1. Time-Varying Angular Rate Sensing for a MEMS Z-Axis Gyroscope Mohammad Salah †, Michael McIntyre †, Darren Dawson †, and John Wagner ‡ Mohammad Salah †, Michael McIntyre †, Darren Dawson †, and John Wagner ‡ Departments of Electrical † and Mechanical ‡ Engineering, Clemson University, Clemson, SC 29634-0921, E-mail: msalah@ces.clemson.edu The technical report is posted online at http://www.ece.clemson.edu/crb/publictn/CU-CRB-2-24-06-1.pdf Assumption 1: The gyroscope’s parameters and are unknown and assumed to be constants with respect to time Assumption 2: The damping ratios and, and the stiffness and are equal to zero. The estimation development could be extended such that these parameters could also be estimated Assumption 3: The time-varying angular rate and its first two time derivatives are bounded;. Abstract ●An off-line parameter estimation strategy is first developed that places the gyroscope in a condition of zero angular rate. A reference input is used to excite both axes such that a subsequent required persistence of excitation condition is met ●An adaptive least-squares algorithm is utilized to estimate the unknown model parameters ●Based on the exact knowledge of the model parameters, an on-line active controller/observer is then developed for time varying angular rate sensing. For this method, a Lyapunov- based nonlinear control algorithm is designed ●The dynamics can be written in the Cartesian coordinate system as where Gyroscope Dynamics and Assumptions Non-ideal simple spring-mass model of the MEMS z-axis gyroscope is the displacement of the gyroscope’s reference point, and is the control input M, D, and K denote the inertia effect, damping ratio, and spring constant, respectively is the centripetal-Coriolis effect and is the gyroscope’s time-varying angular rate Off-Line Parameter Estimation ●The system is configured such that the angular rate is equal to zero ( ), hence, the gyroscope dynamics become ●The reference input is designed to be a bounded, piecewise continuous function ●Because is not measurable while and are measurable, a torque filtering and linear parameterization techniques are utilized to obtain the expression where is the convolution operator, and is the impulse response of a linear stable, strictly proper filter that can be defined as a first-order filter as ● is a vector of the unknown constant parameters and is a known filtered regression matrix, and they are defined as ●The following adaptive rule can be generated using the least-squares estimation method where is the estimate of the gyroscope’s unknown constant parameters, and the following persistence of excitation condition should be satisfied along with some other conditions On-Line Angular Rate Estimation ●The Objectives of the angular rate estimator are two; (i) to ensure that the reference point’s displacement tracks the desired trajectory and (ii) to ensure that the estimated angular rate converges to the actual angular rate ●To achieve the mentioned objectives, the following control input is designed based on a Lyapunov stability analysis where are constants ●By developing the closed-loop error system, the following expression can be obtained Simulation Results The parameters were estimated after 125 seconds within ±0.3% of their actual values with the following simulation setup Off-Line Parameter Estimator On-Line Angular Rate Estimator The estimates obtained from the off-line parameter estimator were used to estimate the time-varying angular rate with the following simulation setup The angular rate error is within ±0.5% of the actual value after 0.775 seconds then the time-varying angular rate can be estimated as are constants