Abstract: This paper is interested in two difficulties encountered in practice when observing a continuous time process. The first is that we cannot observe a process over a time interval; we only take discrete observations. The second is the process frequently observed with a constant additive error. It is important to give an estimator of the spectral density of such a process taking into account the additive observation error and the choice of the discrete observation times. In this work, we propose an estimator based on the spectral smoothing of the periodogram by the polynomial Jackson kernel reducing the additive error. In order to solve the aliasing phenomenon, this estimator is constructed from observations taken at well-chosen times so as to reduce the estimator to the field where the spectral density is not zero. We show that the proposed estimator is asymptotically unbiased and consistent. Thus we obtain an estimate solving the two difficulties concerning the choice of the instants of observations of a continuous time process and the observations affected by a constant error.
Abstract: This work focuses on the symmetric alpha stable process with continuous time frequently used in modeling the signal with indefinitely growing variance, often observed with an unknown additive error. The objective of this paper is to estimate this error from discrete observations of the signal. For that, we propose a method based on the smoothing of the observations via Jackson polynomial kernel and taking into account the width of the interval where the spectral density is non-zero. This technique allows avoiding the “Aliasing phenomenon” encountered when the estimation is made from the discrete observations of a process with continuous time. We have studied the convergence rate of the estimator and have shown that the convergence rate improves in the case where the spectral density is zero at the origin. Thus, we set up an estimator of the additive error that can be subtracted for approaching the original signal without error.
Abstract: This paper describes identification of the two poles
unstable SOPDT process, especially with large time delay. A new
modified relay feedback identification method for two poles unstable
SOPDT process is proposed. Furthermore, for the two poles unstable
SOPDT process, an additional Derivative controller is incorporated
parallel with relay to relax the constraint on the ratio of delay to the
unstable time constant, so that the exact model parameters of
unstable processes can be identified. To cope with measurement
noise in practice, a low pass filter is suggested to get denoised output
signal toimprove the exactness of model parameter of unstable
process. PID Lead-lag tuning formulas are derived for two poles
unstable (SOPDT) processes based on IMC principle. Simulation
example illustrates the effectiveness and the simplicity of the
proposed identification and control method.