remarkRemark hypothesisHypothesis problemProblem claimClaim Finite time
stabilizationS. Zheng, P. Shi, S. Nahavandi and X. Zong ex_supplement

# Finite time stabilization for nonlinear systems with unknown control
directions and unstructured uncertainties

Shiqi Zheng School of Automation, China University of Geosciences,Wuhan
430074, China; Hubei key Laboratory of Advanced Control and Intelligent
Automation for Complex Systems, Wuhan 430074, China ().
<EMAIL_ADDRESS>Peng Shi School of Electrical and Electronic
Engineering, The University of Adelaide, Adelaide, SA 5005, Australia ().
<EMAIL_ADDRESS>Saeid Nahavandi Institute for Intelligent Systems
Research and Innovation, Deakin University, Waurn Ponds VIC 3216, Australia
<EMAIL_ADDRESS>Xiaofeng Zong Corresponding author. School
of Automation, China University of Geosciences,Wuhan 430074, China; Hubei key
Laboratory of Advanced Control and Intelligent Automation for Complex Systems,
Wuhan 430074, China<EMAIL_ADDRESS>
###### Abstract

This paper investigates the finite time stabilization problem for a class of
nonlinear systems with unknown control directions and unstructured
uncertainties. The unstructured uncertainties indicate that not only the
parameters but also the structure of the system nonlinearities are uncertain.
The contributions are mainly in the following two aspects: First, a new
adaptive control method is proposed for the considered system. Logic-based
switching rule is utilized to tune the controller parameters online to
stabilize the system in finite time. Different from the existing adaptive
controllers for structured/parametric uncertainties, a new switching barrier
Lyapunov method and supervisory functions are introduced to overcome the
obstacles caused by unstructured uncertainties and unknown control directions.
Second, based on a time-varying backstepping scheme, an extension is made to
the proposed method such that all the system states can be regulated to zero
in prescribed finite time. Moreover, a new analysis tool is presented to show
the boundedness of the control signals. Simulations are conducted to verify
the effectiveness of the proposed methods.

###### keywords:

finite time stabilization, barrier Lyapunov functions, unknown control
directions, backstepping, time-varying feedback

##  1 Introduction

###  1.1 Background and motivations

Finite time stabilization problem has attracted increasing attention in the
past few years. Finite time stabilization means that by designing a proper
feedback controller, all the states of the closed loop systems will become
exact zero after finite time [40, 34]. However, for asymptotic stabilization,
the states will converge to zero in an infinite time. Lots of works [3, 7, 18,
33] have shown that finite time control has some promising features in
contrast with asymptotic control. These may lie in: 1) Faster convergence rate
and higher precision; 2) Possibility to decouple the stabilization problem
from other control objectives [32].

Many interesting results have been obtained for finite time control. The works
of [40] have studied finite time output feedback stabilization for strict
feedback nonlinear systems. [6] have extended the finite time control to high
order stochastic nonlinear systems. A time-varying feedback method is proposed
in [36] to achieve prescribed finite time control performance. Namely, the
finite convergence time can be determined a prior and is independent of the
initial conditions. Recently, the finite time control problem has been
investigated for multi-agent and networked systems [8]. Moreover, several real
practical applications, such as robot manipulators [2] and servo motor systems
[14, 41], have been considered for finite time control.

Unknown control directions are often encountered in real engineering world. It
means that the sign of control coefficient is unknown. This will bring
difficulties to the controller design because a control effort with wrong
direction can drive the states away from the equilibrium point. Nussbaum-gain
technique, which was originally introduced in [25], is a common way to handle
unknown control direction. Plentiful works [4, 21, 23, 27, 38] have been done
on the control of nonlinear systems by incorporating Nussbaum-gain function.
Nevertheless, as discussed in [30] and [16], the Nussbaum-gain technique could
only achieve asymptotic stability because the constructed Lyapunov function
cannot be negative definite.

In fact, there are very few works concentrating on finite time stabilization
of nonlinear systems with unknown control directions. Lately, in the framework
of backstepping method [24], a new adaptive control strategy is proposed in
[30, 15, 16] to solve this problem. The idea of the method is to adopt a
logic-based switching rule to tune the controller parameters online according
to a well-defined supervisory function. Finite time stability can then be
achieved despite unknown control directions.

The aforementioned works [30, 15], however, only consider the finite time
stabilization problem for nonlinear systems suffering from
structured/parametric uncertainties. This means that the structures of the
nonlinear uncertain functions are available, but contain some unknown
parameters. The structured uncertainties are mainly used to describe the
parameter variations in the systems. However, the nonlinear uncertainties are
often very complicated in practical systems. Hence, it may be difficult or
impossible to obtain the exact form of the uncertainties, and express the
uncertainties in a parametric way. This class of uncertainties is often
referred to as unstructured/nonparametric uncertainties, which can represent
those unknown nonlinearities caused by complex system dynamics and modeling
errors. Therefore, a natural question arises:

How to solve the finite time stabilization problem for nonlinear systems with
unknown control directions and unstructured uncertainties?

To the best of our knowledge, little effort has been made to answer the above
issue. The main challenges may lie in the following aspects:

1) Due to the structure of the nonlinearities is uncertain, the nonlinearities
cannot be parameterized. Hence, it is difficult to directly extend the
adaptive control scheme presented in [30] and [11] to solve the above problem.
Consequently, the design procedures become involved.

2) As previously mentioned, a logic-based switching mechanism has to be
adopted to achieve finite time stability due to the possible limitations of
the Nussbaum-gain technique. Therefore, the entire closed-loop system will
exhibit hybrid feature, which introduces difficulties to the controller design
and stability analysis.

3) The problem becomes more challenging if the finite convergence time needs
to be prescribed.

###  1.2 Contributions

Motivated by the above thought, this paper focuses on the finite time
stabilization problem for a class of nonlinear systems with unknown control
directions and unstructured uncertainties. The contributions are mainly in the
following aspects.

First, a new switching adaptive control method is proposed for the considered
system. Logic-based switching rule is used to tune the controller parameters
online. The proposed method includes two novel techniques:

  * •

Novel switching barrier Lyapunov functions are constructed for the controller
design. The barrier will switch according to the logic-based switching rule
(see Remark 3.9).

  * •

By designing some special auxiliary systems, new supervisory functions are
presented to guide the logic-based switching (see Remark 3.11).

Based on the above two new techniques, the states of the _hybrid_ closed-loop
systems will be constrained in a compact set despite multiple unknown control
directions. Then, constant bounds will be obtained for the unstructured
uncertainties. This contributes to the feasibility of the adaptive control
scheme such that all the states will reach exact zero in finite time.

Second, based on the above control method, a new time-varying switching
adaptive control method is presented such that all the states can be regulated
to zero in prescribed finite time. The novelties of this method are as
follows:

  * •

A time-varying backstepping scheme is proposed to design the adaptive
controller (see Remark 4.5).

  * •

A new analysis tool is presented to show the boundedness of the control
signals (see Remark 4.9).

Thanks to the above techniques, the proposed method is suitable for more
general systems than the existing prescribed finite time control methods,
where only parametric uncertainties and known control directions are
considered.

Last but not least, the proposed methods have some promising features, such as
fast convergence speed, small control overshoot, and low complexity (see
Appendix E).

###  1.3 Organizations

The organization of the paper is as follows. Problem formulation and
preliminaries are presented in Section 2. Section 3 concentrates on the finite
time stabilization problem. Section 4 presents the prescribed finite time
control method. Simulations are conducted in Section 5. Section 6 concludes
the paper. Proofs and some additional discussions are provided in Appendices.

Notations. Given a real number $x$ and a positive constant
$\alpha=\frac{p}{q}$ where $p,q$ are coprime. If $p$ is a positive odd
integer, then $x^{\alpha}=\mathrm{sign}(x)|x|^{\alpha}.$ If $p$ is a positive
even integer, then $x^{\alpha}=|x|^{\alpha}.$ Let $a,b\in\mathbb{R}$, then
$a\triangleq b$ means $a$ is defined as $b$. $a:=b$ means $a$ is set as $b$,
which is used in algorithm.

##  2 Main results

###  2.1 Problem formulation

Consider the following system

$\displaystyle\dot{x}_{i}$
$\displaystyle=h_{i}(\overline{x}_{i})x_{i+1}+f_{i}(\overline{x}_{i}),\thinspace
i=1,2,...,n-1$ (1) $\displaystyle\dot{x}_{n}$
$\displaystyle=h_{n}(\overline{x}_{n})u+f_{n}(\overline{x}_{n}),$
$\displaystyle y$ $\displaystyle=x_{1}$

where
$\overline{x}_{i}=(x_{1},x_{2},...,x_{i})^{\mathrm{T}}\in\mathbb{R}^{i},\thinspace
i=1,2,...,n$ are the system states, $y$ is the system output.
$f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})(i=1,...,n)$ are all unknown
continuously differentiable nonlinear functions. $f_{i}(\overline{x}_{i})$
represents the system nonlinearities and uncertainties such that
$f_{i}(0,0,...,0)\equiv 0$. $h_{i}(\overline{x}_{i})$ are the control gains
such that their signs are unknown and satisfy $|h_{i}(\overline{x}_{i})|>0$.
$u$ denotes the control input.

######  Remark 2.1.

_(More general systems)_ Note that system (1) is more general than the
existing works [30, 13, 10, 16, 41] due to the following reasons:

1) $f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})(i=1,2,...,n)$ represent
unstructured uncertainties, i.e., not only parameters in
$f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})$ but also the form of
$f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})$ are uncertain. In fact, the
continuously differentiable nonlinearities
$f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})$ only need to satisfy
$f_{i}(0,0,...,0)\equiv 0,|h_{i}(\overline{x}_{i})|>0$. This is much more
general than structured uncertainties in [30, 15, 11]. In these references,
$f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})$ need to satisfy
$|f_{i}(\overline{x}_{i})|\leq(|x_{1}|+\cdots+|x_{i}|)\psi_{i}(\overline{x}_{i},\theta)$
and
$0<\underline{h}_{i}\leq|h_{i}(\overline{x}_{i})|\leq\overline{h}_{i}(\overline{x}_{i},\theta)$
where $\psi_{i}(\overline{x}_{i},\theta)$ and
$\overline{h}_{i}(\overline{x}_{i},\theta)$ are known smooth functions,
$\theta,\underline{h}_{i}$ are unknown parameters.

2) System (1) contains multiple unknown control directions, i.e., for $\forall
i=1,2,..,n$, the sign of $h_{i}(\overline{x}_{i})$ is unknown. Moreover,
compared with [31, 27], $|h_{i}(\overline{x}_{i})|$ only needs to be larger
than zero, not a positive constant.

According to the above analysis, we can see that very little information is
needed for $f_{i}(\overline{x}_{i}),h_{i}(\overline{x}_{i})$. This will bring
many difficulties to the controller design. In addition, note that by
unstructured/nonparametric uncertainties, it means that it is difficult to
obtain the exact form of the uncertainties, and the uncertain functions cannot
be parameterized by unknown parameters. However, some crude information of the
uncertainties may need to be known. For instance, the nonlinear function
$f_{i}(\overline{x}_{i})$ needs to satisfy $f_{i}(0,0,...,0)\equiv 0$. Yet, we
can see the structures of the nonlinearities are still uncertain because many
kinds of nonlinear functions satisfy $f_{i}(0,0,...,0)\equiv 0$.

In Appendix E, we will discuss the situation where $f_{i}(0,0,...,0)\neq 0$.
That is $f_{i}(\overline{x}_{i})$ can be any continuously differentiable
functions. Meanwhile, note that our method can be naturally extended to the
case when system (1) contains power orders, non-feedback linearizable and
time-varying dynamics like [11, 26].

Now, we are ready to describe the finite time stabilization problem.

######  Problem 2.2.

(Finite time stabilization problem) Develop an adaptive controller $u(t)$ for
system (1) such that

1) All the control signals in the closed loop system are bounded, and;

2) All the states will converge to zero in finite time, $i.e.$, there exists a
finite time $T$ such that $x_{i}(t)\rightarrow 0(i=1,2,...,n)$ as
$t\rightarrow T^{-}$.

######  Remark 2.3.

The finite time stability of a system means that the system satisfies the
above two conditions. If the finite time $T$ can be prescribed, the above
problem becomes the prescribed finite time stabilization problem. We will
consider the finite time and prescribed finite time stabilization problem in
Sections 3 and 4 separately. In addition, for prescribed finite time control,
the control signals do not include time-varying gains.

###  2.2 Technical lemmas

Some useful lemmas will be presented, which will be used in the controller
design.

######  Lemma 2.4.

[10] Consider the following Young’s inequality

$|x|^{a}|y|^{b}\leq\frac{a}{a+b}\zeta(x,y)|x|^{a+b}+\frac{b}{a+b}\zeta^{-a/b}(x,y)|y|^{a+b}$

where $x,y\in\mathbb{R}$, $a,b$ are positive constants, $\zeta(x,y)>0$ is any
real valued function.

######  Lemma 2.5.

[26, 10] Given a real constant $p\geq 1$ being a ratio of two odd integers and
real numbers $x,y,z_{i}(i=1,2,...,n)$, we have:

$|x-y|^{p}\leq 2^{p-1}|x^{p}-y^{p}|;$ $|x^{1/p}-y^{1/p}|\leq
2^{1-1/p}|x-y|^{1/p};$
$\left(\sum_{i=1}^{n}|z_{i}|\right)^{1/p}\leq\sum_{i=1}^{n}|z_{i}|^{1/p}\leq
n^{1-1/p}\left(\sum_{i=1}^{n}|z_{i}|\right)^{1/p}.$

######  Lemma 2.6.

(Lemma 11.1 in [5]; [22]) Given a continuously differentiable nonlinear
function $f(\overline{x}_{n}):\mathbb{R}^{n}\rightarrow\mathbb{R}$ where
$\overline{x}_{n}=(x_{1},x_{2},...,x_{n})^{\mathrm{T}}\in\mathbb{R}^{n}$ and
$f(0,0,...,0)\equiv 0$, there exists a non-negative smooth function
$\psi(\overline{x}_{n}):\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that

(2) $|f(\overline{x}_{n})|\leq(|x_{1}|+\cdots+|x_{n}|)\psi(\overline{x}_{n}).$

######  Lemma 2.7.

Given four time-varying continuous functions
$x(t),y(t),a(t),b(t):[t_{0},t_{1})\rightarrow\mathbb{R}$ such that

(3) $\displaystyle\dot{x}(t)$ $\displaystyle=-a(t)x^{\gamma}(t)+b(t),$ (4)
$\displaystyle\dot{y}(t)$ $\displaystyle\leq-a(t)y^{\gamma}(t)+b(t)$

for $\forall t\in[t_{0},t_{1})$ where $x(t_{0})=y(t_{0})+\varepsilon$.
$\varepsilon\geq 0$ and $0<\gamma\leq 1$ are constants with $\gamma$ being a
ratio of odd integers, $a(t)>0$ on $[0,+\infty)$. Then, $x(t)\geq y(t)$ for
$\forall t\in[t_{0},t_{1})$.

######  Proof 2.8.

Please see Appendix A for detailed proof.

##  3 Finite time stabilization

This section will focus on the finite time stabilization of system (1). It is
divided into three parts. In Section 3.1, we will mainly present the
controller structure, which contains some adaptive parameters. Section 3.2
will focus on the logic-based switching rule for tuning the adaptive
parameters. Main result and stability analysis for the hybrid closed-loop
system will be given in Section 3.3.

###  3.1 Controller design

It is noted that the high order system (1) can be regarded as a cascade of $n$
first order subsystems. The controller design for this class of system is
inspired by backstepping method [24, 11]. The controller $u$ is recursively
determined by the following equations:

(5) $\displaystyle s_{1}$ $\displaystyle\triangleq x_{1},$ (6) $\displaystyle
x_{i+1}^{*}$
$\displaystyle\triangleq\hat{\Theta}_{i}(t)\left[-K_{i}s_{i}^{q_{i+1}}-\frac{U_{i}s_{i}^{q_{i+1}}}{(\hat{\chi}_{i}^{2}(t)-s_{i}^{2})^{1+2\alpha}}\right],$
(7) $\displaystyle s_{i+1}$ $\displaystyle\triangleq
x_{i+1}^{1/q_{i+1}}-\left(x_{i+1}^{*}\right)^{1/q_{i+1}}(i=1,2,...,n-1),$ (8)
$\displaystyle x_{n+1}^{*}$ $\displaystyle\triangleq
u\triangleq\hat{\Theta}_{n}(t)\left[-K_{n}s_{n}^{q_{n+1}}-\frac{U_{n}s_{n}^{q_{n+1}}}{(\hat{\chi}_{n}^{2}(t)-s_{n}^{2})^{1+2\alpha}}\right].$

$x_{i+1}^{*}(i=1,2,...,n-1)$ are the virtual control efforts for the first
order subsystem
$\dot{x}_{i}=h_{i}(\overline{x}_{i})x_{i+1}+f_{i}(\overline{x}_{i})$ in (1).
$s_{i}(i=1,2,...,n)$ are the virtual control errors which will be regulated to
zero in finite time.

The design parameters $q_{i+1}(i=1,2,...,n)$ are determined by
$q_{i+1}=\alpha-1+q_{i}$ where $\alpha\in(\frac{1}{2},1)$ is a ratio of odd
integers and $q_{1}\triangleq 1$. These parameters are accounting for the
power of Lyapunov function which is the key for the finite time stabilization.
$K_{i},U_{i}(i=1,...,n)$ are positive design parameters, $\hat{\chi}_{i}(t)$
and $\hat{\Theta}_{i}(t)$ are adaptive parameters explained as follows:

$\hat{\chi}_{i}(t):[0,+\infty)\rightarrow\mathbb{R}(i=1,2,...,n)$ are
piecewise constant signals. It will be updated according to the logic-based
switching rule in Section 3.2. $\hat{\chi}_{i}(t)(i=1,...,n)$ are used to
constrain all the virtual control errors $s_{i}$ and states in a compact set,
which is beneficial for dealing with the unstructured uncertainties.

$\hat{\Theta}_{i}(t):[0,+\infty)\rightarrow\mathbb{R}$ is determined by a
switching signal $\sigma_{i}(t)$.

(9) $\hat{\Theta}_{i}(t)=(-1)^{\sigma_{i}(t)}\theta_{i}(\sigma_{i}(t))$

where $\sigma_{i}(t):[0,+\infty)\rightarrow\mathbb{N}$ is also a piecewise
constant signal, $\theta_{i}(\sigma_{i}):\mathbb{N}\rightarrow\mathbb{R}$ is
an increasing function with respect to $\sigma_{i}$ such that
$\theta_{i}(0)>0$ and $\theta_{i}(\sigma_{i})\rightarrow+\infty$ as
$\sigma_{i}\rightarrow+\infty$. A typical example of $\theta_{i}(\sigma_{i})$
is $\theta_{i}(0)=1,\theta_{i}(1)=2,\theta_{i}(2)=3,...$ The idea of the
tuning rule (9) for $\hat{\Theta}_{i}$ is that by changing its sign
repeatedly, one may expect to find a correct control direction, $i.e.,$
dealing with the unknown sign of $h_{i}(\overline{x}_{i})$ in (1). The detail
switching rule for $\sigma_{i},\hat{\Theta}_{i}$ is also given in Section 3.2.

Next, a $n$ step Lyapunov functions analysis will be given according to
$x_{i+1}^{*}(i=1,2,...,n)$. This analysis will be helpful for understanding
the controller design idea and results in Sections 3.2-3.3. It should be noted
that all the adaptive parameters $\hat{\chi}_{i},\hat{\Theta}_{i}$ will be
assumed to be constants in the following results. This will become clear in
Section 3.3.

Step 1. Consider the following Lyapunov function

(10)
$V_{1}=\frac{1}{2}\ln\left(\frac{\hat{\chi}_{1}^{2}(t)}{\hat{\chi}_{1}^{2}(t)-s_{1}^{2}}\right)$

where $\hat{\chi}_{1}$ is defined in (6) or (8).

######  Remark 3.1.

When $\hat{\chi}_{1}$ is a positive constant, $V_{1}$ becomes a standard
barrier Lyapunov function [19, 17, 31] such that if $|s_{1}|<\hat{\chi}_{1}$,
then $V_{1}\rightarrow+\infty$ as $|s_{1}|\rightarrow\hat{\chi}_{1}$. The
parameter $\hat{\chi}_{1}$ acts as a barrier for the virtual control error
$s_{1}$. The purpose of adopting the barrier Lyapunov function is to constrain
$s_{1}$ in the interval $(-\hat{\chi}_{1},\hat{\chi}_{1})$. We can see that if
$V_{1}$ is bounded, $s_{1}\in(-\hat{\chi}_{1},\hat{\chi}_{1})$. In addition,
in Section 3.3 we will show the parameter $\hat{\chi}_{1}$ remains to be a
constant.

By (10), we have the following result.

######  Proposition 3.2.

Suppose $\hat{\chi}_{1}(t)$ is a positive constant and
$|s_{1}|<\hat{\chi}_{1}$. Then, by using (6) with $i=1$, $\dot{V}_{1}$ can be
expressed as

$\displaystyle\dot{V}_{1}\leq$ $\displaystyle-
a_{1}V_{1}^{\frac{1+\alpha}{2}}-Q_{1}s_{1}^{1+\alpha}-\frac{h_{1}(x_{1})K_{1}\hat{\Theta}_{1}s_{1}^{1+\alpha}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
(11)
$\displaystyle+\frac{U_{1}(F_{1}(x_{1},\hat{\chi}_{1})-h_{1}(x_{1})\hat{\Theta}_{1})s_{1}^{1+\alpha}}{(\hat{\chi}_{1}^{2}-s_{1}^{2})^{2+2\alpha}}+c_{12}s_{2}^{1+\alpha}$

where $a_{1},Q_{1},c_{12}$ are positive parameters,
$F_{1}(x_{1},\hat{\chi}_{1})$ is an unknown non-negative function which comes
from the unstructured uncertainties in (1).

######  Proof 3.3.

Under the assumption that $\hat{\chi}_{1}(t)$ is a positive constant,
differentiating $V_{1}$ with respect to time and using (1) and Lemma 2.6, we
have

$\displaystyle\dot{V}_{1}\leq$
$\displaystyle\frac{s_{1}(h_{1}x_{2}^{*}+f_{1}(x_{1}))}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{s_{1}h_{1}(x_{2}-x_{2}^{*})}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
(12) $\displaystyle\leq$
$\displaystyle\frac{s_{1}h_{1}x_{2}^{*}+s_{1}^{2}\psi_{1}(x_{1})}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{s_{1}h_{1}(x_{2}-x_{2}^{*})}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$

where $\psi_{1}(x_{1})$ is an unknown function.

By the inequalities in Lemmas 2.4-2.5 and (7), it follows that

$\displaystyle\dot{V}_{1}\leq$
$\displaystyle\frac{s_{1}h_{1}x_{2}^{*}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{U_{1}s_{1}^{2}\left(\psi_{1}/U_{1}\right)}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
$\displaystyle+\frac{h_{1}^{1+q_{2}}c_{11}s_{1}^{1+q_{2}}}{(\hat{\chi}_{1}^{2}-s_{1}^{2})^{1+q_{2}}}+c_{12}s_{2}^{1+q_{2}}$
(13) $\displaystyle=$
$\displaystyle\frac{s_{1}h_{1}x_{2}^{*}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{U_{1}s_{1}^{1+q_{2}}\overline{F}_{1}(x_{1},\hat{\chi}_{1})}{(\hat{\chi}_{1}^{2}-s_{1}^{2})^{2+2\alpha}}+c_{12}s_{2}^{1+\alpha}$

where $c_{11},c_{12}$ are positive known constants,
$\overline{F}_{1}(x_{1},\hat{\chi}_{1})=s_{1}^{1-q_{2}}\psi_{1}(\hat{\chi}_{1}^{2}-s_{1}^{2})^{1+2\alpha}/U_{1}+h_{1}^{1+q_{2}}c_{11}(\hat{\chi}_{1}^{2}-s_{1}^{2})^{1+\alpha}$
is an unknown function.

With $i=1$, substituting (6) into (13), we have

$\displaystyle\dot{V}_{1}\leq$
$\displaystyle-\frac{K_{1}^{\prime}s_{1}^{1+\alpha}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}-\frac{h_{1}K_{1}\hat{\Theta}_{1}s_{1}^{1+\alpha}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
(14)
$\displaystyle+\frac{U_{1}(F_{1}(x_{1},\hat{\chi}_{1})-h_{1}\hat{\Theta}_{1})s_{1}^{1+\alpha}}{(\hat{\chi}_{1}^{2}-s_{1}^{2})^{2+2\alpha}}+c_{12}s_{2}^{1+\alpha}$

where $K_{1}^{\prime}$ is a positive parameter and
$F_{1}(x_{1},\hat{\chi}_{1})=\overline{F}_{1}(x_{1},\hat{\chi}_{1})+K_{1}^{\prime}(\hat{\chi}_{1}^{2}-s_{1}^{2})^{1+\alpha}/U_{1}$.

Note that $V_{1}\leq\frac{s_{1}^{2}}{2(\hat{\chi}_{1}^{2}-s_{1}^{2})}$ [21]
and $\frac{1}{\hat{\chi}_{1}^{2}-s_{1}^{2}}\geq\frac{1}{\hat{\chi}_{1}^{2}}$,
then there exists a sufficiently large $K_{1}^{\prime}$ such that
$-\frac{K_{1}^{\prime}s_{1}^{1+\alpha}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}\leq-
a_{i}V_{i}^{\frac{1+\alpha}{2}}-Q_{i}s_{i}^{1+\alpha}.$ Using this for (14),
we can complete the proof.

Step i($2\leq i\leq n$). Consider the following Lyapunov function

(15)
$V_{i}(\overline{x}_{i},t)=\int_{x_{i}^{*}}^{x_{i}}\frac{\upsilon_{i}^{2-q_{i}}(\tau)}{\hat{\chi}_{i}^{2}(t)-\upsilon_{i}^{2}(\tau)}d\tau$

where $\upsilon_{i}(\tau)=\tau^{1/q_{i}}-x_{i}^{*1/q_{i}}$,
$\hat{\chi}_{i}(t)\in[0,+\infty)\rightarrow\mathbb{R}$ is the adaptive
parameter in (6) or (8). Meanwhile, $V_{i}$ has the following properties.

######  Proposition 3.4.

[41] Suppose $\hat{\chi}_{i}(t)$ is a positive constant and
$|s_{i}|<\hat{\chi}_{i}$. Then, $V_{i}$ has the following properties:

1)

(16)
$0\leq\frac{2^{1-1/q_{i}}}{\hat{\chi}_{i}^{2}}(x_{i}-x_{i}^{*})^{2/q_{i}}\leq
V_{i}\leq\frac{2\hat{\chi}_{i}^{2}}{\hat{\chi}_{i}^{2}-s_{i}^{2}};$

2) If $x_{i}^{*}$ is bounded, then as $|s_{i}|\rightarrow\hat{\chi}_{i}$,
$V_{i}\rightarrow+\infty$.

######  Remark 3.5.

According to the above result, it can be seen that when $\hat{\chi}_{i}(t)$ is
a constant, the Lyapunov function in (15) is also a barrier Lyapunov function
like (10). The barrier $\hat{\chi}_{i}$ is used to constrain the virtual
control error $s_{i}$.

Next, similar to (11), we have the following result for $\dot{V}_{i}$.

######  Proposition 3.6.

Suppose $\overline{\hat{\Theta}}_{i}(t),\overline{\hat{\chi}}_{i}(t)$ are both
constant vectors and $|s_{i}|<\hat{\chi}_{i}$ where
$\overline{\hat{\Theta}}_{i}\triangleq(\hat{\Theta}_{1},\hat{\Theta}_{2},...,\hat{\Theta}_{i})^{\mathrm{T}}$,
$\overline{\hat{\chi}}_{i}\triangleq(\hat{\chi}_{1},\hat{\chi}_{2},...,\hat{\chi}_{i})^{\mathrm{T}}$.
Then, by using (6) or (8), $\dot{V}_{i}$ can be expressed as

$\displaystyle\dot{V}_{i}\leq$ $\displaystyle-
a_{i}V_{i}^{\frac{1+\alpha}{2}}-Q_{i}s_{i}^{1+\alpha}-\frac{h_{i}(\overline{x}_{i})K_{i}\hat{\Theta}_{i}s_{i}^{1+\alpha}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle+\frac{U_{i}s_{i}^{1+\alpha}(F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})-h_{i}(\overline{x}_{i})\hat{\Theta}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2+2\alpha}}$
(17)
$\displaystyle+\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+c_{i,i+1}s_{i+1}^{1+\alpha}$

where $a_{i},Q_{i},c_{ij}(j=1,2,...,i-1,i+1)$ are positive design parameters,
$s_{n+1}\triangleq 0$.
$F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown non-negative function which comes from the unstructured
uncertainties in (1).

######  Proof 3.7.

First, under the assumption that
$\overline{\hat{\Theta}}_{i}(t),\overline{\hat{\chi}}_{i}(t)$ are both
constant vectors, by resorting to [41], we have

$\displaystyle\dot{V}_{i}\leq$
$\displaystyle\frac{s_{i}^{2-q_{i}}h_{i}x_{i+1}^{*}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}+\frac{U_{i}s_{i}^{1+\alpha}\overline{F}_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2+2\alpha}}$
(18)
$\displaystyle+\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+c_{i,i+1}s_{i+1}^{1+\alpha}$

where
$\overline{F}_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown function. Then, substituting (6) or (8) into (18), we get

$\displaystyle\dot{V}_{i}\leq$
$\displaystyle-\frac{K_{i}^{\prime}s_{i}^{1+\alpha}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}-\frac{h_{i}K_{i}\hat{\Theta}_{i}s_{i}^{1+\alpha}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle+\frac{U_{i}s_{i}^{1+\alpha}(F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})-h_{i}\hat{\Theta}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2+2\alpha}}$
(19)
$\displaystyle+\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+c_{i,i+1}s_{i+1}^{1+\alpha}$

where $K_{i}^{\prime}$ is a positive design parameter. By (16) and
$\frac{1}{\hat{\chi}_{i}^{2}-s_{i}^{2}}\geq\frac{1}{\hat{\chi}_{i}^{2}}$,
there exists a sufficiently large $K_{i}^{\prime}$ such that
$-\frac{K_{i}^{\prime}s_{i}^{1+\alpha}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}\leq-
a_{i}V_{i}^{\frac{1+\alpha}{2}}-Q_{i}s_{i}^{1+\alpha}.$ Using this for the
above inequality, we can complete the proof.

######  Remark 3.8.

(Controller design idea) Note that $V_{i}(i=1,2,...,n)$ are all barrier
Lyapunov functions. By using these functions, we expect to constrain all the
virtual control errors $s_{i}$, states and adaptive parameters in a compact
set. Thus, there exist unknown positive constants
$\overline{F}_{i},\underline{h}_{i}$ such that
$F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})\leq\overline{F}_{i}$
and $|h_{i}(\overline{x}_{i})|\geq\underline{h}_{i}>0$. Then, (17) can be
written as

$\displaystyle\dot{V}_{i}\leq$ $\displaystyle-
a_{i}V_{i}^{\frac{1+\alpha}{2}}-Q_{i}s_{i}^{1+\alpha}-\frac{h_{i}(\overline{x}_{i})K_{i}\hat{\Theta}_{i}s_{i}^{1+\alpha}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle+\frac{U_{i}s_{i}^{1+\alpha}(\overline{F}_{i}-h_{i}(\overline{x}_{i})\hat{\Theta}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2+2\alpha}}+\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+c_{i,i+1}s_{i+1}^{1+\alpha}$

The unknown functions/unstructured uncertainties
$F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
become an unknown parameter $\overline{F}_{i}$. Then, according to (9), there
exists a sufficiently large $\sigma_{i}$ such that
$-h_{i}(\overline{x}_{i})K_{i}\hat{\Theta}_{i}<0$ and
$\overline{F}_{i}-h_{i}(\overline{x}_{i})\hat{\Theta}_{i}<0$. Hence, the
uncertainties can be canceled (see Section 3.3 for details about how the
constant bounds for the unknown functions are obtained). In addition, since
the unknown complicated function
$F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is replaced by an unknown parameter $\overline{F}_{i}$, the complexity of the
controller is reduced considerably.

######  Remark 3.9.

(Switching barrier Lyapunov function) From the above remark, we can see that
the key for the controller design is to constrain all the virtual control
errors and states. However, our case is much more difficult than the existing
methods [19, 17, 35] where only continuous dynamics are considered. In these
references, the state trajectories, controller and adaptive law are all
continuous with respect to time. Therefore, by using barrier Lyapunov
functions with constant barriers, it is not hard to constrain the states. Yet,
in our case, the closed loop nonlinear system is a hybrid system which
contains logic-based switching (discrete dynamics). Thus, the existing barrier
Lyapunov methods will not be applicable. In fact, very few works have
considered the barrier Lyapunov method for hybrid systems.

Specifically, by (7) the virtual control error $s_{i}$ is expressed as
$s_{i}=x_{i}^{1/q_{i}}-x_{i}^{*1/q_{i}}$. Since $x_{i}^{*}$ contains
$\hat{\Theta}_{i-1}(t)$ by (6), it is discontinuous with respect to time. This
implies that $s_{i}$ is also discontinuous with respect to time. Therefore, it
is possible that at some time instants, $s_{i}$ will jump outside the barrier
$\hat{\chi}_{i}$ if $\hat{\chi}_{i}$ remains to be a constant (see Fig 3 in
the simulation example). This implies that the traditional barrier Lyapunov
method is not applicable.  Therefore, we propose the new Lyapunov functions
defined in (15). The novelty here is that the barriers $\hat{\chi}_{i}$ are
regarded as adaptive parameters and will be tuned by the logic-based switching
rule (see Algorithm 3.1-4)-b) in Section 3.2). We refer to this kind of
Lyapunov functions as switching barrier Lyapunov functions. The logic-based
switching rule will guarantee that the barrier $\hat{\chi}_{i}$ is always
larger than the virtual control error $s_{i}$. This is an essential difference
with the existing barrier Lyapunov methods [19, 17, 35], where the virtual
control error $s_{i}$ is continuous and the barrier is a constant.

In Section 3.3, we will prove the switching barrier $\hat{\chi}_{i}$ is
bounded and the virtual control errors are constrained, _i.e.,_
$|s_{i}|<\hat{\chi}_{i}(i=1,2,...,n)$.

###  3.2 Logic-based switching rule

We will present the algorithm for tuning adaptive parameters
$\hat{\Theta}_{i},\hat{\chi}_{i}$ for $i\in\\{1,2,...,n\\}$. First, for
$i\in\\{1,2,...,n\\}$, define the following new supervisory functions
$\mathcal{S}_{i}(t)$.

(20) $\mathcal{S}_{i}(t)=V_{i}(\overline{x}_{i},t)-\eta_{i}(t),$ (21)
$\dot{\eta}_{i}=-a_{i}\eta_{i}^{\frac{1+\alpha}{2}}-Q_{i}s_{i}^{1+\alpha}+\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+c_{i,i+1}s_{i+1}^{1+\alpha}$

where $V_{i}$ is given by (10) and (15), $s_{i}$ is from (5) or (7) with
$s_{n+1}\triangleq 0$, $\eta_{i}$ is an auxiliary variable.
$c_{ij}(j=1,2,...,i-1,i+1)$ and $a_{i},Q_{i}$ are positive design parameters.

Based on the above supervisory functions, the logic-based switching rule is
shown in Algorithm 3.1 in Table I.

Next, we will give some remarks on the algorithm.

######  Remark 3.10.

(Idea of Algorithm 1) The idea of Algorithm 3.1 is as follows. At each time
instant $t$, we verify whether or not the supervisory functions
$\mathcal{S}_{i}(t)>0$. If $\mathcal{S}_{i}(t)\leq 0$, the adaptive parameters
remain the same; otherwise parameters need to be updated. The update is
conducted in the following way. First, $\sigma_{i},\hat{\Theta}_{i}$ are
updated. The switching signal $\sigma_{i}$ is increased by one and the
adaptive parameter $\hat{\Theta}_{i}$ is updated by (9). Second,
$\hat{\chi}_{i}$ needs to be updated. This is because as long as
$\hat{\Theta}_{i}$ is updated, the virtual control error $s_{i}$ will change
accordingly. This may make $s_{i}$ jump at the switching time. Therefore, the
barrier $\hat{\chi}_{i}$ also needs to be updated to guarantee that
$\hat{\chi}_{i}$ is always larger than $s_{i}$ when $\hat{\Theta}_{i}$
finishes its updating (see Remark 3.9). Third, we reset $\eta_{i}$ to make
sure it is larger than $V_{i}$ after $\hat{\Theta}_{i},\hat{\chi}_{i}$ have
been updated. This will make the supervisory functions $\mathcal{S}_{i}(t)<0$.
Then, the above procedures will be repeated. Fig. 1 shows one possible
variations of $\hat{\chi}_{i},\sigma_{i},\eta_{i},V_{i}$.

Next, we will explain why the adaptive parameters
$\hat{\Theta}_{i},\hat{\chi}_{i}$ are pieceswise constant signals. Note that
$\hat{\Theta}_{i},\hat{\chi}_{i}$ are only updated at the switching time,
$i.e.$, the time instant that the event $\mathcal{S}_{i}(t)>0$ occurs. When
$\mathcal{S}_{i}(t)\leq 0$, $\hat{\Theta}_{i},\hat{\chi}_{i}$ will keep
constant. Keeping this in mind, let $t_{s}^{m}(m=0,1,2,...)$ denote the proof.
At $t_{s}^{m}$, all the adaptive parameters will be updated and $\eta_{i}$
will be reset to make $\mathcal{S}_{i}(t_{s}^{m})<0$ for $i=1,2,...,n$ (see
Algorithm 3.1-4). Hence, in the later time the adaptive parameters
$\hat{\Theta}_{i},\hat{\chi}_{i}$ will keep constant until the next event
$\mathcal{S}_{i}(t)>0$ occurs. Then, parameters will be updated and $\eta_{i}$
will be reset again to make $\mathcal{S}_{i}(t)<0$ (see Fig 1). This indicates
that $\hat{\Theta}_{i},\hat{\chi}_{i}$ are pieceswise constant signals. In
addition, since $\mathcal{S}_{i}(t_{s}^{m})<0(i=1,2,...,n)$ after the reset of
$\eta_{i}$, there exists a small time interval
$[t_{s}^{m},t_{s}^{m}+\iota^{m})$ such that $\mathcal{S}_{i}(t)\leq 0$ holds
where $\iota^{m}>0$ is a small constant. That is the adaptive parameters
$\hat{\Theta}_{i},\hat{\chi}_{i}$ will not change on
$[t_{s}^{m},t_{s}^{m}+\iota^{m})$. This also indicates that there exists an
increasing switching time sequence. See [30, 16, 9, 12] for similar idea.

The purpose of Algorithm 3.1 is to let
$\mathcal{S}_{i}(t)=V_{i}(\overline{x}_{i},t)-\eta_{i}(t)\leq 0(i=1,...,n)$
hold forever after a finite number of switchings. Then, we have
$\eta_{i}(t)\geq V_{i}(\overline{x}_{i},t)\geq 0(i=1,2,...,n)$ are all non-
negative (The finite number of switchings and $V_{i}\geq 0$ will be shown in
Claim 1b and its proof in Section 3.3). Then, let
$\overline{\eta}\triangleq\sum_{i=1}^{n}\eta_{i}$, from (21) we have

$\displaystyle\dot{\overline{\eta}}=$
$\displaystyle-\sum_{i=1}^{n}a_{i}\eta_{i}^{\frac{1+\alpha}{2}}-\sum_{i=1}^{n}Q_{i}s_{i}^{1+\alpha}$
$\displaystyle+\sum_{i=1}^{n}\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+\sum_{i=1}^{n}c_{i,i+1}s_{i+1}^{1+\alpha}$
(22) $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}a_{i}\eta_{i}^{\frac{1+\alpha}{2}}-\sum_{i=1}^{n}\left(Q_{i}-\sum_{j=i+1}^{n}c_{ji}-c_{i-1,i}\right)s_{i}^{1+\alpha}$

where $s_{n+1}\triangleq 0$, $\sum_{j=n+1}^{n}c_{jn}\triangleq 0$,
$c_{0i}\triangleq 0$. It can be seen that when $Q_{i}$ is sufficiently large
such that $Q_{i}-\sum_{j=i+1}^{n}c_{ji}-c_{i-1,i}>0$, by Lemma 2.5, we have

(23) $\displaystyle\dot{\overline{\eta}}\leq$
$\displaystyle-\sum_{i=1}^{n}a_{i}\eta_{i}^{\frac{1+\alpha}{2}}\leq-a^{\prime}\overline{\eta}^{\frac{1+\alpha}{2}}$

where $a^{\prime}>0$ is a positive constant. This means $\overline{\eta}$ may
converge to zero in finite time. Since $\eta_{i}\geq V_{i}\geq 0$ for $\forall
i=1,2,...,n$, this implies that $\eta_{i}$ and $V_{i}$ may also converge to
zero in finite time. By (5)-(8), (10), (15) and (16), we can see that all the
states will converge to zero in finite time. Please see Claim 1c and its proof
in Section 3.3 for details.

Initialization At $t=0$,

  1. 1.

Set initial values. Set design parameters $\varsigma,\varepsilon$ where
$\varsigma>0$ and $\varepsilon>0$ is an arbitrary small constant. Set $m:=0$
and the switching time $t_{s}^{m}:=0$. For $i=1,2,...,n$, set
$\sigma_{i}(0):=1$ and compute $\hat{\Theta}_{i}(0)$ by (9); set
$\hat{\chi}_{i}(0)>s_{i}(0)$ and $\eta_{i}(0):=V_{i}(0)+\varepsilon$.

  2. 2.

Output the current $\sigma_{i},\hat{\Theta}_{i},\hat{\chi}_{i}$.

Switching logic

At each time instant $t>0$,

  1. 1.

Compute supervisory functions. Obtain the current states
$\overline{x}_{n}(t)$. For $i=1,2,...,n$, set
$\sigma_{i}(t):=\sigma_{i}(t^{-})$,
$\hat{\Theta}_{i}(t):=\hat{\Theta}_{i}(t^{-})$,
$\hat{\chi}_{i}(t):=\hat{\chi}_{i}(t^{-})$ and compute
$V_{i},\eta_{i},\mathcal{S}_{i}$ by (10), (15), (21) and (20);

  2. 2.

Verify parameters update conditions. For $i=1,2,...,n$, check whether or not
$\mathcal{S}_{i}(t)>0$ for some $i\in\\{1,2,...,n\\}$;

  3. 3.

If $\mathcal{S}_{i}(t)\leq 0$ for $\forall i=1,2,...,n$, $\hat{\Theta}_{i}$
and $\hat{\chi}_{i}(i=1,2,...,n)$ are not updated and keep constant. Goto 5)
to output parameters directly;

  4. 4.

If $\mathcal{S}_{i}(t)>0$ for some $i\in\\{1,2,...,n\\}$, $\hat{\Theta}_{i}$
and $\hat{\chi}_{i}(i=1,2,...,n)$ need to be updated. Set the current time
instant to the switching time, $i.e.$, set $m:=m+1,t_{s}^{m}:=t$. Then, do the
following:

    1. (a)

Update $\hat{\Theta}_{i}$. For $i=1,2,...,n$, if
$\mathcal{S}_{i}(t_{s}^{m})>0$, set
$\sigma_{i}(t_{s}^{m}):=\sigma_{i}(t_{s}^{m})+1$ and update
$\hat{\Theta}_{i}(t_{s}^{m})$ by (9); otherwise $\sigma_{i},\hat{\Theta}_{i}$
are not updated;

    2. (b)

Update $\hat{\chi}_{i}$. For $i=1,2,...,n$, recompute $s_{i}(t_{s}^{m})$, if
$|s_{i}(t_{s}^{m})|\geq|\hat{\chi}_{i}(t_{s}^{m})|$, update
$\hat{\chi}_{i}(t_{s}^{m}):=|s_{i}(t_{s}^{m})|+\varsigma$ to make
$|s_{i}(t_{s}^{m})|<\hat{\chi}_{i}(t_{s}^{m})$; otherwise $\hat{\chi}_{i}$ is
not updated;

    3. (c)

Reset $\eta_{i}$. For $i=1,2,...,n$, recompute
$V_{i}(\overline{x}_{i}(t_{s}^{m}),t_{s}^{m})$, if
$V_{i}(\overline{x}_{i}(t_{s}^{m}),t_{s}^{m})\geq\eta_{i}(t_{s}^{m})$, reset
$\eta_{i}(t_{s}^{m}):=V_{i}(\overline{x}_{i}(t_{s}^{m}),t_{s}^{m})+\varepsilon$
to make $\mathcal{S}_{i}(t_{s}^{m})<0$; otherwise $\eta_{i}$ is not updated;

  5. 5.

Output the current $\sigma_{i},\hat{\Theta}_{i},\hat{\chi}_{i}$.

Algorithm 1 Logic-based switching rule.  Figure 1: One possible variation of
$\hat{\chi}_{i},\sigma_{i},\eta_{i},V_{i}$. $\hat{\chi}_{i},\sigma_{i}$ will
keep constant when $V_{i}\leq\eta_{i}$. As long as the event $V_{i}>\eta_{i}$
occurs, the switching will happen and the parameters
$\hat{\chi}_{i},\sigma_{i}$ will be updated. Then, $\eta_{i}$ is reset to make
$V_{i}<\eta_{i}$ and the previous procedures will be repeated. (For
simplicity, $\hat{\Theta}_{i}$ and other variables are not shown in this
figure.)

######  Remark 3.11.

(New supervisory functions) In order to deal with the unstructured
uncertainties $h_{i}(\overline{x}_{i}),f_{i}(\overline{x}_{i})$ in each
channel of the system (1). The proposed virtual/real control effort
$x_{i+1}^{*}$ in (6) or (8) contains adaptive parameters
$\hat{\chi}_{i},\hat{\Theta}_{i}$ for $i=1,2,...,n$. Therefore, to guarantee
the stability, we need to successively show the boundedness of the parameters
$\hat{\chi}_{i},\hat{\Theta}_{i}$ and state $x_{i}$ for $i=1,2,...,n$. This
makes the logic-based switching rule in the existing works invalid, where the
adaptive parameters may only exist in the last control effort $x_{n+1}^{*}=u$
(Chen et al., 2018). Hence, we propose the new supervisory functions (20)-(21)
to guide the logic-based switching. The new supervisory functions have the
following two major differences from the existing methods, which are important
for the boundedness of the adaptive parameters:

1) In the existing works [30, 11], the Lyapunov function is compared with a
pre-specified time-varying function. However, for (20), the Lyapunov function
$V_{i}$ is compared with a dynamic variable $\eta_{i}$ which is determined by
the constructed auxiliary system (21). It relies on the current state
information.

2) In [30], only one single supervisory function is used to guide the
switching for the adaptive parameters. Yet, in our case, we have used $n$
different supervisory functions $\mathcal{S}_{i}(t)$ to guide the switching
for the adaptive parameters in every virtual control effort $x_{i+1}^{*}$.

Note that the new supervisory functions and switching barrier Lyapunov
function explained in Remark 3.9 make the the proposed method has some
substantial differences with the existing methods, $e.g.$, [17, 30, 11]. This
is reflected in the stability analysis in Section 3.3, where we propose a new
3-Claims procedure to show the finite time stability.

###  3.3 Main result and stability analysis

Based on the analysis in Sections 3.1 and 3.2, we have the following main
result.

######  Theorem 3.12.

Consider the nonlinear system in (1). Then, the controller (5)-(8) with
Algorithm 3.1 can guarantee that:

1) All the signals in the closed-loop system are bounded for $\forall
t\in[0,+\infty)$, and;

2) All the states will converge to zero in finite time.

The proof for the above result will be presented in this subsection. According
to Remark 3.10, we can define a switching time sequence
$\\{0=t_{s}^{0}<t_{s}^{1}<...<t_{s}^{m}<...\leq+\infty\\}$ such that

(24) $t_{s}^{m+1}=\inf\\{t|t\geq
t_{s}^{m},\mathcal{S}_{i}(t)>0,i\in\\{1,2,...,n\\}\\}.$

During time interval $[t_{s}^{m},t_{s}^{m+1})$, the supervisory function
satisfies $\mathcal{S}_{i}(t)=V_{i}(\overline{x}_{i},t)-\eta_{i}(t)\leq 0$ for
$\forall i=1,2,...,n$. Meanwhile,
$\sigma_{i},\hat{\Theta}_{i},\hat{\chi}_{i}(i=1,2,...,n)$ are all constants on
$[t_{s}^{m},t_{s}^{m+1})$, _i.e._ , $\sigma_{i}(t)=\sigma_{i}(t_{s}^{m})$,
$\hat{\Theta}_{i}(t)=\hat{\Theta}_{i}(t_{s}^{m})$,
$\hat{\chi}_{i}(t)=\hat{\chi}_{i}(t_{s}^{m})$ for $\forall
t\in[t_{s}^{m},t_{s}^{m+1})$.

The proof will be obtained by proving the following three claims, i.e., Claims
1a, 1b and 1c. Claim 1a tries to show the boundedness of signals in the system
if the number of switchings is finite. Next, Claim 1b attempts to show the
number of switchings is indeed finite. Finally, Claim 1c proves the finite
time stability.

Claim 1a. For any finite integer $m$, we have

1) The closed loop nonlinear system has continuous solution
$\overline{x}_{n}(t)$ on $[t_{s}^{0},t_{s}^{m+1})$;

2) There exists a positive constant $\delta_{i}^{m}$ such that
$|s_{i}(t)|\leq\hat{\chi}_{i}(t_{s}^{m})-\delta_{i}^{m}$ for $\forall
i=1,2,...,n$ on $[t_{s}^{m},t_{s}^{m+1})$;

3) All the signals in the system are bounded on $[t_{s}^{m},t_{s}^{m+1})$.

######  Proof 3.13.

According to Algorithm 3.1 and Remark 3.10, we know during each time interval
$[t_{s}^{m},t_{s}^{m+1})$, the initial condition satisfies
$|s_{i}(t_{s}^{m})|<\hat{\chi}_{i}(t_{s}^{m})$ and the barrier
$\hat{\chi}_{i}$ remains to be constant. Hence, (10) and (15) become
traditional barrier Lyapunov functions on each $[t_{s}^{m},t_{s}^{m+1})$.
Therefore, according to the theory of barrier Lyapunov functions [19] and the
fact that $V_{i}(\overline{x}_{i},t)\leq\eta_{i}(t)$ on each
$[t_{s}^{m},t_{s}^{m+1})$, we can show the virtual control error $s_{i}(t)$ is
constrained for each $[t_{s}^{m},t_{s}^{m+1})$, $i.e.$,
$|s_{i}(t)|\leq\hat{\chi}_{i}(t_{s}^{m})-\delta_{i}^{m}(i=1,2,...,n)$ for
$\forall t\in[t_{s}^{m},t_{s}^{m+1})$. Detail proofs are put in Appendix C.

Claim 1b. 1) The number of switchings is finite;

2) The closed loop nonlinear system admits continuous solution
$\overline{x}_{n}(t)$ on $[0,+\infty)$;

3) All the signals in the system are bounded on $[0,+\infty)$.

######  Proof 3.14.

Note that from Remark 3.10 and Claim 1a, we know during each time interval
$[t_{s}^{m},t_{s}^{m+1})$, the adaptive paramters keep constant, and
$|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{m})=\hat{\chi}_{i}(t)(i=1,...,n)$ with
$t\in[t_{s}^{m},t_{s}^{m+1})$. Thus, $V_{i}\geq 0(i=1,...,n)$ by (10) and
Proposition 3.4. Meanwhile, the propositions in Section 3.1 are all valid for
each $[t_{s}^{m},t_{s}^{m+1})$. Keeping this in mind, we will first prove the
number of switchings is finite. The proof is divided into the following steps.
These steps correspond to the $n$ steps Lyapunov functions analysis in Section
3.1.

Step 1. We will prove $\hat{\chi}_{1},\hat{\Theta}_{1}$ have finite numbers of
switchings.

1) Show $\hat{\chi}_{1}(t)$ does not switch on $[t_{s}^{0},t_{s}^{m+1})$.

According to Claim 1a, we know
$|s_{1}(t)|\leq\hat{\chi}_{1}(t_{s}^{m})-\delta_{1}^{m}$ on
$[t_{s}^{m},t_{s}^{m+1})$ for any finite integer $m$, and $s_{1}(t)=x_{1}(t)$
is continuous on $[t_{s}^{0},t_{s}^{m+1})$. Then, according to Algorithm 3.1-4
in Switching logic, $\hat{\chi}_{1}$ will not be updated on
$[t_{s}^{0},t_{s}^{m+1})$ since $s_{1}(t)$ will never transgress the barrier
$\hat{\chi}_{1}(t_{s}^{0})$.

2) Show $\hat{\Theta}_{1}(t)$ has a finite number of switchings.

This is proved by contradiction. If this is not true, then $\hat{\Theta}_{1}$
will switch infinite times.

From Claim 1a and the fact that $\hat{\chi}_{1}(t)=\hat{\chi}_{1}(t_{s}^{0})$
with $t\in[t_{s}^{0},t_{s}^{m+1})$, we have
$|s_{1}(t)|=|x_{1}(t)|<\hat{\chi}_{1}(t_{s}^{0})$ is bounded on
$[t_{s}^{0},t_{s}^{m+1})$. It follows that on $[t_{s}^{0},t_{s}^{m+1})$, the
unstructured uncertainties $h_{1}(x_{1})$ and $F_{1}(x_{1},\hat{\chi}_{1})$ in
(11) satisfy

$|h_{1}(x_{1})|\geq\underline{h}_{1}>0,$ $0\leq
F_{1}(x_{1},\hat{\chi}_{1})\leq\overline{F}_{1}$

where $\underline{h}_{1},\overline{F}_{1}$ are unknown constants irrelevant
with the number of switchings $m$.

Then, from the tuning rule (9) and the assumption of infinite switchings, we
can conclude that there exists a sufficiently large finite integer $m_{1}$
such that at switching time $t_{s}^{m_{1}}$, we have

$\displaystyle\mathrm{sgn}(\hat{\Theta}_{1}(t_{s}^{m_{1}}))$
$\displaystyle=\mathrm{sgn}(h_{1}(x_{1})),$ $\displaystyle-
h_{1}K_{1}\hat{\Theta}_{1}(t_{s}^{m_{1}})$
$\displaystyle\leq-\underline{h}_{1}K_{1}|\hat{\Theta}_{1}(t_{s}^{m_{1}})|<0,$
$\displaystyle
F_{1}(x_{1},\hat{\chi}_{1})-h_{1}\hat{\Theta}_{1}(t_{s}^{m_{1}})$
$\displaystyle\leq\overline{F}_{1}-\underline{h}_{1}|\hat{\Theta}_{1}(t_{s}^{m_{1}})|<0$

This implies that at switching time $t_{s}^{m_{1}}$, (11) will become

(25) $\displaystyle\dot{V}_{1}\leq$ $\displaystyle-
a_{1}V_{1}^{\frac{1+\alpha}{2}}-Q_{1}s_{1}^{1+\alpha}+c_{12}s_{2}^{1+\alpha}$

with $a_{1},Q_{1}$ defined in (21).

On the other hand, the auxiliary variable $\eta_{1}(t)$ in (21) satisfies

(26) $\displaystyle\dot{\eta}_{1}=$ $\displaystyle-
a_{1}\eta_{1}^{\frac{1+\alpha}{2}}-Q_{1}s_{1}^{1+\alpha}+c_{12}s_{2}^{1+\alpha}$

where $\eta_{1}(t_{s}^{m_{1}})\geq V_{1}(x_{1}(t_{s}^{m_{1}}),t_{s}^{m_{1}})$
according to Algorithm 3.1-4-(c) in Switching logic.

From Lemma 2.7 and (25)-(26), we know $V_{1}(x_{1}(t),t)\leq\eta_{1}(t)$ will
hold on $[t_{s}^{m_{1}},t_{s}^{m+1})$ for any $m+1>m_{1}$ without resetting
$\eta_{1}(t)$. This means that $\hat{\Theta}_{1}$ will not be updated after
$t_{s}^{m_{1}}$ which contradicts the fact that $\hat{\Theta}_{1}$ has an
infinite number of switchings.

Step 2. We will prove $\hat{\chi}_{2},\hat{\Theta}_{2}$ have finite numbers of
switchings.

The proof will be conducted on $[t_{s}^{m_{1}},t_{s}^{m+1})$ where
$t_{s}^{m_{1}}$ denotes the time instant when
$\hat{\chi}_{1},\hat{\Theta}_{1}$ stop switching and keep constant.

1) Show $\hat{\chi}_{2}(t)$ does not switch on $[t_{s}^{m_{1}},t_{s}^{m+1})$.

Note that $\hat{\Theta}_{1}$ is not updated on $[t_{s}^{m_{1}},t_{s}^{m+1})$.
Meanwhile, according to Claim 1a, we know $x_{2}$ is continuous on
$[t_{s}^{m_{1}},t_{s}^{m+1})$. Then, by (6) and (7), $s_{2}$ is continuous on
$[t_{s}^{m_{1}},t_{s}^{m+1})$. Also by Claim 1a, we have
$|s_{2}(t)|\leq\hat{\chi}_{2}(t_{s}^{m})-\delta_{2}^{m}$ on
$[t_{s}^{m},t_{s}^{m+1})$ for any finite integer $m$. Therefore, according to
Algorithm 1-4 in Switching logic, $\hat{\chi}_{2}$ will not be updated on
$[t_{s}^{m_{1}},t_{s}^{m+1})$ since $s_{2}(t)$ will never transgress the
barrier $\hat{\chi}_{2}(t_{s}^{m_{1}})$.

2) Show $\hat{\Theta}_{2}(t)$ has a finite number of switchings.

This is proved by contradiction. We suppose $\hat{\Theta}_{2}(t)$ will switch
infinite times.

First, since $\hat{\chi}_{2}$ is not updated on $[t_{s}^{m_{1}},t_{s}^{m+1})$,
we have $|s_{2}(t)|<\hat{\chi}_{2}(t_{s}^{m_{1}})$ with
$t\in[t_{s}^{m_{1}},t_{s}^{m+1})$ by Claim 1a. In addition, from Claim 1a, we
know $\hat{\chi}_{2}(t_{s}^{m_{1}})$ is bounded.

Using (26) in Step 1 we know $\eta_{1}$ satisfies

$\displaystyle\dot{\eta}_{1}=$ $\displaystyle-
a_{1}\eta_{1}^{\frac{1+\alpha}{2}}-Q_{1}s_{1}^{1+\alpha}+c_{12}s_{2}^{1+\alpha}$
$\displaystyle\leq$ $\displaystyle-
a_{1}\eta_{1}^{\frac{1+\alpha}{2}}+c_{12}\hat{\chi}_{2}^{1+\alpha}(t_{s}^{m_{1}})$

on $[t_{s}^{m_{1}},t_{s}^{m+1})$.

Therefore, it can be concluded that $0\leq V_{1}\leq\eta_{1}$ is bounded by a
constant irrelevant with $m$ (see Corollary 1 in [2]). Then, from the barrier
Lyapunov function (10), we know
$|s_{1}(t)|=|x_{1}(t)|\leq\hat{\chi}_{1}(t_{s}^{0})-\delta_{1}$ with a
positive constant $\delta_{1}$ irrelevant with $m$. Also from (6) and (7), we
know $x_{2}^{*},x_{2}$ are both bounded by constants irrelevant with $m$.

Hence, we conclude that on $[t_{s}^{m_{1}},t_{s}^{m+1})$,
$h_{2}(\overline{x}_{2})$ and
$F_{2}(\overline{x}_{2},\hat{\Theta}_{1},\hat{\chi}_{1})$ in (17) satisfy

$|h_{2}(\overline{x}_{2})|\geq\underline{h}_{2}>0,$ $0\leq
F_{2}(\overline{x}_{2},\hat{\Theta}_{1},\overline{\hat{\chi}}_{2})\leq\overline{F}_{2}$

where $\underline{h}_{2},\overline{F}_{2}$ are positive constants irrelevant
with $m$. Here, we also use the fact that
$\hat{\chi}_{1},\hat{\chi}_{2},\hat{\Theta}_{1}$ are constants on
$[t_{s}^{m_{1}},t_{s}^{m+1})$.

Then, from tuning rule (9), there exists a finite integer $m_{2}\geq m_{1}$
such that at switching time $t_{s}^{m_{2}}$, we have

$\displaystyle\mathrm{sgn}(\hat{\Theta}_{2}(t_{s}^{m_{2}}))$
$\displaystyle=\mathrm{sgn}(h_{2}(\overline{x}_{2})),$ $\displaystyle-
h_{2}K_{2}\hat{\Theta}_{2}(t_{s}^{m_{2}})$
$\displaystyle\leq-\underline{h}_{2}K_{2}|\hat{\Theta}_{2}(t_{s}^{m_{2}})|<0,$
$\displaystyle
F_{2}(\overline{x}_{2},\hat{\Theta}_{1},\overline{\hat{\chi}}_{2})-h_{2}\hat{\Theta}_{2}(t_{s}^{m_{2}})$
$\displaystyle\leq\overline{F}_{2}-\underline{h}_{2}|\hat{\Theta}_{2}(t_{s}^{m_{2}})|<0$

This implies that (17) will become

(27) $\displaystyle\dot{V}_{2}\leq$ $\displaystyle-
a_{2}V_{2}^{\frac{1+\alpha}{2}}-Q_{2}s_{2}^{1+\alpha}+c_{21}s_{1}^{1+\alpha}+c_{23}s_{3}^{1+\alpha}$

at switching time $t_{s}^{m_{2}}$ with $a_{2},Q_{2}$ defined in (21).

On the other hand, the auxiliary variable $\eta_{2}(t)$ in (21) satisfies

(28) $\displaystyle\dot{\eta}_{2}=$ $\displaystyle-
a_{2}\eta_{2}^{\frac{1+\alpha}{2}}-Q_{2}s_{2}^{1+\alpha}+c_{21}s_{1}^{1+\alpha}+c_{23}s_{3}^{1+\alpha}$

where $\eta_{2}(t_{s}^{m_{2}})\geq
V(\overline{x}_{2}(t_{s}^{m_{2}}),t_{s}^{m_{2}})$ according to Algorithm 3.1-5
in Switching logic.

From Lemma 2.7 and (27)-(28), we know
$V_{2}(\overline{x}_{2}(t),t)\leq\eta_{2}(t)$ will hold on
$[t_{s}^{m_{2}},t_{s}^{m+1})$ for any $m+1>m_{2}$ without resetting. This
means that $\hat{\Theta}_{2}$ will not be updated after $t_{s}^{m_{2}}$ which
contradicts the fact that $\hat{\Theta}_{2}$ has an infinite number of proof.

Step i($3\leq i\leq n$). By repeating the above procedures, we can show all
the parameters $\hat{\Theta}_{i},\hat{\chi}_{i}(i=1,2,...,n)$ have finite
numbers of switchings. Statement 1) in Claim 1b is proved

Next, for Statements 2)-3) in Claim 1b, according to Claim 1a, they hold
naturally when the number of switchings is finite. In fact, there must exist a
finite integer $m_{n}$ such that the switching time $t_{s}^{m_{n}+1}=+\infty$.
The proof is completed.

Claim 1c. All the states will converge to zero in finite time.

######  Proof 3.15.

From Claim 1a, we know
$|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{m})=\hat{\chi}_{i}(t)(i=1,...,n)$ for
$\forall t\in[t_{s}^{m},t_{s}^{m+1})$. Then, we have $V_{i}\geq 0(i=1,...,n)$
by (10), (15) and Proposition 3.4. From Claim 1b, we know the number of
switchings is finite. This implies that there exists a finite switching time
$t_{s}^{m_{n}}$ where $m_{n}$ is a finite integer and
$t_{s}^{m_{n}+1}=+\infty$. When
$t\in[t_{s}^{m_{n}},t_{s}^{m_{n}+1})=[t_{s}^{m_{n}},+\infty)$, $0\leq
V_{i}\leq\eta_{i}$ holds for $\forall i\in\\{1,2,...,n\\}$ without resetting
$\eta_{i}$. Then, on $[t_{s}^{m_{n}},+\infty)$, by (21)-(23) we have

$\dot{\overline{\eta}}\leq-a^{\prime}\overline{\eta}^{\gamma}$

where $a^{\prime}>0$ is a positive parameter,
$\gamma\triangleq\frac{1+\alpha}{2}\in(0,1)$.

Therefore, during time interval $[t_{s}^{m_{n}},+\infty)$, we have

$\displaystyle{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}0\leq}}$
$\displaystyle{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\sum_{i=1}^{n}V_{i}\leq\overline{\eta}}}(t)$
$\displaystyle\leq$
$\displaystyle[\overline{\eta}^{1-\gamma}(t_{s}^{m_{n}})-a^{\prime}(1-\gamma)(t-t_{s}^{m_{n}})]^{\frac{1}{1-\gamma}}.$

Note that by Claim 1b, all the signals including
$\overline{\eta}(t_{s}^{m_{n}})$ are bounded on $[0,+\infty)$. Hence, we can
conclude that after finite time
$t_{s}^{m_{n}}+\frac{\eta^{1-\gamma}(t_{s}^{m_{n}})}{a^{\prime}(1-\gamma)}$,
$\overline{\eta}$ and $V_{i}(i=1,2,...,n)$ will converge to zero. By (5)-(8),
(10), (15) and (16), we know $s_{1}=x_{1},x_{2}^{*}$,
$x_{2},s_{2},x_{3}^{*},x_{3},...,x_{n}$ will also converge to zero in finite
time. The proof is completed.

##  4 Prescribed finite time stabilization

In this section, we will give an extension to the controller designed in
Section 3. The states will be regulated to zero in prescribed finite time. In
the following, the initial time is set zero for simplicity.

###  4.1 Controller design

In this subsection, we will propose a new time-varying backstepping scheme for
the controller design. The scheme includes the design of time-varying
controller and its corresponding Lyapunov functions analysis. The time-varying
controller is successively determined by:

(29) $\displaystyle s_{1}$ $\displaystyle\triangleq x_{1},$ (30)
$\displaystyle x_{i+1}^{*}$
$\displaystyle\triangleq\hat{\Theta}_{i}(t)\left[-K_{i}s_{i}-\frac{\mu^{\beta_{i}}U_{i}s_{i}}{\hat{\chi}_{i}^{2}(t)-s_{i}^{2}}\right],$
(31) $\displaystyle s_{i+1}$ $\displaystyle\triangleq
x_{i+1}-x_{i+1}^{*}(i=1,2,...,n-1),$ (32) $\displaystyle x_{n+1}^{*}$
$\displaystyle\triangleq
u\triangleq\hat{\Theta}_{n}(t)\left[-K_{n}s_{n}-\frac{\mu^{\beta_{n}}U_{n}s_{n}}{\hat{\chi}_{n}^{2}(t)-s_{n}^{2}}\right]$

where $x_{i+1}^{*}(i=1,2,...,n)$ are the virtual or real control efforts.
$K_{i},U_{i}$ and $\beta_{i}\in(1,+\infty)$ are positive design parameters.
$\mu(t):[0,T)\rightarrow\mathbb{R}$ is a time-varying gain by

$\mu(t)=\frac{T}{T-t}\geq 1$

with a prescribed positive constant $T>0$. $\hat{\chi}_{i}(t)$ and
$\hat{\Theta}_{i}(t)$ are piecewise constant adaptive parameters like those in
(6) or (8). They will be updated by the logic-based switching rule in Section
4.2. Next, a $n$ step Lyapunov functions analysis will be given according to
$x_{i+1}^{*}(i=1,2,...,n)$.

Step 1. Take the following barrier Lyapunov function

(33)
$V_{1}=\frac{1}{2}\ln\left(\frac{\hat{\chi}_{1}^{2}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}\right)$

where $\hat{\chi}_{1}(t)$ is defined in (30). Then, we have the following
result.

######  Proposition 4.1.

Suppose $\hat{\chi}_{1}(t)$ is a positive constant and
$|s_{1}|<\hat{\chi}_{1}$. Then, by using (30) with $i=1$, $\dot{V}_{1}$ can be
expressed as

$\displaystyle\dot{V}_{1}\leq$ $\displaystyle-
a_{1}\mu^{\beta_{1}}V_{1}-Q_{1}\mu^{\beta_{1}}s_{1}^{2}-\frac{h_{1}K_{1}\hat{\Theta}_{1}s_{1}^{2}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
(34)
$\displaystyle+\frac{U_{1}(\mu^{\beta_{1}}F_{1}(x_{1},\hat{\chi}_{1})-\mu^{\beta_{1}}h_{1}\hat{\Theta}_{1})s_{1}^{2}}{(\hat{\chi}_{1}^{2}-s_{1}^{2})^{2}}+\frac{c_{12}s_{2}^{2}}{\mu^{\beta_{1}}}$

where $a_{1},Q_{1},c_{12}$ are positive parameters,
$F_{1}(x_{1},\hat{\chi}_{1})$ is an unknown non-negative function which comes
from the unstructured uncertainties in (1).

######  Proof 4.2.

By Lemma 2.4, (33) can be expressed as:

$\displaystyle\dot{V}_{1}\leq$
$\displaystyle\frac{s_{1}h_{1}x_{2}^{*}+s_{1}^{2}\psi_{1}(x_{1})}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{h_{1}s_{1}s_{2}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
$\displaystyle=$
$\displaystyle\frac{s_{1}h_{1}x_{2}^{*}+s_{1}^{2}\psi_{1}(x_{1})}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{h_{1}\left(\mu^{\frac{\beta_{1}}{2}}s_{1}\right)\left(\mu^{-\frac{\beta_{1}}{2}}s_{2}\right)}{\hat{\chi}_{1}^{2}-s_{1}^{2}}$
(35) $\displaystyle\leq$
$\displaystyle\frac{s_{1}h_{1}x_{2}^{*}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}+\frac{\mu^{\beta_{1}}U_{1}s_{1}^{2}\overline{F}_{1}(x_{1},\hat{\chi}_{1})}{(\hat{\chi}_{1}^{2}-s_{1}^{2})^{2}}+\frac{c_{12}s_{2}^{2}}{\mu^{\beta_{1}}}$

where $U_{1},c_{12}$ and $\beta_{1}\in(1,+\infty)$ are positive design
parameters, $\overline{F}_{1}(x_{1},\hat{\chi}_{1})$ is an unknown function.

Substituting (30) into (35), we have (34). It completes the proof.

Step i($2\leq i\leq n$). Take the following switching barrier Lyapunov
function

(36)
$V_{i}=\frac{1}{2}\ln\left(\frac{\hat{\chi}_{i}^{2}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}\right)$

where $\hat{\chi}_{i}(t)$ is defined in (30) or (32).

Then, we have following result:

######  Proposition 4.3.

Suppose $\overline{\hat{\Theta}}_{i}(t),\overline{\hat{\chi}}_{i}(t)$ are
positive constant vectors and $|s_{i}|<\hat{\chi}_{i}$ where
$\overline{\hat{\Theta}}_{i}\triangleq(\hat{\Theta}_{1},\hat{\Theta}_{2},...,\hat{\Theta}_{i})^{\mathrm{T}}$,
$\overline{\hat{\chi}}_{i}\triangleq(\hat{\chi}_{1},\hat{\chi}_{2},...,\hat{\chi}_{i})^{\mathrm{T}}$.
Then, by using (30) or (32), $\dot{V}_{i}$ can be expressed as

$\displaystyle\dot{V}_{i}\leq$ $\displaystyle-
a_{i}\mu^{\beta_{i}}V_{i}-Q_{i}\mu^{\beta_{i}}s_{i}^{2}-\frac{h_{i}K_{i}\hat{\Theta}_{i}s_{i}^{2}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle+\frac{\mu^{\beta_{i}}U_{i}s_{i}^{2}(F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})-h_{i}\hat{\Theta}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2}}$
(37)
$\displaystyle+\sum_{j=1}^{i-1}c_{ij}\mu^{\beta_{j}}s_{j}^{2}+\frac{c_{i,i+1}s_{i+1}^{2}}{\mu^{\beta_{i}}}$

where $a_{i},Q_{i},c_{ij}(j=1,2,...,i-1,i+1)$ are all positive design
parameters, $s_{n+1}\triangleq 0$.
$F_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown non-negative function which comes from the unstructured
uncertainties in (1).

######  Proof 4.4.

The proof is somewhat involved and put in Appendix D.

######  Remark 4.5.

(Time-varying backstepping scheme) The prescribed finite time control method
was initially proposed in [36], where a time-varying state transformation
technique was used to design the controller. However, in our case, the system
contains unstructured uncertainties and unknown control directions. It is very
difficult to integrate the switching barrier Lyapunov functions or the logic-
based switching rule into the time-varying state transformation technique. The
reason may be the nonlinear characteristics of the barrier Lyapunov functions
and the discrete dynamic of the logic-based switching. Therefore, we propose a
new time-varying backstepping scheme explained above. Compared with existing
backstepping methods and [36], the proposed scheme has two distinguishing
features:

1) Based on the skillful use of Lemma 2.4, the derivatives of the Lyapunov
functions are transformed into suitable forms given by (37) and (34), which
contain time-varying gains both in the numerators and denominators. This form
is important for the tuning rule design and stability analysis (see Lemma 4.7
in Section 4.3);

2) Time-varying gains and switching adaptive parameters are both considered in
the virtual and real control efforts to achieve prescribed finite time
stability.

###  4.2 Logic-based switching rule

We will present the logc-based switching rule for tuning adaptive parameters
$\hat{\Theta}_{i}$ and $\hat{\chi}_{i}$ for $i\in\\{1,2,...,n\\}$. It is very
similar to the algorithm in Section 3.2 but with time-varying gains.

For $i\in\\{1,2,...,n\\}$, define the following new supervisory functions
$\mathcal{S}_{i}(t)$.

(38) $\mathcal{S}_{i}(t)\triangleq V_{i}(\overline{x}_{i},t)-\eta_{i}(t),$
$\displaystyle\dot{\eta}_{i}=$ $\displaystyle-
a_{i}\mu^{\beta_{i}}\eta_{i}-Q_{i}\mu^{\beta_{i}}s_{i}^{2}$ (39)
$\displaystyle+\sum_{j=1}^{i-1}c_{ij}\mu^{\beta_{j}}s_{j}^{2}+\frac{c_{i,i+1}s_{i+1}^{2}}{\mu^{\beta_{i}}}(i=1,2,...,n),$

where $V_{i}$ is given by (33) and (36), $\eta_{i}$ is an auxiliary variable,
$a_{i},Q_{i}>0$ are design parameters.

Based on the above supervisory functions, the algorithm is also given by Table
I. The difference is the computation of $V_{i}$, $\eta_{i}$ and
$\mathcal{S}_{i}$, which are from (33), (36), (39) and (38). The algorithm is
referred to as Algorithm 3.2.

Similar to Algorithm 3.1 (see Remark 3.10), the purpose of the algorithm is
also to let $\mathcal{S}_{i}(t)=V_{i}(\overline{x}_{i},t)-\eta_{i}(t)\leq 0$
hold forever after a finite number of switchings. In fact, when
$\mathcal{S}_{i}(t)=V_{i}(\overline{x}_{i},t)-\eta_{i}(t)\leq 0$ holds, let
$\overline{\eta}\triangleq\sum_{i=1}^{n}\eta_{i}$, from (39), we can conclude
that with sufficiently large $Q_{i}$,

(40) $\displaystyle\dot{\overline{\eta}}\leq$
$\displaystyle-\sum_{i=1}^{n}a_{i}\mu^{\beta_{i}}\eta_{i}\leq-\mu^{\beta^{\prime}}a^{\prime}\overline{\eta}$

where $\beta^{\prime}\in(1,+\infty)$, $a^{\prime}>0$ are positive constants.
According to [36], it implies that $\overline{\eta}$ may become zero in
prescribed finite time. Due to
$\mathcal{S}_{i}(t)=V_{i}(\overline{x}_{i},t)-\eta_{i}(t)\leq 0$,
$V_{i},x_{i}$ will also become zero in prescribed finite time.

###  4.3 Stability analysis

######  Theorem 4.6.

Consider the nonlinear system in (1). Then, the controller (29)-(32) with
Algorithm 3.2 can guarantee that:

1) All the signals except for the time-varying gain $\mu$ in the closed-loop
system are bounded on $[0,T)$, and;

2) All the states will be regulated to zero in prescribed finite time $T$.

The proof for the above result will be presented in this subsection. Similar
to Theorem 3.12, define a time sequence
$\\{0=t_{s}^{0}<t_{s}^{1}<...<t_{s}^{m}<...\leq T\\}$ with
$m\in\\{0,1,2,...\\}$. $t_{s}^{m}$ represents the switching time. Then, the
proof will be divided into three claims.

Claim 2a. For a given finite integer $m$, we have

1) The closed loop nonlinear system has continuous solution
$\overline{x}_{n}(t)$ on $[t_{s}^{0},t_{s}^{m+1})$;

2) There exists a positive constant $\delta_{i}^{m}$ such that
$|s_{i}(t)|\leq\hat{\chi}_{i}(t_{s}^{m})-\delta_{i}^{m}$ for $\forall
i=1,2,...,n$ on $[t_{s}^{m},t_{s}^{m+1})$

3) All the signals except for $\mu$ in the system are bounded on
$[t_{s}^{m},t_{s}^{m+1})$.

Claim 2b. 1) The number of switchings is finite;

2) The closed loop nonlinear system admits continuous solution
$\overline{x}_{n}(t)$ on $[0,T)$;

3) All the signals except for $\mu$ in the system are bounded on $[0,T)$.

Claim 2c. All the states will converge to zero in prescribed finite time $T$.

The proof for the above three claims follows the line of the proof of Theorem
3.12. Yet, in order to handle the time-varying gain, an additional lemma will
be used to show the boundedness of the signals and prescribed finite time
stability.

######  Lemma 4.7.

Given two time-varying non-negative continuous functions
$\xi(t),b(t):[t_{1},T)\rightarrow\mathbb{R}$ such that

(41) $\displaystyle\dot{\xi}(t)$
$\displaystyle\leq-a\mu^{\alpha}(t)\xi(t)+b(t),$ (42) $\displaystyle b(t)$
$\displaystyle\leq\frac{\chi}{\mu^{\beta}(t)}$

where $t_{1},T,a,\chi,\alpha,\beta$ are positive constants with $0\leq
t_{1}<T$, $\alpha\in(1,+\infty)$ and $\mu(t)=\frac{T}{T-t}.$

Meanwhile, there exists a time-varying continuous function
$x(t):[t_{1},T)\rightarrow\mathbb{R}$ such that

(43)
$\ln\left(\frac{\overline{\chi}^{2}}{\overline{\chi}^{2}-x^{2}(t)}\right)\leq\xi(t)$

where $\overline{\chi}$ is a positive constant and
$|x(t_{1})|<\overline{\chi}$.

Then, we have $\xi(t),$ $x^{2}(t)\mu^{\alpha+\beta}(t)$ are both bounded on
$[t_{1},T)$.

Particularly, if $b(t)=0$, then $\underset{t\rightarrow
T^{-}}{\lim}x^{2}(t)\mu^{\gamma}(t)=0$ with a positive constant $\gamma$.

######  Proof 4.8.

The proof is put in Appendix B.

######  Remark 4.9.

(New analysis tool) The above lemma serves as a new analysis tool to show the
prescribed finite time stability. As shown in [36] a key aspect for the
prescribed finite time control is to show the boundedness of the control
signals since the time-varying gain $\mu(t)$ will grow to infinite when
$t\rightarrow T$. However, as stated in Remark 3.9, the proposed controller
structure is quite different from the existing prescribed finite time control
methods [36]. Hence, the analysis method in [36] is not applicable for our
case. Therefore, we propose a new analysis tool to deal with the stability
problem. Then, the boundedness of the control signals can be proved directly
from the properties of the Lyapunov and supervisory functions. The proposed
analysis tool lay the foundation for showing prescribed finite time stability
of nonlinear systems with unstructured uncertainties (see the following Step 2
and Corollary 2 in Appendix F.1 for its applications).

Next, we will prove Claims 2b and 2c. Claim 2a can be proved similarly by
resorting to the proof of Claim 1a.

We will first show there are only a finite number of switchings. We will
successively prove $\hat{\chi}_{i},\hat{\Theta}_{i}(i=1,2,...,n)$ only have
finite numbers of switchings.

Step 1. We will prove $\hat{\chi}_{1},\hat{\Theta}_{1}$ have finite numbers of
switchings.

1) Show $\hat{\chi}_{1}(t)$ does not switch on $[t_{s}^{0},t_{s}^{m+1})$.

This can be proved by following the line of Claim 1b and using Claim 2a. We
can show $\hat{\chi}_{1}$ will not be updated on $[t_{s}^{0},t_{s}^{m+1})$,

2) Show $\hat{\Theta}_{1}$ has a finite number of switchings.

This is proved by contradiction. If this is not true, then $\hat{\Theta}_{1}$
will switch infinite times.

Due to $\hat{\chi}_{1}$ is not updated, the unstructured uncertainties
$h_{1}(x_{1})$ and $F_{1}(x_{1},\hat{\chi}_{1})$ in (34) satisfy

$|h_{1}(x_{1})|\geq\underline{h}_{1}>0,$ $0\leq
F_{1}(x_{1},\hat{\chi}_{1})\leq\overline{F}_{1}$

where $\underline{h}_{1},\overline{F}_{1}$ are positive unknown constants
irrelevant with the number of switchings $m$.

Then, from the definition of $\hat{\Theta}_{1}$ and (34), we can conclude that
there exists a sufficiently large finite integer $m_{1}$ such that at time
instant $t_{s}^{m_{1}}$, we have

(44) $\displaystyle\dot{V}_{1}\leq$
$\displaystyle-\mu^{\beta_{1}}a_{1}V_{1}-Q_{1}\mu^{\beta_{1}}s_{1}^{2}+\frac{c_{12}s_{2}^{2}}{\mu^{\beta_{1}}}.$

On the other hand, the auxiliary variable $\eta_{1}(t)$ in (39) satisfies

(45) $\displaystyle\dot{\eta}_{1}=$
$\displaystyle-\mu^{\beta_{1}}a_{1}\eta_{1}-Q_{1}\mu^{\beta_{1}}s_{1}^{2}+\frac{c_{12}s_{2}^{2}}{\mu^{\beta_{1}}}$

where $\eta_{1}(t_{s}^{m_{1}})\geq V_{1}(x_{1}(t_{s}^{m_{1}}),t_{s}^{m_{1}})$.

From Lemma 2.7, we know $V_{1}(x_{1}(t),t)\leq\eta_{1}(t)$ will hold on
$[t_{s}^{m_{1}},t_{s}^{m+1})$ for any $m+1>m_{1}$ without resetting
$\eta_{1}(t)$. This means that $\hat{\Theta}_{1}$ will not be updated after
$t_{s}^{m_{1}}$.

Step 2. We will prove $\hat{\chi}_{2},\hat{\Theta}_{2}$ have finite numbers of
switchings.

The proof will be conducted on $[t_{s}^{m_{1}},t_{s}^{m+1})$ such that
$t_{s}^{m_{1}}$ denotes the time instant when
$\hat{\chi}_{1},\hat{\Theta}_{1}$ stop switching.

1) Show $\hat{\chi}_{2}(t)$ does not switch on $[t_{s}^{m_{1}},t_{s}^{m+1})$.

This can be proved by following the line of Claim 1b. We can show
$\hat{\chi}_{2}$ will not be updated on $[t_{s}^{m_{1}},t_{s}^{m+1})$,

2) Show $\hat{\Theta}_{2}$ has a finite number of switchings.

This is proved by contradiction. If this is not true, then $\hat{\Theta}_{2}$
will switch infinite times.

Since $\hat{\chi}_{2}$ will not be updated on $[t_{s}^{m_{1}},t_{s}^{m+1})$,
we have $|s_{2}(t)|<\hat{\chi}_{2}(t_{s}^{m_{1}})$ on
$[t_{s}^{m_{1}},t_{s}^{m+1})$. In addition, from Claim 2a, we know
$\hat{\chi}_{2}(t_{s}^{m_{1}})$ is bounded.

Note that from (45) in Step 1, we know $\eta_{1}$ satisfies

$\displaystyle\dot{\eta}_{1}\leq$
$\displaystyle-\mu^{\beta_{1}}a_{1}\eta_{1}-Q_{1}\mu^{\beta_{1}}s_{1}^{2}+\frac{c_{12}s_{2}^{2}}{\mu^{\beta_{1}}}$
$\displaystyle\leq$
$\displaystyle-\mu^{\beta_{1}}a_{1}\eta_{1}+\frac{c_{12}\hat{\chi}_{2}^{2}(t_{s}^{m_{1}})}{\mu^{\beta_{1}}}$

on $[t_{s}^{m_{1}},t_{s}^{m+1})$. Based on Lemma 4.7 and (33), we know
$\eta_{1}$ and $\mu^{2\beta_{1}}s_{1}^{2}$ are bounded by some constants
irrelevant with the number of switchings $m$ (let $\xi(t)=\eta_{1}$,
$\chi=c_{12}\hat{\chi}_{2}^{2}(t_{s}^{m_{1}})$, $x(t)=s_{1}(t)$,
$\overline{\chi}=\hat{\chi}_{1}$ in Lemma 4.7).

Since $V_{1}\leq\eta_{1}$ is bounded, by (33) we know
$|s_{1}(t)|\leq\hat{\chi}_{1}(t_{s}^{0})-\delta_{1}$ with a positive constant
$\delta_{1}$ irrelevant with $m$. Meanwhile, due to
$\mu^{2\beta_{1}}s_{1}^{2}$ is bounded, by (30) and (31) we obtain
$x_{2}^{*}$, $x_{2}$ are both bounded by some constants irrelevant with $m$.

Hence, we conclude that on $[t_{s}^{m_{1}},t_{s}^{m+1})$,
$h_{2}(\overline{x}_{2})$ and
$F_{2}(\overline{x}_{2},\hat{\Theta}_{1},\overline{\hat{\chi}}_{2})$ in (37)
satisfy

$h_{2}(\overline{x}_{2})\geq\underline{h}_{2}>0,$ $0\leq
F_{2}(\overline{x}_{2},\hat{\Theta}_{1},\overline{\hat{\chi}}_{2})\leq\overline{F}_{2}$

where $\underline{h}_{2},\overline{F}_{2}$ are positive constants irrelevant
with $m$.

Then, for (37) we can conclude that there exists a finite integer $m_{2}\geq
m_{1}$ such that at switching time $t_{s}^{m_{2}}$, we have

$\displaystyle\dot{V}_{2}\leq$
$\displaystyle-\mu^{\beta_{2}}a_{2}V_{2}-Q_{2}\mu^{\beta_{2}}s_{2}^{2}+c_{21}\mu^{\beta_{1}}s_{1}^{2}+\frac{c_{23}s_{3}^{\beta_{2}}}{\mu^{\beta_{2}}}.$

On the other hand, the auxiliary variable $\eta_{2}(t)$ in (21) satisfies

$\displaystyle\dot{\eta}_{2}=$
$\displaystyle-\mu^{\beta_{2}}a_{2}\eta_{2}-Q_{2}\mu^{\beta_{2}}s_{2}^{2}+c_{21}\mu^{\beta_{1}}s_{1}^{2}+\frac{c_{23}s_{3}^{\beta_{2}}}{\mu^{\beta_{2}}}$

where $\eta_{2}(t_{s}^{m_{2}})\geq
V_{2}(\overline{x}_{2}(t_{s}^{m_{2}}),t_{s}^{m_{2}})$.

From Lemma 2.7, we know $V_{2}(\overline{x}_{2}(t),t)\leq\eta_{2}(t)$ will
hold on $[t_{s}^{m_{2}},t_{s}^{m+1})$ for any $m+1>m_{2}$. This means that
$\hat{\Theta}_{2}$ will not be updated after $t_{s}^{m_{2}}$.

Step i($3\leq i\leq n$). By repeating the above procedures, we can show all
the parameters $\hat{\Theta}_{i},\hat{\chi}_{i}(i=1,2,...,n)$ have finite
numbers of switchings. This proves Statement 1) in Claim 2b.

Next, for Statements 2)-3), according to Claim 2a, they hold naturally when
the number of switchings is finite. Claim 2b is proved.

To prove Claim 2c, from Claim 2b, we know the number of switchings is finite.
This indicates that there exists a switching time $t_{s}^{m_{n}}$ such that
when $t\in[t_{s}^{m_{n}},T)$, $V_{i}(\overline{x}_{i}(t),t)\leq\eta_{i}(t)$
holds for $\forall i\in\\{1,2,...,n\\}$. Then, on $[t_{s}^{m_{n}},T)$, by (40)
and Lemma 4.7, we know all the states will converge to zero in prescribed
finite time.

######  Remark 4.10.

Note that though theoretically we can prove the state can reach zero in
prescribed finite time, as stated in [36] the utilization of an unbounded
time-varying gain $\mu$ may result in some numerical issues in real practice.
This issue can be avoided by setting $T$ larger than the desired finite time
but with a sacrifice of control accuracy. Also the complexity of the
controller will increase when using a time-varying gain compared with the
controller in Section 3. This might bring some difficulties to the
implementation of the controller.

##  5 Simulations

In this section, an illustrative example is presented. Note that some further
discussions about convergence speed, control overshoot, controller parameters
selection, comparison with existing asymptotic control methods and control of
third order nonlinear systems are put in Appendix E.

Given a second order nonlinear system by (1) with $n=2$, we consider the
following six cases:

$\displaystyle\mathrm{Case}\thinspace\mathrm{A}:$ $\displaystyle
h_{1}=1,h_{2}=0.8,f_{1}=\overline{f}_{1},f_{2}=\overline{f}_{2};$
$\displaystyle\mathrm{Case}\thinspace\mathrm{B}:$ $\displaystyle
h_{1}=1,h_{2}=-0.8,f_{1}=\overline{f}_{1},f_{2}=\overline{f}_{2};$
$\displaystyle\mathrm{Case}\thinspace\mathrm{C}:$ $\displaystyle
h_{1}=1,h_{2}=0.8,f_{1}=5\overline{f}_{1},f_{2}=5\overline{f}_{2};$
$\displaystyle\mathrm{Case}\thinspace\mathrm{D}:$ $\displaystyle
h_{1}=-1,h_{2}=0.8,f_{1}=\overline{f}_{1},f_{2}=\overline{f}_{2};$
$\displaystyle\mathrm{Case}\thinspace\mathrm{E}:$ $\displaystyle
h_{1}=-1,h_{2}=-0.8,f_{1}=\overline{f}_{1},f_{2}=\overline{f}_{2};$
$\displaystyle\mathrm{Case}\thinspace\mathrm{F}:$ $\displaystyle
h_{1}=-1,h_{2}=0.8,f_{1}=f_{1}^{*},f_{2}=f_{2}^{*}$

where $\overline{f}_{1}\triangleq 0.1\sin(x_{1})x_{1}$;
$\overline{f}_{2}\triangleq-4.9\sin(x_{1})+0.05\sin(x_{1})e^{-x_{2}}+0.1\sin(x_{2})x_{2}^{2}.$
$f_{1}^{*}\triangleq-3x_{1}^{2}/2-x_{1}^{3}/2$ and $f_{2}^{*}\triangleq
0.1\sin(x_{2})$. Case A is the nominal case. It can be used to describe the
dynamics of a single link robot manipulator (Xing, Wen, Liu, Su, & Cai, 2017).
Other five situations represent the variation of control directions and
modeling uncertainties. Specifically, Case F indicates that the whole system
has been changed into another form. The initial conditions are
$x_{1}(0)=0.1,x_{2}(0)=0.2.$ For the controller design, we assume
$h_{1},h_{2},f_{1},f_{2}$ are all unknown.

1) Effectiveness of the logic-based switching

The controller is designed by (5)-(8). The controller parameters are set as:
$K_{1}=U_{1}=K_{2}=U_{2}=1$, $\alpha=41/49$,
$\hat{\chi}_{1}=\hat{\chi}_{2}(0)=2$. $\theta_{1}(0)=\theta_{2}(0)=0.1$ and
for $\sigma_{i}(t)\geq 1(i=1,2)$, we have
$\theta_{i}(\sigma_{i})=(-1)^{\sigma_{i}(t)}(\iota_{i1}+\iota_{i2}\sigma_{i})$
where $\iota_{i1}=\iota_{i2}=1.$
$\hat{\Theta}_{1},\hat{\Theta}_{2}\text{,$\hat{\chi}_{1}$,$\hat{\chi}_{2}$ }$
are updated by Algorithm 1 with $\varsigma=4,\varepsilon=0.01$,
$a_{1}=a_{2}=0.2,$ $Q_{1}=Q_{2}=c_{11}=c_{21}=1$. The control performance is
shown in Fig. 2. It can be seen that both states converge to zero in a very
short time for the above six situations. This implies that the finite time
stability has been achieved despite multiple unknown control directions and
unstructured uncertainties.

Next, set $\hat{\chi}_{2}(0)=0.8,\hat{\chi}_{1}(0)=2$. The variations of
$s_{i},\hat{\Theta}_{i},\hat{\chi}_{i}(i=1,2)$ are shown in Fig. 3. Note that
$s_{1}$ is constrained into the tube $[-2,2]$. This shows the validity of the
barrier Lyapunov function $V_{1}$ in (10). Also we can see $s_{2}$ is
constrained into the tube $[-0.8,0.8]$ during time interval $[0,1.35)$ and is
larger than $\hat{\chi}_{2}(0)=0.8$ at time instant $1.35s$. Therefore,
$\hat{\chi}_{2}$ jumps to $4.8$ to contain $s_{2}$ in the later time. The
reason for $s_{2}$ jumping outside $0.8$ is that $\hat{\Theta}_{1}$ is updated
at $1.35s$. Note that though $s_{2}$ transgresses over the barrier
$\hat{\chi}_{2}(0)=0.8$, $s_{1},s_{2}$ still can converge to zero in finite
time. All these show the effectiveness of the switching barrier Lyapunov
function.

2) Time-varying feedback

We consider the control performance by the controller designed in Section 4.
The controller is given by (29)-(32). The controller parameters are selected
the same as before except $\beta_{1}=1.4$, $\beta_{2}=4.2$. The prescribed
finite time $T=4.5s.$ Fig. 4 shows that the states are regulated to zero
before the prescribed time $4.5s$ for the considered six cases. This verifies
the vadility of the presented method.

Figure 2: Control performance for second order system. Figure 3: Variations of
$s_{i},\hat{\Theta}_{i},\hat{\chi}_{i}(i=1,2)$. It can be seen that $|s_{2}|$
is constrained in the barrier $\hat{\chi}_{2}(t)=0.8$ during time interval
$[0,1.35)$ and jump outside the barrier $\hat{\chi}_{2}(t)$ at time instant
$1.35s$ due to $\hat{\Theta}_{1}$ is updated. Then, by the proposed method,
$\hat{\chi}_{2}$ will switch to $4.8$ to contain $s_{2}$ in the later time to
guarantee the finite time stability.  Figure 4: Control performance for time-
varying feedback.

##  6 Conclusions

In this paper, two kinds of logic-based switching adaptive controllers are
proposed. The finite time stability can be guaranteed for the nonlinear
systems suffering from multiple unknown control directions and unstructured
uncertainties. Future work will be focused on extending the presented design
approach to general hybrid systems.

##  Appendix A Proof of Lemma 2.7

######  Proof A.1.

This is a variation of Comparison Principle [20]. Subtracting (3) from (4), we
have:

(46) $\dot{e}(t)\leq-a[y^{\gamma}(t)-x^{\gamma}(t)]$

where $e(t)\triangleq y(t)-x(t).$ Then, we only need to prove $e(t)\leq 0$ for
$\forall t\in[t_{0},t_{1})$. In fact, if this is not true, there must exist a
time instant $t^{*}\in[t_{0},+\infty)$ such that $e(t^{*})>0$. Since
$e(t_{0})=y(t_{0})-y(t_{0})-\varepsilon=-\varepsilon\leq 0$, we can conclude
that there exists a time interval $[\tau_{1},\tau_{2}]\subseteq[t_{0},t^{*}]$
such that $e(\tau_{1})=0$, $e(t)>0$ for $\forall t\in(\tau_{1},\tau_{2}]$. By
Mean Value Theorem, there exists a time instant
$\tau_{3}\in[\tau_{1},\tau_{2}]$ such that $e(\tau_{3})>0$ and
$\dot{e}(\tau_{3})>0$. However, this contradicts (46). It completes the proof.

##  Appendix B Proof of Lemma 4.7

######  Proof B.1.

By solving the differential equation (41), we obtain

$\displaystyle\xi(t)\leq$ $\displaystyle
e^{-a\overline{\mu}(t)}\xi(t_{1})+e^{-a\overline{\mu}(t)}\int_{t_{1}}^{t}e^{a\overline{\mu}(\tau)}b(\tau)d\tau$

where $t\in[t_{1},T)$,

$\overline{\mu}(t)=\int_{t_{1}}^{t}\mu^{\alpha}(\tau)d\tau=c_{1}\mu^{\alpha-1}(t)-c_{2}\geq
0$

with positive constants $c_{1},c_{2}$ satisfying
$c_{1}\mu^{\alpha-1}(t_{1})=c_{2}.$

Then, we have

$\displaystyle\mu^{\alpha+\beta}(t)\xi(t)\leq$ $\displaystyle
A_{1}(t)+A_{2}(t)$

where

(47) $A_{1}(t)=\mu^{\alpha+\beta}e^{-a\overline{\mu}(t)}\xi(t_{1}),$ (48)
$A_{2}(t)=\mu^{\alpha+\beta}e^{-a\overline{\mu}(t)}\int_{t_{1}}^{t}e^{a\overline{\mu}(\tau)}\frac{\chi}{\mu^{\beta}(\tau)}d\tau.$

Next, we will show $A_{1}(t)$ and $A_{2}(t)$ are both bounded. For $A_{1}(t)$,
note that by Taylor expansion
$e^{a\overline{\mu}(t)}=\sum_{n=0}^{+\infty}\frac{(a\overline{\mu})^{n}}{n!}.$
Hence, there exists a positive constant $c_{3}$ such that

(49) $e^{a\overline{\mu}(t)}\geq 1+c_{3}(c_{1}\mu^{\alpha-1}(t)-c_{2})^{n}$

where $n$ can be any finite positive integers.

Using (49) for (47), we have

(50) $\displaystyle A_{1}(t)$
$\displaystyle\leq\xi(t_{1})\frac{\mu^{\alpha+\beta}}{1+c_{3}(c_{1}\mu^{\alpha-1}-c_{2})^{n}}$

Hence, if $n$ is large enough such that $n(\alpha-1)>\alpha+\beta$, then
$\mu^{n(\alpha-1)}$ will dominate other terms in (50) when $t\rightarrow
T^{-}$. As a result,

$\underset{t\rightarrow
T^{-}}{\lim}\xi(t_{1})\frac{\mu^{\alpha+\beta}}{1+c_{3}(c_{1}\mu^{\alpha-1}-c_{2})^{n}}=0.$

Then, we can conclude that $A_{1}(t)$ is bounded.

For $A_{2}(t)$, note that it is continuous on $[0,T)$. Hence, we only need to
show when $t\rightarrow T^{-}$, $A_{2}(t)$ is bounded. Then, $A_{2}(t)$ is
bounded on $[0,T)$. By L’Hospital rule,

$\displaystyle\lim_{t\rightarrow T^{-}}A_{2}(t)$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow
T^{-}}\frac{\chi\int_{t_{1}}^{t}\frac{e^{a\overline{\mu}(\tau)}}{\mu^{\beta}(\tau)}d\tau}{\frac{e^{a\overline{\mu}(t)}}{\mu^{\alpha+\beta}(t)}}=\lim_{t\rightarrow
T^{-}}\frac{\chi\frac{e^{a\overline{\mu}(t)}}{\mu^{\beta}(t)}-c_{4}}{\left(\frac{e^{a\overline{\mu}(t)}}{\mu^{\alpha+\beta}(t)}\right)^{\prime}}$
(51) $\displaystyle=$ $\displaystyle\lim_{t\rightarrow T^{-}}\frac{\chi
e^{a\overline{\mu}}\mu^{2\alpha+2\beta}-\chi
c_{4}\mu^{2\alpha+3\beta}}{c_{5}e^{a\overline{\mu}}\mu^{2\alpha+2\beta}-c_{6}e^{a\overline{\mu}}\mu^{\alpha+2\beta}-c_{7}e^{a\overline{\mu}}\mu^{\alpha+2\beta+1}}$

where $c_{4},c_{5},c_{6},c_{7}$ are positive constants.

According to Taylor expansion, we know
$e^{a\overline{\mu}(t)}\mu^{2\alpha+2\beta}$ will dominate the other terms in
(51) when $t\rightarrow T^{-}$. Hence, $\lim_{t\rightarrow
T^{-}}A_{2}(t)=\frac{\chi}{c_{5}}$ which is bounded. This indicates that
$\mu^{\alpha+\beta}(t)\xi(t)$ and $\xi(t)$ are both bounded on $[0,T)$.

From (43), we know when $\xi(t)$ is bounded, there exists a positive constant
$\delta$ such that $|x^{2}(t)|\leq\overline{\chi}^{2}-\delta$. Let $X=x^{2}$,
by Mean Value Theorem,

$\ln\left(\frac{\overline{\chi}^{2}}{\overline{\chi}^{2}-X}\right)=\left.\frac{\partial\ln\left(\frac{\overline{\chi}^{2}}{\overline{\chi}^{2}-X}\right)}{\partial
X}\right|_{X\in[0,\overline{\chi}^{2}-\delta]}X\geq c_{8}x^{2}$

where $c_{8}$ is a positive constant.

Therefore, using (43) we obtain $x^{2}(t)\mu^{\alpha+\beta}(t)$ is bounded on
$[0,T)$. Finally, when $b(t)=0$, similar to (50), we can show $\xi(t)$ will
become zero when $t\rightarrow T^{-}$. This completes the proof.

##  Appendix C Proof of Claim 1a

The proof is divided into two parts. For the first part, we will show Claim 1a
holds with $m=0$. The second part will show Claim 1a is true for any finite
integer $m$.

Part I

We will prove the following claim.

Claim 1a′. Suppose $|s_{i}(t_{s}^{0})|<\hat{\chi}_{i}(t_{s}^{0})$ and
$\eta_{i}(t_{s}^{0})>V_{i}(\overline{x}_{i}(t_{s}^{0}),t_{s}^{0})$ for
$\forall i=1,2,...,n$, then we have

1) The considered closed loop nonlinear system admits continuous solution
$\overline{x}_{n}(t)$ on $[t_{s}^{0},t_{s}^{1})$;

2) There exists a positive constant $\delta_{i}^{0}$ such that
$|s_{i}(t)|\leq\hat{\chi}_{i}(t_{s}^{0})-\delta_{i}^{0}$ for $\forall
i=1,2,...,n$ on $[t_{s}^{0},t_{s}^{1})$;

3) All the signals in the system are bounded on $[t_{s}^{0},t_{s}^{1})$.

######  Proof C.1.

The proof is divided into following phases.

1) Prove that the closed loop system admits continuous solution
$\overline{x}_{n}(t)$ on $[t_{s}^{0},t^{*})$ with
$|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{0})(i=1,2,...,n)$ and $t_{s}^{0}<t^{*}\leq
t_{s}^{1}$.

According to the Initialization in Algorithm 3.1, we know
$|s_{i}(t_{s}^{0})|<\hat{\chi}_{i}(t_{s}^{0})$ and
$\eta_{i}(t_{s}^{0})>V_{i}(\overline{x}_{i}(t_{s}^{0}),t_{s}^{0})$. Meanwhile,
nonlinear functions $h_{i}(\overline{x}_{i}),f_{i}(\overline{x}_{i})$ in (1)
are continuously differentiable and adaptive parameters
$\hat{\chi}_{i},\hat{\Theta}_{i}$ are constants on $[t_{s}^{0},t_{s}^{1})$.
Then, according to [19, 29], we know the above statement is true.

2) Prove $t^{*}$ can be extended to $t_{s}^{1}$. That is the closed loop
system admits continuous solution $\overline{x}_{n}(t)$ on
$[t_{s}^{0},t_{s}^{1})$ with
$|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{0})(i=1,2,...,n)$.

This is proved by contradiction. If this is not true, then $t^{*}<t_{s}^{1}$
and there exists an integer $i^{\prime}\in\\{1,2,...,n\\}$ such that as
$t\rightarrow t^{*}$,
$|s_{i^{\prime}}(t)|\rightarrow\hat{\chi}_{i^{\prime}}(t_{s}^{0})$. Meanwhile,
$|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{0})$ for $\forall i=1,2,...,n$ on
$[t_{s}^{0},t^{*})$.

Note that from (23), we know
$\dot{\overline{\eta}}\leq-a^{\prime}\overline{\eta}^{\frac{1+\alpha}{2}}\leq
0$ where we have used the fact that $\eta_{i}\geq V_{i}\geq 0(i=1,...,n)$ on
$[t_{s}^{0},t^{*})$. This means that all the Lyapunov functions
$V_{i}(i=1,...,n)$ are bounded. Next, analysis will be taken on each step in
the controller design to seek a contradiction.

Step 1. Since $V_{1}$ is bounded, from (10) we conclude that there exists a
positive constant $\delta_{1}^{0}$ such that
$|s_{1}|\leq\hat{\chi}_{1}(t_{s}^{0})-\delta_{1}^{0}$. Using (6) and (7), we
know $x_{2}^{*},x_{2}$ are both bounded.

Step i($2\leq i\leq n$). Due to $x_{i}^{*},V_{i}$ are bounded, from
Proposition 3.4 we know there exists a positive constant $\delta_{i}^{0}$ such
that $|s_{i}|\leq\hat{\chi}_{i}(t_{s}^{0})-\delta_{i}^{0}$. Using (6)-(8), we
conclude that $x_{i+1}^{*},x_{i+1}$ are both bounded with $x_{n+1}\triangleq
0$.

Therefore, we have $|s_{i}|\leq\hat{\chi}_{i}(t_{s}^{0})-\delta_{i}^{0}$ for
$i=1,2,...,n$. This contradicts the fact that there exists an integer
$i^{\prime}\in\\{1,2,...,n\\}$ such that when $t\rightarrow t^{*}$,
$|s_{i^{\prime}}(t)|\rightarrow\hat{\chi}_{i^{\prime}}(t_{s}^{0})$. Then, we
can conclude that the system has continuous solution on
$[t_{s}^{0},t_{s}^{1})$ with $|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{0})$.

3) Prove Claim 1a′ is true.

This can be proved by following the line of the above procedures and using the
fact that $|s_{i}(t)|<\hat{\chi}_{i}(t_{s}^{0})(i=1,2,...,n)$ on
$[t_{s}^{0},t_{s}^{1})$.

Part II

We will prove Claim 1a holds for any finite integer $m$.

According to Claim 1a′, we know the system admits continuous solution on
$[t_{s}^{0},t_{s}^{1})$. Then, by Algorithm 3.1 and Remark 3.10, it is not
hard to show at switching time $t_{s}^{1}$, we have

1)
$s_{i}(t_{s}^{1}),\hat{\chi}_{i}(t_{s}^{1}),\eta_{i}(t_{s}^{1}),V_{i}(\overline{x}_{i}(t_{s}^{1}),t_{s}^{1})$
are all bounded for $\forall i=1,2,...,n$;

2) $|s_{i}(t_{s}^{1})|<|\hat{\chi}_{i}(t_{s}^{1})|$ and
$\eta_{i}(t_{s}^{1})>V_{i}(\overline{x}_{i}(t_{s}^{1}),t_{s}^{1})$ for
$\forall i=1,2,...,n$.

Then, by regarding $t_{s}^{1}$ as a new initial time and repeating the
procedures in Claim 1a′, we can prove Claim 1a holds with $m=1$. Similarily,
we can prove the result for any finite $m$. The proof is completed.

##  Appendix D Proof of Proposition 4.3

######  Proof D.1.

Taking the derivative of Lyapunov function (36) and using (1) and (31), we
have

(52) $\displaystyle\dot{V}_{i}\leq$
$\displaystyle\frac{s_{i}(h_{i}x_{i+1}^{*}+f_{i}-\dot{x}_{i}^{*})}{\hat{\chi}_{i}^{2}-s_{i}^{2}}+\frac{h_{i}s_{i}s_{i+1}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}.$

First, we will have some analysis on the term
$\frac{s_{i}f_{i}(\overline{x}_{i})}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$. By the
coordinate transformation (31), we can obtain

$\displaystyle\frac{s_{i}f_{i}(\overline{x}_{i})}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle\leq\frac{|s_{i}|(|x_{1}|+|x_{2}|+\cdots+|x_{i}|)\psi_{i}(\overline{x}_{i})}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle\leq\sum_{j=1}^{i}\frac{|s_{i}|\cdot|s_{j}|\psi_{i}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}+\sum_{j=2}^{i}\frac{|s_{i}|\cdot|x_{j}^{*}|\psi_{i}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}.$

Note that $x_{i}^{*}$ is given by (30). Using Lemma 2.6 for the above
inequality, we have

(53)
$\displaystyle\frac{s_{i}f_{i}(\overline{x}_{i})}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle\leq\sum_{j=1}^{i-1}\frac{c_{ij}\mu^{\beta_{j}}s_{j}^{2}}{2}+\frac{\mu^{\beta_{i}}U_{i}s_{i}^{2}G_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2}}$

where
$G_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown function.

Second, we will have some analysis on the term
$\frac{s_{i}\dot{x}_{i}^{*}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$ in (52). We will
show

(54) $\displaystyle\left|\dot{x}_{i}^{*}\right|\leq$
$\displaystyle\mu^{\overline{\beta}_{i}}\sum_{j=1}^{i}|s_{j}|H_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$

where $\overline{\beta}_{i}\in(1,+\infty)$ is a positive design parameter,
$H_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown function.

This can be proved by an induction method.

For $i=2$, we have

$\displaystyle\left|\dot{x}_{2}^{*}\right|=$
$\displaystyle\left|\frac{\partial x_{2}^{*}}{\partial
s_{1}}(h_{1}x_{2}+f_{1}(x_{1}))+\frac{\partial
x_{2}^{*}}{\partial\mu}\dot{\mu}\right|$ $\displaystyle\leq$
$\displaystyle\left|\frac{\partial x_{2}^{*}}{\partial
s_{1}}\right|(\left|h_{1}s_{2}\right|+\left|h_{1}x_{2}^{*}\right|+|s_{1}|\cdot|\psi_{1}|)+\left|\frac{\partial
x_{2}^{*}}{\partial\mu}\dot{\mu}\right|$ (55) $\displaystyle\leq$
$\displaystyle\mu^{\overline{\beta}_{2}}(|s_{1}|+|s_{2}|)H_{2}$

where $\overline{\beta}_{2}\in(1,+\infty)$ is a positive design parameter.

Then, suppose
$\left|\dot{x}_{i-1}^{*}\right|\leq\mu^{\overline{\beta}_{i-1}}\sum_{j=1}^{i-1}|s_{j}|H_{i-1}$
where $\overline{\beta}_{i-1}\in(1,+\infty)$ is a positive design parameter.

Then, we have

$\displaystyle\left|\dot{x}_{i}^{*}\right|=$
$\displaystyle\left|\frac{\partial x_{i}^{*}}{\partial
s_{i-1}}(h_{i-1}x_{i}+f_{i-1}-\dot{x}_{i-1}^{*})+\frac{\partial
x_{i}^{*}}{\partial\mu}\dot{\mu}\right|$ $\displaystyle\leq$
$\displaystyle\left|\frac{\partial x_{i}^{*}}{\partial
s_{i-1}}\right|(\left|h_{i-1}s_{i}\right|+\left|h_{i-1}x_{i}^{*}\right|+\sum_{j=1}^{i-1}|x_{j}|\psi_{j})$
$\displaystyle+\left|\frac{\partial x_{i}^{*}}{\partial
s_{i-1}}\right|\left(\mu^{\overline{\beta}_{i-1}}\sum_{j=1}^{i-1}|s_{j}|H_{i-1}\right)+\left|\frac{\partial
x_{i}^{*}}{\partial\mu}\dot{\mu}\right|$ (56) $\displaystyle\leq$
$\displaystyle\mu^{\overline{\beta}_{i}}\sum_{j=1}^{i}|s_{j}|H_{i}.$

Substituting (56) into
$\frac{s_{i}\dot{x}_{i}^{*}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$ and using Lemma
2.6, we have

$\displaystyle\frac{s_{i}\dot{x}_{i}^{*}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}\leq$
$\displaystyle\frac{\mu^{\overline{\beta}_{i}}|s_{i}|}{\hat{\chi}_{i}^{2}-s_{i}^{2}}\sum_{j=1}^{i}|s_{j}|$
(57) $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{i-1}\frac{c_{ij}\mu^{\beta_{j}}s_{j}^{2}}{2}+\frac{\mu^{\beta_{i}}U_{i}s_{i}^{2}\overline{G}_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2}}$

where
$\overline{G}_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown function. Then, substituting (57) and (53) into (52) and using
Lemma 2.6,

$\displaystyle\dot{V}_{i}\leq$
$\displaystyle\frac{h_{i}s_{i}x_{i+1}^{*}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}+\sum_{j=1}^{i-1}c_{ij}\mu^{\beta_{j}}s_{j}^{2}$
$\displaystyle+\frac{\mu^{\beta_{i-1}}U_{i}s_{i}^{2}(G_{i}+\overline{G}_{i})}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2}}+\frac{h_{i}s_{i}s_{i+1}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{h_{i}s_{i}x_{i+1}^{*}}{\hat{\chi}_{i}^{2}-s_{i}^{2}}+\frac{\mu^{\beta_{i}}U_{i}s_{i}^{2}\overline{F}_{i}}{(\hat{\chi}_{i}^{2}-s_{i}^{2})^{2}}+\sum_{j=1}^{i-1}c_{ij}\mu^{\beta_{j}}s_{j}^{2}+\frac{c_{i,i+1}s_{i+1}^{2}}{\mu^{\beta_{i}}}$

where
$\overline{F}_{i}(\overline{x}_{i},\overline{\hat{\Theta}}_{i-1},\overline{\hat{\chi}}_{i})$
is an unknown function.

Finally, substituting (30) or (32) into the above inequality, we can complete
the proof.

##  Appendix E Discussions and comparisons

###  E.1 Extensions

Our method can be naturally extended to the case when system (1) contains
power orders, non-feedback linearizable dynamics and time-varying dynamics as
in [11, 26]. Specifically, (1) can be revised into

$\displaystyle\dot{x}_{i}$
$\displaystyle=h_{i}(\overline{x}_{i},t)x_{i+1}^{p_{i}}+f_{i}(\overline{x}_{i},t),\thinspace
i=1,2,...,n-1$ (58) $\displaystyle\dot{x}_{n}$
$\displaystyle=h_{n}(\overline{x}_{n},t)u+f_{n}(\overline{x}_{n},t)$

where $p_{i}$ is a ratio of positive odd integers,
$h_{i}(\overline{x}_{i},t),f_{i}(\overline{x}_{i},t)$ are unknown functions
such that $|h_{i}(\overline{x}_{i},t)|>0$ and
$|f_{i}(\overline{x}_{i},t)|\leq(|x_{1}|^{\lambda_{1}}+\cdots+|x_{i}|^{\lambda_{i}})\psi_{i}(\overline{x}_{i})$
where $\lambda_{i}>0$ are some specified positive constants and
$\psi_{i}(\overline{x}_{i})$ is unknown. However, it is difficult to extend
the existing works [27, 21, 23, 39] to this kind of systems.

Moreover, there are several ways to relax the condition $f_{i}(0)=0$ in (1).
The first way is to consider practical finite time stability, $i.e.$, the
tracking error will converge to a small neighborhood around origin in finite
time [18]. This can be done by replacing the auxiliary systems (21) with
$\dot{\eta}_{i}(t)=-a\eta_{i}^{(1+\alpha)/2}-Q_{i}s_{i}^{2}+\sum_{j=1}^{i-1}c_{ij}s_{j}^{1+\alpha}+c_{i,i+1}s_{i+1}^{1+\alpha}+\zeta$
where $a,\zeta$ are positive design parameters. Then Algorithm 3.1 in Section
3 is referred to as Algorithm 3.1′. We have the following result.

######  Corollary E.1.

Consider the nonlinear system in (1) with $f_{i}(0)\neq 0(i=1,2,...,n)$. Then
the controller (5)-(8) with Algorithm 3.1′ can guarantee that:

1) All the signals in the closed-loop system are bounded, and;

2) Practical finite time stability is achieved, $i.e.$, there exists a finite
time $T_{0}$ such that $\underset{t\rightarrow
T_{0}}{\lim}|V_{i}(\overline{x}_{i},t)|\leq 2n\zeta/a$ where $V_{i}$ are given
by (10) and (15).

Note that since $\zeta,a$ are freely chosen design parameters, the states can
be regulated to a very small neighborhood around zero in finite time.

The second way is to revise (39) into
$\dot{\eta}_{n}=-a_{n}\mu^{\beta_{n}}\eta_{n}-Q_{n}s_{n}^{2}+\sum_{j=1}^{n-1}c_{ij}\mu^{\beta_{j}}s_{j}^{2}+\zeta_{n}/\mu^{\beta_{n}}$
with positive constant $\zeta_{n}$. Then the controller designed in Section 4
can make the states converge to zero in prescribed finite time when
$f_{n}(0)\neq 0$. To this end, Algorithm 3.2 in Section 4 is referred to as
Algorithm 3.2′ and we have the following corollary.

######  Corollary E.2.

Consider the nonlinear system in (1) with $f_{n}(0)\neq 0$. Then, the
controller (29)-(32), with Algorithm 3.2′ can guarantee that:

1) All the signals except for the time-varying gain $\mu$ in the closed-loop
system are bounded on $[0,T)$, and;

2) All the states will be regulated to zero in prescribed finite time $T$.

######  Proof E.3.

The proofs follows the line of Theorem 4.6. Similar to (40), we can conclude
that with sufficiently large $Q_{i}$,
$\dot{\overline{\eta}}\leq-\mu^{\beta^{\prime}}a^{\prime}\overline{\eta}+\zeta_{n}/\mu^{\beta_{n}}.$Then,
based on Lemma 4.7, we can see $\overline{\eta}$ will also converge to zero in
prescribed finite time.

###  E.2 Convergence speed

Finite time stability can be obtained by the proposed method despite unknown
control directions. However, the works in [23, 21, 27] etc. can only achieve
asymptotic stability meaning the convergence time is infinite. It has been
shown in many references [37, 3, 18] that finite time control has a smaller
settling time, faster convergence rate and higher precision than infinite time
control methods. The reasons are as follows:

1) According to (21), we know for our method there exists a Lyapunov function
$V$ such that $V\leq\eta$ and

$\dot{\eta}\leq-a\eta^{\gamma}=-a\frac{1}{\eta^{1-\gamma}}\eta$

with $\eta$ is an auxiliary internal variable, $\gamma\in(0,1)$, $a>0.$ For
exponential or asymptotic stability, the Lyapunov function $V$ of the system
may satisfy $\dot{V}\leq-a^{\prime}V$ with $a^{\prime}>0$. It can be seen that
due to the additional fractional power $\gamma$ in finite time control, the
closer to the equilibrium point, the faster the convergence rate of the system
states.

2) We have extended our method to obtain prescribed finite time control
performance in Section 4. This will further reduce the settling time.

3) The auxiliary system in (21) can be easily modified into the form as
$\dot{\eta}\leq-a^{\prime}\eta-a\eta^{\gamma}$ with positive constant
$a^{\prime}$. This modification can render fast finite time control as
discussed in [37], which can achieve a faster convergence rate than
asymptotic/exponential control strategies.

4) Since the control gain is updated in a discrete manner, according to [12],
the proposed method could obtain a higher control performance than traditional
continuous adaptive control method.

###  E.3 Control overshoot

The proposed method may obtain a smaller control overshoot than Nussbaum
method. The large overshoot phenomenon of Nussbaum method has been recognized
in many references [1, 28]. Moreover, as stated in [9], the logic-based
switching controller is more sensitive to the mismatch of the unknown control
directions than Nussbaum method. Therefore, it may detect the right control
directions more quickly than Nussbaum method, which will result in a smaller
control overshoot. The simulations in Section 6 have verified the above claim.

Next, we take the following simple Nussbaum controller as an example to
further explain this.

(59) $\displaystyle u$ $\displaystyle=\mathcal{N}(\xi)x,\dot{\xi}=V(x)$

where $x$ represents the state, $\xi$ is an internal variable,
$\mathcal{N}(\xi)=\xi^{2}\cos(\xi)$ or
$\mathcal{N}(\xi)=e^{\xi^{2}}\cos(\xi\pi/2)$. $V(x)=x^{2}\geq 0$ is a Lyapunov
function. The Nussbaum gain $\mathcal{N}(\xi)$ will switch its sign when $\xi$
increases to some threshold values, $i.e.$, zeros of the cosine function in
$\mathcal{N}(\xi)$.

For the logic-based switching controller, the control direction is detected
according to the supervisory function, which is selected as
$\mathcal{S}(t)=V-\eta$ where $\dot{\eta}=-a\eta^{\gamma}$ and
$\eta(0)=V(0)+\varepsilon$ with a small constant $\varepsilon$. The control
gain will be switched when $\mathcal{S}>0$. At the initial phase, if the
states of the system are driven away by a wrong control direction, $i.e.,$ $V$
is increasing, then the supervisory function $\mathcal{S}$ will quickly become
larger than zero and the control gain will be switched. Thus, the right
control direction will be found. If the control direction is right, then $V$
will decrease and we can also identify a right control direction.

However, for Nussbaum gain, since $\xi$ in (59) is an integration of the
Lyapunov $V$, it will always increase whether the states are driven away or
converge to zero. This implies that a longer time may be needed to find the
right control direction.

In addition, the polynomial or exponential increasing terms
$\xi^{2},e^{\xi^{2}}$ in $\mathcal{N}(\xi)$ may lead to large control effort.

###  E.4 Control parameters selection

It is noted that theoretically the states will converge to zero in finite time
if all the design parameters are positive and
$Q_{i}-\sum_{j=i+1}^{n}c_{ji}-c_{i-1,i}>0$ in (22). This is due to the
adaptive feature of our proposed method. The theoretical results imply that
the parameters selection can be very flexible and straightforward. This can be
also seen from the design procedures in Sections 3.1 and 4.1 where Young’s
inequalities are extensively used. In fact, from the simulations in Section 5,
we can see that many of the design parameters are the same and set to one. It
is also noted that appropriate design parameters can result in smaller
settling time and better transient performance. Next, we present some
guidelines for the parameters selections.

Basically, larger control gain $K_{i},U_{i}$ and smaller $q_{i}$ may result in
a faster convergence speed, but bigger overshoot and control effort. By
increasing the value $a_{i}$ in the supervisory functions, we can obtain a
smaller settling time. The initial value $\theta_{i}(0)$ should be set small
to prevent the control effort from becoming very large in the beginning.
Meanwhile, the initial value $\hat{\chi}_{i}(0)$ should be set slightly large
to reduce the number of switchings, thus resulting in a quicker response.

Moreover, by resorting to the time-varying gain technique in Section 4, one
can prescribe a convergence time or transient performance for the closed loop
systems. In fact, from the stability analysis, we know the output satisfies
$|s_{1}|<\hat{\chi}_{1}$ where $\hat{\chi}_{1}$ is a constant. One can replace
this constant with some time-varying performance function, and then a
transient performance can be prescribed. This will bring more convenience to
the parameters selection.

###  E.5 Comparison with Nussbaum-gain method

Consider the system described by Example 5. Given a finite time controller by
(5)-(8) with the same parameters in Example 1, and another controller designed
by Nussbaum-gain technique:

$\displaystyle u$ $\displaystyle=\mathcal{N}_{2}(\xi_{2})v_{2},$
$\displaystyle v_{2}$
$\displaystyle=K_{2}s_{2}+\frac{U_{2}s_{2}}{\hat{\chi}_{2}^{2}-s_{2}^{2}},$
$\displaystyle\dot{\xi}_{2}$
$\displaystyle=\frac{s_{2}v_{2}}{\hat{\chi}_{2}^{2}-s_{2}^{2}}$

where

$\displaystyle s_{2}$ $\displaystyle=x_{2}-x_{2}^{*},$ $\displaystyle
x_{2}^{*}$ $\displaystyle=\mathcal{N}_{1}(\xi_{1})v_{1},$ $\displaystyle
v_{1}$
$\displaystyle=K_{1}s_{1}+\frac{U_{1}s_{1}}{\hat{\chi}_{1}^{2}-s_{1}^{2}},$
$\displaystyle\dot{\xi}_{1}$
$\displaystyle=\frac{s_{1}v_{1}}{\hat{\chi}_{1}^{2}-s_{1}^{2}}.$

$\mathcal{N}_{i}(\xi_{i})(i=1,2)$ are Nussbaum functions such that
$\mathcal{N}_{i}(\xi_{i})=\xi_{i}^{2}\cos(\xi_{i}).$
$K_{1},U_{1},K_{2},U_{2}$, $\hat{\chi}_{1},\hat{\chi}_{2}$ are positive design
parameters. The parameters are set as:
$K_{1}=U_{1}=1,K_{2}=U_{2}=4,\hat{\chi}_{1}=2,\hat{\chi}_{2}=1$. Using these
parameters, a satisfactory control performance can be obtained in the nominal
case. This controller is a variation of the method in [23], which adapts to
the unstructured uncertainties.

Fig. 5 shows the state trajectories and variation of control effort in Case A.
We can see that the finite time control method has a faster convergence speed
and higher precision than Nussbaum-gain method. In fact, the states by finite
time control is around $3.75\times 10^{-5}$ after $6.5s$, while states by
Nussbaum-gain method is $0.01$. Moreover, the proposed method has a smaller
overshoot and control effort than Nussbaum-gain method. The reason for this
may be that the proposed method can find a proper control direction quickly by
switching logic.

Next, with the same controller parameters, we consider the control performance
in Cases B-F. Fig. 6 shows the state trajectories in Cases B and C. We can see
that in both cases, the proposed method has a superior performance with
smaller overshoot and faster convergence rate. Fig. 7 demonstrates the state
trajectories in Cases D and E, it can be seen that control performance of
Nussbaum-gain method deteriorates a lot. For Case F, the Nussbaum-gain method
has become highly unstable. All these show the stronger robustness of the
proposed method.

###  E.6 Control of third order nonlinear systems

Consider a third order nonlinear system by (1) with $n=3$. Let $h_{1}=\pm 1$,
$f_{1}=-1.8x_{1}+0.15\sin(x_{1})$, $h_{2}=\pm 1$,
$f_{2}=0.7x_{2}-4.4x_{1}+0.1\sin(x_{2})x_{1}^{2}$, $h_{3}=\pm 3$,
$f_{3}=-7x_{2}+0.1\sin(x_{3})$. The system can be used to describe some
circuit systems. The initial conditions are
$x_{1}(0)=0.1,x_{2}(0)=0.2,x_{3}(0)=0.5$.

The controller is given by (29)-(32) with parameters selected as:
$K_{i}=U_{i}=1$, $\alpha=41/49$;
$\hat{\chi}_{1}=2,\hat{\chi}_{2}(0)=5,\hat{\chi}_{3}(0)=8$;
$\theta_{i}(0)=0.1(i=1,2,3)$. For $\sigma_{i}(t)\geq 1$, let
$\theta_{i}(\sigma_{i})=(-1)^{\sigma_{i}(t)}(\iota_{i1}+\iota_{i2}\sigma_{i})$
where $\iota_{i1}=1,\iota_{i2}=1.$ $\varsigma=5,\varepsilon=0.01$,
$a_{i}=0.2,$ $Q_{1}=Q_{2}=2,Q_{3}=1$, $c_{12}=c_{21}=c_{23}=c_{31}=c_{32}=1.$
Fig. 8 shows the control performance for this controller. According to the
sign of the control direction, there are total $2\times 2\times 2=8$
situations. Each blue line in Fig. 8 represents one of these situations. We
can see that all the states converge to zero in finite time for all these
eight cases. This demonstrates the validity and robustness of the proposed
method.

Figure 5: Control performance comparison for Case A. Figure 6: Control
performance comparison for Cases B and C. Figure 7: Control performance
comparison for Cases D and E. Figure 8: Control performance for third order
system.

## References

  * [1] Y. Z. C. C. C. Chen, Z. Liu and S. Xie, Saturated nussbaum function based approach for robotic systems with unknown actuator dynamics, IEEE Transactions on Cybernetics, 46 (2016), pp. 2311–2322.
  * [2] W. H. J. N. Z. L. C. Yang, Y. Jiang and B. Xu, Adaptive parameter estimation and control design for robot manipulators with finite-time convergence, IEEE Transactions on Industrial Electronics, 65 (2018), pp. 8112–8123.
  * [3] C. Chen and Z. Sun, A unified approach to finite-time stabilization of high-order nonlinear systems with an asymmetric output constraint, Automatica, 111 (2020), p. 108581.
  * [4] Z. Chen, Nussbaum functions in adaptive control with time-varying unknown control coefficients, Automatica, 102 (2019), pp. 72–79.
  * [5] Z. Chen and J. Huang, Stabilization and regulation of nonlinear systems: A Robust and adaptive approach, Springer, 2015.
  * [6] J. L. G. Zhao and S. Liu, Finite-time stabilization of weak solutions for a class of non-local lipschitzian stochastic nonlinear systems with inverse dynamics, Automatica, 98 (2018), pp. 285–295.
  * [7] W. H. H. Li, S. Zhao and R. Lu, Adaptive finite-time tracking control of full state constrained nonlinear systems with dead-zone, Automatica, 100 (2019), pp. 99–107.
  * [8] K. C. H. Lin and R. Lin, Finite-time formation control of unmanned vehicles using nonlinear sliding mode control with disturbances, International Journal of Innovative Computing, Information and Control, 15 (2019), pp. 2341–2353.
  * [9] C. Huang and C. Yu, Tuning function design for nonlinear adaptive control systems with multiple unknown control directions, Automatica, 89 (2018), pp. 259–265.
  * [10] S. Huang and Z. Xiang, Finite-time stabilization of switched stochastic nonlinear systems with mixed odd and even powers, Automatica, 73 (2016), pp. 130–137.
  * [11] R. M. J. Fu and T. Chia, Adaptive finite-time stabilization of a class of uncertain nonlinear systems via logic-based switchings, IEEE Transactions on Automatic Control, 62 (2017), pp. 5998–6001.
  * [12] D. L. J. Hespanha and A. Morse, Hysteresis-based switching algorithms for supervisory control of uncertain systems, Automatica, 39 (2003), pp. 263–272.
  * [13] W. W. J. Huang, C. Wen and Y. Song, Design of adaptive finite-time controllers for nonlinear uncertain systems based on given transient specifications, Automatica, 69 (2016), pp. 395–404.
  * [14] Y. Z. X. Y. J. Mishra, L. Wang and M. Jalili, A novel mixed cascaded finite-time switching control design for induction motor, IEEE Transactions on Industrial Electronics, 66 (2019), pp. 1172–1181.
  * [15] G. D. Z. J. Wu, J. Li and W. S. Chen, Global finite-time adaptive stabilization of nonlinearly parameterized systems with multiple unknown control directions, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), pp. 1405–1414.
  * [16] W. C. J. Wu and J. Li, Global finite-time adaptive stabilization for nonlinear systems with multiple unknown control directions, Automatica, 69 (2016), pp. 298–307.
  * [17] H. Y. J. Yu, L. Zhao and C. Lin, Barrier lyapunov functions-based command filtered output feedback control for full-state constrained nonlinear systems, Automatica, 105 (2019), pp. 71–79.
  * [18] P. S. J. Yu and L. Zhao, Finite-time command filtered backstepping control for a class of nonlinear systems, Automatica, 92 (2018), pp. 173–180.
  * [19] S. G. K. Tee and E. Tay, Barrier lyapunov functions for the control of output-constrained nonlinear systems, Automatica, 45 (2009), pp. 918–927.
  * [20] H. K. Khalil, Nonlinear systems, Springer, 3 ed., 2002.
  * [21] F. Li and Y. Liu, Control design with prescribed performance for nonlinear systems with unknown control directions and nonparametric uncertainties, IEEE Transactions on Automatic Control, 63 (2018), pp. 3573–3580.
  * [22] W. Lin and Q. Gong, A remark on partial-state feedback stabilization of cascade systems using small gain theorem, IEEE Transactions on Automatic Control, 48 (2003), pp. 497–500.
  * [23] Y. Liu and S. Tong, Barrier lyapunov functions for nussbaum gain adaptive control of full state constrained nonlinear systems, Automatica, 76 (2017), pp. 143–152.
  * [24] I. K. M. Sontag and P. Kokotovic, Nonlinear and adaptive control design, 1995.
  * [25] R. Nussbaum, Some remarks on a conjecture in parameter adaptive control, Systems & Control Letters, 3 (1983), pp. 243–246.
  * [26] C. Qian and W. Lin, Non-lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization, Systems & Control Letters, 42 (2001), pp. 185–200.
  * [27] J. Q. S. Yin, H. Gao and O. Kaynak, Adaptive fault-tolerant control for nonlinear system with unknown control directions based on fuzzy approximation, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), pp. 1909–1917.
  * [28] A. Scheinker and M. Krstic, Minimum-seeking for clfs: Universal semiglobally stabilizing feedback under unknown control directions, IEEE Transactions on Automatic Control, 58 (2013), pp. 1107–1122.
  * [29] E. D. Sontag, Mathematical control theory, 1998.
  * [30] C. W. W. Chen and J. Wu, Global exponential/finite-time stability of nonlinear adaptive switching systems with applications in controlling systems with unknown control direction, IEEE Transactions on Automatic Control, 63 (2018), pp. 2738–2744.
  * [31] T. L. X. Liu, D. Zhai and Q. Zhang, Fuzzy-approximation adaptive fault-tolerant control for nonlinear pure-feedback systems with unknown control directions and sensor failures, Fuzzy Sets and Systems, 356 (2019), pp. 28–43.
  * [32] D. C. Y. Cao, W. Ren and C. Schumacher, Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown lipschitz terms, IEEE Transactions on Automatic Control, 61 (2016), pp. 1700–1705.
  * [33] H. B. Y. Chitour, R. Ushirobira, Stabilization for a perturbed chain of integrators in prescribed time, SIAM Journal of Control and Optimization, 58 (2020), pp. 1022–1048.
  * [34] G. F. Y. Hong, Z. Jiang, Finite-time input-to-state stability and applications to finite-time control design, SIAM Journal of Control and Optimization, 48 (2010), pp. 4395–4418.
  * [35] S. T. X. C. C. L. C. Y. Lin, S. Lu and D. Li, Adaptive control-based barrier lyapunov functions for a class of stochastic nonlinear systems with full state constraints, IEEE Transactions on Automatic Control, 87 (2018), pp. 83–93.
  * [36] J. H. Y. Song, Y. Wang and M. Krstic, Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time, Automatica, 83 (2017), pp. 243–251.
  * [37] Y. S. Z. Sun and C. Chen, Fast finite-time stability and its application in adaptive control of high-order nonlinear system, Automatica, 69 (2019), pp. 298–307.
  * [38] J. Zhang and G. Yang, Prescribed performance fault-tolerant control of uncertain nonlinear systems with unknown control directions, IEEE Transactions on Automatic Control, 62 (2017), pp. 6529–6535.
  * [39] J. Zhang and G. Yang, Low-complexity tracking control of strict-feedback systems with unknown control directions, IEEE Transactions on Automatic Control, (2019), https://doi.org/10.1109/TAC.2019.2910738.
  * [40] Z. Zhao and Z. Jiang, Finite-time output feedback stabilization of lower-triangular nonlinear systems, Automatica, 96 (2018), pp. 259–269.
  * [41] S. Zheng and W. Li, Fuzzy finite time control for switched systems via adding a barrier power integrator, IEEE Transations on Cybernetics, 49 (2019), pp. 2693–2706.