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# On a new methodology for adaptive sliding mode control with self-tuning threshold: application to active vibration control

Jonathan Rodriguez, Antoine Carvalho, Simon Chesné

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Jonathan Rodriguez, Antoine Carvalho, Simon Chesné. On a new methodology for adaptive sliding mode control with self-tuning threshold: application to active vibration control. International Conference on Noise and Vibration Engineering, Sep 2024, Louvain, Belgium. hal-05457686

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# On a new methodology for adaptive sliding mode control with self-tuning threshold: application to active vibration control

J. Rodriguez, A. Carvalho, S. Chesné

INSA Lyon, CNRS, LaMCoS, UMR5259,
69621 Villeurbanne, France.

## Abstract

The control of structures with uncertainties is one of the main challenges in modern control. Thus, robust control methods such as sliding mode control (SMC) have been widely investigated in the past decade. Since chattering is the main drawback of SMC, adaptive SMC methods have also been proposed to avoid the over-estimation of the discontinuous control term. However, majority of the ASMC methods need arbitrary tuning of the real sliding mode boundaries which is dependent on the structure's uncertainty and also the unknown exogenous perturbation. The following proceeding presents a vibration control application of a new method for ASMC with real-time boundary adaptation of the discontinuous term based on chattering detection. The method is experimentally tested on a rotating steel disc with piezoelectric transducers. A contact roller creates parametric uncertainties on the system response and also generates an external disturbance. The performance of the proposed ASMC formulation is addressed for different angular velocities of the disc.

## 1 Introduction

Sliding Mode Controllers (SMC) theoretically allow to drive the closed-loop dynamics of a system to the desired dynamics in finite time and independently of the parametric uncertainties and external perturbations [1, 2]. Such property is achieved by defining a sliding variable function of the tracking error. Then, a discontinuous term forces the states of the system to reach this sliding surface.

However, the performance and robustness of SMC come at the cost of high-frequency switching phenomena in the control signal once the sliding motion is established known as chattering [3] that can damage electronic components and actuators.

Thus, Adaptive Sliding Mode Controllers (ASMC) propose to adapt the switching gain in real-time to the minimum necessary value to maintain the sliding motion [4, 5, 6]. Plestan et al. [7] proposed a new adaptive strategy that guaranteeing the finite time convergence of the sliding variable to a given range defined by a threshold parameter $\epsilon$. In addition, a tuning method for this parameter based on the sampling frequency was provided. Later, Edwards et al. [8] presented an adaptive formulation for conventional, super-twist and higher order sliding mode control based on equivalent control. Again, an arbitrary user-defined parameter $\alpha$ defines the convergence domain of the sliding variable. More recently, Roy et al. [9] developed an ASMC method considering state-dependent uncertainties that are not bounded a priori. Once again, the convergence domain of the sliding variable or vector is conditioned by a user-defined parameter $\alpha_i$.

In this paper, we propose a new ASMC formulation using a pseudo-delay filter acting as a chattering detector to adapt the switching gain accordingly. Here, no a priori upper bound of the disturbance nor any user-defined parameter predetermines the final gain of the discontinuous injection term. The defined ASMC formulation is then implemented on an active vibration control experimental setup including time-varying parametric uncertainties.
## 2 Adaptive sliding Mode formulation

Let us consider the a linear dynamical system with scalar exogenous disturbance and a control input:

$$ \dot{x}(t) = Ax(t) + Bu(t) + B_w w(t) \quad (1) $$

with $x \in \mathbb{R}^n$ as the state vector, $A \in \mathbb{R}^{n \times n}$ Hurwitz stable, $B \in \mathbb{R}^n$, $B_w \in \mathbb{R}^n$, $u \in \mathbb{R}$ is the control input, and $w \in \mathbb{R}$ is the unknown bounded perturbation.

A sliding variable $\sigma(t) \in \mathbb{R}$ is designed such that $\sigma(t) = Sx(t)$ with $S \in \mathbb{R}^n$. $S$ is defined following the standard method of Optimal Sliding Mode Control (OSMC) with $SB$ invertible [2, 10, 11].

The following ASMC law is now defined:

$$ u(t) = -(SB)^{-1}(\rho(t)\operatorname{sgn}(\sigma(t)) + k_l\sigma(t) + SAx(t)) \quad (2) $$

$$ \dot{\rho}(t) = k(|\sigma(t)| - \epsilon(t)) \quad (3) $$

where $\rho(t) \ge 0 \forall t > 0$, $\epsilon(t) \ge 0 \forall t$, $k_l > 0$, $(SB)^{-1}SAx(t)$ corresponding to the equivalent control, $k > 0$ and $\rho(0) = 0$. The proposed approach defines $\epsilon$ variable such that once the real sliding mode occurs (chattering arises) in the signal $\sigma$, the adaptive gain $\rho(t)$ stabilizes. The discrete formulation of $\epsilon(t)$ is:

$$ \dot{\epsilon}_k = -k_\epsilon|\sigma_k|\operatorname{sgn}(\sigma_k)\operatorname{sgn}(\sigma_{k-1}) \quad (4) $$

with $k$ as the sample index, and $k_\epsilon > 0$. The chattering phenomenon is detected if the sliding variable oscillates at the highest possible frequency i.e $F_s/2$, with $F_s$ the controller's sampling frequency, giving $\operatorname{sgn}(\sigma_k)\operatorname{sgn}(\sigma_{k-1}) = -1$. Then, $\epsilon$ increases to reach the value of $|\sigma|$, causing the decrease of the discontinuous gain $\rho(t)$. The continuous formulation of the dynamics of $\epsilon$ can be approximated by the following filter:

$$ \dot{\epsilon}(t) = -k_{\epsilon}(t)|\sigma(t)|\operatorname{sgn}(\sigma(t))\operatorname{sgn}(\eta(t)) \quad (5) $$

$$ \ddot{\eta}(t) = -\omega_0^2\eta(t) - 2\omega_0\dot{\eta}(t) + \omega_0^2\sigma(t) \quad (6) $$

where $\eta$ is the approximation of a one sample delayed variable $\sigma$. $\omega_0$ is chosen such that $H(s)$ from (6) behaves as close as possible to the unit sample delay transfer function $\exp\{-j\omega/F_s\}$ in phase. In particular, the following constraint applies:

$$ \arg\{H(j\pi F_s)\} < -\pi/2 \quad (7) $$

The designed filter is shown in Figure 1 in parallel with the unit sample delay for $\omega_0 = 0.3F_s$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Unit sample delay transfer function and chosen 2^nd^ order approximation.</figcaption></figure>
# 3 Experimental validation on uncertain structure

The following section presents the experimental setup used to validate the proposed ASMC formulation and the results obtained.

## 3.1 Presentation of the test-rig

The test rig consists in a rotating steel spoked disc 2 driven by an electric motor. Three piezoelectric transducers are located radially on the disc, fixed within the rotor frame and connected to the controller through a slip ring. Figure 3 shows a schematic representation of the experimental setup. The transducers 1 and 2 are collocated actuator and sensor, and transducer 3 applies the perturbation if rotation is null. A contact roller generates cyclic boundary conditions and also an external disturbance during the rotation. The controller hardware is a dSpace Microlabbox with a sampling frequency of 10kHz.

<figure><img src="image_4.png" /><figcaption>Figure 2: Picture of the test-rig.</figcaption></figure>

## 3.2 Angle dependent system response

The main objective of this subsection is to highlight the impact of the static angular position of the roller $\theta$ with respect to the transducers 1 and 2 on the system response i.e the transfer function $H_1(s) = Y(s)/U(s)$. Hence, a band-limited white noise with a sampling frequency of 10kHz is applied to the control transducer 1 and the response $y$ is measured for different static angular positions $\theta \in [0; \pi]$.

The Figure 4 displays $H_1(f)$ for $\theta = \{0, \pi/4, \pi/2, 3\pi/4, \pi\}$. One can notice immediately that the first visible mode (55Hz - disc pumping mode) is impacted by the parameter $\theta$ in frequency and also amplitude. Two phenomena can explain this modification of the frequency response function. Firstly, as the transducers move away from the roller due to rotation, the pumping mode frequency may decrease slightly due to softer boundary conditions. In addition, since the pitching mode (the first 1 diameter mode of the disc) is theoretically very close to the pumping mode, it is possible that the actuating transducer excites one or the other mode, or a combination of both, depending on its angular position with respect to the contact roller. As mentioned in the introduction, robust control, such as the proposed ASMC formulation, has the ability to maintain performance and robustness in the presence of such time-varying parametric uncertainties.
<figure><img src="image_5.png" /><figcaption>Figure 3: Schematic representation of the experimental setup.</figcaption></figure>

### 3.3 Identification and Control results

According to Figure 4, the frequency response function $H_1(f)$ for $\theta = \pi$ is selected as reference to establish a transfer function model of the system response. The Figure 5 now displays the aforementioned measured and simplified model of the collocated system. Only the first mode of the structure has been included in the model in addition with a 2nd order low-pass filter and two real poles to adjust the fitting. The final state space representation of $H_1$ is:

$$ \dot{x}(t) = A(t) + Bu(t) \quad (8) $$

$$ y(t) = Cx(t) \quad (9) $$

with

$$ A = 10^4 \begin{bmatrix} -1.95 & -1.24 & -0.55 & -0.14 & -0.06 & -0.05 \\ 0.82 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.41 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.05 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.05 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 128 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \\ C = \begin{bmatrix} 0 & 0 & -63.81 & -2.75 & -6.63 & -0.88 \end{bmatrix} \quad (10) $$

A Kalman observer is designed thereafter to estimate the states $X$:

$$ \dot{\hat{x}}(t) = A\hat{x}(t) + Bu(t) + L(y(t) - C\hat{x}(t)) \quad (11) $$

with $L = [6.67 \ 2.09 \ -27.77 \ -2.82 \ -12.21 \ 26.95]^T$. It is important to recall that since the observer remains linear, the closed-loop performance is also dependent on the state estimation error.

Considering the controller, the numerical parameters are $k_l = 0$ and equivalent control $SAx(t)$ null, $S = [0.011 \ 0.013 \ 0.015 \ 0.013 \ 0.017 \ 0.008]$, $k = 200$, $k_\epsilon = 100$, $\omega_0 = 0.3F_s$, $F_s = 1.10^4$Hz, $\rho(0) = 0$, $\epsilon(0) = 0$. The tuning method is not extensively developed here and the parameters $k$ and $k_\epsilon$ have been chosen
<figure><img src="image_6.png" /><figcaption>Figure 4: Measured frequency response function $H_1(f)$ (a) for different static angular positions, (b) average vector and standard deviation for the measured frequency response function $H_1(f)$ (9 different angular positions).</figcaption></figure>

for the control to display reasonable behavior in terms of convergence time. Besides, the main advantage of the proposed method is that the numerical values of $k$ and $k_\epsilon$ do not affect the final value of the gain $\rho(t)$.

The first experiment is realized without rotation of the disc ($N = 0$ rpm) at $\theta = \pi$. The external perturbation is again a band-limited white noise with a sampling frequency of 10kHz and is applied through piezoelectric transducer 3. The control action start at $t=2$s and the results are shown in Figure 6. $\rho(t)$ quickly increases until chattering is detected, causing the increase of $\epsilon(t)$. Thus, the gain $\rho(t)$ rapidly stabilizes around the value of $5.10^{-4}$. Variations are still visible around the convergence value due to the tuning of ($k, k_\epsilon$) and the random excitation amplitude. However, a finer tuning could reduce these oscillations. Finally the PSD (Power Spectral Density) of the measure $y(t)$ highlights the control performance around the considered disc mode on the bandwidth [40-60]Hz.
<figure><img src="image_7.png" /><figcaption>Figure 5: Measured and identified frequency response function $H_1(f)$ for $\theta = \pi$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Results of experiment 1: (a) adaptive gain $\rho(t)$, (b) sliding variable $\sigma(t)$ and dynamic threshold $\epsilon(t)$, (c) PSD of measured error signal $y(t)$.</figcaption></figure>

The second experiment consists in applying the proposed ASMC when the disc is rotating at different angular velocities $N = \{20, 40, 60, 100\}$ rpm with the same controller parameters from previous experiment. Here,
the external disturbance comes directly from the contact between the roller and the disc. The Figure 7 displays the adaptive gain $\rho(t)$ and the PSD of the measured error signal $y(t)$ for all angular velocities, with and without control. One can observe first the increase of the perturbation level with the velocity, highlighted by the increase of the adaptive gain $\rho(t)$ convergence value. Since the proposed ASMC automatically adapts its gain to the best possible value, the vibration mitigation level is maintained even when the angular velocity increases. Besides, the robustness properties of SMC also guarantee the control performance throughout the rotation of the disc in presence of the induced parametric uncertainties highlighted in Figure 4. However, an increase of the vibration response is visible for higher frequencies with respect to the results obtained without rotation (Figure 6), certainly due to the modeling errors (shifting of the structure modes due to modification of the boundary conditions) associated with a simple linear observer here.

<figure><img src="image_9.png" /><figcaption>Figure 7: Results of experiment 2: (a) adaptive gain $\rho(t)$ with stationary disc rotation, (b) PSD of measured error signal $y(t)$ for different angular velocities of the disc.</figcaption></figure>

The third experiment consists in varying the control parameter $k$ to highlight the main advantage the proposed
method i.e the convergence value of $\rho(t)$ only depends on the perturbations level, and is independent of any user-defined parameter. Thus, the disc rotates at the constant velocity of 100rpm and $k = \{25, 50, 100, 200\}$, the other controller parameters are maintained from the previous experiment. The results are displayed in Figure 8. As expected, the adaptive gain $\rho(t)$ approximately converges around 0.09 for all values of $k$ and reduces the amplitude of $\sigma(t)$ accordingly.

<figure><img src="image_10.png" /><figcaption>Figure 8: Results of experiment 3: (a) adaptive gain $\rho(t)$ with stationary disc rotation of 100rpm, and (b) sliding variable $\sigma(t)$ for $k = \{25, 50, 100, 200\}$.</figcaption></figure>

The final experiment consists in varying the velocity command of the motor to observe in real-time the adaptation of the gain $\rho(t)$. The results are displayed in Figure 9, one can observe immediately the adaptive gain value following the profile of the disc angular velocity. All the experiments performed confirm the performance of the proposed ASMC approach and its ability to converge to the best possible gain regarding the perturbation, exogenous or endogenous (from parametric uncertainties).
<figure><img src="image_11.png" /><figcaption>Figure 9: Results of experiment 4: (a) motor angular velocity, (b) adaptive gain $\rho(t)$, (c) sliding variable $\sigma(t)$, (d) dynamic threshold $\epsilon(t)$.</figcaption></figure>
# 4 Conclusion

This manuscript addresses the challenge of ASMC convergence to the best possible gain minimizing the chattering phenomenon and maintaining the sliding motion. The proposed formulation of ASMC detects high frequency oscillations in the sliding variable signal using a filter that approximates the unit-sample delay at $F_s/2$, and decreases the adaptive gain accordingly. The main advantage of such method is that no user-defined parameter sets a priori the final convergence value of the adaptive gain. The switching gain simply converges to the best possible value for the current disturbance and level of uncertainties. An experimental application in active vibration control is performed to highlight the main features of the proposed ASMC and its practical benefits.

# References

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[2] Y. Shtessel, C. Edwards, L. Fridman, A. Levant et al., *Sliding mode control and observation*. Springer, 2014, vol. 10.

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[7] F. Plestan, Y. Shtessel, V. Bregeault, and A. Poznyak, “New methodologies for adaptive sliding mode control,” *International journal of control*, vol. 83, no. 9, pp. 1907–1919, 2010.

[8] C. Edwards and Y. B. Shtessel, “Adaptive continuous higher order sliding mode control,” *Automatica*, vol. 65, pp. 183–190, 2016.

[9] S. Roy, S. Baldi, and L. M. Fridman, “On adaptive sliding mode control without a priori bounded uncertainty,” *Automatica*, vol. 111, p. 108650, 2020.

[10] J. Rodriguez, M. Collet, and S. Chesné, “Active vibration control on a smart composite structure using modal-shaped sliding mode control,” *Journal of Vibration and Acoustics*, vol. 144, no. 2, p. 021013, 2022.

[11] J. Rodriguez, L. Mesny, and S. Chesné, “Sliding mode control for hybrid mass dampers: Experimental analysis on robustness,” *Journal of Sound and Vibration*, vol. 575, p. 118241, 2024.