Data-Driven Analytic Differentiation via High Gain Observers and Gaussian Process Priors Biagio Trimarchi1 Lorenzo Gentilini1 Fabrizio Schiano2 and Lorenzo Marconi1

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Data-Driven Analytic Differentiation via High Gain Observers and

Gaussian Process Priors

Biagio Trimarchi1, Lorenzo Gentilini1, Fabrizio Schiano2, and Lorenzo Marconi1

Abstract— The presented paper tackles the problem of

modeling an unknown function, and its ﬁrst r−1derivatives,

out of scattered and poor-quality data. The considered setting

embraces a large number of use cases addressed in the

literature and ﬁts especially well in the context of control

barrier functions, where high-order derivatives of the safe set

are required to preserve the safety of the controlled system.

The approach builds on a cascade of high-gain observers and

a set of Gaussian process regressors trained on the observers’

data. The proposed structure allows for high robustness

against measurement noise and ﬂexibility with respect to the

employed sampling law. Unlike previous approaches in the

ﬁeld, where a large number of samples are required to ﬁt

correctly the unknown function derivatives, here we suppose

to have access only to a small window of samples, sliding in

time. The paper presents performance bounds on the attained

regression error and numerical simulations showing how the

proposed method outperforms previous approaches.

I. INTRODUCTION

Autonomous systems have gained a lot of interest in the

last decades, and we witness each year a big effort to increase

their autonomy and capabilities. This effort was motivated by

both an increase in computational resources and a reduction

in Size, Weight, Power, and Cost (SWaPC) of such sys-

tems. The consequent advancements in autonomous systems

technologies unlocked the use of data-driven techniques on

real systems. In the ﬁeld of data-driven techniques, Gaussian

Process (GP) regression [1] is gaining popularity thanks to

its non-parametric nature, the analytical tractability, and the

existence of analytical bounds on the estimate error [2]. This

property makes them particularly suitable for safety-critical

applications, where uncertainty and noisy information could

lead to a critical failure. In the same context, another im-

pactful advancement of the last years is the so-called control

barrier functions [3], which are able to act as a ﬁlter for the

control input of an autonomous system to guarantee that the

safety requirements are always satisﬁed. [3]. Applications of

control barrier functions can be seen in various contexts such

as: quad-copters teleoperation [4], multi-robot systems [5],

and adaptive cruise control [6]. Recently, researchers tried to

merge together GPs and CBFs giving birth to a new learning

control paradigm [7]. Such a solution succeeds in all those

cases when an analytic formulation of the safe-set is a priori

1Biagio Trimarchi, Lorenzo Gentilini, and Lorenzo Marconi are with the

Center for Research on Complex Automated Systems (CASY), Department

of Electrical, Electronic and Information Engineering (DEI), University

of Bologna, Bologna, Italy e-mails: {biagio.trimarchi2,

lorenzo.gentilini6, lorenzo.marconi}@unibo.it

2Fabrizio Schiano is with Leonardo S.p.a., Leonardo Labs, Rome, Italy

e-mail: fabrizio.schiano.ext@leonardo.com

HGO GP

y(t) ˆzk(t)ˆz(k)∼Lk

Fig. 1. Structure of the proposed approach: the high gain observer generates

the data needed to ﬁt the Gaussian process.

not known, e.g. in the case of exploration of an unknown

environment. The main drawback of this last approach is that

an overestimate of the barrier function could compromise the

safety of the system. Moreover, the barrier condition relies

on the knowledge of the barrier functions derivatives, that, in

this learning scenario, is difﬁcult to retrieve. As a matter of

fact, GPs regressors suffer from loss of information during

differentiation [8] which makes them not suitable for such

an application. On the other hand, an accurate estimate of

time derivative can be generated using High Gain Observers

(HGOs), which provide practical convergence even in the

case of model uncertainty for high enough gains [9].

Motivated by these works, in this paper we take a step back

from control barrier functions and focus on proposing a novel

approach to estimate the derivative of an unknown function

of which we have scarce measurements. We estimate the

derivative of this function by combining Gaussian processes

and high gain observers [9] as depicted in Figure 1. The

idea of combining of HGOs and GPs is not novel since it

was already proposed in [10], however, our overarching goal

is different. In [10] the system dynamics are predicted out

of collected data. Instead, in this paper, we focus mainly

on reproducing a state-dependent unknown function, and

its derivatives, regressing only on scattered and noisy mea-

surements. In summary, our contribution is a technique to

obtain an analytic estimate of the directional derivative of an

unknown function out of very scattered and noisy data. We

show that the proposed approach has provable convergence

guarantees and we compare our solution, through numerical

simulations, to the naive approach of deriving a regressor

ﬁtted directly to the scarcely available measurements. We

chose the context of autonomous systems and CBFs to offer

the reader an example of a situation in which our approach

could be adopted. However, we highlight that our approach

is general and could be applied to any context in which one

wants to estimate the derivative of an unknown function from

scarce measurements. The paper unfolds as follows. Section

II reviews the basics of Gaussian processes inference and the

theory of high gain state observation. Section III describes

the general problem along with our assumptions and our

arXiv:2210.15528v1 [eess.SY] 27 Oct 2022

proposed approach. IV describes numerical simulations to

corroborate our approach and V concludes the paper and

describes some future work.

II. PRELIMINARIES

Notation

Consider a nonlinear system of the form ˙x=f(x), with

state x∈Rn. Moreover consider a function h(x) : Rn→

R, we denote with Lfh(x)the Lie derivative Lfh(x) =

∂h

∂x f(x).

The operator k·k :Rn→Rdenotes the stan-

dard Euclidean norm. The set denoted as B(x) =

{¯x∈Rn:k¯x−xk ≤ 1}is the unit ball centered in x∈Rn.

Moreover, if w:R+→R, we set kwk∞= maxt≥0kw(t)k.

Gaussian Process Regression

Let x∈ X ⊂ Rnx. A GP is a stochastic process such that

any ﬁnite number of outputs is assigned a joint Gaussian

distribution with a prior mean function m:Rnx7→ Rand

covariance deﬁned through the kernel κ:Rnx×Rnx7→ R

[1]. While there are many possible choices of mean and

covariance functions, in this work we keep the formulation

of κgeneral, with the only constraint expressed by As-

sumption 2 below. Thus, when we assume that a function

f:X ⊂ Rnx→Ris described by a Gaussian process with

mean mand covariance κwe write

f∼ GP (m(·), κ (·,·)) .

In the following we force, without loss of generality, m(x) =

0nxfor any x∈ X.

Let us denote a time window of N∈Ntime instants

tk∈R>0with S={t1, t2, . . . , tN}. Supposing to have

access to a data-set of samples DS ={(x(tk), y(tk)) ∈

X × R, tk∈ S}, with each pair (x, y)∈ DS obtained

as y(tk) = f(x(tk)) + ε(tk)with ε(tk)∼ N(0, σ2

nI1×r)

white Gaussian noise with known variance σ2

n, the regression

is performed by conditioning the prior GP distribution on

the training data DS and a test point x. Denoting x=

(x(t1), . . . , x(tN))>and y= (y(t1), . . . , y(tN))>, the con-

ditional posterior distribution of f, given the data-set, is still

a Gaussian process with mean µand variance σ2given by

[1]

µ(x) = κ(x)>K+σ2

nIN−1y,

σ2(x) = κ(x, x)−κ(x)>K+σ2

nIN−1κ(x),(1)

where K∈RN×Nis the Gram matrix whose (k, h)-th entry

is Kk,h =κ(xk,xh), with xkthe k-th entry of x, and

κ(x)∈RNis the kernel vector whose k-th component is

κk(x) = κ(x, xk).

Remark 1: The assumption of measurements perturbed by

Gaussian noise is commonly used in learning-based control

since it is caused, for example, by numerical differentiation

(see [11])

From now on we suppose that the following standing as-

sumptions hold (see [10] , [12])

Assumption 1: The funciton µiis Lipschitz continuous

with Lipschitz constant Lµ, and its norm is bounded by µmax.

Assumption 2: The kernel function κis Lipschitz contin-

uous with constant Lκ, with a locally Lipschitz derivative of

constant Ldκ, and its norm is bounded by κmax.

Although any kernel fulﬁlling Assumption 2 can be a

valid candidate, in the following, we exploit the commonly

adopted squared exponential kernel as prior covariance func-

tion, which can be expressed as

κ(x, x0) = exp −(x−x0)>Λ−1(x−x0)(2)

for all x, x0∈Rnx, where Λ = diag(2λ2

x1,...,2λ2

xnx),

λxi∈R>0is known as characteristic length scale relative

to the i-th signal, and is usually called amplitude [1].

Assumption 3: Each component of the unknown map f

has a bounded norm in the RKHS1Hgenerated to the kernel

κ, in Equation (2).

Remark 2: Assumption 3 is asking some Lipschitz conti-

nuity property of the unknown function that makes it well-

representable by means of a Gaussian process prior. Never-

theless, it represents a very strong assumption, difﬁcult to be

checked even if the unknown function is known. Assump-

tion 3 can be relaxed to the condition that each component ¯

is a sample from the Gaussian process GP (0, κ (·,·)), which,

in turn, leads to a larger pool of possible unknown functions

and it is easier to be checked. As an example, the pool

generated by the squared exponential kernel Equation (2)

is equal to the space of continuous functions.

We recall a result based on [12].

Lemma 1: Consider a zero-mean Gaussian process de-

ﬁned through a kernel κ:X × X 7→ R, satisfying As-

sumption 2 on the compact set X. Furthermore, consider a

continuous unknown function f:X 7→ Rwith Lipschitz

constant Lf, and N∈Nobservations yi=fxi+εi,

with εi∼ N(0, σ2

nIny). Then the posterior mean µand

posterior variance σ2conditioned on the training data DS =

x1, y1,...,xN, yN are continuous with Lipschitz

constants Lµand Lσ2on X, respectively, satisfying

Lµ≤Lκ√N



K+σ2

nIN−1y



,

Lσ2≤2ρLκ1 + N



K+σ2

nIN−1



max

x,x0∈X κ(x, x0),

with x= (x1, . . . , xN)>and y= (y1, . . . , yN)>. Moreover,

pick δ∈(0,1),ρ > 0and set

β(ρ) = 2 log M(ρ, X)

δ,

α(ρ)=(Lf+Lµ)ρ+pβ(ρ)Lσ2ρ,

1Reproducing Kernel Hilbert Space

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Data-DrivenAnalyticDifferentiationviaHighGainObserversandGaussianProcessPriorsBiagioTrimarchi1,LorenzoGentilini1,FabrizioSchiano2,andLorenzoMarconi1AbstractThepresentedpapertacklestheproblemofmodelinganunknownfunction,anditsrstr1derivatives,outofscatteredandpoor-qualitydata.Theconsideredsettingemb...

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Data-Driven Analytic Differentiation via High Gain Observers and Gaussian Process Priors Biagio Trimarchi1 Lorenzo Gentilini1 Fabrizio Schiano2 and Lorenzo Marconi1

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