3-DOF-PID nonlinear ℋ_∞ estimators

Abstract

In this article, we present the one-degree-of-freedom (1-DOF) or proportional (P), the 2-DOF or proportional-integral (PI) and the 3-DOF or proportional-integral-derivative (PID) ℋ_∞ nonlinear filters as counterparts of the popular P, PI and PID controllers, respectively. We use the idea of certainty-equivalence principle in which the worst-case disturbance or noise is determined as a function of the estimated state and under the assumption that this converges to the actual value asymptotically. Sufficient conditions for the solvability of the various filter gains are derived in terms of new Hamilton–Jacobi–Isaacs equations, which do not involve the system states. Simulation results are also presented to support the theory.

Keywords:

• nonlinear system
• nonlinear filtering
• ℒ₂-gain
• PID-filters
• Hamilton–Jacobi–Isaacs equations

Notes

Note

1. For a manifold M, TM is the tangent bundle of M and T ^⋆ M is the cotangent bundle of M.