Optimization of a 10-DOF Vehicle model in Simulink

I have built a 10-DOF vehicle model for estimation of the lateral dynamics. Next task is to optimize it. For example, I want to find the optimal values of the Yaw moment of inertia (input) that maintain the Yaw Rate under a certain threshold for a given maneuver (Step steer).

The problem is that since the model is very complex, there is not a transfer function between these two.

Ideally, I want to use a genetic algorithm for these task.

Any suggestions on how to resolve this?

[Aquatris]

Parametrize your model so that you can run simulations with different design parameter values easily, formulate a cost function with penalty terms if necessary, use any nonlinear optimization algorithm, including genetic algorithm, to minimize your cost function by varying the design parameters.

You can also use the simulink tools for this: here is a link.

Where are you stuck exactly?

[PanPan]

Thank you for your answer.

My problem lies mainly on the fact that since the equations are rather complicated, I am not able to build a cost function that relates the desired output with the input. Am I missing something here?

I have tried the tool you propose but it didn't work for my model.

[Aquatris]

If I were you I would start with something like this,

• parametrize the model so that the design parameters can be changed easily.
• Define the model inputs via the 'from workspace' block so that any input can be applied to the model from script.
• create a cost function where the inputs are the design parameters, the function runs the simulation with the preselected model inputs and given design parameters, and outputs the sum of the desired performance metrics with some thought out gains so that one would not dominate (if expected magnitude of one metric is around 5 and the other is around 5000 than weights should be selected accordingly): e.g., max(abs(yawRate))*W_1 + max(abs(verticalDisplacementofWheelCenters))*W_2
• Give the cost function to any nonlinear optimization algorithm and hope for the best

Since it is a relatively complex problem, you would need to iterate the cost function probably to achieve a good trade off between different metrics. You can also use penalty method to put constraints on the design parameters if you do not have strict constraints.

[PanPan]

I don't seem to understand how to create the Cost Function.

Let's suppose that these are the equations describing the lateral model and we want to see how the front damping coefficient Cφf (Design Variable) affects the Yaw Rate γ.dot (Objective Function) and how can Cφf be varied to reduce γ.dot for a predefined maneuver.

How can I create the Cost Function between these two?

I have used Genetic Algorithms in the past but the relationship between the Design Variables and the Objective Function was pretty clear. In this case though, it gets me really confused.

[Aquatris]

So you create your simulation such that it gets the design parameter as input and output a cost function that relates to minimizing the resulting yaw rate. Then you give this function to a solver you desire. One simple cost function is the maximum yaw rate magnitude:

function y = myCost(Cphif)
%% simulate the model using Cphif and the predetermined manevour to get yaw rate
yawRate = mysim(Cphif); 

%% define a cost function based on what you want
y = max(abs(yawRate));
end

This might not be exactly what you want since this indicates that the yaw rate might be constant throughout the whole manevour, for some odd reason :D. So you might want to add the 2-norm of the yaw rate signal, which is the sqrt of the integral of the yaw rate signal that represent a form of size as:

y = max(abs(yawRate))+norm(yawRate,2);

However the order of magnitude of maximum of the yaw rate and norm of the yaw rate might be quite different, e.g., maximum might be around 10 where as norm is around 10k, then you would need to use weights to adjust their contribution to the cost function so that solver does not favor one of them.

y = w1*max(abs(yawRate))+w2*norm(yawRate,2);

Since this is a complex problem, the form of your cost function will probably require some iteration to come up with a good solution..

[Sam Chak]

Hi @PanPan

If you and your research team wish to investigate how the front damping coefficient affects the Yaw Rate , it would be prudent to run multiple simulations across a range of reasonable values. From the results, you should be able to observe the minimum and maximum Yaw Rates that correspond to the values.

For instance, you could run the simulation iteratively for values ranging from 1 to 10, with a step size of 1. If the minimum Yaw Rate in the dataset corresponds to a particular value , you can then use @Aquatris's suggested Cost Function to search for the "best" by setting the lower and upper bounds on the design variable.