# Super-twisting based sliding mode control of hydraulic actuator without
velocity state

Manuel A. Estrada  Michael Ruderman  Leonid Fridman  Facultad de Ingeniería,
Universidad Nacional Autónoma de México (UNAM), 04510 Mexico City, Mexico.
<EMAIL_ADDRESS><EMAIL_ADDRESS> Department of Engineering
Sciences, University of Agder, 4879-Grimstad, Norway.
<EMAIL_ADDRESS>
###### Abstract

This paper provides a novel surface design and experimental evaluation of a
super-twisting algorithm (STA) based control for hydraulic cylinder actuators.
The proposed integral sliding surface allows to track a sufficiently smooth
reference without using the velocity state which is hardly accessible in the
noisy hydraulic systems. A design methodology based on LMI’s is given, and the
STA gains are designed to be adjusted by only one free parameter. The
feasibility and effectiveness of the proposed control method are shown on a
standard hydraulic test bench with one linear degree of freedom and passive
load, where a typical motion profile is tracked with a bounded average error
below $1\%$ from the total drive effective output.

###### keywords:

sliding mode control, robust control design, hydraulic actuator, motion
control

††thanks: Authors accepted manuscript, Oct 2023

##  1 Introduction

Hydraulic actuators are one of the most common options to work in various
applications where high forces in harsh environments are required for a robust
operation, cf. [19]. In that sense, motion control of such actuators remain a
challenging task due to their complex nonlinear behavior and uncertainties,
cf. e.g. [32]. Notwithstanding, most of today’s hydraulic applications still
rely on a few standard linear feedback controllers, like e.g. PID, see for
instance [6].

On the other hand, there are also extensions to nonlinear-type PID controllers
with extra compensators, such as for example feed-forward compensation of the
dead-zone in valves which was shown to be useful for improving the
performance, see e.g. [16]. Apart from that, there are experimental
evaluations on how the uncertainties and process noise affect the controlled
response of such systems [21, 1]. One can also notice that there exist several
remarkable challenges in the implementation of such actuators in real world
applications due to environmental conditions [23]. More references related to
the controlled hydraulic applications can also be found in e.g. [18].

Over and above that, the control problem is not only attributed to the
mechanical properties of the hydraulic actuators. It is rather hydraulic by-
effects which can cause large uncertainties in the whole system dynamics.
Then, it appears imperative that the hydro-dynamical behavior must also be
taken into account, from the control design point of view, see e.g. [27, 25].

Per se, sliding mode control has proven to be an efficient control solution
against uncertainties and external disturbances, see e.g. [10, 9, 7] as
related to the design tools used in the present work. Therefore, it appears
suitable to control also the hydraulic systems, as it was presented in some
previous works, e.g. [5] where a sliding mode observer was designed to
estimate the so-called equivalent control and use it to enforce the sliding
mode [31]. In [17], a sliding surface combined with an $H_{\infty}$ approach
was presented to robustly track a desired trajectory. On the other hand, an
application of the integral sliding mode based approach was shown in
industrial scenario in [14]. In that work, an input-output linearization
technique with the design of an integral sliding surface was proposed to
control the pressure of a die-cushion (hydraulic) cylinder drive, and a
comparison with standard PI and PID controllers was provided.

Although several previously mentioned solutions obtained a sufficiently good
performance, their approach relies often on the first-order (discontinuous)
sliding mode control. The use of such discontinuous control produces the so-
called chattering effect, that means undesired high frequency oscillations
which can damage actuators and the plant. Note that despite chattering appears
generally as unavoidable for all-types of real sliding mode controllers (due
to additional parasitic actuator lag and neglected sub-dynamics in feedback
loop), here it is especially a discontinuous control action which is most
challenging for hydro-mechanical actuators and plants, like control valves and
pressurized cylinders.

For substituting a discontinuous control by a continuous one, [15] introduced
for instance the super-twisting algorithm (STA). As application of STA in
hydraulic systems, an observer-based sliding mode control of low-power
hydraulic cylinder, in the case of unknown load forces, can be found for
example in [13]. Also, an STA-based observer of the large unknown load forces
in hydraulic drives was presented experimentally in [28].

Despite the above mentioned attempts and references therein, an optimal
solution to robustly track a given trajectory while generating a continuous
control signal remains an sufficiently open problem for hydraulic systems. In
particular, the lacking measurement of the system states constitutes a large
practical issue. While the output relative displacement (i.e. stroke) of
cylinder drives is often measured by the attached (yet noisy) linear encoders,
the relative velocity remains mostly unknown. It can also be hardly estimated
(for feedback control) from the stroke measurement with a relatively low SNR
(signal-to-noise ratio). Similarly, the measured pressure signals, if
available, contain often a high measurement and process noise.

Against the background mentioned above, the contribution of the present work
is summarized as follows.

  * •

A novel integral sliding surface is proposed, such that, the pressure state
works as a virtual control for mechanical sub-system, achieving the robust
tracking of the desired trajectory without any measurement or estimation of
the velocity signal.

  * •

An STA-based control law in combination with equivalent-type control produces
the required differential pressure, as a desired virtual control, while
generating the continuous control signal.

  * •

An easy to follow design of the control gains is provided for application.

  * •

The proposed approach is evaluated experimentally on a full-scale nonlinear
system, such that the feasibility of the results is shown in presence of the
complex system dynamics and uncertainties.

The structure of the paper is as follows. In section 2, the nonlinear dynamics
of the hydraulic cylinder actuator is summarized. Section 3 presents a
linearization of the model and formulate the control problem. In section 4,
the solution of this problem is proposed and the control design is left for
section 5. The reader may be interested in the application of an output based
control law using an estimator for the missing state, therefore, in section 6
a comparison with the approach presented in [22] is presented. The
experimental evaluations are provided in section 7. Finally, the conclusions
are drawn in section 8, and the proofs are left in the Appendix.

Notation. For $x\in\mathbb{R}^{n}$, $\|x\|$ denotes the Euclidean norm and
$\mathbb{C}^{n}$ stands for the set of $n$-times continuous differentiable
functions. The sign function is defined on $\mathbb{R}$ by
$\text{sign}(z)=\frac{z}{|z|}$ for $z\neq 0$ and
$\text{sign}(0)\in\left[-1,1\right]$. The following notation is used
throughout the paper, for $s\in\mathbb{R}$ and $q\in\mathbb{R}$, $\lceil
s\rfloor^{q}=\left|s\right|^{q}\text{sign}(s)$. For a matrix
$G\in\mathbb{R}^{n\times n}$, $\lambda_{min}(G)$ (resp. $\lambda_{max}(G)$)
denotes the minimum (resp. maximum) eigenvalue of $G$. The identity matrix of
an appropriate dimension is denoted by $\mathbb{I}$. Solutions for systems
with discontinuous right-hand side are understood in the Filippov sense [11].

##  2 Dynamics of hydraulic actuator

The dynamics of the hydraulic cylinder actuator is briefly summarized for the
sake of clarity. For more details on hydraulic circuits we refer to [19], and
on those related to the used type of linear-stroke actuators to e.g. [27].

###  2.1 Servo-valve approximation

The servo-valve used to control the flow into the cylinder has the own
dynamics corresponding to electro-magnetic actuation of a solenoid which moves
the spool inside the valve. The internal dynamics of the controlled servo-
valve can be well approximated by the second-order

$\ddot{\nu}+2\zeta_{v}\omega_{0}\dot{\nu}+\omega_{0}^{2}\nu=\omega_{0}^{2}U(t),$
(1)

where the output $\nu(t)$ is the relative spool position, and the input value
$U(t)$ is the applied valve control reference. The parameters $\omega_{0}$ and
$\zeta_{v}$ are the eigen-frequency and damping ratio, respectively, of the
low-level controlled closed-loop system. The parameter values have been
experimentally identified in the work [24] and are inline with the valve’s
manufacturer data-sheet. Additionally, the relation between the spool position
and the input flow rate is not linear due to an overlap in the spool area and
saturation effects inside the servo-valve. These output nonlinearities lead to
the whole actuator dynamics (1) augmented by [27]

$g(t)=\begin{cases}c_{s}\text{sign}(\nu(t))\,,\quad\quad\quad\quad|\nu(t)|\geq
c_{s}+c_{d}\\\
\,0\,,\quad\quad\quad\quad\quad\quad\quad\quad\,\,|\nu(t)|<c_{d}\\\
\nu(t)-c_{d}\text{sign}(\nu(t))\,,\quad\text{otherwise},\end{cases}$ (2)

where $g(t)$ is the valve’s orifice opening which is governing the flow
through it. The constants $c_{d}$ and $c_{s}$ are the dead-zone size and
saturation level, respectively.

###  2.2 Orifice and continuity equations

As described in [27], with the load related pressure between the both chambers
$P=P_{A}-P_{B}$, and under the hypothesis of a closed hydraulic circuit, that
means the absolute value of the orifice equations for both chambers are
essentially the same, one has the single orifice equation

$Q=gK_{f}\sqrt{\frac{1}{2}\left(P_{S}-\text{sign}(g)P\right)}\,.$ (3)

Here $Q$ is the overall (load) flow rate of the hydraulic medium, and $P_{S}$
is the supply pressure. $K_{f}$ is the valve flow coefficient, available from
the manufacturer data of the servo-valve. Then, one can assume that the
pressure in each chamber satisfy the following

$P_{A}=\frac{P_{S}+P}{2}\,\quad P_{B}=\frac{P_{S}-P}{2}\,,$ (4)

provided zero pressure in the tank, which is a reasonable assumption for
various hydraulic drive systems. With this in mind, the load pressure gradient
has the form

$\dot{P}=\frac{4E}{V_{t}}\left(Q-A\dot{q}-C_{L}P\right)+\delta_{P}(t)\,,$ (5)

where $V_{t}$ is the total volume in the hydraulic circuits, i.e. between the
pressurized inlet and zero-pressure outlet of the control servo-valve,
including the cylinder with piping. The average effective piston area of the
cylinder is $A=0.5\left(A_{A}+A_{B}\right)$, being $A_{A}$ and $A_{B}$ the
effective cross-sections in the chambers $A$ and $B$ respectively. The
coefficient $C_{L}$ corresponds to some internal leakage (if any) between the
chambers of cylinder, and $\delta_{P}(t)$ constitutes the all (summarized)
uncertainties in the pressure dynamics. The latter are related to the model
simplification and side-effects, like changes in the coefficients due to
varying temperature and external effects, manufacturing tolerances, wear, and
others.

###  2.3 Mechanical sub-system

The mechanical part corresponds to one linear degree of freedom (1DOF)
mechanical system, i.e. moving cylinder piston, of the form

$m\ddot{q}(t)=AP-f(\dot{q})-F_{L}(t)\,,$ (6)

where $m$ is the total mass of the moving piston rod with payload, $F_{L}(t)$
is an external perturbation interfering in the mechanical part, and $q$ is the
linear displacement of cylinder. On the other hand, a weakly known friction
term can comply with [27]

$f(\dot{q})=\tanh(\vartheta\dot{q})\left[F_{c}+(F_{s}-F_{c})\text{exp}\left(-|\dot{q}|^{\iota}\chi^{-\iota}\right)\right]+\sigma\dot{q}$
(7)

which contains an approximation of the Coulomb friction with the smoothing
parameter $\vartheta\gg 1$ for motion reversals. The linear viscous friction
coefficient is $\sigma>0$, the coefficients related to the Stribeck effect are
$F_{s}>F_{c}$, $\chi>0$, and $\iota\neq 0$. Note that while (7) provides
certain structure for the friction-related perturbations, the numerical
parameter values and thus the whole $f(\cdot)$ can remain weakly or even fully
unknown during the operation.

##  3 Linearized model and problem statement

It is worth emphasizing, right from the beginning, that the servo-valve
dynamics is significantly faster comparing to the hydro-mechanical sub-system
and is well specified by the manufacturer. It can be seen from the previous
work [24] that it is at least two orders of magnitude faster in terms of the
time constants. Therefore, it is assumed that the input $g$ is not subject to
additional dynamics, cf. with section 2.1, while the effective parasitic input
dynamics (of the valve as actuator) is respected when designing the STA
control gains, cf. [26]. Recall the orifice equation

$Q=\frac{gK_{f}}{\sqrt{2}}\sqrt{P_{S}-P\text{sign}(g)}$

and consider $\Omega\equiv\sqrt{P_{S}-P\text{sign}(z)}$ as an input gaining
factor that can have variations depending on the hydro-mechanical load. By a
physical reasoning that the load pressure $P$ can not be larger than the
supply pressure $P_{S}$, it is satisfying
$0<\epsilon_{\Omega}\leq\Omega\leq\sqrt{P_{S}}$. Now, considering the
operation points $g_{0}$ and $P_{0}$ of the orifice opening and load pressure,
respectively, one can linearize the system and obtain the coefficients
$C_{qp}=\left.\frac{\partial Q}{\partial P}\right|_{g=g_{0}}$ and
$C_{q}=\frac{1}{\sqrt{2}}K_{f}\Omega$. Then, with the new state variables

$x^{T}=\begin{bmatrix}x_{1}&x_{2}&x_{3}\end{bmatrix}\equiv\begin{bmatrix}q&\dot{q}&P\end{bmatrix}$

one obtains the following linearized model

$\displaystyle\dot{x}_{1}$ $\displaystyle=x_{2},$ (8)
$\displaystyle\dot{x}_{2}$
$\displaystyle=-\frac{\sigma}{m}x_{2}+\frac{A}{m}x_{3}+\delta_{2}(t,x_{2})+F_{L}(t),$
$\displaystyle\dot{x}_{3}$
$\displaystyle=-\frac{4EA}{V_{t}}x_{2}-\frac{4EC_{qp}}{V_{t}}x_{3}+\frac{4EC_{q}}{V_{t}}u(t)+\delta_{3}(t)\,.$

where under the assumption of the servo-valve dynamics having a unitary gain
and a low phase lag, the control input is supposed to be $u(t)=\nu(t)$. Here,
perturbation $\delta_{3}(t)$ is equivalent to the perturbation in the load
pressure $\delta_{P}(t)$, and $\delta_{2}(t,x_{2})$ is a vanishing
perturbation which includes also the unknown friction term (7). Introduce the
tracking error variables as

$\displaystyle e_{1}$ $\displaystyle=x_{1}-r(t),$ (9) $\displaystyle e_{2}$
$\displaystyle=x_{2}-\dot{r}(t),$ $\displaystyle\eta$ $\displaystyle=\tau
x_{3},$

where $r(t)\in\mathbb{C}^{4}$ is the reference signal to be tracked, and
$\tau>0$ is a positive scaling constant for the measurable $x_{3}$ state.
Defining the perturbation terms as

$\displaystyle\bar{\delta}_{2}(t,e)$
$\displaystyle=\delta_{2}(t,x_{2})-\delta_{2}(t,\dot{r}(t))\,,$ (10)
$\displaystyle\beta(t)$
$\displaystyle=\delta_{2}(t,\dot{r}(t))-\ddot{r}(t)-\frac{\sigma}{m}\dot{r}(t)+F_{L}(t)\,,$
$\displaystyle\bar{\delta}_{3}(t)$
$\displaystyle=\tau\delta_{3}(t)-\frac{4\tau EA}{V_{t}}\dot{r}(t)\,,$

the error dynamics can be written as

$\displaystyle\dot{e}_{1}$ $\displaystyle=e_{2}\,,$ (11)
$\displaystyle\dot{e}_{2}$ $\displaystyle=-\frac{\sigma}{m}e_{2}+\frac{A}{\tau
m}\eta+\bar{\delta}_{2}(t,e)+\beta(t)\,,$ $\displaystyle\dot{\eta}$
$\displaystyle=-\frac{4\tau
EA}{V_{t}}e_{2}-\frac{4EC_{q}}{V_{t}}\eta+\frac{4\tau
EC_{q}}{V_{t}}u(t)+\bar{\delta}_{3}(t)\,.$

Control problem: For some positive constants $\overline{e}$ and $T_{e}$, it is
required to enforce the error states to a vicinity of the origin
$(e_{1},e_{2})=0$,  i.e.
$\left\|\left[e_{1}(t),e_{2}(t)\right]^{T}\right\|\leq\overline{e}$ and that:
(i) for all $t>T_{e}$, (ii) despite the presence of perturbations, and (iii)
ensuring a continuous control signal.

##  4 Proposed control solution

The proposed control design consists of two consecutive steps:

  1. 1.

To control the error in the mechanical sub-dynamics
$(\dot{e}_{1},\dot{e}_{2})$ using the internal state $\eta$ as a virtual
control input, and to compensate for perturbations $\bar{\delta}_{2}(t)$.

  2. 2.

To design a sliding surface $s(e_{1},\eta)$, which depends only on two
measurable states, so as to enforce $\eta$ to control the mechanical sub-
system despite all perturbations by using the super-twisting algorithm (STA).

######  Assumption 1.

For all $\bigl{(}t,x(t)\bigr{)}\in\mathbb{R}_{+}\times\mathbb{R}^{3}$, the
following condition is satisfied for some known non-negative constants $\Psi$
and $L_{3}$:

$\begin{gathered}\left|\frac{\bar{\delta}_{2}(t,e)}{x_{2}(t)-\dot{r}(t)}\right|\leq\Psi\,,\quad\left|\frac{d}{dt}\bar{\delta}_{3}(t)\right|\leq
L_{3}\,,\end{gathered}$ (12)

Note that the above condition on $\bar{\delta}_{2}(t,x)$ is satisfied for
structured frictional perturbation in the mechanical systems, cf. (7). Then
one can bring the system (11) into the following form

$\displaystyle\dot{e}_{1}$ $\displaystyle=e_{2}\,,$ (13)
$\displaystyle\dot{e}_{2}$ $\displaystyle=-\frac{\sigma}{m}e_{2}+\frac{A}{\tau
m}\eta+\psi(t,e)e_{2}+\beta(t)\,,$ $\displaystyle\dot{\eta}$
$\displaystyle=-\frac{4\tau
EA}{V_{t}}e_{2}-\frac{4EC_{qp}}{V_{t}}\eta+\frac{4\tau
EC_{q}}{V_{t}}u(t)+\bar{\delta}_{3}(t)\,,$

with $\psi(t,e)=\bar{\delta}_{2}(t,e)/e_{2}$. It is worth mentioning that
because of Assumption 1, it is implied that $\psi(t,e)$ is always bounded and
$\lim_{e_{2}\rightarrow 0}\psi(t,e)$ exists and is well defined.

In the following, the virtual control law will be designed. First, the nominal
case (i.e. stabilization) when $\beta(t)=0$ is considered, that is, when the
reference signal is constant and there are no external forces. In this case,
it can be proved that $e_{1}$ and $e_{2}$ converge asymptotically to zero.
However, in the case of $\beta\neq 0$, only an uniformly ultimate boundedness
can be achieved. Both results are presented separately for the sake of
clearness.

###  4.1 Virtual control for mechanical sub-system

In this section, the design of a virtual control without requiring the
velocity is presented. Assuming that the mechanical sub-system’s input is the
scaled pressure state $\eta(t)$, let us propose a virtual control design. The
following virtual control is proposed as

$\eta(t)=\int v(e_{1},\eta)dt-\left(\kappa+\frac{4\tau
EA}{V_{t}}\right)e_{1}\,,$ (14)

where $v(e_{1},\eta)=-\gamma_{1}e_{1}-\gamma_{2}\eta$, and
$\kappa,\gamma_{1},\gamma_{2}>0$ are the design parameters. Let us assume that
$\beta(t)=0$, and consider the new virtual state:

$e_{3}=\frac{A}{\tau m}\left(\int v(e_{1},\eta)dt-\kappa e_{1}-\frac{4\tau
EA}{V_{t}}e_{1}\right)\,.$ (15)

Then, the error dynamics of the mechanical sub-system can be represented as

$\displaystyle\dot{e}_{1}$ $\displaystyle=e_{2}\,,$ (16)
$\displaystyle\dot{e}_{2}$ $\displaystyle=-\frac{A}{\tau m}\left(\frac{4\tau
EA}{V_{t}}+\kappa\right)e_{1}-\frac{\sigma}{m}e_{2}$
$\displaystyle+\frac{A}{\tau m}\int v(e_{1},\eta)dt+\psi(t,e)e_{2}\,.$

Noting that $\int v(e_{1},\eta)dt=\eta+\kappa e_{1}+\frac{4\tau
EA}{V_{t}}e_{1}$, then $\eta=\frac{\tau m}{A}e_{3}$ such that

$\displaystyle\dot{e}_{1}$ $\displaystyle=e_{2}\,,$ (17)
$\displaystyle\dot{e}_{2}$
$\displaystyle=-a_{22}e_{2}+e_{3}+\psi(t,e)e_{2}\,,$
$\displaystyle\dot{e}_{3}$ $\displaystyle=a_{23}v\left(e_{1},\frac{\tau
m}{A}e_{3}\right)-\frac{A}{\tau m}\left(\frac{4\tau
EA}{V_{t}}+\kappa\right)e_{2}\,,$

One can recognize that the whole error dynamics system becomes

$\dot{e}=(A_{n}-BR)e+\psi(t,e)He,$ (18)

with $e^{T}=\begin{bmatrix}e_{1}&e_{2}&e_{3}\end{bmatrix}$ and

$\begin{gathered}A_{n}=\begin{bmatrix}0&1&0\\\ 0&-\frac{\sigma}{m}&1\\\
0&0&0\end{bmatrix}\,,\quad B=\begin{bmatrix}0\\\ 0\\\ 1\end{bmatrix}\,,\quad
H=\begin{bmatrix}0&0&0\\\ 0&1&0\\\ 0&0&0\end{bmatrix}\end{gathered}$ (19)

The vector $R$ is now to be designed, noting that the poles of the matrix
$(A_{n}-BR)$ can always be collocated by selecting the vector
$R=\begin{bmatrix}r_{1}&r_{2}&r_{3}\end{bmatrix}=\begin{bmatrix}a_{23}\gamma_{1}&a_{23}\left(\kappa+\alpha\right)&\gamma_{2}\end{bmatrix}$.
Then it comes from controllability of the pair $(A_{n},B)$ that one can design
$\gamma_{1}=\frac{\tau m}{A}r_{1}$, $\gamma_{2}=r_{3}$ and $\kappa=\frac{\tau
m}{A}r_{2}-\alpha$ and therefore, assign the dynamics of the extended sub-
system via (14) without requiring the velocity variable.

######  Theorem 1.

Assume that $\beta(t)=0$ and Assumption 1 is fulfilled. The error variables
$e(t)\rightarrow 0$ as $t\rightarrow\infty$ if and only if for some
$M=M^{T}>0$ the following LMI holds:

$MA_{n}+A_{n}^{T}M-R^{T}B^{T}M-MBR+\Psi H^{T}M+\Psi MH<0\,.$ (20)

Proof. The proof is given in Appendix A.

######  Remark 2.

The dynamic behavior can be assigned through LMI approach so that to satisfy
(20) by solving it with respect to the matrix $M$.

The analysis is different when the unmatched perturbation $\beta(t)\neq 0$.
This can happen when attempting a robust tracking of the hydraulic actuator in
presence of the external mechanical forces $F_{L}(t)$. This is discussed
below.

######  Lemma 3.

Suppose that Assumption 1 is satisfied and there exists $\beta_{M}>0$ such
that $|\beta|\leq\beta_{M}$. If (20) is satisfied, then the trajectories of
$e(t)$ satisfies the following bound:

$\|e(t)\|\leq{\frac{2(\lambda_{max}(M))^{3/2}\beta_{M}}{\theta_{V}\mu(M)(\lambda_{min}(M))^{1/2}}}\,,$
(21)

for positive constants $\mu(M)$ (dependent on $M$) and $0<\theta_{V}<1$.

Proof. The proof is given in Appendix B.
In Lemma 3, a free parameter $\theta_{V}$ is introduced in order to estimate
the ultimate bound of the error (21). It is worth to be mentioned that
$\theta_{V}$ is a variable meant to ensure the negative definiteness of the
Lyapunov derivative of $V=e^{T}Me$ whenever (21) is satisfied. The bigger the
$\theta_{V}$, the best the estimation (21). However, it can be seen that as
long as $\theta_{V}$ approaches to $1$, the negativeness of the Lyapunov
derivative is compromised, and therefore the right hand side (21) tends to
infinity. One can therefore establishes $\theta_{V}=\frac{1}{2}$ and then
algorithmically make it smaller (conversely, greater) depending on the
Lyapunov derivative and the estimated ultimate bound.

###  4.2 STA-based sliding mode control design

As shown in the previous section, if the variable $\eta(t)$, as in (14),
emulates the virtual, then the control objective for mechanical sub-system
will be achieved. In order to accomplish this task, let us consider the
sliding variable

$s(e)=\eta-\int v(e_{1},\eta)dt+\kappa e_{1}+\frac{4\tau EA}{V_{t}}e_{1}$ (22)

such that if $s(e)=0$ then $\eta=\int v(e_{1},\eta)dt-\kappa e_{1}-\frac{4\tau
EA}{V_{t}}e_{1}$. Its dynamics results to

$\displaystyle\dot{{s}}(e)$
$\displaystyle=\dot{\eta}-v(e_{1},\eta)+\kappa\dot{e}_{1}+\frac{4\tau
EA}{V_{t}}\dot{e}_{1}$ (23) $\displaystyle=\gamma_{1}e_{1}+\kappa
e_{2}+\left(\gamma_{2}-\frac{4EC_{qp}}{V_{t}}\right)\eta$
$\displaystyle+\frac{4\tau EC_{q}}{V_{t}}u(t)+\bar{\delta}_{3}(t).$

Before introducing the proposed control law, let us prove the reachability of
such sliding surface in the following Lemma.

######  Lemma 4.

Assume that there exist a positive value $C_{e_{2}}$ such that $|e_{2}(t)|\leq
C_{e_{2}}$. Then, there always exist a control law (that does not depend on
the perturbations) so that the trajectories of system (13) converge to the
manifold $s(t)\equiv 0$ in a finite time $T_{s}$, and are kept in such
manifold for all $t\geq T_{s}$.

Proof. The proof is given in Appendix C.
Let us now introduce the STA-based control law of the form:

$\displaystyle u(t)=$ $\displaystyle-\frac{V_{t}}{4\tau
EC_{q}}\left[k_{1}\rho\lceil
s\rfloor^{1/2}+\gamma_{1}e_{1}+\left(\gamma_{2}\frac{4EC_{q}}{V_{t}}\right)\eta\right]$
(24) $\displaystyle-\frac{V_{t}}{4\tau EC_{q}}\left(\int k_{2}\rho^{2}\lceil
s\rfloor^{0}dt\right)\,,$

with the positive constants $k_{1}$, $k_{2}$ selected so as to obtain the
desired behavior of STA, and $\rho$ to be a scaling factor for dealing with
perturbations. It is important to notice that the discontinuous part of
control is responsible for robustness properties of the STA against the
perturbations whose derivative is bounded. Moreover, discontinuity in the
control signal is avoided since such term is integrated. Therefore, in order
to analyze conditions posed on the perturbations, and the stability of the
control algorithm, it is necessary to investigate also the dynamics of
integral part. Setting a new variable

$z=-\frac{V_{t}}{4\tau EC_{q}}\left(\int k_{2}\rho^{2}\lceil
s\rfloor^{0}dt\right)+\bar{\delta}_{3}(t)+\kappa e_{2}$

leads to

$\displaystyle\dot{{s}}(e)$ $\displaystyle=-\rho k_{1}\lceil
s\rfloor^{1/2}+z\,,$ (25) $\displaystyle\dot{z}$
$\displaystyle=-\rho^{2}k_{2}\lceil
s\rfloor^{0}+\dot{\bar{\delta}}_{3}(t)+\kappa\dot{e}_{2}\,,$

where the whole perturbation in the second line is
$\delta_{z}=\dot{\bar{\delta}}_{3}(t)+\kappa\dot{e}_{2}$. Note that an exact
convergence of the variable $z$ to zero implies the theoretically exact
compensation of perturbations. This is because in order to force $z$ to
converge, it is required that the integral term of the control law has
theoretically exactly the same value as the perturbation $\delta_{z}$. Then,
the system becomes a standard STA form

$\displaystyle\dot{{s}}(e)$ $\displaystyle=-k_{1}\rho\lceil
s\rfloor^{1/2}+z\,,$ (26) $\displaystyle\dot{z}$
$\displaystyle=-k_{2}\rho^{2}\lceil s\rfloor^{0}+\delta_{z}(t)\,.$

######  Assumption 5.

For all $t\in\mathbb{R}_{+}$, there exists a non-negative constant $L$ such
that

$\displaystyle|\delta_{z}(t,e)|$ $\displaystyle\leq L\,.$ (27)

######  Lemma 6.

Given the Assumption 5 is satisfied for the sub-system (26). Design the
$k_{1}$, $k_{2}$ gains such that

$A_{k}=\begin{bmatrix}-k_{1}&1\\\ -k_{2}&0\end{bmatrix}$ (28)

is a Hurwitz matrix. Let $M_{k}$ be the solution of
$A^{T}_{k}M_{k}+M_{k}A_{k}=-\mathbb{I}$, if the scaling factor satisfies

$\rho>{2L\lambda_{max}(M_{k})}\,.$ (29)

Then, the trajectories of the dynamic system (26) converges to the origin in a
finite time.

Proof. The proof is given in Appendix D.

######  Remark 7.

Note that with this approach, one can set the gains $k_{1}$, $k_{2}$ so as to
obtain a certain desired behavior, which can then be scaled by the variable
$\rho$ as in [29], [15], and [20]. One option is to set the gains such that it
optimizes a certain criteria in the sense of [26].

##  5 Control synthesis

###  5.1 Surface Design

For the surface design, recall the Lyapunov-type inequality

$MA_{n}+A_{n}^{T}M-K^{T}B^{T}M-MBK+\Psi H^{T}M+\Psi MH<0$ (30)

and consider the change of variable $Y=M^{-1}$ such that multiplying with $Y$
both sides of the above expression yields

$Y(A_{n}+\Psi H)^{T}+(A_{n}+\Psi H)Y+YK^{T}B^{T}+BKY<0\,.$ (31)

Afterwards, setting the new variable to be found as $N=KY$ such that one can
compute $K=NM$, the inequality becomes

$Y(A_{n}+\Psi H)^{T}+(A_{n}+\Psi H)Y+N^{T}B^{T}+BN<0\,.$ (32)

Additionally, let us consider the following constraints on the LMI regions to
be computed as

$\begin{gathered}Y\mathcal{W}^{T}(A_{n},H)+\mathcal{W}(A_{n},H)Y+\mathcal{N}(N,B)+2h_{1}Y<0\\\
Y\mathcal{W}^{T}(A_{n},H)+\mathcal{W}(A_{n},H)Y+\mathcal{N}(N,B)+2h_{2}Y>0\\\
\begin{bmatrix}\sin(\theta)C_{11}&\cos(\theta)C_{12}\\\
\cos(\theta)C^{T}_{12}&\sin(\theta)C_{22}\end{bmatrix}<0,\end{gathered}$

with $\mathcal{W}(A_{n},H)=A_{n}+\Psi H$, $\mathcal{N}(N,B)=BN+N^{T}B^{T}$,
and

$\displaystyle C_{11}$
$\displaystyle=\left[YW^{T}(A_{n},H)+\mathcal{W}(A_{n},H)Y+\mathcal{N}(N,B)\right]$
$\displaystyle C_{12}$
$\displaystyle=\left[YA_{n}^{T}-N^{T}B^{T}-A_{n}Y+BN\right]$ $\displaystyle
C_{22}$
$\displaystyle=\left[Y\mathcal{W}^{T}(A_{n},H)\mathcal{W}(A_{n},H)Y+\mathcal{N}(N,B)\right]\,.$

The above formulated constraints generate a region in the complex plane as
schematically depicted in Fig. 1.

Figure 1: Desired LMI region in the complex plain.

The next to be analyzed is selection of the constants $h_{1}$, $h_{2}$ and
$\theta$ for the LMI-based surface design.

####  5.1.1 Assignment of $h_{2}$:

In order to assign a proper boundary value $h_{2}$, one can consider an
approximated first-order response of a linear system of the form
$c_{h_{2}}(t)=1-\exp(t/T_{h_{2}})$. In order to obtain a time constant such
that the first-order response to the step as
$\lim_{t\rightarrow\infty}c_{h_{2}}(t)=\bar{y}_{2}$, the response should reach
$63.2\%$ of the total desired amplitude. Then, from a straightforward
geometrical reasoning, the associated time constant can be approximated by

$T_{h_{2}}\approx\sqrt{(0.632y_{d})^{2}-(\bar{y_{2}})^{2}}\,.$ (33)

and consequently $h_{2}=1/T_{h_{2}}$.

####  5.1.2 Assignment of $h_{1}$:

Consider the mechanical sub-system and assume that the pressure difference
between the chambers is the given input. Then, in terms of the pressure to
control the second-order dynamics

$\frac{d}{dt}\begin{bmatrix}x_{1}\\\ x_{2}\end{bmatrix}=\begin{bmatrix}0&1\\\
0&-\frac{\sigma}{m}\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}+\begin{bmatrix}0\\\ \frac{1}{\tau m}\end{bmatrix}AP\,,$
(34)

and assuming $AP=-k_{h}x_{1}-\sigma_{h}x_{2}$ with the feedback constants in a
hypothetical closed-loop

$\frac{d}{dt}\begin{bmatrix}x_{1}\\\ x_{2}\end{bmatrix}=\begin{bmatrix}0&1\\\
-\frac{k_{h}}{\tau m}&-\frac{\sigma\tau+\sigma_{h}}{\tau
m}\end{bmatrix}\begin{bmatrix}x_{1}\\\ x_{2}\end{bmatrix},$ (35)

or its summarized compact form
$\frac{d}{dt}\begin{bmatrix}x_{1}&x_{2}\end{bmatrix}^{T}=A_{h}\begin{bmatrix}x_{1}&x_{2}\end{bmatrix}^{T}$,
one can compute the eigenvalues as

$\lambda_{1,2}(A_{h})=-\frac{\sigma\tau+\sigma_{h}}{2\tau m}\pm\frac{1}{2\tau
m}\sqrt[2]{\left(\tau\sigma+\sigma_{h}\right)^{2}-4k_{h}\tau m}\,.$ (36)

The critically damped case is when $(\tau\sigma+\sigma_{h})^{2}-4k_{h}\tau
m=0$, and both eigenvalues have the same real part. Then, the step response of
the position as output has a form
$x_{1}(t)=1-\exp^{-\omega_{n}t}(1+\omega_{n}t)$ where
$\omega_{n}=\frac{\sigma\tau+\sigma_{h}}{2\tau m}$. At the same time, if one
approximates the step response by a first-order behavior, one can compute the
time constant as

$T_{h_{1}}=\frac{2m\tau}{\sigma\tau+\sigma_{h}}\,,$ (37)

which implies that for a desired $T_{h_{1}}$, one can determine

$\sigma_{h}=\frac{2m\tau}{T_{h_{1}}}-\sigma\tau\,.$ (38)

Then, for a desired behavior of the mechanical sub-system, being related to
its time constant $T_{h_{1}}$, one obtains

$h_{1}=-\frac{\sigma\tau+\sigma_{h}}{2\tau m}$ (39)

####  5.1.3 Parameter $\theta$:

For many practical reasons, it is convenient to have only real negative
eigenvalues such that the mechanical response does exhibit an oscillating
response. This can be done if the parameter $\theta=0$, and therefore the
cone-constrained LMI region collapses only into the real line. However, this
can cause computational problems or the feasibility of the LMI’s may be
compromise. Parameter $\theta$ is introduced such that it can be adjusted in
order to find a solution to the LMI, by choosing it as small as possible.

####  5.1.4 Bound for $\psi(t,e)$:

Moreover, assume that the largest mechanical perturbation is due to nonlinear
fiction term, cf. (7), and rewrite it as

$\psi(t,x_{2})=\frac{\tanh(\vartheta
x_{2})\left[F_{c}+(F_{s}-F_{c})\text{exp}\left(-|x_{2}|^{\phi}\chi^{-\phi}\right)\right]}{x_{2}}\,,$
(40)

which has its maximum as $x_{2}$ tends to zero. Then, one can find the bound
as

$\Psi=\sup\lim_{x_{2}\rightarrow 0}\psi(t,x_{2})\,.$ (41)

###  5.2 STA design

In order to compute the $L$ value, take the perturbation $\delta_{z}$ and
calculate its bound as

$\displaystyle|\delta_{z}|$
$\displaystyle\leq|\bar{\delta}_{3}(t)|+\kappa|\dot{e}_{2}(t)|\,.$ (42)

Assuming that the system will be in a compact set, there exists a constant
positive value $\overline{\ddot{q}}$ such that
$|\dot{e}_{2}|\leq\overline{\ddot{q}}$. This way, the value of $L$ can be
determined by

$L=L_{3}+\kappa\overline{\ddot{q}}$ (43)

##  6 Comparison with other SMC strategies

For the sake of clearness in the comparison presented in this section, the
approach proposed in this paper will be called IS-STA.

A qualitative comparison with respect to the approach in [22] will be
performed, taking into account that both of the approaches are based on the
STA algorithm. Additionally, for both methodologies the velocity measurement
is not needed in order to apply the control law.

The approach of [22] consists in three main steps: first, a tracking reference
model is designed, secondly, a variable gain Levant’s differentiator is
implemented, and finally, a variable gain STA based control law is designed.
The variable gains for the differentiator and STA are updated using a norm
observer. The sliding surface designed depends on the output and its
derivatives, which are estimated by the variable gain Levant’s differentiator.
It is important to remark, that the parameters used [22] are tuned so that the
tracking performance for both of the approaches are similar.

According to [22], the following reference model is proposed:

$y_{m}=\frac{(25)^{2}}{(s+25)^{2}}r\,.$ (44)

The chosen Input to Output state variable filters have the form:

$w_{u}(s)=\frac{1}{(s+5)^{2}}u\,,\quad w_{y}(s)=\frac{1}{(s+5)^{2}}q\,.$ (45)

The variable gain differentiator implemented is the following

$\displaystyle\dot{\hat{e}}_{1}$
$\displaystyle=-3L_{vgst}^{1/4}(t,\hat{x})\lceil\hat{e}_{1}-e_{1}\rfloor^{3/4}-2(\hat{e}_{1}-e_{1})+\hat{e}_{2}\,,$
(46) $\displaystyle\dot{\hat{e}}_{2}$
$\displaystyle=-2.5L_{vgst}^{1/3}(t,\hat{x})\lceil\hat{e}_{1}-e_{1}\rfloor^{2/3}-3(\hat{e}_{1}-e_{1})+\hat{e}_{3}\,,$
$\displaystyle\dot{\hat{e}}_{3}$
$\displaystyle=-1.5L_{vgst}^{1/2}(t,\hat{x})\lceil\hat{e}_{1}-e_{1}\rfloor^{1/2}-2(\hat{e}_{1}-e_{1})+\hat{e}_{4}\,,$
$\displaystyle\dot{\hat{e}}_{4}$
$\displaystyle=-1.1L_{vgst}(t,\hat{x})\lceil\hat{e}_{1}-e_{1}\rfloor^{0}-(\hat{e}_{1}-e_{1})\,.$

where the gains comes from:

$\displaystyle\dot{\hat{x}}$
$\displaystyle=-0.8\hat{x}(t)+10+1.5\left\|\begin{bmatrix}w_{u}&w_{y}\end{bmatrix}^{T}\right\|\,,$
(47) $\displaystyle L_{vgst}(t,\hat{x})$
$\displaystyle=1.2|\hat{x}|+2|u|+7\,.$

The control strategy consists of the following sliding surface and controller

$\displaystyle\hat{\sigma}(t)$
$\displaystyle=\hat{e}_{3}+2(25)\hat{e}_{2}+(25)^{2}e_{1}$ (48) $\displaystyle
u_{vgsta}$
$\displaystyle=-\tilde{k}_{1}(t,\hat{x})\phi_{1}(\hat{\sigma})-\int_{0}^{t}\tilde{k}_{2}(\tau,\hat{x})\phi_{2}(\hat{x})d\tau\,,$
$\displaystyle\phi_{1}(\hat{\sigma})$
$\displaystyle=\lceil\hat{\sigma}\rfloor^{1/2}+\hat{\sigma}\,,$
$\displaystyle\phi_{2}(\hat{\sigma})$
$\displaystyle=\frac{1}{2}\lceil\hat{\sigma}\rfloor^{0}+\frac{3}{2}\lceil\hat{\sigma}\rfloor^{1/2}+\hat{\sigma}\,,$

where the gains $\tilde{k}_{1}(t,\hat{x})$ and $\tilde{k}_{2}(t,\hat{x})$ are
designed according to [22], taking into account the following bounds of the
perturbations $\rho_{1}=0$ and $\rho_{2}=10|\hat{e}_{2}|+5|e_{1}|+\hat{x}+1$
and the constants $\delta_{k}=0.01$, $\varepsilon_{k}=1\times 10^{-3}$,
resulting in:

$\displaystyle\tilde{k}_{1}$
$\displaystyle=\delta_{k}+\frac{1}{4\varepsilon_{k}}\rho_{2}^{2}+2\varepsilon_{k}\rho_{2}+\varepsilon_{k}+2\varepsilon_{k}\left(1+4\varepsilon_{k}^{2}\right)\,,$
$\displaystyle\tilde{k}_{2}$
$\displaystyle=1+4\varepsilon_{k}^{2}+2\varepsilon_{k}\tilde{k}_{1}\,.$

The simulations were performed using a band limited white noise in the
measurements, similar to the one presented in the real plant sensors, with
noise power of $1\times 10^{-9}$ and a sample time of $5\times 10^{-6}$. The
details about the designed trajectories are presented in the experimental
evaluation.

Figure 2: Comparison of the position $q$ and tracking error $e_{1}$. $r(t)$
corresponds to the reference to be tracked, $q(t)$\- IS STA is the proposed
approach and $q(t)$\- Oliveira et al. 2018 is the approach presented in [22].

It can be observed in Figure 2, that the tracking is well performed by the two
methodologies. In Figure 2-(b) the errors are presented. Although the IS STA
appears to be better in the first $3$ seconds, the approach of [22] works
similarly well in second stage of the trajectory, from $t=3$ to $t=5$. During
the transient of the step response, the IS STA slightly outperforms [22].

Figure 3: Comparison of the pressure variable $P$ and control signal $u$
corresponds to the reference to be tracked, IS STA is the proposed approach
and Oliveira et al. 2018 is the approach presented in [22].

In Figure 3 the pressure diference $P$ and the control signal $u(t)$ are
presented. From the control signal $u(t)$ it is clear that $u$-IS STA is
smaller in amplitude than the one from the $u$-Oliveira et al. 2018. This can
be caused by the noise in the measurement which is propagated not only in the
Variable Gain Super-Twisting, but also via the third order differentiator. One
may decrease gains in order to reduce the effect of the noise in the control
input. However, this also make the performance to decrease.

##  7 Experimental Evaluation

###  7.1 Experimental setup

In this section, the experimental validation of the proposed control approach
is provided. The presented control solution is implemented in the laboratory
test bench [24], see laboratory view shown in Fig. 4. The hydraulic setup is
composed by a one-side-rod cylinder with a moving load carrier, mounted in a
linear slide bearing, and one translational degree of freedom $q$.

Figure 4: Experimental test bench of hydraulic actuator (laboratory view).

The cylinder is connected via a 4 ways / 3 positions servo-valve with an
internal closed-loop spool control, connected to the hydraulic power unit, cf.
schematic diagram of the hydraulic circuit shown if Fig. 5.

Figure 5: Schematic diagram of the hydraulic circuit of the actuator system.

The supply parameters are set to $10$ MPa pressure and $40$ liter/min
volumetric flow. More details on the hardware components can be found in [24].
Since the low-level controlled servo-valve receives the input reference signal
in percentage of the orifice opening, the control signal $u$ and
correspondingly $g$ (cf. with section 6.3) is inherently unitless i.e.
$u\in[-1,+1]$. Worth noting is also that the control algorithms are
implemented in Simulink®Real-Time on a Speedgoat™platform with 2 kHz sampling
rate.

###  7.2 Motion trajectories

The trajectories were designed as piece-wise continuous functions for
different time instants. From the time $t=0$ to $t=2$, a quintic smooth
polynomial trajectory is tracked, starting from $q(0)=0$ m to $q(2)=0.1$ m in
a fashion of smooth robotic trajectories, cf. [30]. Then, the value is
maintained until $t=3$ s and then switched to another quintic polynomial
trajectory from $q(3)=0.1$ m to $q(5)=0.02$ m. Then a ramp is assumed from
$t=6$ s to $t=8$ s and, finally, a step from $q(9)=0.1$ m is imposed. Worth
noting is that although the final step is not a smooth trajectory, as required
for $r(t)$, the proposed control approach is able to track it as well.

###  7.3 Implemented control scheme

The first implementation issue of the applied control scheme is the dead-zone
effect in the servo-valve opening orifice. This effect, as shown in [24], is
(nearly) static around the $10\,\%$ of the opening range. Therefore, the dead-
zone is compensated in feed-forwarding via an inverse map of the form:

$\tilde{u}=\frac{1}{2}D_{s}\text{sign}(u)+u$ (49)

where $D_{s}>0$ is the dead-zone size.

Second, in the control related modeling and analysis, the servo valve dynamics
was not taken into account. Such actuator dynamics can be pre-compensated in
the same way as dead-zone effect. From the previous work [24], the parameters
of the servo-valve input-output behavior of the form (1) are obtained. Then, a
low-pass filter is implemented in order to count for the fast (but still not
zero) dynamics of the servo valve. The overall control value has then the form

$g=\frac{1}{\left(\mu_{c}\varphi+1\right)^{2}}\left[\frac{1}{2}D_{s}\text{sign}(u)+u\right]$
(50)

where $\varphi$ is the Laplace variable and $\mu_{c}>0$ is a design parameter
chosen to be equal to the servo-valve actuator time constant. The identified
parameters are given in [24]. Moreover, we stress that due to the dead-zone
compensation the chattering effect may be increased in the control zero
crossing. In that sense, the set low-pass filter complies along with the
servo-valve bandwidth, reducing the chattering of the effective control value
$g(t)$. The overall control strategy implemented is presented in Figure 6.

Figure 6: Schematic of the overall control strategy implemented.

###  7.4 Auxiliary frequency domain analyze

Based on the experimental evaluations, the authors notice some practical
issues that are worth to be addressed as below. The stability of the sliding
dynamics can also be lost by changing the design parameters $h_{1}$ and
$h_{2}$. In this section, based on the concept of phase deficit [4], a
possible explanation to this phenomena is given. In a nutshell, the phase
deficit represents a certain measure of robustness for a finite-time
controller, similar to concept of the phase margin usable for asymptotic
linear controllers. It can be seen as the angular difference of the reciprocal
of negative inverse Describing Function (DF) of a nonlinear control law and
the plant’s Nyquist plot [4].

Let us analyze the output of the system that is assumed to have the form:
$y(t)=a_{y}\sin(\Omega t+\phi_{y})$, where $\Omega$ is the frequency and
$a_{y}$ is the amplitude of steady oscillations, and $\phi_{y}$ is the phase
lag. Assuming that the dynamic sliding surface is a periodic input
$s(t)=a_{s}\sin(\omega t)$ being $a_{s}>0$ the amplitude of the oscillations
in the surface $s$, it is possible to find the oscillatory reaction $y(t)$,
and more important, an explanation on how it is affected by the design of the
variable $h_{2}$. Consider that the transfer function function of the closed
loop has the form

$G_{f}(\varphi)=\frac{1}{\frac{1}{T_{s}}\varphi+N_{f}(a_{y})}\,$ (51)

where the positive constant $T_{s}$ is a time constant dependent on the design
parameters $h_{1}$ and $h_{2}$, and $N_{f}$ is the DF of the STA

$N_{f}(a_{y})=\frac{2k_{1}L\gamma(1.25)}{\sqrt{\pi}\gamma(1.75)}a_{y}^{-1/2}=\gamma_{a}a_{y}^{-1/2}\,,$
(52)

where $\gamma(\cdot)$ is the gamma-function, as it is presented in [3].
Setting $a_{1}=\frac{4k_{2}L^{2}}{\pi\omega}$ and following [3], one can
compute the amplitude of the propagated oscillations as:
$a_{y}=a_{1}|G_{f}(jw)|$, or equivalently:

$\frac{w^{2}}{T_{s}^{2}}a_{y}^{2}+\gamma^{2}a_{y}-a_{1}^{2}=0\,,$ (53)

whose solution is given by

$a_{y}=\frac{T_{s}^{2}}{2w^{2}}\left(-\gamma^{2}+\sqrt{\gamma^{4}+4T_{s}^{2}\omega^{2}a_{1}^{2}}\right)\,.$
(54)

Computing its phase deficit it yields [3] :

$\phi_{d}=\frac{\pi}{2}-\arctan\left(\frac{1}{\sqrt{2}}\sqrt{-1+\sqrt{1+\frac{64k_{2}^{2}\rho^{2}}{T_{s}^{4}\pi^{2}\gamma^{4}}}}\right)\,,$
(55)

ensuring that the phase deficit approaches to zero as $T_{s}$ tends to zero as
well. This gives us an explanation why the nominal system’s response can not
be made arbitrarily fast without compromising the system’s stability, and
implies the selection of $h_{2}$ margin should be additionally restricted, cf.
with section 5.1.1.

###  7.5 Experimental results

For the control evaluation, the computed parameters for the LMI conditions are
$h_{2}=-5$ and $h_{1}=-1$, these from the assigned time constants
$T_{h_{2}}=1/5$ s and $T_{h_{1}}=1$ s. The parameter $\theta$ is left to be
small enough such that the LMI is still feasible but, at the same time, tries
to restrict the imaginary part of the resulting poles. In this work,
$\theta=\pi/20$ was chosen. From the aforementioned analysis one can get
$\Psi=0.5$. Finally, for the STA, we assume the gains which minimize the
amplitude of chattering as $k_{1}=1.1$ and $k_{2}=2.028$, see [26], so that
the scaling condition becomes $\rho>2(1.6303)L$. The computed boundary
constant is $L=1.347$, and the assigned scaling factor is $\rho=10$. A summary
of the parameters used for this experimental results is presented in table 1
with the corresponding methodology section.

Table 1: Summary of the parameters and its corresponding section. Parameter: | Value: | Design section:
---|---|---
$h_{1}$ | 1110.1 | Section 5.1.2
$h_{2}$ | 10.1 | Section 5.1.1
$\theta$ | 23.113231 | Section 5.1.3
$k_{1}$ | 1.1 | c.f. [26]
$k_{2}$ | 2.028 |
$\rho$ | 10 | Section 5.1.4

The designed motion trajectory is evaluated experimentally, as shown by the
dotted line in Fig. 7 (a). It can be seen that the cylinder position $q(t)$
follows the desired trajectory, being uniformly close to the reference value
$r(t)$. The measured piston position is shown by blue solid line and the
reference is shown by the dotted black line in Fig. 7 (a). It can be
recognized that the position is tracked accurately even in the case of a
discontinuous (step) change at $t=9\,\,\text{s}$. The corresponding tracking
error is also shown in Fig. 9 (b). Although the velocity is not available and,
thus, not used in the proposed control approach, it is estimated using robust
exact differentiator toolbox [2] for the sake of a better visualization of the
controlled motion. Figure 7 (b) discloses that the estimated velocity is well
in line with derivative of the reference value.

Figure 7: Measured position (versus reference) in (a), zoom from $t=0$ to
$t=2.5$ seconds is presented in (b), zoom on the tracking position from
$t=8.5$ to $t=12$ seconds is presented in (c) and the estimated velocity of
the controlled motion trajectory is shown in (d).

Figure 8 shows the differential pressure, i.e. measured pressure difference
between the right and left chambers, in (a) and the resulted control signal
which is applied to the servo-valve in (b). Worth recognizing is that the
control signal is continuous. Some chattering pattern, where the control
signal appears to oscillate between $\pm\,0.1$, appears at several phases of
steady-state and corresponds to the dead-zone effect in the servo-valve.
Recall that the latter is pre-compensated in feed-forward by the dead-zone
inverse. Also note that the servo-valve blocks energizing the hydraulic
circuits within the dead-zone, so that a low-amplitude chattering is slightly
outside of $\pm\,0.1$. Since the second-order actuator dynamics provides the
corresponding low-pass filtering, such dead-zone related chattering is
irrelevant for loading of the hydraulic circuits. From Fig. 8 (a), the
pressure is acting as virtual control for the sub-mechanical system. It is
important to recall that the $P$ variable is the differential pressure so that
its negative values have reasoning for the selected motion direction. Also
worth recalling that within dead-zone, the valve is locking the hydraulic
circuit from one side and, consequently, one of the chambers remains
pressurized. This is explaining the non-zero $P(t)$ values where there is no
apparent motion of the cylinder piston, cf. Fig. 7 and Fig. 8 (a). The load
pressure magnitude remain below $40\,\,\text{bar}$ while the supply pressure
is $100\,\,\text{bar}$. Figure 8 (b) discloses also the continuous control
signal (apart from the dead-zone chattering) and that the control signal is
never saturated.

Figure 8: Measure differential pressure (as virtual control) in (a) and the
control signal in (b).

In Fig. 9, zoom-in plot of the output tracking error and the sliding variable
are shown in (a) and (b), respectively. Important to notice is that the
sliding mode may be temporary lost when the assumed conditions about
perturbations, cf. (11) and (13), are violated, like in case of the applied
step reference at $t=9$ s. Even then, the sliding mode is subsequently
recovered and the tracking is well performed.

Figure 9: Measured output tracking error in (a) and the sliding mode variable
in (b).

It is important to mention that the convergence velocity may be increased by
the scaling factor $\rho$. In Figure 10 the step response with respect to
different values of $\rho$ is presented. However, the bigger the gain $\rho$,
the bigger the control input. Another important issue with increasing the gain
is the amplitude of the chattering effect. Increasing the gain may
considerable increase the chattering produced by the noise.

Figure 10: Comparison of the step response with different values of $\rho$.

A brief video of the presented experimental results is available under
https://youtu.be/_dnwD7Ens-k.

###  7.6 Quantitative measures of performance

In this section, some quantitative measures of the experimental results are
given. Let us define the following performance indices:

  1. 1.

Maximal absolute value of the tracking error

$M_{e}=\max_{i=1,2,\ldots,N}\left\\{|e_{1}(i)|\right\\}\,,$ (56)

being $N$ the number of obtained values recorded.

  2. 2.

Average tracking error

$\mu_{e}=\frac{1}{N}\sum_{i=1}^{N}|e_{1}(i)|\,.$ (57)

  3. 3.

Standard deviation performance index

$\sigma_{e}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(|e_{1}(i)|-\mu_{e}\right)^{2}}\,.$
(58)

The maximal absolute value of the tracking error can be considered as an index
of the tracking accuracy. Meanwhile, the average tracking error and the
standard deviation performance show the average tracking performance and the
deviation of the tracking errors, respectively. Finally, the Integral Square
Error (ISE) will be taken into account.

The experimentally obtained values for every index are presented in table 2,
taking the error in the last step response, that is, from $t=10$ to $t=14$.

Table 2: Values of perfomance indices Index: | Value:
---|---
$M_{e}$ | 0.0064
$\mu_{e}$ | 0.0011
$\sigma_{e}$ | 3.4598 $\times 10^{-6}$
ISE | $333.7830$
Percentage of steady state error | $0.55\%$

The percentage of the steady state error is obtained taking the average
tracking error as a percentage of the total cylinder drive, that is $0.20$ m.

##  8 Conclusion

A new practical control approach, based on the sliding mode STA, is introduced
and applied to a hydraulic cylinder actuator. The proposed method is proved to
track a smooth trajectory in presence of both matched and unmatched
perturbations, producing an acceptably small and ultimately bounded tracking
error. Important to notice is that via the designed integral sliding surface,
the velocity state is fully excluded and needs to be neither measured nor
estimated for implementing the proposed controller. The design methodology for
such surface is developed in terms of an LMI. The feasibility of the proposed
approach is shown via experimental results achieved on a standard industrial
hydraulic actuator in laboratory setting, without additional measurements and
extensive modeling and identification.

Leonid Fridman and Manuel A. Estrada are grateful for the financial support of
CONACyT (Consejo Nacional de Ciencia y Tecnología): CVU 833748; PAPIIT-UNAM
(Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica):
Project IN106622.

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##  Appendix A Proof of Theorem 1

Assume the following Lyapunov function candidate $V(e)=e^{T}Me$. Its
derivative along the trajectories of $\dot{e}=(A_{n}-BR)e+\psi(t,e)He$ has the
form

$\displaystyle\dot{V}(e)$
$\displaystyle=e^{T}(t)\left[MA_{n}+A_{n}^{T}M+\psi(t,e)\left(H^{T}M+MH\right)\right]e(t)$
(59) $\displaystyle-e(t)^{T}\left(K^{T}B^{T}M+MBK\right)e(t)$
$\displaystyle\leq e^{T}(t)\left[MA_{n}+A_{n}^{T}M-K^{T}B^{T}M-MBK\right.$
$\displaystyle\left.+\Psi\left(H^{T}M+MH\right)\right]e(t)$

Then, as shown in [8], the system is quadratically stable if there exists such
$R\in\mathbb{R}^{3}$ and $M=M^{T}>0$ that (20) is satisfied. The necessity
comes from the quadratic stabilizability. If the system is not quadratically
stabilizable, then there is no solution of (20). This concludes the proof.

##  Appendix B Proof of Lemma 3

If $\beta(t)\neq 0$, the closed loop of mechanical sub-system with the virtual
control has the form $\dot{e}(t)=(A_{n}-BK)e+\psi(t,x)He+\beta(t)D$, where
$D^{T}=\begin{bmatrix}0&1&0\end{bmatrix}$. Taking the Lyapunov function
candidate $V=e^{T}Me$, its derivative is

$\displaystyle\dot{V}(e)$
$\displaystyle=e^{T}(t)\left[MA_{n}+A_{n}^{T}M+\psi(t,e)\left(H^{T}M+MH\right)\right]e(t)$
$\displaystyle-e(t)^{T}\left(K^{T}B^{T}M+MBK\right)e(t)$
$\displaystyle+\beta(t)\left(D^{T}Me+e^{T}MD\right)$ $\displaystyle\leq
e^{T}(t)\left[MA_{n}+A_{n}^{T}M-K^{T}B^{T}M-MBK\right.$
$\displaystyle\left.+\Psi\left(H^{T}M+MH\right)\right]e(t)+2\lambda_{max}(M)\beta_{M}\|e(t)\|\,.$

If the LMI (20) is satisfied, then there exists $\mu(M)>0$ such that

$\dot{V}(e)\leq-\mu(M)\|e(t)\|^{2}+2\lambda_{max}(M)\beta_{M}\|e(t)\|$ (60)

Assuming a constant $0<\theta_{V}<1$, the following is satisfied

$\dot{V}(e)\leq-(1-\theta_{V})\mu(M)\|e(t)\|^{2}\,,\,\|e(t)\|\geq\frac{2\lambda_{max}(M)\beta_{M}}{\theta_{V}\mu(M)}\,.$

By Theorem 4.18 from [12], the system’s trajectories are then ultimately
uniformly bounded and the bound (21) is satisfied.

##  Appendix C Proof of Lemma 4

In order to show the reachability of the sliding surface (22), consider the
following control

$\displaystyle u(t)=$ $\displaystyle-\frac{V_{t}}{4\tau
EC_{q}}K_{s}\text{sign}(s)$ (61) $\displaystyle+\frac{V_{t}}{4\tau
EC_{q}}\left[\gamma_{1}e_{1}-\left(\gamma_{2}+\frac{4EC_{qp}}{V_{t}}\right)\eta\right]$

for a positive constant $K_{s}$. Therefore, the reachability condition has the
form:

$\displaystyle s\dot{s}$
$\displaystyle=s\left[-K_{s}\text{sign}(s)+\bar{\delta}_{3}(t)+\kappa
e_{2}\right]$ (62) $\displaystyle\leq-
K_{s}|s|+|s||\bar{\delta}_{3}(t)|+|s|\kappa|e_{2}|$
$\displaystyle\leq-|s|\left(K_{s}+L_{3}+\kappa|e_{2}|\right)\,,$

since $|e_{2}|\leq C_{e_{2}}$, there always exist $K_{s}>L_{3}+\kappa
C_{e_{2}}$, such that with $\bar{K}_{s}=K_{s}-L_{2}-\kappa C_{e_{2}}$, and the
following is satisfied:

$s\dot{s}\leq-\bar{K}_{s}|s|<0\,,$ (63)

proving that there always exist a control law (indepedent of the uncertainties
and perturbations) that ensures the trajectories converges to the manifold
$s(t)\equiv 0$, and therefore, they remain in such manifold.

##  Appendix D Proof of Lemma 6

To prove the stability of $s=0$ and $z=0$, consider the following change of
coordinates introduced in [20]:
$\zeta^{T}=\begin{bmatrix}\zeta_{1}&\zeta_{2}\end{bmatrix}=\begin{bmatrix}\lceil
s\rfloor^{1/2}&z\end{bmatrix}$. Computing its dynamics yields

$\dot{\zeta}=\frac{\partial\zeta_{1}}{\partial
s}\begin{bmatrix}-k_{1}\rho&1\\\
-k_{2}\rho^{2}&0\end{bmatrix}\zeta+\begin{bmatrix}0\\\
\delta_{z}(t,e)\end{bmatrix}=\frac{\partial\zeta_{1}}{\partial
s}A_{s}\zeta+G\delta_{z}(t,e)\,,$ (64)

where $G^{T}=\begin{bmatrix}0&1\end{bmatrix}$. Note that this term is always
positive and bounded from below as $\left|\frac{\partial\zeta_{1}}{\partial
s}\right|\geq 1$. Consider the following scaling of the states

$\xi=\begin{bmatrix}\xi_{1}\\\
\xi_{2}\end{bmatrix}=\Gamma^{-1}\zeta\quad\text{with}\quad\Gamma=\begin{bmatrix}\rho&0\\\
0&\rho^{2}\end{bmatrix}\,,$ (65)

so that the dynamics has the form $\dot{\xi}=\frac{\partial\xi_{1}}{\partial
s}\Gamma^{-1}M_{s}\xi+\Gamma^{-1}G\delta_{z}(t,e)$ . Then one has

$\frac{d\xi}{dt}=\frac{\partial\xi_{1}}{\partial s}\left(\rho
A_{k}\xi+G\frac{\delta_{z}(t,e)}{\frac{\partial\xi_{1}}{\partial
s}\rho}\right)\,.$ (66)

Consider the Lyapunov function candidate $V_{s}=\xi^{T}M_{k}\xi$, whose
derivatives along the trajectories of the $\dot{\xi}$ vector field has the
form

$\displaystyle\frac{dV_{s}(t)}{dt}$
$\displaystyle=\dot{\xi}^{T}M_{k}\xi+\xi^{T}M_{k}\dot{\xi}$ (67)
$\displaystyle=$ $\displaystyle\rho\frac{\partial\xi_{1}}{\partial
s}\xi^{T}\left(A_{k}^{T}M_{k}+M_{k}A_{k}\right)+2\frac{\partial\xi_{1}}{\partial
s}\xi^{T}M_{k}G\frac{\delta_{z}(t,e)}{\frac{\partial\xi_{1}}{\partial
s}\rho}\,.$

Note that for any Hurwitz matrix $A_{k}$, there exists always a solution
$M_{k}$ of $A_{k}^{T}M_{k}+M_{k}A_{k}=-\mathbb{I}$, so that

$\frac{dV_{s}(t)}{dt}=-\frac{\partial\xi_{1}}{\partial
s}\left(\rho\xi^{T}\xi-2\xi^{T}M_{k}G\frac{\delta_{z}(t,e)}{\frac{\partial\xi_{1}}{\partial
s}\rho}\right)\,.$ (68)

For the assumed $|\delta_{z}|\leq L|\xi_{1}|$, one can obtain the following
bound

$\frac{dV_{s}(t)}{dt}\leq-\frac{\partial\xi_{1}}{\partial
s}\left(\rho-2L\lambda_{max}(M_{k})\right)\|\xi\|^{2}\,.$ (69)

It is sufficient that only the gain $\rho$ satisfies (29). Then, the stability
of the surface’s origin and control law of the closed loop are ensured for a
sufficiently large value of $\rho$.