Input-Output Linearization with Delay Cancellation for Nonlinear Delay Systems: the Problem of the Internal Stability

This paper investigates the issue of the internal stability of nonlinear delay systems controlled with a feedback law that performs exact input-output linearization and delay cancelation. In previous works the authors showed that, differently from the case of systems without state delays, when the relative degree is equal to the number of state variables and the output is forced to be identically zero, delay systems still possess a non trivial internal state dynamics. Not only: in the same conditions delay systems are also characterized by a non trivial input dynamics. Obviously, both internal state and input dynamics should give bounded trajectories, otherwise the exact input-output linearization and delay cancelation technique cannot be applied. This paper studies the relationships between the internal state and input dynamics of a controlled nonlinear delay system. An interesting result is that a suitable stability assumption on the internal state dynamics ensures that, when the output is asymptotically driven to zero, both the state and control variables asymptotically decay to zero.