Adaptive stabilization of non-necessarily inversely stable first-order systems by using estimates modification based on testing the Sylvester determinant

摘要

This paper presents an adaptive control scheme for first-order continuous-time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. The singularities are avoided through a modification of the a priori estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be non-singular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold. In addition, the use of a hysteresis switching function is not required for a modification of the estimates. The global asymptotic stability of the closed-loop scheme is ensured. It is also proved that the problem is well-posed even if chattering motion exists due to the modification process. The results are also extended to the case of presence of uncertainties consisting of bounded noise and a class of unmodelled dynamics so that global stability of the scheme is ensured. In this last case, relative adaptation zones are used on the a priori estimates for robust stabilization purposes so that the estimation process is frozen when the size of the prediction error is small compared to the contribution of the unmodelled dynamics to the plant output.