Second-order state estimation experiments using acceleration measurements

The estimation of dynamic states for feedback control of structural systems using second-order differential equations and acceleration measurements is described. The formulation of the observer model and the design of the observer gains are discussed in detail. It is shown that the second-order observer is highly stable because the stability constraints on the observer gains are model independent. The limitation of the proposed observer is the need for "nearly" collocated actuators and accelerometers. Experimental results using a control-struct ure interaction testbed are presented that show that the second-order observer provided more stability than a Kalman filter estimator without decreasing closed-loop performance. F LEXIBLE-BODY vibration control of large structural systems often requires estimation of dynamic states that cannot be measured directly. The state estimator or observer plays a key role in the overall stability of the control system. Reference 1 has shown that the linear quadratic regulator (LQR) technique provides 60 deg of phase margin and infinite gain margin when the full state is available. However, when an observer must be used in conjunction with LQR, no guaran- tees can be made on the robustness properties due to state reconstruction errors.2'3 Observer state reconstruction suffers from model errors, measurement noise, and unmodeled dy- namics.4 Although much work has been done on optimal state estimation using first-order Kalman filters, the closed-loop stability remains a strong function of the observer model accuracy. Although there is more design freedom when using a first- order model, second-order observers provide a great deal of physical insight when scientists attempt to perform state esti- mation of a second-order system. Second-order observer mod- els have received recent attention in the literature. Reference 5 utilized a Kalman filter for discretized (time and space) sec- ond-order structure models. In addition, robust computa- tional procedures for solution of the Kalman filter estimation error covariance matrices have been developed for second-or- der models in Ref. 6. A dissipative observer for use with velocity measurements that possessed the second-order form was introduced in Ref. 7. This observer was analogous to the collocated controller whereby symmetric positive definite feedback gains were designed to insure stability. Recently, the authors of Ref. 8 developed the conditions for optimal estima- tion using the second-order form assuming velocity measure- ments were available. Accelerometers are frequently used to measure the dynamic response of structural systems because of their low cost and their light weight and because they provide an inertial mea- surement. The first-order observer filters acceleration mea- surements in both space and time to derive the proper phase information to reconstruct the unmeasured states. Spatial fil- tering, which projects the measurement data onto the system coordinate basis, requires accurate dynamic models. The re- duced design freedom available with the second-order ob- server has usually required velocity measurements since nei- ther spatial nor temporal filtering of the measurement is performed. In this work, collocated acceleration measurements and control actuators are used for state estimation by augmenting the second-order observer with additional states. (The addi- tional states provide temporal filtering of the acceleration measurements to insure asymptotic convergence of the esti- mated state error.) By using collocated sensors and actuators, the observer does not require spatial filtering, which signifi- cantly reduces the sensitivity to model errors. Equations for the typical first-order observer and a logical form for the second-order observer are presented. Stability and performance considerations for the first- and second-or- der observers are discussed. A design method for second-order observer gains with acceleration measurements is then de- scribed. Practical aspects of the observers are examined, and both numerical and experimental results are presented.