Most existing work on direct data-driven stabilization considers the equilibrium at the origin. When the desired equilibrium is not the origin, existing data-driven approaches often require performing coordinate transformation, or adding integrator action to the controller. As an alternative, this work addresses data-driven state feedback stabilization of any given assignable equilibrium via dissipativity theory. We show that for a polynomial system, if a data-driven stabilizer can be designed to render the origin globally asymptotically stable, then by modifying the stabilizer, we obtain a stabilizer for any given assignable equilibrium.

This work proposes a novel nonlinear Proportional-Integral (PI) controller, which utilizes a generalized first-order reset element. The proposed element can achieve similar magnitude-characteristics as its linear counterpart but with less phase lag at the open-loop crossover frequency (i.e. the control bandwidth), according to a sinusoidal-input describing function (SIDF) analysis. The same can be achieved with an existing reset-based integrator, the Clegg integrator (CI). However, it is known that a Proportional-CI (PCI) element can excessively generate higher-order harmonics of its input, which are neglected in the SIDF-analysis. Furthermore, a PCI can cause a limit cycle when placed in closed-loop with certain types of plants. The novel PI controller proposed in this work can prevent the limit cycle and can reduce the generation of higher-order harmonics, while retaining the beneficial phase advantage that is associated with existing reset-based PI controllers.