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It is shown that persistent excitation (PE) conditions are overly stringent for ensuring exponential convergence of the tracking error in overparametrized adaptive feedforward systems.
Consideration is given to the concept of a reference signal which is sufficient to cause the parameter error in an adaptive control system to converge to zero. Simple and necessary conditions for the convergence of the parameter error are established for a model reference adaptive control (MRAC) system having a relative degree of one. The exponential stability of the MRAC under persistent excitation (PE) is also considered. Full proofs of the theoretical arguments are given in a companion paper.
Many practical adaptive feedforward systems are overparametrized and for this reason will not satisfy persistent excitation (PE) conditions. For these systems, a weaker PE condition is proposed under which it is shown that the tracking error (the error between the desired and estimated outputs) converges exponentially.
Many practical adaptive feedforward systems are overparametrized and for this reason will not satisy persistent excitation (PE) conditions. For these systems, a weaker PE condition is poposed under which it is shown that the tracking error (the error between the desired and estimated outputs) converges exponentially.
Here, the dynamic mode decomposition (DMD) technique extracts the dominant modes characterizing the innate dynamical behavior of the system within the measurement data. For appropriate identification of dominant modes from the measurement data, the DMD algorithm necessitates ensuring the quality of the input measurement data sequences. On that account, for validating the usability of the dataset for the DMD algorithm, the paper proposed two conditions: Persistence of excitation (PE) and the Granger Causality Test (GCT). The virtual data sequences are designed with the hankel matrix representation such that the dimensions of the subspace spanning the essential system modes are increased with the addition of new state variables. The PE condition provides the lower bound for the trajectory length, and the GCT provides the order of the model. Satisfying the PE condition enables estimating an approximate linear model, but the predictability with the identified model is only assured with the temporal causation among data searched with GCT. The proposed methodology is validated with the application for coherency identification (CI) in a multi-machine power system (MMPS), an essential phenomenon in transient stability analysis. The significance of PE condition and GCT is demonstrated through various case studies implemented on 22 bus six generator system.
The behavior of a standard identifier is analyzed for the case in which the plant contains additional dynamics, called unmodeled dynamics, which invalidate the known order assumption. An input richness condition is obtained that does not depend on the order of the unmmodeled dynamics to guarantee persistency of excitation (PE) of the regressor. It is shown that the PE condition leads to a BIBO stability property for the identifier. The method of averaging is used to define formally the notion of tuned parameters as the equilibrium of the identifier-averaged system. It is shown that the tuned parameters always exist and that the actual parameters converge to some neighborhood of the tuned parameters. An explicit expression is derived to calculate and interpret them as the fixed parameter values that minimize the mean-squared output error.