Adaptive stabilization of continuous-time systems guaranteeing the controllability of a modified estimation model

This paper presents an indirect adaptive control scheme for continuous-time systems. The estimated plant model is controllable while the estimation model is free from singularities. Such singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. This property is achieved by ensuring that the absolute value of its determinant does not lie below a prescribed positive threshold. A switching rule is used in the estimates modification algorithm to ensure the controllability of the modified estimated model while avoiding possible chattering. For that purpose, the switching rule takes values at two possible distinct prefixed thresholds. In the event when the Sylvester determinant takes the current value of the switching function then that one switches to the alternative threshold. The convergence of both the unmodified and modified estimates to finite limits guarantees that switching ends in finite time. Thus, the solution to the controlled plant exist so that all the signals within the loop are well-posed.