Function approximation technique (FAT)-based nonlinear control strategies for smart thin plates with cubic nonlinearities

The current work introduces three nonlinear control solutions for the regulation of a vibrating nonlinear plate considering model uncertainty. These solutions are feedback linearization control (FBL), virtual velocity error-based control (VVEC), and backstepping control (BSC). In the FBL control, a nonlinear control law is designed with linear closed-loop dynamics such that dynamic stability is ensured. Whereas, by the VVEC (or so-called passivity-based approach in the robotics community) the limitations of the feedback linearization are overcome. On the other side, the BSC selects virtual control variables with stabilized intermediate control laws based on Lyapunov theory. Systematic modeling for the target vibrating plate with piezo patches is described. In effect, considering the nonlinear influence makes the resulted mode shapes for the vibrating structure are highly coupled and careful control design is required. Using the Galerkin approach, the partial differential equation for the smart plate is transformed into definite ordinary differential equations; the multi-input multi-output model is established. The aforementioned control strategies are evaluated and investigated in detail. In essence, they are powerful tools for dealing with nonlinear dynamic systems, however, the VVEC could be considered superior in comparison with the FBL control and the BSC since the designed control structure does not include inertia inverse matrix and modal coordinate acceleration that could make computational problems. As a result, simulation experiments were focused on the VVEC strategy, and the latter was implemented on a simply supported thin plate structure with collocated piezo-patches. The results show the validity of the proposed control architecture.