# A survey of modularized backstepping control design approaches to nonlinear
ODE systems

Zhengru Ren
Institute for Ocean Engineering
Shenzhen International Graduate School
Tsinghua University
Shenzhen, 518055, China
<EMAIL_ADDRESS>

###### Abstract

Backstepping is a mature and powerful Lyapunov-based design approach for a
specific set of systems. Throughout the development over three decades,
innovative theories and practices have extended backstepping to stabilization
and tracking problems for nonlinear systems with growing complexity. The
attractions of the backstepping-like approach are the recursive design
processes and modularized design. A nonlinear system can be transferred into a
group of simple problems and solved it by a sequential superposition of the
corresponding approaches for each problem. To handle the complexities,
backstepping designs always come up with adaptive control and robust control.
The survey aims to review the milestone theoretical achievements among
thousands of publications making the state-feedback backstepping designs of
complex ODE systems to be systematic and modularized. Several selected elegant
methods are reviewed, starting from the general designs, and then the finite-
time control enhancing the convergence rate, the fuzzy logic system and neural
network estimating the system unknowns, the Nussbaum function handling unknown
control coefficients, barrier Lyapunov function solving state constraints, and
the hyperbolic tangent function applying in robust designs. The associated
assumptions and Lyapunov function candidates, inequalities, and the deduction
key points are reviewed. The nonlinearity and complexities lay in state
constraints, disturbance, input nonlinearities, time-delay effects, pure
feedback systems, event-triggered systems, and stochastic systems. Instead of
networked systems, the survey focuses on stand-alone systems.

_Keywords_ Backstepping, adaptive control $\cdot$ robust control $\cdot$
nonlinear system $\cdot$ Lyapunov methods $\cdot$ strict-feedback system
$\cdot$ pure-feedback system $\cdot$ input nonlinearity

##  1 Introduction

First developed in 1991, backstepping approach [1, 2] is a Lyapunov-based
recursive design procedure for triangular strict-feedback form systems. In
brief, the main idea is to divide the entire system into several subsystems
and to recursively design a global controller according to the former
subsystem. Adaptive control is involved when uncertainties are included in the
system. The approach has been widely used in a wide scope of nonlinear
systems. The controller and adaptive update laws can be designed
simultaneously which improves the transient performance.

Over three decades of research and development, a significant amount of
research papers extend the application field of backstepping intensively. In
the first five years, the fundamental backstepping schemes and redesigns with
parametric uncertainties attracted great attention. Later, the research
focuses are the methodology improvement and more advanced parametric
uncertainties. In the 2000s, the emphasis is laid on systems with uncertainty
and nonsmoothness, neural adaptive control, time-delay effects, etc.
Innovative design techniques provide solutions to various nonsmooth effects
and complex uncertainties. In the recent decade, the scope of backstepping
design methodology has been broadened to more complex systems, e.g., system
with several aforementioned uncertainties, switched systems, stochastic
systems, time-triggered systems, multi-agent systems, etc.

Survey papers and textbooks on backstepping and adaptive control are reported
in literature, e.g., [3, 4, 5, 6]. However, a considerable amount of studies
have been conducted thereafter, such as, actuator and sensor nonlinearities
[7], nonsmooth system and input nonlinearities [8], partial differential
equation (PDE) [9], neural adaptive control [10], fault-tolerant control [11].
There still lacks a systematically organized summary of state-of-the-art
advances. Although thousands of papers are available online, a limited number
of significant contributions can be extracted from them. Each method handles a
specific form of nonlinear systems. The innovations normally rely on a series
of assumptions. Since a lot of research combines several previous approaches,
most pages for these works focus on some repeated assumptions, lemmas,
definition, Lyapunov function candidates (LFC), and recursive deductions. In
this survey, we try to point out the essence of different methods with
designer-friendly characteristics.

Therefore, we believe that it is timely and helpful to present a pervasive
survey of state-of-the-art backstepping-like techniques to ordinary
differential equation (ODE) systems with various nonlinearities. Another
target of this survey is to help the interesting readers find suitable
approaches to handle complex nonlinear system in a systematical and
modularized way. A complex nonlinear system can be separated into several
simplified problems and solved with the integration of basic methodologies and
the corresponding Lyapunov function components. The selection of specific
approaches is determined by the constructive Lyapunov function. The design
procedures are summarized as follows: (i) divided it into several typical mini
problems and transfer each of the mini problems into a typical form; (ii)
select the reasonable approach to every mini problem, as well as the
corresponding assumptions and Lyapunov function candidates; (iii) design the
backstepping-like control recursively by integrating all approaches.
Admittedly, the references provided in this survey do not give an exhausted
list. Some conceptually deeper methods in the 1990s are not included since
they are difficult to be categorized systematically, and the complexities in
deduction and proof prevent their modularized applications. But it is believed
to contain many important approaches that are welcomed by the successors in
the academic domain.

The majority of the present survey was written in the final phase of my PhD
study in 2019 and was slightly revised in 2020. It has been used as a lecture
note in a PhD course at the Norwegian University of Science and Technology,
Norway. Very positive feedback was received from the PhD candidates, who
claimed that understanding advanced backstepping algorithms can never be so
simple.

The rest of this paper is organized as follows. In Section II, several widely-
used elegant methods are reviewed, including the integrator backstepping, the
dynamic surface control, command filtered backstepping, finite-time control,
fuzzy logic system, neural network, Nussbaum function, barrier Lyapunov
function, and hyperbolic tangent function. In section III, key theoretical and
methodological innovations to a class of systems are presented and discussed,
i.e., state constraints, disturbance, input nonlinearities, and time-delay
effects, pure-feedback, and stochastic systems. Most complexities are
transferred to the forms which can be solved by the elegant methods. In
Section IV, the advancements are summarized and discussed.

To keep the review short and organized, some simplifications will be adopted
hereafter without any specifications. Firstly, only single-input-single-output
(SISO) systems are reviewed since the design methods for multiple-input-
multiple-output (MIMO) systems are always similar. The results from MIMO
systems can be adjusted accordingly. Furthermore, the state estimator design
[12, 13], which is needed for output-feedback control, is neglected since it
is not the emphasis of this survey. The estimated error should be involved
additionally in the LFC to prove the stability of the entire system.

###  1.1 Notations

The Euclidean norm of a vector $x$ and Frobenius norm of a matrix $A$ are
denoted by $|x|$ and $|A|_{F}$, respectively. Blackboard bold variables denote
sets and domains, e.g., $\mathbb{R}^{n}$, $\mathbb{R}_{+}$,
$\mathbb{R}^{n\times m}$ are the n-dimensional real space, nonnegative real
number set, and $n\times m$ real space, respectively. $a^{(b)}$ stands for the
$b^{th}$ derivative of $a$. $C^{i}$ denotes the set of all functions with
continuous $i^{th}$ differential and and $C^{i,1}$ stands for the family of
all nonnegative functions $V(x,t,r)$ on
$\mathbb{R}^{n}\times\mathbb{R}_{+}\times S$ which are $C^{i}$ in $x$ and
$C^{1}$ in $t$. The identify matrix and zero matrix with a size $m\times n$
are denoted by $I_{m\times n}$ and $0_{m\times n}$, respectively. Specially,
$I_{n}$ and $0_{n}$ are the identity matrix and zero matrix short for
$I_{n\times n}$ and $0_{n\times n}$, respectively. A set of indices is defined
as $\mathcal{I}=\\{1,2,\cdots,n-1\\}$. Lie derivative of function $V$ in
respect of $f$ is defined as $L_{f}V=\frac{\partial V}{\partial x}f(x)$.

Variables with hat and tilde operators stand for the estimated values and
estimate errors, respectively. Their relation is, e.g., given by
$\tilde{\theta}=\theta-\hat{\theta}$. When $\theta$ is a constant,
$\dot{\tilde{\theta}}=-\dot{\hat{\theta}}$. Overlines and underlines (e.g.,
$\overline{c}$ and $\underline{c}$), are the maxima and minima of a variable
$c$. The sign function is given by

$\text{sgn}(a)=\begin{cases}1,&\text{if }a>0\\\ 0,&\text{if }a=0\\\
-1,&\text{if }a<0.\end{cases}$

###  1.2 Preliminary

Since the unknown nonlinearities are difficult to be fully estimated and
canceled, the most widely-used stability lemma is updated as follows.

###### Lemma 1

A LFC $V(x)$ is bounded if the initial condition $V(0)$ is bounded, $V(x)$ is
positive definite and continuous and if a Lyapunov-like inequality holds,
i.e.,

$\dot{V}(x)\leq-\gamma V(x)+\delta,$ (1)

where $\gamma>0$ and $\delta>0$. Define $\rho:=\delta/\gamma$,

$0\leq V(t)\leq\rho+(V(0)-\rho)\exp(-\gamma t).$ (2)

And it implies that

$V(t)\leq e^{-\gamma t}V(0)+\int_{0}^{t}e^{-\gamma(t-\tau)}\rho(\tau)d\tau,\
\forall t\geq 0,$ (3)

for any finite constant $\gamma$.

  * Proof

Times $\exp(-\gamma t)$ to both sides of (1), yields $\dot{V}\exp(-\gamma
t)+\gamma V\exp(\gamma t)\leq\delta\exp(\gamma t)$. Integrating both side
yields $\frac{d}{d\,t}V\exp(\gamma
t)\leq\frac{\delta}{\gamma}\frac{d}{d\,t}\exp(\gamma t)$. Therefore, we have
$V(t)\exp(\gamma t)-V(0)\leq\rho(\exp(\gamma t)-1)$.

Lemma 1 is significant to prove the global semi-stability which is the
foundation of a great number of publications. Many works in this survey
utilize Lemma 1 to prove the final stability. The tracking errors converge to
a small neighborhood of the origin, instead of exactly stays at the
equilibrium point. The value of $\gamma$ can be found using a superficial
inequality $c_{\min}\sum_{i=1}^{n}x_{i}^{2}\leq
c_{1}x_{1}^{2}+\cdots+c_{n}x_{n}^{2}\leq c_{\max}\sum_{i=1}^{n}x_{i}^{2}$,
where $c_{\min}=\min\\{c_{1},c_{2},c_{3},\cdots c_{n}\\}$ and
$c_{\max}=\max\\{c_{1},c_{2},c_{3},\cdots c_{n}\\}$. From (3), the selection
of $\gamma$ not only influences the final tracking error limitation, but also
determines the convergence rate. There are several transformations are
developed for more complex systems in similar forms.

Besides, there are many other method to prove the boundness; however, Lemma 1
is the most widely-used with a low deduction complexity and relatively loose
restrictions, and it inspires many other systematic methods.

##  2 Elegant methodologies

Backstepping approaches can be categorized into cancellation-based
backstepping and passivity-based backstepping [14]. According to the number of
publications, the former method stands for the ruling position. In this
section, several elegant design methodologies from which thousands of
publications benefit are reviewed; see Fig. 1. These methods perform as basics
to a significant amount of publications. Each of them could be applied to
various complexities and nonlinearities. We consider a method is elegant if it
can be applied to various problems and cooperate with other design approaches.
The development of each method starts from an integrator chain and is extended
to increasingly complex systems, e.g., from linear systems to nonlinear
systems, from fully known systems to unknown and disturbed systems, and from
second- or third-order systems to recursive design for large-scale systems.
Since most nonlinearities cannot be fully canceled, most of the methods are
prevalent to Lemma 1, and the design procedures and stability proof follow a
similar methodology.

There are two dominant directions to do the cancellation. The first one is to
cancel the nonlinearity as much as possible, resulting in a robustness
approach. The second way is to approximate everything using online neural
networks (NNs) or fuzzy logic systems (FLSs) and cancel the estimates.

Figure 1: Elegant modularized methods used in backstepping designs.

###  2.1 General design approaches

Being the starting point of backstepping designs, the strict-feedback
nonlinear system with a triangular structure is given by

$\displaystyle\dot{x}_{i}$
$\displaystyle=f_{i}(\bar{x}_{i})+g_{i}(\bar{x}_{i})x_{i+1}+\phi_{i}(\bar{x}_{i})^{\top}\theta,i\in\mathcal{I}$
(4a) $\displaystyle\dot{x}_{n}$
$\displaystyle=f_{n}(\bar{x}_{n})+g_{n}(\bar{x}_{n})u+\phi_{n}(\bar{x}_{n})^{\top}\theta,$
(4b) $\displaystyle y$ $\displaystyle=x_{1},$ (4c)

where $x_{1},\cdots,x_{n}$ are the states, $f_{1},\cdots,f_{n}$,
$g_{1},\cdots,g_{n}$ are smooth functions, $g_{i}$ is the control coefficient
function, $\bar{x}_{i}=[x_{1},x_{2},\cdots,x_{i}]^{\top}$,
$\theta\in\mathbb{R}^{p}$ is an unknown constant vector,
$\phi_{i}(\cdot):\mathbb{R}^{i}\to\mathbb{R}^{p}$ is assumed to be known, and
$u$ and $y$ are the input and output respectively. The terms
$\theta^{\top}\phi_{i}(\bar{x}_{i})$ are the parametric uncertainties, namely,
the line-in-parameters condition [15]. Without specific declarations, the
output signal is $y=x_{1}$ hereafter. When $f_{1}=\cdots=f_{n}=0$ and
$g_{1}=\cdots=g_{n}=1$, the system is simplified to be an integrator chain,
or, namely, the Brunovsky form. The error states is defined as
$z_{1}=x_{1}-x_{1d}$ and $z_{i+1}=x_{i+1}-\alpha_{i}$ where $x_{id}$ is the
desired trajectory and $\alpha_{i}$ is the virtual control law to be defined
later. The control objective is to track $y$ to $x_{1d}$ as $t\to\infty$,
i.e., $z_{1}=x_{1}-x_{1d}\to 0$ as $t\to\infty$. New variables are defined as
$\bar{x}_{1d}=[x_{1d}^{(1)},x_{1d}^{(2)},\cdots,x_{1d}^{(i)}]^{\top}$,
$\bar{z}_{i}=[z_{1},z_{2},\cdots,z_{i}]^{\top}$, and
$\bar{z}_{i:j}=[z_{i},z_{i+1},\cdots,z_{j}]^{\top}$.

A common assumption for the backstepping tracking control is the sufficiently
smooth desired trajectory, i.e., the reference signal $x_{d}(t)$ and its
derivatives up to the required number of order are known, bounded, and
continuous. In addition, the signs of $g_{1},\cdots,g_{n}$ are assumed to be
known and constant, i.e., $|g_{i}|>0$ for all $t$. Three fundamental methods
will be reviewed in this section, including (adaptive) integrator
backstepping, adaptive backstepping with tuning functions, and dynamic surface
control/command filtered backstepping. Asymptotically tracking adaptive
backstepping design for a system (4) could be achieved.

Research on each method starts from the integrator chain, where the virtual
control coefficients are constantly $g_{i}=1$. Later, the algorithms are
enriched with control coefficient $g_{i}$. A common assumption is that the
control coefficients are known with unchanged known signs, i.e., there exist
unknown positive constants $\underline{g}_{i}$ and $\overline{g}_{i}$ s.t.,
$0<\underline{g}_{i}\leq|g_{i}|\leq\overline{g}_{i}$. Hence, the partial
derivatives of $g_{i}$ are strictly either positive or negative. Singularity
problem is also neglected.

####  2.1.1 Integrator backstepping and ISS backstepping

A commonly used quadratic Lyapunov function and its time derivative is given
by $V_{i,QF}=\frac{1}{2}z_{i}^{2}$, and $\dot{V}_{i,QF}=z_{i}\dot{z}_{i}$. The
Lyapunov function in step $i$ is the superposition of that in the former step
and $V_{i,QF}$, i.e., $V_{i}=V_{i-1}+V_{i,QF}$. The global asymptotic
stability of the origin after substituting the control laws is verified in
[1]. When an internal system is included in the system, $L_{g}V$ backstepping
shares similar design procedures based on passivity theory [16].

When the unknown parts can not be completely canceled, ISS stability is
employed to prove the boundness of the tracking error[17]. The gains can be
assigned to ensure the tracking accuracy [18].

The two main drawbacks of integrator backstepping are:

(i) Overparameterization problem: Since separate adaptive laws for
$\dot{\hat{\theta}}_{i}$ are designed with every virtual control $\alpha_{i}$,
an overparameterization problem is aroused. This problem is released by using
tuning functions to estimate all the unknown parameters in the final step,
which significantly reduces the controller’s order [19]. However, the
robustness of this method to nonparametric nonlinearities is weak.

(ii) Explosion of complexity: It results from the repeated differentiation of
virtual control laws. The derivatives of the virtual control signals
$\alpha_{i}$ are required. As the order of the nonlinear system increases, the
calculation of the derivation becomes prohibitive. The analytic derivation is
cumbersome and tedious when the system order $n$ is larger than 3. To avoid
the explosion of complexity, there are a few methods. First of all, neglecting
of high order terms is a possible way to reduce the computational costs in
practical implements; however, the Lyapunov stability can not be guaranteed
theoretically and it even results in instability. Second, state transformation
can be adopted by defining a series of states as $\tilde{x}_{1}=x_{1}-x_{1d}$
and $\tilde{x}_{i}=\dot{\tilde{x}}_{i-1}$ with the control objective
$\tilde{x}_{1}\to 0$ as $t\to\infty$. Then, the system
$\text{col}(\tilde{x}_{1},\cdots,\tilde{x}_{n})$ is in a integrator-chain
form. The error states are defined $z_{1}=\tilde{x}_{1}$ and
$z_{i}=\tilde{x}_{i}-\alpha_{i-1}$ [20]. However, the complex time
differentiation of $f_{i}$ is still inevitable. Thirdly, efforts, such as
dynamic surface control (DSC) [21] and command filtered backstepping [22, 23]
have been made to avoid such problem systematically.

####  2.1.2 Cascade backstepping

Instead of integrator backstepping, cascade backstepping is an alternative
[24]. For the dynamics $\dot{z}_{i}$, $z_{i+1}$ is considered as an input. The
virtual control is designed based on Lyapunov function candidate
$V_{i}=V_{i,QF}$. If $z_{i}=0$ and system $(\dot{z}_{1},\cdots,\dot{z}_{i-1})$
is stable, and if $z_{i+1}=0$ and the stability of system $\dot{z}_{i}$ is
proved to be stable, then the stability of system
$(\dot{z}_{1},\cdots,\dot{z}_{i})$ is proved as a cascade system.

####  2.1.3 Dynamic surface control and command filtered backstepping

At the early stage, dynamic surface control is an extension of multiple
surface sliding control overcoming the explosion of terms satisfying semi-
global boundedness of the tracking error. The studies focus on a less general
system. Later, command filtered backstepping is proposed in a more general
triangular form. The control law is similar to that uses tuning functions. The
difference between these two approaches remains in the stability proofs. The
error states also have similar definitions, i.e., $\tilde{x}_{1}=x_{1}-x_{d}$
and $\tilde{x}_{i}=x_{i}-x_{i}^{c}$, where $x_{i}^{c}$ is a filtered signal of
$\alpha_{i}$. First-order lowpass filters are utilized to synthetic input at
each step of the design process. The total number of the lowpass filter is
$n-1$. The derivatives of the virtual control signal are replaced by the
filtered signals; hence, analytic differentiation and the constant-$g_{i}$
assumption are released. Moreover, non-Lipschitz systems can be stabilized.
The DSC methods can global exponentially stabilize the system and arbitrarily
bounded tracking for Lipschitz systems, and semi-globally arbitrarily bounded
regulation and tracking for non-Lipschitz system [25].

The commanded filters have various structures, for instance,

  * •

First-order lowpass [22]

$\dot{x}_{i}^{c}=-\omega_{i}(x_{i}^{c}-\alpha_{i-1}),$ (5)

  * •

Second-order lowpass filter [23]

$\displaystyle\dot{\varphi}_{i,1}$ $\displaystyle=-\omega_{n}\varphi_{i,2},$
(6a) $\displaystyle\dot{\varphi}_{i,2}$
$\displaystyle=-2\zeta\omega_{n}\varphi_{i,2}-\omega_{n}(\varphi_{i,1}-\alpha_{i}),$
(6b)

  * •

First-order Levant differentiator with finite-time convergent property [26]

$\displaystyle\dot{\varphi}_{i,1}$
$\displaystyle=-a_{i}|\varphi_{i,1}-\alpha_{i}|^{1/2}\text{sign}(\varphi_{i,1}-\alpha_{i})+\varphi_{i,2},$
(7a) $\displaystyle\dot{\varphi}_{i,2}$
$\displaystyle=-b_{i}\text{sgn}(\varphi_{i,2}-\dot{\varphi}_{i,1}),\
i\in\mathcal{I},$ (7b)

where $a_{i}$ and $b_{i}$ are tuned coefficients, $\zeta$ is the damping
ratio, $\omega_{n}$ is the natural frequency, and $x_{i+1}^{c}=\varphi_{i,1}$.

The virtual control laws and adaptive laws $\alpha_{i}$ are simplified with
$\dot{x}_{i}^{c}$. The stability is proved with the LFC
$V=\sum_{i=1}^{n}\frac{1}{2}v_{i}^{2}+\frac{1}{2}\tilde{\theta}^{\top}\Gamma^{-1}\tilde{\theta}$
with $v_{i}=z_{i}-\xi_{i}$ and $\xi_{i}$ is commonly defined as

$\displaystyle\dot{\xi}_{1}$
$\displaystyle=-k_{1}\xi_{1}+g_{1}\xi_{2}+(x_{2}^{c}-\alpha_{2})g_{1},$ (8a)
$\displaystyle\dot{\xi}_{i}$
$\displaystyle=-k_{i}\xi_{i}+g_{i}\xi_{i+1}-g_{i-1}\xi_{i-1}+(x_{i+1}^{c}-\alpha_{i})g_{i},$
(8b) $\displaystyle\dot{\xi}_{n}$
$\displaystyle=-k_{n}\xi_{n}+g_{n-1}\xi_{n-1}.$ (8c)

DSC and command filter backstepping can be used combining with other
techniques. Instead of feedback cancellation, adding one power integrator
technique is another possible solution [27, 28].

####  2.1.4 Parameter separation

Young’s inequality is the most widely used parameter separation technique in
backstepping design. If there exists a term $z_{i}a_{i}$ in $\dot{V}_{i}$
where $a_{i}$ is a bounded variable or a constant, then we have
$z_{i}a_{i}\leq\frac{1}{2}\varepsilon_{i}^{2}z_{i}^{2}+\frac{1}{2}\frac{a_{i}^{2}}{\varepsilon_{i}^{2}}$
with $\varepsilon_{i}>0$ according to Young’s inequality. The term
$\frac{1}{2}\varepsilon_{i}^{2}z_{i}^{2}$ can be easily canceled by the
virtual control law $\alpha_{i}$, and
$\frac{1}{2}\frac{a_{i}^{2}}{\varepsilon_{i}^{2}}$ is left to the final
Lyapunov function $V_{n}$ as
$\delta=\sum_{i=1}^{n}\delta_{i}=\sum_{i=1}^{n}\frac{1}{2}\frac{a_{i}^{2}}{\varepsilon_{i}^{2}}\geq
0$. Hence, the stability can be proved when the parameters in $\delta$ are
bounded. According to Lemma 1, when choosing sufficient large gains $c_{i}>0$,
$\gamma$ is increased to reduce the boundary of $V_{n}$ and $\rho$ is reduced
to ensure a smaller region for the errors. The selection of $\varepsilon_{i}$
does not influence the overall control gain
$\bar{c}_{i}=c_{i}+\frac{\varepsilon_{i}^{2}}{2}$ and the final tracking
performance. Smaller $\varepsilon_{i}$ results in lower
$\frac{\varepsilon_{i}^{2}}{2}$ and larger $\delta_{i}$. To ensure the same
boundary, higher $c_{i}$ is required to achieve higher $\gamma$ in (1), and
vice versa.

###  2.2 Finite-time control

It is known that the asymptotic stability implies the system trajectory
converges to the equilibrium as $t\to\infty$ resulting in a slow convergence
rate near the equilibrium. Sometimes, fast response and high-prevision
tracking performance are preferred rather than asymptotic stability. Comparing
with the typical backstepping approaches which grantees asymptotic
convergence, the finite-time control technique has a faster response and
better disturbance-rejection ability, which ensure the convergence to the
equilibrium in a finite time, i.e., $\lim\limits_{t\to T}x(t,x_{0})=0$ and
$x(t,x_{0})=0$ for any $t\geq T$ where $T$ is the settle time. In addition,
the finite-time stability has a higher disturbance-rejection capacity and
allows a system converges to the origin even more than one equilibrium exist.
At the early stage, local finite-time design approaches homogeneous
approximation are limited to two- or three-dimensional systems [29], e.g.,
terminal sliding mode (TSM) control [30]. Then, [31] extends the results to
the global domain.

The origin is said to be a finite-time-stable equilibrium of an autonomous
system $\dot{x}=f(x(t))$, $f(0)=0$, $x\in\mathbb{R}^{n}$ if the finite-time
convergence and Lyapunov stability hold. A Lyapunov-based finite-time
stability theorem is proposed in [32]. The origin is a finite-time-stable
equilibrium if there exists a continuous positive definite function $V(x)$,
real numbers $\gamma>0$, and $r\in(0,1)$, such that,

$\dot{V}(x)\leq-\gamma V^{r}(x),\forall x\in\mathbb{N}\backslash\\{0\\},$ (9)

where $\gamma>0$ and $\mathbb{N}$ is an open neighborhood of the origin [33].
The settling-time function is a function of the initial value of the LF
$T\leq\frac{1}{\gamma(1-r)}V(x_{0})^{1-r},x\in\mathbb{N}$. The selection of
$r$ determines the convergence rate of the Lyapunov function. Compared to the
scenario $r=1$ (Lemma 1), the convergence rate for $r\in(0,1)$ is faster in
the close neighborhood of the origin, while it is slower in the region away
from the origin. In addition, the extended forms of (9) are

  * $\bullet$ $\dot{V}(x)\leq-\gamma_{1}V(x)-\gamma_{2}V^{r}(x)$ [30] (10)

  * $\bullet$ $\dot{V}(x)\leq-\gamma_{1}V^{r^{\prime}}(x)-\gamma_{2}V^{r}(x)$ [34] (11)

where $\gamma_{1}>0$, $\gamma_{2}>0$ and $r^{\prime}>1$. They ensure higher
convergence rates when the tracking error are far away from the origin, and
the settle times are
$T=\frac{1}{\gamma_{1}(1-r)}\ln\frac{\gamma_{1}V^{1-r}(x_{0})+\gamma_{2}}{\gamma_{2}}$
for (2.2) and
$T=\frac{1}{\gamma_{2}(1-r)}+\frac{V^{1-r^{\prime}}(x_{0})-1}{\gamma_{1}(1-r^{\prime})}$
for (2.2). For a stochastic system, the finite-time stability in probability
guaranteed if $\mathcal{L}V(x)\leq-\gamma V^{r}(x)$ holds [35, 36].

The finite-time stability could be extended by involving a finite constant
$\delta>0$ in $\dot{V}$. Similar to (1), Practical finite-time stability is
guaranteed if $\dot{V}(x)\leq-\gamma_{1}V(x)-\gamma_{2}V^{r}(x)+\delta$ holds
[26]. However, the definition “practical” in finite-time stability could be
problematic when considering all systems with asymptotical stability could
reach a certain value in a finite time [37]. Asymptotically stability only
fails to converge to zero in a finite time, and the convergence speed could be
tuned by assigning the control gains.

Similar to the typical backstepping methods, the finite-time backstepping
approach requires the terms to be canceled by the fractional-order terms.
Hence, one of the following assumptions is always required.

  * •

$|f_{i}|\leq(\sum\limits_{j=1}^{i}|x_{j}|)\rho_{i}(\bar{x}_{i})$ [31]

  * •

$|f_{i}|\leq\frac{1}{2}|x_{i+1}|^{r_{i}}+\sum\limits_{j=1}^{i}|x_{j}|\rho_{i}(\bar{x}_{i}))\sigma$
[38]

  * •

$|f_{i}|\leq\sigma\rho_{i}(\bar{x}_{i}))$ [39]

  * •

$|f_{i}|=\varphi(t)\sum\limits_{j=1}^{i}|x_{j}|^{m_{ij}}+\sigma\sum\limits_{j=1}^{i}|x_{j}|^{n_{ij}}$
(time-varying system) [40]

where $\rho_{i}(\bar{x}_{i})$ are smooth known $C^{1}$ positive functions and
$\sigma\geq 1$ is an uncertain constant. The forth assumption is used to
handle a time-varying system. The orders of all systems should be decided
before the design process. The designed parameters are always
$r_{1}>\cdots>r_{n}$ since the higher-dimension dynamics should react faster
than the lower dimension.

The selection of LFCs for finite-time stability could be

  * •

[FT1]
$V_{i,FT}=\int_{\alpha_{i}}^{x_{i}}\left(s^{1/q_{k}}-\alpha_{i}^{1/q_{i}}\right)ds$

  * •

[FT2]
$V_{i,FT}=\int_{\alpha_{i}}^{x_{i}}\left(s^{\beta_{i-1}}-\alpha_{i-1}^{\beta_{i-1}}\right)ds$
[39]

  * •

[FT3]
$V_{i,FT}=\int_{\alpha_{i}}^{x_{i}}\left(s^{1/q_{k}}-\alpha_{i}^{1/q_{i}}\right)^{2-q_{k}}ds$
[31]

  * •

[FT4]
$V_{i,FT}=\int_{\alpha_{i}}^{x_{i}}\left(s^{1/q_{k}}-\alpha_{i}^{1/q_{i}}\right)^{2-q_{k}-\tau}ds$
(order-1) where $\tau$ is a ratio of two numbers.

Besides the recursive design method, the finite-time stabilization is always
designed with an inductive approach, i.e., the virtual control laws
$\alpha_{i}$ are designed accordingly with assumed known form $\dot{V}_{i-1}$
and let $\dot{V}_{i}$ and $\dot{V}_{i-1}$ have a similar inductive form [38].
This is because of the complex forms of the control law and yielded time
derivative of the Lyapunov function. Most designs are according to the
Young’s-inequality-like inequalities in [41]. Another application of the
finite-time stability and recursive design is the finite time disturbance
observer [42].

###  2.3 Estimation-based backstepping - neural network and fuzzy logic system

It is always difficult to design functions $\phi_{i}$ in (4). The estimation-
based approach is a solution to this problem. The most widely used
approximation theories using together with backstepping are the NNs and FLSs.
Their orientations and theories are different. However, their deduction in
backstepping, mathematically, are very similar. Hence, both methods are
presented together in this section.

NNs are widely used in various backstepping designs to approximate the
uncertainties, namely, neural adaptive control. According to the universal
approximation property, any smooth function in a compact set can be
approximated by an NN with an arbitrarily small error by a sufficiently large
number of nodes. In other words, the arguments of an unknown function starting
from any initial compact superset $\Omega^{0}$ remains within a compact set
$\Omega$. Briefly, the idea is to estimate all nonlinearities by NNs and
cancel them by the virtual control laws.

Two NNs are widely used in literature, they are:

  * •

Two-layer radial boundary function NN (RBFNN) [43]

$f(z)=W^{\top}S(z)+\varepsilon,$ (12)

  * •

Multilayer neural networks (MNN) [44, 45] and three-layer wavelet NN (WNN)
[46]

$f(z)=W^{\top}S(D^{\top}z+T)+\varepsilon=W^{\top}S(U^{\top}\bar{z})+\varepsilon,$
(13)

where $W=[w_{1},\cdots,w_{l}]^{\top}\in\mathbb{R}^{l}$ and
$V=[v_{1},\cdots,v_{l}]^{\top}\in\mathbb{R}^{q\times l}$ are the first-to-
second and second-to-third layer weight vectors,
$S(z)=[s_{1}(z),\cdots,s_{l}(z)]^{\top}$,
$T=[t_{1},\cdots,t_{l}]^{\top}\in\mathbb{R}^{l}$, and $\varepsilon$ is the
corresponding reconstruction error. MNN and WNN share a similar format by
using state transformation $U^{\top}=[D^{\top},T]$ and
$\bar{z}=[z^{\top},1]^{\top}$. The difference among various NNs is the
selection of basis function $s_{i}$. For a RBFNN, $s_{i}(z)$ is defined as
$s_{i}(z)=\exp\left[-\frac{(z-\mu_{i})^{\top}(z-\mu_{i})}{r_{i}^{2}}\right]$,
where $\mu_{i}=[\mu_{i1},\cdots,\mu_{in}]^{\top}$ is the center of the
receptive field and $r_{i}$ is the width of the Gaussian function. Then,
function $f$ is approximated as $\hat{f}=\hat{W}^{\top}S(Z)$ and
$\hat{W}^{\top}S(\hat{U}^{\top}\bar{z})$. The fundamental assumption for the
neural adaptive control is the boundness of the weight matrix $W$.

The ideal weights $W$ are assumed to be bounded with known $W_{m}$, i.e.,
$\|W\|_{F}\leq W_{m}$. The NN approximation is compensated by the virtual
control. Define the error vector of weights as $\tilde{W}=W-\hat{W}$, a
quadratic term $\frac{1}{2}\tilde{W}^{\top}\tilde{W}$ is added to the LFC. To
ensure the value of $\gamma$ in Lemma 1 is sufficiently large, a
$\sigma$-modification is adopted, i.e., an extra $-\Gamma\sigma\hat{W}$ with
$\sigma>0$ is added in the adaptive update law $\dot{\hat{W}}$ which is
similar to $\dot{\hat{\theta}}$ in integrator backstepping. The crucial
inequalities in the final stability proof are
$\tilde{W}^{\top}\hat{W}=\tilde{W}^{\top}W-\tilde{W}^{\top}\tilde{W}\leq\frac{1}{2}(-\tilde{W}^{\top}\tilde{W}+W^{\top}W)$
and
$\text{tr}\\{\tilde{W}^{\top}\hat{W}\\}\leq\frac{1}{2}(-|\tilde{W}|_{F}+|W|_{F})$,
which are proved by
$\tilde{W}^{\top}W\leq\frac{1}{2}(\tilde{W}^{\top}\tilde{W}+{W}^{\top}W)$.
Combining with the assumption of bounded $W$, the terms
$-\frac{1}{2}\sigma\tilde{W}^{\top}\tilde{W}$ and $\frac{1}{2}\sigma
W^{\top}W$ are merged into $-\gamma V$ and $\delta$ in Lemma 1, respectively.
The relation also holds to $\theta$, that is,
$\tilde{\theta}^{\top}\hat{\theta}\leq\frac{1}{2}(-\tilde{\theta}^{\top}\tilde{\theta}+\theta^{\top}\theta)$.
Again, they are merged to $-\gamma V$ and $\delta$ in Lemma 1, respectively.

FLS is a ratio of a linear combination of the input variables membership
functions and the sum of the membership functions, given by
$f(x)=\theta^{\top}\varphi(x)+\varepsilon$ or $f(x)=W^{\top}S(x)+\varepsilon$
and $\varepsilon\leq\bar{\varepsilon}$ where $\bar{\varepsilon}$ is a bounded
constant, $\theta=[\bar{y}_{1},\cdots,\bar{y}_{N}]^{\top}$,
$\bar{y}_{l}=s_{i}=\max\limits_{y\in\mathbb{R}}\mu_{G^{l}}(y)$, the fuzzy-
membership function
$\mu_{G^{l}}(x_{i})=\exp[-\frac{x_{i}-a_{i}^{l}}{b_{i}^{l}}]$,
$\varphi^{\top}=[\varphi_{1},\cdots,\varphi_{N}]$, and the fuzzy basic
function
$\varphi_{l}=\frac{\prod_{i=1}^{N}\mu_{i}^{l}(x_{i})}{\sum_{l=1}^{N}\prod_{i=1}^{N}\mu_{i}^{l}(x_{i})}$
[47]. Function $\varphi$ can be the Fourier series expansion which has a
similar form to three-layer NNs [48]. The important properties are
$0\leq\varphi^{\top}\varphi\leq 0$ and
$z_{i}\theta^{\top}\varphi\leq\frac{z_{i}^{2}}{4\lambda}|\theta|^{2}+\lambda$.

The drawbacks of the estimation-based methods are the long learning time
resulting from the significant number of NN nodes and adaption parameters in
order to receive sufficient approximation accuracy. An explosion of states
occurs to high-order systems resulting in difficult implementations. Most
examples in the case studies are second-order systems. To reduce the size of
the adaptive control question, $|W|$ can be used instead of $W$ [49].
Additionally, the design procedure is based on the assumed existed upper
bounds of the weight matrices. The stability is restricted to be local as the
NN approximation is only valid in the specific compact sets. Another
difficulty is that the term
$\frac{\partial\alpha_{i}}{\partial\hat{\theta}_{i}}\dot{\hat{\theta}}_{i}$
for MIMO systems is a function of all states. It cannot be merged into other
terms which to be estimated by the following NNs.

Another shortage of using NNs and FLSs is the lack of capacity to extract the
underlying structures of the nonlinear functions, though the estimation fits
well with the nonlinear function. The typical adaptive control can be applied
to identify the system by using some preset important governing terms. If the
basis function library is well guessed, the function structure can be
identified explicitly. On the other hand, the estimate of an NN or FLS does
not require a well-guesses basis function library.

###  2.4 Nussbaum function

In the aforementioned sections, a tricky assumption for adaptive backstepping
design is the unchanged sign of the known control gain, namely, constant
control direction. Nussbaum function was firstly proposed in [50] and
introduced to adaptive backstepping design by [51]. It has been used to design
control laws for systems with unknown $g_{i}$, for example, with a lack of
priori knowledge of control direction.

A function $\mathcal{N}(\chi)$ is said to be a Nussbaum-type function if
$\lim\limits_{s\to\infty}\sup\frac{1}{s}\int_{0}^{s}\mathcal{N}(\chi)d\chi=+\infty$
and
$\lim\limits_{s\to\infty}\inf\frac{1}{s}\int_{0}^{s}\mathcal{N}(\chi)d\chi=-\infty$
hold. Its integral is defined as
$\mathcal{M}(\chi):=\int_{0}^{\chi}\mathcal{N}(\tau)d\tau$. .

There are two types of Nussbaum-type functions, i.e., the amplitude-elongation
and the time-elongation[52]. Amplitude-elongation Nussbaum-type functions are
commonly adopted which are the products of a class $\mathcal{K}_{\infty}$
function and a trigonometric function, for example,

  * •

$\mathcal{N}(\chi)=\chi^{2}\cos(\chi)$,

  * •

$\mathcal{N}(\chi)=\chi^{2}\sin(\chi)$,

  * •

$\mathcal{N}(\chi)=\exp(\chi^{2})\cos(\chi\pi/2)$ [50],

  * •

$\mathcal{N}(\chi)=\frac{1}{2}e^{-\sigma\chi}\cos\chi-\frac{1}{2}e^{\sigma\chi}\cos\chi$
[53],

  * •

$\mathcal{N}(\chi)=\cosh(\lambda\xi)\sin(\xi)$ [54],

  * •

$\mathcal{N}(\chi)=\exp(\xi^{2}/2)(\xi^{2}+2)\sin(\chi)$ [55].

The virtual control law is designed to change the time derivative of LFC to a
form, for example, in $V(t)\leq
c_{0}+e^{-c_{1}t}\int_{0}^{t}(g_{i}(t)\mathcal{N}(\chi)+1)\dot{\chi}e^{c_{1}\tau}d\tau$.
By multiplying $e^{c_{1}t}$ to both sides, the boundedness of term
$g_{1}z_{1}z_{2}$ can be proved with the Lemmas proposed in [51, 56]. Similar
to Lemma 1, an extension show the boundness on $[0,t_{f})$ of $V(t)$,
$\chi_{i}(t)$, and
$\sum_{i=1}^{n}\eta_{i}\int_{0}^{t}\mathcal{N}(\chi_{i}(\tau))\dot{\chi}_{i}(\tau)d\,\tau$
is given [54], if the following inequality holds

$\displaystyle V(t)\leq$
$\displaystyle\sum_{i=1}^{n}\eta_{i}\int_{0}^{t}\mathcal{N}(\chi_{i}(\tau))\dot{\chi}_{i}(\tau)d\,\tau$
(14)
$\displaystyle+\sum_{i=1}^{n}a_{i}\int_{0}^{t}\dot{\chi}_{i}(\tau)\,\tau+\delta,\
\forall t\in[0,t_{f}),$

where $a_{i}$ and $\eta_{i}$ are constants with $a_{i}>0$, $\eta_{i}$ has the
same sign, and $|\eta_{i}|\in[\eta_{\min},\eta_{\max}]$.

The design is achieved by replacing $g_{i}\alpha_{i}$ by
$g_{i}\alpha_{i}=g_{i}\mathcal{N}(\chi)\bar{\alpha}_{i}=(g_{i}\mathcal{N}(\chi)-1)\bar{\alpha}_{i}+\bar{\alpha}_{i}$
and canceling the nonlinearities by designing $\alpha_{n}$. Then the update
law of $\chi$ is assigned to be $\dot{\chi}=z_{i}\bar{\alpha}_{i}$ and
stability is proved. The variable $\chi$ does not enter the LFC.

The Nussbaum gain technique has been successfully applied in many problems
presented in section 3. A drawback of Nussbaum gain is that the disturbances
can not be compensated directly.

###  2.5 Barrier Lyapunov function

Inspired by [57] and proposed in [58], barrier Lyapunov function (BLF) is very
popular LFC tool in backstepping-like design. It is compatible with nearly all
techniques in this survey. Significant amounts of works use BLF to handle the
asymptotic stabilization problem with partial or full state constraints.
Unlike a typical Lyapunov function holding a globally radically unbounded
property, a BLF goes to infinity when any state approaches its barriers.

A BLF is a scalar function $V_{BLF}(x):\mathbb{D}_{x}\to\mathbb{R}_{+}$,
defined with respect to the system $\dot{x}=f(x)$ on an open region
$\mathbb{D}_{x}$ containing the origin, that is (i) continuous and positive
definite, (ii) has continuous first-order partial derivatives at every point
of $\mathbb{D}_{x}$, (iii) has the property $\dot{V}(x)\to\infty$ as $x$
approaches the boundary of $\mathbb{D}_{x}$, and (iv) satisfies
$V_{BLF}(x)\leq b$ for all $t\geq 0$ along the solution of $\dot{x}=f(x)$ for
$x(0)\in\mathbb{D}_{x}$ and some positive constant $b$. The BLFs are presented
in Figure 2. For any positive constants $k_{ai}$ and $k_{bi}$, let
$\mathbb{D}_{zi}:=\\{z_{i}\in\mathbb{R}|-k_{ai}<z_{i}<k_{bi}\\}\subset\mathbb{R}$,
the BLF $V_{i,BLF}(z_{i})\to\infty$, as $z_{i}\to-k_{ai}$ or $z_{i}\to
k_{bi}$. The tracking error $z_{i}(t)$ is proved to be remained in the open
set $z_{i}\in\mathbb{D}_{zi}$ for all $t\geq 0$ if $\dot{V}\leq 0$ holds.
Again, $z_{i}(t)$ remains bounded for all $t\geq 0$ if the inequality
$\dot{V}\leq-\gamma V+\delta$ holds [58].

The are several ways to construct BLFs depending on the type of the
constraints. Three widely-used BLFs and their time derivatives are listed as
follows:

  * •

[BLF1] Symmetric barriers ($k_{ai}=k_{bi}$)

    * –

$V_{i,BLF}(z_{i})=\frac{1}{2}\log\frac{k_{bi}^{2}}{k_{bi}^{2}-z_{i}^{2}}=\frac{1}{2}\log\frac{1}{1-\xi_{b}^{2}}$,

    * –

$\dot{V}_{i,BLF}(z_{i})=\frac{z_{i}}{k_{bi}^{2}-z_{i}^{2}}\dot{z}_{i}$,

  * •

[BLF2] Asymmetric barriers, ($k_{ai}\neq k_{bi}$)

    * –

$V_{i,BLF}(z_{i})=\frac{q}{p}\log\frac{k_{bi}^{p}}{k_{bi}^{p}-z_{i}^{p}}+\frac{1-q}{p}\log\frac{k_{a}^{p}}{k_{ai}^{p}-z_{i}^{p}}$,

    * –

$\dot{V}_{i,BLF}(z_{i})=\frac{qz_{i}^{p-1}}{k_{bi}^{p}-z_{i}^{p}}\dot{z}_{i}+\frac{(1-q)z_{i}^{p-1}}{k_{ai}^{p}-z_{i}^{p}}\dot{z}_{i}$,

  * •

[BLF3] Time-varying constrain situation ($\dot{k}_{ai}\neq 0$ and
$\dot{k}_{bi}\neq 0$)

    * –

$V_{i,BLF}(z_{i})=\frac{q}{p}\log\frac{1}{1-\xi_{b}^{p}}+\frac{1-q}{p}\log\frac{1}{1-\xi_{a}^{p}}$,

    * –

$\dot{V}_{i,BLF}(z_{i})=\frac{q\xi_{b}^{2p-1}}{k_{bi}(1-\xi_{b}^{2}p)}(\dot{z}_{i}-\frac{z_{i}}{k_{bi}}\dot{k}_{bi})+\frac{(1-q)\xi_{a}^{2p-1}}{k_{ai}(1-\xi_{a}^{2}p)}(\dot{z}_{i}-\frac{z_{i}}{k_{ai}}\dot{k}_{ai})$,

where $\log(\cdot)$ is the natural logarithm, $p\geq n$ is an even integer,
switching function $q(z_{i})=\begin{cases}1,&if\ z_{i}>0\\\ 0,&if\ z_{i}\leq
0\end{cases}$, $\xi_{a}=\frac{z_{i}}{k_{ai}}$, and
$\xi_{b}=\frac{z_{i}}{k_{bi}}$. Then, BLF3 can be rewritten into a simpler
form $V_{i,BLF}(z_{i})=\frac{1}{2p}\log\frac{1}{1-\xi^{2p}}$, where
$\xi=q\xi_{b}+(1-q)\xi_{a}$. Novel BLF functions are given by
$V_{i,BLF}=\frac{k_{bi}^{2}}{\pi}\tan(\frac{\pi z_{i}^{2}}{2k_{bi}^{2}})$ [59]
and $V_{i,BLF}=\cot\frac{\pi}{2}(1-(\frac{z_{i}}{k_{bi}})^{2})$ [60].

Figure 2: Barrier Lyapunov functions.

The BLF-based method can be applied based on two assumptions. First, the
control coefficients $g_{i}$ is constantly positive or negative without
changing sign when $x_{1}(t)\in\mathbb{D}_{x_{1}}$, i.e.,
$|g_{i}(\bar{x_{i}})|>g_{0}>0$ where $g_{0}$ is a positive constant. Second,
there exist positive constants $A_{0}$, $\underline{x}_{1d}$,
$\overline{x}_{1d}$, and $\overline{x}_{id}$, $i=2,\cdots,n$ satisfying
$\max\\{\underline{x}_{1d},\overline{x}_{1d}\\}\leq A_{0}\leq k_{c1}$ and
$-\underline{x}_{1d}\leq x_{1d}(t)\leq\overline{x}_{1d}$ and
$|x_{1d}^{(i)}(t)|<\overline{x}_{id}$ for all $k_{c1}>0$ and $t\geq 0$ [58,
61]. In the design procedures, a BLF, for example in step $i$, is used when
the error state $z_{i}$ is constrained i.e., $V_{i}=V_{i-1}+V_{i,BLF}$.
Otherwise, quadratic LFC is adopted.

For a system which is not perfectly known, an important inequality, which
holds for all $|\xi|<1$ and any positive integer $p$ [61], is given by

$\log\frac{1}{1-\xi^{2p}}<\frac{\xi^{2p}}{1-\xi^{2p}}.$ (15)

For any positive constant $k_{bi}$, substituting $\xi=\frac{z_{i}}{k_{bi}}$
yields
$\log\frac{k_{bi}^{2}}{k_{bi}^{2}-z_{i}^{2}}<\frac{z_{i}^{2}}{k_{bi}^{2}-z_{i}^{2}}$
and
$\log\frac{k_{bi}^{2p}}{k_{bi}^{2p}-z_{i}^{2p}}\leq\frac{z_{i}^{2p}}{k_{bi}^{2p}-z_{i}^{2p}}$.
Eq. (15) simplifies the control law by removing the logarithm operator and
avoid the over-overparameterization problem [62]. When system unmodeled
dynamics exists, applying the inequality removes $k_{b1}^{2}-z_{1}^{2}$ in the
virtual control and simplified the control law. The numerator
$-\frac{c_{i}z_{i}^{2}}{k_{bi}^{2p}-z_{i}^{2p}}$ in $V_{n}$ is replaced by
$-c_{i}\log\frac{z_{i}^{2}}{k_{bi}^{2}-z_{i}^{2}}=-c_{i}V_{i,BLF}$. Hence, the
boundness can be proved according to Lemma 1 [61].

For an MIMO system, any positive constant vector $b_{i}\in\mathbb{R}^{n_{i}}$,
the following inequality holds for any vector $z_{i}\in\mathbb{R}^{n_{i}}$ in
the interval $|z_{i}|<|k_{bi}|$,
$\log\frac{k_{bi}^{\top}k_{bi}}{k_{bi}^{\top}k_{bi}-z_{i}^{\top}z_{i}}\leq\frac{z_{i}^{\top}z_{i}}{k_{bi}^{\top}k_{bi}-z_{i}^{\top}z_{i}}$
[63]. Then, the error signals are bounded,
$|z_{2}|,\cdots,|z_{n}|\leq\sqrt{2V(0)\exp(-\rho t)}$, and
$k_{a1}(t)(1-e^{-2pV(0)\exp(-\rho t)})^{\frac{1}{2p}}\leq z_{1}\leq
k_{b1}(t)(1-e^{-2pV(0)\exp(-\rho t)})^{\frac{1}{2p}}$. As $t\to\infty$,
$|z_{1}(t)|\leq k_{b1}\sqrt{1-\exp(-2(\rho+(V(0)-\rho)\exp(-\gamma t)))}\leq
k_{b1}\sqrt{1-\exp(-2\rho)}$. Hence, the upper limit of the tracking error
$z_{1}$ decreases with $\rho$. Therefore, $|z_{1}|$ can be arbitrary small
when $\gamma$ is sufficiently large.

Large control action may result when the states approach the boundary of the
boundaries. Another drawback of BLF is the upper and lower limits are assumed
to be known which are probably unknown in practical applications. Another
concern is the conservative initial state. The initial states have to stay in
the constraints, i.e., $x_{i}(0)\in(-k_{ai},k_{bi})$. At the region near the
barrier, $V_{i}$ will be extremely large. Integral barrier Lyapunov functional
(iBLF) $V_{i,iBLF}=\int_{0}^{z_{i}}\frac{\sigma
k_{ai}^{2}}{k_{ai}^{2}-(\sigma+\alpha_{i-1})^{2}}d\sigma$ significantly
relaxes the feasibility conditions [64]. The properties of iBLF are (i)
$\frac{z_{i}^{2}}{2}\leq V_{i,iBLF}\leq z_{i}^{2}\int_{0}^{1}\frac{\beta
k_{ai}^{2}}{k_{ai}^{2}-(\beta z_{i}+\text{sgn}(z_{i})A_{i-1})^{2}}d\beta$ and
(ii)
$\dot{V}_{i,iBLF}=\frac{k_{ai}^{2}z_{i}}{k_{ai}^{2}-x_{i}^{2}}\dot{z}_{i}+z_{i}(\frac{k_{ai}^{2}}{k_{ai}^{2}-x_{i}^{2}}-\rho_{i})\dot{\alpha}_{i-1}$,
where $A_{i}<k_{ai}$ is the upper bound of $|x_{id}|$ and
$\rho_{i}=\frac{k_{ai}}{2z_{i}}\ln\frac{(k_{ai}+z_{1}+x_{id})(k_{ai}-x_{id})}{(k_{ai}-z_{i}-x_{id})(k_{a1}+x_{id})})\dot{x}_{id}$.

###  2.6 Hyperbolic tangent function

Backstepping-like cancellation is convenient to cancel the terms of states,
e.g., $z_{i}^{2}$. However, it fails to cancel the unknown scalar terms, e.g.,
the external disturbance $d_{i}$. Compared with a complex projection operator
used in early studies, an innovative inequality is adopted to handle the
parametric bounded disturbance. If the disturbance is bounded, the
backstepping design process main relays on an inequality for any
$\varepsilon>0$ and $\eta\in\mathbb{R}$, i.e.,

$0\leq|\eta|-\eta\tanh(\frac{\eta}{\varepsilon})\leq k_{p}\varepsilon,$ (16)

where $k_{p}$ is a constant that satisfies $k_{p}=e^{-(kp+1)}$, e.g.,
$k_{p}=0.2758$. The aim of the inequality is to transfer the product of a
scalar and the error state, i.e., $d_{i}z_{i}$, into a form with a term of
explicit $z_{i}$ and bounded constant in $\dot{V}_{i}$. The first part can be
easily canceled and the later part can be integrated into $\delta$ in (1). The
error between the absolute value and its hyperbolic tangent approximation is
presented in Figure 3. The introduced $\tanh(\cdot)$ is compensated by the
virtual control law in the recursive design, and the final LFC can be proved
to be bounded using Lemma 1.

The system uncertainties and model errors can be equivalent to a disturbance
$d_{i}$, for example, the input-deadzone effects. In addition, a
$\tanh(\cdot)$ function is used to approximate the symmetric input saturation
and absolute operator to prevent the nonsmoothness in the time derivatives.

Figure 3: tanh($\cdot$) function ($e=0.001$).

The singularity problem challenges the backstepping design. If a positive
unknown term $a$ exists in $\dot{V}_{i}$ resulting in $-\frac{a}{z_{i}}$ in
the virtual control, the system faces singularity problem when $z_{i}=0$,
resulting in the term $-\frac{a}{z_{i}}$ is impossible to be estimated. To
prevent such issue, a property is given, i.e., for any constant $\eta>0$ and
variable $z_{i}\in\mathbb{R}$, $\lim_{z_{i}\to
0}\frac{1}{z_{i}}\tanh^{2}(\frac{z_{i}}{\eta})=0$ [65]. The term $a$ can be
modified to be
$a=\frac{2}{z_{i}}\tanh^{2}(z_{i}/\eta)az_{i}+\left(1-2\tanh^{2}(z_{i}/\eta)\right)a$.
The former part $\frac{2}{z_{i}}\tanh^{2}(z_{i}/\eta)az_{i}$ can be estimated.
An addition useful property is $1-2\tanh^{2}(z_{i}/\eta)\leq 0$ when
$|z_{i}|\geq 0.8814\eta$. Besides, $z_{i}$ is already bounded by $\eta$ if
$|z_{i}|<0.8814\eta$.

Another property of $\tanh(\cdot)$ function is that inequality
$1-16\tanh(\frac{z_{i}}{v_{i}})\leq 0$ holds for $|z_{i}|\leq 0.2554v_{i}$
[65]. It is used to compensate the radical unbounded term in $\dot{V}_{i}$,
e.g.,
$z_{i}^{2}=[1-16\tanh(\frac{z_{i}}{v_{i}})]z_{i}^{2}+16\tanh(\frac{z_{i}}{v}_{i})z_{i}^{2}$
and the latter term $16\tanh(\frac{z_{i}}{v_{i}})z_{i}^{2}$ can be
approximated by the NNs.

Furthermore, $\tanh(\cdot)$ function prevents singularity which will be
discussed later.

##  3 Systematic backstepping designs to various nonlinearities

In this section, backstepping designs applying to a class of systems are
discussed. Systems with various nonlinearities are transferred to specific
forms that are suitable for the aforementioned elegant approaches. A summary
is given in Fig. 4.

Figure 4: A summary of nonlinearities and corresponding design method.

###  3.1 State constraints

Due to the existence of actuator physical limitation and operational limits,
there is a clear boundary that the states or the tracking error can reach in
some applications, i.e., $x(t)\in\mathcal{D}_{x}$ or
$z_{1}(t)=y(t)-y_{d}(t)\in\mathcal{D}_{z1}$. The state constraints are due to
physical limitations or performance requirements. For example, in system (4),
the output $y(t)$ is required to remain in the set $|y|\leq k_{c1}$ for all
$t\geq 0$, where $k_{c1}$ is a positive constant. However, typical
backstepping design methods cannot ensure state constraints. Besides model
predictive control, there are a few Lyapunov methods to handle state
constraints.

BLF is the most popular approach due to its systematic deduction and simple
form. Hundreds of paper adopt BLF to handle state constraints. This has been
intensively reviewed in Section 2.5.

The reference governor is another method to handle the state constraint
problem. To avoid violating the system constraints, the reference signal is
modulated using online optimization [66]. However, this method is not specific
for adaptive backstepping design.

The commended filter can be used by regulating the changing rate of
$\alpha_{i}$ the states. This is achieved by modifying the filters with
several saturation operators [67]. However, state constraints are not proved.
The saturation operators in the filter may deteriorate the stability to the
entire system through critical damping is selected.

State constraints can be canceled by introducing a nonlinear mapping, which
transfers a constrained nonlinear system into a pure-feedback form without
outputs or state constraints [68].

###  3.2 Unknown disturbance

Disturbance is a very common unmodeled dynamics in a practical system due to
the external environment. Besides, the estimate error due to modeling
uncertainty is always modeled as unknown disturbance. A typical disturbed
nonlinear system is given by

$\displaystyle\dot{x}_{i}$
$\displaystyle=f_{i}(\bar{x}_{i})+g_{i}(\bar{x}_{i})x_{i+1}+d_{i},\
i=\mathcal{I},$ (17a) $\displaystyle\dot{x}_{n}$
$\displaystyle=f_{n}(\bar{x}_{n})+g_{n}(\bar{x}_{n})u+d_{n},$ (17b)

where $d_{i}$ ($i=1,\cdots,n-1$) and $d_{n}$ are the mismatched and matched
disturbances, respectively. Disturbances are impossible to be directly
measured, resulting in the robustness violation and failure of the direct
compensation approaches. A disturbance is unmatched when the disturbance
$d_{i}$ enters the system with a different state from the control input $u$;
otherwise it is matched. A system with unmatched disturbance can be converted
into one with matched disturbance using equivalent input disturbance technique
[69].

The designs commonly depend on bounded-disturbance assumptions. The widely-
used assumptions are summarized as follows:

  * •

$|d_{i}|\leq\bar{d}_{i}$ [70],

  * •

$|d_{i}|\leq\rho_{i}(\bar{x}_{i})\theta_{i}$ [71],

  * •

$|d_{i}|\leq(\rho_{i1}(\bar{x}_{i})+\rho_{i2}(\bar{z}_{i}))\theta_{i}$ [72],

The first one assumes a upper limits $\overline{d}_{i}$ for $d_{i}$. Using the
second and third assumptions, the disturbance can be separated into unknown
bounded constants $\theta_{i}$ and known smooth functions
$\rho_{i}(\bar{x}_{i})\in\mathbb{R}_{+}$ for $t>t_{0}$. Disturbance rejection
control can be categorized into robustness-based and estimation-based (or
approximation-based) methods.

####  3.2.1 Robustness-based methods

Projection operator is widely used at the early stage of the robust adaptive
control. A projection operator for two vectors $\theta,\ y\in\mathbb{R}^{k}$
is defined as

$\text{Proj}(\theta,y,f)=\begin{cases}y-\frac{\bigtriangledown
f(\theta)\bigtriangledown f(\theta)^{\top}}{\|\bigtriangledown
f(\theta)\|^{2}}yf(\theta),\quad\\\ \ \ \ \ \text{if}\ f(\theta)>0\bigwedge
y^{\top}\bigtriangledown f(\theta)>0,\\\ y,\qquad\text{otherwise},\end{cases}$
(18)

where $f:\mathbb{R}^{k}\to\mathbb{R}$ is a convex function and
$\bigtriangledown f(\theta_{b})=[\frac{\partial
f(\theta)}{\theta_{1}}\cdots\frac{\partial f(\theta)}{\theta_{k}}]^{\top}$.
The core property of the projection operator
$\tilde{\theta}(\text{Proj}(\theta,y,f)-y)\leq 0$ can be used to design
adaptive update law. Besides the basic projection operator, there are several
different projection operators, such as $\Gamma-$Projection [73] and parameter
projection [74, 75]. The exact structure of the convex function $f$ is of no
importance. When using projection operators in the backstepping design,
sufficient differentiability is required. The projection operators cooperate
with other elegant methods. However, it is no longer popular anymore after the
2000s. A reason is the complex calculation of the terms with Laplace operator
$\bigtriangledown f$.

Combining with Lemma 1, simpler control algorithms are proposed, e.g., [70]. A
more popular method is the $\tanh(\cdot)$ function presented in Section 2.6.
When the bounded uncertain term $d_{i}$ appear, a major part of the term
$|d_{i}||z_{i}|$ in $\dot{V}_{i}$ can be compensated and the uncancellable
parts are merged into $\delta$ in (1).

####  3.2.2 Estimation-based methods

Though the system disturbance cannot be directly measured, several techniques
are available to estimate the disturbance with measurements. The overall
stability is proved with a combination of the control law and observer. Hence,
the estimate error should be included in the Lyapunov function.

As mentioned before, NNs and FLSs can be used to approximate all sort of
bounded nonlinearities. Hence, they have been intensively investigated and
widely applied in literature. We skip here since they have been reviewed them
in Section 2.3.

Several high-gain disturbance observers are designed based on linear system
theories with similar forms [76, 77]. In another way, a lowpass filter is used
as an disturbance observer by filtering the error
$\dot{z}_{i}-(f_{i}+g_{i}-\dot{\alpha}_{i-1})$. However, $\dot{z}_{i}$
measurements are not always available. A possible solution is to replace
$\dot{z}_{i}$ with $-\mu_{i}z_{i}$, and the estimated disturbance
$\hat{d}_{i}$ is the output from the lowpass filter plus $\mu_{i}z_{i}$ [78].
The LFC for Step $i$ is modified to be
$V_{i}=V_{i-1}+V_{i,QF}+\frac{1}{2}\tilde{d}_{i}^{2}$, and $\hat{d}_{i}$ is
canceled by the virtual control law directly.

According to the internal model principle, deterministic disturbances with a
priori known waveform can be expressed in a form an exosystem [79], the
disturbance can be compensated by invoking the exosystem in the feedback loop.
The problem is firstly solved by proposing an nonlinear observer of the
unknown external disturbance and using a state-feedback adaptive regulator
[80]. Different from NN/FLS, parameter-dependent observer for unknown
disturbance based on exosystem has a simpler structure. The bounded
disturbance can be presented as $d=\vartheta+\delta_{d}$, where $\delta_{d}$
is a bounded function, $\vartheta$ is the output of a linear exosystem given
by

$\displaystyle\dot{\chi}$ $\displaystyle=\Gamma\chi,$ (19a)
$\displaystyle\vartheta$ $\displaystyle=c^{\top}\chi,$ (19b)

where $\chi\in\mathbb{R}^{q}$ is the state vector, the constant $q\times q$
matrix $\Gamma$ has all its eigenvalues on the imaginary axis, and $c$ is a
constant vector. Without loss of generality, the pair $(c^{\top},\Gamma)$ is
assumed to be observable. Here, the dimension of the exosystem $q$ is a design
parameter, $c$ and $\Gamma$ are unknown. Similar to the estimation-based
approach, higher-order exosystems are adopted when complex disturbances are
involved. The application of this disturbance observer technique can be found
in both unmatched and matched disturbances [81, 82].

A specific case is the periodic disturbance. Periodic disturbances often exist
in many mechanical control systems such as industrial robots and marine
systems. Fourier series expansion can be adopted to model the disturbance,
i.e., $d(t)=B^{\top}\Phi(t)+\delta_{d}(t)$ with bounded truncation error
$|\delta_{d}(t)|\leq\overline{\delta}_{d}$, where
$\Phi=[\phi_{1},\cdots,\phi_{q}]^{\top}$, $\phi_{1}=1$,
$\phi_{2j}=\sqrt{2}\sin(2\pi jt/T)$ and $\phi_{1}=1$,
$\phi_{2j+1}=\sqrt{2}\cos(2\pi jt/T)$ [48].

However, a common challenge for the observer-based approaches is the process
to model a rapidly changing disturbance. The observer may react not fast
enough to estimate the high-frequency disturbances. It is argued that if the
observer dynamics are much faster than the system dynamics, the disturbance is
very likely to be estimated and canceled by the controller. The training time
in addition depends on system complexity. The stability of the entire closed-
loop system can be deteriorated due to the large model mismatch.

###  3.3 Input nonlinearities

For a system with perfect actuator performance, the actuator output $u$ equals
to the control signal $v$, i.e., $u(t)=v(t)$. However, $u$ is a function of
$v$ due to the actuator characteristics and limitations in the form of
$u(v(t-\tau_{d}(t)))$ where $\tau_{d}$ is the time delay. In this section,
several input nonlinearities are reviewed, assuming $\tau_{d}=0$.

####  3.3.1 Input saturation

Physical actuators surely have their limits, also called input constraints.
When the control input remains within the boundness, the input saturation
effects are negligible. The main challenges for input saturation is the
nonsmoothness at the breakpoints and limited scope of input. To stabilize a
system with bounded control input has significant meaning to practical
industrial applications. A general form of input saturation is given by

$u(v)=\text{sat}(v)=\begin{cases}u_{min},&\text{if }v\leq u_{min},\\\
g_{s}(v),&\text{if }u_{min}<v<u_{max},\\\ u_{max},&\text{if }v\geq
u_{max},\end{cases}$ (20)

where $g_{s}$ is a smooth function, and $u_{min}$ and $u_{max}$ are the
minimum and maximum values for $u$. For a simplest linear saturation with
symmetric limits, i.e., $g_{s}(v)=v$ and $-u_{min}=u_{max}=u_{m}>0$, the input
saturation is simplified to be

$u(v(t))=\text{sat}(v(t))=\begin{cases}v(t),&\text{if }|v|<u_{m},\\\
\text{sgn}(v(t))u_{m},&\text{if }|v|\geq u_{m}.\end{cases}$ (21)

Smooth approximations are adopted to overcome the nonsmoothness.

  * •

The symmetric saturation (21) is approximated by [83, 84]

$\text{sat}(v)\approxeq\eta_{sat}(v)=u_{m}\tanh(\frac{v}{u_{m}})=u_{m}\frac{e^{v/u_{m}}-e^{-v/u_{m}}}{e^{v/u_{m}}+e^{-v/u_{m}}}.$
(22)

  * •

The approximation for the asymmetric saturation is given by

$\text{sat}(v)\approxeq\eta_{sat}(v)=\frac{2\overline{u}}{\pi}\arctan\left(\frac{\pi
v}{2\overline{v}}\right),$ (23)

where $\overline{u}=u_{max}$, $\overline{v}=v_{max}$ if $u_{max}/v_{max}\geq
u_{min}/v_{min}$ and $\overline{u}=u_{min}$, $\overline{v}=v_{min}$ if
$u_{max}/v_{max}\leq u_{min}/v_{min}$ [60].

The approximation difference $d_{sat}(v)=\text{sat}(v)-\eta_{sat}(v)$ is a
bounded function with a upper limit $|d_{sat}(v)|\leq
u_{m}(1-\tanh(1))=\overline{d}_{sat}$. There are a few approaches to continue
after the smooth approximation, i.e., (i) Consider $d_{sat}$ as a bounded
disturbance, then the robustness of the system can be proved with Lemma 1
using disturbance rejection control shown in Section 3.2. (ii) The system is
augmented with an extra state equation $\dot{v}=-cv+w$ and $z_{n+1}=u-v$. A
Nussbaum function is employed to handle the nonlinearity caused by
$\frac{\partial\eta_{sat}(v)}{\partial v}$ which is impossible to be used in
cancellation-based design procedures [85].

Figure 5: Saturation approximation ($u_{m}=2$).

If the saturation model is known, an auxiliary design system is introduced to
modify the control law from normally backstepping to
$v=g_{n}^{-1}[g_{n-1}z_{n-1}+f_{n}+\dot{\alpha}_{n-1}-\phi_{n}^{\top}\hat{\theta}_{n}-c_{n}(z_{n}-e)]$,
with an auxiliary state $e$. The state dynamics $\dot{e}$ is updated with
$\Delta u=u-v$ when $e$ is greater than a specific boundary [86].

####  3.3.2 Input deadzone

Deadzone occurs frequently in industrial applications, e.g., gear transmission
servo system, DC motor, hydraulic aircraft elevator control system, and valve.
It is a memoryless nonlinearity, a non-differential function, and insensitive
to a small control input. Consequently, the effect of the deadzone is the
undesired chattering, which is a problem in high-precision control. In
practice, the deadzone barriers are normally unknown.

There are several methods to handle the deadzone effects, such as deadzone
inverse, adaptive control with NN, and robust control with a combination of
linear input and bounded disturbance.

A general expression of deadzone operator is presented in Figure 6 and given
by

$u=\text{Dead}(v)=\begin{cases}g_{r}(v),&\text{if }v\geq b_{r},\\\ 0,&\text{if
}b_{l}<\theta<b_{r},\\\ g_{l}(v),&\text{if }v\leq b_{l},\end{cases}$ (24)

where $g_{r}$ and $g_{l}$ are the functions in the right and left parts,
$b_{l}$ and $b_{r}$ are the barriers in the right and left parts. The
challenges of the deadzone nonlinearity is to find the deadzone inverse, i.e.,
$v=\text{Dead}^{-1}(u)$. The parameters for a deadzone operator are the
breaking points $b_{r}$ and $b_{l}$, slopes of $g_{r}$ and $g_{l}$. Hereafter,
$b_{r}>0$ and $b_{l}<0$. The breaking points can be known or unknown,
symmetric or asymmetric. The slopes of $g_{r}$ or $g_{l}$ can be constant or a
function, and they can be the same or not in the positive and negative
domains. From simple to complex, three types of deadzones are discussed.

Figure 6: General deadzone operator.

We start from the simplest form with linear slopes in both sides, i.e.,

$u=\text{Dead}(v)=\begin{cases}k_{r}(v-b_{r}),&\text{if }v\geq b_{r},\\\
0,&\text{if }-b_{l}<\theta<b_{r},\\\ k_{l}(v-b_{l}),&\text{if }v\leq
b_{l},\end{cases}$ (25)

where $k_{r}$ and $k_{l}$ are the slopes in the right and left sides,
respectively. When all the parameters are known, a direct deadzone inverse [7]
can be applied, given by

$v=\text{Dead}^{-1}(u)=\begin{cases}u/k_{r}+b_{r},&\text{if }u\geq 0,\\\
0,&ifu=0,\\\ u/k_{l}+b_{l},&\text{if }v\leq 0.\end{cases}$ (26)

However, the deadzone inverse (26) is not a smooth function. The asymmetric
deadzone inverse approximation with constant slopes and known width can be
$v\approxeq\eta_{dead}(u)=\frac{u+k_{r}b_{r}}{k_{r}}\phi_{r}+\frac{u+k_{l}b_{l}}{k_{l}}\phi_{l}$,
where $\phi_{r}=\frac{e^{u/e_{0}}}{e^{u/e_{0}}+e^{-u/e_{0}}}$,
$\phi_{l}=\frac{e^{-u/e_{0}}}{e^{u/e_{0}}+e^{-u/e_{0}}}$, and $e_{0}$ is a
designed parameter [87]. More general deadzone inverse models are tabulated in
Table 1 [88]. The approximation error, defined by
$d_{dead}(v)=\text{Dead}(v)-\eta_{dead}(v)$, is bounded and can be made
arbitrarily small if the parameter $\rho_{dz}$ is sufficiently large, i.e.,
$\lim_{\rho_{dz}\to\infty}|\eta_{dead}(v)|=0$. For the second equation in
Table 1, $d_{dead}(v)<k_{dz}\ln\frac{2}{\rho_{dz}}$. However, a large
$\rho_{dz}$ results in aggressive close-loop response which may degrade the
system.

Table 1: Differential asymmetric deadzone approximation model [88]. $k_{r}=k_{l}=k_{dz}$ | $b_{r}=b_{l}=b_{dz}$ | $\eta_{dead}(v)=k_{dz}v+\frac{k_{dz}}{2\rho_{dz}}\ln\frac{\cosh{\rho_{dz}(\theta_{dz}-b_{dz})}}{\cosh{-\rho_{dz}k_{dz}(v+b_{dz})}}$
---|---|---
$k_{r}=k_{l}=k_{dz}$ | $b_{r}\neq b_{l}$ | $\eta_{dead}(v)=k_{dz}\left(v+\frac{b_{l}-b_{r}}{2}\right)+\frac{k_{dz}}{2\rho_{dz}}\ln\frac{\cosh{\rho_{dz}(v-b_{r})}}{\cosh{-\rho_{dz}k_{dz}(v+b_{l})}}$
$k_{r}\neq k_{l}$ | $b_{r}\neq b_{l}$ | $\eta_{dead}(v)=\frac{1}{\rho_{dz}}\ln\frac{1+e^{\rho_{dz}k_{r}(v-b_{r})}}{1+e^{-\rho_{dz}k_{l}(v+b_{l})}}$
Figure 7: Deadzone approximation with different soft degrees: $k_{l}=1$,
$k_{r}=2$, $a_{l}=0.5$, and $a_{r}=1.5$. (a) $\rho_{dz}=5$. (b)
$\rho_{dz}=10$. [88]

In practical applications, the deadzone breakpoints are always unknown. Early
adaptive solutions assume that the slopes and breakpoints are unknown
parameters, namely, adaptive deadzone inverse. Adaptive laws are adopted to
estimate parameters $\hat{k}_{r}$, $\hat{k}_{l}$, $\widehat{k_{r}b_{r}}$, and
$\widehat{k_{l}b_{l}}$ [89]. The deadzone is divided into separate smooth
regions and the LFC is
$V=V_{n}+\tilde{\theta}_{r}^{\top}\Gamma_{r}^{-1}\tilde{\theta}_{r}+\tilde{\theta}_{l}^{\top}\Gamma_{l}^{-1}\tilde{\theta}_{l}$,
where the adaptive states are $\theta_{r}=[k_{r},k_{r}b_{r}]^{\top}$ and
$\theta_{l}=[k_{l},k_{l}b_{l}]^{\top}$, and $\Gamma_{r}$ and $\Gamma_{l}$ are
positive definite matrices.

Alternatively, robust adaptive control approaches are proposed using Lemma 1
[90, 91].When $k_{l}$, $k_{r}$, $b_{l}$ and $b_{r}$ are unknown, but stay
within known ranges, the deadzone function can be separated into a linear term
and a bounded disturbance,

$\text{Dead}(v)=k_{dz}v+d_{dead}(v)$ (27)

where $d_{dead}(v)=\begin{cases}-k_{r}b_{r},&\text{if }v\geq b_{r},\\\
-k_{dz}v,&\text{if }-b_{l}<v<b_{r},\\\ -k_{l}b_{l},&\text{if }v\leq
b_{l}.\end{cases}$ and $k_{dz}=\begin{cases}k_{r}&\text{if }v\geq b_{r}\\\
k_{l}&\text{if }v\leq b_{l}\end{cases}$. Based on the assumption that
$k_{dz}\in[k_{\min},k_{\max}]$, $b_{r}\in[b_{r\min},b_{r\max}]$, and
$b_{l}\in[b_{l\min},b_{l\max}]$, we have
$|d_{dead}|\leq\max\\{k_{\max}b_{r\max},\allowbreak-k_{\max}b_{l\max}\\}$.
Besides robust adaptive control, the disturbance $d_{dead}$ can be handled
with estimation-based methods.

A more complicated case is the unknown nonlinear $g_{r}$ and $g_{l}$. When the
deadzone width parameters $b_{r}$ and $b_{l}$ are bounded constants, $g_{r}$
and $g_{l}$ are smooth functions with bounded slopes, a separate model [92]
similar to (27) is given by $\text{Dead}(v)=K^{\top}\Phi v+d_{dead}$ where the
denotations of $K$, $\Phi$, and $d$ can be found in [93]. Since $d_{dead}$ is
bounded, the deadzone is transferred to the problems relevant to bounded
disturbance with unknown time-varying control coefficient. NN/FLS-based
approximation approaches are used to estimate the deadzone dynamics with
unknown $g_{r}$ and $g_{l}$, as well as unknown $b_{r}$ and $b_{l}$ [94]. The
deadzone inverse is $v=\text{Dead}^{-1}(u)=u+u_{NN}$, where

$u_{NN}=\begin{cases}g_{l}^{-1}(u),&\text{if }u<0\\\ 0,&\text{if }u=0\\\
g_{r}^{-1}(u),&\text{if }u>0\end{cases}.$

####  3.3.3 Time-varying control coefficient

If the control coefficients $g_{i}$ is an unknown nonzero constant with known
sign, $V_{i}=\frac{1}{g_{i}}z_{i}$ can be used as the LFCs. In $\dot{V}_{i}$,
the resulting term $\frac{1}{g_{i}}f_{i}$ is similar to a typical adaptive
backstepping design, where $\theta_{gi}=\frac{1}{g_{i}}$ [95, 96]. Singularity
problem can be avoided with this modifications.

A more challenging scenario is that $g_{i}$ is only partly known, e.g.,
$\dot{x}_{i}=f_{i}+(g_{i}+\Delta g_{i})x_{i+1}$ where $\Delta g_{i}$ is the
unknown control coefficients. The basic assumption is that the uncertain parts
$\Delta g_{i}$ is bounded, i.e., there exist positive constants
$\overline{g}_{i\Delta}$ s.t. $|\Delta
g_{i}(\bar{x}_{i})|\leq\overline{g}_{i\Delta}$ [86]. The aim is to design
virtual control laws in case of the singularity issues caused by the $g_{i}$
part. In addition, this assumption also ensures the sign of $g_{i}+\Delta
g_{i}$ is unchanged. The virtual control law is modified to
$\alpha_{1}=(g_{1}+\gamma_{1})^{-1}(-c_{1}z_{1}-f_{1}+\dot{x}_{1d})$, where
$\gamma_{1}>\overline{g}_{1}$ is selected to avoid the singularity problem of
$(g_{1}+\gamma_{1})^{-1}$.

To avoid the singularity problem when $g_{i}(\bar{x}_{i}(t))$ is unknown,
integral Lyapunov functionals can be used, such as

  * •

$V_{1,I}=\int_{0}^{z_{1}}\sigma\beta_{i}(\sigma+x_{1d})d\sigma$

  * •

$\dot{V}_{1,I}=z_{1}\beta_{1}\dot{z}_{1}+\int_{0}^{z_{1}}\sigma\frac{\partial\beta_{i}(\sigma+x_{1d})}{\partial
x_{1d}}\dot{x}_{1d}d\sigma$

  * •

$V_{i,I}=\int_{0}^{z_{i}}\sigma\beta_{i}(\bar{x}_{i-1},\sigma+\alpha_{i-1})d\sigma,\
i=2,\cdots,n$

where $\sigma=\theta z_{1}$ and $\beta_{i}=\frac{\overline{g}_{i}}{|g_{i}|}$.
The integral Lyapunov functionals is applicable to autonomous systems since
$\frac{1}{2}z_{i}^{2}\leq
V_{i,I}\leq\frac{z_{i}^{2}}{\underline{g}_{i}}\int_{0}^{1}\theta\overline{g}(\theta
z_{i}+\alpha_{i-1})$ [97].The design continues using the Nussbaum function and
NN/FLS.

For the cases, the control coefficients $g_{i}$ is non-positive or non-
negative. For example, absolute is a quite commonly used operator in hydraulic
systems. When the absolute operator appears in $g_{i}(x)$ in Eq. (4), $g_{i}$
is no longer a first-order continuous function. Consequently, a singularity
problem is aroused, i.e., $1/g_{i}\to\infty$ when $g\to 0$. Sign function is
usually applied to remove the absolute operator, $|a|=sgn(a)a$. An
approximation $sgn(a)\approxeq tanh(a/\delta)$ transfers $g(x)$ into a
continuously Lipschitz function where $\delta$ is a small enough positive
constant [98, 83]. Another way to avoid $1/g_{i}$ a singularity problem is to
use $g_{i,new}=\frac{g_{i}^{2}}{g_{i}+\varepsilon_{i}}$ or
$g_{i,new}=g_{i}+\varepsilon_{i}$ during the control design, where $\delta$ is
a small value.

A more general problem is that the sign of $g_{i}$ is unknown. Nussbaum
function presented in Section 2.4 is widely used to handle such an issue.

####  3.3.4 Actuator failure

Actuator failures change the output and parameters, introduce additional
system uncertainties and disturbances, and result in performance deterioration
and even accidents. A failure is normally uncertain in time and often
unrecoverable. Modern researches on fault-tolerant control are in two
directions. (i) Firstly, the model-based redundancy approach for fault-
tolerant control is based on a bank of residual signals generated by multiple
online monitoring modules running in parallel with specific possible failures.
If the failures are not contained in the bank, the performance is unreliable.
(ii) Another direction is the adaptive failure compensation design without
explicit failure detection, which has a simpler structure. The objective is to
compensate for the effects of reasoning from the actuator failures, and
meanwhile, to ensure the asymptotic tracking performance with a bounded error.
The controller remains the same structure through the running; therefore, it
is easier to implement. In this survey, we focus on (ii) adaptive failure
compensation design.

Failures are categorized into the total loss of effectiveness (TLOE) and
partial loss of effectiveness (PLOE). Suppose that there are in total $m$
actuators in the system, i.e., $u=[u_{1},\cdots,u_{m}]^{\top}$ For a TLOE
scenario, redundant actuators are required. A common assumption is that the
remaining actuators are fully actuated, and the desired control objective is
still achievable for up to $m-1$ actuator faults for an SISO system [99, 100].
This assumption ensures the controllability of the plant with the remaining
actuation power and the existence of a normal solution for the actuator
failure compensation problem.

In literature, there are three types of actuator failure models. From simple
to complex, they are listed as follows.

  * •

If failure for the $j^{th}$ actuator occurs at $t_{j}$, i.e.,

$u_{j}(t)=\bar{u}_{j},\ \forall t\geq t_{j},\ j=1,\cdots,m,$ (28)

where $\bar{u}_{j}$ and $t_{j}$ are unknown. The widely-used actuator fault
model is $u=\sigma\bar{u}+(I_{m}-\sigma)v$, where
$\sigma=diag\\{\sigma_{1},\cdots,\sigma_{m}\\}$ is a matrix containing failure
patterns $\sigma_{j}$ and $\sigma_{j}=\begin{cases}1&\text{if the
}j^{th}\text{ actuator fails},\\\ 0&\text{otherwise}.\end{cases}$ Then
$\sigma\bar{u}$ and $(I_{m}-\sigma)$ can be estimated using adaptive laws
[101].

  * •

Actuator model with both gain fault and bias fault is modeled by [102, 103],
i.e.,

$u_{j}=\rho_{j}v_{j}+b_{uj}.$ (29)

The model has four modes as follows.

    * –

Failure-free: $\rho_{j}=1$ and $b_{uj}=0$;

    * –

PLOE: $\rho_{j}\in(0,1)$ and $b_{uj}=0$;

    * –

TLOE: $\rho_{j}=0$ and $b_{uj}=0$;

    * –

Bias fault: $\rho_{j}=0$ and $b_{uj}\neq 0$.

  * •

First- [104], second-[105], and third-order [106] dynamic actuator failure
models are proposed. The first-order model is given by

$\dot{u}_{j}=-(1-\sigma_{j})\lambda_{j}(u_{j}-k_{j}v_{j})$ (30)

and the second-order model is given by

$\displaystyle\dot{u}_{1j}$ $\displaystyle=u_{2j},$ (31)
$\displaystyle\dot{u}_{2j}$
$\displaystyle=-(\lambda_{2j}+\sigma_{j}\beta_{j})u_{2j}+(1-\sigma_{j})\lambda_{1j}(k_{j}v_{j}-u_{1j})$

where $\lambda_{j}\gg 1$, $\lambda_{1j}\gg 1$, $\lambda_{1j}\gg\lambda_{2j}$,
and $\lambda_{2j}+\beta_{j}\gg 1$. the three modes are

    * –

Failure-free: $\sigma_{j}=0$ and $k_{j}=1$;

    * –

PLOE: $\sigma_{j}=0$ and $k_{j}\in(0,1)$;

    * –

TLOE: $\sigma_{j}=1$.

For a PLOE issue, the control coefficient can be considered as time-varying
unknown parameters, which can be solved by NN-based design [107] and Nussbaum
function [11]. The fault detection and isolation mechanism can be integrated
into the adaptive control to improve the performance, e.g., the deactivation
function for failed actuators is realizable. NNs can be adopted to optimize
the effects of the failure after fault detection but before the isolation
[108]. For the system with healthy actuators for backups, an adaptive state-
feedback fault-tolerant control function can be achieved by combining the BLF
and a monitoring function [109].

To ensure the transient performance of the tracking error, prescribed
performance bounds based control design is used [110]. The tracking error
should stay in a tube of a prescribed decreasing smooth prescribed performance
function $\eta(t)$, i.e.,
$-\underline{\delta}\eta(t)<z_{1}(t)<\overline{\delta}\eta(t)$, where
$0<\underline{\delta}<\overline{\delta}\leq 1$ are given positive constants. A
new error state is defined as $z_{1}=S^{-1}(\frac{z_{1}}{\eta})$, where the
smooth, strictly increasing, and invertible function $S(z_{1})$ satisfies: (i)
$\underline{\delta}<S(z_{1})<\overline{\delta}$, (ii)
$\lim\limits_{z_{1}\to+\infty}S(z_{1})=\overline{\delta}$ and
$\lim\limits_{z_{1}\to-\infty}S(z_{1})=\underline{\delta}$, and (iii)
$S(0)=0$. For example, functions $\eta(t)$ and $S$ are designed as
$\eta(t)=[\eta(0)-\eta(\infty)]\exp(-at)+\eta(\infty)$ and
$S(v)=\frac{\overline{\delta}e^{(v+r)}-\underline{\delta}e^{-(v+r)}}{e^{(v+r)}+e^{-(v+r)}}$,
where $a,\eta(0),\eta(\infty)>0$, $\eta(0)>\eta(\infty)$, and
$r=\ln(\underline{\delta}/\overline{\delta})/2$.

###  3.4 Time-delay effects

Widely existing in chemical systems, biological systems, economic systems, and
hydraulic/pneumatic systems, the time-delay phenomenon usually deteriorates
the performance of a closed-loop system. A system stabilized by a feedback
control law may become unstable after introducing the time-delay effects.
Time-delay effects probably exist everywhere in the system (4), e.g., $f_{i}$,
$x_{i+1}$, $u$, and $y$. The delayed times are often unknown, which can be a
constant value for all parameters, constant but different values for different
parameters, and time-varying. The feedback laws have to be modified to adjust
the time-delay effects. There exist two main types of Lyapunov functions for a
time-delay system, i.e., Lyapunov-Razumikhin approach [111] and Lyapunov-
Krasovskii approach [112]. Most studies on adaptive backstepping adopt
Lyapunov-Krasovskii functionals, and only a few studies rely on Razumikhin
lemma [113]. Lyapunov-Krasovskii approach is a predictor-like technique
according to the Lyapunov-Krasovskii Theorem.

####  3.4.1 State delay

A strict-feedback system with state-delay is given by

$\displaystyle\dot{x}_{i}(t)$
$\displaystyle=f_{i}(\bar{x}_{i}(t))+f_{di}(\bar{x}_{i}(t-\tau_{di}))+g_{i}(\bar{x}_{i}(t))x_{i+1}(t),$
(32a) $\displaystyle\dot{x}_{n}(t)$
$\displaystyle=f_{n}(\bar{x}_{n}(t))+f_{dn}(\bar{x}_{n}(t-\tau_{dn}))+g_{n}(\bar{x}_{n}(t))u(t),$
(32b)

where $\tau_{di}>0$ denotes the delayed time,
$f_{di}(\bar{x}_{i}(t-\tau_{di}))$ are the time-delay terms, and
$\bar{x}_{i}(t-\tau_{di})=[x_{1}(t-\tau_{d1}),x_{2}(t-\tau_{d2}),\cdots,x_{i}(t-\tau_{di})]^{\top}$.

A common assumption for a time-delay system is that the unknown time delays
are bounded by a known constant, i.e., $\tau_{di}\leq\tau_{d,max}$. At the
beginning, strict-feedback form with known equal delay
$\tau_{d1}=\cdots=\tau_{dn}=\tau_{d}$ is handled by robustness-based approach
by assuming all functions are known and bounded [114, 46]. However, the delays
in different states always vary in practical applications, i.e.,
$\tau_{d1}\neq\cdots\neq\tau_{dn}$ [95, 65, 115].

In addition, various assumptions of bounded parametric time-delayed terms are
adopted to make $f_{di}$ to be compensable. The absolute value of the time-
delay term is bounded by known smooth functions $\rho_{i}(\bar{x}_{i})$ in
several parameter-separation forms. Commonly used assumptions are listed as
follows:

  * •

$|f_{di}(\bar{x}_{i}(t-\tau_{di}))|\leq\sum\limits_{j=1}^{n}\rho_{j}(\bar{x}_{j})$
[65],

  * •

$|f_{di}(\bar{x}_{i}(t-\tau_{di}))|\leq\rho_{i}(\bar{x}_{i}(t-\tau_{di}))$
[97],

  * •

$|f_{di}(\bar{z}_{i}(t-\tau_{di}))|\leq\sum\limits_{j=1}^{i}|z_{j}(t-\tau_{dj})|\rho_{ij}(\bar{z}_{j}(t-\tau_{dj}))$
[114, 46],

  * •

$|f_{di}(\bar{x}_{i}(t-\tau_{di}))|=\theta_{di}^{\top}\phi_{di}(\bar{x}_{i}(t-\tau_{di}))+\delta_{di}(\bar{x}_{i}(t-\tau_{di}))$
[95, 116],

where $\rho_{ij}(\cdot)$ is a known continuous and smooth function,
$\theta_{di}\in\mathbb{R}^{n_{i}}$,
$\phi_{di}:\mathbb{R}^{i}\to\mathbb{R}^{n_{i}}$ is a known smooth function
vector, $\delta_{di}$ is a bounded unknown smooth function, i.e.,
$|\delta_{di}(\bar{x}_{i}(t-\tau_{di}))|\leq
c_{di}\rho_{i}(\bar{x}_{i}(t-\tau_{di}))$ where $c_{di}$ is an unknown
constant.

An additional Lyapunov-Krasovskii functional is added to the LFC in the
corresponding step. A basic integrate-type Lyapunov-Krasovskii functional and
its derivative are

$\displaystyle V_{i,LK}=$
$\displaystyle\int_{t-\tau_{d}}^{t}S_{i}(\bar{z}_{i}(\sigma))d\sigma,$ (33a)
$\displaystyle\dot{V}_{i,LK}=$ $\displaystyle S_{i}(t)-S_{i}(t-\tau_{d}),$
(33b)

where $S_{i}(\bar{z}_{i}(t))$ is a positive definite function, e.g.,
$S_{i}(\bar{z}_{i}(t))=\rho_{i}^{2}(\bar{z}_{i}(t))$. The main idea of
Lyapunov-Krasovskii functional is to cancel the delayed term with the
selection of functions $S_{i}$. A virtual control law only compensates the
terms without delay, and the delayed terms are canceled by a selection of
function $S_{i}$. Depending on the number of time delayed components in the
$i^{th}$ step, the general Lyapunov-Krasovskii functional is selected as
$V_{i,LK}=\sum_{\mathcal{J}}\int_{t-\tau_{dj}}^{t}S_{j}(\bar{z}_{j}(\sigma))d\sigma$
when several delays $\tau_{dj}$ exist in the a same dynamics $\dot{x}_{i}$,
with $\mathcal{J}$ is the set containing all indices of the delayed states
[116]. When multiple time delays exist in the states
$x_{1},x_{2},\cdots,x_{n}$, a nonlinear decomposition strategy is adopted,
e.g.,
$f_{di}(\bar{x}_{i}(t-\tau_{di}))\leq\sum_{j=1}^{i}\beta_{ij}(x_{j}(t-\tau_{di}))$.
Then, the time derivative of $V_{i,QF}$ has a part $z_{i}f_{di}$, which can be
separated by the Young’s inequality, i.e.,
$z_{i}f_{di}\leq\frac{i}{2}z_{i}^{2}+\frac{1}{2}\sum_{j=1}^{i}\beta_{ij}^{2}(x_{j}(t-\tau_{dj}))$.
Lyapunov-Krasovskii functionals are designed to compensate
$\frac{1}{2}\sum_{j=1}^{i}\beta_{ij}^{2}(x_{j}(t-\tau_{dj}))$ [117], such as
$V_{i,LK}=\frac{1}{2}\int_{t-\tau_{di}}^{t}\sum_{j=1}^{i}\beta_{ij}^{2}(x_{j}(\sigma))d\sigma$.
The effects of the additional Lyapunov-Krasovskii functional is an addition in
the the control gain $c_{i}$.

Besides, the delayed function $f_{di}$ can be approximated and canceled using
NNs/FLSs [65, 115]. However, such approaches are limited when the
nonlinearities change rapidly.

####  3.4.2 Delayed integrator

A system with delayed integrator is given by

$\displaystyle\dot{x}_{i}$
$\displaystyle=f_{i}(\bar{x}_{i})+g_{i}(\bar{x}_{i})x_{i+1}(t-\tau_{di}),$
(34a) $\displaystyle\dot{x}_{n}$
$\displaystyle=f_{n}(\bar{x}_{n})+g_{n}(\bar{x}_{n})u.$ (34b)

A chain of delayed integrators (i.e., $f_{i}=0$) [118] and linear system
(i.e., $f_{i}=\sum\limits_{j=1}^{i}a_{ij}x_{j}$) [119] have been studied.
However, the control for nonlinear systems with unknown delayed integrators is
still open.

####  3.4.3 Input delay

There are several methods to handle a system with delayed input, which is
given by

$\displaystyle\dot{x}_{i}$
$\displaystyle=f_{i}(\bar{x}_{i})+g_{i}(\bar{x}_{i})x_{i+1},$ (35a)
$\displaystyle\dot{x}_{n}$
$\displaystyle=f_{n}(\bar{x}_{n})+g_{n}(\bar{x}_{n})u(t-\tau_{d}).$ (35b)

First, pade approximation approach can be used to handle time-varying input
delay [120, 121]. One additional state $x_{n+1}$ and a lowpass process is
introduced to cancel the influence of the delayed time, i.e.,

$\displaystyle\dot{x}_{i}$
$\displaystyle=f_{i}(\bar{x}_{i})+g_{i}(\bar{x}_{i})x_{i+1},$ (36a)
$\displaystyle\dot{x}_{n}$
$\displaystyle=f_{n}(\bar{x}_{n})+g_{n}(\bar{x}_{n})(x_{n+1}-u),$ (36b)
$\displaystyle\dot{x}_{n+1}$ $\displaystyle=-\gamma_{d}x_{n+1}+2\gamma_{d}u,$
(36c)

where $\gamma_{d}=\frac{2}{\tau_{d}}$. The last error state is defined as
$z_{n}:=x_{n}-\alpha_{n-1}+\frac{1}{\gamma_{d}}x_{n+1}$. When $\tau_{d}$ is
unknown, $\gamma=\gamma_{n}+\tilde{\gamma}$, where $\gamma_{n}$ is assumed to
be known and $\tilde{\gamma}$ is the bounded error. The Pade approximation
approach is only capable to work in a short delay, but not a long delay.

Moreover, emulation is another approach for the input delay with a short
delay. The controller is first designed without delay, then the upper bound of
the delay is found such that the closed-loop system is still asymptotically
stable [122].

Besides, the unknown arbitrarily long actuator delay length can be handled by
prediction-based boundary control [123] and predictor feedback [124],
particularly used on linear time-invariant, finite-dimensional, and completely
controllable system. The effects of the delay is compensated with an
integration over the delay period. The delay time is estimated by a time-delay
identifier when using the prediction-based boundary control. The main idea of
the predictor feedback approach is to modeled the actuator time delay effects
as a transport partial differential equation [125]. Assuming the distributed
input $u(x,t)=U(t+\tau_{d}(x-1))$, the delayed input is replace by a real-time
input $v$, s.t.,

$\displaystyle\tau_{d}\frac{\partial u}{\partial t}(x,t)$
$\displaystyle=\frac{\partial u}{\partial x}(x,t),\ x\in[0,1),$ (37a)
$\displaystyle u(1,t)$ $\displaystyle=U(t).$ (37b)

However, the distributed terms may not always be easy to compute. Furthermore,
this approach is not applicable to nonlinear system due to the inconvenience
integration over the delay interval.

An alternative way to solve a system with an arbitrarily long input delay is
proposed in [126] without distributed terms. A time-delay system is
transformed into another dynamics with a delayed system state as its input
[127].

####  3.4.4 Measurement delay

In additional to the aforementioned time-delayed systems, the measurement
delay is a few less popular issue, i.e., $y=x_{1}(t-\tau_{d})$, which can be
transformed to a state-delay problem. The zero-order hold delayed measurement
[128] is overcome by predictor-type designs [129] and estimators based on FLS
[130].

###  3.5 Pure feedback system

A limitation of backstepping approach is that it requires the system to be in
a strict-feedback form. A milder state equation, namely nonlinear pure-
feedback system, is given by

$\displaystyle\dot{x}_{i}$ $\displaystyle=f_{i}(\bar{x}_{i},x_{i+1}),\
i\in\mathcal{I},$ (38a) $\displaystyle\dot{x}_{n}$
$\displaystyle=f_{n}(\bar{x}_{n},u).$ (38b)

Comparing to the strict-feedback system in Eq. (4), a pure-feedback system
(38) has no affine appearance of the variables to be used as virtual control.
The basis of the following solutions is to receive affine variables used as
virtual control.

####  3.5.1 Taylor series expansion

If the pure-feedback system has a strong relative degree and can be
transformed into an integrator chain, Taylor series expansion can be adopted
to design an observer and neural adaptive controller [131]. The high-order
terms are modeled as disturbances.

####  3.5.2 Mean value theorem

Mean value theorem is the most widely used method. A pure-feedback system (38)
is transformed to a strict-feedback form with unknown control coefficient
[132], i.e.,

$\displaystyle f_{i}(\bar{x}_{i+1})$
$\displaystyle=f_{i}(\bar{x}_{i},0)+\mu_{i}(\bar{x}_{i},x_{i+1}^{0})x_{i+1},$
(39a) $\displaystyle f_{n}(x,u)$
$\displaystyle=f_{n}(\bar{x}_{n},0)+\mu_{n}(\bar{x}_{n},u^{0})u,$ (39b)

where $\mu_{i}(\bar{x}_{i+1})=\frac{\partial f_{i}(\bar{x}_{i+1})}{\partial
x_{i+1}}$ and $\mu_{n}(\bar{x}_{n},u)=\frac{\partial f_{n}(x,u)}{\partial u}$
exist but are unknown, $x_{i+1}^{0}$ is an unknown point between 0 and
$x_{i+1}$, and $u^{0}$ is an unknown value between 0 and $u$. Hence, the pure-
feedback systems are transferred into a strict-feedback system with unknown
functions $f_{i}$ and control coefficients $\mu_{i}$. Refer to the above-
mentioned methods, the deduction is based on assumptions with bounded control
gains, i.e., $\underline{\mu}_{i}<\mu_{i}<\overline{\mu}_{i}$. The error
dynamics is then transferred to

$\dot{z}_{i}=f_{i}(\bar{x}_{i},\alpha_{i}^{*})+g_{i\mu_{i}}(x_{i+1}-\alpha_{i}^{*}),$
(40)

where
$g_{i\mu_{1}}=g_{i}(\bar{x}_{i},\mu_{i}x_{i+1}+(1-\mu_{i})\alpha_{i}^{*})$,
with $\mu_{i}\in(0,1)$, $\alpha_{i}^{*}$ is the ideal control input such that
$f_{1}(x_{1},\alpha_{1}^{*})-\dot{x}_{1d}+k_{1}z_{1}=0$. Assuming that
$|g_{1\mu_{1}}|$ is bounded within a known constant, and $\alpha_{1}^{*}$ is a
function of $x_{1}$ and $x_{1d}$.

First, the system (40) can be solved using NNs to approximate $\alpha_{i}^{*}$
[97]. Semiglobal stability can be proved when the NN stays in some compact
sets which can be made arbitrarily large with a sufficiently large number of
NN nodes. Alternatively, input-to-state stability (ISS) is adopted to divide
the entire interconnected system into two subsystems with overall stability
guaranteed by the small-gain theorem [133]. It is proved that the first error
dynamic system with state $z$, inputs
$\text{col}(\tilde{W}_{1},\tilde{W}_{2},\cdots,\tilde{W}_{n})$ and
$\text{col}(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})$ is ISS.
Furthermore, the NN weight estimator with state
$\text{col}(\tilde{W}_{1},\tilde{W}_{2},\cdots,\tilde{W}_{n})$, inputs $z$ and
$\text{col}(\tilde{W}_{1}^{*},\tilde{W}_{2}^{*},\cdots,\tilde{W}_{n}^{*})$ is
ISS. Then the stability of the interconnected system is guaranteed.

Instead of approximation on all unknown, robustness-based methods can be
developed by assumed bounded uncertainties, e.g.,
$f_{i}(\bar{x}_{i},\alpha_{i}^{*})\leq\sigma_{i}\rho_{i}(\bar{x}_{i})\sum\limits_{j=1}^{i}|x_{j}|$.
A high-gain idea is used with dynamic control gains $\dot{c}_{i}$, and the
final stability is proved with Barbalat’s lemma [134]. However, a rapid
oscillation has been noticed in the control input. Another method with
simplified control law is the prescribed performance control [135]. The system
is bounded by predefined trajectories, where the convergence rate and
exhibiting maximum overshoot ensure preassigned performance. Without the
attempts to cancel all nonlinearity, the control law is similar to that using
BLF, i.e., $u=-c_{n}T_{f}(\frac{x_{n}-\alpha_{n-1}}{\rho_{n}(t)})$ and
$\alpha_{i}=-c_{i}T_{f}(\frac{x_{i}-\alpha_{i-1}}{\rho_{i}(t)})$ where
$\rho_{i}$ is a prescribed performance function and
$T_{f}(a)=\ln\frac{1+a}{1-a}$ with $a\in(0,1)$. The control gains are enlarged
near the performance boundary.

Singular perturbation theory is combined with backstepping design [136, 137].
The control signal is sought as a solution to the fast dynamics and is shown
to asymptotically stabilize the original non-affine system.

When using mean value theorem, the main drawbacks are the requirement of
positive $\frac{\partial f_{i}(\bar{x_{i}},x_{i+1})}{\partial x_{i+1}}$ and
the complexity for high-order system. DSC and minimal learning parameter
techniques are employed to release the requirement and achieve neutral
adaptive control to the system with nondifferential $f_{i}$ [138]. To enhance
the estimation performance, the increasing number of NN nodes results in a
significant number of adjustable parameters. The online learning time can be
very long. A possible solution to reduce the adjustable parameters is to
estimate $|W_{i}^{*}|^{2}$ with a single state $\theta$. Another shortage is
the circular construction problem, since the NN is used to approximate $u$
$\dot{u}$, and the desired virtual and practical controls $\alpha_{i}^{*}$ and
$u^{*}$, simultaneously [132].

####  3.5.3 Brunovsky norm form

System (38) is transferred to an affine strict-feedback form in Brunovsky norm
form by introduce low-pass process $\dot{u}=-au+v$, where $a>0$ [92]. Defining
states $s_{1}:=x_{1}=b_{1}(x_{1})$ and
$s_{i}=\dot{s}_{i-1}=:b_{i}(\bar{x}_{i})$, $i=2,\cdots,n+1$ yields,
$\dot{s}_{i}=b_{i+1}=\sum\limits_{j=1}^{i}\frac{\partial
b_{j}(\bar{x_{j}})}{\partial
x_{j}}\dot{x}_{j}=\sum\limits_{j=1}^{i-1}\frac{\partial
b_{j}(\bar{x}_{j})}{\partial x_{j}}f_{j}(\bar{x}_{j-1},x_{j})+\frac{\partial
b_{i}(\bar{x}_{i})}{\partial x_{i}}f_{i}(\bar{x}_{i-1},x_{i})$ and
$\frac{\partial b_{i+1}(\bar{x}_{i+1})}{\partial x_{i+1}}=\frac{\partial
b_{i}(\bar{x}_{i})}{\partial x_{i}}\frac{\partial
f_{i+1}(\bar{x}_{i},x_{i+1})}{\partial
x_{i+1}}=\prod_{j=1}^{i}g_{j}(\bar{x}_{j-1},x_{j})$ where
$g_{i}=\frac{\partial f_{i+1}(\bar{x}_{i},x_{i+1})}{\partial x_{i+1}}$ and
$g_{n}(\bar{x}_{n},u)=-a\prod_{j=1}^{n}g_{j}(\bar{x}_{j-1},x_{j})$ satisfy the
bounded $g_{i}$ assumption. Consequently,
$\dot{s}=As+B[f_{n}(\bar{x}_{n},u)+g_{n}(\bar{x}_{n},u)v]$, where
$s=[s_{1},\cdots,s_{n+1}]^{\top}$, $A=\begin{bmatrix}0_{n\times 1}&I_{n}\\\
0&0_{1\times n}\end{bmatrix}$, $B=[0_{1\times n},1]^{\top}$, and
$C=[1,0_{1\times n}]$. The final control laws have simpler forms if the linear
systems are used.

###  3.6 Event-triggered systems

In typical backstepping designs, the continuous control is adopted in the
design process and the system updates at discrete-time triggered instants
$t_{k}$, namely, time-triggered system. In contrast, the event-triggered
control, or event-based system, denotes a system updates aperiodically with
the pre-designed event-triggered condition. The execution of control tasks
update after the occurrence of an event, and the control input (or
measurements) is held between two consecutive updates (zero-order hold).
Normally, the monotonically increasing subsequence $t_{k}$ starts from
$t_{0}=0$. It is a more natural sampling way that of a human controller. The
strategy takes advantage of reducing network traffic loads and improving
resource utilization with minor control performance degradation. The event-
triggered problem can be categorized according to which part is event-
triggered.

(i) A major research focuses on the event-triggered control input. We assume
constant control input between two triggered instants, i.e., $u(t)=v(t_{k})$
for all $t\in[t_{k},t_{k+1})$. The design procedure is the same as the time-
triggered execution, except the last step.

The event-triggered conditions are designed as the specific error
$e_{et}:=u(t)-v(t_{k})$ is larger than a preset threshold function. The
thresholds can be fixed and state-dependent; listed as follows.

  * •

Fixed threshold strategy [139, 96]:

$t_{k+1}=\inf\\{t>t_{k}||e_{et}(t)|>\delta_{et}\\}$ (41)

  * •

State-dependent threshold 1 [140]:

$t_{k+1}=\inf\\{t>t_{k}||e_{et}(t)|>\kappa_{et}|u(t)|\\}$ (42)

  * •

State-dependent threshold 2 [141]:

$t_{k+1}=\inf\\{t>t_{k}||e_{et}(t)|>\kappa_{et}|u(t)|+\delta_{et}\\}$ (43)

where $\kappa_{et}\in(0,1)$ and $\delta_{et}>0$.

Removing the absolute operators and substituting them into $\dot{V}_{n}$,
$v(t)$ can be designed to be a function of the last virtual control
$\alpha_{i}$. For example, the control input under event-triggered condition
(43) becomes
$u(t)=\frac{v(t)}{1+\eta_{et,1}(t)\kappa_{et}}-\frac{\eta_{et,2}(t)\delta_{et}}{1+\eta_{et,1}(t)\kappa_{et}}$
where $|\eta_{et,1}|\leq 1$ and $|\eta_{et,2}|\leq 1$. Therefore, the an
unknown bounded external disturbance
$-\frac{\eta_{2}(t)\delta_{et}}{1+\eta_{1}(t)\kappa_{et}}$ is introduced to
state equation $\dot{x}_{n}=f_{n}(\bar{x}_{n})+g_{n}(\bar{x}_{n})u$. The
problem can be easily solved by the $\tanh(\cdot)$ function [141].

(ii) Alternatively, if the state-measurements are event-triggered, namely
event-sampled, the measurements are considered as a jump, i.e.,
$\hat{x}_{i}(t)=x_{i}(t_{k})$, for all $t\in[t_{k},t_{k+1})$. The error in the
event-triggered conditions then becomes $e_{et,i}(t):=x_{i}-\hat{x}_{i}$ for
event-triggered sampling. The jump can be estimated with the newly-defined
errors $e_{z_{i}}=z_{i}-\hat{z}_{i}$ [142]. To solve the problem by observer-
based approach, the weight updates of the NN and FLS between the trigger
instants are zero, i.e., $\dot{\hat{W}}_{i}=0$, for all $t\in(t_{k},t_{k+1})$
[143].

The nonlinearity introduced by the event-triggered mechanism is difficult to
be fully canceled; hence, only boundness is guaranteed in most studies. The
control should satisfy ISS conditions which are difficult to check concerning
the error. The high control input $u$ results in longer sampling intervals and
degrades the control performance. To improve the control precisely, a
switching threshold strategy is proposed based on the amplitude of tracking
error [139]. The tracking error converges to a domain. Exponential convergence
and finite-time stability are studied in [144].

Though the bandwidth is reduced for the input and state measurements,
continuously monitoring is required by the trigger mechanism, and it is
impossible for digital platforms. To further release the hardware resources,
periodic event-triggered control at fixed equidistant time instants is
proposed, in which the event-triggered function is evaluated periodically.
Borgers et. al.[145] redesigns the event function for an available continuous
ETC using convex overapproximation techniques.

###  3.7 Stochastic system

A practical system is normally influenced by the external environments; hence,
stochastic nonlinear systems arouse broad interests. According to Itô lemma,
stochastic different equations are adopted to present stochastic processes,
such as,

$dx=f(x,u)dt+h_{w}(x,u)dw(t),$ (44)

where $w\in\mathbb{R}^{r}$ is r-dimensional independent standard Wiener
process, $f(x,u):\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n}$ and
$h_{w}(x,u):\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n\times r}$ are
locally Lipschitz for all $t\geq 0$ satisfying $f(0,0)=0$ and $h_{w}(0,0)=0$.

Different from the disturbance $d_{i}$ in system (17) whose amplitude is
assumed to be bounded, the magnitude of the disturbance in a stochastic system
can be arbitrarily large in sufficiently long period. Hence, stabilities and
properties are defined in probability. Unlike deterministic systems, it is
impossible to prove the boundness. The equilibrium point $x(0)=0$ of (44) is
said to be

  1. 1.

Stable in probability if, for every $\varepsilon>0$ and $\delta>0$, there
exists an $r$ s.t. if $t>t_{0}$, $|x_{0}|<r$ and $i_{0}\in S$, then
$P\\{|x(t)|>\varepsilon\\}<\delta$;

  2. 2.

Asymptotically stable in probability if it is stable in probability and, for
each $\varepsilon>0$, $x\in\mathbb{R}^{n}$ and $i_{0}\in S$, there is
$\lim\limits_{t\to\infty}P\\{|x(t)|>\varepsilon\\}=0$;

  3. 3.

Bounded in probability if the random variable $|x(t)|$ are bounded in
probability uniformly in $t$, i.e.,
$\lim\limits_{t\to\infty}\sup\limits_{t>t_{0}}P\\{|x(t)|>R\\}=0$ [146].

The error state is defined as $dz_{1}=dx_{1}-dx_{1d}=dx_{1}-\dot{x}_{1d}dt$.
To prove the stability of a stochastic system, an infinitesimal generator for
a stochastic system is adopted, given by

$\mathcal{L}V(x)=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial
x}f+\frac{1}{2}\text{Tr}[h_{w}^{\top}\frac{\partial^{2}V}{\partial
x^{2}}h_{w}],$ (45)

where the higher-order Hessian term
$\frac{1}{2}\text{Tr}[h_{w}^{\top}\frac{\partial^{2}V}{\partial x^{2}}h_{w}]$
is due to the stochastic noises and $\text{Tr}(\cdot)$ is the trace operator.
To adjust the newly-involved term $\frac{\partial^{2}V}{\partial x^{2}}$,
quartic Lyapunov functions are always chosen, e.g., $\frac{1}{4}z_{i}^{4}$ and
$\frac{1}{4}\log\left(\frac{k_{bi}^{4}}{k_{bi}^{4}-z_{i}^{4}}\right)$. While
the functions accounting for the errors of parameter estimates are still
quadratic, e.g., $\frac{1}{2}\tilde{\theta}_{i}^{\top}\tilde{\theta}_{i}$.
Consequently, $z_{1}^{2}$ appears in $\mathcal{L}V$. Applying Young’s
inequality generates a term with $z_{1}^{4}$ which can be compensated by the
virtual control law $\alpha_{i}$.

Suppose a positive definite, radically unbounded, twice continuously
differentiable function $V$ exists s.t. $\mathcal{L}V(x)$ is negative
definite, then the equilibrium $x=0$ is globally asymptotically stable in
probability [147]. Additionally, the solution of the system, similar to Lemma
1, is bounded in probability suppose there exist two constants $\gamma>0$ and
$\delta>0$ such that $\mathcal{L}V(x)\leq-\gamma V(x,t)+\delta$, for all
$x\in\mathbb{R}^{n}$ and $t>t_{0}$ and $E[V(x,t)]\leq V(x_{0})e^{-\gamma
t}+\frac{\delta}{\gamma}$ [148]. If Nussbaum-type function is adopted,
$\mathcal{L}V(x)\leq-\gamma
V(x,t)+\frac{1}{c}(g_{n}\mu_{1}\mathcal{N}(\chi)+1)\dot{\chi}\delta$ results
in $E[V(x,t)]\leq V(x_{0})e^{-\gamma
t}+\frac{\delta}{\gamma}+\frac{\sigma}{c}$ where
$\sigma=\sup\int_{0}^{t}|E(g_{n}\mu_{1}\mathcal{N}(\chi)+1)\dot{\chi}e^{\gamma\tau}|d\tau$
[149]. The finite-time stability is guaranteed by
$\mathcal{L}V+cV^{\alpha}(x)\leq 0$ [150].

##  4 Discussion and conclusion

Backstepping design approaches have been significantly developed since the
first appearance. Numerous modifications and innovations are proposed to
achieve recursive Lyapunov-based controller design to increasingly complex and
uncertain systems. This survey reviews the state-of-the-art development of
several widely applied elegant methods and adaptive control using
backstepping-like approaches to various nonlinearities are presented. The
complex nonlinear systems are transferred into the integration of a number of
simple problems, and each sub-problem is solved by the corresponding
assumptions and approaches. To sum up, backstepping, together with its
extensions, is a systematic and modularized control design approach.

Cancellation is the keyword in the backstepping-like methodology. The
nonlinearities result in weak convergence, undesired inaccuracy and
oscillation, and even instability. To cancel the uncertain and unknown parts,
each method relays on some specific assumptions on boundness and parametric
separation. Increasingly restrictive assumptions are needed for higher
nonlinearities. The Lyapunov function for each step can be a sum of several
Lyapunov functions for specific complexities, including tracking errors,
approximation errors, smoothness errors, etc. When the appropriate assumptions
and Lyapunov functions are invoked, the virtual control laws, adaptive update
laws, and final control input can be designed to compensate the state-relevant
terms in the time derivative of the LFC.

In practical applications, there are many systems other than the strict-
feedback form. When the simplifications are not satisfied, nonlinearities and
complexities appear. Since the system uncertainties and complexities are not
perfectly known, the Lyapunov stability criteria in a form of Lemma 1 play
important roles in the recursive design procedures and stability proofs.
Noncancelable positive parts are moved to $\delta$ in (1) and $\gamma$ is
chosen as the minimum value of groups of control and update gains. As to
exactly cancel all unknown terms is impossible, boundness is always guaranteed
rather than the asymptotic or exponential stabilities at the equilibrium
point. With growing complexities and uncertainties, the magnitude of $\delta$
in (1) increases with a sum of positive components. The control gains have to
be sufficiently large to ensure a satisfying tracking performance. However,
the real value of $\delta$ is probably much less than $\delta$ we received,
resulting in a conservative design and a waste of actuator capability.

Most developments belong to two foremost approaches, i.e., robustness-based
approach and estimation-based approach.

  * •

The main idea of the robustness-based method is to largely compensate all
nonlinearities by some parameter-separation assumptions, combing with
$\dot{V}=-\gamma V+\delta$ in Lemma 1. The positive terms, which cannot be
canceled but bounded, are accumulated in the final $\delta$. Then, the effects
of system uncertainties can be reduced with high control gains and adaptive
coefficients. However, such aggressive control would require expensive
hardware, and cause instability. Hence, the value of $\gamma$ must be enhanced
to achieve similar performance. Consequently, the control and adjust laws are
growingly aggressive with the increasing system dimension $n$. To enhance the
adaption speed, large gains in the adaptive update laws may result in
instabilities.

  * •

The approximation-based approach employs estimators based on the existed
measurements. All nonlinearities are assumed to be approximated. Increasing
the order of the estimator is the solution to increasingly complex
nonlinearities. Suppose that the unknown parts are well estimated, the
nonlinearities can be canceled. However, exosystem, NN, and FLS all fail to
estimate the high-frequency uncertainties. Though the NN/FLS-based approaches
have high flexibility to approximate all sorts of bounded smooth functions,
the learning process is time-consuming.

The primary drawback of the backstepping approach is the complexity of the
controller design and stability analysis, which is caused by increasing the
system dimension and repeated differentiation of the virtual control laws.
Overparameterization is still a very important problem. The case studies in
most works are concentrated on the system with a degree equal and less than
three. Estimation-based approaches, such as NN, and adaptive control laws
introduce significant amounts of states to the entire control system,
consequently, influencing the computation speed, learning time, and reaction
speed. Attempts have been made to overcome the overparameterization problem
during backstepping design procedures, e.g., tuning function, dynamic surface
control, and command filtered backstepping.

The unknown terms are assumed to be either bounded or a product of unknown
constant vectors and known functions. Systematic methods to find
$\phi_{i}(\bar{x}_{i})$ is still challenging. Though the appearance of
$\phi_{i}(\bar{x}_{i})$ can be received with careful tuning, it is difficult
to ensure the proposed functions fit all the uncertainties. Additionally, the
design procedures would be cumbersome and the computational time boosts if too
many unnecessary components invoked. Approaches with lower gains in control
laws and adjusting laws will be valuable.

Another phenomenon of studies on backstepping is the combination-based
innovations. Recalling the memory of cocktail and mixology, the recipe gets a
new name with various mixtures of ingredients. However, they are still
beverages without distinction in the cooking methods. This is similar to
today’s publications on backstepping design. Some elegant methods are
developed. Notwithstanding, most studies are just never-appeared-before
combinations of several complexities using an integration of existing
approaches that handle the corresponding nonlinearities. On the other hand,
this is a preferred feature by the control designers, which makes the design
modularized.

Backstepping has shown a great promise in numerous applications since initial
development. Novel methods have extended its applications to increasingly
complex systems. We believe that it will be applied to new future challenges
by the involvement of theoretical innovations in relevant realms.

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