Active control of flexible systems

Since mechanically flexible systems are distributed-parameter systems, they are infinite-dimensional in theory and, in practice, must be modelled by large-dimensional systems. The fundamental problem of actively controlling very flexible systems is to control a large-dimensional system with a much smaller dimensional controller. For example, a large number of elastic modes may be needed to describe the behavior of a flexible satellite; however, active control of all these modes would be out of the question due to onboard computer limitations and modelling error. Consequently, active control must be restricted to a few critical modes. The effect of the residual (uncontrolled) modes on the closed-loop system is often ignored. In this paper, we consider the class of flexible systems that can be described by a generalized wave equation,u _tt+Au=F, which relates the displacementu(x,t) of a body Θ inn-dimensional space to the applied force distributionF(x,t). The operatorA is a time-invariant symmetric differential operator with a discrete, semibounded spectrum. This class of distributed parameter systems includes vibrating strings, membranes, thin beams, and thin plates. The control force distribution $$F(x,t) = \sum\limits_{i = 1}^M { \delta (x - x_i )f_i (t)} $$ is provided byM point force actuators located at pointsx _i on the body. The displacements (or their velocities) are measured byP point sensorsy _i(t)=u(z _j,t), oru _t(z _j,t),j=1, 2, ...,P, located at various pointsz _j along the body. We obtain feedback control ofN modes of the flexible system and display the controllability and observability conditions required for successful operation. We examine the control and observation spillover due to the residual modes and show that the combined effect of spillover can lead to instabilities in the closed-loop system. We suggest some remedies for spillover, including a straightforward phase-locked loop prefilter, to remove the instability mechanism. To illustrate the concepts of this paper, we present the results of some numerical studies on the active control of a simply supported beam. The beam dynamics are modelled by the Euler-Bernoulli partial differential equation, and the feedback controller is obtained by the above procedures. One actuator and one sensor (at different locations) are used to control three modes of the beam quite effectively. A fourth residual mode is simulated, and the destabilizing effect of control and observation spillover together on this mode is clearly illustrated. Once observation spillover is eliminated (e.g., by prefiltering the sensor outputs), the effect of control spillover alone on this system is negligible.