Stable Feedback Control of Linear Distributed Parameter Systems: Time and Frequency Domain Conditions

AbstractLarge space structures, or any mechanically flexible structures, are inherently distributed parameter systems (DPSs) whose dynamics are modeled by partial, rather than ordinary, differential equations. Such DPSs are described by operator equations on an infinite-dimensional Hilbert (or Banach) space. However, any feedback controller for such a DPS must be a finite-dimensional (and discrete-time) system to be implemented with on-line digital computers and a finite (small) number of actuators and sensors. There are many ways to synthesize such controllers; we will emphasize the Galerkin or finite-element approach. Although the overall performance of finite-dimensional controllers is important, the first consideration is their stability in closed-loop with the actual DPS. The analysis of DPSs makes use of the theory of semigroups on the infinite-dimensional state space. We will present stability bounds in both the time and frequency domains for infinite-dimensional systems. Currently, the frequency domain approach appears to yield more easily tested stability conditions than the time domain approach; however, we will show some relationships between these two methods and emphasize the role played by the DPS semigroup and its properties. It seems to us that such stability conditions are essential for the planning and successful operation of complex systems like large aerospace structures