# Static output-feedback \(H_\infty\) control for T–S fuzzy vehicle lateral dynamics

- 148 Downloads

**Part of the following topical collections:**

## Abstract

This paper deals with static output-feedback \(H_\infty\) control for a system of the vehicle lateral dynamics, represented by Takagi–Sugeno (T–S) fuzzy models. Sufficient conditions of the existence of \(H_\infty\) control based on the static output-feedback are presented. The bilinear matrix inequalities are converted to a set of linear matrix inequalities, with the aid of some special derivations. Simulation results demonstrate the effectiveness of the proposed method.

## Keywords

Static output control \(H_\infty\) Control Vehicle lateral dynamics T–S fuzzy systems Bicycle model## 1 Introduction

In the field of road safety, the most important is to control the vehicle in critical driving situations. With the new control laws and the existence of artificial intelligence in today’s vehicles, vehicles are becoming easier to control. Nevertheless, their efficiency, their stability limits and their technical possibilities will be more and more developed. Changing the vehicle’s path in difficult driving situations is a major challenge for drivers. For this, several works are devoted to the control of vehicles. To this end, several driver assistance systems have emerged in recent years, such as the anti-lock braking system (ABS), the traction control system (TCS) and the electronic stability control (ESC) [1, 2, 3]. They depend on the service provided by the electronic systems for various activities of the vehicle in hazardous driving situations. In the event of a deviation in an undesirable direction, the computer makes immediate decisions to react with corrective actions to ensure the stability of the vehicle. The influence of the vehicle speed and the variation of road adhesion is presented in the works [4, 5]. The control based on the estimated state feedback is widely studied in recent years as in [7, 8], The authors have proposed an approach to stabilize the system of vehicle dynamics in the presence of disturbances. In [6, 8, 9], the authors discuss a control method that tolerates sensor and actuator faults for vehicle dynamics. The \(H_{\infty }\) Control For Vehicle Active Suspension Systems In Finite Frequency Domain is studied in [11]. The authors propose a technique to estimate the vehicle states and the crosswind in [10, 12], In order to improve the stability of the vehicle as well as its safety despite the presence of parametric/modeling disturbances.

Often system states are not always fully accessible, designing the controller using other methods is very necessary. Notice that the most suitable control is based on static output feedback because, at a lower cost, it can be easily implemented. However, the issue of stabilization through the static output-feedback control of vehicle dynamics systems is rarely investigated, although in practice, it is important and useful. This motivates our work. In this study, a new \(H_\infty\) control law for the vehicle’s lateral dynamics system is developed to ensure vehicle stability and avoid slippage in critical driving situations. The lateral dynamics of the vehicle in the presence of disturbances is expressed by the fuzzy model of Takagi–Sugeno, which is much studied to solve the control question for complex non-linear systems [13, 14]. The proposed design makes it possible to design a static output-feedback based controller, avoiding the problem of bi-linearity and without any equality constraint, such as the technique used in [15]. This planning has a superiority compared to some techniques in the bibliography like [19, 21]. Using the Lyapunov’s quadratic function, sufficient asymptotic stability conditions, that do not require transformation matrices or equality constraints are given as linear matrix inequalities (LMIs). Thus, it is easy to recover the gains of the controller by using the LMI-Toolbox solver [16].

This paper is organized as follows. The second section deals with the T–S fuzzy modeling of the vehicle lateral dynamics; while the third section presents the main results. The fourth section is devoted to the numerical illustration. Finally, a conclusion is given in the last section.

## 2 T–S fuzzy vehicle model and problem formulation

if \(|\alpha _f(t)|\) is \(M_1\) and \(|\alpha _r(t)|\) is \(N_1\) then \({\left\{ \begin{array}{ll} \dot{x}(t)=A_{1}x(t)+B_{21}u(t)+B_{11}\omega (t)\\ y(t)=C_{21}x(t)+D_{21}\omega (t)\\ z(t)=C_{11}x(t)+D_{11}u(t) \end{array}\right. }\)

if \(|\alpha _f(t)|\) is \(M_1\) and \(|\alpha _r(t)|\) is \(N_2\) then \({\left\{ \begin{array}{ll} \dot{x}(t)=A_{2}x(t)+B_{22}u(t)+B_{12}\omega (t)\\ y(t)=C_{22}x(t)+D_{22}\omega (t)\\ z(t)=C_{12}x(t)+D_{12}u(t) \end{array}\right. }\)

if \(|\alpha _f(t)|\) is \(M_2\) and \(|\alpha _r(t)|\) is \(N_1\) then \({\left\{ \begin{array}{ll} \dot{x}(t)=A_{3}x(t)+B_{23}u(t)+B_{13}\omega (t)\\ y(t)=C_{23}x(t)+D_{23}\omega (t)\\ z(t)=C_{13}x(t)+D_{13}u(t) \end{array}\right. }\)

if \(|\alpha _f(t)|\) is \(M_2\) and \(|\alpha _r(t)|\) is \(N_2\) then \({\left\{ \begin{array}{ll} \dot{x}(t)=A_{4}x(t)+B_{24}u(t)+B_{14}\omega (t)\\ y(t)=C_{24}x(t)+D_{24}\omega (t)\\ z(t)=C_{14}x(t)+D_{14}u(t) \end{array}\right. }\)

*u*(

*t*) is an assistant steering angle \(\delta _c(t)\) added to the driver’s steering angle \(\delta _f(t)\). \(A_i\), \(B_{1i}\), \(B_{2i}\), \(D_{1i}\), \(D_{2i}\), \(C_{1i}\) and \(C_{2i}\) are constant matrices with compatible dimensions.

### Lemma 1

*Given a symmetric matrix*\(\varSigma \in \mathbb {R}^{p\times p}\)

*and two matrices*

*X*,

*Z*

*of column dimension*

*p*,

*there exists a matrix*

*Y*

*such that the LMI*

*holds if and only if the following two projection inequalities with respect to*

*Y*

*are satisfied:*

*where*\(X^{\bot }\)

*and*\(Z^{\bot }\)

*are arbitrary matrices whose columns form a basis of the null spaces of*

*X*

*and*

*Z*,

*respectively.*

### Lemma 2

*For a positive definite matrix*\(R\in \mathbb {R}^{n\times n}\),

*matrices X and Y with appropriate dimensions, the following inequality holds:*

### Lemma 3

*For a positive definite matrix*\(R\in \mathbb {R}^{n\times n}\),

*a square matrix*\(X\in \mathbb {R}^{n\times n}\),

*and a scalar*\(\alpha\),

*the following inequality holds:*

## 3 Main results

In this section, we will produce an \(H_{\infty }\) control based on static output-feedback for the T–S systems of the vehicle lateral dynamics.

### Theorem 1

*Consider a scalar*\(\gamma > 0\),

*the closed-loop fuzzy system*(13)

*is asymptotically stable and satisfies the*\(H_{\infty }\)

*performances, if there exist symmetric matrices*\(P>0\),

*and matrices*

*S*

*and*

*G*,

*such that the following LMI is satisfied.*

*where,*

### Proof

*V*(

*t*) results in

In the following theorem, we will propose new LMI conditions to guarantee the asymptotic stability of the closed-loop T–S fuzzy system (13) with the design of the \(H_\infty\) control based on the static output-feedback.

### Theorem 2

*Consider a scalar*\(\gamma > 0\),

*the closed-loop fuzzy system*(13)

*is asymptotically stable and satisfies the*\(H_{\infty }\)

*performances, if for known scalar parameter*\(\alpha\)

*there exist symmetric positif matrices*

*P*

*and*

*R*

*and other matrices*

*S*,

*G*,

*H*,

*M*,

*and*\(N_{i}\),

*such that the following LMIs hold.*

*where*

*Furthermore, the static output-feedback controller gain matrices are given by*\(F_{j}=M^{-1}N_{j}\), \(j=1,2,\ldots ,4\).

### Proof

Assume that the inequality (27) holds, that means that \(I-H-H^{T}<0\) and \(\alpha ^{2}R -\alpha M -\alpha M^T<0\) (with \(\alpha ^{2}R>0\)), which guarantees the non singularity of the matrices *M* and *H*.

This completes the proof. \(\square\)

## 4 Numerical illustration

Parameters for the vehicle simulation

Constants | Value | Unit |
---|---|---|

\(m_v\) | 1832 | kg |

\(v_x\) | 50 | \({\text {m}}\,{\text {s}}^{-1}\) |

\(I_z\) | 2988 | \({\text {kg}}\,{\text {m}}^2\) |

\(C_{f1}\) | 55234 | \({\text {N}}\,{\text {rad}}^{-1}\) |

\(C_{f2}\) | 15544 | \({\text {N}}\,{\text {rad}}^{-1}\) |

\(C_{r1}\) | 49200 | \({\text {N}}\,{\text {rad}}^{-1}\) |

\(C_{r2}\) | 13543 | \({\text {N}}\,{\text {rad}}^{-1}\) |

\(l_r\) | 1.77 | m |

\(l_f\) | 1.18 | m |

The understeer performance of the vehicle is presented for the steering maneuvers considered in Fig. 2. We can see in Figures 3 and 4, that the \(H_\infty\) controller based on the proposed static output-feedback, with disturbances rejection, helps to keep and improve stability and safety under difficult driving conditions. We also notice that the additional disturbances have been dismissed.

## 5 Conclusion

In this manuscript, we have developed a method of \(H_\infty\) control based on the static output-feedback, for the vehicle lateral dynamics system represented by the Takagi–Sugeno fuzzy model. The stability of the whole system is studied with the Lyapunov’s quadratic function; appropriate conditions for the existence of this control are analyzed and their conceptions are expressed in the form of linear matrix inequalities (LMI). This study answers to some problems related to vehicle dynamics control. The simulation results tested and validated the efficacy in a crucial driving situation of the proposed solution for controlling the vehicle dynamics. As perspectives, we aimed to work on the problems of uncertainties and delay, and make an application on the CarSim software.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Winner H, Hakuli S, Lotz F et al (eds) (2016) Handbook of driver assistance systems: basic information, components and systems for active safety and comfort. Springer, BerlinGoogle Scholar
- 2.Zakaryukin V, Kryukov A, Cherepanov A (2017) Intelligent traction power supply system. In: Murgul V, Popovic Z (eds) Energy management of municipal transportation facilities and transport. Springer, Berlin, pp 91–99Google Scholar
- 3.Papelis YE, Watson GS, Brown TL (2010) An empirical study of the effectiveness of electronic stability control system in reducing loss of vehicle control. Accid Anal Prev 42(3):929–934CrossRefGoogle Scholar
- 4.Ma B, Lv C, Chen Liu Y, Zheng M, Yang Y, Ji X (2018) Estimation of road adhesion coefficient based on tire aligning torque distribution. J Dyn Syst Meas Control 140(5):051010CrossRefGoogle Scholar
- 5.Dahmani H, Pagès O, El Hajjaji A (2015) Observer-based state feedback control for vehicle chassis stability in critical situations. IEEE Trans Control Syst Technol 24(2):636–643Google Scholar
- 6.Youssfi N El, Oudghiri M, Bachtiri R El, Chafouk H (2018) \(H_\infty\) yaw and roll sensors fault tolerant control for vehicle lateral dynamics. In: 2018 international conference on electronics, control, optimization and computer science (ICECOCS), IEEE, pp 1–6Google Scholar
- 7.El Youssfi N, Oudghiri M, El Bachtiri R (2018) Control design and sensors fault tolerant for vehicle dynamics (a selected paper from SSD’17). Int J Dig Signals Smart Syst 2(1):50–67Google Scholar
- 8.Ichalal D, Marx B, Ragot J, Mammar S, Maquin D (2016) Sensor fault tolerant control of nonlinear Takagi–Sugeno systems. Application to vehicle lateral dynamics. Int J Robust Nonlinear Control 26(7):1376–1394MathSciNetCrossRefGoogle Scholar
- 9.Mrazgua J, Tissir EH, Ouahi M (2019) Fuzzy fault-tolerant \(H_{\infty }\) control approach for nonlinear active suspension systems with actuator failure. Procedia Comput Sci 148:465–474CrossRefGoogle Scholar
- 10.Yakub F, Abu A, Sarip S, Mori Y (2016) Study of model predictive control for path-following autonomous ground vehicle control under crosswind effect. J Control Sci Eng 2016, 6752671MathSciNetCrossRefGoogle Scholar
- 11.Mrabah L, Tissir EH, Ouahi M (2019) \(H_{\infty }\) control for vehicle active suspension systems in finite frequency domain. In: 2019 5th international conference on optimization and applications (ICOA), IEEE, pp 1-5. MLAGoogle Scholar
- 12.Youssfi N El, Oudghiri M, Bachtiri R El (2019) Vehicle lateral dynamics estimation using unknown input observer. Procedia Comput Sci 148:502–511CrossRefGoogle Scholar
- 13.Ahmad Taher AZAR (ed) (2010) Fuzzy systems. BoD—Books on Demand, NorderstedtGoogle Scholar
- 14.Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Systems Man Cybern 1:116–132CrossRefGoogle Scholar
- 15.Chaibi R, Rachid I Er, Tissir EH, Hmamed A (2019) Finite-frequency static output feedback \({H_\infty }\) control of continuous-time T–S fuzzy systems. J Circuits Syst Comput 28(02):1950023CrossRefGoogle Scholar
- 16.Boyd S, Ghaoui L El, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaCrossRefGoogle Scholar
- 17.Zhang H, Wang J (2015) Vehicle lateral dynamics control through AFS/DYC and robust gain-scheduling approach. IEEE Trans Vehicular Technol 65(1):489–494CrossRefGoogle Scholar
- 18.Wang R, Zhangi H, Wang J, Yan F, Chen N (2015) Robust lateral motion control of four-wheel independently actuated electric vehicles with tire force saturation consideration. J Frankl Inst 352(2):645–668CrossRefGoogle Scholar
- 19.Chaibi R, Hmamed A (2017) Static output feedback control problem for polynomial fuzzy systems via a sum of squares (SOS) approach. In: 2017 intelligent systems and computer vision (ISCV), IEEE, pp 1–6Google Scholar
- 20.Tuan VL, Hajjaji A El (2018) Robust observer-based control for TS fuzzy models application to vehicle lateral dynamics. In : 2018 26th mediterranean conference on control and automation (MED), IEEE, pp 1–6Google Scholar
- 21.Chen J, Lin C, Chen B, Zhang D (2016) Stabilisation of TS fuzzy systems via static output feedback: an iterative method. In: 2016 35th Chinese Control Conference (CCC), IEEE, pp 3710–3715Google Scholar
- 22.Jeung ET, Lee KR (2014) Static output feedback control for continuous-time TS fuzzy systems: an LMI approach. Int J Control Autom Syst 12(3):703–708CrossRefGoogle Scholar
- 23.Gahinet P, Apkarian P (1994) A linear matrix inequality approach to \(H_\infty\) control. Int J Robust Nonlinear Control 4(4):421–448MathSciNetCrossRefGoogle Scholar