Joint State and Dynamics Estimation With High-Gain Observers and Gaussian Process Models

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Mona Buisson-Fenet, Valery Morgenthaler, Sebastian Trimpe, Florent Di

Meglio

To cite this version:

Mona Buisson-Fenet, Valery Morgenthaler, Sebastian Trimpe, Florent Di Meglio. Joint State and

Dy-namics Estimation With High-Gain Observers and Gaussian Process Models. IEEE Control Systems

Letters, IEEE, 2021, 5 (5), pp.1627-1632. �10.1109/LCSYS.2020.3042412�. �hal-03098157�

Joint state and dynamics estimation with high-gain observers and

Gaussian process models

Mona Buisson-Fenet

1,2,3

_{, Valery Morgenthaler}

2

_{, Sebastian Trimpe}

3

_{, Florent Di Meglio}

1

Abstract—

With the rising complexity of dynamical

sys-tems generating ever more data, learning dynamics models

appears as a promising alternative to physics-based

mod-eling. However, the data available from physical platforms

may be noisy and not cover all state variables. Hence, it

is necessary to jointly perform state and dynamics

estima-tion. In this paper, we propose interconnecting a high-gain

observer and a dynamics learning framework, specifically

a Gaussian process state-space model. The observer

pro-vides state estimates, which serve as the data for training

the dynamics model. The updated model, in turn, is used

to improve the observer. Joint convergence of the observer

and the dynamics model is proved for high enough gain,

up to the measurement and process perturbations.

Simulta-neous dynamics learning and state estimation are

demon-strated on simulations of a mass-spring-mass system.

Index Terms—

Machine Learning, Nonlinear systems

identification, Observers for nonlinear systems

I. I

NTRODUCTION

W

ITH the recent advances in control engineering, more

and more complex dynamical systems are being

con-sidered, for which it is often difficult to derive physics-based

models. Because of the availability of experimental data, as well

as novel algorithms to process it, learning dynamics models

directly from that data is an attractive alternative. However, this

experimental data often originates from sensor measurements

on the physical system, which can be noisy and may not

reflect all state variables. On the one hand, reconstituting

the full state from noisy, partial measurements falls into the

area of state estimation and observer design. For nonlinear

systems, this remains a challenging task for which knowledge

of the dynamics is usually required [1]. On the other hand,

most existing approaches for learning dynamics models require

knowing the full state [2]. Therefore, joint state estimation and

dynamics learning is needed, a problem often referred to in

the machine learning community as inference and learning [3].

The design of observers for nonlinear systems is a complex

task for which various approaches have been investigated

(see [1] for an overview). We focus on the fairly large class of

systems that can be expressed in the so-called observable

canonical form (see Sect. II). For these systems, one can

design High-Gain Observers (HGOs), which rely on a triangular

1

_{Centre Automatique et Syst `emes, Mines ParisTech, PSL University,}

Paris, France,

florent.di meglio@mines-paristech.fr

,

mona.buisson@mines-paristech.fr

2

_{ANSYS Research Team, ANSYS France, Villeurbanne, France,}

valery.morgenthaler@ansys.com

3

_{Institute for Data Science in Mechanical Engineering, RWTH Aachen}

University, Aachen, Germany,

trimpe@dsme.rwth-aachen.de

structure with increasing gain power to compensate for the

nonlinearity farther from the measurement. HGOs have been

used for a wide variety of applications [4], [5]. In particular,

they provide robustness to model uncertainty, as practical

convergence can be proved for high enough gain, given only

an upper bound on the nonlinearity [6].

Learning dynamics models is also an active research topic.

In particular, Gaussian process (GP) state-space models are

increasingly used [7], [8]. These nonparametric models exhibit

many advantageous properties for learning dynamical systems:

they are flexible, data-efficient, probabilistic, and can easily

incorporate prior knowledge (see [9] for details). Thanks to their

analytical formulation, GPs also allow for theoretical guarantees

[10], [11], which is rarely the case for nonparametric machine

learning frameworks.

The problem of joint inference and learning for GP

state-space models is tackled in its most general form in [3] using

variational inference, by modeling the latent states as extra

hyperparameters of the GP. The expectation maximization

(EM) algortihm is applied: in the first step, measurements

are collected and the posterior distribution of the GP is

computed. In the second step, all hyperparameters, including

the pseudo inputs and outputs representing the evolution of

the latent states, are optimized to maximize the data log

likelihood. Improvements of this approach have been proposed,

e.g., by using more sophisticated loss functions or incorporating

additional structure [12], [13]. However, the optimization

procedure remains high-dimensional and non-convex. This

leads to a high computational burden and a risk of overfitting,

which can make the models difficult to train. Furthermore, no

theoretical guarantees are yet provided for such methods.

Recent works tackle this problem by combining observer

design and data-driven dynamics learning with universal

approximators. The model is learned with a neural network

using smooth, continuous-time weight update laws [14], [15]

or a basis expansion [16], then added to an observer built as a

copy of the system plus linear output injection terms. Limited

theoretical guarantees have been shown [14], [16], but joint

convergence has only been proved if suitable gains can be

found by solving a large set of linear matrix inequalities [15].

However, this yields an unusual neural network model for

f

and an observer with a high number of parameters left to tune,

limiting the practical use of the framework.

In this work, we combine the predictive power of machine

learning with existing convergence results for state estimation.

Our main contribution is the design of a framework for

simultaneous state and dynamics estimation, by combining

a HGO that estimates the full state from measurements and a

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.

The final version of record is available at

http://dx.doi.org/10.1109/LCSYS.2020.3042412

GP model that learns the unknown nonlinearity. Convergence

guarantees for both the observer and the dynamics model are

provided; practical applicability is discussed and demonstrated

on simulations. This builds upon the scheme proposed in

[17], in which the nonlinearity is considered as a state

with partially known dynamics in an extended HGO, and

is learned by an identifier satisfying certain requirements.

The key difference of our approach is to directly learn a

discrete model of the nonlinearity instead of differentiating

it and extending the observer. This enables us to deal with

controlled systems and input-dependent nonlinearities, and to

decrease the error in the data used for regression. Furthermore,

it decreases the dimensionality of the observer, which reduces

noise amplification by the HGO. We also show that more

flexible, non-parametric models such as GPs can learn the

input-dependent dynamics while satisfying the smoothness

assumptions which are necessary to prove joint convergence.

After formalizing the problem, we present the proposed

framework in Sect. II. In Sect. III we show joint convergence

of both state and dynamics estimation, then demonstrate our

approach on a numerical example in Sect. IV. Comparison to

related work and limitations are discussed in Sect. V, before

concluding in Sect. VI.

II. P

ROBLEM FORMULATION AND PROPOSED FRAMEWORK

We consider a dynamical system of state

x

_{∈ R}

n

_{, output}

y

∈ R, and bounded control input u ∈ R

m

_{, where}

_{n, m}

_{∈ N.}

For ease of notation we focus on the single-output case, but

all results extend to multiple outputs by concatenation. We

assume the following observable canonical form

˙x = Ax + Bf (x, u) + D(u) + d,

y = Cx + ,

(1)

with

f an unknown nonlinearity acting on the n

th

_state

_x

n

,

while the rest of the dynamics follows a chain of integrators:

A =

0

n−1

I

n−1

0

0

|

_n−1



,

B =

0

n−1

1



,

C =



1

0

n−1



|

.

The input function

D : R

m

→ R

n

_{is continuous and known}

while

d

∈ R

n

_,

_

_{∈ R are unknown disturbances, typically}

considered deterministic (see Remark 2). All vectors are

column vectors;

0

n

denotes a vector of

n zeros, while I

n

is the identity matrix of size

n. A broad class of systems

can be transformed into this canonical form without knowing

f , e.g., all differentially observable systems [1, Sect. 7.1].

Our aim is to compute an estimate

ˆ

x of the full state from

measurements

y, while jointly learning a model ˆ

f of f . We

make the following assumptions on (1).

Assumption 1:

The true nonlinearity

f is Lipschitz

continu-ous of constant

L

f

. There exist compact sets

X and U such

that

x(t)

∈ X , u(t) ∈ U ∀ t ≥ 0. Since f is continuous on a

compact space, its norm is also bounded by

f

max

.

The proposed observer follows a cyclic structure, illustrated

in Figure 1. During cycle number

j

_{∈ N}

∗

_{, the observer produces}

an estimated state trajectory based on measurements and on

the current dynamics model ˆ

f

j−1

. This data is sampled and

saved. At the end of the cycle, the model is updated based on

the available estimated data. It produces an estimate ˆ

f

j

, which

is then used by the observer for the next cycle.

13

Update observer

Update GP

Final model

Data y

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ˆ

x

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ˆ

f

j

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Fig. 1

: Structure of the framework: after each cycle, an updated

model ˆ

f

j

is computed and the observer is adapted.

A. High-gain observer

During cycle number

j, the HGO performs state estimation

using the current model of the dynamics ˆ

f

j−1

:

˙ˆx = Aˆx + B ˆ

f

j−1

(ˆ

x, u) + D(u) + Λ(g)(y

− C ˆx).

(2)

The gain is denoted

g > 1, while Λ(g) is the gain matrix

following a standard high-gain construction:

Λ(g) := gL

1

g

2

L

1

· · · g

n

L

n

|

,

(3)

with

L = L

1

· · · L

n

|

∈ R

n

_{such that}

_A

−LC is Hurwitz.

Equation (2) corresponds to the observer block in Figure 1.

We make the following assumption, which is ensured by our

dynamics model described hereafter.

Assumption 2:

For all

j

_{∈ N, ˆ}

f

j

is continuous and its norm

is bounded by ˆ

f

max

.

Hence, we can pick

_{X large enough such that ˆx(t) ∈ X}

∀ t ≥ 0. With Assumptions 1–2, the error on the nonlinearity

ˆ

f (x, u)

_{− f(x, u) is bounded. Then, as proved in [6] and used}

in the literature on HGOs, practical convergence in finite time

can be shown in the absence of disturbances: for a given error

level

ν > 0 and a given time ¯

t > 0, there exists a gain g high

enough to ensure that for all

t

≥ ¯t, kˆx(t) − x(t)k ≤ ν. Hence,

no matter how bad the approximation ˆ

f of f is, as long as an

upper bound of the difference is known, practical convergence

of the observer can be guaranteed for high enough gain. We

leverage this property to build our method.

B. Preliminaries on Gaussian processes

In this paper we focus on Gaussian processes. However, any

learning algorithm that satisfies our assumptions can be used.

A Gaussian process (GP) is a collection of random variables,

any finite subset of which is jointly normally distributed (see

[9] for an overview). It is fully characterized by its mean

function

m(

·) and its covariance function k(·, ·). Without loss

of generality, we assume

m

≡ 0. Any prior information can

be included by substracting it from the output data in order

to learn the residuals of this prior. Function properties such

as smoothness are encoded in the choice of the kernel

k(

_{·, ·),}

which acts as a similarity measure for the values of

f (

_{·). At}

an unobserved point

x, given a dataset (X, Y ) with Gaussian

noise of variance

σ

2



on the output, the GP prediction of

f (x)

is normally distributed with posterior mean and variance

µ(x

|X, Y ) = k(x)

|

(K + σ

2



I)

−1

Y

(4)

σ

2

_(x

|X, Y ) = k(x, x) − k(x)

|

(K + σ

2



I)

−1

k(x),

(5)

where

I is the identity matrix, K = (k(x

i

, x

j

))

x

i

,x

j

∈X

is the

kernel

k usually depends on some hyperparameters, often

obtained by maximizing the data marginal log likelihood.

Assumption 3:

The kernel

k is differentiable, Lipschitz

continuous of constant

L

k

, and its norm is bounded by

k

max

.

This is the case for most commonly used covariance functions,

such as the squared exponential. GPs are increasingly used for

learning dynamics thanks to their flexibility, data efficiency and

analytical formulation. In this work, we use the GP posterior

mean as a function approximator to estimate

f .

C. Learning method

The state trajectory estimated by the observer is used to

learn the dynamics model ˆ

f through batch updates. For each

update indexed by

j

_{∈ N}

∗

_{, a dataset of length}

_{N is constructed}

by sampling this trajectory with period

∆t, starting from the

last sample collected at

t

j

:

X

j

= (ˆ

x(t

j−N

), u(t

j−N

))

· · · (ˆx(t

j−1

), u(t

j−1

))

|

Y

j

= ˆ

x

n

(t

j−N +1

)

· · · ˆx

n

(t

j

)

|

.

(6)

Since the nonlinearity acts on the

n

th

_dimension,

_x

n

is the only

output that needs to be collected. The

j

th

_{update ˆ}

_f

j

is learned

from inputs

X

j

and outputs

Y

j

, then used in the observer (2)

for the next cycle. It can be updated offline since this may

necessitate more computing power, and the updates are not

necessarily periodic. The model learned from (6) is

µ

j

(

·|X

j

, Y

j

) = k

j

(

·)

|

(K

j

+ σ

2



I

N

)

−1

Y

j

(7)

with

K

j

= (k(x

i

, x

l

))

x

i

,x

l

∈X

j

,

k

j

(x) = (k(x

i

, x))

x

i

∈X

J

. For

the formulation of the GP posterior, Gaussian measurement

noise of variance

σ

2



is assumed. However, this need not be the

case in reality as we only use the GP as a function approximator

and

σ

2



, similarly to other hyperparameters, is chosen in practice

for calibration purposes. No assumption on

 is needed for the

provided theoretical guarantees.

We perform nonlinear regression to estimate the mapping

µ :

(ˆ

x, u)

_{7→ ˆx}

n

(t + ∆t). However, ˆ

f in the observer corresponds

to the continuous time derivative of

x

ˆ

n

. Hence, we form ˆ

f

j

with a Euler differentiation step:

ˆ

f

j

(ˆ

x, u) =

1

∆t

(µ

j

(ˆ

x, u

|X

j

, Y

j

)

− ˆx

n

).

(8)

To guarantee the boundedness of ˆ

f , we saturate it directly in the

observer by imposing

_{k ˆ}

f

_{k ≤ ˆ}

f

max

. The model (8) corresponds

to the GP block in Figure 1; given the continuity of (7) and

this saturation, it satisfies Assumption 2.

Remark 1:

The choice of

∆t results from a trade-off: small

enough to keep the numerical error from (8) low, large enough

to see a real difference between two samples. Learning a

discrete model from the estimated trajectory enables us to

avoid extending the observer, contrarily to [17].

III. T

HEORETICAL GUARANTEES

As stated previously, HGOs are robust to model uncertainty

for systems in the observable canonical form. This enables us

to decouple the procedures of state estimation and dynamics

learning. Indeed, thanks to this robustness, convergence

guar-antees can still be obtained even in the worst-case scenario,

i.e., with maximal but bounded model error, at the cost of a

high gain. These convergence properties are then transferred

to the dynamics model through its smoothness with respect to

the dataset used for learning. Both practical and asymptotic

convergence results are provided.

In the following proofs, we focus on the

`

2

norm for vectors

and matrices, but equivalent bounds can be obtained for any

vector norm and its induced matrix norm. We first give a

technical result on the smoothness of GP models, showing

that the posterior mean is Lipschitz continuous not only with

respect to the test point but also to the training dataset.

Lemma 1:

Under Assumptions 1–3, the dynamics model ˆ

f

as defined in (8) is Lipschitz continuous with respect to each

of its variables:

(x, u)

_{7→ ˆ}

f (x, u

_{|X, Y ) with constant L}

x

, and

(X, Y )

_{7→ ˆ}

f (x, u

_{|X, Y ) with constant L}

z

.

Sketch of proof:

Assumptions 1–3 yield upper bounds on

k

and

Y . These and the Lipschitz continuity of k directly yield

the first claim. The sensitivity of the entries of

K with respect

to each entry of

X is then bounded by elementary algebraic

inequalities, and the second claim is obtained similarly.

_

Since the dataset

(X

j

, Y

j

) is constructed from state

esti-mation samples, the error in this data directly depends on

the state estimation error. This corresponds to the stability

requirement in [17]. Then, Lemma 1 guarantees that any state

estimation error in the dataset is smoothly transferred to the

obtained model. This is essential to obtain stability guarantees,

and corresponds to the regularity requirement in [17]. While

we focus on GPs, any learning algorithm satisfying Lemma

1 based on a dataset constructed as in (6) can be used in our

framework, to produce an observer and a dynamics model that

can both be used for further control tasks.

A. Practical convergence

We denote

d

g

= g

−1

d

1

· · · g

−n

d

n

, ¯

x = (x, u), ˆ

x =

¯

(ˆ

x, u) and

_kxk

¯

t

= max

{kx(t)k | t ≤ ¯t}. At a given j ∈ N

∗

,

ˆ

f

∗

j

is the nonlinearity that would have been learned if the true

data

X

∗

j

,

Y

j

∗

had been available, i.e., if the state

x(t) had been

directly available for sampling according to (6) instead of its

estimate

x(t). The prediction error is written as

ˆ

ε

j

(

·) = ˆ

f

j

(

·) − f(·),

(9)

while

ε

∗

j

(

·) = ˆ

f

j

∗

(

·) − f(·) is the optimal prediction error that

would have been obtained if the true state had been available

instead of an estimate. We now state a first result on the

practical convergence of the proposed framework.

Theorem 1:

For system (1)–(2) with (8) under Assumptions

1–3, for any given error level

ν > 0, any time ¯

t > 0, there

exists a gain

g

∗

_{large enough such that for all}

_g

≥ g

∗

_,

_t

≥ ¯t

and

j

_{∈ N}

∗

_{such that ¯}

_t

≤ t

j

≤ t, in the absence of disturbances

(d

_{≡  ≡ 0), we have for any fixed ¯x ∈ X × U:}

max

_{kˆx(t) − x(t)k, k ˆ}

f

j

(¯

x)

− ˆ

f

j

∗

(¯

x)

k ≤ ν.

(10)

Proof:

The proof follows three steps. During a given cycle

indexed by some fixed

j

∈ N

∗

_{, with}

_t

j

≥ t ≥ t

j−1

, the model

error can be bounded by the maximal error:

_{k ˆ}

f

j−1

(

·)−f(·)k ≤

ˆ

f

max

+f

max

. Using existing proofs of the practical convergence

of HGOs under bounded model uncertainty [6], [17], we show

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.

The final version of record is available at

http://dx.doi.org/10.1109/LCSYS.2020.3042412

that there exist constants

ρ

1

, ρ

2

, ρ

3

> 0 and a gain g

∗

> 0

such that for all

g

≥ g

∗

_,

_i

_{∈ {1, ..., n}:}

kˆx

i

(t)

− x

i

(t)

k ≤ max

g

i−1

ρ

0

e

−ρ

1

gt

kˆx(0) − x(0)k,

ρ

2

g

i−n−1

, ρ

3

g

i−1

k(d

g

, )

k

t

:= B

i

(t).

(11)

The key then lies in bounding the error on

X

j

and

Y

j

:

kX

j

− X

j

∗

k ≤

j−1

X

l=j−N

n

X

i=1

kˆx

i

(t

l

)

− x

i

(t

l

)

k,

(12)

kY

j

− Y

j

∗

k ≤

j

X

l=j−N +1

kˆx

n

(t

l

)

− x

n

(t

l

)

k.

(13)

Therefore, the error on the input and output datasets used to

learn ˆ

f

j

decreases as the state estimation error decreases, with

a delay of

N time steps corresponding to the time before earlier

samples with a larger error are forgotten. Applying Lemma 1

then (12)-(13) and (11) yields for any

¯

x

_{∈ X × U:}

k ˆ

f

j

(¯

x)

− ˆ

f

j

∗

(¯

x)

k ≤L

z

N (n + 1) max

ρ

3

g

n−1

k(d

g

, )

k

t

j

,

ρ

2

g

, g

n−1

_ρ

0

e

−ρ

1

gt

j−N

kˆx(0) − x(0)k

.

(14)

Combining (11) and (14) as such leads to a joint practical

convergence result, with an additional term corresponding to

the disturbances. In the absence of disturbances, i.e., with

d

_{≡  ≡ 0, this concludes the proof.}

B. Asymptotic convergence

Theorem 1 shows that the practical convergence guarantees

obtained for HGOs with bounded nonlinearity extend to the

complete error system. Both the state estimation error and

the error made by the dynamics model due to seeing only

estimated instead of true data can be made arbitrarily small

arbitrarily fast, up to the disturbances, at the cost of a high

gain. We now present an asymptotic convergence result arising

from the practical convergence of HGOs in the presence of

model uncertainty, by bounding this uncertainty depending on

the data, the test point, and the optimal

ε

∗

_.

Theorem 2:

For system (1)–(2) with (8) under Assumptions

1–3, there exist constants

c, c

0

_{, c}

00

_{> 0 and a gain g}

∗

_{> 0 such}

that

_{∀ g ≥ g}

∗

,

_{∀ i ∈ {1, ..., n} and for any fixed test point}

¯

x

∈ X × U, we have

lim sup

t→∞

kˆx

i

(t)

− x

i

(t)

k ≤ c max



g

i−n−1

lim sup

t+j→∞

kε

∗

j

(x, u)

k, g

i−1

lim sup

t→∞

k(d

g

, )

k

, (15)

lim sup

t+j→∞

k ˆ

f

j

(¯

x)

− ˆ

f

∗

j

(¯

x)

k ≤ max



c

0

g

−1

lim sup

t+j→∞

kε

∗

j

(x, u)

k, c

00

g

n−1

lim sup

t→∞

k(d

g

, )

k

. (16)

Proof:

First, using definition (9) and Lemma 1, we show

that for any fixed

j

_{∈ N}

∗

_,

_{(x, ˆ}

_{x, u)}

∈ X × X × U, there exists

c

1

> 0 such that

k ˆ

f

j−1

(ˆ

x, u)

− f(x, u)k

≤ k ˆ

f

j−1

(ˆ

x, u)

− ˆ

f

j−1

∗

(x, u)

k + kε

∗

j−1

(x, u)

k

≤ c

1

kˆx − xk + kX

j−1

− X

j−1

∗

k + kY

j−1

− Y

j−1

∗

k

+

_kε

∗

j−1

(x, u)

k

.

(17)

_m

1

m

2

x

Fig. 2

: Schematics of the mass-spring-mass system.

Then, using (17) in the proof of practical convergence of the

HGO with

g large enough yields a bound similar to (11), but

involving

_kX

j−1

− X

j−1

∗

k + kY

j−1

− Y

j−1

∗

k + kε

∗

j−1

(x, u)

k

instead of the constant term in

ρ

2

. Going to the limit in both

time and number of cycles and using (12)–(13) yields the first

claim. Lemma 1 yields the second claim.

Theorem 2 bounds the difference between ˆ

f and ˆ

f

∗

_that

would have been obtained using true instead of estimated data

for learning, without any assumption on the fit of the GP. If

ε

∗

is zero, both the state estimation and the prediction error are

input-to-state stable with respect to the disturbances. If there are

also no disturbances, they converge asymptotically. Since GPs

with universal kernels are universal function approximators,

ε

∗

can get very small (it converges up to the numerical errors due

to the Euler differentiation) as the number of samples grows

to infinity, if the samples are densely distributed over

_{X × U,}

i.e., if the state-action space is well explored. However, if the

hyperparameters of the GP do not enable a good fit or if the

data is not rich enough, then

ε

∗

_{may be large. We further note}

that it is also possible to bound

_{k ˆ}

f

j

(¯

x)

− f(¯x)k similarly to

(16) using the same error decomposition as (17): a term

ε

∗

j

(¯

x)

appears, representing the optimal error due to the GP model

at the test point

x.

¯

Remark 2:

Nowhere do we use the form of

d and . In

principle, they could also be realizations of stochastic processes.

However, Theorem 2 may then not be as meaningful as,

depending on the process,

lim sup may not exist.

Remark 3:

The proposed framework is modular: any

regres-sor ˆ

f can be used seamlessly instead of the GP. If Assumptions

1–2 and Lemma 1 are satisfied, the same theoretical guarantees

will hold. This is the case for models of form ˆ

f (

_{·) = θ}

|

σ(

_·),

where

θ is a parameter to be learned from data following a

Lipschitz procedure (such as recursive least squares), and

σ

is a known, Lipschitz feature vector. Other observer designs

providing guarantees similar to HGOs could also be considered,

e.g., sliding mode observers [18].

Remark 4:

In practice, one can learn the residuals model

µ

res

: (ˆ

x, u)

7→ ˆx

n

(t + ∆t)

− ˆx

n

− ∆tD

n

(u) instead of µ, and

use

µ(ˆ

x, u) = µ

res

(ˆ

x, u) + ˆ

x

n

+ ∆tD

n

(u) for prediction. This

eases the training process by incorporating prior knowledge

into the regression problem, while minimally changing the

form of

Y

j

, which leads to a factor

(n + 2) instead of (n + 1)

in (14) and maintains the theoretical guarantees.

IV. N

UMERICAL SIMULATION

We demonstrate the performance of the proposed approach

on a mass-spring-mass system with a nonlinear spring, as

illustrated in Figure 2, and motivated by series-elastic actuator,

e.g., [19]. We assume the system can be described by

where

x

1

,

x

2

are the positions of the two objects,

m

1

= m

2

= 1

are their masses, and

f

k

(

·) is some unknown nonlinear function

representing the spring dynamics.

Assuming system (18) is differentially observable of order

4 (see [1] and references), it can be transformed into the

observable canonical form (1). This is done by introducing a

new state

z

∈ R

4

_{, taking}

_z

1

= x

1

and computing the successive

derivatives of

z

1

. As in (1), this yields

˙z =Az + Bf (z, u) + d

(19)

y = Cz + ,

where

f is an unknown nonlinearity. The proposed approach

can directly be applied to (19) without further knowledge

about

f . In our simulations, we use f

k

(

·) = k

1

(

·) + k

2

(

·)

3

with

k

1

= 0.3, k

2

= 0.1 the spring constants. This yields for

the true system in observable canonical form:

f (z, u) =

3k

2

m

1

m

2

(u

_{− (m}

1

+ m

2

)z

3

)v

2

1

+

6k

2

m

1

v

1

v

2

2

+

k

1

m

1

m

2

(u

− (m

1

+ m

2

)z

3

),

(20)

α =

3

v

u

u

t

m

1

z

3

2k

2

+

s

 k

1

3k

2



3

+

 m

1

z

3

2k

2



2

,

β =

3

v

u

u

t

m

1

z

3

2k

2

−

s

 k

1

3k

2



3

+

 m

1

z

3

2k

2



2

,

v

1

= α + β,

v

2

=

z

4

k

1

m

1

+

3k

2

m

1

v

2

1

.

We simulate (19) with

d,  Gaussian noise of standard

deviation

σ

d

= σ



= 10

−4

for ten cycles of

15 seconds

each, sampled at

∆t = 0.06s. We set u(t) = 0.4 cos(1.2t),

N = 3000, g = 10, L = (5, 5, 3, 1)

|

, and ˆ

f

0

≡ 0. We use a

squared exponential kernel whose hyperparameters are fixed

by maximizing the marginal log likelihood on a subset of data

offline. We run 10 simulations from 10 initial conditions with

x

1

∈ [0, 0.1], x

2

∈ [0.1, 0.2], and ˙

x

1

, ˙

x

2

∈ [−0.005, 0.005]

1

.

For each, we start by precomputing a grid of random states

and inputs, along with

50 test trajectories of 200 time steps,

using a random initial state and one of three control strategies:

random control,

u(t) = 0.4 cos(1.2t), or u(t) = 0.

In each simulation, we evaluate both the observer and the

model at the end of every cycle. The observer (2) containing

the current model is evaluated by computing the root mean

square error (RMSE) between true and estimated trajectory

over the last cycle, but also over the test trajectories, given

ˆ

z(t

0

) = (y(t

0

), 0, 0, 0)

|

and

y(t). The model (8) is evaluated

by computing the RMSE of one step ahead predictions over

the precomputed grid, and the RMSE of the predicted test

trajectories, given the initial state but no measurements.

We observe joint convergence of the observer and the

dynamics model. The error in all considered metrics decreases

over time, as depicted in Figure 3. The numbers themselves are

not necessarily meaningful, but their decreasing behavior and

1

_{Code available at https://github.com/monabf/joint_state_}

dynamics_estimation_HGOs_GPs.git

0

500

1000

1500

2000

Time steps

0.02

0.04

0.06

0.08

0.10

0.12

RMSE

(a) State estimation over last cycle

500

1000

1500

2000

2500

Time steps

0.010

0.011

0.012

0.013

RMSE

(b) GP predictions over a grid

0

1000

2000

Time steps

0.575

0.600

0.625

0.650

0.675

0.700

RMSE

(c) Estimation of test trajectories

0

1000

2000

Time steps

0

5

10

15

20

25

RMSE

(d) Prediction of test trajectories

Fig. 3

: RMSE of metrics over 10 simulations of (19)

(mean

_{± 2 standard deviations).}

0

50

100

150

200

t

−2

−1

0

1

2

z

4

True

Estimated

(a) Before learning

0

50

100

150

200

t

−3

−2

−1

0

1

2

z

4

(b) After ten cycles

Fig. 4

: Estimation of a test trajectory of

z

4

(random control).

the visual results before and after a few cycles are significant.

The variance is due to having different initial conditions,

evaluation grids and test trajectories for each run rather than

a different performance of the method. The remaining error

is caused by the measurement and process noise, along with

the irreducible model error given the available data. A test

trajectory as estimated by the observer is presented in Figure 4.

Before the model is learned (a), the observer’s estimates are

delayed compared to the true state, because the observer has

to wait for correction from the measurements. Once the model

has been learned (b), the observer can anticipate and produce

accurate estimates without delay. The phase portrait of another

test trajectory predicted by the dynamics model is also depicted

in Figure 5. It shows the final model can predict the first 100

time steps accurately, then slowly deviates.

Note that the same simulations were also run with the method

proposed in [17]. By avoiding to differentiate ˆ

f , our method

introduces less bias in the data used for regression, and the

HGO has a lower dimensionality. As expected, the results seem

to indicate that a discrete model learned directly from sampled

trajectories yields a better dynamics model, which in turn yields

a better observer. This difference grows with the dimensionality

and the complexity of the considered nonlinearity.

V. D

ISCUSSION

In this paper, we propose a framework for joint state and

dynamics estimation of dynamical systems in the observable

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.

The final version of record is available at

http://dx.doi.org/10.1109/LCSYS.2020.3042412