A new perspective on stable inversion of non-minimum phase nonlinear systems

Alexey Pavlov,Kristin Y Pettersen

doi:10.4173/mic.2008.1.3

Abstract

Abstract We present a new perspective on the problem of stable inversion of nonlinear non-minimum phase systems. It is based on the notion of convergent systems. The machinery of convergent systems allows us to obtain novel qualitative and quantitative conditions for solving this problem. These conditions provide insight into the dynamics behind the stable inversion problem and make it possible to treat this problem in a non-local way. Qualitatively, they cover the conditions for the stable inversion of non-minimum phase nonlinear systems previously reported in literature and allow us to solve this problem for a broader class of systems. The proposed approach is supported with a novel computational method.

Highlights

In output tracking control of nonlinear systems x = f (x) + g(x)u, x ∈ Rn, u ∈ R (1)y = h(x), y ∈ R, with sufficiently smooth f (x), g(x) and h(x), one often has to find a bounded input ud(t) such that system (1) with this input has a bounded solution xd(t) and the corresponding output equals h(xd(t)) ≡ yd(t), where yd(t) is a given sufficiently smooth bounded reference output trajectory
For non-minimum phase systems the tracking dy- In Section 4 we discuss the case of the tracking dynamics (4) are unstable, and finding a bounded namics with periodic inputs and propose a numerical solution ξ(t) or even proving its existence becomes prob- method for solving the stable inversion problem
We have presented a new approach to solving the problem of stable inversion of non-minimum phase nonlinear systems

Summary

Introduction

For non-minimum phase systems the tracking dy- In Section 4 we discuss the case of the tracking dynamics (4) are unstable, and finding a bounded namics with periodic inputs and propose a numerical solution ξ(t) or even proving its existence becomes prob- method for solving the stable inversion problem. In this paper we propose a new approach to prove the existence of a bounded solution of the time-varying tracking dynamics (4) This approach provides novel qualitative and quantitative conditions for the stable inversion problem. The weakness condition is formulated as a small gain condition In addition to this result, we bring attention to the old yet overlooked fact due to (Demidovich, 1967) that for a time-varying nonlinear system with a unique bounded on R solution, like for instance the tracking dynamics (4), periodicity of the right-hand side with respect to time implies periodicity of this bounded solution. We use this fact for numerical computation of the bounded solution of the tracking dynamics

Convergent systems and a steady-state operator

Main results

Conclusions

Example