Sufficient conditions for global integral action via incremental forwarding for input-affine nonlinear systems

In this article, we study the problem of con- stant output regulation for a class of input-affine multi-input multi-output nonlinear systems, which do not necessarily admit a normal form. We allow the references and the dis- turbances to be arbitrarily large and the initial conditions of the system to range in the full-state space. We cast the problem in the contraction framework, and we rely on the common approach of extending the system with an integral action processing the regulation error. We then present suf- ficient conditions for the design of a state-feedback control law able to make the resulting closed-loop system incre- mentally stable, uniformly with respect to the references and the disturbances. Such a property ensures uniqueness and attractiveness of an equilibrium on which output regu- lation is obtained. To this end, we develop an incremental version of the forwarding (mod {Lg V }) approach. Finally, we provide a set of sufficient conditions for the design of a pure (small-gain) integral-feedback control. The proposed approach is also specialized for two classes of systems that are linear systems having a Lipschitz nonlinearity and a class of minimum-phase systems whose zero dynamics are incrementally stable.