How can I use fmincon to solve problems with ODEs

I would like to use fmincon to analytically find the maximum lateral acceleration acheivable by a car, modeled using the single track bicycle model. I've solved simple problems with linear equations using fmincon (max surface area of a cube) but this one is over my head because one of the equations is an ODE.

The 2 Degree of Freedom (DOF) bicycle model is widely used as a simplified model of a car, used to study lateral dynamics.

The inputs to the simple 2 DOF model are the steering angle (df) and the longitudinal velocity (vx).

Using some constants to describe the car and the tires,

a, b and Iz = geometric constants

m = car mass = constant

Cyf and Cyr = cornering stiffnesses = constants

The car's lateral acceleration is modeled by,

vy_dot = (Fyf/m)*cos(df) + Fyr/m - vx*r;

where,

Fyf = Cyf * Af;

Fyr = Cyr * Ar;

and,

Af = (vy + a*r)/vx - df;

Ar = (vy - b*r)/vx

The car's yaw acceleration is modeled by,

r_dot = time derivative of yaw velocity r = (a/Iz)*Fyf*cos(df) - (b/Iz)*Fyr;

So it would all be simple for me to maximize vy_dot using fmincon if r was not the integral of r_dot. I simply don't know how to represent this r_dot = d/dt (r) in the functions for fmincon.

Thanks in advance for your help!

Answer 1

[J. Alex Lee]

I presume you mean to use r at some specific time, where r(0) (or some other time) is known - a standard 1D ODE.

with some initial condition, so either see if your r_dot expression is analytically integrable, or integrate it numerically.

Look into "integral" or "ode45".

[Justin Taylor]

Yes perhaps I can use ode45 (or variant) in the ceq portion of the nonl argument. I'll give it a go, thanks!