Technical communiqueA larger family of nonlinear systems for the repetitive learning control☆
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 (curvilinear abscissa), in the presence of the uncertain periodic -function and of the positive real , are . 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 (i.e. there exists a neighbourhood of the origin and a positive real such that the previous system is convergent in for all inputs on whose modulus is less than for any ). 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 of the nonlinear time-invariant single input-single output system ( and are suitable uncertain smooth vector fields on , while is a suitable uncertain smooth function) is required to track a smooth periodic reference signal (with known period ): In particular, we assume that the global relative degree is known and well defined for (1) and that system (1) is
Stability proof
Consider the system According to Lemma 2.30 in Pavlov et al. (2006), assumption (i) implies that there exists (only depending on and and thus possibly depending on ) such that for any and for any , the inequality holds with . In particular, the preceding inequality holds for . 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 . 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 )
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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Cited by (16)
Virtual control strategy and conditioning technique for tracking and learning controls under input restrictions
2022, Annual Reviews in ControlCitation Excerpt :In respect to this, the application of the resulting adaptation of the virtual control technique to such systems is shown not only to own strong connections with the conditioning technique, but also to achieve remarkable convergence properties that decouple the behaviour of the unfaulty variable from the faulty input. On the other hand, when a transposition of such a derived adaptation is applied to one-relative-degree single-input, single-output systems in output feedback form under input constraints (referred to as class II, Section 3) — the reader is referred to Marino et al. (2012), Verrelli (2016, 2020) and references therein for the detailed analysis & design of RLC for the tracking of periodic output references in the absence of input saturation —, the RLC problem under input saturation is innovatively solved. The output dynamics of the aforementioned kinds of systems complies with the case of the speed dynamics of a (non-salient-pole) Permanent Magnet Synchronous Motor under constant load torque, for which commercial drives typically employ a saturated control action and nested loops performing time-scale separation (see Verrelli and Tomei (2020) for a definite solution in the special case of output regulation to constant references), while the same structure of such systems in suitable coordinates also complies with the second-order system appearing in Scalzi, Tomei, and Verrelli (2012) and Paradiso, Pietrosanti, Scalzi, Tomei, and Verrelli (2013), for which an RLC has been successfully designed to regulate the heart rate to a given constant value.
Saturated repetitive control for a class of nonlinear systems: A contraction mapping method
2018, Systems and Control Letters
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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.