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Digital PID control requires a differencing operation to implement the Dgain. In order to suppress the effects of noisy data, the traditional approachis to filter the data, where the frequency response of the filter is adjustedmanually based on the characteristics of the sensor noise. The present paperconsiders the case where the characteristics of the sensor noise change overtime in an unknown way. This problem is addressed by applying adaptivereal-time numerical differentiation based on adaptive input and stateestimation (AISE). The contribution of this paper is to extend AISE to includevariable-rate forgetting with exponential resetting, which allows AISE to morerapidly respond to changing noise characteristics while enforcing theboundedness of the covariance matrix used in recursive least squares.

Adaptive Real-Time Numerical Differentiation with Variable-Rate Forgetting and Exponential Resetting