This paper addresses the design problem of observer-structured controller, which has been recently proposed as an extension of Luenberger observer-based controllers to the case of controlling Linear Parameter-Varying (LPV) systems, and demonstrates its effectiveness in three practical design examples. The first one is model-based visual feedback controller design for a cart-driven inverted pendulum, the second one is data-based flight controller design, and the last one is model-data-concurrent-based flight controller design. Observer-structured controller has been proposed to embed plant state estimation ability to a priori designed Linear-Time-Invariant (LTI) or LPV controllers, similarly to existing methods in literature to embed plant state estimation ability to a priori designed LTI controllers which are designed for controlling LTI plant systems. However, the demonstrations of its applicability and usefulness have not been sufficiently illustrated. This paper therefore shows them directly by designing observer-structured controllers for model-based design problem, in which only mathematical plant models and a priori designed controller state-space matrices are available, for data-based design problem, in which only operational data and a priori designed controller state space matrices are available, and for model-data-concurrent-based design problem, in which operational data as well as mathematical plant models and a priori designed controller state-space matrices are available. In particular, the controller designed in the first formulation demonstrates the practical state estimation performance by lab-level experiments, and the controller in the second formulation demonstrates the usefulness of operational data to improve plant state estimation ability but simultaneously illustrates the lack of theoretical performance guarantee, and the controller in the last formulation well demonstrates the effectiveness of operational data to improve plant state estimation performance while rigorous estimation performance is guaranteed by using models.
We consider the problem of transporting an open liquid-filled container along a straight line in minimum time while preventing liquid spilling. The container is transported by a planar motor with three degrees of freedom: horizontal translation, vertical translation, and pitching angle. Under the assumptions of linear sloshing, small liquid oscillations, and an axisymmetric container, the liquid dynamics are represented by an equivalent damped pendulum. The resulting minimum-time motion planning problem is non-convex. To obtain a convex formulation, the nonlinear terms are approximated using first-order Taylor expansions and McCormick relaxations. A general proportional relation between the vertical displacement of the liquid free surface and the pendulum angle is also derived for axisymmetric containers. The final time is fixed and the minimum feasible time is found through a bisection procedure, allowing the global optimum of the approximated problem to be obtained. Experiments using different axisymmetric container shapes and sloshing limits demonstrate the effectiveness of the approach. The approximation error is evaluated by measuring the constraint violation of the approximated solution with respect to the original non-convex problem, showing small violations in all considered cases. The contribution of vertical translation and pitching is also evaluated, showing that vertical translation has a negligible impact on the optimal transport time, while pitching can provide significant time reductions, especially for low sloshing limits, up to 76.26%. Finally, the effect of container geometry is investigated, showing that the minimum transport time is little sensitive to geometry changes, while the optimal control depends on the geometry.
