Evolutionary Multi-Objective Optimisation of an Automotive Active ...

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School of Engineering, Manchester Metropolitan University, Manchester, United Kingdom ... control engineers in the automotive sector – primarily moti- vated by a desire to improve .... set of diverse trade-off solutions within the search space of.
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Evolutionary Multi-Objective Optimisation of an Automotive Active Steering Controller Shahin Rostami, Peter Delves, Alex Shenfield School of Engineering, Manchester Metropolitan University, Manchester, United Kingdom [email protected] [email protected] [email protected]

Abstract—Many real-world engineering design problems involve the satisfaction of multiple conflicting objectives. In this case it is unlikely that a single ideal solution will exist. Instead, the solution of an Multi-Objective Optimisation problem will lead to a family of Pareto optimal points, where any improvement in one objective will result in the degradation of one or more of the other objectives. This paper investigates the use of Evolutionary Multi-objective Optimization (EMO) to optimise the performance of a closed loop feedback Proportional Integral (PI) vehicle yaw controller on a non-linear vehicle. This is done by comparing results against traditional empirical tuning methods relating to rise time, settling time, overshoot and steady-state error. The EMO showed improvement on the original control tuning and also brings light to the difficulty control engineers have with objective interaction for complex problems.

However, the use of feedback yaw rate control, as shown in Figure 1, requires less control effort from the driver whilst still providing good side-slip behaviour, so is more widely implemented. Segawa et al. [2] have proposed a mixed feed-forward and feedback controller (see Figure 2) that, as well as providing yaw rate feedback control as in Figure 1, also helps to reduce the driver effort at low speeds by providing Variable Gain Steering assistance (the gain of the feed-forward part of the controller is scheduled with speed – at low speeds more assistance is provided by the controller into the steering system).

I. ACTIVE STEERING FOR PASSENGER VEHICLES Dynamic vehicle control has been an important area of research and development over the past 20 years. This can be seen in the widespread implementation of Electronic Stability Control, Anti-lock Braking Systems and Traction Control Systems onto almost all modern cars. These systems are designed to improve handling and vehicle stability by controlling the vehicle’s dynamic states. Chassis control schemes are a major area of research for control engineers in the automotive sector – primarily motivated by a desire to improve the lateral handling characteristics of vehicles. Many solutions to this lateral stability control problem have been explored in [1]: from simple feed-forward control of yaw rate and lateral acceleration using active front steering to reference model yaw rate feedback control (see Figure 1).

Fig. 1.

Active steering using feedback yaw rate control.

In the simple feed-forward control scheme, a supplementary steer effort, dm, is provided into the vehicle temporarily to reduce the time lag between the application of the steering effort and the reaction of the vehicle. Under limit handling manoeuvres this provides an improvement on no control at all, but still requires the driver to apply much of the control effort.

Fig. 2.

Active steering using both feedback and feed-forward control.

However, a key challenge in implementing the kind of feedback control shown in Figures 1 and 2 is in the choice of the controller structure and controller gains. One possible approach to this problem is to use Evolutionary Multiobjective Optimisation (EMO) methods to simultaneously optimise multiple aspects of the controller’s performance. EMO methods have previously been used in the tuning of Proportional Integral Differential (PID) controllers for active magnetic bearings [3], as well as the design of loop-shaping H-Infinity controllers for industrial control applications [4]. The vehicle model used in this paper is an 8 Degree of Freedom (D.O.F) non-linear vehicle model based on the model used in [5]. It represents longitudinal velocity u lateral velocity v, yaw rate r (Figure 3) and roll rate φ of the vehicle body, with the other 4 D.O.F represented by the rotational speed of a wheels. This model influences the lateral and yaw behaviour of the vehicle through manipulation of the front wheel steer angles which are fed into a non-linear combined slip Pacejka tyre model. The simulation is as follows: The vehicle is travelling at a longitudinal velocity of 29m/s and enters a corner at time t = 1s where a step steer is applied to the front wheels. At this speed the model is working in its non-linear region near its limit of adhesion. This is a non-linear multi-objective problem since there are many objectives for the performance of the final system

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Fig. 3.

Vehicle dynamics.

including minimizing response time, settling time, overshoot, and control error. This is performed through closed loop negative feedback control using a PI controller. Traditional tuning of PI controllers are performed using empirical tuning rules such as the Ziegler-Nichols approach to select desirable P and I terms. This is a good approach for quick and effective tuning of a controller but can be a laborious process for fine tuning. EMO is advantageous as it allows more accurate control gains to be identified as it generates and evaluates populations of PI value combinations at a time optimising P and I terms at each subsequent generation.

Fig. 4.

The solution space of a multi-objective problem.

then checked to see if the maximum number of generations has been reached or any of the solutions are satisfactory to stop the optimisation process, otherwise it will continue onto selection of the fittest individuals from the population. The selected candidate solutions are then used for recombination to exploit the best solution information, and mutation to allow for exploration of the search space beyond the available solution information present in the population and prevent the possibility of getting stuck in a local optima.

II. EVOLUTIONARY MULTI-OBJECTIVE OPTIMISATION Many real-world engineering design problems involve the satisfaction of multiple conflicting objectives. In this case it is unlikely that a single ideal solution will exist. Instead, the solution of an Multi-Objective Optimisation (MOO) problem will lead to a family of Pareto1 optimal points, where any improvement in one objective will result in the degradation of one or more of the other objectives. Evolutionary Algorithms (EAs) are particularly well suited to this kind of MOO, because they search a population of candidate solutions. This enables the EA to find multiple solutions which form the Pareto optimal set (see Figure 4). EAs are often able to find superior solutions to realworld problems than conventional optimisation techniques (i.e. constraint satisfaction). This is due to the difficulty that conventional optimisation techniques have when searching in the noisy or discontinuous solution spaces that real-world problems often have. The execution order of a general and basic EA is shown in Figure 5. The optimisation process begins by generating an initial population of random candidate solutions which are then evaluated using objective functions and assigned a fitness value based on the objective value and potentially other values introducing population bias. A termination criteria is 1 This notion of ”Pareto” optimality was originally proposed by Francis Edgeworth in 1881 [6] and was later developed by the Italian economist Vilfredo Pareto in 1896 who used the concept in his studies of economic efficiency and income distribution [7].

Fig. 5.

General execution order of a basic Evolutionary Algorithm.

The quality of EMO candidate solution sets can be measured by their proximity, diversity and pertinence. Proximity is a measure of the distance between the approximation set and the true Pareto-optimal front whilst diversity is a measure of the distribution of solutions along that front in multiobjective space. An ideal MOO converges to solutions that are uniformly spread along the true Pareto-optimal front [8]. In real-world optimisation problems this approximation set must also be pertinent [9] (relevant to the preferences expressed by the Decision Maker (DM) i.e. an engineer). A good Multi-Objective Evolutionary Algorithm satisfies these goals adequately, presenting the DM with an approximation set of diverse trade-off solutions within the search space of their specified Region Of Interest (ROI). These measures of performance have been illustrated in Figure 6. This paper will investigate the use of the Covariance Matrix Adaptation Pareto Archived Evolution Strategy (CMA-PAES) [10] in determining both the controller structure and the controller gains of a yaw rate feedback controller in an active steering system.

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Fig. 6. Proximity, diversity, and pertinence characteristics in an approximation set for a bi-objective problem.

Fig. 7. Plot of the yaw response of the empirical (solid line), EMO 10-90 (dashed line) and EMO 0-100 (dotted line), cropped for clarity.

III. R ESULTS

IV. C ONCLUSIONS AND F URTHER W ORK

In this results section a comparison of two differently threshold (rise time measure between 10-90% of steady-state and the other 0-100% ) sets of EMO parameters and objective values are presented and evaluated in relation to empirical method. Table I compares the empirical tuning values against two sets of EMO values and shows the percentage difference in their response compared with those of the original tuning method. It can be seen that the EMO generates more precise gains to more than 5 decimal places where as the empirical method has only been tuned to one. To get to this level of precision empirically by trial and error would be a lengthy process and as an engineer not a very cost effective use of time. It is interesting how low the proportional term is and that the integral term ends up being larger than the proportional gain which is against the empirical tuning convention but could be a contributing factor to the slightly more oscillator behaviour.

Preliminary analysis of EMOs on controller optimisation yielded promising results, showing that the use of EMOs can save an engineer time and also provide optimised simulation parameters. This initial work will allow easier optimization of more complex control structures increasing the number of objectives. For example a disturbance could be applied to the model (such as strong side wind) on top of the problem. Another component that could be added to the problem is generate an optimization matrix for a speed range which can automatically look up the pre-calculated optimal value for a given speed and desired yaw rate.

TABLE I TABLE OF RESULTS FROM THE EMPIRICAL AND EMO(CMA-PAES) OPTIMISATION . optimiser P I Empirical 2 0.1 EMO 10-90% 0.179648 0.579897 EMO 0-100% 0.139840 0.542717 optimiser rise t settling t overshoot Empirical 0.73603 2.67000 0.08526 EMO 10-90% 0.72955 2.49000 0.07530 EMO 10-90% difference 0.881% 6.742% 11.677% EMO 0-100% 0.72965 2.49000 0.07344 EMO 0-100% difference 0.867% 6.742% 13.869%

error 0.02196 0.02191 0.225% 0.02193 0.159%

Figure 7 plots the yaw response of the empirical, EMO 10-90 and EMO 0-100. Both configurations of the EMO perform similarly reducing overshoot by over 10% and both improve steady state error slightly. Oscillatory behaviour has been introduced but this results in a quicker settling time within the set 5% threshold so is a tolerable feature. These results show that the algorithm is very flexible delivering similar performance with different thresholds producing far more precise gains than any empirical method could whilst optimising all objectives.

R EFERENCES [1] W. Manning and D. Crolla, “A review of yaw rate and sideslip controllers for passenger vehicles,” Transactions of the Institute of Measurement and Control, vol. 29, no. 2, pp. 117–135, 2007. [2] M. Segawa, K. Nishizaki, and S. Nakano, “A study of vehicle stability control by steer by wire system,” in Proceeding of AVEC, vol. 5, pp. 22– 24, 2000. [3] P. Schroder, B. Green, N. Grum, and P. Fleming, “On-line genetic autotuning of pid controllers for an active magnetic bearing application,” in Preprints of the 5th IFAC Workshop on Algorithms and Architectures for Real-Time Control AARTC, vol. 98, 1998. [4] I. Griffin, P. Schroder, A. Chipperfield, and P. Fleming, “Multi-objective optimization approach to the alstom gasifier problem,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 214, no. 6, pp. 453–469, 2000. [5] P. Delves, W. Manning, and A. Shenfield, “A torque vectoring approach to post incident control,” in The 11th International Symposium on Advanced Vehicle Control, pp. 1–6, AVEC 12, 2012. [6] F. Y. Edgeworth, Mathematical psychics: An essay on the application of mathematics to the moral sciences. No. 10, CK Paul, 1881. [7] C. A. Coello Coello, “Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art,” Computer methods in applied mechanics and engineering, vol. 191, no. 11, pp. 1245–1287, 2002. [8] K. Deb, “Multi-objective optimization,” Multi-objective optimization using evolutionary algorithms, pp. 13–46, 2001. [9] R. C. Purshouse and P. J. Fleming, “On the evolutionary optimization of many conflicting objectives,” Evolutionary Computation, IEEE Transactions on, vol. 11, no. 6, pp. 770–784, 2007. [10] S. Rostami and A. Shenfield, “Cma-paes: Pareto archived evolution strategy using covariance matrix adaptation for multi-objective optimisation,” in Computational Intelligence (UKCI), 2012 12th UK Workshop on, pp. 1–8, IEEE, 2012.