Non-overshooting sliding mode for UAV control

Xinhua Wang 1, and Xuerui Mao 2

1 University of Nottingham, Aerospace Engineering, Nottingham, United Kingdom

Email<EMAIL_ADDRESS>
2 Beijing Institute of Technology, Advanced Research Institute of
Multidisciplinary Sciences, Beijing, China

_Abstract:_ For a class of uncertain systems, a non-overshooting sliding mode
control is presented to make them globally exponentially stable and without
overshoot. Even when the unknown stochastic disturbance exists, and the time-
variant reference trajectory is required, the strict non-overshooting
stabilization is still achieved. The control law design is based on a desired
second-order sliding mode (2-sliding mode), which successively includes two
bounded-gain subsystems. Non-overshooting stability requires that the system
gains depend on the initial values of system variables. In order to obtain the
global non-overshooting stability, the first subsystem with non-overshooting
reachability compresses the initial values of the second subsystem to a given
bounded range. By partitioning these initial values, the bounded system gains
are determined to satisfy the robust non-overshooting stability. In order to
reject the chattering in the controller output, a tanh-function-based sliding
mode is developed for the design of smoothed non-overshooting controller. The
proposed method is applied to a UAV trajectory tracking when the disturbances
and uncertainties exist. The control laws are designed to implement the non-
overshooting stabilization in position and attitude. Finally, the
effectiveness of the proposed method is demonstrated by the flying tests.

_Keywords:_ Non-overshooting sliding mode; global non-overshooting stability;
non-overshooting reachability; smoothed non-overshooting controller; UAV
trajectory tracking

Nomenclature

${\small e(t)}$ | system variable of error form
---|---
${\small e}^{(n)}{\small(t)}$ | the ${\small n}$-th derivative of ${\small e(t)}$
${\small f(\cdot)}$ | system function
${\small g(\cdot)}$ | system function
${\small k}_{p}$ | proportional gain of PID controller
${\small k}_{i}$ | integral gain of PID controller
${\small k}_{d}$ | derivative gain of PID controller
${\small e}_{1}{\small(t)}$ | sliding variable or error variable
${\small e}_{2}{\small(t)}$ | sliding variable or error variable
${\small k}_{1}$ | parameter of sliding mode or controller parameter
${\small k}_{2}$ | parameter of sliding mode or controller parameter
${\small\rho}$ | parameter of sliding mode or controller parameter
${\small d(t)}$ | disturbance or system uncertainty
${\small L}_{d}$ | upper bound of ${\small d(t)}$
${\small\sigma}{\small(t)}$ | sliding function
${\small x}_{1}{\small(t)}$ | system state
${\small x}_{2}{\small(t)}$ | system state

${\small u(t)}$ | controller
---|---
${\small\delta}{\small(t)}$ | system uncertainty or disturbance

${\small x}_{d}{\small(t)}$ | reference
---|---
${\small\dot{x}}_{d}{\small(t)}$ | derivative of reference
${\small m}$ | mass of UAV
${\small g}$ | gravity of acceleration
${\small l}$ | rotor distance to gravity center
${\small J}_{\phi}$ | moment of inertia about roll
${\small x}$ | position in ${\small x}$ direction
${\small y}$ | position in ${\small y}$ direction
${\small z}$ | position in ${\small z}$ direction
${\small\phi}$ | roll angle
${\small\theta}$ | pitch angle
${\small\psi}$ | yaw angle
${\small J}_{\theta}$ | moment of inertia about pitch
${\small J}_{\psi}$ | moment of inertia about yaw
${\small b}$ | rotor force coefficient
${\small k}$ | rotor torque coefficient
${\small F}_{i}$ | thrust force by rotor ${\small i}$
${\small Q}_{i}$ | reactive torque of rotor ${\small i}$
${\small k}_{x}$ | drag coefficients of UAV in ${\small x}$ direction
${\small k}_{y}$ | drag coefficients of UAV in ${\small y}$ direction
${\small k}_{z}$ | drag coefficients of UAV in ${\small z}$ direction
${\small k}_{\phi}$ | drag coefficients of UAV about roll
${\small k}_{\theta}$ | drag coefficients of UAV about pitch
${\small k}_{\psi}$ | drag coefficients of UAV about yaw
${\small\Delta}_{x}$ | uncertainty in ${\small x}$ direction
${\small\Delta}_{y}$ | uncertainty in ${\small y}$ direction
${\small\Delta}_{z}$ | uncertainty in ${\small z}$ direction
${\small\Delta}_{\phi}$ | uncertainty about roll
${\small\Delta}_{\theta}$ | uncertainty about pitch
${\small\Delta}_{\psi}$ | uncertainty about yaw

##  1 Introduction

This paper considers robust non-overshooting stabilization for a class of
dynamical systems with stochastic disturbance and application to UAV flight
control. Control without overshoot is very important for many industrial
control systems [1, 2, 3], for example, aircraft safe landing, automated
vehicle safety control, and manufacturing process control, etc. In a control
system, overshoot makes the actual behavior exceed its target, and it may
bring the devastating results. Therefore, in order to guarantee the safety
control, a known reference needs to be tracked without overshoot, i.e., non-
overshooting stabilization is required. For control systems, in addition to
reduce the overshoot and oscillations in the system outputs, the effect from
the disturbance or uncertainty also needs to be avoided. Furthermore, a
smoothed control law is helpful to improve the system response and reduce the
actuator chattering.

PID control is very popular for many industrial control systems because of its
simplicity and its acceptable control performance [4, 5]. However, PID control
has some disadvantages, for example: sensitive to the disturbance and
uncertainty types, adverse effect by the time-variant references, overshoot
existence because of integral windup or integration saturation. Theoretically,
PID control can completely reject the effect from an unknown constant
disturbance because of the integration term. If the disturbance is slowly
time-varying, through increasing the control gains, the disturbance effect can
be reduced to some extent. For a fast time-varying disturbance or nonlinear
system uncertainty, PID control performance is affected obviously. In
addition, PID control is usually used to control a system with step reference
for good performance. If the time-variant reference is required, the
feedforward term including reference derivatives information should be added
in the PID controller. For PID or PI control, the large initial error may
bring the phenomenon of integral windup, and a relatively long-time overshoot
exits in the system output. There are some methods to reduce overshoot in
system output: the pole-placement or pole-zero configuration methods [6, 7],
the optimization approach to minimize overshoot [8], the compensation-based
method [9, 10], the characteristic ratio assignment method [11], the iterative
technique based on gradient descent-like procedure [12], the eigenvector
placement technique to construct an invariant set [13]. They are mainly used
for the step reference non-overshooting tracking in the linear systems.
However, they still cannot overcome the effect from the time-varying
disturbances or nonlinear system uncertainties, and these time-varying
disturbances may bring overshoot or oscillations in the system outputs.

Sliding mode is also widely used in many industrial applications due to its
strong robustness against the bounded stochastic disturbances or
uncertainties, especially for aircraft navigation and control: the robust
controller [14], the robust observer [15], and the signal corrector [16].
However, chattering in sliding mode affects its output performance, and
overshoot is inevitable for the usual sliding mode control systems. Some
methods were proposed to reduce overshoot and chattering, and keep the
robustness property of sliding mode control [17, 18, 19, 20]. In [17], for the
systems with the parameter uncertainties and the matched disturbances, an
adaptive sliding mode was designed to create the non-overshooting responses
over the selected output variables. In [18], a cascade sliding mode-PID
control was presented to get the non-overshooting time responses for a step
reference. In [19, 20], for a linear time-invariant systems with the matched
disturbance, the integral sliding mode technique along with the Moore’s
eigenstructure assignment was used to make the system stable with non-
overshooting behavior. For the above methods, not only the upper bound of
disturbance should be known, but also the upper bound information of
disturbance derivatives is required. For some cases, the non-overshooting
controllers with the observers were presented for the systems with the
disturbances [21, 22]. The observers were designed to estimate the
disturbances, and the estimations were used for the non-overshooting
stabilization. However, only the step reference was considered, and the
disturbance was assumed to be constant. Furthermore, the overshoot may exist
in the estimation from the observer, and the non-overshooting control
performance is affected adversely.

Recent years, non-overshooting stabilization for the nonlinear systems with
disturbance was developed [23, 24, 25, 26]. In [23], a non-overshooting
control was presented for a class of nonlinear system under the condition that
the initial value of the system output was strictly required below the initial
value of the reference trajectory. The approximately non-overshooting
performance was achieved by appropriately choosing the control gains under the
deterministic disturbances. In [24], a controller was designed for a class of
nonlinear systems to make the mean of the system output asymptotically track a
given trajectory without overshoot, i.e., the mean-nonovershooting tracking
was achieved. Furthermore, the initial value of the system output has the same
constraint as that in [23]. In [25, 26], the non-overshooting stabilizers were
designed for a class of nonlinear systems that were input-output linearizable
with a full relative degree, and the matched disturbances were considered. The
bounded measurable disturbances should satisfy the two inequalities including
the initial values of variables, and it was required to be continuous in the
system variables [26]. In addition, the above non-overshooting control methods
are locally stable.

In this paper, for a class of uncertain systems, a method of globally
exponentially stable control without overshoot is proposed. The design of
control law is based on a global non-overshooting 2-sliding mode of error
form. Non-overshooting stability requires that the system gains depend on the
initial values of system variables, making the system locally stable. In order
to achieve the global non-overshooting stability and avoid the excessively
large system gains, the 2-sliding mode consists of two bounded-gain
subsystems. The implementation of global non-overshooting stability is through
the successive connection of the first subsystem with non-overshooting
reachability and the second subsystem with locally non-overshooting stability.
The first subsystem enables the sliding variables to reach a given bounded
range without overshoot within a finite time. The boundary of this range
serves as the initial values for the second subsystem. Through partitioning
these initial values, the sliding variables are analytically expressed, and in
each zone, the bounded system parameters are determined to achieve the non-
overshooting stability. For the second subsystem, the sliding variables are
attracted without overshoot onto a non-overshooting sliding surface (i.e., the
sliding variables are made to satisfy a linear non-overshooting convergence
law). Thus, the sliding variables converge exponentially to zero, and there is
no overshoot for the first sliding variable. The influence of the bounded
stochastic disturbance or uncertainty is rejected completely due to the
sliding mode gain coverage. The disturbance or uncertainty is only required to
be bounded. Also, the 2-sliding mode can separate the measurement noise from
the sliding variables, and it makes the variables smoothed. To implement the
trajectory non-overshooting tracking, it is only required that the second-
order derivative of reference trajectory is bounded. In order to reject the
chattering in the controller output, a tanh-function-based sliding mode is
developed to get the smoothed non-overshooting controller.

The rationality of the proposed non-overshooting sliding mode controller in
this paper incudes: 1) The structure of the two successive subsystems, which
have the non-overshooting reachability and the non-overshooting stability
respectively, can achieve global non-overshooting stability; 2) The initial
value partitioning of the second subsystem is to determine the analytical
expressions for the sliding variables, so that the conditions on non-
overshooting stability can be obtained; 3) The sliding mode gains can be
assigned to completely eliminate the influence of bounded stochastic
disturbance. The advantages of the proposed method are: 1) the robust and
global non-overshooting stability even in the presence of bounded stochastic
disturbance; 2) no restriction on the system initial values; 3) the bounded
and smoothed controller to be easily performed by the actuators.

For flight control, for example, spacecraft, hypersonic vehicle or UAV
control, sliding mode control plays an important role [27,28,29]. However,
overshoot or oscillations exist for the traditional sliding mode control
methods. In the high-speed or high-maneuverability flight conditions,
overshoot or oscillations in attitude control may affect the flight
performance and cause the safety issues. In fact, at a supersonic flying
speed, the angles of attack of the aerodynamic surfaces are usually very
small. The overshoot of the system output will cause these angles of attack
large and irregular, resulting in the unstable flight. Therefore, the non-
overshooting control is necessary for these aircrafts to implement the safe
and maneuvering flight. The robust non-overshooting control proposed in this
paper can overcome the overshoot issue, and the strict non-overshooting
stability can be achieved even in the presence of bounded stochastic
disturbance or system uncertainty. Importantly, the bounded-gain and smoothed
sliding controller is fit for many actuators.

The proposed method is applied to a UAV non-overshooting control. In the UAV
flight test, some adverse situations are considered: only a simple model is
constructed; the system uncertainties and the bounded unknown stochastic
disturbances exist; noise is in the measurements of position and attitude; and
the reference includes the multi-segment trajectory. The control laws based on
the non-overshooting sliding mode are designed to drive the UAV to achieve the
flight mission. The control system can implement the agile and non-
overshooting tracking for the complex reference trajectory. Furthermore, even
when the reference suddenly changes or jumps, i.e., the reference trajectory
is discontinuous, the controller parameters can be updated to keep the non-
overshooting stabilization through the parameter regulation conditions.

Compared with the research results in the existing relevant literature, the
contributions of this paper include: 1) Even with the presence of stochastic
disturbances and the requirement of time-variant reference trajectories, the
strict non-overshooting stability can still be achieved; 2) The stability is
global, and the system gains are bounded; 3) The stochastic disturbances are
only required to be bounded; 4) There is no restriction on the initial values
of the systems; 5) Due to the filter-corrector property of the sliding mode
surface, the noise can be fully separated from the sliding variables, even
when the frequency bands of the variables and noise overlap; 6) The controller
output is bounded and is smoothed, and it is easily performed by the
actuators.

##  2 Problem description and analysis

The problem considered in this paper is for system safety control to implement
non-overshooting stabilization, even the unknown stochastic disturbance
exists, and the time-variant reference is required. Overshoot means that
signal passes over or exceeds its target, and it affects the safety control
adversely.

We know that, the control performance of a system is determined by its closed-
loop error system after a controller is selected. Equivalently speaking,
controller design is to determine a control law to turn the open-loop error
system into a desired stable system of error form. The control performance is
determined by the desired stable system. The controller is a connection
between the open-loop error system and the desired stable system.

Therefore, when it is difficult to design a controller to make a system stable
and without overshoot, we can construct a desired stable system with non-
overshooting and robust properties. Then, from the relation between the open-
loop error system and the desired stable system, the controller is solved.

A conclusion on controller design based on desired stable system is introduced
as follows.

_2.1 Controller design and desired stable system_

Conclusion 2.1 (_Controller design based on desired stable system_):

A dynamical system of error form is considered as follows:

$e^{\left(n\right)}\left(t\right)=g\left(e\left(t\right),\dot{e}\left(t\right),\cdots,e^{\left(n-1\right)}\left(t\right),t\right)+u\left(t\right)+d\left(t\right)$
(1)

where, $e\left(t\right)$ is the system error variable, and
$\dot{e}\left(t\right),\cdots,e^{\left(n\right)}\left(t\right)$ are the
derivatives of $e\left(t\right)$; $g\left(\cdot\right)$ is the known function;
$u\left(t\right)$ is the control input; and $d\left(t\right)$ is the unknown
disturbance or uncertainty of any kind. If the system

$e^{\left(n\right)}\left(t\right)=f\left(e\left(t\right),\dot{e}\left(t\right),\cdots,e^{\left(n-1\right)}\left(t\right),t\right)+d\left(t\right)$
(2)

is already stable even disturbance $d\left(t\right)$ exists, i.e.,

$\underset{t\rightarrow\infty}{\lim}e^{\left(i\right)}\left(t\right)=0\text{,
}i=0,1,\cdots,n-1$ (3)

where, $f\left(\cdot\right)$ is the known function, then, in order to make
system (1) stable, the controller can be selected as

$u\left(t\right)=f\left(e\left(t\right),\dot{e}\left(t\right),\cdots,e^{\left(n-1\right)}\left(t\right),t\right)-g\left(e\left(t\right),\dot{e}\left(t\right),\cdots,e^{\left(n-1\right)}\left(t\right),t\right)$
(4)

In fact, system (2) is already stable without control. If we can select
controller $u\left(t\right)$ to turn system (1) into system (2), then, the
system (1) will become stable. Because the left sides of (1) and (2) are same,
we just make the right sides of (1) and (2) equal, i.e.,

$g\left(e\left(t\right),\dot{e}\left(t\right),\cdots,e^{\left(n-1\right)}\left(t\right),t\right)+u\left(t\right)+d\left(t\right)=f\left(e\left(t\right),\dot{e}\left(t\right),\cdots,e^{\left(n-1\right)}\left(t\right),t\right)+d\left(t\right)$
(5)

Then, through solving the equality (5), the disturbance $d\left(t\right)$ is
canceled out, and we get the controller (4).

Remark 2.1: For system (1) with controller (4), system (2) determines its
stability, transient process and robustness. Therefore, for a dynamical
control system, we can use a desired stable system to determine the controller
and analyze the performance of control system. Importantly, for safety
control, if the desired stable system has the non-overshooting property, then
the dynamical control system has the same performance.

_2.2 PID/PI desired stable system_

PID/PI control is popular for many industrial control systems. Ideally, PID or
PI control can completely reject the effect of constant disturbance when
stabilizing a system. The desired stable system in PID form is a third-order
system, and the overshoot often happens. Especially, a large overshoot exists
due to the windup effect or the integration saturation. We have the following
two Lemmas.

Lemma 2.1 (_PID desired stable system with unknown constant disturbance_): For
system

$\ddot{e}\left(t\right)=-k_{p}e\left(t\right)-k_{i}\int_{0}^{t}e\left(\tau\right)d\tau-
k_{d}\dot{e}\left(t\right)+d$ (6)

where, $d$ is any unknown constant disturbance, if $k_{p}$, $k_{i}$ and
$k_{d}$ are selected to make the real parts of all the roots of the
characteristic equation $s^{3}+k_{d}s^{2}+k_{p}s+k_{i}=0$ negative, then,
system (6) is exponentially stable, and

$\underset{t\rightarrow\infty}{\lim}e\left(t\right)=0\text{,
}\underset{t\rightarrow\infty}{\lim}\dot{e}\left(t\right)=0\text{, and
}\underset{t\rightarrow\infty}{\lim}k_{i}\int_{0}^{t}e\left(\tau\right)d\tau=d$
(7)

The proof of Lemma 2.1 is presented in Appendix. $\blacksquare$

Remark 2.2 (_PID control for second-order systems_): Ideally, PID controller
is used for the control of a second-order system with unknown constant
disturbance (but not time-varying disturbance). For the second-order system of
error form

$\ddot{e}\left(t\right)=-g\left(t\right)-u\left(t\right)+d$ (8)

the system (6) is selected as the desired stable system. We select the right
sides of (6) and (8) equal, i.e.,

$-g\left(t\right)-u\left(t\right)+d=-k_{p}e\left(t\right)-k_{i}\int_{0}^{t}e\left(\tau\right)d\tau-
k_{d}\dot{e}\left(t\right)+d$ (9)

Then, solving the equality (9), the disturbance $d$ is canceled out, and we
get the controller:

$u\left(t\right)=k_{p}e\left(t\right)+k_{i}\int_{0}^{t}e\left(\tau\right)d\tau+k_{d}\dot{e}\left(t\right)-g\left(t\right)$
(10)

From (68), the closed-loop error system is a third-order system, the
combination of three roots of $s^{3}+k_{d}s^{2}+k_{p}s+k_{i}=0$ determines the
stability and transient process, and the overshooting response often happens.

For PID control, if the large initial error of system output variable exist,
then a relatively large long-time overshoot may happen due to the integrator
windup. PID control can only completely reject the effect of constant
disturbance and make $\underset{t\rightarrow\infty}{\lim}e\left(t\right)=0$
and $\underset{t\rightarrow\infty}{\lim}\dot{e}\left(t\right)=0$. In addition,
for a time-varying disturbance, PID control can only make the system
approximately stable. Even the parameter regulation methods are used to
implement the non-overshooting stabilization, the disturbance is still assumed
to be constant. If disturbance is fast time-varying, then the control
performance becomes worse, and the overshoot and oscillations deteriorate.

Lemma 2.2 (_PI desired stable system with unknown constant disturbance_): For
system

$\dot{e}\left(t\right)=-k_{p}e\left(t\right)-k_{i}\int_{0}^{t}e\left(\tau\right)d\tau+d$
(11)

where, $d$ is any unknown constant disturbance, if $k_{p}$ and $k_{i}$ are
selected to make the real parts of all the roots of the characteristic
equation $s^{2}+k_{p}s+k_{i}=0$ negative, then system is exponentially stable,
and

$\underset{t\rightarrow\infty}{\lim}e\left(t\right)=0\text{, and
}\underset{t\rightarrow\infty}{\lim}k_{i}\int_{0}^{t}e\left(\tau\right)d\tau=d$
(12)

The proof of Lemma 2.2 is presented in Appendix. $\blacksquare$

Remark 2.3: Ideally, PI controller is used for the control of a first-order
system with unknown constant disturbance. For the first-order system of error
form

$\dot{e}\left(t\right)=-g\left(t\right)-u\left(t\right)+d$ (13)

the system (11) is selected as the desired stable system. We select the right
sides of (11) and (13) equal, i.e.,

$-g\left(t\right)-u\left(t\right)+d=-k_{p}e\left(t\right)-k_{i}\int_{0}^{t}e\left(\tau\right)d\tau+d$
(14)

Then, solving the equality (14), the disturbance $d$ is canceled out, and we
get the controller:

$u\left(t\right)=k_{p}e\left(t\right)+k_{i}\int_{0}^{t}e\left(\tau\right)d\tau-g\left(t\right)$
(15)

From (74), the closed-loop error system is a second-order system, the
combination of two roots of $s^{2}+k_{p}s+k_{i}=0$ determines the stability
and transient process, and sometimes overshooting response happens.
Furthermore, due to the integral windup, the overshoot happens. Strictly, PI
control can only reject the effect of constant disturbance and make
$\underset{t\rightarrow\infty}{\lim}e\left(t\right)=0$. For a time-varying
disturbance, PI control can only make the system approximately stable.

_2.3 Performance metrics to evaluate the control methods_

In the view of the problems under consideration, here, we give the performance
metrics for evaluating the proposed control methods: 1) Global non-
overshooting stability; 2) Robustness against the bounded stochastic
disturbances; 3) Precision and maneuverability for reference trajectory
tracking; 4) Non restriction on the system initial values.

##  3 Robust non-overshooting 2-sliding mode

(a)

(b)

Figure 1: Configuration of globally non-overshooting 2-sliding mode. (a) Flow
chart of 2-sliding mode. (b) Convergence process of sliding variables.

_3.1 Configuration of robust non-overshooting 2-sliding mode with global
stability_

Before we present the design of non-overshooting control for a class of
uncertain systems, we create a desired non-overshooting 2-sliding mode system.
Non-overshooting stability requires that the system gain depends on the
initial values of the system variables. When the initial value amplitudes are
large, the system gain also becomes large. In order to achieve the global non-
overshooting stability and the bounded system gain, the 2-sliding mode
consists of two successive subsystems. The first subsystem with non-
overshooting reachability enables the sliding variables to reach a given
bounded range. This range boundary serves as the initial condition of the
second subsystem and makes the determined system gain bounded. The second
subsystem has the property of local non-overshooting stability, such that the
sliding variables converge to zero, and no overshoot exists for the first
sliding variable. The flow chart of 2-sliding mode is shown in Fig.1(a).

The configuration of non-overshooting 2-sliding mode with two subsystems is
expressed by

$\displaystyle\dot{e}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)$ $\displaystyle\dot{e}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}f_{1}\left[e_{1}\left(t\right),e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right]+d(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
f_{2}\left[e_{1}\left(t\right),e_{2}\left(t\right)\right]+d(t),\text{ if
}\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (18)

where, the gains of functions $f_{1}\left[\cdot\right]$ and
$f_{2}\left[\cdot\right]$ are bounded; $d(t)$ is the bounded disturbance;
$e_{1c}>0$ and $e_{2c}>0$ make the system gains bounded. The sliding variables
$e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ experience the following
convergence process:

$\left\\{\begin{array}[]{l}e_{1}\left(t\right)\text{ reaches
}e_{1c}\text{sign}\left(e_{1}\left(t\right)\right)\text{ without overshoot and
}e_{2}\left(t\right)\rightarrow-
e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\text{, if
}\left|e_{1}\left(t\right)\right|>e_{1c}\\\ e_{1}\left(t\right)\rightarrow
0\text{ without overshoot and }e_{2}\left(t\right)\rightarrow 0\text{, if
}\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (19)

The convergence process of sliding variables is shown in Fig.1(b):

1) The first subsystem (_with robust non-overshooting reachability_): its gain
is bounded; it makes $e_{1}\left(t\right)$ reach the given range
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$ without overshoot, and
$e_{2}\left(t\right)$ approaches to the assigned
$-e_{2c}$sign$\left(e_{1}\left(t\right)\right)$ which is opposite to
$e_{1}\left(t\right)$; the bounded disturbance $d(t)$ can be rejected
completely; and then the first subsystem switches to the second subsystem,
which it has the initial values $e_{1}\left(t_{c}\right)$ and
$e_{2}\left(t_{c}\right)$.

2) The second subsystem (_with robust non-overshooting stability_): its
bounded gain is determined from the initial values $e_{1}\left(t_{c}\right)$
and $e_{2}\left(t_{c}\right)$; the sliding variables $e_{1}\left(t\right)$ and
$e_{2}\left(t\right)$ converge to zero, and no overshoot exists for
$e_{1}\left(t\right)$; moreover, the bounded disturbance $d(t)$ can be
rejected completely.

Two Theorems on the explicit forms of robust non-overshooting 2-sliding mode
are presented as follows.

_3.2 Design of non-overshooting sliding mode system_

Theorem 3.1 (_Robust non-overshooting 2-sliding mode_): The 2-sliding mode
system is as follows:

$\displaystyle\dot{e}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)$ $\displaystyle\dot{e}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}-k_{c}\text{sign}\left[e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right]+d(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
-k_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]+d(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (22)

where, $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ are the sliding
variables; the bounded unknown disturbance $d(t)$ satisfies
$\sup_{t\in\left[0,\infty\right)}\left|d(t)\right|\leq L_{d}<\infty$;
$k_{c}>L_{d}$; $e_{1c}\in\left(0,k_{2M}-L_{d}\right)$,
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$, and
$k_{2M}>L_{d}$ is the up-bound of $k_{2}$ from the system gain limitation;
$e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial values
of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$; and

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left\\{\begin{array}[]{l}\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\infty\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(0,\infty\right),\text{ if others}\end{array}\right.$ (26) $\displaystyle
k_{2}$ $\displaystyle>$
$\displaystyle\left\\{\begin{array}[]{l}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$ (29)

Then, we get the linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ (i.e., sliding surface
$e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=0$), i.e., the system (18) is
globally exponentially stable, and

$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0\text{ and
}\underset{t\rightarrow\infty}{\lim}e_{2}\left(t\right)=0$ (30)

In addition, the convergence of variable $e_{1}\left(t\right)$ is non-
overshooting.

The proof of Theorem 3.1 is presented in Appendix. $\blacksquare$

Remark 3.1 (_Performance analysis of sliding mode (18) with conditions (19)
and (20)_):

_1) Globally exponential stability and no overshoot._ During the whole
transient process, both $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$
converge to zero, and no overshoot exists in $e_{1}\left(t\right)$:

i) Firstly, there exists a finite time $t_{c}>0$, for $t\geq t_{c}$, the first
subsystem enables $e_{1}\left(t\right)$ to reach
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$ without overshoot; meanwhile,
$e_{2}\left(t\right)$ gets to $e_{2c}$sign$\left(e_{1}\left(t\right)\right)$;
then, the first subsystem switches to the second subsystem.

ii) Secondly, for the second subsystem within the range
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$, from its initial values
$e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$, there exists a finite
time $t_{s}>0$, for $t\geq t_{c}+t_{s}$, the variables $e_{1}\left(t\right)$
and $e_{2}\left(t\right)$ are attracted without overshoot onto the sliding
surface $\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=0$,
and the sliding function $\sigma\left(t\right)$ and the sliding variable
$e_{1}\left(t\right)$ are non-overshooting during
$t\in\left[t_{c},t_{c}+t_{s}\right)$; because
$\dot{e}_{1}\left(t\right)=e_{2}\left(t\right)$, the linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ is achieved; therefore,
for $t\geq t_{c}+t_{s}$, $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ are
exponentially convergent, and no overshoot exists in $e_{1}\left(t\right)$.
Therefore, it is a non-overshooting convergence process.

_2) Complete rejection of bounded stochastic disturbance/uncertainty._
Disturbances or uncertainties exist in many dynamic systems. For example, in a
UAV flight, the aerodynamic disturbance exists from the crosswind, and the
influences of unmodelled dynamic uncertainties are not avoidable in modeling.
They perform to be stochastic and bounded. Therefore, we can define the
disturbance or uncertainty $d(t)$ to satisfy
$\sup_{t\in\left[0,\infty\right)}\left|d(t)\right|\leq L_{d}<\infty$. For the
sliding mode (18), even the bounded disturbance $d\left(t\right)$ exists, the
strict non-overshooting and exponential stability is still achieved, and
$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0$ and
$\underset{t\rightarrow\infty}{\lim}e_{2}\left(t\right)=0$. In fact, from
(86)$\sim$(88) in the proof of Theorem 3.1, due to $k_{c}>L_{d}$ and
$k_{2}>L_{d}$ in the sliding mode, the influence of bounded disturbances or
uncertainties can be completely rejected. Furthermore, for this sliding mode,
the condition on the stochastic disturbance $d\left(t\right)$ is relax, and
$d\left(t\right)$ is only required to be bounded.

_3) Bounded gain of the sliding mode._ From (20), the selection of
$e_{1c}\in\left(0,k_{2M}-L_{d}\right)$ (i.e.,
$e_{1c}=\left|e_{1}\left(t_{c}\right)\right|$) and
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$ (i.e.,
$e_{2c}=\left|e_{2}\left(t_{c}\right)\right|$) makes
$\left|e_{1}\left(t_{c}\right)\right|$, $\left|e_{2}\left(t_{c}\right)\right|$
and
$\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}$
all bounded. Therefore, $k_{2}$ is bounded, and the non-overshooting
performance is guaranteed.

_4) Chattering phenomenon in the sliding variable_ $e_{2}\left(t\right)$_._
For the sliding mode (18), due to the switching functions in the
$e_{2}\left(t\right)$ dynamic equation, chattering may exist in
$e_{2}\left(t\right)$.

Remark 3.2 (_Parameter conditions that cause slow convergence_): For the
selection of $k_{1}$ and $k_{2}$, we can use the partitioning
$e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0$ and
$e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)\geq 0$, and we get

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left\\{\begin{array}[]{l}\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0;\\\
\left(0,\infty\right),\text{ if others}\end{array}\right.$ (33) $\displaystyle
k_{2}$ $\displaystyle>$
$\displaystyle\left\\{\begin{array}[]{l}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0;\\\
\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$ (36)

Thus, the system (18) is also globally exponentially stable, and no overshoot
exists in $e_{1}\left(t\right)$.

However, for the second subsystem, when the initial values
$e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ of sliding variables
are in the zones II-1 and IV-1, i.e., in
{$\left(e_{1}\left(t_{c}\right),e_{2}\left(t_{c}\right)\right)|e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\
$and
$\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|$},
the slow convergence exists for sliding mode (18) with conditions (22) and
(23), but it does not happen for sliding mode (18) with conditions (19) and
(20). In fact:

1) Slow convergence for sliding mode (18) with conditions (22) and (23): For
the second subsystem, when the initial values of sliding variables are in the
zones II-1 and IV-1, from (22), we know that
$k_{1}\in\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right)$.
$k_{1}\in\left(0,1\right)$ should be satisfied when
$\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|$.
Therefore, the $e_{1}\left(t\right)$ convergence may be slow due to the
convergence law $\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$.

2) Fast convergence for sliding mode (18) with conditions (19) and (20): For
the second subsystem, when the initial values of sliding variables are in the
zones II-1 and IV-1, from (19), we know that
$k_{1}\in\left(\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\infty\right)$.
Because
$\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|$,
we need to select $k_{1}\geq 1$ for the convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ to get a fast
convergence.

_3.3 Design of smoothed non-overshooting sliding mode system_

In order to avoid $e_{2}\left(t\right)$ chattering in the sliding mode, the
continuous functions are used in the sliding mode, and the following Theorem
is presented.

Theorem 3.2 (_Smoothed non-overshooting sliding mode_): The tanh-function-
based 2-sliding mode system is as follows:

$\displaystyle\dot{e}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)$ $\displaystyle\dot{e}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}-k_{c}\text{tanh}\left[\rho_{c}\left(e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right)\right]+d(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
-k_{2}\text{tanh}\left[\rho(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right))\right]+d(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (39)

where, $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ are the sliding
variables; the bounded unknown disturbance $d(t)$ satisfies
$\sup_{t\in\left[0,\infty\right)}\left|d(t)\right|\leq L_{d}<\infty$;
$e_{1c}\in\left(0,k_{2M}-L_{d}\right)$,
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$, and
$k_{2M}>L_{d}$ is the up-bound of $k_{2}$ from the system gain limitation;
function

$\text{tanh}\left(\rho\cdot x\right)=\frac{\text{e}^{\rho\cdot
x}-\text{e}^{-\rho\cdot x}}{\text{e}^{\rho\cdot x}+\text{e}^{-\rho\cdot
x}}=1-\frac{2}{\text{e}^{2\rho\cdot x}+1}$ (40)

$e_{1c}>0,\text{ }e_{2c}>0,\text{ }k_{c}>L_{d},\text{
}\rho_{c}\gg\frac{1}{2}\ln\frac{k_{c}+L_{d}}{k_{c}-L_{d}}$ (41)

$e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial values
of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$; and

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left\\{\begin{array}[]{l}\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\infty\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(0,\infty\right),\text{ if others}\end{array}\right.$ (45) $\displaystyle
k_{2}$ $\displaystyle>$
$\displaystyle\left\\{\begin{array}[]{l}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$ (48)

$\rho\gg\max\left\\{\frac{1}{2k_{1}},1\right\\}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$
(49)

and

$e_{2\max}=\max\left\\{\left|e_{2}\left(t_{c}\right)\right|,\frac{k_{1}}{3}\left[\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right]\right\\}$
(50)

Then:

(i) The effect of disturbances is rejected, and the variables
$e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ of system (24) are in the
bounds as follows:

$\underset{t\rightarrow\infty}{\lim}\left|e_{1}\left(t\right)\right|\leq\frac{1}{2\rho
k_{1}}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}\text{,
and
}\underset{t\rightarrow\infty}{\lim}\left|e_{2}\left(t\right)\right|\leq\frac{1}{\rho}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$
(51)

In addition, the convergence of variable $e_{1}\left(t\right)$ is non-
overshooting.

(ii) Specially, if $\rho$ is large enough, i.e., $\rho\rightarrow+\infty$,
then the system (24) becomes the ideal sliding mode (18), and we get

$\underset{t\rightarrow\infty}{\lim}\underset{\rho\rightarrow+\infty}{\lim}e_{1}\left(t\right)=0\text{
and
}\underset{t\rightarrow\infty}{\lim}\underset{\rho\rightarrow+\infty}{\lim}e_{2}\left(t\right)=0$
(52)

The proof of Theorem 3.2 is presented in Appendix. $\blacksquare$

Remark 3.3 (_Smoothed and non-overshooting convergence for sliding mode (24)
with (25)_ $\sim$_(30)_):

In addition to the non-overshooting convergence, the smoothed sliding mode
system (24) has the following properties:

1) Universal approximation: Because
$\underset{\rho\rightarrow+\infty}{\lim}$tanh$\left(\rho\cdot
x\right)=$sign$(x)$, the sliding mode system (24) is the smoothed
approximation of ideal sliding mode system (18).

2) Smoothed outputs of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$: Due to
continuity in the sliding mode system (24), the outputs of both
$e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ are smoothed. For
$-k_{c}$tanh$\left[\rho_{c}\left(e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right)\right]+d(t)$
in (24), the function sign$\left(e_{1}\left(t\right)\right)$ does not change
its sign due to continuity of $e_{1}\left(t\right)$ when
$\left|e_{1}\left(t\right)\right|>e_{1c}$. Therefore, no chattering happens.

Remark 3.4 (_Parameters determination for the sliding mode system (24)_)

For the algorithm calculation, we need to turn the inequality expressions of
the parameters into the corresponding equalities, and the calculated maximum
or minimum values of $k_{1}$ and $k_{2}$ are multiplied by the corresponding
coefficients. The determination steps of $e_{1c}$, $e_{2c}$, $k_{c}$,
$\rho_{c}$, $k_{1}$, $k_{2}$ and $\rho$ are presented as follows.

Step 1: Get the initial errors $e_{1}\left(0\right)$ and
$e_{2}\left(0\right)$.

Step 2: Select

$\displaystyle e_{1c}$ $\displaystyle\in$
$\displaystyle\left(0,k_{2M}-L_{d}\right),e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$
$\displaystyle k_{c}$ $\displaystyle>$ $\displaystyle L_{d},\text{
}\rho_{c}=\rho_{c0}\frac{1}{2}\ln\frac{k_{c}+L_{d}}{k_{c}-L_{d}}$ (53)

where, $k_{2M}>L_{d}$ is the up-bound of $k_{2}$ limited from the system gain;
and $\rho_{c0}>1$

Step 3: Determine the parameters $k_{1}$, $k_{2}$ and $\rho$ through the
following calculations.

$k_{1}=\left\\{\begin{array}[]{l}\beta_{11}\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\in\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\beta_{12}\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\in\left(\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\infty\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
\beta_{13}\in\left(0,\infty\right),\text{ if others}\end{array}\right.$ (54)

$k_{2}=\left\\{\begin{array}[]{l}\beta_{2}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\beta_{2}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$ (55)

where, $e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial
values of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$; $\beta_{11}\in(0,1)$,
$\beta_{12}>1$ and $\beta_{2}>1$.

_Adjustment of_ $k_{1}$_:_

$k_{1}=1\text{ is selected if the calculated }k_{1}\in\left(0,1\right)$ (56)

[_Note:_ If $k_{1}$ is calculated to be $k_{1}\in\left(0,1\right)$, we can
select $k_{1}=1$ for the convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ to get a fast
convergence. We find that $k_{1}=1$ always holds from the condition (27).]

For $\rho$, we select

$\rho=\rho_{0}\max\left\\{\frac{1}{2k_{1}},1\right\\}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$
(57)

where, $\rho_{0}>1$, and

$e_{2\max}=\max\left\\{\left|e_{2}\left(t_{c}\right)\right|,\frac{k_{1}}{3}\left[\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right]\right\\}$
(58)

##  4 Simulation examples on non-overshooting sliding mode

We use two examples to demonstrate the stability of the two non-overshooting
sliding mode systems.

For sliding modes (18) and (24), we suppose:

the initial sliding variables $e_{1}\left(0\right)=100$,
$e_{2}\left(0\right)=-10$;

the disturbance
$d\left(t\right)=3+2\sin\left(0.3t\right)\sin\left(1.6t\right)$, and its upper
bound $L_{d}=5$.

Suppose the system gain $k_{2}\leq k_{2M}=20$.

Figure 2: Example 4.1 Sliding variables $e_{1}\left(t\right)$ and
$e_{2}\left(t\right)$ of sliding mode (18).

Example 4.1: _(Sliding mode (18) from Theorem 3.1):_

From $e_{1c}\in\left(0,k_{2M}-L_{d}\right)=\left(0,15\right)$, we select
$e_{1c}=2$. Then, we get
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]=\left(2,\sqrt{\left(20-5\right)\times
2}\right]=\left(2,5.5\right]$. We select $e_{2c}=5$. Select $k_{c}=6>L_{d}$.

_Determination of sliding mode parameters_ $k_{1}$_and_ $k_{2}$_according to
the parameter determination steps (34)_ $\sim$_(36):_

$k_{1}=\left\\{\begin{array}[]{l}0.5\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
2.3\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
1\in\left(0,\infty\right),\text{ if others}\end{array}\right.$

where, $e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial
values of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$.

_Adjustment of_ $k_{1}$_:_ $k_{1}=1$ is selected if the calculated
$k_{1}\in\left(0,1\right)$; and

$k_{2}=\left\\{\begin{array}[]{l}1.5\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
1.5\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$

From the algorithm calculation for the above equations, we can obtain
$k_{1}=1.25$ and $k_{2}=16.93$.

Figure 2 shows the plots of sliding variables $e_{1}\left(t\right)$ and
$e_{2}\left(t\right)$. Even the time-varying disturbance exists, the non-
overshooting convergence is implemented: the sign of $e_{1}\left(t\right)$ is
always unchanged, and
$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0$ and
$\underset{t\rightarrow\infty}{\lim}e_{2}\left(t\right)=0$ hold. Also, we find
that, small chattering exists in $e_{2}\left(t\right)$.

Example 4.2: _(Smoothed sliding mode (24) from Theorem 3.2):_

Similar to Example 4.1, select $e_{1c}=2$, $e_{2c}=5$, and $k_{c}=6>L_{d}$.
Then, we get

$\rho_{c}=50\frac{1}{2}\ln\frac{k_{c}+L_{d}}{k_{c}-L_{d}}=59.95$

_Determination of sliding mode parameters_ $k_{1}$_,_ $k_{2}$ _and_ $\rho$
_according to the parameter determination steps (34)_ $\sim$_(38):_

Firstly, for determination of $k_{1}$ and $k_{2}$, we use the same algorithm
steps as example 3.1, and we can obtain $k_{1}=1.26$ and $k_{2}=16.95$.
Secondly, for $\rho$, we have

$e_{2\max}=\max\left\\{\left|e_{2}\left(t_{c}\right)\right|,\frac{k_{1}}{3}\left[\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right]\right\\}$

$\rho=20\max\left\\{\frac{1}{2k_{1}},1\right\\}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$

From the algorithm calculation for the above equations, we can read
$\rho=12.19$.

Figure 3: Example 4.2 Sliding variables $e_{1}\left(t\right)$ and
$e_{2}\left(t\right)$ of sliding mode (24).

Figure 3 describes the plots of sliding variables $e_{1}\left(t\right)$ and
$e_{2}\left(t\right)$. Even the time-varying disturbance exists, the sliding
variables $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ converge to zero,
and $e_{1}\left(t\right)$ convergence is non-overshooting. In addition, both
$e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ are smoothed.

##  5 Non-overshooting control for uncertain systems

_5.1 Model of uncertain systems_

The following uncertain system has a minimum number of states and inputs but
retains the essential features that must be considered when designing control
laws for many dynamical systems (e.g., UAV dynamics):

$\displaystyle\dot{x}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
x_{2}\left(t\right)$ $\displaystyle\dot{x}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle h(t)+u(t)-\delta\left(t\right)$ (59)

where, $x_{1}\left(t\right)$ and $x_{2}\left(t\right)$ the system states;
$h(t)$ is the known function; $u(t)$ is the control input; and
$\delta\left(t\right)$ is the unknown time-varying disturbance or system
uncertainty, and $\sup_{t\in\left[0,\infty\right)}\left|\delta(t)\right|\leq
L_{\delta}<\infty$. We consider to design a controller for the uncertain
system (39) when the reference is time variant, and $x_{1}\left(t\right)$
tracking the reference is required to be non-overshooting.

_5.2 Non-overshooting control for uncertain systems_

Theorem 5.1 (_Non-overshooting control based on sliding mode (18)_): For the
uncertain system (39) with the time-variant reference $x_{d}\left(t\right)$,
if the controller is selected as

$u(t)=\left\\{\begin{array}[]{l}k_{c}\text{sign}\left[e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c}\\\
k_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (60)

then, the system is globally exponentially stable, $x_{1}\left(t\right)$
tracking $x_{d}\left(t\right)$ is non-overshooting, and

$\underset{t\rightarrow\infty}{\lim}x_{1}\left(t\right)=x_{d}\left(t\right)\text{
and}\underset{t\rightarrow\infty}{\lim}x_{2}\left(t\right)=\dot{x}_{d}\left(t\right)$
(61)

where, $e_{1}\left(t\right)=x_{d}\left(t\right)-x_{1}\left(t\right)$ and
$e_{2}\left(t\right)=\dot{x}_{d}\left(t\right)-x_{2}\left(t\right)$;
$\sup_{t\in\left[0,\infty\right)}\left|\ddot{x}_{d}\left(t\right)\right|\leq
L_{x}<\infty$, and
$\sup_{t\in\left[0,\infty\right)}\left|\delta(t)\right|+\sup_{t\in\left[0,\infty\right)}\left|\ddot{x}_{d}\left(t\right)\right|\leq
L_{d}<\infty$; $k_{c}>L_{d}$; $e_{1c}\in\left(0,k_{2M}-L_{d}\right)$,
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$, and
$k_{2M}>L_{d}$ is the up-bound of $k_{2}$ from the system gain limitation;
$e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial values
of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$; and

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left\\{\begin{array}[]{l}\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\infty\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(0,\infty\right),\text{ if others}\end{array}\right.$ (65) $\displaystyle
k_{2}$ $\displaystyle>$
$\displaystyle\left\\{\begin{array}[]{l}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$ (68)

The proof of Theorem 5.1 is presented in Appendix. $\blacksquare$

Remark 5.1 (_Non-overshooting control (40)_):

_1) Non-overshooting of_ $x_{1}\left(t\right)$_tracking_
$x_{d}\left(t\right)$_:_ The sliding mode (18) is the desired stable system of
error form, therefore,
$e_{1}\left(t\right)=x_{d}\left(t\right)-x_{1}\left(t\right)$ converges to
zero without overshoot. Thus, $x_{1}\left(t\right)$ tracking
$x_{d}\left(t\right)$ is non-overshooting.

_2) Complete rejection of the influence from bounded disturbance and time-
variant reference:_ For the uncertain system (39), the disturbance $\delta(t)$
and the second-order derivative of reference $x_{d}(t)$ are bounded, and
$\sup_{t\in\left[0,\infty\right)}\left|\delta(t)\right|+\sup_{t\in\left[0,\infty\right)}\left|\ddot{x}_{d}(t)\right|\leq
L_{d}<\infty$ are satisfied. The parameters of controller (40) satisfy
$k_{c}>L_{d}$ and (43). When selecting the controller (40), from (173) and
(174) in the proof of Theorem 5.1, the closed-loop error system for (39) is
the robust non-overshooting 2-sliding mode (18). Therefore,
$\underset{t\rightarrow\infty}{\lim}x_{1}\left(t\right)=x_{d}\left(t\right)$
and
$\underset{t\rightarrow\infty}{\lim}x_{2}\left(t\right)=\dot{x}_{d}\left(t\right)$,
and there is no overshoot for $x_{1}\left(t\right)$ tracking
$x_{d}\left(t\right)$.

_3) Smoothed_ $x_{1}(t)$_:_ Due to the integral-chain structure of second-
order sliding mode (18), $e_{1}\left(t\right)$ is smoothed. Therefore, for the
control system, $x_{1}\left(t\right)$ is smoothed.

_4) Chattering in the variable_ $x_{2}\left(t\right)$_and the controller_
$u\left(t\right)$_:_ For system (39), due to the controller of switching
function exists in the $x_{2}\left(t\right)$ dynamic equation, chattering
happens in $x_{2}\left(t\right)$. Also, the controller $u\left(t\right)$ in
(40) is discontinuous, therefore, chattering exists in the controller output.
The chattering in controller output will affect actuator performance
adversely.

_5.3 Smoothed non-overshooting control for uncertain systems_

In order to reject chattering in the outputs of controller $u\left(t\right)$
and and variable $x_{2}\left(t\right)$, we present a smoothed control based on
the sliding mode (24), and a Theorem is presented as follows.

Theorem 5.2 (_Non-overshooting control based on smoothed sliding mode (24)_):
For the uncertain system (39) with the time-variant reference
$x_{d}\left(t\right)$, if the controller is selected as

$u(t)=\left\\{\begin{array}[]{l}k_{c}\text{tanh}\left[\rho_{c}\left(e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
k_{2}\text{tanh}\left[\rho(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right))\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (69)

then, the system is globally exponentially stable, $x_{1}\left(t\right)$
tracking $x_{d}\left(t\right)$ is non-overshooting, and

$\underset{t\rightarrow\infty}{\lim}\left|e_{1}\left(t\right)\right|\leq\frac{1}{2\rho
k_{1}}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}\text{,
and
}\underset{t\rightarrow\infty}{\lim}\left|e_{2}\left(t\right)\right|\leq\frac{1}{\rho}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$
(70)

where, $e_{1}\left(t\right)=x_{d}\left(t\right)-x_{1}\left(t\right)$ and
$e_{2}\left(t\right)=\dot{x}_{d}\left(t\right)-x_{2}\left(t\right)$;
$\sup_{t\in\left[0,\infty\right)}\left|\ddot{x}_{d}\left(t\right)\right|\leq
L_{x}<\infty$, and
$\sup_{t\in\left[0,\infty\right)}\left|\delta(t)\right|+\sup_{t\in\left[0,\infty\right)}\left|\ddot{x}_{d}\left(t\right)\right|\leq
L_{d}<\infty$;$e_{1c}\in\left(0,k_{2M}-L_{d}\right)$,
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$, and
$k_{2M}>L_{d}$ is the up-bound of $k_{2}$ from the system gain limitation;

$k_{c}>L_{d},\text{ }\rho_{c}\gg\frac{1}{2}\ln\frac{k_{c}+L_{d}}{k_{c}-L_{d}}$
(71)

$e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial values
of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$; and

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left\\{\begin{array}[]{l}\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\infty\right),\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
\left(0,\infty\right),\text{ if others}\end{array}\right.$ (75) $\displaystyle
k_{2}$ $\displaystyle>$
$\displaystyle\left\\{\begin{array}[]{l}\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$ (78)

$\rho\gg\max\left\\{\frac{1}{2k_{1}},1\right\\}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$
(79)

with

$e_{2\max}=\max\left\\{\left|e_{2}\left(t_{c}\right)\right|,\frac{k_{1}}{3}\left[\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right]\right\\}$
(80)

The proof of Theorem 5.2 is presented in Appendix. $\blacksquare$

Remark 5.2 (_Smoothed non-overshooting control (44)_):

1) Non-overshooting of $x_{1}\left(t\right)$ tracking $x_{d}\left(t\right)$:
The sliding mode (24) is the desired stable system of error form, therefore,
no overshoot happens in
$e_{1}\left(t\right)=x_{d}\left(t\right)-x_{1}\left(t\right)$. Thus,
$x_{1}\left(t\right)$ tracking $x_{d}\left(t\right)$ is non-overshooting.

2) Smoothed $x_{1}\left(t\right)$ and $x_{2}\left(t\right)$: Due to the use of
continuous functions in sliding mode (24), both $e_{1}\left(t\right)$ and
$e_{2}\left(t\right)$ are smoothed. Therefore, for the control system,
$x_{1}\left(t\right)$ and $x_{2}\left(t\right)$ are smoothed.

3) Smoothed controller $u(t)$: For system (39), the controller $u(t)$ in (44)
is smoothed, and it is fit for the implementation by many actuators.

Remark 5.3 (_Rejection of measurement noise_):

For system (39) with the controller (40) or (44), the measurement noise is
rejected on the sliding surface because of its filter-corrector property. Even
when the frequency bands of the variables and noise overlap, the outputs of
sliding surface are smoothed and are accurate.

In fact, suppose noise $n_{1}\left(t\right)$ and $n_{2}\left(t\right)$ exist
in the measurements of $x_{1}\left(t\right)$ and $x_{2}\left(t\right)$,
respectively. Then, for the sliding surface, we get

$\dot{x}_{d}\left(t\right)-x_{2}\left(t\right)-n_{2}\left(t\right)+k_{1}\left[x_{d}\left(t\right)-x_{1}\left(t\right)-n_{1}\left(t\right)\right]=0$
(81)

i.e.,

$e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=k_{1}n_{1}\left(t\right)+n_{2}\left(t\right)$
(82)

Define the Laplace transforms
$E_{1}\left(s\right)=L\left[e_{1}\left(t\right)\right]$,
$E_{2}\left(s\right)=L\left[e_{2}\left(t\right)\right]$,
$N_{1}\left(s\right)=L\left[n_{1}\left(t\right)\right]$, and
$N_{2}\left(s\right)=L\left[n_{2}\left(t\right)\right]$. Taking Laplace
transform for (52), we get

$sE_{1}\left(s\right)+k_{1}E_{1}\left(s\right)=k_{1}N_{1}\left(s\right)+N_{2}\left(s\right)$
(83)

i.e.,

$E_{1}\left(s\right)=\frac{k_{1}}{s+k_{1}}\left(N_{1}\left(s\right)+\frac{1}{k_{1}}N_{2}\left(s\right)\right)$
(84)

where, $\frac{k_{1}}{s+k_{1}}$ is the form of first-order filter, and $k_{1}$
is the cut-off frequency of the filter. For the filter, the input is the
noise, and the output is the system error. As long as $k_{1}$ is less than the
minimum frequency of the noise, the noise will be rejected. From $k_{1}$
selection conditions, $k_{1}$ can be neither too large nor too small, for
example, $k_{1}=1$ or $k_{1}=2$. Therefore, the noise
$N_{1}\left(s\right)+\frac{1}{k_{1}}N_{2}\left(s\right)$ is rejected
sufficiently, and the system error is reduced to be small enough.

Remark 5.4 (_Parameters regulation of the controller_):

1) The parameter selection conditions (42) and (43) (i.e., (47) and (48)) make
the system non-overshooting stable.

2) $k_{c}>L_{d}$. If $\left|e_{2}\left(0\right)\right|$ is large, $k_{c}$
should increase to reduce $\left|e_{2}\left(t\right)\right|$ effectively.

2) $k_{1}$ determines the convergence rate of linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$; from (54), $k_{1}$ also
determines the filtering frequency band of the sliding mode. Therefore,
$k_{1}$ should not be too small to keep a convergence rate of linear
convergence law; and $k_{1}$ should not be too large to get a suitable
frequency band for noise rejection.

3) $k_{2}$ determines the convergence rate of the second subsystem; $k_{2}$
also keeps the signs of sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$ and
sliding variable $e_{1}\left(t\right)$ unchanged for
$t\in\left[t_{c},\infty\right)$; also, $k_{1}$ affects the selection of
$k_{2}$.

4) Because $\left|e_{1}\left(t_{c}\right)\right|=e_{1c}$ and
$\left|e_{2}\left(t_{c}\right)\right|=e_{2c}$ when $t=t_{c}$, the selection of
$e_{1c}\in\left(0,k_{2M}-L_{d}\right)$ and
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$ makes
$\left|e_{1}\left(t_{c}\right)\right|$,
$\left|e_{2}\left(t_{c}\right)\right|$,
$\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}$
and
$\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}$
are all bounded. Then, the bounded $k_{1}$ and $k_{2}$ are determined from
(47) and (48).

5) The selection of $\rho$ from (49) affects the smoothness of system
variables and controller output. Also, from (45), $\rho$ affects the control
precision. Therefore, the selection of $\rho$ should balance the smoothness of
variables and controller and the precision of control performance.

Figure 4: Flow chart of non-overshooting controller design.

Remark 5.5 (_Steps on determination of non-overshooting controller_)

For the system (39), the flow chart of controller design is explained in Fig.
4. Furthermore, the the steps on controller determination are described as
follows.

Step 1: Measure the system initial states, and get the initial errors
$e_{1}\left(0\right)$ and $e_{2}\left(0\right)$.

Step 2: Select $e_{1c}\in\left(0,k_{2M}-L_{d}\right)$ and
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]$,
respectively, and

$k_{c}>L_{d},\text{
}\rho_{c}=\rho_{c0}\frac{1}{2}\ln\frac{k_{c}+L_{d}}{k_{c}-L_{d}}$ (85)

where, $k_{2M}>L_{d}$ is the up-bound of $k_{2}$ from the system gain
limitation; $\rho_{c0}>1$.

Step 3: Determine the controller parameters $k_{1}$, $k_{2}$ and $\rho$
through the calculations in Equations (34)$\sim$(38). [Note: When the known
reference jumps or suddenly changes, the parameters $k_{1}$, $k_{2}$ and
$\rho$ are updated]

Step 4: Controller output:

$u(t)=\left\\{\begin{array}[]{l}k_{c}\text{tanh}\left[\rho_{c}\left(e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
k_{2}\text{tanh}\left[\rho(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right))\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$ (86)

##  6 Simulation examples on non-overshooting control of uncertain systems

We use two examples to illustrate the non-overshooting control presented in
Theorems 5.1 and 5.2, respectively. Consider the uncertain system:

$\displaystyle\dot{x}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
x_{2}\left(t\right)$ $\displaystyle\dot{x}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle h(t)+u(t)-\delta\left(t\right)$

where, $h(t)=5x_{1}^{\frac{1}{3}}\sin\left(0.5t\right)$, and the unknown
disturbance or uncertainty $\delta\left(t\right)=1+0.3\sin(0.3t)\sin(1.6t)$.

The initial conditions of states: $x_{1}\left(0\right)=10$,
$x_{2}\left(0\right)=-1$

The reference: $x_{d}\left(t\right)=2+0.5\sin(0.8t)$. Therefore, we know that
$\dot{x}_{d}\left(t\right)=0.4\cos(0.8t)$ and
$\ddot{x}_{d}\left(t\right)=-0.32\sin(0.8t)$; $x_{d}\left(0\right)=2$, and
$\dot{x}_{d}\left(0\right)=0.4$.

The upper bound of disturbance/uncertainty:
$\sup_{t\in\left[0,\infty\right)}\left|\delta\left(t\right)\right|+\sup_{t\in\left[0,\infty\right)}\left|\ddot{x}_{d}\left(t\right)\right|=1.3+0.32=1.62$,
and we can select $L_{d}=1.62$.

Define system errors
$e_{1}\left(t\right)=x_{d}\left(t\right)-x_{1}\left(t\right)$ and
$e_{2}\left(t\right)=\dot{x}_{d}\left(t\right)-x_{2}\left(t\right)$. Then, the
error system is:

$\displaystyle\dot{e}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)$ $\displaystyle\dot{e}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle-h(t)-u(t)+\ddot{x}_{d}\left(t\right)+\delta\left(t\right)$ and
the initial errors are
$e_{1}\left(0\right)=x_{d}\left(0\right)-x_{1}\left(0\right)=-8$, and
$e_{2}\left(0\right)=\dot{x}_{d}\left(0\right)-x_{2}\left(0\right)=1.4$.

Suppose the system gain $k_{2}\leq k_{2M}=10$.

Example 6.1: _(Non-overshooting control from Theorem 5.1):_

_1) Selection of the desired stable error system_

The sliding mode (18) is selected as the desired stable error system, and

$\displaystyle\dot{e}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)$ $\displaystyle\dot{e}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}-k_{c}\text{sign}\left[e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right]+\ddot{x}_{d}\left(t\right)+\delta\left(t\right),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
-k_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]+\ddot{x}_{d}\left(t\right)+\delta\left(t\right),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$

_2) Determination of parameters_ $k_{1}$_and_ $k_{2}$

From $e_{1c}\in\left(0,k_{2M}-L_{d}\right)=\left(0,8.38\right)$, we select
$e_{1c}=1$. Then, we get
$e_{2c}\in\left(e_{1c},\sqrt{\left(k_{2M}-L_{d}\right)e_{1c}}\right]=\left(1,\sqrt{\left(10-1.62\right)\times
1}\right]=\left(1,2.9\right]$. We select $e_{2c}=2$. Select $k_{c}=2.5>L_{d}$.

According to the parameter determination steps (34)$\sim$(38), we get

$k_{1}=\left\\{\begin{array}[]{l}0.5\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
2.3\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|;\\\
2\in\left(0,\infty\right),\text{ if others}\end{array}\right.$

_Adjustment of_ $k_{1}$_:_ $k_{1}=1$ if the calculated
$k_{1}\in\left(0,1\right)$; and

$k_{2}=\left\\{\begin{array}[]{l}1.5\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}+L_{d}\right\\},\text{
if }e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0\text{ and
}\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|;\\\
1.5\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{k_{1}^{2}}{3}\left(\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right)+L_{d}\right\\},\text{
if others}\end{array}\right.$

where, $e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are the initial
values of $e_{1}\left(t\right)$ and $e_{2}\left(t\right)$ respectively when
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$. From the algorithm calculation
for the above equations, we can read $k_{1}=1$ and $k_{2}=5.44$.

_3) Controller design_

According to the controller (40), we get

$u(t)=\left\\{\begin{array}[]{l}k_{c}\text{sign}\left[e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c}\\\
k_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$

where, $e_{1c}=1$, $e_{2c}=2$, $k_{c}=2.5$, $k_{1}=1$, $k_{2}=5.44$, and
$h\left(t\right)=5x_{1}^{\frac{1}{3}}\sin\left(0.5t\right)$.

(a)

(b)

(c)

Figure 5: Example 6.1 Non-overshooting sliding mode control. (a) $x_{1}$. (b)
$x_{2}$. (c) Controller $u\left(t\right)$.

(a)

(b)

(c)

Figure 6: Example 6.2 Smoothed non-overshooting sliding mode control. (a)
$x_{1}$. (b) $x_{2}$. (c) Controller $u\left(t\right)$.

_4) Analysis of simulation results_

The control performance based on the ideal sliding mode is presented in Fig.
5\. Figure 5(a) describes the system variable $x_{1}$ tracking the reference
$x_{d}\left(t\right)$. $x_{1}$ tracking $x_{d}\left(t\right)$ is smoothed and
non-overshooting, even the unknown time-varying disturbance exists, and the
reference is also time variant. Figure 5(b) presents the variable $x_{2}$
convergence to the reference derivative $\dot{x}_{d}\left(t\right)$. At the
beginning, $x_{2}$ increases to speed up the finite-time convergence, then it
makes $x_{1}$ in the linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$, and
$\underset{t\rightarrow\infty}{\lim}x_{1}\left(t\right)=x_{d}\left(t\right)$
and
$\underset{t\rightarrow\infty}{\lim}x_{2}\left(t\right)=\dot{x}_{d}\left(t\right)$.
Figure 5(c) shows the controller $u\left(t\right)$ output. Even the controller
can make the system stable and non-overshooting, chattering happens in the
controller output, and it may increase actuator trembling.

Example 6.2: _(Smoothed non-overshooting control from Theorem 5.2):_

_1) Selection of the desired stable error system_

The smoothed sliding mode (24) is selected as the desired stable error system,
and

$\displaystyle\dot{e}_{1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)$ $\displaystyle\dot{e}_{2}\left(t\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l}-k_{c}\text{tanh}\left[\rho_{c}\left(e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right)\right]+\ddot{x}_{d}\left(t\right)+\delta\left(t\right),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
-k_{2}\text{tanh}\left[\rho(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right))\right]+\ddot{x}_{d}\left(t\right)+\delta\left(t\right),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$

_2) Determination of parameters_ $k_{1}$_,_ $k_{2}$_and_ $\rho$ _according to
the parameter determination steps (34)_ $\sim$_(38):_

Select $e_{1c}=1$, $e_{2c}=2$, and $k_{c}=2.5$. Then, we get
$\rho_{c}=20\frac{1}{2}\ln\frac{k_{c}+L_{d}}{k_{c}-L_{d}}=15.44$.

Firstly, for determination of $k_{1}$ and $k_{2}$, we use the same algorithm
steps to example 6.1, and we can read $k_{1}=1.01$ and $k_{2}=5.49$.

Secondly, for $\rho$, we have

$e_{2\max}=\max\left\\{\left|e_{2}\left(t_{c}\right)\right|,\frac{k_{1}}{3}\left[\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right]\right\\}$

$\rho=20\max\left\\{\frac{1}{2k_{1}},1\right\\}\ln\frac{k_{2}+k_{1}e_{2\max}+L_{d}}{k_{2}-k_{1}e_{2\max}-L_{d}}$

From the algorithm calculation for the above equations, we can read
$\rho=32.19$.

_3) Controller design_

According to the smoothed controller (44), we get

$u(t)=\left\\{\begin{array}[]{l}k_{c}\text{tanh}\left[\rho_{c}\left(e_{2}\left(t\right)+e_{2c}\text{sign}\left(e_{1}\left(t\right)\right)\right)\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|>e_{1c};\\\
k_{2}\text{tanh}\left[\rho(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right))\right]-h(t),\text{
if }\left|e_{1}\left(t\right)\right|\leq e_{1c}\end{array}\right.$

where, $e_{1c}=1$, $e_{2c}=2$, $k_{c}=2.5$, $\rho_{c}=15.44$, $k_{1}=1.01$,
$k_{2}=5.33$, $\rho=32.19$, and
$h\left(t\right)=5x_{1}^{\frac{1}{3}}\sin\left(0.5t\right)$.

_4) Analysis of simulation results_

Figure 6 presents the control performance based on the smoothed sliding mode.
Figure 6(a) describes $x_{1}$ tracking the reference $x_{d}\left(t\right)$,
and Figure 6(b) presents $x_{2}$ convergence to the reference derivative
$\dot{x}_{d}\left(t\right)$. Even time-varying disturbance exists, and time-
variant reference is required, $x_{1}$ tracking $x_{d}\left(t\right)$ is
smoothed and non-overshooting. Comparing to $x_{2}$ in Example 6.1, $x_{2}$ in
Example 6.2 is smoother. Figure 6(c) shows the smoothed controller
$u\left(t\right)$ output. The smoothed controller output is beneficial for
actuator implementation, and it reduce the actuator trembling.

##  7 UAV control application

A quadrotor UAV prototype is used [30], which is shown in Fig. 7, and the
forces and torques of UAV are described. The system parameters are introduced
in Table I.

Figure 7: Forces and torques in UAV [30].

Table I. UAV Parameters [30]

Symbol | Quantity | Value
---|---|---
$m$ | mass of UAV |  $2.01$kg
$g$ | gravity acceleration |  $9.81$m/s2
$l$ | rotor distance to gravity center |  $0.2$m
$J_{\phi}$ | moment of inertia about roll |  $0.25$kg$\cdot$m2
$J_{\theta}$ | moment of inertia about pitch |  $0.25$kg$\cdot$m2
$J_{\psi}$ | moment of inertia about yaw |  $0.5$kg$\cdot$m2
$b$ | rotor force coefficient | $2.923\times 10^{-3}$
$k$ | rotor torque coefficient | $5\times 10^{-4}$

_7.1 Model of UAV flight dynamics [30]_

The inertial and fuselage frames are denoted by
$\Xi_{g}=\left(E_{x},E_{y},E_{z}\right)$ and
$\Xi_{b}=\left(E_{x}^{b},E_{y}^{b},E_{z}^{b}\right)$, respectively; $\psi$,
$\theta$ and $\phi$ are the yaw, pitch and roll angles, respectively.
$F_{i}=b\omega_{i}^{2}$ is the thrust force by rotor $i$, and its reactive
torque is $Q_{i}=k\omega_{i}^{2}$. The sum of the four rotor thrusts is
$F=\sum\limits_{{i=1}}^{{4}}F_{i}$. The motion equations of the UAV flight
dynamics can be expressed by

$\displaystyle\dot{x}_{\ast 1}$ $\displaystyle=$ $\displaystyle x_{\ast 2}$
$\displaystyle\dot{x}_{\ast 2}$ $\displaystyle=$ $\displaystyle
h_{\ast}(t)+\bar{u}_{\ast}\left(t\right)+\delta_{\ast}(t)$ (91)

where, $\ast=x,y,z,\psi,\theta,\phi$; $x_{x1}=x$, $x_{y1}=y$, $x_{z1}=z$,
$x_{\psi 1}=\psi$, $x_{\theta 1}=\theta$, $x_{\phi 1}=\phi$; $h_{x}(t)=0$,
$h_{y}(t)=0$, $h_{z}(t)=-g$, $h_{\psi}(t)=0$, $h_{\theta}(t)=0$,
$h_{\phi}(t)=0$; $\delta_{x}(t)=m^{-1}(-k_{x}\dot{x}+\Delta_{x})$;
$\delta_{y}(t)=m^{-1}(-k_{y}\dot{y}+\Delta_{y})$;
$\delta_{z}(t)=m^{-1}(-k_{z}\dot{z}+\Delta_{z})$;
$\delta_{\psi}(t)=J_{\psi}^{-1}(-k_{\psi}\dot{\psi}+\Delta_{\psi})$;
$\delta_{\theta}(t)=J_{\theta}^{-1}(-lk_{\theta}\dot{\theta}+\Delta_{\theta})$;
$\delta_{\phi}(t)=J_{\phi}^{-1}(-lk_{\phi}\dot{\phi}+\Delta_{\phi})$; $k_{x}$,
$k_{y}$, $k_{z}$, $k_{\psi}$, $k_{\theta}$ and $k_{\phi}$ are the unknown drag
coefficients; $(\Delta_{x},\Delta_{y},\Delta_{z})$ and
$(\Delta_{\psi},\Delta_{\theta},\Delta_{\phi})$ are the uncertainties in
position and attitude dynamics, respectively;
$J=diag\\{J_{\psi},J_{\theta},J_{\phi}\\}$ is the matrix of three-axial moment
of inertias; $c_{\theta}$ and $s_{\theta}$ are expressed for $\cos\theta$ and
$\sin\theta$, respectively; and

$\displaystyle\bar{u}_{x}\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{x}\left(t\right)/m=(c_{\psi}s_{\theta}c_{\phi}+s_{\psi}s_{\phi})F/m$
$\displaystyle\bar{u}_{y}\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{y}\left(t\right)/m=(s_{\psi}s_{\theta}c_{\phi}-c_{\psi}s_{\phi})F/m$
$\displaystyle\bar{u}_{z}\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{z}\left(t\right)/m=c_{\theta}c_{\phi}F/m$
$\displaystyle\bar{u}_{\psi}\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{z}\left(t\right)/J_{\psi}=\frac{k}{b}\left(\sum\limits_{{i=1}}^{{4}}(-1)^{i+1}F_{i}\right)/J_{\psi}$
$\displaystyle\bar{u}_{\theta}\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{\theta}\left(t\right)/J_{\theta}=(F_{3}-F_{1})l/J_{\theta}$
$\displaystyle\bar{u}_{\phi}\left(t\right)$ $\displaystyle=$ $\displaystyle
u_{\phi}\left(t\right)/J_{\phi}=(F_{2}-F_{4})l/J_{\phi}$ (92)

_7.2 Measurements_

A Vicon system provides position and velocity, and a Doppler radar sensor
measures height and vertical velocity. An IMU gives the attitude angle and
angular velocity. The sensor outputs are:

$y_{\ast 1}\left(t\right)=x_{\ast 1};\text{ }y_{\ast 2}\left(t\right)=x_{\ast
2}$ (93)

where, $\ast=x,y,z,\psi,\theta,\phi$.

_7.3 Controller design_

In this section, the control laws are derived for UAV trajectory tracking and
attitude stabilization.

_1) Error systems_

The control laws are designed to stabilize the UAV flight. For the desired
trajectory ($x_{d}\left(t\right)$, $y_{d}\left(t\right)$,
$z_{d}\left(t\right)$) and attitude angle ($\psi_{d}\left(t\right)$,
$\theta_{d}\left(t\right)$, $\phi_{d}\left(t\right)$), the error systems of
position and attitude dynamics can be expressed, respectively, by

$\displaystyle\dot{e}_{\ast 1}\left(t\right)$ $\displaystyle=$
$\displaystyle\dot{e}_{\ast 2}\left(t\right)$ $\displaystyle\dot{e}_{\ast
2}\left(t\right)$ $\displaystyle=$ $\displaystyle-
h_{\ast}(t)-\bar{u}_{\ast}\left(t\right)+\ddot{\ast}_{d}\left(t\right)-\delta_{\ast}(t)$
(94)

where, $e_{\ast 1}\left(t\right)=\ast_{d}\left(t\right)-x_{\ast 1}$, $e_{\ast
2}\left(t\right)=\dot{\ast}_{d}\left(t\right)-x_{\ast 2}$;
$\ast=x,y,z,\psi,\theta,\phi$.

_2) Controller design_

From (24), we select the smoothed non-overshooting sliding mode as the desired
stable, i.e.,

$\displaystyle\dot{e}_{\ast 1}\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{\ast 2}\left(t\right)$ $\displaystyle\dot{e}_{\ast 2}\left(t\right)$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{l}-k_{\ast
c}\text{tanh}\left[\rho_{\ast c}\left(e_{\ast 2}\left(t\right)+e_{\ast
2c}\text{sign}\left(e_{\ast
1}\left(t\right)\right)\right)\right]+\ddot{\ast}_{d}\left(t\right)-\delta_{\ast}(t),\text{
if }\left|e_{\ast 1}\left(t\right)\right|>e_{\ast 1c};\\\ -k_{\ast
2}\text{tanh}\left[\rho_{\ast}(e_{\ast 2}\left(t\right)+k_{\ast 1}e_{\ast
1}\left(t\right))\right]+\ddot{\ast}_{d}\left(t\right)-\delta_{\ast}(t),\text{
if }\left|e_{\ast 1}\left(t\right)\right|\leq e_{\ast 1c}\end{array}\right.$
(97)

In order to turn the error system (62) into the desired stable sliding mode
(63), we select

$\displaystyle-
h_{\ast}(t)-\bar{u}_{\ast}\left(t\right)+\ddot{\ast}_{d}\left(t\right)-\delta_{\ast}(t)$
(100) $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{l}-k_{\ast
c}\text{tanh}\left[\rho_{\ast c}\left(e_{\ast 2}\left(t\right)+e_{\ast
2c}\text{sign}\left(e_{\ast
1}\left(t\right)\right)\right)\right]+\ddot{\ast}_{d}\left(t\right)-\delta_{\ast}(t),\text{
if }\left|e_{\ast 1}\left(t\right)\right|>e_{\ast 1c};\\\ -k_{\ast
2}\text{tanh}\left[\rho_{\ast}(e_{\ast 2}\left(t\right)+k_{\ast 1}e_{\ast
1}\left(t\right))\right]+\ddot{\ast}_{d}\left(t\right)-\delta_{\ast}(t),\text{
if }\left|e_{\ast 1}\left(t\right)\right|\leq e_{\ast 1c}\end{array}\right.$

Therefore, we get the controller as follows:

$\bar{u}_{\ast}\left(t\right)=\left\\{\begin{array}[]{l}k_{\ast
c}\text{tanh}\left[\rho_{\ast c}\left(e_{\ast 2}\left(t\right)+e_{\ast
2c}\text{sign}\left(e_{\ast
1}\left(t\right)\right)\right)\right]-h_{\ast}(t),\text{ if }\left|e_{\ast
1}\left(t\right)\right|>e_{\ast 1c};\\\ k_{\ast
2}\text{tanh}\left[\rho_{\ast}(e_{\ast 2}\left(t\right)+k_{\ast 1}e_{\ast
1}\left(t\right))\right]-h_{\ast}(t),\text{ if }\left|e_{\ast
1}\left(t\right)\right|\leq e_{\ast 1c}\end{array}\right.$ (101)

where, $\ast=x,y,z,\psi,\theta,\phi$. Thus, for the UAV system (59), when the
controller (65) is selected, the system is stable, and the system variables
$x_{\ast 1}$ (where, $\ast=x$, $y$, $z$, $\psi$, $\theta$, $\phi$) are non-
overshooting.

##  8 Experiment on UAV non-overshooting control

Figure 8: Control system hardware.

In this section, an experiment on a quadrotor UAV is presented to demonstrate
the proposed non-overshooting control in practice. The UAV prototype shown in
Figure 7 is used for the flight test. The flight control system implementation
on the hardware is shown in Figure 8, whose elements include: A Gumstix and
Arduino Mega 2560 (16MHz) are selected as the driven boards; Gumstix is to
collect data from measurements; Arduino Mega is to run control algorithm,
which has multiple PWM output channels; A XsensMTI AHRS (10 kHz) provides the
3-axial attitude angles and the angular velocities. A microwave Doppler radar
sensor (24GHz) is to detect the height and its vertical velocity. The Vicon
system provides position and velocity.

(a)

(b)

Figure 9: UAV 3D flight trajectories. (a) Reference trajectory. (b) Flight
trajectory comparison.

_Flight reference trajectory:_ The UAV reference trajectory includes: 1) take
off vertically and hover at the height of 1m; 2) then cruise along a
horizontal line and keep the height; 3) then climb and cruise in a circle with
the radius 5m and the height 2.5m. The 3D reference trajectory is shown in
Fig. 9(a).

In the experiment, considering the disturbance (e.g., the crosswind from a
swinging electric fan) and the modelling uncertainty in the UAV flight
dynamics, the UAV is controlled to track the reference trajectory. The
position and velocity are obtained from the Vicon, the height and its vertical
velocity are detected by the microwave Doppler radar sensor, and the attitude
angle and the angular velocity are measured by the IMU. The controller (65)
drives the UAV to track the reference trajectory.

(a)

(b)

(c)

Figure 10: Control performance in $x$-direction. (a) $x_{1}$. (b) $x_{2}$. (c)
Controller $u_{x}\left(t\right)$.

(a)

(b)

(c)

Figure 11: Control performance in $y$-direction. (a) $y_{1}$. (b) $y_{2}$. (c)
Controller $u_{y}\left(t\right)$.

(a)

(b)

(c)

Figure 12: Control performance in $z$-direction. (a) $z_{1}$. (b) $z_{2}$. (c)
Controller $u_{z}\left(t\right)$.

_8.1 Controller parameters determination:_

Through testing the crosswind from the electric fan, and considering the
system uncertainties, we estimate the upper bound of disturbances and
uncertainties to be within $L_{\delta}=4.5N$. The up-bound of $k_{2}$ is
$k_{2M}=8N$ through the motor tests.

_Steps on determination of controller parameters according to (34)_
$\sim$_(38):_

Step 1: Measure the initial states, and determine the initial errors.

the initial states:

$x\left(0\right)=0.3$, $\dot{x}\left(0\right)=-0.02$;

$y\left(0\right)=0.2$, $\dot{y}\left(0\right)=-0.01$;

$z\left(0\right)=0.05$, $\dot{z}\left(0\right)=0.01$;

the initial reference:

$x_{d}\left(0\right)=0$, $\dot{x}_{d}\left(0\right)=0$;

$y_{d}\left(0\right)=0$, $\dot{y}_{d}\left(0\right)=0$;

$z_{d}\left(0\right)=1$, $\dot{z}_{d}\left(0\right)=0$;

then, the initial errors:

$e_{x1}\left(0\right)=-0.3$, $e_{x2}\left(0\right)=0.02$;

$e_{y1}\left(0\right)=-0.2$, $e_{y2}\left(0\right)=0.01$;

$e_{z1}\left(0\right)=0.95$, $e_{z2}\left(0\right)=-0.01$.

Step 2: Select $e_{\ast 1c}$ and $e_{\ast 2c}$ and determine $k_{\ast c}$ and
$\rho_{\ast c}$.

From $e_{\ast 1c}\in\left(0,k_{\ast 2M}-L_{\delta}\right)=\left(0,3.5\right)$,
we select $e_{\ast 1c}=1$. Then, we get $e_{\ast 2c}\in\left(e_{\ast
1c},\sqrt{\left(k_{\ast 2M}-L_{\delta}\right)e_{\ast
1c}}\right]=\left(1,\sqrt{\left(8-4.5\right)\times
1}\right]=\left(1,1.87\right]$. We select $e_{\ast 2c}=1.2$. Select $k_{\ast
c}=5.5>L_{\delta}$. Then, $\rho_{\ast c}=6\frac{1}{2}\ln\frac{k_{\ast
c}+L_{\ast d}}{k_{\ast c}-L_{\ast
d}}=6\frac{1}{2}\ln\frac{5.5+4.5}{5.5-4.5}=8.83$, where,
$\ast=x,y,z,\psi,\theta,\phi$.

Step 3: Determine the controller parameters $k_{\ast 1}$, $k_{\ast 2}$ and
$\rho_{\ast}$ (where, $\ast=x,y,z,\psi,\theta,\phi$) through the following
calculations:

$k_{\ast 1}=\left\\{\begin{array}[]{l}0.5\frac{\left|e_{\ast
2}\left(t_{c}\right)\right|}{\left|e_{\ast 1}\left(t_{c}\right)\right|},\text{
if }e_{\ast 1}\left(t_{c}\right)e_{\ast 2}\left(t_{c}\right)<0\text{ and
}\left|e_{\ast 1}\left(t_{c}\right)\right|<\left|e_{\ast
2}\left(t_{c}\right)\right|;\\\ 2.3\frac{\left|e_{\ast
2}\left(t_{c}\right)\right|}{\left|e_{\ast 1}\left(t_{c}\right)\right|},\text{
if }e_{\ast 1}\left(t_{c}\right)e_{\ast 2}\left(t_{c}\right)<0\text{ and
}\left|e_{\ast 1}\left(t_{c}\right)\right|\geq\left|e_{\ast
2}\left(t_{c}\right)\right|;\\\ 2\in\left(0,\infty\right),\text{ if
others}\end{array}\right.$

where, $e_{\ast 1}\left(t_{c}\right)$ and $e_{\ast 2}\left(t_{c}\right)$ are
the initial values of $e_{\ast 1}\left(t\right)$ and $e_{\ast
2}\left(t\right)$ respectively when $\left|e_{\ast 1}\left(t\right)\right|\leq
e_{\ast 1c}$.

_Adjustment of_ $k_{\ast 1}$_:_ $k_{\ast 1}=1$ if the calculated $k_{\ast
1}\in\left(0,1\right)$;

$k_{\ast 2}=\left\\{\begin{array}[]{l}1.5\max\left\\{k_{\ast 1}\left|e_{\ast
2}\left(t_{c}\right)\right|+L_{d},\frac{e_{\ast
2}^{2}\left(t_{c}\right)}{2\left|e_{\ast 1}\left(t_{c}\right)\right|}+L_{\ast
d}\right\\},\text{ if }e_{\ast 1}\left(t_{c}\right)e_{\ast
2}\left(t_{c}\right)<0\text{ and }\left|e_{\ast
1}\left(t_{c}\right)\right|<\left|e_{\ast 2}\left(t_{c}\right)\right|;\\\
1.5\max\left\\{k_{\ast 1}\left|e_{2}\left(t_{c}\right)\right|+L_{\ast
d},\frac{k_{\ast 1}^{2}}{3}\left(\left|e_{\ast
1}\left(t_{c}\right)\right|+\sqrt{e_{\ast
1}^{2}\left(t_{c}\right)+3\left(\frac{e_{\ast 2}\left(t_{c}\right)}{k_{\ast
1}}\right)^{2}}\right)+L_{\ast d}\right\\},\text{ if
others}\end{array}\right.$

$e_{\ast 2\max}=\max\left\\{\left|e_{\ast
2}\left(t_{c}\right)\right|,\frac{k_{\ast 1}}{3}\left[\left|e_{\ast
1}\left(t_{c}\right)\right|+\sqrt{e_{\ast
1}^{2}\left(t_{c}\right)+3\left(\frac{e_{\ast 2}\left(t_{c}\right)}{k_{\ast
1}}\right)^{2}}\right]\right\\}$

$\rho_{\ast}=3\max\left\\{\frac{1}{2k_{\ast 1}},1\right\\}\ln\frac{k_{\ast
2}+k_{\ast 1}e_{\ast 2\max}+L_{\ast d}}{k_{\ast 2}-k_{\ast 1}e_{\ast
2\max}-L_{\ast d}}$

Step 4: Controller output (65):

$\bar{u}_{\ast}(t)=\left\\{\begin{array}[]{l}k_{\ast
c}\text{tanh}\left[\rho_{\ast c}\left(e_{\ast 2}\left(t\right)+e_{\ast
2c}\text{sign}\left(e_{\ast
1}\left(t\right)\right)\right)\right]-h_{\ast}(t),\text{ if }\left|e_{\ast
1}\left(t\right)\right|>e_{1\ast c};\\\ k_{\ast
2}\text{tanh}\left[\rho_{\ast}(e_{\ast 2}\left(t\right)+k_{\ast 1}e_{\ast
1}\left(t\right))\right]-h_{\ast}(t),\text{ if }\left|e_{\ast
1}\left(t\right)\right|\leq e_{\ast 1c}\end{array}\right.$

Because the reference trajectory jumps once at the $6$th second, the algorithm
in Remark 5.5 updates the controller parameters at this time. From the program
algorithm calculation, we can read the two groups of controller parameters as
follows:

1) Parameters for takeoff and hovering for $0\leq t<6$ (sec):

$k_{x1}=1$, $k_{x2}=7.05$, $\rho_{x}=4.83$

$k_{y1}=1$, $k_{y2}=6.95$, $\rho_{y}=4.83$

$k_{z1}=1$, $k_{z2}=7.70$, $\rho_{z}=4.83$

2) Parameters for climbing and cruise in a circle for $t\geq 6$ (sec):

$k_{x1}=1$, $k_{x2}=8.75$, $\rho_{x}=4.83$

$k_{y1}=2$, $k_{y2}=6.81$, $\rho_{y}=4.83$

$k_{z1}=2$, $k_{z2}=6.94$, $\rho_{z}=4.83$

_8.2 Analysis of UAV control performance:_

Figure 9(b) gives the UAV 3D flight trajectory comparison. Figures 10$\sim$12
show the UAV flight trajectories, the tracking errors and the controller
outputs in the three directions, respectively: Figure 10 describes the control
performance in $x$-direction; Figure 11 shows the control performance in
$y$-direction; and Figure 12 presents the height control performance in
$z$-direction. From the error outputs, the position errors were within
$0.04$m, and the velocity errors were within $0.2$m/s.

The position ($x,y,z$) of the UAV was controlled to track the reference
($x_{d}\left(t\right)$, $y_{d}\left(t\right)$, $z_{d}\left(t\right)$). From
the position trajectory outputs in Figures 10$\sim$12, the smooth and accurate
trajectory tracking was achieved with almost no chattering and no overshoot.
Even in the presence of time-varying disturbance (i.e., the crosswind from a
swinging electric fan), the position and velocity tracking errors remained
very small, and there was no overshoot. In addition, by reading the program,
we found that although the reference trajectory jumped at the 6th second, the
control system re-made the range judgement and the initial value partitioning,
and the control parameters were updated at this moment. The UAV remained in
the safe flight status throughout the flight.

In this indoor flight test, the bounded time-varying crosswind was generated
by a swinging electric fan. Because the up-bound of disturbance was within a
certain range, the influence of disturbance was rejected sufficiently by the
proposed controller. The experimental environment and test equipment are shown
in [30]. Since the proposed control method only requires that the magnitude of
disturbance is within a given range, and it is unrelated to other aspects of
the disturbance. For the outdoor flight tests, as long as the magnitude of
outdoor wind is within the required range, this control method will be equally
effective in suppressing the disturbance.

_8.3 Limitations of the proposed method:_

Through the theoretical analysis (see Theorem 3.1 and Remark 5.3) and the
experimental results, we found that the high-frequency noise in the
measurements and the bounded stochastic disturbances in the system dynamics
almost do not affect the control performance. However, the low-frequency
disturbances or errors in the measurements cannot be suppressed effectively by
the sliding mode system, resulting in the reduced control accuracy. The
potentially effective methods, such as utilizing the signal fusion with
multiple sensors, can be used to obtain the relatively accurate measurements.

##  9 Conclusions

In this paper, a non-overshooting sliding mode control has been presented to
stabilize a class of uncertain systems, and the global non-overshooting
stability has been implemented. Even when the bounded stochastic disturbance
exists, and the time-variant reference is required, the strict non-
overshooting stabilization is still achieved. The performance of the proposed
control method was demonstrated by two simulation examples, and it was applied
successfully to a UAV flying test: 1) The high-precision and non-overshooting
trajectory tracking was performed; 2) The bounded stochastic disturbances were
rejected sufficiently; 3) The bounded and smoothed controller outputs were
easily performed by the actuators; 4) The UAV trajectory tracking experiments
verified the high maneuverability control capability and non-overshooting
performance for the proposed control. The merits of the control method include
its global non-overshooting stability, strong robustness and no restriction on
the system initial values. Our future work is to optimize the parameters in
the proposed controller.

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Appendix

_Proof of Lemma 2.1:_

Define a new variable

$z\left(t\right)=\int_{0}^{t}e\left(\tau\right)d\tau-\frac{d}{k_{i}}$ (102)

Then, we get

$\dot{z}\left(t\right)=e\left(t\right),\text{
}\ddot{z}\left(t\right)=\dot{e}\left(t\right),\text{
}\dddot{z}\left(t\right)=\ddot{e}\left(t\right)$ (103)

Substituting the relations (66) and (67) into system (6), we can rewrite the
system (6) as

$\dddot{z}\left(t\right)+k_{d}\ddot{z}\left(t\right)+k_{p}\dot{z}\left(t\right)+k_{i}z\left(t\right)=0$
(104)

The characteristic equation of system (68) can be expressed by

$s^{3}+k_{d}s^{2}+k_{p}s+k_{i}=0$ (105)

According to the stability conditions, for system (69), if $k_{p}$, $k_{i}$
and $k_{d}$ are selected to make the real parts of all the roots of the
characteristic equation (69) negative, then system is exponentially stable,
and we get

$\underset{t\rightarrow\infty}{\lim}z\left(t\right)=0\text{,
}\underset{t\rightarrow\infty}{\lim}\dot{z}\left(t\right)=0\text{, and
}\underset{t\rightarrow\infty}{\lim}\ddot{z}\left(t\right)=0$ (106)

For (70), from the relations (66) and (67), we get

$\underset{t\rightarrow\infty}{\lim}\int_{0}^{t}e\left(\tau\right)d\tau=\frac{d}{k_{i}}\text{,
}\underset{t\rightarrow\infty}{\lim}e\left(t\right)=0\text{, and
}\underset{t\rightarrow\infty}{\lim}\dot{e}\left(t\right)=0$ (107)

This concludes the proof. $\blacksquare$

_Proof of Lemma 2.2:_

Define a new variable

$z\left(t\right)=\int_{0}^{t}e\left(\tau\right)d\tau-\frac{d}{k_{i}}$ (108)

Then, we get

$\dot{z}\left(t\right)=e\left(t\right),\text{
}\ddot{z}\left(t\right)=\dot{e}\left(t\right)$ (109)

Substituting the relations (72) and (73) into system (11), we can rewrite the
system (11) as

$\ddot{z}\left(t\right)+k_{p}\dot{z}\left(t\right)+k_{i}z\left(t\right)=0$
(110)

The characteristic equation of system (74) can be expressed by

$s^{2}+k_{p}s+k_{i}=0$ (111)

According to the stability conditions, for system (74), if $k_{p}$ and $k_{i}$
are selected to make the real parts of all the roots of the characteristic
equation (75) negative, then system is exponentially stable, and we get

$\underset{t\rightarrow\infty}{\lim}z\left(t\right)=0\text{, and
}\underset{t\rightarrow\infty}{\lim}\dot{z}\left(t\right)=0$ (112)

For (76), from the relations (72) and (73), we get

$\underset{t\rightarrow\infty}{\lim}\int_{0}^{t}e\left(\tau\right)d\tau=\frac{d}{k_{i}}\text{,
and }\underset{t\rightarrow\infty}{\lim}e\left(t\right)=0$ (113)

This concludes the proof. $\blacksquare$

_Proof of Theorem 3.1:_

_Proof introduction._ The proof on the robust and global non-overshooting
stability of the proposed 2-sliding mode is divided into three steps:

A. Robust non-overshooting reachability of the first subsystem. It proves that
the first subsystem can make the sliding variables reach a given bounded range
and without overshoot. That is to say, the initial values of the second
subsystem are compressed to a given bounded range by the first subsystem with
robust non-overshooting reachability.

B. Robust non-overshooting stability of the second subsystem. It proves that:
conditions on finite-time stability; analytical expressions of sliding
variables through partitioning the initial values; and the determination of
the bounded parameters for the robust non-overshooting stability.

C. Existence and determination of the initial value range for the second
subsystem to obtain the bounded system gains and the non-overshooting
stability.

A. Robust non-overshooting reachability of the first subsystem

Case one: If $e_{1}\left(t\right)>e_{1c}$, for system (18), we get

$\frac{d\left(e_{2}\left(t\right)+e_{2c}\right)}{dt}=-k_{c}\text{sign}\left[e_{2}\left(t\right)+e_{2c}\right]+d(t)$
(114)

A Lyapunov function candidate is selected as

$V_{c}=\frac{1}{2}\left(e_{2}\left(t\right)+e_{2c}\right)^{2}$ (115)

Then, taking the derivative for $V_{c}$, we get

$\displaystyle\dot{V}_{c}$ $\displaystyle=$
$\displaystyle\left(e_{2}\left(t\right)+e_{2c}\right)\left\\{-k_{c}\text{sign}\left[e_{2}\left(t\right)+e_{2c}\right]+d(t)\right\\}$
(116) $\displaystyle\leq$
$\displaystyle-\left(k_{c}-L_{d}\right)\left|e_{2}\left(t\right)+e_{2c}\right|=-\sqrt{2}\left(k_{c}-L_{d}\right)V_{c}^{\frac{1}{2}}$

We know that $k_{c}>L_{d}$. Therefore, there exists a finite time $t_{c1}>0$,
for $t\geq t_{c1}$, such that $e_{2}\left(t\right)=-e_{2c}$. According to
$\dot{e}_{1}=e_{2}$, we get that $\dot{e}_{1}\left(t\right)=-e_{2c}$ for
$t\geq t_{c1}$. Therefore, there exists a finite time $t_{c}>0$, for $t\geq
t_{c}$, such that $e_{1}\left(t\right)\leq e_{1c}$.

Case two: If $e_{1}\left(t\right)<-e_{1c}$, for system (18), we get

$\frac{d\left(e_{2}\left(t\right)-e_{2c}\right)}{dt}=-k_{c}\text{sign}\left[e_{2}\left(t\right)-e_{2c}\right]+d(t)$
(117)

Similar method to case $e_{1}\left(t\right)>e_{1c}$, when we select the
Lyapunov function candidate as
$V_{c}=\frac{1}{2}\left(e_{2}\left(t\right)-e_{2c}\right)^{2}$, we can get
that $e_{1}\left(t\right)\geq-e_{1c}$.

Combining cases one and two, we can get that
$\left|e_{1}\left(t\right)\right|\leq e_{1c}$ for $t\geq t_{c}$.

B. Robust non-overshooting stability of the second subsystem

In the following, we discuss the case $\left|e_{1}\left(t\right)\right|\leq
e_{1c}$.

_Finite-time stability conditions_

For the sliding mode (18) when $\left|e_{1}\left(t\right)\right|\leq e_{1c}$,
a Lyapunov function candidate is selected as

$V_{1}=\frac{1}{2}\sigma\left(t\right)^{2}$ (118)

where, the sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$. Then, we
get

$\displaystyle\dot{V}_{1}$ $\displaystyle=$
$\displaystyle\left(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right)\left\\{-k_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]+d(t)+k_{1}e_{2}\left(t\right)\right\\}$
(119) $\displaystyle=$ $\displaystyle-
k_{2}\left|e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right|+\left(k_{1}e_{2}\left(t\right)+d(t)\right)\left(e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right)$
$\displaystyle\leq$ $\displaystyle-
k_{2}\left|e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right|+\left(k_{1}\left|e_{2}\left(t\right)\right|+L_{d}\right)\left|e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right|$
$\displaystyle=$
$\displaystyle-\left(k_{2}-k_{1}\left|e_{2}\left(t\right)\right|-L_{d}\right)\left|e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right|$
$\displaystyle=$
$\displaystyle-\sqrt{2}\left(k_{2}-k_{1}\left|e_{2}\left(t\right)\right|-L_{d}\right)V^{\frac{1}{2}}$

If $k_{2}>k_{1}\left|e_{2}\left(t\right)\right|+L_{d}$, then the sliding mode
is finite-time stable. That means there exists a finite time $t_{s}>0$, for
$t\geq t_{c}+t_{s}$, the sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=0$. Then,
we get the linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$, and
$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0$.

_Trajectories of_ $e_{1}\left(t\right)$,__ $e_{2}\left(t\right)$_and_
$\sigma\left(t\right)$ _for_ $t\in\left[t_{c},t_{c}+t_{s}\right)$

For the sliding mode (18), according to the differential inclusion theory, we
get

$\dot{e}_{2}\left(t\right)\in-\left[k_{2}-L_{d},k_{2}-L_{d}\right]\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]$
(120)

That is to say, there exists
$\bar{k}_{2}\in\left[k_{2}-L_{d},k_{2}-L_{d}\right]$, such that

$\dot{e}_{2}\left(t\right)=-\bar{k}_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]$
(121)

We will determine $k_{1}$ and $k_{2}$ to make the sign of sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$ unchanged
for $t\in\left[t_{c},t_{c}+t_{s}\right)$, i.e.,
sign$\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]=$sign$\left[e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)\right]$
for $t\in\left[t_{c},t_{c}+t_{s}\right)$. Then, we can get the simple solution
to (85) for $t\in\left[t_{c},t_{c}+t_{s}\right)$, as follows:

$e_{2}\left(t\right)=e_{2}\left(t_{c}\right)-\bar{k}_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]\left(t-t_{c}\right)$
(122)

From $\dot{e}_{1}\left(t\right)=e_{2}\left(t\right)$ in sliding mode (18) and
$e_{2}\left(t\right)$ expression in (86), we get

$e_{1}\left(t\right)=e_{1}\left(t_{c}\right)+e_{2}\left(t_{c}\right)\left(t-t_{c}\right)-\frac{1}{2}\bar{k}_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]\cdot\left(t-t_{c}\right)^{2}$
(123)

and

$\displaystyle\sigma\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)-\left(\bar{k}_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]-k_{1}e_{2}\left(t_{c}\right)\right)\left(t-t_{c}\right)$
(124)
$\displaystyle-\frac{1}{2}k_{1}\bar{k}_{2}\text{sign}\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]\cdot\left(t-t_{c}\right)^{2}$

_Non-overshooting convergence conditions_

1) For the 2-sliding mode (18), we will determine $k_{1}$ and $k_{2}$ to
generate the two convergence laws: 1) the finite-time convergence law to get
the sliding surface
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=0$ after a
finite time $t_{c}+t_{s}$; 2) the linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ to make
$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0$.

2) For the 2-sliding mode (18), in order to get the simple form of sliding
variables assumed in (86)$\sim$(88), we hope that, for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, the sign of sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$ unchanged
as
$\sigma\left(t_{c}\right)=e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)$.
Thus, function sign$\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]$
becomes constant $1$ or $-1$ for $t\in\left[t_{c},t_{c}+t_{s}\right)$. Then,
we can get $\dot{e}_{2}\left(t\right)=-\bar{k}_{2}$ or
$\dot{e}_{2}\left(t\right)=\bar{k}_{2}$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$.

3) For being non-overshooting,
sign$\left[e_{1}\left(t\right)\right]=$sign$\left[e_{1}\left(t_{c}\right)\right]$
needs always hold until
$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0$.

We know that, the sliding variable $e_{1}\left(t\right)$ is non-overshooting
when it is in the convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ for $t\geq t_{c}+t_{s}$.
Therefore, in order to make $e_{1}\left(t\right)$ non-overshooting during the
whole transient process, we only need to guarantee $e_{1}\left(t\right)$ non-
overshooting in the finite-time convergence law for
$t\in\left[t_{c},t_{c}+t_{s}\right)$.

_Partitioning of_ $e_{1}\left(t_{c}\right)$_and_ $e_{2}\left(t_{c}\right)$_in
coordinate_

Figure 13: Partitioning of $e_{1}\left(t_{c}\right)$ and
$e_{2}\left(t_{c}\right)$ in coordinate.

We consider $e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ in the
zones shown in Fig. 13:

1) Zones II-2 and IV-2: $e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0$
with
$\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|$

2) Zones IV-1 and II-1: $e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0$
with
$\left|e_{1}\left(t_{c}\right)\right|\geq\left|e_{2}\left(t_{c}\right)\right|$

3) Zones III-1 and I-1: $e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)\geq 0$
with
$\left|e_{1}\left(t_{c}\right)\right|\leq\left|e_{2}\left(t_{c}\right)\right|$

4) Zones III-2 and I-2: $e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)\geq 0$
with
$\left|e_{1}\left(t_{c}\right)\right|>\left|e_{2}\left(t_{c}\right)\right|$

The right side of the sliding mode (18) is the odd function about the origin,
therefore, we only consider the convergence performance for the zones (II-2,
IV-1, III-1 and III-2). The corresponding zones (IV-2,II-1, I-1 and I-2) about
the origin have the same stability performance to the zones (II-2, IV-1, III-1
and III-2), respectively.

Therefore, in the following, we will consider the convergence performance for
the following zones:

1) Zone II-2: $-e_{2}\left(t_{c}\right)>e_{1}\left(t_{c}\right)>0$

2) Zone IV-1: $-e_{1}\left(t_{c}\right)\geq e_{2}\left(t_{c}\right)>0$

3) Zone III-1: $-e_{2}\left(t_{c}\right)\geq-e_{1}\left(t_{c}\right)\geq 0$

4) Zone III-2: $-e_{1}\left(t_{c}\right)>-e_{2}\left(t_{c}\right)\geq 0$

_Analytical expressions of sliding variables by assuming the unchanged sign
of_ $\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$_for_
$t\in\left[t_{c},t_{c}+t_{s}\right)$

In the selected zones (II-2, IV-1, III-1 and III-2), we will determine $k_{1}$
and $k_{2}$ to make
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)<0$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$. Then,
sign$\left[e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)\right]=$sign$\left[e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)\right]=-1$.
Therefore, from (86), (87) and (88), for $t\in\left[t_{c},t_{c}+t_{s}\right)$,
we get the expressions of variables $e_{2}\left(t\right)$,
$e_{1}\left(t\right)$ and function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$,
respectively, as follows:

$e_{2}\left(t\right)=e_{2}\left(t_{c}\right)+\bar{k}_{2}\left(t-t_{c}\right)$
(125)

$e_{1}\left(t\right)=e_{1}\left(t_{c}\right)+e_{2}\left(t_{c}\right)\left(t-t_{c}\right)+\frac{1}{2}\bar{k}_{2}\left(t-t_{c}\right)^{2}\overset{\text{define}}{=}c_{1}+b_{1}\left(t-t_{c}\right)+a_{1}\left(t-t_{c}\right)^{2}$
(126)

$\displaystyle\sigma\left(t\right)$ $\displaystyle=$ $\displaystyle
e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)+\left(\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)\right)\left(t-t_{c}\right)+\frac{1}{2}k_{1}\bar{k}_{2}\left(t-t_{c}\right)^{2}$
(127)
$\displaystyle\overset{\text{define}}{=}c+b\left(t-t_{c}\right)+a\left(t-t_{c}\right)^{2}$

From (90), $e_{1}\left(t\right)$ is a segment of a parabola for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, and we get its axis of symmetry:

$-\frac{b_{1}}{2a_{1}}=-\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}$ (128)

its vertex:

$\frac{4a_{1}c_{1}-b_{1}^{2}}{4a_{1}}=\frac{2\bar{k}_{2}e_{1}\left(t_{c}\right)-e_{2}^{2}\left(t_{c}\right)}{2\bar{k}_{2}}$
(129)

and the time instant when $e_{1}\left(t\right)=0$:

$\left.t\right|_{e_{1}\left(t\right)=0}=\frac{-b_{1}+\sqrt{b_{1}^{2}-4a_{1}c_{1}}}{2a_{1}}+t_{c}=-\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}+\sqrt{\left(\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}\right)^{2}-\frac{2e_{1}\left(t_{c}\right)}{\bar{k}_{2}}}+t_{c}$
(130)

From (91), function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$ is a
segment of a parabola for $t\in\left[t_{c},t_{c}+t_{s}\right)$, and we get its
axis of symmetry:

$-\frac{b}{2a}=-\frac{\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)}{k_{1}\bar{k}_{2}}$
(131)

its vertex:

$\frac{4ac-b^{2}}{4a}=\frac{2k_{1}\bar{k}_{2}\left(e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)\right)-\left(\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)\right)^{2}}{2k_{1}\bar{k}_{2}}$
(132)

and the time instant when $\sigma\left(t\right)=0$, i.e., the settling time
$t_{c}+t_{s}$:

$t_{c}+t_{s}=t_{c}+\frac{-b+\sqrt{b^{2}-4ac}}{2a}=t_{c}-\left(\frac{1}{k_{1}}+\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}\right)+\sqrt{\frac{1}{k_{1}^{2}}+\left(\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}\right)^{2}-\frac{2e_{1}\left(t_{c}\right)}{\bar{k}_{2}}}$
(133)

In the following, for any zone of $e_{1}\left(t_{c}\right)$ and
$e_{2}\left(t_{c}\right)$, we will determine the parameters $k_{1}$ and
$k_{2}$ to make the sliding mode finite-time stable and $e_{1}\left(t\right)$
non-overshooting for $t\in\left[t_{c},t_{c}+t_{s}\right)$. We know the linear
convergence law $\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ makes
the system non-overshooting stable automatically.

Figure 14: Arranged trajectories of $e_{1}\left(t\right)$,
$e_{2}\left(t\right)$ and $\sigma\left(t\right)$ for $e_{1}\left(t\right)$
non-overshooting convergence.

_Trajectories arrangement of sliding variables:_ In general, for the different
zones of $e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ shown in Fig.
13, the parameters $k_{1}$ and $k_{2}$ will be determined to make sliding
variables $e_{1}\left(t\right)$, $e_{2}\left(t\right)$ and function
$\sigma\left(t\right)$ generate the desired convergent trajectories with
$e_{1}\left(t\right)$ non-overshooting, shown in Fig. 14:

1) Fig. 14 (a)-1 and (a)-2 are for the zones II-2 and IV-2 in Fig. 13,
respectively;

2) Fig. 14 (b)-1 and (b)-2 are for the zones IV-1 and II-1 in Fig. 13,
respectively;

3) Fig. 14 (c)-1 and (c)-2 are for the zones III-1 and I-1 in Fig. 13,
respectively; and Fig. 13 (d)-1 and (d)-2 are for zones III-2 and I-2 in Fig.
13, respectively.

_Conditions on robust non-overshooting stability_

_1) For zone II-2:_ $-e_{2}\left(t_{c}\right)>e_{1}\left(t_{c}\right)>0$

From Figure 13, for zone II-2, we know that the corresponding symmetrical zone
is IV-2.

Figure 15: Arranged trajectories of $e_{1}\left(t\right)$ and
$\sigma\left(t\right)$ for $e_{1}\left(t\right)$ non-overshooting convergence
for $t\in\left[t_{c},t_{c}+t_{s}\right)$ in range II-2:
$\emph{-}e_{2}\left(t_{c}\right)>e_{1}\left(t_{c}\right)>0$.

For zone II-2, we can get the conditions of non-overshooting convergence for
$t\in\left[t_{c},t_{c}+t_{s}\right)$:

1) $\bar{k}_{2}>k_{1}\left|e_{2}\left(t\right)\right|$ [finite-time stability
to get convergence law $\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$];

2) the sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)<0$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, and
$\sigma\left(t_{c}+t_{s}\right)=e_{2}\left(t_{c}+t_{s}\right)+k_{1}e_{1}\left(t_{c}+t_{s}\right)=0$;

3) $e_{1}\left(t\right)>0$ for $t\in\left[t_{c},t_{c}+t_{s}\right)$ [
$e_{1}\left(t\right)$ does not go beyond zero];

4) $e_{2}\left(t\right)<0$ for $t\in\left[t_{c},t_{c}+t_{s}\right)$ [
$e_{2}\left(t\right)$ does not go beyond zero].

In the following, we will determine $k_{1}$ and $k_{2}$ to satisfy these
conditions.

From (90), for $e_{1}\left(t\right)$ (a segment of a parabola) for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, we get its axis of symmetry

$-\frac{b_{1}}{2a_{1}}=-\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}>0$ (134)

because $e_{2}\left(t_{c}\right)<0$. In order to make $e_{1}\left(t\right)>0$
for $t\in\left[t_{c},t_{c}+t_{s}\right)$ (See $e_{1}\left(t\right)$ in Figure
15(a)), the vertex needs to satisfy

$\frac{4a_{1}c_{1}-b_{1}^{2}}{4a_{1}}=\frac{2\bar{k}_{2}e_{1}\left(t_{c}\right)-e_{2}^{2}\left(t_{c}\right)}{2\bar{k}_{2}}>0$
(135)

Because $e_{1}\left(t_{c}\right)>0$ and $e_{2}\left(t_{c}\right)<0$, for (99),
the positive $\bar{k}_{2}$ should satisfy

$\bar{k}_{2}>\frac{e_{2}^{2}\left(t_{c}\right)}{2e_{1}\left(t_{c}\right)}$
(136)

Therefore, from $\bar{k}_{2}$ in condition (100), we can get that
$e_{1}\left(t\right)>0$ for $t\in\left[t_{c},t_{c}+t_{s}\right)$.

From (91), for the sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$ (a segment
of a parabola) for $t\in\left[t_{c},t_{c}+t_{s}\right)$, we get its axis of
symmetry

$-\frac{b}{2a}=-\frac{\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)}{k_{1}\bar{k}_{2}}<0$
(137)

because of the finite-time convergence condition

$\bar{k}_{2}>-k_{1}e_{2}\left(t_{c}\right)$ (138)

In order to make
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)<0$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$ (See $\sigma\left(t\right)$ in Figure
15(b)), the vertex needs to satisfy

$\frac{4ac-b^{2}}{4a}=\frac{2k_{1}\bar{k}_{2}\left(e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)\right)-\left(\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)\right)^{2}}{2k_{1}\bar{k}_{2}}<0$
(139)

In order to get the relation in (103), we can let
$\sigma\left(t_{c}\right)=e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)<0$,
i.e., $k_{1}e_{1}\left(t_{c}\right)<-e_{2}\left(t_{c}\right)$. Therefore, we
get the condition on $k_{1}$, as follows:

$k_{1}<\frac{-e_{2}\left(t_{c}\right)}{e_{1}\left(t_{c}\right)}$ (140)

For $e_{2}\left(t\right)$ in (89), we get the time instant when
$e_{2}\left(t\right)=0$ as follows:

$\left.t\right|_{e_{2}\left(t\right)=0}=\frac{-e_{2}\left(t_{c}\right)}{\bar{k}_{2}}+t_{c}$
(141)

Comparing the settling time $t_{c}+t_{s}$ in (97) and
$\left.t\right|_{e_{2}\left(t\right)=0}$ in (105), we can get

$t_{s}+t_{c}<\left.t\right|_{e_{2}\left(t\right)=0}$ (142)

Then, it follows that

$e_{2}\left(t\right)<0\text{ for }t\in\left[t_{c},t_{c}+t_{s}\right)$ (143)

We know that zones II-2 and IV-2 have the same convergence performance because
of the odd function property in the sliding mode. Therefore, combing (100),
(102) and (104), for the zones II-2 (where,
$-e_{2}\left(t_{c}\right)>e_{1}\left(t_{c}\right)>0$) and IV-2 (where,
$e_{2}\left(t_{c}\right)>-e_{1}\left(t_{c}\right)>0$), we get the non-
overshooting convergence conditions, when
$e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0$ and
$\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|$,
as follows:

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right)$
(144) $\displaystyle\bar{k}_{2}$ $\displaystyle>$
$\displaystyle\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|,\frac{e_{2}^{2}\left(t_{c}\right)}{2\left|e_{1}\left(t_{c}\right)\right|}\right\\}$
(145)

Furthermore, considering the disturbance $d\left(t\right)$, the conditions of
parameters selection, when $e_{1}\left(t_{c}\right)e_{2}\left(t_{c}\right)<0$
and
$\left|e_{1}\left(t_{c}\right)\right|<\left|e_{2}\left(t_{c}\right)\right|$,
are expressed as follows:

$\displaystyle k_{1}$ $\displaystyle\in$
$\displaystyle\left(0,\frac{\left|e_{2}\left(t_{c}\right)\right|}{\left|e_{1}\left(t_{c}\right)\right|}\right)$
(146) $\displaystyle k_{2}$ $\displaystyle>$
$\displaystyle\max\left\\{k_{1}\left|e_{2}\left(t_{c}\right)\right|+L_{d},\frac{e_{2}^{2}\left(t_{c}\right)}{2e_{1}\left(t_{c}\right)}+L_{d}\right\\}$
(147)

Therefore, when $k_{1}$ and $k_{2}$ are selected from (110) and (111), the
sliding mode is non-overshooting stable for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, and
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=0$ holds
for $t\geq t_{c}+t_{s}$. Then, the linear convergence law
$\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$ makes
$\underset{t\rightarrow\infty}{\lim}e_{1}\left(t\right)=0$ without overshoot.
In addition, from
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)=0$ for
$t\geq t_{c}+t_{s}$, we get
$\underset{t\rightarrow\infty}{\lim}e_{2}\left(t\right)=\underset{t\rightarrow\infty}{\lim}\left[-k_{1}e_{1}\left(t\right)\right]=0$.

In general, when $e_{1}\left(t_{c}\right)$ and $e_{2}\left(t_{c}\right)$ are
in zones II-2 and IV-2, the system is exponentially stable, and no overshoot
exists for the variable $e_{1}\left(t\right)$. This confirms the convergence
curves of sliding variables $e_{1}\left(t\right)$, $e_{2}\left(t\right)$ and
function $\sigma\left(t\right)$ described in Figures 14 (a)-1 and (a)-2 for
the zones II-2 and IV-2, respectively.

_2) For zone IV-1:_ $-e_{1}\left(t_{c}\right)\geq e_{2}\left(t_{c}\right)>0$

From Figure 13, for zone IV-1, we know that its corresponding symmetrical zone
is II-1.

For zone IV-1, we can get the conditions of non-overshooting convergence for
$t\in\left[t_{c},t_{c}+t_{s}\right)$:

1) $\bar{k}_{2}>k_{1}\left|e_{2}\left(t\right)\right|$ [finite-time stability
to get convergence law $\dot{e}_{1}\left(t\right)=-k_{1}e_{1}\left(t\right)$]

2) the sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)<0$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, and
$\sigma\left(t_{c}+t_{s}\right)=e_{2}\left(t_{c}+t_{s}\right)+k_{1}e_{1}\left(t_{c}+t_{s}\right)=0$;

3) $e_{1}\left(t\right)<0$ for $t\in\left[t_{c},t_{c}+t_{s}\right)$ [
$e_{1}\left(t\right)$ does not go beyond zero];

4) $e_{2}\left(t\right)>0$ for $t\in\left[t_{c},t_{c}+t_{s}\right)$ [
$e_{2}\left(t\right)$ does not go beyond zero].

In the following, we will determine $k_{1}$ and $k_{2}$ to satisfy these
conditions.

From (90), for $e_{1}\left(t\right)$ (a segment of a parabola) for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, we get its axis of symmetry

$-\frac{b_{1}}{2a_{1}}=-\frac{e_{2}\left(t_{c}\right)}{\bar{k}_{2}}<0$ (148)

because $e_{2}\left(t_{c}\right)>0$. Also, its vertex satisfies

$\frac{4a_{1}c_{1}-b_{1}^{2}}{4a_{1}}=\frac{2\bar{k}_{2}e_{1}\left(t_{c}\right)-e_{2}^{2}\left(t_{c}\right)}{2\bar{k}_{2}}<0$
(149)

because $e_{1}\left(t_{c}\right)<0$ (See $e_{1}\left(t\right)$ in Figure
16(a)).

Figure 16: Arranged trajectories of $e_{1}\left(t\right)$ and
$\sigma\left(t\right)$ for $e_{1}\left(t\right)$ non-overshooting convergence
for $t\in\left[t_{c},t_{c}+t_{s}\right)$ in range IV-1:
$-e_{1}\left(t_{c}\right)\geq e_{2}\left(t_{c}\right)>0$.

From (91), for the sliding function
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$, its axis of symmetry satisfies

$-\frac{b}{2a}=-\frac{\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)}{k_{1}\bar{k}_{2}}<0$
(150)

because of the finite-time stability condition:

$\bar{k}_{2}>k_{1}\left|e_{2}\left(t_{c}\right)\right|$ (151)

In order to make
$\sigma\left(t\right)=e_{2}\left(t\right)+k_{1}e_{1}\left(t\right)<0$ for
$t\in\left[t_{c},t_{c}+t_{s}\right)$ (See $\sigma\left(t\right)$ in Figure
16(b)), the vertex needs to satisfy

$\frac{4ac-b^{2}}{4a}=\frac{2k_{1}\bar{k}_{2}\left(e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)\right)-\left(\bar{k}_{2}+k_{1}e_{2}\left(t_{c}\right)\right)^{2}}{2k_{1}\bar{k}_{2}}<0$
(152)

To satisfy (116), we can make
$\sigma\left(t_{c}\right)=e_{2}\left(t_{c}\right)+k_{1}e_{1}\left(t_{c}\right)<0$
i.e., $-k_{1}e_{1}\left(t_{c}\right)>e_{2}\left(t_{c}\right)$. Therefore, we
get the condition on $k_{1}$, as follows:

$k_{1}>\frac{e_{2}\left(t_{c}\right)}{-e_{1}\left(t_{c}\right)}$ (153)

Comparing the settling time $t_{c}+t_{s}$ in (97) and
$\left.t\right|_{e_{1}\left(t\right)=0}$ in (94), we get

$t_{c}+t_{s}<\left.t\right|_{e_{1}\left(t\right)=0}$ (154)

Therefore, we get

$e_{1}\left(t\right)<0\text{ for }t\in\left[t_{c},t_{c}+t_{s}\right)$ (155)

For $e_{2}\left(t\right)$, when $t=t_{c}+t_{s}$, from (89) and (97), we get

$\left.e_{2}\left(t\right)\right|_{t=t_{c}+t_{s}}=e_{2}\left(t_{c}\right)+\bar{k}_{2}t_{s}=-\frac{\bar{k}_{2}}{k_{1}}+\sqrt{e_{2}^{2}\left(t_{c}\right)+\left(\frac{\bar{k}_{2}}{k_{1}}\right)^{2}-2\bar{k}_{2}e_{1}\left(t_{c}\right)}$
(156)

For $t\in\left[t_{c},t_{c}+t_{s}\right)$, considering the finite-time
convergence condition, we need

$\bar{k}_{2}>k_{1}\max\left\\{\left|e_{2}\left(t\right)\right|\right\\}=k_{1}\left|\left.e_{2}\left(t\right)\right|_{t=t_{c}+t_{s}}\right|$
(157)

Combining (120) and (121), we get

$\bar{k}_{2}>k_{1}\left|\left.e_{2}\left(t\right)\right|_{t=t_{c}+t_{s}}\right|=k_{1}\left|-\frac{\bar{k}_{2}}{k_{1}}+\sqrt{e_{2}^{2}\left(t_{c}\right)+\left(\frac{\bar{k}_{2}}{k_{1}}\right)^{2}-2\bar{k}_{2}e_{1}\left(t_{c}\right)}\right|$
(158)

Therefore, $\bar{k}_{2}$ should satisfy

$\bar{k}_{2}>\frac{k_{1}^{2}}{3}\left[\left|e_{1}\left(t_{c}\right)\right|+\sqrt{e_{1}^{2}\left(t_{c}\right)+3\left(\frac{e_{2}\left(t_{c}\right)}{k_{1}}\right)^{2}}\right]$
(159)