The design of controllers for structural systems, particularly those associated with large space structures, has received a considerable amount of attention in the past few years. The usual approach to designing these controllers is to apply modern control theory to a reduced linear system obtained from finite element analysis or from a truncated modal analysis. In most of these designs, the sensor signal must be processed to separate out the contributions from each mode so that it may be sent to the appropriate actuators. The analysis presented here, on the other hand, obtains exact solutions for a selected set of sensor and actuator positions for simple structural elements. Sensor signals are fed back directly to the actuators with appropriate gains. The method of analysis is that of classical control theory using Laplace transforms and the associated open and closed-loop transfer functions. Single-input-single-output feedback control is applied to various flexible cable and beam configurations. Root-loci for various values of gain are constructed and the system characteristics and the global system stability are determined.

Although the procedure outlined above can be carried out for basic structural elements, more complex structures and control configurations are synthesized using the dynamic stiffness matrix method. With this method, the exact relationships of the basic elements can be combined to allow analysis of multi-input-multi-output control of more complex structures. Using this approach, examples for flexible cable and beam configurations are presented. It was found that exact solutions can be obtained using a finite number of sensors and actuators. It was also determined that a single co-located sensor-actuator at the boundary of a fixed-free cable or beam can control all the vibrational modes of the cable or beam. Also, pure signals from a perfect sensor can be used without any additional signal processing. The multi-input-multi-output investigation demonstrates that, even without cross-gain feedback, there is interaction between the sets of co-located sensor-actuator pairs. It appears that this interactive effect needs to be included in any multi-input-multi-output control design. By starting with fundamental elements of beams and cables, it was shown that reasonably sophisticated systems can be modeled. Finally, considerable insight is offered by analyzing the control of flexible structures using exact transfer function relationships.