangular control regions. We establish the positive result thanks to the fictitious control method and the negative one by interpreting the associated observability inequality as an L2 estimate on complex polynomials.

Systematic design of robust controller for electro-hydraulic system with model uncertainties

A systematic method is developed to design new robust tracking control for an electro-hydraulic system. In this article, the model dynamics is represented in strict feedback form to be linearly parameterized with regard to model parameters to consider its uncertainties and the separate error dynamics are derived for the tracking of states. The proposed method is intuitively similar to a previously developed integrator backstepping approach since it designs fictitious controls as desired states for recursive procedures. However, unlike previous works that have relied exclusively on Lyapunov analysis using the sum of the squares of state errors, this work utilizes a passivity approach with an energy function for stability. The advantage of the proposed method is that the fictitious control can be designed in a decoupled form and an L2-gain analysis can be applied to guarantee the robustness to model uncertainties. For the separate error dynamics, one part of the fictitious control is designed using stability analysis with a strict passivity formulation for error convergence; this leads to a separation of the other fictitious control design for robustness from the effects of other equations of dynamics. Hence, this method makes it possible to treat the terms caused by the model uncertainties as disturbance and to design the fictitious robust control using the L2-gain analysis for the disturbance attenuation, which does not require the bound function of the disturbance to be known. The validity of the proposed method is shown through simulations.