Abstract

1. Abstract • 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 • 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 Non-ideal simple spring-mass model of the MEMS z-axis gyroscope are constants Gyroscope Dynamics and Assumptions On-Line Angular Rate Estimation • The dynamics can be written in the Cartesian coordinate system as where • 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 is the control input is the displacement of the gyroscope’s reference point, and 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 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; . then the time-varying angular rate can be estimated as Simulation Results Off-Line Parameter Estimator On-Line Angular Rate Estimator The parameters were estimated after 125 seconds within ±0.3% of their actual values with the following simulation setup The estimates obtained from the off-line parameter estimator were used to estimate the time-varying angular rate with the following simulation setup 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 The angular rate error is within ±0.5% of the actual value after 0.775 seconds