Nonlinear analysis of vehicle control actuations based on controlled invariant sets

Németh, Balázs; Gáspár, Péter; Péni, Tamás

doi:10.1515/amcs-2016-0003

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.

References

Beal, C.E. and Gerdes, J.C. (2013). Model predictive control for vehicle stabilization at the limits of handling, IEEE Transactions on Control Systems Technology21(4): 1258–1269.10.1109/TCST.2012.2200826
Search in Google Scholar
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 Technology21(4): 1236–1248.10.1109/TCST.2012.2198886
Search in Google Scholar
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.
Search in Google Scholar
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 Control40(3): 419–425.10.1109/9.376053
Search in Google Scholar
Grip, H., Imsland, L., Johansen, T., Fossen, T., Kalkkuhl, J. and Suissa, A. (2008). Nonlinear vehicle side-slip estimation with friction adaptation, Automatica44(11): 611–622.10.1016/j.automatica.2007.06.017
Search in Google Scholar
Gustafsson, F. (1997). Slip-based tire-road friction estimation, Automatica33(6): 1087–1099.10.1016/S0005-1098(97)00003-4
Search in Google Scholar
Jarvis-Wloszek, Z. (2003). Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization, Ph.D. Thesis, University of California, Berkeley, CA.
Search in Google Scholar
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.
Search in Google Scholar
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.
Search in Google Scholar
Khelassi, A., Theilliol, D. and Weber, P. (2011). Reconfigurability analysis for reliable fault-tolerant control design, International Journal of Applied Mathematics and Computer Science21(3): 431–439, DOI: 10.2478/v10006-011-0032-z.10.2478/v10006-011-0032-z
Search in Google Scholar
Kiencke, U. and Nielsen, L. (2000). Automotive Control Systems for Engine, Driveline and Vehicle, Springer, New York, NY.10.1088/0957-0233/11/12/708
Search in Google Scholar
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 Technology15(3): 474–482.10.1109/TCST.2007.894641
Search in Google Scholar
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.
Search in Google Scholar
Kritayakirana, K. and Gerdes, J. (2012a). Using the centre of percussion to design a steering controller for an autonomous race car, Vehicle System Dynamics50(Supp1): 33–51.10.1080/00423114.2012.672842
Search in Google Scholar
Kritayakirana, K. and Gerdes, J.C. (2012b). Autonomous vehicle control at the limits of handling, International Journal of Vehicle Autonomous Systems10(4): 271–296.10.1504/IJVAS.2012.051270
Search in Google Scholar
Lasserre, J.B. (2007). Sum of squares approximation of nonnegative polynomials, SIAM Journal on Optimization49(4): 651–669.10.1137/070693709
Search in Google Scholar
Löfberg, J. (2009). Pre- and post-processing sum-of-squares programs in practice, IEEE Trans. on Automatic Control54(5): 1007–1011.
Search in Google Scholar
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.
Search in Google Scholar
Mastinu, G., Babbel, E., Lugner, P., Margolis, D., Mittermayr P. and Richter, B. (1994). Integrated controls of lateral vehicle dynamics, Vehicle System Dynamics23(Supp1): 358–377.10.1080/00423119308969527
Search in Google Scholar
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.
Search in Google Scholar
Németh, B. and Gáspár, P. (2013). Analysis of vehicle actuators based on reachable sets, European Control Conference, Zurich, Switzerland, pp. 3137–3142.
Search in Google Scholar
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 Dynamics44(2): 139–151.10.1080/00423110500385790
Search in Google Scholar
Pacejka, H.B. (2004). Tyre and Vehicle Dynamics, Elsevier Butterworth-Heinemann, Oxford.
Search in Google Scholar
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.
Search in Google Scholar
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 Dynamics50(6): 861–882.10.1080/00423114.2012.666353
Search in Google Scholar
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. 23–43.10.1007/10997703_2
Search in Google Scholar
Parrilo, P. (2003). Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming B96(2): 293–320.10.1007/s10107-003-0387-5
Search in Google Scholar
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 Practice16(12): 1519–1534.10.1016/j.conengprac.2008.05.002
Search in Google Scholar
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.
Search in Google Scholar
Sadri, S. and Wu, C. (2013). Stability analysis of a nonlinear vehicle model in plane motion using the concept of Lyapunov exponents, Vehicle System Dynamics51(6): 906–924.10.1080/00423114.2013.771785
Search in Google Scholar
Scherer, C.W. and Hol, C.W.J. (2006). Matrix sum-of-squares relaxations for robust semi-definite programs, Mathematical Programming107(1): 189–211.10.1007/s10107-005-0684-2
Search in Google Scholar
Sontag, E.D. (1989). A “universal” construction of Artstein’s theorem on nonlinear stabilization, Systems & Control Letters13(2): 117–123.10.1016/0167-6911(89)90028-5
Search in Google Scholar
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 Control23(10): 1115–1135.10.1002/rnc.2948
Search in Google Scholar
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 Control53(2): 565–571.10.1109/TAC.2007.914221
Search in Google Scholar
Topcu, U. and Packard, A. (2009). Local robust performance analysis for nonlinear dynamical systems, American Control Conference, St. Louis, MO, USA, pp. 784–789.
Search in Google Scholar
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 Science22(1): 211–223, DOI: 10.2478/v10006-012-0016-7.10.2478/v10006-012-0016-7
Search in Google Scholar
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.
Search in Google Scholar
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.
Search in Google Scholar

Nonlinear analysis of vehicle control actuations based on controlled invariant sets

Abstract

Paradigm

My account