Prescribed-Time Stabilization Robust to Measurement Disturbances

Abstract

Prescribed-time stabilization employs time-varying gains that grow and multiply states that decay. Such feedback structures have unprecedented properties of regulation in user-prescribed finite time, independent of the initial condition, and with zero asymptotic gains to process right-hand side disturbances (perfect disturbance rejection), regardless of the disturbance size. However, when the state measurement is itself subject to a disturbance, the multiplication with growing gains threatens to result in unbounded control inputs. In this paper we present results—for linear systems in the controllable canonical form and for nonlinear high-dimensional Euler-Lagrange systems that describe high-degree-of-freedom robotic manipulators—which carry no such risk: the sum of the state and the measurement disturbance is still driven to zero, the input remains bounded, and a particular ISS property relative to the disturbance is guaranteed. The price we pay for such strong and fairly unexpected results is a structural condition we impose on the disturbance, which is met in practical applications that rely on accelerometer, gyroscope, or encoder measurements.