Direct Optimal Vibration Control of a Piezoelastic Plate

An optimal control method for distributed parameter systems is developed and applied to active vibration control of a plate. Piezoelectric material is used as an actuator. The structural dynamics are modeled using the space—time assumed mode expansion technique (a Rayleigh—Ritz method applied to both time and space variables). In particular, the generalized coordinates (both mechanical and electrical) and all generalized forces are described in terms of expansion functions. Using Hamilton's law of varying action along with the space—time assumed mode expansion results in algebraic equations of motion. These are then used as the constraint equations in the optimal controller design. Using space—time expansion, the usual variational optimal control problem is transformed into an equivalent algebraic problem. Optimal solutions are then obtained in a closed form and the solution is global within the time period considered. The solution procedure does not lead to a Riccati equation as is the case in the conventional optimal control solutions. The direct optimal control problem of a vibrating plate is illustrated through simulations where a parametric study is undertaken to check the controller performance. This study shows that the developed optimal control method is simple and the structure—control interaction is an inherent property of the method.