Certainty equivalence implies detectability 1

It is shown that any stabilizing, certainty equivalence control used within an adaptive control system, causes the familiar interconnection of a controlled process and associated output estimator to be detectable through the estimator's output error ep, for every frozen value of the index or parameter vector p upon which both the estimator and controller dynamics depend. The fact that certainty equivalence implies detectability has been known for some time – this has been shown to be so whenever the process model is linear and the controller and estimator models are also linear for every frozen value of p. In this paper, use is made of recently introduced concepts of input-to-state stability and detectability for nonlinear systems to prove that the same implication is valid in a more general, nonlinear setting.