On modified parameter estimators for identification and adaptive control. A unified framework and some new schemes

https://doi.org/10.1016/j.arcontrol.2020.06.002Get rights and content

Abstract

A key assumption in the development of system identification and adaptive control schemes is the availability of a regression model which is linear in the unknown parameters (of the plant and/or the controller). Applying standard— e.g., gradient descent-based—parameter estimators leads to a linear time-varying equation for the parameter errors, whose stability relies on the usually stringent persistency of excitation assumption. As suggested in Kreisselmeier (1977) and Lion (1967), with the inclusion of linear filters, it is possible to generate alternative regression models, whose parameter error equations have different stability properties. In Duarte and, Narendra (1989), Panteley, Ortega,and Moya, (2002) and Slotine and Li, (1989) estimators that combine tracking and identification errors, to generate new parameter error equations, were proposed. The main objectives of this paper are: first, based on the two key developments mentioned above, provide a unified framework for the analysis and design of parameter estimators and, in particular, show that they lie at the core of some modified schemes recently proposed in the literature. Second, extend the realm of application of these estimators to the class of nonlinear systems considered in Panteley et al. (2002). Third, use this framework to propose some new schemes with relaxed conditions for convergence and improved transient performance. Particular attention is given to the task of obviating the persistency of excitation assumption, which is rarely verified in applications and is, certainly not, the only way to ensure robustness of the schemes.

Introduction

The starting point for the development of many system identification and adaptive control schemes is the construction of a regression model which is linear in the unknown parameters (of the plant and/or the controller). That is, it is necessary to prove the existence of relationships of the form Y=ϕθ, with Y(t)R,ϕ(t)Rq measurable signals and θRq a constant vector of unknown parameters—that we call in the sequel linear regression equation (LRE) with ϕ the regressor vector. In the case of system identification (or indirect adaptive control) θ are the plant parameters, while for direct adaptive control they are the parameters of the controller, while ϕ consists of filtered versions of the systems input and output and, eventually, the output reference signal. In the next section we derive the LRE associated to some control problems, see also Ioannou and Sun (1996), Narendra and Annaswamy (1989) and Sastry and Bodson (1989) for more details and a historical review of the material reported in the 80s and 90s.

The identification or adaptive control design is completed adding a parameter adaptation algorithm that generates online estimates θ^(t)Rq of the unknown parameters. Very often, standard— e.g., gradient descent-based or least-squares—parameter estimators are used. This leads to a linear time-varying (LTV) dynamical system that describes the behavior of the estimation errors, called parameter error equations (PEE), that have been extensively studied in the literature. A fundamental result is that a necessary and sufficient condition for the global exponential stability (GES) of the PEEs is that the regressor vector satisfies a persistency of excitation (PE) condition—which is a uniform observability property for the associated LTV system. Here we underscore the qualifier “exponential” because (non-uniform) global asymptotic stability can be ensured under weaker conditions (Barabanov, Ortega, 2017, Praly, 2017).

Thanks to the “self tuning property” (Astrom, Wittenmark, 1973, Monopoli, 1974), it has been shown that for a class of linear time-invariant (LTI) systems it is possible to ensure global tracking of a well-defined class of arbitrary signals in model reference adaptive control (MRAC) without parameter convergence. However, as vividly shown in Rohrs, Valavani, Athans, and Stein (1985), see also Ioannou and Kokotovic (1984), the stability property of these schemes is fragile. In spite of intensive research efforts (Ioannou, Sun, 1996, Narendra, Annaswamy, 1989) the various fixes that have been introduced in the estimators—that include projections, deadzones and integrator leakages—have only partially alleviated this problem. Indeed, it has only been possible to establish a “continuity” property with respect to unmodeled dynamics and the preservation of signal boundedness in the face of noise. More precisely, it has been proven that there exists a sufficiently small bound on the norm of the error dynamics such that (some kind of) stability is preserved—but, unfortunately, this bound is not quantifiable from the data of the problem (Khargonekar and Ortega, 1989). Regarding the presence of noise only signal boundedness, again with no uniform bound, is established. It should be furthermore added that all of this “robustified” schemes rely on the introduction of the dynamic normalization introduced in Ortega, Praly, and Landau (1985) that slows-down the adaptation.

It is reasonable to expect that if we can guarantee that the parameters converge to their desired value a robust behavior will be achieved. Unfortunately, the PE property that ensures parameter convergence for standard estimators—which imposes a “spanning behavior” to the signals—is rarely satisfied in applications, where the task is often to drive the signals to some constant value. Hence the interest to propose new adaptation algorithms that ensure parameter convergence without PE. This research line has been intensively pursued in the last few years and some recent adaptive schemes, where the PE assumption is obviated, have been reported in the literature. Interestingly, although not always acknowledged, instrumental for these developments are two—40+ years old and somehow overlooked—fundamental contributions. First, the inclusion of linear filters to generate alternative LRE and, consequently, new PEE with different stability properties, originally proposed in Kreisselmeier (1977) and Lion (1967). Second, the use of estimators that combine tracking and identification errors to generate different PEE, which was first suggested in Duarte and Narendra (1989), Panteley, Ortega, and Moya (2002) and Slotine and Li (1989).

The main objectives of this paper are:

  • O1.

    Provide a unified regression-based framework for the analysis and design of estimators, which concentrates on the generation of new LRE, leading to their associated PEE.

  • O2.

    Reveal the central role played by the developments of Kreisselmeier (1977) and Slotine and Li (1989) in some of the new schemes proposed in the literature.

  • O3.

    Extend some of these new results, reported for LTI systems, to the class of nonlinear systems considered in Panteley et al. (2002).

  • O4.

    Use the proposed framework, to provide generalizations of the new estimators and to suggest new ones, whose stability properties can be established without PE and that exhibit better performance.

The remainder of the paper is organized as follows. In Section 2 we briefly recall how LRE are derived in classical identification and adaptive control tasks. In Section 3 we give a brief review of the gradient estimator and its PEE stability properties—underscoring the central role played by the PE condition. Section 4 is devoted to recall the two key modification mentioned above, namely, the regressor extensions of Kreisselmeier (1977) and Lion (1967) and the combined estimators of Duarte and Narendra (1989), Panteley et al. (2002) and Slotine and Li (1989). In Section 5 we critically review four modified adaptive controllers—recently proposed in the literature—that ensure parameter convergence without the PE condition. The key modification of regressor mixing, first reported in Aranovskiy, Bobtsov, Ortega, and Pyrkin (2017), is presented in Section 6. The paper is wrapped-up with concluding remarks and future research in Section 7. Two appendices that discuss related, but somehow particular, topics are also included.

Caveat Emptor. To simplify the presentation and concentrate on the main issues at hand, we consider throughout the paper single-input single-output (SISO), continuous-time systems and basic regulation tasks. This simplification is done without loss of generality, as most of the presented results can be verbatim generalized to multivariable plants, as well as to discrete-time systems and tracking objectives.

Notation. Iq is the q × q identity matrix. For xRn, we denote the Euclidean norm square as |x|2 ≔ xx. All mappings are assumed smooth. Given a function V:RnR we define the differential operator V:=(Vx). The action of an operator H on a signal u is denoted H[u](t), when clear from the context the brackets and the time argument are omitted.

Section snippets

Linear regression equations in adaptive systems

Instrumental for the development of most system identifiers and adaptive controllers is the availability of a LRE of the formY=ϕθ+εt,where Y(t)R,ϕ(t)Rq are measurable signals, θRq is a constant vector of unknown parameters and εt is a (generic) exponentially decaying signal, stemming from the effect of the initial conditions of various filters. In this section we, first, give some examples of identification and adaptive control problems where this model is used. A discussion on the critical

Classical gradient estimator: The PE condition

In this section we recall the well-known gradient estimator, derive its PEE and recall its stability properties. Although this material is very well-known, it is included to make the document self-contained.

A gradient-descent estimator for the parameters of (1) is given byθ^˙=γϕ(ϕθ^Y)with γ > 0 the adaptation gain.1 Defining the parameter estimation error θ˜:=θ^θ, and neglecting the term εt

Two precursor modifications

In this section we present—and slightly extend—two key modifications to adaptive systems that, with the motivation to improve the stability properties, were introduced more than 40 years ago. As shown later, these two modifications lie at the core of some estimators recently reported in the literature.

Four adaptive controllers recently proposed in the literature

In this section four adaptive controllers that have been recently reported in the literature are presented using the framework developed above. The two main objectives of the section are, first, to show the close connections of them with the two key developments presented in the previous section—a fact that is, unfortunately, not always recognized. Second, to comment on some drawbacks of these schemes.

Generating scalar LREs via mixing

In Aranovskiy et al. (2017) a key modification to the LRE of the DRE procedure was proposed and given the name “DRE and mixing” (DREM). The DREM procedure gave rise to PEE with—rigorously proven— stronger transient stability properties and identified a necessary and sufficient condition for parameter convergence that does not rely on PE. Although this material has already been reported before, for the sake of completeness, in this section we review DREM and propose to add the mixing step to the

Concluding remarks and future research

A theoretical framework for the analysis and design of parameter estimators has been proposed. The framework is based on two principles proposed 40+ years ago:

  • (i)

    the modification, via the inclusion of linear operators (Lion, 1967) or algebraic manipulations (Kreisselmeier, 1977), of the original LRE (1);

  • (ii)

    the use of two parameterizations in the estimators, as done in composite control (Duarte, Narendra, 1989, Panteley, Ortega, Moya, 2002, Slotine, Li, 1989).

We have leveraged this framework to treat,

Romeo Ortega He obtained his BSc in Electrical and Mechanical Engineering from the National University of Mexico, Master of Engineering from Polytechnical Institute of Leningrad, USSR, and the Docteur D‘Etat from the Politechnical Institute of Grenoble, France in 1974, 1978 and 1984 respectively. He then joined the National University of Mexico, where he worked until 1989. He was a Visiting Professor at the University of Illinois in 1987-88 and at the McGill University in 1991–1992, and a

References (69)

  • S. Roy et al.

    A UGES switched MRAC architecture using initial excitation

    IFAC-PapersOnLine

    (2017)
  • V. Adetola et al.

    Finite-time parameter estimation in adaptive control of nonlinear systems

    IEEE Transactions on Automatic Control

    (2008)
  • Anderson, B., Bitmead, R., Johnson, C., Kokotovic, P., Kosut, R., Mareels, I., Praly, L., Riedle, B. Stability of...
  • S. Aranovskiy et al.

    Parameter identification of linear time-invariant systems using dynamic regressor extension and mixing

    International Journal of Adaptive Control and Signal Processing

    (2019)
  • S. Aranovskiy et al.

    Performance enhancement of parameter estimator via dynamic regressor extension and mixing

    IEEE Transactions on Automatic Control

    (2017)
  • A. Astolfi et al.

    Nonlinear and adaptive control with applications

    (2008)
  • N.E. Barabanov et al.

    On global asymptotic stability of x˙=ϕ(t)ϕ(t)x with ϕ(t) bounded and not persistently exciting

    Systems and Control Letters

    (2017)
  • N. Barabanov et al.

    On global asymptotic stability of spr adaptive systems without persistent excitation

    2017 IEEE 56th annual conference on decision and control (CDC), Melbourne, Australia

    (2017)
  • H. Berghuis et al.

    A robust adaptive robot controller

    IEEE Transactions on Robotics and Automation

    (1993)
  • J. Chen et al.

    Adaptive robust dynamic surface control with composite adaptation laws

    International Journal on Adaptive Control Signal Processing

    (2010)
  • N. Cho et al.

    Composite MRAC with parameter convergence under finite excitation

    IEEE Transactions on Automatic Control

    (2018)
  • G. Chowdhary et al.

    Concurrent learning adaptive control of linear systems with exponentially convergent bounds

    International Journal of Adaptive Control and Signal Processing

    (2013)
  • M.K. Ciliz

    Adaptive backstepping control using combined direct and indirect adaptation

    Circuits Systtem Signal Processing

    (2007)
  • M.A. Duarte et al.

    Combined direct and indirect approach to adaptive control

    IEEE Transactions on Automatic Control

    (1989)
  • Gaudio, J., Annaswamy, A.M., Lavretsky, E., Bolender, M. Parameter estimation in adaptive control of time-varying...
  • D. Gerasimov et al.

    Performance improvement of discrete MRAC by dynamic and memory regressor extension

    European Control Conference, Naples, Italy

    (2019)
  • D. Gerasimov et al.

    Performance improvement of MRAC by dynamic regressor extension

    2018 IEEE Conference on Decision and Control (CDC), USA, FL, Miami

    (2018)
  • D. Gerasimov et al.

    Improved adaptive compensation of unmatched multisinusoidal disturbances in uncertain nonlinear plants

    Proeedings of American Control Conference (ACC) 2020

    (2020)
  • D. Gerasimov et al.

    Improved adaptive servotracking for a class of nonlinear plants with unmatched uncertainties

    IFAC-PapersOnLine

    (2020)
  • A. Goel et al.

    Recursive least squares with variable-direction forgetting

    IEEE Control Systems Magazine

    (2020)
  • N. Hovakimyan et al.

    L1–Adaptive control for safety-critical systems

    Control Systems Magazine

    (2011)
  • P. Ioannou et al.

    L1Adaptive control: Stability and robustness properties and misperceptions

    IEEE Transactions on Automatic Control

    (2014)
  • P. Ioannou et al.

    Instability analysis and improvement of robustness of adaptive control

    Automatica

    (1984)
  • P. Ioannou et al.

    Robust adaptive control

    (1996)
  • Cited by (96)

    View all citing articles on Scopus

    Romeo Ortega He obtained his BSc in Electrical and Mechanical Engineering from the National University of Mexico, Master of Engineering from Polytechnical Institute of Leningrad, USSR, and the Docteur D‘Etat from the Politechnical Institute of Grenoble, France in 1974, 1978 and 1984 respectively. He then joined the National University of Mexico, where he worked until 1989. He was a Visiting Professor at the University of Illinois in 1987-88 and at the McGill University in 1991–1992, and a Fellow of the Japan Society for Promotion of Science in 1990–1991. He has been a member of the French National Researcher Council (CNRS) since June 1992. Currently he is in the Laboratoire de Signaux et Systemes (SUPELEC) in Paris. His research interests are in the fields of nonlinear and adaptive control, with special emphasis on applications. Dr. Ortega has published three books and more than 380 scientific papers in international journals, with an h-index of 84. He has supervised 36 PhD thesis. He is a Life Fellow Member of the IEEE since 1999 and a Fellow Member of IFAC since 2016. He has served as chairman in several IFAC and IEEE committees and editorial boards. Currently, he is Editor in Chief of Int. J. on Adaptive Control and Signal Processing.

    Vladimir Nikiforov received Electrical Engineering Degree in 1986 from Leningrad Institute of Fine Mechanics and Optics (present name is Saint-Petersburg State University of Information Technologies, Mechanics and Optics — ITMO University). In 1991 he received Ph.D. Degree and in 2001 Doctor of Science Degree from ITMO University. Vladimir Nikiforov is currently Vice-Rector for Research of ITMO University, Professor in the Department of Control Systems and Robotics at the same university, Editor-in-Chief of the journal “Control Engineering Russia”. The list of his works published includes more than 150 titles including four books.

    Dmitry Gerasimov received Electrical Engineering Degree in 2005 from Saint-Petersburg University of Fine Mechanics and Optics (present name is Saint-Petersburg State University of Information Technologies, Mechanics and Optics — ITMO University), received Ph.D. Degree in 2009 from the same university. Since 2009 Dmitry Gerasimov has been with the Department of Control Systems and Informatics at ITMO University and, currently, he holds a position of associate professor. His present interests include adaptive and robust control, identification theory, nonlinear and delayed systems. He published more than 40 papers in national and international journals.

    This work was financially supported by Government of Russian Federation (Grant 08-08) and by the Ministry of Science and Higher Education of Russian Federation (goszadanie no. 2019-0898).

    View full text