Elsevier

Automatica

Volume 71, September 2016, Pages 38-43
Automatica

Technical communique
A larger family of nonlinear systems for the repetitive learning control

https://doi.org/10.1016/j.automatica.2016.04.021Get rights and content

Abstract

A generalization of the PIDρ1 control has been recently presented in Marino et al. (2012), which guarantees the output tracking of smooth periodic reference signals (with known period) for uncertain minimum phase nonlinear systems in output feedback form with any relative degree ρ1. In this note, we show that the repetitive learning control proposed in Marino et al. (2012) also solves the same output tracking problem for a larger class of uncertain nonlinear systems in normal form, provided that the inverse system dynamics satisfy the Demidovich condition on suitable compact sets.

Introduction

Considerable research efforts have been spent in the last decades with the aim of addressing the existence of solutions to the output tracking problem for uncertain nonlinear systems.

In the unfavourable case in which the uncertainties are unstructured, the repetitive learning control approach (see  Xu & Tan, 2003 and  Ahn, Chen, & Moore, 2007) can be successfully followed when the output reference signals belong to the family of periodic time functions with known period.

In this respect, the most recent results for special classes of single-input single-output nonlinear systems can be found in (i)  Marino and Tomei (2009) for systems which are partially feedback linearizable by state feedback with linear exponentially stable inverse system dynamics; (ii)  Bifaretti, Tomei, and Verrelli (2012) for nonlinear systems with matching uncertainties; (iii)  Marino, Tomei, and Verrelli (2012) for minimum phase systems in output feedback form with output dependent nonlinearities; (iv)  Jin and Xu (2013) for nonlinear systems in normal form with no inverse system dynamics.

In this note, the above design techniques are used in conjunction with the notion of exponentially convergent systems (see  Pavlov, van de Wouw, & Nijmeijer, 2006) in order to solve the output tracking problem for the larger class of uncertain nonlinear systems in normal form with inverse system dynamics satisfying the Demidovich condition on suitable compact sets. Differently from  Jin and Xu (2013), the presence of uncertain inverse system dynamics is allowed, while in contrast to  Marino et al. (2012), Marino and Tomei (2009), and Bifaretti et al. (2012) the inverse system dynamics are not restricted to be linear.

The resulting control design along with the related convergence analysis are not straightforward since nonlinearities appear and they are not exclusively output-dependent; nevertheless, the existence of a suitable periodic solution is no longer straightforward to be proved.

Those specific features appear in  Consolini and Verrelli (2014) when a learning control for autonomous vehicles reproducing the human driver behaviour (which thus represents a special application of the general theory presented in this note) is designed to track planar curves with uncertain periodic curvature.1 The first order inverse dynamics, which are expressed in the independent variable s (curvilinear abscissa), in the presence of the uncertain periodic s-function κ(s) and of the positive real l, are η(s)=sin(η(s)+y(s)l)lκ(s). According to Theorem 2.41 and Definition 2.20 in  Pavlov et al. (2006), the above dynamics represent a locally exponentially convergent system for the class of bounded inputs on [0,+) (i.e. there exists a neighbourhood Z of the origin and a positive real ρ such that the previous system is convergent in Z for all inputs on [0,+) whose modulus is less than ρ for any s). Additional technical difficulties arise since an uncertain nonlinear state function (in contrast to  Bifaretti et al., 2012, Marino and Tomei, 2009, Marino et al., 2012) is allowed, in this note, to multiply the control input.

Section snippets

Problem statement

We address the output tracking problem in which the output y of the nonlinear time-invariant single input-single output system (f() and g() are suitable uncertain smooth vector fields on Rn, while h():RnR is a suitable uncertain smooth function) ẋ=f(x)+g(x)uy=h(x) is required to track a smooth periodic reference signal y(t) (with known period T): y(t+T)=y(t),tT. In particular, we assume that the global relative degree ρn is known and well defined for (1) and that system (1) is

Stability proof

Consider the system ż=ϕ(z,ξ). According to Lemma 2.30 in  Pavlov et al. (2006), assumption (i) implies that there exists β>0 (only depending on P and Q and thus possibly depending on BMξ) such that for any ξBBMξ(0)¯ and for any z1,z2Rnρ, the inequality (z1z2)TP(ϕ(z1,ξ)ϕ(z2,ξ))βz1z2P2 holds with z1z2P2=(z1z2)TP(z1z2). In particular, the preceding inequality holds for ξ=ξ. Therefore, according to the first part of the proof of Theorem 2.29 in Pavlov et al. (2006), any solution of

An illustrative example

In this section, we consider an illustrative nonlinear example with relative degree ρ=2. The aim is to confirm the effectiveness of the proposed approach beyond the relevant real application being reported in the Introduction, while actually showing that it is possible to enlarge the class of systems which (7) successfully applies to.

The system dynamics are as in (3) with (the second order inverse system dynamics is adapted from  Pavlov et al. (2006) by setting w=ξ12+ds1ξ22) ϕ(z,ξ)=[z1+(ξ12+ds

Conclusions

The repetitive learning control (7), which has been recently designed in  Marino et al. (2012), solves the nonlinear tracking problem stated in Sections  2 Problem statement, 3 Stability proof for the larger class of nonlinear systems (3) including (i) the presence of uncertain, nonlinear inverse system dynamics (the related problem of the existence of a periodic solution for the tracking dynamics is solved through the Demidovich condition (4)); (ii) an uncertain nonlinear state function

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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Chunjiang Qian under the direction of Editor André L. Tits.

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