# Nonlinear analysis of vehicle control actuations based on controlled invariant sets

• Volume: 26, Issue: 1, page 31-43
• ISSN: 1641-876X

top

## Abstract

top
In the paper, an analysis method is applied to the lateral stabilization problem of vehicle systems. The aim is to find the largest state-space region in which the lateral stability of the vehicle can be guaranteed by the peak-bounded control input. In the analysis, the nonlinear polynomial sum-of-squares programming method is applied. A practical computation technique is developed to calculate the maximum controlled invariant set of the system. The method calculates the maximum controlled invariant sets of the steering and braking control systems at various velocities and road conditions. Illustration examples show that, depending on the environments, different vehicle dynamic regions can be reached and stabilized by these controllers. The results can be applied to the theoretical basis of their interventions into the vehicle control system.

## How to cite

top

Balázs Németh, Péter Gáspár, and Tamás Péni. "Nonlinear analysis of vehicle control actuations based on controlled invariant sets." International Journal of Applied Mathematics and Computer Science 26.1 (2016): 31-43. <http://eudml.org/doc/276550>.

@article{BalázsNémeth2016,
abstract = {In the paper, an analysis method is applied to the lateral stabilization problem of vehicle systems. The aim is to find the largest state-space region in which the lateral stability of the vehicle can be guaranteed by the peak-bounded control input. In the analysis, the nonlinear polynomial sum-of-squares programming method is applied. A practical computation technique is developed to calculate the maximum controlled invariant set of the system. The method calculates the maximum controlled invariant sets of the steering and braking control systems at various velocities and road conditions. Illustration examples show that, depending on the environments, different vehicle dynamic regions can be reached and stabilized by these controllers. The results can be applied to the theoretical basis of their interventions into the vehicle control system.},
author = {Balázs Németh, Péter Gáspár, Tamás Péni},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {vehicle dynamics; sum-of-squares programming; Lyapunov method},
language = {eng},
number = {1},
pages = {31-43},
title = {Nonlinear analysis of vehicle control actuations based on controlled invariant sets},
url = {http://eudml.org/doc/276550},
volume = {26},
year = {2016},
}

TY - JOUR
AU - Balázs Németh
AU - Péter Gáspár
AU - Tamás Péni
TI - Nonlinear analysis of vehicle control actuations based on controlled invariant sets
JO - International Journal of Applied Mathematics and Computer Science
PY - 2016
VL - 26
IS - 1
SP - 31
EP - 43
AB - In the paper, an analysis method is applied to the lateral stabilization problem of vehicle systems. The aim is to find the largest state-space region in which the lateral stability of the vehicle can be guaranteed by the peak-bounded control input. In the analysis, the nonlinear polynomial sum-of-squares programming method is applied. A practical computation technique is developed to calculate the maximum controlled invariant set of the system. The method calculates the maximum controlled invariant sets of the steering and braking control systems at various velocities and road conditions. Illustration examples show that, depending on the environments, different vehicle dynamic regions can be reached and stabilized by these controllers. The results can be applied to the theoretical basis of their interventions into the vehicle control system.
LA - eng
KW - vehicle dynamics; sum-of-squares programming; Lyapunov method
UR - http://eudml.org/doc/276550
ER -

## References

top
1. Beal, C.E. and Gerdes, J.C. (2013). Model predictive control for vehicle stabilization at the limits of handling, IEEE Transactions on Control Systems Technology 21(4): 1258-1269.
2. Cairano, S., Tseng, H.E., Bernardini, D. and Bemporad, A. (2013). Vehicle yaw stability control by coordinated active front steering and differential braking in the tire sideslip angles domain, IEEE Transactions on Control Systems Technology 21(4): 1236-1248.
3. Carvalho, A., Palmieri, G., Tseng, H., Glielmo, L. and Borrelli, F. (2013). Robust vehicle stability control with an uncertain driver model, European Control Conference, Zurich, Switzerland, pp. 440-445.
4. de Wit, C.C., Olsson, H., Astrom, K.J. and Lischinsky, P. (1995). A new model for control of systems with friction, IEEE Transactions on Automatic Control 40(3): 419-425. Zbl0821.93007
5. Grip, H., Imsland, L., Johansen, T., Fossen, T., Kalkkuhl, J. and Suissa, A. (2008). Nonlinear vehicle side-slip estimation with friction adaptation, Automatica 44(11): 611-622. Zbl1283.93265
6. Gustafsson, F. (1997). Slip-based tire-road friction estimation, Automatica 33(6): 1087-1099. Zbl0884.93022
7. Jarvis-Wloszek, Z. (2003). Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sumof-Squares Optimization, Ph.D. Thesis, University of California, Berkeley, CA.
8. Jarvis-Wloszek, Z., Feeley, R., Tan, W., Sun, K. and Packard, A. (2003). Some controls applications of sum of squares programming, 42nd IEEE Conference on Decision and Control, Maui, HI, USA, Vol. 5, pp. 4676-4681.
9. Jianyong, W., Houjun, T., Shaoyuan, L. and Wan, F. (2007). Improvement of vehicle handling and stability by integrated control of four wheel steering and direct yaw moment, 26th Chinese Control Conference, Zhangjiajie, China, pp. 730-735.
10. Khelassi, A., Theilliol, D. and Weber, P. (2011). Reconfigurability analysis for reliable fault-tolerant control design, International Journal of Applied Mathematics and Computer Science 21(3): 431-439, DOI: 10.2478/v10006-011-0032-z. Zbl1234.93037
11. Kiencke, U. and Nielsen, L. (2000). Automotive Control Systems for Engine, Driveline and Vehicle, Springer, New York, NY.
12. Kim, D., Peng, H., Bai, S. and Maguire, J. (2007). Control of integrated powertrain with electronic throttle and automatic transmission, IEEE Transactions on Control Systems Technology 15(3): 474-482.
13. Korda, M., Henrion, D. and Jones, C.N. (2013). Convex computation of the maximum controlled invariant set for discrete-time polynomial control systems, Conference on Decision and Control, Firenze, Italy, pp. 7107-7112.
14. Kritayakirana, K. and Gerdes, J. (2012a). Using the centre of percussion to design a steering controller for an autonomous race car, Vehicle System Dynamics 50(Supp1): 33-51.
15. Kritayakirana, K. and Gerdes, J.C. (2012b). Autonomous vehicle control at the limits of handling, International Journal of Vehicle Autonomous Systems 10(4): 271-296.
16. Lasserre, J.B. (2007). Sum of squares approximation of nonnegative polynomials, SIAM Journal on Optimization 49(4): 651-669. Zbl1129.12004
17. Löfberg, J. (2009). Pre- and post-processing sum-of-squares programs in practice, IEEE Trans. on Automatic Control 54(5): 1007-1011.
18. Lu, J. and Filev, D. (2009). Multi-loop interactive control motivated by driver-in-the-loop vehicle dynamics controls: The framework, Conference on Decision and Control, Shanghai, China, pp. 3569-3574.
19. Mastinu, G., Babbel, E., Lugner, P., Margolis, D., Mittermayr P. and Richter, B. (1994). Integrated controls of lateral vehicle dynamics, Vehicle System Dynamics 23(Supp1): 358-377.
20. Németh, B. and Gáspár, P. (2011). Design of actuator interventions in the trajectory tracking for road vehicles, Conference on Decision and Control, Orlando, FL, USA, pp. 7434-7439.
21. Németh, B. and Gáspár, P. (2013). Analysis of vehicle actuators based on reachable sets, European Control Conference, Zurich, Switzerland, pp. 3137-3142.
22. Ono, E., Hattori, Y., Muragishi, Y. and Koibuchi, K. (2006). Vehicle dynamics integrated control for four-wheel-distributed steering and four-wheel-distributed traction/braking systems, Vehicle System Dynamics 44(2): 139-151.
23. Pacejka, H.B. (2004). Tyre and Vehicle Dynamics, Elsevier Butterworth-Heinemann, Oxford. Zbl0997.74044
24. Palmieri, G., Barc, M., Glielmo, L., Tseng, E.H. and Borrelli, F. (2011). Robust vehicle lateral stabilization via set-based methods for uncertain piecewise affine systems: Experimental results,50th IEEE Conference on Decision and Control, Orlando, FL, USA, pp. 3252-3257.
25. Palmieri, G., Baric, M., Glielmo, L. and Borrelli, F. (2012). Robust vehicle lateral stabilisation via set-based methods for uncertain piecewise affine systems, Vehicle System Dynamics 50(6): 861-882.
26. Papachristodoulou, A. and Prajna, S. (2005). Analysis of non-polynomial systems using the sum of squares decomposition, in D. Henrion and A. Garulli (Eds.), Positive Polynomials in Control, Springer-Verlag, Berlin/Heidelberg, pp. 23-43. Zbl1138.93391
27. Parrilo, P. (2003). Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming B 96(2): 293-320. Zbl1043.14018
28. Poussot-Vassal, C., Sename, O., Dugard, L., Gáspár, P., Szabó, Z. and Bokor, J. (2008). A new semi-active suspension control strategy through LPV technique, Control Engineering Practice 16(12): 1519-1534.
29. Prajna, S., Papachristodoulou, A. and Wu., F. (2004). Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach, 5th IEEE Asian Control Conference, Melbourne, Australia, Vol. 1, pp. 157-165.
30. Sadri, S. and Wu, C. (2013). Stability analysis of a nonlinear vehicle model in plane motion using the concept of Lyapunov exponents, Vehicle System Dynamics 51(6): 906-924.
31. Scherer, C.W. and Hol, C.W.J. (2006). Matrix sum-of-squares relaxations for robust semi-definite programs, Mathematical Programming 107(1): 189-211. Zbl1134.90033
32. Sontag, E.D. (1989). A “universal” construction of Artstein's theorem on nonlinear stabilization, Systems & Control Letters 13(2): 117-123. Zbl0684.93063
33. Summers, E., Chakraborty, A., Tan, W., Topcu, U., Seiler, P., Balas, G. and Packard, A. (2003). Quantitative local l₂-gain and reachability analysis for nonlinear systems, International Journal of Robust and Nonlinear Control 23(10): 1115-1135. Zbl1286.93032
34. Tan, W. and Packard, A. (2008). Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming, IEEE Transactions on Automatic Control 53(2): 565-571.
35. Topcu, U. and Packard, A. (2009). Local robust performance analysis for nonlinear dynamical systems, American Control Conference, St. Louis, MO, USA, pp. 784-789.
36. Yetendje, A., Seron, M.M. and De Doná, J.D. (2012). Robust multisensor fault tolerant model-following MPC design for constrained systems, International Journal of Applied Mathematics and Computer Science 22(1): 211-223, DOI: 10.2478/v10006-012-0016-7. Zbl1273.93059
37. Yu, F., Li, D. and Crolla, D. (2008). Integrated vehicle dynamics control: State-of-the art review, IEEE Vehicle Power and Propulsion Conference, Harbin, China, pp. 1-6.
38. Zhang, S., Zhang, T. and Zhou, S. (2009). Vehicle stability control strategy based on active torque distribution and differential braking, Conference on Measuring Technology and Mechatronics Automation, Zhangjiajie, Hunan, China, pp. 922-925.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.