Typical Section Problems for Structural Control Applications

Two low order problems are studied which capture some of the important fundamental physics associated with the control of structures. The order of the problems is kept low to allow the derivation of a closed-form solution. This identifies the dependency of the solution on the basic parameters of the problem. These two problems derive the optimal H ₂ and H _∞ control for a spring/mass system described by a second order, ordinary differential equation. The H ₂ solution is compared with the closed-form H ₂ solution to the optimal regulator for an infinite rod and beam whose behaviors are described by second order, partial differential equations. This comparison identifies the analogies between the typical section problem and a simple structural control problem. The optimal control solutions for the H ₂ and H _∞ problems are expanded upon by optimizing passive parameters. This reduces total closed-loop cost and is analogous to a control/structure optimization problem. It is found that certain levels of finite passive damping and stiffness are desirable even if they are available at no cost in the H ₂ problem, and that additional stiffness (any amount) and damping (up to ζ = 1/√2) enhances performance in the H _∞ problem. It is shown that while the structure and control must be designed simultaneously to optimize an H ₂ performance metric, sequential design will work in some cases which use an H _∞ performance metric. These simple problems reveal important properties of more complex structural control problems.