# Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons

^{*}

## Abstract

**:**

## 1. Introduction

- -
- The ABS dynamics in braking mode are extremely nonlinear and many parameters of the underlying model change with time.
- -

## 2. The Anti-Lock Braking System Model

_{1}(t) is the braking torque evolving according to the third differential equation, where constants b

_{1}and b

_{2}model the actuator dead-zone dynamics whose threshold is u

_{0}, while $S(\lambda )=\frac{s\mu (\lambda )}{L(\mathrm{sin}\varphi -s\mu (\lambda )\mathrm{cos}\varphi )}$ depends on the constant angle $\varphi $ and $\mu (\lambda )={w}_{4}{\lambda}^{p}/(a+{\lambda}^{p})+{w}_{3}{\lambda}^{3}+{w}_{2}{\lambda}^{2}+{w}_{1}\lambda $ is the friction coefficient ($\mu $)-slip coefficient ($\lambda $) static characteristic with ${w}_{i},i=\overline{1,4},a,p$ being known constants. Notice at this point that the model (1) is not obtained directly, but its coefficients depend on many physical parameters of the ABS system and can be recovered in the original documentation [15]. Some of them are properly presented throughout the paper when needed.

_{1}, r

_{2}are the radii of the two wheels in contact. For the given laboratory model, since the two radii are approximately equal, it will be considered that r

_{1}= r

_{2}= 0.99 [m], leading to a simpler slip in (2), i.e., $\lambda =({x}_{2}-{x}_{1})/{x}_{2}$.

_{1}, the friction torque in the upper bearing and the friction torque among the wheels. There are two torques acting on the lower wheel: the friction torque in the lower bearing and the friction torque among the wheels. Besides these, two gravity force of the upper wheel and the pressing force of a shock absorber of the upper wheel force the contact between the two wheels [15].

## 3. Sliding Mode Control Design

#### 3.1. Lyapunov-Based Sliding Mode Control Design with Included Uncertainties in the Process Model

_{1}and v

_{2}are additive terms accounting for unmodeled dynamics, external disturbances and parametric uncertainties. A switching variable is defined as $g(t)=\lambda (t)-{\lambda}_{d}(t)$, with ${\lambda}_{d}(t)$ the desired slip. We notice that since the simplified slip in (2) is $\lambda =1-{x}_{1}/{x}_{2}$, then $g(t)=\lambda -{\lambda}_{d}=1-{x}_{1}/{x}_{2}-{\lambda}_{d}=0$ is equivalent to the sliding surface line equation ${x}_{1}-(1-{\lambda}_{d}){x}_{2}=0$ which is consistent with the more familiar definition of the sliding surface in the form $g(x,t)=0$.

Algorithm 1. The LSMC Algorithm |

Selected initial parameters $\Delta ,\delta ,{v}_{\mathrm{max}}$ 1. Read the desired slip value ${\lambda}_{d}(t)$ and the angular speed values ${x}_{1},{x}_{2}$. 2. Calculate $\lambda (t)$ from (2), then calculate $g(t)=\lambda (t)-{\lambda}_{d}(t)$. Find $F(t)$, $G(t)$ from (6), calculate $\tau (t)$ from (7). 3. Calculate $u(t)$ from (8) and send it to the process. 4. After the current integration step is finished, go to step 1. |

**Theorem**

**1.**

**Proof of Theorem**

**1.**

#### 3.2. Reaching Law Sliding Mode Control Design with Partial Process Model Uncertainty

- $g>0$ involves $\dot{g}=-k\xb71\Rightarrow \dot{g}<0$, meaning that $g$ is decreasing to zero.
- $g<0$ involves $\dot{g}=k\Rightarrow \text{}\dot{g}0$, meaning that $g$ is increasing to zero.

- The case $g>0$. For a finite reaching time, it is necessary that $\text{}\dot{g}-\eta $, with $\eta >0$ as a positive constant. In this case, it is necessary that $\widehat{f}-f-k\le -\eta \iff k\ge \widehat{f}-f+\eta $. Since $\left|\widehat{f}-f+\eta \right|\le $ $\left|\widehat{f}-f\right|+\eta \le {U}_{F}+\eta $, a value $k\ge {U}_{F}+\eta $ ensures a finite reaching time at most equal to $\left|g(0)\right|/\eta $.
- The case $g<0$. In similar reasoning, a value as $k\ge {U}_{F}+\eta $ ensures $\dot{g}>\eta $ which involves that the attainment of the finite reaching time is $\left|g(0)\right|/\eta $ at most.

Algorithm 2. The RSMC Algorithm |

Selected initial parameter $k$ 1. Read the desired slip value ${\lambda}_{d}(t)$ and the angular speed values ${x}_{1},{x}_{2}$. 2. Calculate $\lambda (t)$ from (2), then calculate $g(t)=\lambda (t)-{\lambda}_{d}(t)$. Find $f(t),b(t)$ from (12). Compute the derivative ${\dot{\lambda}}_{d}(t)$. 3. Calculate $u(t)$ as in (13) then send it to the process. 4. After the current integration step is finished, go to step 1. |

**Theorem**

**2.**

**Proof of Theorem**

**2.**

## 4. Validation Case Study

^{–3}seconds (s), which is consistent with the real-world execution times. The sampling time corresponding to the fixed step solver allows for data acquisition in order to collect the discrete-time values of $\lambda (t),{\lambda}_{d}(t)$ as $\lambda (k),{\lambda}_{d}(k)$, respectively. Which in turn allows the computation of the following performance index

_{i}is the sampling instant index corresponding to the start of the braking process, and N samples of the controlled slip and desired slip are measured, until the angular speed x

_{2}of the lower wheel (the “car”) drops below 10 rad/s towards the end of the braking cycle.

^{2}], the viscous frictions ${d}_{1}=120\times {10}^{-6},{J}_{2}=225\times {10}^{-6}$ are measured in [kg m

^{2}/s], the static frictions in the upper and lower wheel are ${M}_{10}=3\times {10}^{-3},{M}_{20}=93\times {10}^{-3}$ in [Nm]. The terms $\phi (\lambda )=\mathrm{sin}({C}_{x}\mathrm{arctan}({B}_{x}\lambda )),\theta =\mu {D}_{x}$ are used to approximate the original term $\mu {F}_{n}(\lambda )=\theta \phi (\lambda )$ related to the normal force F

_{n}from the original ABS equations [1], with the so-called Pacejka’s magic formula, for the parameter’s values ${C}_{x}=1.68,{B}_{x}=28,{D}_{x}=22.9,\mu =0.95$.

_{1}, thus neglecting the actuator dynamics. To make the controller (16) compatible with controllers (8) and (13), M

_{1}is transformed back into control input u by using the simplified actuator model, that is $u(t)={M}_{1}(t)/\chi $. To be consistent with the control inputs calculated by the LSMC and RSMC laws (8) and (13), respectively, the braking torque ${M}_{1}(t)$ is further saturated to [–9;9] after being calculated from (16).

**A**and

**B**, ensuring that $CB\ne 0$. Possible exogeneous disturbances, unmodeled dynamic as well as parametric uncertainties are modeled under terms ${v}_{1}(t),{v}_{2}(t)$ who enter the larger disturbance term $d(t)$ for which a simplifying upper bound $\overline{d}$ is found. In the following, the control law (17) from [38] is implemented based on Equations (5), (14), (16), (18) and (23) from the same work. This control scheme is designed for output model reference tracking control, it includes dynamical filters for the underlying uncertain controlled system missing full state measurement. The proposed PSSMC structure was tested in [38] on a linear, simplified, first order ABS model. Although the control system structure is simple, there are many equations and parameters involved in calculating the redesigned modulating function $\rho (t)$ from Equation (23) in [38]. The tuning parameters were selected as ${\underset{\_}{k}}_{p}=0.2$, ${c}_{1}=0.5$, the parameters of the reference model transfer function $1/(0.01s+1)$ were set as ${a}_{m}=100$, ${k}_{m}=100$, then $\beta =100$, $\overline{\theta}=5$, $\delta =0.1$, $\Lambda =-100$, $g=0.1$, ${\varphi}_{1}={\varphi}_{2}=0$, $\overline{d}=2$, ${c}_{d}=100$, ${\lambda}_{d}=100$. The three most influential parameters of this PSSMC scheme were found ${\lambda}_{1}=21.57,{\lambda}_{2}=9.13$ and $\epsilon =0.012$ using a GA to minimize the performance index ${I}_{test}$ on the test scenario under control input saturation in [–1;1]. While the others parameters have been determined starting from the values in [38]. The tuned parameters were searched within the domain $[0;100]\times [0;100]\times [0;10]$. Except for ${a}_{m}$, ${k}_{m}$, there is a number of 15 tuning parameters to search for, a large number with respect to the other controllers used for comparisons. The GA solver parameters were set to ensure convergence to approximately the same solution on five runs.

_{LSMC}= 1272, N

_{RSMC}= 1272, N

_{ADC}= 1262, N

_{RL}= 1330, N

_{MFSMC}= 1275, N

_{PSSMC}= 1298. Revealing that the LSMC and RSMC controllers lead to very similar braking scenarios in terms of the car speed and measured performance.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 3.**The dependence ${k}_{1}({\lambda}_{d})$ for the reaching-law-based sliding mode controller (RSMC) stability analysis case.

**Figure 5.**The slip feedback control system where the slip controller is replaceable with any of the following designed controllers.

**Figure 6.**The slip control results obtained with the Lyapunov-based sliding mode controller (LSMC), RSMC, active dynamic controller (ADC), reinforcement learning (RL) and model-free sliding-mode controller (MFSMC) structures.

**Figure 7.**The slip control result obtained with the periodic switching sliding mode controller (PSSMC) structure.

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**MDPI and ACS Style**

Chereji, E.; Radac, M.-B.; Szedlak-Stinean, A.-I.
Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons. *Algorithms* **2021**, *14*, 2.
https://doi.org/10.3390/a14010002

**AMA Style**

Chereji E, Radac M-B, Szedlak-Stinean A-I.
Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons. *Algorithms*. 2021; 14(1):2.
https://doi.org/10.3390/a14010002

**Chicago/Turabian Style**

Chereji, Emanuel, Mircea-Bogdan Radac, and Alexandra-Iulia Szedlak-Stinean.
2021. "Sliding Mode Control Algorithms for Anti-Lock Braking Systems with Performance Comparisons" *Algorithms* 14, no. 1: 2.
https://doi.org/10.3390/a14010002