Matt Bement

Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with an in-domain nonlinearity is considered first. For this system a nonlinear feedback law, based on gain scheduling, is derived explicitly, and a proof of local exponential stability, with an estimate of the region of attraction, is presented for the closed-loop system. Control designs (without proofs) are then presented for a string PDE and a shear beam PDE, both with Kelvin–Voigt (KV) damping and free-end nonlinearities of a potentially destabilizing kind. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization based design. [DOI: 10.1115/1.4004065]

Motion planning and tracking for tip displacement and deflection angle for flexible beams

We present results for motion planning and tracking for flexible beams with Kelvin-Voigt damping, when the goal is to track sinusoidal reference signals for the displacement and deflection angle at the free-end of the beam using only actuation at the base. We present the solution to the motion planning problem for the string model, and a method of leveraging the string solution with PDE backstepping theory to solve the motion planning problem for the shear beam. We then present state-feedback boundary controllers that stabilize their respective systems around the motion planning solution.

Gain Scheduling-Inspired Control for Nonlinear Partial Differential Equations

We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with a destabilizing in-domain nonlinearity is considered first. For this system a nonlinear feedback law based on gain scheduling is derived explicitly, and a statement of stability is presented for the closed-loop system. Control designs are then presented for a string and shear beam PDE, both with Kelvin-Voigt damping and potentially destabilizing free-end nonlinearities. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization-based design.Copyright

Parametrization of Reduced Order MIMO Tracking Prefilters With Optimality Considerations

A primary disadvantage of using an internal model to achieve multivariable tracking is the high order of the internal model. In situations where it is known that each output is to track only its associated reference input, the internal model formulation results in an overdesign of sorts. A method is presented through which a prefilter may be constructed to achieve asymptotic tracking of only the required reference inputs. It is shown that obtaining the prefilter requires the solution of a polynomial matrix equation. Conditions for existence of a solution to this equation, as well as an algorithm for its construction, are presented. Since existence of a solution implies an infinite number of solutions, the algorithm provides a means of parametrizing all solutions of a given order. Unlike prefilter techniques such as plant inversion, the method presented may be applied to nonminimum phase systems and results in proper, physically realizable systems. Since an infinite number of solutions exist, criteria for defining and obtaining the optimal solution are presented. In fact, it is shown that obtaining the optimal prefilter reduces to solving a set of linear equations. A multivariable system is used to demonstrate the effectiveness of the optimization procedure. In addition, the tracking is shown to be robust with respect to certain structured plant perturbations.