# Fixed-Time Parameter Adaptation for Safe Control Synthesis

Mitchell Black<EMAIL_ADDRESS>Ehsan Arabi<EMAIL_ADDRESS>Dimitra Panagou
<EMAIL_ADDRESS>Department of Aerospace Engineering, University of
Michigan, Ann Arbor, MI 48109, USA  Ford Research and Advanced Engineering,
Ford Motor Company, Dearborn, MI 48124, USA

###### Abstract

We propose a fixed-time stable parameter adaptation law that guarantees that
an additive, parameter-affine disturbance appearing in the dynamics of a class
of nonlinear, control-affine systems is learned within a fixed time. We then
provide an upper bound on the parameter estimation error as an explicit
function of time, and use such knowledge to formulate a control barrier
function condition to guarantee safety of the system trajectories at all
times. We further show that our proposed parameter adaptation law is robust
against bounded perturbations to the system dynamics in that the resulting
estimated parametric disturbance converges to a known neighborhood of the true
parametric disturbance within a fixed time. To illustrate the advantages of
our proposed approach, we conduct a comparative numerical study against
various controllers from the literature. Finally, we validate our approach on
a trajectory-tracking problem subject to safety requirements using a 6 degree-
of-freedom quadrotor model in an unknown wind field.

###### keywords:

control of constrained systems; robust control of nonlinear systems;
autonomous systems; safety-critical systems; fixed-time stability.

theoremstyle AMStheoremstyle

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author is Mitchell Black.

, ,

##  1 Introduction

Synthesizing controllers for dynamical systems subject to uncertainty is in
general a challenging task. The rise of control barrier functions (CBFs) [49,
2, 38, 15], a tool proven to be effective for enabling safety and
stabilization objectives online via quadratic programs (QPs), has contributed
significantly to the state-of-the art in the field. However, utilizing an
uncertain or inaccurate system model for control synthesis may lead to safety
violations. As such, control engineers often seek to compensate for model
uncertainty in one of the following ways: by taking a robust approach [50, 22,
25], assuming some worst-case bound on the perturbation to the system
dynamics; by using adaptive techniques [44, 27, 51], defining parameter-
adaptation laws according to which estimates of unknown system parameters are
modified online; or by employing learning algorithms [45, 23, 21], using data
to learn a representation of system behavior.

Categorically, these approaches to safe control design possess both strengths
and weaknesses. In the context of safety-critical systems, robust-CBFs (rCBFs)
[22] protect against bounded perturbations to the system dynamics in exchange
for conservatism. Controllers designed using adaptive-CBFs (aCBFs) and
parameter estimators may adjust quickly to changes in the system dynamics
online, but can lead to undesirable chattering [44] or require strict
assumptions on persistently excited (PE) signals in the system dynamics [31].
And while learning-based control algorithms have found success empirically in
certain scenarios, they lack guarantees of safety and stabilization [23]. In
this paper, we seek to design a control framework that leverages the fast
reactivity of adaptation-based controllers and the strong robustness
properties of rCBFs to guarantee the following: 1) that uncertainty
represented by a class of additive, parameter-affine perturbations in the
system dynamics is learned within a finite time independent of the initial
parameter estimates, and 2) that the system trajectories remain safe at all
times. Integral to our proposed framework is the notion of fixed-time
stability.

While asymptotic (AS) and exponential (ES) stability imply convergence of the
trajectories of a dynamical system to an equilibrium point as time tends
toward infinity, the notions of finite- (FTS) [8] and fixed-time stability
(FxTS) [35] guarantee convergence to the equilibrium within a finite time.
Specifically, if an equilibrium of a dynamical system is FxTS then the
trajectories are guaranteed to converge to the equilibrium within a finite
time independent of the initial conditions. Much recent work has been
dedicated to the area of FxTS, both for control design [18, 20, 36, 37] and
estimation [9, 13, 47, 48], though little that synthesizes FxT-adaptation of
unknown system parameters in a safety-critical control framework. In the
following, we highlight important works in parameter estimation and safe
control under uncertainty, and detail how our safe control framework with
fixed-time guarantees advances the state-of-the-art.

###  1.1 Literature Review

A significant amount of research has been devoted to the problem of finite-
and fixed-time parameter identification and estimation in recent years [13,
30, 31, 39]. For example, [30] and [31] design a filtering-based approach to
finite-time parameter estimation of nonlinear and robotic manipulator systems
respectively. In [43], the same filtering scheme is used to learn neural
network parameters in order to approximate an unknown disturbance to the
system dynamics in fixed-time. The authors of [13] design an adaptive
controller for manipulator systems that guarantees convergence of the
parameter estimates to a neighborhood of their true values in fixed-time, but
do not consider safety in the adaptation process. In [39] both FT and FxT
estimation laws are designed and analyzed for perturbed linear regression
systems, but they do not consider control design during the parameter learning
process. Recently, the dynamic regressor extension and mixing (DREM) algorithm
has received significant attention in the field of parameter estimation for
its promise to ensure monotonicity of each element of the parameter error
vector while relaxing the often restrictive persistence of excitation
condition [5, 33, 4]. Recent theoretical results involve the design of finite-
time [32] and discrete-time [7] estimators, and the algorithm has even been
utilized in the design of safety-critical controllers [48]. It is unclear,
however, under what conditions the requirements of DREM actually relax the PE
condition versus when they are similarly restrictive.

In the context of safe control for systems with additive, parameter-affine
uncertainty in the system dynamics, previous work has typically focused on
safety at the exclusion of stabilization. The authors of [44] introduced the
idea of the aCBF, but performed adaptation for the sole purpose of preserving
safety rather than to learn the disturbance in the system dynamics. In [27],
this open problem is addressed via the synthesis of aCBFs with a set
membership identification (SMID)-based parameter update scheme that uses data
to tighten the admissible parameter set. Although their approach led to
improved stabilization results, it lacks any guarantees of convergence of the
parameter estimates. To the best knowledge of the authors, the first joint
design of safety-critical fixed-time adaptation and control laws was proposed
by [9], though the robustness of the approach was not investigated. The
authors of [48] built upon this previous work by leveraging finite-time DREM
and introducing a switching CBF condition that chooses between a novel CBF
result and the one developed by [9], and also performed a robustness analysis
of their approach. Their method, however, inherits the restrictions imposed by
DREM and the bound on convergence time of their parameter estimates depends on
the initial conditions. In what follows, we explain how the work of this paper
seeks to address these gaps.

###  1.2 Summary of Contributions

  * •

We introduce a parameter adaptation law for a class of nonlinear, control- and
parameter-affine dynamical systems with guarantees that the parametric
disturbance is learned within a fixed time, independent of both the initial
conditions and the PE condition.

  * •

We specify a monotonically-decreasing upper bound on the parameter estimation
error and leverage such knowledge to specify a CBF condition that is both
robust against this error and adaptive to the change in parameter estimates,
and use this condition to guarantee safety at all times.

  * •

We use a known condition for FxTS to characterize the robustness of our
proposed adaptation law in the presence of bounded measurement error, and show
that the trajectories of the parameter estimation error are guaranteed to
converge to a neighborhood of the origin within a fixed-time under such
circumstances.

  * •

We use a single-integrator example to highlight the advantages of fixed-time
estimation in a safe control framework over various approaches from the
literature, and further demonstrate the utility of our proposed approach for
safe trajectory tracking via simulations of a quadrotor dynamical model in an
unknown wind field.

The paper is organized as follows. Section 2 reviews some preliminaries and
introduces the problem at hand. Section 3 contains our main results on FxTS
parameter adaptation, while Section 4 analyzes their robustness against a
class of additive, bounded perturbations to the system dynamics and proposes a
CBF condition guaranteeing safety during parameter adaptation. Section 5
highlights the case studies: a comparative single-integrator example and a
quadrotor tracking problem. We conclude in Section 6 with summaries and
directions for future work.

##  2 Mathematical Preliminaries

In the rest of the paper, $\mathbb{R}$ denotes the set of real numbers. The
ones matrix of size $n\times m$ is denoted $\bm{1}_{n\times m}$. We use
$\|\cdot\|$ to denote the Euclidean norm and $\|\cdot\|_{\infty}$ to denote
the sup norm. We denote the minimum and maximum eigenvalue of a matrix
$\bm{M}$ as $\lambda_{min}(\bm{M})$ and $\lambda_{max}(\bm{M})$, and its
minimum nonzero singular value as $\underaccent{\bar}{\sigma}(\bm{M})$. We
also refer to its rowspace and nullspace as $\textrm{row}(\bm{M})$ and
$\mathcal{N}(\bm{M})$ respectively. We write $\partial S$ for the boundary of
the closed set $S$, and $\textrm{int}(S)$ for its interior. The Lie derivative
of a function $V:\mathbb{R}^{n}\rightarrow\mathbb{R}$ along a vector field
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ at a point $x\in\mathbb{R}^{n}$ is
denoted as $L_{f}V(x)\triangleq\frac{\partial V}{\partial x}f(x)$.

In this paper, we consider the following class of nonlinear, control- and
parameter-affine systems:

$\displaystyle\dot{\bm{x}}$
$\displaystyle=f(\bm{x}(t))+g(\bm{x}(t))\bm{u}(\bm{x}(t))+\Delta(\bm{x}(t))\bm{\theta}^{*},$
(1) $\displaystyle\bm{x}(0)$ $\displaystyle=\bm{x}_{0},$

where $\bm{x}\in\mathbb{R}^{n}$ denotes the state, which we assume is known at
all times, $\bm{u}\in\mathcal{U}\subset\mathbb{R}^{m}$ the control input, and
$\bm{\theta}^{*}\in\Theta\subset\mathbb{R}^{p}$ a vector of unknown, static
parameters. The sets $\mathcal{U}$ and $\Theta$ are the input constraint and
admissible parameter sets respectively, where both are known and $\Theta$ is a
polytope. We assume that the drift vector field
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, control matrix field
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$, and regressor
$\Delta:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times p}$ are known,
continuous, and bounded for bounded inputs, such that for a continuous control
input, $\bm{u}(\bm{x})$, (1) admits a unique solution. We note that the
product of the regressor and parameter vector, i.e.
$d(\bm{x})=\Delta(\bm{x})\bm{\theta}^{*}$, may describe disturbances or
unmodelled phenomena in the system dynamics, and that nothing is assumed about
the rank of $\Delta(\bm{x})$. Accordingly, although $\bm{\theta}^{*}$
represents the true parameter vector there may be other vectors $\bm{\theta}$
that satisfy $\Delta(\bm{x})\bm{\theta}=d(\bm{x})$. Thus, we define the
following state- and parameter-dependent set,
$\Omega(\bm{x},\bm{\theta}^{*})$, containing such vectors $\bm{\theta}$:

$\Omega(\bm{x},\bm{\theta}^{*})=\\{\bm{\theta}\in\Theta:\bm{\theta}=\bm{\theta}^{*}+\mathcal{N}(\Delta(\bm{x}))\\}.$
(2)

In the rest of the paper, we write $\Omega$ in place of
$\Omega(\bm{x},\bm{\theta}^{*})$ for conciseness. We now define
$\hat{\bm{\theta}}$ as the estimated parameter vector, and
$\tilde{\bm{\theta}}=\bm{\theta}^{*}-\hat{\bm{\theta}}$ as the parameter
estimation error vector. As such, the parameter estimation error dynamics are

$\dot{\tilde{\bm{\theta}}}=-\dot{\hat{\bm{\theta}}},\;\;\tilde{\bm{\theta}}(0)=\tilde{\bm{\theta}}_{0}.$
(3)

It is worth noting that if
$\tilde{\bm{\theta}}\in\mathcal{N}(\Delta(\bm{x}))$, then
$\Delta(\bm{x})\bm{\theta}^{*}=\Delta(\bm{x})\hat{\bm{\theta}}$, which implies
that $\hat{\bm{\theta}}\in\Omega$.

###  2.1 Set Invariance

Let us also define a set of safe states as

$\displaystyle S$ $\displaystyle=\left\\{\bm{x}\in\mathbb{R}^{n}:h(\bm{x})\geq
0\right\\},$ (4a) $\displaystyle\partial S$
$\displaystyle=\left\\{\bm{x}\in\mathbb{R}^{n}:h(\bm{x})=0\right\\},$ (4b)
$\displaystyle\textrm{int}(S)$
$\displaystyle=\left\\{\bm{x}\in\mathbb{R}^{n}:h(\bm{x})>0\right\\},$ (4c)

where $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is continuously differentiable.
Then, the following Lemma, known as Nagumo’s Theorem [11], provides a
necessary and sufficient condition for the forward invariance of the set (4)
under the system dynamics (1).

######  Lemma 1.

Suppose that $\bm{u}(\bm{x}):S\rightarrow\mathcal{U}$ is a continuous control
input such that the closed-loop trajectories of (1) are uniquely determined in
forward time. The set $S$ is forward-invariant for (1) if and only if the
following holds, for all $\bm{x}\in\partial S$:

$L_{f}h(\bm{x})+L_{g}h(\bm{x})\bm{u}(\bm{x})+L_{\Delta}h(\bm{x})\bm{\theta}^{*}\geq
0.$ (5)

We note that (5) may be written alternatively as
$L_{F}h(\bm{x})+L_{g}h(\bm{x})\bm{u}\geq 0$, where
$L_{F}h(\bm{x})=L_{f}h(\bm{x})+L_{\Delta}h(\bm{x})\bm{\theta}^{*}$. One way to
render a set, $S$, forward-invariant is to use CBFs in the control design.

######  Definition 1.

[2, Definition 5] Given a set $S\subset\mathbb{R}^{n}$ defined by (4) for a
continuously differentiable function $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$,
the function $h$ is a control barrier function (CBF) defined on a set $D$,
where $S\subseteq D\subset\mathbb{R}^{n}$, if there exists an extended class
$\mathcal{K}$ function $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ such that

$\sup_{\bm{u}\in\mathcal{U}}\left\\{L_{F}h(\bm{x})+L_{g}h(\bm{x})\bm{u}\right\\}\geq-\alpha(h(\bm{x})),\;\forall\bm{x}\in
D.$ (6)

The authors of [27] ensure that the closed-loop trajectories of (1) are safe
with respect $S$ by enforcing that a shrunken set, $S_{r}$, is forward-
invariant. Before formally defining $S_{r}$ we make the following assumption.

######  Assumption 1.

The sup norm of the parameter estimation error is non-increasing, i.e.
$\frac{d}{dt}\|\tilde{\bm{\theta}}(t)\|_{\infty}\leq 0$, $\forall t\geq 0$.

Since $\Theta$ is a polytope, it follows from Assumption 1 that if
$\hat{\bm{\theta}}(0)\in\Theta$, then we can define an upper bound
$\vartheta\coloneqq\sup_{\bm{\theta}_{1},\bm{\theta}_{2}\in\Theta}(\|\bm{\theta}_{1}-\bm{\theta}_{2}\|_{\infty})$
such that $\|\tilde{\bm{\theta}}(t)\|_{\infty}\leq\vartheta$, for all $t\geq
0$. It is worth noting that Assumption 1 may be satisfied via the appropriate
design of a parameter adaptation law, a trivial example of which is
$\dot{\hat{\bm{\theta}}}=\bm{0}_{p\times 1}$. We let
$\bm{\vartheta}=\vartheta\cdot\bm{1}_{p\times 1}$, and are now ready to
present the definition for the shrunken set, $S_{r}$, as defined in [27].

$\displaystyle S_{r}$
$\displaystyle=\left\\{\bm{x}\in\mathbb{R}^{n}:h(\bm{x})\geq\frac{1}{2}\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\bm{\vartheta}\right\\}$
(7a) $\displaystyle\partial S_{r}$
$\displaystyle=\left\\{\bm{x}\in\mathbb{R}^{n}:h(\bm{x})=\frac{1}{2}\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\bm{\vartheta}\right\\}$
(7b) $\displaystyle\textrm{int}(S_{r})$
$\displaystyle=\left\\{\bm{x}\in\mathbb{R}^{n}:h(\bm{x})>\frac{1}{2}\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\bm{\vartheta}\right\\}$
(7c)

where $\bm{\Gamma}$ is a constant, positive-definite matrix such that
$h(\bm{x}_{0})\geq\frac{1}{2}\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\bm{\vartheta}$.
We then define a new function,

$h_{r}(\bm{x},\bm{\vartheta})=h(\bm{x})-\frac{1}{2}\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\bm{\vartheta},$
(8)

which, according to Definition 1, is a CBF for $S_{r}$ defined over $D$ if

$\sup_{\bm{u}\in\mathcal{U}}\left\\{L_{F}h_{r}(\bm{x})+L_{g}h_{r}(\bm{x})\bm{u}\right\\}\geq-\alpha(h_{r})+\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\dot{\bm{\vartheta}},$
(9)

holds for all $\bm{x}\in D$. Note that if $\dot{\bm{\vartheta}}=0$ and
$\bm{\vartheta}\neq 0$, then (9) becomes $\dot{h}_{r}\geq-\alpha(h)+\gamma$,
where $\gamma>0$, which is a rCBF condition [22]. We now make the following
assumption to guarantee that $S_{r}$ can be rendered forward-invariant for
(1).

######  Assumption 2.

For all $\bm{x}\in\partial S_{r}$, there exists a control input
$\bm{u}\in\mathcal{U}$ such that the following condition holds:

$L_{F}h_{r}(\bm{x})+L_{g}h_{r}(\bm{x})\bm{u}-\bm{\vartheta}^{T}\bm{\Gamma}^{-1}\dot{\bm{\vartheta}}\geq
0.$ (10)

######  Remark 1.

Since $S_{r}\subseteq S$, it follows that for all $\bm{x}_{0}\in S_{r}$, the
satisfaction of (9) implies that $\bm{x}(t)\in S$ for all $t\geq 0$.

###  2.2 Finite- and Fixed-Time Stability

We now review finite- and fixed-time stability of an equilibrium point of the
dynamical system

$\displaystyle\dot{\bm{x}}$
$\displaystyle=f(\bm{x}(t)),\quad\bm{x}(0)=\bm{x}_{0},$ (11)

where $\bm{x}\in\mathbb{R}^{n}$, $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
is continuous such that (11) admits a unique solution, and it is assumed
without loss of generality that the origin is an equilibrium, i.e. $f(0)=0$.
Though a definition for FTS first appeared in [8], we use the following
definition given in [35] because it is directly used to define the notion of
fixed-time stability.

######  Definition 2.

[35, Definition 1]] The origin of (11) is globally finite-time stable (FTS) if
it is stable in the sense of Lyapunov and any solution $\bm{x}(t,\bm{x}_{0})$
of (11) reaches the origin at some finite time moment, i.e.
$\bm{x}(t,\bm{x}_{0})=0$, $\forall t\geq T(\bm{x}_{0})$, where
$T:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ is the settling-time
function.

######  Definition 3.

[35, Definition 2] The origin of (11) is fixed-time stable (FxTS) if it is
globally FTS and the settling-time function $T(\bm{x}_{0})$ is bounded, i.e.
$\exists T_{b}>0:T(\bm{x}_{0})\leq T_{b}$,
$\forall\bm{x}_{0}\in\mathbb{R}^{n}$.

We now review conditions for FxTS.

######  Theorem 1.

[35, Lemma 1] Suppose that there exists a continuous, positive definite,
radially unbounded function $V:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$
for the system (11) such that

$\displaystyle\dot{V}(\bm{x})\leq-
c_{1}V^{\gamma_{1}}(\bm{x})-c_{2}V^{\gamma_{2}}(\bm{x}),$ (12)

for all $\bm{x}\in\mathbb{R}^{n}\setminus\\{0\\}$, where $c_{1},c_{2}>0$,
$\gamma_{1}>1$, and $0<\gamma_{2}<1$. Then, the origin of (11) is fixed-time
stable with settling time, $T(\bm{x}_{0})$, bounded for all
$\bm{x}_{0}\in\mathbb{R}^{n}$ by

$\displaystyle T(\bm{x}_{0})\leq
T_{b}=\frac{1}{c_{1}(\gamma_{1}-1)}+\frac{1}{c_{2}(1-\gamma_{2})}.$ (13)

We now introduce the following Lemma, which is an important ingredient in the
proof of one of our main results.

######  Lemma 2.

Suppose that $c_{1},c_{2}>0$, $c_{3}\in\mathbb{R}$,
$\gamma_{1}=1+\frac{1}{\mu}$, $\gamma_{2}=1-\frac{1}{\mu}$, $\mu>1$, $0<k<1$,
and that $\bar{V}=1$ and $V_{0}>1$. Consider the following integral,

$I\triangleq\int_{V_{0}}^{\bar{V}}\frac{dV}{-c_{1}V^{\gamma_{1}}-c_{2}V^{\gamma_{2}}+c_{3}V}.$
(14)

Then, the following holds:

  1. (i)

if $c_{3}<2\sqrt{c_{1}c_{2}}$, then for all $V_{0}>1$

$I\leq\frac{\mu}{c_{1}k_{1}}\left(\frac{\pi}{2}-\tan^{-1}k_{2}\right),$ (15)

where

$\displaystyle k_{1}$
$\displaystyle=\sqrt{\frac{4c_{1}c_{2}-c_{3}^{2}}{4c_{1}^{2}}}$ (16)
$\displaystyle k_{2}$
$\displaystyle=\frac{2c_{1}-c_{3}}{\sqrt{4c_{1}c_{2}-c_{3}^{2}}}.$ (17)

  2. (ii)

if $c_{3}\geq 2\sqrt{c_{1}c_{2}}$, and $1\leq V_{0}^{\frac{1}{\mu}}\leq
k\frac{c_{3}-\sqrt{c_{3}^{2}-4c_{1}c_{2}}}{2c_{1}}$, we have

$I\leq\frac{\mu k}{(1-k)\sqrt{c_{1}c_{2}}}.$ (18)

The proof follows directly by changing the upper integration bound from
$\bar{V}=0$ in [17, Lemma 2] to $\bar{V}=1$ here.

###  2.3 Persistence of Excitation

We now review two concepts from the adaptive control literature:
identifiability and persistence of excitation. First, let (1) be expressed as
$\dot{\bm{x}}=\Sigma(\bm{x},\bm{u},\bm{\theta})$ for some state, control, and
parameter tuple $(\bm{x},\bm{u},\bm{\theta})$.

######  Definition 4.

[26, Definition 4.6] The system (1) is identifiable at
$(\bm{x},\bm{u},\bm{\theta}^{*})$ if

$\Sigma(\bm{x},\bm{u},\bm{\theta})=\Sigma(\bm{x},\bm{u},\bm{\theta}^{*})\implies\bm{\theta}=\bm{\theta}^{*}.$

######  Definition 5.

The system (1) is unidentifiable at $(\bm{x},\bm{u},\bm{\theta}^{*})$ if it is
not identifiable.

######  Definition 6.

[12, Definition 2.5.3] A vector or matrix function,
$\phi:[0,\infty)\rightarrow\mathbb{R}^{p\times r}$, is persistently excited
(PE) if there exist $T>0$, $\epsilon_{1},\epsilon_{2}>0$, such that

$\epsilon_{1}I\leq\int_{t}^{t+T}\phi(r)\phi^{T}(r)dr\leq\epsilon_{2}I,\quad\forall
t\geq 0$

######  Remark 2.

While the PE condition is generally difficult to verify in practice for a
given vector or matrix, especially a priori, the regressor, $\Delta$, of (1)
cannot be PE if (1) is unidentifiable for all $(\bm{x},\bm{u},\bm{\theta})$ in
$(\mathbb{R}^{n},\mathcal{U},\Theta)$ [3]. It is worth noting that if
$\mathcal{N}(\Delta(\bm{x}))$ is nontrivial for all $\bm{x}\in\mathbb{R}^{n}$,
i.e. $\Delta(\bm{x})$ is globally rank-deficient, then the system is globally
unidentifiable.

In the context of control design, however, it is not necessary to learn the
exact parameters, $\bm{\theta}^{*}$, appearing in the system dynamics. An
estimate $\hat{\bm{\theta}}\in\Omega$, where $\Omega$ is given by (2), is
sufficient to satisfy
$\Delta(\bm{x})\hat{\bm{\theta}}=\Delta(\bm{x})\bm{\theta}^{*}$. In this
paper, we show how an adaptation law capable of learning such an estimate
within a fixed time adds value to a class of safe controllers. We now
formalize the problem under consideration.

######  Problem 1.

Consider a dynamical system of the form (1). Design adaptation and control
laws, $\dot{\hat{\bm{\theta}}}=\tau(\bm{x},\hat{\bm{\theta}})$ and
$\bm{u}=k(\bm{x},\hat{\bm{\theta}})$ respectively, such that the following
conditions are satisfied:

  1. 1.

The estimated parameter vector, $\hat{\bm{\theta}}$, is rendered fixed-time
stable to the set $\Omega$ given by (2) within a fixed-time $T<\infty$,
independent of the initial estimate $\hat{\bm{\theta}}(0)$, i.e.
$\hat{\bm{\theta}}(t)\rightarrow\Omega$ as $t\rightarrow T$ and
$\hat{\bm{\theta}}(t)\in\Omega$ for all $t\geq T$, independent of
$\hat{\bm{\theta}}(0)$.

  2. 2.

The system trajectories remain safe for all time, i.e. $\bm{x}(t)\in S$,
$\forall t\geq 0$.

The following section contains our proposed solution to Problem 1.

##  3 Parameter Adaptation

Before introducing our first main result, a FxTS adaptation law suitable for
synthesizing a safe control law under uncertainty, we review a result on
exponentially stabilizing (ES) adaptation from the literature [31]. This
requires the following assumption.

######  Assumption 3.

For all time $t\geq 0$ there is a known vector, $\bm{v}(t)\in\mathbb{R}^{q}$,
and a known, full column rank matrix, $\bm{M}(t)\in\mathbb{R}^{q\times p}$,
such that the parameter estimation error, $\tilde{\bm{\theta}}(t)$, is a
solution to

$\bm{M}(t)\tilde{\bm{\theta}}(t)=\bm{v}(t).$ (19)

We note that Assumption 3 is common in the literature [30, 31, 39], but is
infrequently stated so explicitly. Its utility is in enabling the design of
adaptation laws that provide certain convergence guarantees when (19) is
perturbed, i.e. when

$\bm{M}(t)\tilde{\bm{\theta}}(t)=\bm{v}(t)-\bm{\delta}(t),$ (20)

for some unknown $\bm{\delta}\in\mathbb{R}^{q}$. For example, in [39] the
authors seek to design an adaptation law for a class of systems of the form
(20), and assume that $\bm{M}(t)$ satisfies the property
$\sigma_{min}(\bm{M}(t))>\sigma>0$ for all $t\geq 0$, which is equivalent to
the identifiability condition and analogous to Assumption 3 in the
disturbance-free case, i.e. when $\delta(t)\equiv 0$, which they treat
explicitly. The authors of [31] further note that Assumption 3 can be
satisfied when the regressor, $\Delta$, of system (1) satisfies the PE
condition.

We now review the ES adaptation law introduced in [31], modified to conform to
our notation.

######  Theorem 2.

[31, Theorem 2.1] Consider a system (1), for which it is known that Assumption
3 holds. Then, under the ensuing adaptation law,

$\dot{\hat{\bm{\theta}}}=\bm{\Gamma}\bm{v},$ (21)

where $\bm{\Gamma}>0$ is a constant, diagonal gain matrix, the parameter
estimation error, $\tilde{\bm{\theta}}$, converges exponentially to the
origin, i.e. $\tilde{\bm{\theta}}\rightarrow 0$ as $t\rightarrow\infty$.

In what follows, we design an adaptation law for the system (1) that
guarantees convergence of parameter estimates to the set $\Omega$ given by (2)
within a fixed-time, independent of the initial estimates, without
requirements on system identifiability or persistent excitation. Before
stating our result, we introduce a relaxed version of Assumption 3.

######  Assumption 4.

For all $t\geq 0$ there is a known vector, $\bm{v}(t)\in\mathbb{R}^{n}$, and a
known matrix, $\bm{M}(t)\in\mathbb{R}^{n\times p}$, that jointly satisfy the
following properties:

  1. 1.

the parameter estimation error, $\tilde{\bm{\theta}}(t)$, is one solution to
(19)

  2. 2.

$\bm{v}(t)$ and $\bm{M}(t)$ are bounded for bounded $\bm{x}(t)$

  3. 3.

the nullspace of $\bm{M}(t)$ is equal to the nullspace of $\Delta(\bm{x}(t))$,
i.e $\mathcal{N}(\Delta(\bm{x}(t)))=\mathcal{N}(\bm{M}(t))$.

######  Remark 3.

In contrast to Assumption 3, it is not required by Assumption 4 that $\bm{M}$
be full column-rank. Instead, the condition is that
$\mathcal{N}(\bm{M})=\mathcal{N}(\bm{\Delta})$, which relaxes restrictions
imposed by previous works [30, 31, 39].

Since it is not immediately obvious whether Assumption 4 is reasonable for the
class of systems under consideration, we now introduce two possible
measurement schemes, the first for illustrative and the second for practical
purposes, which lead to its satisfaction.

###  3.1 Rate Measurements

For the illustrative scheme, the following assumption is required:

######  Assumption 5.

The time derivative of the state, $\dot{\bm{x}}$, is bounded and accessible
for measurement.

Under Assumption 5, we can rewrite the perturbed dynamics (1) to obtain a
linear system of equations,

$\displaystyle\underbrace{\Delta(\bm{x})}_{\text{$\bm{M}$}}\tilde{\bm{\theta}}=\underbrace{\dot{\bm{x}}-f(\bm{x})-g(\bm{x})\bm{u}-\Delta(\bm{x})\hat{\bm{\theta}}}_{\text{$\bm{v}$}},$
(22)

all of whose terms are either known or measured except $\tilde{\bm{\theta}}$,
and to which at least one solution (albeit not necessarily unique) exists by
construction. It is also evident that the second facet of Assumption 4 is
satisfied via assumptions on the components of (1), and that the third is
satisfied due to $\bm{M}=\Delta(\bm{x})$. While exact measurements of
$\dot{\bm{x}}$ may not be obtainable in practice, it may be approximated using
state measurements and numerical differentiation. This may introduce some
approximation error into (22) such that it takes the form (20). We address the
effect of this disturbance, $\bm{\delta}(t)$, in Section 4.

###  3.2 State Predictor

We now present a predictor-based technique adapted from [1] which also yields
a linear system of equations that satisfies Assumption 4 under certain
conditions. First, consider the following state predictor:

$\displaystyle\dot{\bm{z}}=f(\bm{x})+g(\bm{x})\bm{u}+\Delta(\bm{x})\hat{\bm{\theta}}+k_{e}(\bm{x}-\bm{z})+\bm{W}\dot{\hat{\bm{\theta}}}$
(23) $\displaystyle\bm{z}(0)=\bm{x}(0),$

where $\bm{z}\in\mathbb{R}^{n}$ is the predicted state,
$\hat{\bm{\theta}}\in\mathbb{R}^{p}$ is the estimated parameter vector,
$\dot{\hat{\bm{\theta}}}$ is its time derivative to be defined later, $f$,
$g$, and $\Delta$ are as in (1), $k_{e}>0$ is an error gain, and $\bm{W}$ is a
matrix described by the following dynamics:

$\dot{\bm{W}}=-k_{e}\bm{W}+\Delta(\bm{x}),\quad\bm{W}(0)=\bm{0}_{n\times p}.$
(24)

Then, by defining the observation error as $\bm{e}=\bm{x}-\bm{z}$, the error
dynamics are described by:

$\dot{\bm{e}}=\Delta(\bm{x})\tilde{\bm{\theta}}-k_{e}\bm{e}-\bm{W}\dot{\hat{\bm{\theta}}}.$
(25)

Next, define an auxiliary variable,

$\bm{\xi}=\bm{e}-\bm{W}\tilde{\bm{\theta}},$ (26)

the dynamics of which become

$\dot{\bm{\xi}}=\dot{\bm{e}}-\dot{\bm{W}}\tilde{\bm{\theta}}-\bm{W}\dot{\tilde{\bm{\theta}}},$
(27)

which, by substituting in (24) and (25), amounts to

$\dot{\bm{\xi}}=-k_{e}\bm{e}+k_{e}\bm{W}\tilde{\bm{\theta}}=-k_{e}\bm{\xi},\quad\bm{\xi}(0)=\bm{e}(0),$
(28)

a linear system with an exponentially stable equilibrium point at the origin.
Then, by using the initial conditions from (23) and (24), it follows that
$\bm{\xi}(0)=\bm{0}$ and therefore $\dot{\bm{\xi}}=\bm{0}$ for all $t\geq 0$,
i.e. $\bm{\xi}(t)=\bm{\xi}(0)=\bm{0}$. As such, (26) implies that

$\bm{W}\tilde{\bm{\theta}}=\bm{e},$ (29)

which takes the form (19) and satisfies Assumption 4 in the following way: the
first part is satisfied by construction; the second holds since, according to
(24), $\bm{W}(t)=\int_{0}^{t}e^{-k_{e}(t-s)}\Delta(\bm{x}(s))ds$, and thus
$\bm{W}$ is bounded for bounded $\Delta(\bm{x})$ (which is bounded for bounded
$\bm{x}$); and the third part is satisfied if $\mathcal{N}(\Delta(\bm{x}))$ is
constant, though this is a sufficient and not a necessary condition.

###  3.3 Fixed-Time Adaptation

We are now ready to introduce an adaptation law which renders
$\hat{\bm{\theta}}$ FxTS to the set (2).

######  Theorem 3.

Consider a nonlinear, control-affine system of the form (1) and suppose that
Assumption 4 holds. If the nullspace of the regressor $\Delta(\bm{x}(t))$ is
constant over the interval $[0,T]$, i.e.
$\mathcal{N}(\Delta(\bm{x}(t)))=\mathcal{N}(\Delta(\bm{x}(0)))$, $\forall
t\in[0,T]$, where

$T=\frac{\mu\pi}{2k_{V}^{2}\sqrt{ab}},$ (30)

with $a,b>0$, $\mu>2$, and

$k_{V}=\underaccent{\bar}{\sigma}(\bm{M})\sqrt{2\lambda_{max}(\bm{\Gamma})},$
(31)

where $\bm{\Gamma}\in\mathbb{R}^{p\times p}$ is a constant, positive-definite,
gain matrix and $\underaccent{\bar}{\sigma}(\bm{M})>0$ denotes the smallest
nonzero singular value of $\bm{M}$ over the time interval, then under the
ensuing adaptation law,

$\dot{\hat{\bm{\theta}}}=\bm{\Gamma}\bm{M}^{T}\bm{v}\left(a\|\bm{v}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{v}\|^{\frac{2}{\mu}}}\right),$
(32)

the estimated parameter vector, $\hat{\bm{\theta}}$, converges to $\Omega$
defined by (2) within a fixed time $T$, i.e.
$\hat{\bm{\theta}}(t)\rightarrow\Omega$ and
$\tilde{\bm{\theta}}(t)\rightarrow\mathcal{N}(\Delta(\bm{x}(t))$ as
$t\rightarrow T_{c}(\hat{\bm{\theta}}(0))\leq T$,
$\forall\hat{\bm{\theta}}(0)\in\mathbb{R}^{p}$.

We first show that (32) is well-defined, and then prove that
$\hat{\bm{\theta}}$ converges to the set $\Omega$ within fixed-time $T$. It is
obvious by boundedness of $\bm{v}$ and $\bm{M}$ that (32) is bounded whenever
$0<\|\bm{v}\|<\infty$, so we focus on what happens when $\|\bm{v}\|\rightarrow
0$. Consider that (32) can be rewritten as
$\dot{\hat{\bm{\theta}}}=G_{1}(\bm{M},\bm{v})+G_{2}(\bm{M},\bm{v})$, where
$G_{1}(\bm{M},\bm{v})=a\bm{\Gamma}\bm{M}^{T}\bm{v}\|\bm{v}\|^{\frac{2}{\mu}}$
and
$G_{2}(\bm{M},\bm{v})=b\bm{\Gamma}\bm{M}^{T}\frac{\bm{v}}{\|\bm{v}\|^{\frac{2}{\mu}}}$.
It is evident that $G_{1}(\bm{M},\bm{v})\rightarrow 0$ as
$\|\bm{v}\|\rightarrow 0$, however, for $G_{2}(\bm{M},\bm{v})$ it follows that

$\displaystyle\|G_{2}(\bm{M},\bm{v})\|^{2}=\frac{b^{2}}{\|\bm{v}\|^{4/\mu}}\bm{v}^{T}\left(\bm{M}\bm{\Gamma}^{T}\bm{\Gamma}\bm{M}^{T}\right)\bm{v}.$

Then, defining $\bm{N}=\bm{M}\bm{\Gamma}^{T}\bm{\Gamma}\bm{M}^{T}$, we obtain
that

$\displaystyle\frac{b^{2}\sigma_{min}(\bm{N})}{\|\bm{v}\|^{4/\mu}}\|\bm{v}\|^{2}\leq\|G_{2}(\bm{M},\bm{v})\|^{2}\leq\frac{b^{2}\sigma_{max}(\bm{N})}{\|\bm{v}\|^{4/\mu}}\|\bm{v}\|^{2}.$

Since $0\leq\sigma_{min}(\bm{N}),\sigma_{max}(\bm{N})<\infty$, it follows that
if $\mu>2$, then

$\lim_{\|\bm{v}\|\rightarrow 0}\|G_{2}(\bm{M},\bm{v})\|=0.$

Thus, we require that $\mu>2$, which results in
$G_{2}(\bm{M},\bm{v})\rightarrow 0$ as $\|\bm{v}\|\rightarrow 0$ and therefore
(32) is well-defined.

For the convergence of $\hat{\bm{\theta}}$ to the set $\Omega$ in fixed-time,
we begin by observing that $\tilde{\bm{\theta}}$, $\dot{\tilde{\bm{\theta}}}$,
and $\dot{\hat{\bm{\theta}}}$ may always be expressed as the following linear
combinations:

$\displaystyle\tilde{\bm{\theta}}$
$\displaystyle=\tilde{\bm{\theta}}_{R}+\tilde{\bm{\theta}}_{N},$ (33)
$\displaystyle\dot{\tilde{\bm{\theta}}}$
$\displaystyle=\dot{\tilde{\bm{\theta}}}_{R}+\dot{\tilde{\bm{\theta}}}_{N},$
(34) $\displaystyle\dot{\hat{\bm{\theta}}}$
$\displaystyle=\dot{\hat{\bm{\theta}}}_{R}+\dot{\hat{\bm{\theta}}}_{N}$ (35)

where
$\tilde{\bm{\theta}}_{R},\dot{\tilde{\bm{\theta}}}_{R},\dot{\hat{\bm{\theta}}}_{R}\in\textrm{row}(\bm{M})$
and
$\tilde{\bm{\theta}}_{N},\dot{\tilde{\bm{\theta}}}_{N},\dot{\hat{\bm{\theta}}}_{N}\in\mathcal{N}(\bm{M})$.
By rank-nullity and the fact that row$(\bm{M})$ and $\mathcal{N}(\bm{M})$ are
orthogonal complements, such a decomposition always exists [41]. Next, from
(32) and Assumption 4 we observe that

$\displaystyle\dot{\hat{\bm{\theta}}}$
$\displaystyle=\bm{\Gamma}\bm{M}^{T}\bm{v}\left(a\|\bm{v}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{v}\|^{\frac{2}{\mu}}}\right)$
$\displaystyle=\bm{\Gamma}\bm{M}^{T}\bm{M}\tilde{\bm{\theta}}\left(a\|\bm{M}\tilde{\bm{\theta}}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{M}\tilde{\bm{\theta}}\|^{\frac{2}{\mu}}}\right).$

Since
$\bm{M}\tilde{\bm{\theta}}=\bm{M}(\tilde{\bm{\theta}}_{R}+\tilde{\bm{\theta}}_{N})=\bm{M}\tilde{\bm{\theta}}_{R}$,
it follows that

$\displaystyle\dot{\hat{\bm{\theta}}}$
$\displaystyle=\bm{\Gamma}\bm{M}^{T}\bm{M}\tilde{\bm{\theta}}_{R}\left(a\|\bm{M}\tilde{\bm{\theta}}_{R}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{M}\tilde{\bm{\theta}}_{R}\|^{\frac{2}{\mu}}}\right)$
$\displaystyle=\dot{\hat{\bm{\theta}}}_{R}.$

Then, from (35) we obtain that $\dot{\hat{\bm{\theta}}}_{N}=\bm{0}_{p\times
1}$ and by (3)

$\displaystyle\dot{\tilde{\bm{\theta}}}_{R}$
$\displaystyle=-\dot{\hat{\bm{\theta}}}_{R},\;\;\tilde{\bm{\theta}}_{R}(0)=\tilde{\bm{\theta}}_{R,0},$
(36) $\displaystyle\dot{\tilde{\bm{\theta}}}_{N}$
$\displaystyle=\bm{0}_{p\times
1},\;\;\tilde{\bm{\theta}}_{N}(0)=\tilde{\bm{\theta}}_{N,0}.$ (37)

By Assumption 4, it follows that
$\tilde{\bm{\theta}}_{N}(0)\in\mathcal{N}(\Delta(\bm{x}(0)))$, and by
$\mathcal{N}(\Delta(\bm{x}(t)))=\mathcal{N}(\Delta(\bm{x}(0)))$ for all
$t\in[0,T]$ we then have that $\hat{\bm{\theta}}\in\Omega$ whenever
$\tilde{\bm{\theta}}_{R}=0$. It is sufficient, therefore, to show that
$\tilde{\bm{\theta}}_{R}(t)\rightarrow 0$ as $t\rightarrow
T_{c}(\hat{\bm{\theta}}(0))\leq T$.

Consider the Lyapunov function candidate
$V=\frac{1}{2}\tilde{\bm{\theta}}_{R}^{T}\bm{\Gamma}^{-1}\tilde{\bm{\theta}}_{R}$.
Its time derivative along the trajectories of (36) is

$\displaystyle\dot{V}$
$\displaystyle=-\tilde{\bm{\theta}}_{R}^{T}\bm{\Gamma}^{-1}\dot{\hat{\bm{\theta}}}_{R}$
$\displaystyle=-\tilde{\bm{\theta}}_{R}^{T}\bm{M}^{T}\bm{M}\tilde{\bm{\theta}}_{R}\left(a\|\bm{M}\tilde{\bm{\theta}}_{R}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{M}\tilde{\bm{\theta}}_{R}\|^{\frac{2}{\mu}}}\right)$
$\displaystyle=-a\|\bm{M}\tilde{\bm{\theta}}_{R}\|^{2+\frac{2}{\mu}}-b\|\bm{M}\tilde{\bm{\theta}}_{R}\|^{2-\frac{2}{\mu}}$

When $\tilde{\bm{\theta}}_{R}\neq 0$, we have that
$\|\bm{M}\tilde{\bm{\theta}}_{R}\|>0$, and therefore

$0<\underaccent{\bar}{\sigma}(\bm{M})\|\tilde{\bm{\theta}}_{R}\|\leq\|\bm{M}\tilde{\bm{\theta}}_{R}\|\leq\sigma_{max}(\bm{M})\|\tilde{\bm{\theta}}_{R}\|.$

Then, since
$\frac{1}{2}\lambda_{min}(\bm{\Gamma}^{-1})\tilde{\bm{\theta}}_{R}^{T}\tilde{\bm{\theta}}_{R}\leq
V\leq\frac{1}{2}\lambda_{max}(\bm{\Gamma}^{-1})\tilde{\bm{\theta}}_{R}^{T}\tilde{\bm{\theta}}_{R}$
and $\lambda_{min}(\bm{\Gamma}^{-1})=1/\lambda_{max}(\bm{\Gamma})$, it follows
then from the definition of $V$ that
$\|\tilde{\bm{\theta}}_{R}\|\leq\sqrt{2V\lambda_{max}(\bm{\Gamma})}=L(V)$.
Using this, we obtain

$\displaystyle\dot{V}$
$\displaystyle\leq-a\Big{(}\underaccent{\bar}{\sigma}(\bm{M})L(V)\Big{)}^{2+\frac{2}{\mu}}-b\Big{(}\underaccent{\bar}{\sigma}(\bm{M})L(V)\Big{)}^{2-\frac{2}{\mu}}$
$\displaystyle\leq-c_{1}V^{1+\frac{1}{\mu}}-c_{2}V^{1-\frac{1}{\mu}}$ (38)

which takes the form (12), where

$\displaystyle c_{1}=ak_{V}^{2+\frac{2}{\mu}}$ (39) $\displaystyle
c_{2}=bk_{V}^{2-\frac{2}{\mu}},$ (40)

and $k_{V}$ is given by (31). Thus, the requirements for FxTS provided in
Theorem 1 are met insofar as $\tilde{\bm{\theta}}_{R}$ is concerned.
Therefore, $\tilde{\bm{\theta}}_{R}\rightarrow 0$ in fixed-time, i.e. as
$t\rightarrow T_{c}(\hat{\bm{\theta}}(0))\leq T_{b}$ where $T_{b}$ is given by
(13) and $\gamma_{1}=1+\frac{1}{\mu}$ and $\gamma_{2}=1-\frac{1}{\mu}$. With
$\gamma_{1},\gamma_{2}$ in this form, it follows from [34, Lemma 2] that the
settling time function $T_{c}$ is bounded from above by $T$ given by (30).
Thus, we have that $\hat{\bm{\theta}}\rightarrow\Omega$ within time $T$,
independent of the initial estimates, $\hat{\bm{\theta}}(0)$. This completes
the proof. The adaptation law (32) is usable as long as there exists a way to
satisfy Assumption 4. The measurement and prediction schemes outlined in
Sections 3.1 and 3.2 are two avenues toward realizing this goal, though there
may exist others. While it is not guaranteed that
$\hat{\bm{\theta}}\rightarrow\bm{\theta}^{*}$ unless
$\mathcal{N}(\Delta(\bm{x}(t)))=\\{0\\}$ for all $t\in[0,T]$, it is sufficient
from a control-oriented perspective to learn some $\hat{\bm{\theta}}$ such
that $\Delta(\bm{x})\hat{\bm{\theta}}=\Delta(\bm{x})\bm{\theta}^{*}$.

######  Remark 4.

The bound on settling time given by (30) requires knowledge of the minimum
nonzero singular value of the matrix $\bm{M}$. While the exact value may not
be known a priori, domain knowledge can be used to obtain a lower bound over
the relevant region of the state space. If using the predictor-based
measurement scheme, a warm-up time, $T_{w}$, may be required for
$\underaccent{\bar}{\sigma}(\bm{W})$ to eclipse such a lower bound. This warm-
up time must be added to the convergence time derived in (30).

With knowledge of the parameter estimates converging to $\Omega$ in fixed-
time, we derive an expression for the upper bound on the sup norm of the
parameter error as a function of time.

######  Corollary 1.

Suppose that the premises of Theorem 3 hold. If, in addition, the following
conditions hold

  1. (i)

The initial estimated parameter vector lies within the known admissible
parameter set, $\Theta$, i.e. $\hat{\bm{\theta}}(0)\in\Theta$,

  2. (ii)

The estimated parameter update law, $\dot{\hat{\bm{\theta}}}$, is given by
(32), where $\bm{\Gamma}$ is constant, positive-definite, and also diagonal,

  3. (iii)

A lower bound on $\underaccent{\bar}{\sigma}(\bm{M})>s>0$, is known,

then $\forall t\in[0,T]$, where $T$ is given by (30), the following expression
constitutes a monotonically decreasing upper bound on
$\|\tilde{\bm{\theta}}_{R}(t)\|_{\infty}$:

$\|\tilde{\bm{\theta}}_{R}(t)\|_{\infty}\leq\sqrt{2\lambda_{max}(\bm{\Gamma})\left(\sqrt{\frac{c_{1}}{c_{2}}}\tan\left(A(t)\right)\right)^{\mu}}\coloneqq\eta(t),$
(41)

where $c_{1}$ and $c_{2}$ are defined by (39) and (40), and

$\displaystyle A(t)$
$\displaystyle=\max\left\\{\Xi-\frac{\sqrt{c_{1}c_{2}}}{\mu}t,0\right\\},$
(42) $\displaystyle\Xi$
$\displaystyle=\tan^{-1}\left(\sqrt{\frac{c_{2}}{c_{1}}}\left(\frac{1}{2}\bm{\eta}(0)^{T}\Gamma^{-1}\bm{\eta}(0)\right)^{\frac{1}{\mu}}\right)$
(43)

with $\bm{\eta}(t)=\eta(t)\cdot\bm{1}_{p\times 1}$, and
$\|\tilde{\bm{\theta}}_{R}(t)\|_{\infty}=0$, $\forall t>T$.

Similar to the proof of Theorem 3, we begin by considering the Lyapunov
function candidate
$V=\frac{1}{2}\tilde{\bm{\theta}}_{R}^{T}\Gamma^{-1}\tilde{\bm{\theta}}_{R}$,
the time-derivative of which is given by (38). Then, from (38), we rearrange
terms and use the change of variables $x=V^{\frac{1}{\mu}}$ and
$dx=\frac{1}{\mu}V^{\frac{1}{\mu}-1}dV$ to obtain

$\displaystyle t$
$\displaystyle\leq\int_{V_{0}^{\frac{1}{\mu}}}^{V^{\frac{1}{\mu}}}\frac{\mu
x^{\mu-1}dx}{-c_{1}x^{\mu-1}-c_{2}x^{\mu+1}}=\int_{V_{0}^{\frac{1}{\mu}}}^{V^{\frac{1}{\mu}}}\frac{\mu
dx}{-c_{1}-c_{2}x^{2}}$
$\displaystyle\leq-\frac{\mu}{\sqrt{c_{1}c_{2}}}\left(\tan^{-1}\left(\sqrt{\frac{c_{2}}{c_{1}}}V^{\frac{1}{\mu}}\right)-\tan^{-1}\left(\sqrt{\frac{c_{2}}{c_{1}}}V_{0}^{\frac{1}{\mu}}\right)\right).$

By rearranging terms, we obtain

$\displaystyle V(t)$
$\displaystyle\leq\left(\sqrt{\frac{c_{1}}{c_{2}}}\tan\left(\tan^{-1}\left(\sqrt{\frac{c_{2}}{c_{1}}}V_{0}^{\frac{1}{\mu}}\right)-\frac{\sqrt{c_{1}c_{2}}}{\mu}t\right)\right)^{\mu},$

where, by observing that
$V_{0}=\frac{1}{2}\tilde{\bm{\theta}}_{R}(0)^{T}\Gamma^{-1}\tilde{\bm{\theta}}_{R}(0)\leq\frac{1}{2}\bm{\eta}(0)^{T}\Gamma^{-1}\bm{\eta}(0)$,
we obtain

$\displaystyle V(t)$
$\displaystyle\leq\left(\sqrt{\frac{c_{1}}{c_{2}}}\tan\left(\Xi-\frac{\sqrt{c_{1}c_{2}}}{\mu}t\right)\right)^{\mu}.$

We observe that in the limit $\Xi\rightarrow\frac{\pi}{2}$ as
$\|\bm{\eta}(0)\|\rightarrow\infty$. With $\hat{\bm{\theta}}(0)\in\Theta$,
however, it will not occur that $V_{0}\rightarrow\infty$, and therefore, since
the bound (30) holds for arbitrarily large $V_{0}$, it is possible that
$\Xi-\frac{\sqrt{c_{1}c_{2}}}{\mu}t<0$ for some $t<T$. To account for this, we
define $A(t)=\max\left\\{\Xi-\frac{\sqrt{c_{1}c_{2}}}{\mu}t,0\right\\}$ and
obtain

$V(t)\leq\left(\sqrt{\frac{c_{1}}{c_{2}}}\tan\left(A(t)\right)\right)^{\mu}.$
(44)

Now, since $\Gamma$ diagonal we can express
$V=\frac{1}{2}(\Gamma^{-1}_{11}\tilde{\theta}_{R,1}^{2}+...+\Gamma^{-1}_{pp}\tilde{\theta}_{R,p}^{2})$,
and observe that
$V\geq\frac{1}{2}\lambda^{-1}_{max}(\Gamma)\|\tilde{\bm{\theta}}_{R}\|^{2}\geq\frac{1}{2}\lambda^{-1}_{max}(\Gamma)\|\tilde{\bm{\theta}}_{R}\|_{\infty}^{2}$.
Then, we substitute (44) in this inequality and rearrange terms to recover
(41). Then, for

$0\leq
t\leq\frac{\mu}{\sqrt{c_{1}c_{2}}}\textrm{tan}^{-1}\left(\sqrt{\frac{c_{2}}{c_{1}}}\left(\frac{1}{2}\bm{\eta}(0)^{T}\Gamma^{-1}\bm{\eta}(0)\right)^{\frac{1}{\mu}}\right)$

we have that (41) decreases monotonically to zero. This completes the proof.

######  Remark 5.

Since we assume that $\bm{\theta}^{*}\in\Theta$ at $t=0$, we can as a
consequence of (41) tighten the set of admissible parameters at time $t$, i.e.
it is known that $\bm{\theta}^{*}\in\Lambda(t)\subset\Theta$, where
$\Lambda(t)=\\{\bm{\lambda}\in\Theta:\|\bm{\lambda}_{R}-\hat{\bm{\theta}}_{R}\|_{\infty}\leq\eta(t)\\}$,
for all $t\in[0,T]$, where $\bm{\lambda}=\bm{\lambda}_{R}+\bm{\lambda}_{N}$,
with $\bm{\lambda}_{R},\hat{\bm{\theta}}_{R}\in\textrm{row}(\Delta)$,
$\bm{\lambda}_{N},\hat{\bm{\theta}}_{N}\in\mathcal{N}(\Delta)$, and
$\Lambda(t)=\\{\hat{\bm{\theta}}+\mathcal{N}(\Delta)\\}$ for all
$t\in(T,\infty)$. As a special case, if $\Delta$ is full column rank, then
$\Lambda(t)=\\{\hat{\bm{\theta}}\\}$ for all $t\in[T,\infty)$.

In the following section, we formalize results on the robustness of the
proposed adaptation law (32) and show how they can improve control design
under parametric uncertainty via robust, adaptive CBF-based control.

##  4 Robust Adaptive Safety

Consider the system (1), now subject to an additive, bounded disturbance,

$\dot{\bm{x}}=f(\bm{x})+g(\bm{x})\bm{u}+\Delta(\bm{x})\bm{\theta}^{*}+d(\bm{x},t),$
(45)

which captures the effects of external perturbations. As such, the rate
measurement (22) and state predictor (23) schemes do not produce a relation of
the form (19). Instead, for rate measurements we have that

$\displaystyle\underbrace{\dot{\bm{x}}-f(\bm{x})-g(\bm{x})\bm{u}-\Delta(\bm{x})\hat{\bm{\theta}}}_{\text{$\bm{v}$}}=\underbrace{\Delta(\bm{x})}_{\text{$\bm{M}$}}\tilde{\bm{\theta}}+\underbrace{d(\bm{x},t)}_{\text{$\bm{\delta}$}},$
(46)

and for the state predictor scheme,

$\dot{\bm{\xi}}=-k\bm{\xi}+d(t,\bm{x}),\quad\bm{\xi}(0)=\bm{e}(0),$ (47)

which implies that $\bm{e}=\bm{W}\tilde{\bm{\theta}}+\bm{\alpha}$ for all
$t>0$, where $\bm{\alpha}\in\mathbb{R}^{n}$ is unknown. We observe that these
new relations may be expressed in the form (20), and as such we introduce the
following assumption.

######  Assumption 6.

For all $t\geq 0$ there is a known vector, $\bm{v}(t)\in\mathbb{R}^{n}$, a
known matrix, $\bm{M}(t)\in\mathbb{R}^{n\times p}$, and an unknown vector,
$\bm{\delta}(t)$ that jointly satisfy the following properties:

  1. (i)

the parameter error, $\tilde{\bm{\theta}}(t)$, is one solution to (20),

  2. (ii)

there exists a known $\Upsilon>0$ such that

$\sup_{t\in\mathbb{R}_{\geq 0}}\|\delta(t)\|\leq\Upsilon,$

  3. (iii)

and items (ii) and (iii) of Assumption 4 hold

We note that part (ii) of Assumption 6 is reasonable given that it is assumed
that the disturbance $d$ in (45) is bounded. We now introduce the following
Lemma, a requirement for the proof of our next result.

######  Lemma 3.

Suppose that $\bm{x},\bm{y}\in\mathbb{R}^{n}$, and define

$P(\bm{x},\bm{y})\coloneqq\bm{x}^{T}\bm{y}\left(a\|\bm{x}+\bm{y}\|^{\frac{2}{m}}+\frac{b}{\|\bm{x}+\bm{y}\|^{\frac{2}{m}}}\right).$
(48)

If $a,b>0$, $m>2$, and there exists $B_{y}>0$ such that $\|\bm{y}\|\leq B_{y}$
and $\|\bm{x}\|>2B_{y}$, then it holds that

$P(\bm{x},\bm{y})\geq-
B_{y}\left(a\|\bm{x}\|^{1+\frac{2}{m}}+2^{\frac{2}{m}}b\|\bm{x}\|^{1-\frac{2}{m}}\right).$
(49)

Provided in Appendix A. We are now ready to present a result on the robustness
of the FxTS adaptation law proposed in (32).

######  Theorem 4.

Consider a perturbed dynamical system of the form (45). Suppose that
Assumption 6 holds, and that
$\mathcal{N}(\Delta(\bm{x}(t)))=\mathcal{N}(\Delta(\bm{x}(0)))$, $\forall
t\in[0,T]$. Then, under the adaptation law (32), with
$\lambda_{min}(\bm{\Gamma})\geq
2(\frac{\Upsilon}{\underaccent{\bar}{\sigma}(\bm{M})})^{2}$, there exist
neighborhoods $D_{0}$ and $D$ of $\Omega$ such that for all
$\hat{\bm{\theta}}(0)\in D_{0}$, the trajectories of (36) satisfy
$\tilde{\bm{\theta}}_{R}(t)\in D_{0}$ for all $t\geq 0$, and reach $D$ within
a fixed time $T$, where
$D=\\{\tilde{\bm{\theta}}_{R}\;|\;V(\tilde{\bm{\theta}}_{R})\leq 1\\}$, and

$\displaystyle D_{0}$
$\displaystyle=\begin{cases}\mathbb{R}^{p};&\Upsilon<Y,\\\
\left\\{\tilde{\bm{\theta}}_{R}\;|\;V(\tilde{\bm{\theta}}_{R})\leq\left(k\frac{\alpha_{3}-\sqrt{\alpha_{3}^{2}-4\alpha_{1}\alpha_{2}}}{2\alpha_{1}}\right)^{\mu}\right\\};&\Upsilon\geq
Y,\end{cases}$ (50) $\displaystyle T$
$\displaystyle\leq\begin{cases}\frac{\mu}{\alpha_{1}k_{1}}\left(\frac{\pi}{2}-\tan^{-1}k_{2}\right);&\Upsilon<Y,\\\
\frac{\mu k}{(1-k)\sqrt{\alpha_{1}\alpha_{2}}};&\Upsilon\geq Y,\end{cases}$
(51)

where $\mu>2$, $0<k<1$, $k_{1}$ and $k_{2}$ are given by (16) and (17)
respectively, and

$\displaystyle\alpha_{1}$
$\displaystyle=2^{\frac{-2}{\mu}}ak_{V}^{2+\frac{2}{\mu}},$ (52)
$\displaystyle\alpha_{2}$
$\displaystyle=2^{\frac{2}{\mu}}bk_{V}^{2-\frac{2}{\mu}},$ (53)
$\displaystyle\alpha_{3}$ $\displaystyle=a\Upsilon
k_{V}^{1+\frac{2}{\mu}}+2^{\frac{2}{\mu}}b\Upsilon k_{V}^{1-\frac{2}{\mu}},$
(54)

with $k_{V}$ defined by (31), and

$Y=2\frac{k_{V}^{2}\sqrt{ab}}{ak_{V}^{1+\frac{2}{\mu}}+2^{\frac{2}{\mu}}bk_{V}^{1-\frac{2}{\mu}}}.$
(55)

Consider the Lyapunov function candidate
$V=\frac{1}{2}\tilde{\bm{\theta}}_{R}^{T}\bm{\Gamma}^{-1}\tilde{\bm{\theta}}_{R}$.
Let $\bm{q}=\bm{M}\tilde{\bm{\theta}}_{R}$. We have under the adaptation law
(32) that

$\dot{V}=-\bm{q}^{T}\left(\bm{q}+\bm{\delta}\right)\left(a\|\bm{q}+\bm{\delta}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{q}+\bm{\delta}\|^{\frac{2}{\mu}}}\right),$

where $a,b>0$ and $\mu>2$. By definition of the vector inner product, it
follows that $\bm{q}^{T}\bm{q}>|\bm{q}^{T}\bm{\delta}|$ as long as
$\|\bm{q}\|>\|\bm{\delta}\|$. Thus, when $\|\bm{q}\|>\|\bm{\delta}\|$ it holds
that $\bm{q}^{T}(\bm{q}+\bm{\delta})>0$ and therefore that $\dot{V}<0$. By
Assumption 6 and the vector triangle inequality, we also obtain that
$\|\bm{q}+\bm{\delta}\|>\Upsilon$ whenever $\|\bm{q}\|>2\Upsilon$. We then
observe that $\|\bm{q}+\bm{\delta}\|>\frac{1}{2}\|\bm{q}\|$ for all
$\|\bm{q}\|>2\Upsilon$. As such, we have from Theorem 3 that whenever
$\|\bm{q}\|>2\Upsilon$ the following holds:

$\displaystyle\dot{V}\leq$
$\displaystyle-\bm{q}^{T}\bm{q}\left(a\left(\frac{\|\bm{q}\|}{2}\right)^{\frac{2}{\mu}}+b\left(\frac{2}{\|\bm{q}\|}\right)^{\frac{2}{\mu}}\right)-F(\bm{q},\bm{\delta})$
$\displaystyle\leq$
$\displaystyle-\alpha_{1}V^{1+\frac{1}{\mu}}-\alpha_{2}V^{1-\frac{1}{\mu}}-F(\bm{q},\bm{\delta})<0,$

where $\alpha_{1}$ and $\alpha_{2}$ are given by (52) and (53), and
$F(\bm{q},\bm{\delta})=\bm{q}^{T}\bm{\delta}\left(a\|\bm{q}+\bm{\delta}\|^{\frac{2}{\mu}}+b\|\bm{q}+\bm{\delta}\|^{-\frac{2}{\mu}}\right)$.
Then, using Lemma 3 it follows that

$F(\bm{q},\bm{\delta})\geq-\Upsilon\left(a\|\bm{q}\|^{1+\frac{2}{\mu}}+2^{\frac{2}{\mu}}b\|\bm{q}\|^{1-\frac{2}{\mu}}\right),$

and therefore that

$\displaystyle\dot{V}$
$\displaystyle\leq-\alpha_{1}V^{1+\frac{1}{\mu}}-\alpha_{2}V^{1-\frac{1}{\mu}}+a\Upsilon\|\bm{q}\|^{1+\frac{2}{\mu}}+2^{\frac{2}{\mu}}b\Upsilon\|\bm{q}\|^{1-\frac{2}{\mu}},$
$\displaystyle\leq-\alpha_{1}V^{1+\frac{1}{\mu}}-\alpha_{2}V^{1-\frac{1}{\mu}}+\beta_{1}V^{\frac{1}{2}+\frac{1}{\mu}}+\beta_{2}V^{\frac{1}{2}-\frac{1}{\mu}},$

where $\beta_{1}=a\Upsilon k_{V}^{1+\frac{2}{\mu}}$ and
$\beta_{2}=2^{\frac{2}{\mu}}b\Upsilon k_{V}^{1-\frac{2}{\mu}}$, and which, for
all $V\geq 1$, we observe obeys

$\dot{V}\leq-\alpha_{1}V^{1+\frac{1}{\mu}}-\alpha_{2}V^{1-\frac{1}{\mu}}+\alpha_{3}V,$
(56)

where $\alpha_{3}=\beta_{1}+\beta_{2}$. Now, observe that
$\frac{1}{2}\lambda_{max}^{-1}(\bm{\Gamma})\|\tilde{\bm{\theta}}\|^{2}\leq
V\leq\frac{1}{2}\lambda_{min}^{-1}(\bm{\Gamma})\|\tilde{\bm{\theta}}\|^{2}$,
and
$\underaccent{\bar}{\sigma}^{2}(\bm{M})\|\tilde{\bm{\theta}}\|^{2}\leq\|\bm{q}\|^{2}\leq\sigma_{max}^{2}(\bm{M})\|\tilde{\bm{\theta}}\|^{2}$,
and thus

$V_{min}(\tilde{\bm{\theta}}_{R})=\frac{1}{2}g_{1}\|\bm{q}\|^{2}\leq
V(\tilde{\bm{\theta}}_{R})\leq\frac{1}{2}g_{2}\|\bm{q}\|^{2}=V_{max}(\tilde{\bm{\theta}}_{R}),$

where $g_{1}=(\lambda_{max}(\bm{\Gamma})\sigma_{max}^{2}(\bm{M}))^{-1}$ and
$g_{2}=(\lambda_{min}(\bm{\Gamma})\underaccent{\bar}{\sigma}^{2}(\bm{M}))^{-1}$.
Then, since (56) holds for all $V\geq 1$ and we have shown that $\dot{V}<0$
for all $\|\bm{q}\|>2\Upsilon$, we have that $\dot{V}<0$ for all $V>1$ if
$V_{max}(\tilde{\bm{\theta}}_{R})\leq 1$ when $\|\bm{q}\|=2\Upsilon$. To
satisfy this condition, we enforce that $\lambda_{min}(\bm{\Gamma})\geq
2(\frac{\Upsilon}{\underaccent{\bar}{\sigma}(\bm{M})})^{2}$.

We now have that for all $\|\bm{q}\|>2\Upsilon$, both $\dot{V}<0$ and (56)
hold. Thus, it follows from using Lemma 2 in combination with [17, Theorem 1]
that for all $\tilde{\bm{\theta}}_{R}(0)\in D_{0}$, the trajectories of (36)
satisfy $\tilde{\bm{\theta}}_{R}(t)\in D_{0}$ for all $t\geq 0$, and reach the
set $D$ within a fixed time $T$, where we have rearranged the terms
$\alpha_{1},\alpha_{2},\alpha_{3}$ to rewrite the conditions $r<1,r\geq 1$
from [17, Theorem 1] as $\Upsilon<Y,\Upsilon\geq Y$, where $Y$ is given by
(55). This completes the proof.

We now formalize the robust-adaptive CBF (raCBF) condition for forward-
invariance of sets $S$ and $S_{r}$, defined in (4) and (7).

######  Theorem 5.

Consider a dynamical system belonging to the class of nonlinear, control-
affine systems described by (1), and a subset of the safe set, $S_{r}\subseteq
S$, defined by (7). Under the premises of Corollary 1, the set $S_{r}$ is
rendered forward-invariant if the parameter estimates, $\hat{\bm{\theta}}$,
are adapted according to (32) and there exists a control input, $\bm{u}$, for
which the following condition holds $\forall\bm{x}\in S_{r}$, $\forall t\geq
0$:

$\displaystyle\sup_{\bm{u}\in
U}\left\\{L_{f}h_{r}(\bm{x})+L_{g}h_{r}(\bm{x})\bm{u}\right\\}\geq-\alpha\left(h_{r}(\bm{x}))\right)+\bm{r}(t,\hat{\bm{\theta}})$
(57)

where
$\bm{r}(t,\hat{\bm{\theta}})=\mathrm{Tr}(\bm{\Gamma}^{-1})\eta(t)\dot{\eta}(t)+\bm{\nu}(\hat{\bm{\theta}})$,
$\eta(t)$ is given by (41), and

$\displaystyle\dot{\eta}(t)$
$\displaystyle=-c_{1}\sqrt{\frac{\lambda_{max}(\bm{\Gamma})}{2}}\left(\sqrt{\frac{c_{1}}{c_{2}}}\tan\left(A(t)\right)\right)^{\frac{\mu}{2}-1}\sec^{2}\left(A(t)\right)$
(58) $\displaystyle\bm{\nu}(\hat{\bm{\theta}})$
$\displaystyle=\sum_{i=1}^{p}\min\left\\{C_{i}P_{\Theta}(\hat{\theta}_{i}-\eta),C_{i}P_{\Theta}(\hat{\theta}_{i}+\eta)\right\\}$
(59)

where $A(t)$ is given by (42) and $C_{i}$ denotes the $i^{th}$ column of
$L_{\Delta}h_{r}(\bm{x})$ for $i\in\\{1,\ldots,p\\}$, with
$P_{\Theta}:\mathbb{R}^{p}\rightarrow\Theta$ the vector projection operator
onto $\Theta$, as defined in [28].

We begin by recalling that $S_{r}$ is rendered forward-invariant if (9) holds
for all $\bm{x}\in S_{r}$. Now, we replace the constant $\bm{\vartheta}$ from
(8) with the time-varying $\bm{\eta}(t)$ (and henceforth omit the argument).
It follows from (8) and $\bm{\Gamma}$ being diagonal that

$\displaystyle\frac{\partial h_{r}}{\partial\bm{x}}\dot{\bm{x}}$
$\displaystyle=L_{f}h_{r}(\bm{x})+L_{g}h_{r}(\bm{x})\bm{u}+L_{\Delta}h_{r}(\bm{x})\bm{\theta}^{*},$
(60) $\displaystyle\frac{\partial h_{r}}{\partial\bm{\eta}}\dot{\bm{\eta}}$
$\displaystyle=-\bm{\eta}^{T}\Gamma^{-1}\dot{\bm{\eta}}=-\mathrm{Tr}(\Gamma^{-1})\eta\dot{\eta},$
(61)

where $\dot{\eta}$ given by (58) is obtained by differentiating (41). We note
that (60) includes the unknown term, $L_{\Delta}h_{r}(\bm{x})\bm{\theta}^{*}$,
and now must prove that $\bm{\nu}$ given by (59) compensates for this unknown,
and thus that (57) implies (9).

First, we have that
$L_{\Delta}h_{r}(\bm{x})\bm{\theta}^{*}=L_{\Delta}h_{r}(\bm{x})\tilde{\bm{\theta}}+L_{\Delta}h_{r}(\bm{x})\hat{\bm{\theta}}=L_{\Delta}h_{r}(\bm{x})\tilde{\bm{\theta}}_{R}+L_{\Delta}h_{r}(\bm{x})\hat{\bm{\theta}}$,
where $\tilde{\bm{\theta}}_{R}$ is defined as in (33). Thus, it follows that
(57) implies (9) if $L_{\Delta}h_{r}(\bm{x})\bm{\theta}^{*}\geq
L_{\Delta}h_{r}(\bm{x})\tilde{\bm{\theta}}_{R}+L_{\Delta}h_{r}(\bm{x})\hat{\bm{\theta}}$.
Second, since (41) defines an upper bound on the error at time $t$,
$\|\tilde{\bm{\theta}}_{R}(t)\|_{\infty}=\eta(t)$, let the set of admissible
unknown parameters at time $t$ be denoted as $\Phi(t)$, where

$\Phi(t)=\Theta\cap\\{\bm{\theta}\in\mathbb{R}^{p}:\;\|\bm{\theta}_{R}-\hat{\bm{\theta}}_{R}(t)\|_{\infty}\leq\eta(t)\\},$
(62)

where $\bm{\theta}=\bm{\theta}_{R}+\bm{\theta}_{N}$, with
$\bm{\theta}_{R}\in\textrm{row}(\bm{M})$ and
$\bm{\theta}_{N}\in\mathcal{N}(\bm{M})$. By definition, $\Phi(t)$ is compact
and convex. As such, consider the following minimization problem:

$J_{\phi}=\min_{\bm{\phi}\in\Phi(t)}L_{\Delta}h_{r}(x)\bm{\phi},$ (63)

which is a constrained linear program over a compact domain. It is well known
that for this class of problems there exists a minimum,
$J_{\phi}=L_{\Delta}h_{r}(\bm{x})\bm{\phi}^{*}$.

Now, since we can express
$L_{\Delta}h_{r}(x)\bm{\phi}=\sum_{i=1}^{p}C_{i}\phi_{i}$ with $C_{i}$ as the
ith column of $L_{\Delta}h_{r}(x)$, we have that
$\bar{\phi}_{1},\;\ldots\;,\bar{\phi}_{p}$ are the minimizers of the following
$p$ constrained linear programs:

$\min_{\hat{\theta}_{i}-\eta\leq\phi_{i}\leq\hat{\theta}_{i}+\eta}C_{i}\phi_{i},\quad\forall
i\in\\{1,\ldots,p\\}$

Thus, for all $C_{i}\neq 0$ it follows that
$\bar{\phi}_{i}=\hat{\theta}_{i}-\eta$ or
$\bar{\phi}_{i}=\hat{\theta}_{i}+\eta$, and $\bar{\phi}_{i}$ can take any
value for $C_{i}=0$. Since it is possible that $\bar{\phi}_{i}\notin\Theta$,
we apply the projection operator to $\bar{\phi}_{i}$ and write that

$\zeta_{i}^{*}=P_{\Theta}\bar{\phi}_{i}.$

By linearity, it holds that

$\displaystyle C_{i}\zeta_{i}^{*}$
$\displaystyle=\min\left\\{C_{i}P_{\Theta}(\hat{\theta}_{i}-\eta),C_{i}P_{\Theta}(\hat{\theta}_{i}+\eta)\right\\}\leq
C_{i}\phi_{i},$

for all $\phi_{i}\in[\hat{\theta}_{i}-\eta,\hat{\theta}_{i}+\eta]$. As such,
we denote $\bm{\nu}=\sum_{i=1}^{p}C_{i}\zeta_{i}^{*}$, for which it holds that
$\bm{\nu}\leq L_{\Delta}h(\bm{x})\bm{\theta}^{*}$. Thus, we have that (57)
implies (9). It follows, then, that as long as there exists a control input,
$\bm{u}$, which satisfies (57), the set $S_{r}$ defined by (7) is forward-
invariant. This completes the proof. In the next section, we demonstrate the
effectiveness of our proposed adaptation law and adaptive safety condition in
two very different situations: a goal-reaching example which requires vehicle
to maneuver through a narrow gap, and a quadrotor trajectory-tracking problem.

##  5 Numerical Case Studies11footnotemark: 1

22footnotetext: All simulation parameters, code, and data are available online
at https://github.com/6lackmitchell/fxt_adaptation

###  5.1 “Shoot the Gap”: A Comparative Study

In our first numerical study, we investigate whether our parameter adaptation
law improves controller performance for a single-integrator system with
parametric uncertainty, whose regressor may or may not be full-rank, under the
rate measurement scheme from Section 3.1 relative to other techniques in the
literature. Table 1 provides the legend codes used to refer to these other
works.

Table 1:

Controllers from the Literature
---
Authors | Citation | Legend Code
Black et al. | [9] | BLA
Black et al. | [10] | BLR
Lopez et al. |  [27] (w/o SMID) | LOP
Lopez et al. |  [27] (w/ SMID) | LSM
Taylor et al. | [44] | TAY
Zhao et al. | [51] | ZHA
Proposed Method |  | PRO

Note: [27] presents RaCBF-based control formulations with and without SMID for
parameter estimation. We consider both.

Specifically, we consider the problem where a point mass must pass through a
narrow gap between two unsafe regions in order to reach the goal. We call this
the “Shoot the Gap” scenario, and depict its setup in Figure 1.

Figure 1: Problem Setup for the Shoot the Gap scenario.

Our system model is

$\displaystyle\dot{z}=\begin{bmatrix}\dot{x}\\\ \dot{y}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}\begin{bmatrix}u_{x}\\\
u_{y}\end{bmatrix}+\Delta(z)\begin{bmatrix}\theta_{1}\\\
\theta_{2}\end{bmatrix},$ (64)

where $z=[x\;y]^{T}$ is the state vector comprised of $x$ and $y$, the lateral
and longitudinal position coordinates with respect to an inertial frame,
$\theta_{1}$, $\theta_{2}$ are constant parameters that are unknown within
some bounds, and $\Delta$ is the regressor. We first investigate controller
performance when the regressor is full-rank, i.e. $\Delta=\Delta_{F}$, where

$\Delta_{F}(z)=K_{\Delta}\begin{bmatrix}1+\sin^{2}(2\pi f_{1}x)&0\\\
0&1+\cos^{2}(2\pi f_{2}y)\end{bmatrix},$

with $K_{\Delta},f_{1},f_{2}>0$, and subsequently examine performance under a
rank-deficient regressor, $\Delta=\Delta_{D}$, where

$\Delta_{D}(z)=\begin{bmatrix}-ax&-2ax\\\ -\frac{a}{2}x&-ax\end{bmatrix},$

and $a=1/2$.

(a) States

(b) Control inputs

(c) Minimum CBF value

Figure 2: Evolutions of the states, control inputs, and control barrier
functions for the full-rank regressor “Shoot the Gap” example.

(a) States

(b) Control inputs

(c) Minimum CBF value

Figure 3: Evolutions of the states, control inputs, and control barrier
functions for the rank-deficient regressor “Shoot the Gap” example.

To encode the goal-convergence criterion we define the following control
Lyapunov function (CLF):

$\displaystyle V(z)=K_{V}(x^{2}+y^{2}),$ (65)

where $K_{V}=1$. The safe states are those residing outside of the two
ellipses shown in Figure 1, which results in the following CBFs:

$\displaystyle
h_{i}(z)=\frac{(x-x_{i})^{2}}{a^{2}}+\frac{(y-y_{i})^{2}}{b^{2}}-1,$ (66)

for $i\in\\{1,2\\}$, where $x_{1}$, $x_{2}$, $y_{1}$, $y_{2}$, $a$, and $b$
are parameters that define the location, size, and shape of the ellipses.

We choose the CLF-CBF-QP framework ([50, 2, 10]) for computing the control
inputs. To more fairly compare our proposed approach with the controllers from
the literature, we tested the FxT-CLF condition in all controllers in addition
to their individual approaches to stabilization. Because we found no
meaningful differences between the native and FxT-CLF implementations in their
abilities to “shoot the gap”, we present results for the latter cases. Our
control framework is then:

$\displaystyle\min_{\bm{u},\delta_{0},\delta_{1},...,\delta_{q}}\frac{1}{2}\bm{u}^{T}Q\bm{u}$
$\displaystyle+q_{0}\delta_{0}^{2}+\sum_{i=1}^{q}p_{i}\delta_{i}^{2}$ (67a)
s.t. $\displaystyle-\bar{u}_{j}\leq$ $\displaystyle u_{j}\leq\bar{u}_{j}$
(67b) $\displaystyle 1$ $\displaystyle\leq\delta_{i}$ (67c) $\displaystyle
L_{f}V+L_{g}V\bm{u}$
$\displaystyle+\phi_{V}\leq\delta_{0}-c_{1}V^{\gamma_{1}}-c_{2}V^{\gamma_{2}}$
(67d) $\displaystyle L_{f}h_{i}+L_{g}h_{i}\bm{u}$
$\displaystyle+\phi_{h}\geq-\delta_{i}h_{i}$ (67e)

$\forall i\in\\{1,\ldots,q\\}$ and $\forall j\in\\{1,\ldots,m\\}$, where we
have omitted the arguments of $V$ and $h_{i}$ for conciseness. For this
problem $u_{1}=u_{x}$ and $u_{2}=u_{y}$, $\delta_{0}$ is a relaxation
parameter (penalized by $q_{0}>0$) on the performance objective whose
inclusion aids the feasibility of the QP, $\delta_{i}$ allows for larger
negative values of $\dot{h}_{i}$ away from the boundary of the safe set, and
$p_{i}>0$ penalizes values of $\delta_{i}$. The terms
$\phi_{V}=\phi_{V}(\bm{x},\Delta(\bm{x}),\hat{\bm{\theta}},\eta)$ and
$\phi_{h}=\phi_{h}(\bm{x},\Delta(\bm{x}),\hat{\bm{\theta}},\eta)$ are left
intentionally vague and are used as placeholders that represent the specific
way in which each work from the literature treats the parametric uncertainty
in the system dynamics. For example, in this paper we propose the fixed-time
adaptation law (32) to learn the uncertain term in the system dynamics,
whereas in [10] rCBFs are used to protect against the worst-case disturbance.
The $\phi_{V}$ and $\phi_{h}$ terms encode these differences between the
considered works. To summarize, (67b) enforces input constraints, (67c)
imposes a lower bound on the relaxation coefficients $\delta_{i}$, (67d)
encodes FxT convergence to the goal, and safety is guaranteed by (67e).

The simulated state, control, and CBF trajectories for the case of the full-
rank regressor, $\Delta=\Delta_{F}$, are shown in Figure 2. It is worth noting
from Figure 2(a) that the only controllers that successfully reach the goal
while maintaining safety at all times are those that closely estimate the
unknown parameters in the system dynamics, i.e. the BLA, LSM, and PRO
approaches. In fact, we observe from Figure 4 that, in accordance with Theorem
3, the parameter estimates, $\hat{\theta}$, converge to their true values
within fixed-time $T$ given by (30). Figure 2(b) further highlights that our
technique accomplishes safe goal-reaching while avoiding the undesirable
chattering control behavior exhibited by the TAY approach [44].

When a rank-deficient regressor appears in (64), i.e. $\Delta=\Delta_{D}$,
however, both the BLA and LSM methods that succeed under a full-rank regressor
fail; each requires the inversion of a non-invertible matrix. As illustrated
by Figure 3, our proposed controller succeeds in safely reaching the goal by
learning a parameter vector, $\hat{\bm{\theta}}$, within a fixed time that
satisfies
$\Delta_{D}(\bm{x})\hat{\bm{\theta}}=\Delta_{D}(\bm{x})\bm{\theta}^{*}$, as
supported by Figure 5. Under these dynamics, the proposed approach is the only
controller that successfully reaches the goal.

In the case of the full-rank regressor, it is evident that the ability of the
adaptation law (32) to learn the system uncertainty in fixed-time allowed our
controller to succeed in driving the state to the goal where others did not.
Doing so despite a rank-deficient regressor further demonstrates that our
adaptation law adds value to controllers for the class of systems under
consideration, especially when none of the considered approaches from the
literature were capable of succeeding under such conditions.

Figure 4: Parameter estimate trajectories for the full-rank regressor “Shoot
the Gap” example. Figure 5: Parameter estimate trajectories for the rank-
deficient regressor “Shoot the Gap” example.

###  5.2 Quadrotor with unknown drag coefficients

Our next study examines a simulated quadrotor in a wind field. We consider a
quadrotor whose objective is to track a Gerono lemniscate (figure-eight)
trajectory while obeying attitude and altitude constraints despite wind
perturbing the nominal system dynamics.

For our nominal system model, we consider the 6-DOF rigid-body dynamic model
of the quadrotor as described in [6]. We denote
$\bm{\chi}=[x\;y\;z\;u\;v\;w\;\phi\;\theta\;\psi\;p\;q\;r]^{T}$ as the state,
where $x$, $y$, and $z$ are the position coordinates (in m) with respect to an
inertial frame, $u$, $v$, and $w$ are the translational velocities (in m/s)
with respect to the body-fixed frame, $\phi$, $\theta$, and $\psi$ (in rad)
are the roll, pitch, and yaw Euler angles defining a ZYX rotation from the
inertial frame to the body-fixed frame, and $p$, $q$, and $r$ are the roll,
pitch, and yaw rates (in rad/s) defined with respect to the body-fixed frame.
To model the wind, we note that much attention has been given to its effect on
the nominal dynamics of a quadrotor system [42, 46, 19]. Specifically, the
phenomena of blade flapping, induced velocity, and aerodynamic drag have been
studied extensively. We neglect the effects of blade flapping and induced
velocity, and instead focus on aerodynamic drag.

####  5.2.1 Constant Wind Field

We first assume that a constant, known wind field perturbs the quadrotor,
whose drag coefficient vector $\bm{C}_{d}=[C_{x}\;C_{y}\;C_{z}]^{T}$ is
unknown, where $C_{x}$, $C_{y}$, and $C_{z}$ are the components in the
principal body-fixed directions. The system is then described by:

$\displaystyle\begin{pmatrix}\dot{x}\\\ \dot{y}\\\ \dot{z}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}c\theta c\psi&s\phi s\theta c\psi-c\phi
s\psi&c\phi s\theta c\psi+s\phi s\psi\\\ c\theta s\psi&s\phi s\theta
s\psi+c\phi c\psi&c\phi s\theta s\psi-s\phi c\psi\\\ s\theta&-s\phi
c\theta&-c\phi c\theta\end{pmatrix}\begin{pmatrix}u\\\ v\\\ w\end{pmatrix}$
$\displaystyle\begin{pmatrix}\dot{u}\\\ \dot{v}\\\ \dot{w}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}rv-qw\\\ pw-ru\\\ qu-
pw\end{pmatrix}+\begin{pmatrix}-gs\theta\\\ gc\theta s\phi\\\ gc\theta
c\phi\end{pmatrix}+\frac{1}{M}\begin{pmatrix}0\\\ 0\\\
-F\end{pmatrix}+\Delta_{a}(\bm{\chi})\bm{C}_{d}$
$\displaystyle\begin{pmatrix}\dot{\phi}\\\ \dot{\theta}\\\
\dot{\psi}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}1&s\phi t\theta&c\phi
t\theta\\\ 0&c\phi&-s\phi\\\
0&\frac{s\phi}{c\theta}&\frac{c\phi}{c\theta}\end{pmatrix}\begin{pmatrix}p\\\
q\\\ r\end{pmatrix}$ (68) $\displaystyle\begin{pmatrix}\dot{p}\\\ \dot{q}\\\
\dot{r}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{J_{y}-J_{z}}{J_{x}}qr\\\
\frac{J_{z}-J_{x}}{J_{y}}pr\\\
\frac{J_{x}-J_{y}}{J_{z}}pq\end{pmatrix}+\begin{pmatrix}\frac{1}{J_{x}}\tau_{\phi}\\\
\frac{1}{J_{y}}\tau_{\theta}\\\ \frac{1}{J_{z}}\tau_{\psi}\end{pmatrix}$

where $g$ is the acceleration due to gravity (in m/sec2), the functions
$\sin$, $\cos$, and $\tan$ are denoted $s$, $c$, and $t$ for brevity, $M$ is
the mass of the quadrotor (in kg), $J_{x}$, $J_{y}$, and $J_{z}$ are the
principal moments of inertia (in kg$\cdot$m2), $F$ is the thrust of the rotors
(in N), and $\tau_{\phi}$, $\tau_{\theta}$, and $\tau_{\psi}$ are rolling,
pitching, and yawing torques (in N$\cdot$m) respectively due to the rotors.
Since (5.2.1) is affine in both the control,
$\bm{u}=[F\;\tau_{\phi}\;\tau_{\theta}\;\tau_{\psi}]^{T}$, and unknown
parameters, $\bm{C}_{d}$, it can be written as
$\dot{\bm{\chi}}=f(\bm{\chi})+g(\bm{\chi})\bm{u}+\Delta(\bm{\chi})\bm{C}_{d}$,
where $\Delta(\bm{\chi})=[\bm{0}_{3\times
3}\;\Delta_{a}(\bm{\chi})\;\bm{0}_{6\times 3}]^{T}$, with

$\displaystyle\Delta_{a}(\bm{\chi})=K_{\Delta}\frac{\|\bm{v}_{r}\|}{M}\begin{bmatrix}v_{r,1}&0&0\\\
0&v_{r,2}&0\\\ 0&0&v_{r,3}\end{bmatrix}$

where $K_{\Delta}>0$, $\bm{v}_{r}=\bm{R}\bm{v}_{w}-\bm{v}_{q}$ is the relative
wind-velocity vector in the body-fixed frame, with $\bm{v}_{q}=[u\;v\;w]^{T}$
the quadrotor velocity vector in the body-fixed frame, $\bm{v}_{w}$ being the
wind velocity vector in the inertial frame, and $\bm{R}$ being the rotation
matrix from the inertial to the body-fixed frame. Additionally, $v_{r,1}$,
$v_{r,2}$, and $v_{r,3}$ are the principal components of $\bm{v}_{r}$ such
that the term $\Delta(\bm{\chi})\bm{C}_{d}$ models the effect of aerodynamic
drag acting on the center of mass of the quadrotor.

We now define $\bm{v}=[\bm{u}\;\delta_{1}\;\delta_{2}]^{T}\in\mathbb{R}^{6}$
and use the following CBF-QP-based controller [29, 14, 10]:

$\displaystyle\bm{v}^{*}=\operatorname*{arg\,min}_{\bm{v}}\frac{1}{2}\|\bm{u}$
$\displaystyle-\bm{u}_{nom}\|^{2}+p_{1}\delta_{1}^{2}+p_{2}\delta_{2}^{2}$
(69a) s.t. $\displaystyle A\bm{v}$ $\displaystyle\leq\bm{b}$ (69b)
$\displaystyle L_{f}h_{1}(\bm{\chi})+L_{g}h_{1}(\bm{\chi})\bm{u}$
$\displaystyle\geq-\delta_{1}h_{1}(\bm{\chi})-\bm{r}(t,\hat{\bm{C}}_{d})$
(69c) $\displaystyle L_{f}h_{2}(\bm{\chi})+L_{g}h_{2}(\bm{\chi})\bm{u}$
$\displaystyle\geq-\delta_{2}h_{2}(\bm{\chi})-\bm{r}(t,\hat{\bm{C}}_{d})$
(69d)

where $p_{1},p_{2}>0$. The cost function (69a) seeks to minimize the deviation
of $\bm{u}^{*}$ from some nominal control $\bm{u}_{nom}$, and to minimize the
slack variables $\delta_{1}^{*},\delta_{2}^{*}$ penalized by $p_{1},p_{2}>0$;
(69b) encodes input constraints on $\bm{u}^{*}$ and enforces that
$\delta_{1}^{*},\delta_{2}^{*}\geq\underaccent{\bar}{\delta}>0$; and (69c) and
(69d) enforce safety according to the following CBFs,

$\displaystyle h_{1}(\bm{\chi})$
$\displaystyle=1-\left(\frac{z-c_{z}}{p_{z}}\right)^{n_{z}}$ (70)
$\displaystyle h_{2}(\bm{\chi})$
$\displaystyle=\cos(\phi)\cos(\theta)-\cos(\alpha),$ (71)

where $c_{z}=2.5$, $p_{z}=2.5$, and $n_{z}=2$ are parameters such that $h_{1}$
defines the safe altitude range, i.e. between $0$ and $5$ meters, and $h_{2}$
restricts the thrust vectoring angle of the quadrotor to be within $\alpha$
rad of the vertical. In this case, we selected $\alpha=\frac{\pi}{2}$ so that
the quadrotor could not flip upside-down. For the nominal control input
$\bm{u}_{nom}$, we use the tracking controller developed for quadrotors by
[40] and used for safe quadrotor control in [24].

We examined the quadrotor behavior under the proposed adaptation law (32)
using the state predictor scheme (23)-(29) (results denoted SP-CW), which is
defined here as

$\dot{\hat{\bm{C}}}_{d}=\bm{\Gamma}\bm{W}^{T}\bm{e}\left(a\|\bm{e}\|^{\frac{2}{\mu}}+\frac{b}{\|\bm{e}\|^{\frac{2}{\mu}}}\right),$

where $\bm{e}=\bm{\chi}-\bm{z}$, with

$\displaystyle\dot{\bm{z}}$
$\displaystyle=f(\bm{\chi})+g(\bm{\chi})\bm{u}+\Delta(\bm{\chi})\hat{\bm{C}}_{d}+k_{e}\bm{e}+\bm{W}\dot{\hat{\bm{C}}}_{d},$
$\displaystyle\bm{z}(0)$ $\displaystyle=\bm{\chi}(0),$

and

$\dot{\bm{W}}=-k_{e}\bm{W}+\Delta(\bm{\chi}),\quad\bm{W}(0)=\bm{0}_{n\times
p},$

where $a,b,\mu=5$ and $k_{e}=10$. For comparison, we additionally studied the
case where $\dot{\hat{\bm{C}}}_{d}=\bm{0}_{3\times 1}$ (results denoted 0-CW).
The wind-velocity vector $\bm{v}_{w}$ was held constant at
$\bm{v}_{r}=[10,\;-8,\;-5]^{T}$, which resulted in the regressor being full-
rank for all $t\geq 0$. To account for the warm-up time required in the state
prediction scheme, we set $T_{w}=0.1$ sec and added it to the theoretical
bound given by (30). It is possible that a longer warm-up time could result in
a decrease in the minimum non-zero singular value of the regressor, thereby
reducing the upper bound on convergence time, but this is problem-dependent
and can be viewed as a tuning parameter. As such, it may be seen from Figure 6
that the estimates of the coefficients of drag converged to their true values
within the fixed-time bound. Figures 7 and 8 highlight that the controller is
able to safely track the reference trajectory using the adaptation law (32)
with the state prediction scheme, whereas poor tracking, albeit safe, is
achieved without parameter adaptation. Control trajectories are provided in
Figure 9.

Figure 6: Principal coefficient of drag estimates for the quadrotor in a 1)
constant (CW) and 2) gusty (WG) wind field under the proposed controller with
the state prediction (SP) scheme. Figure 7: Quadrotor XY trajectories as the
controller seeks to track the reference trajectory (RT) in a wind field.
Figure 8: CBF trajectories for the quadrotor numerical study. Figure 9:
Control inputs for the quadrotor numerical study.

####  5.2.2 Unknown Wind Field

We also simulate the quadrotor dynamical model (5.2.1) using the controller
(69) and state predictor scheme in an unknown wind field using the wind gust
model from [16] (denoted SP-WG). The wind gusts appear in the dynamics as an
unknown, additive disturbance $\bm{d}(t,\bm{\chi})=[\bm{0}_{3\times
3}\;\bm{d}_{a}(t,\bm{\chi})\;\bm{0}_{6\times 3}]^{T}$ such that the system
takes the form of (45), where

$\bm{d}_{a}(t,\bm{\chi})=\left(K_{\Delta}\frac{\|\bm{v}_{r}^{*}\|}{M}\begin{bmatrix}v_{r,1}^{*}&0&0\\\
0&v_{r,2}^{*}&0\\\
0&0&v_{r,3}^{*}\end{bmatrix}-\Delta_{a}(\bm{\chi})\right)\begin{bmatrix}C_{x}\\\
C_{y}\\\ C_{z}\end{bmatrix},$

with $\bm{v}_{r}^{*}=\bm{R}(\bm{v}_{w}+\bm{v}_{G})-\bm{v}_{q}$ the relative-
wind velocity vector considering both the constant, known wind velocity
$\bm{v}_{w}$ from the previous example and unknown gust velocity $\bm{v}_{G}$
generated using the model from [16], with principal components $v_{r,1}^{*}$,
$v_{r,2}^{*}$, and $v_{r,3}^{*}$.

In the presence of an unknown disturbance $\bm{d}(t,\bm{\chi})$, it is not
guaranteed that the estimates of the drag coefficients $\hat{\bm{C}}_{d}$
converge to the set $\Omega$ but rather to a neighborhood of this set as
specified in Theorem 4. While the specific simulation parameters may be viewed
on Github, they lead to $Y=3.65$ using (55) and $\Upsilon=0.69$ from the wind
gust model. Thus, $\Upsilon<Y$, and we have that $D_{0}=\mathbb{R}^{p}$ and
the fixed-time bound $T$ is given by (51) plus the warm-up time $T_{w}$. In
addition, the choice of $\lambda_{min}(\bm{\Gamma})=63.16$ satisfies the
requirement that
$\lambda_{min}(\bm{\Gamma})>2(\frac{\Upsilon}{\underaccent{\bar}{\sigma}(\bm{M})})^{2}=26.36$,
where $\underaccent{\bar}{\sigma}(\bm{M}(t))\geq 0.19$ for all $t\geq T_{w}$.
We see from Figures 7 and 8 that the controller achieves safe, accurate
tracking despite the parameter estimates not converging to the true values. We
hypothesize that this is because the parameter estimates are capturing the
effect of the unknown perturbation in the system dynamics in addition to the
parameter-affine disturbance, i.e.
$\Delta(\bm{\chi})\hat{\bm{C}}_{d}\approx\Delta(\bm{\chi})\bm{C}_{d}+d(t,\bm{\chi})$,
though we leave the analysis to future work.

##  6 Conclusion

In this study on the merits of fixed-time adaptation for safe control under
parametric model uncertainty, we presented a novel adaptation law that learns
additive, parametric uncertainty associated with a class of nonlinear,
control-affine dynamical systems within a fixed time without requiring that
the regressor meet rank or persistence of excitation conditions. We used a
result on fixed-time stability to show the robustness of the proposed law
against a class of bounded disturbances. Then, we synthesized the adaptation
law with a robust, adaptive CBF in a quadratic program-based control law, and
provided an upper bound on the parameter estimation error as an explicit
function of time. We then studied the performance of our method on a simple,
2D single integrator system in relation to several recent works from the
literature and demonstrated that our contribution succeeds in navigating near
unsafe regions of the state space where others fail. We further demonstrated
the value of our approach on a safe trajectory-tracking problem using the
quadrotor dynamical model in wind fields both known and unknown.

In the future, we intend to address the problem of safe control for parameter-
affine systems in which the uncertain parameters are time-varying, as well as
for systems with nonlinearly parameterized and non-parametric additive
disturbances in the system dynamics.

The authors would like to acknowledge the support of the National Science
Foundation award number 1931982.

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##  Appendix A Proof of Lemma 2

We first observe that $\bm{x}^{T}\bm{y}=\|\bm{x}\|\|\bm{y}\|\cos(\theta_{xy})$
and
$\|\bm{x}+\bm{y}\|^{2}=\|\bm{x}\|^{2}+\|\bm{y}\|^{2}+2\|\bm{x}\|\|\bm{y}\|\cos(\theta_{xy})$,
where $\theta_{xy}$ is the angle between vectors $\bm{x}$ and $\bm{y}$, and
rewrite (48) as $P(\bm{x},\bm{y})=P_{1}(x,y,z)+P_{2}(x,y,z)$, where
$x=\|\bm{x}\|$, $y=\|\bm{y}\|$, $z=\cos(\theta_{xy})$, and

$\displaystyle P_{1}(x,y,z)$
$\displaystyle=axyz\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}},$
$\displaystyle P_{2}(x,y,z)$
$\displaystyle=\frac{bxyz}{\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}}}.$

It is then true that for fixed $x$ and $y$,
$P(\bm{x},\bm{y})\geq\min_{z}P_{1}(x,y,z)+\min_{z}P_{2}(x,y,z)$. As such, we
proceed by solving for $\min_{z}P_{1}(x,y,z)$ and $\min_{z}P_{2}(x,y,z)$, and
omit the arguments when appropriate for conciseness. We first observe that
$x,y\geq 0$, which means that for all $z\geq 0$ it is true that
$P_{1},P_{2}\geq 0$. Additionally, by the AM-GM inequality333The arithmetic
mean-geometric mean inequality states that $x+y\geq 2\sqrt{xy}$, for $x,y>0$.
it is true that $x^{2}+y^{2}+2xyz>0$ for all $z\in(-1,0]$ and
$x^{2}+y^{2}+2xyz\geq 0$ for $z=-1$, thus it is obvious that there exists
$z\in[-1,0]$ such that $P_{1},P_{2}<0$. As such, we focus on the case where
$z\in[-1,0]$.

We have that

$\displaystyle\frac{\partial P_{1}}{\partial
z}=axy\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}}+\frac{2a}{m}x^{2}y^{2}z\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}-1},$

Setting $\frac{\partial P_{1}}{\partial z}=0$, we obtain that the candidate
minimizer is $z_{1}^{*}=-\frac{x^{2}+y^{2}}{2xy(1+\frac{1}{m})}$. Since
$\frac{\partial^{2}P_{1}}{\partial
z^{2}}=\frac{4a}{m}x^{2}y^{2}\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}-1}+(\frac{1}{m}-1)\frac{4a}{m}x^{3}y^{3}z\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}-2}$,
with $a>0$, $m>2$, it follows that $\frac{\partial^{2}P_{1}}{\partial
z^{2}}>0$ for all $x,y>0$ and $z\in(-1,0]$. Therefore, by second order
sufficient conditions for optimality $z_{1}^{*}$ is the unique minimizer for
$P_{1}$ over $z\in(-1,0]$. It also follows from the AM-GM inequality that
$z_{1}^{*}\leq-\frac{m}{m+1}$ and thus from $m>2$ that
$z_{1}^{*}\leq-\frac{2}{3}$. Then, consider in addition that
$\|\bm{x}+\bm{y}\|^{2}\geq x^{2}$ if and only if
$\cos(\theta_{xy})\geq-\frac{y}{2x}$. Since $x>2y\geq 0$ it is true that
$-\frac{y}{2x}\geq-\frac{1}{4}>z_{1}^{*}$, and thus
$x^{2}+y^{2}+2xyz_{1}^{*}<x^{2}$. As such, we obtain that

$\min_{z\in[-1,1]}P_{1}(x,y,z)=axyz_{1}^{*}\left(x^{2}+y^{2}+2xyz_{1}^{*}\right)^{\frac{2}{m}}\geq-
aB_{y}x^{1+\frac{2}{m}}.$ (72)

For $P_{2}$, we have by

$\displaystyle\frac{\partial P_{2}}{\partial
z}=\frac{bxy}{\left(x^{2}+y^{2}+2xyz\right)^{\frac{1}{m}}}-\frac{2bx^{2}y^{2}z}{m\left(x^{2}+y^{2}+2xyz\right)^{1+\frac{1}{m}}},$

and second order sufficient conditions for optimality that
$z_{2}^{*}=-\frac{x^{2}+y^{2}}{2xy(1-\frac{1}{m})}$ is the unique minimizer of
$P_{2}$ for fixed $x,y$. Now, by the AM-GM inequality and $m>2$ we have that
$z_{2}^{*}<-1$. Since $\frac{\partial P_{2}}{\partial z}>0$ for all $z\leq 0$,
it follows that $z_{2}^{*}=-1$ is the minimizer of $P_{2}$ over the domain
$z\in[-1,1]$. With $z_{2}^{*}=-1$ and $x>2y$, it is then true that
$\|\bm{x}+\bm{y}\|\geq\frac{1}{2}x$. As such, we obtain that

$\min_{z\in[-1,1]}P_{2}(x,y,z)=-\frac{bxy}{\left(x^{2}+y^{2}-2xy\right)^{\frac{1}{m}}}\geq-2^{\frac{2}{m}}bB_{y}x^{1-\frac{2}{m}}.$
(73)

Thus, by $P(\bm{x},\bm{y})\geq\min_{z}P_{1}(x,y,z)+\min_{z}P_{2}(x,y,z)$ we
obtain (49) from (72), (73), $x=\|\bm{x}\|$, and $y=\|\bm{y}\|$. This
completes the proof.