Data-driven Output Regulation via Gaussian Processes and Luenberger

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Data-driven Output Regulation via

Gaussian Processes and Luenberger

Internal Models

Lorenzo Gentilini ∗Michelangelo Bin ∗∗ Lorenzo Marconi ∗

∗Dept. of Electrical, Electronic and Information Engineering,

Alma Mater Studiorum University of Bologna, Bologna, Italy

(e-mails: {lorenzo.gentilini6, lorenzo.marconi}@unibo.it).

∗∗ Dept. of Electrical and Electronic Engineering,

Imperial College London, London, UK

(e-mail: m.bin@imperial.ac.uk).

Abstract: This paper deals with the problem of adaptive output regulation for multivariable

nonlinear systems by presenting a learning-based adaptive internal model-based design strategy.

The approach builds on the recently proposed adaptive internal model design techniques based

on the theory of nonlinear Luenberger observers, and the adaptation side is approached as

a probabilistic regression problem. In particular, Gaussian processes priors are employed to

cope with the learning problem. Unlike the previous approaches in the ﬁeld, here only coarse

assumptions about the friend structure are required, making the proposed approach suitable for

applications where the exosystem is highly uncertain. The paper presents performance bounds

on the attained regulation error and numerical simulations showing how the proposed method

outperforms previous approaches.

Keywords: Nonlinear Output Regulation, Adaptive Control Systems, Gaussian Processes,

Nonparametric Methods, Identiﬁcation for Control

1. INTRODUCTION

In this paper we consider a class of nonlinear systems of

the form

˙x=f(x, w, u), e =h(x, w),(1)

with state x∈Rnx, control input u∈Rnu, regulation

error e∈Rne, and with w∈Rnwan exogenous signal.

As customary in the literature of output regulation, we

assume that the exogenous signal wbelongs to the set of

solutions of an exosystem of the form

˙w=s(w),(2)

originating in a compact invariant subset Wof Rnw. For

the class of systems (1) (2), in this paper we consider the

problem of design an output-feedback regulator of the form

˙xc=fx(xc, e), uc=kc(xc, e),

that ensures boundedness of the closed-loop trajectories

and asymptotically removes the eﬀect of wfrom the

regulated output e, thus ideally obtaining e(t)→0 as

t→ ∞. More precisely, the sought regulator must ensure

lim sup

t→∞ ke(t)k ≤ 

with ≥0 possibly a small number measuring the

regulator’s asymptotic performance. In this work we focus

on the speciﬁc case of adaptive approximate regulation,

where the aforementioned control objective is relaxed

to the case with  > 0, and a learning technique is

employed to cope with uncertainties in the exosystem, and

in the plant dynamics. In particular, the adaptation side

is approached in a system identiﬁcation fashion where the

Gaussian process regression is used to infer the internal

model dynamics directly out of the collected data.

Related Works Most of the work in the ﬁeld of output

regulation can be traced-back to Francis and Wonham

(1976) and Davison (1976) who ﬁrstly formalize and solve

the asymptotic regulation problem in the context of linear

systems. Asymptotic results was given also in the ﬁeld

of Single-Input-Single-Output (SISO) nonlinear systems,

ﬁrst in a local context (Byrnes et al. (1997), Isidori and

Byrnes (1990)), and later in a purely nonlinear framework

(Byrnes and Isidori (2003), Marconi et al. (2007)), based

on the “non-equilibrium” theory (Byrnes et al. (2003)).

Recently, asymptotic regulators have been also extended

to some classes of multivariable nonlinear systems (Wang

et al. (2016), Wang et al. (2017)). The major drawbacks

of asymptotic regulators reside in their complexity and

fragility (Bin et al. (2018)). Indeed, suﬃcient conditions

under which asymptotic regulation is ensured are typically

expressed by equations whose analytic solution becomes

a hard task even for relative simple problems. Moreover,

even if a regulator can be build, asymptotic regulation

may be lost at front of exosystem perturbation and plant

uncertainties. The aforementioned problems motivates the

researchers to move toward more robust solutions, intro-

ducing the concept of adaptive and approximate regula-

tion. Among the approaches to approximate regulation it

is worth mentioning Marconi and Praly (2008) and Astolﬁ

et al. (2015), whereas practical regulators can be found

in Isidori et al. (2012) and Freidovich and Khalil (2008).

Adaptive designs of regulators can be found in Priscoli

et al. (2006) and Pyrkin and Isidori (2017), where linearly

parametrized internal models are constructed in the con-

arXiv:2210.15938v1 [eess.SY] 28 Oct 2022

text of adaptive control, in Bin et al. (2019) where discrete-

time adaptation algorithms are used in the context of

multivariable linear systems, and in Forte et al. (2016), Bin

and Marconi (2019), Bin et al. (2020), where adaptation

of a nonlinear internal model is approached as a system

identiﬁcation problem.

Learning dynamics models is also an active research

topic. In particular, Gaussian Processes (GPs) are in-

creasingly used to estimate unknown dynamics (Kocijan

(2016), Buisson-Fenet et al. (2020)). Unlike other nonpara-

metric models, GPs represent an attractive tool in learning

dynamics due to their ﬂexibility in modeling nonlinearities

and the possibility to incorporate prior knowledge (Ras-

mussen (2003)). Moreover, since GPs allow for analytical

formulations, theoretical guarantees on the a posteriori can

be drawn directly from the collected data (Umlauft and

Hirche (2020), Lederer et al. (2019)). Recently, GP models

spread inside the ﬁeld of nonlinear optimal control (Sforni

et al. (2021)), with several applications to the particular

case of Model Predictive Control (MPC) (Torrente et al.

(2021), Kabzan et al. (2019)), and inside the ﬁeld of

nonlinear observers (Buisson-Fenet et al. (2021)).

Contributions In this paper, we propose a data-driven

adaptive output regulation scheme, built on top of the re-

cently published works Bin et al. (2020) and Gentilini et al.

(2022), in which the problem of approximate regulation is

solved by means of a regulator embedding an adaptive

internal model. Unlike previous approaches, here the high

ﬂexibility of Gaussian process priors (Rasmussen (2003))

is used to adapt an internal model unit in a discrete-time

system identiﬁcation fashion, enabling the possibility to

handle a possibly inﬁnite class of input signals needed to

ensure zero regulation error (the so-called friend, Isidori

and Byrnes (1990)). Compared to Bin et al. (2020), where

the identiﬁer is related to a particular choice of class of

functions, to which the friend may (or may not) belongs,

the proposed approach aims to perform probabilistic infer-

ence in a possibly inﬁnite-dimensional space. Unlike Gen-

tilini et al. (2022), the proposed regulator relies on non-

high-gain stabilising actions and Luenberger-like internal

models that lead to a ﬁxed choice of the model order.

The latter property, jointly with the black-box nature of

Gaussian process methods, makes the proposed approach

suitable for those applications where the exosystem dy-

namics is highly uncertain and the friend structure is not

a priori known. Theoretical performance bounds on the

attained regulation error are analytically established.

The paper unfolds as follows. In Section 2 we brieﬂy

describe the problem at hand along with the standing

assumptions over the presented results build. Section 2.1

reviews the most recent advancements in the output regu-

lation ﬁeld, and introduces the barebone regulator adapted

for this work, while Section 2.2 introduces the basics of

Gaussian process inference. In Section 3 we present the

proposed regulator and state the main result of the paper.

Finally, in Section 4 a numerical example is presented.

2. PROBLEM SET-UP & PRELIMINARIES

In this section, we ﬁrst detail the subclass of problems

that this work focuses on, along with the constructive as-

sumptions. Then, a Luenberger-like internal model design

technique is reviewed, together with the adaptive regulator

of Bin et al. (2020). Finally, basic concepts behind the

notion of Gaussian process regression are introduced.

2.1 Approximate Nonlinear Regulation

In this paper, we focus on a subclass of the general

regulation problem presented in Section 1, by considering

systems of the form

˙z=f0(w, z, e),

˙e=Ae +B(q(w, z, e) + b(w, z, e)u),

y=Ce,

(3)

in which z∈Rnztogether with the error dynamics e∈Rne

represent the overall state of the plant. The quantities u∈

Rnyand y∈Rnyare the control input and the measured

output respectively, while w∈Rnwis an exogenous input,

f0:Rnw×Rnz×Rne7→ Rnz,q:Rnw×Rnz×Rne7→ Rny,

b:Rnw×Rnz×Rne7→ Rny×nyare continuous functions,

and A,B, and Care deﬁned as

A=0(r−1)ny×nyI(r−1)ny

0ny×ny0ny×(r−1)ny, B =0(r−1)ny×ny

Iny,

C=Iny0ny×(r−1)ny,

for some r∈N, consisting in a chain of rintegrators of

dimension ny. The aforementioned framework embraces a

large number of use-cases addressed in literature. In par-

ticular, all systems presenting a well-deﬁned vector relative

degree and admitting a canonical normal form, or that are

strongly invertible and feedback linearisable ﬁt inside the

proposed framework. Nevertheless, this approach limits to

systems having an equal number of inputs and controlled

outputs (ny). The results presented in the next sections are

grounded over the following set of standing assumptions.

Assumption 1. The function f0is locally Lipschitz and the

functions qand bare C1functions, with local Lipschitz

derivative.

Assumption 2. There exists a C1map π:P ⊂ Rnw7→

Rnz, with Pan open neighborhood of W, satisfying

L(w)

s(w)π(w) = f0(w, π (w),0) ,

with L(w)

s(w)π(w) = ∂π(w)

∂w s(w), such that the system

˙w=s(w),˙z=f0(w, z, e),

is Input-to-State Stable (ISS) with respect to the input e,

relative to the compact set A={(w, z)∈ W × Rnz:z=π(w)}.

Assumption 3. There exists a known constant nonsingular

matrix b∈Rny×nysuch that the inequality



(b(w, z, e)−b)b−1

≤1−µ0,

holds for some known scalar µ0∈(0,1), and for all

(w, z, e)∈ W × Rnz×Rne.

Remark 2. Although not necessary (see Byrnes and Isidori

(2003)), Assumption 2 is a minimum-phase assumption

customary made in the literature of output regulation

(see Isidori (2017), Pavlov et al. (2006)). In particular, As-

sumption 2 is asking that the zero dynamics

˙w=s(w),˙z=f0(w, z, 0) ,

has a steady-state of the kind z=π(w), compatible with

the control objective y= 0. As a consequence, the ideal

input u?making the set B=A×{0}invariant for (3)

reads as

u?(w, π(w)) = −b(w, π(w),0)−1q(w, π(w),0) .

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Data-drivenOutputRegulationviaGaussianProcessesandLuenbergerInternalModelsLorenzoGentiliniMichelangeloBinLorenzoMarconiDept.ofElectrical,ElectronicandInformationEngineering,AlmaMaterStudiorumUniversityofBologna,Bologna,Italy(e-mails:florenzo.gentilini6,lorenzo.marconig@unibo.it).Dept.ofElectr...

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