# Viability under Degraded Control Authority

Hamza El-Kebir1, Richard Berlin2, Joseph Bentsman3, and Melkior Ornik4 1H. El-
Kebir is with the Department of Aerospace Engineering, University of Illinois
Urbana-Champaign, Urbana, IL, 61801 USA<EMAIL_ADDRESS>Berlin is
with the Department of Trauma Surgery, Carle Hospital and the University of
Illinois Urbana-Champaign, Urbana, IL 61801 USA.3J. Bentsman is with the
Department of Mechanical Science and Engineering, University of Illinois
Urbana-Champaign, Urbana, IL, 61801 USA<EMAIL_ADDRESS>Ornik is with
the Department of Aerospace Engineering and the Coordinated Science
Laboratory, University of Illinois Urbana-Champaign, Urbana, IL, 61801 USA
<EMAIL_ADDRESS>reported in this publication was supported by the
National Institute of Biomedical Imaging and Bioengineering of the National
Institutes of Health under award number R01EB029766. The content is solely the
responsibility of the authors and does not necessarily represent the official
views of the National Institutes of Health.

###### Abstract

In this work, we solve the problem of quantifying and mitigating control
authority degradation in real time. Here, our target systems are controlled
nonlinear affine-in-control evolution equations with finite control input and
finite- or infinite-dimensional state. We consider two cases of control input
degradation: finitely many affine maps acting on unknown disjoint subsets of
the inputs and general Lipschitz continuous maps. These degradation modes are
encountered in practice due to actuator wear and tear, hard locks on actuator
ranges due to over-excitation, as well as more general changes in the control
allocation dynamics. We derive sufficient conditions for identifiability of
control authority degradation, and propose a novel real-time algorithm for
identifying or approximating control degradation modes. We demonstrate our
method on a nonlinear distributed parameter system, namely a one-dimensional
heat equation with a velocity-controlled moveable heat source, motivated by
autonomous energy-based surgery.

##  I Introduction

In control systems, fault detection and mitigation is key in ensuring
prolonged safe operation in safety-critical environments [1]. Any physical
system undergoes gradual degradation during its operational life cycle, for
instance due to interactions with the environment or from within as a result
of actuator wear and tear. Gradual degradation or impairment, as the name
suggests, often reduces the performance of a system in cases when potential
degradation modes were not taken into account during control synthesis. Fault
tolerance is a key property of systems that are capable of mitigating or
withstanding system faults, including gradual degradation.

A number of stochastic approaches to fault identification and mitigation have
been developed in the past, with the main objective of estimating the
_remaining useful life_ (RUL) of a system, and how this metric is influenced
by the controller. Mo and Xie [2] developed an approach to approximate the
loss in effectiveness cause by actuator component degradation using a
reliability value. Their method relies on frequency domain analysis using the
Laplace transform, which is limited to linear systems; in turn, proposed
reliability improvements hinge on the use of a PID controller strategy and
rely on a particle swarm optimization routine, which is highly restrictive
with regard to runtime constraints and convergence guarantees. A similar
approach was developed by Si _et al._ [3], where reliability was assessed
using an event-based Monte Carlo simulation approach, wherein potential
degradation modes are simulated _en masse_ , further limiting the
applicability of this method. This is due to the intractable number of
potential failure modes that may be encountered in practice, which would
demand a very large number of Monte Carlo simulations.

In the deterministic setting, Wang _et al._ [4] considered control input map
degradation and actuator saturation in discrete-time linear systems, where a
fault-tolerant control is developed by solving a constrained optimization
problem. Given the discrete-time linear system setting, [4] uses efficient
linear matrix inequality (LMI) techniques for controller synthesis. However,
the class of actuator degradations considered in [4] is limited to linear
diagonal control authority degradation with input saturation. In the context
of switching systems, Niu _et al._ [5] considered the problem of active mode
discrimination (AMD) with temporal logic-constrained switching, where a set of
known switching modes was known _a priori_. The AMD problem rests on a
nonlinear optimization routine, which depends directly on temporal logic
constraints and known switching modes that are often not known in advance.

In the present work, we consider a class of faults, which we refer to as
_actuator degradation_. The latter may arise as a result of wear and tear,
software errors, or even adversarial intervention. Considering the following
nonlinear control-affine dynamics $\dot{x}(t)=f(x(t))+g(x(t))u(t),$ we define
input degradation modes of the form
$\dot{\bar{x}}(t)=f(\bar{x}(t))+Rg(\bar{x}(t))Pu(t),$ where $P$ and $R$ are
two unknown time-varying maps. We refer to $P$ as a _control authority
degradation map_ (CDM), whereas $R$ is referred to as a _control effectiveness
degradation map_ (CEM). Our focus in this work is on CDMs; a number of common
CDMs are illustrated in Fig. 1. A CDM $P$ effectively acts as a control input
remapping, and can be thought of in the context of control systems with
delegated control allocation, e.g., when an actuator with internal dynamics
takes $u(t)$ and remaps it based on its internal state. Such a setting
includes common degradation modes such as deadband or saturation, or any other
nonlinear transformation due to effects such as friction. In more extreme
cases, it is possible that $P$ maps a control signal $u_{i}(t)$ to another
control signal $u_{j}(t)$ due to incorrect wiring or software design. The
types of control authority degradation maps that we allow for in this work go
beyond linear maps applied to discrete-time finite-dimensional linear systems,
which hitherto been the main focus in prior work. We develop an _efficient
passive algorithm for detection and identification of CDMs_ , with the quality
of the reconstructed CDM monotonically increasing with system run time. Using
this reconstruction of the CDM, we develop a _novel method for viabilizing
control signals_ , with tight approximation error bounds that decrease with
system run time.

We note that we do not consider external disturbances or other unmodeled
dynamics in this work; robustness results regarding the effects of
disturbances will be the subject of future work. The results of this work
allow for _guaranteed approximation of arbitrary control degradation maps_
without the need for knowledge of possible degradation modes or handcrafted
filters, addressing an open problem in the literature The natural next step of
this work, outside of the scope of this letter, is to approximate unviable
control signal with their closest viable counterpart, with robustness bounds
on the maximum trajectory deviation.

Figure 1: Comparison between various classes of control authority degradation
maps.

##  II Preliminaries

###  II-A Notation

We use $\|\cdot\|$ to denote the Euclidean norm. Given two sets
$A,B\subseteq\mathbb{R}^{n}$, we denote by $A+B$ their Minkowski sum
$\\{a+b:a\in A,b\in B\\}$; the Minkowski difference is defined similarly. By
$2^{A}$ we refer to the power set of $A$, i.e., the family of all subsets of
$A$. We denote a closed ball centered around the origin with radius $r>0$ as
$\mathcal{B}_{r}$. By $\mathcal{B}(x,r)$ we denote $\\{x\\}+\mathcal{B}_{r}$.
We denote by $\mathcal{L}(A,B)$ the set of bounded linear operators, and by
$\mathcal{C}(A,B)$ the set of closed linear operators between $A$ and $B$. We
define $\mathbb{R}_{+}:=[0,\infty)$. For two points in a Banach space
$\mathscr{B}\ni a,b$, let $[a,b]$ denote the convex hull of $a$ and $b$, i.e.,
$[a,b]:=\mathrm{conv}\\{a,b\\}$. Given a point $x\in S$ and a set $A\subseteq
S$, we denote $d(x,A):=\inf_{y\in A}d(x,y)$. We define the distance between
two sets $A,B\subseteq\mathbb{R}^{n}$ to be

$d(A,B):=\sup_{a\in A}\inf_{b\in B}\|a-b\|.$ (1)

We denote the Hausdorff distance as

$d_{\mathrm{H}}(A,B):=\max\\{d(A,B),d(B,A)\\},$ (2)

An alternative characterization of the Hausdorff distance reads:

$d_{\mathrm{H}}(A,B)=\inf\\{\rho\geq 0:A\subseteq B_{+\rho},B\subseteq
A_{+\rho}\\},$ (3)

where $X_{+\rho}$ denotes the _$\rho$ -fattening_ of $X$, i.e.,
$X_{+\rho}:=\bigcup_{x\in X}\\{y\in\mathbb{R}^{n}:\|x-y\|\leq\rho\\}$.

We denote by $\partial A$ the boundary of $A$ in the topology induced by the
Euclidean norm. For a function $g:A\to B$, we denote by $g^{-1}$ the inverse
of this function if an inverse exists and otherwise denoting the preimage. By
$\mathrm{dom}(g)$ we refer to the domain of the function (in this case $A$).
We denote by $g^{\dagger}$ the Moore–Penrose pseudo-inverse of a linear
function $g$. We use the Iverson bracket notation $\llbracket\cdot\rrbracket$,
where the value is $1$ if the expression between the brackets is true, and $0$
otherwise.

In this work, we shall consider star-shaped sets, which are defined as
follows:

######  Definition II.1 (Star-shaped Set and MGFs).

We call a closed compact set $K\subseteq\mathscr{B}$ _star-shaped_ if there
exist (i) $\varsigma\in K$, and (ii) a unique function
$\varrho:\mathcal{B}_{1}\to\mathbb{R}_{+}$, such that:
$K=\bigcup_{l\in\mathcal{B}_{1}}[\varsigma,\varsigma+\varrho(l)l]$ where
$\mathcal{B}_{1}$ denotes the unit ball in $\mathscr{B}$. We call $\varrho$ a
_Minkowski gauge function_ (MGF), and $\varsigma$ the _star center_.

###  II-B Problem Formulation

Consider a known nonlinear control-affine system of the form of

$\dot{x}(t)=f(x(t))+g(x(t))u(t),$ (4)

where $x\in X$, $u\in U\subseteq\mathscr{U}$, $X$ and $\mathscr{U}$ are
Hilbert spaces, and $f:X\to X$ and $g:X\to\mathcal{L}(\mathscr{U},X)$. In this
work, we assume $\mathscr{U}=\mathbb{R}^{m}$. In addition, we assume that $U$
is a _star-shaped_ subset of $\mathbb{R}^{m}$ such that $\mathrm{span}\
U=\mathbb{R}^{m}$. Finally, we assume that the full-state of the degraded
system,

$\dot{\bar{x}}(t)=f(\bar{x}(t))+Rg(\bar{x}(t))Pu(t),$ (5)

is known without error.

In system (5), a control action degradation map $R$ can model changes in the
control allocation function $g$, which may include actuator reconfiguration,
such as a change in the trim angle on aircraft control surfaces, or
misalignment of actuators due to manufacturing imperfections or wear and tear.
Since $R$ acts after $g$, it does not directly remap the control signal
$u(t)$, but it changes the action of a control input on the system; we
therefore talk about control effectiveness, as opposed to control authority in
the case of $P$, which acts before $g$. Changes in the drift dynamics
$f(x(t))$ will not be treated in this work.

In addition to identifying or approximating CDM $P$, we are interested in
‘undoing’ the effects of control authority degradation as much as possible. In
particular, we are interested in the set of control signals (4) that can still
be replicated in (5) when the CDM is acting; we call this the set of _viable
control inputs_ , $U_{\mathrm{v}}$. With knowledge of $P$, we develop in this
work a method to obtain, for $u_{\mathrm{cmd}}\in U_{\mathrm{v}}$,
$u_{\mathrm{v}}$ such that $Pu_{\mathrm{v}}=u_{\mathrm{cmd}}$; here,
$u_{\mathrm{cmd}}$ and $u_{\mathrm{v}}$ are called commanded and viabilized
control inputs, respectively. This approach is closely related to a technique
known in the literature as _fault hiding_ [8]. Fault hiding is achieved by
introducing an output observer based on the output of the degraded system, and
augmenting the nominal system model by introducing so-called virtual
actuators, which requires a nonlinear reconfiguration block that is strongly
dependent on the underlying problem structure and failure modes [8, §3.6, p.
42]. In the setting considered in this work, we show that we can adopt the
fault hiding philosophy under much less stringent constraints for a general
class of systems and degradation modes.

In this work, we are interested in modeling unknown degraded system dynamics
(5) for a time-invariant control authority degradation map (CDM)
$P:U\to\bar{U}$, and no control effectiveness degradation (i.e., $R=I$). This
amounts to reconstructing, or _identifying_ , $P$:

######  Problem 1 (Identifiability of Control Authority Degradation Maps).

For a class of time-invariant CDMs $\mathcal{P}\in P$, if possible, identify
$P$ based on a finite number of full state, velocity, and control input
observations ($\bar{x}(t)$, $\dot{\bar{x}}(t)$, $u(t)$) of the degraded
system.

Ideally, we would like to identify general nonlinear CDMs with known bounds on
the approximation error. We illustrate the control authority degradation modes
that are covered in this work in Fig. 1.

We now proceed by solving Problem 1 for an unknown multi-mode affine CDMs,
which allows for approximating Lipschitz continuous nonlinear CDMs with
bounded error.

##  III Identifiability of Control Authority Degradation Maps

We now consider Problem 1. Let us assume that for $U$, the Minkowski gauge
function $\varrho$ is known. Let $P:U\to\bar{U}$ be an unknown control
authority degradation map (CDM). We assume that $\bar{U}$ is also a star-
shaped set, providing conditions on $P$ and $U$ under which this holds. It
bears mentioning that star-shaped sets are more general than convex sets; most
results presented in this work will apply to star-shaped sets, which include
polytopes, polynomial zonotopes, and ellipsoids.

Before we provide any results on the identifiability of control authority
degradation modes, we pose the following key assumption on the nominal system
dynamics (4). We allow for an _infinite-dimensional_ state-space $X$, that is
to say, $X$ is a set of _functions_ , but $X=\mathbb{R}^{n}$ is also captured:

######  Assumption 1.

For system (5), assume that

  1. i.

$g(x)$ has closed range for all $x\in X$;

  2. ii.

$g(x)$ is injective for all $x\in X$, i.e., $\mathrm{ker}(g(x))=\\{0\\}$;

  3. iii.

$\dot{x}$ is known at some $x\in X$ with $u=0$.

######  Remark 1.

In the case of finite-dimensional systems, i.e., $X\subseteq\mathbb{R}^{n}$,
the first two conditions of Assumption 1 can be stated as:

  1. i.

The system is not overactuated, i.e., $m\leq n$;

  2. ii.

$g(x)$ is of full-column rank for all $x\in X$.

We shall consider the case of multiple control degradation modes acting
throughout the space $U$. The simplest of the so-called _conditional control
authority degradation modes_ (c-CDMs) acts only on a compact subset of $U$; we
refer to these c-CDMs as _partial control authority degradation modes_
(p-CDMs). Consider two compact star-shaped sets $\check{U},\hat{U}\subseteq
U$, and two p-CDMs

$P_{\check{U}}(u):=u+\llbracket u\in\check{U}\rrbracket(P-I)u,$ (6)
$P_{\hat{U}}(u):=u+\llbracket u\not\in\hat{U}\rrbracket(P-I)u,$ (7)

for some control degradation map $P$. Here, $P_{\check{U}}$ is an _internally
acting_ partial CDM (i.e., acting inside $\check{U}$), whereas $P_{\hat{U}}$
is an _externally acting_ partial CDM (acting outside $\hat{U}$); when this
distinction is immaterial, we use a combined hat and check symbol (e.g.,
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$), where
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ is simply called the
_affected set_ of control inputs.

In reconstructing an $N$-mode c-CDM, we face the problem of discerning which
control inputs belong to which conditional degradation mode. To make this
problem tractable, we pose the following assumption:

######  Assumption 2.

Let the internally acting $N$-mode c-CDM satisfy the following properties:

  1. i.

The number of modes $N$ is known;

  2. ii.

$\check{\mathcal{U}}$ is a family of convex sets;

  3. iii.

$\check{\mathcal{P}}$ is a family of affine maps denoted by
$Q_{i}=p_{i}+P_{i}$.

  4. iv.

There exists a known $\delta>0$, such that for all $i\neq j$,
$d_{\mathrm{H}}\left((\check{U}_{i},\check{P}_{i}\check{U}_{i}),(\check{U}_{j},\check{P}_{j}\check{U}_{j})\right)\geq\delta$.

We are also interested in obtaining outer-approximations of $\check{U}$ and
inner-approximations of $\hat{U}$ for each degradation mode, as illustrated in
Fig. 2, so that we can restrict control inputs to regions that are guaranteed
to be unaffected. Since we only have access to a finite number of control
input samples, we pose the following assumption regarding the regularity of
the MGF associated with $P\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$.

######  Assumption 3.

Assume that $\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ has star center
$\ThisStyle{\stackengine{0pt}{\SavedStyle\varsigma}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{\varsigma}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{\varsigma}}}{O}{c}{F}{T}{L}}=0$, and assume
that the MGF
$\ThisStyle{\stackengine{0pt}{\SavedStyle\varrho}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{\varrho}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{\varrho}}}{O}{c}{F}{T}{L}}$ associated with
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ is Lipschitz continuous,
i.e., there exists a known $\ThisStyle{\stackengine{0pt}{\SavedStyle
L}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{L}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{L}}}{O}{c}{F}{T}{L}}$ such that
$|\ThisStyle{\stackengine{0pt}{\SavedStyle\varrho}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{\varrho}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{\varrho}}}{O}{c}{F}{T}{L}}(l)-\ThisStyle{\stackengine{0pt}{\SavedStyle\varrho}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{\varrho}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{\varrho}}}{O}{c}{F}{T}{L}}(l^{\prime})|\leq\ThisStyle{\stackengine{0pt}{\SavedStyle
L}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{L}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{L}}}{O}{c}{F}{T}{L}}\|l-l^{\prime}\|,$ for all
$l,l^{\prime}\in\mathcal{B}_{1}$.

We now proceed to show that Assumption 3 holds for the image of Lipschitz
star-shaped sets under affine maps.

######  Lemma 1.

Given a star-shaped set $U$ characterized by a Lipschitz MGF $\varrho$ and
star center $\varsigma$, the range of $U$ under an affine map $Qu:=p+Pu$ is
also a star-shaped set with Lipschitz MGF.

We can now pose a key result on the guaranteed approximation of Lipschitz MGFs
from a finite set of samples.

######  Proposition 1.

Assume that Assumption 3 holds for the unknown MGFs $\check{\varrho}$ and
$\hat{\varrho}$. Then, for some given
$\check{u},\check{u}^{\prime}\in\check{U}$ and
$\hat{u},\hat{u}^{\prime}\not\in\hat{U}$, we have for all $\mu\in[0,1]$:

$\begin{split}&\check{\varrho}\left(\frac{\mu\check{l}+(1-\mu)\check{l}^{\prime}}{\|\mu\check{l}+(1-\mu)\check{l}^{\prime}\|}\right)\leq\\\
&\min\left\\{\|\check{u}\|+(1-\mu)\check{L}\|l-l^{\prime}\|,\|\check{u}^{\prime}\|+\mu\check{L}\|\check{l}-\check{l}^{\prime}\|\right\\},\end{split}$
(8)

and

$\begin{split}&\hat{\varrho}\left(\frac{\mu\hat{l}+(1-\mu)\hat{l}^{\prime}}{\|\mu\hat{l}+(1-\mu)\hat{l}^{\prime}\|}\right)\geq\\\
&\max\left\\{0,\|\hat{u}\|-(1-\mu)\hat{L}\|\hat{l}-\hat{l}^{\prime}\|,\|\hat{u}^{\prime}\|-\mu\hat{L}\|\hat{l}-\hat{l}^{\prime}\|\right\\},\end{split}$
(9)

where $\ThisStyle{\stackengine{0pt}{\SavedStyle l}{\SavedStyle\hskip
1.38885pt\hat{\vphantom{l}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{l}}}{O}{c}{F}{T}{L}}:=\ThisStyle{\stackengine{0pt}{\SavedStyle
u}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{u}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{u}}}{O}{c}{F}{T}{L}}/\|\ThisStyle{\stackengine{0pt}{\SavedStyle
u}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{u}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{u}}}{O}{c}{F}{T}{L}}\|$ and
$\ThisStyle{\stackengine{0pt}{\SavedStyle l}{\SavedStyle\hskip
1.38885pt\hat{\vphantom{l}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{l}}}{O}{c}{F}{T}{L}}^{\prime}:=\ThisStyle{\stackengine{0pt}{\SavedStyle
u}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{u}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{u}}}{O}{c}{F}{T}{L}}^{\prime}/\|\ThisStyle{\stackengine{0pt}{\SavedStyle
u}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{u}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{u}}}{O}{c}{F}{T}{L}}^{\prime}\|$.

###### Proof.

This result follows directly from non-negativity of the MGF and the mean value
theorem, given the Lipschitz continuity of
$\ThisStyle{\stackengine{0pt}{\SavedStyle\varrho}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{\varrho}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{\varrho}}}{O}{c}{F}{T}{L}}$ as assumed in
Assumption 3. ∎

The results given in Proposition 1 allow for direct inner-approximation of
$\hat{U}$ and outer-approximation of $\check{U}$; these results will allow us
to restrict closed-loop control inputs to a subset of $U$ that is _guaranteed
to be unaffected by $P$_ as illustrated in provided in Fig. 2. The method for
approximating $\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ will be rigorized in the
next theorem.

Figure 2: Comparison between inner- and outer-approximations of $\hat{U}$ and
$\check{U}$ respectively, based on Proposition 1 and Theorem 1 for a 1-mode
c-CDM. The region with top-right-pointing hatching indicates the set in which
the control input is unaffected; the red-colored region indicates the affected
set. The respective approximations of
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.22997pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ allow one to find
regions in which control inputs are guaranteed to be unaffected. In the left
image, the set indicated by top-left-pointing hatching is an inner-
approximation of $\hat{U}$, and in the right image this set is an outer-
approximation of $\check{U}$.

We now pose the main result on the identifiability of $N$-mode conditional
control authority degradation modes (c-CDMs), where multiple affine CDMs act
on disjoint subsets of $U$; this will allow us to approximate of Lipschitz
continuous CDMs as shown at the end of the next section.

######  Theorem 1 (Reconstructing $N$-mode Affine c-CDMs).

Consider system (5) and Assumptions 1–2. Assume that the c-CDM is represented
by $N$ unknown internally acting _affine_ maps $Q_{i}$, each acting on
mutually disjoint unknown star-shaped sets $\check{U}_{i}\subseteq U$, giving
$Q_{\check{\mathcal{U}}}$ as the p-CDM. Let there be a given array of distinct
state–input pairs $[(\bar{x}[i],u[i])]_{i=1}^{N}$, and a corresponding array
of degraded velocities $[\dot{\bar{x}}[i]]_{i=1}^{N^{\prime}}$ obtained from
system (5), with $N^{\prime}\geq N(m+1)$. Let there also be a given array of
undegraded state–input pairs $[(x_{*}[i],u_{*}[i])]_{i=1}^{M}$, with $M\geq
m$. Assume that there exist $m$ state–input pairs indexed by $\jmath$ and
$\jmath_{*}$, such that the arrays of input vectors
$\\{u[\jmath_{j}]\\}_{j=1}^{m}$ and $[u_{*}[\jmath_{*,j}]]_{j=1}^{m}$ are
linearly independent.

Cluster the array $[(u[i],\ThisStyle{\stackengine{0pt}{\SavedStyle
u}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{u}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{u}}}{O}{c}{F}{T}{L}}[i])]_{i=1}^{N^{\prime}}$
into $N$ clusters with a Hausdorff distance of at least $\delta$ between each
pair of clusters. If each cluster $i$ contains at least $m$ vectors $u[i]$
that are linearly independent, then $Q_{\check{\mathcal{U}}}$ can be
approximated as follows:

$\tilde{Q}_{\check{\mathcal{U}}}u=\begin{cases}u&u\not\in\check{U}_{\mathrm{outer}},\\\
\sum_{i=1}^{N}\llbracket u\in\check{U}_{i,\mathrm{inner}}\rrbracket
Q_{i}u\quad&u\in\check{U}_{\mathrm{inner}},\\\
\mathrm{inconclusive}&u\in\check{U}_{\mathrm{outer}}\setminus\check{U}_{\mathrm{inner}},\end{cases}$
(10)

where
$\check{U}_{\mathrm{inner}}:=\bigcup_{i=1}^{N}\check{U}_{i,\mathrm{inner}}$
and
$\check{U}_{\mathrm{outer}}:=\bigcup_{i=1}^{N}\check{U}_{i,\mathrm{outer}}$.
Each $Q_{i}$ is obtained by considering for each cluster
$v_{i}:=g^{\dagger}(\bar{x}[j])(\dot{\bar{x}}[j]-f(\bar{x}[j]))$ where index
$j$ is not part of the array of linearly independent inputs indexed by
$\jmath$,
$\bm{\mathbf{u}}:=[\begin{smallmatrix}u[\jmath_{1}]-v_{i}&\cdots&u[\jmath_{m}]-v_{i}\end{smallmatrix}]$
and
$\Delta\bm{\mathbf{u}}:=\left[g^{\dagger}(\bar{x}[\jmath_{j}])(\dot{\bar{x}}[\jmath_{j}]-f(\bar{x}[\jmath_{j}]))-u[\jmath_{j}]-v_{i}\right]_{j=1}^{m}.$
Linear operator $P_{i}$ is obtained as

$P_{i}=(\bm{\mathbf{u}}+\Delta\bm{\mathbf{u}})\bm{\mathbf{u}}^{\mathsf{T}}(\bm{\mathbf{u}}\bm{\mathbf{u}}^{\mathsf{T}})^{-1}.$
(11)

The translation $p_{i}$ is obtained as $p_{i}=v_{j}-P_{i}u[j]$, which yields
the $i$’th mode affine CDM $Q_{i}$:

$Q_{i}u:=p_{i}+P_{i}u.$ (12)

Here, each affected set is approximated as follows: In case
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$ is internally acting
(i.e., $\check{U}_{i}=\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$), (8) yields an
outer-approximation to $\check{\varrho}_{i}$ by taking a convex combination of
the $m$ basis vectors $\\{\ThisStyle{\stackengine{0pt}{\SavedStyle
l}{\SavedStyle\hskip 1.38885pt\hat{\vphantom{l}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{l}}}{O}{c}{F}{T}{L}}_{*}[\jmath_{*,j}]=u_{*}[\jmath_{*,j}]/\|u_{*}[\jmath_{*,j}]\|\\}_{j=1}^{m}$
and their values. Similarly, for externally acting
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$ (i.e.,
$\hat{U}_{i}=\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$), (9) yields an
inner-approximation to $\hat{\varrho}_{i}$ using $m$ basis vectors
$\\{\ThisStyle{\stackengine{0pt}{\SavedStyle l}{\SavedStyle\hskip
1.38885pt\hat{\vphantom{l}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{l}}}{O}{c}{F}{T}{L}}[\jmath_{j}]\\}_{j=1}^{m}$.
Inner- and outer-approximations satisfy the relation
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i,\mathrm{inner}}\subseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}\subseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i,\mathrm{outer}}$ (cf.
Fig. 2).

###### Proof.

We first consider a globally acting affine CDM. We obtain the closed-form
expression of $P_{i}$, (11), by solving the quadratic program
$\min_{P\in\mathcal{L}(U,\bar{U})}\|P\bm{\mathbf{u}}-(\bm{\mathbf{u}}+\Delta\bm{\mathbf{u}})\|^{2}$,
which yields a unique linear map $P$ that maps $\bm{\mathbf{u}}$ to
$\bm{\mathbf{u}}+\Delta\bm{\mathbf{u}}$ as desired. The translation term
$p_{i}$ can be verified by direct substitution in (12), yielding the affine
map $Q_{i}$.

In (11), since the inverse of $\bm{\mathbf{u}}^{\mathsf{T}}\bm{\mathbf{u}}$
must be taken, we require both that $\bm{\mathbf{u}}$ is a square matrix, and
$\bm{\mathbf{u}}^{\mathsf{T}}\bm{\mathbf{u}}$ is invertible. This is achieved
by considering $\bm{\mathbf{u}}\in\mathbb{R}^{m\times m}$ of full column rank,
as guaranteed by the linear independence hypothesis.

Regarding $g^{\dagger}(x)$, the Moore–Penrose pseudo-inverse is defined for a
general Hilbert space $X$, provided that $\mathrm{range}(g(x))$ is closed for
all $x\in X$ [10, §4.2, p. 47]. For $g^{\dagger}(x)$ to be a left-inverse, a
necessary condition is that $g(x)$ be injective, i.e.,
$\mathrm{ker}(g(x))=\\{0\\}$ for all $x\in X$ [10, Cor. 2.13, p. 36]. Finally,
the translation term $p$ is accounted for as well (12).

To approximate the $i$’th affected set,
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$, we require a
spanning set of basis vectors that lie within
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$, as provided for in
the hypotheses. The unknown MGF associated with
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}_{i}$ can be obtained
according to Proposition 1 using (8)–(9), where an inner-approximation is
desired for internally acting p-CDMs, and outer-approximations for externally
acting p-CDMs. These approximations are obtained through repeated convex
combinations and the corresponding inequality given in (8)–(9), for a total of
$m$ times; an explicit expansion of the resulting expression is omitted here
for the sake of space. ∎

######  Remark 2.

This result incorporates p-CDMs that map a set $\check{U}$ to a constant,
e.g., $Q_{\check{\mathcal{U}}}\check{U}=p$. To highlight the utility of this
result, it should be noted that the _hypotheses given here allow for commonly
encountered degradation modes such as deadzones and saturation to be modeled_
(see Fig. 1(4)). Additionally, Theorem 1 allows for _discontinuous_ control
authority degradation modes, a property that is rarely present in prior work.

We can now consider the case in which $P$ is a Lipschitz continuous CDM. We
consider an approximation of $P$ by an $N$-mode affine c-CDM $\tilde{P}$, for
which we derive an explicit error bound given that the Lipschitz constant of
$P$, $L_{P}$, is known.

######  Theorem 2 (Approximating Lipschitz continuous CDMs by $N$-mode Affine
c-CDMs).

Let the hypotheses of Theorem 1 hold, with the exception that
$P:=Q_{\check{\mathcal{U}}}$ is now a Lipschitz continuous CDM with Lipschitz
constant $L_{P}$ and Assumption 2 is now dropped. If $N$ clusters that satisfy
the linear independence requirements of Theorem 1 are identified, then the
resulting $N$-mode affine c-CDM approximation $\tilde{P}$ has the following
error:

For all $u\in\check{U}_{i,\mathrm{inner}}$ and all $i=1,\ldots,N$,

$\|Pu-\tilde{P}u\|\leq\|\min_{j=1,\ldots,m}\varepsilon_{i,j}+L_{P}\|u[i,j]-u\|,$
(13)

where
$\varepsilon_{i,j}:=\|P\bm{\mathbf{u}}_{i}[j]-\check{P}_{i}\bm{\mathbf{u}}_{i}[j]\|$,
and $u[i,j]:=\bm{\mathbf{u}}_{i}[j]$, where $\bm{\mathbf{u}}_{i}$ is an array
composed of all control inputs in the $i$’th cluster.

###### Proof.

The proof is similar to that of Theorem 1, with the error bound (13) following
an application of the triangle inequality in combination with the Lipschitz
continuity of $P$, the properties of the affine maps $\check{P}_{i}$, and the
known samples of $(u,Pu)$. ∎

We can now pose a convergence result on the $N$-mode affine c-CDM
approximation $\tilde{P}$ of a Lipschitz continuous CDM $P$.

######  Corollary 1.

Error bound (13) is monotonically decreasing in the the number of samples
$N^{\prime}$ and the number of c-CDM modes $N$. In the limit of the
$N^{\prime},N\to\infty$, error bound (13) converges to zero.

###### Proof.

In (13), $\varepsilon_{i,j}$ monotonically converges to zero, because the
operator norm $\|P-\check{P}_{i}\|$ restricted to the $i$’th cluster converges
monotonically to zero; this fact follows by considering that the diameter of
each cluster converges to zero for a greater number of samples and clusters,
similarly to the proof of Lemma 2, as well as the fact that $P$ is Lipschitz
continuous, meaning that the total variation of $P$ on this restriction
decreases monotonically as well. Another consequence of the diminishing
cluster diameter is that $\|u[i,j]-u\|$ converges monotonically to zero. ∎

In the results given above, we find that it is in general impossible to
uniquely determine each $\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ from finitely many
samples. Intuitively, given a greater number of distinct points inside
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ and
$U\setminus\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$, it should be possible
to more tightly approximate $\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$. This idea is
illustrated in Fig. 3. We now state a lemma on the convergence of inner- and
outer-approximations of the affected set
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$.

######  Lemma 2.

Consider $\epsilon>0$, such that a given set of $N_{\epsilon}\geq m$ distinct
pairs $(u,P_{\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
0.95665pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}}u)$ denoted by
$\mathcal{G}U_{N,\epsilon}$, satisfies Assumptions 1–3, where
$P_{\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
0.95665pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}}$ is (i) an $N$-mode
affine c-CDM, or (ii) a Lipschitz continuous CDM. Let $\mathcal{G}U_{N}$ be
such that for each $u_{i}$ in $\mathcal{G}U_{N,\epsilon}$,
$\bigcup_{i=1}^{N_{\epsilon}}\mathcal{B}_{\epsilon}(u_{i})\supseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$; i.e., $\epsilon$-balls
centered at each sampled control input form a cover of
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$. Let
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{inner}}$
and $\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{outer}}$
denote the corresponding inner- and outer-approximations of
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ using the procedure
given in Theorem 1 from $\mathcal{G}U_{N,\epsilon}$. Then, we have
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{inner}}\subseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon^{\prime}}}_{\mathrm{inner}}\subseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ and
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}\subseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon^{\prime}}}_{\mathrm{outer}}\subseteq\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{outer}}$
for all $\epsilon^{\prime}<\epsilon$. In addition, we have $\lim_{\epsilon\to
0}\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{inner}}=\lim_{\epsilon\to
0}\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{outer}}=\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}.$

###### Proof.

Since it is assumed that the pairs in $\mathcal{G}U_{N,\epsilon}$ are
distinct, the approximations of $\check{\varrho}$ and $\hat{\varrho}$ obtained
in Theorem 1 will become increasingly tight for decreasing $\epsilon$, since
the expressions derived in Theorem 1 will rely increasingly less on the
Lipschitz bound assumption. Since
$d_{\mathrm{H}}(\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{inner}},\ThisStyle{\stackengine{0pt}{\SavedStyle
U}{\SavedStyle\hskip 1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}^{N_{\epsilon}}_{\mathrm{outer}})$
is monotonically decreasing for decreasing $\epsilon$, in the limit of
$\epsilon\to 0$, both sequences will converge to
$\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ in the Hausdorff
distance. This follows from the fact that the Hausdorff distance between the
boundary of $\ThisStyle{\stackengine{0pt}{\SavedStyle U}{\SavedStyle\hskip
1.36664pt\hat{\vphantom{U}}\kern
2.2pt\LMpt+.7pt\check{\vphantom{U}}}{O}{c}{F}{T}{L}}$ and the sampled points
$u$ decreases monotonically with decreasing $\epsilon$, leading to tighter
approximations of $\hat{\varrho}$ and $\check{\varrho}$ as per Proposition 1.
∎

Figure 3: Comparison between inner- and outer-approximations of $\hat{U}$ and
$\check{U}$ respectively, based on Proposition 1 and Theorem 1 for an
increasing number of samples for a 1-mode c-CDM. Clearly, for a larger number
of points of sufficiently dispersed points, increasingly tight approximations
are obtained as formalized in Lemma 2.

######  Remark 3.

In Lemma 2, note that the $\epsilon$-covering argument is required to ensure
that the distinct points are sufficiently dispersed; simply considering
$N\to\infty$ does not ensure convergence of the Hausdorff distance between the
inner- and outer-approximation to zero. This fact can also be observed when
looking at Fig. 3.

##  IV Application

We consider an infinite-dimensional system based on a 3D model of tissue
thermodynamics during electrosurgery [El-Kebir2023c]:

$\begin{split}\dot{z}(t,\xi)&=a\nabla^{2}z(t,\xi)+q(\xi)u_{1},\\\
\dot{d}(t)&=z(t,1)u_{2}\end{split}$ (14)

where $u\in[0,10]\times[0,1]$. The unit heat source is modeled as
$q(\xi)=\frac{1}{\epsilon}\llbracket\xi\in[0,\epsilon]\rrbracket$, for some
known $\epsilon>0$. This model approximates a slab of tissue with the state
representing the surface temperature; $u_{1}$ denotes the input power and
$u_{2}$ denotes the needle depth.

Figure 4: Illustration of the domain on which (14) acts. There are three
regions of degradation, of which the burned and the vascularized region form
two; the last region is not shown, but saturates the $u_{1}$ to 5.

For simplicity, we set the input power $u_{1}=1$, and consider only the needle
depth $u_{2}$ as the free control input. We can express system (14) as
affected by a CDM $P$ as:

$\dot{x}(t,\xi)=\begin{bmatrix}\dot{z}(t,\xi)\\\
\dot{d}(t)\end{bmatrix}=\begin{bmatrix}a\nabla^{2}x_{1}(t,\xi)\\\
0\end{bmatrix}+\begin{bmatrix}q(\xi)&0\\\
0&1\end{bmatrix}P\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix}.$

We consider a CDM of the form $Pu=\begin{bmatrix}I&P_{12}\\\
0&I\end{bmatrix}u,$ where $P_{12}$ is the map to be identified. We are
interested in a 3-mode piecewise linear CDM $P_{12}$, with
$\check{U}_{1}=[0,0.25]$, $\check{U}_{2}=[0.5,0.75]$, and
$\check{U}_{3}=[0.75,1]$; these regions are illustrated in Fig. 4. Region 1
corresponds to a charred region at the top of the tissue where the needle does
not fully contact the tissue. Region 2 is a layer of pristine tissue, where
the original dynamics act. Region 3 is a layer of highly vascularized tissue,
in which a large fraction of heat that is added to the system gets transported
away. We consider a piecewise linear function

$\begin{split}P_{12}p:=\ &(0.25+3p)\llbracket p<0.25\rrbracket+\llbracket
0.25\leq p\leq 0.75\rrbracket\\\ +\ &(2.5-2p)\llbracket
p>0.75\rrbracket.\end{split}$

We consider a sinusoidal control signal for the probe depth with a period of
0.3 seconds, $u_{2}(t)=(1-\cos(20\pi t/3))/2$, and a state–input sampling
frequency of 20 Hz. We assume stochastic sampling periods, where the time is
perturbed with a uniform 0.01 second error to model signal processing delays.
The underlying goal of this application is to perform passive probing of the
affected tissue layers and reconfigure the thermodynamics model to account for
tissue damage, as is commonly encountered in electrosurgery. Fig. 5 shows on
the left the Hausdorff distance between each affected region over time to show
that approximations become tighter with time, according to the decreasing
minimal covering radius $\epsilon$ (right), as shown in Lemma 2. After three
samples in each region, we uniquely identify the appropriate affine map, but
the inner-approximation of the affected region is refined passively over time.

Figure 5: Hausdorff distance error for the inner approximations (left) and
minimal covering radius (right) of the three affected regions on which the
c-CDM of (14) acts as a function of time.

##  V Conclusion

In this work, we have introduced the concept of a _control authority
degradation map_ (CDM). We have proved conditions on the identifiability of a
broad class CDMs, including $N$-mode affine CDMs and Lipschitz continuous
CDMs, for a class of affine-in-control nonlinear systems. Based on the
identifiability results, we have formulated a constructive method for
reconstruction or approximating CDMs, with explicit bounds on the
approximation error. Our CDM identification method is executable in real time,
and is guaranteed to monotonically decrease in error as more full-state
observations become available. We apply our methods of CDM identification and
viabilization of control signals to a controlled partial differential equation
motivated by an electrosurgical process, showing how our guaranteed CDM
reconstruction quality improves over time.

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