Output regulation for linear systems with unknown exosystem order

The regulator problem is considered for 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 2r + 1 and its parameters {ω₁, ..., ω_r} are unknown. The proposed regulator makes use of two observers and contains an adaptive internal model which adapts 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_M, the regulation error tends exponentially into a closed ball whose radius is proportional to ε_M, provided that ε_M is sufficiently small so that singularities are avoided.