Disturbance cancellation for linear systems by adaptive internal models

The adaptive disturbance cancellation problem is considered for single-input, single-output, linear systems when the linear exosystem which generates the disturbances to be rejected and/or the reference signals to be tracked is unknown, that is both its order 2 r + 1 and its parameters { ω 1 , … , ω r } are unknown. The proposed regulator makes use of two observers and contains an adaptive internal model which tunes m internal parameters on the basis of the output regulation error. It is shown that: if m ≥ r , that is the adaptive internal model can reproduce the required input, the regulation error tends exponentially to zero; if m < r , that is the adaptive internal model can only approximate the required input within an approximation error ϵ , the regulation error tends exponentially into a closed ball whose radius is proportional to ϵ , provided that ϵ is sufficiently small so that singularities are avoided.