Adaptive stabilization of first-order systems using estimates modification based on a Sylvester determinant test

This note presents an adaptive control scheme for first-order continuous-time systems subject to bounded noise and unmodelled dynamics. The estimated plant model is controllable and then the adaptive scheme is free from singularities. The estimation scheme involves the use of a relative dead zone which freezes the estimation process when the size of the prediction error is small compared to the contribution of the unmodelled dynamics. The singularities are avoided through the use of a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a prefixed positive threshold. In addition, the use of a hysteresis switching function is not used for a modification of the estimates while the absence of chattering is guaranteed in the eventual case when the Sylvester determinant tends to zero. The global stability of the closed-loop system is also guaranteed.