Modal Control of Certain Flexible Dynamic Systems

Interest has increased in the active control of vibrations in mechanically flexible systems, e.g. attitude control of flexible spacecraft, ride quality improvement of air and surface transportation, and active optics. To insure satisfactory performance of such systems, their distributed parameter nature must be taken into account in control system design. In this paper, we obtain feedback control of N nodes of a flexible system and treat the problem of control “spillover” into the uncontrolled modes. We consider the class of flexible systems that can be described by a generalized wave equation, uₜₜ + Au = F, which relates the displacement $u(x,t)$ of a body Ω in n-dimensional space to the applied control forces $F(x,t)$. The operator A is a time-invariant, symmetric differential operator with a discrete semibounded spectrum and the control forces F(x,t) = ∑i = 1M bᵢ (x)fᵢ (t) are provided by M actuators with influence functions bᵢ (x). The displacements are measured by P sensors yⱼ (t), j = 1,2, ⋯ ,P, located at various points along the body. This class of distributed parameter systems includes interior and boundary control of vibrating strings, membranes, thin beams, and thin plates. A feedback control is found by applying known state variable methods to a lumped parameter approximation of the above system. Such a discretization can be obtained by expanding the system state in the eigenmodes of the operator A and truncating the expansion. The N controlled modes are selected and the resulting finite dimensional approximate system must meet certain controllability and observability conditions which we display. The feedback control so obtained will stabilize the N selected modes of the flexible system, and the energy in the uncontrolled modes is usually ignored in the design. However, in all but some very special situations, the actuators will excite these uncontrolled modes. In vibrating systems, such control “spillover” can severely degrade the system performance. We prove a result that guarantees suitable system performance with the above modal feedback control, provided that the controlled modes are controllable and observable, the remaining modes are unobservable, and the control spillover meets an a priori bound determined by the desired performance. This result justifies the use of a simple and direct modal feedback control for flexible systems which satisfy the spillover condition, and it issues a warning for modal control of systems which have excessive control spillover. To illustrate these results, a modal feedback control is obtained for a simple beam with pinned ends and the performance is evaluated.