I am relatively new to this topic. I understand the basics of optimization of control systems using a cost function and constraints and solve it as a minimization or maximization problem. I also understand that the cost function should be expressed as a function of parameters that I can change or control [for optimal output]. All the examples i have gone through in the theory take a simplistic control system like an inverted pendulum and come up with cost function for example as a function of initial position based on simple physics equations.

My question here is how can this be done for complex control systems like a Braking control systems in a vehicle. If braking distance is considered as a performance metric to be optimized to be as minimum as possible. How to come up with a cost function for this as a function of some input parameter? Is there a formal mathematical way of doing it?


1 Answer 1


I can imagine in your problem, you have to minimise the braking distance by arriving at an optimal PWM for the ABS.

1.The case can be broken down into a simpler one if you have the differential equation for the break actuation to the breaking distance. In that case the the breaking distance will turn out to be function of the PWM itself. And then looking at the function, one can go about minimising it.

  1. If you already have a model with you which can simulate vehicle braking, then you can hit and try with various PWM values and then choose the one that fits you best. Here the distance to stop itself can be called the error or the cost function. In the ideal case it should be minimised to zero.

  2. There are interesting minimising options if you already have the desired performance curves. Data like that can be fed to engines like matlab to solve for the optimal solution.

The process depends on the application you are intending and the resource available at hand.

  • $\begingroup$ I was aiming mostly for option 1. Having a Differential equation with controller parameters is ideal for my case. I wanted to know if there is a mathematical way of deriving one. $\endgroup$ Mar 22, 2019 at 8:50
  • $\begingroup$ Mathematical way of deriving the differential equation? You can ise system identification to get a transfer function/DE out of the model you have to start with. $\endgroup$
    – Utkarsh
    May 31, 2019 at 12:52

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