Indirect Adaptive Control Using a Static Update Law

The update law in the indirect adaptive control scheme can be extended to include feedthrough of an error term. This reduces undesired oscillations of the calculated weights. When the $σ$-modification is used for achieving robustness against unstructured uncertainties, the gain of the feedthrough in the update law cannot be chosen arbitrarily. Compared to our previous result, we show stability of the closed loop for a larger parameter-range for the gain of the feedthrough in the update law. This parameter-range includes a configuration for which the influence of the integration in the update law diminishes over time, i.e. for which the adaptation for large times is governed solely by the feedthrough in the update law. By initializing at zero, this allows for removing the integration from the update law, resulting in a static update law. For the purely linear case, the adaptation acts like a disturbance observer. Frequency-domain analysis of the closed loop with a second order plant shows that removing the integration from the update law with $σ$-modification and feedthrough affects how precisely disturbances in the low-frequency band are observed. If the damping injected into the adaptation process by the $σ$-modification exceeds certain bounds, then the precision is increased by using the static update law.