# Energy-saving sub-optimal sliding mode control with bounded actuation

Michael Ruderman<EMAIL_ADDRESS>Alessandro Pisano Elio Usai
Department of Engineering Sciences, University of Agder, Norway Department of
Electrical and Electronic Engineering, University of Cagliari, Italy

###### Abstract

The second-order _sub-optimal_ sliding mode control (SMC), known in the
literature for the last two decades, is extended by a control-off mode which
allows for saving energy during the finite time convergence. The systems with
relative degree two between the sliding variable and switching control with
bounded actuation are considered, while the matched upper-bounded
perturbations are not necessarily continuous. Detailed analysis of the
proposed energy-saving sub-optimal SMC is performed with regard to the
parametric conditions, reaching and convergence time, and residual steady
oscillations if the parasitic actuator dynamics is added. Constraints for both
switching threshold parameters are formulated with respect to the control
authority and perturbations upper bound. Based on the estimated finite
convergence time, the parameterization of the switching thresholds is solved
as constrained minimization of the derived energy cost function. The total
energy consuming control-on time is guaranteed to be lower than the upper-
bounded convergence time of the conventional sub-optimal SMC. Numerical
evaluations expose the properties of the proposed energy-saving sub-optimal
SMC and compare it with conventional sub-optimal SMC in terms of the fuel
consumption during the convergence.

###### keywords:

second-order sliding modes , robust control , sub-optimal sliding mode control
, energy saving

††journal: ArXiv

##  1 Introduction

Sliding mode control (SMC), see [1], [2], [3] for fundamentals, is one of the
most promising robust control techniques. In SMC, the control authority aims
to force an uncertain system dynamics onto a specified manifold $\sigma$ and
then maintain the system behavior on it, this way ensuring a reduced-order
dynamics and insensitivity to a certain class of perturbations. Being in
sliding mode implies $\sigma=0$ for all times $t>T_{c}$ after the convergence
at $0<T_{c}<\infty$. In second-order sliding modes [4], [5], also the time
derivative of the sliding variable is forced to zero, i.e.
$\sigma=\dot{\sigma}=0$, thus allowing for robust control of uncertain systems
with relative degree two between the sliding variable and control signal.
Worth recalling is that a second-order SMC belongs to a more generic class of
higher-order sliding mode approaches, which have been actively developed over
the last two decades for uncertain dynamic systems with relative degree two
and higher, see e.g. [3] and references therein.

Among the second-order SMC issues, two of them can be highlighted as
particularly relevant for several applications. First, the time derivative of
the sliding variable may be hardy available in the systems under control.
Second, a discontinuous SMC implies a high-frequency (theoretically infinite
frequency) commutation of the control variable, that can be both hardware-
fatiguing and energy-inefficient for different system plants. Addressing the
first above mentioned issue, the so called sub-optimal SMC [6], [7], which is
in focus of our present work, was developed based on the bang-bang control
principles. Recall that the single commutation in an unperturbed system
characterizes the optimal bang-bang control. In turn, the sub-optimal SMC
converges robustly and in finite time to origin of the $(\sigma,\dot{\sigma})$
plane by executing a commutating control sequence $u(t)$ with increasing
frequency. The corresponding cumulative energy of a sub-optimal SMC is
proportional to the convergence time, since a discontinuous commutating
sequence implies the control signal is always on, i.e. $|u(t)|\neq 0$ for
$0<t\leq T_{c}$. Here we explicitly point out that the control behavior after
the convergence, i.e. for $t>T_{c}$, is not the subject of our recent
discussion, while a related remark is given at the end of the article. To the
best of our knowledge, an energy saving approach which allows also for $u=0$
mode during convergence was not elaborated for second-order SMC. At the same
time, one should recall that an equivalent fuel-optimal problem is well known
for unperturbed second-order systems [8].

Against the above background, and being motivated by the previous works [8],
[9], we propose an extension of the sub-optimal SMC which allows for energy
saving during the entire convergence phase. Here the systems with relative
degree two and discrete bounded actuation are targeted. The latter implies the
control value to be in the finite set $u\in\\{-U,\,0,\,U\\}$. The original
sub-optimal SMC is modified by introducing an additional switching threshold
which enables for a mandatory $u=0$ phase between two consecutive extreme
values of the sliding variable, cf. with [7]. The proposed extension in the
control law is rather simple, while the associated analysis developed in
detail is essential for parameterization and guaranteed energy-saving
operation of SMC.

The paper is organized as follows. The preliminaries of the sub-optimal SMC
are given in Section 2, in close accordance with its basic notations from [7].
This section provides also the problem under consideration and the formulated
requirements for an energy-saving modification of the sub-optimal SMC to be
introduced. Section 3 contains the main results. The proposed energy-saving
sub-optimal SMC is formulated and analyzed in terms of the convergence
conditions and a rigorous estimation of the worst-case reaching- and
convergence-time. Based on that, the constrained optimization problem of
energy-saving parametrization is formulated, and the switching threshold
parameters are determined so as to minimize the fuel consumption. Also the
well-known problem of residual steady-state oscillations (i.e. chattering)
owing to the additional parasitic actuator dynamics is addressed in Section 4.
This is done by applying the describing function analysis of harmonic balance.
Detailed numerical examples, revealing and visualizing the main properties of
the energy saving sub-optimal SMC and also comparing it to the original sub-
optimal SMC, can be found in section 5. Finally, section 6 provides main
concluding remarks.

##  2 Preliminaries

Considered is a class of uncertain dynamic systems with the matched bounded
perturbations, for which the well-defined sliding variable

$\sigma(t)=\sigma\bigl{(}\mathbf{x}(t),t\bigr{)}$ (1)

has the relative degree $r=2$ with respect to the control variable
$u\in\mathbb{R}$. The measurable vector of the system states is
$\mathbf{x}\in\mathbb{R}^{n}$, with $n\in\mathbb{N}$ and $2\leq n<\infty$. The
dynamics of the sliding variable can be written in the normal form as

$\ddot{\sigma}(t)=f(\cdot,t)+g(\cdot,t)u(t),$ (2)

where $f$ is a perturbation function and $g$ is the uncertain input coupling
function. Both are satisfying the global boundedness conditions

  * (i)

$\bigl{|}f(\cdot,t)\bigr{|}\leq\Phi$, with $\Phi>0$ is a known real constant;

  * (ii)

$0<\Gamma_{m}\leq g(\cdot,t)\leq\Gamma_{M}$, with $\Gamma_{m},\Gamma_{M}$ to
be known.

The sub-optimal second-order sliding mode control (SMC), proposed initially in
[6], can be written in the following form, cf. [7]

$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle-\alpha(t)U\mathrm{sign}(\sigma-\beta\sigma_{M}),$ (3)
$\displaystyle\alpha(t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1,&\hbox{ if
}(\sigma-\beta\sigma_{M})\sigma_{M}\geq 0\\\\[2.84526pt] \alpha^{*},&\hbox{ if
}(\sigma-\beta\sigma_{M})\sigma_{M}<0,\\\ \end{array}\right.$ (6)
$\displaystyle\beta$ $\displaystyle\in$ $\displaystyle[0;\,1).$ (7)

Here $U>0$ is the minimal control magnitude and $\alpha^{*}\geq 1$ and $\beta$
are the modulation and anticipation factors, correspondingly. The dynamic
state $\sigma_{M}$ constitutes the last _extreme value_ of the sliding
variable during the control (3) operates on (2). The extreme value refers to
the value of $\sigma$ at the last time instant at which a local maximum or
horizontal flex point of $\sigma(t)$ has occurred, cf. [7]. Following the
previous works on the sub-optimal SMC, see in survey [7], the control
parameters must be set so as to satisfy

$\displaystyle U$ $\displaystyle>$ $\displaystyle\frac{\Phi}{\Gamma_{m}},$ (8)
$\displaystyle\alpha^{*}$ $\displaystyle\in$
$\displaystyle[1;\infty)\cap\Biggl{(}\frac{2\Phi+(1-\beta)\Gamma_{M}U}{(1+\beta)\Gamma_{m}U};\infty\Biggr{)}.$
(9)

The condition (8) represents the _control authority_ , which is required so as
to overcome the unknown perturbations bounded by $\Phi$ and, therefore, to
decide the sign of $\ddot{\sigma}$. The condition (9) determines and ensures
the convergence of $\sigma(t)$ to the origin, where the second-order sliding
mode appears in its proper sense. In addition, a stronger inequality than (9)
can be imposed by

$\alpha^{*}\in[1;\infty)\cap\Biggl{(}\frac{\Phi+(1-\beta)\Gamma_{M}U}{\beta\Gamma_{m}U};\infty\Biggr{)},$
(10)

which is required for a monotonic convergence to zero, cf. [7]. The latter
means $\sigma(t)$ will experience at most one zero-crossing, depending on the
initial conditions $(\sigma,\dot{\sigma})(0)$ before the first
$\sigma_{M}$-state occurs. For the sake of better distinguishing we will
denote the parametric conditions (8), (9) by the _twisting convergence_ , and
(8), (10) by the _monotonic convergence_ conditions. Either of both conditions
ensure the establishment of the second-order sliding mode of (2) with (3) in a
finite time.

Worth recalling is also that $\dot{\sigma}$ is not available, so that it is
impossible to determine $\sigma_{M}$ by observing zero-crossing of the
$\dot{\sigma}$ state variable. Notwithstanding, according to the ’Algorithm 2’
provided in [6], a recursive detection of the extreme values of $\sigma(t)$
takes place simultaneously with execution of the control algorithm (3).

### Problem under consideration

The problem under consideration requires a bounded actuation $|u|\leq U$ with
only three discrete values, i.e. $u\in\\{-U,\,0,\,U\\}$, where $U>0$ is a
known parameter of the control system. For such systems, we relax the above
uncertainty condition (ii) and assume a constant input coupling, meaning
$\Gamma_{m}=\Gamma_{M}=g=1$. This implies the maximal control magnitude $U$
and leads to $\alpha=1$ to be substituted instead of (6). Under the above
assumptions, the inequalities (8), (9) and (10) result in a set of the
parametric conditions

$\displaystyle U$ $\displaystyle>$ $\displaystyle\Phi,$ (11)
$\displaystyle\beta$ $\displaystyle>$
$\displaystyle\frac{\Phi}{U}\qquad\qquad\hbox{: twisting convergence},$ (12)
$\displaystyle\beta$ $\displaystyle>$
$\displaystyle\frac{\Phi+U}{2U}\qquad\hbox{: monotonic convergence}.$ (13)

Now, we are in the position to specify an _energy-saving sub-optimal_ SMC with
the bounded actuation.

Given the fact of a bounded control authority $U$, we are interested to modify
the control (3) with $\alpha=1$ so as to allow for energy-saving control
phases with $u=0$. Within each reaching cycle, i.e. between two successive
extreme values $\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$ with
$i\in\mathbb{N}<\infty$, the new control policy must admit the following. The
control is on, i.e. $|u|=U$, on the intervals
$\sigma\in[\beta_{1}\sigma_{M_{i}};\sigma_{M_{i}}]$ and
$\sigma\in[\sigma_{M_{i+1}};\beta_{2}\sigma_{M_{i}}]$. And the control is off,
i.e. $u=0$, on the interval
$\sigma\in[\beta_{2}\sigma_{M_{i}};\beta_{1}\sigma_{M_{i}}]$, while
$\beta_{2}\neq\beta_{1}$ is required for an energy-saving operation. The case
where $\beta_{2}=\beta_{1}=\beta$ should correspond to the sub-optimal SMC (3)
with $\alpha=1$, thus satisfying (11) and either (12) or (13). This
benchmarking control will be denoted as _conventional sub-optimal_ SMC. For
the energy-saving sub-optimal SMC fulfills its task, following must be
guaranteed:

  * (a)

convergence in finite time $T_{c}<\infty$, which implies
$|\sigma_{M_{i+1}}|<|\sigma_{M_{i}}|$ for all $1\leq i<\infty$;

  * (b)

fuel- and thus energy-saving, which implies

$\int\limits_{0}^{T_{c}}\bigl{|}u(\beta_{1},\beta_{2})\bigr{|}dt<\int\limits_{0}^{\hat{T}_{c}}\bigl{|}\hat{u}(\beta_{1})\bigr{|}dt,$

where the variables with hat (like $\hat{\circ}$) are used for the
conventional sub-optimal SMC.

##  3 Energy-saving sub-optimal SMC

###  3.1 Proposed control algorithm

The proposed modification of the second-order sub-optimal SMC algorithm is

$u(t)=-0.5U\mathrm{sign}(\sigma-\beta_{1}\sigma_{M})-0.5U\mathrm{sign}(\sigma-\beta_{2}\sigma_{M}).$
(14)

In addition to (11), the thresholds relationship $\beta_{1}>\beta_{2}$ is
required, while the convergence and energy-saving conditions on
$(\beta_{1},\beta_{2})$ parameters are derived and analyzed in the following.
An initializing control action, cf. [7],

$\bar{u}(t)=-U\mathrm{sign}\bigl{(}\sigma(t)-\sigma(0)\bigr{)},\quad\forall\,t\in[0;t_{M_{1}}]$
(15)

is also required for speeding up the reaching of the first extreme value at
$t_{M_{1}}$.

###  3.2 Proof of convergence and parametric conditions

Using the general formula

$\sigma=\sigma_{0}+0.5\,\frac{\dot{\sigma}^{2}-\dot{\sigma}^{2}_{0}}{v},$ (16)

for a double-integrator with $v=\pm U\pm\Phi$, the
$(\sigma,\dot{\sigma})$-trajectories can be calculated for an arbitrary
initial state $(\sigma_{0},\dot{\sigma}_{0})$. Then, without loss of
generality, consider the first extreme value $\sigma_{M_{i}}>0$, while the
preceding reaching phase is always ensured by the initialization control (15).
After applying first the control value $u=-U$, according to (14), the state
trajectory proceeds in the IV-th quadrant, cf. Figure 2. While both limiting
cases of the perturbation $f=\pm\Phi$ are effective, we focus here on the
worst-case scenario, that is essential for convergence, in which the sign of
$f$ is negative. Note that this renders the system to be underdamped. Using
$-U-\Phi$ in the denominator of (16) and substituting the threshold value
$\beta_{1}\sigma_{M_{i}}$ on the left-hand side of (16) one obtains the first
peaking point (labeled by $P^{\prime\prime}$) of the $\dot{\sigma}$-state as

$\dot{\sigma}_{P^{\prime\prime}}=-\sqrt{2\sigma_{M_{i}}(U+\Phi)(1-\beta_{1})},$
(17)

At this point, the control is switching from $-U$ to $0$. Then, evaluating the
parabolic trajectory (16) between the $\beta_{1}\sigma_{M_{i}}$ and
$\beta_{2}\sigma_{M_{i}}$ points, with $u=0$ and worst-case $f=-\Phi$, one
obtains

$\beta_{2}\sigma_{M_{i}}=\beta_{1}\sigma_{M_{i}}+0.5\,\frac{\dot{\sigma}^{2}-\dot{\sigma}_{P^{\prime\prime}}^{2}}{-\Phi}.$
(18)

Substituting (17) into (18) and solving (18) with respect to $\dot{\sigma}$
results in

$\dot{\sigma}_{P_{2}}=-\sqrt{2\sigma_{M_{i}}\bigl{(}U(1-\beta_{1})+\Phi(1-\beta_{2})\bigr{)}},$
(19)

where the second peaking point $P_{2}$ is the one where the control is
switching back from $0$ to $U$. Continuing this line of calculations, one
considers also the parabolic trajectory (16) after passing the
$\beta_{2}\sigma_{M_{i}}$-threshold as

$\sigma=\beta_{2}\sigma_{M_{i}}+0.5\,\frac{\dot{\sigma}^{2}-\dot{\sigma}_{P_{2}}^{2}}{U-\Phi}.$
(20)

Here again, $f=-\Phi$ is assumed as worst-case from a convergence point of
view. Substituting the left-hand-side of (20) by the next extreme value
$\sigma_{M_{i+1}}$, at which $\dot{\sigma}=0$, results in

$\sigma_{M_{i+1}}=\beta_{2}\sigma_{M_{i}}+0.5\,\frac{-\dot{\sigma}_{P_{2}}^{2}}{U-\Phi}.$
(21)

Note that for convergence, it is strictly required that

$|\sigma_{M_{i+1}}|<|\sigma_{M_{i}}|$

for all pairs of the consecutive extreme values, indexed by $i$ and $i+1$.
Since $\sigma_{M_{i}}>\sigma_{M_{i+1}}$ is guaranteed for all trajectories in
the IV-th and III-rd quadrants, cf. Figure 2, the above convergence inequality
can be written, with respect to (21), as

$\beta_{2}\sigma_{M_{i}}-0.5\,\frac{\dot{\sigma}_{P_{2}}^{2}}{U-\Phi}>-\sigma_{M_{i}}.$
(22)

Substituting (19) into (22) yields

$1+\beta_{2}>\frac{\Phi+U-\Phi\beta_{2}-U\beta_{1}}{U-\Phi},$ (23)

which results in the strict parameters condition

$\beta_{1}+\beta_{2}>\frac{2\Phi}{U}.$ (24)

Furthermore, one needs to ensure

$\displaystyle 0$ $\displaystyle\leq\beta_{1}<$ $\displaystyle 1,$ (25)
$\displaystyle-1$ $\displaystyle<\beta_{2}<$ $\displaystyle\beta_{1},$ (26)

in order to realize the both control switchings between two successive extreme
values. Note that (25) is inherited from the conventional sub-optimal SMC, cf.
(7). Here we emphasize that the parametric conditions (24), (25), (26) must be
satisfied for the energy-saving sub-optimal SMC to yield the global
convergence, for $U>\Phi$ and irrespective the perturbations $|f(t)|<\Phi$.

The imposed $\beta_{1},\beta_{2}$ constraints (24), (25), (26) have an
illustrative graphical interpretation as shown in Figure 1.

Figure 1: Parametric constraints (24), (25), (26) in $(\beta_{1},\beta_{2})$
plane.

The $\beta_{1},\beta_{2}$ values admissible for convergence are inside of a
grey-shadowed triangle, while on the $\beta_{1}=\beta_{2}$ edge the energy-
saving control (14), (15) reduces to the conventional sub-optimal SMC.
Obviously, the $\Phi/U$ ratio determines the size of an admissible
$\\{\beta_{1},\beta_{2}\\}$ set, including whether the negative
$\beta_{2}$-values are allowed. It is also worth noting that the opposite
$\beta_{2}=2\Phi/U-\beta_{1}$ edge maximizes the $\beta_{1}-\beta_{2}$
distance. However, this can largely increase the convergence time (including
$T_{c}\rightarrow\infty$), similar as in case of a minimal-fuel control with
free response time of an unperturbed double-integrator analyzed in detail in
[8].

Figure 2: Phase-plane of the energy-saving sub-optimal SMC for worst-case
scenario of the reaching time.

###  3.3 Reaching and convergence time

Consider first the reaching time between two consecutive extreme values
$\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$. Recall that a conservative approach
is taken so that a worst-case scenario is calculated for all control phases,
in terms of a slowest time. The associated trajectories of one reaching phase
are schematically drawn in Figure 2. Similar as before, and without loss of
generality, consider the first extreme value $\sigma_{M_{i}}>0$ so that the
trajectories proceed in the III-rd and IV-th quadrants. We recall that the
threshold value $\beta_{2}$ can be either positive or negative, cf. Figure 1.
Evaluating first the peaking point $P^{\prime}$ one obtains the corresponding
$\dot{\sigma}$-value as

$\dot{\sigma}_{P^{\prime}}=-\sqrt{2\sigma_{M_{i}}(U-\Phi)(1-\beta_{1})},$ (27)

which is reached by the slowest trajectory driven by $-U+\Phi$. For the
subsequent phase with control-off, i.e. $u=0$, one needs distinguishing
between two boundary trajectories, one driven by $\Phi$ and another driven by
$-\Phi$. Also we notice that both control-off phases are considered starting
from the same $P^{\prime\prime}$ peaking point. Indeed, one can recognize that
starting from $\beta_{1}\sigma_{M_{i}}$, the $\Phi$-driven trajectory reaches
the $\sigma$-abscissa later from $P^{\prime\prime}$ than from $P^{\prime}$
peaking point, see Figure 2. For the control-off phase between the
$\beta_{1}\sigma_{M_{i}}$ and $\beta_{2}\sigma_{M_{i}}$ thresholds, starting
from $P^{\prime\prime}$ and ending in $P_{1}$ peaking point, one obtains

$\dot{\sigma}_{P_{1}}=-\sqrt{2\sigma_{M_{i}}\bigl{(}U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})\bigr{)}}.$
(28)

Note that the second term under the square root in (28) must be positive, so
here $\beta_{2}$ has to be larger than the point where the $\Phi$-driven
trajectory reaches the $\sigma$-abscissa (see the dotted arc from $P_{1}$ in
Figure 2).

With the determined $\dot{\sigma}$ values at all relevant peaking points, i.e.
$P^{\prime}$, $P^{\prime\prime}$, $P_{1}$ and $P_{2}$, one can analyze the
slowest time for each phase of a reaching cycle individually.

The slowest time of the first control-on phase is

$T^{\prime}\equiv
t_{P^{\prime}}-t_{\sigma_{M_{i}}}=\frac{|\dot{\sigma}_{P^{\prime}}|}{U-\Phi}=\frac{\sqrt{2\sigma_{M_{i}}}}{U-\Phi}\sqrt{(U-\Phi)(1-\beta_{1})}.$
(29)

In a similar way, one obtains the slowest time of the subsequent control-off
phase driven by $\Phi$ as

$\begin{split}T_{1}^{*}&\equiv
t_{P_{1}}-t_{P^{\prime\prime}}=\frac{|\dot{\sigma}_{P^{\prime\prime}}-\dot{\sigma}_{P_{1}}|}{\Phi}=\frac{\sqrt{2\sigma_{M_{i}}}}{\Phi}\,\times\\\
&\Bigl{(}\sqrt{(U+\Phi)(1-\beta_{1})}-\sqrt{U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})}\Bigr{)}.\end{split}$
(30)

One can recognize that if $\beta_{2}=\beta_{1}$ (yielding the conventional
sub-optimal SMC), then both square root terms in (30) become equal, and
$T_{1}^{*}$ becomes zero. One can also recognize that the upper bound of
$T_{1}^{*}$ is when the second square root term in (30) becomes zero. This
results in the maximal control-off time of this phase equal to

$T^{*}_{1,\max}=\frac{\sqrt{2\sigma_{M_{i}}}}{\Phi}\sqrt{(U+\Phi)(1-\beta_{1})}.$
(31)

When evaluating the control-off phase driven by $-\Phi$, cf. Figure 2, we
write

$\dot{\sigma}_{P_{2}}=\dot{\sigma}_{P^{\prime\prime}}-\Phi T^{*}_{2}.$ (32)

Substituting (17) and (19) into (32) and solving it with respect to
$T^{*}_{2}$ results in

$\begin{split}T^{*}_{2}=&\frac{\sqrt{2\sigma_{M_{i}}}}{\Phi}\,\times\\\
&\Bigl{(}\sqrt{U(1-\beta_{1})+\Phi(1-\beta_{2})}-\sqrt{(U+\Phi)(1-\beta_{1})}\Bigr{)}.\end{split}$
(33)

Here, it can also be recognized that $T^{*}_{2}$ has its maximal value when
the second square root term in (33) becomes zero. Thus, the maximal control-
off time for this phase is

$T^{*}_{2,\max}=\frac{\sqrt{2\sigma_{M_{i}}}}{\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-\beta_{2})}.$
(34)

Comparing both maximal time values, one can show that

$T^{*}_{1,\max}<T^{*}_{2,\max}$

holds always due to $\beta_{2}<\beta_{1}$. Despite this fact, which determines
the slowest control-off phase during one reaching cycle, we recall that the
overall convergence time depends also on the number of cycles. Therefore, both
(31) and (34) are taken into account in further calculations.

Evaluating both successive control-on phases, starting once from $P_{1}$ and
once from $P_{2}$ peaking point, we obtain

$\begin{split}T_{1}^{**}\equiv&\>t_{\sigma^{*}_{M_{i+1}}}-t_{P_{1}}=\frac{\sqrt{2\sigma_{M_{i}}}}{U-\Phi}\,\times\\\
&\sqrt{U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})}\end{split}$ (35)

and

$T_{2}^{**}\equiv
t_{\sigma^{**}_{M_{i+1}}}-t_{P_{2}}=\frac{\sqrt{2\sigma_{M_{i}}}}{U-\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-\beta_{2})},$
(36)

respectively. Also here one can show, cf. (35), (36), that

$T_{1}^{**}<T_{2}^{**}$

holds always due to $\beta_{2}<\beta_{1}$.

For analyzing the total worst-case (i.e. largest) reaching time between two
consecutive extreme values $\sigma_{M_{i}}$ and $\sigma_{M_{i+1}}$, one needs
to distinguish between two pathes: $P^{\prime\prime}\rightarrowtail
P_{1}\rightarrowtail\sigma^{*}_{M_{i+1}}$ and $P^{\prime\prime}\rightarrowtail
P_{2}\rightarrowtail\sigma^{**}_{M_{i+1}}$, cf. Figure 2. In the first case,
the overall maximal reaching time is

$\begin{split}T_{1,\max,i}=&\>T^{\prime}+T^{*}_{1,\max}+T_{1}^{**}=\sqrt{2\sigma_{M_{i}}}\,\Omega_{1}=\\\
&\sqrt{2\sigma_{M_{i}}}\,\Bigl{(}\bar{\Omega}_{1}+\tilde{\Omega}_{1}\Bigr{)},\\\
\hbox{with }\;\Omega_{1}=&\frac{1}{\Phi}\sqrt{(U+\Phi)(1-\beta_{1})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{(U-\Phi)(1-\beta_{1})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})}.\end{split}$
(37)

In the second case, respectively, the overall maximal reaching time is

$\begin{split}T_{2,\max,i}=&\>T^{\prime}+T^{*}_{2,\max}+T_{2}^{**}=\sqrt{2\sigma_{M_{i}}}\,\Omega_{2}=\\\
&\sqrt{2\sigma_{M_{i}}}\,\Bigl{(}\bar{\Omega}_{2}+\tilde{\Omega}_{2}\Bigr{)},\\\
\hbox{with
}\;\Omega_{2}=&\frac{1}{\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-\beta_{2})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{(U-\Phi)(1-\beta_{1})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-\beta_{2})}.\end{split}$ (38)

Note that in both above equations $\bar{\Omega}$ denote the terms where the
control is off, i.e. $u=0$, while $\tilde{\Omega}$ denote the terms where the
control is on, i.e. $|u|=U$.

Once the convergence of the energy-saving sub-optimal SMC is guaranteed, being
analyzed in Section 3.2, the overall accumulated worst-case reaching time
results in

$T_{\max,\sum}=\sqrt{2}\,\max\\{\Omega_{1},\Omega_{2}\\}\,\sum\limits_{i=1}^{\infty}\sqrt{|\sigma_{M_{i}}|}.$
(39)

Because the convergence implies

$|\sigma_{M_{i+1}}|=\eta|\sigma_{M_{i}}|\quad\hbox{with}\quad 0<\eta<1,$ (40)

the infinite sum in (39) turns out into a geometric series

$\sum\limits_{i=1}^{\infty}\sqrt{|\sigma_{M_{i}}|}=|\sigma_{M_{1}}|^{\frac{1}{2}}\sum\limits_{n=0}^{\infty}\eta^{\frac{1}{2}n}=|\sigma_{M_{1}}|^{\frac{1}{2}}\frac{1}{1-\eta^{\frac{1}{2}}},$
(41)

which is convergent. This allows concluding that the upper bound of the finite
time convergence is

$T_{c}\leq
t_{M_{1}}+\frac{\sqrt{2}\,\max\\{\Omega_{1},\Omega_{2}\\}}{1-\sqrt{\eta}}\,\sqrt{|\sigma_{M_{1}}|}.$
(42)

Note that the initial reaching time of the first extreme value is
$t_{M_{1}}\leq|\dot{\sigma}(0)|/(U-\Phi)$, cf. (15). Furthermore, we note that
depending on the branching of trajectories, as shown above for the control-off
phase, also a maximum value $\eta=\max\\{\eta_{1},\eta_{2}\\}$ must be taken
into account in (42).

For determining $\eta_{1}$ and $\eta_{2}$, we evaluate both subsequent extreme
values and obtain, based on (16), the equations

$\begin{split}\sigma^{*}_{M_{i+1}}=&\beta_{2}^{*}\sigma_{M_{i}}+0.5\frac{-\dot{\sigma}_{P_{1}}^{2}}{U-\Phi}=\\\
&\sigma_{M_{i}}\Biggl{(}\beta_{2}-\frac{U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})}{U-\Phi}\Biggr{)}\end{split}$
(43)

and

$\begin{split}\sigma^{**}_{M_{i+1}}=&\beta_{2}^{**}\sigma_{M_{i}}+0.5\frac{-\dot{\sigma}_{P_{2}}^{2}}{U-\Phi}=\\\
&\sigma_{M_{i}}\Biggl{(}\beta_{2}-\frac{U(1-\beta_{1})+\Phi(1-\beta_{2})}{U-\Phi}\Biggr{)},\end{split}$
(44)

Comparing (43) and (44) with (40), results in

$\eta_{1}=\Biggl{|}\beta_{2}-\frac{U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})}{U-\Phi}\Biggr{|}<1$
(45)

and

$\eta_{2}=\Biggl{|}\beta_{2}-\frac{U(1-\beta_{1})+\Phi(1-\beta_{2})}{U-\Phi}\Biggr{|}<1.$
(46)

Following the same line of calculations as above, one can evaluate the
reaching and convergence time of the conventional sub-optimal SMC, for which
$\beta\equiv\beta_{1}$. Considering the $P^{\prime}$ and $P^{\prime\prime}$
peaking points results in

$\begin{split}\hat{T}_{\max,i}=&T^{\prime}+\hat{T}^{**}=\sqrt{2\sigma_{M_{i}}}\,\hat{\Omega},\\\
\hbox{with }\;\hat{\Omega}=&\frac{1}{U-\Phi}\sqrt{(U+\Phi)(1-\beta_{1})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{(U-\Phi)(1-\beta_{1})}.\end{split}$ (47)

Then, the upper bound of the finite time convergence of the conventional sub-
optimal SMC results in

$\hat{T}_{c}<t_{M_{1}}+\frac{\sqrt{2}\,\hat{\Omega}}{1-\sqrt{\hat{\eta}}}\,\sqrt{|\sigma_{M_{1}}|},$
(48)

where $0<\hat{\eta}<1$, cf. with (40). For evaluating $\hat{\eta}$, consider
the peaking point $P^{\prime\prime}$ and obtain

$\hat{\eta}=\Biggl{|}\beta_{1}-\frac{(U+\Phi)(1-\beta_{1})}{U-\Phi}\Biggr{|}<1.$
(49)

Note that both (45) and (46) reduce to (49) if assuming $\beta_{2}=\beta_{1}$.
Also worth noting is that (49) coincides with the convergence condition (24)
if $\beta_{2}=\beta_{1}$ and requires

$\beta_{1}>\frac{\Phi}{U}.$ (50)

We will denote $\Omega_{1},\,\Omega_{2},\,\hat{\Omega}$ as _amplification
factor_ s of the convergence time, and $\eta_{1},\,\eta_{2},\,\hat{\eta}$ as
_contraction factor_ s of the convergence. Inspecting (42) and (48), one can
recognize that the former are proportionally increasing the convergence time
while the latter can drastically slow down the overall convergence once
$\eta,\hat{\eta}$ become closer to one.

###  3.4 Energy-saving control parametrization

For the energy-saving sub-optimal SMC becomes effective, one needs to ensure
that its total energy consumption is less than that of the conventional sub-
optimal SMC, cf. condition (b) at the end of Section 2, while assuming the
$U$, $\Phi$, and $\beta_{1}$ values are the same. In terms of the both
control-on phases, i.e. $u\neq 0$, the amplification factors to be taken into
account are

$\begin{split}\tilde{\Omega}_{1}=&\frac{1}{U-\Phi}\sqrt{(U-\Phi)(1-\beta_{1})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-2\beta_{1}+\beta_{2})}\end{split}$
(51)

and

$\begin{split}\tilde{\Omega}_{2}=&\frac{1}{U-\Phi}\sqrt{(U-\Phi)(1-\beta_{1})}\,+\\\
&\frac{1}{U-\Phi}\sqrt{U(1-\beta_{1})+\Phi(1-\beta_{2})},\end{split}$ (52)

cf. (37), (38). Taking out of consideration the initial reaching phase in (42)
and (48), we obtain two energy cost functions

$J(\beta_{1},\beta_{2},U,\Phi)=\frac{\max\\{\tilde{\Omega}_{1},\tilde{\Omega}_{2}\\}}{1-\max\\{\sqrt{\eta_{1}},\sqrt{\eta_{2}}\\}}$
(53)

and

$\hat{J}(\beta_{1},U,\Phi)=\frac{\hat{\Omega}}{1-\sqrt{\hat{\eta}}},$ (54)

for the energy-saving and conventional sub-optimal SMC, respectively. In order
to find an energy-saving pair of the $(\beta_{1},\beta_{2})$ threshold
parameters for with some fixed $U,\Phi$ process values satisfying $U>\Phi$, we
formulate the

$\underset{\beta_{1},\beta_{2}}{\min}\,\Bigl{[}J(\beta_{1},\beta_{2},U,\Phi)-\hat{J}(\beta_{1},U,\Phi)\Bigr{]}$
(55)

minimization problem under the hard constraint

$J(\beta_{1},\beta_{2},U,\Phi)-\hat{J}(\beta_{1},U,\Phi)<0.$ (56)

In addition, in order to avoid $\beta_{1}\rightarrow 1$, which can lead to
$\hat{T}_{c}\rightarrow\infty$, see [8] for details, we specify the upper
bound

$\hat{J}(\beta_{1},U,\Phi)<\hat{J}_{\max}.$ (57)

This implies restriction on the convergence time $\hat{T}_{c}$ and, this way,
forces a reasonable $\beta_{1}<1$ in course of solving the minimization
problem (55).

Figure 3: Convergence time cost function $\hat{J}$ in dependency of
$\beta_{1}$.

The convergence time cost function $\hat{J}$ is exemplary visualized in Figure
3 in dependency of $\beta_{1}$, for two cases assuming $U=1$ and
$\Phi=\\{0.3,\,0.6\\}$. Note that $\hat{J}\rightarrow\infty$ for both boundary
values $\beta_{1}\rightarrow\Phi/U$, cf. (50), and $\beta_{1}\rightarrow 1$.

Note that the hard constraint (56) is required to ensure that the total
control-on time of the energy-saving sub-optimal SMC is shorter than that of
the conventional one. Moreover, one can recognize that (55) ensures the
maximal possible energy-saving, due to the control-off phases throughout the
convergence, comparing to the conventional sub-optimal SMC, both having the
same $\beta_{1}$ value. That means for any fixed $\beta_{1}$ complying with
(57), there is an optimal $\beta_{2}$ counterpart. The latter represents the
solution of the constrained minimization problem (55). The results of the
constrained optimization are exemplary visualized in Figure 4, for the
perturbation to control ratio equal to $\Phi/U=0.3$.

Figure 4: Constrained objective function $(J-\hat{J})$ of the $\beta_{1}$,
$\beta_{2}$ parameters, for exemplary perturbation to control ratio
$\Phi/U=0.3$.

It should be noted that the formulated optimization problem does not take
explicitly into account those transient phases where the control is off, cf.
(37), (38) and (51), (52). However, this does not imply any issues since the
finite time convergence of the energy-saving sub-optimal SMC is upper bounded
by (42).

##  4 Chattering due to actuator dynamics

If the system under control is subject to an additional (parasitic) actuator
dynamics, the second-order sliding mode experiences the residual steady-state
oscillations, also known as _chattering_ , see e.g. [10, 11]. For the first-
order actuator dynamics with a time constant $\mu>0$, the control variable
$u(t)$ in (2) must be substituted by a new control variable $v(t)$, which is
the solution of

$\mu\dot{v}(t)+v(t)=u(t).$

Assume that the stable residual steady-state oscillations of the system (2)
with the new input channel $v$ and control (14) are established. Then, the
input-output map of the energy-saving sub-optimal SMC (14) takes the form of a
three-state hysteresis relay as shown in Figure 5. For steady-state
oscillations, the switching thresholds keep the constant values and are
symmetric with respect to zero. Note that the amplitude of oscillations is
determined by the cyclic extreme value $\sigma_{M_{i}}\equiv\sigma_{A}$.
Furthermore, we note that if $\beta_{1}=\beta_{2}$, the input-output map
reduces to a standard two-state hysteresis relay, as it was used for harmonic
balance analysis of the original sub-optimal control, cf. [12, chapter 5.2].

Figure 5: Hysteresis relay representation of the control (14) at residual
steady-state oscillations due to the parasitic actuator dynamics.

Before examining the residual oscillating behavior of the energy-saving sub-
optimal SMC, recall the principle procedure of the describing function (DF)
analysis, see e.g. [13], here in context of the second-order SMC and, in
particular, when using the sub-optimal SMC. For the given DF of the SMC in
feedback, denoted by $N(\sigma_{A})$, the harmonic balance equation

$N(\sigma_{A})\,W(j\omega)+1=0$ (58)

must have a real solution, in terms of the amplitude $\sigma_{A}$ and angular
frequency $\omega$, for the residual steady-state oscillations to exist.
Considering the double-integrator plant augmented by the first-order actuator
dynamics, we write

$W(j\omega)=\frac{1}{s^{2}(\mu
s+1)}\Biggr{|}_{s=j\omega}=\frac{-1}{\omega^{2}+j\cdot\omega^{3}\mu}.$ (59)

For the conventional sub-optimal SMC, parameterized by $U$ and $\beta_{1}$,
the DF is given by, cf. [12],

$\hat{N}(\sigma_{A})=\frac{4U}{\pi\sigma_{A}}\Bigl{(}\sqrt{1-\beta_{1}^{2}}+j\beta_{1}\Bigr{)}.$
(60)

Note that also for the generalized sub-optimal SMC, the DF analysis was
provided in [14], cf. with (60). The negative reciprocal of (60) yields

$-\hat{N}(\sigma_{A})^{-1}=\frac{\pi\sigma_{A}}{4U}\Bigl{(}-\sqrt{1-\beta_{1}^{2}}+j\beta_{1}\Bigr{)}.$
(61)

One can show that the locus of (61), that is its graphical interpretation in
the harmonic balance (58), is a straight line starting in the origin and
forming a clockwise angle $\hat{\phi}$, see Figure 6. The latter can be
directly calculated as

$\hat{\phi}=\arctan\Biggl{(}-\frac{\beta_{1}}{\sqrt{1-\beta_{1}^{2}}}\Biggr{)}.$
(62)

One can recognize that larger $\beta_{1}$-parameter values lead to a larger
$|\hat{\phi}|$ and, therefore, shirt the locus intersection towards higher
frequencies and lower amplitudes.

Figure 6: Negative reciprocal of DF of the conventional and energy-saving sub-
optimal SMC crossing the Nyquist plot of $W(j\omega)$.

Since the energy-saving sub-optimal SMC constitutes a linear combination of
two negative hysteresis relays, cf. (14), its DF can be represented as a
summation of two DFs of the type (60), this way resulting in

$N(\sigma_{A})=\frac{2U}{\pi\sigma_{A}}\Bigl{(}\sqrt{1-\beta_{1}^{2}}+\sqrt{1-\beta_{2}^{2}}+j(\beta_{1}+\beta_{2})\Bigr{)}.$
(63)

After evaluating the negative reciprocal of (63), similar to (61), we obtain

$\phi=\arctan\Biggl{(}-\frac{\beta_{1}+\beta_{2}}{\sqrt{1-\beta_{1}^{2}}+\sqrt{1-\beta_{2}^{2}}}\Biggr{)}.$
(64)

In Figure 6, the negative reciprocal of both DFs are depicted, once of the
conventional sub-optimal SMC with DF (60) and once of the energy-saving sub-
optimal SMC with DF (63), together with the Nyquist plots of $W(j\omega)$ and
double integrator $(j\omega)^{-2}$. The energy-saving sub-optimal SMC has
always a lower angle $|\phi|$ for the same fixed $\beta_{1}$ value, while for
$(\beta_{1}-\beta_{2})\rightarrow 0$ the angle $\phi\rightarrow\hat{\phi}$.
One can also recognize that unlike the double integrator, the Nyquist plot of
$W(j\omega)$ has always an intersection point with the negative reciprocals of
DFs. This implies an existence of $(\sigma_{A},\omega)$ solutions of the
harmonic balance equation (58). The corresponding steady-state oscillations
have a higher amplitude and lower frequency for the energy-saving sub-optimal
SMC compared to those of the conventional sub-optimal SMC.

Equating the phase angle of (59) and (64)

$\frac{\mathrm{Im}\\{W\\}}{\mathrm{Re}\\{W\\}}=-\frac{\beta_{1}+\beta_{2}}{\sqrt{1-\beta_{1}^{2}}+\sqrt{1-\beta_{2}^{2}}}$

and solving it with respect to $\omega$ results in

$\omega_{c}=\mu^{-1}\frac{\beta_{1}+\beta_{2}}{\sqrt{1-\beta_{1}^{2}}+\sqrt{1-\beta_{2}^{2}}}$
(65)

of the chattering. The corresponding amplitude of the steady-state
oscillations is then obtained as

$\sigma_{A}=\frac{\sqrt{\mu^{2}\omega_{c}^{2}+1}}{\omega_{c}^{2}\bigl{(}\mu^{2}\omega_{c}^{2}+1\bigr{)}}.$
(66)

##  5 Numerical examples

This section demonstrates the numerical examples which characterize the main
properties of the energy-saving sub-optimal SMC formulated and analyzed above.
Implemented is a double-integrator system (2) with $g=1$ and the energy-saving
sub-optimal SMC (14). For simulating the conventional sub-optimal SMC, the
threshold value is set $\beta_{2}=\beta_{1}$, cf. section 2. All simulation
results are obtained by using the simple first-order Euler-type numerical
solver with the fixed-step size of 0.001 sec.

###  5.1 Convergence behavior

The convergence behavior of the energy-saving sub-optimal SMC is exemplary
shown for examining the parametric condition (24). The constraining parameters
are chosen so that $\Phi/U=0.5$ with $U=1$ and $\mu=0$, while for a worst-case
scenario of the bounded perturbation $f=0.5\,\mathrm{sign}(\dot{\sigma})$ is
assumed. Note that the assumed $f$ constitutes an always ’co-acting’
perturbation signal of the maximal amplitude $\Phi$, a worst-case scenario
that challenges the contraction during each reaching cycle and, at large, the
total convergence to $\sigma=\dot{\sigma}=0$ equilibrium. Figure 7 discloses
the converging behavior for $\beta_{1}=0.8$, $\beta_{2}=0.25$ versus the
diverging behavior for $\beta_{1}=0.8$, $\beta_{2}=0.19$. Note that here, the
threshold values distance $\beta_{1}-\beta_{2}>0.6$ is required for (24) is
fulfilled.

Figure 7: Exemplary response of the controlled sliding variable for
$\beta_{1},\beta_{2}$ once fulfilling (in blue) and once violating (in red)
the parametric convergence condition (24); $U=1$, $\Phi=0.5$,
$f=0.5\,\mathrm{sign}(\dot{\sigma})$.

###  5.2 Energy-saving versus conventional sub-optimal SMC

The energy-saving sub-optimal control behavior is analyzed in comparison to
the conventional sub-optimal one, with regard to the recorded convergence time
$T_{c}$ and overall energy consumption

$\int_{0}^{T_{c}}|u(t)|dt.$

In order to determine the numerical convergence instant, the $T_{c}$ value is
obtained once the Euclidian norm of the state vector
$\bigl{(}\sigma(t),\dot{\sigma}(t)\bigr{)}$ decreases below some assigned low
residual constant, i.e. $\bigl{\|}(\sigma,\dot{\sigma})\bigr{\|}(t)<0.004$.
The residual constant is determined out from the numerical simulations, with
respect to the solver type and step-size and the resulted numerical pattern
around $\sigma=0$ after the principal dynamics of $\sigma(t)$ has converged.
For the assigned $\Phi/U=0.3$ ratio with $U=1$ and $\mu=0$, three
configurations of the perturbing signal are considered: $f=0$,
$f=0.3\,\mathrm{sign}(\dot{\sigma})$, $f=-0.3\,\mathrm{sign}(\dot{\sigma})$.
Note that the second one is ’co-acting’ to the control signal. This has,
however, an adverse impact on contraction of the state trajectories, cf.
Figure 2. The third one acts, on the contrary, as a maximal possible system
damping and can negatively affect the control-off phases in terms of the
reaching time, cf. section 3.3. Two parameter sets, obtained from the
constrained optimization, $[\beta_{1}=0.7,\beta_{2}=0.55]$ and
$[\beta_{1}=0.83,\beta_{2}=0.32]$ cf. with Figure 4, are used. All simulation
results are summarized in Table 1. Recall that for the conventional sub-
optimal SMC the $\beta_{2}=\beta_{1}$ is always assigned.

Table 1: Convergence time and energy consumption |  $\beta_{1}=0.7$, $\beta_{2}=0.55$ |  $\beta_{1}=0.83$, $\beta_{2}=0.32$
---|---|---
Configuration |  $T_{c}$ |  $\int|u|dt$ |  $T_{c}$ |  $\int|u|dt$
conventional
$f=0$ |  0.420 |  0.391 |  0.643 |  0.580
energy-saving
$f=0$ |  0.345 |  0.292 |  0.332 |  0.176
conventional
$f=0.3\,\mathrm{sign}(\dot{\sigma})$ |  0.342 |  0.285 |  0.552 |  0.448
energy-saving
$f=0.3\,\mathrm{sign}(\dot{\sigma})$ |  0.288 |  0.247 |  0.435 |  0.278
conventional
$f=-0.3\,\mathrm{sign}(\dot{\sigma})$ |  0.517 |  0.441 |  0.645 |  0.548
energy-saving
$f=-0.3\,\mathrm{sign}(\dot{\sigma})$ |  0.442 |  0.325 |  3.718 |  0.155

One can recognize a superior performance of the energy-saving sub-optimal SMC
in all above given cases, except the last row in Table 1 (see light red
shadowed). Here the energy saving is largely achieved, but the convergence
time is significantly increased comparing to the conventional sub-optimal SMC.
This is related to a relatively slowly ’drifting’ trajectory between the lying
apart $\beta_{1}=0.83$ and $\beta_{2}=0.32$, being driven by the
$f=-0.3\,\mathrm{sign}(\dot{\sigma})$ perturbation. The situation with $T_{c}$
improves, still guaranteing for energy saving, when both threshold parameters
are lying closer to each other like in the shown case $\beta_{1}=0.7$ and
$\beta_{2}=0.55$, see Table 1.

For better interpreting the listed results, some of the above configurations
are shown below in the plots of $\sigma(t)$ and $u(t)$. All configurations
with $f=0$ are depicted in Figures 8 and 9.

Figure 8: Convergence of $\sigma$ with $f=0$ for the parameters
$[\beta_{1}=0.7,\beta_{2}=0.55]$ in (a) and $[\beta_{1}=0.83,\beta_{2}=0.32]$
in (b).

One can see that the energy-saving sub-optimal SMC converges faster,
correspondingly stops to switch earlier than its conventional counterpart. One
can also notice that both controllers here disclose a monotonic convergence
without twisting, cf. section 2.

Figure 9: Control $u$ with $f=0$ for the parameters
$[\beta_{1}=0.7,\beta_{2}=0.55]$ in (a) on the left, and
$[\beta_{1}=0.83,\beta_{2}=0.32]$ in (b) on the right.

The convergence in case of $f=0.3\,\mathrm{sign}(\dot{\sigma})$ is also
exemplary shown in Figure 10. Here, the conventional sub-optimal SMC has a
monotonic convergence, in accord with (13). On the contrary, the energy-saving
sub-optimal SMC changes into twisting mode here, because it allows for control
phases driven only by the perturbation quantity.

Figure 10: Convergence of $\sigma$ with $f=0.3\,\mathrm{sign}(\dot{\sigma})$
for the parameters $[\beta_{1}=0.7,\beta_{2}=0.55]$ in (a) and
$[\beta_{1}=0.83,\beta_{2}=0.32]$ in (b).

###  5.3 Residual steady-state oscillations

For evaluating the residual oscillations (i.e. chattering) owing to a
parasitic actuator dynamics, cf. section 4, different parameter sets with
$\mu\neq 0$ are compared. Three parameter configurations used are shown in
Table 2. The amplitude and frequency of the residual steady-state oscillations
are computed based on the harmonic balance analysis provided in section 4,
i.e. using (65), (66), and compared with those out from the numerical
simulations. Certain, yet acceptable, error of the harmonic balance analysis
is not surprising, since DF solely approximates the first main harmonic, cf.
with results in [14].

Table 2: Harmonic balance analysis versus simulation results | Harmonic balance | Numerical simulation
---|---|---
Parameters |  $\sigma_{A}$ |  $\omega_{c}$ |  $\sigma_{A}$ |  $\omega_{c}$
$\mu=0.03$, $\beta_{1}=0.8$, $\beta_{2}=0.2$ |  0.0019 |  21.1 |  0.0025 |  20.0
$\mu=0.01$, $\beta_{1}=0.6$, $\beta_{2}=0.0$ |  0.00085 |  33.3 |  0.0012 |  30.2
$\mu=0.01$, $\beta_{1}=0.8$, $\beta_{2}=0.2$ |  0.00021 |  63.3 |  0.00029 |  59.3

Figure 11 exemplifies the appearance of residual steady-state oscillations
when the actuator dynamics is in place. The time response of the sliding
variable $\sigma(t)$ is shown above for all three parameter sets, cf. Table 2.
The control value $u(t)$, i.e. of the control (14), is exemplary shown below
for the parameter set $\mu=0.03$, $\beta_{1}=0.8$, $\beta_{2}=0.2$.

Figure 11: Exemplary response of the residual steady-state oscillations,
sliding variable (above) and control value (below).

Table 2 and Figure 11 confirm the chattering analysis of the energy-saving
sub-optimal SMC, developed in section 4, and disclose how the amplitude and
frequency change depending on the control and actuator parameters $\beta_{1}$,
$\beta_{2}$, and $\mu$.

##  6 Conclusions

The problem of saving energy by the control-off phases in the second-order
sliding modes was addressed. More specifically, the conventional sub-optimal
SMC, see [7] for survey, was extended for a class of the systems (2) with
$g=1$ and $u\in\\{-U,\,0,\,U\\}$. This way, a well parameterizable $u=0$ phase
appears within each reaching cycle, during the entire finite-time convergence.
The latter was estimated for the worst-case scenario of an upper-bounded
perturbation occurring in each control phase. The analyzed contraction and
maximal reaching- as well as total convergence-time served to formulate a
constrained minimization problem for an energy cost function. Two switching
threshold parameters represent the solution of a complex nonlinear
optimization, while the perturbation-boundary to control-authority ratio, i.e.
$\Phi/U$, is crucial for convergence and energy-saving conditions. Also the
impact of an additional parasitic actuator dynamics, typical for real sliding
modes in terms of chattering, was analyzed based on the harmonic balance. The
overall analysis performed and the dedicated numerical evaluations argue in
favor of using the proposed energy-saving sub-optimal SMC for $\Phi/U<0.35$.
In applications, a resettable control-off mode can also be added for
$\bigl{\|}(\sigma,\dot{\sigma})\bigr{\|}(t)<\mathrm{const}$, this way ensuring
energy saving after convergence and a reactivation in case of perturbations in
the equilibrium.

## References

  * [1] V. Utkin, Sliding modes in control and optimization, Springer, 1992.
  * [2] C. Edwards, S. Spurgeon, Sliding mode control: theory and applications, CRC Press, 1998.
  * [3] Y. Shtessel, C. Edwards, L. Fridman, A. Levant, Sliding mode control and observation, Springer, 2014.
  * [4] A. Levant, Sliding order and sliding accuracy in sliding mode control, Inter. Journal of Control 58 (6) (1993) 1247–1263.
  * [5] L. Fridman, A. Levant, Higher order sliding modes, Marcel Dekker New York, 2002, pp. 53–102.
  * [6] G. Bartolini, A. Ferrara, E. Usai, Output tracking control of uncertain nonlinear second-order systems, Automatica 33 (12) (1997) 2203–2212.
  * [7] G. Bartolini, A. Pisano, E. Punta, E. Usai, A survey of applications of second-order sliding mode control to mechanical systems, Inter. Journal of Control 76 (9–10) (2003) 875–892.
  * [8] M. Athans, Fuel-optimal control of a double integral plant with response time constraints, IEEE Transactions on Applications and Industry 83 (73) (1964) 240–246.
  * [9] G. Bartolini, A. Ferrara, A. Levant, E. Usai, On second order sliding mode controllers, Springer, 1999, pp. 329–350.
  * [10] G. Bartolini, A. Ferrara, E. Usai, Chattering avoidance by second-order sliding mode control, IEEE Transactions on Automatic Control 43 (2) (1998) 241–246.
  * [11] I. Boiko, L. Fridman, A. Pisano, E. Usai, Analysis of chattering in systems with second-order sliding modes, IEEE Transactions on Automatic Control 52 (11) (2007) 2085–2102.
  * [12] I. Boiko, Discontinuous control systems: frequency-domain analysis and design, Birkhaeuser, 2009.
  * [13] D. Atherton, Nonlinear Control Engineering - Describing Function Analysis and Design, Workingam Beks, 1975.
  * [14] I. Boiko, L. Fridman, R. Iriarte, A. Pisano, E. Usai, Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators, Automatica 42 (5) (2006) 833–839.