#  Robust Control Barrier Functions with Uncertainty Estimation

Ersin Daş1, Skylar X. Wei1, Joel W. Burdick1 *This work was supported by DARPA
under the LINC program.1E. Daş, S. X. Wei, and J. W. Burdick are with the
Department of Mechanical and Civil Engineering, California Institute of
Technology, Pasadena, CA 91125, USA.
<EMAIL_ADDRESS>
###### Abstract

This paper proposes a safety controller for control-affine nonlinear systems
with unmodelled dynamics and disturbances to improve closed-loop robustness.
Uncertainty estimation-based control barrier functions (CBFs) are utilized to
ensure robust safety in the presence of model uncertainties, which may depend
on control input and states. We present a new uncertainty/disturbance
estimator with theoretical upper bounds on estimation error and estimated
outputs, which are used to ensure robust safety by formulating a convex
optimization problem using a high-order CBF. The possibly unsafe nominal
feedback controller is augmented with the proposed estimator in two frameworks
(1) an uncertainty compensator and (2) a robustifying reformulation of CBF
constraint with respect to the estimator outputs. The former scheme ensures
safety with performance improvement by adaptively rejecting the matched
uncertainty. The second method uses uncertainty estimation to robustify
higher-order CBFs for safety-critical control. The proposed methods are
demonstrated in simulations of an uncertain adaptive cruise control problem
and a multirotor obstacle avoidance situation.

##  I Introduction

Safety-critical controller synthesis is a challenging problem in the
deployment of autonomous and cyber-physical systems [1, 2]. Control barrier
functions (CBFs) ensure safety by certifying forward invariance of a safe set
[3]. Thanks to the linearity of the CBF constraint for nonlinear control-
affine systems, safety requirements can be enforced with a CBF-QP to optimize
for pointwise control input [3].

Because known dynamics models are needed to synthesize the CBF constraints,
the standard formulation may be sensitive to inevitable model uncertainties
and disturbances. Numerous robust CBF approaches have been proposed to address
this issue [4, 5, 6]. However, these methods use a bound to represent the
unmodelled dynamics, which may be difficult to estimate in practice, and
usually result in undesired conservativeness and a reduction in performance.

Recently, connections between disturbance observer-based techniques, a well-
studied robust control tool [7], and CBFs have been made to actively estimate
and compensate for external disturbance to guarantee robust safety [8, 9, 10,
11, 12]. Despite promising results in joining CBFs with disturbance observers,
prior studies have been limited to state-dependent external disturbances.
However, in many applications, uncertainties may depend on both states and
control. Furthermore, these studies have not used active
disturbance/uncertainty compensation capabilities to purposefully design
observers that improve control performance and guarantee safety.

Motivated by these limitations, we propose a more general
uncertainty/disturbance estimation-based robust CBF framework for control-
affine nonlinear systems with state and control input dependent uncertainties.
We introduce a new uncertainty/disturbance estimator to observe unmodeled
dynamics in multiple-input-multiple-output nonlinear systems. Upper bounds for
the estimation error and estimated output are developed under boundedness and
local Lipschitz continuity assumptions on the uncertainty. Then, two
uncertainty estimator-based robust CBFs methods are proposed. Method 1: when a
matched uncertainty/disturbance dynamics condition is met, estimated values
are used to compensate for these dynamics and adaptively modify the CBF
condition against unmodelled system dynamics (see Fig. 1, top). Method 2
integrates the proposed estimator with higher-order CBFs to guarantee
robustness for high relative degree safety constraints and to obtain an
augmented safe set (see Fig. 1, bottom). To showcase our results, we simulate
an uncertain adaptive cruise control example (Method 1) and a multirotor
obstacle avoidance scenario (Method 2).

Figure 1: A block diagram of the proposed uncertainty estimator-based safe
control frameworks. Augmenting a given, and potentially unsafe nominal
controller with an error-bounded uncertainty estimator and safety filter,
guarantees that the uncertain nonlinear system states remain in an inner safe
set.

This paper is organized as follows. After preliminaries are introduced in
Section II, Section III states our problem of safety-critical control under
unmodeled dynamics. Section IV proposes and analyzes an uncertainty estimator,
and introduces estimator-based robust CBFs schemes. Simulations are presented
in Section IV, and Section V concludes the paper.

##  II Preliminaries

Notation: ${\mathbb{R},\mathbb{R}^{+},\mathbb{R}^{+}_{0}}$ represent the set
of real, positive real, and non-negative real numbers, respectively. The
Euclidean norm of a matrix is denoted by $\|\cdot\|$. For a given set
${\mathcal{C}\subset\mathbb{R}^{n}}$, ${\partial\mathcal{C}}$ and
Int$(\mathcal{C})$ denote its boundary and interior, respectively.

A continuous function
${\alpha:\mathbb{R}^{+}_{0}\rightarrow\mathbb{R}^{+}_{0}}$ belongs to
class-${\mathcal{K}}$ (${\alpha\in\mathcal{K}}$) if it is strictly
monotonically increasing and ${\alpha(0)=0}$. Further, a continuous function
${\alpha:\mathbb{R}^{+}_{0}\rightarrow\mathbb{R}^{+}_{0}}$ belongs to
class-${\mathcal{K}_{\infty}}$ (${\alpha\in\mathcal{K}_{\infty}}$) if it is
strictly increasing, ${\alpha(0)=0}$, and ${\alpha(r)\rightarrow\infty}$ as
${r\rightarrow\infty}$. A continuous function
${\alpha:\mathbb{R}\rightarrow\mathbb{R}}$ belongs to the set of extended
class-$\mathcal{K}$ functions (${\alpha\in\mathcal{K}_{\infty,e}}$) if it is
strictly monotonically increasing, ${\alpha(0)=0}$,
${\alpha(r)\rightarrow\infty}$ as ${r\rightarrow\infty}$, and
${\alpha(r)\rightarrow-\infty}$ as ${r\rightarrow-\infty}$. Lastly, a
continuous function
${\beta(\cdot,\cdot):\mathbb{R}^{+}_{0}\times\mathbb{R}^{+}_{0}\to\mathbb{R}^{+}_{0}}$
belongs to class-${\mathcal{KL}}$ denoted by (${\beta\in\mathcal{KL}}$), if
for every ${s\in\mathbb{R}^{+}_{0}}$,
${\beta(\cdot,s):\mathbb{R}^{+}_{0}\to\mathbb{R}^{+}_{0}}$ is a
class-$\mathcal{K}$ function and for every ${r\in\mathbb{R}^{+}_{0}}$,
${\beta(r,\cdot)}$ is decreasing and ${\lim_{s\to\infty}\beta(s,r)=0}$.

In this work, we consider a nonlinear control affine system:

$\dot{x}=f(x)+g(x)u,$ (1)

where ${x\in X\subset\mathbb{R}^{n}}$ is the states, ${u\in
U\subset\mathbb{R}^{m}}$ is the control input, and
${f:X\rightarrow\mathbb{R}^{n}}$, ${g:X\rightarrow\mathbb{R}^{n\times m}}$ are
locally Lipschitz continuous functions. Given an initial state, ${x_{0}\in X}$
and a locally Lipschitz continuous controller ${u=\mathbf{k}:X\to U}$, which
yields a locally Lipschitz continuous closed-loop system, there exists a
unique solution ${x(t),~{}\forall t\geq 0}$ satisfying the closed-loop
dynamics and initial state. We assume ${f(0)=0}$. Throughout this paper, we
call (1) the actual (uncertain, real) model.

###  II-A Control Barrier Functions

We consider a set ${\mathcal{C}\subset X}$ defined as a 0-superlevel set of a
continuously differentiable function ${h:X\rightarrow\mathbb{R}}$ as

$\displaystyle\mathcal{C}\triangleq\left\\{x\in
X\subset\mathbb{R}^{n}:h(x)\geq 0\right\\},$ (2)
$\displaystyle\partial\mathcal{C}\triangleq\left\\{{x\in
X\subset\mathbb{R}^{n}}:h(x)=0\right\\},$ (3)
$\displaystyle\text{Int}(\mathcal{C})\triangleq\left\\{{x\in
X\subset\mathbb{R}^{n}}:h(x)>0\right\\}.$ (4)

This set is forward invariant if, for every initial condition
${x(0)\in\mathcal{C}}$, the solution of (1) satisfies
${x(t)\in\mathcal{C},~{}\forall t\geq 0}$. Then, the closed-loop system (1) is
safe on the set $\mathcal{C}$ if $\mathcal{C}$ is forward invariant [3].

######  Definition 1 (Control Barrier Function [3]).

Let ${\mathcal{C}\subset X}$ be the 0-superlevel set of a continuously
differentiable function ${h:X\rightarrow\mathbb{R}}$. Then, $h(x)$ is a
Control Barrier Function for system (1) on $\mathcal{C}$ if ${\frac{\partial
h}{\partial x}\neq 0}$ for all ${x\in\partial\mathcal{C}}$ and there exists
$\alpha\in\mathcal{K}_{\infty,e}$ such that $\forall x\in\mathcal{C}$:

$\displaystyle\sup_{u\in
U}\\!\Big{(}\\!{\dot{h}(x,u)}\\!=\\!\underbrace{\dfrac{\partial h(x)}{\partial
x}\\!f(x)}_{L_{f}h(x)}\\!+\\!\underbrace{\dfrac{\partial h(x)}{\partial
x}g(x)}_{L_{g}h(x)}\\!u\\!\Big{)}\\!\geq\\!-\alpha(h(x)).$ (5)

######  Theorem 1.

[3] If $h(x)$ is a valid CBF for (1) on $\mathcal{C}$, then any Lipschitz
continuous controller ${u=\mathbf{k}:X\to U}$ satisfying

$\dot{h}\left(x,\mathbf{k}(x)\right)\geq-\alpha(h(x)),~{}~{}\forall
x\in\mathcal{C}$ (6)

results in the safety of set $\mathcal{C}$ for (1).

Given a nominal (potentially unsafe) locally Lipschitz continuous controller
$\mathbf{k_{d}}:X\to U$ and a CBF ${h(x)}$ for system (1), the safety can be
ensured by solving the following CBF-Quadratic Program (CBF-QP): |s| u
∈U∥u-k_d(x)∥^2 k= ˙h(x, u) ≥- α(h(x))

###  II-B Model Uncertainty

Consider the nominal control affine system:

$\dot{x}=\hat{f}(x)+\hat{g}(x)u,$ (7)

where ${\hat{f}:X\rightarrow\mathbb{R}^{n}}$,
${\hat{g}:X\rightarrow\mathbb{R}^{n\times m}}$ are locally Lipschitz
continuous are known for controller design. To define the discrepancies
between the actual model and nominal model, we add and subtract (7) to (1):

$\dot{x}=\hat{f}(x)+\hat{g}(x)u+\underbrace{{f}(x)-\hat{f}(x)}_{\Delta
f(x)}+\underbrace{({g}(x)-\hat{g}(x))}_{\Delta g(x)}u,$ (8)

where ${\Delta f:X\to\mathbb{R}^{n},~{}\Delta g:X\to\mathbb{R}^{n\times m}}$
are the unmodelled system dynamics, i.e., model uncertainties. Now, we have
the time derivative of CBF for the uncertain system as:

$\\!\\!{\dot{h}(x,u)}\\!=\\!\dfrac{\partial h}{\partial
x}(\hat{f}(x)\\!+\\!\hat{g}(x)u)\\\ \\!+\\!\dfrac{\partial h}{\partial
x}\Delta f(x)\\!+\\!\dfrac{\partial h}{\partial x}\Delta g(x)u.$ (9)

######  Assumption 1.

The augmented uncertainty function

$\Delta(x,u):=\Delta f(x)+\Delta g(x)u,$ (10)

is locally Lipschitz continuous for all ${t\geq 0}$ with Lipschitz constant
$\delta_{L}$, and bounded by ${\|\Delta(x,u)\|\leq\delta_{b}}$ for all
${(x,u)\in X\times U}$. Further, we assume
${\delta_{L},\delta_{b}\in\mathbb{R}^{+}}$ are known constants.

######  Definition 2 (Input Relative Degree).

The input relative degree (IRD) of a sufficiently differentiable output
function $h:X\rightarrow\mathbb{R}$ of system (1) is defined as an integer
${r\leq n}$ if ${\forall{x\in X}}$:

$\displaystyle\begin{split}{L_{{g}}L_{{f}}^{r-1}h(x)\neq 0},\\\
L_{{g}}h(x)=L_{{g}}L_{{f}}h(x)=\cdots=L_{{g}}L_{{f}}^{r-2}h(x)=0,\end{split}$
(11)

######  Definition 3 (Disturbance Relative Degree).

The disturbance relative degree (DRD) of a sufficiently differentiable output
function $h:X\rightarrow\mathbb{R}$ of the uncertain system (8) with the
augmented uncertainty function (10) is defined as ${v\leq n}$ if
${\forall{x\in X}}$:

$\displaystyle\begin{split}{\frac{\partial L_{\hat{f}}^{r-1}h(x)}{\partial
x}\Delta(x,u)\neq 0},\\\ \frac{\partial h(x)}{\partial
x}\Delta(x,u)=\cdots=\frac{\partial L_{\hat{f}}^{r-2}h(x)}{\partial
x}\Delta(x,u)=0,\end{split}$ (12)

The uncertainty in (8) is mismatched if DRD < IRD and matched if DRD = IRD.
Practically speaking, the matching condition implies the disturbance, and the
control input appears on the same output channel of the system [13].

######  Assumption 2.

System (1) with a sufficiently differentiable output function
$h:X\to\mathbb{R}$ satisfies IRD = DRD, i.e., matched uncertainty.

###  II-C Higher Order Control Barrier Functions

The definitions of CBFs in Section II-A require a relative degree one
${h(x)}$. However, for some safety requirements, we may need to differentiate
function $h(x)$ with respect to the system (1) until the control input
appears. In such scenarios, we can utilize higher order CBFs [14], a general
form of exponential CBFs [15].

######  Definition 4 (High Order Control Barrier Function [14]).

For a differentiable function $h:X\to\mathbb{R}$ with IRD = $m$, consider a
sequence of functions ${\phi_{i}:X\to\mathbb{R},i=1,2,\ldots,m}$, as

$\displaystyle\phi_{0}(x):=h(x),\phi_{i}(x):=\dot{\phi}_{i-1}(x)+\alpha_{i}({\phi}_{i-1}(x)),$
(13)

where ${\alpha_{i}\in\mathcal{K}_{\infty}}$ is a ${(m-i)^{th}}$ order
differentiable function. We also define the sets ${\mathcal{C}_{i}:=\\{x\in
X:{\phi}_{i-1}(x)\geq 0\\}}$. Then, $h(x)$ is a high order control barrier
function (HOCBF) for system (1) on
${\mathcal{C}_{1}\cap\mathcal{C}_{2}\cap\ldots\mathcal{C}_{m}}$ if there
exists ${\alpha_{m}\in\mathcal{K}_{\infty}}$ and
${\alpha_{i}\in\mathcal{K}_{\infty}}$ such that ${\forall
x\in{\mathcal{C}_{1}\cap\mathcal{C}_{2}\cap\ldots\mathcal{C}_{m}}}$:

$\displaystyle\sup_{u\in U}\Big{(}{{h}^{m}(x,u)}=L_{f}^{m}h(x)$
$\displaystyle+L_{g}L_{f}^{m-1}h(x)u+\mathcal{O}(h(x))\Big{)}$
$\displaystyle\geq-\alpha_{m}(h_{m-1}(x)),$ (14)

where ${{h}^{m}(x,u)}$ is the ${m^{th}}$ time derivative of $h(x)$,
${L_{f}^{m}h(x)\triangleq\frac{\partial L_{f}^{m-1}h(x)}{\partial x}f(x)}$,
${L_{g}L_{f}^{m-1}h(x)\\!\triangleq\\!\frac{\partial L_{f}^{m-1}h(x)}{\partial
x}g(x)}$, and
${\mathcal{O}(h(x))\triangleq\sum_{i=1}^{m-1}L_{f}^{i}(\alpha_{m-i}\circ{\phi}_{m-i-1})}$.

Given an HOCBF $h(x)$ with ${\alpha_{m}\in\mathcal{K}_{\infty}}$ and
${\alpha_{i}\in\mathcal{K}_{\infty}}$ for system (1), the set of safe
controllers given as

$K_{\text{HOCBF}}(x)\triangleq\\{u\in
U\big{|}{{h}^{m}(x,u)}\geq-\alpha_{m}(h^{m-1}(x))\\}.$ (15)

##  III Problem Statement and Motivation

Real-world safety-critical control systems suffer from not completely modelled
uncertainties that may deteriorate the safety guarantees of controllers that
are designed considering nominal models. We must consider the actual models
when synthesizing safe controllers to address this robustness problem, as
exemplified in the following example.

######  Example 1.

Consider an adaptive cruise control (ACC), whose controlled vehicle dynamics
take the form (8):

$\underbrace{\begin{bmatrix}\dot{v}_{f}\\\
\dot{D}\end{bmatrix}}_{\dot{x}}=\underbrace{\begin{bmatrix}\frac{-F_{r}(v_{f})}{M}\\\
v_{l}-v_{f}\end{bmatrix}}_{\hat{f}(x)}+\underbrace{\begin{bmatrix}\frac{1}{M}\\\
0\end{bmatrix}}_{\hat{g}(x)}u+\overbrace{\underbrace{\begin{bmatrix}\delta_{F_{r},M}\\\
0\end{bmatrix}}_{\Delta f(x)}+\underbrace{\begin{bmatrix}\delta_{M^{-1}}\\\
0\end{bmatrix}}_{\Delta g(x)}u}^{\Delta(x,u)},$ (16)

where ${v_{l}~{}[m/s]}$ and ${v_{f}~{}[m/s]}$ are respectively the velocities
of the lead car and following (controlled) car, ${D~{}[m]}$ is the distance
between the lead and following cars,
${F_{r}=f_{0}+f_{1}v_{f}+f_{2}v_{f}^{2}~{}[N]}$ is the aerodynamic drag,
${M~{}[kg]}$ is the following car’s mass, and
${\delta_{F_{r},M},~{}\delta_{{M}^{-1}}}$ are parametric uncertainties (for
example ${1/M:=\hat{M}^{-1}\pm\delta_{{M}^{-1}}}$) about
${{-F_{r}(v_{f})}/{M},1/M}$.

For safety, the following car must maintain a safe distance behind the lead
car: ${D\geq v_{f}\tau_{d}}$, where $\tau_{d}$ is a time interval. Therefore,
we choose a CBF: ${h(x)\triangleq D-v_{f}\tau_{d}}$, which encodes a safe
stopping distance. This CBF choice results in

$\displaystyle L_{\hat{f}}h(x)$
$\displaystyle=\tau_{d}\frac{F_{r}(v_{f})}{M}+v_{l}-v_{f},$ (17)
$\displaystyle L_{\hat{g}}h(x)$ $\displaystyle=-\frac{\tau_{d}}{M},$ (18)
$\displaystyle\frac{\partial h}{\partial x}\Delta(x,u)$
$\displaystyle=-\tau_{d}(\delta_{F_{r},M}+\delta_{M^{-1}}~{}u).$ (19)

The CBF ${h(x)}$ has IRD of 1, and DRD of 1, which satisfies the matching
condition. Thus, we choose a CBF following Definition 1 by selecting
${-\alpha(h(x))=-\alpha h(x),~{}\alpha\in\mathbb{R}^{+}}$ for simplicity.
Suppose $h(x)$ is a valid CBF for the nominal system (without uncertainty
${\Delta(x,u)}$):

$\displaystyle\tau_{d}\frac{F_{r}(v_{f})}{M}+v_{l}-v_{f}-\frac{\tau_{d}}{M}u\geq-\alpha
h(x).$ (20)

Then, we use the CBF constraint (6) for the uncertain model:

$\displaystyle\begin{split}\overbrace{\tau_{d}\frac{F_{r}(v_{f})}{M}+v_{l}-v_{f}-\frac{\tau_{d}}{M}u+\alpha
h(x)}^{\geq 0}\geq\\\
\tau_{d}(\delta_{F_{r},M}+\delta_{M^{-1}}~{}u)\end{split}$ (21)

The right-hand side of (21) is the effect of the uncertainty on the CBF
constraint. If it is greater than 0, (e.g.: if the real vehicle mass is less
than the nominal mass and if $u>0$, then the term ${\delta_{M^{-1}}u}$ will be
positive), safety may be violated.

To remedy this problem, we propose to estimate the uncertainty with a
quantified estimation error bound and incorporate the effects into the CBF
constraints.

##  IV Uncertainty Estimator

To estimate the uncertainty ${\Delta(x,u)}$ defined in (8), (10) which we
denote as $\hat{\Delta}$, we propose a novel uncertainty estimator that has
the structure:

$\displaystyle\hat{\Delta}(t)$ $\displaystyle={\Lambda}x-\xi(t),$ (22)
$\displaystyle\dot{\xi}(t)$
$\displaystyle=\Lambda\left(\hat{f}(x)+\hat{g}(x)u+\hat{\Delta}(x,\xi(t))\right),$
(23)

where ${\xi\in\mathbb{R}^{n}}$ is an auxiliary state vector and
${0\prec\Lambda\in\mathbb{R}^{n\times n}}$ is a diagonal positive definite
estimator design matrix, i.e.,
${\Lambda\triangleq\operatorname{diag}(\lambda_{1},\ldots,\lambda_{n})}$.
Without loss of generality, the initial values of ${\hat{\Delta}(t)}$ and
${\xi(t)}$ are set to zero, i.e., ${\hat{\Delta}(0)={\bf 0}}$,
${\xi(0)=\Lambda x(0)}$. Note that this estimator is an extension of the
disturbance observer studied in [9, 12] to multiple-input multiple-output
(MIMO) systems.

The following Lemma characterizes the input-to-state stability (ISS) of the
estimation error dynamics around zero; it implies that the estimation error is
bounded.

######  Lemma 1.

Consider uncertain system (8) with a continuously differentiable function
$\Delta(x,u)$ that satisfies Assumption 1 with a Lipschitz constant
${\delta_{L}}$, upper bound ${\delta_{b}}$, and the uncertainty estimator
(22)-(23) with a diagonal and positive definite matrix
${\Lambda\in\mathbb{R}^{n\times n}}$. The uncertainty estimator’s error
dynamics, ${\dot{e}(t)=\dot{{\Delta}}(t)-\dot{\hat{\Delta}}(t)}$, are ISS
around ${e=0}$.

###### Proof.

We define the uncertainty estimation error as

$\displaystyle e(t)={\Delta}(t)-\hat{\Delta}(t),~{}e(t_{0})=e_{0}$ (24)

From (8), (10), (22) and (23), we have

$\displaystyle\dot{e}(t)$ $\displaystyle=\dot{{\Delta}}(t)-\Lambda
e(t),~{}e(t_{0})=e_{0}.$ (25)

Consider a candidate Lyapunov function
${V_{e}(e)\triangleq\frac{1}{2}e^{T}e}$. Taking a time derivative of ${V_{e}}$
along the trajectory of (25) yields

$\displaystyle\begin{split}\dot{V}&=e^{T}(\dot{{\Delta}}-\Lambda
e)\leq-e^{T}\Lambda
e+\|e^{T}\|\underbrace{\|\dot{{\Delta}}\|}_{\leq\delta_{L}}\\\
&\leq-e^{T}\Lambda
e+\|e^{T}\|\delta_{L}\leq-\lambda_{min}\|e\|^{2}+\|e^{T}\|\delta_{L}\end{split}$
(26)

The last inequality arises from the fact that since matrix $\Lambda$ is a real
symmetric positive definite, Rayleigh’s inequality holds:
${\lambda_{min}\|e\|^{2}\leq e^{T}\Lambda e\leq\lambda_{max}\|e\|^{2}}$, where
${\lambda_{min},\lambda_{max}}$ are the minimum and maximum eigenvalues of
$\Lambda$. Introduce an auxiliary non-negative constant,
$\left(\sqrt{\lambda_{min}}\|e\|-\dfrac{\delta_{L}}{\sqrt{\lambda_{min}}}\right)^{2}\geq
0$. We replace ${\|e^{T}\|\delta_{L}}$ with the following upper bound:

$\displaystyle\|e^{T}\|\delta_{L}\leq\dfrac{\lambda_{min}}{2}\|e\|^{2}+\dfrac{\delta_{L}^{2}}{2\lambda_{min}}.$
(27)

Finally, substituting (27) into (26) yields

$\displaystyle\dot{V}(e,\dot{e})\leq-\dfrac{\lambda_{min}}{2}\|e\|^{2}+\dfrac{\delta_{L}^{2}}{2\lambda_{min}},$
(28)

which ensures ISS of the observer error dynamics near ${e=0}$:

$\displaystyle\dot{V}(e,\dot{e})\leq-2\mu_{e}\|e\|^{2}+\gamma(\|\delta_{L}\|),$
(29)

with ${\mu_{e}\triangleq\frac{\lambda_{min}}{4}}$, and a class-${\mathcal{K}}$
function
${\gamma(\|\delta_{L}\|)\triangleq\frac{\delta_{L}^{2}}{2\lambda_{min}}}$. ∎

From Lemma 1, the ISS property ensures the boundedness of error dynamics
${\|e(t)\|}$. This motivates us to derive an explicit tight bound, which
results from applying Lemma 2 to the estimation error and output.

######  Lemma 2.

(Adapted from [16]) Consider a Hurwitz matrix ${{\rm A}\in\mathbb{R}^{n\times
n}}$. The norm ${\|{\rm e}^{{\rm A}t}\|}$ is bounded by

${\|{\rm e}^{{\rm A}t}\|}\leq\left(\sqrt{\|P^{-1}\|\|P\|}\right){\rm
e}^{-\left(\dfrac{t}{2\|P\|}\right)},$ (30)

where ${P}$ is the Hermitian matrix solution that solves the Lyapunov
equation: ${P{\rm A}+{\rm A}^{T}P=-I}$.

######  Lemma 3.

Under the same assumptions in Lemma 1, the following bounds hold for
estimation error ${e(t)\\!=\\!{\Delta}(t)\\!-\\!\hat{\Delta}(t)}$ and
estimated uncertainty ${\hat{\Delta}(t)}$, respectively.

$\displaystyle\|e(t)\|$
$\displaystyle\leq\mathcal{P}\left(\delta_{b}-2{\delta_{L}}{\|P\|}\right){\rm
e}^{-\left(\dfrac{t}{2\|P\|}\right)}+2\mathcal{P}{\|P\|}\delta_{L}$ (31)
$\displaystyle\|\hat{\Delta}(t)\|$ $\displaystyle\leq
2\mathcal{P}\delta_{b}\|\Lambda\|{\|P\|}\Bigg{(}1-{\rm
e}^{-\left(\dfrac{t}{2\|P\|}\right)}\Bigg{)}$ (32)

where ${P{\rm\Lambda}+{\rm\Lambda}^{T}P=-I}$, and
${\mathcal{P}\triangleq\sqrt{\|P^{-1}\|\|P\|}}$.

###### Proof.

From (8), (10), (22), (23) and (24), we have

$\displaystyle\dot{\hat{\Delta}}(t)$
$\displaystyle=\Lambda(\Delta(x,u)-\hat{\Delta}(t)).$ (33)

Under Assumption 1, we have ${\|\Delta(x,u)\|\leq\delta_{b}}$, and
${\|\dot{\Delta}(t)\|\leq\delta_{L}}$. Since we set ${\hat{\Delta}(0)={\bf
0}}$, the initial error will be bounded by ${\|e_{0}\|\leq\delta_{b}}$. The
solution to ${e(t)}$ in (25) with the state transition matrix ${{\rm
e}^{-\Lambda t}}$ and initial condition ${e_{0}}$ is given by

$\displaystyle{e}(t)$ $\displaystyle={\rm e}^{-\Lambda
t}e_{0}+\int_{0}^{t}{\rm e}^{-\Lambda(t-\tau)}\dot{{\Delta}}(\tau)d\tau.$

Taking norms and integrating the right-hand side yield

$\displaystyle\|{e}(t)\|$ $\displaystyle\leq\big{\|}{\rm e}^{-\Lambda
t}\big{\|}\underbrace{\|e_{0}\|}_{\leq\delta_{b}}+\int_{0}^{t}\big{\|}{\rm
e}^{-\Lambda(t-\tau)}\big{\|}\underbrace{\|\dot{{\Delta}}(\tau)\|}_{\leq\delta_{L}}d\tau$
$\displaystyle\leq\delta_{b}\mathcal{P}{\rm
e}^{-\left(\dfrac{t}{2\|P\|}\right)}+\delta_{L}\int_{0}^{t}\mathcal{P}{\rm
e}^{-\left(\dfrac{t-\tau}{2\|P\|}\right)}d\tau$
$\displaystyle=\mathcal{P}\left(\left(\delta_{b}-2{\delta_{L}}{\|P\|}\right){\rm
e}^{-\left(\dfrac{t}{2\|P\|}\right)}+2{\|P\|}\delta_{L}\right),$

which yields the first theorem statement. The second inequality follows from
Lemma 2. Note that the derived bound is in the form of an input-to-state
stable system requirement:

$\displaystyle\|e(t)\|$
$\displaystyle\leq\beta(\|e_{0}\|,t)+\gamma(\|\delta_{L}\|),$

where ${\beta\in\mathcal{KL},\gamma\in\mathcal{K}}$.

To derive the second statement of the theorem, we consider the integration of
(33) with ${\hat{\Delta}(0)={\bf 0}}$:

$\displaystyle{\hat{\Delta}}(t)$ $\displaystyle={\rm e}^{-\Lambda
t}\underbrace{\hat{\Delta}(0)}_{0}+\int_{0}^{t}{\rm
e}^{-\Lambda(t-\tau)}\Lambda{{\Delta}}(\tau)d\tau$
$\displaystyle=\int_{0}^{t}{\rm
e}^{-\Lambda(t-\tau)}\Lambda{{\Delta}}(\tau)d\tau$

Taking norms and integrating the right-hand side yield

$\displaystyle\|{\hat{\Delta}}(t)\|$
$\displaystyle\leq\delta_{b}\|\Lambda\|\int_{0}^{t}\Big{\|}{\rm
e}^{-\Lambda(t-\tau)}\Big{\|}d\tau$
$\displaystyle\leq\mathcal{P}\delta_{b}\|\Lambda\|\int_{0}^{t}{\rm
e}^{-\left(\dfrac{t-\tau}{2\|P\|}\right)}d\tau$
$\displaystyle=2\mathcal{P}\delta_{b}\|\Lambda\|{\|P\|}\Bigg{(}1-{\rm
e}^{-\left(\dfrac{t}{2\|P\|}\right)}\Bigg{)}$

which is the second statement of the theorem. ∎

######  Remark 1.

The first item of Lemma 3 states the input-to-state stability of the observer
error dynamics around ${e=0}$, and it consists of transient and steady-state
parts. The second item shows that if the uncertainty and its derivative are
bounded, then the proposed estimator results in a bounded output.

Lemma 1 presents a property of the estimator’s time derivative, leveraging
Lyapunov functions, while Lemma 3 presents the time-dependent convergence of
the estimator. The following section introduces two methods to assert safety
with the estimator (22)-(23) using these results.

##  V Main Results

To guarantee the robustness of CBFs, we must incorporate the uncertainty
${\Delta(x,u)}$, but it is not directly accessible in actual implementations.
A possible solution to this issue is to replace ${\Delta(x,u)}$ with the
estimated uncertainty term ${\hat{\Delta}(t)}$ and the lower bound of the
associated estimation error ${\|{e}(t)\|}$.

###  V-A Method 1

This section uses the proposed estimator’s active uncertainty attenuation
capability to reject uncertainty via input augmentation. We also show that
this method provides a performance improvement on top of guaranteed robust
safety.

The disturbance decoupling problem [17] (Theorem, 9.20) states that finding a
state feedback controller for system (8), such that the disturbance has no
effect on the system output is possible iff IRD = DRD. Therefore, most
disturbance observer-based robust control methods apply only to the
compensation of matched disturbance inputs. While a few studies have proposed
nonlinear disturbance observers that eliminate system output disturbances,
they ignore the transient dynamics under disturbance effects [13] or consider
specific classes of nonlinear systems, e.g., lower triangular systems.
Therefore, for simplicity, we assume IRD = DRD = 1 and uncertainty satisfies
the matching condition.

We utilize Lemma 1 in this section. Suppose that there exists a matrix
${Q\in\mathbb{R}^{m\times n}}$ that satisfies

$\hat{g}(x)Q(x)\Delta(x,u)=\Delta(x,u),~{}\forall t\geq 0,$ (34)

which matches the lumped uncertainty into the input channel:

$\displaystyle\begin{split}\dot{x}&=\hat{f}(x)+\hat{g}(x)u+\Delta(x,u)\\\
&\equiv\hat{f}(x)+\hat{g}(x)(u+Q(x)\Delta(x,u)).\end{split}$ (35)

Finding a valid matrix $Q(x)$ for equality (35) is equivalent to decoupling
the disturbance from the system states, which is a well-studied problem, [18].
However, to our knowledge, no formulation exists for general MIMO nonlinear
systems.

While a general formula is beyond the scope of this study, we define a useful
matrix ${Q(x)}$. If ${rank(\hat{g}(x))=m}$, for all ${x\in X}$, and its
pseudo-inverse ${(\hat{g}(x)^{T}\hat{g}(x))^{-1}\hat{g}^{T}(x)}$ exists for
all ${x\in X}$, then

$Q(x)\triangleq(\hat{g}(x)^{T}\hat{g}(x))^{-1}\hat{g}^{T}(x).$ (36)

We revisit the ACC problem of Example 1 to demonstrate how to find a ${Q(x)}$
matrix:

$\displaystyle\begin{split}\underbrace{\begin{bmatrix}\frac{1}{M}\\\
0\end{bmatrix}}_{\hat{g}(x)}Q(x)&\underbrace{\begin{bmatrix}\delta_{F_{r},M}+\delta_{M^{-1}}u\\\
0\end{bmatrix}}_{\Delta(x,u)}=\begin{bmatrix}\delta_{F_{r},M}+\delta_{M^{-1}}u\\\
0\end{bmatrix},\\\ \implies
Q(x)&=\left(\begin{bmatrix}\dfrac{1}{M}&0\end{bmatrix}\begin{bmatrix}\dfrac{1}{M}\\\
0\end{bmatrix}\right)^{-1}\begin{bmatrix}\dfrac{1}{M}&0\end{bmatrix}\\\
&=\begin{bmatrix}M&0\end{bmatrix}.\end{split}$ (37)

Since Lemma 1 holds for the uncertainty estimation error ${e(t)}$ for the
proposed estimator with Lyapunov function ${V_{e}(e)=\frac{1}{2}e^{T}e}$, we
have ${\dot{V}_{e}(e,\dot{e})\leq-2\mu_{e}V_{e}+\gamma(\|\delta_{L}\|)}$,
which is given in (29). Using this property, we incorporate the uncertainty
estimator (22), (23) into the CBF setup with the augmentation of the given CBF
and the control input.

To compensate for the uncertainty, we incorporate the uncertainty estimator
into the feedback control law as

$\displaystyle u=\tilde{u}-\underbrace{Q(x)\hat{\Delta}}_{u_{\hat{\Delta}}},$
(38)

where ${\tilde{u}=\mathbf{k}(x)}$ is the feedback controller and parameter
${u_{\hat{\Delta}}}$ is for the uncertainty attenuation (See Fig. 1 top).
Then, we obtain the closed-loop dynamics:

$\displaystyle\begin{split}\dot{x}&=\hat{f}(x)+\hat{g}(x)\left(\tilde{u}-u_{\hat{\Delta}}\right)+\Delta(x,u)\\\
&=\hat{f}(x)+\hat{g}(x)\tilde{u}+\underbrace{\Delta(t)-\hat{\Delta}(t)}_{e(t)},\end{split}$
(39)

which depends on the uncertainty estimation error.

Then, inspired by [19], we modify the CBF and its associated safe set to
provide robustness against estimation error $e(t)$:

$h_{V}(x,e)=h(x)-\sigma_{V}V_{e}(e),~{}\sigma_{V}\in\mathbb{R}^{+}.$ (40)
$\mathcal{C}_{V}:=\left\\{x\in X~{}|~{}h_{V}(x,e)\geq 0\right\\},$ (41)

which is a subset of the safe set ${\mathcal{C}:=\left\\{x\in X~{}|~{}h(x)\geq
0\right\\}}$ since ${V_{e}(e)}$ is a Lyapunov function. The following theorem
relates the controllers designed for the safety of the nominal system to the
safety of the uncertain system by considering the estimator dynamics. Note for
the following theorem; we use ${-\alpha(h(x))=-\alpha
h(x),~{}\alpha\in\mathbb{R}^{+}}$ for simplicity, and we drop the arguments of
functions for convenience in the proof.

######  Theorem 2.

Consider the uncertain system (8), a valid CBF $h(x)$ defining the set
$\mathcal{C}$ as ${\mathcal{C}:=\left\\{x\in X~{}|~{}h(x)\geq 0\right\\}}$
such that ${\frac{\partial h}{\partial x}\neq 0}$, ${\forall
x\in\mathcal{C}}$, with a continuously differentiable uncertainty function
${\Delta(x,u)}$ defined by (10) that satisfies Assumption 1 with a Lipschitz
constant ${\delta_{L}}$, upper bound ${\delta_{b}}$, the uncertainty estimator
that satisfies (29) and ${\hat{\Delta}(0)={\bf 0}}$, and a Lipschitz
continuous controller ${u}$. If there exist a constant
${\mu_{h}\in\mathbb{R}^{+}}$ such that

$\mathcal{D}\triangleq 4\sigma_{V}\mu_{e}-2\sigma_{V}\mu_{h}>0,$ (42)

and the following condition holds

$L_{\hat{f}}h(x)+L_{\hat{g}}h(x)\tilde{u}\geq-\mu_{h}h(x)+\dfrac{\left\|\dfrac{\partial
h}{\partial x}\right\|^{2}}{\mathcal{D}}+\sigma_{V}\gamma(\|\delta_{L}\|),$
(43)

then ${x_{0}\in\mathcal{C}\implies x(t)\in\mathcal{C}}$.

###### Proof.

Our goal is to obtain ${\dot{h}_{V}\geq-\mu_{h}h_{V},~{}\forall t\geq 0}$,
which implies ${\dot{h}\geq-\mu_{h}h,~{}\forall t\geq 0}$. The time derivative
of ${h_{V}(x,e)}$ given in (40) satisfies:

$\displaystyle\dot{h}_{V}$ $\displaystyle=\dot{h}-\sigma_{V}\dot{V}_{e}$
$\displaystyle\geq\dot{h}-\sigma_{V}(-\mu_{e}e^{2}+\gamma(\|\delta_{L}\|))$
$\displaystyle=L_{\hat{f}}h(x)+L_{\hat{g}}h(x){u}+\frac{\partial h}{\partial
x}\Delta+\sigma_{V}\mu_{e}e^{2}-\sigma_{V}\gamma(\|\delta_{L}\|)$
$\displaystyle=L_{\hat{f}}h(x)+L_{\hat{g}}h(x)\tilde{u}+\frac{\partial
h}{\partial
x}\underbrace{({\Delta}-\hat{\Delta})}_{:=e}-\sigma_{V}\gamma(\|\delta_{L}\|)$
$\displaystyle~{}~{}+\left(\sigma_{V}\mu_{e}-\frac{\sigma_{V}\mu_{h}}{2}\right)e^{2}+\frac{\sigma_{V}\mu_{h}}{2}e^{2}$
$\displaystyle=L_{\hat{f}}h(x)+L_{\hat{g}}h(x)\tilde{u}-\sigma_{V}\gamma(\|\delta_{L}\|)$
$\displaystyle~{}~{}+\left(\frac{\sqrt{\mathcal{D}}e}{2}+\dfrac{{\frac{\partial
h}{\partial x}}}{\sqrt{\mathcal{D}}}\right)^{2}-\dfrac{\left\|\frac{\partial
h}{\partial x}\right\|^{2}}{\mathcal{D}}+\frac{\sigma_{V}\mu_{h}}{2}e^{2}$
$\displaystyle\geq\underbrace{L_{\hat{f}}h(x)\\!+\\!L_{\hat{g}}h(x)\tilde{u}\\!-\\!\dfrac{\left\|\frac{\partial
h}{\partial
x}\right\|^{2}}{\mathcal{D}}\\!-\\!\sigma_{V}\gamma(\|\delta_{L}\|)}_{\geq-\mu_{h}h(x)}\\!+\frac{\sigma_{V}\mu_{h}e^{2}}{2}$
$\displaystyle\implies\dot{h}_{V}\geq-\mu_{h}h(x)+\frac{\sigma_{V}\mu_{h}e^{2}}{2}=-\mu_{h}h_{V}$

This leads to ${h(x(t))\geq 0}$, that is, ${x(t)\in\mathcal{C}}$, ${\forall
t\geq 0}$. ∎

The second proof line uses a lower bound for ${V_{e}}$. In the fourth line, we
replaced $u$ with ${\tilde{u}-u_{\hat{\Delta}}}$ (see (38)) to obtain a term
that explicitly depends on ${e}$. In the next line, we use a square to define
a new lower bound. Finally, using the condition in the theorem, we obtain the
main statement.

Then, given an $h(x)$, $\mu_{h}\in\mathcal{K}_{\infty,e}$, ${\sigma_{V}}$ for
system (8), we define the set of robust safe controllers as

$K_{\text{UE}}(x)\triangleq\left\\{\tilde{u}\in
U\big{|}L_{\hat{f}}h(x)+L_{\hat{g}}h(x)\tilde{u}\geq\mathcal{S}(x)\right\\},$
(44)

where ${\mathcal{S}(x)=-\mu_{h}h(x)+\dfrac{\left\|\dfrac{\partial h}{\partial
x}\right\|^{2}}{\mathcal{D}}+\sigma_{V}\gamma(\|\delta_{L}\|)}$.

Finally, we have the following robust CBF-QP: |s| ~u ∈U ∥~u -k_d(x) ∥^2 ~u
^*(x)= L_^f h(x) + L_^g h(x) ~u ≥S(x)

###  V-B Method 2

Method 1 composed the estimated signals and the feedback controller. Even
though Method 1 is useful for disturbance attenuation, it may not be useful
for high dimensional systems as it depends on the existence of matrix
${Q(x)}$. Furthermore, Method 1 is mainly aimed at CBFs with relative degree
one. This section proposes an alternative method to address model uncertainty
for HOCBFs by integrating the proposed estimator with CBFs only for the safety
condition.

We construct a robust safety constraint using the estimated part of the
uncertainty and an associated error bound inspired by [9, 11]. Specifically,
we utilize the proposed estimator with the main outputs of Lemma 3 to
robustify the given CBFs for the nominal systems. We incorporate the
uncertainty estimator into the CBF constraint (9) by plugging in for
${\Delta(x,u)}$:

$\displaystyle\\!\\!\\!{\dot{h}(x,u)}\\!=\\!\dfrac{\partial h}{\partial
x}(\hat{f}(x)\\!+\\!\hat{g}(x)u)\\!+\\!\frac{\partial h}{\partial
x}\\!\underbrace{{\Delta(x,u)}}_{=\hat{\Delta}+e}\\!\geq\\!-\alpha(h(x)).$
(45)

Due to uncertainty in the estimation error, we can not directly replace
${\Delta(x,u)}$ in (45) with ${\hat{\Delta}+e}$, which requires the exact
information about the estimation error ${e}$ and the uncertainty. Therefore,
to compensate for the uncertainty in (45), we use the derived time-dependent
estimator bounds.

######  Theorem 3.

Under the same assumptions of Theorem 2, if ${\frac{\partial h}{\partial
x}\neq 0}$, ${\forall x\in\mathcal{C}}$ and there exists
${\alpha\in\mathcal{K}_{\infty,e}}$ such that ${\forall x\in\mathcal{C}}$

$\displaystyle\\!\\!\\!\sup_{u\in U}\\!\Bigg{(}\\!\dfrac{\partial h}{\partial
x}(\hat{f}(x)\\!+\\!\hat{g}(x)u\\!+\\!\hat{\Delta})\\!-\\!\left\|\frac{\partial
h}{\partial x}\right\|\\!\|e\|\\!\Bigg{)}\\!\geq\\!-\alpha(h(x)),$ (46)

then ${x_{0}\in\mathcal{C}\implies x(t)\in\mathcal{C}}$.

###### Proof.

A lower bound for the time derivative of $h(x)$ can be derived using (24), (9)
and (45) as

$\displaystyle\\!\\!\\!{\dot{h}(x,u)}$ $\displaystyle\\!=\\!\dfrac{\partial
h}{\partial x}(\hat{f}(x)+\hat{g}(x)u)+\frac{\partial h}{\partial
x}({\hat{\Delta}+e})$ (47)
$\displaystyle\\!\geq\\!L_{\hat{f}}h(x)+L_{\hat{g}}h(x)u+\frac{\partial
h}{\partial x}{\hat{\Delta}-{\left\|\frac{\partial h}{\partial
x}\right\|\|e\|}},$ (48)

where the term ${\left\|\frac{\partial h}{\partial x}\right\|\|e(t)\|\geq 0}$
is to provide a lower bound against the effect of the estimation error $e(t)$
on the CBF constraint, which results in ${x(t)\in\mathcal{C}}$, ${\forall
t\geq 0}$. ∎

Method 2 also applies to HOCBFs, as supported by the following corollary.

######  Corollary 1.

Consider a sufficiently differentiable and valid HOCBF ${h(x)}$ for the
uncertain model (8) with a sequence of functions
${\phi_{i}:X\to\mathbb{R},i=1,2,\ldots,m}$, and assume that IRD = DRD = $m$.
Also consider a continuously differentiable function ${\Delta(x,u)}$ defined
by (10) that satisfies Assumption 1 with a Lipschitz constant ${\delta_{L}}$,
upper bound ${\delta_{b}}$, the uncertainty estimator that satisfies (29) and
${\hat{\Delta}(0)={\bf 0}}$, and a Lipschitz continuous controller ${u}$.
Then, ${h(x)}$ is a HOCBF if there exists
${\alpha_{m}\in\mathcal{K}_{\infty}}$ and
${\alpha_{i}\in\mathcal{K}_{\infty}}$ such that ${\forall
x\in{\mathcal{C}_{1}\cap\mathcal{C}_{2}\cap\ldots\mathcal{C}_{m}}}$:

$\displaystyle\begin{split}\sup_{u\in
U}\Big{(}L_{\hat{f}}^{m}h(x)+L_{\hat{g}}L_{\hat{f}}^{m-1}h(x)u+\frac{\partial
L_{\hat{f}}^{m-1}h(x)}{\partial x}\hat{\Delta}\\\
-\Bigg{\|}{\frac{L_{\hat{f}}^{m\\!-\\!1}\\!h(x)}{\partial
x}}\Bigg{\|}\|e(t)\|\\!+\\!\mathcal{O}(h(x))\Big{)}\\!\geq\\!-\alpha_{m}(h_{m-1}(x)),\end{split}$
(49)

###### Proof.

Follows from Theorem 3 by replacing the uncertainty term with the associated
estimation and error bound. ∎

###  V-C Discussion

One of the differences between the proposed methods is that Method 2
explicitly depends on the estimation error, whereas in Method 1, we consider a
subset of the original safe set with a design parameter $\sigma_{V}$. As can
be understood from (40), choosing a larger $\sigma_{V}$ may lead to a
conservative safe set. Therefore, the tuning process for this parameter is a
crucial step for Method 1. On the other hand, there is no additional tuning
parameter for Method 2. To take this advantage and leverages the idea of
uncertainty compensation, we can incorporate the estimation error dynamics
into the CBF constraint for the closed-loop system (39) with the following
corollary.

######  Corollary 2.

Under the same assumptions of Theorem 2, if ${\frac{\partial h}{\partial
x}\neq 0}$ ${\forall x\in\mathcal{C}}$ and there exists
${\alpha\in\mathcal{K}_{\infty,e}}$ such that ${\forall x\in\mathcal{C}}$

$\displaystyle\\!\\!\\!\sup_{\tilde{u}\in U}\\!\Bigg{(}\\!\dfrac{\partial
h}{\partial
x}(\hat{f}(x)\\!+\\!\hat{g}(x)\tilde{u})\\!-\\!\left\|\frac{\partial
h}{\partial x}\right\|\\!\|e\|\\!\Bigg{)}\\!\geq\\!-\alpha(h(x)),$ (50)

then ${x_{0}\in\mathcal{C}\implies x(t)\in\mathcal{C}}$.

###### Proof.

Since the term ${(\\!-\\!\left\|\frac{\partial h}{\partial
x}\right\|\\!\|e\|)}$ is a lower bound for the effect of the uncertainty
estimation error on the time derivative of CBF, (50) results in
${x(t)\in\mathcal{C}}$, ${\forall t\geq 0}$. ∎

Using Corollary 2, we define an alternative robust CBF-QP for Method 1: |s| ~u
∈U∥~u -k_d(x) ∥^2 ~u ^*(x)= ∂h∂x ( ^f(x) + ^g (x) ~u ) - ∥∂h∂x ∥
∥e∥ ≥ -α(h(x)) which explicitly depends on the error dynamics of the
uncertainty estimator.

##  VI Simulations

The parameters of the simulation are taken from [9] (Table 1).

Figure 2: Simulations for the ACC Example with uncertainty. (Left) Evaluation
of CBF h(x). The proposed controller maintains safety in the presence of
unmodelled dynamics. (Middle) The closed-loop system performance is improved
by disturbance rejection (Method 1). (Right) The estimation error and output
satisfy the theoretical bounds.

This section completes the ACC example with the proposed uncertainty
estimator, and uses a multi-rotor example to illustrate Method 2.

###  VI-A Adaptive Cruise Control Example

The parameters of the simulation are taken from [9] (Table 1). The closed-loop
control performance requires cruise at a constant speed encoded into the QP
via control Lyapunov function (CLF) ${V(x)=(v_{f}-v_{d})^{2}}$. We augment
robust CBF-QP with the related CLF constraint; hence we have the following
robust CLF-CBF-QP: |s| u ∈U, δ_c ∈R∥u ∥^2 + p_c δ_c^2 ~u ^*(x)= L_^f h(x) +
L_^g h(x) u ≥S(x) ˙V(x, u) ≤-λV(x) + δ_c where ${\delta_{c}}$ is a relaxation
variable, ${p_{c}=100}$, we pick ${\lambda=0.7}$.

We set the first row of ${\Delta(t)=(\delta_{F_{r},M}+\delta_{M^{-1}}~{}u)}$
given in (16), as ${\Delta(t):=\frac{2sin(2\pi
t)}{M}+\frac{0.2F_{r}(v_{f})}{M}+\frac{1}{2M}u}$, which represents both the
drag and mass uncertainties and violates the safety as discussed in (21). To
obtain ${\delta_{L},\delta_{b}\in\mathbb{R}^{+}}$ constants for this
uncertainty, we run the simulation for possible scenarios. Then, we have
${\delta_{L}=26,~{}\delta_{b}=12}$. We choose
${\Lambda=\operatorname{diag}(100,100)}$ to design the proposed uncertainty
with a small estimation error; therefore,
${\mu_{e}:=\frac{\lambda_{min}}{4}=25}$. For Method 1, we pick ${\mu_{h}=1}$
and ${\sigma_{V}=0.1}$, so ${\mathcal{D}\triangleq
4\sigma_{V}\mu_{e}-2\sigma_{V}\mu_{h}>0}$.

Fig. 2 shows the evaluation of CBF $h$, the states for an initial set
${x(0)=[18\ 12\ 24]^{T}}$, and the uncertainty estimation performance of the
proposed observer, respectively. As seen from Fig. 2-(Left), the unmodelled
dynamics cause safety violations without the estimator. Method 2 provides a
safety guarantee but does not compensate for the effect of disturbance on the
states (${v_{f},D}$), as observed from Fig. 2-(Middle) Method 1 achieves the
disturbance attenuation requirement. Finally, it is observed from Fig.
2-(Right) that the proposed uncertainty estimator can effectively estimate
actual modelling uncertainties within the quantified bounds.

###  VI-B Multirotor Trajectory Tracking Example

The multirotor states are its global position
${\mathrm{p}=[\mathrm{x},\mathrm{y},\mathrm{z}]\in\mathbb{R}^{3}}$, and Euler
angle parameterization of rotation matrix
${R(\phi,\theta,\psi)\in\text{SO(3)}}$ where ${\phi}$, ${\theta}$, and
${\psi}$ are roll, pitch, and yaw angles. Rotation rates ${p,~{}q}$, and $r$
are defined with respect to the body-fixed $\mathrm{x}$, $\mathrm{y}$, and
$\mathrm{z}$ axes. If $T$ denotes the total axial thrust on the multirotor, we
have the following dynamics (nominal) where gravity is compensated to ensure
${f(0)=0}$:

$\displaystyle\ddot{p}$
$\displaystyle=TR(\phi,\theta,\psi)e_{3},\qquad\dot{T}=u_{1},$ (51)
$\displaystyle\begin{bmatrix}\dot{\phi}\\\ \dot{\theta}\\\
\dot{\psi}\end{bmatrix}$
$\displaystyle=\underbrace{\begin{bmatrix}1&\sin(\phi)\tan(\theta)&\cos(\phi)\tan(\theta)\\\
0&\cos(\phi)&-\sin(\phi)\\\
0&\sin(\phi)\cot(\theta)&\cos(\phi)\cot(\theta)\end{bmatrix}}_{W(\phi,\theta,\psi)}\begin{bmatrix}u_{2}\\\
u_{3}\\\ u_{4}\end{bmatrix},$ (52)

where $\mathrm{m}$ is the vehicle mass, ${e_{3}=[0,0,1]^{T}}$.
${u=[u_{1},u_{2},u_{3},u_{4}]}$ are the control inputs where $u_{1}$ is the
rate of change of mass normalized thrust, ${u_{2},u_{3}}$, and $u_{4}$ are the
body rotation rates with respect to the body-fixed $\mathrm{x}$, $\mathrm{y}$,
and $\mathrm{z}$ axes. The system outputs are
${s=[\mathrm{x},\mathrm{y},\mathrm{z},\psi]^{T}}$, and the goal is to track a
desired differential trajectory ${s_{d}:[t_{0},t_{f}]\to\mathbb{R}^{4}}$. Let
${e_{p}=p-[x^{d},y^{d},z^{d}]^{T}}$ be the position tracking errors and
${e_{\psi}=\psi-\psi^{d}}$ be the yaw angle tracking error. To obtain an error
dynamics in control-affine form, we perform a time-derivative on (51) to
obtain the following:

$\displaystyle\frac{d}{dt}\underbrace{\begin{bmatrix}e_{p}\\\ \dot{e}_{p}\\\
\ddot{e}_{p}\\\
e_{\psi}\end{bmatrix}}_{\eta}=\mathcal{F}\eta+\underbrace{\begin{bmatrix}0_{6\times
4}\\\ I_{4\times
4}\end{bmatrix}}_{\mathcal{G}}\underbrace{\underbrace{\begin{bmatrix}Re_{3}&-TRe_{3}^{\vee}\\\
0&e_{3}^{T}W\end{bmatrix}}_{B_{u}}\begin{bmatrix}u_{1}\\\ u_{2}\\\ u_{3}\\\
u_{4}\end{bmatrix}}_{v},\vspace{-1mm}$ (53)

where ${\mathcal{F}=\left[\begin{smallmatrix}0_{6\times 3}&I_{6\times
6}&0_{6\times 1}\\\ 0_{4\times 3}&0_{4\times 6}&0_{4\times
1}\end{smallmatrix}\right]}$. For simplicity, we assume full-state
observability. The output has a vector relative degree
${\vec{\gamma}=[3,3,3,1]}$. We aim to find a family of tracking controllers
that would exponentially stabilize the error dynamics (53) to the origin.
Consider a feedback control law in terms of the auxiliary input ${v=K\eta}$,
the goal to stabilize the closed-loop system
${\dot{\eta}=(\mathcal{F}+\mathcal{G}K)\eta=A_{cl}\eta}$. Let
${V_{\eta}=\eta^{T}P\eta}$ where the positive-definite matrix $P$ is the
solution to the Continuous-Time Algebraic Equation (CARE)
${\mathcal{F}^{T}P+P\mathcal{F}-P\mathcal{G}\mathcal{G}^{T}P=-Q}$, where
${Q=I}$ to ensure maximum regions of attraction. Nominally, we obtain a
stabilizing controller by solving a standard CLF-QP.

For safety, known static spherical obstacles must be avoided regardless of the
reference trajectory. Let
${p_{i}^{obs}=[x_{i}^{obs},y_{i}^{obs},z_{i}^{obs}]^{T}}$ and $r_{i}$ denote
the position of the $i^{th}$ static spherical obstacle in global/inertial
frame and the radius of the obstacle, respectively. We use the following CBF
for obstacle avoidance:${~{}h_{i}=\|p-p^{obs}_{i}\|-r_{i}-r_{0},}$ where
$r_{0}$ is the radius of the multirotor. To match the relative degree, we
employ a
HOCBF:${~{}h_{e,i}\\!=\\!L_{f}^{2}h_{i}\\!+\\!\alpha_{2}L_{f}h_{i}\\!+\\!\alpha_{1}h_{i},}$
where ${\alpha_{1},\alpha_{2}\\!\in\\!\mathbb{R}^{+}}$ are tunable constants.

Figure 3: Simulated multirotor obstacle avoidance scenarios. Despite the
nominal CBF being able to avoid the obstacle, the actual system with
uncertainty (54) failed as shown in figure (E2a). Using the proposed
uncertainty estimator (method 2), the augmented CBF can avoid the obstacle in
10 variations of uncertainties

An optimization-based safety filter can be obtained by solving (1). We
introduce state and input-dependent uncertainty in the system dynamics to
model physical phenomena like aerodynamic interactions, input delays,
compliance, etc. Therefore, error dynamics (53) is more accurately

$\dot{\eta}=\underbrace{\mathcal{F}\eta}_{\hat{f}(\eta)}+\underbrace{\mathcal{G}B_{u}u}_{\hat{g}(\eta)u}+\underbrace{\Delta_{A}(\eta)+\Delta_{B}(\eta)u}_{\Delta(\eta,u)}.$
(54)

In particular, we simulated the presence of the following state-dependent
disturbance:

$\displaystyle\Delta_{A}(\eta)=[\,0_{1\times
6},\,c_{d}\tanh{(\dot{p}^{T})},\,0_{1\times 4}]^{T},$ (55)

where $c_{d}$ can be interpreted as drag coefficients. Moreover, we also
inject input uncertainty as
${\Delta_{B}(\eta)=\mbox{diag}([\delta_{u_{1}},\delta_{u_{2}},\delta_{u_{3}},\delta_{u_{4}}]),}$
where ${\delta_{u_{i}}\in(-1,0]}$ is the percent reduction of input
representing delays. Since higher-order barrier functions are adopted for this
scenario, we employ Method 2 to address model uncertainty. Incorporating the
proposed disturbance observer (22)-(23) estimator into the CBF constraint, we
have ${\dot{h}_{e,i}=\dot{\hat{h}}_{e,i}+\frac{\partial
h_{e,i}}{\partial\eta}(\hat{\Delta}+e)}$, where ${\dot{\hat{h}}_{e,i}}$
represents the time derivative of CBF along with the nominal system dynamics,
and one can verify symbolically that ${\frac{\partial
h_{e,i}}{\partial\eta}\neq 0}$. Therefore, we leverage Lemma 3 and Corollary 1
to obtain a robust control barrier constraint with uncertainty estimation.

For simulation, we choose parameters such that the uncertainty Lipschitz
constant ${\delta_{L}=0.1}$ and the global upper bound ${\delta_{b}=0.1}$, and
the results are depicted in Fig. 3. The nominal system with a CLF-QP
controller filters with a barrier-based safety filter can avoid the obstacle
(shown in (E2a)), and the disturbed system under a nominal approach is unable
to main safety. We then incorporated the proposed disturbance observer
(22)-(23) and augment the barrier constraint using (43). The resulting
controller can avoid the obstacle with different disturbances (different
coefficients/parameters while satisfying ${\delta_{L}}$ and ${\delta_{b}}$) as
presented in (E2b), showcasing the versatility of the proposed uncertainty
estimator.

##  VII Conclusions and Future Work

We proposed a novel uncertainty estimator for control affine systems with
unmodeled, state, and input-dependent uncertainty to improve system
robustness. Moreover, the bounds for the estimation error and output were
introduced. By incorporating the estimator with CBFs, robust safety conditions
were derived. We showcase two methods to assert safety with theoretical
analysis and simulation validations.

Future work aims to remove the knowledge of the uncertainty bound
$\delta_{b}$, and Lipschitz bound $\delta_{L}$ by using machine learning
techniques like Gaussian processes [20] to estimate these parameters from
data. We also seek to combine state observer-based CBF methods [21] with our
methods. Lastly, hardware validation and comparison with our prior work [22]
will further showcase the performance of these methods.

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