Suboptimal Bilinear Control Methods Applied to Suppressing Car Vibrations

Abstract

Methods for suboptimal bilinear control of semiactive, smart damping elements are developed in this study by use of extensions of linear-quadratic optimal control. These damping elements are considered in the form of continuously variable dampers, which are capable of adapting their damping forces to reasonably match full- or limited-state control (inputs) forces generated by broadband actuators. During the operation of these dampers, simultaneous broadband force generators control some other locations in the vibratory system. This leads to a class of bilinear vibratory systems with disjunct (active) linear and (semiactive) multiplicative control inputs. Generally, these disjunct control inputs might be contributing to control the same vibratory modes or not. A generalized quadratic performance index is considered for minimizing variances of the system state variables, the broadband control forces, and the damping factors of semiactive dampers. This performance index is constrained by (i) the inability to measure all the system state variables that are necessary for the operation of the broadband control input and (ii) the necessity to consider the switching states of the semiactive damping elements. Methods for full-state bilinear control and limited-state bilinear control are first derived. Then an application is made to an in-plane 4-DOF car model of front suspension unit having semiactive damper and rear suspension unit having broadband ideal actuator, or vice versa, as two different suspension schemes. Time-domain simulation of the vehicle response to (a hole followed by a bump) deterministic and stochastic road inputs is made. Comparisons with full-state control and limited-state control designs show the effectiveness of the bilinear control configurations derived in this work.