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Two Design Variable Optimisation, MATLAB Model Only
With such reasonable results obtained using the simple model for the computation of gradient information, it is necessary to justify the use of the complete MSC.ADAMS vehicle model for the function value in the optimisation process. The same optimisation was done as above but using only the simple Matlab model for the optimisation procedure. From the results in Figure 6.7 it can be seen that the function values are not the same as the MSC.ADAMS simulation values (calculated at iteration 5 and 25) and that the optimisation algorithm will converge to an infeasible point, when considering the constraints. Thus the use of the full MSC.ADAMS vehicle model is necessary in order to ensure the optimisation algorithm terminates at a feasible minimum. Although the simplified model has very similar trends, the absolute values are not always the same, especially when considering the tyre hop constraints. This explains why the converged solution may not be feasible.
Four Design Variable Optimisation
The four design variable ride comfort optimisation, was started from the optimum achieved from the two design variable optimisation. The optimisation process worked equally well as in the previously considered cases, although only small improvements are visible from the starting point, as can be seen from the MSC.ADAMS gradient history in Figure 6.8, and the Matlab gradient history in Figure 6.9. It is observed that although both methods converge to equally feasible solutions, the front and rear spring characteristics should differ in absolute value as can be seen by design variables x2 and x4. The result of this is that if the front gas volume is larger the front seated passengers will experience better ride comfort than the rear passengers, and the opposite if the rear spring gas volume is larger. From the above studies it is concluded that the optimisation process, making use of the simplified Matlab models for gradient information, produces equally feasible results in substantially less computational time. It will now be assumed that these models are sufficiently representative of the system for gradient information.
Definition of Design Variables
The front and rear static gas volumes are kept as design variables, defining the non-linear spring stiffness. The design variables that define the damper characteristics are redefined in order to achieve a more accurate description of the required damper characteristics. The standard rear damper characteristic is used, and redefined in terms of piecewise quadratic approximations, as illustrated in Figure 7.1. This gives a very accurate approximation to the measured damper characteristics. The damping force is primarily generated as a result of oil flow through an orifice, and for this reason the quadratic approximation is used to describe the characteristics.
Handling Optimisation
The handling optimisation was performed using the middle of the design space as a starting point, the opposite of the 4 design variable optimum (i.e. the infeasible point), and a random point in the design space. The results indicated that design variables 3, 4, 10 and 11 (the scale factors of fit2 and fit3, front and rear) all converged to the maximum boundary value of one, while design variables 7 and 14 (the gas volumes) converged to the minimum boundary value of almost zero. However, the other design variables did not change from their initial starting value. Yet when using different values starting values for these variables, different minima f∗(x), less than the above result were obtained. This indicates that design variables 1, 2, 5, 6, 8, 9, 12 and 13 do have an effect on the local minimum found, yet not as strong as 3, 4, 7, 10, 11 and 14. Figure 7.2 indicates the objective function convergence history and the relative summed change in the design variables, and objective function, from one iteration to the next. It could be argued that the convergence/termination criteria are not strict enough allowing premature termination of the optimisation algorithm on a non-optimum point. The termination criteria were then made 10% of previous, yet the optimisation still converged to the same points. This indicates that the design variable’s current scaling flattens out their effect, resulting in an almost zero gradient, or low sensitivity.
This low sensitivity could be overcome by one of two methods. Firstly by rescaling the particular variables that they have the same magnitudes but over a much smaller range, effectively increasing their sensitivity.
Alternatively by using a much larger perturbation of the design variables when calculating the gradient by central finite differences. However, Figure 7.3 indicates that the relative change in the design variable is so small, that, a change in the perturbation when calculating the gradient will not work. When observing the change in the normalised objective function values with respect to, for example design variable x2 (Figure 7.4), it can be seen that there is a definite minimum of the objective function with respect to x2.
The objective function with respect to these design variables, or the design variables themselves has to be rescaled. This highlights that the scaling of the design variables between zero and one, as suggested by Snyman (2005b) does not necessarily guarantee good convergence to a minimum.
1 Introduction and Background
1.1 Ride Comfort vs. Handling
1.2 Development of the 4S4
1.3 The need for Optimisation
1.4 Summary
2 Mathematical Optimisation
2.1 The Use of Mathematical Optimisation
2.2 The SQPMethod
2.3 The Dynamic-QMethod
2.4 Gradient ApproximationMethods
2.5 Conclusions
3 Full Vehicle Simulation Model
3.1 Initial VehicleModel
3.2 Refined VehicleModel
3.3 Vehicle Speed Control
3.4 DriverModel For Steering Control
3.5 Conclusions
4 Finite Difference Gradient Information
4.1 Optimisation Algorithms
4.2 Gradient ApproximationMethods
4.3 Optimisation
4.4 Conclusion
5 Simplified Vehicle Models
5.1 Optimisation Procedure
5.2 Definition of Optimisation Parameters
5.3 Simplified VehicleModels
5.4 Conclusions
6 Multi-Fidelity Optimisation
6.1 Optimisation Procedure
6.2 Handling Optimisation Results
6.3 Ride Comfort Optimisation Results
6.4 Summary of Results
6.5 Conclusions
7 Numerous Design Variables
7.1 Definition of Design Variables
7.2 Handling Optimisation
7.3 Ride Comfort Optimisation
7.4 Conclusions
8 Automatic Scaling of Design Variables
8.1 Formulation of Unconstrained Automatic Scaling
8.2 Concept Test
8.3 Modification for Constrained Problems
8.4 Implementation in the Vehicle Suspension Problem
8.5 Conclusions
9 Combined Optimisation
9.1 Handling Followed by Ride Comfort
9.2 Maximum of Ride Comfort and Handling
9.3 Pareto Optimal Front
9.4 Summary of Results
9.5 Conclusions
10 Conclusions
11 Discussion of Future Work
