European Journal of Control (2008)5:432–436 # 2008 EUCA DOI:10.3166/EJC.14.432–436
Discussion on: ‘‘Hybrid Parameter-varying Model Predictive Control for Autonomous Vehicle Steering’’ Jahan Asgari1, Francesco Borrelli2, and H. Eric Tseng1 1 2
Ford Research Laboratories, Dearborn, Michigan 48124, USA Department of Mechanical Engineering, University of California, Berkeley, 94720–1740, USA
1. Introduction The paper by T. Besselmann and M. Morari approximate a nonlinear vehicle model presented in [1,2,5,13] to obtain a hybrid parameter varying (HPV) model of the system. This model is used for designing several types of Model Predictive Controllers (MPC) with the objective of studying possible reduction of online computations and associated degradation of control performance. A comparison between controllers was made using prediction models varying from the full nonlinear model, as an indication forthe maximal achievable performance, to a linear model. Two benchmark scenarios extracted from [1,2] are investigated, the displacement of a car on an icy road due to a side wind gust, and a double lane-change maneuver performed autonomously on snow covered road surface. These scenarios were originally developed to investigate and compare different strategies for optimal integrated vehicle controls and have been previously utilized in [12, 3]. The present paper’s contribution nicely complements the previous work of MPC application on Autonomous Vehicle Steering, which started from [3, 5, 12, 13] using the NLMPC (Non-Linear MPC) with the corresponding real-time implementation in [6]. Although the NLMPC can achieve good performance and constraints fulfillment, its computational burden, in general, does not allow a real-time implementation for high speed maneuvers. In [7] a suboptimal MPC E-mails:
[email protected],
[email protected], htseng@ ford.com
scheme based on successive on-line linearization of the non-linear vehicle model was proposed to decrease the complexity of the controller. At each time step, the system model was linearized at the current operating condition and a linear MPC controller is designed for the resulting linear time-varying (LTV) system. The method stems from a direct analysis of the vehicle nonlinearities, the constraints and the performance index in the optimal control problem. The simulation results showed a significant reduction of the controller complexity, with minor performances degradation compared to a NLMPC controller. The experimental results of LTV MPC demonstrated that the controller can stabilize the vehicle even at high speed on slippery surfaces [8]. In [9] a sufficient stability condition for such LTV MPC scheme was presented for a general class of nonlinear discrete time systems and can be incorporated as an additional convex constraint to be included in the LTV MPC design.
2. Discussion The authors approach to MPC complexity reduction is based on Hybrid Parameter-Varying models (HPV models). A general HPV model is xðt þ 1Þ ¼ Ai ðÞxðtÞ þ Bi ðÞuðtÞ þ f i ðÞ if xðtÞ 2 P i
i ¼ 1; . . . ; s
ð1Þ
system (1) is a hybrid For a fixed parameter ¼ , system with known matrices and s modes (also called switched affine system). For a fixed mode j, system (1)
433
Discussion on: ‘‘HPV-MPC for Autonomous Vehicle Steering’’
is an LPV system xðt þ 1Þ ¼ Aj ðÞxðtÞ þ B j ðÞuðtÞ þ f j ðÞ . Summarizing, the mode j depends on a linear combination of the states (through the polyhedral set P i), while the parameter affects the matrices A and B in each mode. In the proposed HPV vehicle model the states x are lateral position Y, speed Y_ , yaw angle and yaw rate _ . The six parameters ¼ ½1 ; . . . ; 6 are uniquely determined by longitudinal slip sf, road friction coefficient and wind speed vw. The switching surface (describing P i in system (1) ) is a function of the sideslip angle f which is approximated to be a linear combination of the states x. In conclusion, for a fixed set of parameters sf, , vw the proposed HPV model becomes a switched affine system, while for a fixed side slip angle f the proposed HPV model becomes a linear parameter varying model where the matrices are a function of sf, , vw. The following list summarizes the controllers from the present paper, along with our related comments: 1. Nonlinear MPC. This is used for indication for the maximal achievable performance. In other words, for a given performance index and set of constraints this approach leads to optimal performance which then can serve as a benchmark for other (sub-optimal) approaches. 2. LPV MPC. Online, measure/estimates the parameters , substitute them in system (1) and design a linear MPC based on the corresponding linear model. This approach is less computationally demanding than the one presented in [7, 9] and previously discussed. However, as a drawback, in order to obtain the HPV model the authors introduce additional assumptions which are not required in [7, 9] (such as the ‘‘small angle assumption’’ on ). 3. Hybrid MPC. Off-line, fix a set of parameters sf, , vw. System (1) becomes a switched linear system for which the explicit solution can be computed offline and then implemented on-line as a set of tables. This approach has the drawback that parameters sf, , vw cannot be updated online and the advantage that no on-line optimization is required. 4. Linear MPC. Consider one mode of system (1) (corresponding to the non-saturated the tire model) and a fixed set of parameters sf, , vw For such a linear system the explicit solution can be computed. Parameters sf, , vw cannot be updated online. As an additional remark the results of linear versus hybrid explicit solution seem counter intuitive since linear model performs better than HPV. This might be
the result of authors effort in implicitly testing the robustness of the scheme by fixing the coefficient of friction ¼ 0.1 while simulation is performed with ¼ 0.3. It would be very interesting to see the performance of the nominal case when pushing the stability envelope through gradually increasing vehicle speed.
3. Conclusion The simulation results show that the HPV modeling seems a very interesting and flexible approach to tackle the problem of complexity reduction in MPC, however it may face the following challenges and restrictions: 1. Explicit solution algorithms for HPV models are not currently available and this limits the MPC implementation either to LTV models where the parameter is updated online but no prediction about the switching behavior is possible or to switched affine models where the parameters cannot be updated online. We remark that the explicit solution to LTV MPC is not currently available as well. 2. Deriving HPV models in a systematic way might be a challenge for higher dimensional systems. It would be of interest to investigate how the proper HPV would extend to the case on combined steering and braking actuation. Simulation results for LTV MPC [10] show further performance enhancement when combined braking and steering are used instead of steering only. This was further tested in experiments [11] for the double lane change maneuver on a snow covered road at high speed. The authors may consider the extension to multiple actuators on their future work with the HPV MPC approach.
References 1. Asgari J, Hrovat D. First/‘‘baby’’ steps toward using MPC/hybrid for vehicle dynamic control. Internal notes, Ford Motor Company, Dearborn, MI, December 2003 2. Asgari J, Hrovat D. Double lane change maneuver for vehicle dynamic control. Internal notes, Ford Motor Company, Dearborn, MI, January 2005 3. Asgari J, Tran JM, Hrovat D. Method and apparatus for four wheel steering control. US Patent No. 5488555, 1996 4. Borrelli F, Bemporad A, Fodor M, Hrovat D. An MPC/ hybrid system approach to traction control. IEEE Trans Control Syst Technol 2006; 14(3): 541–552 5. Borrelli F, Falcone P, Keviczky T, Asgari J, Hrovat D. MPC-based approach to active steering for autonomous vehicle systems. Int J Veh Auton Syst 2005; 3(2–4): 265–291
434 6. Falcone P, Borrelli F, Asgari J, Tseng HE, Hrovat D. Towards Real-time model predictive control approach for autonomous active steering. 8th International Symposium on Advanced Vehicle Control, Taipei, Taiwan, 2006. 7. Falcone P, Borrelli F, Asgari J, Tseng HE, Hrovat D. A real-time model predictive control approach for autonomous active steering. First IFAC Workshop on Nonlinear Model Predictive Control for Fast Systems, 2006 8. Falcone P, Borrelli F, Asgari J, Tseng HE, Hrovat D. Predictive active steering control for autonomous vehicle systems. IEEE Trans Control Syst Technol 2007; 15(3): 566–580 9. Falcone P, Borrelli F, Tseng HE, Asgari J, Hrovat D. Linear time-varying model predictive control and its application to active steering systems: stability analysis and experimental validation. Int J Robust Nonlinear Control 2008; 18(8): 862–875
Discussion on: ‘‘HPV-MPC for Autonomous Vehicle Steering’’
10. Falcone P, Borrelli F, Asgari J, Tseng HE, Hrovat D. A model predictive control approach for combined braking and steering in autonomous vehicles. 15th Mediterranean Conference on Control and Automation, Athens, June 2007. 11. Falcone P, Borrelli F, Tseng HE, Asgari J, Hrovat D. Integrated braking and steering model predictive control approach in autonomous vehicles. Fifth IFAC Symposium on Advances of Automotive Control, Berkeley, CA, USA, August 2007. 12. Hrovat D, Asgari J, Fodor M. Automotive mechatronics systems. In: Leondes CT (ed.) Mechatronics Systems Techniques and Applications Handbook, Gordon and Breach Science Publishers, 2000, pp. 1–98 13. Keviczky T, Falcone P, Borrelli F, Asgari J, Hrovat D. Predictive control approach to autonomous vehicle steering. American Control Conference, Minneapolis, Minnesota, June 14–16, 2006.
Discussion on: ‘‘Hybrid Parameter-varying Model Predictive Control for Autonomous Vehicle Steering’’ Rajesh Rajamani Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA
This paper focuses on autonomous steering for vehicle lateral control using a hybrid parameter varying model predictive control system. The challenges in designing a control system for this application come from the fact that the plant has nonlinear dynamics, significant saturation constraints, varying parameters in terms of changing tire–road friction coefficient and varying side wind disturbance forces. The approach taken by the authors is to approximate the original nonlinear dynamics of the system with a hybrid parameter varying model and to utilize a model predictive control system to determine the steering control input. A comparison between the performances of controllers using four different predictive models that include the original nonlinear model, a hybrid parameter varying model, hybrid constant parameter model and a simple linear time invariant (LTI) model is provided. Two scenarios consisting of driving on a straight icy road in the presence of a side wind gust and of executing a double lane change maneuver on a snowy road are used for the evaluation.
E-mail:
[email protected]
1. Scope for Additional Studies The paper is very interesting and presents a new approach for lateral vehicle control that can handle disturbances and varying road conditions. In the case of the scenario with wind disturbance on ice, all four of the predictive models perform well and have very small errors (of the order of a few mm). In the case of the scenario involving a double lane change on snow, all four of the controllers perform poorly, with lateral position errors between 2.5 and 3.1 m. The nonlinear and hybrid parameter varying models perform only marginally better than the other two models. A surprising result is that the LTI predictive model actually performs better than the hybrid constant parameter model in both of the investigated scenarios in the paper. The above simulation results raise a number of questions about whether there is value in using a hybrid parameter varying model. It is possible to conduct simulation studies for a number of additional scenarios which could provide a more comprehensive picture of the relative benefits of the different predictive models.
