Model Reference Gaussian Process Regression Data-Driven Output Feedback Controller Hyuntae Kim Hamin Chang and Hyungbo Shim

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Model Reference Gaussian Process Regression:

Data-Driven Output Feedback Controller

Hyuntae Kim, Hamin Chang, and Hyungbo Shim

Abstract— Data-driven controls using Gaussian process re-

gression have recently gained much attention. In such ap-

proaches, system identiﬁcation by Gaussian process regression

is mostly followed by model-based controller designs. However,

the outcomes of Gaussian process regression are often too

complicated to apply conventional control designs, which makes

the numerical design such as model predictive control employed

in many cases. To overcome the restriction, our idea is to

perform Gaussian process regression to the inverse of the

plant with the same input/output data for the conventional

regression. With the inverse, one can design a model reference

controller without resorting to numerical control methods. This

paper considers single-input single-output (SISO) discrete-time

nonlinear systems of minimum phase with relative degree one.

It is highlighted that the model reference Gaussian process

regression controller is designed directly from pre-collected

input/output data without system identiﬁcation.

I. INTRODUCTION

Gaussian process regression (GPR) [1], one of the most

well-known regression tools for nonlinear functions, has

been extensively used in various ﬁelds by virtue of the

following properties [2]. First, since it is a nonparametric

method, it has some ﬂexibility to deal with a large amount

of data. Secondly, prior knowledge of the regression target

can easily be incorporated. Finally, it gives some conﬁdence

information about the regression result, which can be utilized

to measure the regression error.

Particularly in control systems, GPR has been mainly

applied for identifying unknown nonlinear systems using

input/output or even state data before designing a model-

based controller for the identiﬁed model. For instance, [3]

and [4] show that a model predictive controller can be

designed based on the model identiﬁed by GPR. Moreover,

its real world applications are presented in [5] and [6]

for quadrotors and mobile robots, respectively. In addition,

combining the prior knowledge of a nominal model, [7] and

[8] demonstrate the utility of such Gaussian process-based

model predictive control (GP-MPC) method in autonomous

racing systems by identifying a residual model instead of the

full dynamical system. Also, [9] presents real world exper-

iments of quadrotors controlled by the GP-MPC approach,

where only aerodynamic effects on quadrotors are modeled

by Gaussian process. On the other hand, [10] and [11]

propose a feedback linearization controller for the system,

which is identiﬁed by GPR. They also provide Lyapunov

This work was supported by the grant from Hyundai Motor Company’s

R&D Division.

All authors are with ASRI, Department of Electrical and Computer

Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul,

08826, Korea. Corresponding author: hshim@snu.ac.kr

stability analysis of the controlled system based on the result

concerning the error of the identiﬁcation in [12]. On the other

hand, [13] and [14] propose event triggered online learning

of GPR in order to increase data efﬁciency.

However, most of these studies focus only on the system

identiﬁcation capability of the GPR in the sense that con-

trollers should be designed only after the system identiﬁca-

tion by using the GPR is completed. This leads to a problem

with controller design because even if the given system has

a relatively simple analytic formula, its identiﬁed model by

the GPR can be too complicated consisting of a summation

of as many terms as the number of data points. This

complexity has often restricted applicable control methods,

so that numerical methods such as MPC (model predictive

control) are typically employed for the identiﬁed model.

Since numerical control methods require a certain amount

of online computation resource, its utility can sometimes be

limited.

To overcome the restriction, we propose identiﬁcation of

the inverse of the given system by GPR, with the purpose

of using it for model reference control. In this way, we can

bypass numerical controls and can combine classical controls

resulting in a data-driven controller, which we call model

reference Gaussian process regression (MR-GPR) control

in this paper. Since it is natural to assume that we have

access to only input/output measurements of the plant, we

propose the MR-GPR controller in the form of an output

feedback control. Therefore, the GPR is performed only with

input/output data of the system. Since our approach is based

on input/output inversion in some sense, a few limitations

naturally follow such as causality and minimum phase issues.

In this paper, we assume that the system has relative degree

one to resolve the causality issue, which is not very restrictive

because a sampled-data system of a continuous-time system

generically has relative degree one. Moreover, we assume

that the system is of minimum phase.

This paper is organized as follows. The problem formu-

lation with a class of nonlinear systems under consideration

and a couple of assumptions on the class of systems are in

Section II. In Section III, we propose the data-driven MR-

GPR controller and explain how to design it using the GPR.

Also, a stability analysis of the closed-loop system with the

MR-GPR controller is presented. An illustrative example that

demonstrates the usefulness of the MR-GPR controller is

given in Section IV. Finally, this paper is summarized and

concluded in Section V.

Notation: For column vectors aand b,[a;b]denotes

[aT, bT]T. For discrete-time vector sequences y(t)and z(t),

arXiv:2210.02494v1 [eess.SY] 5 Oct 2022

we deﬁne a vector

z[k,k+T]:= [z(k); z(k+ 1); · · · ;z(k+T)],

and a set

{(y(t), z(t))}k+T

t=k

:= {(y(k), z(k)),· · · ,(y(k+T), z(k+T))}.

II. PROBLEM FORMULATION

Consider a single-input single-output (SISO) nonlinear

discrete-time control-afﬁne system with relative degree one

in Byrnes-Isidori normal form [15]:

y(t+ 1) = f(z(t), y(t)) + g(z(t), y(t))u(t)(1a)

z(t+ 1) = h(z(t), y(t)) (1b)

where u(t)∈Ris the input, z(t)∈Rn−1is the state of the

zero dynamics, and y(t)∈Ris the output. It is assumed that

the functions f(·,·),g(·,·), and h(·,·)are unknown and only

the input/output of the system are available as measurements.

Also, we assume that the functions f(·,·),g(·,·), and h(·,·)

are smooth. In addition, the following assumption is given.

Assumption 1: The system (1) satisﬁes the followings:

(a) The system has global relative degree one, or equiv-

alently, g(z, y)6= 0 for all (z, y)∈Rn. Also, the

system dimension nand the global relative degree one

are known.

(b) The internal dynamics (1b) is input-to-state stable with

the input being y.

If the plant to be controlled is a continuous-time physical

system, then its discretization generically yields a discrete-

time system of relative degree one [16]. Therefore, the

system description of (1) may not be too restrictive. Now,

we assume observability of the system (i.e., observability for

the state z) as follows.

Assumption 2: There exists a smooth mapping O:

R2n−1→Rn−1that determines the state z(t)as

z(t) = O([y[t,t+n−1];u[t,t+n−2]])

for any pair of input u[t,t+n−2] and output y[t,t+n−1] of the

system (1). 

Example 1: For simplicity, let us write y(t)by ytin this

example. When the system (1) has the form of

yt+1 =fz(zt) + fy(yt) + ut

zt+1 =h(zt, yt)(2)

then Assumption 2 holds if, for any input/output trajectory

u[t,t+n−2] and y[t,t+n−1] of (1), there exists a unique solution

z∗∈Rn−1to the equations

fz(z∗) = yt+1 −fy(yt)−ut,

fz(h(z∗, yt)) = yt+2 −fy(yt+1)−ut+1,

fz(h(h(z∗, yt),fy(yt) + fz(z∗) + ut))

=yt+3 −fy(yt+2)−ut+2,

fz(h(· · · (h(z∗, yt),fy(yt) + fz(z∗) + ut),· · · ))

=yt+n−1−fy(yt+n−2)−ut+n−2

which is derived directly from the system (2). In this case,

z(t) = z∗.

On the other hand, let us consider a stable reference model

given by

yr(t+ 1) = fr(yr(t)) ∈R(3)

which satisﬁes the additional assumption that

yr(t+ 1) = fr(yr(t)) + η(t)

is input-to-state stable when ηis viewed as an input. In

order to make the controlled system (1) become the reference

model (3), the controller should be

u(t) = fr(y(t)) −f(z(t), y(t))

g(z(t), y(t)) .(4)

For designing the controller (4), however, not only the

functions f(·,·)and g(·,·)are needed, but also the state z(t)

needs to be measured. In this paper, we present a method to

construct the controller (4) by using only the input/output

data of the system (1).

III. MAIN RESULT

In this section, we design a data-driven controller that can

produce almost the same control input as (4) by using GPR

trained by input/output data of the system (1).

We ﬁrstly show that the state z(t)can be expressed by the

input/output history of the system (1). For this, let

ζ0(t) := [y[t−n+1,t−1];u[t−n+1,t−1]]∈R2(n−1).(5)

Lemma 1: Under Assumption 2, there exists a smooth

function θ:R2(n−1) ×R→Rn−1, such that the state z(t)

of (1) is given by

z(t) = θ(ζ0(t), y(t))

for all time step t.

Proof: Since there exists a smooth mapping Osuch

that

z(t−n+ 1) = O([y[t−n+1,t];u[t−n+1,t−1]])

by Assumption 2, it follows that

z(t) = h(z(t−1), y(t−1))

=h(h(z(t−2), y(t−2)), y(t−1))

=h(h(h(z(t−3), y(t−3)), y(t−2)), y(t−1))

=h(· · · (h(z(t−n+ 1), y(t−n+ 1)),· · · ), y(t−1))

=h(· · · (h(O([y[t−n+1,t];u[t−n+1,t−1]]),

y(t−n+ 1)),· · · ), y(t−1))

=: θ(ζ0(t), y(t))

which completes the proof.

Let us deﬁne the vectors

ζ1(t) := [ζ0(t); y(t)] ∈R2n−1

ξ(t) := [ζ1(t); y(t+ 1)] ∈R2n

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ModelReferenceGaussianProcessRegression:Data-DrivenOutputFeedbackControllerHyuntaeKim,HaminChang,andHyungboShimAbstractData-drivencontrolsusingGaussianprocessre-gressionhaverecentlygainedmuchattention.Insuchap-proaches,systemidenticationbyGaussianprocessregressionismostlyfollowedbymodel-basedcontr...

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