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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.
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.