2011 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM2011) Budapest, Hungary, July 3-7, 2011
Enhanced Slip Control Performance Using Nonlinear Passive Suspension System Samuel John, Jimoh O. Pedro, and Claudiu R. Pozna Abstract— Antilock Brake System (ABS) controller maintains or controls the slip between tyre and road to maximize the braking torque to achieve a shorter braking distance and control of the steering wheel. This paper presents a PID slip controller performance that incorporates nonlinear passive suspension dynamics. Three scenarios were compared The first scenario is the performance of the controller in a vehicle model without any suspension dynamics, the second scenario incorporates linear passive vehicle suspension system (LPVSS) and the third scenario incorporates nonlinear passive vehicle suspension system (NLPVSS). The incorporation of the passive suspension dynamics enhanced the ABS performance.
hampers the performance of the ABS. Secondly, several road tests are required in order to get the controller tuned, which is usually carried out by a trial and error method. This process is expensive and time consuming [1].
Key words: PID, ABS, slip control, Friction model, passive suspension, road disturbance
The proportional-integral-derivative (PID) controller has been a success story in industrial applications and has been applied to ABS controller [4][5][6]. Solyom [1] proposed a slip tracking approach, in which the design objective is for each wheel to follow a reference trajectory for the longitudinal wheel slip. A quarter-car model was used for the analysis and a gain scheduled PI(D) controller was implemented.
Current research is based on slip control. The aim of the controller is to continuously monitor the slip value and by manipulating the braking pressure it is possible to avoid a slip value of 100% (wheel lock) and maintain the slip at about the optimum value which is estimated for most road conditions to be between 10% and 20% [3].
I. INTRODUCTION In an emergency braking, usually the driver slams the brake pedal in panic and this leads to the locking of the wheel while the vehicle body‘s momentum is still high, this causes the car to skid. During skidding the driver loses the control of the steering wheel and the outcome can be disastrous. The antilock braking system (ABS) is a device that senses when the wheels of a car are about to lock while braking and it releases the brakes so that locking does not occur. This reduces the braking distance and also enhances the driver‘s control of the steering wheel.
On the other hand, the PID controller has been known to behave poorly when systems are highly non-linear. This has led to the proposal of variations of the classical PID for non linear systems. Jiang and Gao [4] for example, have proposed a ‗nonlinear PID‘ (NPID) for the ABS. This so called NPID incorporates two non linear functions to the linear PID. The method of gain scheduling implemented for the NPID is same for the linear PID. A comparison between the two control methods by Jiang and Gao [4] revealed that the NPID has a better robustness compared with linear PID with respect to vehicle stopping distance, road conditions and tyre conditions. The NPID performance was rated to have achieved 25% improvement over the linear PID when tested on trucks. With respect to tuning, the NPID requires the tuning of five parameters as against the four parameters in the current work.
The application of ABS to road vehicles started in the early 50s [1] and now it has become a standard component in automobiles. Most commercial ABS has an algorithm that is based on complicated logic rules, which attempt to capture all possible operating scenarios, conditions and un-modelled dynamics. These rules are executed by means of a control computer that switches on and off solenoid valves to ensure the right pressures are delivered to the wheels while avoiding slippage [2]. This control algorithm, has a number of short-comings: firstly, the slip tends to oscillate around the peak friction point, creating a chattering effect and this
In all the above cited cases none incorporates the suspension dynamics, and the simulation results are mainly for braking on flat terrains. The incorporation of the suspension dynamics as an integrated ABS system has become an area of focus for researchers and industry. Most focus has been on linear active suspension dynamics. An example is the work of Alleyne [7], in which he uses the advantage of an active suspension system to optimise the normal tire force to
Manuscript received _______, 2011. S. John. is with the Polytechnic of Namibia, and currently he is a PhD student in the Department of Mechanical Engineering, University of the Witwatersrand, Johannesburg, South Africa (corresponding author: phone: +264-61-207-2548; fax: +264-61-207-2521; e-mail:
[email protected]). J. O. Pedro, is with the School of Mechanical, Industrial and Aeronautical Engineering at the University of the Witwatersrand, Johannesburg, South Africa (e-mail:
[email protected]). C. R. Pozna is with the Department of Informatics, Faculty of Engineering Sciences, Széchenyi István University, Győr, Hungary on leave from Department of Product Design and Robotics, Transilvania University, Brasov Romania (e-mail:
[email protected]) 978-1-4577-0837-4/11/$26.00 ©2011 IEEE
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achieve shorter braking distance. His design was tested on dry and wet asphalt conditions using a ‗bang-bang‘ controller. He achieved 8% improvement with the integrated system over and above the ABS without the active suspension system. In another case, Ting and Lin [8] also achieved 8% improvement on stopping distance with the incorporation of an active suspension system to the ABS system. The goal of the current work is to investigate the effect of the linear and nonlinear passive suspension dynamics on ABS performance. The aim is to enhance the performance of the slip controller. We have therefore compared the performance of a PID ABS controller in three vehicle model scenarios. The first scenario is the performance of the controller in a vehicle model without any suspension dynamics; this is considered to be the benchmark model. The second scenario incorporates linear passive vehicle suspension system (LPVSS) and the third scenario incorporates nonlinear passive vehicle suspension system (NLPVSS). The PID controller gains, were kept the same for the three scenarios, to enable the evaluation of ABS migration which is a problem identified with current ABS technology.
where is the total mass of the quarter-car and the wheel, is the longitudinal velocity of the vehicle, is the rotational inertia of the wheel, is the radius of the tyre, is the angular velocity of the wheel, is the normal force exerted on the wheel, is the friction coefficient. Further more is the effective braking torque, is the braking gain, is the braking pressure, and is the hydraulic time constant. and are the viscous friction of the wheel and the vehicle aerodynamic friction coefficient respectively. The longitudinal wheel slip
is defined by the equation:
The tyre slip as defined by equation (4) shows that a locked wheel will occur when and free accelerating wheel will mean .
The rest of the paper is arranged as follows: Section two presents the mathematical background for the quarter-car model with a passive suspension system, section three discusses the PID controller design while the simulation results are presented in section four. Finally in section five we draw some conclusions from the results obtained.
Several friction models exist in the literature representing the relationship for different road conditions, the famous being the so-called ‗magic formulae‘ [10] which was used to generate the curves shown in Fig. 2. Detailed reviews of various static and dynamic friction models used in developing vehicle dynamics for ABS control are presented in [11] [12] [13].
II. SYSTEM MODELLING A. Vehicle and Wheel Dynamics The quarter-car model shown in Fig. 1 was used to develop the mathematical equations governing the vehicle and wheel dynamics. This model has been adopted due to its extensive use by researchers [1][5][9].
As stated in the introduction, the objective of the controller is to regulate the slip around the optimum slip value. From Fig. 2 this optimum slip value is about for all road conditions.
Fig. 1: Quarter car braking model
Fig. 2:
From Newton‘s second law of motion the equations describing the vehicle, tyre and road interaction dynamics during braking are given as [8]: 278
relationship for different road conditions
The friction coefficient between the road and the tyre influences the braking or traction of the vehicle as can be deduced from equation (5).
nonlinear
suspension
spring
constants
respectively,
bsl and bsnl are the linear and non-linear damping coefficients
of the suspension respectively, bssym is the suspension symmetrical damping coefficient, zs is the car body heave zr is the road surface disturbance and zu is the vertical displacement of the unsprung mass.
The friction model used for the approximation of the friction coefficient as proposed by [9] is given by equation (5) and adopted in the current work.
A deterministic sudden road disturbance was introduced to excite the suspension. This excitation is a bump with a profile described as [17]: where is optimal slip ratio, which gives maximum friction coefficient value . For dry asphalt and for icy road [8] B. Passive Suspension System where Am is the height of the bump, v is the velocity of the vehicle and l is the half wavelength of the sinusoidal excitation (Fig.3).
The passive vehicle suspension system (PVSS) consists of springs and dampers usually placed parallel to one another at each corner of the chassis of the vehicle (see Fig. 1). Their main purpose is to either store or dissipate energy [16] There are primarily two types of external disturbances; road and load disturbances. The road disturbance is characterised by high amplitude with low frequency e.g. sinusoidal-likeroughness or a bump. The suspension system therefore insulates the road and load disturbances by minimizing the levels of vibration transmitted to the vehicle. Unlike the active suspension system the passive system does not have energy source to regulate the stiffness of the system. The effect of the suspension on the normal force transmitted by the vehicle to the road surface will affect the torques generated and hence the operation of the ABS [9].
Fig. 3: Bump profile
Applying Newton‘s laws of motion to the suspension dynamics of Fig.1 gives [14][15][9].
III. CONTROLLER DESIGN A. Controller performance specifications Current research focuses on slip control[1][5][8], and the goal of the controller is to follow a pre-determined trajectory of the slip, this will be our reference input, and the performance of the controller will be measured against its ability to meet the following criteria: I. good tracking of the desired slip [3] II. rise time for the slip not greater than 0.2 sec III. maximum overshoot not greater than 5% IV. stopping distance from initial speed of [18] V. nominal stability VI. the controller should minimize the performance index given by [19]:
If
is the total mass of the quarter car, then;
where ms and mu are the sprung (vehicle) and un-sprung
where ratio,
masses (tyre) respectively. K sl and K snl are the linear and 279
is the final simulation time, is the desired slip is the maximum allowable slip ratio, which in
this case is 0.18, and is the maximum allowable braking pressure (control input).
Dry asphalt road condition with optimum friction coefficient was used for the simulations. The desired slip value was set at 18% and braking operation commenced at an initial velocity without any steering manoeuvring. The simulations were terminated at speeds of . This is because as the wheel speed approaches zero, the slip becomes unstable therefore the ABS must disengage at low speed to allow the vehicle to come to a stop. Hydraulic brakes were used for the braking operation.
B. PID Controller model The PID controller used for this work, is a modified classical series PID controller [19] and its structure is as given by [20]:
Other numerical parameters used for the simulations are as given in Table 2. where E ( s) and U ( s) are the error signal, and plant input signal respectively, K p is the proportional gain, Td is the derivative time constant, Ti is the integral step-time and the lag factor in the derivative component of the PID.
Table 2. Parameters of the quarter-car model
Parameters
is
Table 1 presents the PID gains used for the simulations, the tuning of the PID controller was carried out for the vehicle model without any suspension system; this was then migrated without any further tuning for use on the vehicle model incorporating the linear and nonlinear passive suspension system. These controller gains were tuned using the Zigler-Nichols tuning method [21].
The simulation results presented in Figures 5 to 7 show the stopping distance, wheel and vehicle deceleration and the slip tracking. Figure 5 is for the case of ABS without suspension system, used as the benchmark model. The controller was designed and tuned for this model before it was migrated to the models with suspension system.
Table 1: PID Gains Gains Parameter Kp 120 Ti 0.01 Td 0.25 0.025
Figure 6 presents the results for the ABS model incorporating linear passive suspension dynamics and Figure 7 presents the results for the ABS with nonlinear passive suspension dynamics. Table 3 provides the summary of the ABS controller performance in the three cases with respect to the performance criteria set-out in section III-A.
Plant
PID
Parameters
Table 3: Performance evaluation Specified ABS ABSvalue LPVSS ≤ 50 26.67 24.7
ABSNLPVSS 20.3
Rise time (sec)
0.2
0.04
0.087
0.045
Overshoot (%)
5
0.73
0
1.3
Min.
0.004477
0.02832
0.4403
Performance parameters Stopping distance (m)
Estimator
Fig.4: Schematic diagram of ABS
The schematic diagram of the ABS is shown in Figure 4. This was implemented in Matlab®/Simulink® and the simulations results are presented in section IV.
Performance index (J)
IV. SIMULATION RESULTS AND DISCUSSIONS In performance based design, the simulation phase is very critical in investigating the effect of external disturbances and of other dynamics on the controller to ascertain its ability to disturbance attenuation. In the current work, our focus is on the suspension dynamics and road disturbance. 280
Fig.5 ABS Performance without PVSS
Fig.7 ABS Performance with NLPVSS
V. CONCLUSIONS In this work we have designed a robust PID controller and demonstrated the possibilities of enhancing ABS controller performance by incorporating passive suspension system. We had a 7.5% improvement in stopping distance with linear passive suspension dynamics and a 23.9% improvement with the nonlinear suspension dynamics. The performance of the controller when migrated to a new system demonstrated its robustness to handle uncertainties coming from the road disturbance and suspension effects. In order for designers to achieve a more robust ABS system designs, it will require the incorporation of the nonlinear characteristics of the suspension system rather than the current popular method of approximating them to linear components only. The poor performance indexes (J) for the LPVSS and NLPVSS controllers are attributed to the introduction of the road disturbance, and due to the fact that re-tuning of the controllers for these cases were not carried out. This demonstrated the non-adaptive nature of the PID. This is however expected. Future work by the authors will further reveal the importance of the nonlinear suspension system with respect to some important parameters like ride comfort of passengers, suspension travel and road holding characteristics.
Fig.6 ABS Performance with LPVSS 281
REFERENCES 1.
2. 3. 4. 5. 6.
7. 8. 9.
10. 11.
12.
13. 14. 15. 16. 17. 18. 19. 20. 21.
S. Solyom, ―Synthesis of a Model-based tyre Slip Controller‖. Technical Report Licentiate thesis ISRN LUTFD2/TFRT-3228-SE, Department of Automatic Control, Lund Institute of Technology, Sweden, June 2002. P.E. Wellstead and N.B.O.L. Pettit, ―Analysis and redesign of antilock brake system controller‖ IEEE Proc. Control Theory Appl., Vol. 144, No. 5, (1997 pp 413-425. F. Chikhi, A. El-Hadri and J.C. Cadiou, ―ABS control design based on wheel-slip peak localization‖, Fifth Int. Workshop on Robot Motion and Control, June 23-25, 2005. F. Jiang and Z. Gao, ―An application of nonlinear PID control to a class of truck ABS problems‖, Proceedings of the 40th IEEE conference on decision and control, 2001, pp 516 - 521C.M. D.K. Yoo, ―Model based wheel Slip Control via constrained optimal algorithm‖, MSc Thesis, School of Electrical and computer Engineering, RMIT University, Melbourne Australia, 2006. J.O. Pedro, O.T.C Nyandoro and S. John, ―Neural Network Based Feedback Linearization Slip Control of an Anti-lock Braking system‖, Proceedings of the 7th Asian Control Conference (ASCC2009), (Hong Kong, 2009) pp1251 – 1257. A. Alleyne, ―Improved Vehicle Performance Using Combined Suspension and Braking Forces‖, Vehicle Systems Dynamics. 27(4) (1997) 235 – 265. J. Lin and W. Ting, ―Nonlinear Control Design of Anti-lock Braking Systems with Assistance of Active Suspensions‖, IET Control Theory Appl., Vol. 1, No. 1, January 2007. pp343 – 348. I. Petersen, ―Wheel Slip Control in ABS Brakes Using Gain Scheduled Optimal Control With Constrains‖, PhD Thesis, Department of Engineering Cybernetics, Norwegian University of Science and Technology Trondheim, Norway, 2003. E. Bakker, H.B. Pacejka and L. Lidner, ―A New Tyre Model with an Application in Vehicle Dynamics Studies‖, SAE 890087, 1989 pp.8395. S. John and J. Pedro, ―Review of Tyre/Road Friction Coefficient Estimation for ABS Controller Design‖ Fifth South African Conference on Computational and Applied Mechanics, 2006, pp 432 – 440. L. Li, F-Y. Wang and Q. Zhou, ―Integrated longitudinal and lateral tire/road friction modelling and monitoring for vehicle motion control‖, IEEE Trans. On Intelligent Transportation systems, Vol. 7, No 1, March 2006. J. Svendenius, ―Tyre models for use in braking applications‖, PhD Thesis. Department of Automatic Control, Lund Institute of Technology, Lund Sweden. 2003. I. Fiahlo and G. J. Balas, ―Road Adaptive Active Suspension Using Linear Parameter Varying Gain Scheduling‖, IEEE Transaction on Control System Technology Vo.10 No 1, 2002 pp 43 -54. P. Gaspar, I. Szaszi and J. Bokor, ―Active Suspension Design Using Linear Parameter Varying Control‖, International Journal of Autonomous Systems (IJVAS). Vol. 1 No 2, 2003 pp 206-221. F. Wang, ―Design and synthesis of active and passive vehicle suspensions‖, PhD Thesis, Department of Engineering, Control group, Queens’ College, University of Cambridge, U.K., Sept. 2001. J.O. Pedro, O.A. Dahunsi, ―Neural Network-based Feedback Linearization Control of a Servo-Hydraulic Vehicle Suspension Sysytem‖, Int. J. Appl. Math. Comput. Sci., Vol.21 No 1, 2011. K. Dietsche and M. Klingebiel, Automotive Handbook’ 7th Edition, Robert Bosch GmbH, Plochingen, July 2007 pp 820 -849. A. O‘Dwyer, Handbook of PI and PID Controller tuning Rules, Imperial College Press, London. 2006, pp 525-543 M. Norgaard, O. Ravn, N.K. Poulsen and L.K. Hansen, Neural Networks for Modelling and Control of Dynamic Systems, A Practitioner’s Handbook, Springer, Boston, MA, 2003 pp150-152. J.G. Ziegler and N.B. Nichols, ―Optimum settings for automatic controllers‖, Trans. A.S.M.E., Vol. 64, Nov. 1942, pp 759-765.
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