# Third order sliding mode controller based on adaptive integral sliding mode concept: experimental application to an electropneumatic actuator

Mohammed Taleb, Xinming Yan, Franck Plestan

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Mohammed Taleb, Xinming Yan, Franck Plestan. Third order sliding mode controller based on adaptive integral sliding mode concept: experimental application to an electropneumatic actuator. 53rd IEEE Conference on Decision and Control (CDC 2014), Dec 2014, Los Angeles, CA, United States. 10.1109/CDC.2014.7039699. [hal-05299475](https://hal.science/hal-05299475v1)

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# Third order sliding mode controller based on adaptive integral sliding mode concept: experimental application to an electropneumatic actuator

M. Taleb, X. Yan, and F. Plestan

**Abstract**— This paper presents an adaptive third order sliding mode controller based on integral sliding mode concept. This new controller is designed for the position control of an electropneumatic actuator, on which bounded external perturbations with unknown magnitude are applied. The efficiency of the controller is experimentally evaluated.

**Keywords.** Higher order sliding mode, adaptive control, uncertain nonlinear systems, electropneumatic actuator.

## I. INTRODUCTION

The objective of this paper is to present a new controller for the position control of an electropneumatic system which is perturbed by external forces whose magnitudes and dynamics are totally unknown. Given the presence of external perturbations and also of numerous uncertainties on the modeling of the experimental set-up, the use of robust control approach is required.

Sliding mode control [19] is considered as one of the most popular robust nonlinear control laws with respect to uncertainties and perturbations. However, the “sign” function used in the controller induces a high frequency control switching phenomenon, the so-called “chattering”, which could cause dangerous damages to the actuators [5].

Many solutions have been proposed these last decades in order to reduce this phenomenon. Among them, one can cite the use of higher order sliding mode strategies [9]. This class of controllers consists in driving the sliding variable and a finite number of its consecutive time derivatives to zero in a finite time. By this way, the discontinuous control acts on high order time derivative of the sliding variable which leads to chattering reduction. Another solution consists in combining higher order sliding mode control with dynamically adapted gain [12], [13].

The gain adaptation allows a gain adjustment with respect to a predefined criterion without any information of uncertainties and perturbations bounds. This approach has been applied to first order sliding mode control [12], [13] and successfully used for the position control of an electropneumatic actuator. In [1], [17], [14], the adaptive versions of “twisting” and “super-twisting” have been proposed; these adaptive controllers have been also applied to the control of electropneumatic actuator.

In [16], [11], adaptive higher (i.e. larger than second) order sliding mode controllers have been proposed. A drawback

of ([11]) is inherent to the gain adaptation law: the gain increases until a higher order sliding mode is detected; then, the gain approximately stays constant even if the uncertainties and perturbations are reduced. The gain adaptation law is not totally efficient because it is not sufficiently reduced versus perturbations/uncertainties effects.

In [16], the gain of discontinuous control increases and decreases according to an adaptation law: by this point-of-view, it is more efficient than [11]. However, the gain tuning is not optimized because it does not follow by a precise manner the variations of uncertainties and perturbations. The approach proposed in the sequel paper allows that the gain is approximately tracking the variation of uncertainties and perturbations. By this way, the gain is “just sufficient” to counteract perturbations and uncertainties effects: it induces a reduced chattering phenomenon and, then, a better accuracy than previous results. The result of [16] has been improved in [18] by changing the adaptation law based now on [1], which induces a more accurate adaptation of the gain versus perturbation/uncertainties effects.

In this paper, the control solution developed in [18] is used for the position control of an actuator. Note that many results have been published since several years on the design of second order sliding mode control solutions for electropneumatic systems (for example, [6], [8], [12], [13], [14], [15]), but more rarely on the design of third order sliding mode controllers [7]. Some works have been made also on standard sliding mode control [2]. However, there is a real interest to use third order sliding mode controller especially for a more accurate tracking. It is the first time, to authors’ best knowledge, that an adaptive version of a third order sliding mode controller is experimentally evaluated on a physical system.

Section II presents the experimental set-up. Section III describes the adaptive high order sliding mode controller and its design. In order to show the applicability of this approach, the application to position control of a perturbed pneumatic actuator is presented in Section IV.

## II. EXPERIMENTAL SET-UP

### A. Model

The electropneumatic system (see Figure 1-2) is composed of two actuators. The first one, named the “main” one (left hand side), is a double acting electropneumatic actuator controlled by two servodistributors and is composed of two chambers denoted *P* and *N*. Piston diameter is 80 mm and rod diameter 25 mm. With a source pressure equal to 7 bar, the maximum force developed by the actuator is 2720 N. The

<footnote>M. Taleb, X. Yan and F. Plestan are with LUNAM Université, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, France. M. Taleb is also with Moulay Ismail University, Meknes, Morocco. **Corresponding author:** F. Plestan, Franck.Plestan@irccyn.ec-nantes.fr.</footnote>
air mass flow rates $q_m$ entering the chambers are modulated by two three-way servodistributors. The pneumatic jack horizontally moves a load carriage of mass $M$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Scheme of electropneumatic system</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. **Photo of electropneumatic system.** On the left hand side is the “main” actuator whose position and pressure in a chamber can be controlled. On the right hand side is the “perturbation” actuator which provides varying load force.</figcaption></figure>

The carriage is coupled with the second electropneumatic actuator, the so-called “perturbation” one. The goal of this latter is to produce a dynamic load force on the main actuator. The actuator has the same mechanical characteristics as the main one, but the air mass flow rate is modulated by a single five-way servodistributor.

The electropneumatic plant model is obtained from three physical laws: the mass flow rate through a restriction, the pressure behaviour in a chamber with variable volume and the fundamental mechanical equation.

From $V_P(y) = V_0 + S_p \cdot y$, $V_N(y) = V_0 - S_p \cdot y$ (with $V_0$ the half-cylinder volume, $S_p$ the piston surface and $y$ the piston position) and from standard assumptions,

one gets [3]

$$ 
\begin{aligned}
\dot{p}_P &= \frac{kr_g T}{V_P(y)} [\varphi_P + \psi_P \cdot u - \frac{S_p}{r_g T} p_P v] \\
\dot{p}_N &= \frac{kr_g T}{V_N(y)} [\varphi_N + \psi_N \cdot u + \frac{S_p}{r_g T} p_N v] \\
\dot{v} &= \frac{1}{M} [S_p(p_P - p_N) - b_v v - F_{ext}] \\
\dot{y} &= v
\end{aligned}
\qquad (1) $$

where $p_P$ and $p_N$ are respectively the pressures of P- and N-chambers, and $v$ the piston velocity. $r_g$ is the perfect gas constant, $b_v$ the viscous friction coefficient, $T$ the supply temperature, $k$ the polytropic constant and $F_{ext}$ the external force produced by the “perturbation” actuator. The control input $u$ is applied to each servodistributor. By denoting $x = [p_P \ p_N \ v \ y]^T$, one gets

$$ \dot{x} = f(x) + g(x)u \qquad (2) $$

with

$$ f(x) = \begin{bmatrix} \frac{kr_g T}{V_P(y)} [\varphi_P - \frac{S_p}{r_g T} p_P v] \\ \frac{kr_g T}{V_N(y)} [\varphi_N + \frac{S_p}{r_g T} p_N v] \\ \frac{1}{M} [S_p(p_P - p_N) - b_v v - F_{ext}] \\ v \end{bmatrix}, \quad (3) $$

and

$$ g(x) = \begin{bmatrix} \frac{kr_g T}{V_P(y)} \psi_P \\ \frac{kr_g T}{V_N(y)} \psi_N \\ 0 \\ 0 \end{bmatrix}. \qquad (4) $$

## B. Problem statement

The purpose of this paper is focused on the position control of the “main” actuator, the “perturbation” one being controlled by a force PID controller provided by the experimental set-up constructor. Then, the control problem consists in controlling the piston position $y$ in spite of uncertainties on parameters

* mass $M$ - it could vary if additive load is put on the carriage;

* temperature $T$ - it is supposed constant whereas there are variations in standard applications;

* functions $\varphi_*$ and $\psi_*$ representing the mass flow rate - the models of this latter suffer of uncertainties inherent to the identification process;

* viscous friction coefficient $b_v$;

and external perturbations $F_{ext}$ supposed unknown. Note that the magnitude of the variation of perturbations and uncertainties are supposed to be bounded, but unknown. The fact that bounds on the uncertainties/variations of the parameters/external perturbations are unknown, implies that
the control approach must not require these informations,
which is the case in the adaptive control scheme developed
in the sequel.

### III. CONTROL DESIGN

#### A. Problem statement

The problem consists in designing a robust controller that
drives the output (the position $y$) of the electropneumatic
actuator to follow a desired trajectory (see Figure 3-top)
in spite of bounded external perturbations with unknown
bounds. At first, a square form external force has been
applied to the main actuator (see Figure 3-bottom).

<figure><img src="image_4.png" /><figcaption>Fig. 3. Top. Position reference $y_{ref}(t)$ ($m$) versus time ($s$). Bottom. External perturbation ($N$) versus time ($s$).</figcaption></figure>

Consider the nonlinear uncertain system (2). Given that the
control objective consists in controlling the piston position,
one defines

$$ \sigma = y - y_{ref}(t). \quad (5) $$

In the context of sliding mode control theory, the variable
(5) is called “sliding variable”.

**Property 1.** The relative degree of system (2) with respect to $\sigma$ equals 3 and is constant.

**Definition 1:** ([9]) Consider the nonlinear system (2) and the sliding variable $\sigma$; consider also an integer $r \ge 1$. Assume that the successive time derivatives $\sigma, \dot{\sigma}, \dots, \sigma^{(r-1)}$ are continuous functions. The manifold defined as $(\tau$ being the sampling period of the control law)

$$ \Sigma_*^r = \{x \mid |\sigma| \le \mu_0 \tau^r, \dots, |\sigma^{(r-1)}| \le \mu_{r-1} \tau\} $$

with $\mu_i \ge 0$ (with $0 \le i \le r-1$), is called “real $r^{th}$-order sliding mode set”, which is non-empty and is locally an integral set in the Fillipov sense ([4]). The motion on $\Sigma_*^r$ is called “real $r^{th}$-order sliding mode” with respect to the sliding variable $\sigma$.

In the sequel, let us state $r = 3$, which means that the designed controller is a third order sliding mode one. From (2)-(5), one gets

$$ \sigma^{(3)} = a(x, t) + b(x)u \quad (6) $$

with both functions $a$ and $b$ derived from $f, g$ and $\sigma$ (see [7], [8] for details). Suppose that $a$ and $b$ have a nominal known part (named $a_N$ and $b_N$ respectively) and an unknown bounded part (named $\Delta a$ and $\Delta b$ respectively) such that [8]

$$ a = a_N + \Delta a \qquad b = b_N + \Delta b. \quad (7) $$

Also, the function $b$ fullfills $b_N > 0$ and $|\Delta b|/b_N < 1$ [14]. Define the control input (with $w$ the “new” control input)

$$ u = \frac{1}{b_N} (-a_N + w) \quad (8) $$

and apply this latter to (6), which gives

$$ \sigma^{(3)} = \underbrace{\Delta a - \frac{\Delta b}{b_N} a_N}_{\psi(x, t)} + \underbrace{\left(1 + \frac{\Delta b}{b_N}\right)}_{\varphi(x)} w. \quad (9) $$

**Property 2.** For the electropneumatic system (2), there exist positive constants $C$, $\varphi_m$ and $\varphi_M$ such as

$$ |\psi(x, t)| < C \quad 0 < \varphi_m < \varphi(x) < \varphi_M. \quad (10) $$

A $3^{rd}$-order sliding mode controller for system (2) with respect to sliding variable $\sigma$ is equivalent to finite time stabilization of

$$ \begin{aligned} \dot{z}_1 &= z_2 \\ \dot{z}_2 &= z_3 \\ \dot{z}_3 &= \psi(x, t) + \varphi(x)w, \end{aligned} \quad (11) $$

where $z = [z_1 z_2 z_3]^T = [\sigma \ \dot{\sigma} \ \ddot{\sigma}]^T$. Consider the so-called “integral sliding variable” [16], [17] defined as $(t_0$ being the initial time)

$$ s(z(t)) = z_3(t) - z_3(t_0) - \int_{t_0}^{t} w_{nom}(\tau) d\tau \quad (12) $$

with $w_{nom}$ a control law stabilizing in a vicinity of zero, in a finite time,

$$ \begin{aligned} \dot{z}_1 &= z_2 \\ \dot{z}_2 &= z_3 \\ \dot{z}_3 &= w_{nom} + \gamma(t) \end{aligned} \quad (13) $$

with $\gamma(t)$ a small bounded perturbation. Note that $s(z(t_0)) = 0$: then, the system is evolving early from the initial time on the integral real sliding mode set [18]

$$ \Sigma_*^r = \{z \mid |s(z(t))| \le \mu\tau\}. $$

**Property 3.** The control $w_{nom}$ is bounded.

#### B. Adaptive controller

As detailed in [18], the control law forces, via the integral sliding variable $s$, the system trajectories to evolve on $\Sigma_*^r$; a (first order) real sliding mode behavior with respect to $s$ is then established. Then, the system dynamics are governed by the finite time convergent controller $w_{nom}$; by this way, a $3^{rd}$-order sliding mode with respect to $\sigma$ is established.
The dynamics of $s$ reads as

$$ 
\begin{aligned}
\dot{s} &= \sigma^{(3)} - w_{nom} \\
&= \psi(x,t) - w_{nom} + \varphi(x,t)w
\end{aligned}
\quad (14) $$

which gives

$$ \sigma^{(3)} = w_{nom} + \dot{s}. \quad (15) $$

Then, system (11) becomes

$$ 
\begin{aligned}
\dot{z}_1 &= z_2 \\
\dot{z}_2 &= z_3 \\
\dot{z}_3 &= w_{nom} + \dot{s}
\end{aligned}
\quad (16) $$

By a practical point-of-view, a realistic objective consists in ensuring that, in a finite time, $\dot{s}$ is sufficiently small. In this case, the control law $w_{nom}$ has to be sufficiently robust in order to stabilize, in a finite time, system (16) in spite of the term $\dot{s}$. The first time derivative of $s$ can be written as

$$ \dot{s} = \Gamma(x,t) + \varphi(x,t)w \quad (17) $$

with

$$ \Gamma(x,t) = \psi(x,t) - w_{nom}. \quad (18) $$

The objective of the adaptive first order sliding mode controller is then to establish a first order real sliding mode with respect to $s$ and to ensure the convergence of the first time derivative of sliding variable $\dot{s}$ to a vicinity of zero in a finite time $t_F$. A solution has been proposed in [1], in which the adaptation consists in increasing the control gain $K$ until that a real first order sliding mode with respect to $s$ is detected, and in gradually reducing $K$ until the sliding mode is lost. This strategy allows to dynamically adapt the gain value: it implies the use of a **sliding mode detector** and a **gain adaptation law**.

**Sliding mode detector** [1]. Let $\tau > 0$ be the sampling period. Given that a first order sliding mode provides an accuracy proportional to $\tau$, consider a natural number $N$ and some $\mu > 0$. Given the sampling time $t_i$ ($i \in \mathbb{N}$), define the following criterion

$$ \alpha(t) = \begin{cases} 1 & \text{if } \forall t_j \in [t_i, t_i + N\tau] : |s(t_j)| \le \mu K(t_j)\tau \\ -1 & \text{if } \exists t_j \in [t_i, t_i + N\tau] : |s(t_j)| > \mu K(t_j)\tau \end{cases} \quad (19) $$

This criterion consists in verifying over the sliding window $[t_i, t_i + N\tau]$ if all values of $s$ are less than a number proportional to the sampling period $\tau$.

**Adaptation law algorithm** [1]. As previously developed, the gain must decrease if $\alpha = 1$ whereas, if $\alpha = -1$, the gain will increase. Introduce the constants $K_m$ and $q$ satisfying

$$ K_m > 0, \quad q > 1 \quad (20) $$

and let the adaptation law be

$$ \dot{K} = \begin{cases} -\alpha\lambda K & \text{if } K > K_m \\ \lambda & \text{if } K \le K_m \end{cases} \quad (21) $$

where $\lambda$ is a positive adaptation parameter. In order to decrease the convergence time, a gain jump is forced if the first order sliding mode criterion is violated which gives

$$ K(t_i) = \begin{cases} qK(t_{i-1}) & \text{if } \alpha(t_{i-1}) = 1 \text{ and } \alpha(t_i) = -1 \\ K(t_i) & \text{if } \alpha(t_{i-1}) \neq 1 \text{ or } \alpha(t_i) \neq -1 \end{cases} \quad (22) $$

By using the previous sliding mode detector and adaptation law, one gets the following result.

**Theorem 1 ([1]):** Consider the nonlinear system (17) with $w$ reading as

$$ w = -K(t)\mathrm{sign}(s). \quad (23) $$

Consider also the adaptation law (21)-(22) with $\lambda, \mu$ and $N$ sufficiently large. Then, for a given $q$, the gain $K(t)$ is bounded and the accuracy $|s| \le \eta\tau K(t)$, with $\eta > \mu$, is obtained in finite time.

In [1], it is proved that the gain $K(t)$ defined by (21)-(22) approximately tracks $|\Gamma|/\varphi$: the gain is then “just sufficient” to counteract uncertainties and perturbations.

**Theorem 2 ([18]):** Consider system (2) and the sliding variable $\sigma$ (5). Consider also the control input (8) with

$$ w = -s - K(t)\mathrm{sign}(s), \quad (24) $$

the gain $K(t)$ adapted according to (21)-(22), and the integral sliding variable $s$ defined by (12)

$$ s = \ddot{\sigma}(t) - \ddot{\sigma}(0) - \int_{0}^{t} w_{nom} d\tau \quad (25) $$

with $w_{nom}$ a finite time convergent controller sufficiently robust. Then, an $3^{rd}$-order sliding mode versus $\sigma$ is established in a finite time.

**Remark 1.** the control $w$ is composed of two terms

* a discontinuous term $-K(t)\mathrm{sign}(s)$ to force the establishment of a real first order sliding mode w.r.t $s$;

* a continuous term $-s$ to drive $\dot{s}$ exponentially to a vicinity of 0 [18].

## IV. EXPERIMENTAL RESULTS

The control law is developed under Matlab/Simulink (The Mathworks Co.). The use of this software allows to use a DS1104 board (dSpace Co.) on which the control law is implemented. In the sequel, the experimental results have been obtained with a sampling period equal to 0.5 ms. The control law $w_{nom}$ is defined as<fnref n="1" /> [10]

$$ w_{nom} = \begin{cases} -\alpha_v\mathrm{sign}(\ddot{\sigma} + 2(|\dot{\sigma}|^3 + |\sigma|^2)^{1/6} \\ \mathrm{sign}(\dot{\sigma} + |\sigma|^{2/3}\mathrm{sign}(\sigma))) \end{cases} \quad (26) $$

In practice, the acceleration (required in $\ddot{\sigma}$) is not measurable directly whereas an estimation of the acceleration is required. For the experimental tests, the high order sliding mode

<footnote n="1">¹This choice of control strategy for $w_{nom}$ is arbitrary. The absolutely required feature is the finite time convergence one.</footnote>
differentiator proposed in [9] is used to evaluate this second order time derivative.

From the physical parameters of the actuator, and given the evaluation of the uncertainties from the identification procedure, the controller parameters have been tuned as

$$ 
\begin{aligned}
\tau &= 0.5 \text{ ms}, \quad \mu = 0.15, \quad \lambda = 10, \\
q &= 1.2, \quad K_m = 0.05, \quad \alpha_v = 4.
\end{aligned} 
\qquad (27) 
$$

Figure 4 displays the measured position and the associated tracking error with respect to the desired trajectory. Note that the influence of the perturbation is correctly rejected given that no information on the perturbation (magnitude, dynamics, ...) is taken into account for the tuning of the controller. Figure 5 displays the behavior of the gain $K(t)$: it is clear that there is gain adaptation when there is the change of sign of the perturbation force. Figure 5 shows also that the control input is not saturated and its magnitude is quite reduced.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Top. Position (m) (y-solid line) and its reference ($y_{ref}(t)$-dotted line) versus time (s). Bottom. Tracking position error $y(t) - y_{ref}(t)$ (m) versus time (s).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Top. Control input $u(V)$ versus time (s). Bottom. Gain $K(t)$ versus time (s).</figcaption></figure>

The second tests have been made with a sinusoidal perturbation force (Figure 6-bottom). In this case, there is less stress on the system given that the applied force has a smoother behavior. The experimental results are displayed by Figures 7-8: the main remark concerns the behavior of the gain which is adapted, but by a smoother way than previously: it is clearly shown the link between the perturbation nature and the gain behavior.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Top. Position reference $y_{ref}(t)$ (m) versus time (s). Bottom. External sinusoidal perturbation (N) versus time (s).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Top. Position (m) (y-solid line) and its reference ($y_{ref}(t)$-dotted line) versus time (s). Bottom. Tracking position error $y(t) - y_{ref}(t)$ (m) versus time (s) with sinusoidal perturbation.</figcaption></figure>

## V. CONCLUSION

In this paper, an adaptive-gain third order sliding mode controller is developed. The control law is composed by a third other sliding mode controller and an adaptive first order controller with respect to the so-called integral sliding variable. This controller is applied to the electropneumatic system, and experimental results have shown the feasibility of this approach. Future works will consist in reducing the use of high order time derivatives of sliding variable in the controller (for example, for a third order sliding mode controller, an objective would be to use only the first time derivative) - by a practical point-of-view, this reduction will appear as an opportunity to simplify the design of the controller, and to reduce the effect of the noise).

## ACKNOWLEDGEMENTS

The authors acknowledge the support of the ANR grant CHASLIM (ANR-11-BS03-0007).

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