Optimal sensor/actuator placement for active vibration control using explicit solution of algebraic Riccati equation

This paper deals with an optimal placement problem of sensors and actuators for active vibration control of flexible structures. For undamped structures with collocated rate sensors and actuators, two solutions of generalized algebraic Riccati equations (generalized control algebraic Riccati equation, GCARE; generalized filtering algebraic Riccati equation, GFARE) are obtained explicitly. Employing these explicit solutions, we can obtain a stabilizing H-infinity controller based on the normalized coprime factorization approach without solving any algebraic Riccati equations numerically. Generally, in a optimal sensor/actuator placement problem with a model-based control law (LQG or H-infinity), the feedback controller needs to be obtained for all candidates of the optimal placement (which may be derived with some numerical optimization techniques) by solving algebraic Riccati equations numerically. Therefore, the amount of computation required to determine the optimal sensor/actuator placement and the controller increases rapidly for large-scale structures which have many pairs of sensor/actuators. The H-infinity controller in this paper can be obtained just by addition and multiplication of several matrices. Furthermore, a closed-loop property on H-infinity norm is automatically bounded for all candidates of the optimal placement. Hence, we can formulate the optimal sensor/actuator placement problem to optimize other closed-loop properties (norm of the closed-loop system) with less computational requirement than the model-based method mentioned above. The gradient of the H-2 norm of the closed-loop system, which is necessary for a descent-based optimization technique, is derived. Using this sensitivity formula, we obtain the optimal placements of two pairs of sensors and actuators which minimize the H-2 norm of the closed-loop system for a simply supported beam by the quasi-Newton method. The simulation results show the effectiveness of the proposed design method.