#  Safe continual learning in MPC with prescribed bounds on the tracking error
111We gratefully acknowledge funding by the German Research Foundation (DFG;
project numbers 471539468 and 507037103).

Lukas Lanza<EMAIL_ADDRESS>Dario Dennstädt  dario.dennstaedt@uni-
paderborn.de Thomas Berger<EMAIL_ADDRESS>Karl Worthmann
<EMAIL_ADDRESS>
###### Abstract

Recently, robust funnel Model Predictive Control (MPC) was introduced, which
consists of model-based funnel MPC and model-free funnel control for its
robustification w.r.t. model-plant mismatches, bounded disturbances, and
uncertainties. It achieves output-reference tracking within prescribed bounds
on the tracking error for a class of unknown nonlinear systems. We extend
robust funnel MPC by a learning component to adapt the underlying model to the
system data and hence to improve the contribution of MPC. Since robust funnel
MPC is inherently robust and the evolution of the tracking error in the
prescribed performance funnel is guaranteed, the additional learning component
is able to perform the learning task online – even without an initial model or
offline training.

###### keywords:

Data-based control, funnel control, model predictive control, robust control,
safe learning

††journal: Engineering Applications of Artificial Intelligence

[label_tuilm]organization=Technische Universität Ilmenau, Institute of
Mathematics, Optimization-based Control Group,addressline=Weimarer Str. 25,
city=98693 Ilmenau, country=Germany

[label_upb]organization=Universität Paderborn, Institut für
Mathematik,addressline=Warburger Str. 100, city=33098 Paderborn,
country=Germany

##  1 Introduction

Model Predictive Control (MPC) is a well-established control technique for
constrained linear as well as for nonlinear multi-input multi-output systems.
Using a model of the system to be controlled, the idea is to iteratively solve
Optimal Control Problems (OCPs) based on predictions about the system behavior
on a finite-time horizon, see e.g. the textbook [1], and the references
therein. MPC is nowadays widely used in various applications, see e.g. [2] and
the references therein.

Since MPC heavily relies on the availability and accuracy of the underlying
model of the actual system, some research effort has been made to compensate
for model-plant mismatches and external disturbances. One research branch
focuses on the MPC algorithm itself and its robustification, see e.g. [3, 4,
5, 6, 7], and the references therein, respectively. For instance, in [5] a
robust MPC framework was developed for nonlinear discrete-time systems. The
uncertainties and disturbances are compensated by introducing tightening tubes
around the input and output constraints to ensure robust satisfaction of the
constraints. A similar tube-based technique was used in [7]. In [3] it was
shown that combining funnel MPC, which achieves reference tracking with
guaranteed error performance, cf. [8], with an additional robust feedback
controller leads to a control scheme, which achieves a tracking objective even
in case of a severe model-plant mismatch.

A different research branch focuses on the idea to achieve robust constraint
satisfaction of the actual system via adaption of the underlying model, see
e.g. [9, 10, 11] and the references therein. For instance, in [10] the notion
of persistently exciting data, cf. [12, 13], is exploited to update the model
and show initial and recursive feasibility of the resulting MPC scheme. In
[14, 15] combining MPC with Gaussian process-based learning schemes was
proposed to achieve predictive control with stability guarantees. In [15] a
NARX model was incorporated in the learning scheme. The proposed controller in
[14] is formulated to address satisfaction of chance constraints, and its
functioning is demonstrated with an autonomous racing vehicle. A similar
approach was used in [16] to perform safe learning-based control in robotics.
The issue of malfunctioning of the machine learning scheme is considered in
[17], where the underlying neural network is forced to stay close to a
predefined nominal model, and hence the MPC scheme safely achieves the control
objective. Predictive safety filters are a MPC variant which is also closely
related to tube-based approaches and allows the application of learning-based
control techniques while guaranteeing compliance with constraints. The core
idea is that the predictive safety filter verifies a control input signal
proposed by a learning algorithm against a model. If the proposed control
signal is deemed safe it is applied to the system, otherwise it is modified as
little as necessary to guarantee constraint satisfaction, see e.g. [18, 19,
20]. A comprehensive overview of the application of various safe learning
methods in MPC can be found in [21] and the references therein.

A particular control task, which is considered in the present article, is
output-reference tracking with prescribed performance. To achieve this
objective, funnel MPC was introduced in [22] and further developed in [8, 23,
24]. Funnel MPC incorporates a particular choice of the stage cost inspired by
funnel control. The latter is a model-free adaptive high-gain feedback
controller, see e.g. [25] and the references therein. By penalizing the
distance of the tracking error to the prescribed error boundary via a “funnel-
like” stage cost, the funnel MPC scheme achieves output tracking for systems
with global strict relative degree one and bounded-input bounded-state stable
internal dynamics, where the input and output are of the same dimension.
Within this framework it was shown that the tracking objective is achieved and
funnel MPC is initially and recursively feasible, without incorporating state
constraints, terminal conditions, or the requirement of a sufficiently long
prediction horizon. Although not proven yet mathematically, numerical results
indicate that this control algorithm can successfully be applied to systems
with higher relative degree, cf. [8, 22], as well as Section 5.2 in the
present article. Extensions to arbitrary relative degree based on different
stage cost functions are discussed in [23, 24] and initial and recursive
feasibility is proved.

Many MPC schemes assume access to the full state of the system to be
controlled, however this is not satisfied in general. Therefore, in [3] a
distinction between the system to be controlled and the model to be used in
the MPC algorithm is made, and only access to the model’s state and to output
measurements (but no state measurements) of the real system is used to
initialize the model. In the present article we make use of that distinction,
too, see Section 3. A different approach is pursed in [26, 27] with
uncertain/disturbed linear discrete-time systems under consideration, where
the state is estimated using a Luenberger observer. Combining the observer
structure with a tube-based MPC scheme, robust constraint satisfaction and
feasibility of the control algorithm were shown.

Along the lines of the second research branch discussed above, namely to
update the model using measurement data from the system, we build on the
results of [3], where a two component controller was used, consisting of
funnel MPC in combination with a model-free funnel control feedback loop. We
extend this approach by introducing a general online learning framework in
order to continually improve the model utilizing the past system data, meaning
system output, past model-based predictions, and applied control signals, both
from the model-based and the model-free controller component. This not only
allows to learn and fine-tune parameters of an already detailed model, but it
is even possible to learn an unknown system without an initially given model.
Continually improving the model and thereby the prediction capability required
in MPC, the predictive funnel MPC control signal achieves the control
objective for the unknown plant.

The present article is organized as follows. In Section 2 we provide the
problem formulation. The control objective, as well as the controller
components are introduced. In Section 3 we formally introduce the system class
and the model class under study. Anticipating the later considerations, we
highlight that the actual system to be controlled, and the model of that
system can be quite different. E.g., it is possible to have a nonlinear system
and a linear surrogate model on which the MPC algorithm operates. Section 4
contains the main result of the article. We introduce the combined three-
component controller, establish the corresponding control 1, and formulate the
main result Theorem 4.2, which yields initial and recursive feasibility of the
proposed control algorithm. Furthermore, we present a particular learning
scheme, and rigorously prove its feasibility. In the last Section 5 we
illustrate the control 1 with two numerical simulations.

Nomenclature:
$\mathcal{B}_{\eta}:=\\{x\in\mathds{R}^{n}|\left\|x\right\|\leq\eta\\}$ is the
closed ball with radius $\eta>0$ around the origin in $\mathds{R}^{n}$. ${\rm
Gl}_{n}(\mathds{R})$ is the group of invertible $\mathds{R}^{n\times n}$
matrices. For an interval $I\subseteq\mathds{R}$,
$L^{\infty}(I,\mathds{R}^{n})$ is the Lebesgue space of measurable,
essentially bounded functions $f\colon I\to\mathds{R}^{n}$ with norm
$\|f\|_{\infty}=\operatorname*{ess\,sup}_{t\in I}\|f(t)\|$, and
$W^{k,\infty}(I,\mathds{R}^{n})$ is the Sobolev space of all functions
$f:I\to\mathds{R}^{n}$ with $k$-th order weak derivative $f^{(k)}$ and
$f,f^{(1)},\ldots,f^{(k)}\in L^{\infty}(I,\mathds{R}^{n})$; and $f|_{J}$
denotes the restriction of a function $f:I\to\mathds{R}^{n}$ to the interval
$J\subseteq I$.

##  2 Problem formulation

The overall task is output-reference tracking within prescribed bounds on the
tracking error. To this end, we apply a model predictive controller to achieve
a superior controller performance while maintaining input and output
constraints. However, since model-plant mismatches are unavoidable, we
safeguard the MPC controller by an additional component, i.e., the model-free
funnel controller, to guarantee satisfaction of the output-tracking criterion.
The third component, besides MPC and funnel control, is a learning one, which
repeatedly updates the model to reduce the model-plant mismatch and, thus,
improves the overall controller performance. A major challenge is to ensure
proper functioning of the interplay of these three components, which requires
a consistency condition (see Section 3) for the model updates – the key
novelty of our approach in comparison to the recently proposed robust funnel
MPC (which combines the first two components) in [3].

The control objective is output reference tracking within prescribed transient
error bounds. This means that, for an unknown control system of the form

$\dot{z}(t)\\!=\\!F(z(t),d(t))\\!+\\!G(z(t),d(t))u(t),\ \,y(t)\\!=\\!H(z(t)),$
(1)

where $z(t)\in\mathds{R}^{Z}$, bounded disturbance $d\in
L^{\infty}(\mathds{R}_{\geq 0},\mathds{R}^{D})$, and $y(t)\in\mathds{R}^{m}$
for $Z,D,m\in\mathds{N}$, we seek an input $u(t)\in\mathds{R}^{m}$ such that
the output $y$ tracks a given reference signal $y_{\rm ref}$. Specific
properties of the system parameters $F,G,H$ are introduced in Section 3.
Moreover, this tracking task is asked to be satisfied with a given precision,
i.e. the tracking error $e(t):=y(t)-y_{\rm ref}(t)$ should evolve within
(possibly time-varying) boundaries given by a so-called funnel function
$\psi$, prescribed by the user. To be precise, the tracking error shall evolve
within the funnel

$\mathcal{F}_{\psi}:=\left\\{\ \\!(t,e)\in\mathds{R}_{\geq
0}\times\mathds{R}^{m}\\!\ \left|\,\vphantom{\\!(t,e)\in\mathds{R}_{\geq
0}\times\mathds{R}^{m}\\!}\\!\left\|e\right\|<\psi(t)\\!\,\right.\right\\},$

which is determined by the choice of $\psi$ belonging to the set

$\mathcal{G}=\left\\{\ \\!\psi\in W^{1,\infty}(\mathds{R}_{\geq
0},\mathds{R})\\!\ \left|\,\vphantom{\\!\psi\in W^{1,\infty}(\mathds{R}_{\geq
0},\mathds{R})\\!}\text{inf}_{t\geq 0}\,\psi(t)>0\\!\,\right.\right\\}.$

Note that by $\psi\in\mathcal{G}$ the tracking error is not forced to converge
asymptotically to zero. For a function $y_{\rm ref}\in
W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$ we denote the set of all
functions which evolve in the performance funnel defined by
$\psi\in\mathcal{G}$ around $y_{\rm ref}$ by

$Y(y_{\rm ref},\\!\psi):=\left\\{\
\\!\\!y\\!\in\\!\mathcal{C}(\mathds{R}_{\geq 0},\mathds{R}^{m})\\!\\!\\!\
\left|\,\vphantom{\\!\\!y\\!\in\\!\mathcal{C}(\mathds{R}_{\geq
0},\mathds{R}^{m})\\!\\!\\!}\\!\\!\\!\
\forall\,t\\!\geq\\!\\!0:\\!\|y(t)\\!-\\!y_{\rm
ref}(t)\|\\!<\\!\psi(t)\\!\\!\,\right.\right\\}\\!.$

Robust funnel MPC. To solve the described problem for the unknown system (1),
robust funnel MPC was proposed in [3]. This control scheme consists of two
components, one model-based and one model-free, see Figure 1.

Model-based controller componentModel-free controller componentLearning
componentSystemFunnel Controller$+$$+$Funnel
MPC$y$$y_{\operatorname*{ref}}$Model$\begin{array}[]{c}\text{Machine}\\\
\hskip 15.6491pt\text{learning}\end{array}$$u_{\rm FMPC}$$u_{\rm FMPC}$$y_{\rm
M}$$y_{\rm M}$$y$$+$$-$$y$$y-y_{\rm M}$$u_{\rm FC}$$u=u_{\rm FMPC}+u_{\rm
FC}$$\begin{array}[]{l}\text{Model}\\\ \text{update}\end{array}$$u_{\rm FC}$

Figure 1: Design of the proposed three-component controller. The grey box
(containing the red (funnel MPC) and the blue (funnel control) structures)
represents the two-component controller _robust funnel MPC_ proposed in [3].
The green box represents the learning component, which receives the four
signals: system output $y$, model output $y_{\rm M}$, funnel MPC control
signal $u_{\rm FMPC}$, and funnel control signal $u_{\rm FC}$.

The model-based funnel MPC component (red box in Figure 1) uses a model of the
form

$\displaystyle\dot{x}(t)$ $\displaystyle=f(x(t))+g(x(t))u(t),\quad y_{\rm
M}(t)=h(x(t))$ (2)

where $x(t)\in\mathds{R}^{n}$ and $y_{\rm M}(t),u(t)\in\mathds{R}^{m}$, as an
approximation of the system (1). Specific properties of the model parameters
$f,g,h$ are introduced in Section 3. At time instances
$t_{k}\in\delta\mathds{N}_{0}$ with $\delta>0$, the current output $y_{\rm
M}(t_{k})$ of the model (2) is measured and predictions of the future model
behavior are computed over the next time interval of length $T=N\delta>0$,
$N\in\mathbb{N}$. Using the time-varying stage cost $\ell:\mathds{R}_{\geq
0}\times\mathds{R}^{m}\times\mathds{R}^{m}\to\mathds{R}_{\geq
0}\cup\\{\infty\\}$ where $\ell(t,y_{\rm M},u)$ is defined by

$\displaystyle\ell(t,y_{\rm M},u)=\begin{dcases}\frac{\left\|y_{\rm
M}-y_{\operatorname*{ref}}(t)\right\|^{2}}{\psi(t)^{2}-\left\|y_{\rm
M}-y_{\operatorname*{ref}}(t)\right\|^{2}}+\lambda_{u}\left\|u\right\|^{2},&\left\|y_{\rm
M}-y_{\operatorname*{ref}}(t)\right\|\neq\psi(t)\\\
\infty,&\text{else},\end{dcases}$ (3)

with design parameter $\lambda_{u}\geq 0$, a control signal $u_{\rm FMPC}\in
L^{\infty}([t_{k},t_{k}+T],\mathds{R}^{m})$ is computed as a solution of an
optimization problem. Additionally, the corresponding predicted output $y_{\rm
M}(\cdot;t_{k},x_{k},u_{\rm FMPC})={h(x(\cdot;t_{k},x_{k},u_{\rm FMPC}))}\in
L^{\infty}([t_{k},t_{k}+T],\mathds{R}^{m})$ is calculated. Here
$x(\cdot;t_{k},{x_{k}},u)$ denotes the unique solution of (2) with initial
condition $x(t_{k})=x_{k}$, which is well-defined for appropriate $f,g,h$ (the
model is specified in Definition 3.2) on the interval $[t_{k},t_{k}+T]$ for
appropriate control input $u$.

The model-free funnel control component (blue box in Figure 1) computes an
instantaneous control signal $u_{\rm FC}$ based on the deviation between the
output $y$ of system (1) and the funnel MPC-based predicted $y_{\rm M}$. The
sum $u(t)=u_{\rm FMPC}(t)+u_{\rm FC}(t)$ is then applied to the actual system
(1) at time instant $t$. The signal $u_{\rm FC}$ from the funnel controller is
used to compensate for occurring disturbances, uncertainties in the model (2)
and unmodelled dynamics. It is solely determined by the instantaneous values
of the system output $y$, the funnel function $\psi$, and the prediction
$y_{\rm M}$. Therefore, the model-free component cannot plan ahead. This may
result in large control values and a rapidly changing control signal if the
actual output significantly deviates from its predicted counterpart, where the
term _significant_ is to be understood in comparison to the current funnel
size. Numerical simulations in [8, 22] show that funnel MPC exhibits a
considerably better controller performance than pure funnel control. In order
to reduce the control effort of the funnel controller, a continuous
_activation function_ $\beta$ is incorporated and has the following effect:
whenever the deviation between predicted $y_{\rm M}$ and plant output $y$ is
acceptable (designer’s choice), only $u_{\rm FMPC}$ is applied. A reasonable
and simple choice for $\beta$ is a ReLU-like map, which is zero below a given
threshold and linear above.

Learning the model is the third component of the overall task addressed in the
present article. Since funnel MPC exhibits better controller performance, and
robust funnel MPC is able to compensate for model-plant mismatches, it is
desirable to improve the model so that, preferably, the control $u_{\rm FMPC}$
is sufficient to achieve the tracking task with prescribed performance for the
unknown system while satisfying the input constraints – in other words, it is
desirable that the funnel controller is inactive most of the time. In
Definition 3.2 we identify and establish properties of the learning component
such that learning and updating the model preserves the structure necessary
for robust funnel MPC [3]. The particular robustness w.r.t. model-plant
mismatches of robust funnel MPC even allows to start with “no model”, e.g.,
only an integrator chain, and then learn the remaining drift-dynamics. This is
considered in a numerical simulation in Section 5.

##  3 System class and model class

In this section we formally introduce the class of systems to be controlled as
well as the class of models to be used in the MPC part. We consider nonlinear
multi-input multi-output systems. Since in our later analysis we mainly refer
to the results of [3, 8], where the system is considered in so-called Byrnes-
Isidori normal form, we assume that the system (1) can be transformed (by a
transformation which does not need to be known) into input-output normal form

$\displaystyle\dot{y}(t)$
$\displaystyle\\!=\\!P(y(t),\zeta(t),d(t))\\!+\\!\Gamma(y(t),\zeta(t),d(t))u(t),$
$\displaystyle y(0)\\!=\\!y^{0}\\!\\!,$ (4a) $\displaystyle\dot{\zeta}(t)$
$\displaystyle\\!=\\!Q(y(t),\zeta(t),d(t)),$
$\displaystyle\zeta(0)\\!=\\!\zeta^{0}\\!\\!,$ (4b)

with control input $u:\mathds{R}_{\geq 0}\to\mathds{R}^{m}$, a bounded
internal or external disturbance $d:\mathds{R}_{\geq 0}\to\mathds{R}^{a}$, and
output $y:\mathds{R}_{\geq 0}\to\mathds{R}^{m}$. The system dynamics are
governed by _unknown_ nonlinear functions
$P\in\mathcal{C}(\mathds{R}^{m}\times\mathds{R}^{\kappa}\times\mathds{R}^{a},\mathds{R}^{m})$
and
$\Gamma\in\mathcal{C}(\mathds{R}^{m}\times\mathds{R}^{\kappa}\times\mathds{R}^{a},\mathds{R}^{m\times
m})$, where the latter is the so-called high-gain matrix. The last equation in
(4) with
$Q\in\mathcal{C}^{1}(\mathds{R}^{m}\times\mathds{R}^{\kappa}\times\mathds{R}^{a},\mathds{R}^{q})$
describes the internal dynamics, i.e. the dynamics within the system, which
cannot be directly measured at the output, with dimension
$\kappa:=Z-m\in\mathds{N}$ ($Z$ is the state dimension in (1)). Referring to
system (4) above, we formally introduce the class of systems under
consideration in the present article.

######  Definition 3.1.

For a given funnel function $\psi\in\mathcal{G}$ and reference $y_{\rm ref}\in
W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$ , we say that the system (4)
belongs to the system class $\mathcal{N}$, written
${(P,\Gamma,Q,d)}\in\mathcal{N}$, if $d\in L^{\infty}(\mathds{R}_{\geq
0},\mathds{R}^{a})$, the symmetric part of the high gain matrix $\Gamma$ is
positive definite, i.e. $\Gamma(\cdot)+\Gamma(\cdot)^{\top}>0$, and the
internal dynamics satisfy the following bounded-input bounded-state property

$\displaystyle\forall\,c_{0}>0\ \exists\,c_{1}>0\
\forall\,\zeta^{0}\in\mathds{R}^{\kappa}\ \forall\,d\in
L^{\infty}(\mathds{R}_{\geq 0},\mathds{R}^{a})$ $\displaystyle\forall\,y\in
Y(y_{\operatorname*{ref}},\psi):$
$\displaystyle\|\zeta^{0}\|+\|y\|_{\infty}+\|d\|_{\infty}\leq
c_{0}\Rightarrow\|\zeta(\cdot;0,\zeta^{0},y,d)\|_{\infty}\leq c_{1},$

where $\zeta(\cdot;0,\zeta^{0},y,d):\mathds{R}_{\geq 0}\to\mathds{R}^{\kappa}$
denotes the unique global solution of (4b).

We remark that the globality of the solution $\zeta$ is ensured by the imposed
conditions. Note that it is also possible to allow for a negative definite
matrix $\Gamma+\Gamma^{\top}$. Then we may simply change the sign in the
control (9) defined below. The last condition, namely bounded-input bounded-
state stability of the internal dynamics, is a common condition for control
systems, cf. [28, 29, 30]. In our particular case this ensures that the
internal states, which cannot be measured directly, are bounded if the output
of the system evolves within a funnel around the reference signal and the
disturbance is bounded. For linear systems the function $Q(\cdot)$ consists of
three matrices $Q_{\zeta}\in\mathds{R}^{\kappa\times\kappa}$,
$Q_{y}\in\mathds{R}^{\kappa\times m}$ and $Q_{d}\in\mathds{R}^{\kappa\times
a}$ ($\dot{\zeta}=Q_{\zeta}\zeta+Q_{y}y+Q_{d}d$), and the system satisfies the
bounded-input bounded-state property, if the matrix $Q_{\zeta}$ is Hurwitz
(all eigenvalues have negative real part). Such systems are called _minimum
phase_ , cf. [31, 32].

For the unknown system (4) we consider a surrogate model

$\displaystyle\dot{y}_{\rm M}(t)$ $\displaystyle=p(y_{\rm
M}(t),\eta(t))+\gamma(y_{\rm M}(t),\eta(t))u(t),$ $\displaystyle y_{\rm
M}(0)=y_{\rm M}^{0},$ (5a) $\displaystyle\dot{\eta}(t)$
$\displaystyle=q(y_{\rm M}(t),\eta(t)),$ $\displaystyle\ \
\,\eta(0)=\eta^{0},$ (5b)

where properties of the functions $p,\gamma$ and $q$ are specified in
Definition 3.2. Equation (5b) describes the internal dynamics. Referring to
(5), we introduce the class of feasible models. This class is parameterized by
the value $\bar{u}$, which ensures that for each member of the class the
tracking task can be performed successfully by robust funnel MPC for a given
reference signal and funnel boundary with the same upper bound $\bar{u}$ for
the maximal control value.

######  Definition 3.2.

Let $\bar{u}>0$. For given $\psi\in\mathcal{G}$ and $y_{\rm ref}\in
W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$ a model (5) is said to
belong to the model class $\mathcal{M}_{\bar{u}}$, written
$(p,\gamma,q,\eta^{0})\in\mathcal{M}_{\bar{u}}$, if there exist
$\nu\in\mathds{N}_{0}$ and $\bar{\eta}\geq 0$ such that the vector
$\eta^{0}\in\mathds{R}^{\nu}$ and the functions
$p\in\mathcal{C}^{1}(\mathds{R}^{m}\times\mathds{R}^{\nu},\mathds{R}^{m})$,
$\gamma\in\mathcal{C}^{1}(\mathds{R}^{m}\times\mathds{R}^{\nu},\mathds{R}^{m\times
m})$, and
$q\in\mathcal{C}^{1}(\mathds{R}^{m}\times\mathds{R}^{\nu},\mathds{R}^{\nu})$
satisfy the following conditions:

  1. (M.1)

$\gamma(\rho,\eta)\in{\rm Gl}_{m}(\mathds{R})$ for all
$(\rho,\eta)\in\mathds{R}^{m}\times\mathds{R}^{\nu}$, that is (5) has global
strict relative degree one,

  2. (M.2)

the solutions of the internal dynamics (5b) are bounded by $\bar{\eta}$, that
is

$\ \forall\,\eta^{0}\\!\in\\!\mathds{R}^{\nu}\ \forall\,y_{\rm
M}\\!\in\\!Y(y_{\operatorname*{ref}},\psi):\,\left\|\eta(\cdot;0,\eta^{0},y_{\rm
M})\right\|_{\infty}\\!\leq\\!\bar{\eta},$

where $\eta(\cdot;0,\eta^{0},y_{\rm M}):\mathds{R}_{\geq
0}\to\mathds{R}^{\nu}$ denotes the global solution of (5b),

  3. (M.3)

for the a-priori fixed maximal control input value $\bar{u}$ the model
parameters, reference signal and funnel function satisfy the estimate

$\displaystyle
G_{\max}\left(P_{\max}+\|{\dot{\psi}}\|_{\infty}+\left\|\dot{y}_{\operatorname*{ref}}\right\|_{\infty}\right)\leq\bar{u},$

where

$\displaystyle G_{\max}$
$\displaystyle=\max_{(\rho,\eta)\in\Psi\times\mathcal{B}_{\bar{\eta}}}\left\|\gamma(\rho,\eta)^{-1}\right\|,$
$\displaystyle P_{\max}$
$\displaystyle=\max_{(\rho,\eta)\in\Psi\times\mathcal{B}_{\bar{\eta}}}\\!\\!\left\|p(\rho,\eta)\right\|,$
$\displaystyle\Psi$ $\displaystyle=\bigcup_{t\in\mathds{R}_{\geq
0}}\\{\rho\in\mathds{R}^{m}|\left\|\rho-
y_{\operatorname*{ref}}(t)\right\|\leq\psi(t)\\}.$

Note that, although present in (5), the initial value $y_{\rm M}^{0}$ is not
part of the tuple $(p,\gamma,q,\eta^{0})\in\mathcal{M}_{\bar{u}}$. The reason
is that initializing the model output $y_{\rm M}(t_{k})$ with the measured
system output $y(t_{k})$ at time instances $t_{k}$ is part of 1.

Condition (M.2) seems quite technical at first glace. It’s intention is to
ensure the following. If the internal state is bounded at every beginning of
the MPC cycle, and the model output evolves within the funnel boundaries
around the reference signal, then the internal dynamics evolve within an
a-priori-fixed compact set. This condition is used in the feasibility proof of
the funnel MPC algorithm, cf. [8, Lem. 4.8, Prop. 4.9], and also for the
robust funnel MPC algorithm, cf. [3, Prop. 5.1, Thm. 3.13]. Condition (M.3)
ensures that the a-priori prescribed maximal control effort $\bar{u}$ is
sufficient to keep the tracking error within the funnel boundaries. Since
$\bar{u}$ is fixed in advance, the models determined by the learning component
must be amenable to the achievement of the tracking objective under control
constraints $\|u\|_{\infty}\leq\bar{u}$.

There is some difference between the two classes $\mathcal{N}$ and
$\mathcal{M}_{\bar{u}}$. Under certain technical assumptions, cf. [33, Cor.
5.7], system (4) as well as model (5) may be obtained via a transformation
from a state-space representation (1) or (2), respectively. To avoid these
technicalities, we restrict ourselves to the input-output representations (4)
and (5). The system class $\mathcal{N}$ encompasses systems with bounded
external or internal disturbances $d\in L^{\infty}(\mathds{R}_{\geq
0},\mathds{R}^{a})$, while no such disturbances are allowed in the model class
$\mathcal{M}_{\bar{u}}$. The high-gain matrix $\Gamma$ of the system is
required to have a positive definite symmetric part; in contrast, the high-
gain matrix $\gamma$ in the model is only required to be invertible.

######  Remark 3.3.

The dimension $\kappa$ of the internal dynamics of system (4) is unknown but
fixed. In contrast, the dimension $\nu\in\mathds{N}_{0}$ of the model’s
internal dynamics can be considered as a parameter in the learning step. This
means, in order to improve the model such that it “explains” the system
measurements, the dimension of the internal state can be varied. Note that
$\nu=0$ (no internal dynamics) is explicitly allowed for the model.

######  Remark 3.4.

One way to verify satisfaction of (M.2) is to apply [34, Thm. 4.3], which
states the following. If there exists
$V\in\mathcal{C}^{1}(\mathds{R}^{\nu},\mathds{R}_{\geq 0})$ such that
$V(\eta)\to\infty$ as $\|\eta\|\to\infty$, and for
$q\in\mathcal{C}(\mathds{R}^{\nu}\times\mathds{R}^{m},\mathds{R}^{\nu})$ we
have $V^{\prime}(\eta)\cdot q(\eta,\rho)\leq 0$ for all
$\rho\in\mathds{R}^{m}$ with $\|\rho\|\leq\|\psi\|_{\infty}+\|y_{\rm
ref}\|_{\infty}$ and all $\eta\in\mathds{R}^{\nu}$ with $\|\eta\|>\alpha$ for
a predefined $\alpha>0$. Then $\|\eta(t;0,\eta^{0},y_{\rm
M})\|_{\infty}\leq\max\\{\|\eta^{0}\|,\alpha\\}$ for all $t\geq 0$,
$\eta^{0}\in\mathds{R}^{\nu}$ and all $y_{\rm M}\in
Y(y_{\operatorname*{ref}},\psi)$. Hence, fixing $V(\cdot)$ and $\alpha>0$ in
advance can be used to restrict choices of $q(\cdot)$ satisfying (M.2). We
will make use of this fact later in Sections 4.2 and 4.3, where we discuss a
particular learning scheme.

##  4 Learning-based robust funnel MPC

In this section we develop the control algorithm and present our main result.
First, we establish the control methodology to achieve the control objective
introduced in Section 2. For the sake of completeness and readability, we
recall the robust funnel MPC algorithm proposed in [3, Alg. 3.7], adapted to
our framework. Before, we define an abstract learning scheme $\mathcal{L}$,
which is incorporated in the algorithm. The idea of the learning component is
to use measurement data from the model output $y_{\rm M}$, the system output
$y$, the funnel MPC signal $u_{\rm FMPC}$ and the funnel control signal
$u_{\rm FC}$ to improve the model used for computation of $u_{\rm FMPC}$ in
every MPC sampling interval (cf. Figure 1), under the condition that the
updated model is still a member of the model class $\mathcal{M}_{\bar{u}}$
given in Definition 3.2.

For example, let $(y_{\rm M},y,u_{\rm FMPC},u_{\rm FC})|_{I_{0}^{k}}$ be given
data collected up to ${t=t_{k}}$ over the interval $I_{0}^{k}:=[0,t_{k}]$.
This data is then used to choose
$(p,\gamma,q,\eta^{0})\in\mathcal{M}_{\bar{u}}$, i.e.,

$\mathcal{L}\big{(}(y_{\rm M},y,u_{\rm FMPC},u_{\rm
FC})|_{I_{0}^{k}}\big{)}=(p,\gamma,q,\eta^{0})\in\mathcal{M}_{\bar{u}}.$

The function $\mathcal{L}$ receives the data, which are available up to the
time instant $t\in\mathds{R}_{\geq 0}$, and maps it to suitable model
functions $(p,\gamma,q,\eta^{0})\in\mathcal{M}_{\bar{u}}$. Note that
$\mathcal{L}$ does not necessarily use all available data, see the discussion
in Section 4.3. Since all four signal types, the outputs $y$ and $y_{\rm M}$
and the control signals $u_{\rm FMPC}$ and $u_{\rm FC}$, are bounded, the
space of data is $L^{\infty}$; moreover, all of these four signals have the
same dimension $m\in\mathds{N}$. This motivates the following definition.

######  Definition 4.1.

For $\bar{u}>0$, we call a function $\mathcal{L}:\bigcup_{t\geq
0}L^{\infty}([0,t],\mathds{R}^{m})^{4}\to\mathcal{M}_{\bar{u}}$ a feasible
learning scheme for robust funnel MPC with respect to the model class
$\mathcal{M}_{\bar{u}}$, in short notation $\mathcal{M}_{\bar{u}}$-feasible.

###  4.1 Control algorithm and main result

Now, we summarize the reasoning so far in the following algorithm, which
achieves the control objective formulated in Section 2. It is a modification
of [3, Alg. 3.7]. Here, the proper re-initialization of the model at every
time step done in [3] is substituted by the learning component $\mathcal{L}$.
Concerning notation, for an input $u\in
L^{\infty}([t_{k},t_{k}+T],\mathds{R}^{m})$, we denote by $y_{\rm
M}(\cdot;t_{k},y_{k}^{0},\eta_{k}^{0},u)$ the first component of the unique
maximal solution of (5) under the initial condition $(y_{\rm
M}(t_{k}),\eta(t_{k}))=(y_{k}^{0},\eta_{k}^{0})\in\mathds{R}^{m+\nu}$.

######  Algorithm 1 (Learning-based robust funnel MPC).

Input: instantaneous measurements of the output $y(t)$ of system (4),
reference signal $y_{\operatorname*{ref}}\in W^{1,\infty}(\mathds{R}_{\geq
0},\mathds{R}^{m})$, funnel function $\psi\in\mathcal{G}$, an initial model
$(p_{0},\gamma_{0},q_{0},\eta_{0}^{0})\in\mathcal{M}_{\bar{u}}$, and an
$\mathcal{M}_{\bar{u}}$-feasible learning scheme $\mathcal{L}$.
Initialisation: Set time shift $\delta>0$, prediction horizon $T\geq\delta$,
and index $k:=0$. Define the time sequence $(t_{k})_{k\in\mathds{N}_{0}}$ by
$t_{k}:=k\delta$.
Steps:

  1. (a)

Initialize the model (5) given by
$(p_{k},\gamma_{k},q_{k},\eta_{k}^{0})\in\mathcal{M}_{\bar{u}}$ with the data
$(y_{\rm M}(t_{k}),\eta(t_{k}))=(y(t_{k}),\eta_{k}^{0})$.

  2. (b)

Funnel MPC
For $\ell$ as in (3) compute a solution $u_{\rm FMPC}$ of the optimal control
problem

$\mathop{\operatorname{minimize}}_{\begin{subarray}{c}u\in
L^{\infty}(I^{k},\mathds{R}^{m}),\\\
\left\|u\right\|_{\infty}\leq\bar{u}\end{subarray}}\int_{I^{k}}\ell(t,y_{\rm
M}(t;t_{k},{y(t_{k})},\eta_{k}^{0},u),u(t))\,\mathrm{d}t,$ (6)

over the interval $I^{k}:=[t_{k},t_{k}+T]$. Predict the output $y_{\rm
M}(t)=y_{\rm M}(t;t_{k},{y(t_{k})},\eta_{k}^{0},u_{\rm FMPC})$ of the model
(5) on the interval $[t_{k},t_{k+1}]$ and define the adaptive funnel
$\varphi:[t_{k},t_{k+1}]\to\mathds{R}_{>0}$ by

$\varphi(t):=\psi(t)-\|y_{\rm M}(t)-y_{\operatorname*{ref}}(t)\|.$ (7)

  3. (c)

Funnel control
Define the funnel control law with $y_{\rm M}|_{[t_{k},t_{k+1}]}$ and funnel
$\varphi$ as in (7) by

$u_{\rm FC}(t,y(t)):=-\beta\left(\tfrac{\|y(t)-y_{\rm
M}(t)\|}{\varphi(t)}\right)\tfrac{\varphi(t)(y(t)-y_{\rm
M}(t))}{\varphi(t)^{2}-\|y(t)-y_{\rm M}(t)\|^{2}},$ (8)

for $t\in[t_{k},t_{k+1})$, where the activation function
$\beta\in\mathcal{C}([0,1],[0,\beta^{+}])$ with $\beta^{+}>0$ is such that
$\beta(1)=\beta^{+}$. Apply the feedback control
$\mu:[t_{k},t_{k+1})\times\mathds{R}^{m}\to\mathds{R}^{m}$ to system (4),
given by

$\mu(t,y(t))=u_{\rm FMPC}(t)+u_{\rm FC}(t,y(t)).$ (9)

  4. (d)

Continual learning
Increment $k$ by $1$, find a feasible model for the next sampling interval

$\\!\mathcal{L}\big{(}(y_{\rm M},y,u_{\rm FMPC},u_{\rm
FC})|_{I_{0}^{k}}\big{)}=(p_{k},\gamma_{k},q_{k},\eta_{k}^{0})$

with $I_{0}^{k}:=[0,t_{k}]$. Then go to Step (a).

Now we are in the position to formulate the main result of the present
article, which extends [3, Thm. 3.13] by the learning component.

######  Theorem 4.2.

Consider a system (4) with ${(P,\Gamma,Q,d)}\in\mathcal{N}$ and initial values
$y^{0}\in\mathds{R}^{m}$ and $\zeta^{0}\in\mathds{R}^{\kappa}$. For given
${\bar{u}>0}$, choose an initial model (5) with
$(p_{0},\gamma_{0},q_{0},\eta_{0}^{0})\in\mathcal{M}_{\bar{u}}$. Let a
reference signal $y_{\operatorname*{ref}}\in W^{1,\infty}(\mathds{R}_{\geq
0},\mathds{R}^{m})$ and a funnel function $\psi\in\mathcal{G}$ be given such
that $\|y^{0}-y_{\rm ref}(0)\|<\psi(0)$. Further, let $\mathcal{L}$ be an
$\mathcal{M}_{\bar{u}}$-feasible learning scheme. Then, the learning-based
robust funnel MPC 1 with $\delta>0$ and $T\geq\delta$ is initially and
recursively feasible, i.e., at every time instance $t_{k}:=k\delta$ for
$k\in\mathds{N}_{0}$ the OCP (6) has a solution $u_{k}^{*}\in
L^{\infty}([t_{k},t_{k}+T],\mathds{R}^{m})$, and the closed-loop system
consisting of system (4) and the feedback (9) has a global solution
$(y,\zeta):[0,\infty)\to\mathds{R}^{m}\times\mathds{R}^{\kappa}$. In
particular, each global solution $(y,\zeta)$ satisfies:

  1. (i)

all signals are bounded, in particular, we have $u_{\rm FMPC},u_{\rm FC},y\in
L^{\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$,

  2. (ii)

the tracking error $y-y_{\rm ref}$ evolves within prescribed boundaries, i.e.,

$\forall\,t\geq 0\,:\ \|y(t)-y_{\rm ref}(t)\|<\psi(t).$

###### Proof.

To begin with, we show initial feasibility of 1. By construction of
$\mathcal{M}_{\bar{u}}$, and since $\|y^{0}-y_{\rm ref}(0)\|<\psi(0)$,
invoking [8, Prop. 4.8] there exists a control $u\in
L^{\infty}([0,T],\mathds{R}^{m})$ with $\|u\|_{\infty}\leq\bar{u}$ such that
$\|y_{\rm M}(t;0,y^{0},\eta_{0}^{0},u)-y_{\rm ref}(t)\|<\psi(t)$ for all
$t\in[0,T]$. Then [8, Thm. 4.5] yields that the OCP (6) has a solution.
Moreover, $\varphi(t)=\psi(t)-\|y_{\rm M}(t)-y_{\rm ref}(t)\|>0$ for all
$t\in[0,\delta]\subset[0,T]$. Therefore, the closed-loop system (4), (8) has a
solution on $[0,\delta]$ according to [35, Thm. 7]. Then the result [3, Thm.
3.13] is applicable and yields the existence of a solution of the closed-loop
system (4) and (9) on $[0,\delta]$, where the initialization strategy in [3]
is substituted by the initial choice of the model in 1. Recursive feasibility
can then be obtained as follows. We aim to apply [3, Prop. 5.1], which yields
that the OCP in (6) has a solution for all $k\in\mathds{N}$, if $y_{\rm ref}$
and $\psi$ are defined globally, and the bound $\bar{u}>0$ is sufficiently
large for the actual model. The first two conditions are satisfied since
$y_{\rm ref}\in W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$ and
$\psi\in\mathcal{G}$, and the last since the learning scheme is chosen to be
$\mathcal{M}_{\bar{u}}$-feasible.

Next we have to take care about the possible induced discontinuities in the
function $\varphi$ in (7). These discontinuities, however, occur in both cases
either by re-initialization as in [3], or due to the learning step in 1.
Therefore, the existence of a global solution of the closed loop system (4)
and (9) follows with the same reasoning as in [3, Thm. 3.13]. Namely,
$\|y(t_{k})-y_{\rm ref}(t_{k})\|<\varphi(t_{k})$ is satisfied for each
$k\in\mathds{N}$ by the previous argument. Then, [35, Thm. 7] yields the
existence of a solution of the closed-loop system (4), (8) on
$[t_{k},t_{k}+\delta]$ for all $k\in\mathds{N}$. This also yields assertion
(i). Assertion (ii) follows by the estimation $\|y(t)-y_{\rm
ref}(t)\|=\|y(t)-y_{\rm M}(t)+(y_{\rm M}(t)-y_{\rm ref}(t))\|<\psi(t)-\|y_{\rm
M}(t)-y_{\rm ref}(t)\|+\|y_{\rm M}(t)-y_{\rm ref}(t)\|=\psi(t)$ for all $t\geq
0$. ∎

###  4.2 Learning scheme

In this section we derive sufficient conditions on the parameters of models to
be learned, which then guarantee that a respective model is contained in
$\mathcal{M}_{\bar{u}}$ for a fixed $\bar{u}$. Later, we invoke these
conditions to discuss the $\mathcal{M}_{\bar{u}}$-feasibility of various
learning programs. Since in many applications a linear model may serve as a
good prediction model, we derive sufficient conditions on the parameters of
linear systems

$\displaystyle\dot{y}_{\rm M}(t)$ $\displaystyle=Ry_{\rm
M}(t)+S\eta(t)+D_{1}+\gamma u(t),$ $\displaystyle y_{\rm M}(0)$
$\displaystyle=y_{\rm M}^{0},$ (10) $\displaystyle\dot{\eta}(t)$
$\displaystyle=Q\eta(t)+Py_{\rm M}(t)+D_{2},$ $\displaystyle\eta(0)$
$\displaystyle=\eta^{0}.$

In the following, we denote by $\lambda_{Q}^{+}<0$ the largest eigenvalue of a
symmetric negative definite matrix $Q=Q^{\top}<0$. For a given reference
signal $y_{\rm ref}\in W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$ and
funnel boundary $\psi\in\mathcal{G}$ let
$\bar{\rho}:=\max_{\rho\in\Psi}\left\|\rho\right\|$ for the set $\Psi$ defined
in (M.3). Furthermore, for given numbers
$\bar{\eta},\bar{u},\bar{r},\bar{s},\bar{\gamma},\bar{p}\geq 0$ we define the
following set of matrices, where we do not indicate the dependence on the
parameters. Let

$\bar{\mathcal{K}}\\!:=\\!\mathds{R}^{m\times
m}\\!\times\mathds{R}^{m\times\nu}\\!\times{\rm
Gl}_{m}(\mathds{R})\\!\times\mathds{R}^{m}\\!\times\mathds{R}^{\nu\times\nu}\\!\times\mathds{R}^{\nu\times
m}\\!\times\mathds{R}^{\nu}\\!\times{\mathds{R}^{\nu}},$

and define

$\mathcal{K}:=\left\\{\
(R,S,\gamma,D_{1},Q,P,D_{2},\eta^{0})\in\bar{\mathcal{K}}\
\left|\,\vphantom{(R,S,\gamma,D_{1},Q,P,D_{2},\eta^{0})\in\bar{\mathcal{K}}}\eqref{eq:ConditionsMatrices}\,\right.\right\\},$
(11)

where

$\displaystyle\|R\|$ $\displaystyle\leq\bar{r},$ (12a) $\displaystyle\|S\|$
$\displaystyle\leq\bar{s},$ (12b) $\displaystyle\|\gamma^{-1}\|$
$\displaystyle\leq\bar{\gamma},$ (12c)
$\displaystyle\bar{r}\bar{\rho}+s\bar{\eta}+\|D_{1}\|$
$\displaystyle\leq\frac{\bar{u}}{\bar{\gamma}}-\|\dot{\psi}\|_{\infty}-\|\dot{y}_{\rm
ref}\|_{\infty},$ (12d) $\displaystyle\lambda_{Q}^{+}$
$\displaystyle\leq-\frac{\bar{p}\bar{\rho}+\|D_{2}\|}{\bar{\eta}},$ (12e)
$\displaystyle\|P\|$ $\displaystyle\leq\bar{p},$ (12f)
$\displaystyle\|\eta^{0}\|$ $\displaystyle\leq\bar{\eta}.$ (12g)

For $(R,S,\gamma,D_{1},Q,P,D_{2},\eta^{0})\in\bar{\mathcal{K}}$ we define the
functions

$\displaystyle p_{R,S,D_{1}}:\mathds{R}^{m}\times\mathds{R}^{\nu}$
$\displaystyle\to\mathds{R}^{m},$ $\displaystyle(y,\eta)$
$\displaystyle\mapsto Ry+S\eta+D_{1},$ (13) $\displaystyle
q_{Q,P,D_{2}}:\mathds{R}^{m}\times\mathds{R}^{\nu}$
$\displaystyle\to\mathds{R}^{\nu},$ $\displaystyle(y,\eta)$
$\displaystyle\mapsto Py+Q\eta+D_{2}.$

Then, we may derive the following statement.

######  Proposition 4.3.

Let $y_{\rm ref}\in W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$ and
$\psi\in\mathcal{G}$ be given and
$\bar{\rho}:=\max_{\rho\in\Psi}\left\|\rho\right\|$. Moreover, let parameters
$\bar{\eta},\bar{u},\bar{r},\bar{s},\bar{\gamma},\bar{p}\geq 0$ be given. Then
the set $\mathcal{K}$ defined in (11) satisfies the implication

$(R,S,\gamma,D_{1},Q,P,D_{2},\eta^{0})\in\mathcal{K}\implies(p_{R,S,D_{1}},\gamma,q_{Q,P,D_{2}},\eta^{0})\in\mathcal{M}_{\bar{u}},$

where $p_{R,S,D_{1}}$ and $q_{Q,P,D_{2}}$ are given in (13).

###### Proof.

The proof consists of three parts, with one part devoted to each condition
(M.1) – (M.3) in Definition 3.2, respectively.
_Step one(M.1)._ The choice $\gamma\in{\rm Gl}_{m}(\mathds{R})$ trivially
satisfies condition (M.1).
_Step two(M.2)._ We show for the given parameter ${\bar{\eta}\geq 0}$, that
for any tuple of parameters contained in $\mathcal{K}$, the solution of (10)
satisfies $\|\eta(t;0,\eta^{0},y_{\rm M})\|\leq\bar{\eta}$ for all $t\geq 0$
and all $y_{\rm M}\in Y(y_{\rm ref},\psi)$. Note that any such $y_{\rm M}$
satisfies $\|y_{\rm M}\|_{\infty}\leq\bar{\rho}$. In virtue of Remark 3.4 we
calculate for $t\geq 0$ that

$\displaystyle\tfrac{\text{d}}{\text{d}t}\tfrac{1}{2}\|\eta(t;0,\eta^{0},y_{\rm
M})\|^{2}$ $\displaystyle=\eta(t;0,\eta^{0},y_{\rm
M})^{\top}\Big{(}Q\eta(t;0,\eta^{0},y_{\rm M})+Py_{\rm M}(t)+D_{2}\Big{)}$
$\displaystyle\leq\|\eta(t;0,\eta^{0},y_{\rm
M})\|\Big{(}\lambda_{Q}^{+}\|\eta(t;0,\eta^{0},y_{\rm
M})\|+\bar{p}\bar{\rho}+\|D_{2}\|\Big{)},$

which by $\lambda_{Q}^{+}<0$ is non-positive for $\|\eta(t;\eta^{0},y_{\rm
M})\|\geq(\bar{p}\bar{\rho}+\|D_{2}\|)/|\lambda_{Q}^{+}|$. Therefore, [34,
Thm. 4.3] yields

$\ \forall\,t\geq 0:\ \|\eta(t;0,\eta^{0},y_{\rm
M})\|\leq\max\\{(\bar{p}\bar{\rho}+\|D_{2}\|)/|\lambda_{Q}^{+}|,\|\eta^{0}\|\\}.$

By assumption (12) we have $\|\eta^{0}\|\leq\bar{\eta}$ and
$|\lambda_{Q}^{+}|\geq(\bar{p}\bar{\rho}+\|D_{2}\|)/\bar{\eta}$, hence it
follows $\|\eta(t;0,\eta^{0},y_{\rm M})\|\leq\bar{\eta}$ for all $t\geq 0$.
_Step three(M.3)._ We may estimate

$\displaystyle G_{\max}$
$\displaystyle=\left\|\gamma^{-1}\right\|\leq\bar{\gamma},$ $\displaystyle
P_{\max}$
$\displaystyle=\max_{(\rho,\eta)\in\Psi\times\mathcal{B}_{\bar{\eta}}}\left\|p_{R,S,D_{1}}(\rho,\eta)\right\|\leq\bar{r}\bar{\rho}+\bar{s}\bar{\eta}+\|D_{1}\|,$

and hence

$\displaystyle
G_{\max}\\!\left(P_{\max}+\|{\dot{\psi}}\|_{\infty}+\left\|\dot{y}_{\operatorname*{ref}}\right\|_{\infty}\\!\right)$
$\displaystyle\leq\bar{\gamma}\Big{(}\bar{r}\bar{\rho}+\bar{s}\bar{\eta}+\|D_{1}\|+\|{\dot{\psi}}\|_{\infty}+\left\|\dot{y}_{\operatorname*{ref}}\right\|_{\infty}\Big{)}\stackrel{{\scriptstyle\eqref{eq:CondUSatisfied}}}{{\leq}}\bar{u}.$

This completes the proof. ∎

###  4.3 A particular learning scheme

With the set of parameters $\mathcal{K}$, the functions
$p_{R,S,D_{1}},q_{Q,P,D_{2}}$ defined in (13), and Proposition 4.3, we may
define a learning scheme $\mathcal{L}$ mapping from $\bigcup_{t\geq
0}L^{\infty}([0,t],\mathds{R}^{m})^{4}$ to the subset

$\left\\{\ \\!(p_{R,S,D_{1}},\gamma,q_{Q,P,D_{2}},\eta^{0})\\!\
\left|\,\vphantom{\\!(p_{R,S,D_{1}},\gamma,q_{Q,P,D_{2}},\eta^{0})\\!}(R,S,\gamma,D_{1},Q,P,D_{2},\eta^{0})\\!\in\\!\mathcal{K}\\!\,\right.\right\\}$

of $\mathcal{M}_{\bar{u}}$, defined by

$\displaystyle\mathcal{L}:\ $ $\displaystyle((y_{\rm M},y,u_{\rm FMPC},u_{\rm
FC})|_{[0,t]})\mapsto(p_{R,S,D_{1}},\gamma,q_{Q,P,D_{2}},\eta^{0})$

for some $t\geq 0$, where $(p_{R,S,D_{1}},\gamma,q_{Q,P,D_{2}},\eta^{0})$ is
determined by the solution of an optimization problem involving discrete
measurements of the system data $y$ and the applied control signals $u_{\rm
FMPC}$ and $u_{\rm FC}$ at time instances $i\tau$ with $\tau>0$,
$i\in\mathds{N}$ and $i\tau\leq t$ of the form

$\displaystyle\mathop{\operatorname{minimize}}_{\begin{subarray}{c}(R,S,\gamma,D_{1},Q,P,D_{2},\eta^{0})\in\mathcal{K}\end{subarray}}\
J((y,z)|_{[0,t]})$ (14) s.t.
$\displaystyle{\makebox[0.0pt]{$z(0)$}}\\!=\\!z^{0}\text{and for all }i\leq
t/\tau:$ $\displaystyle z(i\tau)\\!=\\!\chi(\tau;z((i-1)\tau),(u_{\rm
FMPC}+u_{\rm FC})((i-1)\tau)),$

where $J(\cdot)$ is a suitable cost function, $z=(\tilde{y}_{\rm M},\eta)$
denotes the states of the linear model (10), and the expression
$\chi(\cdot;z((i-1)\tau),(u_{\rm FMPC}+u_{\rm FC})((i-1)\tau))$ denotes its
solution under the initial condition $\chi(0)=z((i-1)\tau)$ and with constant
control $u(\cdot)\equiv(u_{\rm FMPC}+u_{\rm FC})((i-1)\tau)$. In the following
we discuss some possible choices for the cost function $J(\cdot)$.

  1. (i)

$J((y,z)|_{[0,t]}):=\sum_{i=0}^{\lfloor t/\tau\rfloor}\xi_{i}\|\tilde{y}_{\rm
M}(i\tau)-y(i\tau)\|^{2}$ with weights $\xi_{i}\geq 0$. The idea is to find a
model in the set $\mathcal{K}$ which minimizes the weighted squared measured
output errors. The weights $\xi_{i}$ reflect the relative importance of the
measurements $y(i\tau)$. In certain cases it might be beneficial to weight
data points that are far in the past lower than current data points. By
choosing $\xi_{i}>0$ for all $i>0$ all measured past data is taken into
account. With increasing runtime of the algorithm, this results in a growing
complexity of the optimization problem, computation time, and required memory
space for the measurements. Therefore, this is not suitable in practice. Thus,
it is beneficial to use a moving horizon estimation approach and only take the
last $N$ measurements into account and set $\xi_{i}=0$ for $i<\lfloor
t/\tau\rfloor-N$. In application, one has to find a good balance between
considering many data points (large $N$), thus having a probably more accurate
model, and low computation time and memory requirements (small $N$).

  2. (ii)

If the computation of the solution of the optimization problem has to be done
very quickly, it is also possible to only consider the last measurement
$y(\lfloor t/\tau\rfloor\tau)$. Thus, one might choose the cost function
$J((y,z)|_{[0,t]}):=\|\tilde{y}_{\rm M}(\lfloor t/\tau\rfloor\tau)-y(\lfloor
t/\tau\rfloor\tau)\|^{2}$. The idea is to find a model, which best explains
the last MPC period in terms of output error, i.e., a model on the prediction
interval $[t_{k},t_{k+1}]$, so that with $\tau=\delta$ the error
$\|\tilde{y}_{\rm M}(t_{k+1})-y(t_{k+1})\|$ at the end of the interval is
minimal.

  3. (iii)

In addition, it is worth considering to include regularization terms for the
model parameters in the cost function. For the parameters
$\mathcal{K}_{i}=(R_{i},S_{i},\gamma_{i},D_{1,i},Q_{i},P_{i},D_{2,i}\eta^{0}_{i})\in\mathcal{K}$
one could either penalize the weighted distance of $\mathcal{K}_{i}$ to an a
priori known tuple of parameters
$\mathcal{K}^{\star}=(R^{\star},S^{\star},\gamma^{\star},D^{\star},Q^{\star},P^{\star},D_{2}^{\star},{\eta^{0}}^{\star})$
and thus allow only small adaptions of the a priori known model or penalize
the change of parameters $\mathcal{K}_{i}$ such that the model does only
change slightly between two learning steps. This results in a cost function of
the form $J((y,z)|_{[0,t]}):=\sum_{i=0}^{\lfloor
t/\tau\rfloor}\big{(}\xi_{i}\|\tilde{y}_{\rm
M}(i\tau)-y(i\tau)\|^{2}+\sum_{j=1}^{8}k_{i}^{j}\|(\mathcal{K}_{i}^{j}-\tilde{\mathcal{K}}^{j})\|\big{)}$,
where $\tilde{\mathcal{K}}=\mathcal{K}^{\star}$ or
$\tilde{\mathcal{K}}=\mathcal{K}_{i-1}$ and with weights
$\xi_{i},k_{i}^{j}\geq 0$. Here the expressions
$\mathcal{K}_{i}^{j},\tilde{\mathcal{K}}_{i}^{j}$ with $j=1,\ldots,8$ refer to
the $j^{\rm th}$ entry of the tuple $\mathcal{K}_{i}$,
$\tilde{\mathcal{K}}_{i}$, respectively; for instance,
$\mathcal{K}_{i}^{2}=S_{i}$.

##  5 Numerical simulation

In this section we provide two numerical simulations to illustrate 1. In the
first example, we simulate the tracking task for a system of relative degree
one, which is contained in the system class $\mathcal{N}$, and the
corresponding surrogate model belongs to $\mathcal{M}_{\bar{u}}$, for a given
$\bar{u}$. The second example goes beyond the system class $\mathcal{N}$, it
is a system of relative degree two. We show that, although no theoretical
results on its functioning are available yet, 1 is successful in this case.

###  5.1 Relative degree one.

To illustrate the application of 1, we consider a model of an exothermic
chemical reaction which was also used in [8] to study funnel MPC and in [3] to
study robust funnel MPC. The dynamics for one reactant $\zeta_{1}$, a product
$\zeta_{2}$, and reactor temperature $y$ is described by a system (4) where
$m=1$, $\kappa=2$, $a=0$, $\Gamma=1$, $P(y,\zeta)=b_{1}\alpha(y,\zeta)-b_{2}y$
and $Q(y,\zeta)=\left[\begin{smallmatrix}c_{1}&0\\\
0&c_{2}\end{smallmatrix}\right]\alpha(y,\zeta)+d(\zeta^{\rm in}-\zeta)$, with
the Arrhenius law $\alpha:\mathds{R}_{>0}\times\mathds{R}_{\geq
0}^{2}\to\mathds{R}_{\geq 0}$ given by
$\alpha(y,\zeta):=k_{0}\mathrm{e}^{-k_{1}/y}\zeta_{1}$, and parameters
$k_{0}$, $k_{1}$, $b_{1}$, $d$, $b_{2}>0$, $c_{1}<0$, $c_{2}\in\mathds{R}$,
$\zeta^{\rm in}\in\mathds{R}_{\geq 0}^{2}$. The control objective is to steer
the reactor’s temperature $y$ to a desired constant value
$y_{\operatorname*{ref}}\equiv\bar{y}$ within prescribed funnel boundaries
$\psi\in\mathcal{G}$, i.e., $\left\|y(t)-\bar{y}\right\|<\psi(t)$ for all
$t\geq 0$. For the funnel MPC component of 1 we consider linear models of the
form (10) with $R,D_{1}\in\mathds{R}$,
$S,D_{2}^{\top},P^{\top}\in\mathds{R}^{1\times 2}$, and
$Q\in\mathds{R}^{2\times 2}$. We assume $\gamma=1$ and as initial model we
choose $R=D_{1}=0\in\mathds{R}$,
$S=D_{2}^{\top}=P^{\top}=0\in\mathds{R}^{1\times 2}$,
$Q=0\in\mathds{R}^{2\times 2}$, and $\eta^{0}=(0.02,0.9)$. To improve this
model over time, we adapt the matrices over a compact set $\mathcal{K}$ as in
(11) at every fifth time step $t_{k}$ by minimizing the plant-model mismatch
based on the data of the last system output $y(t_{k-1})$, i.e., we solve the
optimization problem

$\displaystyle\\!\mathop{\operatorname{minimize}}_{\begin{subarray}{c}(R,S,1,D_{1},Q,P,D_{2},\zeta(0))\in\mathcal{K}\end{subarray}}\quad\left\|y_{\rm
M}(t_{k})-y(t_{k})\right\|^{2}$ s.t.
$\displaystyle\makebox[0.0pt]{$\tfrac{\text{d}}{\text{d}t}\left(\begin{smallmatrix}y_{\rm
M}(t)\\\ \eta(t)\end{smallmatrix}\right)$}=\left[\begin{smallmatrix}R&S\\\
P&Q\end{smallmatrix}\right]\left(\begin{smallmatrix}y_{\rm M}(t)\\\
\eta(t)\end{smallmatrix}\right)+\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]u(t)+\left[\begin{smallmatrix}D_{1}\\\
D_{2}\end{smallmatrix}\right]$
$\displaystyle\makebox[0.0pt]{$\left(\begin{smallmatrix}y_{\rm M}(t_{k-1})\\\
\eta(t_{k-1})\end{smallmatrix}\right)$}=\left(\begin{smallmatrix}y(t_{k-1})\\\
\zeta(0)\end{smallmatrix}\right),$

where $u(t)=u_{\rm FMPC}(t_{k-1})+u_{\rm FC}(t_{k-1})$ which was applied to
the model at the last time step $t_{k-1}$ and
$\zeta(0)=(\zeta_{1}(0),\zeta_{2}(0))$ is the vector of initial concentrations
of the substances $\zeta_{1}$ and $\zeta_{2}$.

For the simulation we choose the funnel function
$\psi(t)=100\mathrm{e}^{-2t}+1.5$. As in [3, 8], the initial data is
$(y^{0},\zeta_{1}(0),\zeta_{2}(0)):=(270,0.02,0.9)$, the reference signal is
$y_{\operatorname*{ref}}\equiv 337.1$, and the parameters are $c_{1}=-1$,
$c_{2}=1$, $k_{0}=\mathrm{e}^{25}$, $k_{1}=8700$, $d=1.1$, $b_{1}=209.2$,
$b_{2}=1.25$, $\zeta_{1}^{\rm in}=1$, $\zeta_{2}^{\rm in}=0$. In this example
we restrict the funnel MPC control to $\left\|u_{\rm
FMPC}\right\|_{\infty}\leq\bar{u}:=735$, choose $\lambda_{u}=10^{-4}$,
prediction horizon $T=1$, and time shift $\delta=0.1$. In accordance with
Proposition 4.3, we choose for the set $\mathcal{K}$ as in (11) the parameters
as $\bar{r}=1.3$, $\bar{s}=1.4$, $\bar{\eta}=0.91$, $\bar{\rho}=408.6$,
$\bar{\gamma}=1$, $\bar{p}=1/400$,
$\|D_{1}\|\leq\frac{\bar{u}}{\bar{\gamma}}-\|\dot{\psi}\|_{\infty}-\|\dot{y}_{\rm
ref}\|_{\infty}-\bar{r}\bar{\rho}-\bar{s}\bar{\eta}=2.546$, and
$\left\|D_{2}\right\|\leq 3$ so that (12) is satisfied. Thus, the learning
scheme is ${\mathcal{M}_{\bar{u}}}$–feasible. Due to discretisation, only step
functions with a constant step length of $0.1$ were considered to solve the
OCP (6). The activation function of the funnel controller is constant
$\beta\equiv 10$.

Figure 2: Application of learning based robust funnel MPC to the exothermic
chemical reactor system

.

Figure 2 shows the control signals and the system and model output errors,
respectively. It is evident that both $y_{\rm M}-y_{\operatorname*{ref}}$ and
$y-y_{\operatorname*{ref}}$ remain within the predefined funnel boundaries
$\psi$. Before the first learning step ($t\in[0,0.5)$) the tracking error
$y-y_{\operatorname*{ref}}$ and the predicted error $y_{\rm
M}-y_{\operatorname*{ref}}$ diverge due to the poor quality of the initial
model. However, since the tracking error is not close to the funnel boundary,
the funnel controller remains inactive in the beginning and only reacts when
the tracking error is close to the boundary. After the first learning step,
the general direction of the predicted tracking error is consistent with the
actual tracking error. The funnel controller still has to compensate for the
model inaccuracies in order to guarantee that the tracking error remains
within the boundaries, but with a significantly smaller contribution to the
control signal. After each learning step, the model output jumps to the system
output due to the newly updated model. The control signal $u_{\rm FC}$ is zero
after each learning step since the system and model output coincide, and it
becomes larger afterwards to compensate for the model inaccuracy. After
$t=1.5$ the system output is close to the desired constant reference signal.
Thenceforth, the linear model is adequate to predict the system behavior and
the control signal computed by funnel MPC is sufficient to achieve the
tracking objective. The funnel controller only has to slightly compensate for
model errors. We note that this example merely serves to illustrate that
robust funnel MPC can be combined with $\mathcal{M}_{\bar{u}}$–feasible
learning techniques. We do not claim that the learning algorithm used is
superior to other methods.

###  5.2 Second numerical example: higher relative degree.

In this section, we present some promising preliminary results on extending
the learning-based robust funnel MPC 1 to a larger system class. Numerical
simulations in [8] suggest that the funnel MPC algorithm is also applicable to
systems of higher relative degree, meaning systems of the form

$\displaystyle y^{(r)}(t)$
$\displaystyle=P(y(t),\dot{y}(t),\ldots,y^{(r-1)}(t),\zeta(t))+\Gamma(y(t),\dot{y}(t),\ldots,y^{(r-1)}(t),\zeta(t))u(t),$
$\displaystyle\dot{\zeta}(t)$
$\displaystyle=Q(y(t),\dot{y}(t),\ldots,y^{(r-1)}(t),\zeta(t)),$

with $r>1$. Restricting the class of admissible funnel functions, utilizing an
adapted cost function, and incorporating so-called feasibility constraints in
the optimization problem, feasibility of funnel MPC for this system class was
proved in [23]. It is an open problem whether it is sufficient to utilize the
far simpler optimization (6) with cost function (3) and without the mentioned
constraints. Accordingly, only the case of a relative degree one system was
considered for robust funnel MPC in [3]. A generalisation to systems with
higher relative degree has yet to be found.

To illustrate that, nevertheless, the learning-based robust funnel MPC 1 shows
promising results for this lager system class with fixed relative degree
$r>1$, we consider the example of a mass-on-car system from [36] which was was
also used in [8]. The mass $m_{2}$ is mounted on a car with mass $m_{1}$ via a
spring and damper system with spring constant $k>0$ and damper constant $d>0$
and moves on a ramp which is inclined by the angle
$\vartheta\in[0,\tfrac{\pi}{2})$. The car can be controlled via the force $u$
acting on it. The situation is depicted in Figure 3.

Figure 3: Mass-on-car system. The figure is based on the respective figures in
[25], and [36].

The system can be described by the equations

$\displaystyle\small\begin{bmatrix}m_{1}+m_{2}&m_{2}\cos(\vartheta)\\\
m_{2}\cos(\vartheta)&m_{2}\end{bmatrix}\begin{pmatrix}\ddot{z}(t)\\\
\ddot{s}(t)\end{pmatrix}\\!+\\!\begin{pmatrix}0\\\
ks(t)+d\dot{s}(t)\end{pmatrix}\\!=\\!\begin{pmatrix}u(t)\\\ 0\end{pmatrix}.$
(15)

The horizontal position of the car is $z(t)$ and the relative position of the
mass on the ramp at time $t$ is $s(t)$ . The output $y$ of the system is

$y(t)=z(t)+s(t)\cos(\vartheta),$

the horizontal position of the mass on the ramp. For the system we chose the
same parameters $m_{1}=4$, $m_{2}=1$, $k=2$, $d=1$, $\vartheta=\pi/4$, and
initial values $z(0)=s(0)=\dot{z}(0)=\dot{s}(0)=0$ as in [8]. Then, as shown
there, the system has a relative degree of $r=2$. The objective is tracking of
the reference signal $y_{\operatorname*{ref}}(t)=\cos(t)$ such that the
tracking error $y(t)-y_{\operatorname*{ref}}(t)$ evolves within the prescribed
performance funnel given by the function $\psi\in\mathcal{G}$ with
$\psi(t)=5\mathrm{e}^{-2t}+0.2$.

For the model-based component of the controller, we solve the optimal control
problem (6) with stage cost (3). The prediction horizon and time shift are
selected as $T=0.6$ and $\delta=0.06$, resp. We restrict the funnel MPC
control to $\left\|u_{\rm FMPC}\right\|_{\infty}\leq 25$ and choose the
parameter $\lambda_{u}=10^{-2}$ for the stage cost $\ell$. Due to
discretisation, only step functions with a constant step length of $0.06$ were
considered to solve the OCP (6). For the model-free component of the
controller, we use the funnel controller for systems with relative degree two
from [25] instead of (8). This component takes the form

$\displaystyle w(t)$ $\displaystyle=\tfrac{\dot{e}_{\rm
M}(t)}{\varphi(t)}+\alpha\left(\tfrac{e_{\rm
M}(t)^{2}}{\varphi(t)^{2}}\right)\tfrac{e_{\rm M}(t)}{\varphi(t)},$
$\displaystyle u_{\rm FC}(t)$ $\displaystyle=-\alpha(w(t)^{2})w(t),$

with $e_{\rm M}(t)=y_{\rm M}(t)-y(t)$ and $\alpha(s)=\frac{1}{1-s}$ for
$s\in[0,1)$. The controller is applied to the system with a step size
$0.6\cdot 10^{-5}$. Similar to [22], where this problem was studied in the
context of model identification during runtime, for the learning component we
assume some knowledge about the structure of the system, but only limited
information about the parameters and the initial value. We assume to know
$\vartheta\in(0,2\pi]$, $m_{1}\in[1,6]$, $m_{2}\in[0.5,1.5]$, $k\in[1,3]$,
$d\in[0.5,1.5]$ and
$z^{0}:=(x(0),\dot{x}(0),s(0),\dot{s}(0))\in[-2.5,3.5]\times[-2,2]\times[-2.75,3.25]\times[-2,2]$.
As initial model all model parameters where chosen equal to $1$ and the
initial state $z^{0}=(0,1,0,1)$. To learn the system parameters, we take
measurements of the input-output data $((u_{\rm FMPC}+u_{\rm
FC})(i\tau),y(i\tau))$ for $\tau=0.006$ and $i\in\mathds{N}_{0}$ and at every
time $jT=100j\tau$ for $j\in\mathds{N}$ we solve the optimization problem

$\displaystyle\\!\mathop{\operatorname{minimize}}_{\vartheta,m_{1},m_{2},k,d,z^{0}}\quad\sum_{i=0}^{100j}\left\|\tilde{y}_{\rm
M}(i\tau)-y(i\tau)\right\|^{2}$ s.t.
$\displaystyle\makebox[0.0pt]{$z(0)$}\\!=\\!z^{0}\text{ and for all
}i=1,\ldots,100j:$ $\displaystyle
z(i\tau)\\!=\\!\chi(\tau;z((i-1)\tau),(u_{\rm FMPC}+u_{\rm FC})((i-1)\tau)),$
$\displaystyle\tilde{y}_{\rm
M}(i\tau)\\!=\\![1,\cos(\vartheta),0,0]\,z(i\tau),$

where $z=(x,\dot{x},s,\dot{s})$ denotes the state of the mass on car system
(15) and $\chi(\cdot;z((i-1)\tau),(u_{\rm FMPC}+u_{\rm FC})((i-1)\tau))$
denotes its solution under the initial condition $\chi(0)=z((i-1)\tau)$ and
with constant control $u(\cdot)\equiv(u_{\rm FMPC}+u_{\rm FC})((i-1)\tau)$.
Since only interval $[0,4]$ is considered for the simulation, the entire
history of input-output data is considered in the optimization problem instead
of a moving horizon approach as discussed in Section 4.3 (i).

All simulations are performed on the time interval $[0,4]$ with Matlab and the
toolkit Casadi and are depicted in Figure 4.

Figure 4: Simulation of system (15) under learning-based robust funnel MPC 1.

It is evident that the control scheme is feasible and achieves the control
objective. Both errors $y_{\rm M}-y_{\operatorname*{ref}}$ and
$y-y_{\operatorname*{ref}}$ evolve within the funnel boundaries given by
$\psi$. Overall, a similar behavior as in the first simulation (initial
divergence, small funnel control signals afterwards, jumps at each learning
step, etc.) can be observed. We like to emphasize that already after the first
learning step, the quality of the model is apparently quite good such that the
funnel controller only has to slightly compensate for model errors and the
control signal mainly consists of the control $u_{\rm FMPC}$ generated by the
model-based controller component.

##  6 Conclusion

We extended the recently developed robust funnel MPC algorithm [3] by a
learning component in order to improve the model-based control component by
learning the model from system data. Based on the characterization of the
model class and corresponding feasible learning structures, we showed that the
interplay between pure feedback control, model predictive control and learning
is consistent and successful; in particular, the control objective of output
reference tracking with prescribed performance is achieved. Future research
will focus on several particular aspects of the presented approach. Questions
to be answered are, among other things, what it means to have a _good_
learning scheme; which existing techniques (Willem’s fundamental lemma [13],
Koopman framework [37, 38], Neural Networks, etc.) can be used to exploit the
data; how can knowledge about the system be incorporated into the learning
scheme; to name but three aspects. Of special interest is the question of
rigorously proving $\mathcal{M}_{\bar{u}}$-feasibility for more sophisticated
learning algorithms.

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