Disturbance Attenuation by Measurement Feedback in Nonlinear Systems via Immersion and Algebraic Conditions

In this paper, we consider the problem of disturbance attenuation with internal stability for nonlinear, input-affine systems via measurement feedback. The solution to the above-mentioned problem has been provided, three decades ago, in terms of the solution to a system of coupled nonlinear, first-order partial differential equations (PDEs). As a consequence, despite the rather elegant characterisation of the solution, the presence of PDEs renders the control design synthesis almost infeasible in practice. Therefore, to circumvent such a computational bottle-neck, in this paper we provide a novel characterisation of the exact solution to the problem that does not hinge upon the explicit computation of the solution to any PDE. The result is achieved by considering the immersion of the nonlinear dynamics into an extended system for which locally positive definite functions solving the required PDEs may be directly provided in closed form by relying only on the solutions to Riccati-like, state-dependent, algebraic matrix equations.